NBS

PUBLICATIONS

NATL INST. OF STAND & TECH

AlllDb 3fi7bTtD

\ NBS TECHNICAL NOTE 1086

U.S. DEPARTMENT OF COMMERCE / National Bureau of Standards

100

.U5753

No. 1086

1985

ftedby

letand

ripi1\i. Itano

|«its C, Bergqutst

i iifhii J. Bollinger
TM he National Bureau of Standards' was established by an act of Congress on March 3, 1901. The

Jl Bureau's overall goal is to strengthen and advance the nation's science and technology aind facilitate their effective application for public benefit. To this end, the Bureau conducts research and provides: (1) a

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MD, unless otherwise noted; mailing address
SATIONAL BUREAU OF STANDARDS

LIBRARY

Trapped Ions and Laser Cooling

Selected publications of the Ion Storage Group of the Time and Frequency Division, NBS, Boulder, Colorado

Edited by

David J. Wineland

Wayne IVI. Itano

James C. Bergquist

Joiin J. Bollinger

Time and Frequency Division

Center for Basic Standards

National Measurement Laboratory

National Bureau of Standards

Boulder, Colorado 80303

Supported in part by

U.S. Office of Naval Research

800 North Quincy

Arlington, VA 22217

and

U.S. Air Force Office of Scientific Research

Boiling Air Force Base, DC 20332

/

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued June 1985
National Bureau of Standards Technical Note 1086

Natl. Bur. Stand. (U.S.), Tech Note 1086, 192 pages (July 1985)

CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1985

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC 20402
PREFACE

This collection of papers represents the work on stored atomic ions

(initiated in the Fall of 1977) by the Time and Frequency Division of the

National Bureau of Standards. Although the primary goal of this research

has been the development of techniques necessary for achieving better time

and frequency standards, we have also been able to investigate related

areas of research.

Papers 15, 20, 21, 24, 25, A5, A8 and A13 were intended to be more

general reviews of the subject and include discussion of experiments by

other groups. Papers listed with the prefix A are not included here but

can be obtained on request. We intend to update this publication periodically to include new work not contained here. We hope this collection of

papers will be useful to our colleagues in this and related fields.

We gratefully acknowledge the support of the U.S. Office of Naval

Research and the U.S. Air Force Office of Scientific Research.

David J. Wineland

Wayne M. Itano

James C. Bergquist

John J. Bollinger

Boulder, Colorado

July 1985

1X1

CONTENTS

Page

1. "Radiation-Pressure Cooling of Bound Resonant Absorbers,"

D. J. Wineland, R. E. Drullinger, and F. L. Walls TN-1

2. "Laser Cooling of Atoms,"

D. J. Wineland and W. M. Itano TN-5

3. "Laser-to-Microwave Frequency Division Using Synchrotron

Radiation,"

D. J. Wineland TN-25

4. "Laser to Microwave Frequency Division Using Synchrotron

Radiation II,"

J. C. Bergquist and D. J. Wineland TN-30

5. "Frequency and Time Standards Utilizing Laser Cooled Ions,"

W. M. Itano and D. J. Wineland TN-34

6. "Laser Induced Magnetron Compression (Expansion) of Ions

Stored in a Penning Trap,"

D.J. Wineland, R.E. Drullinger, J.C. Bergquist, and W.M. Itano.. TN-35

7. "High- Resolution Optical Spectra of Laser Cooled Ions,"

R. E. Drullinger, D. J. Wineland, and J. C. Bergquist TN-36

8. "Double-Resonance and Optical-Pumping Experiments on Electromagnetically Confined, Laser Cooled Ions,"

D.J. Wineland, J.C. Bergquist, W.M. Itano, and R.E. Drullinger.. TN-40

9. "Spectroscopy of a Single Mg Ion,"

D. J. Wineland and W. M. Itano TN-43

10. "Proposed Stored ^^-kg"*" Ion Frequency Standards,"

D. J. Wineland, W. M. Itano, J. C. Bergquist, and F. L, Walls... TN-47

11. "Precision Measurement of the Ground-State Hyperfine Constant

of 25Mg+,"

W. M. Itano and D. J. Wineland TN-57

12. "Laser Cooling of Ions Stored in Harmonic and Penning Traps,"

W. M. Itano and D. J. Wineland TN-67

13. "Shift of ^S, , 2 Hyperfine Splittings Due to Blackbody Radiation,"

W. M. Itano, 'l. L. Lewis, and D. J. Wineland TN-87

14. "Laser Cooled, Stored Ion Experiments at NBS and Possible

Applications to Microwave and Optical Frequency Standards,"

D. J. Wineland, J. C. Bergquist, R. E. Drullinger, H. Hemmati,

W. M. Itano, and F. L. Walls TN-90

V
Page

15. "Spectroscopy of Stored Ions,"

D. J. Wlneland TN-97

16. "Magnetic Field Dependence of (s,£,) Electron Configurations,"

D. J. Wineland and W. M. Itano TN-107

17. "Laser Cooling and Double Resonance Spectroscopy of Stored

Ions,"

W. M. Itano and D. J. Wineland TN-108

18. "Generation of Continuous-Wave 194-nin Radiation by SumFrequency Mixing in an External Ring Cavity,"

H. Heiranati, J, C. Bergquist, and W. M. Itano TN-117

19. "Laser Fluorescence Mass Spectroscopy,"

D. J. Wineland, J. J. Bollinger, and W. M. Itano TN-120

20. "Time and Frequency Standards Based on Charged Particle

Trapping,"

W. M. Itano, D. J. Wineland, H. Hemmati, J. C. Bergquist, and

J. J. Bollinger TN-125

21. "Frequency Standard Research Using Stored Ions,"

D. J. Wineland, W. M. Itano, J. C. Bergquist, J. J. Bollinger,

and H. Hemmati TN-128

22. "Precision Measurements of Laser Cooled 9Be+ Ions,"

J. J. Bollinger, D. J. Wineland, W. M. Itano, and J. S. Wells... TN-132

23. "Hyperfine Structure of the 2p P, ,„ State in ^Be+>"

J. J. Bollinger, J. S. Wells, D. J. Wineland, and W. M. Itano... TN-137

24. "Trapped Ions, Laser Cooling, and Better Clocks,"

D. J. Wineland TN-141

25. "Spectroscopy of Stored Atomic Ions,"

D. J. Wineland, W. M. Itano, J. C. Bergquist, J. J. Bollinger,

and J. D. Prestage TN-147

26. "Strongly Coupled Nonneutral Ion Plasma,"

J. J. Bollinger and D. J. Wineland TN-172

27. "Laser-Cooled-Atomic Frequency Standard,"

J. J. Bollinger, J. D. Prestage, W. M. Itano, and

D. J. Wineland TN-176

28. "Limits for Spatial Anisotropy by Use of Nuclear- SpinPolarized ^Be+ Ions,"

J. D. Prestage, J. J. Bollinger, W. M. Itano, and

D. J. Wineland TN-180

VI
Additional publications of the Time and Frequency

Division Ion Storage Group not included in this technical note.

A1 . "New Possibilities for Frequency Standards Using Laser Cooling and

Detection of Stored Ions," F. L. Walls, D. J. Wineland, and R. E.

Drullinger. Proc. 32nd Annual Symp. on Frequency Control, June f978, p.

453-459. (Copies available from: Annual Frequency Control Symposium,

c/o Electronic Industries Assoc, 2001 Eye St., Washington, DC, 20006).

A2. "Laser Cooling of Ions Bound to a Penning Trap," R. E. Drullinger and

D. J. Wineland, in: Laser spectroscopy IV , Eds. H. Walther and K. W.

Rothe, (Springer, Berlin, 1979) p. 66-72. Same title by D. J. Wineland

and R. E. Drullinger, Proc. 6th Vavilov Conf. on Non-Linear Optics,

Novosibirsk, Springer, June 1979.

A3. "The Isolated Electron," Philip Ekstrom and David Wineland, Sci.

American 243 , no. 2, p. 105-121, August 1980.

A4. "Shift of ^S^/2 Hyperfine Splittings Due to Blackbody Radiation and

Its Influence on Frequency Standards/' Wayne M. Itano, L. L. Lewis, and D. J. Wineland, J. de Physique, Colloque C8, suppl. to no. T2, vol. H2_, Dec.

1981 , p. C8-283-287.

A5. "Prospects for Stored Ion Frequency Standards," D. J. Wineland, Proc.

13th Ann. Precise Time and Time Interval (PTTI) Applications and Planning

Meeting, Naval Research Laboratory, Washington, DC, Dec. 1981. NASA Conf.

Publ. 2220., p. 579-592.

A6. "Progress Toward a Stored Ion Frequency Standard at NBS," Wayne M.

Itano, D. J. Wineland, J. C. Bergquist, and F. L. Walls. Proc. Conf. on Precision Measurements and Fundamental Constants; Gaithersburg, MD June

1981; Precision Measurement and Fundamental Constants II, B.N. Taylor and

W. D. Phillips, eds. National Bureau of Standards (US) Spec. Publ. 617

(1984), p. 93-97.

A7. "High Power Second Harmonic Generation of 257 nm Radiation in an External Ring Cavity," J. C. Bergquist, H. Hemmati, and W. M. Itano, Opt.

Commun. _43, 437-442 (1982).

A8. "High Resolution Spectroscopy of Stored Ions," D. J. Wineland, Wayne

M. Itano and R. S. Van Dyck Jr., in Advances in Atomic and Molecular

Physics , Vol. 19, ed. by Bederson & Bates, (Academic Press, Aug. 1983), p.

135-186.

A9. "Laser Cooled ^Be"^ Accurate Clock," J. J. Bollinger, Wayne M. Itano,

and D. J. Wineland, Proc. 37th Annual Symp. Freq. Control, 1983, p. 37-41

(copies available from Systematics General Corp., Brinley Plaza, Rt. 38,

Wall Township, NJ 07719.)

A10. "Spectroscopy of Stored Ions Using Fluorescence Techniques," D. J.

Wineland, Wayne M. Itano, J. C. Bergquist, and H. Hemmati, S.P.I.E., Vol.

426-Laser Based Ultrasensitive Spectroscopy and Detection V , Society of

Photo-Optical Instrumentation Engineers, 1983, p. 65-70.

vii
All. "Frequency and Time Standards Based on Stored Ions," J. J.

Bollinger, D. J. Wineland, W. M. Itano, J. C. Bergquist and J. D.

Prestage, Proc. I6th Ann. Precise Time and Time Interval (PTTI)

Applications and Planning Meeting, Naval Research Laboratory, Washington,

DC, Dec. 1984, to be published.

A12. "Measurements of the gj - Factors of the 6s ^\/2 ^^<^ ^P '^1/2

States in ^^^Hg"^," W. M. Itano, J. C. Bergquist, and D. J. Wineland, J.

Optical Soc. Am., to be published.

AI3. "Optical Pumping of Stored Atomic Ions," D. J. Wineland, W. M.

Itano, J. C. Bergquist, J. J. Bollinger, and J. D. Prestage, Proc. Symp.

Alfred Kastler, Paris, Jan. t985. , to be published.

A14. "Angular Momentum of Trapped Atomic Particles," D. J. Wineland, J.

J. Bollinger, W. M. Itano, and J. D. Prestage, J. Opt. Soc. Am. B, Special

issue on "The Mechanical Effects of Light", to be published.

vili
Volume 40, Number 25 PHYSICAL REVIEW LETTERS 19 June 1978

Radiation-Pressure Cooling of Bound Resonant Absorbers

D. J. Wineland, R. E. Drullinger, and F. L. Walls

Time and Frequency Division, National Bureau of Standards , Boulder, Colorado 80303

(Received 26 April 1978)

We report the first observation of radiation-pressure cooling on a system of resonant absorbers which are elastically bound to a laboratory fixed apparatus. Mg ii ions confined in a Penning electromagnetic trap are cooled to < 40 K by irradiating them with the 8-/iW output of a frequency doubled, single-mode dye laser tuned to the low- frequency

side of the Doppler profile on the ^Siy5—*^P3/2 (Mj = + 4*-*^Wj =+

transitions. Cooling to approximately 10'^ K should be possible.

iorM^ = -i—A/,=-i)

Any scheme to improve fundamentally the resolution and accuracy of high-resolution spectroscopy (rf as well as optical) beyond the present

limits must incorporate sub-Doppler techniques

and a means to suppress substantially the secondorder and residual first-order Doppler effects. In the following we report the first experimental

evidence that bound absorbers can be cooled via

radiation pressure thereby fundamentally reducing

both the first- and second-order Doppler ef- fects. The use of radiation pressure for cooling was independently suggested for the case of a

gas of neutral atoms^ and for ions bound in an

electromagnetic trap^ and has also been incorporated into the suggested schemes for optical

trapping.^' ^ Methods for significantly cooling

electrons and ions contained in electromagnetic

traps via radiation damping or collisions with

cold gas have been known for some time^"^; however, these techniques do not provide cooling of atomic ions much below room temperature. It has recently been demonstrated^ that the magnetron motion of electrons in a Penning trap can be

cooled by nonlinear excitation of the other degrees of freedom. The method of radiation-pres- sure cooling demonstrated here offers the possiblity to cool substantially all degrees of freedom.

The method is outlined for the case of a harmonically bound resonant absorber (resonance

frequency i^q) which is constrained to move along

the X axis. We assume that its velocity is given

\3yv,.=v (jcos 2Tiu^t where u^ is its vibrational frequency and that the natural linewidth (Ai^) is less

than f„. The observed spectrum of the absorber

in the laboratory contains the central resonance

line with first-order-Doppler- effect- generated

sidebands separated by i^„ having intensity Jn^^vJ

cuj) (with n a positive or negative integer) when

observed in the direction of motion; here, J„ is the Bessel function of order n. If we irradiate

the absorber with photons of frequency t'x, =t'o

+nu^, the frequencies of the resonantly scattered

photons occur at u^ and symmetrically around v^ at the sideband frequencies t'o± i^ „, Uo±2u^, .... Therefore, although photons of energy h (j^q +nv„) are absorbed, on the average photons of energy

Hvq are reemitted; when n is negative, this energy difference causes the kinetic energy of the absorber to decrease by khi/^l per scattering event.

In our experiments, Ay»i^„; however, the above

conclusion is still valid. An alternative explanation is that when Vl'^v^, the absorber predominantly interacts with the incident radiation when

it moves towards the source of radiation and Doppler shifts the frequency into resonance, i.e.,

when Vj^{\ +vjc) =Vq. In the absorption process,

the photon momentum is first transferred to the

absorber causing its momentum to change by h/X

(A.i^-^ -c) and since the emission occurs symmetrically in the ±x directions, the net effect is to change the velocity of the absorber by i^v^ =h/M\

(M =mass of absorber). If Ay, «f,, then the

kinetic energy of the absorber decreases by an amount Mv^ At; ^ =nhu^. Through the harmonic

binding force, virtually all of the photon recoil

momentum is eventually transferred to the "lattice" (trap structure in this case); however,

since the mass of this lattice is very large, the

energy change occurs in the vibrational motion.

Note that, when Ai'<i'„, scattering of photons

with energy hu^ is equivalent to the Mossbauer

effect. In our experiments we store approximately 5 X 10^ Mg II ions (density approximately 2 x W/

cm^) in a copper Penning tap with hyperbolic electrodes of characteristic dimensions^'® rQ = 1.642(, = 0.63 cm. Typical operating parameters are v^

= 7 V and B^ = 1.3 T. The motion of an ion in the

trap is comprised of a harmonic oscillation along

the axial (z) direction (frequency uj plus a composite of circular cyclotron motion (frequency Vg')

superimposed upon a circular magnetron motion

(frequency vj in the x-y plane. In the notation of

© 1978 The American Physical Society TN-1 1639
Volume 40, Number 25 PHYSICAL REVIEW LETTERS 19 June 1978

Byrne and Farago,*^

Tc expCtw^'O+r exp(za)^), (1)

v.'here u) =2Tfu. For the above operating parameters, u,^ 205 kHz, u^'^198 kHz, and u„^ 26

kHz.

Tho trap apparatus is mounted in an enclosure

at high vacuum allowing observed ion thermalizadon times due to collisions with the background

g-us as long as 30 min and storage times of approximately 1 day. Mg is emitted from an oven and ions are formed by an electron beam coincicient v;ith the trap axis. Because the oven is approximately 0.5 cm from the trap, the temperature of the trap and therefore of the background

gas is elevated above room temperature. Thermal motion of the ions induces currents in the

trap electrodes yielding a signal proportional to NT (iV-ion number, T =ion temperature).^'^

Since the time for the cooling experiments is

much less than the ion storage time and since

observations confirm that the laser does not

eject ions from, the trap, N can be assumed constant. Therefore the signal provides a direct

measurement of ion temperature.

The radiation for the expe Mments is derived

from the frequency- doubled output of a singlemode, C.W., Rhodamine-110 dye laser. In the

spectral region of interest (~560 nm), this laser

has a power output of approximately 250 mW in a 1-MHz bandwidth when pumped with 3 W of 514.5-

nm radiation from an Ar"^ laser. The dye laser can be continuously tuned across 30 GHz and is frequency doubled into the uv with a 90°-phasematched AD*P (deuterated ammonia dihydrogen

phosphate) crystal. This allows us to tune across

the ^S^/2 — ^P3/2 (Mj=± ^~M^ = ±|) transitions

in one sweep. The uv output (up to 40 mW) is confined in a 1-2-mm-diam beam and is polarized

perpendicular to the trap/magnet axis. It is inti'oduced to and exits from, holes in the ring electrode of the trap.

In a magnetic field the ^S^/j-— ^Pg/, transition

splits according to the electronic Zeeman effect. The laser radiation is nearly uncoupled from the

AM =0 transitions due to the polarization; moreover, all of the \Mj\=j-^ \Mj\='2 transitions

optically pump the ground state. However, the

Mj = ^z'-'M J =±2 transitions can be driven many

times in succession and provide the transitions

for cooling. Figure 1(a) shows an experimental

trace of ion temperature versus uv frequency

FIG. 1. Ion temperature vs swept laser frequency.

Arrows above the traces show the direction of the sweep. The two resonance curves correspond to the

•^1/2 '^3/2 (M^--| —Mj=-iandMj= + i •M

= + |) transitions. When the laser frequency sweep is positive, the transitions are indicated by a temperature

decrease; when the sweep is negative the transitions are indicated by an increase in temperature. Curve b

is displaced upward and the baseline is not shown.

when the laser is swept from low to high frequencies through ihe ^Sj^'^'^Psh manifold. A decrease

in temperature is observed at two frequencies cor- responding to the separation of the +j — +| and

-2 — -2 transitions. As the radiation frequency

is swept up through the low-frequency side of the

Doppler profile on either transition, it cools the

ions. When line center is reached, cooling stops;

and as the laser sweeps through the high-frequency side of the Doppler profile, heating occurs. Conversely, if the laser frequency is swept down

through the transitions, heating appears first and

then cooling. [See Fig. 1(b).]

The rate of heat extraction (addition) per ion

is given by

dQ/dt = Ch {v^ - v^)o,{vr}I{vL)/hvL

,

(2) where I (vl) is the laser flux (W/cm^) and C is a

factor <1 describing the overlap of the laser beam

with the ion cloud. The cross section (Ts{i^l) is

given by the convolution of the natural line shape

with the Doppler profile. When Ai/ is much less

than the Doppler width, (Jsi^^j) can be given by

<^5K)^(7o-r— exp

Vj-Vr

(3)

where Oq is the resonance scattering cross section (l"'^^ for the transitions here) and where v^

= 0fr/MY'-vJc. The monitored signal is proportional to the temperature of the strongly

coupled cyclotron and axial motions but not to the magnetron motion.^'^ Moreover the laser

only weakly couples to the magnetron motion (see

below) and therefore each ion has a specific heat

of 2k (/? =Boltzman's constant). Using Eqs. (2)

1640 TN-2
Volume 40, Number 25 PHYSICAL REVIEW LETTERS 19 June !9"<S

and (3) the rate of temperature change is therefore

dt 2k dt ^viivji

^ (4)

When the laser frequency is swept across the line so that I'x -u^-'vt (v=dvL/dt) and when /(^x,) equals a constant value /q, then in the limit of small cooling (heating) where u^ is nearly constant, Eq. (4) can be integrated from - °° to ^ to give

T{t)=T{v^-u,)^^CI^o, Vti al/l^D

8^^'o/? exp - (5)

where the sign is negative for sweeping up and

positive for sweeping down. The maximum cooling rate is expected when Vj^^v^- v^ffl. Note that for small temperature changes the observed line

shape (temperature versus frequency) should re- produce the normal Doppler line shape. For the curves in Fig. 1, /g- 4 MW/mm^ for the - | 1 and 6 /iW/mm^ for the +1-^ + 2 transitions, C

-0.5, v' ^Q.\ GHz/ sec, and'° A:^ =^ 43 MHz, yielding a factor 250 K in front of the exponential in

Eq, (5). The ambient (trap) temperature was

measured to be approximately 3 50 ± 50 K by infrared means. The observed cooling (heating)

(~ 100 K) is not as strong as predicted which may

be caused by the observed presence of other ions

in trap which reduces the net effect or by too high an estimate of C. The observed splitting of the

lines (- 37 GHz) agrees with the value predicted

based on magnetic field measurements to about 3%.

The mean frequency of the two lines agrees with

the reported value" to within 0.02 cm"' by comparison of the dye-laser output to the spectrum of

/g. The observed width of the lines (4.5 GHz) is

larger than that predicted from the ambient temperature (2.9 GHz) and is partially caused by the

magnetron-motion Doppler broadening and possibly the slightly shifted resonances of the ^^Mg

and "'^Mg isotopes (about 10% each). We note,

however, that if we cool the ions to one- half of their equilibrium temperature, and then sweep

through the transition, the linewidth decreases

by a factor of approximately 1.5 as expected.

In Fig. 2 a plot of temperature versus time is

shown for the case when Vj^ - v^ is constant and

the laser is turned on for a fixed time as shown.

The ions had previously been heated with the la- ser to about 700 K. Laser cooling is then implemented for a fixed time showing the ion temperature approaching K with an upper bound of 40 K determined by the noise. After the laser

is turned off, the ions rethermalize as shown.

With/o = 8/iW, Ul-Vq- -{2±l) GHz, we measure

a cooling rate of 5 K/sec but predict a rale ol 15±^ K/sec based on Eq. (4) evaluated at T ^35U

K. For this trace the background gas pressure was purposely made higher to reduce the thermaiization time.

The laser can couple to the magnetron motion

and we must estimate this effect. If we use Eq.

(1) and its time derivative we can relate a given

velocity change, ^v , to aj;hange in magnetron

radius by the expression Ar„ - lAv/ioo^' - (jjJ.

For each scattering event Av -xh/MX which causes

the magnetron orbit size to change by lAr;„l- 1 X 10'*^ cm. If we irradiate the ion cloud uniformly,

Ir^l increases in a radom walk with step size

I Ar„l; this is a small effect. Moreover, if one

irradiates the cloud preferentially on the side of the trap where the magnetron motion recedes from

the laser, the magnetron vector should become

smaller. This can be used to overcome the dif- fusion of ions out ci the trap due to collisions and

therefore indefinite confinement should be possible. We have mapped the ion cloud shape by

measuring the cooling (heating) signal strengths a as a function of laser beam position and founii r"„(max)- 1 mm and 2 (max)- 0.5 mm.

LASER ON

h~ 5 6 min-H IONS REMQVFD

FIG. 2. Ion temperature vs time when laser cooling

is applied for fixed Vj^-v^. The ions were initially

heated above equilibrium temperature with the lasor. Laser cooling was then applied on the - |*^ - ; transition for a fixed time until a temperature approaching

K (<40 K) was achieved. After the laser is turned

off, the ions rethermalize to the ambient temperature.

TN-3 1541
Volume 40, Number 25 PHYSICAL REVIEW LETTERS 19 June 1978

The ultimate limit on cooling is determined by

a competition of the damping rate on the vibrational energ>' due to laser cooling and the "noise"

excitation of the vibrational energy due to the

random occurrence (in time) of the photon impulses. In the limit where i^^ « Ai/ and when one

tunes for maximum cooling (u^ -Vq = - Iai^), the resulting kinetic energy is approximately equal

toh^u/8 corresponding to T- 0.5x10"^ K for the

transitions discussed here. One can use the scat- tered photons from the cooling process as a monitor in a double- resonance experiment. Since the

laser is not needed for trapping, it can be turned

off for a relatively long time while the resonance

of interest is probed. The possiblity also exists

to use a mixture of ions—one kind which can be

laser cooled and by collisions cools the other

kind which are the ions of spectroscopic interest.

We note that the cooling method is quite general

and can in principle be applied to other cases

such as ions or nuclei bound in solid lattice. The authors wish to acknowledge the help of the other members of the Time and Frequency

Division, in particular the support and leadership of H. Hellwig, and electronics construction

by C. Klepper. The basic trap structure was

built by one of the authors (D.J.W.) and R. S. VanDyck, Jr., at the University of Washington and

was used with the kind permission of H, G. Dehmelt. The cooling limit was estimated with key

input from E. M. Pure ell and D.J. Larson, Harvard University. This work was funded in part

by the U. S. Office of Naval Research under Contract No. N00014-77-F-0046.

T. W. Hansch and A. L. Schawlow, Opt. Commun.

13, 68 (1975). 'd. J. Wineland and H. Dehmelt, Bull. Am. Phys.

Soc. 20, 637 (1975). ^A. Ashkin, Phys. Rev. Lett. 24, 156 (1970), and 25,

1321 (1970), and 40, 729 (1978). ^V. S. Letokhov, Comments At. Mol. Phys. 6, 119

(1977), and many earlier references contained herein.

'H. G. Dehmelt and F. L. Walls, Phys. Rev. Lett. 21, 127 (1968); H. Dehmelt, in Advances in Atomic and

Molecular Physics, edited by D. R. Bates and I. Ester- man (Academic, New York, 1967, 1969), Vols. 3 and 5; D. J. Wineland and H. G. Dehmelt, J. Appl. Phys. 46,

919 (1975). ^R. A. Heppner, F. L. Walls, W. T. Armstrong, and

G. H. Dunn, Phys. Rev. A 13, 1000 (1976).

'd. a. Church and H. G. Dehmelt, J. Appl. Phys. 40,

3421 (1969).

^R. S. VanDyck, Jr., P. B. Schwinberg, and H. G. Dehmelt, in Proceedings of Orbis Scientiae, Coral

Gables, 1978 (Plenum, New York, to be published).

^J. Byrne and P. S. Farago, Proc. Phys. Soc. (London) 86, 801 (1965). '"W. L, Wiese, M. W, Smith, and B. M. Miles, Atomic

Transition Probabilities, U. S. National Bureau of Standards, National Standards Reference Data Series

—22 (U. S. GPO, Washington, D. C, 1969), Vol. IL

"S. Bashkin and J. O. Stoner, Jr., Atomic Energy

Levels and Grotrian Diagrams (North- Holland, Amsterdam, 1975).

1642 TN-4
PHYSICAL REVIEW A VOLUME 20, NUMBER 4 OCTOBER 1979

Laser cooling of atoms

D. J. Wineland

Frequency and Time Standards Group, National Bureau of Standards, Boulder, Colorado 80303 Wayne M. Itano*

Department of Physics, Harvard University, Cambridge, Massachusetts 02138

(Received 16 May 1979)

Various aspects of the laser cooling of atoms are investigated theoretically. More generally, the authors

investigate a process through which the kinetic energy of a collection of resonant absorbers can be reduced by irradiating these absorbers with near-resonant electromagnetic radiation. The process is described here as anti-Stokes spontaneous Raman scattering. Cooling mechanisms, rates, and limits are discussed for both free and bound atoms.

L INTRODUCTION

In the past few years, there has been increasing

interest in the use of near-resonant photon scattering to cool a collection of atoms, ions, or molecules. This interest is motivated in part by

the practical need to reduce first- and secondorder Doppler shifts in ultra-high-resolution

spectroscopy and in part by the esthetic appeal of controlling the positions and velocities of a collection of atomic particles to within the limits

imposed by quantum fluctuations. Recent proposals and experiments using narrow-band tunable

lasers suggest that such control may soon become

a reality. It is not difficult to imagine that the concepts and techniques which are being developed

may have application in a variety of areas not

initially anticipated.

Current interest in the possibility of cooling began with independent proposals to reduce the temperature of a gas of neutral atoms^ or ions which

are bound in an electromagnetic "trap"^ with nearresonant laser radiation. This method of cooling

has subsequently been incorporated into the in- teresting schemes for trapping of particles using

near-resonant optical fields.^ The first demonstration* of cooling using the basic techniques

described here was made for a slightly modified

situation; specifically, the magnetron motion of an electron bound in a Penning trap was "cooled"

by a technique called motional sideband excitation,^ which is formally equivalent to the laser

cooling of atoms. Cooling of ions bound in an

electromagnetic trap was more recently demonstrated.®"^ The cooling which is potentially

achievable should permit spectroscopy of unprecedented resolution and accuracy.

As discussed below, the technique can variously

be described in terms of radiation pressure, motional sideband excitation, optical pumping, or

anti-Stokes spontaneous Raman scattering; this last concept is the one primarily used here be- cause of its generality. It should be mentioned

that cooling by Raman scattering is not a new

idea; "lumino-refrigeration" was hypothesized

as early as 1950 by Kastler.^ We also note that other cooling processes are possible; for example, one could use optical pumping followed by

coUisional relaxation, as discussed in Sec. II, or cooling by collisionally aided fluorescence.®

The process described in this paper is, however, more direct and does not rely on atom-atom collisions to alter the atom kinetic energy. The paper is divided as follows. In Sec. II we

describe the general aspects of the cooling process and treat the problem combining simple

classical and quantum ideas. Section III introduces

the concepts and notation of the quantum-mechanical treatment which is then applied to free atoms

in Sec. IV and bound atoms in Sec. V. In order to make the problem somewhat more tractable, we

limit the discussion to simple systems which exhibit the salient features of the process.

II. SIMPLE DESCRIPTION OF THE COOLING PROCESS

A. Analogy with optical pumping

The basic features of the cooling process have

been outlined previously (Refs. 1, 2, 5, 6, 7). The

attempt is made here to describe the qualitative

aspects of the problem more completely; however, the general problem becomes quite complicated, and therefore several limiting cases wiU be treated.

First, recall that in optical pumping we have a way of drastically altering the temperature of a

specific degree of freedom in an atom or molecule. Assume, for example, that we have an

alkali-like atom which has ground-state "hyperfine" structure. This atom can also have many

20 1521 © 1979 The American Physical Society

TN-5
1522 D. J. WINELAND AND WAYNE M . ITANO 20

excited electronic states, but we assume that we

have a narrow-band laser which can excite the atom from one of the ground states to only one

excited electronic state. The relevant levels are shown in Fig. 1. Without laser irradiation, the atoms eventually

reach thermal equilibrium by collisions or interaction with the background blackbody radiation.

Therefore, the ratio of the number of atoms in state 2 to those in state 1 is given by the Boltz- mann law: N^/N, = ex^[- {E^- E,)/ksT\ , (1) where /jb = Boltzmann's constant, T = temperature,

and E^, E2 and iV^, A'^2 ^^^ the respective energies

and numbers of atoms in the states. For simplicity we assume that the ground state has only two nondegenerate energy levels; hence statistical weight factors are absent in Eq. (1).

If we now apply the laser radiation to the atoms,

optical pumping occurs. Atoms are excited to level 3, but can decay into either ground state.

If we neglect the finite frequency widths of the laser and of the optical transition, this process

continues until all of the atoms are in level 1. They remain there until another process (say, the

collisions) depopulates this level. However, in the pumping process, N^/N^-O, and, via Eq. (1), we may say that T^ also. In this simple example, we see that we can cool an internal degree of freedom of the atom (the hyperfine structure) by

optical pumping. In principle, we could continue

this optical pumping process and, using collisions

to transfer kinetic energy to the internal degree

of freedom, could reduce the translational temperature of the gas if there were sufficient isolation from the rest of the environment. The process of laser cooling discussed below is very

similar to the optical pumping case except that the translational degrees of freedom are optically

pumped directly.

B. Laser cooling of free atoms Assume that we have an unbound gas of atoms

(or resonant absorbers in general) which possess

electronic excited state

ENERGY

Loser driven transition

tiyperfine states

FIG. 1. Levels of interest in a hypothetical alkalilike atom. Optical pumping into state 1 occurs while

driving the 2 *—' 3 transition with a laser.

a resonant electric dipole transition (frequency,

1^0) in some convenient spectral region with radiative linewidth y/2-n (full width at half-intensity

points). Now suppose that we irradiate these

atoms with monochromatic, directed, low intensity radiation tuned near, but slightly lower than, the resonance frequency. We assume that the in- tensity is well below that which would cause saturation (the case of saturation is treated in Ref.

10), and that the thermalizing collision rate y^ between atoms is much less than the natural line- width y, but is larger than the optical absorption

rate (y »yc » absorption rate, see Sec. VF).

Those atoms of a particular velocity class moving

against the radiation are Doppler shifted toward

the resonant frequency v^ and scatter the incoming

light at a higher rate than those atoms moving with

the radiation which are Doppler shifted away from

resonance. For each scattering event, the atom

receives a momentum impulse ^k (k is the photon wave vector) in the absorption process. For an atom which is moving against the radiation, this impulse retards its motion. This retardation can

also be described in terms of radiation pres1,10 sure The average momentum per scattering event transferred to the atom by the reemitted

photons is zero, because of the randomness of the photons' directions (if we neglect terms of second

order in |v|/c, where v is the atom velocity and c

is the speed of light). The average net effect then

is that the atomic velocity is changed by an amount

Av = ^k/M per scattering event, where M is the atomic mass. When v and k are antiparallel, this leads to a net cooling, provided |v + Av| < |v|.^

(See Fig. 2.) In a practical cooling experiment it would be desirable to irradiate the atoms from all sides with radiation that covered the entire lower

half of the Doppler profile.^ Alternatively, nar- row-band laser schemes might be employed where

the laser frequency is swept from some very low

value to a value approaching the rest frequency.^

This requirement is substantially relaxed if the

ABSORPTION

-fik

REEMISSION

FIG. 2. Qualitative description of radiation-pressure

cooling. In the absorption process, the atomic velocity

is changed (reduced for k'v<0) by an amount Av = ^k/Af.

In the reemission process, the average change in velocity is zero. Therefore in the overall scattering process, the kinetic energy can be reduced.

TN-6
20 LASER COOLING OF ATOMS 1523

atoms are bound, as is described later. (See

Sec. lie.)

The above argument gives the essence of the cooling process but neglects residual heating ef- fects due to recoil. For example, in the re- emission process, as explained below, the average

momentum transfer is zero (neglecting terms of order |v|Vc^); however, the atom undergoes a random walk in momentum space because of the

finite momentum transfer in each emission. Thus,

the limiting kinetic energy for a single atom in the cooling process must be at least the recoil

energy R = {HkY/2M, and, as we will see below,

may be considerably larger than the recoil energy. To be more quantitative, we first write

the resonance absorption and spontaneous emission frequencies, which are given by:

, = ct;o + ke^ • v' - iu^o^" -R/n ,

(2a)

(2b)

where a)o = 27ri^o, jk^^,^ \=ijj^^Jc, \^,J =y,Jc,v

= atom velocity in the ground state, v' = atom

velocity in excited state, c = speed of light, ^

= |v|/c, H -Zti - Planck's constant. These equations, valid to order /3^ in the atom velocity, and

to first order R/Hug in the recoil energy, can be

obtained by invoking conservation of energy and

momentum in the absorption and emission processes. The second term on the right-hand side

of these equations is the ordinary first-order

Doppler shift, the third term is the second-order

Doppler or time dilation shift, and the last term

is usually called the recoil shift. We will use the nonrelativistic approximation and therefore neglect the third term in Eq. 2. Also, as explained

in Sec. Ill, the net effect of averaging over all possible directions of reemission, is equivalent

to setting the second term in Eq. 2(b) equal to zero (neglecting terms of order |3^ and higher).

Hence, the average energy change of the photon

per resonant spontaneous scattering event is given

by

A£(photon) = H (u;^„ - a.,,^) = -Hk^^^-v -2R . Here we note that this equation applies to both resonant and nonresonant scattering and can also

be simply derived in general by considering conservation of energy and momentum in the overall

scattering process. For this more general case,

k^bj is replaced by k. The change in photon energy is accounted for by a change in atom kinetic energy per scattering event

^Ejc(sitom) = HV,-v + 2R . (3) This leads to a net average cooling as long as Hk-v<-2R, and net heating for the opposite condition. As an example, suppose we have an atom

with M = 100 amu, i^o= 5x 10" Hz(X = 600 nm) at thermal energies (T = 300 K), then v,^^ = 2.2 x 10" cm/s, ^|k|y,„3 = 2.5x10-^8 ergs, 2/? = 7.3x lO'^' ergs. Clearly, the recoil heating does not play

a significant role until very low temperatures are

achieved. Of course, at higher frequencies Uq it becomes more important. If we interpret the energy changes in terms of temperature changes

by the relation A£'^ = -|feB AT, then AT s 0.012 K

per scattering event at T = 300 K, and it therefore

takes on the order of lO'' scattering events to do

substantial cooling at optical frequencies.

J. Cross section and cooling rates The atomic cross section for absorption, neglecting saturation,^" has the form

a(a)) = (To(iy)V[(u)-a.,^)^+(iy)^], (4) where Oq is the resonance scattering cross section {Oq-2vX^ for unpolarized atoms). Therefore,

if the radiation is incident along the x direction,

the rate of kinetic energy change is, using Eq.

(3): dE

dt Hoj

aiuj){Hkv^ + 2R), (5)

where / is the energy flux of the laser beam

(ergs/sec cm^) and where a is given by Eq. (4). (This expression will be more fully justified in Sec. III.) Equation (5) gives the cooling (heating) rate for a single atom. If we consider an en- semble of atoms we must average y^j over the velocity distribution. If, for example, the distribution is given by a Maxwell-Boltzmann distribution,

Eq. (4) becomes a Voigt profile:

J-oo V TT M

dfv l+[{2/y){u:-^',-kv,)Y'

where u^ilk^T/MY'^ and u}'o= iVo+R/ftSimilarly, Eq. (5) becomes

(6)

dE lOc Hkv. + 2R

dt

- ^w^

J_„

f —l+[(2/y)(a)-a)^-feJ]'

exp[- (vJuY]

ItjU

-dv.. (7)

Equation (7) can also be written in the form

+ |(c.^-u.-^)lmZ(i^,Q)],

where -F=(a,J-a;)/(feM), Q = y/{2ku), andZ(f', Q)

IN7
1524 D. J. WINELAND AND WAYNE M. ITANO 20

is sometimes called the plasma dispersion functions^ defined as

ZiF,Q) = r'-''' [

dx e' X -F -H

for Q>0. We can numerically solve Eq. (7) for dE/dt; however, when y and R/H are much less than the Doppler width we can approximate the in- tegral and the cooling rate by

dE

dt |J[.U-.„)..]|^ '""^[{"^-f}]'

(8)

where ij:p= ^o^/c.

For the case of unbound atoms, radiation inci- dent along only one direction would retard the motion only in that direction and heat the motion

in the other directions due to recoil. Therefore,

in a practical situation we might irradiate the atoms with six narrow-band laser beams directed

along the ±x, ±y, and ±z directions of a Cartesian

coordinate system. In this case we must sum

terms like Eq. (5) for each laser beam. If (i) the frequencies of the lasers are equal giving standing waves, or (ii) if the waves are not plane, we must

consider the effect of the induced dipole forces

responsible for optical trapping.^ In order to disregard this effect we will assume that the incident

(plane -wave) radiation has low enough power —that

is, the limiting kinetic energy is larger than the potential energy "hills" created by the standingwave-induced dipole force. In this limit, the in- duced forces average to zero. When the atoms

are bound this complication need not arise, since one laser beam may be sufficient to cool all degrees of freedom.

2. Cooling limit We can estimate the limit on cooling with the following argument whose general features were

advanced by PurceU.^^ Assume that substantial

cooling has already taken place so that the Doppler

width of the atomic transition is smaller than the natural linewidth y. We further assume that we

irradiate the unpolarized atoms with six laser

beams as described above, which are tuned near

the lower half-power point of the optical resonance, (ij-a;o = -^y. Considering the rate of energy change in the x direction, we can write

dE^

dt Hu) [aARkVx + 2R) + a.{-fikv,: + 2R}] , (9)

where / is the energy flux of each laser beam and

where CT+(a_) is the scattering cross section for the laser light moving with (against) the atomic

motion.

With the approximation that the Doppler width

is much less than the natural width, we can show

that the maximum cooling rate is achieved when

(xi'o-zy- In this case Eq. (9) reduces to dE^_ 2/a, dt Hufl) \ y /

* (10) We find the minimum kinetic energy iE^) when

dE^/dt = 0, in which case (E,J^.^-\ny, (11) which is independent of intensity. In a real sitvia- tion, of course, other causes of heating would be

present, making high intensity desirable. (See,

however. Sec. VF). Note that Eq« (11) holds for a

single atom or an ensemble of weakly thermalized

atoms. If the atoms have a Maxwell-Boltzmann

distribution, then Eicx'^^ksT. If y = 27r • 10 MHz, we find T„i„ = 2.4x10"^ K. From Eq. (11) the rms velocity v^,- {{vDY^'^ can be used to arrive at an approximate Doppler

broadening Aw^iS u^gy^^yc. In deriving Eq. (11), we have assumed that this Doppler broadening

is much less than the natural linewidth, which

then implies that R « Hy. In our example, Hy

s0.7x 10"^^ ergs»i?^4x 10~" ergs, and our as- sumption is justified. However, for weakly al- lowed transitions in light atoms, this condition

will be violated and the minimum kinetic energy

will be limited as shown below to approximately

the recoil energy R. When the condition R «Hy is not satisfied we

must accurately evaluate Eq. (7) to find the cooling limit. Assuming the thermalizing collision rate between atoms is large compared to the optical absorption rate, we can evaluate the plasma

dispersion function to find the average energy

2R

niin versus ^(cl)o — w)/i? for four values of R/Hy . We have assumed that the thermalizing

collision rate (y^) is much greater than the optical

absorption rate.

TN-8
20 LASER COOLING OF ATOMS 1525

which makes Eq. (7) vanish for particular values

of ui'o-ijj, R, and y. (We note that cooling is possible only if (wq - o)) > 2R/fi,) In Fig. 3 we show

the minimum energy thus obtained for various val- ues of Hiuj'o - u})/R and R/Hy. We note that it is possible to cool to an average energy less than the recoil energy; however, we must also note the

difficulty in obtaining the proper conditions under

which these results hold. In particular, free atoms

cannot really be thermalized amongst themselves.

Therefore some kind of confinement is required,

in which case the required degree of thermal isolation from the environment would be difficult to achieve. (Ideally, we would like a transparent

box whose walls have zero heat capacity.) This

situation can be nearly realized for electromagnetically bound ions, in which case we might be

able to obtain temperatures less than R/kg, when

y<R/H. For the atoms described in the previous

example (iJs4xlO"^^ ergs), this would imply r < 2.7 X 10"'' K, when the natural linewidth of the transition y/2v was less than 6 kHz. We also note

that practical limitations on laser power would

occur for the case of free atoms since the dif- fusion of the atoms during the cooling process

would require large volumes (» 1 cm^).

C. Laser cooling of bound atoms For the case of bound atoms we consider the special situation where they are harmonically

bound in all dimensions. The assumption of harmonic binding is not essential, but it allows a simple interpretation and exhibits the most important features found in any bound system. For

simplicity, we consider an atom constrained in the X, y, and z directions by springs of negligible mass as shown in Fig. 4. Of course, in a real

situation, the springs might be replaced by electromagnetic restoring forces. We assume that the springs are attached to a massive block and

that they act independently and may have different

spring constants giving rise to different frequencies of oscillation J2,, 0,^, and J2^ in the three directions, where we assume {fij}«u)o, i^Af, 3^,2.

/. Strong binding

When the binding is "strong", (y«{n,}), the gross features of the absorption spectrum are

significantly altered. We can see this by considering the optical electrical field seen by a particular atom. If the incident (plane-wave) radiation is directed along the x axis.

Ea,„„, = EoSin(fe>f-a)i), (12) where .v is the atomic x coordinate and k and a; are the wave vector and frequency of the incident

radiation. Now

x=Xa sin(n,tf + 0j) , where x^ is the oscillation amplitude and (p.^ is a phase factor. Choosing 0, = 0, we have

E„ Eq sin(feCa sinn,ci - oj/)

.

(13)

This expression is familiar as the signal derived from an oscillator of frequency w which is frequency modulated at frequency fi, with modulation index kxg- Equation (13) can be expanded

in terms of a series of Bessel functions and gives

rise to the spectrum shown in Fig. 5(a) when kx^

= 1.5. We note that the atom sees a spectrum

comprised of a "carrier" at frequency w with

equally spaced first-order Doppler effect genla) IccJm C^'a'

UKrZSX,

springsofom {mass=M)

block (moss=Mg»M)

photons FIG. 4. Idealized representation of bound atom. In

this picture the binding forces are due to (massless)

springs; in an actual experiment these springs might be replaced by electromagnetic restoring forces.

(b)

/ [\ '1 \

//

* /

, 1

\ y—

/

\-«— / I '

1

/

\ 1 \ 1

\ .-r^. /

\ 1

V ,-T-.. ._

FIG. 5. Radiation spectrum when fi, »y. Part (a) illustrates the spectrum of an incident wave E= EoSin(/?x — iot) as seen by an atom which oscillates according to x = Xi^ cosQ.^t, where we have chosen kx^=l.^. Part (b) shows the emission spectrum as observed in the laboratory including the broadening due to the lifetime of the upper electronic state.

TN-9
1526 D . J . W I N E L A N D AND WAYNE M . I T A N 20

erated sidebands at the frequencies ijj + 77iClj (m =± 1,±2,±3, . V. ) having intensities proportional to J ^(feca). Similarly, the spectrum of the atom as seen by an obser^/er in the laboratory

will have the same character as in Fig. 5(a), but each component will be broadened by the natural

linewidth > as shown in Fig. 5(b). To approximate the cooling rate, we note that when > «fij we can tune the incident radiation to one of the resolved lower sidebands. Thus, an atom can be made to absorb photons predominantly

of energy /7(a)o-*-'-wfij) where ?» is a (negative) in- teger. Photons of average energy approximately

equal to /Tcuq are reemitted, leading to a net cooling.^ One can, therefore, approximate the cooling rate to be: dE

dt {rmSl,)^JUkxJ (14)

In Sec. V we will see that this expression is strictly valid only when the recoil effect is negligible. For this approximation, as an example,

suppose n^/27i = 12.5 MHz,.'l/ = 100 amu, A. = 600 nm,

andm = -28. Let >'/277 = 1.0 MHz, and suppose

that the intensity is such that aJ/Jlu>o= 5 MHz. If the X motion initially has an energy corresponding

to room temperature, so that iA/n|(x^)

= i.\m^xl = ikBTiT = 300 K) then x, = 2.8x10-' cm,

feCa = 30, J'^gCfeXo)^ 0,046 and dE/dt=0.3 eV/sec,

1-2.1 mW/cm^. We therefore see that when the atom is bound, rather substantial cooling can be

achieved with modest incident power tuned to a

single frequency. (We have, however, chosen a strong sideband). Additionally, we can simultaneously cool all degrees of freedom with a single

laser beam if the incident radiation is directed

along, for example, the i+j + k direction when the vibration frequencies are related by rational numbers (but are not equal), or if, as in a cloud of ions, the long-range electrostatic forces tend to thermalize all degrees of freedom. This technique" has been called "cooling by motional sideband excitation."'

In a collection of bound atoms that are weakly

thermalized (i.e., the collision rate >c«>) a more

realistic estimate of the cooling can be obtained

if we average Eq. (14) over a thermal distribution. We have

[ksT J explir4rV-^m(fevJd£V ksT

Since E = \MQ.lxl the factor in parentheses can be

integrated to give the result

^^)=(/^mnj|jexp(-fe^(x2»

X4(fe2(x2», (15) where /„, is the modified Bessel function of order

m and k^{x^) =2RkBT/{n^^)-. If we assume that we irradiate somewhere within the Doppler profile, then we can show [see Eq. (41)] that \m\ s k{x^}^^^. In the high-temperature or shortwavelength limit, k({x^))'-^^»l. Therefore we

can use the following asymptotic expansion" of /„:

IJz)~ {2vz)-^'^exp{z -m-/22) z»m»l (16)

and we obtain

dt (fimn,)^ n_ ^w /7t^,

exp (^^^). (17)

where a) - ^g = ?nn^. The last factor shows the Doppler profile. For the above conditions,

{dE/dt) = 0.051 eV/sec.

The above description is not quite correct be- cause we have not considered in detail momentum

and energy conservation in the process of absorption and reemission. It happens that the absorption

and emission spectra are also altered by recoil. Qualitatively we can say that in the emission spectrum, for example, components occur at fre- quencies x)o and oj^ + ni^^, but the spectrum is now

slightly skewed. For example, suppose we look

at the emissions at aJo±n>t when kXa<l. For photon emission at frequency uJq-^i, conservation

of energy tells us that the energy of oscillatory

motion must increase; hence, during the emission,

kXa increases and so also does the amplitude of the spectral component at oJo-fi,- The opposite

argument holds for the component at Wg + fij and,

consequently, the average energy spectrum is shifted to a value slightly below Jiijo^. The solution to this problem is straightforward when quantum mechanics is used and, therefore, we defer

discussion to Sec. V. It has been treated by a semiclassical approach" when kxa« 1. A further complication arises when we ask for

the cooling limit when y«Q,^. We find that the minimum kinetic energy achieved from a classical treatment is less than the zero-point energy of the harmonic oscillator and therefore we defer

this discussion to the quantum -mechanical treat- ment. At this point we can see the similarity of this problem (when y«-^x) to the Mossbauer effect. Indeed, when kXa"^ 1, the central peak or "carrier"

becomes dominant; excitation at this central peak

is equivalent to absorption on the Mossbauer

line —the main difference being that at >-ray energies the recoil plays a much larger role. As

TN-10
20 LASER COOLING OF ATOMS 1527

we will see below, the quantum-mechanical treat- ment of laser cooling can use some of the same

formalism as the treatment of the Mossbauer effect. Finally, another picture which provides a connection with a familiar situation is given by considering the "molecule" formed by the block and

the atom bound to it. Cooling of the vibrational

motion is achieved by anti-Stokes Raman scattering. However, the ordinary selection rules for vibrational transitions do not apply in general

since the "molecular" dimensions may be considerably larger than the optical wave length.

2. Weak binding

When->'»ni, the period of the atomic center- of-mass oscillation is much larger than the optical

decay time. Therefore, during one oscillation

period the atomic velocity gradually reaches the value where the incident radiation is shifted into resonance; scattering occurs, and for each scat- tering event Eq. (3) holds. In this limit we have

treated the interaction of the radiation with the atom as occurring at an instant of time. This is reasonable since the atom loses optical phase

memory during one oscillation period and therefore coherence during successive oscillations

(which gives rise to the resolved sidebands) can

be neglected. An important difference from the free-atom case is that the momentum of the photon is transferred to the center-of-mass motion

of the block and atom. In the limit that the mass

of the block becomes infinite, the change in velocity of the center of mass, per scattering event,

goes to zero and the energy change must occur

in the kinetic energy of the bound atom.

Therefore, in the weak binding limit we may

apply Eq. (5) and find the average cooling rate by

averaging v^ over one center-of-mass oscillation

period. In an ensemble of weakly thermalized

atoms Eqs. (7) and (8) therefore also apply. Neglecting recoil, we obtain Eqs. (2) and (3) of Ref.

6. However, if we assume only one laser beam

is present to do the cooling we must assume that the transition rate is small enough to avoid heating by recoil in directions perpendicular to the laser beam. (See Sec. VF). If three mutually

perpendicular laser beams are used to do the cooling this restriction is not necessary, but the power must be held low enough to avoid the effects

of the induced dipole force. Note that the esti- mated cooling limit given in Ref. 6 is too low by

a factor of 2. This is because only the recoil

heating in the absorption process was considered

and not the contribution from reemission.

D. Generalization of the cooling process

In this paper we address a rather specific problem, that is, the laser cooling of atoms. Abstracting the problem slightly, we have excited one "oscillator" —the atom's internal electronic resonance at frequency Wq—with radiation at a lower frequency o). In the frame of the harmonically bound atom, this excitation occurs due to one

or more of the upper sidebands shown in Fig. 5(a). For the free atom these "sidebands" become a

single line. Instead of the internal atomic "oscillator," consider an electron (or in general a charged particle) which is constrained like the atom shown in Fig. 4. Further suppose that the resonance fre- quencies fiy,n^ are very high so that ^Sl^, /zfi^ »/Ifij. If incident radiation is polarized along the

z axis and is directed along the x axis at frequency

n^ -J2j, then the z oscillation is excited much the same way that the electronic transition in the atom was excited by the frequency Wo-flx- Thus, cooling of the X motion can be achieved at the expense

of driving (heating) the z motion. This is the es- sence of "magnetron" cooling which was demonstrated for an electron bound in a Penning trap.** Treating this problem classically is reasonably

straightforward,^' "'^^ but a general quantummechanical treatment is complicated by two things,

(a) Recall that for the laser cooling of atoms,

the electronic states can be assumed to comprise

a two-level system which is not saturated; this

is a good approximation for many practical situations. However, when the atomic transition is replaced by the harmonic oscillator, many levels are involved and saturation between levels can

occur, (b) In addition, the condition k^T

«HQ,^,HQ,^ is not generally satisfied and one must

contend with the effects of thermal background

radiation. Nevertheless, certain cases of this more general problem can be treated. ^'^ Fortunately, these two problems usually do not occur

for the laser cooling of atoms and many aspects

of the quantum- mechanical treatment are quite

straightforward.

We remark that the velocity selectivity provided

by the narrow-band (laser) source might be provided by other means. For example, assume the atom is polarized in the z direction and is constrained to move along the x axis. If white light

irradiates the atom at an angle with respect to the X axis, then, because the absorption rate is dependent on the angle the radiation is received (in the frame of the atom), we obtain a differential

effect which is dependent on the atom velocity as

is required above. This is analogous to the state selectivity provided by optical polarization in the

TN-11
1528 D. J. WINELAND AND WAYNE M. ITANO 20

case of ordinary optical pumping.

We note that in terms of the above description we are discussing the case of a nearly ideal refrigerator: for each scattering event (i) we extract an amount of energy fiQ.^ from the "x reservoir" at temperature T^, (ii) we supply an amount

of energy HiSl^ -n^), and (iii) we transfer the amount of energy Ji(Q^ - n^) + liSl^ = fi^^ to the "z reservoir" at temperature T^. The second law

of thermodynamics then implies that the minimum

obtainable temperature T,. is given by the relation^''r./r, > n,/n,

.

(18) The competing heating mechanism which gives

rise to this limit arises because the thermally

excited z motion (frequency flj modulates the Lorentz force due to the incoming radiation (frequency fij -fii) and gives a component at frequency n, which then excites the x motion."

In the following we will assume that T^ = 0; equivalently, we will assume that the atoms are normally in their ground electronic state and the background blackbody radiation is negligible.

Then the minimum temperature achieved will be

determined by recoil effects. We must, however,

keep the thermodynamic limit in mind in certain cases. We further remark that in terms of the above

description, the cooling process is entirely

analogous to the Overhauser effect where the

flipping of an electron spin in a solid is accompanied by a flipping of the nuclear spin system.^''

We also note that cooling by the technique of collisionally aided fluorescence^ shares many similarities with cooling described here. When collisions are present, the atoms' absorption spectrum is broadened, and, if the collisions were

very regular like the ion "collisions" with the electric restoring forces in an electromagnetic

trap, then the spectrum might appear as in Fig.

5(b). Of course, for the general case of collisions

the line broadening is not so regular; however,

the cooling principle still applies, so that if we

irradiated the atom at a frequency lower than the center of gravity of its emission spectrum, cooling would occur.

Finally, we observe that the cooling process

can be described in terms of radiation pressure.

The radiation pressure force on atoms has usually

been developed in terms of scattering^' ^°; however, we note that an alternative approach sometimes used in elementary texts^^ describes the radiation pressure force on a conductor in terms

of the Lorentz force interaction of the oscillating

magnetic field with the induced current. For the atomic case, a dynamic polarization is established

which is usually given in terms of the susceptibility. This atomic polarization interacts with

the magnetic component of the radiation field via

the Lorentz force leading to a force in the direction of the radiation:

F = f =fark. dt fiuj The effects of recoil must be added separately in this approach. The treatment of the problem as

scattering has the advantage of completeness and

is therefore used in the following. We finally re- mark that the interest in radiation pressure forces

has had a long history. Some of the very early

experimental and theoretical papers have been

summarized by Nichols and Hull.^° The first ex- periment observing radiation pressure on atoms was reported by Frisch.^^

III. QUANTUM-MECHANICAL TREATMENT-GENERAL

ASPECTS

For the quantum-mechanical treatment we will make some simplifying assumptions in order to illustrate the basic features of the problem. We

can, however, assume that the atom (or molecule, or resonant absorber in general) moves in three dimensions. We assume that the motions

in the different directions are independent, al- though it will sometimes be convenient to assume

for a collection of atoms that they are thermalized with each other by some relatively weak

long-range forces yielding a collision rate yc«yIt is also assumed that the time required for the kinetic energy to thermalize with the outside en- vironment is extremely long. This situation is closely approximated for a cloud of ions stored

in an electromagnetic trap, for example. We

further assume that the internal structure of the atom comprises a two-level system with groundstate energy Eg and excited-state energy E^ such

that Eg -Eg»kgT , where T is the ambient temperature. Within the limits imposed by thermodynamics (Sec. II D) this allows us to neglect the effects of background blackbody radiation; that is, in the absence of laser irradiation, we assume

that the atoms are in the ground state. Finally, we assume that the incident radiation is highly

monochromatic (spectral width «>) and that the

intensity is well below saturation.

In the following we will assume that the Hamiltonian for the system can be written

where H ^ is the unperturbed Hamiltonian and H'

describes the interaction of the radiation with the atom. We have

TN-12
20 LASER COOLING OF ATOMS 1529

FIG. 6. Pictorial representation of electronic [ground

(g) and excited (e)] energy states and translatibnal en- ergy states (denoted by integers) for an atom. When the atom is very weakly bound or unbound the spacing be- tween translational energy levels goes to zero. The

cooling can be described as anti-Stokes Raman scattering or optical pumping where the frequency {co^) of the scattered photon is greater than the frequency (uj^) of the "laser" photon.

where H^ is the Hamiltonian for the two internal states of the atom, H ^ is the Hamiltonian for the translational degrees of freedom, and H^^^ is the Hamiltonian of the radiation field. Assuming that the various degrees of freedom are not coupled

in the absence of H' , we write the total wave function as a direct product:

1

lip) = I int) I trans) I rad) .

In the absence of the radiation field, the internal

and translational energy levels are depicted in Fig. 6. In the nonrelativistic limit, and assuming

low intensity, we make the usual approximation

//'-(e/mc)P,-A(R^,),

where P^, and R^., are the momentum and position

of the optically active electron, A is the vector

potential of the radiation field evaluated at the position of the electron, and m is the electron mass. We have neglected the A^ term in the Hamiltonian.^^ For the cooling problem, we are primarily in- terested in a scattering process that changes the translational and radiation states but leaves the internal atomic state unchanged. In particular, we can formulate the problem by asking for the cross section for scattering a photon of wave vector k and polarization e where the translational

energy has changed from £, (trans) to £y: (trans)

and where the scattered photon has wave vector

kj and polarization e^. The differential cross section for this process can be obtained as a particular result from the Kramers-Heisenberg

formula, which can be derived from time-dependent perturbation theory. ^^ Specifically, the dif- ferential scattering cross section for scattering

from initial state / to final state/ can be written

dWjdJ^ 4^ a) \mti)

(/|e, • Peiexp(-zks-Rei)|;)<;|€ • Pciexp(zk- Rei)|/)

(Dj -ij) -ziy

x&{(jj-(jOs- [5:^: (trans) -£,(trans)lA)

,

where cKl is the differential element of solid angle

into which the photons are scattered, cj is the in- cident photon frequency (|k| =u>/c), HiOj = ^a)o + £j(trans) -£, (trans), r^ is the classical electron

radius e^/mc'^ , subscript s denotes the two possible

directions of the polarization of the scattered

photon, j denotes the intermediate state, and

where the wave functions now describe only the electronic and translational degrees of freedom

Ij) = |int(;))|trans(;)) . The 5 function in Eq. (19) ensures conservation

of energy for the scattering process; that is, ^(w^ - u;) = £j (trans) - E^ (trans) . The operator in the matrix elements of Eq. (19) is proportional to E' . From this equation we see qualitatively that the process is described as the absorption of a photon which promotes the atom to a distribution

of possible intermediate (virtual) states {;'} (where

(19)

energy need not be conserved) followed by spontaneous reemission. The denominator indicates

that a resonance is involved in the absorption.

This is the main cause for the cooling, since a resonance can occur when the kinetic energy of the atom changes, that is, when u; = a)o+ [£j (trans)

-£,(trans)]/^. Equation (19) can, of course, also

describe ordinary Raman and Rayleigh scattering.^^

To evaluate the matrix elements in Eq. (19), it is useful to express the electron position and momentum as functions of the atomic center-of-mass

position and momentum x and P, the position of the electron relative to the atomic core r, and the velocity of the electron relative to the atomic core p/m^ {rrij in this expression is the electron re- duced mass given by m{\+m/Mi.)~^, where M^ is the mass of the core. In the approximation that m «M^ we have:

TN-13
1530 D. J. WINELAND AND WAYNE M. ITANO 20

Re, =x + r, (20a)

(20b)

where M =M^ + ni. Before applying Eq. (19), we make the qualitative

observation that the cooling process can be represented as in Fig. 6, where we have illustrated a process involving specific initial, intermediate,

and final states. In general we must, of course, sum over all possible intermediate states, aver- age over the initial-state distribution, and sum

over all final states to get the net result. We see, however, that the cooling process can be described as spontaneous anti-Stokes Raman scat- tering or as a type of optical pumping as described

in Sec. U.

IV. LASER COOLING OF FREE ATOMS

A. Matrix elements For a free atom we have

Ht = 2Af and the spatial eigenfunctions are

I trans) = C^ exp(iK- x) = |K) , where C^ is a normalization constant and

<K| P|K) = UK , (K|i/,|K) = [JiKY/lM , <K'|K)=5(K',K),

where 6 denotes the Kronecker 6. Using Eqs. (20), the matrix elements in Eq. (19) take the form:

(j|e-P^,exp(fk-R„)|0

= K' M

P exp(zk'x) K)<e|exp(zk-r)k)

where e and ^ denote the internal atomic excited

and ground states, respectively.

Qualitatively, the first term in Eq. (21) represents a transition of the center-of-mass motion

which is caused by a field component resulting

from the laser field "modulated" by the internal

electronic oscillation. The second term represents an electronic transition which is caused by

a field component of the laser field which has been

Doppler shifted by the motion. For both terms,

however, a simultaneous change in electronic and

kinetic energy state has occurred.

We have

[e • P, exp(f k • x)] = (Coulomb gauge)

.

Therefore,

(K'le • ^Pexp(zk- x)|K) = (m/M)^K- e5(K + k, K')

,

M

(22a)

(22b)

(22c)

(22d)

<K'|exp(/k-x)|K) = 6(K + k,K'),

<e I exp(fk • r) \g) =^ z k • (r ) ,

{e\e •pexp(zk- r)|^) -imujol -(r) ,

+ (K'|exp(zk-x)|K)(e|e-pexp(fk-r)|g-) , (21)

where we have made the usual dipole approximation (1/1 k| » atomic dimensions) and have defined

(r) ={e\'v\g) . The ratio of the first term to the second term in Eq. (21) can be written:

^(K-e)k-(r) ^i-' M^oC • (r) c

'

In the nonrelativistic approximation used here,

the first term is therefore negligible and Eq. (19) becomes:

df^i-'f -r V ^.si^ n\W^ /r>|2 V (K^|exp(-fk3-x)|K,-)(Kj|exp(zk-x)|K,)

or equivalently:

6(u) - uj, - {ri/2M){K^ -Kj))

da_ duj

i^ = rV^le (r>Plf (r)\' ^^^^ ^"^^' ^' ^"^^^^^ " ^^ ~ i^/^''^)(Kf-^'))

(23)

where C = {r^m (j^l/n)^ . The 5 functions in this equation express conservation of momentum and

energy in the scattering process. We set K^

= K, tk-kj. After some manipulation, which is outlined in Ref. 23, Eq. (23) can be put in the form

do,^LiL

dUgdSl

_ cE,(u>3/a,)|€-(r)P|e-(r)|^_^.V6(u),-u>J

(23a)

TN-14
20 LASER COOLING OF ATOMS 1531

where N is an angular factor of order unitj which

depends on the angles between k, k^, and K,. An

exact expression for A'^ is given in Ref. 23. Neglecting terms of order R/Hu)^, /3^ and higher, we

have

N^ l + j3,cos0,s (23b)

where 9,^ is the angle between K, and k^ and |3, = HKi/Mc. (x)+ is the frequency of the scattered

photon^' and, neglecting the same order of terms

as above, is given approximately by

j+ == oj - {2R/n){l - cose^) -i3,a)(cos6i, - cosfljj ,

(23c)

where 0^ is the angle between k and kj, and 6», is the angle between k and K,.

B. Cross sections

The total cross section for scattering of the laser light by an atom in the translational state

|K,), obtained by integrating the differential cross

section obtained from Eq. (23a) over all angles

of the scattered photon, is (neglecting terms of order R/Hujq, jS^, and (w - wJ/oIq)

J

rfn d<j)^ da

_ CiA^'{r)V-\{r)\^

(Wabs-^)' + i>'" where w^bsi^ given by Eq. (2a) (neglecting the /3^ term). Since y is equal to 4<?^|(r)pa)y(3fe-^), then

a,-677^^|e-<r)p(iy)V[(a)abs-a;)^+(ir)'], (24) where {r) is the unit vector corresponding to (r). If the orientation of the atom with respect

to the polarization vector e is random, the aver- age value of |e '(r)!^ is i, and we arrive at Eq.

(4). However, in a particular experimental situation, the orientation of (f ) and e may be fixed,

in which case this averaging should not be performed.

We will assume as in Sec. II that we have six

laser beams of equal intensity directed along the ±x,±y,±z directions. To avoid the effects of the induced dipole forces we must make the same ap- proximations as in the classical treatment. In this case the initial wave function describing the radiation field becomes:

|n(k)) = |«(k,),n(-kj,«(k,),

n(-k,),n(kj,«(-kj)

where the occupation numbers are the same and

\^i\- k. Equation (23) then becomes

dCTi- L^Idoj.dSl -cZt\^s-rr)\^Z-

(r) |^6(K, + k,, K, + kj6(u> - u,, - {n/2M){K', - K]))

(25)

where the sum over tn represents the sum over

the directions of the six laser beams.

C. Cooling rates and limits For each scattering event, the average change

in the kinetic energy of an atom initially in the state |K,) is the negative of the average change

in energy of the scattered photon and is equal to [with a, given by Eq. (24)]

/

dOj ^(co - 0)3) ^^^ /a, = 2/? + ^k- V, (26)

in agreement with Eq. (3). Again, smaller terms

of order jS^^Wq, /3,/?, and higher have been neglected. This same result was obtained in Sec. II by averaging over all possible directions of the scattered photon. The net cooling rate is given by

ld{Ef-Ei)\

-I-jdnJ^P,iE,-E,)

dO. (27)

where / is the intensity of each laser beam, P,

is the probability for an atom to be in the initial state |K,), and do,^f/dSl is given by Eq. (25). With the help of Eq. (26), we can write

/d{E,-E,)\ _ la, y, ,,,

(-^^%^)-..

where

(i/r')G m,l [wo + {n/2M){kl + 2k „

• K,) - o) - yy]

and y' =i(l -i)y. If we assume that the Doppler

width is smaller than the natural width

(^k„' K,/M«y) and write u) = (jj^ + R/n -^y + Liji we can write when Aa)« y

G„.,^l + Aa;/y'-(^k„-K,)/y'M.

Assuming P, is isotropic, the cooling rate is given by Eq. (10) (maximum cooling for Aa) = 0) and the corresponding limit by Eq. (11). Similarly if the Doppler width is not negligible comTN-15
1532 D. J. WINELAND AND WAYNE M . ITANO 20

pared to the natiiral width, we obtain Eq. (7), if P, is given by a Maxwell-Boltzmann distribution.

Consequently, the limits of Fig. 3 are also obtained. Thus the fully quantum-mechanical treat- ment is not necessary to describe the cooling for

the case of free atoms. We shaU see, however,

that it is necessary for an accurate description

of bound atoms.

V. LASER COOLING OF BOUND ATOMS

A. Matrix elements

To study the cooling process when the atoms

are bound we will assume for simplicity that the atoms are harmonically bound in three dimensions

as discussed in Sec. II. This case may not be

realized in practical systems, but it may be a

close approximation; moreover, it illustrates the important features of the problem. Thus we have

The solution for the eigenstates of a harmonic

oscillator appears in many texts; using the operator formalism we have

where a^ and aj are the lowering and raising op- erators for the states satisfying [aj,aj] -^u- The

position and momentum operators are given by

x^^{K/2M^,)"\ (28) and similarly for y and z . We have * = ^j^^^^,

where in the above representation

ai\ni) = {niY''^\ni - 1> -^ =\n^)\n^)\n,)

a]\ni)^ («j + ly^'^Wi + 1) i,j =x, y,

z

and in the Schrodinger representation^'':

In,) = (2'««j7ri/2)-^/2exp(-Mn,,xV2/2)

^U^\{M^JnY'H'\, (29) where H„^ is the Hermite polynomial of order n^. From Eqs. (19) and (20) we will be interested

in matrix elements of the form

(w'le • (w/M)Pexp(z'k'x)|n)(e|exp(fk' r)|^)

+ (n'|exp(fk-x)|n)<e|€ •pexp(fk- r)|^) , (30) and therefore we are interested in harmonic oscillator matrix elements of the form

(w'|exp(2k- x)|m)

= <ni|exp(ffe,;ic)|w,)(«y|exp(zfey3)|ny)

x(n;|exp(?fe^2)|n2) .

Therefore, in general, we are interested in matrix elements of the type (n'|exp(ffec)|K), where we have dropped the subscripts.

These matrix elements can be straightforwardly

evaluated from the explicit form of the wave functions given in Eq. (29) or by the following operator

method. Since

when

[A,[A,5]] = [£,[A,B]] = 0, we have

exp(zfe(:) = exp[ffexro(a^ +a)]

= exp[- \ (fevo)^ exp(27aofl^) X exp(2fet:(3a)

.

Since r

a"'|n) = <[w!/(w -m)!]^^^|n -m) m^n

^ m>n

then

exp(zfexoa)|«)

{ihx^— It ^\ \n-m)

\{n -m)l/

Thus the matrix elements can be written

{n'\exp(ikx)\n)

x{n'\exp(ikx^^)exp{ikX(fl)\n} .

Operating to the right with exp(ikx^) and to the

left with expiikx^^) and using orthogonality, we

obtain

{n'\exp{ikx)\n)

= exp[-Ukx,r]{n\n'lV%kx^^

,, ^

i-irikx^)""

^oml(m+An)l{nt -m)l ' where An - \n' -«| and «< is the lesser of n and

n' . From the explicit form of the generalized

Laguerre polynomial

iH \n-m

we have

mjm !

TN-16
20 LASER COOLING OF ATOMS 1533

{n'\exp{ikx)\n)

= exp[-i(feCon[nJ/(«.+A«)!]^/2

(31)

of the operator:

+ (k,-<?)e,-P)/(McOo), (32)

In the following we will see that the cooling problem for bound absorbers shares many similarities

with the Mbssbauer effect; therefore, some of the same mathematics^^'^^ can be employed. In the study of the Mbssbauer effect, one typically looks

at the absorption and emission processes separately; for the cooling problem we recall that we

are primarily interested in a scattering process.

Nevertheless, it will be useful to look at the absorption and emission processes separately in the cooling problem as well. With the help of Eqs. (22c) and (22d), Eq. (30) reduces to

im a;o(«' I exp(ik • x) (e • ( r ) - ^^e • P) | «)

and this expression can be substituted into Eq.

(19) to obtain the cross section.

B. Spontaneous emission With the approximation of low laser intensity

used here, the scattered photon results from

spontaneous emission. Therefore, if we look at the emission process separately, we are in- terested in amplitudes which are matrix elements

(nj KiHt,A,,] In,) = i? (e, • (r ) )^ + r{^^) ( (e. -^P)^, -

Strictly speaking, we must sum over aU amplitude

contributions of the intermediate states nj as in Eq. (19); however, if only one intermediate state

Wj is excited, which is a good approximation when

y«fij, then the probability for reemission is proportional to Kn^lAspIn,)!^.

It is instructive to first look at the average en- ergy change of the translational energy states in the emission process. We have

(p _r)- U^T^I^.AE, - E,)\{nf\Ajn,)\'

^""^ ''^^- ;dns^j<n/|A.p|«,)|^

where we can write

T{Ef-Ej)\{nf\Ajnj)\''

- y,(rij\Al\nf}{nf\[H„A,,]\nj)

= {nj\Al[H„A,^]\nj}.

Using the expression

[/^(,exp(-zk,-x)]

= [exp(- ik, x)/2M] {h^kl -2nk, •P) we have

-^/s-ir}l,'Cr}{{e,'P)ils-P))j

(33)

£.(k.-rr))^(?..P

\m=i,y,z I

(34)

where, for an operator J,

<J-),^<n,|j|V,

and Ik^l - |ki.

The third term in Eq. (34) is approximately

equal to Kr)P(fl£^Awo)-^WcJ3^Kr)|%

and the fourth term is approximately equal to

|<r)|^i?(n/a;o)^- The fourth term is clearly negligible with respect to the first, and the third term

is negligible when £^«^a;o, which is reasonable

and is assumed here.

With the above approximations and noting that

{Ef-Ei)^R, (36)

XiK«/|Asp|«,)| (35) we have

which is independent of the intermediate state. This is a familiar result from the studies of the Mbssbauer effect and shows that a fundamental

limit to the cooling process is caused by the recoil heating upon reemission. It also agrees with

Eq. (2b) (averaged over emission angles) for the free atom case (neglecting /3^ terms). At this point we note that

[(k,-<r))/Mc^o]^((e,.P)^>-Kr)|^i3%

and therefore we will neglect the second term in Eqs. (32) and (35) in what follows. We can arrive at a better understanding of the emission spectrum by asking for the second moment of translational energy change in the emission process. This is interesting because it gives the spread of photon energies in the emission process. We have

TN-17
1534 D. J. WINELAND AND WAYNE M . ITANO 20

{{i^EY)={{Ef-E,-{E,-E,))^

= {{E,-Ej)')-R\

where we have used Eq. (36). We have

[Ef - EjV) = ( dn^(Ef- £j^K«/|exp(- ?k, • x)\nj)\y J

dSlV K«/|exp(- il^ • x)\nj)\'' .

Writing

{Ef -Ej)%nf\exp(-ik,-x)\nj)\^

= !<«/ 1 [Ht, exp(- ik^ x)]\nj) | '

,

using Eqs. (28) and (33), and assuming the atoms

to be unpolarized, we obtain

<(A£)^) =!/?<//,>. (37)

If the motion is characterized by the MaxwellBoltzmann distribution, this is just the expression

for broadening due to the Doppler effect. Below we will show that in the classical limit the profile

of the emission spectrum observed in a particular

direction is a shifted Gaussian. In this case, for

the recoil energy R approximately equal to the atom kinetic energy, the spectrum is shown in Fig. 7. We note that Eqs. (36) and (37) are also

valid for the case of the free atom (neglecting

terms of order j3^), since Eq. (33) holds for both cases; however, the spectrum is continuous and

is shown by the dotted line in Fig. 7.

C. Absorption cross section For the case of bound atoms, Eq. (19) becomes

d(v.dSl

<7)|'|e-<r>| E

(«^|exp(- ik^ • x)\nj){nj\exp{ik x)\ni}

UJ0-0J+ {Ej -Ei)/n-^iy

,(„-„.-A^ (38)

o-M

(a)

-fix -i'o^R/ti u<o<3n. i;^

P(m)

(b) y- ' \

\ /

/ \

/ \ / \

/ \

/ \ \ y v,^ ..-r r-..

a>o-R/'fi uiQ^^n Wo+34ix FIG. 7. Atomic spectra in classical limit (KQ^ «kgT)

when R % Kio p. Part (a) shows the absorption cross section for a laser directed along the x axis for the case when y «n^ (giving the discrete lines) and when il^ — (dashed curve) which is also the case for free atom. Part (b) shows the emission spectrum observed along

the X direction for the same two cases.

We can obtain the absorption cross section by

summing over the final states and integrating over

possible scattered photon states. Following the procedure used to obtain Eq. (24), we have

a, = 6TrX^|e '{r)\

{nj\exp{-ik-x)\ni)^y

a;o-ct'+ {Ej -E,)/fi-jiy (39)

y. Strong binding

Equation (39) clearly shows the sideband structure in the absorption spectrum when J2( »y. For

this case, essentially only one term contributes

to the sum over j when the resonance condition

[oJq-(jj + {Ej - Ei)/H= 0] is met. For simplicity, assume that the atoms are unpolarized and that the laser radiation is incident along the x axis, so that a; = ix>Q + mQ,.c{m =0,±1,±2,...) that is, we

are tuned to the wth sideband. Then Eq. (39) be- comes

CT, ^aoK«( + w|exp(/k' x)|«,)| (40) where we have dropped the x subscripts in the wave functions.

It is useful to examine Eq. (40) when R « kgT

and where «, is approximately equal to the mean

occupation number, i.e..

TN-18
20 LASER COOLING OF ATOMS 1535

ni=n*^{n)=(Ht)/{n^J-i.

For the case of interest, the laser frequency must

lie somewhere within the Doppler profile of the atomic resonance (i.e., \m\fiQ,^%^H(jjQ). In this case, we can show that \m\ «n*. We have

and so

(41)

2\l/2

K n* is large (HQ,^/E^ « 1), we can approximate

the Laguerre polynomial in the matrix element by

its asymptotic form.^'' Dropping the * superscript, we have

exp[-i(feXo)T(feXo)''"'Ll::|„,((fec„)^)

^

r(n+l)J|„,l[feCo(4«-2|m I + 2)^/^1

(n, -"llml +^)l'"l/2(n - |m|)!

Since « » |m|, then

{n - \m\)l = nl [n{n - l)(n - 2). . . {n - \m\ + 1)]'^

and

SO that

-«l'"l/V|„|(2n^/^fex„).

Therefore

{n* - \?n\\exp(ik- x)\n*}

-(0''"'jM(2«*^/^(fe^o))

and since

where x^ is the amplitude of the classical oscillatory motion, then Eq. (40) can be written

v/hich is equivalent to the cross section used in Eq. (14). In general, we must average the absorption cross section over a distribution of initial state values. This averaged cross section is then given

by:

a(m)=X)^i^/' (42)

where a, is given by Eq. (40) and where P, is the probability of the atom initially being in the state

I. In this more general case we can arrive at an expression for the cross section by assuming

that P, is given by a thermal distribution. That

is,

P, = [1 - exp(- HClJkBT)]exp{-n,HaJkBT)

,

so that when m «

a(^)=ao[l-exp(-^)] |:_^exp(:"-ff^)K«, + mk"-k)|

-o[-exp(^)]exp[-(^J^.(^)](^,-^^i:exp(.-nttU^ \ n\

kaT j{n-m)\ [L^-ikX)]',

where we have used Eq. (31). We obtain a similar

result for m s 0. These sums can be evaluated

with the generating function^^

/2

If we let

z -expi-H^^/kBT) , x~y-{kx^^, a-\m\

and noting that !„-!_„, then

a(m)=aoexp -m /zfi. 2 k.T -2{kx,f

""(,2 l-ex^{-nnJksT)j\

(2{kx^yexp(-^m,/kBT)

^^""V\ 1-' lexp(-^S2,ABT)

Noting that

{x')^2xl({n)+^),

where (n) is the mean occupation number

(n>=2:«^TM = [exp(|^)-l]", (43)

then

TN-19
1536 D. J. WINELAND AND WAYNE M. ITANO 20

a(m) =aoexp[|w (n^JksT) - k\x^)]

(44) Examination of Eq. (44) shows the skewing of the cross section to high-energy values, i.e., m>0.

This is plausible because, for example, in the low-temperature limit {k^T -^ R) the atom more

easily gains energy rather than losing it in the scattering process. (When T = it is impossible

for it to lose energy.)

A result similar to Eq. (44) is obtained when

one looks for the probability of emission at a certain sideband frequency. From Sec. VB we

write a general expression for the probability

of emission of a photon of frequency LOQ + m^^,

which we call P{m). We have

Pi.m)= X]l^«j -w|exp(-/k^'x)|«j)|^Pj , i

where P, is the probability of the atom being in the translational energy level; when in the excited

state. P{m) is normalized to unity as can be

verified by summing over m . If we observe the emitted light in the x direction, then k^ • x— k^.

If Pj is given by a Maxwell -Boltzmann distribution then we find P{m) = exp[- im [U^JksT) - k\x')]

-Kl^{exp{ri^j2kBT)2(kx,)\n)] . (45)

This same result was derived by an alternative

method and is given by Eq. (3.66a) of Ref. 26. Equation (45) clearly shows the skewing to lowenergy values.

In a high-resolution experiment with lasers used

to probe a bound atom, we might be particularly

interested in the cross section for absorption on

the unshifted carrier given by Eq. (44) for m =0.

We have

a{m=0)=(}oexp{-k\x^))

x/„ [exp(/fn,/fesr)2(fex-,)2(«)] . A special case is when (n) — 0, yielding the cross

section:

o{0-'0) = aoexp{-k^xl).

The exponential factor is known as the DebyeWaller factor and is familiar in studies of x-ray

scattering and the Mbssbauer effect. It shows the

suppression of the cross section due to zero-point

vibrations.

It is interesting to evaluate Eq. (44) in the classical limit, that is, when /znj« feflT. In this case,

if we apply Eq. (16) we obtain

a(m) = {a^J/¥ujj)) exp{- [(a; - u}'o)/(VdV} , (46) where w - u;o = m9,^ - R/ti. This expression is analogous to the cross section used in Eq. (17), but now shows the dependence on recoil. We note

that the envelope of the cross section for the sidebands has a normal Doppler shape; however,

the center of gravity of this Doppler profile is shifted to the frequency (jj'q = io^ + R/H. Similarly, we can also find the expression for the emission

spectrum which then has the same form as Eq.

(46) except that Wq- uJo = Wq -R/H. Thus, we obtain a Doppler profile whose center of gravity is shifted to the frequency uj^-R/H as indicated in Fig. 7.

2. Weak binding

When the condition y «^x is not satisfied, we

must return to the cross section given in Eq. (42), where a, is given by Eqs. (38) or (39). Assuming

the atoms are thermalized and unpolarized, we

have:

a(a,)= E Y7

a{m)

[(2/y)(wo-a) + mnJ]^

(47)

where a{m) is given by Eq. (44). In the classical

limit {ti^„ « feflT), we can use Eq. (46) and convert the sum in Eq. (47) to an integral to obtain

exp[-(^/a)j)']

[(2/y)(u);-w+n)J^

(M,

which is identical to Eq. (6). Thus we see that the absorption cross section for a bound atom

approaches that of the free atom in the limit of weak binding.

D. Cooling rate The cooling rate can be derived from the dif- ferential cross section by means of Eq. (27), where (Kl,^f/cSl is given by Eq. (38). After some

simplification, and assuming the atoms are unpolarized, we can write

d{E,-Ej

dt l£o_Y P f(R + F F) \{n,\expUk-x)\n;)\^

Hoj tf'^'y^'^ ''' '''^l+{(2/y)K-u;+(£.-£,)A]P (48)

TN-20
20 LASER COOLING OF ATOMS 1537

In this form we have separated out the effect due

to reemission (R term) and the effect due to absorption {Ej -E, term).

J. Strong binding

In the limit y«Ci:^, and assuming we are tuned

to resonance (w = u)Q + mQ,j,), then only one term

of the sum in Eq. (48) is important, and we obtain/diEf-E,)\ _ \ dt Im

la {m

)

Ha) [R+m HQ,^ (49)

where o{m} is given by Eq. (42) or by Eq. (44) when P, is given by a Maxwell-Boltzmann distribution. In the classical limit a{m) is given

by Eq. (46) and we obtain

ldiEf-E,)\

\ dt I lo^.H/vijOQij^o

[R + m no,^]

xexp[-(a;-wJ)Vw|], (50)

which is analogous to Eq. (17) but now includes

the effect due to recoil.

2. Weak binding

When the condition y«Q,^ is not satisfied, we

must perform the sum in Eq. (48). In the classical limit h9,^ « k^T we obtain

/d{Ef-E,)\

dt I ^/jra)o<^i>

/:

{2R + nn) exp[-(fiVt^D)]<^^

l + [(2/y)(u;i-co + n)]^ ' which is identical to Eq. (7).

E. Cooling limits

It is also interesting to estimate the limits to the cooling processes; that is, what are the minimum temperatures which can be achieved by the technique? As explained in Sec. II, the limit is obtained when the heating rate due to recoil equals

the cooling rate.

1. Strong binding

We first examine the case when Q,,. »y. We will assume that substantial coo'ling has already taken

place, and therefore the spectrum of the atom

consists of the "carrier" at frequency a;,, and sidebands at Woifij; that is, k{x^)^'^«l. Note also

that for this model to work, we must require R

«fiSl^ or else the reemission spectrum does not

satisfy the simple picture of a "carrier" and two

adjacent sidebands; however, this seems to be a most interesting case in the optical domain. The

case when R »H^^ could be easily treated. For

maximum cooling the incident radiation is tuned

to u} = u)o-^y,, i.e., the lower sideband. In Eq.

(49) we have included the effect of only the resonant sideband in arriving at a cooling rate. However, to obtain the cooling limit, we will in- clude the effect of both upper and lower sidebands

and the carrier. From Eq. (48) we therefore find near the cooling limit: 'dE\_Ia^^( 5 y' \ (51)

which is independent of initial-state distribution

but assumes that the y and z degrees of freedom

are thermalized with the x degree of freedom and

therefore have the same distribution. We might

be tempted to apply Eq. (51) to the case of a single

isolated atom, but we note that if the laser is di- rected along the x axis, then the y and z degrees

of freedom are heated without bound by the recoil

upon reemission. If the frequencies {fij} are degenerate then this argument applies regardless

of the direction of the laser since there are no unique axes for the bound atom. If the frequencies {fij} are unequal, then Eq. (51) cannot hold

for a single laser directed along, say, the i +j + k direction, because the laser frequency

cannot simultaneously coincide with the first lower

sidebands for the three independent oscillations. Each specific case is easily treated; for simplicity, however, we will assume as above that we have a collection of weakly thermalized atoms

such that the collision rate between atoms is much

larger than the optical transition rate. (See Sec. VF). To find the minimum energy possible, we

set {dE/dt) = and obtain

(«x)=Tl(yVn.^)«i.

Note that this expression differs from that of Ref. 7. The difference appears to be due to the neglect of the heating effect due to recoil in the direction perpendicular to the laser beam in Ref.

7. That is, if the unshifted emission in the di- rection perpendicular to the laser beam is considered (the source of the "1+ 1" factor), then

the heating due to recoil in this direction must

also be considered.

The minimum energy is of course limited to the zero-point energy i/?fii; however, this can presumably be accurately determined if ft^. is known, and therefore the uncertainty in energy

will be on the order of {ti^}fiQ,^. In this limit, if the atoms have a Maxwell-Boltzmann distribution, then the temperature of the atoms is given

by Eq. (43). If 0^277 = 12.5 MHz and y/fi^ = 0.1,

then T = 10"* K; however, the fractional uncertainty in this minimum energy is about 3 X 10"^. In this simple case where the absorption

TN-21
1538 D. J. WINELAND AND WAYNE M. ITANO 20

spectrum consists of a strong carrier with weak

adjacent sidebands {k{x^)'-^^«l), we remark that

if the incident radiation were tuned to u>q we would

see in the scattered light a strong component at frequency lUo, a weak Stokes line at (wg-fJi), and

a weak anti-Stokes line at (wo+fij). However,

since the incident radiation is tuned to u;,, - n^

,

the scattered light will contain a component at this frequency, but a much stronger component at fre- quency Wq. Because of this we might regard the cooling process as anti-Stokes or inverse -Stokes

spontaneous resonance Raman scattering.

2. fVea,v binding

When the condition y«^:, is not satisfied, there are two interesting cases to discuss. For both cases we will assume that ><. is much larger than

the transition rate or that we irradiate the atoms

with three mutually perpendicular laser beams.

(See Sec. VF.) In the first case we will assume

that substantial cooling has already taken place so that the Doppler width is much less than the natural width (.which as before requires R«}iy).

Maximum cooling will be obtained when uj -Wg

^-^y and we can expand the denominator in Eq.

(48) to obtain

dE ^/ d{Ef-E,)

dt \ at

la^5]^X«il^(k)^i?A(k)

2 /To)

+ [i/„A(k)])|«,>,

where

A (k) = exp(zk • x) + (l//z y') [exp(f k • x), H^]

After some simplification we obtain

dE

dt

Ia_H^-'w-y

which does not depend on the distribution of initial states and therefore holds for an isolated atom

if we use three mutually perpendicular laser

beams. Minimum kinetic energy is obtained when

dE/dt = 0, and we obtain Eq. (11) ((i:^^)^„ = i/fy). It should be noted here that the total energy of a harmonic oscillator is, on the average, equally

divided between the kinetic and potential energies.

The second case to discuss is when R »Hy.

In this limit we have shown that Eq. (7) applies

when P, is given by a Maxwell -Boltzmann distribution leading to the limits in Fig. 3.

F. Energy distribution for low thermalization rate We are interested in the kinetic energy distribution when we use a single laser beam to cool a

collection of bound atoms. For simplicity we

discuss a particular example; other cases could

be similarly treated. Suppose we have a single laser beam incident

along the x axis and consider the limit y»{p.^.

If we ask for the cooling limit, then we want

a; - u;o--^y when R «P!y. For this case the equation analogous to Eq. (10) is

dt "^'l" riy ^3^j + yc|^2

2 -^^j'

where y, =/ao/(2^u;) and where we have included

the transfer of energy between degrees of freedom (via the "collision" rate y^) and have assumed

for simplicity that the recoil upon reemission is distributed equally over the three degrees of freedom. Note, however, that this simple assumption

is violated even for unpolarized atoms. For the y and 2 directions we have

dt yr

R

+ yc

dE^ _ R

~df^'"J

2

2

-£,

-E,

In steady state, dEi/dt = Q, and we have E^-^Hy,

which is identical to Eq. (11), and ^^ = £^ = ^^y

+ ^Ryi/y^. Hence if yc«y,, the other degrees of freedom are substantially heated by recoil; for a

single isolated atom they are heated without

bound.

VI. SUMMARY

We have discussed laser cooling of atoms from

the standpoint of anti-Stokes spontaneous Raman

scattering. To isolate the essential features of the problem we have made several simplifying

assumptions. We have assumed that the internal states of the atoms comprise a simple two-level

structure and that radiative transitions between

these two levels proceed via pure electric dipole

transitions. We assume that the plane-wave radiation source (laser) has spectral width «y and

power much less than that required for saturation.

We also assume that the power is small enough

to avoid the effects of the induced dipole force^;

this is particularly important for the free-atom

case where it is desirable to have several laser

beams in order to cool in all directions (Sec.

IIB-Z). Finally, we assume that the laser power

is low enough to avoid the effects of recoil heating

in directions perpendicular to the laser beam when

only one beam is used. The treatment has been

nonrelativistic; hence we have neglected terms of order (S^/zoJq and fiR in the energy. The cooling rates for free atoms are given in Eq. (5) (one laser beam in -v direction), Eq. (7)

TN-22
20 LASER COOLING OF ATOMS 1539

(one laser beam in x direction, atoms thermalized), and Eq. (8) [approximation of Eq. (7) when

y,i?/^«a;J. These formulas are justified in Sec.

IV. For bound atoms, the most general expression for the cooling rate (for one laser beam) is given in Eq. (48). Cooling rates for various approximations are given in Eq. (49) (y«nx> one

laser beam directed along the x axis and tuned to

C(j = u)o + wfi,, where m is an integer), Eq. (50) [Eq. (49) in classical limit ^nj(«/?flT], and Eq.

(7) (classical limit when y»n,, one laser beam

directed along the x axis).

As an example, for M = 100 amu, i^o" ^x 10"

Hz(X = 600 nm),y = 277X10 MHz,/ =50 mW/cm^

ao= 5.7 X 10"^° cm^ the cooling rate for "free"

or weakly bound atoms (f2x«y) is given by Eq.

(8) when the atoms are thermalized. This gives a rate dE/dt-- 1.4 eV/sec. We have assumed

that the laser is tuned to w - a)o= - Wc/VT; this

is the condition for maximum cooling when y

«<x>D- This result is independent of temperature

as long as y «a)fl. (Note, however, that the laser

must be swept closer to the line center as the atoms cool to satisfy the condition oj - u)o= - w^/

/2.) Equation (8) will also be invalid when u)o ^ lOy in this example due to saturation of the optical transition.

When the atom is tightly bound {Q,^ » y) we may

apply Eq. (50) in the classical limit (^fi, «/?flT).

We note that for the same approximate detuning

(i.e., assuming the laser is tuned to the sideband

which is closest to satisfying the condition tnHQ,^

= -u) o/{2), the cross section is equal to the cross

section in Eq. (8) times 2Q,^/{T\y), which results

from the compression of the full Doppler profile

into discrete sidebands. The cooling rate is in- creased by the same factor.

It is particularly interesting to examine the cooling limits for various cases. These are summarized in Table I. In each case the mechanism

which limits the cooling is the recoil effect; however, this may not be apparent from the expressions. The limits given in Table I assume that there are no other heating mechanisms present;

in particular, we have avoided the thermodynamic

limits caused by blackbody radiation (Sec. II D)

by assuming that blackbody radiation from the surroundings is absent.

For our example atom above, since R/H «y,

the first limit for both free and bound atoms applies. (The assumption is that n «27rx 10 MHz,

which is usually the case for electromagnetically

TABLE I. Cooling limits for various limiting cases.

Case Limit Assumptions

Free

atoms

Bound atoms

<J?ir,)min =\^y

kaTS^R (Fig. 3)

<-Eifx>mm =Hy

ksT^R (Fig. 3)

R/H«y. Six laser beams in ±x,

±3;, and ±z directions, tuned to

u) = u)Q + R/H-y/2. (Atoms need

not be thermalized via collisions.)

y/2iT is the full width at half-maximum for the optical transition. R/H~y. Ensemble of atoms assumed

to be thermalized with MaxwellBoltzmann distribution.

R/K,U «y. Laser(s) tuned to w = Wo+flA-r/2. If atoms are thermalized only one laser beam need be present. If atoms are not thermalized we

must have 3 mutually perpendicular

laser beams or one laser beam

along i+j-^k direction when

Same as for free atom. (y»S2^)

y,R/fi«^y.. One laser beam

directed along x axis and

tuned to ojq — fi^, atoms thermalized. Same limit for noncoUiding atoms if 9,^=0,^ = ^^ and 3 mutually perpendicular

beams are used.

TN-23
1540 D. J. WINELAND AND WAYNE M. ITA1\0 20

confined ions, for example.) We have (£x^x)min ^ 10"* eV or, if the atoms are thermalized, T^i„

== 2.4 X 10"* K. If the atomic transition was only

very weakly allowed (given by the condition y %R/H), then the limits given in Fig. 3 apply.

This would imply T ^ 2.7 x 10"'' K for M = 100 amu,

!/(,= 5x 10" Hz. However, for this limit to apply we must have y/27r<6 kHz. This condition could

be realized on an intercombination line, for ex- ample. Finally, if the atom was tightly bound, we

could realize the last limit in Table I. In this case the minimum energy is given by the zeropoint energy, but as discussed in Sec. VE, the uncertainty in this energy could be quite small.

ACKNOWLEDGMENTS

The authors are particularly indebted to J. C. Bergquist (National Bureau of Standards), J. Cooper (University of Colorado), D. J. Larson

(University of Virginia), and F. L. Walls (National

Bureau of Standards) for the critical reading of the manuscript and several important suggestions

for improvement. We thank S. Jarvis, Jr. for the computer solution of the plasma dispersion function and E. DeWeese for careful manuscript

preparation. One of us (D.J.W.) wishes to acknowledge partial support by the Office of Naval Research.

Present address: Frequency and Time Standards Group, National Bureau of Standards, Boulder, Col. 80303.

'T. W. Hansch and A. L. Schawlow, Opt. Commun. 13, 68 (1975). ^D. J. Wineland and H. Dehmelt, Bull. Am. Phys. Soc.

20, 637 (1975). ^See, for example, J. E. Bjorkholm, R. R. Freeman,

A. Ashkln, and D. B. Pearson, Phys. Rev. Lett. 41, 1361 (1978); A. Ashkin, Phys. Rev. Lett. 40, 729 a978); V. S. Letokhov, V. G. Minogin, and B. D. Pavlik, Zh. Eksp. Teor. Fiz. 72, 1328 (1977) [Sov. Phys. JETP

45, 698 (1977)]; A. P. Kazantsev, Usp. Fiz. Nauk. 124. 113 (1978) ISov. Phys. Usp. 21, 58 (1978)]; A. Ashkin

and J. P. Gordon, Opt. Lett. 4, 161 (1979). ^R. S. Van Dyck, Jr., P. B. Schwinberg, and H. G. Dehmelt, in New Frontiers in High-Energy Physics,

edited by B. M. Kursunoglu, A. Perlmutter, and L. F.

Scott (Plenum, New York, 1978), p. 159.

^H. G. Dehmelt, Nature 262, 777 (1976); D. Wineland and

H. Dehmelt, Int. J. Mass Spectrometry Ion Phys. 16, 338 (1975); 19, 251(E) (1976).

^D. J. Wineland, R. E. DruUinger, and F. L. Walls,

Phys. Rev. Lett. 40, 1639 (1978). 'W. Neuhauser, M. Hohenstatt, P. Toschek, and H. Dehmelt, Phys. Rev. Lett. 41, 233 (1978). ^A. Kastler, J. Phys. Radium 11, 255 (1950). ^For a summary, see P. R. Berman and S. Stenholm,

Opt. Commun. 24, 155 (1978).

'"a. Ashkin, Phys. Rev. Lett. 24, 156 (1970); 25, 1321

(1970). "b. D. Fried and S. D. Conte, The Plasma Dispersion

Function (Academic, New York, 1961).

'^E. M. Purcell (private communication).

"W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for Special Functions of Mathematical Physics (Springer- Verlag, New York, 1966),

p. 140.

^^D. J. Wineland, J. Appl. Phys. 50, 2528 (1979). '^Ya. B. Zel'dovich, Pis'ma Zh. Eksp. Teor. Fiz. 19, 120 (1974) [JETP Lett. 19, 74 (1974)].

'^S. P. Vyatchanin, Dokl. Akad. Nauk. SSSR 234, 1295

(1977); Sov. Phys. Dokl. 22, 321 (1977).

'^See, for example, F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York,

1965).

'%. G. Dehmelt (private communication).

*'See, for example, D. R. Corson and P. Lorrain, In- troduction to Electromagnetic Fields and Waves (Free- man, San Francisco, 1962), p. 396 or D. Halliday and

R. Resnick, Physics, Part 11, 2nd ed. (Wiley, New

York, 1962), p. 995.

2°E. E. Nichols and G. F. Hull, Phys. Rev. 17, 26 (1903). "R. Frisch, Z. Phys. 86, 42 (1933).

2i.

^^See, for example, Rodney Loudon, The Quantum Theory of Light (Oxford University, London, 1973) or W. Heltler, The Quantum Theory of Radiation, 3rd ed. (Clarendon, Oxford, 1954).

^'j. F. Lam and P. R. Berman, Phys. Rev. A 14, 1683

(1976). See, for example, D. Bohm, Quantum Theory (Prentice-Hall, New Jersey, 1951), p. 296.

^^See, for example, H. Frauenfelder, The Mossbauer

Effect (Benjamin, New York, 1962).

^^H. J. Lipkin, Quantum Mechanics (North- Holland

,

Amsterdam, 1973), Chaps. 2-4.

'^''Higher Transcendental Functions, edited by A. Erdelyi (McGraw-Hill, New York, 1953), Vol. H, p. 199.

28.See Ref. 27, p. 189.

TN-24
Laser-to-microwave frequency division using synclirotron radiation

D. J. Wineland

Time & Frequency Division, National Bureau of Standards, Boulder, Colorado 80303

(Received 28 July 1978; accepted for publication 25 September 1978)

Calculations are made to demonstrate the feasibility of obtaining one-step frequency

division from optical or infrared laser frequencies to a subharmonic in the microwave

spectral region. The cyclotron orbit of a single relativistic electron in a Penning trap is driven with a Gaussian laser beam focused to a spot diameter ~\; the laser subharmonic frequency is measured from the electron synchrotron radiation. The

uncertainty in orbit dimensions is limited to X/2 by radiative cooling and the technique

of motional sideband excitation. PACS numbers: 42.65.Cq, 42.60. -v

I. INTRODUCTION

The need for precise frequency measurement in the in- frared and optical regions is well known.' Present successful techniques' use the method ofharmonic mixing to intercompare optical and infrared laser frequencies and to compare

these laser frequencies with the more conventional micro- wave frequency standards. It would of course be desireable

to simplify this process as much as possible; for example, if the most stable and accurate frequency sources are eventually realized in the infrared or optical regions, one needs to

translate the frequency to very low values to obtain precise

and accurate timing. With this in mind, it is suggested that frequency translation from the infrared or optical region to the microwave region might be accomplished by a single

device which is a type of frequency divider as described

below.

II. PRINCIPLE OF FREQUENCY DIVISION

The more conventional techniques of frequency multiplication or harmonic mixing use the harmonic generation

and mixing properties resulting from a nonlinear device

(e.g., crystal with nonlinear response for harmonic generation) which is driven by one or more linear excitations.' It is noted, however, that harmonic response can also be obtained

by driving a purely harmonic oscillator by a nonlinear (spatially inhomogeneous) field. This technique has been used in high-energy particle accelerators for many years.^ For example, particles in synchrotrons are sometimes driven by radi- ation from microwave cavities (localized to a small portion

of the cyclotron orbit) at frequencies which are higher harmonics of the cyclotron orbit frequency. It is suggested then,

to carry this technique to its practical limit; that is, one can

drive the cyclotron orbit of a charged particle (electron) in a magnetic field at a very high harmonic of the cyclotron fre- quency. The power supplied by the harmonic excitation is balanced by the synchrotron radiation. If the orbit is stable,

then the cyclotron frequency is an exact submultiple of the

driving frequency and can be measured from the emitted

synchrotron or cyclotron radiation. To illustrate this technique, consider the example of

Fig. 1. Radiation from a collimated Gaussian laser beam

(polarized in the x direction) travels in the — z direction and

is focused to a spot diameter So (beam-waist diameter at focal

point). The center of the beam waist is made to coincide with

the orbit path of a single electron confined to the x,y plane

(see Sec. III). As the electron passes through this region it experiences an electric field whose amplitude in the x direc- tion can approximately' be given by

where co, = 2irc/A = Ittvi is the laser angular frequency, t is the time (where r = coincides with the electron crossing

the;' axis), S is the phase factor, and (f> is the amplitude factor

(in volts). Assuming the change in energy (A W) for one pass through the laser field is small compared to the total electron energy and that the spot size is small compared to the orbit diameter, we have

A W^ r eE^ dx = \/2e<t> cosS exp[ -(—- ^V , J_ oo I \2 V AJ 1 (2)

SPOT SIZE = >.

SYNCHROTRON RADIATION

FIG. 1. Schematic depicting the principle of the synchrotron frequency

divider. The electron-cyclotron orbit is stabilized by balancing the synchrotron radiation (frequency v^) with power supplied by the focused laser beam (frequency v,). For the phase-lock condition, the measured cyclo- tron frequency is an exact submultiple (k ) of the laser frequency.

2528 J. Appl. Phys. 50(4), April 1979 2528 TN-25
where the substitution t = x/v (u is the electron velocity) has

been used. We can write the incident laser power in the form

and

477- J^J: =

rrr,\=r,= [Hiq + hV^i^c - 2v^)r'no]

EldAdt^-^\

Stt

If P, = 1 mW, then ^ = 0.87 V, and for y = 0.8c and ^o = A, AW—A.\X 10"^' (cos^) J/pass. If the cyclotron frequency

v^ = 50 GHz (r,. = 0.076 cm), then the power (PJ absorbed

by the electron is P^ = 2 X 10"'°(cos5) W. In steady state, this same power is radiated by the electron, and it is assumed that a substantial fraction can be coupled out at frequency v^. Note in Eq. (2) that A JF depends critically on the ratio cSo/ipX ); this expresses the fact that AW'xs, small unless the

electron experiences a nearly constant phase ofthe laser field as it passes through the interaction region. If the extent of

this interaction region could be made substantially less than

A (for example with some kind of cavity), then the speed of

the electron could be made correspondingly less.

III. ELECTRON CONFINEMENT

The electron-cyclotron orbit can be provided by a static magnetic field B = B^ (277v^ = eBJmc, m is the electron mass). However, deviations from the circular electron orbit path must be S\A, or A Waverages to zero after several

passes. This can be accomplished by confining the electron

in a Penning trap with hyperbolic electrodes." The electric potential inside such a trap is given by

iP=Vo[r^-2iz'-Zl)]/iRl + 2Zl), where r is the radial coordinate, Ro and Zo are the characteristic dimensions of

the trap," and Vo is the voltage applied between the trap elec- trodes. The axially symmetric trap is formed with two "endcap" electrodes which conform to ^Z- = equipotentials and a

"ring" electrode which conforms to the equipotential

ip = Vo. In the nonrelativistic limit, the motion is comprised

of purely harmonic motion along the axial (z) direction at frequency v^ and the motion in the xy plane is described by

the sum ofthe two vectors r^ and r„, which rotate at frequencies v^ and v^. These frequencies are given by

v^ = [eVo/m(R I + 2Zo)] "Vir, and vj. and v„ are solutions of the equation v^ — v^v + ^v^ — 0. When v^>v^, then

Vn.^vl/'^v^ and v^^^v^. To include relativistic effects, consider only the case where | r^ | andz 5 ^A, which can be accomplished by radiatively cooling" the axial (z) motion and suppressing the magnetron motion by the techinque of motional sideband excita- tion." The electron orbit is then nearly circular with

frequency vj.; however, the above expressions must take into account that the electron mass is the relativistic mass m = ymo„ where nio is the electron rest mass and

7 = [1 — (v/cyy^K Note that this special solution is possible

when I r^ | and zS\A only because the electron velocity

(and therefore y) are nearly constant. Treating the problem quantum mechanically, one can show that the amplitudes of the axial and magnetron motion

are given by '

where n and q are the axial and magn'etron-oscillation quan- tum numbers, respectively. Expressed in terms of temperature (rj we have (when hv^4kT^)

hvln + i) = kT, = m(o\<fy

The axial motion can be thermalized to the ambient temperature, which could be that of liquid He. The magnetron

motion could be "cooled" by the technique of motional side- band excitation.'-' Treating the problem classically (see the Appendix) the magnetron radius can be shrunk to a value

ri-2(6;„/^,)<z^>,

where <z^> is the mean-square thermally excited axial amphtude. Assuming R^ = 1.6Zo = 0.1 cm, Vo = 10 kV,

v/c = 0.8, So = A,v, = 50 GHz, and T^ = 4K, we have

v^ = 7.8 GHz, «^10.3,Zio.3 = 175 nm, r^^50nm, and

Bo^3.0 T. Hence the path of the orbit can be confined to < ^A provided A ^ 700 nm in this example.

IV. CYCLOTRON ORBIT STABILITY

Under the proper conditions, the cyclotron orbit will phase lock to a subharmonic of the laser frequency. This is generally true if the condition ^ir> 5 > is satisfied and if the laser has rather modest frequency and amplitude stability. Consider that S = ^tt initially. Then P,s 1.4 x 10"'° W,

and this same power must be dissipated by the synchrotron

radiation. The power radiated by an electron orbiting in a magnetic field in free space is^

PfAnv^

e) rz„= [fi(n + ^)/iTv,ymo]^

For the example conditions discussed above, /y = 2.5 X 10'" W. To achieve the necessary increase in damping rate, it would be necessary to split the ring electrode along a plane

containing the z axis and couple the two halves to a microstrip which is then made to resonate at frequency v^ (2~500). This circuit would then be coupled into frequency-measurement electronics to determine v^. One could al- ternatively reduce the laser power but it is desirable to couple

the maximum power out in order to maximize the signal-tonoise ratio.

Stability is achieved because if the energy ofthe electron

increases slightly, due for example to increased laser power

or reduction in Q ofthe microwave resonator, then the cyclo- tron frequency of the electron decreases slightly (due to the

increase in relativistic mass) and it arrives at the laser spot

experiencing a slightly advanced phase on the next pass, i.e., 5 in Eq. (1) increases slightly. From Eq. (2) this implies 4 ^

decreases slightly and therefore the electron energy decreases. In this sense the electron-cyclotron frequency phase

locks to the laser frequency. To be more specific, suppose

that the division factor Vj/v]. = k. (In the above example if A = 600 nm, k—\0 000.) We now consider the response of

the electron cyclotron frequency to various instabilities in the system.

2529 J. Appl. Phys,, Vol. 50, No. 4, April 1 979 TN-26 D.J. Wineland 2529
A. Laser frequency stability requirements

An obvious upper limit for the phase fluctuations in the

laser is given by the condition that \ S \S^irina time less than the cyclotron orbit time. A stronger limit is, however,

given by the condition that the cyclotron frequency be able

to track the laser frequency as it jitters or drifts. An approximate limit on how fast the cyclotron frequency can slew is given by the rate at which the cyclotron energy decays. (This

is also approximately equal to the maximum rate at which

the energy can increase, which is the condition when 5—>0.) In a more thorough treatment of the problem one must consider the effect of stimulated emission.' This would actually create a stronger tendency to lock as the cyclotron frequency

became less; for simplicity we neglect stimulated emission

here. We have

E '

dvc _ Vc dE

dt~ E dt where Eis the total relativistic electron energy. In the phaselock condition, v, — kv[.; therefore to maintain phase lock, the maximum rate of frequency change that can be tolerated by the laser is dvi

dt -k^P ~E'

which equals 5x10" Hz/sec for the example parameters

chosen above. Stabilized lasers' have shown frequency slew

rates as small as 50-500 kHz/sec, and therefore this condition is satisfied.

B. Laser amplitude stability requirements

Similarly, rather modest amplitude stability requirements are imposed on the laser. When the laser input power

is "matched" to the extracted power, ( Sc^^tt) then phase

lock is maintained except for extremely large amplitude fluc- tuations in a time < 1/v^. Over longer times, if the laser power decreases by more than a factor of 2 from the matched

condition [^ < v 2^(matched)], then A Pf becomes too

small to maintain phase lock.

In the phase-locked condition, we note that amplitude

changes cause phase fluctuations given by the condition that A Wmust remain constant. From Eq. (2), we note that changes in ^ must therefore be accompanied by changes in 5. This implies that the change in phase at the output frequency

(v^) is equal to A5/k.

C. Magnetic field stability requirements

If the magnetic field strength changes, then phase lock

will still be maintained since the cyclotron radius (rj will change to keep v], constant. This is true to the extent that the cyclotron orbit path continues to pass through the laser spot. From the expression a^ = eB /(ymoc) we find that for v^ to remain constant we have

r, \c ) Bo

If we require dr^ < 0. lA, then for the example discussed

above we must have dBo/Bo < lA X 10'', which is easily satisfied. The limit on how fast Bo can change is given by how fast the energy can change for constant a>^. We have

\ dBo^ I dE ^l

p

Bo dt ~ E dt^ E

"'

For the example conditions, the maximum fractional rate of

change of Bo is equal to lOVsec, which is easily satisfied. Similar stability requirements are imposed on the trap electric potentials, but the requirements are relaxed by the ratio

D. Determination of k

The division factor k may be determined two ways: (1) by a prior knowledge of v, to an accuracy of better than l/k

and a measurement of v^, or (2) a measurement of v^ for two

diflFerent values ofk. It is interesting to ask if a given geometrical arrangement can support different values of ^. This is true if the cyclotron orbit passes through the laser spot for

different k values. From the expression co^ = v/r^, it follows

that dr^_ d(o], /c\2

r^ co' \vJ

In the above example, for k to change by 1, dcojco^ — 10'" and we have dr^ =119 nm < So. Therefore the same geometrical configuration will support division by a few values of A:.

E. Radiation pressure

In addition to the force on the electron along the x di- rection as it passes through the laser field, there are forces along the other directions as well. For example, the distor- tion of the laser fields for tight focusing' causes a force along

the z direction. A larger force in the z direction is due to radiation pressure; this is the force due to the v X B term

experienced by the electron passing through the laser spot. The corresponding energy gained per pass, A W^, is much

smaller than A Jf because the electron axial velocity is ex- tremely small. For the example above, A W^^fuo^; moreover,

this force is periodic with frequency v^, not v^, and therefore

its effect is negligible.

F. Background gas pressure

Clearly, any collisions with background gas will cause

the electron to either be scattered out of the trap or, for ex- ample, receive sufficient axial energy that A Jf averages to zero over many passes and phase lock is lost. In an apparatus

at liquid-helium temperatures, pressures of 5 1.3 X 10" Pa

(10'" Torr) should be obtainable, and the primary background gas constituent should be He. For v/c = 0.8, the electron-He total scattering cross section (a) is estimated to

be'" 2.4 X 10'" cmS yielding a total "destructive" collision rate (R ) of

R = n(He)vaS2.2x lO'Vsec,

where n(He) is the He density. Therefore, continuous operation for a time ^ 4x 10" sec could be expected, which is long enough to complete a precise frequency measurement.

2530 J. Appl. Phys., Vol. 50, No. 4, April 1979 D.J. Wineland 2530 TN-27
V. SCENARIO OF OPERATION

A single electron could be provided in the Penning trap

by using established procedures."'* The relativistic cyclotron

orbit could be initialized by using an external drive (across the split ring electrodes) at frequency v^. Note that this exter- nal drive frequency must be swept down as the electron ener- gy increases due to the relativistic mass shift; for the example

above v^(final) = 0.6v^(initial). With microwave electric

field strengths of IV/cm at the cyclotron orbit, this "run- up" time can be less than 1 msec. With the external cyclotron

drive held at i'^(final), the inhomogeneous excitation at fre- quency v^ + v,„ must then be applied in order to "freeze out"

the magnetron oscillations. The laser can then be turned on and the v^ drive turned off and phase locking should occur. Note that the inhomogeneous excitation at v^ + v,„ must be

continuously applied or the magnetron motion will be excited by the background thermal radiation. The focused la- ser beam could be projected through the trap if, for example,

the endcaps are made of mesh with hole size >A.

VI. OPERATION AT LONGER LASER

WAVELENGTHS

Experimental conditions (traps size, etc.) are considerably relaxed by using lower laser frequencies and therefore reducing confinement constraints. For example, if v^ = 25

GHz, Ro = I.6Z0 = 0.2 cm, Vo = 500 V, v/c = 0.8, S^ = A, and r, = 4 K, then v, = 866 MHz, « = 96, z„ = 1.57 ^m,

and r^ = 200 nm, which is suitable for CO2 laser wavelengths (10 /urn).

VII. DISCUSSION

The use of a type of frequency divider as discussed here

potentially has important advantages over presently used

techniques. First, the divider could replace in a single device

the elements in a frequency synthesis chain which may use

several multipliers and intermediate oscillators. It should be

able to divide over a continuous range of laser frequencies

being limited only by the frequency sensitivity of the focus- ing lens. Also, we note that in frequency multipliers of order

k, the noise spectral density increases as k ^ relative to the

carrier. For high-order multiplication this may mean that the carrier becomes completely lost in the noise unless an

oscillator with very high spectral purity is used as the input

oscillator. With a type of frequency divider as discussed

here, this problem does not occur. The noise spectral density decreases relative to the carrier, being limited by the added

instrumental noise in the v^ detection electronics. ACKNOWLEDGMENTS

The author wishes to thank his colleagues R. Barger, J. Bergquist, R. Drulhnger, G. Dunn, K. Evenson, J. Hall, H.

Hellwig, D. Jennings, J. Levine, K. Persson, R. Petersen, E. Smith, S. Stein, and F. Walls for useful comments and dis- cussion. The author also thanks J. Baker and E. DeWeese for manuscript preparation.

APPENDIX

Doppler-effect-generated optical sideband coohng* "

and magnetron cooling'' are formally identical, but for the sake of.conciseness only the specific case of magnetron cool- ing is examined. Briefly, the technique works as follows: To

reduce the magnetron oscillation amplitude, the axial motion can be driven by an inhomogeneous rf field at the side- band frequency v^ + v,„; however, the axial motion pre- dominantly reradiates at frequency v^. The energy difference per scattered microwave photon

h [v^ — (v^ + v„,)] = — hv^ must come from the magnetron energy; therefore, the oscillation amplitude decreases.

Qualitatively, the cooling is limited because the magnetron

motion is excited by the noise generated from the thermal

axial motion via the inhomogeneous rf field. '^ This noise has

the same spectral shape as that of the axial oscillation, but is centered around frequency v^.

To estimate the cooling limit classically, assume that the electron in the Penning trap is subjected to an inhomogeneous rf field E(0 = E(r) cos((y, + (y Jf,

where

(Al)

E(r) = E„ + z dx

x +

dz dz

')

and the x coordinate of the electron motion can be written x = r^ cos(6J> 4- (t)c) + r„ cos(co^t + (p). (A2)

Substituting Eq. (A2) into (Al) we find that the axial motion is driven by a resonant rf electric field dE^r^

EXt ) = ^- V cos(&>/ - <I>J, dx 2 and the driven axial resonance has the form"

2^(0 = *z cos(coJ -(f>„- \tt), (A3)

where b^ = eE^T/(2mco^), and r^ is the amplitude-damping

time constant. The power delivered to the external damping

resistance is"

dE,

dt

mb zo) 2,>2 {eE^

{Am)

r,(16w)-'.

The corresponding rate of magnetron energy change is dE^ _ 0)^ dE^ co„( dE \^ ~ ~dt' ~ " r,(16m)-'. (A4)

dt co^ dt 0)^ \ dx

Note that for the magnetron motion, r^ becomes smaller as £,„ increases since most of the magnetron energy is electric

field potential energy. The axial motion can be described by

z(t) = z,{t) + z„, (A5)

where Zj(t ) is given by Eq. (A3) and z„ is the thermally ex2531 J. Appl. Phys., Vol. 50, No. 4, April 1979 D.J. WIneland 2531 TN-28
cited noise oscillation. Substituting Eq. (A5) into Eq. (Al) we find that the magnetron motion is driven by the rf electric

field

^.(0 = El

dz

(Zj(t) + z„) co%{co, + co„) t. (A6)

The first term is the "damping" term which resulted in the

magnetron energy change given by Eq. (A4). The second

term is responsible for the magnetron noise excitation. To

estimate the effect of this term we consider the following: If the undamped magnetron motion is driven by a resonant

electric rf field the resulting amphtude contains a term

which grows according to'" rJt)cos{coJ + ^) = ; z—cosicoJ + C)- 2m (^icoi-col,)

When excited by noise, the amplitude grows in this fashion

only for a time r^, where r^ is the coherence time ofthe noise. In this case the coherence time is determined by the bandwidth of the axial oscillation, and we have r,. = 2r^. After

t /Tc coherence times the amplitude has grown in a randomwalk fashion and we have

\lmi\(o\-oP-j) r. The total magnetron energy can be written as

Using Eqs. (A6)-(A8) we obtain

dE„ rAeidEJdz)]\zW„

dt \6m(,\o)] - Oil,)

(A7)

(A8)

(A9)

Balance between "cooling" [Eq. (A4)] and "heating" [Eq.

(A9)] is obtained when

/i =

(^z^m)

(K col)

<z'y, (AlO)

where we have assumed that the rf magnetic field is neghgible and have used Faradays law. When co^'^co^ we have

r„^[(2co„/coX^^y]''\ (All)

'K.M. Evenson, D.A. Jennings, F.R. Peterson, and J.S. Wells, Proc. Third

Int. Con/, on Laser Spectroscopy. Edited by J.L. Hall and J.L. Carlsten

(Springer-Verlag, Heidelberg, 1977), Vol. 7, p. 56; D.J.E. Knight and P.T. Woods, J. Phys. E 9 898 (1976). ^H. Semat and J.R. Albright, Introduction to Atomic and Nuclear Physics, 5th ed. (Holt, Rinehart and Winston, New York, 1972), Chap. 12; B.A. Mamyrin, N.N. Aruev, and S.A. Alekseenko, Zh. Eksp. Teor. Fiz. 63, 3 (1972); Sov. Phys.-JETP 36, 1 (1973). 'For the case of tight focusing discussed here the fields become distorted near the focal point; however, this distortion is small enough that it is disregarded here. See D.J. Innes and A.L. Bloom, Spectra Physics Laser

Tech. Bull. No. 5, 1966 (unpublished) 'H.G. Dehmelt and F.L. Walls, Phys. Rev. Lett. 21, 127 (1968); H. Dehmelt, in Advances in Atomic and Molecular Physics, edited by D.R. Bates and L Esterman (Academic, New York, 1967, 1969), Vols. 3 and 5; D.J. Wineland, P. Ekstrom, and H. Dehmelt, Phys. Rev. Lett. 31, 1279 (1973). Note that in general one needs only axially symmetric electrodes; for simplicity the case of hyperbolic electrodes is discussed here. 'D. Wineland and H. Dehmelt, Bull. Am. Phys. Soc. 20, 637 (1975); H.G.

Dehmelt, Nature 262, 777 (1976),

'R.S. VanDyck, Jr., P.B. Schwinberg, and H.G. Dehmelt, to appear in Proc. ofOrbis Scientiae (Plenum, New York, to be published).

'A. Sokolov and Y. Pavlenko, Opt. Spectrosc, 22, 1 (1967); G. Graff, E. Klempt, and G. Werth, Z. Phys., 222, 201 (1969).

»J. Schneider, Phys. Rev. Lett. 2, 504 (1959).

'J.L. Hall, Proc. Esfahan Symp. on Fund. andAppl. Laser Phys., edited by

M.S. Feld, A. Javan, and N.A. Kumitt (Wiley, New York, 1973), p. 469. "The total cross section is the sum of elastic (a^,) and inelastic {a,^ cross

sections. We have <Tj|~7.3 X 10" cm^ See: M. Inokuti and M.R.C. McDowell, J. Phys. B. 7, 2382 (1974); a,„~2.35 X 10" cm'; see M. Inokuti, Y.K. Kim, and R.L. Platzman, Phys. Rev. 164, 55 (1967). "D.J. Wineland, R.E. Drullinger, and F.L. Walls, Phys. Rev. Lett. 40, 1639

(1978); W. Neuhauser, M. Hohenstatt, P. Toschek, and H. Dehmelt,

Phys. Rev. Lett. 41, 233 (1978). ''H.G. Dehmelt (private communication). "L.D. Landau and E.M. Lifshitz, Mechanics (Pergamon, New York, 1960), Chap. V.

'"S. Jarvis, Jr. (unpublished).

2532 J. Appl. Phys., Vol. 50, No. 4, April 1979 TN-29

D.J. Wineland 2532
LASER TO MICROWAVE FREQUENCY DIVISION

USING SYNCHROTRON RADIATION II

J. C. Bergquist and D. J. Wineland

Frequency and Time Standards Group

National Bureau of Standards

Boulder, CO 80303

(303) 499-1000, ext. 4459

Abstract

We present a review of theoretical calculations which demonstrate the feasiblity of obtaining

one step frequency division from optical or infrared laser frequencies to a subharmonic in the

microwave spectral region, and include current ex- perimental designs toward a practical realization

of this goal. We plan to drive the cyclotron orbit

of a single relativistic electron, which is con- fined in a Penning ion trap, with a laser beam

focused to a spot diameter "^ \. Th,is method is an extension of a common technique used in cyclotrons

and synchrotrons where the orbit of high energy

particles is driven at a harmonic of the orbit fre- quency. Our experiment is designed to measure this

orbit frequency which is then a subharmonic of the driving (laser) frequency. This technique requires

that the uncertainty in the electron orbit dimensions be limited to < \/2, which is possible by

radiative cooling and the method of motional sideband excitation. The possibility of a unified

optical wavelength/frequency standard is evident.

Introduction

It is our intent in this paper to summarize

and extend the theoretical analysis of a broadband

laser to microwave frequency divider proposed

earlier.^ We also give a brief description of

experimental designs being considered for the

realization of this division. The importance of accurate freuqency division from the optical spec- trum derives primarily from frequency (time) and

wavelength metrology and the extreme likelihood

that the most accurate, reproducible, and stable

oscillators may be realized in that part of the

electromagnetic spectrum. Certainly, there is great value of such a device in the area of atomic

and molecular spectroscopy.

Principle of Operation

The proposed synchrotron divider is based on

the principle that a purely harmonic oscillator can

be coherently driven (pumped) by a spatially non- uniform, higher order harmonic driving field. This

method has been used for years in cyclotrons,

synchrocyclotrons, and synchrotrons to accelerate

charged particles to relativistic speeds. In these

devices, particles are frequently driven by radiation from microwave cavities, which is localized to

a small portion of the cyclotron orbit, at frequencies integrally related to the cyclotron orbit

Proc. 33rd Annual Symp. on Frequency

Control, 1979, p. 494-497 . Copies

available from: Annual Frequency

Control Symposium, c/o Electronic

Industries Assoc, 2001 Eye St.,

Washington, D.C., 20006.

frequency. For a significant transfer of energy

from the RF driving field into the cyclotron orbit

it is necessary that the interaction time of the

charged particles with the radiation field be

comparable to or less than one-half period of the RF frequency. Longer interaction times quickly

average the transfer of energy to zero.

It would seem possible to straightforwardly

extend this technique to driving fields of shorter

wavelengths in order to drive the cyclotron orbit

of a single electron in a magnetic field at a very

high harmonic of the cyclotron frequency. Theoretically, this is the case, however, shorter wavelengths entail special practical considerations

discussed below. If the orbit is stable then the power absorbed by the electron from the harmonic

excitation is balanced by the emitted synchrotron

radiation. The cyclotron frequency is then a phaselocked submultiple of the driving frequency.

We illustrate the important features of the

proposed method in fig. 1.

SPOT SIZE = X

SYNCHROTRON RADIATION

SYNCHROTRON FREQUENCY DIVIDER

Radiation from a well collimated, Gaussian laser

beam, linearly polarized in the x direction and

traveling in the -z direction is focused into a spot with diameter "v \ by a lens. If it is pos- sible to provide that the electron orbit, which is confined to lie in the x,y plane, pass through this

focus, then the electron will absorb energy from

M

TN-30
the radiation field. Assuming that the envelope

for the electric field vector is approximately

Gaussian and that the absorbed energy per pass (AW)

is small compared to the total electron energy, we have

SB AW = f eE dx = V2 eA cos 6 exp

TicS 2v\

(1)

whereA = field amplitude factor

6 = phase of laser field V = electron's velocity

S = actual diameter of focused line (.'^ \)

c = speed of light

Note that AW shows a double exponential dependence

on the ratio cS /kv. Thus the energy transferred

to the electron is extremely small unless this

ratio is approximately equal to one. This is because the oscillating driving field will rapidly

average the energy transfer to zero unless we can arrange that the electron spend approximately one cycle or less in the radiation field. If the extent of the interaction region could be made

substantially less than X then the speed of the

electron could be made correspondingly less while

maintaining an effective energy transfer. A pos- sibility may be a dielectric waveguide with trans- verse dimensions less than X that is projected into

the trap and is cut off within about k of the

electron orbit.

Confinement of the circular electron orbit

path to < \/2, which is necessary to prevent AW •* 0, is accomplished by trapping the electron in a Penning trap with hyperbolic electrodes immersed in

a static magnetic field B = 6 z. The electrostatic potential is given by (|) = V (r^-2(z^-z ^))/

(r * + 2z ^) where r is the radial coordinate,' z is the axial coordinate, r and z are the character- ' o o Istic dimensions of the trap,^ and V the voltage

applied between the trap electrodes. The axially

symmetric, static electromagnetic trap is thus

formed w^h two endcap electrodes which are con- structed 'to lie on the tj) = equipotential surface

and a ring electrode which conforms to the equipotential 4> = V . The motion of the electron is given by the synthesis of three separate oscillations. First there is the harmonic motion parallel

to the z axis at frequency v = (eV /an^ir ^ +

2z ')) due solely to the applied electric field.

In the radial plane ^he notion is-comprised of the SUB of two vectors r and r which rotate at frec D

quencies v ' and v . These frequencies are ap- proximated by V = V */(2v ) and v ' = v = B Z C C C (eB /27imc), when v » v . In order to satisfy our requir^Bent to confine the electron orbit path to < X/2, Ir

I and z Bust be held to < X/4. This can be

accomplished by radiatively cooling the axial (z)

motion and suppressing the magnetron (r ) motion by

the method of motional sideband excitation. ^ ' ^ The

electron orbit is then nearly circular with cyclo- tron frequency V ' .

The minimum r is given by^

r 2 s 2 (u) /u) ) <z2> m m z (2)

S = X, B ^ IT with

o ' o

where <z^> is the mean square thermally excited

a'xial amplitude. As a quantitative example, assume

that r S 1.6 z = 0.1 cm, V = 10 kV, v/c = 0.8,

o ' o ' ' V -v 15 GHz, and T =4 K.

c ' z These conditions will give v =7.8 GHz, ^<z^> =

175 nm, and r "^ 150 nm and an orbit confined to m

less than X/2 provided X >. 700 nm. We note that V

is extremely large in order to confine the axial z excursions to < X/4. This requirement is greatly

reduced if the focus is a cylindrical line parallel

to z rather than a spot of dimensions X on all

sides. This focus (X x X x 20X) could be obtained

with a softly focused beam followed by a strong

focusing cylindrical lens. A smaller value of V

reduces the likelihood of spurious field emission

currents which could give rise to extraneous trapped electrons,

reduction of V

Since <z^> « v

" and V a V ^, a o z by 10^ increases ^<z^> by 10 and

reduces V by the same factor. From Eq. 2, the

minimum value of r increases by only a factor of

three. Thus, this change in V to 100 V, allowed

by the cylindrical lens focus, gives v = 780 MHz,

V<z^> "^ 1.75 pm, and r =0.45 pm, permitting X > 600 nm. (It is further noted that the electron

orbit confinement problem is greatly reduced by

choosing longer wavelength lasers).

If we now provide an energy balance between

the energy absorbed by the electron and the energy

lost through synchrotron radiation, which is efficiently coupled out to the frequency measurement

electronics, the the cyclotron frequency will phase

lock to a subharmonic of the laser frequency.

Given that the laser has only minimum amplitude and

frequency stability,^ the phase lock condition will occur for 0<6<7i/2. Consider that 6 = 7i/4 initially

and that the laser power incident on the cylindrical lens is approximately 20 mW. The power absorbed and subsequently radiated by the electron,

confined as described above, is about 1.4 x 10

W.

Feedback

Stability of the phase lock is automatically

achieved since, for example, an increase in the

energy of the electron results in a corresponding

495 TN-31
decrease in the cyclotron frequency owing to the

increase in relativistic mass. The electron thus

experiences a slightly advanced phase of the laser

on the next pass through the focus. This implies

that AW decreases slightly and therefore the elec- tron energy decreases. It is in this manner that

the electron phase locks to the laser frequency.

We note that if initially, or at any later

time, an energy imbalance exists between the energy

absorbed and the energy radiated, then the auto- matic tracking of the electron to the balance point

is not critically damped. Rather, the electron

slews to the lock point in a slowly damped oscillatory fashion with a time constant equal to the trap

coupled radiative decay time^ of approximately 1 ms

.

To be more quantitative, we write the change

of phase of the microwave field (4>) as Sip = i(t, , „^.

where 0(t) is the frequency deviation from the

normal "locked" cyclotron frequency -U) '. We have

(t»(t) = Auu(t) = Au)^(t) + AiUj(t) (3)

where Aw^^ is the is the instantaneous frequency

deviation of the electron from u) ' due to the

c energy imparted by the laser, and Aiu, is the fre- quency deviation due to the radiation decay.

Writing 6 = 7T/2 - and assuming we are in the high

power limit of the laser then the nominal value of

6 - nil so that Eq. (1) becomes AW = W 6, where W

= V2 eA exp (-(ncS /2v\)2]. Now 6 = kcji where k is the division factor v„/v ' so that AW = W kill. 1 z o

^

Noting that

dUa^^_

dt «s) fb

where E is the total energy of the electron, then

(4)

d(Aa)^)

To estimate

dt wd we first note that

dt

d(AiJu,) u)

' d ^

dt t{f)

An accurate expression for the energy decay

due to synchrotron radiation (dE/dt) , in the ab- sence of coupling structures is given in Ref. 1. Using this expression, we can find the dependence

of d(Au),)/dt on the difference frequency (Aiu) and

obtain

d(Au».) Ae^Y^w '^ d .^1 c dt 3Ec Aw = - - Au) (5)

where t is approximately equal to the damping time

at low energy. We will assume that this expression

is also valid when T is decreased by coupling the

electron to the trap electrodes. Differentiating

Eq. 3 with respect to time and using Eqs . (4) and

(5) we obtain

$ + ^ (t) + (U) )2 = (6)

where uj. (Vc'V^)""- Thus phase oscillations

around the nominal "locked" phase occur at fre- quency u). and are damped with approximately the

radiation damping time. For a laser power of 200 mW (\ = 3.39 |Jm) in the above example W = 1.8 x

10 ergs, and u)./2n = 1.7 x 10® Hz

Spread of the electron wave packet

To estimate the restrictions placed on the

model by quantum mechanics we start with the un- certainty relation A(|» An i 1, where A(t) is the

uncertainty in phase of the cyclotron orbit of the

electron and An represents the corresponding un- certainty in energy for the electron. Neglecting

the electric field and electron spin, the electron

energy is given by

E = [(mc2)2 + mc^hu) (n + ^)1^ (7)

We want to build a wave packet which has its phase

defined with an accuracy A(|i <. (2ti\/ 10) / (Znt ) -4 ^ = 1.3 X 10 rad. From the uncertainty relation

this requires a spread in energy quantum number of

4 An < 0.75 X 10 . From Eq. (7) this corresponds to

a range of natural frequencies Au) = u) ' An/n = 1.3

-6 ^ X 10 u) ' . Classically, if we assume that the

initial conditions for Eq. 6 are given by these

values of A(|i and Aut, then we see that the phase is bound and initially oscillates with amplitude 1.75 -4

X 10 rad. Quantum mechanically, it therefore

seems likely that the electron will phase lock if

the initial wave packet has a limited range of

values of n and (j). Clearly this treatment is not rigorous and a more careful analysis of the quantum

fluctuations must be made.

Acknowledgments

The authors acknowledge input from others; in

particular we thank Wayne M. Itano and P. L. Bender

for useful discussions concerning the uncertainty

problem and S. R. Stein for helpful input on the

electronics problems.

References

1. D. J. Wineland, J. Appl. Phys. 50, 2528

(1979).

496 TN-32
H. G. Dehmelt and F. L. Walls, Phys . Rev. Lett. 21, 127 (1968); H. Dehmelt, in Advances

in Atomic and Molecular Physics, D. R. Bates

and 1. Esterma*n, eds . (Academic, New York,

1967, 1969), Vols. 3 and 5. D. J. Wineland,

P. Ekstrom and H. Dehmelt, Phys. Rev. Lett.

31, 1279 (1973). Note that in general one needs only axially symmetric electrodes, for

simplicity the case of hyperbolic electrodes

is discussed here.

R. S. Van Dyck, Jr., P. B. Schwinberg, and H.

G. Dehmelt, in New Frontiers in High-Energy

Physics , Kursunoglu, Perlmutter, and Scott,

eds., (Plenum, New York, 1978) p. 159.

497

TN-33
Abstract Submitted

for the Houston DEAR Meeting of the

American Physical Society

10-12 December, 1979

Physics and Astronomy

Classification Scheme

Number .32

Suggested title of session

in which paper should be pic

Atom-Photon Interactions

Bull. Am. Phys. Soc.24, 1185 (1979).

Frequency and Time Standards Utilizing LaserCooled Ions .* Wayne M. Itano and D. J. Wineland, Frequency and Time Standards Group, N. time) standards

B.S . --Frequency (and

could be based on microwave transitions

(where precise synthesis is currently possible, as opposed to optical transitions) in stored ions. Laser

cooling to ^ 0.1 K reduces the second-order Doppler

shift to < 10"!^. Transitions can be detected with near

unit efficiency by repeated laser scattering [1].

Schemes using a single laser for both cooling and detection are possible only for ions with a strong optical

transition and no intermediate metastable level e.g.,

He"^, Be"*", Mg"*", Zn"^, and Cd"^. A simple scheme could use the (M,=l/2, MT=5/2)e^(Mj=l/2, Mi=3/2) transition in

67Zn+(\>-lGHz) which has an extremum at H -80 kG with

|6v/vo|-10"2(AH/Hq)2. The ions could be cooled and simultaneously optically pumped into the (Mt=5/2) level.

Depletion of this level by microwaves would be detected

as a decrease in the photon scattering rate. For 10°

ions the shot-noise-limited S/N ratio is 10^. For an interrogation time T of 100 s, QrHTv^^SxlO'^ , and a^(lOOs)

-IQ(S/N)]-T~3xl0-15. Experimental work is being initiated on Cd+. *Supported in part by AFOSR.

[1] D.J. Wineland, R.E. Drullinger, and F.L. Walls, Phys,

Rev. Lett. 40, 1639 (1978).

( ) Prefer Poster Session

(;<) Prefer Standard Session

( ) No Preference

^;

David J. I'^neland

Time and Frequency Division

National Bureau of Standards

Boulder, CO 80303

Address

TN-34
Abstract Submitted

for the Houston DEAR Meeting of the

American Physical Society

10-12 December, 1979

Physics and Astronomy

Classification Scheme

Number 35

Suggested title of session

in which paper should be p"

Atom-Photon Interactions

Bull. Am. Phys. Soc.24, 1185 (1979).

Laser Induced Magnetron Compression (Expansion)

of Ions Stored in a Penning Trap .* D.J.Wineland, R.E.

Drullinger, J.C. Bergquist, and Wayne M. Itano, Time and

Frequency Division, N.B.St -As conjectured earlier [1]; we

have been able to shrink (or expand) the magnetron orbits

of ions stored in a quadrupole Penning trap with radiation pressure. In the axial ly symmetric trap,_^the radial

motion of ;^he ions can be expressed as r(t) = rc(0)exp

(-ico^t) + r|y](0)exp(-ica^t) where the x(y) coordinate corresponds to the real (imaginary) part in this notation

and where w"^ and w are the cyclotron and magnetron frequencies, when COfji: m

w. « Y(the natural linewidth of the

ion optical transition) the scattering of a photon may be

viewed a^ a p^int-like interaction which imparts momentum

change ^p = Tfk^to the ion. (k is_^the photon wave vector).

Hence Ar^^ = iTii<(M(a}ni-aj^))~' and Ar^, = iTik(M(a)^-a)f^))~^ per

scattering event. We present data_^illustrating the re- sulting shrj_nking or expanding of r^; the corresponding

effect for r was described earlier [1]. Such effects

could also be realized by other means of momentum transfer

(e.g. atomic beam or coupling r to negative resistance),

*Supported in part by ONR. "^ [IJ D.J. Wineland, R.E. Drullinger, F.L. Walls, Phys.

Rev. Lett. 40, 1639 (1978).

( ) Prefer Poster Session

(x) Prefer Standard Session

( ) No Preference

David v3. W'tneland

Time and Frequency Division

National Bureau of Standards

Boulder, CO 80303

Address

TN-35

J
Appl. Phys. 22, 365-368 (1980) Applied Physics

© by Springer- Verlag 1980

High-Resolution Optical Spectra of Laser Cooled Ions

R. E. Drullinger. D. J. Wineland, and J. C. Bergquist

Time and Frequency Division, National Bureau of Standards, Boulder, CO 80303, USA

Received 25 February 1980/Accepted 29 April 1980

Abstract. We obtain essentially Doppler free spectra of the naturally occuring isotopes of

Mg"^, which are bound in a Penning trap, by using a frequency stabilized laser to continuously cool the ions, while the scatter rate from a second, frequency swept laser is monitored. We show that the magnetron motion as well as the cyclotron and axial motion

can be minimized. Line position measurements yielding resonance transition energy, isotope and hyperfine shifts are reported.

PACS: 32

Radiation-pressure cooling to substantially reduce

Doppler broadening, shifts, and residual velocity effects to all orders [1, 2] represents an important step in the study of ultra-narrow, lifetime limited transitions. The

method reported here investigates the full Doppler

width of the quantum absorber studied ; but, because

the absorbers are substantially cooled by radiation

pressure, the line width can be dominated by the natural resonance width. Furthermore, in contrast to conventional sub-Doppler techniques, this method di- rectly reduces all velocity effects. Recently, cooling of a sample of bound resonant

absorbers by radiation pressure has been demonstrated [1, 2]. In [1] Mg^ ions were confined to a room

temperature Penning trap and were irradiated near the ^Sp2 — ^^3 -> resonance transition (280 nm) with the output of a frequency doubled cw dye laser. In (such) a

two-level system, the process of absorption followed by

re-emission returns the atom to the original internal

state; but, in general, a different kinetic energy state. After many such scattering events, the collective mo- mentum of the absorbed photons can be made to cancel the thermal momentum of the absorber, whereas the recoil momenta from the randomly re-emitted

photons cancel to first order [3]. In this paper, we

describe some of the interesting features of electromagnetically trapped ions which are cooled by this tech- nique and also report the first high-resolution spectra

obtained on a laser cooled species. Preliminary results of this work are reported in [4].

Briefly, the eigenmotions of ions in a Penning trap [5] are a simple harmonic oscillation in the electrostatic well along the trap axis, a cyclotron orbit in the plane

perpendicular to the trap axis, and a magnetron drift orbit also in that plane. The axial vibration and the cyclotron orbital motions are thermal and coupled

through the long range coulomb collisions between

ions in the trap, whereas the magnetron motion is nonthermal and not strongly coupled to the other

motions [1, 5]. In fact, the magnetron motion is in an

unstable equilibrium in the trap, but the confinement

time of unperturbed ions is observed to be days.

Experiment

In our present experiments, the laser beam (diameter

=5: 100 jam) crosses the trap normal to the horizontal

trap axis (B field axis) and is also linearly polarized

perpendicular to the trap axis. With this configuration,

the laser directly interacts with both the cyclotron and

magnetron motion. These motions can be represented

as [6]

:x + iy = r, + r„ = r. (1)

where r^ and r„ are the cyclotron and magnetron

radius vectors, respectively, and co^, co^ the associated

angular frequencies. If the ions undergo an incremental

change in velocity, A\, per scattering event, we find

0340-3793 /80/0022/0365/$01.00

TN-36
366 R. E. Druliinger et al.

o

o

Time —

Fig. 1. Scatter rate vs. time after the laser has been blocked for 5 min. On unblocking the laser, the ion cloud which has begun to spread and heat, yields a low scatter rate at first, but returns to its previous

value in approximately 1 min

that there is a corresponding incremental change in the radius vector given by

Ar^

C0, — 0)„ A\ and At„CO. - aj„ Ay. (2)

By irradiating the side of the ion cloud which recedes

from the laser with photons of frequency co^ < cOq, both

r^ and r^ can be made smaller. This effect is demonstrated below.

A cloud of < lO"^ Mg^ ions is loaded into the trap. The

static magnetic field is 0.976 tesla and the electric potential is % 7 V. The laser beam is initially positioned about 600 )am below the trap center; that is, where the magnetron motion carries the ions in the

direction away from the laser. The laser frequency is swept up to and held at approximately the half- intensity point of the resonance transition which is Doppler shifted by the magnetron motion. Photons

which are backscattered into an //4 cone are collected and counted. The overall detection efficiency is a; 3 X 10""^. The scatter rate is observed to decrease in time indicating that the ions move closer to trap

center. This process is repeated by moving the laser beam toward trap center in 100 |am steps. When the beam reaches a position of :^100(im below trap

center, the scatter rate remains constant, which in- dicates a stable cloud configuration (dia: ^200^m)

has been obtained.

Additionally, if the laser radiation is blocked, the ions

begin to heat and spatially spread; however, the low

entropy ion cloud can be reconstituted at trap center by simply unblocking the laser. This is indicated in Fig.

1, where a plot of scatter rate vs. time is shown, after

the laser had been blocked for 5 min. The signal rises and then stabilizes at the previous scatter rate, indicat- ing that the ions have returned to the preblocked

configuration at trap center. If the laser beam is positioned above trap center, the cloud is "blown-out"

to a large radius, but still remains confined to the trap.

Once again, we can reform a small, cold ion cloud by

spatially stepping the laser below trap center and

repeating the above procedure.

If the ions are localized and cooled and the laser beam

maintained about 50 [im below trap center, an estimate

of ion temperature can be obtained by rapidly scan- ning the laser through the optical transition and

recording the scatter rate as a function of laser fre- quency. The broken curve (circles) in Fig. 2 shows a

typical plot of scatter rate vs. laser frequency. As the frequency is swept into resonance, the ions are maintained cold and a nearly Lorentzian line shape is produced. However, even at our low laser power

(*8|a.W) and rapid sweep (700MHz/s), the scattering process at laser frequencies blue of resonance significantly heats the ions which results in the observed

line asymmetry. The half-width at half-maximum

(HWHM) of the cooled side of the line is 3=40MHz

corresponding to a temperature of JtiO.S K. It is to be

noted however, that the laser beam, which has nonzero extent, interacts with a spread of magnetron

orbits and velocities. The magnitude of the resulting

Doppler broadening is a function of the cloud size and

charge density, but it is expected to result in the major

portion of the observed 40 MHz half width. This

indicates that the cyclotron and axial motions may be

substantially colder than 0.5 K. This residual Doppler

effect due to magnetron motion could be avoided

by interrogating the ions with the laser directed

along the trap axis. To help overcome the effects of heating as the laser is scanned through the line, a second, frequency-doubled,

ring dye laser is introduced into the trap. The ring laser

is frequency-stabilized by locking to a saturated absorption hyperfine feature in iodine whose doubled

frequency is nearly coincidental with but to the low

frequency side of ^"^Mg"^. The stabilized laser serves to keep the sample cold as the first laser is scanned at reduced power. For reasons not fully understood at

this time, the half line-widths obtained in this fashion

are a factor of two broader than those obtained with one laser if all other parameters are unchanged. If the

static magnetic field is increased from the previous

s;1.0 to % 1.4 Tesla, the two laser scheme also pro- duces narrow lines corresponding to temperatures

^0.5 K. In Fig. 2, we show the line profile of the two

laser generated signal (solid curve).

Equally important, the second fixed frequency laser allows us to do high-resolution spectroscopy of like or

TN-37
High-Resolution Optical Spectra of Laser Cooled Ions 367

dissimilar ionic species. The magnesium used in these experiments had a natural isotopic mixture containing

approximately 80% ^^Mg^ and 10% each ^^'-^Mg^.

The ^^--^Mg"^ lines are positioned in the high fre- quency wings of ^'^Mg^. Thus, scanning these lines heats ^'"'Mg'^, which in turn heats ^^Mg"^ and ^^Mg"*"

by Coulomb collisions. This is in addition to the heating caused by scanning to the high frequency side of ^^Mg"^ and -^Mg"^. These heating effects are largely

eliminated by the frequency stabilized laser. Whereas

^''^Mg'^ is cooled directly by the fixed frequency laser, ^^Mg"^ and ^^Mg"^ are cooled and maintained cold by

Coulomb interactions with the cooled ^"'Mg^. This

confirms an earlier conjecture [1] that one ion (in this case, ^"'Mg"^) can be used to cool other ions of

spectroscopic interest. Monitoring the scatter rate of the low power, fre- quency swept laser results in the spectrum shown in Fig. 3. ^*Mg* and -^Mg"^ have zero nuclear spin and

yield single lines, while -^Mg^ has nuclear spin 7 = 5/2

and would ordinarily exhibit six lines. However, the nuclear hyperfine components are optically pumped

such that nearly all of the ^^Mg ions are in the Mi= —5/2 state. Hence, ^^Mg'^ is also observed as a

single line. The relative positions of the observed lines depend upon the isotope shifts and (in the case of

'^^Mg'^) the hyperfine coupling constants. The isotope

shift of the ^Si/2 — ^^^3/2 lii^^ ir* Mgll is measured to be

zlv(2*Mg-^^Mg) = 3.050±0.1GHz. (3) Combining this result with the mass shift formula,

which is a good approximation for low Z elements

such as Mg"^, we calculate that the isotope shift for "Mg+ is

Zlv(^^Mg- "Mg)= 1.461 ±0.1 GHz. (4) The frequency difference between ^^Mg(;?i;= —5/2, mj = - 3/2^m^ = - 5/2, mj=- 1/2) and ^^Mg(Wj = - 3/2

*m, 1/2) optical transitions is

zl/ = 772.1 ±60 MHz. (5) From this result, we obtain the hyperfine coupling

constant, a^s, for the ^S^:2 state of ^^Mg"^

^35=- 607.8 ±50 MHz. (6) This is in fair agreement with earlier works of

Crawford [7] and Weber [8]. For this calculation, we

first estimated the ^Pjj2 coupling constant to be a^p =

— 19.1 ±5 MHz using the Goudsmit formula and the known fine structure separation [9]. The uncertainty

in this value of ±5 MHz contributes an uncertainty

± 15 MHz in the value calculated for Aj^. We expect to substantially improve the accuracy in the a value for both states with a double resonance technique presently being pursued.

Fig. 2. Scatter rate vs. laser frequency. The curve shovk'n in circles is with one laser only. The solid curve uses one laser to produce cooling

while a second scans the line profile

-3GHzMgH RESONANCE

TRANSITIONS

{3s S, , my

Fig. 3. Spectra of one Zeeman component of laser cooled

24.25,26j^g+ jj^g room temperature Doppler width of these lines is about 3 GHz. Only the ^^Mg"^ is directly laser cooled. The •^^Mg'^ hyperfine structure has been optically pumped resulting in the observation of a (m^= — 1/2, m^ = — 5/2)<-*(w_, = — 3/2, m,= —5ll)

component

By measuring the frequency difference between the center of the Doppler broadened /, line and the

particular hyperfine component to which the stabilized ring laser is locked, we are able to reference the ^'''Mg'^

3^Si,2-«-^3^P3,2 resonance transition to the known

iodine spectra [10]. Subtracting out the linear Zeeman

shift, we determine the 3^Si/2'*-^3^P3,2 resonance transition to be 35,760.834 ±0.004 cm "^ This compares

favorably to the accepted value of 35,760.88 cm" ^

[11].

Conclusion

In conclusion, we have presented high-resolution spec- troscopy of laser cooled Mg"^ ions electromagnetically

confined to a Penning trap. We have demonstrated a

TN-38
368 R. E. Drullinger et al.

general two laser scheme which provides both ra- References

diation pressure cooling and high-resolution spectro- i scopy on the cooled species. Additionally, we have

demonstrated sympathetic cooling of a second species 2. by long range Coulomb cooling (i.e. ^^Mg'^ and

^^Mg"^ by ^"^Mg"^). We have minimized the motions of 4 the trapped ions yielding a temperature <0.5 K; but,

most likely, substantially colder for the axial and

cyclotron motions. We make the observation that the ^

present experiments are limited in accuracy and resolu- tion by the broadening and shift of the optical re- sonances due to residual magnetron motion. This

limitation can be removed by probing the cooled ions 6. with a laser directed along the trap axis.

Acknowledgements. The authors wish to thank W. M. Itano for 8. assistance in parts of the experiment. We thank Dr. Itano and other 9. members of the Frequency and Time Standards Group for helpful discussions durmg the course of this work. We acknowledge partial support by the Office of Naval Research and the Air Force Office of 10. Scientific Research. 11.

D.J.Wineland, R.E. Drullinger, F.L.Walls: Phys. Rev. Lett. 40, 1639-1642 (1978) W.Neuhauser, M.Hohenstatt, P.Toschek, H.Dehmelt: Phys. Rev. Lett. 41, 233-236(1978)

D.J.Wineland, W.M. Itano: Phys. Rev. A20, 1521-1540 (1979) R.E. Drullinger, D.J.Wineland: In Laser Spectroscopy IV, ed. by H.Walther and K.W.Rothe, Springer Series Opt. Sci. 21

(Springer, Berlin, Heidelberg, New York 1979) H.G.Dehmelt, F.L.Walls: Phys. Rev. Lett. 21, 127-130 (1968); H.Dehmelt: In Advances in Atomic and Molecular Physics, ed. by D. R. Bates and I. Esterman (Academic Press, New York 1967,

1969), Vols. 3, 53-72 and 5, 109-154;

D.J.Wineland, H.G.Dehmelt : J. Appl. Phys. 46, 919-930 (1975) J.Byrne, P.S.Farago: Proc. Phys. Soc. (London 86, 801-815

(1965) M.F.Crawford, F.M.Kelly, A.L.Schawlow, W.M.Gray: Phys.

Rev. 76, 1527-1528 (1949) E.Weber: University of Heidelberg, private communication

Average of values obtained from Eqs. (III. 84) and (III. 86) of N.F.Ramsey: Molecular Beams (Oxford University Press 1956)

p. 75 S.Gerstenkorn, P.Luc: Rev. Phys. Appl. 14, 791-794 (1979) P.Risberg: Ark. Fys. 9, 483 (1955)

TN-39
June 1980 / Vol. 5, No. 6 / OPTICS LETTERS 245

Double-resonance and optical-pumping experiments on

electromagnetically confined, laser-cooled ions

D. J, Wineland, J. C. Bergquist, Wayne M. Itano, and R. E. DruUinger

Frequency and Time Standards Group, National Bureau of Standards, Boulder, Colorado 80303

Received March 3, 1980

Experiments illustrating advantages and unique features of double-resonance and optical pumping on electromagnetically confined, laser-cooled ions are discussed. In certain cases, scattered light from the cooling transition can

be used as a monitor in double-resonance experiments to give nearly 100% detection efficiency. Nonradiative re- laxation rates are extremely small for stored ions, permitting nearly complete optical pumping, even in extremely

weak pumping schemes.

If one hopes to improve accuracy and resolution substantially in spectroscopic studies beyond the present

state of the art,^ then one must address the problems of

observation time (i.e., line Q), and first- and secondorder Doppler effects. With this in mind, experiments

have been carried out at the National Bureau of Standards^ and Heidelberg University'^ to cool substantially

by radiation pressure (to less than 1 K) ions that are stored in room-temperature, macroscopic electromagnetic traps. In this Letter, we point out further advantages of using laser-cooled, trapped ions in highresolution, double-resonance schemes and demonstrate some of these features with preliminary data. In some

cases, high-Q microwave or optical transitions can be

detected with nearly 100% efficiency; in addition, negirly

complete optical pumping can be obtained, even in very weak pumping schemes.

More than a decade ago, Dehmelt and his co-workers

pointed out advantages of using stored ions for highresolution spectroscopy.'* Storage times of ions can be

extremely long (days), and perturbations to the measured frequency from ion-ion collisions and confinement can be very small (less than 1 part in 10^^). This

is to be contrasted with experiments on confined neutral

atoms, in which spin exchange, buffer gas, and wall

shifts can limit the accuracy achieved at the parts-in- 10^^ level. Progress on stored-ion spectroscopy has

been slowed somewhat because the number of ions that can be stored is rather small (typically ;S10^), thus

yielding a small signal-to-noise ratio in double-reso- nance experiments. Because the ions are confined, the

Dicke criterion is satisfied at microwave wavelengths;

thus first-order Doppler shifts and broadening can be

eliminated. Unfortunately, the temperature of ions in the trap is often above the ambient temperature, thus

causing significant second-order Doppler effects. The

cooling that has been achieved^-^ minimizes the sec- ond-order Doppler shift, and, as discussed below, the

use of lasers in high-resolution, double-resonance ex- periments can increase the signal-to-noise ratio, limited

only by the statistical fluctuations in the number of ions

that have made the transition.

For high-resolution spectroscopic experiments on

laser-cooled ions, the use of double-resonance schemes

may be a natural choice. This is because, for cooling, one prefers to drive a strongly allowed electric dipole

transition repeatedly, whereas for the high-resolution

spectroscopy, a microwave or weakly allowed optical

transition is desired. In a few interesting cases, the ions are also forced by selection rules to return to the original

ground-state sublevel on re-emission from the excited

state. Thus the cooling can be performed without the

necessity for mixing ground-state sublevels. Although

this type of repopulation pumping has been reeilized in conventional optical-pumping experiments using, for example, circularly polarized light, the experiments

discussed below have additional advantages.

Small clouds (10^ < N < 10*; cloud diameter ~200

jum to ~1 mm) of Mg+ ions are stored in a Penning

trap.^ The confining axial (z) magnetic field (0.978 T)

splits the ^'^Mg"'' energy levels, as shown in Fig. 1. Light

(~5 juW focused to a spot diameter s 100 ^m; X s 280

nm) from a frequency-doubled dye laser incident along

the y axis and polarized in the x direction is tuned

slightly lower (~100 MHz) than the (3p ^Pan, Mj =

—3/2) *— (3s ^Si/2, Mj = —1/2) transition frequency.

This cools the ions (to less than 1 K for the smallest

clouds) and simultaneously pumps them into the Mj = — 1/2 ground state. This repopulation pumping can be

explained as follows: Ions in the Mj = +1/2 ground

SCATTBffiD UGKT wrasnYiBvpti

BACKGROUND —

27413

MICnOWAVE FREQUENCY IMHzl

Fig. 1. Microwave-optical double resonance in ^'Mg"'". Inset shows relevant microwave and optical pumping transitions.

Reprinted from Optics Letters, Vol. ii nacrp 94.'s .Tnnp IPSO Copyright © 1980 by the Optical Society of America and i TN-40

f the copyright owner.
246 OPTICS LETTERS / Vol. 5. No. 6 / June 1980

state are driven weakly in the wings of the (—1/2) -^

(+1/2) transition (we denote the excited- and groundstate Mj values by the numbers in the first and second

parentheses, respectively). On re-emission, the ion

decays with 2/3 probability to the ( — 1/2) ground state, which thus tends to populate this state. Once in the

(—1/2) ground state, the ion is driven with high probability to the (—3/2) excited state, from which it can only

decay to the ( — 1/2) ground-state level. Depopulation

of the ( — 1/2) ground state can occur because we are pumping in the wings of the (+1/2) -^ (—1/2) transition. Even though this transition frequency is about 36.6 GHz

away from the laser frequency, it is driven at a rate that

far exceeds the nonradiative relaxation rate in the

ground state. However, the (—1/2) -^ (+1/2) transition

frequency is only about 9.15 GHz away from the laser frequency, and therefore we should expect the population in the ( — 1/2) ground state to be about 16 times

larger than that in the ( + 1/2) ground state. This is confirmed by a simple rate-equation analysis.

If we now drive a high-Q transition (here called a

clock transition), which has as its ground state the same

repopulated level from the laser cooling, this level will be depleted and the transition will be indicated by a decrease in the fluorescence scattering on the cooling

transition.^ This simple double-resonance principle

is illustrated in our experiments, in which the clock

transition is represented by the ground-state Zeeman

transition (Fig. 1). In these experiments, we observed

the backscattered light in an //4 cone; the net collection

efficiency was 3 X 10~^. The poor resolution in the

present experiment is due to poor magnetic-field sta- bilization, that is, the field fluctuates about 10 parts in 10^ in the time taken to trace the resonance. In a similar fashion, we have observed the (—1/2) -«— (—1/2) and

(+1/2) -— (—1/2) optical transitions by driving them

with a second, low-power, frequency-swept laser. We can observe the repopulation pumping process by first equalizing the ground-state populations with

microwave radiation (indicated by the fluorescence

signal's dropping to about 50% of its initial value). We

then turn off the microwave radiation and observe that

the fluorescence signal returns to its original value with

a time constant characteristic of the repopulation

pumping described above (Fig. 2). A simple rate- equation analysis (assuming we are well below saturation) indicates that the time constant for this is given

by T = 24/{nBR), where B is the rate fot scattering on

SCATTB1H) UGHT INTHenY

MICROWAVES ON

BACKGROUND^ —

MICROWAVES Off

Fig. 2. Fluorescence scattering rate versus time, showing

repumping into the Mj = —1/2 ground state on ^'^Mg"'". The

laser is tuned to the {Mj = -3/2) <- (Mj = -1/2) optical

transition.

the (—1/2) - (+1/2) transition for an ion in the laser beam [recall that the laser is tuned to the (—3/2) -^

(—1/2) transition] and R is the geometrical fractional

overlap of the laser beam with the ion cloud. For the

conditions in Fig. 2, we estimated thatB s 30/sec, R ^

0.5 X 10-2 {N ^ 103 to 104), resulting in r s 10 sec. Although they are similar to conventional optical

pumping schemes, double-resonance experiments on

laser-cooled, stored ions differ in some important re- spects:

1. Laser cooling, in addition to reducing Doppler

shifts on the clock transition.^s can reduce the Doppler

broadening of the cooling-optical-pumping transition

to less than the natural width. As a consequence, the

scattering cross section for all ions is approximately

equal to the resonant cross section, thus greatly in- creasing the scattering rate. This is important for many

types of ions that have first resonance lines in the ul- traviolet, where only small power is available. These

high scatter rates can permit the observation of many

cooling-pumping photons for each microwave (or optical) photon absorbed on the clock transition.^ (Actually, in the above scheme, we observe the absence of

many scattered cooling-pumping photons when the

clock transition is driven, but the optimum signal-tonoise ratio will be the same.) In schemes in which depopulation pumping occurs in the ground state,^ the

ratio of scattered (detection) photons to absorbed

photons on the clock transition is somewhat limited; for the case in which the ion is forced by selection rules to return to the original ground state, as discussed here,

the ratio can be »10^. This large ratio can help to make

up for the losses in detection that are due to small col- lection solid angle, small detection quantum efficiency,

etc., so that the high-Q transition can be monitored with

nearly unit-detection efficiency.

2. Relaxation times of ions in the ion trap are es- sentially indefinite (>1 h). This allows one to realize nearly complete optical pumping in weak opticalpumping schemes. For the case of ^^Mg"*" discussed

above, the repopulation pumping occurs because we are driving weakly in the wings of the (—1/2) -^ (+1/2)

transition. This rate is rather small (Fig. 2), but it far exceeds any other relaxation between ground-state

levels. Perhaps more importantly, we have observed nearly

complete optical pumping of the (Mj = —1/2, Mj =

—5/2) ground state in ^^Mg"^. In this case, pumping

into the Mj = —1/2 ground states occurs as in the case

of 24Mg+. Pumping into the Mj = —3/2 excited states occurs from ground states that to a first approximation

have the same value of M/. However, because of the

hyperfine coupling, the excited states (except for the Mj

= —5/2 state) have a small admixture of nuclear states with lower M/;jience, when they decay, there is a slight

probability (about 10"^ to 10"^ at a field of ~1 T) that

Ml is reduced. After many scattering events, the ions

become "trapped" in the M/ = —5/2 state. This

pumping into individual hyperfine states should permit

a double-resonance measurement of the ^^Mg"*" hyperfine structure and possibly the realization of certain

types of frequency standards.^

TN-41
June 1980 / Vol. 5. No. 6 / OPTICS LETTERS 247

3. The extremely small spatial extent of the ion

clouds possible with the stored-ion technique^-^ can be

a benefit in many spectroscopic studies; we mention the

additional advantage for Zeeman frequency measure- ments that the problem of field homogeneity is greatly

reduced.

The example of ^''Mg''" is perhaps not so interesting

from the high-resolution standpoint; however, it illus- trates the general features of a class of experiments that

hold promise for significantly improving accuracy and

resolution in the future. Conceptually simple

schemes^'^ for high-resolution, double-resonance ex- periments are not without practical difficulties; however, some of the advantages discussed here can be re- alized in other experiments without undue complication.

This work was supported in part by the U.S. Air Force

Office of Scientific Research and the Office of Naval

Research.

References

1. J. L. Hall, "Stabilized lasers and precision measurements,"

Science 202, 147 (1978); A. L. Schawlow, "Laser spectro- scopy of atoms and molecules," Science 202, 141 (1978); H.

Hellwig, K. M. Evenson, and D. J. Wineland, "Time, fre- quency, and physical measurement," Phys. Today 31(12), 23 (1978); D. J. Wineland, "Limitations on long-term stability and accuracy in atomic clocks," in Proceedings of 11th Annual Precise Time and Time Interval Application

and Planning Meeting (Goddard Space Flight Center,

Greenbelt, Md., 1980), to be published.

2. D. J. Wineland, R. E. DruUinger, and F. L. Walls, "Radiation pressure cooling of bound resonant absorbers," Phys. Rev. Lett. 40, 1639 (1978); R. E. DruUinger and D. J. Wineland, "Laser cooling of ions bound to a Penning trap,"

in Laser Spectroscopy IV, H. Walther and K. W. Rothe,

eds. (Springer-Verlag, Heidelberg, 1979), p. 66.

3. W. Neuhauser, M. Hohenstatt, P. Toschek, and H. Dehmelt, "Optical-sideband cooling of visible atom cloud

confined in parabolic well," Phys. Rev. Lett. 41, 233 (1978); "Preparation, cooling, and spectroscopy of single, localized ions," in Laser Spectroscopy IV, H. Walther and K. W.

Rothe, eds. (Springer-Verlag, Heidelberg, 1979), p. 73.

4. H. G. Dehmelt, "Radio frequency spectroscopy of stored

ions, I and II," in Advances in Atomic and Molecular

Physics, D. R. Bates and I. Esterman, eds. (Academic, New

York, 1967 and 1969), Vols. 3 and 5.

5. To avoid light shifts and broadening, the laser could be

turned off while the clock transition is driven.

6. F. L. Walls, D. J. Wineland, and R. E. DruUinger, "New

possibilities for frequency standards using laser cooling and

detection of stored ions," in Proceedings of 32nd Annual

Symposium on Frequency Control (U.S. Army Electronics Command, Fort Monmouth, N.J., 1978), p. 453.

7. W. M. Itano and D. J. Wineland, "Frequency and time

standards utilizing laser cooled ions," Bull. Am. Phys. Soc.

24, 1185 (1979).

8. H. G. Dehmelt, "Proposed 10^^ Av < p laser fluorescence spectroscopy on T1+ mono-ion oscillator, I and II," Bull. Am. Phys. Soc. 18, 1521 (1973); 20, 60 (1975).

TN-42
Volume 82A, number 2 PHYSICS LETTERS 9 March 1981

SPECTROSCOPY OF A SINGLE Mg+ lON^

D.J. WINELAND and Wayne M. ITAND

Frequency and Time Standards Group, Time and Frequency Division, National Bureau of Standards, Boulder, CO 80303, USA

Received 1 December 1980

A single ^"^Mg* ion has been laser cooled in a Penning trap and its optical spectrum observed by a double resonance

technique. Residual Doppler line broadening implies a cyclotron- magnetron temperature of 50 ± 30 mK.

Laser cooling of ions bound in electromagnetic traps was first demonstrated [1] in 1978. More recently,

this technique has been used in experiments that detect the fluorescence of, photograph, and observe a Raman transition on a single, laser-cooled Ba"*" ion in an RF trap [2] . This abiHty to localize single cold

particles approaches the experimentalist's ideal of obtaining unperturbed systems at rest. In this letter, we

report the isolation and double resonance detection

[3] of a single, laser-cooled Mg"*" ion in a Penning trap. The primary differences of this work from that of

ref. [2] are: (1) the requirements for localization and

cooUng of an ion in a Penning trap are substantially

different than for the case of the RF trap, (2) we

derive a kinetic "temperature" for the ion from the measured Doppler broadening of the optical spectrum

rather than from the cloud size, (3) the intensity of

fluorescence from a single ion is determined by de- struction of individual ions by charge exchange rather than by creation by ionization. The basic configuration of the experiment has been

described elsewhere [1,3,4]. Briefly, in the present

experiments, approximately three ^^Mg"*" ions are loaded into the trap {Vq =» 2 V, ^q === 1 T), laser cooled,

and localized [4] at the center of the trap. Vq and

^0 are the applied static trap potential and magnetic

field. The laser beam (power approximately 5 /xW) is weakly focused near the center of the trap to spot size Wq ^ 25 ^tm and is incident at an angle Q = 82° with

* Contribution of National Bureau of Standards.

respect to the z {B field) axis. This provides both

axial and radial laser cooling. Typically, the laser beam is positioned approximately 15 ;um radially

from trap center in a direction such that the magnetron velocity is in the same direction as the laser beam. Typical detuning of the laser frequency is about —25 MHz from the non-Doppler-shifted

(3p 2p3/2, Mj = -3/2) - (3s 28^2- ^/ = -1/2)

24 Mg"*" transition. This combination of frequency

detuning and spatial positioning ensures both magnetron and cyclotron cooling [4]. Fluorescence from the 24Mg''" ions is observed while

a

25 Mg oven is heated at a low level and the resulting

atomic beam directed at the 24 Mg'^ ions. The fluo- rescence is observed to decrease as a function of time

as shown in fig. 1 . We attribute the step decreases in fluorescence intensity to the occurrence of single

charge exchange events (24Mg"^ + 25]v[g -» ^^y[% +

25 Mg"^) eliminating the 24 Mg"*" ions, one at a time. The

trapped 25 Mg"*" ions are ejected from the trap in about

0.2 s by cyclotron—magnetron RF excitation. We note that the steps are of unequal size, but this

is not unexpected since the magnitude of the fluores- cence will depend on the spatial overlap of the ions and laser beam (which depends on ion number) and

the magnetron Doppler shift, which depends on the Coulomb interaction between ions as well as applied

trap voltage. We also remark that fig. 1 is not a "typical" curve because it shows only charge exchange events.

Elastic collisions (24Mg'*" + 25Mg -> 24Mg+ + 25 Mg) can

also be observed. These are indicated by a sudden drop

75

TN-43
Volume 82A, number 2 PHYSICS LETTERS 9 March 1981

tm

z

bJZ /

— 100 s

/ b- ij- A BACKGROUND

4- [ Z^Mg"+"Mg— "Mg^ + "Mg _/--^^"^

^^n/^v,^/l^—

TIMEFig. 1. ^'^Mg'*' ion fluorescence versus time when a ^^Mg atomic beam is directed at trap center. Fluorescence is integrated for 10 s on each experimental point. The large steps are attributed

to single charge exchange events (^'*Mg'^ + ^^Mg - 24 Mg

+ ^^Mg*) which remove the ^^Mg* ions, one at a time. The

last plateau is due to a single ^'*Mg'^ ion at trap center.

in fluorescence (a step down) as a ^'^Mg'*' ion is heated

by the collision, causing its fluorescence to decrease due to its large Doppler width and orbit sizes. However, after some time (up to a few hundred seconds),

it is again cooled down and localized at trap center and we see a step increase in the fluorescence to the former

value. More important, one might suspect that the steps

could be due to two or more ions: i.e., one ^'^Mg"'' ion

is removed by charge exchange and during the time

the 25 Mg"*" ion is still in the trap, it has some probabihty

(although estimated to be very small) that it heats the remaining ^'^Mg"^ ions by Coulomb collisions. Since the goal of the experiment was to observe a

single ^^Mg"^ ion, the above difficulties could be avoid- ed as follows: The ^^Mg"*" loading conditions were

scaled down so that after careful searching, fluorescence was observed on only 20% of the loads. Statistically, this meant that if any fluorescence was observed, it was, with high probabihty, from a single ion. At this time, the double resonance experiments (described

below) were performed, after which, we looked for a

single step after the ^^Mg oven was turned on. Immediately after this step was observed, the ^^Mg oven was turned off and a careful search was made to find any remaining ^'^Mg"'" ions. Since none were found, we

concluded that, with high probability, we had a single ion.The features of the optical spectrum of a single

ion (fig. 2) could be recorded by using a simple optical pumping, double resonance scheme [3]. Briefly, a "high-power" laser (power approximately equal to 5 /xW) is used to keep the ion cold and at trap center;

76

-200 -100

RELATIVE LASER FREQUENCY (MHz)

Fig. 2. Double resonance curve of a single ^''Mg* ion. On the

vertical axis is the fluorescence from a fixed frequency laser (power approximately 5 /iW) tuned to the (.^^3/2' ^J- -3/2)

«-(^Si/2,^/= -1/2) transition. Each point representsa 10

s integration; the connecting lines are only for clarity. The

horizontal axis is the frequency of the low-power (<5 fiW)

laser which is continuously scanned across the (^Ps/j > Mj

= -1/2) •<- (^Sj/2, M_/= -1/2) transition. The dashed curves

are simulations of fluorescence at 7" = K and 100 mK (no added noise). The solid curve is experimental data. From these

data, we conclude r= 50 + 30 mK.

this laser also optically pumps the ion [3] so that it spends about 94% of its time in the (^S^i2,Mj = -\/2)

ground state Zeeman sublevel and about 6% of its time in the (^Sii2,Mj= +1/2) level. If the frequency of

a second lower-power (<5 juW) laser beam (coincident

spatially with the first laser beam) is swept through the (2p3/2'^/" -1/2) ^(^Si/2,^/ = -1/2) transition, the resonance condition will be indicated by a decrease

in fluorescence due to the first laser. Specifically, the observed fiuorescence is proportional to the fraction of

time, /_ 2^2' ^he ion spends in the (Mj = —1/2) ground

state sublevel, which is approximately equal to

/-V2 = T-7[l + 8i?i/(17/?2)]-l, (1)

where i?i is the (^?2/2'^J = -1/2) ^ 0Sii2,Mj

= —1/2) transition rate, due to the low-power laser, and R2 is the i^?2/2> Mj = -1/2) <- (^$112, Mj = +112)

transition rate, due to off-resonance scattering by the high-power laser. This method is superior to a previous

method [4] , since the signal is due to the higher scat- tering by the high-power laser; also the ion temperature

is not significantly perturbed by the low-power laser. The experimental resonance curve in fig. 2 can be compared to the two other curves which were simulated,

TN-4A
Volume 82A, number 2 PHYSICS LETTERS 9 March 1981

assuming the ion kinetic energy had a Maxwell—

Boltzmann distribution in time. The width of the K

curve is due to the 43 MHz natural width [5] . It is shghtly broadened because of the detailed nature of

the fluorescence intensity as given by eq. (1) and be- cause the laser is continuously swept during the inte- gration period. It is slightly asymmetric since the in- tegration periods are not symmetric about line center. From this kind of data, we conclude that the cyclotron—magnetron "temperature" was 50 ± 30 mK. The

axial temperature, T,, can be estimated by lowering

the axial well depth by lowering Vq and noting the decrease in fluorescence as the axial excursions become

comparable to the laser spot size. With Fg = 1 V, the fluorescence signal drops to about half its value for Kq === 7 V, which impHes T^ % 600 mK. The Doppler

broadening information is relatively insensitive to the

axial temperature because 9 is close to 90°. Assuming

the cyclotron—magnetron temperature is zero, this yields an axial temperature of 2.6 ± 1.5 K. Thus, we

conclude that the Doppler broadening is primarily a measure of cyclotron—magnetron temperature. We

also note that experimentally, for a single ion, the

radial extent is significantly less than the laser spot

diameter.

Cyclotron—axial cooling and magnetron "cooling"

in a Penning trap are fundamentally different in nature. Here, cooling is defined as reducing the kinetic energy

in the three (cyclotron, axial, and magnetron) degrees

of freedom. Previous treatments [6] apply to the harmonic axial motion. If we define the potential ener- gy to be zero at trap center, then we can write the total energy (kinetic and potential) in the magnetron and

cyclotron degrees of freedom as ^m"I^4(^m-i"z)

£, = iM/-2(a;;2_i^2)^

where

.2 _

co," = 4eFo/M(r^+2z^),

(2)

(3)

cj^, cj|,, and co^ are the magnetron, cyclotron, and

axial frequencies and r^ and r^ are the magnetron and

cyclotron radii. The dependence of the potential ener- gy [second terms in eq. (2)] on the applied trap potential Vq and trap dimensions Vq and Zg is apparent in eq. (3). Since co^^ < co^/V^ < a;[, for trapping, then

E^ > 0, but E^ < 0. To reduce r^^ and thereby the magnetron kinetic energy, we must therefore increase the total magnetron energy. Conversely, to reduce r^ and thereby the cyclotron kinetic energy, we must decrease the total cyclotron energy. (The cyclotron case

is analogous to the harmonic binding case described in

ref. [6] .) If we view laser scattering events as point

interactions (valid since cj^, oj^, cj'^ <y, where 7

is the optical linewidth) then the requirements for cooling are that we preferentially scatter photons when

the magnetron motion recedes from the laser and the cyclotron motion approaches the laser. Since the

direction of circulation is the same for both degrees

of freedom, it might appear that these conditions cannot simultaneously be satisfied, but they can if the

laser intensity is higher on the side of trap center where

the magnetron motion recedes from the laser [4,7].

This is accomplished in our experiments by working on the slope of the nearly gaussian laser beam profile.

We have theoretically modeled the coohng of a

single ion in a Penning trap [7] , and when applied to our experimental conditions, these calculations can be

used to predict minimum temperatures. If we define T^ = Mc^yj(2k) , T, = Mj^^rll(2k)

T,=M{v])lk,

where k is Boltzmann's constant, then we find mini- mum temperatures T^^\3X 10"^ K, T^. ^ 0.99

mK, r^ % 1 1 mK. At this time, we do not understand

why our measured temperature appears higher than

these predictions, but we have ruled out obvious ef- fects, such as background gas collisions, laser frequency

jitter, etc. We note, however, that if we conservatively

estimate /•j^(max) = 15 /nm, this gives Tj^(max) «= 1 mK. Therefore, the observed temperature would ap- pear to be due to "hot" cyclotron motion. We also note that the predicted and experimental values of

T^IT^ approximately agree. One serious shortcoming of the present apparatus

[3,4] is that the net detection efficiency of the scat- tered light (fraction of scattered photons which are counted) is only about 10~^, therefore limiting sig- nal-to-noise ratio. A new apparatus with significantly

improved detection efficiency is being constructed. This will improve signal-to-noise ratio in the double

resonance experiments; moreover, it should allow

simpler identification of a single ion because, over short

77

TN-45
Volume 82A, number 2 PHYSICS LETTERS 9 March 1981

time intervals, the fluorescence should maintain a steady value when the ion is in the Mj = — 1/2 ground

state and be totally absent when the ion is in the Mj = +1/2 ground state. We also note that when the Doppler width of the optical Hnes is less than the natural width, the spectrum of the scattered light is broadened only by the Doppler effect and not the

natural width. If the laser linewidth is assumed negligible, the spectrum of the scattered light is composed

of discrete lines, one at the laser frequency, and side- bands at pcj^ + qojj^ + 5cu^ where p, q, and s are positive and negative integers. For example, if we assume

that only the cyclotron motion is excited and has a Maxwell-Boltzmann distribution in time, then the

profile of these sidebands for Hght scattered in the backward direction is a gaussian with full width at half maximum = 4(2 In 2 • kT/M\^)^/^ where X is the wavelength of the hght (this is twice the normal

Doppler broadening). The centroid is shifted from the

laser frequency by Au> = —AR/fi where R= h^j

2MX^. This spectrum could then give a more sensitive measure of the temperature *'.

** The case of resolved sidebands in the Lamb-Dicke regime

has been treated by Javanainen [8].

We gratefully acknowledge the contributions of

J.C. Bergquist in the laboratory and in the course of numerous discussions. We appreciate a critical reading

of the manuscript by J.C. Bergquist, L. Lewis and

G. Dunn. We also acknowledge the support of the Air Force Office of Scientific Research and the Office of

Naval Research. One of us (W.M.I.) held an NRC

Postdoctoral Research Associateship during this work.

References

[1] D.J. Wineland, R.E. Drullinger and F.L. Walls, Phys. Rev.

Lett. 40 (1978) 1638; W. Neuhauser, M. Hohenstatt, P. Toschek and H. Dehmelt,

Phys. Rev. Lett. 41 (1978) 233.

[2] W. Neuhauser, M. Hohenstatt, P.E. Toschek and H. Dehmelt, Phys. Rev. A22 (1980) 1137.

[3] D.J. Wineland, J.C. Bergquist, W.M. Itano and R.E.

Drullinger, Opt. Lett. 5 (1980) 245.

[4] R.E. Drullinger, D.J. Wineland and J.C. Bergquist, Appl.

Phys. 22 (1980) 365.

[5] W.L. Wiese, M.W. Smith and B.M. Miles, Atomic transi- tion probabilities, U.S. National bureau of Standards, National Standards Reference Data Series 22, Vol. II (US GPO, Washington, DC, 1969).

[6] D.J. Wineland and W.M. Itano, Phys. Rev. A20 (1979)

1521;

J. Javanainen and S. Stenholm, Appl. Phys. 21 (1980)

283;

J. Javanainen, Appl. Phys. 23 (1980) 175.

[7] W.M. Itano and D.J. Wineland, to be published.

[8] J. Javanainen, Opt. Commun. 34 (1980) 375.

78

TN-46
Proc. 35th Ann. Freq. Control Symposium, USAERADCOM, Ft. Monmouth, NJ 07703, May 1981

PROPOSED STORED ^oiHg ION FREQUENCY STANDARDS*

D. J. Wineland, Wayne M. Itano, J. C. Bergquist,

and F. L. Walls

Frequency and Time Standards Group

Time and Frequency Division

National Bureau of Standards

Boulder, Colorado 80303

Summary

In this paper, we discuss the performance

potential and the problems of implementing a microwave frequency (and time) standard |nd an optical frequency standard utilizing ^^^Hg ions

stored in a Penning trap. Many of the discussions

apply to ion storage-based frequency standards in general. Laser cooling, optical pumping, and

optical detection of the microwave or optical

clock transition could be achieved using narrowband

radiation at the 194.2 nm 6p ^p <- 6s ^s. tran- sition, while selectively mixing xhe ground-state

hyperfine levels with appropriate microwave radiation. A first-order field-independent microwave

clock transition, which is particularly wellsuited to the use of the Penning ion trap is the

25.9 GHz (F,Mp) = (2,1) ^ (1,1) hyperfine transition at a mSgnetic field of 0.534 T. The two- photon Doppler-free Sd^ Ss^ 2O5 j *^ Sd^° 6s 2$

transition at 563 nm is a possible candidate for

an optical frequency standard. Both standards

have the potential of achieving absolute accura- cies of better than one part in 10^^ and frequency

stabilities of less than 10-^^.

Introduction

In this paper, a specific proposal is made

for a ^°^Hg stored ion microwave frequency (and

time) standard which could have an absolute fractional uncertainty of less than 10-^^. We also

discuss the possibilities for a 2°^Hg optical

frequency standard. A future stored ion frequency

standard may not take the exact form described

here; nevertheless, it is useful to investigate a

specific proposal, since many of the same generic

problems will be encountered in any standard based

on stored ions. Since the pioneering work of Dehmelt and

co-workers, who first developed the stored ion

"Contribution of the National Bureau of Standards

not subject to copyright.

method for high-resolution spectroscopy, it has been clear that these techniques provide the basis

2-10

for an excellent time and frequency standard.

This conjecture is based primarily on the ability

to confine the ions for long periods of time

without the usual perturbations associated with

confinement (e.g., "wall shift" as in the hydrogen

maser). Starting with the work of Major and

5 7 Werth reported in 1973, groups at Mainz and

Orsay and at least one commercial company

have sought to develop a frequency standard based

on ^^®Hg ions stored in an rf trap. The choice

of Hg ions for a microwave stored ion frequency

standard is a natural one because its ground-state

hyperfine structure is the largest of any ion

which might easily be used in a frequency standard,

and its relatively large mass gives a small secondorder Doppler shift at a given temperature. This

work has been developed to a fairly high level; the group at Orsay has made a working standard

whose stability compares favorably to that of a commercial cesium beam frequency standard. How- ever, the full potential of the stored ion techniques has yet to be realized; this appears to be due to two problems: (1) Historically, it has been difficult to cool the ions below the ambient

temperature; this is made more difficult in the rf trap by "rf heating" —a process not clearly

understood, but one that makes it difficult to

12 cool even to the ambient temperature. For both the rf and Penning traps, the inability to cool below the ambient temperature means that one must contend with the frequency shift from the

second-order Doppler or time-dilation effect.

Although it is possible to calculate this shift

from the measured Doppler (sideband) spectra, to

602 TN-47
do so with the required accuracy may be difficult

for ions near room temperature. (2) A second

problem is that the number of ions that can be

stored in a restricted volume (dimensions < 1 cm)

is typically rather small (< 10^). This, coupled

with the somewhat poor signal-to-noise ratios

realized with conventional lamp sources, causes the short-term stability in a frequency standard

based on ions to be degraded, even though the Q's

realized are quite high.

In the past two or three years, both of these

problems have been addressed in experiments and

the results suggest viable solutions. In 1978, groups at the National Bureau of Standards (NBS)

13 14 and Heidelberg demonstrated ' that radiation

pressure from lasers could be used to cool ions to temperatures < 1 K, thereby reducing the secondorder Doppler shifts by 2-3 orders of magnitude

below the room temperature case. As discussed

below, the cooling is most favorable for very

small numbers of ions (down to one ion), so that

there is a trade-off between the maximum number of

ions we can use and the minimum second-order

Doppler shift that can be achieved. To improve signal to noise, we note that in certain optical -pumping, double-resonance experiments, it is possible to scatter many optical

photons from each ion for each microwave photon

g absorbed. This can allow one to make up for

losses in detection efficiency due to small solid

angle, small quantum efficiency in the photon

detector, etc., so that the transition probability

for each ion can be measured with unity detection

efficiency. This means that the signal-to-noise

ratio need be limited only by the statistical

fluctuations in the number of ions that have made

15 the transition. This is discussed in a simple

example in Appendix A. More recently, the narrow linewidths anticipated for the stored ion technique have been

observed. A resonance linewidth of about 0.012 Hz

Ramsey interference method was implemented by

applying two rf pulses of 1 s duration separated

by 41 s). These narrow linewidths should be

preserved with hyperfine transitions of higher

requency, such as in 2°^Hg , but, of course, more attention must be paid to field homogeneity and

stability.

These results have encouraged us to begin

studies on the ^°^Hg system and although this ion

may not provide the "final answer," it appears to provide a case where inaccuracies significantly

smaller than 10-^^ can be achieved. The discussion here is largely devoted to a microwave fre- quency standard with a design goal of accuracy better than 10-^^; however, the possibilities for

a stored ion optical frequency standard in 2°^Hg

17 32 are also briefly included. ' For a given

interaction time, the Q of a transition will scale

with the frequency. Therefore, in principle, an optical frequency standard would have clear advantages over a microwave frequency standard. Our decision to work on a microwave frequency standard

(as well as an optical standard) is motivated

largely by practical considerations: (1) Before

the full potential of an ion-storage optical

frequency standard could be realized, tunable

lasers with suitable spectral purity must become

available. This problem may be nearing solu20 tion. (2) If an optical frequency standard is to provide time, the phase of the radiation must

be measured. This appears to present a much more

18 19 formidable problem. ' Both of these problems

are already solved in the microwave region of the

spectrum—thus, the attraction for investigating

a microwave frequency standard.

201 Hg Stored Ion Microwave Frequency Standard

at 292 MHz has been observed for the (mj, m,) = (^/2> ^/z) *~* ("Vai ^^2) hyperfine transition of

^^Mg at a magnetic field of about 1.24 T where

the first derivative of the transition frequency trap (where "rf heating" occurred), they can be

2°iHg ions will be stored in a Penning trap.

The choice of the Penning trap over the rf trap is motivated primarily because it appears easier to cool a cloud of ions in a Penning trap than in an

rf trap. Residual heating mechanisms in the

13 Penning trap are quite small whereas in the rf

with respect to magnetic field is zero

16 (The

14 substantial. Of course, this problem does not

603 TN-48
exist for the rf trap if single ions are used, but

for a microwave frequency standard, It is desirable

to use as many ions as possible in order to in- crease signal-to-noise ratio. Use of a Penning

trap means that one must use transitions that are independent of magnetic field to first or8 9 16 der * *

; this limits the number of transitions

available, but need not be an absolute restriction.

The ^®^Hg isotope is therefore not considered,

since there are no frequency extrema at practical

fields.

21 + The energy level structure of the ^^^Hg

ion vs. magnetic field is shown in Fig. 1. The

1 1 1 1 1

,

20 N- ,„ ^ ' (1.0)

(1.1)

=

. 15 - 201Hg*

10 - -

"m 5 I

o

- -

Energy ^^^32)„.—

-^

-^^L^===:=^ (2.1) 10

(2 0) -

-15

< 1 1 1 1 —

ai 0.2 0.3 0.4 0.5 Bo (T)

Figure 1. Ground state hyperfine energy levels of

2°^Hg vs. magnetic field. States are designated by the (F, Mp) representation.

Three transitions are indicated at the

fields where the transition frequencies

are independent of magnetic field to first order.

possible field independent "clock transitions" are

the (F. mp) = (2,0) *-* (1,1) and (2,1) ^ (1,0)

transitions at about 0.29 T, the (2,1) *-»• (1,1)

transition at 0.534 T, the (1,1) -> (1,0) transition at 3.91 T and the (1,0) -^ (1,-1) transition

at 28.1 T. In principle, one desires to work at the highest microwave frequency possible (for

highest Q) at the highest attainable magnetic

field (to maximize the nua^er of Ions). The

choice seems to be between the 25.9 GHz (2,1) *-*

(1,1) transition at 0.534 T and the 7.72 GHz

(1,1) -» (1,0) transition at 3.91 T. At these

magnetic fields, where the transition frequency is Independent of magnetic field to first order, the

second-order field dependence is given by Av/v

((2,1) ^ (1,1)) = Ve (AH/H^)2 and Au/v,

((1.1) 1,0)) = .04(AH/Hq)2. The remaining part

of the proposal is modeled around the (2,1) -»• (1,1)

transition at 0.534 T because of its higher fre- quency. From the second-order field dependence

noted above, it is necessary to control the field

stability and homogeneity over the ion cloud to better than 10-' in order to achieve 10-^^ accuracy. This is an important problem, of course, but

not insurmountable—we note that the free-running

stability of the magnetic field must be better

than 10-' over the clock transition time; in longer term it can be stabilized to this level by

locking the field to a Zeeman transition in the

ions.^SWe will assume that a laser can be tuned to the 6p 2p <- 6s 2$, transition at 194.2 nm with

sufficient power to provide laser cooling, optical

pumping, and fluorescence detection. A specific

scheme for observing the clock transition might be

the following: (we will assume that when the ions are cold, the widths of the Doppler-broadened

optical lines are approximately equal to the

natural width (~ 70 MHz)).

(1) We tune the laser about 35 MHz below the

6p 2p (1,-1) - 6s 2$, (2,-2) optical transition.

This laser tuning gives the maximum cooling pos22 sible, but rapidly pumps the ions out of the

(2,-2) ground state. (For example, the (1,-1)

excited state decays to the (1,-1) ground state with a probability equal to 0.329.) For cooling

and detection, we require each ion to scatter many

15 ic po photons. Unfortunately, the simple schemes ' *

for multiple scattering that have been realized

with ions like ^^Mg do not exist for Hg . There604 TN-49
fore, for the initial cooling, we must mix all the

ground states with microwave radiation. This

effectively reduces the cooling by a factor of 8 from the two-level ion case, since only ^/g of the

ions are in the (2,-2) ground state, but should

not be a problem given enough laser power (see

below).

(2) Once cooling is accomplished, we can pump all the ions into one of the clock levels by,

for example, mixing all the ground-state levels

except the (1,1) level. This pumping occurs because we are exciting the ions in the wings of

other optical transitions; for example, we are pumping in the wings of the (1,0) <- (2,-1) optical

transition (about 3.5 GHz away) and the (1,0)

excited state decays into the (1,1) ground state with a probability of 0.039. (Note that the

(1,-1) excited state cannot decay to the (1,1)

ground state.) We remark that we have neglected

the decay of the ions from the ^p excited state

to either the ^0^/2 ^^ ^^s/a states. That this is

a good approximation is shown in Appendix B. The

ratio of the scattering rate on the (1,-1) • (2,-2)

optical transition to the pumping rate into the

(1,1) ground-state level is about 6 x 10^, i.e.,

about 6 X 10* photons are scattered by each ion

before it is pumped into the (1,1) ground- state clock level. This number can be reduced by tuning

the laser to another optical transition; this may

be necessary if the laser power is too small. As discussed below, it is desirable for detection

purposes for this number to be large.

(3) After the pumping is achieved, the laser

and ground-state mixing rf are turned off (to

avoid ac Stark shifts and relaxation) and the

clock transition is driven. If we use the Ramsey

interference method as was done for ^^Mg , the

resulting linewidth is about one-half as wide as if we use continuous excitation (the Rabi method).

Therefore, the Ramsey method will be assumed.

(4) After the rf cycle is complete, the

laser is turned back on as well as the mixing rf (excluding the (1,1) ground-state level) and the

fluorescence scattering is observed. It should

not be too difficult to detect the scattered

photons with about 10-^ overall efficiency.

This would mean that about 60 photons would be

collected for each ion that had made the transition

(before it is repumped to the (1,1) level) insuring

that the noise would only be due to the statistical

fluctuations in the number of ions that had made

the transition (see Appendix A). In order to lock a local oscillator to the center of the clock transition, one would first

complete one of the above cycles with the local oscillator tuned to the half- intensity point

(where the transition probability for each ion is one half) on, say, the low side of the central

Ramsey peak. These photon counts could then be

stored and the cycle repeated with the local oscillator tuned to the high-frequency side of the

line. These resulting counts could then be subtracted from the first to give an error signal

which can be used to force the mean frequency of

the local oscillator to be at the center of the

clock transition. We will assume that the time

for fluorescence observation and repumping is much

less than the clock transition time.

Systematic Frequency Shifts

We have already mentioned the stringent

requirements on magnetic field stability and

homogeneity. Using a superconducting magnet, it

is possible to stabilize the magnetic field to better than 10-''. (This may require locking the

field to an NMR probe adjacent to the trap over

the time of the clock transition). The field

homogeneity requirements will be more difficult to satisfy (assuming a 1 cm diameter spherical working

volume inside the trap), but are still feasible.

The electric fields from the applied trap

potentials and from Coulomb interactions between

ions can cause second-order Stark shifts, but the

resulting fractional frequency shifts are estimated to be much less than 10-^^. The black body

38

ac Zeeman shift is estimated to be Av/v^ =

1.3 X 10-1^ (T/300)2 and is therefore neglected.

39 The black body ac Stark shift is estimated to be Av/v ^ 2 X 10-^® (T/300)* and therefore must be

accounted for or reduced environmental tempera- tures are required.

605

TN-50
In spite of the laser cooling that has been

achieved, we must still be concerned with the

second-order Doppler frequency shift. In the

Penning trap, the cyclotron-axial temperature of

2°^Hg ions needs to be cooled to below 1.45 K to insure that the second-order Doppler shifts (on

these degrees of freedom) is less than 10-^^. These low temperatures should be easy to obtain. A more serious problem exists for the magnetron

rotation of the cloud; the kinetic energy in this

degree of freedom will probably limit how small 24 the second-order Doppler shift can be. In the

limit of very small numbers of ions, the magnetron

25 kinetic energy should be negligible, but in the case discussed here, we would like to use the

maximum number of ions possible and this will cause problems as described below.

The "magnetron" rotation in a Penning trap is simply a form of circular I x I drift; that is,

the radial electric fields in the trap from the

applied potentials on the electrodes and from space charge act in a direction perpendicular to the magnetic field. This causes the ions to drift

in circular "magnetron" orbits about the axis of

the trap. If we assume that we must keep the ions

inside a 1 cm diameter spherical working volume,

then qualitatively, the nature of the problem is as follows: if we add more ions to this volume,

then the magnetron frequency increases due to two effects. First, we must increase the applied trap

potentials to overcome the increased space charge

repulsion along the z axis, which tends to elongate

the cloud in this direction. Consequently, the

magnetron frequency increases due to the increased

potentials and the increased space charge fields

in the radial direction. In Appendix C, we esti- mate the maximum number of ions contained in a

1 cm diameter sphere assuming that the secondorder Doppler shift is 10-^^ for ions on the

perimeter of the cloud at z = 0. We obtain N =

max

8.2 X IC and note that the applied trap voltage

is only' 71 mV for z = r /1.64 = 0.8 cm.

Frequency Stability

At optimum power (transition probability is equal to one at line center), we can closely

approximate the number of detected photon counts

for each experimental cycle as

(1 + cos(uru) ) T\ 2—^-j (4)

where, as in Appendix A, N. is the number of ions

in the trap and n. is the average number of detected photons for each ion that has made the

transition, lu and u) are the frequency of the

applied rf and "clock" center frequency; T is the

time between the rf pulses at the beginning and

end of the rf period. (We assume that the time of

the rf pulses is much less than T. ) As described

above, the interrogating oscillator is switched

between ui - n/2T and u) + n/2T (where we assume

lu) - u)

I << 7t/2T), and the resulting counts subtracted to give an error signal. The sensitivity

to mistuning of lu is given by calculating the

slope of the signal in Eq. 4 at the half- intensity

points (|iu -u) I = 7i/2T). We have

idu)l

N.n, T

7I_ 2T

After one full switching cycle (taking the

difference of the counts from both sides of the

line), the error signal is

6N = 2 1^I au)| X 6iu = rt/2T

where 6iu = w-w . Since N fluctuates statistically,

these fluctuations (6N) give rise to frequency

fluctuations in the locked local oscillator:

(6uj)' rms 6N2

u)-u)

I = n/2T

Maximum frequency stability is thus given by

a2(2T) . 1 ^—V^ y 2 u) 2

Assuming n . » 2 (Appendix A), we have

V'> = 5T=7^ Vr

26

T > 2T

606 TN-51
For N^ = 8.2 X 10* and T = 50 s,

o (x) = 2 X 10-15 ^-h

T > 100 s

and

Q = -^ = 2.6 X 10>

71

Lasers

Perhaps the single reason that such a proposal

has not been made previously is that the required

narrowband tunable laser has not been available at 194.2 nm. However, it appears that two possible

approaches lend themselves to initial experiments

in this system. Briefly, the first approach might

be to use an externally narrowband filtered

(= 100 MHz) pulsed ArF excimer laser. We estimate

that it should be possible to achieve saturating

intensity from such a filtered laser. However,

the pulse lengths of these lasers are quite short

(< 10 ns), so that only about two photons per ion

will be effective in each laser shot to drive the

optical transition. (The lifetime of the upper

2p, (1,-1) state is about 2.3 ns and decays with

0.46 probability to the (2,-2) ground state.) The

potential advantage of excimer laser systems is that the repetition rates can be quite high (KrF

lasers have been built with 1 kHz repetition

rates). However, at the present time, it is probably only feasible to realize 150 Hz repetition

rates for ArF lasers. This should allow reasonable

signal to noise in the above scheme when the ions are cold; however, we estimate cooling times of

order 20 minutes, which is uncomfortably long.

A second scheme using frequency mixing of cw lasers in nonlinear crystals is presently being

pursued at NBS. Tunable coherent radiation in the

194 nm region has previously been produced by

phase-matched sum frequency mixing of pulsed

lasers in a potassium pentaborate (KB5) crys27 + tal. The second harmonic of the 514 nm Ar single frequency cw laser line, when mixed with

radiation near 790 nm from a cw dye laser, will

generate single frequency cw radiation at 194 nm.

The estimated efficiency of this process for a

3 cm crystal is given by

P3 = 8 X 10-5 p^ p^^

where P3 is the output power at 194 nm, Pj is the

input power at 257 nm from the doubled Ar laser,

28 29 and P2 is the input power at 790 nm. ' All powers are expressed in watts. The second harmonic

of 514 nm radiation can be generated in 90° phase

matched temperature tuned KDP or ADP crystals with

an efficiency (for a 5 cm crystal) given by

P, = 2.5 x 10-3 p 2

where P^ is the output power at 257 nm and P is the input power at 514 nm. Thus, about 10 mW can

be produced with a 2 W input. The output power

can be increased considerably by using a cavity to build up the circulating power. As much as 300 mW

of cw power at 257 nm has been produced in this

30

way. Assuming that 200 mW at 257 nm and

500 mW at 790 nm are available, about 8 pW at 194 nm could be produced. From the experience

with Mg ions, ' ' this should be adequate

power for initial experiments. The output could

be further increased by building a cavity around

the KB5 crystal to increase the circulating power

at 790 nm, 257 nm, or both. Finally, the output

power at the ions could be increased by building a ring cavity around the trap. The frequency of the

Ar laser can be stabilized to an Ij absorption.

The dye laser can be stabilized and tuned using

standard techniques. The temperature of the KB5 may have to be shifted slightly from room tempera- ture in order to satisfy the phase-matching conditions. We estimate from the observed temperature

31 tuning at 201.6 nm that the required shift is less than 25 °C.

Two-Photon Optical Frequency Standard in ^"^Hg

We will briefly describe the properties of a 2°iHg optical frequency standard that has been

32 * suggested previously. (^^^Hg should, of course, also be considered because of its simpler

607

TN-52
structure.) Using two-photon Doppler-free spec33 + troscopy, it should be possible to excite ^"^Hg

ions from a ground-state sublevel to a particular

205,2 sublevel using a dye laser tuned to approximately 563.2 nm. An excited state and magnetic

field could be chosen so that the transition

frequency could be independent of magnetic field

to first order. For the case of this optical

transition, the second-order dependence of fractional frequency offset due to magnetic field

would be reduced by approximately the ratio of the

optical frequency (5.33 x 10^* Hz) to the groundstate hyperfine frequency. This represents a reduction in sensitivity of about four orders of

magnitude which would greatly relax the constraints

on magnetic field. The ground and excited states could be chosen such that the ground-state level would be depopulated by the two-photon transition;

therefore, detection could be accomplished in essentially the same manner as described for the

microwave case above. The lifetime of the excited

35 states IS about 0.11 s, so that the Q of this

transition is about 7.4 x 10^*! A drawback to this scheme is the accompanying

33 ac Stark shift ; this shift is formally equivalent to the "light shift" in rubidium frequency

standards. We have roughly estimated the transition probability per unit time to be W = 0.3 P/Sv

where 6v is the larger of the (doubled) laser

linewidth or the natural linewidth (1.4 Hz) and I is the laser intensity in each direction in W/cm^. We have also estimated the accompanying light

shift to be Av = -I (Hz). If the (doubled) laser

linewidth is less than the natural width and if we drive the transition near saturating intensity

(W = 1/s), then I = 2.2 W/cm^, which implies a fractional frequency shift of 2 x 10-^^. 38 The black body ac Stark shift is estimated

to be Au/v^ = 10-16 (T/300)*. The frequency shift*°

due to the interaction of the quadrupole moment of

the atomic D levels with the applied quadrupole

field and fields due to ion-ion collision is estimated to be less than 10-^® for the conditions

described here.

In initial experiments, it will very likely

not be possible to obtain (doubled) laser linewidths less than 1.4 Hz; however, laser linewidths

of a few tens of hertz should be achievable.

Therefore, the above projections may not be too optimistic.

Assuming the same conditions as for the

microwave case, predicted stabilities are also

quite dramatic (assuming laser linewidths are sufficiently narrow). If we assume that the

(Rabi) interaction time is 1 s and that the detection and repumping time is much less than 1 s, this would imply (for N. = 8.2 x 10*)

a (t) ^ 2 X 10-18 T- ^

For N^. = 1, o (t) ^ 6 X 10-i« x

I > 2 s

-H

Conclusions

Some of the problems associated with a storedion frequency standard have been addressed by

making a specific proposal around the ^^^Hg ion. Although other interesting candidates exist, this

system appears feasible enough that experimental

work has begun at NBS. Current efforts are aimed

at producing the 194.2 nm laser light, producing

narrowband 563.2 nm laser light for the two-photon

transition, studying the ion cloud dynamics in order to produce the rather diffuse, spatially

stable ion clouds and making ion traps with significantly increased collection efficiency.

Acknowledgments

The authors wish to acknowledge the continued

support from the Air Force Office of Scientific

Research and the Office of Naval Research. We thank H. Dehmelt, C. Pollock, and H. Robinson for

critical comments and we thank R. Ray and B. Barrett for manuscript preparation.

APPENDIX A

Statistical Fluctuations in Detected Photon Counts

For simplicity, we will assume the conditions

shown in Fig. 2. The actual conditions discussed

in this paper and in Refs. 15 and 16 are somewhat

TN-53

608
Laser pump)

microwave"clock" transition

Figure 2. "Model" 3 level system to show the ef- fects of statistical fluctuations in an optical pumping, double resonance

experiment.

more complicated, but the basic result still

applies. We assume that a laser is tuned to the

3 «- 1 transition wavelength and that we can neglect

excitation in the wings of '^.he 3 <- 2 transition by

this laser. We also assume that the decay branching ratios R32 and R31 are such that R31 >> R32. The basic scheme for an optical-pumping

double-resonance experiment in this simple example

is the following: (1) If the number of scattering

events (for each ion) is sufficiently large, then

essentially all the ions are pumped into the 2 level. (2) The laser is now turned off and the

"clock" (1 •<- 2) transition is driven. (3) The

ions that have made the transition are then de- tected by turning the laser back on and observing

the fluorescence scattering by collecting the

light in a phototube.

For each ion that has made the clock transition, let the average number of detected photons

be n.; the rms fluctuation in the number of de- tected photons per ion is ^fn'.. Because of inefficiencies in collection and detection, typically

n. « R3i/R32. but the interesting case will be when n. is still significantly greater than one. We will assume that the clock transition is driven

with optimum power, so that when we are tuned to

the half- intensity point on the line (for maximum

frequency sensitivity), the probability, p, that

each ion has nade a transition is 0.5. In this case, if the total number of ions is N., then the

average number that have made the transition is pN. = N./2 and the rms fluctuations in the number

of ions that have made the transition on each

experimental cycle is

AN. = ^H. p(l-p) = VN7/2.

For one photon counting cycle, the average number of photons detected is N,q, = n . N./2. The

fluctuations in the number of detected photons

(AN,-,) is due to two causes: (1) The fluctuations

in the number of photons counted (An-j-Qj) . due to fluctuations in the counted photons for each ion. Since these are statistically independent:

^nOT^d n

II

i^n^V = VVV2

(2) The fluctuation in the number of photons

counted (Anj-.,). due to the fluctuations in the

number of ions that have made the transition:

(An^O^). = AN. n^ = n^ ^./2

Since these two processes are also statistically independent, the total fluctuations in N,-,

are given by:

^TOT = V^^OT^d^^nOT^i

d ^ 1 v^

Therefore when n . >> 2, the fluctuations in the

photon counts are given by the statistical fluctuations in the number of ions that have made the

transition. We must note, of course, that the

above arguments assume that the laser intensity

and cloud of ions is stable, and therefore, fluctuations in signal due to changes in laser-cloud

spatial overlap are negligible.

TN-5A

609
2P,

Appendix B

203-2 5/2 Decay Rates

Decay from the 5d^° 6p ^p levels to the

5d® 6s2 2D levels is allowed through configuration

interaction, which can mix some 5d^° 6d amplitude

into the D states and some 5d® 6s 6p amplitude

35 36 into the P states. Crandell , et al . have

determined that the decay rate from the ^^3,2 state to the ^s, ground state is 350 ± 30% times

greater than the decay rate to the ^Ds-j state.

If LS coupling is valid, the ^p, state decays to the ^Dj/j with probability 3 x 10-''. This decay

is highly suppressed because the energy difference

is only 933 cm-^, which corresponds to a wavelength

of 11 |jm. (The decay rate is proportional to the

cube of the energy difference.) Decay of the ^p state to the ^0^,2 state is forbidden by the

electric dipole selection rules, since it requires

J to change by 2. Hyperfine and Zeeman interactions mix different J states, making this decay

slightly allowed. However, we estimate this

probability to be less than 10-^^ at a magnetic

field of 0.5 T. The ^o. state decays to the /2 205,2 state at a rate of about 54 s-^ and to the

2$, state at a rate of about 42 s-^. The ^Ds/j state decays to the 25 at a rate of about

9.5 s _i 35If the laser-induced 25 to 2p

transition is denoted by Yj. the ions reach the

2D3 /2 state at a rate of 3 x 10-'' • y. from which

they decay, with probability about 0.56, to the

2Ds/2 state, where they stay for an average of

about 0.11 s before decaying back to the 2c

state. If Yn is less than 6 x 10^ s-^, the ions

spend less than 10% of their time in the meta- stable D states, so that this trapping does not cause a problem for the cooling. Transitions to the D states occur at a rate which is much less than the optical pumping rates between ground-state

sublevels, so their neglect in the previous

discussion on optical pumping is justified.

Appendix C

Maximum Number of Stored Ions

Since the cyclotron and axial kinetic energies

are assumed cold, we will assume a uniform charge

distribution for the ions. For a spherical ion

cloud, the magnetron rotation frequency of the

cloud (u) ) is given from the equations of motion

as:

^m- 2" -V(?r -(^^) '-,

where lu is the unperturbed ion cyclotron frequency

u) = eB/Mc, tj is the axial oscillation frequency

c z •, derived from the applied trap voltage (V ) as:

u)2 = 4e V /M(r2 + 2z2)

z 00

and p is the space charge density. We have for

the space charge potential from the ions (inside

the cloud)

^. = I

np (r2 + z2)

and the trap potential may be expressed as

Vq (r2 - 2z2)

r„2 + 2z„2

When the axial -cyclotron motion is cold, we

have (|).(z) = 4>j(z) = 0; this implies

(C.2)

^nt the maximum second-order

J less than £, then we require

p

=

. 3

' 47t M

e "4

If we ass r/ we

s Doppler Shi ftl 1

r

u) = c V2e/r„„ where

m c£ c£

is the radius of the

cloud. If we want to maximize the number of ions

N = 4np(r .)3 /(3e), then we want to maximize this

expression subject to the above contraints on u» . Substituting Eqs. (C.l) and (C.2) into this expression for N we have:

N = 2 VTMc

l?r- \i ^ ("'c '"cil c V2e (C.4)

6i0

TN-55
(Values of r . where this expression is negative

are unphysical because we assume that e is a fixed

value on the perimeter of the cloud (at z = 0) no matter what its size is. For very small clouds,

this requires V large enough that the ions are unbounded—the case for N < 0.) From (C.2) and the expression for for u)|,

Vo = ^K * 2z§) p/3 (C.5)

c£ For the conditions assumed in the text, r 0.5 cm, u) = 40.5 kHz, and e = 10-^^ we have N = 8.2 X 10*. Assuming z = r /1. 64 = 0.8 cm, then

V = 0.071 V.

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21. The ground-state hyperfine separation was taken from the measurements of Y. Guern, A. Bideau-Mehu, R. Abjean, and A. JohanninGilles, Physica Scripta 14, 273 (1977).

22. D. J. Wineland and Wayne M. Itano, Phys. Rev. A 20, 1521 (1979).

23. Wayne M. Itano and D. J. Wineland, Bull. Am. Phys. Soc. 24, 1185 (1979).

24. A similar limitation for the rf trap is imposed by the kinetic energy in the forced

"micromotion" of ions not at the center of

the trap. See D. J. Wineland, Proc. 2nd Int. Conf. on Precision Meas. and Fundamental

Constants, Gaithersburg, MD, June 1981, to be published.

25. D. J. Wineland and Wayne M. Itano, Phys.

Lett. 82A, 75 (1981).

26. The basic scheme proposed for ^^^Ba in Ref.

9 should be possible; however, two important

errors in this paper should be noted: first,

the number of scattered photons before re- pumping was overestimated because pumping in the wings of the other optical lines was not considered. Second, although the improvement

in signal-to-noise ratio due to multiple

scattering was noted, it was overestimated

because it was not assumed that the fluctuations would be limited to the statistical

variations in the number of ions that made

the clock transition. (See Appendix A). 27. R. E. Stickel, Jr. and F. B. Dunning, Appl.

Opt. 17, 981 (1978).

28. G. D. Boyd and D. A. Kleinman, J. Appl. Phys.

39, 3597 (1968).

29. F. B. Dunning and R. E. Stickel, Jr., Appl.

Opt. 15, 3131 (1976).

30. P. Huber, Opt. Commun. 15, 196 (1975).

31. R. E. Stickel, Jr. and F. B. Dunning, Appl.

Opt. 16, 2356 (1977).

32. P. L. Bender, J. L. Hall, R. H. Garstang, F. M. J. Pichanick, W. W. Smith, R. L. Barger,

and J. B. West, Bull. Am. Phys. Soc. 21, 599 (1976).

33. See for example: V. S. Letokhov and V. P. Chebotayev, Nonlinear Laser Spectroscopy , (Springer-Verlag, New York) 1977.

34. Ye. V. Baklanov, V. P. Chebotayev, and B. Ya. Dubetskii, Appl. Phys. 11, 201 (1976).

35. R. H. Garstang, National Bureau of Standards

J. Research 68A, 61 (1964).

36. D. H. Crandall , R. A. Phaneuf, and G. H. Dunn, Phys. Rev. All, 1223 (1975).

37. See for example: S. A. Prasad and T. M. O'Neil, Phys. Fluids 22, 278 (1979). T. M. O'Neil and C. F. Driscoll, Phys. Fluids 22,

260 (1979).

38. T. F. Gallagher and W. E. Cooke, Phys. Rev. Lett. 42, 835 (1979).

39. L. L. Lewis, Wayne M. Itano, and D. J. Wineland, to be published.

40. H. G. Dehmelt, paper in these prodeedings.

611 TN-56
PHYSICAL REVIEW A VOLUME 24, NUMBER 3 SEPTEMBER 1981

Precision measurement of the ground-state hyperHne constant of Mg^

Wayne M. Itano and D. J. Wineland

Time and Frequency Division, National Bureau ofStandards, Boulder, Colorado 80303

(Received 13 April 1981) The ground-state hyperfine constant A and the nuclear-to-electronic g-factor ratio g,/gj of "Mg* have been measured by a laser optical-pumping double-resonance technique. The ions were stored in a Penning trap at a magnetic field of about 1 T. The results are /4 = - 596.254 376(54) MHz and g,/gj = 9.299 484(75) X lO"'. The

magnetic field at the ions was stabilized by servoing it to an (J m, =0, Anij = ±1) electronic Zeeman transition. Other hyperfine (Am, = ± 1, Amj = 0) transitions were detected while the field was thus stabilized. The derivative of the {m,,mj] = ( — 3/2, 1/2) to ( — 1/2, 1/2) transition frequency with respect to magnetic field Bq goes to zero at 5o~1.24 T. The corresponding resonance was observed at this field with linewidths as small as 0.012 Hz

{Q = 2.4 X 10'°) by implementing the Ramsey interference method with two coherent rf pulses separated in time by up to 41.4 s.

I. INTRODUCTION

The ground- state magnetic-dipole hyperfine constants {A values) of the neutral alkali atoms have

been measured to high precision by opticalpumping and atomic beam magnetic resonance

techniques.^'^ A variety of theoretical methods

have been used to calculate these quantities.^ By

comparison, much less experimental and theoretical work has been done on the singly ionized alkaline-earth atoms, which are isoelectronic to the

alkalis. Purely optical measurements generally

suffer from low resolution. Direct and indirect

optical- pumping methods have been used to measure the ground-state A values of several alkalineearth ions and some other ions that have a ^S^/g

ground state,^ either in buffer-gas cells [Ba*, Cd*,

Be*, Mg* (Ref. 5)] or in ion traps [He% Kg* (Ref.

6), Ba* (Ref. 7)]. Also, electron-spin resonance

(ESR) spectroscopy has been used to study Mg*

(Ref. 8) and Cd* (Ref. 9) trapped in solid rare- gas

matrices, where the A values may be shifted from

their free- ion values by a significant amount. This

shift is typically a few percent for the neutral alkalis." The ground- state A value of Li- like fluorine

has been measured by the method of perturbed

angular correlations." A few calculations of the ground- state A values of alkalilike positive ions

have been published, especially for the Li

series. ^'^^

In this paper, we report on the first high-precision determination of the ground- state A value

of the free ^^Mg* ion. The nuclear-to-electronic

^g^- factor ratio g/gj was also determined. A preliminary report on these measurements has ap- peared previously."

Our experimental techniques have several unusual features:

(1) The ions were stored in a Penning ion trap

in ultrahigh vacuum. Perturbations to energy

levels caused by collisions with neutral molecules

or other ions or by the trapping fields were very

small, except for the Zeeman shift due to the large applied magnetic field. Relaxation times were many seconds so that very narrow reson- ances could be observed.

(2) Resonant light pressure from a frequencydoubled dye laser was used to reduce the ion temperature to less than 1 K, thus reducing all Doppler effects, and to confine the ions to a small

volume around the trap center, which reduced

magnetic- field inhomogeneity effects. ^''"^^

(3) Laser optical- pumping, double-resonance

techniques can achieve good signal-to-noise ratios, even with very small ion numbers.^'' In fact,

double- resonance signals can be observed from a

single ion.^^'^*

(4) The experiments were carried out in a high

magnetic field, which was needed to confine the

ions. In this case, the transition frequencies depend upon A, the g factors, and the magnetic field. Short-term fluctuations (~1 s) of the magnetic

field imposed the main limitation on the determinations of A and gj/gj.

The details of the experimental method are discussed in Sec. II. The values of A sndg/gj are

calculated from the data and their uncertainties are estimated in Sec. in. In Sec. IV these values are compared with the results of other experiments and with theory, and applications and extensions of our experimental techniques are discussed.

II. EXPERIMENTAL METHOD

A. ^Mg* level structure

Singly ionized Mg is isoelectric to neutral Na. The ground state is labeled 3s^Si^2> ^^'^ ^^^ ^^^^^ excited states are labeled ip^Pj (J = 2,|). Light

of wavelength 279.6 nm was used to drive the

24 1364 © 1981 The American Physical Society

TN-57
24 PRECISION MEASUREMENT OF THE G R O U N D S T A T E . .

.

1365

transition from the ground state to the 3p ^P^/z level. The ^^Mg nucleus has nuclear spin / = |, so the ground and excited states have hyperfine structure. The experiments were carried out in a magnetic field of about 1 T, so that the nuclear and

electronic angular momenta I and J were essentially decoupled. Figure 1 shows the ground- state energy levels as a function of magnetic field. The

hyperfine constants of the 2p ^Pj^z state have not been measured. However, they can be estimated^^

from the known ip'^Pj fine structure splitting^"

and the known magnetic dipole^^ and electric quadrupole^ moments of the ^^Mg nucleus. These val- ues are A =-18 MHz and 5^22 MHz. A value for the effective nuclear charge Z, = 10 was assumed.

B. Apparatus

The basic experimental apparatus has been described previously. ^^'^^ A Penning-style ion trap,

which uses a uniform magnetic field and a quadrupolar electrostatic potential,^^ was used to store

the ions. The characteristic dimensions of the trap were r^ = 1.642o = 0.63 cm. Typical operating parameters were Kg = 7 V and Bq = 1 T. ^^Mg*

ions were loaded into the trap in the following way:

An electron beam coincident with the trap axis

created positive ions by collisions with background

gas, which were confined by the trap. An oven

containing ^^Mg of 98% isotopic purity was heated,

and the ^^Mg neutrals underwent charge- exchange

reactions with the trapped positive ions to yield

trapped ^'^Mg* ions. The magnetron radii of the ions

30 —

-I 1

\ r- (m,,m,)

8o(T) FIG. 1. Ground-state energy levels of ^^Mg* as a function of the magnetic field Bq. Note the transition from low field, where I and J are coupled to form the

total angular momentum F, to high field, where they are decoupled. All of the experiments were done at high field.

increased slowly due to collisions with background

gas. The storage time has been observed to be

about one day without light pressure cooling or

confinement. The number of ions initially stored

could be varied from about 10^ down to about one

by varying the current and duration of the electron

beam. Typical numbers for the experiments re- ported here were about 10 to 100 ions. The rea- son for using such small numbers was that in or- der to get experimentally convenient opticalpumping time constants of no more than a few seconds, relatively high light intensity was required.

Since the total available power was low (<40 mW),

it was necessary to focus the beam to a small spot

size (about 50- /^m diameter). The maximum ion

density is limited by space-charge effects, so in order for the ion cloud to be small enough to have

good overlap with the focused beam, the total ion

number had to be small. During the resonance ex- periments, the background pressure was maintained below 1.3 X 10"^ Pa (lO^^ Torr).

The source of 279.6 nm radiation was the second

harmonic of the output of a single- mode cw rhodamine 110 dye laser. The frequency doubling took

place ina90°-phase-matchedAD*P (deuterated ammonium dihydrogen phosphate) crystal, operated at a temperature of about 117 t. The uv power generated was typically between 5 and 30 jiW. The bandwidth of the dye laser was about 1 MHz and could

be frequency stabilized in a long term to about

±200 kHz by locking it to a saturated absorption

hyperfine feature in Ij. If the laser frequency was

fixed in this way, fine tuning could be accomplished by varying the trap magnetic field to Zee- man shift the Mg* levels. The second harmonic radiation was separated

from the fundamental by a prism and focused into the trap, passing through 3-mm holes drilled into the ring electrode. The beam entered and exited

from the vacuum enclosure through Brewsterangle windows. Resonance fluorescence photons

emitted into the backward direction inside an//4

cone were collected by a flat mirror set at 45° to the incident beam and imaged by lenses onto a

photomultiplier tube (PMT). (The incident beam

passed through a hole in this mirror.) Two stages

of spatial filtering were used to reduce background

due to light scattered from the windows. An interference filter centered at 280 nm and a Corning

7-54 glass filter reduced background due to light

at other wavelengths. The net detection efficiency

for a photon emitted from an ion, taking into ac- count the detection solid angle, mirror reflectivity, transmission of the lenses and filters, and

PMT quantum efficiency, was about 2x 10"^. Data collection was controlled by a minicomputer, which swept the rf or microwave frequency .

TN-58
1366 WAYNE M. ITANO AND D. J. WINELAND 24

used to drive the ^^Mg* ground- state transitions

and recorded the number of photons detected by

the PMT. The output of various frequency synthesizers was applied to a probe near the gap between

the endcap and ring electrode and used to drive the nuclear spin flip (Amj-0, Aw^ = ±l) rf transitions

at about 300 MHz. The output of a klystron was

directed at the trap from the end of a waveguide

positioned outside one of the Brewster windows

and used to drive the electronic spin flip {Amj-±l,

Amj=0) microwave transitions at about 30 GHz.

The klystron was phase locked to the third harmonic of a microwave frequency synthesizer. For

high-accuracy work, the frequency synthesizers were phase locked to the output of a cesium atomic beam frequency standard. The computer also

controlled an rf switch and a light shutter, which were necessary for some of the resonance schemes.

C. Resonance methods The basic principles of the optical-pumping,

double-resonan' methods have been outlined previously.^^ Resonant light pressure was used to cool the ions to roughly 1 K and to confine them

to a small volume near the trap center. The electronic Zeeman splitting was much greater than the Doppler width. With light polarized perpendicular

to the magnetic field and slightly lower in fre- quency than the (^Si/2> mj--j, mj = -^) to (^Pj/z,

m j=-2,'mj = -\) transition frequency, the steadystate population of the (^S^/2» ^/=-l> ^^--2)

level was about 94% and the (^Sj/g, m , = -\, m j =5)

population was about 6%, as a result of competition

between relatively weak off-resonance transition

rates.^'' The frequency detuning from the reson- ance center, typically about 30 to 50 MHz, was

necessary in order to provide cooling. ^*'^^ At the

light intensities used, these weak pumping rates were about 1 s"^, much faster than any other relaxation rate between ground-state sublevels.

The optical-pumping mechanism is unusual in that the [wj--\, nij--^) ground-state sublevel,

which is driven to the excited state at the highest

rate, retains the highest steady -state population.

The optical transition frequencies are shown in Fig. 2. Optical pumping into the Wj = -i groundstate manifold takes place by the same mechanism

as for ^*Mg*, which has zero nuclear spin. The

nij quantum number can be ignored for this part

of the discussion. For light polarized perpendicular to the magnetic field, Wj must change by ± 1 in a transition from the ground state to the excited state. In spontaneous decay from the excited state to the ground state, nij can change by

or ±1. Thus, the -| 1 transition, which is strongly driven by the laser, and the +i — I transition do not optically pump the ground state. The

first and second numbers refer to the ground- and

excited- state trij values. The other allowed transitions, + 1 —i and -i — + 2, do cause optical

pumping because the ion can decay to either

ground- state nij manifold. These transitions are

driven weakly in their Lorentzian wings. Since

the second harmonic of the laser frequency is close to the -i —| transition frequency, its detuning from the -i — + 2 transition is four times

as great as its detuning from the +^ —i transition (see Fig. 2). Hence, the mj = + ^ ground-state

manifold is depopulated 16 times faster than the mj = -^ manifold. In the steady state, the mj = --^ manifold has 16 times the population of the nij= + ^ manifold, provided that the optical-pumping rate

is much faster than competing relaxation rates. Optical pumping into the (w?; = ~|) ground-state

sublevels occurs because of hyperfine coupling in the excited state. The excited- state sublevels,

which are labeled {nij, m j = -|), actually contain

small admixtures of states with lower nij, except

ior {mi=-l, mj = -i), which is pure. For example, the state which is nominally {nij = -i, mj = -\)

contains a small amplitude (about 1.4 x 10'^ at B^

= 1 T) of the mj=-\, mj = -i) state, etc. Hence,

if an ion in the (w^ w_, = f) ground- state sublevel

is driven to the (ra^ mj = -l) excited- state sublevel, it has a small probability of decaying to a ground- state sublevel with lower nij, unless m,

= -|. Eventually, the ions are trapped in the mj

= -| sublevels. The process by which an ion absorbs and emits a photon and is left in a different

ground- state sublevel has been treated as if the absorption and emission were separate events. It would be more accurate to describe this process

in terms of spontaneous resonance Raman scattering, but this distinction is actually of no importance, since neither the coherence nor the time delay between the two events is of interest. The fluorescence level was proportional to the

(^Si/2. '"/=-2' w^ = -i) population. Hence, a transition which changed this population could be detected as a change in fluorescence. In the following, let the (^S^/2> ^r "^j) state be denoted by

{yyij, mj). Both the (-1, -i) to (-|,-i) nuclear

spin-flip transition and the (-2,-2) to (-|,i)

electronic spin-flip transition were detected by a decrease in fluorescence when rf at the resonance

frequency was applied. When the rf transition was

saturated, the fluorescence dropped by about 50%.

This technique was previously used to detect the ground- state electronic spin-flip transition of ^^Mg*, which has zero nuclear spin.^^ When light was tuned close to the (^S^/g. nij=-2, mj = \) to

{^P^,2, m,= l, nij-z) transition, most of the population was pumped into the (|, |) state. The (|,

TN-59
24 PRECISION MEASUREMENT OF THE GROUND-STATE 1367

1.07 X lO'^ / Hz /

4-4

/ 4-4

. 93 GHz

Optical-Pumping

/ Frequency

c

/

3

«

1^1 4-4

\

w

c / '"l

\ , , c

• 1.4 GHz

\ 4 — 4 /z \

'- "- \ \ \ vz

\ \z

\ -i-^i

Electronic Zeetnan

Structure

Hyperfine

Structure

FIG. 2. Structure of the ^*Mg* 3s ^Si/2 to 3p ¥3/2 279.6-nin line at a magnetic field Bo = lT. The quantum numbers

mj and m'j refer to the ground and excited states, respectively. The frequency splitting between adjacent electronic Zeeman components is approximately f^gBo//? . At high magnetic field, m[ is a good quantum number which is conserved in the optical transition process. Hence, each electronic Zeeman component splits into six hyperfine components corresponding to different values of nij. This splitting is shown for the (wj^ — wp= (-| — -|) component. The

hyperfine splitting is drawn to a different scale than the electronic Zeeman splitting. The natural linewidth is 43 MHz,

and the Doppler width is typically 100 MHz when the ions are cold. For most of the optical pumping experiments, the laser was tuned so that its second harmonic was about 30 to 50 MHz lower than the (^S,/2, W;^=-f, rnj=-i) to (^P3/2, >W/=-|, W/=-i) transition, which is marked with an arrow.

2) to (I, 2) and (i, i) to (I, -2) transitions were

then detected by a decrease in fluorescence when

rf was applied.

Other transitions were detected by more complex resonance schemes. The (~|,i) to (-2,2)

transition was detected by a triple resonance

method. The ions were optically pumped into the

(-1, -i) state, the (-1, -j) to (-1, |) transition was saturated by frequency modulating the kyl- stron through the resonance at a rate of about 10 Hz, and a decrease in fluorescence was observed

as a frequency synthesizer was swept through the (-1, i) to (-1, i) resonance. The (-|, i) to (-i, 5) transition was also detected by a quadruple res- onance method. The ions were optically pumped

into the (-|, -\) state and both the (-|, -i) to (-1, 1) transition and the (-|, i) to (-|, |) transition were saturated. A decrease in fluorescence was observed as the frequency of another oscilla- tor was swept through the (-1, 2) to (-2, i) resonance. The reason for going to such lengths to ob- serve this transition is that the first derivative

of its frequency with respect to magnetic field goes to zero at about 1.24 T, so that broadening

and shifts of the resonance due to magnetic -field instability are essentially eliminated. The (-|,i)

to (-1, i) transition frequency becomes field independent at about 1.9 T, which was higher than our electromagnet could reach. These methods could

be extended to observe other transitions with fur- ther reduction in signal.

D. Resonance shifts and broadenings

The rf and microwave resonances can be broadened and shifted by magnetic-field fluctuations and

drift and by interaction with the uv radiation and

with additional rf fields. At T = 20 K, which was

higher than was reached in any of the resonance

experiments, the fractional frequency shift due to the second-order Doppler effect is -1.1 x 10'",

which is negligible. Frequency shifts due to col- lisions with ions and neutral molecules were also

negligible. At low temperatures, the repulsive

TN-60
1368 WAYNE M. ITANO AND D. J. WINELAND 24

Coulomb force kept positive ions from approaching

each other very closely. At a temperature T, the distance of closest approach R is given roughly

hyksT = e^/R. For 7 = 20 K, i? = 8.4x 10"^ cm, so

that spin exchange reactions between ions did not occur. The maximum electric field for this case

is e/R^ = 2\ V/cm, and the rms field is much

lower, so Stark shifts of the hyperfine structure are too small to be observable. Electrons and

negative ions were expelled from the Penning trap

because the potentials were set to trap positive

ions. Collisions with background neutrals took

place at a rate of less than about (5 min)'^, which was determined from the ion temperature relaxation rate. Spin depolarization rates, which would

increase the resonance linewidths, should be much

lower.^ Fractional hyperfine density (pressure)

shifts, estimated from the known shift for the

^^''Ba* ion,'* were less than 10"^^. In general, the magnetic-field instability caused

the greatest uncertainty in determining the reson- ance frequencies. The fractional fluctuations,

measured with an NMR probe, were about 2x10"''

in 1 s and 10'® in several seconds, with occasional

sudden jumps of about 10'®. The electronic spinflip transition frequencies are essentially proportional to the magnetic field B^, so that typically, a 10"^ shift in B^ causes a 30 kHz frequency shift at 30 GHz. The nuclear spin-flip transition fre- quencies have field sensitivities that vary with B^. A 10"^ shift in B^ at B^ = 1.U T causes a 16-Hz

shift of the (-|,-|) to (-i,-i) transition at 287.7

MHz and a 6-Hz shift of the {-|,i) to {-|,i) transition frequency at 286.7 MHz, For some experiments, the magnetic field was stabilized to an NMR probe. Because of the space limitations, the probe had to be located in a region where the field was not very homogeneous. This resulted in a

fractional NMR linewidth of about 10"^ with a sample of diameter approximately 1 cm. The field fluctuations were at best about 1 x 10"^ over short

time periods, as determined by the resonance

linewidths. However, long-term drifts of about

10"^, due primarily to changes with temperature

of the tuned circuit in the NMR probe, were observed over periods of about 1 h. This field stabilization method was used to obtain the data re- ported previously.^^ Another field stabilization

method, based on sensing the electronic spin-flip

transition of the ions, will be discussed later. The light used for state preparation and detection causes resonances to be shifted and broadened.^* These effects can be eliminated by shutting the light off while the transition is driven. When

the light was chopped, the linewidth of the (-|,-i)

to (-§,2) electronic spin-flip transition was observed to be as narrow as 15 kHz, which is consistent with the measured short-term magneticfield stability. The light was chopped on and off with a 50% duty cycle at a rate typically between

10 and 100 Hz. When the light was not chopped,

linewidths as wide as 1.3 MHz were observed.

The narrowing of the resonances when the light was chopped indicates that the transitions were

driven primarily when the light was off. The light-broadened line shape can be understood by applying the Bloch equations^^ to the ef- fective two-level system consisting of the (-1, -i)

and (-2, 2) states. The ions spend very little time

in any of the other ground- state or excited- state sublevels. In the absence of the microwave radiation, the state populations relax to their steadystate values with a time constant Tj, which can be

measured by first saturating the microwave transition, then suddenly shutting off the microwave

power, and observing the fluorescence return to

its normal level, as shown in Fig. 2 of Ref. 16. Typically, T^ was about 1 s, which corresponds

to an average light intensity of about 3.5 mW/cm^.

The transverse relaxation time T2 was much

shorter, since every optical scattering event destroyed the coherence of the effective two-level

system. The resonance linewidth in the limit of zero microwave power is l/lvrTg). This was observed to vary from about 40 kHz or less to 250

kHz, depending upon laser intensity and overlap of the light beam and ion cloud. For steady, uniform

illumination 2/T2 = y^, where 7'^ is the average scat- tering rate. If the overlap of the beam with the cloud

is poor, thenthelight is effectively chopped on and

off as the magnetron motion^^ takes the ions into and out of the beam. In this case, the effective

scattering rate which determines T^ saturates at about the magnetron frequency p^ as the light in- tensity increases.

The resonance linewidths were observed to in- crease with m.icrowave power in accordance with

theory .^^ This increase gave an absolute calibration of the microwave magnetic field amplitude

at the ions. Under typical operating conditions,

this was about 2x lO"" T (0.2 mG). Similar light

broadening effects were seen on the (-|, -|) to (-2,-2) nuclear spin-flip transition, but were

not studied in detail. They were eliminated by

chopping the light, so that linewidths of about 30

Hz were observed, due to magnetic-field fluctua- tions during the time required to sweep the res- onance.

In most cases, light shifts^* of resonances were

difficult to observe because of field fluctuations

and light broadening effects. However, in one

case, a light shift was definitely observed. The

(-2, 2) to (-i,i) field insensitive transition at B^

= 1.24 T was observed by the quadruple resonance

TN-61
24 PRECISION MEASUREMENT OF THE GROUND STATE 1369

method described previously. This transition was

driven with the light on and with it off. In both cases, the rf pulses at the (-2,2) to (-f.i) and

the (-|,i) to (-i,i) transition frequencies were

separated in time in order not to broaden the resonance. The resonance frequency was shifted down by about 3.5 Hz when the light was on relative to its value when the light was off. This shift implies an average light intensity of about 10 mW/

cm^, which is in reasonable agreement with the

value inferred from T^. Bloch-Siegert and other rf line-pulling effects^^ were negligible because of the low powers used.

Line broadenings or shifts due to real or virtual

transitions induced by the auxiliary rf fields in the triple and quadruple resonance methods can be eliminated by shutting them off while the main

transition is driven.

E. Zeeman resonance magnetic-field stabilization Asystem was developed which servoed the magnetic field by keeping the frequency of the (-§,-2)

to (-1, 2) electronic spin- flip transition resonant

with a fixed microwave frequency. Other rf transition frequencies were measured while the field was thus stabilized. The field stabilization scheme

worked as follows: The light was tuned close to resonance with the (^Si/2> »2/ = -|> ynj = -k) ^0

(^-P3/2> "^/=-li Wj = - 2) transition frequency. A

microwave frequency v, with which to stabilize the electronic spin-flip transition, was chosen.

The microwave radiation was shut off for 0.2 s, so that the rf transition could be driven without

broadening from this source. The ions were subjected to microwave radiation of frequency i^ + lO

kHz for 0.8 s. Scattered photons were counted

during this period and the number was stored. The microwave radiation was shut off for 0.2 s.

It was turned on again for 0.8 s at frequency i' - 10 kHz and the scattered photons were counted. This

number was digitally subtracted from the one

stored previously. If the result was positive, the

field was too low, and vice versa. This number

was sent to a digital-to-analog converter which was connected to an analog integrator. The output

of the integrator controlled the magnet power supply and corrected the field. The cycle was then

repeated. Either the (-|, -|) to (-|, -i) or the

(-1, 2) to (-|,i) nuclear spin-flip transitions

could be detected by a decrease in the average

photon scattering rate when rf at the resonance

frequency was introduced. The light was chopped

at about 8 Hz to reduce light broadenings and

shifts of the resonances. A typical nuclear spinflip resonance curve is shown in Fig. 3. Unfortunately, the field stability achieved with this

1 ' • >^ ... • • s

10c

•

• 9

• •

•

• •

_aj • • •

• • ^ _ • -

<u • c

A

O)u — —

to

a> • 3

Ll.

J. -1. -i.

•

1 i

1 1 850 950

rf Frequency - 287 699 000 (Hz) FIG. 3. rf resonance curve for the {mj, nij) = (-f, — ^) to (— f. -2) hyperfine transition. The magnetic field Bo was actively stabilized by keeping the (— |, -;|) to ( — |, +2) electronic spin-flip resonance

centered at 33 466.187 MHz, which corresponds to Bo= 1.1405 T. The 35-Hz linewidth is due primarily

to the remaining short-term fractional fluctuations in Bq of about 10 "^. Each point represents an integration

time of 4 s.

system was limited by the low signal-to-noise

ratio and the relatively poor short-term stability of the magnet. The fractional field fluctuations,

estimated from the widths and reproducibility of the nuclear spin-flip resonances, were about

1 X 10"^. These experiments were all carried out

at B^ = 1.14 T. This was experimentally convenient because the dye laser could be locked to a

strong I2 hyperfine feature.

F. Field independent transition The first derivative with respect to the magnetic

field of the (-|, i) to (-i, 5) transition frequency

goes to zero at a field of approximately 1.24 T. The fractional frequency shift bv/v for a fractional

field shift bB/B from this field is -0.012 {bB/B^.

This transition was detected by the quadruple res- onance scheme outlined briefly in Sec. 11 C. The

(-Ij-i) to (-|,i) electronic spin-flip frequency

for this field could be predicted from previous

data with an uncertainty of less than 10"^. The

microwave source was set to this frequency and

the magnetic field adjusted until the decrease in fluorescence corresponding to the double reson- ance was observed. The field was left unstabilized, but was periodically reset by this method.

The drift, after warmup, was on the order of (1-2) X 10'^ over 30 min, so it was not a serious

problem. The microwave source was frequency

swept at a rate of about 8 Hz by ± 750 kHz, so

that the ions would still be in resonance during

part of the sweep, even if the field drifted by

± 2 X 10"^ This field shift corresponds to a fre- quency shift 61^ of 1.4x10-3 Hz on the (-1,5) to (-2J 2) transition. The necessary dye laser freTN-62
1370 WAYNE M. ITANO AND D. J. WINELAND 24

quency for this field was about 25 MHz away from

the nearest hyperfine feature in I2. The laser

frequency was stabilized to this feature by fre- quency shifting part of the beam by 25 MHz with an acousto-optic modulator and passing it through

the I2 cell. The measurement sequence was as follows: The

microwave source was left on continuously. We

estimate that this caused an rf Stark shift of the (-2,2) to (-2 ,2) transition frequency of less than

10"^ Hz. The light was then blocked with an electromechanical shutter to eliminate light shifts and broadenings. An rf oscillator was switched

on for a period of about 1 s to drive the (-f ,i) to (-2,2) transition. The oscillator was frequency

modulated by about 600 Hz at a rate of about 9 Hz to eliminate any sensitivity to field drifts. After this first rf oscillator was shut off, the (-2,1) to (-i,i) transition was then driven. Both

the Rabi single-pulse method and the Ramsey twopulse method were used.^^ For the Rabi method,

the rf was switched on for a period t. For the Ramsey method, the rf was turned on for a period

T, off for a period T , and on again for a period t. The two pulses were coherent in phase because the oscillator ran continuously during the period T

,

but was attenuated by 130 dB. At the end of this resonance period, the first rf oscillator was

switched on for about 1 s to drive the (-f.i) to (-|,i) transition. The light was then unblocked

and the scattered photons were counted. The

initial fluorescence level was lower if ions had

been driven to the (-i,i) level, than if they had

not. The fluorescence level then rose to the previous steady -state value as the ions were optically

pumped." The initial fluorescence level (first 2 to 4 s) was divided by the final level (next 4 to 10 s) in order to obtain a signal that was less

sensitive to long-term laser intensity fluctuations.

This cycle was repeated several times for each

frequency setting of the (-|,2) to (-2,2) rf oscillator in order to improve the signal-to-noise

ratio. With the Rabi method, one obtains a linewidth

(in Hz) of (1.25^"^ at optimum rf power. Linewidths as small as 40 mHz were observed with t= 20 s. With the Ramsey method with t« T and

optimum rf power, one obtains an oscillating

line shape with a central minimum of width of

(2 T)"\ The central minimum can be identified by the fact that it does not change for different

T or by using the Rabi method, which yields only one deep miminum. Linewidths as small as 12 mHz were observed with T = 41.4 s. Since the frequency was about 292 MHz, this corresponds

to aQ of 2.4 X 10^°. Typical curves are shown

in Figs. 4 and 5. No evidence was seen for relaxations between ground -state sublevels, even

for these long values of T . However, the initial scattering rate (and hence, the signal) became

low for Ts 40 s due to heating of the ions while

the light was off, which increased the Doppler

width.

III. RESULTS

The transition frequencies between {mi,mj)

sublevels are given in closed form by the BreitRabi formula. ^^ There are only three independent

parameters in this formula: A, gi/gj, and

gj[i-B^Jh, where Mb is the Bohr magnetron and

Bq is the magnetic field. Hence, it is sufficient

to measure three different transition frequencies

at the same magnetic field to determine all three

parameters. In order to decorrelate the three

parameters, it is helpful to measure one Am/=0,

Aw7_^ = ±l transition frequency, which depends

mostly Qfs\ gj\i.^Bjh, and two Aw/ = ±1, An2j- = transition frequencies, which depend primarily

on A and secondarily on the other parameters.

One of the two Awz^ = ± 1 , ^rrij = transitions

should have mj= -j and the other should have

mj=^ in order to decorrelate A ^ridgj/gj. The

absolute g factors can be determined if an independent measurement of magnetic field (actually

><

1

1 1 I

1

1

(75 0.023 Hz c

* 0) A cA P\ A

<A A A /ft

- u / \ /

' ' \ h^ V

) c: ) J / A \ 0/ ' n) / a / \ r \ / 1 (J / ^ / X \ \ / ~

0) \°/ )V9 \ V/ \ / oV hJ \J hJ Vo

3

1

t 291996 251.899 Hz

1 1 1

'

1 1.80 1.85 1.90 195 200

rf Frequency -291 996 250 (Hz) FIG. 4. rf resonance curve for the {mj,mj)

= (—2.2) to ( — 2,2) hyperfine transition. Each circle represents the average of four measurements (total detection fluorescence integration time of 8 s). The

solid curve is a theoretical fit. The quadruple reso- nance technique described in the text was used. The

oscillatory line shape results from the use of the Ramsey method to drive the transition. Two coherent rf pulses of duration t=1.02 s separated by T = 20.72 s were applied. The vertical arrow marks the central minimum, which corresponds to the resonance frequency.

The magnetic field Sq was set so that the (- 1, — ^) to (-2,2) electronic spin-flip resonance was in the range

of 36 248.374 ± 0.750 MHz, which corresponds to Bo= 1-2398 T.

TN-63
24 PRECISION MEASUREMENT OF THE GROUND STATE 1371

>^

I 1 1 1

- o -

(7) ,^—^ ^.^ c Cj^\\ /^^^

(U / \ / \ c - / \ / \ - 8 q/ Y0.0I2HZ-/ o\ c 7 \ ad \

a> O / \ / \ o / \ / \

— y °Vc/ 'M) o

3

Ll

-

1

t 291 996 251 899 Hz

1

1 i

-i) to (-|,i) and the (-1,5) to (-i,i) transition

frequencies at B(2). The experimental results

(in MHz) are

1.88 1.89 1.90 1.91 1.92

rf Frequency - 291 996 250 ( Hz

)

FIG. 5. rf resonance curve for the (mi,mj)

= ( — 2.2) to ("212) hyperfine transition. Each circle represents the average of four measurements (total detection fluorescence integration time of 16 s). The

pulse separation time T was 41.40 s. The experimental

conditions were otherwise the same as for the curve shown in Fig. 4. The vertical arrow marks the central minimum.

^igB^/h) is made at the same time. However,

this was not done.

At the time of our first experiments, the value

of A was not very well known.^'^^'^^ The ^^Mg*

(-|,-i) to (-f,i) transition was found after searching a frequency range corresponding to A values from -628 to -595 MHz. The value

of gj^gB^/h had been determined from the electronic spin-flip transition frequency of ^^Mg*,

before loading ^^Mg* into the trap. After allowing

for drift of the magnetic field between measurements, A could be determined with a fractional

uncertainty of 2x10"^ by taking g^ from previous

NMR measurements and assuming gj was equal

to the free-electron g factor. The positions of the other transitions could then

be predicted and were observed by the methods

decribed previously. Several transition frequencies were measured at each of several magnetic

field settings between 1.0 and 1.2 T, and were

used to determined and gj/gj.^^

Our final determinations of A and g^/gj are based on a combination of the (-§,-2) to (-|,

-i) transition frequency measured with the Zee- man resonance field stabilization scheme at magnetic field 5(1) and the (-1,1) to (-1,1) field- independent transition frequency at magnetic

field 5(2). There are thus four parameters to be

determined: A, gj/gj, gjtJ.gB(l)/h, and g^iigB (2)/

h. Four independent equations are provided by

the Breit-Rabi formula expressions for the four

frequencies v(l) through 1^(4). Here f(l) and v{2) are, respectively, the (-f.-i) to (-§,2) and the {-|,-i) to (-2,-2) transition frequencies at B(l) and 1^(3) and v(4) are, respectively, the (-§,

1^(1)= 33 466.187 13,

i/(2)= 287.699 897(48),

v(3)= 36 248.374,

i^(4)= 291.996251 899(3)

(1)

No uncertainty is shown for v{l) or 1^(3), since

they were fixed experimentally. [Uncertainties in f(l) or 1^(3) due to magnetic -field fluctuations would

show up as fluctuations in v(2) or f(4), respectively.] The uncertainty of ^'(2) due to the lack

of reproducibility of the line centers from run to run was only about 19 Hz. The uncertainty of v{2)

has been increased to reflect possible offsets of the field servo system due to light shifts of the electronic spin-flip transition frequency or to the magnetic -field inhomogeneity. These effects were

reduced by chopping the light, but may not have

been completely eliminated, because the micro- wave power and the light were not chopped alternately. Thus, the microwave resonance line shapes may have consisted of a broad, shifted

component, due to transitions made while the

light was on, and a narrow, unshifted component

due to transitions made while the light was off. The microwave power was adjusted so that the linewidths were about 40 kHz, due largely, if not totally, to short-term field fluctuations. We

estimate that the offset from the light-shifted

component should have caused a fractional shift

in the field of less than 10*®, which would cause

a shift in v(2) of less than 16 Hz. The field inhomogeneity over a typical ion cloud diameter of about

150 /xm is estimated to be less than 3x 10"®, and

to a high degree the effect on the resonances is averaged away as the individual ions circulate

through the volume. A direct light shift of the

v{2) resonance would be accompanied by a broadening of several kHz, which we did not observe.

No uncertainty is shown for f (3), which was swept

by ± 750 kHz around this center value. The uncertainty shown for 1^(4) reflects, in about equal parts,

the statistical uncertainty of determining the line center and possible magnetic -field drift within

the limits set by the frequency modulation of 1^(3). The results of the fit are A = -596.254376(54) MHz,

gj/Sj = 9. 299 484(75) X lO'^

.

(2)

These results are in agreement with, and about

a factor of 4 more precise than our previously

reported values. Essentially all of the uncertainty

of the fit is due to the uncertainty of v(2). Also

TN-64
1372 WAYNE M. ITANO AND D. J. WINELAND 24

determined were gjiiBB(l)/h= 31 962.151 585 MHz

and 5^^Mb-B(2)A= 34 745.375 447 MHz. The corresponding value of the magnetic field at the position of the ions in terms of its value at the position

of the NMR probe was not known very precisely.

Hence, we are only able to say that g^ is approximately equal to the g value of the free electron

ge, e.g., ^^ = 2.002(2).

We have estimated that all perturbations to A

SLndgj/gj, such as pressure. Stark, and diamagnetic shifts are negligible compared to the stated

uncertainties. If the diamagnetic shift of A has

a coefficient comparable to the measured value

for ^*Rb,^' then its contribution to u{4) is around

0.3 Hz, which is much greater than the uncertainty

of v(4). However, this does not affect the determinations of A andgj/gj, since their uncertainties are dominated by the uncertainty of 1^(2).

IV. DISCUSSION

The present result for A is generally in agreement with the previous less precise measurements. An observation of the 3s^Si/2 ^° ^P^P3/2 resonance line has been made by Crawford et al., using an atomic beam source and Fabry -Perot

etalons.^^ Their reported value for A is -693(38)

MHz, which differs from ours by 16%. The only

other reported optical measurement of A is that

of Drullinger et al.,^^ using the method of laser

fluorescence, stored ion spectroscopy. The isotope shifts and hyperfine structure of the (3s ^8^/2' mj= -i) to {3p'P^/2> ^j- "t) Zeeman component

in a magnetic field of about 0.98 T were observed.

The value obtained was A= -608(50) MHz, in good

agreement with the present result. Both of the optical determinations depend on assumptions

about the isotope shift (the mass -shift formula

is assumed) and on an estimate of the 'P3/2 hyperfine structure. However, the uncertainties of these effects are small compared with the experimental precision. Weber and Gragel^ measured

A by optical pumping in a buffer-gas cell. Their

result, A = -616(50) MHz, is in good agreement

with the present result. Brom and Weltner^ have

observed the electron spin resonance (ESR) spectrum of ^^Mg* trapped in a solid argon matrix

at 4 K. Their result was A = - 595.0(3) MHz, which

indicates that A is reduced by 0.21(5)% under

these conditions relative to its free-ion value (present determination). This is less than typical

shifts for alkali atoms under similar conditions.'"

Ab initio calculations of A in ^^Mg* should be

possible with about the same accuracy as for "Na,^° since the two systems are isoelectronic.

The only such calculation that has been carried

out so far is by Lindgren,^' who obtains A = -456

MHz in the restricted Hartree-Fock approximation

(- 463 MHz with relativistic corrections) and

A = - 545 MHz when core polarization is included

(-553 MHz with relativistic corrections). The

remaining 7% difference between the relativistic calculation including core polarization and the experimental result is presumably due to pure

correlation effects involving double excitations.

The semiempirical Fermi-Segre formula'^'^^ predicts A = -627.8 MHz, which is 5% too high

in magnitude. The NMR measurement of ^j,^'

uncorrected for diamagnetic effects, is used.

Relativistic correction factors are left out. The

derivative of the quantum defect with respect to quantum number which appears in this formula was evaluated from the extended Ritz formula

given by Risberg.^° If this term is left out, leaving

the Goudsmit formula, the result (A = -606.4 MHz)

is actually in better agreement with experiment.

Veseth has carried out a Hartree-Fock calculation of gj of Mg*." The values obtained for the individual correction terms, in the notation of Ref. 34, are 62=-7.91x 10-^ 63=9.70x10-5, and

6^=_7.66x 10-^ The final result isg^^ = 2.002 260 6,

with an estimated fractional uncertainty of 3x 10"^. Hegstrom, using a simple hydrogenic approximation for the valence electron, obtains gj/gg= (I

31)xl0-% or ^^ = 2.002 257.'' The j^^ value of ^'Mg* trapped in a solid argon matrix at 4 K was

reported by Brom and Weltner® to be 2.006(4).

The shift from the free -ion value is consistent

with theory.

Our result for gj/gj can be combined with Veseth 's calculated value of ^^ to obtain^/ for the free ^^Mg* ion. This can be combined with the NMR result for gj of ^'Mg^* ions in Hfi (Ref. 21)

to obtain the shielding difference

0^*= 1 -g>(NMR)/^/free ion)

= - 9.9(9) X 10-5.

The corresponding shielding differences for the akali atoms (free atoms versus hydrated ions)

have been measured.'^ They also are negative

and increase monotonically in magnitude with

atomic number Z. For "Na, a*= - 6.05(10) x 10-'. The absolute accuracy of our measurements

could, of course, be substantially improved if the magnetic -field stability and homogeneity were

improved. Accurate measurements of ^^ could

be made by comparing the electronic spin-flip

frequency in the ion, gji^B^^/h, to, for example,

the cyclotron frequency of free electrons,'^ 2^gBj

TN-65
24 PRECISION MEASUREMENT OF THE GROUND-STATE 1373

h, in the same magnetic field. These same experimental techniques could be

extended to measure hyperfine constants and g

factors of other ions,^** such as ^Be*. The high

resolution with small perturbations which can be

obtained may make it possible to study small

effects not previously observed, such as the diamagnetic susceptibility of the nucleus, ^^ or to realize new types of frequency standards.

ACKNOWLEDGMENTS

We wish to thank J. C. Bergquist and R. E. Drullinger for experimental assistance during the early part of the work. This work was supported

in part by the Air Force Office of Scientific Research and the Office of Naval Research. One of us (W.M.I.) acknowledges the financial support of the NRC in the form of a Postdoctoral Research

Associateship during this work.

*For reviews, see E. Arimondo, M. Inguscio, and

P. Violino, Rev. Mod. Phys. 49, 31 (1977); G. H. Fuller, J. Chem. Phys. Ref. Data 5, 835 (1976). For recent work on radioactive alkalis, including

francium, see S. Liberman, J. Pinard, H. T. Duong,

P. Juncar, P. PUlet, J.-L. Vialle, P. Jacquinot,

F. Touchard, S. Biittgenbach, C. Thibault, M. de Saint-Simon, R. Klapisch, A. Pesnelle, and G. Ruber,

Phys. Rev. A 22, 2732 (1980) and references therein. For examples of recent work and reviews, see M. Vajed-Samii, S. N. Ray, T. P. Das, and J. Andriessen, Phys. Rev. A 20, 1787 (1979); S. Garpman,

I. Lindgren, J. Lindgren, and J. Morrison, Z. Phys.

A 276, 176 (1976); I. Lindgren, Phys. Scr. 11, 111

(1975). ^For a review, see E. W. Weber, Phys. Rep. 32, 123

(1977). E. W. Weber and R. Gragel, private communication.

^F. G. Major and G. Werth, Phys. Rev. Lett. 30, 1155

(1973); Appl. Phys. 15, 201 (1978); M. D. McGuire,

R. Petsch, and G. Werth, Phys. Rev. A 17, 1999(1978).

'R. Blatt and G. Werth, Z. Phys. A 299, 93 (1981).

*J. M. Brom and W. Weltner, J. Chem. Phys. 58, 5322

(1973). 'p. H. Kasai, Phys. Rev. Lett. 21, 67 (1968). "c. K. Jen, V. A. Bowers, E. L. Cochran, and S. N. Foner, Phys. Rev. 126, 1749 (1962). W. L. Randolph, J. Asher, J. W. Koen, P. Rowe, and

E. Matthias, Hyper. Interact. 1, 145 (1975).

J. Hata and M. Sakai, Phys. Lett. 57A, 419 (1976);

S. N. Ray, J. E. Rodgers, and T. P. Das, Phys. Rev. A^S, 1983 (1976); S. Ahmad, J. Andriessen, K. Raghunathan, and T. P. Das, Bull. Am. Phys. Soc. 26, 481 (1981). W. M. Itano, D. J. Wineland, R. E. Drullinger, and

J. C. Bergquist, Abstracts of the 7th International Conference on Atomic Physics, MIT, Cambridge,

1980 (unpublished), p. 238.

p. 238. ^*W. Neuhauser, M. Hohenstatt, P. E. Toschek, and

H. Dehmelt, Phys. Rev. Lett. 41, 233 (1978); Appl.

Phys. 17, 123 (1978); Phys. Rev. A 22, 1137 (1980).

11

12

13

16

17

18

19

Phys. Rev. Lett. 40, 1639 (1978).

'R. E. Driillinger, D. J. Wineland, and J. C. Bergquist,

Appl. Phys. 22, 365 (1980). D. J. Wineland, J. C. Bergquist, W. M. Itano, and R. E

Drullinger, Opt. Lett. 5, 245 (1980). D. J. Wineland and W. M. Itano, Phys. Lett. 82A, 75

(1981). H. Kopfermann, Nuclear Moments (Academic, New

York, 1958). P. Risberg, Ark. Fys. 9, 483 (1955). O. Lutz, A. Schwenk, and A. Uhl, Z. Naturforsch. 30a , 1122 (1975). 126 , 1768 (1962); J. Bauche,

J. Labarthe, Z. Phys. 270, 311

22 A. Lurio, Phys. Rev

G. Couarraze, and J

(1974). H. Dehmelt, in Advances in Atomic and Molecular

Physics, edited by D. R. Bates and I. Estermann (Academic, New York, 1967, 1969), Vols. 3 and 5. "W. Happer, Rev. Mod. Phys. 44, 169 (1972). A. Abragam, The Principles of Nuclear Magnetism.

(Oxford University Press, London, 1961).

'm. Mizushima, Phys. Rev. 133, A414 (1964). N. F. Ramsey, Molecular Beams (Oxford University

Press, London, 1956).

'm. F. Crawford, F. M. Kelly, A. L. Schawlow, and W. M. Gray, Phys. Rev. 76, 1527 (1949). N. P. Economou, S. J. Lipson, and D. J. Larson, Phys. Rev. Lett. 38, 1394 (1977). T. Lee, N. C. Dutta, and T. P. Das, Phys. Rev. A 1

,

995 (1970).

I.

23

25

Lindgren, private communication.

'"-L. L. Foldy, Phys. Rev. Ill, 1093 (1958).

33 ~~^

L. Veseth, private communication.

"l. Veseth, Phys. Rev. A 22, 803 (1980).

35R. Hegstrom, private communication.

^*A. Beckmann, K. D. Boklen, and D. Elke, Z. Phys. 270 , 173 (1974). ^'See for example, H. G. Dehmelt and F. L. Walls, Phys. Rev. Lett. 21, 127 (1968); R. M. Jopson and D. J. Larson, Bull. Am. Phys. Soc. 25, 1133 (1980). W. M. Itano and D. J. Wineland, Bull. Am. Phys. Soc.

24, 1185 (1979).

38

15 D. J. Wineland, R. E. Drullinger, and F. L. Walls,

TN-66
PHYSICAL REVIEW A VOLUME 25, NUMBER 1 JANUARY 1982

Laser cooling of ions stored in harmonic and Penning traps

Wayne M. Itano and D. J. Wineland

Frequency and Time Standards Group, Time and Frequency Division, National Bureau of Standards,

Boulder, Colorado 80303

(Received 17 August 1981)

Laser (light-pressure) cooling of two-level ions stored in a Penning trap is treated

theoretically, in the limit that the frequencies of motion of the ions are much smaller than the natural linewidth of the optical transition. Rate equations for the mean-squared

amplitudes of motion are derived from a semiclassical analysis, which is confirmed by

quantum-mechanical perturbation theory. Simultaneous cooling of all three modes of motion is shown to require a spatially nonuniform laser-beam profile, unlike the case of

the harmonic trap. Comparison is made with recent experiments. Also, the three- dimensional harmonic trap is treated by two simple methods. One is based on the energy

rate equations and the other on Langevin-type equations. Identical results are obtained by the two methods for the steady-state energies. These results are compared with the works of others.

I. INTRODUCTION

Laser (light-pressure) cooling of atoms and

atomic ions has been demonstrated in recent exper- iments which were stimulated by the original proposals for laser cooling of free atoms' and of ions

bound in electromagnetic traps.^ Cooling is of interest for high-resolution spectroscopy largely be- cause all Doppler effects are fundamentally re- duced. Cooling of Na atoms in an atomic beam

has been reported by Balykin et al?

Two types of electromagnetic traps'* have been

used for the laser-cooling experiments on ions. Cooling of Ba"*" ions stored in an rf quadrupole

trap was demonstrated by a group at Heidelberg^'^

and of Mg"*" ions stored in a Penning trap by a group at the National Bureau of Standards

(NBS).^~^ The rf quadrupole trap uses an alter- nating inhomogeneous electric field, whose average

effect is to create a three-dimensional harmonic

potential well. The Penning trap uses a static uniform magnetic field. The ions undergo harmonic

motion along the magnetic field and a superposition of circular motions in the plane perpendicular

to the field. Laser cooling of neutral atoms has been treated

theoretically by several authors.'' '°~'^ Various

proposals have been made for trapping neutral

atoms in potential wells created by nearly resonant

optical fields,'""'^ by static electric fields,'^ and by

static magnetic fields,'^ all of which include laser cooling. Previous theoretical treatments of laser cooling of trapped ions have been restricted to har- monic potential wells.^'^'^°~^^ The purposes of this paper are twofold. First, we augment the work of Ref. 16 by explicitly in- cluding the effects of light and atomic polarization

and of the angular distribution of scattered photons. Unpolarized atoms and isotropic scattering

of reemitted photons were usually assumed in Ref.

16. Cooling limits are derived from energy rate equations and also from a simple force fluctuation

model. These results are briefly compared with the work of others. Second, we discuss laser cooling of ions stored in a Penning trap. The natural linewidth of the optical transition used for cooling is assumed to be

much greater than any of the frequencies of

motion of the trapped ions. This is the case which

is most easily realized experimentally. Rate equations are obtained for the mean-squared values of

the amplitudes of motion of the ion. Saturation

effects are not included, so the results are valid only for low light intensity. Configurations of

light beams which can cool all three modes of oscillation are described. For the Penning trap, un25 35

TN-67
36 WAYNE M. ITANO AND D. J. WINELAND 25

like the harmonic trap, this requires a spatially

nonuniform laser beam profile. Some of the general methods were discussed in our previous paper,'^ in which they were applied only to laser cooling of free and harmonically bound atoms. Some of the results of the calculations presented

here have been quoted in a previous publication,^

in which the laser cooling of a single Mg"*" ion in a Penning trap was reported. Laser cooling of an atom bound in a harmonic

well is treated in Sec. II, in order to illustrate the

basic ideas in some simple cases. A model of laser cooling of an ion in a Penning trap in which the

motion of the ion is treated classically is given in Sec. III. These results are briefly compared with

experiment. A quantum-mechanical treatment is given in Sec. IV. The results are discussed in Sec. V.

II. LASER COOLING IN A HARMONIC TRAP

A. Formulation of the problem

The problem to be considered here is the laser cooling of a two-level atom bound in a three- dimensional harmonic well. The term atom in- cludes atomic ions. The ground electronic state

I g ) is assumed to be stable; the excited electronic

state I e ) is assumed to decay only to | g ) by a one-photon electric-dipole transition at a rate y. The energy difference between | g ) and | e ) is

Two-level atoms do not exist, but in practice

they can be approximated. In zero external field, either the ground or excited states have some Zee- man degeneracy, since a (7 =0)-to-(y' =0) transi- tion is forbidden for any one-photon process. (/

and J' are the electronic angular momentum quantum numbers in the ground and excited states, respectively.) If a strong magnetic field is applied

so that the Zeeman splitting of the optical transi- tion is greater than either the Doppler or natural broadenings, then the laser can be tuned close to resonance with one Zeeman component. The

corresponding magnetic sublevels of the ground

and excited states will form a two-level system,

provided that the excited-state sublevel cannot de- cay to any other ground-state sublevel. The simplest case involves 7=0 and 7'= 1. More generally, if 7' =7 + 1, then the M =7 ground-state sublevel and the M'=J + 1 excited-state sublevel or

the M = —J ground-state sublevel and the M'= —J — 1 excited-state sublevel will form twolevel systems.^ There are also possibilities when

hyperfine structure is present. If the Zeeman splitting is not large compared to the Doppler broadening, it is still possible to form an effective two-level system, but only for certain polarizations and

directions of propagation of the light. Systems

which are not effectively two level, including those

in which hyperfine and 2^man structure must be

taken into account, can be treated by straightforward extension of the two-level methods. For such

cases, it may be necessary to introduce additional

optical or rf frequencies to ensure that the atoms

continue to interact with the laser beam or beams

used for cooling. For the Ba"*" experiments,^'^ the

magnetic field is zero and the 6s ^Si/2—>- 6p ^P\/2

transition is used (7=7'=y), so four levels are in- volved (for 7=0 isotopes), aside from the metastable D states, but this case can be treated with

only minor modifications to a two-level theory. ^^ The harmonic potential causes the atom, which

has mass M, to oscillate in the x, y, and z direc- tions at angular frequencies il^, Cty, and ft^. These frequencies are assumed to be much less than the natural linewidth y, which is much less than the optical transition frequency coq. This is the "weak binding" case of Ref. 16. This case is easily satisfied in practice, since, for a typical reso- nance line, 7/27r> 20 MHz, and for an rf quadrupole trap, ft,/2ir<2 MHz {i=x,y,z).

The atom interacts with one or more monochromatic polarized laser beams. The electric field of such a beam is assumed to be given by a classi- cal plane-wave solution:

E( r',^) = 6 Re£'oexp(/ k -F— icot ] (1)

where Re stands for the real part, 6 is a unit polarization vector perpendicular to k, and | k | =/: =o}/c.

Consider a scattering event in which a free atom,

moving with velocity v, absorbs a photon of wave

vector k, emits a photon of wave vector kj, and

has its velocity changed to v'. From conservation

of energy and momentum in the nonrelativistic (v/c « 1) limit. Av = v'-v=^(k-k,)/M (2)

=^2(k-k,)V2M+^(k-k,)-v , (3)

=fih ki - ksi ) V2Af + Mhi - k,i )t;,- {i=x,y,z) .

(4)

TN-68
25 LASER COOLING OF IONS STORED IN HARMONIC AND 37

The subscripts / denote Cartesian components of

vectors, except in A£,, which is the kinetic energy due to motion in the / direction. These relations were derived for a free atom, but they also hold for the harmonically bound atom if the scattering takes place in a time much shorter than the period

of oscillation, since the atom behaves as a free particle for such short times. Since resonance scatter- ing takes place in a time roughly equal to 7"', this condition is equivalent to our weak binding as- sumption. The conservation laws which led to Eqs. (2) — (4) do not depend on the photon scatter- ing being near a resonance. However, we wish to consider only the near-resonance case, where o) is within roughly the combined Doppler and natural broadening of the optical transition frequency.

Since it was assumed that v/c <<\ and that Y<<o)q, this implies that ^=wo/c=/co. We will also assume that R <<^7, where R =(^/c)"/2M = (^/co)^/2M. This is easily satisfied for an al- lowed optical electric-dipole transition. For example, for the 6s ^5 1/2^ 6p ^P]/2 493-nm resonance

line in '^^Ba+, /? =/? X5.9 kHz, r = 27rX21

MHz,^'^ and i? /^r = 2. 8 X 10"^ For the

35 ^51/2—> 3p ^Pi/2 280-nm resonance line in ^'^Mg^, R=hX 106 kHz, 7=2;rx43 MHz,-^ and

R /frY=2.5x 10~^. Since the fractional change in k,\k—k,\ /k, is at most about 2v/c + 2-fik /Mc, we can also assume k^k^. Equation (4) can be

rewritten as

A£:,- =R{kl- 2kik,i + kl ) + Hi k^ - A:,,- )i;,- , (5)

where /c, and /c^, are the / components of the unit vectors k = k/k and /c^ = k j //Cj

.

The angular distribution of the scattered photons

depends only on the orientation of the electric- dipole moment of the transition dg^ = (g | d \e).

We note here that the ground and excited states we

consider are single M states and that an effective two-level system has been created by, for example,

applying a magnetic field large enough to resolve the Zeeman structure. The probability that a photon is emitted into a solid angle dH in the k^ direction is

PAK)dil-- :3/87r)X \dge-ej(k. ^dVL , (6)

where

*ge dge/|dg^

and the summation is over the two unit polarization vectors for propagation in the k^ direction. These vectors must satisfy

ek(ksf-€j{k,) = 8kj-,

ekiks)-k,=0 ,

(7)

where 8^ is the Kronecker delta, but are otherwise

arbitrary. The normalization is such that j Ps(ks)dil — \. Equation (6) applies to a non- moving atom, but it is approximately true for a moving atom, provided v/c « I, which has been

assumed. This treatment could be generalized to include other types of multipole radiation, such as magnetic dipole or electric quadrupole. The decay

rates for such transitions are usually so slow

( < 10^ s~') that it becomes difficult to satisfy the weak binding condition, so they are not treated

here. Equation (6) can be evaluated if | g ) and

I e ) are eigenstates of the component of the total angular momentum along a quantization axis defined, for example, by a magnetic field. For a AAf = transition (M is the eigenvalue of the total angular momentum in the B direction) we have

Ps(k,) = {3 /STr)sin% .

For a AM = + 1 transition Ps{0 = (3/\6Tr){\+cos%) .

(8a)

(8b)

Here 6, is defined by B-ks = cos9s. The isotropic

distribution, which has been used in much of the previous work,'^'^'^ corresponds to letting Ps = 1 /4tt. This is not of the form of Eq. (6). Given P^, we can average Eq. (5) over all angles of

the scattered photon

(A£:,),= fP,(k,)AE:dn =R(fi+f,i) + miVi ,

(9)

where

and

f,,^ ^ P,(k,)kld^

(10a)

(10b)

The terms linear in k^i drop out because Psik^) =Ps( —k^ ) for any P^ of the form of Eq. (6) or for the isotropic distribution. The total energy change

per scattering event, averaged over all angles of the

scattered photon, is (A£), = <A£J, + <A£^), + <A£,), = 2R+fik-v .

This is the same as Eq. (3) of Ref. 16.

(11)

TN-69
38 WAYNE M. ITANO AND D. J. WINELAND 25

The average rate of energy change is obtained by

multiplying the scattering rate by the average ener- gy change per scattering event. The scattering rate

Ys is equal to the product of the number of photons per unit area per unit time I /-fko and the ab- sorption cross section for photons of tiiat particular polarization and propagation direction (/

=c I Eq I ^/Stt). For a particular atomic velocity

V, the cross section can be obtained from perturbation theory,'^ provided the intensity is below sa- turation. This is proportional to the Dopplershifted natural line shape

a{co,v) =ao{Y/2)^/[{coo+k-v

+R/-fi-co)^ + {Y/2)^] , where

aQ=6'rrkQ \ e'd,eg kQ=c/coo

(12)

:i3)

This is Eq. (24) of Ref. 16. Terms of order (v/c)^

and (v /c){R /fko) are neglected (see Sees. IV A and

IV B of Ref. 16). If we take | e-d^g \^ = j, which

is the average over all polarizations e, then ao = 2TrkQ. As a more realistic example, for the

^P'^Pi/2 (^J =±y)-^3s^5i/2 (My = ±7) transi- tions in ^"^Mg"*" excited by light polarized perpendicular to the magnetic field, ctq = 37rX^. (Note that the expression given in Ref. 7 is too small by a

factor of 2.) The rate equations for the energies in the X, y, and z vibrational modes, which in the ab- sence of collisions are separately conserved, are

dEi

IT + (I/fio})(a{co,y)[iikiVi +R (/,• +/„)] )„ {i=x,y,z) .

(14) Here the velocity average (g(v))y of an arbitrary

function g( v ) is defined in terms of the velocity

probability function P(v) as {g(\))^= jP(v)

Xg{v)d^v. If a form for P( v ) as a function of

[Ej] is assumed, then Eq. (14) can be solved to obtain j Ej ! as a function of time. This might, for example, be a product of Maxwell-Boltzmann dis- tributions for motion in the x, y, and z directions,

characterized by three temperatures, which need

not be equal, '^'^^ unless there is rapid energy

transfer between modes by collisions between

bound atoms. A special case of Eq. (14), with additional collisional relaxation terms, was discussed

in Sec. VF of Ref. 16. If more than one laser beam is present, this can be taken into account by

adding additional terms of the same form, with

different e, Eq, k, and co, to the right-hand side of

Eq. (14). Equation (14) holds for a single atom or

for the ensemble average of a cloud of noninteracting atoms. If collisions between atoms are important, they can be taken into account by introducing

phenomenological relaxation constants which tend

to equalize the energies in the different modes. '^

If the evolution of the entire velocity distribution,

and not of just the average kinetic energy, is of interest, methods based on the Fokker-Planck or another kinetic equation can be used.'^~'^'^°~'^^'^^

B. Cooling limits derived from energy rate equations

1. One laser beam along x axis

Consider a single laser beam propagating along the x axis {k=kx). Equation (14) becomes

dEr

{I/iuo)(aico,V)[Mv^+Ri\+f^)])^ , dt dEy

~dt

dE,

"di

= {I/fuo){o{co,\)),Rf^ ,

= (I/iicj){aia),y)),Rf^

(15a)

(15b)

;i5c)

For negative laser detuning (co kcoq +R/H^coq), a(co,v) is larger for negative v.^ than for positive v^ so

that the right-hand side of Eq. (15a) is negative, until very low temperatures are reached. This is the basic

principle of laser cooling. If considerable cooling has already taken place, so that the Doppler broadening is much less than the natural linewidth or the detuning, i.e., k{Vx)^^«y/2 or k{Vx)nns«^^o+^ /'^~^h

then Eq. (12) for the cross section can be approximated by

a(a),V)^CTo{y/2)^[(y/2)^ + (coo-co)^]-Hl-2{coo-co)k-V/[(y/2f- + {(Oo~(o)^]] . (16)

IN- 70
25 LASER COOLING OF IONS STORED IN HARMONIC AND 39

R has been dropped, because it is assumed that R/ii«{coo—co). Note that

{a(co,V)),^aoiY/2)\{Y/2)^ + icoo-(o)^]-^ ,

provided P( v )=P( — v ), which we assume to be true. Let y^ be the average scattering rate:

Ys={I/iio})(a(co,v))u .

Equations (15a) — (15c) now become

dt dEy

~dt

dE,

-Ys[-2((OQ-oi)fikHvly,/[{Yn9 + {(OQ-oi)^]+R{\+fs:,)\ ,

= rsRfsy >

= YsRfsz-

(17)

(18)

(19a)

(19b)

(19c)

The motion in the>' and z directions is heated by recoil and does not reach a steady state. '^ The kinetic en- ergy in the x direction reaches a steady-state value when the right-hand side of Eq. (19a) is zero:

=i^(l+Ax)[(r/2)^ + (wo-^)^]/8(«o-«) •

The minimum value is obtained by setting {coQ —(jL)) = y/l:

EK, = {\+f^)-fiy/8.

(20)

(21)

For isotropic scattering, /jx = 7, so the minimum E^^ =^7/6. This agrees with the result of Javanainen

and Stenholm,^° in the unsaturated limit. They treated this situation by using a Fokker-Planck equation. If the atomic electric-dipole moment is oriented along a quantization axis perpendicular to the x axis, corre- sponding to a AM = transition, then from Eqs. (8a) and (10),

.iTT fsx^ii/^Tr) f f

sin^d{sin9cos<f>)^d(f>dcose--

where 6 and (p are spherical polar angles, so the

minimum Ef^^ = Ifty/^O. This is in agreement

with the results obtained by Cook,''* who used a Fokker-Planck equation, for the one-dimensional

cooling of an unbound atom in a weak standing wave, when the dipole is perpendicular to the light

propagation direction. This is essentially the same

problem that we have treated here. For an un- bound atom, a standing wave is required in order

that the average radiation pressure force be zero, so

that is not accelerated continuously in one direction. Cooling from a single running wave can oc- cur for the harmonically bound atom, since a

steady force only shifts the equilibrium point in the potential well. The harmonic restoring force was required in order to ensure that P( V )=/*( — V ), which led to the cancellation of a term in deriving Eq. (16) from Eq. (14). The

method we have used here and in Ref. 16, which is

2T ' (22)

based on calculating the average energy imparted

to the atom in the photon scattering process, takes

into account heating due to both induced fluctuations (recoil on absorption) and spontaneous fluc- tuations (recoil on emission), at least in the lowintensity limit. These two sources of fluctuations were introduced separately in Ref. 14. In our

treatment, the induced fluctuations are associated

with the term proportional to /, and the spontane- ous fluctuations with the term proportional to /j, in Eq. (14). The treatment of the atomic motion has been

essentially classical. This should be valid in the

limit of large harmonic oscillator quantum numbers. The mean occupation number for motion in the X direction is {rij^) ^lEf^^/^^x > y/3^x> so

in the weak binding approximation the motion

remains in the classical regime.

TN-71
40 WAYNE M. ITANO AND D. J. WINELAND 25

2. Laser beams along x, y, and z axes

In Sec. II B 1 it was shown that a single laser beam parallel to one of the principal axes of the harmonic

potential cools motion parallel to the beam but heats the perpendicular motion by recoil. One simple configuration which is capable of cooling all modes of oscillation consists of three laser beams, one along each of

the X, y, and z axes. Denote the laser frequencies i)y w, (/ =x,y,z) and assume that all are tuned below reso- nance (&), <(o,^). Then the atoms are cooled, and, assuming that the Doppler widths are small compared

with either the natural linewidth or the detunings, [^,(y, )niis<<7/2 or /c,(u, )rms<<(&)o— <y/ ). where

dE

dt = -2rsi((oo-coi)fikt{vl),/[(rnY+{coQ-coiY]+R[rsi^irs\oJsi], (23)

where 7j, is the average scattering rate due to the laser beam along the / axis and (y^ )tot == ^sx + y^- + ^c • In the steady state,

=^[i+/./(r.)to./rri][(r/2)'+(«o-^/)']/8(wo-w/) • (24)

This is minimized, [for constant (yi)tot/?'«]> by setting (cjq— w,) = y/2:

£i^,=[i+A(rJ.o./r.,]^r/8 • (25) The total mean kinetic energy can be minimized by adjusting the relative laser powers so that

(r«)/(r.).o,=(A)"V[(/„)'/^+(/,,)'/^+(/„)'/^]

.

(26)

For the isotropic case, the minimum kinetic energy in each mode is Hy/A, which agrees with Ref. 16 (Sec. VE2).

3. One laser beam at oblique angle to axes

Another method of cooling all three modes of oscillation is with a single laser beam which is not parallel to any of the principal axes. Equation (14) can be used to obtain the cooling rate. As has been pointed out

previously, it is necessary that all of the motional frequencies (H^, Hy, and ftz) be different.^' '^'^^ Otherwise

there is at least one mode of oscillation perpendicular to the laser beam which is not cooled, but rather is heated by recoil, like the;' and z modes discussed in Sec. II B 1. In the low Doppler limit [(k-v)^^; <<7/2

or (k-v)rms<<(Wo —^)]'

dE

dt where we have assumed that (y,u^ )i, =0 if i=^j. This should be valid if the frequencies are sufficiently dif- ferent and the intensity is sufficiently low. In the steady state,

£:^,.=^(l+/^.//,)[(y/2)2 + (a)o-a))2]/8(a)o-a)) , (28)

and this is minimized by setting {coQ—(o) = y/2. accompHshed by pointing the laser along the

Then £+>r+z direction, in which case, £'^,=^7/4.

^^,=(l+/5,///)%/8. (29) This is close to the configuration used by

Neuhauser et al} Equation (29) is equivalent to

If one has control over the direction of the beam, the low-intensity limit of Eq. (31) of Javanainen."

then the total average kinetic energy can be minimized if

// = (/.)''V[(/„)'/^ + (/^)'/2 + (/„)'/2] . (30)

C. Cooling limit derived from force fluctuations The steady-state cooling limits derived in Sec. For an isotropic angular distribution, this can be II B can also be obtained from a simple classical

TN-72
25 LASER COOLING OF IONS STORED IN HARMONIC AND 41

model, in which the effect of the scattered photons on the atomic motion is divided into an average damping force and a random, fluctuating force. The advantage of this approach is its great simplicity; it uses concepts which are familiar from the theory of noise in electronic circuits. The one-dimensional cooling case, which was

solved by the energy rate equation method in Sec.

II B 1 , will be treated, in order to demonstrate the

basic method in the simplest case. The method

can also be applied to the other cases treated in Sec. II B. The equation of motion (Newton's second law)

for the X coordinate of the atom is Mi+Milix=FJt) (31)

where F^^it) is the x component of the force on the atom due to photon scattering. Each scattering event imparts a momentum impulse ^( k — k^ ) in a time less than about y~\ which we have assumed

to be very short compared to the period of oscillation. Therefore we can represent F^(t) by a sum of

Dirac delta functions:

F^{t)= '2^it -t,)fi[k -k,^il)] , (32)

where k^^il) is the value of fc„ for the lih scattering. The scattering events take place at random

times at an average rate Ia(co,v)/fko. The timeaveraged force on an atom of velocity v is

{F^)=(I/-fko)a{o),\)iik , (33)

since the k^^il) terms average to zero. In the low

Doppler limit,

{F^)=r,{\-2{a)o-co)ki/[(Y/2)^ + {a}o-o))^]\fik

^y,iik-Mrx . (34) The equation of motion can now be written as MX+MrX+MnlX=F'(t)=F^(t)-{F^) , (35) which describes a damped harmonic oscillator driven by a random force F'(t). We have eliminated the constant force fsfik by shifting the origin, so that X(nev/)=x(o]d) — Ys'fik/MD,l. This equation has the same form as the one which describes

the response of a series RCL (resistancecapacitance-inductance) circuit to a fluctuating

emf.

The fluctuations in X can be calculated by stand- ard methods.^^ Let G (/) be the complex gain

function which describes the steady-state response X to a sinusoidal F' with frequency /, and which is

defined by

X (/)exp( 2mft) = G (/)F'(/)exp( Inift ) . (36) Here the physical quantities X(t) and F'(t) are given by the real parts of A'(/)exp(277-/'/r) and

F'(f)e\p(2mft), respectively. Let w(f) be the

one-sided power spectral density of F'(r), defined

by

w{f) =4 f" {F'{t)F'(t + T))cos2TrfTdT . •^ Then the mean-squared value of X is {X^)= fj \G{f)\^wif)df . (37)

For the system described by Eq. (35), G(/)=l/M[n^-(2i7/)^ + 727rr/] . (38) The spectral density function can be calculated by

following the standard derivation for the power

spectrum of shot noise.^^ The only difference is that the impulses vary in size because of the ran- dom direction of the emitted photon. A derivation

is given in the Appendix. We have

w(f) = 2y,-^H(k-k,,)')s = 2Y,ff(k'+{kl},) =4MYsR(\+L,) , (39)

which is valid for frequencies / less than about

y/2Tr. The fact that Eq. (39) is plausible can be

seen by examining the more familiar expression for the spectral density of current for shot noise,

Wj(f) = 2eI = 2e^N, where / is the average current, and A'^ is the average number of electrons per second passing through a temperature limited

diode. Substituting the rms change in momentum

for e and the average scattering rate for N, we ar- rive at Eq. (39). The term | G (/) | ^ in the integrand in Eq. (37)

is sharply peaked around f — ilx/2Tr, where wif)

is nearly constant, so (X^)=w(n^/2Tr)

f^"' \G(f)\W

4}VR

M (i+/«)X

df

[n^-(27r/)2]^ + (27rr/)2

^rsR(\+fs.)/Mriii (40)

.

The average kinetic energy is

(41)

TN-73
42 WAYNE M. ITANO AND D. J. WINELAND 25

In the last step'Eq. (34) was used to express T in terms of other parameters. This is the same resuh

obtained previously from the energy rate equation

[see Eq. (20)], and the minimum energy is obtained

when {coQ-~Cii) = y/2 [see Eq. (21)]. Equation (35)

is the Langevin equation, with the addition of the

harmonic restoring force. Therefore the theory

developed for the solution of the Langevin equation could be applied to the present problem.'"'"*'

We have assumed that q, Aq, and Bq are all positive. The amplitudes {r„,rcr2), defined to be

greater than or equal to zero, and the phases

(6^,9^,62) are determined by the initial conditions.

The frequencies are the magnetron frequency co„

,

the modified cyclotron frequency co',., and the axial frequency co^, where

fi)z = (4gv4o/M ,1/2 (46a)

III. LASER COOLING IN A PENNING TRAP

(SEMICLASSICAL TREATMENT)

A. Classical motion of an ion in a Penning trap

The classical motion of a single ion in a Penning

trap has been treated previously."''"^ Here we summarize the important results. The idealized Penning trap consists of a uniform

magnetic field B =5oZ along the 2 axis and a quadrupole electrostatic potential of the form

V{x,y,z)=Ao(2z^-x--y' (42)

Consider an ion of mass M and charge q. The

equation of motion for the ion position F is Mv = -qVV(f) + (q/c)vXB (v = r*) (43)

and the equations for the Cartesian components

{x,y,z) of r are

i)^ = {2qAQ/M)x+{ qBo /Mc)Vy , Vy={2qAQ/M)y — {qBo/Mc)Vx ,

v, = -{4qAo/M)z .

The general solutions to these equations are

x(t) = r„cos{co„t +9n) + ri.cos{co't + 6^

(44a)

(44b)

(44c)

(45a)

(45b)

z{t) = r,cos(co,t + 9,) . (45c)

We can also write the expressions for x and y in the complex form"^

x{t) + iy(t) = r„exp[-Hco„t + 9„)]

+ r,[-i{co',t + 9,)]. (45d)

co„ =co,/2-(col/A-col/2)^^^

co'=co,/2 + {col/4-col/2)^^^ .

(46b)

(46c)

Here 0)^ =qBQ/Mc is the ordinary cyclotron fre- quency. These solutions assume that col > ^l'

since otherwise the motion is unstable. A typical

experimental configuration might have M= 25u,

q=e,Bo = \T (\0 kG), and ^0 = 10 V/cml For

these values of the parameters, co2^2TrX 197.7

kHz, 0)^^277X33.7 kHz, and a); = 27rX580.6

kHz. The ion executes simple harmonic motion at frequency co^ in the z direction and a superposition

of circular motions at frequencies co'c and a)„ in the xy plane. Figure 1 shows a typical xy orbit for the case co„ <<co'c and r„>r^. In this case the magnetron motion can be interpreted as an ExB

drift of the center of the cyclotron orbit around the

trap axis.

FIG. 1. Orbit of an ion in a Penning trap, projected onto the xy plane. The magnetic field direction is out of

the drawing and the charge of the ion is assumed to be

positive.

TN-74
25 LASER COOLING OF IONS STORED IN HARMONIC AND . 43

The kinetic energy of the ion is

Ef^ = -jMv=jMr„(i>„ + ^

Mr^co'c +Mr„ r^co^co'^cosio^ t —ci>'ct + 9„—6c) + -jMr^colsin^ico^t + 6^) .

The potential energy is

(47)

(48)

The total energy is a constant of the motion,

E =Ef:+Ep = YMr^a)'^+MD.(o)'crc-~co„r„) ,

where

n = —(a}'c —COm ) •

(49)

(50) ^

AVe are interested primarily in reducing Ef^, rather than E, since this will reduce Doppler shifts. The

time-averaged kinetic energies in the axial, magnetron, and cyclotron modes are

{Efc.) = jMr^col,

( EKm) = T^rio}i

{Et^,) = ^Mr}oi', 2. .'2

c

(51a)

(51b)

(51c)

Hence, we are interested in reducing r^, r^, and r^.. Note, however, that an increase in r„ leads to a de- crease in the total magnetron energy, because of

the potential energy contribution. Therefore, the

orbit is unstable toward an increase of r„ if the ion

is perturbed, for example, by collisions with neutral molecules. In the following, the term "cooling"

will refer to the reduction of the kinetic energy,^

regardless of what happens to the total energy. We note that the canonical momentum p is de- fined by p=Mv + (9/c)A, where A is the vector

potential. We choose the symmetric gauge, where

A(r) = yBxr*. In this gauge, the axial component

of the canonical angular momentum is a constant of the motion given by

L, =xpy -yp^ =Mmri-rc] (52)

B. Effect of photon interactions We treat the effect of a laser beam on the motion of an ion with essentially the same assumptions that were made for the case of a harmonically bound atom. The ion is assumed to have only two internal energy

levels, which are connected by an electric-dipole transition. The decay rate of the upper state is assumed to be much greater than any of the frequencies of motion {co'^, o)„, and o)^). Hence a photon scattering event can be considered to take place instantaneously, as far as the motion of the ion is concerned. In this semclassical treatment, the ion moves according to the classical equations of motion between scattering events, which perturb its velocity. The scattering events occur at a rate determined by quantum-mechanical perturbation theory. We consider only the limit of low laser intensity. The various quantities coq, y, R, fi, fsi, dge, Ps, and CTq are defined as in Sec. II. Consider an event in which an ion moving according to Eqs. (45a) — (45c) absorbs a photon of wave vector k and emits a photon of wave vector kj at time Iq. For t <to, the motion is described by the amplitudes r, and the phases 0, (/ =m,c,z); for t > to they are modified to r,' and 6] . Let Av =^( k — kj )/M. Then

^r^ = {r^)^-r^ = (^v^/co,)^-i2r^^v^/o},)sm(co^tQ + e,) , (53a)

^ri=(r;„)^-ri=[(^v^)^ + (^Vy)']/4n^ + {r„/{^)[sm{co^to + 9^)^v^+cos(co„to + ^^}^Vy] , (53b)

Ar,^ = (r; )^-r^^[(^v^)^ + (^Vy)^]/Aa^-{r,/il)[sm(co',to + 9,)^v^+cos(co',to + ^c)^Vy] . (53c)

These equations are derived from combinations of Eqs. (45a) — (45c) and their time derivatives. Equation

(53a) is, of course, equivalent to Eq. (4), since the axial mode is harmonic. From Eqs. (53b) and (53c) it can

be shown that the magnetron radius is reduced if A v is tangential to and in the same direction as the magnetron motion, while the cyclotron radius is reduced if Av is tangential to and in the opposite direction as the cyclotron motion. The radii are increased for the opposite cases. This behavior has been noted previousTN-75
44 WAYNE M. ITANO AND D. J. WINELAND 25

\yj ^ The requirements for cooling the cyclotron mode are similar to those for a harmonic oscillator, while

those for the magnetron mode are different.

C. Rate equations for mean-squared amplitudes

1. Uniform laser beams

Here we consider the effect of a plane-wave laser beam of the form given by Eq. (1). We assume the in- tensity to be constant over the orbit of the ion. Consider a laser beam parallel to the z axis. If the laser fre- quency CO is less than coq, the axial motion is cooled. Throughout the rest of Sec. Ill we neglect R /H compared with {(Oo —o)) or y. The magnetron and cyclotron modes are heated by recoil. As far as the axial mode is concerned, this is the same as the problem treated in Sec. II B 1 . In the steady state {Eic^)=m+f^)[(r/2)' + (coo-co)^]mcoo-co),

and for the optimum detuning,

((Oo-o}) = Y/2, <£jt,)=(l+/„)^r/8.

The second terms on the right-hand sides of Eqs. (53b) and (53c) average to zero; therefore, the heating rates of the magnetron and cyclotron modes are given by

d{ri) d{r^)

dt dt =rsR(fsx+fsy)/2Mn' (54)

The brackets around r^ and r^ denote ensemble averages. Now consider a laser beam propagating along

the X direction {k = kx). Assume that the system has somehow been cooled, so that k(Vy )^^ «y/2 or kiVy)^^« I coq —co

I . Then, using Eqs. (16), (45a), and (45b), the scattering cross section is

a{co,v) = ao(Y/2)^[{y/2}^ + ((Oo-o))^]~^\l+2(coo-co)k[r„co„sm{(o„t-\-9„)

+rcCOcSin{co'ct + dc)]/[{r/2)^ + (coq-co)^]} . (55)

For an individual ion, the amplitudes and phases {r^,rc,6m,dc ) are perturbed by each scattering event. To

find the average rate of change of {rj),{rli),{rc), we multiply il/Hco) by the cross section [Eq. (55)] and

by the change per scattering [Eqs. (53a) — (53c)] and average over r„, r^, 6^, dc, and k^: d{rj) , dt d{ri)

dt dirl)

dt

= rs\icoQ-(o)1ik^(o^{ri)/Ma[(r/2)^ + {coo-(o)^] + R(\+f^+f^)/2Ma^] ,

=rs[-(o)o-oi)1tk^(o',{r^)/Ma{{Y/29^{o)Q-oi)^]+R{\^fs^^f^)/2Ma^] .

(56a)

(56b)

(56c)

For negative laser detuning, (clxcoq), Eq. (56c)

predicts that { r^ ) is reduced until it reaches a

steady-state value, while Eq. (56b) predicts that (r^ ) increases without limit. This is because the Doppler shift causes the scattering rate to increase

when either the cyclotron or magnetron contribution to the velocity is opposed to k. For positive

laser detuning, [co > coq), the opposite occurs. The

Doppler shift due to the heated mode will eventually cause the approximation for the cross section [Eq. (55)] to break down, which will modify the heating and cooling rates, but not change these

qualitiative features. The axial motion is heated

regardless of detuning. It could be cooled by introducing a laser beam with some component along z, with negative detuning. However, there is no combination of uniform laser beams which is capable

of cooling all three modes.

2. Nonuniform beam along x and uniform beam

along z

Cooling of both the magnetron and cyclotron

modes can be accomplished with a laser beam

whose direction of propagation is in the xy plane

and whose intensity increases on the side of the z

axis on which the magnetron velocity is along the

direction of propagation. The photon scattering

TN-76
25 LASER COOLING OF IONS STORED IN HARMONIC AND 45

then tends to cool the magnetron motion. With

negative laser detuning, the cyclotron motion will be cooled, due to Doppler effect velocity selection. This method has been demonstrated experimentally.*' ' The axial motion can be cooled with a separate beam or by tilting the nonuniform beam

so that it has a component along the z axis. Here we treat the case of a nonuniform beam of

frequency u)\ and wave vector k^x and a uniform

beam of frequency (02 and wave vector /c2Z which

cools the axial motion. The laser intensity of the nonuniform beam is assumed to be a function of y

with value /i at >' =0 and a positive slope /j /yo at y =0 (see Fig. 2). We approximate the intensity

profile by /(>') =/i(l+>'/yo) for \y \ «yo- The

rms magnetron and cyclotron radii are assumed to be much less than yQ, so that this linear approximation is valid. Let I2 be the intensity of the laser beam along z. Let the scattering cross sections for the beams along the x and z axes, which depend

upon the light polarizations through Eq. (13), be

o-i((i)i,v) and ajicoi,^), respectively, with resonant (V =0,co=o)q) cross sections aoi and ao2- The rate equations for the mean-squared amplitudes are

dt = il2/fuo2)(a2((02,V)^rP

+ {I I /ikoi){(l +y /yo)ai(co^,v)^r;) ,

yn y —

FIG. 2. Intensity profile of nonuniform laser beam,

/i is the intensity at y =0 and I\/yo is its slope. The

tangent to the curve is used as an approximation to the actual profile in the calculation.

—-^— = (/,//ko,)<(l+>'/;'o)o^i(t^bV)Ar,')

dt + (/2/^2)^0-2(<y2>'v)A/-/) , (57b)

(57a)

where i =m,c. The Ar^ and Ar/ are evaluated

from Eqs. (53a) — (53c) using the k appropriate to each laser beam. We now make the low Doppler

approximation to the cross section and perform the

ensemble average. The average scattering rates y^i

(/ = 1,2) are given by

r,i = iIiaoi/fio),)(y/2)^/[(y/2)^ + {coo-(o,)^] . The rate equations become

d{r')

(58)

dt dirj)

dt

= -lY,2io)o-o32)1ik\{rl)/M[(Yn)^Mcoo-co2)^] + (lYsiR/Moj,l)[\+{\+rsx/rs2)fsz], (59a)

+(r.,i?/2Mn2)[i+(i4-r.2/r.iHAx+/.v)], (59b)

d{r',)

dt = rs\^k^{r^) /2MayQ-rs\(oio-coOfik]oi',{rl) /M^iiyn)^ ^{oo-co^)^]

-f(r„i?/2Mn2)[i+(i+r.2/r.i)(/.x +/.>-)] (59c)

The axial motion can be cooled if (^2 <^o [see Eq. (59a)]. The first term on the right-hand side of Eq.

(59b) shows cooling of the magnetron motion due the intensity gradient (for yQ > 0), the second shows heat- ing due to Doppler selection (for &>] <(i)q), and the third shows heating due to recoil. For y^yO and o)] <coq, the first term on the right-hand side of Eq. (59c) shows heating of the cyclotron motion due the intensity

gradient, the second shows cooling due to Doppler selection, and the third shows heating due to recoil. Some

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46 WAYNE M. ITANO AND D. J. WINELAND 25

terms have been neglected in Eqs. (59b) and (59c) because they are at most of relative order (k^Vx/yHy/yo)

compared to the recoil heating term. The term in the first set of parentheses is small, by the low Doppler

assumption, while the term in the second set is small because (r^ )^^ and ('c)rms ^re assumed to be much

less than >'o- The condition for simultaneous cooling of the magnetron and cyclotron modes, for yo>0 and

CO] <coo, is ^m <[(7/2)^ + («o —«i)^]/2'ci>'o(wo —^i)<^c • (60) Assuming this condition is satisfied, as can be obtained experimentally, the steady state is given by

{r^}=ii[\+{l+rs,/rs2)fsz][^r/2)- + {coo-o)2)^]/2Mcol{o)o-co2) , (61a)

Hkiyoil + il+Ysi/rsO^fsx+fsyn

{rl) =

(r') =

2Mn{\-2k^yoio)o-cOi)co„/[{Y/2)^ + {o)o-o)^)^]\

iikiyo[\+{l +y,2/rs\)ifsx +fsy)]

(61b)

(61c)

The average kinetic energy can be obtained by in- serting these values into Eqs. (51a) — (51c). The

parameters 7j2/7ii' (^o — (^^i)' i(0o--co2), and >'o can

be adjusted in order to minimize the kinetic energy. In the general case, this is complicated.

We consider a simple case which can be experimentally realized. For this case, the Doppler selec- tion cooling rate of the cyclotron motion is much

greater than the intensity gradient heating rate, and

the Doppler selection heating rate of the magnetron motion is less than and not too close to the

intensity gradient cooling rate. These condition are most easily satisfied when co'c/co^ » U which is obtained by reducing the electric potential. Also,

_Vo must fall within a certain range, so that k\yQO)'c./y» 1 and {l—2k^yQa)„/y) is not too

small, assuming that {ci>o —(0\) = y/2. Then the magnetron kinetic energy is much less than the cy- clotron kinetic energy. We have

X[(y/2)^ + {coo-(02)^]mcoo-co2) .

(62a)

<£^,)=^[i+(i+r.2/r.iK/.x+^)]

(62b)'

and we neglect {Efc^ ). For constant y^i and y^i, the kinetic energies are minimized if we set

(ci)Q — oi2) = {oi>Q —(x)\) = y/2:

<£^,) = [l+(l+r,,/n2)/«]^/8 , (63a)

<£^,)-[l + (l+r,2/r.iH/5x+^)]%/4.

(63b)

The total kinetic energy can be minimized by adjusting the relative laser beam intensities so that

r.i/7,2=[2(/^+/^ )//,,]

1/2 (64)

For the isotropic angular distribution, this condition is 75i/7x2 = 2, in which case, {E^^) ^ -jiEf^c ) =^7/4. This gives the same average ki- netic energy in the x, y, and z directions as for the harmonic trap discussed in Sees. II B 2 and II B 3. The minimum energies are slightly different for an

anisotropic distribution, as is the case for the har- monic trap.

3. Nonuniform beam in xz plane

Cooling of all three modes can be accomplished

by rotating the direction of the nonuniform beam

FIG. 3. Nonuniform laser beam configuration which

is capable of cooling all three modes of motion of an ion

in a Penning trap. The trap coordinate axes are indicated. The ion cloud is at the origin, and the direction of

the magnetron rotation is indicated by an arrow. The

laser light propagates in the k direction, which lies in the xz plane. The laser beam is focused so that it is more intense for y >0 than for y <Q.

TN-78
25 LASER COOLING OF IONS STORED IN HARMONIC AND 47

described in Sec. IIIC2 so that it has a component

along z and ehminating the uniform beam. Let k = A:sin0x+fccos0f; (0< 0<ir/2), be the wave

vector for the nonuniform beam, which has frequency CO. The geometry is shown in Fig. 3. Let

7^ be the average scattering rate. In the low

Doppler approximation,

dt

dkrj)

dt

d{r])

dt

= -2r,(«o-w)^^^cos^^(r,^)/M[(r/2)^ + (6)o-a))2] + (2r,/?/Ma;i)(cos20+/,,) ,

= -r,^fcsin0(r^>/2Ma>'o + r.(^o-w)^^^sin^6»a);„<'-m>/M"[(r/2)^ + (ft)o-w)^]

+ (r,i?/2Mft2)(sin20+/^+/^) ,

= r,/zA:sin6'</-,^>/2Mnj;o-r5(^o-«)^^^sin^0w;(r,^)/Mll[(r/2)2 + (6;o-ft>)^]

+ (7,/?/2Mn^)(sin^0+/«+/^) .

(65a)

(65b)

(65c)

Some small terms have been neglected. The axial motion can be cooled if 6)< &)o- The condition for simul- taneous cooling of the magnetron and cyclotron modes, assuming j^o > ^nd en <coq, is given by Eq. (60), with k %\ViQ substituted for /cj. If this condition is met, which is feasible experimentally, the steady-state

mean-squared amplitudes are given by

(r,^)=^(l+/^/cos20)[(r/2)2 + (a)o-ft>)^]/2Mwi(ft)o-«) ,

<.rl) =

<rl) =

akyoisin^d+f^+f^]

2Mnsmd{\-2ksineyo{(Oo-co)co„/[{Y/2)^ + {coo-co)^]]

^/c;;o(sin^6>+/^+/^)

2Milsmd\2ksineyo(coQ-o})co'c/[(y/2)^ + {coo-co)^]-\}

(66a)

(66b)

(66c)

We consider the same limit as in Sec. Ill C 3, i.e., the Doppler selection cooling rate of the cyclotron motion

is much greater than the intensity gradient heating rate, and the Doppler selection heating rate of the magnetron motion is less than and not too close to the intensity gradient tooling rate. Then

(Efc ) =^( 1 +fs,/cos^d)[(y/2)^ + {ojo-o))^]mo}o-co) , (67a)

(67b)

and ( E](„ ) is negligible. These quantities are minimized by setting (coo —tL>) = y/2:

{Eii,)=(l+f^ /cos^e my/8 , (68a)

<£j^,)-[l+(/,^+A,)/sin20]^y/4 . (68b)

The total kinetic energy is minimized with respect

to d for

tan^0 = 2(/,,+/^)//„ (69) For the isotropic angular distribution, this condition is, = tan-'(\/2)^54.74°, in which case, {Ei^ ) s -jiEf^c ) =^7/4. This is the same limit obtained in Sec. IIIC3.

4. Comparison with experimental data

The most unambiguous comparison between

theory and experiment can be made for the case of

a single ion. The details of energy exchange due to

collisions are not precisely understood for the case of a cloud of ions; moreover the frequencies of oscillation (0^, (o'c, and co„ are all shifted by the pres- ence of space charge. Both of these problems are absent for a single ion, and the simple theory dis- cussed above should apply. Laser cooling of a single ^''Mg"'" ion in a Penning trap has been performed previously.^ The

cooling achieved was such that the Doppler

TN-79
48 WAYNE M. ITANO AND D. J. WINELAND 25

broadening was significantly less than the natural linewidth of the transition, so the results of the last section should apply. For this experiment,^ a sin- gle ^^Mg"*" ion was stored in the trap and cooled. The trap parameters were Bq^I.OI T and

Ao= Vq/ (rl + 2zl) ^2.95 V/cm^ where the in- terelectrode voltage was Fq = 2 V and the electrode dimensions, in the notation of Ref. 4, were such

that rg + lzo = 0.678 cm". This gave approximate

observed frequencies of ty^— 277" X 638. 5 kHz,

fi)z =2irX 106.9 kHz, and co,„ =27rX8.95 kHz.

The laser beam intensity profile in the vicinity of

the ion was measured to be

/(3')=/'exp[— 2(_v —d)^/ wl], where d^\5 /xm

and u;o = 25 ^m.

From this expression, we estimate

_1_

yo

1 di

l(y=0) dy

1 y=0 10.4^m

The transition involved in the laser cooling was the

24

transition at A. = 279. 6 nm; for this case [see Eqs.

2 3 (8b) and (10b)] /j^ = y and f^x +fsy = j- From Eq.

(69), the angle of laser propagation that minimizes

the total kinetic energy is = tan" '(3*'"*) ^52. 77°. Experimentally, 6 could be varied between 82° and

90°, and it was set at 82°. The laser detuning was

such that (coq—co) = 2TrX25 MHz.

From Eqs. (66a) — (66c) we predict (r^^) = (4. 17 XlO-^cm)^ (a-;) = (2.07X10-^ cm)^ and

(fm )=(5.25x 10~^ cm)^. An important parameter in very high resolution spectroscopy is the

second-order Doppler shift Acoo/coq= — j{v^)/c^.

Because this shift is proportional to the kinetic en- ergy of the ions, it is convenient to define a "temperature" in terms of the kinetic energy. The concept of a temperature may not be valid for a single

ion, but it is reasonable to define temperatures for the axial and cyclotron degrees of freedom in terms of the time-averaged kinetic energies as {Ef:^) = ^kBT^ and {Eiic)=kBTc, where k^ is Boltzmann's constant. For the magnetron degree

of freedom, there is an additional problem since the total energy is negative (when the electric potential is defined to be zero at the origin) and de- creases to minus infinity as {r^) goes to plus infinity. However, it is still useful to define, by anal- ogy with the cyclotron motion, {Ef^„ )=kgT„.

Using these definitions, we find for the experiment

of Ref. 9, r^^ 1 1 mK, T, ^ 1.0 mK, and

r^ = 1.3 X 10-^ K. From the residual Doppler

broadening of the optical lines measured in Ref. 9

we estimated that 7^^(50 ±30) mK and T^ < 1 mK. At the present time we do not understand

why the measured temperatures are higher than

predicted. One difficulty not mentioned previously

is the possible collisional heating due to impurity

ions, such as NH3+, H3O+, N2H+, and HCO+,

that might be simultaneously trapped.

IV. LASER COOLING IN A PENNING TRAP

(QUANTUM-MECHANICAL TREATMENT)

A. Quantum states of an ion in a Penning trap

The Hamiltonian operator for a single ion in the same idealized Penning trap which was treated

classically in Sec. II A, is H = {^-(q/c)Mr)]^/2M +qV(r)=H^+H, ,

(70) where

Hxy={p^+p})/2M-^\MilHx^+y^)-{co,L, ,

and

H,=p^/2M+\McoIz^

(71a)

(71b)

Here T and p are operators and L^ is an operator defined by L^ =xpy —ypx- This Hamiltonian

has been dealt with by other authors. ^°'^' Here we

summarize the useful results. H^ is the Hamiltoni- an of a one-dimensional simple harmonic oscillator. Therefore it can be rewritten as H, = {N, + T)-fio},

where

z=Zo(a2 +az) , p^^izoMo^ia^ —Or) ,

Zo = (^/2Ma),)'/^

(72)

(73a)

(73b)

(73c)

(73d)

H^y is the Hamiltonian of a two-dimensional iso- tropic oscillator, plus a term proportional to L^. It can be rewritten as Hxy={N, + { )iko', -iN„ + \ )fuo„ , (74)

where

Ni=aiai (i—c,m)

ac = (ax+iay)/2 1/2

(75a)

(75b)

TN-80
25

am='^0x-i0y)/2

LASER COOLING OF IONS STORED IN HARMONIC AND . . .

(75c)

49

1/2

x=^o{ax+a,)/2'^^ y ^ro(ay+ay)/2

Pj = roM€i{aj -Oj )/V" (j =x,y)

ro = (VMn)'/2 .

1/2

where / is the identity operator. The axial component of the angular momentum operator can be

written in the form

(75d)

(75e)

(75f)

L, = {N^-N^fi. ill)

The three sets of raising and lowering operators

satisfy the commutation relations

[a/,a;]=0,

[ai,a]] = bijl (i^z,c,m)

(76a)

(76b)

(76c)

I

The complete set of eigenstates of H is given by

the set of simultaneous eigenstates of A^^, N„, and

Nc- We label these states | «2,n^,nc ), where w^, n„, and «<. are the eigenvalues of N^, N„, and N^

and can be any set of nonnegative integers. They

can be generated from the | 0,0,0) state by applying the raising operators. The normalized states are

l«z,nm,«c>=(«z!«m!nc!)-'^'(fl/)^ai)'''"(aJ)"MO,0,0) .

the coordinate-space representation of | 0,0,0) is

{x,y,z \0,0,0) =ir-^^\2^^horl)-^^hxp[-z^/2zl-(x^+y^)/2rl] .

(78)

(79)

Explicit expressions for the wave functions for arbitrary quantum numbers have been given by Sokolov and

Pavlenko.^' By comparing the classical expressions for the energy [Eq. (49)] and the axial angular momentum component [Eq. (52)] with the corresponding quantum-mechanical operators, [Eqs. (72), (74), and (77)], one obtains the following correspondences between classical amplitudes of motion and the quantum numbers:

'm~(«m+T^''0 » r^^Mn, + j)zl .

(80a)

(80b)

(80c)

B. Transition rates We treat the interaction of the ion with the electromagnetic field by second-order time-dependent perturbation theory, in the form of the "golden rule." The motion of the ion as a whole, as well as its internal structure, is described by wave functions. In Sec. Ill, only the internal structure was treated quantum

mechanically. Consider the transition rate from | n2,n„,nc)^ | j«'j ) to | [n-^] ) induced by a near- resonant uniform laser beam described by Eq. (1). The ion is in its ground electronic state \g} before and

after the transition. This is a kind of spontaneous resonance Raman scattering, through the intermediate

electronic state \e). The cross section for this process is given by Eq. (19) of Ref. 16. We make the

electric-dipole approximation and sum over all polarizations and directions of the scattered photon.

This cross section is

C7(i«' ,/i

=(7o(r/2)' /

{{n^\ |exp(-/k,-r)| {nJ}){\nJ] |exp(/k-r)| [n'})

(Oo-(o-iy/2+co{{nJ})-a)({n']) Ps{k,)dn , (81)

where co(\n']) = (n2 + j)co2 + inc + j)cL)'c—(n'„ -irj)co„, and the other terms have been defined previously.

We have replaced x of Ref. 16 with Fhere. The transition rate is obtained by multiplying this cross section

by I/fico.

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50 WAYNE M. ITANO AND D. J. WINELAND 25

The transition rate due to a nonuniform laser beam can also be calculated by perturbation theory. The

only difference is that the electric field vector due to the laser perturbation — d-E(F,r), which induces transitions to the intermediate state, is no longer a uniform plane wave of the form of Eq. (1). We consider only

the low-intensity limit, in which case the transition from the intermediate state to the final state is induced

by the zero-point field. Consider a nonuniform beam like the one treated in Sec. IIIC3, which has an intensity gradient in the y direction and has a wave vector k =/c sin0x + ^ cos0z, (0 < < n/l). Let the electric

field be

E(r,?) = e{ 1 +>'/2_Vo)Re£'oexp(/k-F— /(yf ) , which gives the intensity profile

(82)

(83)

for \y I <<>'o- We can use this E in the perturbation matrix element to reproduce the case treated classi- cally in Sec. IIIC3, provided that the wave functions are localized in the region \y \ «yQ. The transition rate is

r([«'j^S«^|)={/o/^)c7*(j«' n^\). (84)

where the effective cross section a* is given by

=ao{Y/2f f

{{n^} |exp(-/k,-r)| {nJ]){{nJ] | ( 1 +>;/2j;o)exp(/k-F) | j«

. coo-co-iy/2+co{lnJ])-co({n']

)

PAK)dil (85)

C. Rate equations for average quantum numbers

We calculate the rate equations for the ensemble averages of the quantum numbers (n^), (n^), and

(«c ). We do not consider coherences between | n^^ny^.n^. ) states. Let P( j«'l ) be the probability that the

ion has the set of quantum numbers j «

'j . We can derive the rate equations for any of the cases considered

in Sec. IIL For the sake of brevity we consider only the case of the nonuniform laser beam propagating in the xz plane [Sec. IIIC3], using the effective cross section defined by Eqs. (84) and (85). The rate equations

are given by'^

diuj)

dt = 2 P({n'])r({n']^\n-^]){rrf~n!),

\n'\,\nf]

(86)

(87)

where

/ =z,m,c . We make the low Doppler approximation. In this limit we can make the approximation

[coo-co-iy/2 +a}{\nJ\)-co({n'])]-^

^(coo-co-ir/2)-^{\ + [co({n'])-a){\nJ])]/(o}o-o}-iY/2)]

in Eq. (85). Equation (87) is valid for

\[co({n'])-co({nJ])]/{coo-co-ir/2)\ «1 .

This condition is violated for some terms in Eq. (85), which correspond to large differences in quantum

numbers, since the summation is over all possible |«-'|, but these terms are of negligible size, because the matrix elements are very small. With this approximation, Eq. (86) can be evaluated by straightforward alge- bra. We outline the important steps here. The method was previously used for the harmonic trap in Ref.

16. We note that

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25 LASER COOLING OF IONS STORED IN HARMONIC AND . 51

{{nJ]\A \\n'])[<o({n'\)-co(\nJ})] = (\nJ]\[A,Hm\\n']} (88)

for any operator A. The summations in Eq. (86) over jn-^j and {«•') can be carried out using the closure

property, and we obtain

din,}

dt =rs^P^W\) S {W\\A\Ni,A]\W])p,{kM^

Here

y4=exp(-/k^-F){(l+;;/2;;o)exp(/k-r) + [(l+j;/2>'o)exp(/k-F), HmoiQ-co-iy/l)]] , and

ys = iIocro/-fico){y/2)^/[(coo-(o)^ + (y/2)^]

is the average scattering rate. We define the ensemble averages (/*/), using Eqs. (80a) — (80c) by

{rf)=rl{ni + j}=rl2Pi{n']nn!+{) U =--m,c)

{n'\

The rate equations for (/•/) (/ =z,m,c), obtained

by evaluating Eq. (89) and dropping small terms,

are identical with Eqs. (65a) — (65c). This is not

surprising, because the average quantum numbers

are large. The other cases treated in Sec. II C can

be treated by the same methods. The rate equations obtained are identical to those obtained from

the classical analysis. Here the following quantities are regarded as small: r„ /yo, r^/yo, oim /y»

ci'c/y, co^/y, R/Hy, ko)„r„/y, kco'cr^/y, and

Ukyo.

The quantities kzQ and /cz-q are assumed to be

not too large, so that when multiplied by a small

quantity, the result is still small.

V. DISCUSSION

We have carried out a semiclassical calculation

of the laser cooling of an ion in a Penning trap, for the case where the frequencies of motion are much

less than the natural linewidth of the optical transition. The results are confirmed by a quantummechanical calculation based on perturbation

theory. Also, some simple calculations of laser cooling in a harmonic trap are presented. Where a

direct comparison can be made, the results agree with those obtained by more formal methods.^'''

^^ Experiments have confirmed the qualitative features of the present analysis.^"^ Cooling to average kinetic energies so low that the Doppler

broadening of the optical transition is smaller than

(89)

(90)

(91)

(92a)

(92b)

the natural linewidth has been observed.^ However, the ultimate limits predicted by the theory have

not been achieved in these experiments.

Useful improvements to the theory would in- clude taking into account Coulomb forces between

ions and assuming a more realistic laser beam profile. Saturation of the optical transition has not been included. However, this is not a severe re- striction, especially for the final stages of cooling,

since we would expect the minimum energies to be

obtained in the low-intensity limit, as for the har- monic trap.^° It would be interesting to apply the

Fokker-Planck formalism to the Penning trap as has been done for the harmonic trap.^°~^^ In this paper, the case of resolved sidebands

{co'c,Ci)2,(Om >>Y) has not been treated but should

be quite straightforward and could follow the

development of Ref. 1 6 and this paper. We re- mark that when the mean occupation numbers

( «, ) are small enough, then all sidebands can be

well resolved. This means that cooling can be

achieved using spatially uniform plane waves pro- vided the lasers are tuned to the correct sidebands.

Motional sideband cooling of the magnetron

motion of an electron in a Penning trap^^ has been

demonstrated and is closely related to laser cooling.

Here the axial resonance, which is analogous to the

optical transition of the two-level ion, is driven by

an inhomogeneous rf field at the sideband frequen- cy a)^ +0)^ . The modes are coupled by the field in such a way that the magnetron motion is cooled.

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52 WAYNE M. ITANO AND D. J. WINELAND 25

We also remark that, in general, the reduction of

the magnetron orbits can be achieved by other means which add (total) energy to the magnetron

degree of freedom.^-^ This could be achieved, for example, by coupling the magnetron motion to a negative resistance or by collisions with atomic

beams that preferentially interact with the side of

the cloud where the magnetron motion is in the same direction as the beam. This effect could be

important in certain collision experiments.^"* All previous detailed theoretical treatments of

laser cooling of trapped ions have assumed har- monic potential wells. The experiments which

most closely approach this situations are per- formed with rf quadrupole traps, in which the motion of an ion consists of the high frequency,

small amplitude "micromotion" superimposed on

the low frequency, large amplitude "secular motion. "* Approximating such a trap by a har- monic potential neglects the micromotion. The

average kinetic energy in the micromotion, aver- aged over an rf cycle, is just an effective potential energy for the secular motion. A formalism which

could take account of the micromotion, either classically or quantum mechanically, might be useful

for the interpretation of the experiments.

Finally, we wish to point out that Einstein, ^^ in the same paper in which the A and B coefficients for spontaneous and stimulated emission were in- troduced, treated a problem which is <:losely related

to laser cooling. He showed that a gas of two-level atoms comes into thermal equilibrium with radiation having the Planck spectrum, as a result of

light pressure forces. In this case, since the radia- tion is broad band and isotropic, the average damping force is mostly due to an effect that we

have neglected: The apparent intensity of a light source increases if the observer moves toward the

source, as a result of the transformation law for the electric and magnetic fields. The Doppler shift and angular aberration of the light must also be

taken into account. The damping force due to the average light pressure competes with the fluctuat- ing force due to photon recoil so that, in the steady

state and in the nonrelativistic limit, EKx^jkgT,

if the radiation has the Planck spectrum for temperature T. Recently, the calculation has been ex- tended to demonstrate that the gas has the relativistic Boltzmann distribution after it comes into thermal equilibrium with the radiation. ^^

This work was supported in part by the Air

Force Office of Scientific Research and the Office

of Naval Research.

APPENDIX: SPECTRAL DENSITY OF

A RANDOM SERIES OF IMPULSES

We calculate here the one-sided (i.e., defined for

positive frequencies) power spectral density of

G(t), where Git) is a stationary, random function

of time consisting of a series of Dirac delta functions. These impulses occur at an average rate y

and they may vary in amplitude. This calculation

follows that of Rice."

First we calculate the correlation function,

R(T) = {G{t)G{t+T)) (Al)

where the angular brackets denote an ensemble

average, and R is independent of t, by the assumption of stationarity. We consider G(t) during the time interval < r < T, where T is arbitrary. We

denote a particular member of the ensemble by

G/d^it), where K is the number of impulses that oc- cur in the period, and k uniquely identifies the member. This function takes the following form:

GKk(t)= J,gKkMt-tf:ki'

/ = i (A2)

The amplitudes g/^^/ can vary, but are not correlat- ed in any way with t/^j^i, K, or /. The / index does

not indicate an order in time. For fixed K and /,

tnki can be anywhere in the interval from to T

with equal probability. We define ensemble aver- ages over the index k as

(gKkih = (g> , (gL)k^(g'),

(A3)

(A4)

where (g ) and (g^) are independent of / and K. The product Giii^{t)Gf:^i^(t -\-t) takes the form

K K GKkit)G,ik(t+T)= X ^ gKMgKkV^^t -ttcM)Ut + T-tickV

/ = !/'=!

where

=5,+S2 > (A5)

TN-84
25 LASER COOLING OF IONS STORED IN HARMONIC AND . . 53

5,= 2igKkl)^&it-tf:kim+T-tfck,)

1 = 1 and

^2 = 2 SKklgKkl'^i t-tfckt)^(t + T- tKkf

)

(A6)

(A7)

Now we carry out the ensemble averages over the index k for fixed K: K T

= 2 i(g^)/T)d{T)=K{{g^)/T)8{T) (A8)

/ = i and

iS2)k= 2 «g>/T)'f^5{t-tfcki)dtfck, f^^it+r-tKkr)dttckr = 2 ({g)/T)^=K(K-l){{g)/T)^ .

The ensemble average over K is carried out using the Poisson distribution function

p(K) = (rTfe-^'^/K\,

where/? (AT) is the probability that K impulses occur during the period T. Finally, we have

R{t)= ^ p{K)({s,)k + {S2)k) = r{g^)^ir)^r'igy

K=0

(A9)

(A 10)

(All)

The spectral density is given by

u;{/) =4/^"/?(r)cos27r/rc?r=2r<g^)+2r(g)W) , (A12)

which is defined for frequencies / > 0. The first (white noise) term corresponds to the spectral density cal- culated in Eq. (39). The second (dc) term comes from the average value of G, which correspond to (F^ ) de- fined in Eq. (33).

'T. W. Hansch and A. L. Schawlow, Opt. Commun.

13, 68 (1975). 2d. J. Wineland and H. Dehmelt, Bull. Am. Phys. Soc.

20, 637 (1975). ^V. I. Balykin, V. S. Letokhov, and V. I. Mishin, Pis'ma

Zh. Eksp. Teor. Fiz. 29, 614 (1979) [JET? Lett. 29, 560 (1979)]; Zh. Eksp. Teor. Fiz. 78, 1376 (1980)

[Sov. Phys.—JETP 51, 692 (1980]. ^H. G. Dehmelt, in Advances in Atomic and Molecular

Physics, edited by D. R. Bates and I. Esterman

(Academic, New York, 1967), Vol. 3. ^W. Neuhauser, M. Hohenstatt, P. Toschek, and H. Dehmeh, Phys. Rev. Lett. 41, 233 (1978); Appl.

Phys. 17, 123 (1978). *W. Neuhauser, M. Hohenstatt, P. Toschek, and H.

Dehmelt, Phys. Rev. A 22, 1137 (1980). ^D. J. Wineland, R. E. Drullinger, and F. L. Walls,

Phys. Rev. Lett. 40, 1639 (1978). *R. E. Drullinger, D. J. Wineland, and J. C. Bergquist,

Appl. Phys. 22, 365 (1980). ^D. J. Wineland and W. M. Itano, Phys. Lett. 82A, 75

(1981). 'OA. Ashkin, Phys. Rev. Lett. 40, 729 (1978); A. Ashkin and J. P. Gordon, Opt. Lett. 4, 161 (1979); A.

Ashkin, Science 210, 1081 (1980). "J. P. Gordon and A. Ashkin, Phys. Rev. A 21, 1606

(1980). '2V. S. Letokhov, V. G. Minogin, and B. D. Pavlik, Zh.

Eksp. Teor. Fiz. 72, 1328 (1977) [Sov. Phys.—JETP

45, 698 (1977)]; V. S. Letokhov and V. G. Minogin,

ibid. 74, 1318 (1978) [/Wc?. £7,690(1978)].

13R. J. Cook, Phys. Rev. A 20, 224 (1979); 21, 268

(1980); 22, 1078 (1980). '4R. J. Cook, Phys. Rev. Lett. 44, 976 (1980).

TN-85
54 WAYNE M. ITANO AND D. J. WINELAND 25

'^J. Javanainen and S. Stenholm, Appl Phys. 21, 35 and

163 (1980). '^D. J. Wineland and W. M. Itano, Phys. Rev. A 20, 1521 (1979). i^A. P. Kazantsev, Usp. Fiz. Nauk. 124, 113 (1978)

[Sov. Phys. Usp. 21, 58 (1978)]. 18W. H. Wing, Phys. Rev. Lett. 45, 631 (1980). '^V. S. Letokhov and V. G. Minogin, Opt. Commun.

35, 199 (1980).

20j. Javanainen and S. Stenholm, Appl. Phys. 21, 283

(1980).

2

'J. Javanainen and S. Stenholm, Appl. Phys. 24. 71

(1980).

22j. Javanainen and S. Stenholm, Appl. Phys. 24, 151

(1981).

23j. Javanainen, Appl. Phys. 23, 175 (1980). 24F. M. Kelly and M. S. Mathur, Phys. Rev. A 18, 2135

(1978). 25W. L. Wiese, M. W. Smith, and B. M. Miles, Atomic

Transition Probabilities, Vol. II, Natl. Bur. Stand.

(U.S.), Natl. Stand. Ref. Data Ser. 22 (U.S. G.P.O.,

Washington, D. C, 1969), p. 30. ^^For discussions of the classical and quantum FokkerFlanck and Langevin equations, see M. Sargent III, M. O. Scully, and W. E. Lamb, Jr., Laser Physics

(Addison- Wesley, Reading, Mass., 1974).

2^See, for example, F. N. H. Robinson, Noise in Electri- cal Circuits (Oxford University Press, London, 1962), Chap. II.

2*See, for example, Threshold Signals, edited by James

L. Lawson and George E. Uhlenbeck (Dover, New

York, 1965), pp. 79-83.

^''J. Byrne and P. S. Farago, Proc. Phys. Soc, London

86, 801 (1965). ^°G. Graff and E. Klempt, Z. Naturforsch. 22a, I960

(1967); G. Graff, E. Klempt, and G. Werth, Z. Phys.

222, 201 (1969). ^'A. A. Sokolov and Yu. G. Pavlenko, Opt. Spectosc.

(USSR) 22, 1 (1967). ^^H. G. DehmeU, Nature 262, 777(1976); R. S. Van

Dyck, Jr., P. B. Schwinberg, and H. G. Dehmelt, in New Frontiers in High-Energy Physics, edited by B. M. Kursunoglu, A. Perlmutter, and L. F. Scott (Plenum, New York, 1978), p. 159; D. J. Wineland, J. Appl. Phys. 50, 2528 (1979). '^D. J. Wineland, R. E. Drullinger, J. C. Bergquist, and

W. M. Itano, Bull. Am. Phys. Soc. 24, 1185 (1979).

^"^M. D. McGuire and E. N. Fortson, Phys. Rev. Lett. 33, 737 (1974). 35a. Einstein, Phys. Z. 18, 121 (1917); Enghsh translation in Sources of Quantum Mechanics, edited by B.

L. van der Waerden (North-Holland, Amsterdam,

1967). 36V. Ben-Ya'acov, Phys. Rev. D 23, 1441 (1981); C. H.

Braden, R. F. Fox, and H. A. Gersch, ibid. 23, 1455

(1981); A. Peres, ibid. 23, 1458 (1981). "S. O. Rice, Bell Syst. Tech. J. 23, 282 (1944).

TN-86
PHYSICAL REVIEW A VOLUME 25, NUMBER 2 FEBRUARY 1982

Shift of ^Si/2 hyperfine splittings due to blackbody radiation

Wayne M. Itano, L. L. Lewis, and D. J. Wineland

Frequency and Time Standards Group, Time and Frequency Division, National Bureau of Standards. Boulder, Colorado 80303

(Received 27 October 1981)

Frequency shifts of hyperfine splittings of ^S^/2 stales due to the blackbody radiation field are

calculated. It is shown that they can be estimated from the dc hyperfine Stark shifts, which have previously been measured in the ground states of hydrogen and the alkali atoms. The

shift at 300 K is large enough to be significant in primary Cs atomic beam frequency standards, and should be measurable. A simple method of calculating the hyperfine Stark shifts is described, which is based on the Bates-Damgaard method for determining radial matrix ele- ments and the Fermi-Segre formula for determining the contact hyperfine matrix elements. It

is applied to Ba"*" and Hg"*", for which no experimental data are yet available, and which are currently of interest for frequency standards.

The most accurate and stable atomic frequency

standards are based on hyperfine transition frequencies in ^5i/2 ground states, such as in '^^Cs, 'H, and

*^Rb. In this Communication, we estimate the temperature-dependent shift of ^5i/2 hyperfine split- tings due to the blackbody radiation field. We note

that this effect is large enough to be observable in a Cs atomic beam apparatus. The shift of the Cs hyperfine splitting at r = 300 K from the unperturbed

( r = K) value causes a frequency offset which is signiflcant for primary frequency standards. According to the Planck radiation law,

EHw)dw = BH(j>})dcj = (1) n exp{oj/kT) - 1 where E^{b))du)[B^{(jt))dw] is the squared amplitude

of the blackbody electric (magnetic) field in a bandwidth doj around w. [Atomic units (a.u.) are used unless otherwise specified (^ = w^ = e = 1).] The mean-squared fields are {EHt))=\

'0rEHoj)d(^

- = ^aHkTr 15

(8.319 V/cm)2Ir(K)/300]'' (2)

and

{BHi))=~ J"BHc^)dc = (2.775 xlO-*T)2[r(K)/300]' (3)

Gallagher and Cooke' pointed out that these fields induce temperature-dependent shifts of transition fre- quencies in atoms and molecules through the ac Zee- man and Stark effects. ^'^ They estimated the shifts

in high-« Rydberg levels and some other systems.

Blackbody frequency shifts have not yet been ob- served in any system.

The fractional blackbody ac Zeeman shift of the ground-state hyperfine splitting in H or Cs was es- timated in Ref 1 to be about lO"'^ at r = 300 K. We

have derived the following expression for this shift in any ^5i/2 ground state, which is valid at zero dc magnetic field and at temperatures such that the peak of

the blackbody spectrum is at a much higher frequency than the hyperfine frequency. We find

8a» hfs Whfs 24 (gj-gi >'S:

BHc)

d<t)

= -{7rl\%){gj-g,ya'{kTy

= - 1.304 xl0-'^[r(K)/300]2 , (4)

where gj and gi are the electronic and nuclear g factors, respectively. In the last line, we have assumed

that gy = 2 and that \gilgj\ « 1. At laboratory temperatures, the blackbody ac Stark

shift of the hyperfine splitting, which has previously

been neglected, is generally larger than the ac Zee- man shift. The ac hyperfine Stark shift due to an

electric field of frequency w is approximately equal to the dc hyperfine Stark shift due to a static field with

the same rms values, if w « Wres. where Wres is the lowest allowed electric dipole transition frequency.

The correction is of order (w/wres)^ and will be considered in more detail below. For the ground states of any of the alkali atoms, (w/wres)^<3 x 10~\

where w is the frequency corresponding to the peak

of the blackbody spectrum at 300 K. Therefore, at 300 K, the blackbody ac hyperfine shift is approximately equal to the shift caused by a dc field of 8.3 V/cm.

The dc hyperfine Stark shift was first observed in Cs by Haun and Zacharias.'* Later, it was observed in

25 1233

TN-87
1234 RAPID COMMUNICATIONS 25

H (see Ref. 5) and other alkali atoms. ^ These experi- ments measured the Stark shift of the {F = I + j,

Mf = 0)— (f = /-y, Mf = ) transition. They can

be considered to be measurements of the scalar hyperfine polarizabilities, which are independent of Mf

and the orientation of the electric field, since the contributions from the tensor polarizabilities can be

estimated and are less than the experimental uncertainties.' Only the scalar polarizability contributes to the blackbody ac Stark shift, because of the isotropy of the blackbody radiation. The fractional ac Stark

shift of the Cs hyperfine splitting can be estimated

from the measured dc hyperfine polarizability (see

Table I of Ref. 6) and Eq. (2) to be -1.69(4) x lO"'^

[ 7'(K)/300]''. This shift is large enough to affect the

calibration of primary Cs frequency standards and

therefore should be taken into account. For example, the fractional uncertainty of one primary Cs fre- quency standard (Cs 1 of the Physikalisch-Technische

Bundesanstalt) is stated to be 6.5 x 10"'^ but has not been corrected for the blackbody shift.* The theory of the dc hyperfine Stark shift of ^Si/2 ground states is quite well developed, and the calcula- tions are in good agreement with the experiments.

The shift appears in the third order of perturbation

theory, where the electric-field interaction is taken

twice and the hyperfine interaction is taken once. For hydrogenic atoms and ions, an analytic solution has been obtained.'''" Numerical calculations have

been made for the neutral alkali atoms.""''* The

shift in Li has been calculated without using pertur- bation theory by the spin-optimized self-consistent

field method, with the electric field included in the Hamiltonian.'^

We write the third-order perturbation expression

for the scalar fractional dc hyperfine Stark shift of the

ns ^Si/2 state of an alkalilike atom or ion in the fol- lowing form, which is independent of the spin and

magnetic moment of the nucleus,

8(hmJ(x}Ms= - k£^ = - {ki + k2)E^

where

n J 6[iV{n"PJ)- W{nS)V

(5)

(6a)

and

k2 = -

n ,J,n ^ n

{n^SM2\\r\\n"'^Pj)

l[W{n"PJ)- WinS)]

(«%/2llrll«"2Py).// , (0) ' n s

. (6b)

The reduced matrix elements of f, the position

operator of the valence electron, are defined with the conventions of Edmonds.'^ We choose the phases of

the radial wave functions so that they are real. W{n"PJ) and W{nS) are the energies (not including

the hyperfine interaction) of the n'p'^Pj and the

ns^S\/2 states, respectively, and 4>ns(^) is the value

of ns ^S\/2 wave function at the origin. The dc magnetic field is assumed to be so small that the Zeeman

splitting is much less than the hyperfine splitting. These formulas were derived by following the method of Feichtner et al., '^ in which the ns^S]/2 wave functions and energies are calculated to first or- der in the contact hyperfine interaction and the Stark

shifts are then calculated by second-order perturbation theory. We ignore the spin-dipolar and quadrupole hyperfine interactions, since they contribute only

to the tensor hyperfine polarizability. The terms ^1 and /c2 correspond, respectively, to the diagrams

drawn in Figs. 1 (a) and 1 (b) of Lee et a/." and the "hfs" and "wave function" terms of Feichtner

etal.^^ We have developed a simple method of approximately evaluating Eqs. (6a) and (6b). We calculate the radial matrix elements using the Coulomb

(Bates-Damgaard) approximation'^ and the values of

the s-state wave functions at the origin using the Fermi-Segre formula." We have used this method to calculate the scalar fractional ground-state hyperfine

polarizabilities of Li, Na, K, Rb, and Cs and have obtained agreement with experiment to within 12% or

better in all cases. The lowest three p states and the lowest five 5 states were included in the basis. In using the Bates-Damgaard method, the experimental term values were used to calculate the n* values. For the p states, the centers of gravity of the two

fine-structure levels were used for the term values. Table 3 of Ref. 17 was used with linear interpolation. In evaluating Eq. (6b), the phases of the wave functions must be kept consistent. If we define the s- state wave functions to be positive at the origin, then

the sign of the radial integral involving the ns wave

function, obtained from Table 3 of Ref. 17, must be

reversed if n is even. This is because the Coulomb

wave functions used by Bates and Damgaard, which

are good approximations to the true wave functions

outside the core, are positive for r greater than the

last node [see Eq. (10) of Ref. 17] and the true ns wave functions have n - 1 nodes. This method can be used for other atoms, for which no experimental data or calculations have yet

been published, such as the singly ionized alkaline

earths. We have carried out the calculations for the ground states of Hg"^ and Ba^ which are currently of

interest for applications in stored-ion frequency standards. "^° In atomic units, k =37.9 for Hg"^ and

k =902 for Ba"*". The conversion between atomic and

laboratory units of E^ is given by

EH&.u.) =3.782 X 10-2°£^[(V/cm)^] . (7)

TN-88
25 RAPID COMMUNICATIONS 1235

At 300 K, the fractional blackbody ac hyperfine Stark

Shifts are -9.9 x lO"''' and -2.4 x lO"'' for Hg+and

Ba"^, respectively. In rf trap experiments, the ac Stark shift due to the trapping fields may be larger. The ground-state hyperfine shift due to an ac electric field of magnitude E{t) = Eiw) coswt can be obtained by the same method that w.as used to derive

Eqs. (6a) and (6b), except that the formula for ac Stark shift is used. We assume that w « Wres, where

Wres is taken to be the frequency of the center of

gravity of the fine-structure components of the first resonance line. If we assume that only the first excit- ed p state makes a significant contribution, which is a good approximation for the alkali atoms, the shift is

8cuhfs/whfs = -t(A:i[1 +3(a>/a>res)^] + A:2[l +(cu/a),es)^]lf^(w)

= -k{E\t))[\+{lkJk+k2lk){u>lw,,y] (8)

The effect of the frequency distribution of the black- body electric field is thus to increase ISwhfs/whsfl by

the fractional amount

€ = (3ki/k+k2/k){w^)/(oi, (9)

relative to its value for a dc electric field of the same

rms value. The mean-squared frequency of the

blackbody electric field (w^) is given by

(a>2) = J\^EHw)dw/ JJ' E\aj)daj

407r^

21 -(kTV (10)

Among the alkali atoms, e is the largest for Cs,

which has the lowest value of Wres- If we take k] and

/c2 from Ref. 11, then for Cs,

e = 1.4xlO-^[r(K)/300]2 , (11)

which, at 300 K, is smaller than the experimental uncertainty in the dc hyperfine polarizability.

The blackbody shift could be observed in a Cs fre- quency standard which was modified so that the temperature of a tube surrounding the atoms in the reso- nance region could be varied. If the temperature were changed, for example, from 300 to 400 K, the

fractional frequency shift would be 3.7 x 10"''*. If the frequency standard had the same frequency stability as NBS-6, the primary frequency standard of the United States,^' this shift could be determined to 30%

or better in an averaging time of several hours.

This work was supported in part by the Air Force

Office of Scientific Research and the Office of Naval

Research.

'T. F. Gallagher and W. E, Cooke, Phys. Rev. Lett. 42, 835

(1979). 2M. Mizushima, Phys. Rev. UB, A414 (1964).

'C. H. Townes and A. L. Schawlow, Microwave Spectroscopy (McGraw-Hill, New York, 1955), pp. 273-279.

*R. D. Haun and J. R. Zacharias, Phys. Rev. 107, 107

(1957).

^E. N. Fortson, D. Kleppner, and N. F. Ramsey, Phys. Rev.

Lett. 13, 22 (1964); P. C. Gibbons and N. F. Ramsey,

Phys. Rev. A 5, 73 (1972); J. G. Stuart, D. J. Larson, and

N. F. Ramsey, ibid. 22, 2092 (1980).

6J. R. Mowat, Phys. Rev. A 5, 1059 (1972). ^H. Gould, E. Lipworth, and M. C. Weisskopf, Phys. Rev,

188 , 24 (1969). Ki. Becker, IEEE Trans. Instrum. Meas. 29, 297 (1980).

'C. Schwartz, Ann. Phys. (N.Y.) 6, 156 (1959).

'Op. G. H. Sandars, Proc. Phys. Soc. London 92, 857 (1967). "T. Lee, T. P. Das, and R. M. Sternheimer, Phys. Rev. A

n, 1784 (1975). '^. L. Manakov, V. D. Ovsyannikov, and L. P. Rapoport,

Opt. Spektrosk. 38, 424 (1975) [Opt. Spectrocs. (USSR)

38, 242 (1975)].

'^J. D. Feichtner, M. E. Hoover, and M. Mizushima, Phys. Rev. 137, A702 (1965).

"»H. P. Kelly, R. L. Chase, G. R. Daum, and J. J. Chang,

Phys. Rev. A 8, 2777 (1973).

•5U. Kaldor, J. Phys. B 6, 71 (1973).

'*A. R. Edmonds, Angular Momentum in Quantum Mechanics, 2nd ed. (Princeton University, Princeton, N.J., 1974).

'^D. R. Bates and A. Damgaard, Philos. Trans. R. Soc. Lon- don Ser. A 242, 101 (1949).

'*H. Kopfermann, Nuclear Moments (Academic, New York,

1958). "F. G. Major and G. Werth, Phys. Rev. Lett. 30, 1155

(1973); M. D. McGuire, R. Petsch, and G. Werth, Phys. Rev. A 12, 1999 (1978); M. Jardino, M. Desaintfuscien,

R. Barillet, J. Viennet, P. Petit, and C. Audoin, Appl.

Phys. 24, 107 (1981); M. D. McGuire, Bull Am. Phys.

Soc. 26, 615 (1981); D. J. Wineland, W. M. Itano, J. C.

Bergquist, and F. L. Walls, in Proceedings of the 35th Annual Symposium on Frequency Control (U.S. Army Electronics Command, Fort Monmouth, N.J., 1981).

^oR. Blatt and G. Werth, Z. Phys. A 299, 93 (1981); F. L. Walls, D. J. Wineland, and R. E. Drullinger, in Proceedings of the 32nd Annual Symposium on Frequency Control (U.S. Army Electronics Command, Fort Monmouth, N.J., 1978).

2'D. J. Wineland, D. W. Allan, D. J. Glaze, H. W. Hellwig, and S. Jarvis, IEEE Trans. Instrum. Meas. 25, 453

(1976).

TN-89
JOURNAL DE PHYSIQUE

Colloque C8, supplement ecu n°12. Tome 42, deoembre 1981 page C8-307

LASER COOLED, STORED ION EXPERIMENTS AT NBS AND POSSIBLE APPLICATIONS

TO MICROWAVE AND OPTICAL FREQUENCY STANDARDS

D.J. Wineland, J.C. Bergquist, R.E. Drullinger, H. Hennnati, W.M. Itano

and F.L. Walls

Frequenoy and Time Standards Group, Time and Frequency Division, National

Bureau of Standards, Boulder, Colorado 80303, U.S.A.

Abstract . - Research on stored ion frequency standards at the United

States National Bureau of Standards is briefly discussed. We summarize

past work and indicate directions of future research.

Introduction . - The purpose of this paper is to briefly summarize the work at the

National Bureau of Standards (NBS) which has been directed towards realizing a frequency standard based on stored ions. The similar work of other groups [1] is not discussed here. This summary briefly describes past work and indicates

future directions of research at NBS. The stored ion work at NBS was initiated in the Fall of 1977. The goal of this work has been to devise a technique for realizing a frequency standard whose accuracy would be significantly better than that of the Cs atomic beam frequency

standard. With this in mind, the ion storage method [2] was pursued because it can provide long confinement times without the usual perturbations associated

with confinement (e.g. , the wall shift in the H-maser).

The initial work was directed toward realizing laser cooling of ions [3],

since this would provide a way to substantially suppress the frequency shift due

to the second-order Doppler or time-dilation effect. For these initial experiments, Mg ions were stored in a Penning trap. Mg ions were chosen, since the

electronic structure is fairly simple (i.e., like that of neutral alkali atoms)

and required a laser source at 280 nm. The Penning trap was chosen because the residual heating mechanisms (at least for a cloud of ions) are less than for the

rf trap.

In the first experiments [4], the temperature of the ions was monitored

directly by the bolometric technique [5]. Figure 1 shows the Mg ion temperature

vs. time for a fixed laser detuning of about -2 GHz from the 3s ^s, (M, = -h) *

-i J 3p ^^3/2 C^j ~ "^^2) transition frequency. The sensitivity of these temperature measurments was limited by noise in the electronic detection. Therefore, subsequent measurements detected the temperature by observing the fluorescence light

scattered by the ions and measuring the Doppler widths of the optical lines [6].

TN-90
C8-308 JOURNAL DE PHYSIQUE

USER ON

1. Ion temperature vs. time when laser cooling is applied for fixed laser de- tuning of -2 GHz. The ions were initially heated above equilibrium temperature with the laser. Laser cooling was then applied on the 3s ^s, (M, = -H) * 3p ^P3/2 (M, = ~^ 1 2) transition for a fixed time until a temperature

approaching K (< 40 K) was achieved. After the laser is turned off, the ions rethermalize to the ambient temperature.

K- -3GH2-

Mgl RESONANCE

TRANSITIONS

(3s^S, . mr'-''2 )

(3P'P,,

2. Spectra of one Zeeman component of laser cooled ^''.^^.^•'Mg . The room temperature Doppler width of these lines is about 3 GHz. Only the ^''Mg is directly laser cooled. The ^^Mg hyperfine structure has been optically

pumped resulting in the observation of only the (M, = -\, W-. = -^/2)''~* (M, = - ^/j, M^ = -^

1 2) component.

Figure 2 shows the fluorescence light from a "low" power laser which is swept in frequency across the 3s 25 (m = -J5) -^ 3p '^^^,1 (M, = -^/z) optical transitions

of the three naturally occuring Mg isotopes. At the same time, a higher power (approximately 10 times higher power) fixed- frequency laser is tuned to the low

frequency side of the ^^Mg transition to keep the sample cold. (Note that the ^^Mg and ^^Mg isotopes are not directly laser cooled, but are cooled by Coulomb

collisions with the cooled ^"^Mg .)

TN-91
C8-309

Lowest temperatures have so far been obtained for single ions [7] where

cyclotron-magnetron "temperatures" of approximately 0.05 K have been obtained for

a single ^''Mg ion. Figure 3 shows the double-resonance detection of the ^Si (M, = -Jj) ^- 2P3 /2 (Mj = -h) optical transition in a single ^'^Mg ion. Our theore-

-200

•c —

1

/\ / ^^^Nr

\ /^V^'^nV ^^S*'"""

r V V * ^^ ' y^f

t

f y^/ '% 'f /

COz H\ 'l//-"0 K

uj %- i7

liJ J/^ 100 mK 8 ^ f^

q;

* "

1 1 1 r

r- BACKGROUND LEVEL

'

_1 \-Li 1 I 1 I 1 1 I 1 1 I 1 1_ 1 1 1 1 1 1 1 1 1 i 1 1 1 1 1 1 1 i_J -100 100

RELATIVE LASER FREQUENCY (MHz)

200

3. Double-resonance curve of a single ^''Mg ion. On the vertical axis is the fluorescence from a fixed-frequency laser (power approximately 5 pW) tuned

to the (^P3/2, M, = -^/2)-^(^Si , M, = -\) transition. Each point represents

a 10 s integration; the connecting lines are only for clarity. The horizontal axis is the frequency of the low-power (<< 5 pW) laser which is continuously scanned across the (^P3/2. M, = -^)-^(^Si , M, = -%) transition.

The dashed curves are simulations of fluorescence at T = K and 100 mK (no

added noise). The solid curve is experimental data. From these data, we conclude T = 50 ± 30 mK.

tical predictions [8, also Itano, Wayne M. , and Wineland, D. J., to be published]

indicated that lower temperatures should be obtained; possible limitations may be caused by the presence of impurity ions in the trap. Future experimental investigations are planned; a new apparatus with improved fluorescence collection

efficiency is being constructed.

In the Penning trap, ions are unstable with respect to collisions with

background gas; that is, collisions cause the magnetron orbits to increase in size and this leads to ion loss. This process can be reversed by a technique

which is formally equivalent to the usual laser cooling [4,6, also Itano, Wayne

M. , and Wineland, D. J., to be published]; specifically, the magnetron energy is increased and the orbit size reduced by laser scattering by spatially tailoring

the laser beam. The result is that infinite confinement times are in principle

possible in the Penning trap.

TN-92
C8-310 JOURNAL DE PHYSIQUE

The ground-state structure of Mg ions has been measured by double-resonance

schemes where changes in the laser fluorescence can be used to monitor ground- state rf and microwave transitions [9,10]. This has led to a measurement of the ground-state hyperfine constant (A = -596.254 376(54) MHz) and Qt/Qi

(= 9.299 484(75) x 10-^) in ^^Mg"^. For this ion, the derivative of the ground- state (M., , Mj) = (-^/2, h) to (,-h, h) transition with respect to magnetic field

B goes to zero at B s 1.24 T. The corresponding resonance (v = 292 MHz) was ob- served at this field with linewidths as small as 0.012 Hz (Q = 2.4 x 10i°) by im- plementing the Ramsey interference method with two coherent rf pulses separated in time by 41.4 s (see figure 4). We expect that such narrow resonance lines (and even narrower lines) can be observed in other ions (e.g. , Hg ) with higher ground- state transition frequencies. This could then yield extremely high Q (> 10^^) in microwave transitions which would be valuable for frequency standard applications.

>»

1 —1— ' 1 1

o

irt ^„^ ^'-"N. c Ci^^Sn y^

0) / \ /

c _ / \ / -

CJ J Vo.OI2Hz-/ o\ c 7 \ /^

(U O / \ / o " / \ /

~

(A y °Vc/ M) o

3

Ll

1

291 996 251.899 Hz

1 1 1 1.88 1.89 1.90 1.91 1.92 RF Frequency - 291 996 250 ( H z

)

4. Rf resonance curve for the (m^ , m,) = (-^Z,, h) to (-^, h) ground-state

hyperfine transition in ^^Mg . Each circle represents the average of four measurements (total detection fluorescence integration time of 16 s). The oscillatory lineshape results from the use of the Ramsey method to drive the transition. Two coherent rf pulses of duration t = 1.02 s separated by

T = 41.4 s were applied. The vertical arrow marks the central minimum,

which corresponds to the resonance frequency.

In some double-resonance schemes, it is possible to scatter many photons for each microwave or optical "clock" photon absorbed. (In Ref. 10, a factor of about 10^ was achieved.) This "quantum multiplication" should allow the signalto-noise ratio in double-resonance detection schemes to be limited only by the statistical noise in the number of ions that have made the "clock" transition

[9,11]. This will be extremely important for frequency standards based on ions where the number of ions is necessarily rather small.

TN-93
C8-31

Because of the above results, we have initiated work at NBS to realize

microwave [11] and optical [11,12] frequency standards based on Hg ions stored

in a Penning trap. The proposed microwave frequency standard is based on the (F, Mr) = (1,1) <->

(2,1) ground-state hyperfine transition in ^°^Hg , which is field-independent to first order at B = 0.534 T, with frequency = 25.9 GHz (see figure 5). If B can

be controlled to slightly better than 0.1 ppm over the ion cloud, the fractional

frequency shift can be kept below 10-^^. (At the "field-independent" point

Au/uq = (AB/Bo)^/6). The velocity in the magnetron motion will give a secondorder Doppler shift; it should be controllable to 10-^^ (see below). All other

systematic shifts, such as those due to collisions, the trap electric fields, or thermal radiation appear to be less than 10-^^ [11]. It should be possible to observe the transition with a Q of 2.6 x 10^^ or better, by using optical pumping

and detection techniques similar to those demonstrated with ^^Mg . The accuracy

of this standard could be better than 10-^^.

1 1 1 1 1 ___

20 - ^ = 1 _—

' 11,01

t .1,1,

' z

'

15 20, Hg*

10 - -

rsl X

5 - ~

00c

LlJ 6 __________,^_

^

10 ;;i-^=r=rr: (2,11

1 1 1

-

IS

1 1 —

Bo (T)

5. Ground-state hyperfine energy levels of ^°^Hg vs. magnetic field. States

are designated by the (F, Mr) representation. Three transitions are indicated at the fields where the transition frequencies are independent of magnetic field to first order.

The proposed optical frequency standard [11,12] is based on the two-photonallowed 5d'° 6s 2$, ^ 5d^ 6s2 ^0^/2 ^Q transition, which has a natural Q of 7.4 X 10^^. The first-order Doppler effect can be elimated by driving the transiTN-94
C8-312 JOURNAL DE PHYSIQUE

tion with counter-propagating 563.2 nm laser beams. Hyperfine-Zeeman components,

whose magnetic field derivatives vanish at particular values of B , exist in ^^^Hg and ^°^Hg . The two-photon transition can be detected with high efficiency

by using the 194.2 nm fluorescence intensity as a probe of the ground-state

population. Taking full advantage of the high Q transition would require a laser

with linewidth less than 1 Hz, which does not exist at present. However, linewidths < 100 Hz appear feasible and could be used for initial experiments. If the laser linewidth is less than the natural linewidth, then the ac Stark shift

is about 2 x 10-^^ near saturation. All other systematic shifts appear to be less than 10-^^. We note that the ac Stark shift can be made negligibly small by

driving the single photon 2$ -* 205,2 quadrupole transition. In this case, it will be desirable to use a single ion that can be confined to approximately real- ize the Dicke criterion in order to suppress first-order Doppler effects.

The method currently being investigated for generating the required 194.2 nm radiation for laser cooling and optical detection is sum-frequency mixing in a KB5 crystal of the output of a 792 nm single-mode cw ring dye laser and the second harmonic, generated in an ADP crystal, of the output of a 514 nm stabilized, single-mode cw Ar laser. For the microwave frequency standard, it is especially desirable to use the largest possible number of ions in order to increase signal to noise. Unfortunately, as the number of ions is increased, the second-order Doppler shift due

to magnetron rotation also increases, which will, therefore, limit the number of ions to about 10^ or less for 10-^^ accuracy [11]. (A similar problem exists for the rf trap due to the kinetic energy in the micromotion [13]). Therefore, studies

will also be devoted to methods of controlling the density and shape of the ion clouds.We gratefully acknowledge the support of the United States Air Force Office

of Scientific Research and the Office of Naval Research.

References .

[1] See for example the following references and other references therein:

Major, F. G. , and Werth, G. , Phys. Rev. 30 (1973) 1115; Dehmelt, H. , Proc. 35th Ann. Symp. Freq. Control, Philadelphia, Pa., June 1981; McGuire, M.

D. , Petsch, R. , and Werth, G. , Phys. Rev. A. 17 (1978) 1999; McGuire, M.

D. , Bull. Am. Phys. Soc. 26 (1981) 615; Jardino, M. , Desaintfuscien, M. , Barillet, R. , Viennet, J., Petit, P., and Audoin, C. , Appl. Phys. 24 (1981) 1. Neuhauser, W. , Hohenstatt, M. , Toschek, P. E. , and Dehmelt, H.

,

Phys. Rev. A22 (1980) 1137; Javanainen, J., and Stenholm, S. , Appl. Phys. 24 (1981) 151.

[2] Dehmelt, H. G. , Advances At. Mol. Phys. 3 (1967) 53 and 5 (1969) 109.

[3] Wineland, D. J., and Dehmelt, H. G. , Bull. Am. Phys. Soc. 20 (1975) 637; Hansch, T. W. , and Schawlow, A. L. , Opt. Commun. 13 (1975) 68.

[4] Wineland, D. J., Drullinger, R. E. , and Walls, F. L. , Phys. Rev. Lett. 40 (1978) 1639.

TN-95
C8-313

[5] Dehmelt, H. , and Walls, F. , Phys. Rev. Lett. 21 (1968) 127. Wineland, D. J. , and Dehmelt, H. G. , J. Appl . Phys. 46 (1975) 919.

[6] Drullinger, R. E. , Wineland, D. J., Bergquist, J. C. , Appl. Phys. 22 (1980) 365.

[7] Wineland, D. J. , and Itano, Wayne M. , Phys. Lett. 82A (1981) 75.

[8] Wineland, D. J. , and Itano, Wayne M. , Phys. Rev. A20 (1979) 1521.

[9] Wineland, 0. J., Bergquist, J. C. , Itano, Wayne M. , and Drullinger, R. E.

,

Opt. Lett. 5 (1980) 245.

[10] Itano, Wayne M. , and Wineland, D. J., Phys. Rev. A24, (1981) 1364.

[11] Wineland, D. J., Itano, Wayne M. , Bergquist, J. C. , and Walls, F. L. , Proc. 35th Ann. Symp. Freq. Control, Philadelphia, Pa., June, 1981.

[12] Bender, P. L. , Hall, J. L. , Garstang, R. H. , Pichanick, F. M. J., Smith,

W. W. , Barger, R. L. , and West, J. B. , Bull. Am. Phys. Soc. 21 (1976) 599.

[13] Wineland, D. J., Proc. Conf. on Prec. Meas. and Fundamental Constants,

Gaithersburg, Md. June, 1981.

TN-96
Precision Measurement and Fundamental Constants II, B. N. Taylor and W. D. Phillips, Eds.,

Natl. Bur. Stand. (U.S.), Spec. Publ. 617 (1984).

Spectroscopy of Stored Ions

D. J. Wineland

Frequency and Time Standards Group, Time and Frequency Division

National Bureau of Standards, Boulder, CO 80303

The benign environment and long confinement times obtained with ion storage techniques have led

to some unique experiments in the area of precision measurements and fundamental constants. This is perhaps epitomized by the single electron g factor measurements at the University of Washington in which a precision of 4 parts in 10" has been attained. Now, use of lasers to cool stored ions has allowed

the experimentalist to approach the goal of unperturbed atomic ions nearly at rest; most recently, spec- troscopy has been performed on single "cold" trapped ions. Stored ion experiments in the area of precision measurements and fundamental constants will be briefly reviewed. These include experiments on

e"/e* g factors, mass spectroscopy, lifetimes, and atomic spectroscopy. The intent is to emphasize the unique environment provided by ion storage techniques for these measurements. Key words: atomic ion spectroscopy; electron g factor; fundamental constants; ion storage; mass spec- troscopy; precision measurements.

1 . Introduction

The purpose of this paper is twofold. The first purpose is to outline "trapped ion" experiments of the last decade

which have contributed to the area of precision measure- ments and fundamental constants. The second purpose is to emphasize why the stored ion '

' environment' ' is particularly well suited to the area of precision measurements.

The main advantage of the stored ion technique is that

the ideal of an unperturbed species at rest in space is ap- proached to a high degree. Specifically, charged particles such as electrons and atomic ions can be stored for long periods of time (essentially indefinitely) without the

usual perturbations associated with confinement (for ex- ample the perturbations due to collisions with walls or

buffer gasses in a traditional optical pumping experiment). These unique properties were exploited most no- tably by Dehmelt and co-workers [1] before 1970; since

then, that group and others have continued to extend these

techniques with dramatic results. Unfortunately, there is a price to be paid for this property of long storage times with small perturbations—the

number of particles that can be stored is typically small

(slO*' for a "trap" with centimeter dimensions); the

resulting low densities are governed by the competition

between space charge repulsion and the confining elec- tromagnetic forces obtained under normal laboratory conditions. These low numbers, of course, require very sensitive detection techniques and preclude many types of

experiments—for example spectroscopic experiments on complex molecular ions where only a small fraction of the

ions is in a given state. Although some effort has been

made to increase the number of ions by space charge neutralization [2, 3] and trap "arrays" [4], these methods in- troduce other experimental complications. In spite of the

low numbers obtained, sensitive techniques have been developed to detect simple species such as electrons and

atomic ions so that single electrons [5, 6] and ions [7, 8] have been observed.

Since the intent is to discuss experiments relating to precision measurements and fundamental constants.

many interesting experiments using stored ion techniques are not discussed in this paper. The reader is re- ferred elsewhere to experiments on, for example, photodetachment [9, 10], chemical reactions [11], electron-ion

recombination [12], charge transfer [13, 14], and nonneutral plasma studies [15]. Also, more general reviews

are available [1, 16, 17]. In Sec. 2, methods of trapping are briefly reviewed. In

Sees. 3-6, experiments on electrons/positrons, mass

spectroscopy, atomic spectroscopy, and lifetime measure- ments are described with emphasis on the ion's environment. In the last section, speculations about other fu- ture experiments are made.

2. Ion Trapping Methods

Four types of traps have been useful: the rf (or Paul)

trap, the Penning trap, and electrostatic and magnetostatic traps. These traps and their relevant properties are briefly described here. More detailed descriptions are given elsewhere [1, 16].

2.1 Tlie rf Quadrupole Trap

The ideal rf or Paul [18] trap uses hyperbolic elec- trodes in a vacuum apparatus as shown in Fig. 1. These

electrodes are symmetric about the z axis, so we can

describe the potential in cylindrical coordinates. If an al- ternating voltage of frequency O is applied between the

endcaps and ring electrode, then the instantaneous potential inside the trap is given by (following the notation

of Ref. [1]):

(})(r,2) = A(r2 - 2^2), A = Ao cos ^ t. (1) An ion experiences an rf electric field such that its motion (the "micromotion") is 180° out of phase with

respect to the electric force. Because the electric field is inhomogeneous, the force averaged over one period (7" =

2ir/n) of the micromotion is in a direction of weaker field amplitude (independent of the sign of the charge), i.e.,

83

TN-97
(a) rf or Penning trap

"end caps

B = B z (required for Penning trapl

<>(r, z) = A(r2-2z2) A = - 2^9, 2

electrode surfaces generated by 6(r, z) = const.

(b) Orbitron trap

= A(r2-2z2) + Bln(y)

Figure 1. Electrode configuration for rf. Penning, and orbi- tron ion traps. All electrode surfaces are figures of revolution about the axial (z) direction, hi a practical orbitron trap, it is desirable to make the center electrode as thin as possible to reduce ion loss from collisions.

towards the center of the trap. For fl sufficiently high,

this restoring force gives rise to a pseudopotential

'ifir,z) =

MCl^

[(r)2 + 4(2)2]

where r and z are the positions of the ion averaged over

T, e is the ion charge, and M is the ion mass. The result- ing "secular" motion is

Z = Zr, cos (cJzi + 4>z) , - _ - „- 2V2eAo

X = Xq cos (cOr^ + ct)j-) (^i^ = 2(0^ -

y ^ yo cos (air< + <t>i/) In more detail, we have for the z motion:

MCI

20 1 + cos n t cos coji, a

where we require fl/coj » 1. For ease of comparison with the Penning trap, it will be useful to consider the case of a spherical pseudopotential well in the rf trap. This is accompUshed by

simultaneously applying a static potential between the

ring and endcaps so that

<i> = iUo + Ao cos at)ir^ - 222).

If we choose

Ur

^(r, 2) =

MCl^

2eAi

Mil^

Ao, then

(r)2 + (2)2 (2)

In this case:

2 = 2o

2(1), 1 +

-"Y"

cos Clt cos (ii,t

Xo

yo

1 + —^ cos Clt cos (jXrt , (3)

where 0>r 2eAo

MCI

Approximate experimental values for stored atomic ions are such that ro — 1 cm, A^ is a few hundred volts/cm^,

n/2TT is a few MHz, and Cl/io, = 10. If the secular motion is "cold" then we expect the

maximum density of ions to be given when the force from

the space charge potential 4), cancels that due to the

trap. In this case, for the pseudo-potential given by Eq.

(2), the ion cloud is a uniformly charged sphere of charge

density p with internal potential

<\>, = (2/3) irp (r2 + z^)

.

Therefore for a well with 2eAi/MCl^ = 10 V/cm^, the

condition <i)i - - ^(r, z ) gives a density n = 3.3 x

lOVcm^. In practice, attainable densities are approximately 1 to 2 orders of magnitude lower due to "rf

heating" —a process which couples kinetic energy from

the micromotion into the secular motion. One way this happens is through collisions with background gas. However, at high vacuum other coupling mechanisms can

occur partly due to the presence of impurity ion s or imperfections in the trap electrodes. This process can

give kinetic energies of ions of as much as electron volts and shortened storage times of less than 1 s. If one uses

a light buffer gas such as He to provide viscous damping

[1] then storage times of many days can be achieved

[19, 20]. However, this increases perturbations due to

collisions; therefore, for experiments on large numbers of ions in an rf trap, there appears to be a practical tradeoff between the temperature of the ions and possible per- turbations due to collisions wdth a coohng buffer gas. However, this is not a fundamental limitation and one hopes it will be solved in the future. We note that in principle, densities could be increased

by increasing the rf field strengths; in practice, higher

densities (close to lO^/cm^ [21]) have been achieved by de- creasing the electrode dimensions. This, of course, may

not increase the ion number and it also may increase the

rf heating because the field imperfections may be rela- tively larger.

It is interesting to note that the methods proposed for trapping of neutral particles in laser beams [22] are very

analogous to the rf trapping of charged particles. In the

laser beam case, the outer atomic electron experiences

net forces in the (inhomogeneous) laser field. But since

the electron is bound to the nucleus, dispersive effects

84

TN-98
occur. If the laser is tuned above the frequency of an

electronic transition, similar to an ion in a rf trap, the

atom is forced to regions of low field intensity. If the

laser is tuned below resonance, the atom is forced to re- gions of high field intensity.

2.2 The Penning Trap

The Penning trap [23] uses the same electrode configuration as the I'f trap (Fig. 1) but now A in Eq. (1) is a constant L^'o such that the charged species see a static potential well along the z axis. We have 4>t =

Uoir^ - 2z^). This causes a repulsive potential in the X -y plane which can be overcome by superimposing a

static magnetic field along z{B ^ Bqz). For a single ion

in the trap, the equations of motion are [1, 24]

z + oiiz = 0,

co| =

4e[7(

M

AeVc

M{ri + 2zh '

T = —0)1 r - ioicT

,

eBo

where

T — X + iy.

Therefore

z = Zocosoi^t and r - r^e''"^' + r,„e~"°"'' where

CJr I 1 J+

ICOm I 2 Some useful expressions are:

(1)^ + (3iz/2 = (jOcO) (o) = COm or (Oc),

(iic^m — U)|/2.

^cu, _ [(co,/2)2-a)2/2]i^^

(4)

(5)

(6a)

(6b)

(6c)

Typical operating conditions in a Penning trap are such

that for electrode spacings about 1 cm, Vq is a few volts and Bo is larger than a few tenths of a tesla. The re- quired large magnetic field can either be an advantage or disadvantage, depending on the experiment. It should be

noted that the magnetron motion (the r„,e

"""' term) is in an unstable equilibrium in the trap. For example, if collisions with background neutrals occur, the ions will diffuse out of the trap. (When the magnetron velocity is much less than the velocity of the neutrals, r,„ random

walks in the x-y plane with step size ~r,..) This is a potential disadvantage when compared to the rf trap for which all three degrees of freedom {x, y, z) are in stable

equilibrium. In practice however, this is not a limitation

because ions can be stored for days in a room temperature apparatus [25] and electrons for weeks [5] in an 80 K

apparatus. Moreover, the technique of sideband or radiation pressure cooling [6, 21, 25, 26] can reverse this diffu- sion process.

If the voltage V^ applied to the electrodes becomes too

high, then the radial electric field is high enough to over- come the ev X B/c magnetic force and the ions strike the

ring electrode in exponentially increasing orbits (argument of square root in Eq. (5) becomes negative). For

singly ionized atoms, the voltage where this occurs is

given by V, - 1200 fi^rg + 2 2^/M where V, is in volts, Bo in tesla, M in u (atomic mass units) and dimensions in cm. This same mechanism limits the densities

achievable in the Penning trap since space charge also gives radial electric fields. If the axial and cyclotron temperatures are low enough, then we can approximate the

equilibrium distribution of the ion cloud [27] by a uniformly charged ellipsoid which has an (internal) potential

of:

<}),(r) = (2/3) irp {ar^ + hz^)

where a and h are constants. The r motion of an individ- ual ion is now given by Eq. (4) with

12 i-nepa

3M

1/2

(7)

where co^ is the axial frequency for a single ion in the

trap. For a spherical distribution, a = b =1; and the

maximum density allowable (argument of square root term kept positive) is given by (using <^i{z) - -<t>7'(2 )):

e <

Sitcom

or < 2.7 X lO^Bi/M

where So is in tesla and M in u. For 5o = 1 T, M =

100 u, n < 2.7 X lO'^/cm^ which is similar to the rf trap

case. In experiments at the National Bureau of Standards (NBS), densities within a factor of about 3 of the

theoretical maximum have been observed as indicated by

the space charge shifted magnetron rotation frequency.

2.3 Electrostatic Traps

The orbitron trap [28] has the advantage of simplicity

and although lifetimes are observed to be relatively

short, precision lifetime and spectroscopic measurements

have been made by Prior and his colleagues at Berkeley

[29, 30]. The orbitron trap is purely electrostatic and has axial

symmetry as shown in Fig. lb. A potential which is useful is given by the general form [31]:

<i> = A (r^ - 2z^) + 51n(r/ri),

because a harmonic well is provided along the z axis. The

equations of motion for the x-y plane are not solvable

analytically, but the motion is basically composed of pre- cessing orbits about the z axis [32]. Storage times in such

a trap should be shorter because a single collision with a background neutral is sufficient to cause an ion to collide with the center electrode and be lost. Also the cooling

discussed in Sec. 6 would be more difficult to realize. Nevertheless, with a trap whose electrodes approximate

equipotentials of this function, trapping times of about one second were achieved by Knight [31] at a pressure of

lO"*^ Pa (=10"* Torr). (The center electrode was a 100

fjLm dia. wire.)

An axial magnetic field superimposed along the z axis

has been investigated theoretically [33]; in this case, storage times should increase dramatically. Ions will eventually be lost by diffusing in towards the center elec- trode; therefore, storage times should be comparable to that of the Penning trap. However, the frequency of the

drift motion about the center of the trap vnll depend on

the distance from the z axis; for the Penning trap this

drift (magnetron) frequency is independent of radial posi85 TN-99
tion (neglecting space charge) and may be part of the reason for the slow evolution of the ion cloud under the

influence of the ion-ion collisions. A possible advantage

over the Penning trap is that experiments may be possi- ble at much lower magnetic fields [33].

2.4 Magnetostatic Traps

Magnetic bottles are of course used extensively for plasma confinement and also in ion spectroscopy. The

disadvantage in terms of spectroscopy appears to be that

rather large magnetic field inhomogeneities are required

for trapping, thus causing inhomogeneities and broadening in paramagnetic ion line shapes. The main application

in terms of precision measurements has been the JVIichi- gan e^/e' experiments [34]; these are reviewed by A. Rich at this conference.

3. e /e^ g-2 Measurements in a Penning

Trap

This experiment has been pursued primarily by the

groups of Dehmelt at the University of Washington, and

Graff at Mainz [35-37]. Only the Washington single elec- tron experiment [6] will be discussed here, since its accu- racy significantly exceeds that of the experiments on

clouds of electrons [35-38]. Experiments on clouds of electrons are susceptible to rather large and difficult to measure space charge shifts. For example, the fractional

shift of the cyclotron frequency of one electron due to another electron a distance r away (in a direction perpendicular to the magnetic field) is approximatelv given by

[39]

Acq

0)0

1 m cor r dr where m is the electron mass, co,. is the cyclotron fre- quency, and Er is the electric field at one electron from

the other. For a magnetic field of 1 T, and r = 100 ixm,

Aco/coo =4.1 X 10"^ which gives a shift of about 4 ppm in g-2. In 1971, the Michigan group had measured the

electron g - 2 to 3 ppm [40]; this result and the problems

associated with the space charge shifts led the Washington group to develop an experiment based on single electrons. A complete account of this experiment is given

in Refs. [6, 41]. A simpler account is given in Ref. [42]. Briefly, a single electron can be detected [5, 42, 43] by

monitoring the currents induced in the electrodes by the

axial (z) oscillation at frequency coj. The harmonically

bound electron is equivalent to a series L-C circuit which

shunts the electrodes [5]; this L-C oscillator can be phase

locked to an external oscillator by feeding back on the

trap voltage Vq [6]. It is useful to think of the single

electron bound to the Penning trap as a one electron

pseudo atom whose "nucleus" is the earth (to which the

Penning trap is fixed), thus the name "geonium" [6]. The quantum mechanical solution for the energy levels of

this "atom" are given by:

E/h = mvs + in + l/2)v'c + ik + l/2)v^ - (q + l/2)v,„

where m is the spin quantum number (m = ±1/2) and n,

k, and q are the cyclotron, axial and magnetron quantum

numbers in, k, q = 0, 1, 2 . . .). (The magnetron energy

is mostly potential energy and is negative.)

In the g-2 and eVe" mass ratio experiments it is necessary to detect the occurrence of spin flip and cyclo- tron transitions. This can be accomplished by superimposing a weak magnetic bottle centered on the trap such

that the field is [6]: B = iBo + ^iz^ - r 2/2)2 - ^zyy - ^zxx . The spin, cyclotron, and magnetron magnetic moments

interact with this magnetic bottle to shift the axial fre- quency by an amount

8v2 = [m + n -f 1/2 -I- ivm/vc)q]b

8 = (xg ^/i2T^mv,)

For V, = 60 MHz and p = 0.012 T/cm^, 8 - 1 Hz, therefore for example, spin flip transitions would show up as 1 Hz changes in Vz or as an accompanying step increase in feedback voltage to the locked axial resonance [6]. For reproducibility, it was desirable to have the elec- tron at the center of the trap. This could be accomplished

by sideband excitation [6, 26, 44, 45] as follows: vdth the

(damped) axial motion driven by an inhomogeneous rf

field of frequency v^ -i- v,,, , the electron sees a sideband

at Vz because of the magnetron motion through the rf

field. Therefore the electron absorbs quanta of energy

h (vj + v„, ) and reradiates (by the damping) quanta of energy hv^. The deficit, hv„j, goes into the magnetron

energy which shrinks the magnetron orbit. In principle,

this technique allows for infinite storage times.

The electron g-2 has now been measured with an ac- curacy of 0.04 ppm [46] and the positron g - 2 to 0.05

ppm [47]. The details of these remarkable experiments

are described in Ref. [6]; rather than give a complete

description of the experiments, only the experimental en- vironment is described here. From Ref. [6], the cyclotron

degree of freedom was radiatively thermalized to 4 K. The axial temperature was somewhat higher. Pressure was estimated to be as low as 10"^^ Pa so that collisions with background gas could be neglected. At temperatures near 4 K, axial excursions of 0.1 mm and cyclotron

orbit sizes of —60 nm would be obtained. During cyclo- tron excitation to n — 15, this gave a corresponding relativistic shift of only -6 x 10"^. Using sideband excitation, r,„ — 14 |jLm. From Eq. (6b), there is, of course a large systematic shift (~6 x 10""* in gr - 2) between the

observed and free space cyclotron frequencies. Due to imperfections in the trap, a direct measurement of v,„ (by

magnetron excitation) and the calculated value from Eq.

(6c) can disagree slightly. However, it can be shown that

the error in v,„ is much less [6, 48] than the above differ- ence which was measured to be 10"^ v,„. Thus the error

in g - 2 was negligible. Errors in the present experiments are estimated from field jitter and drift and dependence of g -2 on applied spin flip power [47]. Ulti- mately, the experiment will be limited by systematic

shifts associated with the superimposed magnetic bottle. To eliminate this problem, new ways have been proposed

to detect the spin flips [6, 49]. For example, it might be

possible to use a switchable magnetic bottle so that while

the cyclotron and g-2 resonances are driven, the bottle would be off and transitions would be detected by turn- ing the bottle on [50]. Even without a magnetic bottle,

the axial motion is still hnked to the cyclotron degees of freedom via the relativistic mass shift [6, 51]. This cou- phng is about an order of magnitude smaller than for the

magnetic bottle; the required increase in axial well depth

stability might be provided with a reference electron in an adjacent trap whose axial frequency is locked to a particular value [52].

86 TN-100
4. Mass Spectroscopy

With the long storage times and high vacuums in the

traps, one could expect extremely narrow motional reso- nances. In the Washington electron experiments cyclotron

excitation resonances reproducible to a few parts in 10^ have been observed [53], and in principle the resolution

should only be limited by effects due to inhomogeneous or time varying fields. Using the methods described in Sec.

3, the electron/positron mass ratio has recently been

measured to about 0.1 ppm [63]. This very important ex- periment represents an improvement by more than two

orders of magnitude over previous experiments. In addition, the cyclotron resonances of both atomic ions and

electrons can be measured in the same magnetic field and

in the same apparatus. This rather unique feature has led

to experiments designed to make a direct measurement of

the electron/proton mass ratio in a Penning trap.

The three current experiments using this method

[53-55] all basically measure the (shifted) cyclotron fre- quencies of electrons and protons in the same magnetic

field. In the experiment of Gartner and Klempt [54], resonances were detected by electron/proton loss; the

largest uncertainty was due to a space charge shift in the

proton cyclotron resonance. In the Van Dyck and

Schwinberg experiment, resonances were detected in several (non-destructive) ways [53]; the limit here was

the uncertainty in the relative positions of the electrons

and protons in the trap and also possibly ion number

(space charge) dependent shifts. In the experiment of

Graff, Kalinowsky, and Traut, resonances were detected

by an energy dependent time of flight technique [55]. The

experiment of Van Dyck and Schwinberg has the smallest

uncertainty (0.14 ppm) and substantial improvements can be expected.

With regard to space charge shifts, it is worth noting

that in an ideal trap described by a quadratic electric potential and uniform magnetic field, if only one species is trapped and if the cyclotron motion is excited by a uni- form electric field, then no space charge shifts can occur because only the center of mass mode is excited [26, 56]. Unfortunately, deviations from the ideal trap fields can

occur, impurity ions can be present, and the excitation

usually occurs from non-uniform fields. All of these

things can lead to (partial) excitation of the internal

modes which then results in space charge shifts. These

systematic shifts increase with the mass of the ion and

therefore are much more important for protons and

atomic ions than for electrons. After the initial single electron work, it became ap- parent that it should be possible to observe single ion cy- clotron resonances by observing the induced currents in the ring electrode if it is divided in half [57] or into quadrants [53]; the key advantage here is that space charge

shifts are totally absent. Since the cyclotron frequency

of protons in a field of 4 T is about 60 MHz, electronic

detection should be similar to the detection of the axial resonance of single electrons. Other, related mass spectroscopy experiments should

be possible. First, using techniques as those referred to above, mass ratios of various atomic (molecular) ions

could be measured. It should also be possible to observe

the cyclotron resonance of single atomic ions by laser techniques. For example, if a single cold Be^ ion is local- ized at the center of the trap [8] with a laser beam whose

diameter is approximately 3 \xm, then it should be possi- ble to observe cyclotron excitation which increases the

orbit radius to about 3 |xm by the decrease in fluores- cence scattering. This corresponds to a temperature

(ksT = (1/2) M {(xic)'ri) of 14 K in a field of 5 T and a rel- ativistic shift of 1.4 x 10 ^^^ rpj^g ^^^j^] ^j^^ magnetron

frequencies could be measured in a similar fashion. If the electron cyclotron frequency were measured in the same magnetic field, the accuracy in Mg/Mp from this in- direct experiment would be limited by the accuracy of Mse/Mp which is about 0.04 ppm [58]. In preliminary experiments at NBS, the axial reso- nance (vj = 200 kHz) of a small cloud (<10) of ^^ Mg"^ ions

has been measured in this way; however significant

broadening (a few parts in 10'*) is observed—presumably

due to impurity ions. For an initial experiment on Be"^, the electron cyclotron frequency can be measured in- directly by measuring the spin flip frequency on the same

ion; since gj (Be"^) may be calculable to as good as a few

parts in 10^ [59], the error in Mg/Mp could still be lim- ited by the error in Mse/Mp. Ultimately when M^/M pis

known better by other means, such measurements should

lead to better experimental determinations of gj (Be*)

and AfBe/il^pWe conclude by remarking that the extreme confinement possible with the stored ion techniques considerably relaxes the requirements on magnetic field homogeneity in these experiments.

5. Radiative Lifetime IVIeasurements

Although perhaps not "high precision" in an absolute

sense, the radiative lifetime measurements of Prior and

colleagues at Berkeley should be included in this review

because these measurements, made on simple atomic

ions, can be compared with various theories. Lifetimes of

2^50 Li+ (t - 503 jxs) in a Penning trap [60], 2s He* (t -

1.92 ms) in an orbitron [29], and 2^51 Li* (t - 58.6 s) in an rf trap [61] have been measured. One should also in- clude the 5^Z)3/2 Ba* (t — 17.5 s) measurement in an rf trap by Schneider and Werth [62] and the d^D^,^ Ba* (t —

47 s) measurement in an rf trap by Plumelle et al. [19]. The key advantage of the traps in these measurements is the very long storage times and the benign environment

which minimizes the effects of other relaxations. For the

very long lifetimes, perhaps no other method is available.

6. Atomic Spectroscopy

As discussed in Sec. 1, the small number of ions in- volved makes it desirable to study simple atomic ions,

but this does not preclude the study of simple molecular

ions [1, 64]. As we will see, the potential accuracy and

precision for atomic spectroscopy using ion storage tech- niques is quite remarkable. This was apparent from the

early experiments on the ^He* ground state hyperfine

structure [1, 65] where linewidths of about 10 Hz at 8.7 GHz were obtained. Since that time the 2s ^He* hyperfine structure (hfs) has been measured in an orbitron

trap by Prior and Wang [66]. The importance of this measurement is that a comparison between the Is and 2s

hfs suppresses the effects of nuclear structure corrections and allows an accurate check of the QED corrections; such experimental/theoretical comparisons are most meaningful for hydrogenic ions. A significant interest in stored atomic ion spectroscopy

stems from the desire to make a frequency standard.

Stimulated by the He* experiments, various proposals

[65, 67-78] have been made for microwave and optical

stored ion frequency standards. For a microwave stand87 TN-101
ard, considerable attention has been given to the ^^Hg"^

ion because (1) its ground state hyperfine frequency (—40

GHz) may be the largest of any conveniently usable ion,

(2) its large mass gives a relatively small second order

Doppler shift at a given temperature (~2 x 10"^^ at 300

K), and (3) a ^''^Hg^ lamp source (194 nm) is available for optical pumping. Starting with the work of Major and

Werth [69], groups at Mainz [74], Orsay [76], and at least one commercial company [74] have sought to develop a frequency standard based on ^^Hg"^ ions stored in an rf trap. This work has been developed to a fairly high

level; the group at Orsay [76] has made a working standard whose stability compares favorably with that of a commercial cesium beam frequency standard.

However, the full potential of the stored ion technique

has yet to be realized. Historically, it appears that this has been due to two problems. (1) Because it has been

difficult to cool the ions, second order Doppler or time di- lation shifts can be sizeable (e.g.: —10"^' in Ref. [74], ~4

X 10"^° in Ref. [65]). (2) Signal-to-noise ratio has been

poor due to the small number of stored ions. In the past two or three years, both of these problems

have been addressed. In 1978, groups at NBS and

Heidelberg demonstrated [21,25] that radiation pressure

from lasers [44, 70, 79] could be used to cool ions to temperatures <0.1 K, thereby reducing the second-order

Doppler shift by more than three orders of magnitude

below the room temperature case. As discussed below,

the reduction of kinetic energy is most favorable for very

small numbers of ions (down to one ion), so that there is a trade-off between the maximum number of ions we can

use and the minimum second-order Doppler shift that can be achieved.

With regard to signal-to-noise ratio, in certain opticalpumping, double-resonance experiments, it is possible to scatter many optical photons from each ion for each mi- crowave or optical "clock" photon absorbed [70,

73, 79-81]. This can allow one to make up for losses in detection efficiency due to small solid angle, small quantum efficiency in the photon detector, etc., so that the

transition probability for each ion can be measured with

U7iity detection efficiency. This means that the signal-tonoise ratio need be limited only by the statistical fluctua- tions in the number of ions that have made the transition

[78, 81]. Therefore the signal-to-noise ratio can be maximized for the given number of ions. The ability to sensi- tively detect the fluorescence from ions is emphasized by

experiments on single ions [7, 8] where it is even possible

to make photographs as shown in Fig. 2. More recently, the narrow linewidths anticipated for the stored ion technique have been observed; a resonance

linewidth of about 0.012 Hz at 292 MHz has been ob- served [82] for the (m, , mj) = (-3/2, 1/2) ^-> (-1/2. 1/2)

hyperfine transition of ^^Mg"^ at a magnetic field of about

1.24 T where the first derivative of the transition fre- quency with respect to magnetic field is zero. (The Ramsey interference method was implemented by applying

two rf pulses of 1 s duration separated by 41 s.) These narrow linewidths should be preserved with hyperfine

transitions of higher frequency, such as in Hg^, but, of course, more attention must be paid to field homogeneity

and stability. In any case, the above results allow one to contemplate microwave frequency standards with Q >

10^^ and inaccuracies <10"^^ [78]. Of course, in addition

to the interest in frequency standards, very high precision hyperfine and optical spectroscopy on Hg^ [74], Ba^

[20, 82], Mg^ [25, 82, 83], and other alkali-like ions [75] is possible.

Figure 2. Photograph of fluorescence from single Ba'^ ion in an rf trap fT = 10-36 mK; from Ref. [86]). The schematic di- agram of the rf trap from same view. Inner diameter of ring is 0.7 mm.

The possibilities for an optical frequency standard are even more dramatic [70, 71, 77, 78, 84, 85]. For a given in- teraction time, the Q of a transition will scale with the

frequency. Therefore, in principle, an optical frequency

standard would have clear advantages over a microwave

frequency standard. The increased Q would allow one to work with smaller numbers, even down to one ion

[70,77,78], and therefore obtain the "ultimate" atomic

frequency standard. Example optical frequency standard

candidates are the ^Pq ^- ^So transition in Tl^, Al"^ or Ga"^

[70], the ^P ^ 15 transition in 5^ [85], the D ^P ^ S

Raman transitions in Ba^ [77], the S ^ D two photon

transition in Hg^ [71, 78], and the ^P3/2 <— ^iy2 transition

in Pb^ [72]. All have Q > W\ Unfortunately, before the

full potential of an ion-storage optical frequency standard can be realized, tunable lasers with suitable spectral purity must become available. This problem may be nearing

solution [84]. Also, if an optical frequency standard is to provide time, the phase of the radiation must be measured. This appears to present a much more formidable

problem [84, 85]. Therefore it is expedient to continue to pursue a microwave frequency standard where both of these problems have been solved. Rather than concentrate on the details of the various

experiments it is useful to examine the environment of the stored ions in order to identify possible systematic

effects. We will be primarily concerned with Stark and

second order Doppler shifts. Usually only second order

Stark shifts will be important; therefore we will be in- terested only in <E^>. As an example, for the strongly

polarizable cesium (neutral) atom, the ground state hyperfine shift [88] is equal to: 8v/v = -2.5 x

10"i^ E^ (E

in volts/cm); therefore electric fields of volts/cm become

88

TN-102
important in very high accuracy experiments [90]. Magnetic field shifts due to instabilities and inhomogeneities

are of course important in the Penning trap but can be

made negligible in maqy experiments. (See for example

Refs. [78,82].) Other effects such as electric quadrupole

frequency shifts could be important in certain experiments [70, 78]. For single ions, laser cooling has already achieved temperatures between 10 mK and 100 mK [7, 8]. Theoretically, when the motional oscillation frequencies n,(coj and

o)^ for the rf trap and oj^, Wc, and 00^ for the Penning

trap) are less than the natural linewidth (Av = 7/277) of

the optical cooling transition, then the limiting "temperature" (governed by photon recoil) in each degree of freedom is given by hgT ^ ty/2 [21,25,44,87]. (For a

single ion, the precise minimum temperature depends on

the angle of incidence of the laser beam and on the spatial distribution of recoil photons [87].) For strongly al- lowed transitions as in Ba^ or Mg"^, this limiting temperature is about 1 mK. For more weakly allowed transitions the temperature is correspondingly less but other

limits such as thermodynamic limits can come into play

[44]. When the condition O, » 7 is fulfilled then the

limiting energy [44, 86] is given by Ei = %VLi (<n,> + 1/2) where <n,> = 5^/(160,^). Therefore the

limiting kinetic energy is given by Eki — ^n,/4 but the

uncertainty in energy (and therefore second order

Doppler effect) is much less [44]. For simplicity we will assume only the case fl, « y below, however even

better results are potentially obtained for the opposite

condition. For a single ion in an rf trap, when Uo = the

nonthermal micromotion has an average kinetic energy

equal to that of the secular motion [1]; this is also true in the spherical trap. In the Penning trap the kinetic en- ergy in the nonthermal magnetron motion can be much

less than in the cyclotron or axial modes. Therefore, the

minimum second order Doppler shifts are given approximately by:

Ek _ \^

4

ty/Mc^

ty/Mc^

rf trap

Penning trap

For a single ion in an rf trap (assuming the spherical

trap of Eqs. (2) and (3)), <E^> is primarily due to the

oscillating rf fields and is largest for the z motion. A simple calculation gives <E^>, = M Ct^^y/e^ for maximum

laser cooling or <E'^>2 = 2M D^ksT/e^ for a given temperature in the z secular motion. For a single ion in a Penning trap, it is usually possible to make r,„, Vc « z

[8, 87], therefore Stark shifts from the static fields are primarily due to the z motion. We find <E^>2 -

%y M iiif/{2e'^) for maximum laser cooling or <E'^>^ - kgTM ixi^/e'^ at temperature T. In the Penning trap, a

larger effect can be caused by the motional electric field E = V X B/c. We have <E^>m = tyB§/{Mc^) (maximum laser cooling) and <E^>m = ^kgT B^/iMc"^). In Table 1 are shown examples of the second order Doppler

shift and <E^> for single ions in rf and Penning traps.

For clouds of identical ions, we first consider the electric fields due to collisions between ions. For the rf trap, we neglect the energy in the micromotion since the ions are driven in phase, therefore ion collisional effects in the rf and Penning traps are treated the same. <E^>

will of course depend on the cloud density and temperature but some idea of the magnitude can be given by cal- culating the electric field for one ion on another at the

distance of closest approach. Assuming the maximum en- ergy available for closest approach is given by SfcgT', we

have £'max = 6.7 x 10"^ V/cm {y/2'a - 10 MHz and maxTable 1. Fractional second order Doppler shifts, Stark fields, and classical rms axial amplitudes (z„„J for single ions in rf and Perming traps. When yl2-n is given, we assume maximum theoretical laser cooling ((1, « y). For both traps we assume M = 100 u. For the rf trap fl/2iT (rf drive frequency) = 1 MHz, Aq =

300 V/cm'^. For the Penning trap, io./2-it = "20 kHz, B = 1 T. T is the temperature

of the secular motion for the rf trap and the temperature of the cyclotron and axial motion for the Penning trap. <E^>, is the mean square electric field for motion along the z axis, <E'^>^; is the mean square "motional" electric field for the V X B/c force. Note that z„„s for the Penning trap can be reduced at the expense

of increasing <E'^>^.

y/2-n 10 MHz IkHz

T(K) 300 4 2.4 X 10-4 2.4 X 10-8

Av£,/v^ 8.3 X 10-13 1.1 X 10-1'* 6.6 X 10-19 6.6 X 10-23

rf <E^>,{V^/cm^y 210 2.8 1.7 X 10-4 1.7 X 10-8

2rms(|JLm) 170 20 0.15 1.5 X 10-3 Av^/vq 4.2 X 10-1=^ 5.5 X 10-15 3.3 X

10-i« 3.3 X 10-=^ <E^>, 4.2 X 10-2 5.7 X 10-4 3.4 X 10-8 3.4 X 10-12 Pen. (y2/cm2)

<E^>M 4.99 0.067 4.0 X 10-6 4.0 X 10-10 (V2/cm2)

i 2,TM./Mm) 1300 145 1.1 0.011

89

TN-103
imum laser cooling) and E^^ = 7.4 V/cm at T = 4 K. Therefore at modest temperatures, ion-ion collision in- duced Stark shifts can be quite small. For clouds of ions, other effects can contribute to Stark

and second order Doppler shifts. We will consider only

theoretical limits and therefore neglect effects such as rf heating which may be the real limitation in a practical

experiment. We will assume that the secular motion in an

rf trap and the axial and cyclotron modes in a Penning

trap have been cooled to negligible values. For both

traps we will assume that it is desirable to maximize the

number of ions A'^. Then in an rf trap we must consider the effects of the

micromotion and corresponding electric fields for ions on

the edge of the cloud. We impose the constraint that the

maximum fractional second order Doppler shift not exceed a certain value (e). Therefore for the rf spherical

well we set ^ir,z) - -(}),, and with the condition on e (the maximum second order Doppler effect is due to ions

at 2 - Vj, r - 0) we find Nr 6.48 X 10^^ rMe

when M is in u, and r, is the cloud radius.

For the Penning trap, the maximum second order

Doppler effect is due to the magnetron motion of ions on

the edge of the cloud (r,„ = r,, z = 0). We set <\>i{z) -

- <^Tiz) and for a spherical cloud we find from Eqs.

(5-7),

N, 1.96 X IQi^BVr 440 M vr

B

'

where B is in tesla, M in u. Negative solutions are unphysical because they correspond to parameters where

the magnetron second order Doppler shift cannot be

made as large as e (for a spherical cloud).

For very small values of e, very low density ion clouds are required which implies very small applied potentials.

From the condition cf), = - ^r for the rf trap we have

(for the spherical well):

2eA§

= 7.2 X 10"

A^, V

cm^

and similarly, for the Penning trap we have:

A = 3.6 X 10-8 Nr V

cm'

We can also calculate the corresponding electric fields. As before, for the rf trap, the maximum fields occur on

ions for z = r, and r - and we have

<E^>2 (max) = 8 n Ai - 2MCfNr

In the Penning trap, the electric fields cancel along the z axis. Along the radial direction,

<^^>r(max) = {2Mch/ (er,))^

.

In Table 2 are shovni some representative values of max- imum numbers of stored ions, trap potentials, and Stark

shifts for various values of e and r; on clouds of ions. In

certain configurations, second order Doppler and Stark

shifts could still be a problem; however, with small

enough numbers of ions these can be overcome. We note

that in many cases, the electric fields from black-body ra- diation {<E^>bb — (10 V/cm)^) can be much larger than

those due to trapping conditions [78, 89, 90]. Therefore,

operation at reduced environmental temperatures may

ultimately be required.

Table 2. Maximum num,bers, pseudopotential well depths for rf traps, applied potentials for Penning traps, and electric fields for "cold" spherical ion clouds in rf and Penning traps. A maximum fractional second order Doppler shift e is assumed. The secular motion for the rf trap and the axial and cyclotron motion for the Penning trap are assumed to be frozen

out (i.e., cooled to negligible values), r, = ion cloud radius: M = 100 u, Q/2-n = 1 MHz, B

= 1 T.

10" 10 -12 10" 10-

rf

Pen.

r I (cm)

N•" max 2eA§ f Mil^

V

cm^

N•" max

f V

cm'' V2

0.5 0.01 0.5 0.01

3.2 X 10^ 6500 320 ~6

0.19 470 1.9 X 10-4 0.47

760 760 0.76 0.76

4.5 X 106 1.5x10^ 53

1.3 0.043 1.9

0.14 1.4 X 10-7 3.5 X 10-4

TN-104

90
7. Other Possibilities

With the special features of the ion traps in mind, it is worthwhile speculating on possible future experiments.

A very interesting class of experiments has already been

initiated on multiply charged ions. Electron capture

rates of Ne''^ (1 « 9 « 10) in an orbitron trap have re- cently been measured [14] and Penning traps are being

developed for high Z ion storage [91]. These experiments

may pave the way for interesting spectroscopy on for ex- ample high Z hydrogenic ions where precise theoretical

calculations are available. Various energy separations be- come higher; for example the 235u9i+ ground state hfs transition may be in the visible [92]. In the future, more

exotic species may be available for study such as anti- protons, e"/e" pairs in a Penning trap or e^/e", e'^/p",

e~/p"^ or p"^/p" pairs in an rf trap [92]. The Penning trap may be an ideal place to measure gj

factors in certain atomic ions. First, because very small

samples (<100 (xm) can be used, the effects of inhomogeneous fields are greatly reduced [81]. Second, atomic

Qj factors could be measured in terms of the Bohr magnetron by measuring the cyclotron frequency of free elec- trons in the same magnetic field. This technique has al- ready been used to measure the gj factor in S" [93]. More generally, these features plus the very high resolutions possible [82] should allow precise measurements of

other magnetic-field-dependent structure. For example,

deviations in the Breit-Rabi formula for hfs could be

measured and it may be possible to measure the effects of nuclear diamagnetism [94]. From Tables 1 and 2, it is evident that Stark shifts can be very small on cooled ions; this suggests that the spec- troscopy of Rydberg ions with minimal Stark shifts may

be possible. With this in mind, perhaps a measurement of

the Rydberg along the lines of Kleppner's proposal [95]

is possible in an ion. Finally, ion storage techniques might also be used for measurement devices, for example, to measure magnetic

fields [96] or frequencies of lasers [45, 97]. Quite generally, it is now feasible to think of experiments on stored ions where the perturbations are ex- tremely small. Hopefully, this can lead to many new precision experiments other than the ones discussed here.

The preparation of this paper was supported in part by

the Air Force Office of Scientific Research and the Office

of Naval Research. The author thanks W. M. Itano, G. Dunn, J. C. Bergquist, R. E. Drullinger and F. L. Walls

for a critical reading of the manuscript and R. Ray and

B. Barrett for manuscript typing.

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[21] W. Neuhauser, M. Hohenstatt, P. Toschek, and H. G. Dehmelt, Phys. Rev. Lett. 41, 233 (1978); W. Neuhauser, M. Hohenstatt, P. Toschek, and H. G. Dehmelt, Appl.

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[22] For a review see A. Ashkin, Science 210, 1081 (1980).

[23] F. M. Penning, Physica 3, 873 (1936).

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[26] D. J. Wineland and H. G. Dehmelt, Int. J. Mass. Spectr. Ion

Phys. 16, 338 (1975); and erratum, 19, 251 (1976); H. G. Dehmelt, Nature 262, 777 (1976).

[27] See for example: S. A. Prasad and T. M. O'Neil, Phys.

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[32] The case of a purely logarithmic potential has been investi- gated by R. H. Hooverman, J. Appl. Phys. 34, 3505 (1963). See also R. R. Lewis, J. Appl. Phys. 53, 3975 (1982).

[33] C. E. Johnson, private communication.

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(1969).

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[41] R. S. Van Dyck, Jr., P. B. Schwinberg, and H. G. Dehmelt,

Phys. Rev. Lett. 38, 310 (1977).

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G. Dehmelt, Appl. Phys. Lett. 28, 446 (1976).

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(1979).

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[46] R. S. Van Dyck, Jr., P. B. Schwinberg, and H. G. Dehmelt,

Bull. Am. Phys. Soc. 24, 758 (1979).

91 TN-105
[47] P. B. Schwinberg, R. S. Van Dyck, Jr., and H. G. Dehmelt,

Phys. Rev. Lett. 24, 1679 (1981). Also see these proceedings.

[48] G. Gabrielse and R. S. Van Dyck, Jr., Bull. Am. Phys. Soc.

26, 598 (1981).

[49] H. Dehmelt, R. Van Dyck, P. Schwinberg, and G. Gabrielse,

Bull. Am. Phys. Soc. 24, 675 (1979); and H. G. Dehmelt

and G. Gabrielse, Bull. Am. Phys. Soc. 24, 758 (1979).

[50] P. B. Schwinberg and R. S. Van Dyck, Jr., Bull. Am. Phys.

Soc. 24, 1202 (1979); Bull. Am. Phys. Soc. 26, 598 (1981).

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[52] P. B. Schwinberg and R. S. Van Dyck, Jr., Bull Am. Phys.

Soc. 23, 40 (1978).

[53] R. S. Van Dyck, Jr., P. B. Schwinberg, and S. H. Bailey, in Atomic Masses and Fundamental Constants 6, Ed. by J. A. Nolen, Jr., and W. Benenson (Plenum, New York,

1980), p. 173; R. S. Van Dyck, Jr., and P. B. Schwinberg,

these proceedings.

[54] G. Gartner and E. Klempt, Z. Phys. A 287, 1 (1978).

[55] G. Graff, H. Kalinowsky, and J. Traut, Z. Phys. 296, 35

(1980). Also see these proceedings.

[56] D. J. Wineland and H. G. Dehmelt, J. Appl. Phys. 46, 919

(1975).

[57] D. J. Wineland, Bull. Am. Phys. Soc. 18, 1573 (1973).

[58] A. M. Wapstra and K. Bos, Atomic Data and Nuc. Data

Tables 19, 177 (1977).

[59] R. Hegstrom, private communication.

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(1971).

[61] R. D. Knight and M. H. Prior, Phys. Rev. 21, 179 (1980).

[62] R. Schneider and G. Werth, Z. Phys. 293, 103 (1980).

[63] P. B. Schwinberg, R. S. Van Dyck, Jr., and H. G. Dehmelt,

Phys. Lett. 81A, 119 (1981).

[64] S. C. Menasian and H. G. Dehmelt, Bull. Am. Phys. Soc. 18, 408 (1973); F. J. Grieman, B. H. Mahan, and A. O'Keefe,

J. Chem. Phys. 72, 4246 (1980).

[65] H. A. Schuessler, E. N. Fortson, and H. G. Dehmelt, Phys. Rev. 187, 5 (1969).

[66] M. H. Prior and E. C. Wang, Phys. Rev. Lett. 35, 29 (1975); and Phys. Rev. A 16, 6 (1977).

[67] F. G. Major, NASA Report X-521.69.167, Goddard Space

Flight Center (1969).

[68] H. A. Schuessler, Metrologia 7, 103 (1971).

[69] F. G. Major and G. Werth, Phys. Rev. Lett. 30, 1155 (1973); F. G. Major and G. Werth, Appl. Phys. 15, 201 (1978).

[70] H. Dehmelt, Bull. Am. Phys. Soc. 18, 1521 (1973); H. Dehmelt, Bull. Am. Phys. Soc. 20, 60 (1975); D. J. Wineland and H. Dehmelt, Bull. Am. Phys. Soc. 20, 637 (1975); H. Dehmelt, IEEE Trans. Instrum. Meas. IM-31, 83

(1982).

[71] P. L. Bender, J. L. Hall, R. H. Garstang, F. M. J. Pichanick, W. W. Smith, R. L. Barger, and J. B. West, Bull. Am. Phys. Soc. 21, 599 (1976).

[72] F. Strumia, Proc. 32nd Ann. Symp. Freq. Control, June 1978,

p. 444. (Copies available from Electronic Industries Assn., 2001 Eye St., NW., Washington, DC 20006.)

[73] F. L. Walls, D. J. Wineland, and R. E. DruUinger, Proc. 32nd Ann. Symp. Freq. Control, June 1978, p. 453. (Copies avail- able from Electronic Industries Assn., 2001 Eye St., NW.,

Washington, DC 20006.)

[74] M. D. McGuire, R. Petsch, and G. Werth, Phys. Rev. A 17, 1999 (1978). M. D. McGuire, Bull. Am. Phys. Soc. 26, 615

(1981).

[75] W. M. Itano and D. J. Wineland, Bull. Am. Phys. Soc. 24, 1185 (1979).

[76] M. Jardino, M. Desaintfuscien, R. Barillet, J. Viennet, P.

Petit, and C. Audoin, Appl. Phys. 24, 1 (1981).

[77] P. E. Toschek and W. Neuhauser, in Atomic Physics 7, Ed. by D. Kleppner and F. M. Pipkin (Plenum, New York,

1981), p. 529.

[78] D. J. Wineland, W. M. Itano, J. C. Bergquist, and F. L. Walls,

Proc. 35th Ann. Symp. Freq. Control, June, 1981. (Copies

available from Electronic Industries Assn., 2001 Eve St., NW., Washington, DC 20006.) Also see W. M. Itano, D. J. Wineland, J. C. Bergquist and F. L. Walls, these proceedings.

[79] T. W. Hansch and A. L. Schawlow, Opt. Commun. 13, 68

(1975).

[80] M. H. Prior and R. D. Knight, Opt. Comm. 35, 54 (1980).

[81] D. J. Wineland, J. C. Bergquist, W. M. Itano, and R. E. DruUinger, Opt. Lett. 5, 245 (1980).

[82] W. M. Itano and D. J. Wineland, Phys. Rev. A 24, 1364

(1981).

[83] W. Nagourney and H. Dehmelt, Bull. Am. Phys. Soc. 26

(1981).

[84] J. L. Hall, Science 202, 147 (1978); in Atomic Physics 7, Ed. by D. Kleppner and F. M. Pipkin (Plenum, New York,

1981), p. 267.

[85] D. J. Wineland, Proc. 11th Ann. Precise Time and Time

Interval Meeting (NASA Publ. 2129, Nov. 1979), p. 81. [86] W. Neuhauser, M. Hohenstatt, P. Toschek, and H. Dehmelt,

Proc. 5th Int. Conf. on Spectral Lineshapes, West Berlin,

July 1980.

[87] W. M. Itano and D. J. Wineland, Phys. Rev. A 25, 35 (1982).

[88] R. D. Haun, Jr., and J. R. Zacharias, Phys. Rev. 107, 107

(1957).

[89] T. F. Gallagher and W. E. Cooke, Phys. Rev. Lett. 42, 835

(1979).

[90] W. M. Itano, L. L. Lewis, and D. J. Wineland, Phys. Rev. A

25, 1233 (1982).

[91] D. A. Church, in Physics of Electron and Atomic Collisions, Ed. by S. Datz (North Holland, 1982), p. 533.

[92] H. G. Dehmelt, R. S. Van Dyck, Jr., P. B. Schwinberg, and

G. Gabrielse, Bull. Am. Phys. Soc. 24, 757 (1979).

[93] R. M. Jopson and D. J. Larson, Phys. Rev. Lett. 47, 789

(1981).

[94] N. P. Economou, S. J. Lipson, and D. J. Larson, Phys. Rev.

Lett. 38, 1394 (1977).

[95] D. Kleppner, Bull. Am. Phys. Soc. 20, 1458 (1975).

[96] P. B. Schwinberg and R. S. Van Dyck, Bull. Am. Phys. Soc.

24, 35 (1979).

[97] J. C. Bergquist and D. J. Wineland, Proc. 33rd Ann. Symp.

Freq. Control, June 1979. (Copies available from Electronic

Industries Assn., 2001 Eye St., NW., Washington, DC 20006.)

IN-10

6

92
Abstract Submitted

for the Philadelphia, PA meeting of the

American Physical Society

3-5 Nov. 1982

Physics and Astronomy

Classification Scheme

Number 32

Bull. Am. Phys. Soc.27, 864 (1982)

Suggested title of session

in which paper should be

placed

Spectroscopy, Optics and

Photoionization

Magnetic Field Dependence of (s,l) Electron

Configurations.* D. J. WINELAND and WAYNE M. ITANO, NBS.

Using the modified Breit-Wills theory (for hfs of (s,l

)

electron configurations) in second order perturbation

theory, we have calculated the magnetic field dependence

of L. states. Our main interest is in predicting the

field"^ dependence of intercombination lines of group liUA

ions for possible use in optical frequency standards. '

For example, the {5s^)^So(M.=-9/2)-> (5s 5p)^Po (Mt=-9/2)

15 transition frequency (vo==1.3xl0 Hz) in In II is independent of magnetic field(B) in first order at Bg25±5k62with

second order field dependence of Av/Vq=-2x10~ (aB/B) . This suggests an In II (or other ion) frequency standard in a Penning trap or rf trap with superimposed magnetic field. For In II, coherent radiation

(power >10yW) for laser cooling/optical .pumping could

be accomplished by sum frequency mixing in ADP.

Supported by ONR and AFOSR

46 (1966).

Instr. Meas. IM-31 83

and D. (1982)

1 A. Lurio, Phys. Rev. 142 , 2 H.G. Dehmelt, IEEE Trans.

3 R.L. Barger, J.C. Bergquist, T.C. English,

Glaze, Appl. Phys. Lett. 34, 850 (1979).

4 H. Hemmati , J.C. Bergquist, and Wayne M. Itano, to be

published.

( ) Prefer Standard Session

Signature of APS Member

D. J. Wineland

National Bureau of Standards

325 Broadway

Boulder, CO 80303

TN-107
Laser Cooling and Double Resonance Spectroscopy of

Stored Ions

W.M. Itano and D.J. Wineland

Frequency and Time Standards Group, Time and Frequency Division

National Bureau of Standards, 325 S. Broadway

Boulder, CO 80303, USA

1. Introduction

The use of ion storage techniques for spectroscopy is motivated by the fact

that ions can be confined by electric and magnetic fields for long periods

of time without suffering the large perturbations which usually accompany

other methods of confinement, such as those due to collisions with buffer

gas molecules. Linewidths as small as a few Hz and Q's as high as 10 have

been observed on ground-state hyperfine transitions of atomic ions stored

in rf quadrupole traps [1-3]. The accuracy of these measurements has been

limited largely by the second-order Doppler shift. The signal-to-noise

ratios have been limited by the small number of ions that can be stored

(about 10 -10 ) and by the difficulty of detecting transitions. Hyperfine

and Zeeman transitions have been detected by charge exchange [1], fluorescence [2,3], photodetachment [4], and photodissociation [5].

In this paper we discuss recent work at the National Bureau of Standards

(NBS) in this area. The second-order Doppler shift can be reduced by laser

(resonant light pressure) cooling. This technique has been demonstrated in experiments on stored ions by our group at NBS [6-8] and also by a group at Heidelberg [9]. Other work at NBS has been directed toward the realization

of high-efficiency laser-optical -pumping, double-resonance detection techniques [10]. These techniques have been used to make the first highprecision

25 + hyperfine structure measurements of Mg [11]. We anticipate that the laser

cooling and double resonance techniques will find practical application in the development of frequency standards based on stored ions [12].

Laser Spectroscopy V. eds . A.R.W.

McKellar, T. Oka, and B.P. Stoicheff.

(Springer Verlag, N.Y.,1981).

TN-108
361

2. Apparatus

A block diagram of the apparatus is shown in Fig.l. The Mg ions are confined in a Penning-style ion trap. They are irradiated by light resonant

2 2 with the 3s S, ,p

-* 3p P,,^ 280 nm transition and also by rf and microwave

radiation resonant with various transitions between ground-state sublevels.

Scattered 280 nm photons are collected by e mirror and counted by a photo-

-5 multiplier tube, with a net detection efficiency of about 10 . A Penning trap confines ions by a combination of a uniform static magnetic field, ^ = B z, and a quadrupolar electrostatic potential. The magnetic

field causes the ions to move in circular "cyclotron" orbits in the x-y plane. The electric fields provide a harmonic restoring force for the axial (z) motion and cause the centers of the cyclotron orbits to move in the x-y plane

in circular "magnetron" orbits around the trap symmetry axis. The angular

frequencies corresponding to the axial, cyclotron, and magnetron motions are denoted by u , u'

,

^ z c and u respectively. For typical operating conditions.

J, = 2iT . 200 kHz, co' = 2tt . 800 kHz, and oj = 2tt • 25 kHz. Typically,

z c m

B„ = IT, and a potential '^ V 7 V is applied across the trap electrodes,

which have inside dimensions of about 1 cm. The axial and cyclotron motions

are thermal and stable. The magnetron motion is nonthermal and unstable,

since an increase in the size of the magnetron orbit leads to a decrease in the total (kinetic plus potential) energy. However, storage times of about

1 day have been observed, even without laser cooling. The background pres-

-8 sure is typically < 10 Pa.

Fig. 1. Apparatus for Mg experiments

RF, Microwave

Generator Penning

Trap

PM

Field

TN-109
362

The 280 nm radiation was produced by frequency doubling the output of a single-mode cw Rhodamine 110 dye laser in a 90° phase-matched AD*P crystal.

Between 5 and 30 uW of UV radiation was generated in a bandwidth of about

1 MHz. The dye laser could be long-term stabilized to discrete frequencies

to less than 1 MHz by locking it to saturated absorption features in I-. Fine tuning could be done by varying the trap magnetic field to Zeeman shift

the Mg levels. For efficient laser cooling, the UV linewidth must be less

than the Doppler width of the transition (3 GHz at 300K), but need not be much less than the natural linewidth (43 MHz).

3. Laser Cooling in a Penning Trap

The basic principle of laser cooling is that light pressure can be used to damp the velocity of an atom or ion if the light frequency is tuned slightly

below that of a strong optical transition. The process is easily described

for the case of free or harmonically bound atoms [9,13-16]. Atoms whose

velocities are directed toward the light source see a frequency Doppler

shifted closer to resonance; if their velocities are directed away from the source, the frequency shift is away from resonance. Therefore, the atoms

tend to absorb photons when their velocity is directed toward the source.

This cools them, since the momentum of the absorbed photons reduces the atomic momentum. (The photons are re-emitted in random directions.) Threedimensional cooling can be obtained with six laser beams directed along

the ± X, ± y, and ± z directions. A single beam suffices to. cool all oscillational modes of an atom bound in a three-dimensional harmonic potential,

if the modes are nondegenerate and if the beam is not directed along one of

the principal axes. Cooling of the axial and cyclotron modes of an ion in a Penning trap

takes place by the process just described for free or harmonically bound

atoms, if the light is tuned below resonance [17]. Cooling of the magnetron

mode can be accomplished by focusing the light beam so that it is more in- tense on the side of the trap axis on which the magnetron motion recedes

from the light source [6-8,17]. By "cooling" of a mode, we mean the reduction of the average kinetic energy in that mode, regardless of what happens

to the potential energy. In order to efficiently cool all modes, the angle

between the beam and the z axis should be about 45°; the optimum angle depends on the angular distribution of the scattered photons [17]. We have derived rate equations for the laser cooling of an ion in a Penning trap by

TN-110
363

a single nonuniform light beam [17]. Solving these equations, we find that

it should be possible to cool an ion to a "temperature" T given by kgT = h y/2,

where kg is Boltzmann's constant and y is the natural linewidth of the upper

state. The "temperature" is defined in terms of the mean kinetic energy and may be different for the different modes. For Mg , this minimum temperature

is about 1 mK.

4. Optical Pumping and Double Resonance

The extremely long ground-state relaxation times possible for stored ions

make it possible to observe "i/ery narrow transition linewidths and to observe

4 wery weak optical pumping processes. An ion must scatter about 10 optical

photons in order to be cooled significantly below room temperature. Therefore, even weak depopulation pumping processes would have an adverse effect

on the cooling. Cooling, in turn, influences the optical pumping by reducing

the Doppler broadening until it is less than the natural linewidth. This

makes it possible for a monochromatic laser to interact strongly with all of

the ions and not just with a small velocity class.

I" ^ + +

SCATTBSO UGKT

NTBVSnY(5s/pl|

BACKGROUND -

+ +

_l_ 27412 27413

MICROWAVE FREQUENCY (MHz) 24. Fig. 2. Optical pumping and double resonance of Mg

Generally, the (^5^,2, Mj = -1/2) -^ i^^^/Z' "j " '^^^"^ Zeeman component

was used for laser cooling of Mg"^, which has 1=0 (see Fig. 2). This tran- sition does not lead to depopulation pumping of the ground state, since the

M, = -3/2 excited state sublevel must decay to the Mj = -1/2 ground state

TN-111
364

sublevel. When this transition is driven near resonance with light polarized

2 2 perpendicular to the magnetic field, the ( S, ,p, M, = -1/2) ->

( Pn/o.

Mj = + 1/2) and (^S^^2' '^j = + ^Z^) - (^P3/2' ^j = " 1/2) transitions are driven weakly in their Lorentzian wings (the Zeeman splitting is much

greater than the Doppler broadening). In the steady state, which is achieved

in < Is, 16/17 = 94% of the population is in the M, = -1/2 ground-state sublevel [10].

2 The ground-state Zeeman transition ( S, ,-, M, = -1/2 -* + 1/2) can be de- tected by a decrease in the fluorescence intensity. This "flop-out" detection method can be very efficient, since a transition due to a single microwave photon interrupts the flow of scattered optical photons until the ion

is pumped back to the M, = -1/2 sublevel by weak, off-resonance scattering.

Since the decrease in the number of scattered photons per ion can be very

large, it is possible to make up for poor light collection and detector quantum efficiency, so that the transition can be detected with nearly unit efficiency. Similar detection methods have been proposed previously 18 . The (^5^/2- Mi = -V2, Mj = -1/2) - (^P3/2.

^i

= "5/2, Mj = -3/2) hyperfine-Zeeman component was generally used for laser cooling of Mg (1=5/2).

In the steady state, about 16/17 of the population is pumped into the

(M, = -5/2, M, = -1/2) ground-state sublevel. Pumping into the M, = -1/2

I- ^ 24 + manifold takes place by the same mechanism as in Mg . Pumping into the

Mr = -5/2 sublevel s takes place because of hyperfine coupling in the excited

state [10]. Any transition which decreases the population in the (M, = -5/2,

Mj = - 1/2) ground-state sublevel can be detected by a decrease in the fluorescence intensity.

5. Results

In the Penning trap, low temperatures become easier to achieve as the ion

density is reduced. This is because the radial electric field due to space

charge increases the magnetron velocity. This problem does not exist for a single, isolated ion, and the lowest temperatures were observed for this case [8]. 24 + Figure 3 shows the fluorescence from a small number of Mg ions as a function of time. After the ions were cooled and localized at trap center,

25 an oven containing Mg (98% isotopic purity) was heated in order to induce

24 + 25 24 25 +

the resonant charge exchange reaction ( Mg + Mg ->- Mg + Mg ). The

TN-112
365

n^jTTx/^-^

—BACKGROUND

TIME

24,. + Fig. 3. Fluorescence from 3, 2, and 1 Mg ions

25 + resulting Mg ions were ejected from the trap by resonant cyclotronmagnetron rf excitation. The three large step decreases in fluorescence are due

24 + to loss of the Mg ions, one at a time, and the last plateau above background is the fluorescence from a single ion. The signal was about 50 photons/s

per ion.

24 + The "temperature" of a single Mg ion was determined from the Doppler

width by optical -optical double resonance (see Fig. 4). One laser was tuned

slightly below the (^S^,^, Mj = -1/2) -* (^P3/2. ^j = "3/2) transition, to provide cooling and fluorescence detection. A low power laser was swept con2 2 tinuously across the ( S, .p' ^j = -1/2) -* ( P3/2' ^j ^ "1/2) transition,

which was detected by a decrease in the fluorescence. The resulting lineshape

reflects both the natural and Doppler broadenings. The data points represent

10s integrations; the connecting lines are only for clarity. Simulated curves

are shown for temperatures of K and 1000 mK. We estimate that T = 50 ± 30 mK. Since the light was incident at 82" with respect to the z axis, this is es- sentially a measurement of the cyclotron-magnetron (x-y) temperature. The

axial (z) temperature was estimated to be about 600 mK by probing the axial

excursions with a focused laser beam. According to our calculations, it should be possible to obtain a cyclotron-magnetron temperature of about 1 mK

and an axial temperature of about 11 mK for these conditions [17]. At present, the discrepancy is not understood, but may be due to the presence of

impurity ions in the trap.

Ground-state Zeeman and hyperfine transitions were detected by the optical-pumping, double-resonance methods outlined previously. The only tran24 + sition in Mg is the electronic spin flip transition (see Fig. 2). Several

TN-113
366

-200 -100 100

RELATIVE LASER FREQUENCY (MHz)

200

24 + Fig. 4. Optical-optical double resonance of a single Mg ion

transitions in Mg"*", corresponding to both nuclear and electronic spin

flip transitions, were observed. Most transitions were broadened by the in- stability of B , which was, at best, about 1 ppm in a few seconds. The mag- netic field derivative of the (Mj, Mj) = (- 3/2, + 1/2) ^(- 1/2, + 1/2)

transition goes to zero at B = 1.2398 T. Near this field, the transition

was observed with linewidths as narrow as 0.012 Hz (see Fig. 5). The oscil-

>,

- -

V)c

<u h - -

0)

(

)

q/ VOOIZHz-/ o\ c 7

?^ -° -

l/> / oV^/ 1 y ^ _/ ^-^0 o

3

1

291996 251.899 Hz

1.88 1.89 1.90 i.9l 1.92 RF Frequency - 291 996 250 ( Hz

)

Fig.

5

^^Mg^ (Mj.Mj) = (- 3/2), + 1/2)— (- 1/2, + 1/2]

ground-state hyperfine resonance

TN-114
367

latory lineshape results from the use of the Ramsey interference method [19],

which was implemented by driving the transition with two coherent rf pulses

of 1.02 s duration separated by 41.4 s. The center of the resonance can be determined with an uncertainty of about 10 . Several different transition

frequencies were measured in order to obtain separately the hyperfine con- stant (A = -596.254 376 (54) MHz) and the nuclear-to-electronic g factor

ratio (gj/gj = 9.299 484 (75) x lO'^) [11].

6. Future Possibilities

Work is being initiated at NBS on the development of a microwave frequency

and time standard based on a hyperfine transition in Hg and of an optical frequency standard based on a two-photon transition to a metastable state

in Hg or Hg . Both proposed standards are based on a cloud of Hg

2 2 ions stored in a Penning trap and use the 6s S, ,„ -> 6p P-..^ 194 nm tran- sition for laser cooling.

The proposed microwave frequency standard is based on the (F, Mp)

= (1,1) -<-*-(2,l) transition, which is field-independent to first order at

B^ = 0.534 T, with frequency = 25.9 GHz. If B can be controlled to slightly

better than 0.1 ppm over the ion cloud, the fractional frequency shift can

be kept below 10 15

All other systematic shifts, such as those due to the second-order Doppler effect, collisions, the trap electric fields, or thermal radiation, appear to be less than 10 -15

12 It should be possible to ob- serve the transition with a Q of 2.6x10 or better, by using optical pump25 + ing and detection techniques similar to those demonstrated with Mg . The

-15 accuracy of this standard could be as good as 10 , which is about 100 times

better than that of the best frequency standards now available.

The proposed optical frequency standard is based on the two-photon^al lowed

10 2 9 2 2 +

5d 6s S, ,„ -»• 5d 6s D,-,, Hg transition, which has a natural Q of

7.4 X 10 [12]. The first-order Doppler effect can be eliminated by driving

the transition with counter-propagating 563.2 nm laser beams. Hyperfine-Zeeman

components whose magnetic field derivatives vanish at particular values of

B exist in Hg"*" and Hg"*". The two-photon transition can be detected with

high efficiency by using the 194.2 nm fluorescence intensity as a probe of the ground state population. Taking full advantage of the high Q transition

would require a laser with linewidth less than 1 Hz, which does not exist at present. However, linewidths < 100 Hz appear feasible and could be used for

TN-115
368

initial experiments. If the laser linewidth is less than the natural line10-^ ^-15

15

width, then the ac Stark shift is about 2x10 near saturation. All other

systematic shifts appear to be less than 10 The method currently being investigated for generating the required

194.2 nm radiation is sum-frequency mixing in a KB5 crystal of the output

of a 792 nm single-mode cw ring dye laser and the second harmonic, generated

in an ADP crystal, of the output of a 514 nm stabilized, single-mode cw Ar

laser. The method has been demonstrated previously with pulsed lasers [21].

Further details of these proposed Hg frequency standards are published

elsewhere [12].

Acknowledgments. We wish to acknowledge experimental assistance by Drs. J.C. Bergquist and R.E. Drullinger. This work was supported in part by the Air Force Office of Scientific Research and the Office of Naval Research.

References

1. H.A. Schuessler, E.N. Fortson, H.G. Dehmelt: Phys. Rev. 287, 5-38 (1969)

2. F.G. Major, G. Werth: Phys. Rev. Lett. 20, 1155-1158 (1973)

3. R. Blatt, G. Werth: Z. Phys. A 299, 93 (1981)

4. R.M. Jopson, D.J. Larson: Bull. Am. Phys. Soc. 25, 1133 (1980)

5. C.B. Richardson, K.B. Jefferts, H.G. Dehmelt: Phys. Rev. 165, 80-87

(1968)

6. D.J. Wineland, R.E. Drullinger, F.L. Walls: Phys. Rev. Lett. 40, 1639-

1642 (1978)

7. R.E. Drullinger, D.J. Wineland, J.C. Bergquist: Appl . Phys. 22, 365-368

(1980)

8. D.J. Wineland, W.M. Itano: Phys. Lett. 82l\, 75-78 (1981)

9. W. Neuhauser, M. Hohenstatt, P.E. Toschek, H. Dehmelt: Phys. Rev. Lett.

41, 233-236 (1978); Appl. Phys. 17, 123-129 (1978); Phys. Rev. A 22,

1137-1140 (1980)

10. D.J. Wineland, J.C. Bergquist, W.M. Itano, R.E. Drullinger: Opt. Lett.

5, 245-247 (1980)

11. W.ri. Itano, D.J. Wineland: Phys. Rev. A, to be published

12. D.J. Wineland, W.M. Itano, J.C. Bergquist, F.L. Walls: Proc. 35th Ann.

Symp. on Freq. Control (U.S. Army Electronics Command, Fort Monmouth,

NJ, 1981)

13. T.W. Hansch, A.L. Schawlow: Opt. Commun. U, 68-69 (1975)

14. D.J. Wineland, H. Dehmelt: Bull. Am. Phys. Soc. 20, 637 (1975)

15. D.J. Wineland, W.M. Itano: Phys. Rev. A 20, 1521-1540 (1979)

16. J. Javanainen: Appl. Phys. 23, 175-182 (1980)

17. W.M. Itano, D.J. Wineland: Phys. Rev. A, submitted

18. H. Dehmelt: Bull. Am. Phys. Soc. 20, 60 (1975)

19. N.F. Ramsey: Molecular Beams (Oxford Univ. Press, London 1956) pp. 124-134

20. P.L. Bender, J.L. Hall, R.H. Garstang, F.M.J. Pichanick, W.W. Smith, R.L. Barger, J.B. West: Bull. Am. Phys. Soc. 21, 599 (1976)

21. R.E. Stickel, F.B. Dunning: Appl. Opt. 17, 981-982 (1978)

TN-116
February 1983 / Vol. 8, No. 2 / OPTICS LETTERS 73

Generation of continuous-wave 194-nm radiation by sumfrequency mixing in an external ring cavity

H. Hemmati, }. C. Bergquist, and Wayne M. Itano

Time and Frequency Division. National Bureau of Standards, Boulder, Colorado 80303

Received November 4, 1982

Several microwatts of tunable cw radiation near 194 nm in a linewidth of less than 2 MHz have been generated by

sum-frequency mixing the radiation from a frequency-doubled argon-ion laser with the radiation from a ring dye

laser in a crystal of potassium pentaborate. An external ring cavity resonant with the dye laser gives an enhance- ment factor of about 14 in the sum-frequency-generated radiation power. The Doppler-limited absorption spec- trum of the 6s ^Si/2~6p ^Pi/2 first resonance line of natural Hg II has been resolved, and the vacuum wave number

for the mass-202 isotope has been measured to be 51485.904(20) cm~^

High-resolution spectroscopy of atomic and molecular

lines in the UV region of the spectrum requires tunable

sources of radiation. Many of the tunable UV sources can only be obtained by second-harmonic generation

(SHG) or sum-frequency mixing (SFM). For radiation

pressure cooling and optical pumping of electromagnetically confined mercury ions, which have great potential in optical and microwave frequency standards,^ a narrow-band tunable cw laser near the 6s ~Si/2-6p

^Pi/2 first resonance line (194.2 nm) is required. For

optimum cooling, the frequency, linewidth, and stability

of the radiation source must be less than the natural

linewidth of the resonance line of Hg+ (about 70 MHz)

with a minimum cw power requirement, which is esti- mated, from experiments on Mg+ ions,^ to be about 1 mW.

In this Letter we describe a method for producing

194-nm radiation by SFM, in a potassium pentaborate

(KB5) crystal, the 257-nm second harmonic of the

output of a cw 515-nm argon-ion laser with the output

of a tunable cw dye laser in the 792-nm region. Previously, Stickel and Dunning,'^ using pulsed dye lasers,

generated coherent radiation tunable between 185 and

217 nm by SFM in KB5. However, cw sources of ra- diation can demonstrate better frequency stability and

smaller linewidths. The shortest-wavelength cw ra- diation that was previously produced by SFM in a

crystal is approximately 211 nm."* The only other

technique that has been demonstrated for generation

of coherent cw radiation below 200 nm is four-wave

mixing in strontium vapor, at approximately 170 nm,

by Freeman et al.^ A schematic of the experimental setup is shown as

Fig. 1. The primary radiation sources are a ring dye

laser operating with the dye LD700 near 792 nm and a 257-nm source that is the frequency-doubled output of the 515-nm line of an argon-ion laser. The 257-nm

second harmonic is generated in an ammonium dihydrogen phosphate (ADP) crystal, which is placed in an

external ring cavity that is resonant at the fundamental

frequency.*' Both lasers operate in a single-frequency

mode and are frequency stabilized to separate reference

cavities. For the argon-ion laser, additional long-term

stabilization is provided by locking the reference cavity

to a hyperfine component of a corresponding molecular

iodine line. A 30-mm X 5-mm X 5-mm Brewster-cut

KB5 crystal is used for SFM. To enhance the 792-nm

radiation power inside the crystal, this crystal is placed

inside an external ring cavity, similar to the fre- quency-doubling cavity, which is locked on resonance

with the dye laser. The Brewster-cut crystal minimizes

Fresnel losses for the input beams and compensates for beam astigmatism inside the ring cavity. The 792- and

257-nm input beams are polarized along the a axis of the

crystal. The 194-nm beam is polarized along the c axis,

and all three beams propagate along the b axis. For the

biaxial KB5 crystal, this is the 90° phase-matched

condition. The dye-laser beam is introduced into the

cavity through a partially reflecting mirror {R s 90%).

The sum-frequency radiation is well separated from the

792-nm radiation at 15 cm from the KB5 crystal since

the refraction angles for these radiations differ by ~2.5°.

Because of this, no special mirrors or dichroic beam

splitters are necessary inside the 792-nm enhancement

ring cavity to extract the 194-nm radiation. Maximum

Fig. 1. Experimental setup for generation of 194-nm radia- tion in an external ring cavity.

TN-117

Reprinted from Optics Letters, Vol. 8, page 73, February 198.3 Copyright © 1983 by the Optical Society of America and reprinted by permission of the copyright owner.
74 OPTICS LETTERS / Vol. 8, No. 2 / Februan^ 1983

Fig. 2.

194.5 194,7 Sum Wavelength (nm) Angle tuning in KB5 crystal. The crosses represent our data; the solid curve is from calculations.

enhancement is obtained inside the cavity by stabiHzing

the cavity on resonance, using the polarization technique discussed by Hansch and Couillaud." A similar

enhancement cavity was recently used by Couillaud et

al. for SFM in an ADP crystal.^

As was previously discussed for SHG of the 257-nm

radiation,^ the external ring cavity offers several at- tractive features, including the possibility of achieving

higher enhancement factors than is generally possible

with laser intracavity SFM. In this experiment,

buildup factors as high as 14 were obtained using an

input mirror with a reflectivity of about 90%. This

buildup factor is presumably limited by the quality of

the surface polish of our KB5 crystal.

The KB5 crystal is located inside a small housing that

is equipped with electrical heaters to provide the temperature tuning necessary for the 90° phase-matched

condition. This crystal is positioned in the beam waist

of the cavity, which is about 50 ura in diameter and is close to the optimum spot size for sum-frequency generation. The 257-nm radiation is focused to about the same diameter and superposed with the 792-nm ra- diation for maximum UV generation.

With input powers of 25 and 220 mW at 257 and 792

nm, respectively, and a buildup factor of 12 in the en- hancement cavity, about 2 juW of single-frequency

194-nm radiation is obtained. This corresponds to an

efficiency parameter of about 3 X 10"-'' W~^ for the SFM

process. This is about a factor of 2 smaller than the

value calculated from the previously measured nonlinear coefficients of KB5.^ The output radiation was measured by a calibrated EMR Type 541F-05M-14

solar-blind photomultiplier tube (PMT). About 80

mW of 257-nm and 300 mW of 792-nm radiation were

obtained previously in our experiments. At these input

power levels, a fourfold improvement to 8 /iW of 194-nm

radiation is expected. At the 2-^W power level, we see no degradation of the 194-nm radiation power with

time. Since the 257- and 792-nm radiation beams are

also well separated outside the crystal, again because

of different refraction angles, a second ring cavity may

be added to enhance the power of the 257-nm radiation,

which would further increase the 194-nm radiation power. The linewidth of the 194-nm output is less than

2 MHz, inferred from the linewidth of the primary

beams, each of which is about 1 MHz.

Figure 2 shows the propagation angle for the input

beams in the a-b plane, relative to the b axis, versus the

wavelength for which angle phase-matched SFM occurs

at a temperature of about 25° C. The solid curve is calculated from the published indices of refraction.^^ Figure 3 shows the wavelength for which 90° phasematched SFM occurs, as a function of temperature. In both cases (Figs. 2 and 3) the 257-nm wavelength is fixed

and the 792-nm wavelength is varied. The greatest

conversion efficiency occurs when the 90° condition is met. For the mercury-ion resonance line at 194 nm, the

required 90° phase-matching temperature is about

34°C.

A preliminary study was made of the isotope shift and

hyperfine structure of the 6s 2Si/2-6p ^Pi/2 194.2-nm

transition of singly ionized mercury by absorption

spectroscopy. The argon-ion laser was stabilized at a

fixed frequency while the dye laser was tuned in fre- quency. The dye laser could be continuously scanned

over 40 GHz. The external enhancement ring cavity

is locked and s>Tichronously scanned with the dye laser. Both cavities are frequency scanned by double tipping

Brewster plates, which have nearly zero insertion loss and do not cause any beam displacement as the cavities are tuned.

The absorption spectrum is obtained by probing the

ground state of the singly ionized Hg ions that are created in a glass cell excited by an electrodeless rf discharge, using 150-MHz rf radiation with power of less than 10 W. The cell contained the naturally occurring

Hg isotopes with Xe gas(10^^ cm"^) as the buffer and

was heated to about 70° C to increase the density of the

Hg atoms. To eliminate laser intensity noise, the signal was differentially detected by using two photomultiplier

tubes. A fraction of the 194-nm beam was split off and

directed into one PMT. The remaining radiation was

directed through the Hg cell and subsequently detected

by the second PMT. By using a differential amplifier,

the gains of the two PMT's were adjusted to balance the

intensity fluctuations of the 194-nm radiation. No

attempt was made to normalize the spectrum against

laser-power variations.

To calibrate the frequency scan, a portion of the

792-nm dye laser beam was directed into a 250-MHz

free-spectral-range confocal interferometer. This in194.30

-5 19420-

194.102000 25.00 30.00 35.00 40.00 Temperature (degrees C)

4500

Fig. 3. Temperature tuning in KB5 crystal. The crosses represent our data; the line is a linear least-squares fit.

TN-118
February 1983 / Vol. 8, No. 2 / OPTICS LETTERS 75

200

I I I I I I I

1

i I I I I ill I li '

i I I ' ^ I ' I I I' I I I ill I' I I ^i i I 'i -20 -15 -10 -S 5 10 15 20 FREQUENCY (GHz)

Fig. 4. Isotope and hyperfine-structure components of the 194.2-nm resonance line in natural Hg il.

terferometer gave simultaneous frequency markers with

the recording of the atomic-absorption spectrum as a function of the dye-laser frequency. A single scan of

the isotope and hyperfine structure of the 194-nm

transition recorded in approximately 24 sec is shown in Fig. 4. The most abundant isotopes of mercury are masses 202 (30%), 200 (23%), 199 (17%), 198 (10%), 201

(13%), and 204 (7%), which appear clearly in our scan. The contribution from the 199a hyperfine component,

which is located at about 24.7 GHz, to the blue of the

201a hyperfine component is not included in this figure.

The relative line spacing of the absorption spectra is in good agreement with the previous work by Guern et al. , in which Fabry-Perot interferometric techniques were

used.i^

The wavelength of the 202j^g+ component was determined from the wavelengths of the input lasers. The

argon-ion laser was locked to the 02 component of the

43-0, P(13) transition of ^^^l2, whose vacuum wavelength is known to be 514.67352 nm.^^ ^^ interferometric wavelength meter,^^ which has an accuracy of a few parts in 10', was used to measure the wavelength of the dye laser. The reference for the wavelength meter was a He-Ne laser stabilized to the h component of the

11-5, R{121) transition of ^^'l2, which has a vacuum

wavelength of 632.99137 nm. Proper corrections were made for the refractive index of air for our laboratory

physical conditions, using the formulas of Edlen.^^ The

dye-laser vacuum wavelength necessary to generate UV

radiation in resonance with the ^^^Hg"*" component was measured to be 791.9965 ± 0.0002 nm, where the accuracy is limited by our inability to determine the line center. This corresponds to the UV sum-frequency wave number of 51485.904 ± 0.020 cm~^. The uncertainty in the UV wave number has been increased be- cause of possible line shifts caused by the electric fields of electrons and ions in the discharge. The measured

width of an individual component is 1.8 GHz. If we

assume that the line profile is the convolution of a Lorentzian line shape and a Gaussian Doppler profile,

calculated for the temperature measured at the cell walls, we find that the width of the Lorentzian is 620

MHz, compared with the natural width of 70 MHz.

The line shift can be of the same order. In conclusion, we have demonstrated the possibility

of generating narrow-band tunable cw coherent radiation below 200 nm by SFM in KB5. If the argon-ion

laser were replaced with a tunable dye laser, or if other

ion laser lines were used, tunable cw UV radiation could

be generated at even shorter wavelengths, possibly down

to 185 nm or below. The 194-nm radiation produced

by the SFM was used to resolve the isotope and hyperfine structure of the 6s ^iSi/2-6p ^Pi/2 transition in Hg II and to measure the absolute wave number for this

line in 202Hg+.

We wish to thank D. J. Wineland, R. E. Drullinger,

F. L. Walls, and K. B. Persson for many stimulating and

useful discussions. We also thank J. Faller for the loan

of the National Bureau of Standards iodine-stabilized

He-Ne laser. We gratefully acknowledge the support

of the U.S. Air Force Office of Scientific Research and

the Office of Naval Research.

References

1. D. J. Wineland, W. M. Itano, J. C. Bergquist, and F. L. Walls, in Proceedings of 35th Annual Symposium on Frequency Control (Electronic Industries Associates,

Washington, D.C., 1981), pp. 602-611; F. G. Major and

G. Werth, Phys. Rev. Lett. 30, 1155-1158 (1973); M. D. McGuire, R. Petsch, and G. Werth, Phys. Rev. A 17, 1999-2004 (1978); M. Jardino, M. Desaintfuscien, R.

Barillet, J. Viennet, P. Petit, and C. Audoin, Appl. Phys.

24, 107-112 (1981); L. S. Cutler, R. F. Gifford, and M. D. McGuire, in Proceedings of Thirteenth Annual Precise Time and Time Interval Applications and Planning

Meeting, Washington, D.C., 1981, NASA Conf. Pub. 2220

(National Aeronautics and Space Administration,

Washington, D.C., 1982), pp. 563-578; P. L. Bender, J. L. Hall, R. H. Garstang, F. M. J. Pichanick, W. W. Smith,

R. L. Barger, and J. B. West, Bull. Am. Phys. Soc. 21, 599

(1976).

2. D. J. Wineland, R. E. Drullinger, and F. L. Walls, Phys. Rev. Lett. 40, 1639-1642 (1978).

3. R. E. Stickel, Jr., and F. B. Dunning, Appl. Opt. 17, 981-982 (1978).

4. R. E. Stickel, Jr., S. Blit, G. F. Hildebrandt, E. D. Dahl,

F. B. Dunning, and F. K. Tittle, Appl. Opt. 17, 2270

(1978).

5. R. R. Freeman, G. C. Bjorklund, N. P. Economou, P. F. Liao, and J. E. Bjorkholm, Appl. Phys. Lett. 33, 739-742

(1978).

6. J. C. Bergquist, H. Hemmati, and W. M. Itano, Opt.

Commun. (to be published).

7. T. W. Hansch and B. Couillaud, Opt. Commun. 35, 441-444 (1980).

8. B. Couillaud, Ph. Dabkiewicz, L. A. Bloomfield, and T. W. Hansch, Opt. Commun. 7, 265-267 (1982).

9. H. J. Dewey, IEEE J. Quantum Electron. QE-12, 303-306

(1976).

10. W. R. Cook, Jr., and L. M. Hubby, Jr., J. Opt. Soc. Am. 66, 72-73 (1976); F. B. Dunning and R. E. Stickel, Jr., Appl

Opt. 15,3131-3134(1976).

11. Y. Guern, A. Bideau-Mehu, R. Abjean, and A. Johannin-Gilles, Phys. Scr. 14, 273-276 (1977).

12. F. Spieweck, IEEE Trans. Instrum. Meas. IM-29, 361-363

(1980).

13. J. L. Hall and S. A. Lee, Appl. Phys. Lett. 29, 367-369

(1976).

14. B. Edlen, Metrologia 2, 71-80 (1966).

TN-119
Volume 50, Number 9 PHYSICAL REVIEW LETTERS 28 February 1983

Laser-Fluorescence Mass Spectroscopy

D. J. Wineland, J. J. Bollinger, and Wayne M. Itano Time and Frequency Divisio7i, National Bureau of Standards, Boulder, Colorado 80303

(Received 29 December 1982)

Measurements of ion cyclotron- resonance frequencies in a Penning trap, by a laser fluorescence technique, are described. This technique has been applied to indirect measurements of the proton-to-electron mass ratio and the ^Be* electron gj factor. It is found

that mp/me= 1836.152 38(62) (0.34 ppm) and^j-(^Be*) = 2.002 262 06(42) (0.21 ppm). Ultimately, ion cyclotron-resonance accuracies near 1 part in lO''^ should be possible.

PACS numbers: 06.20.Jr, 07.75. + h, 35.10.-d

The possibility of performing high-resolution mass spectroscopy in a Penning electromagnetic

trap has been realized for some time.^ Recently,

experiments^"^ have determined the proton-electron mass ratio mp/m^ by alternately storing protons and electrons in the same Penning trap apparatus and comparing their cyclotron frequencies.

In Ref. 2, cyclotron resonance is detected by

measuring ion loss from the trap after resonant

excitation. In Ref. 3, resonant excitation of cyclotron motion is detected by the increase of the

ion's orbital magnetic moment; this appears as a change in the time-of -flight spectrum when ions are ejected from the trap into an axially symmetric inhomogeneous magnetic field. In Ref. 4, ion

cyclotron resonance is detected by synchronously

observing induced currents in the ring electrode,

which is split into quadrants. The respective ac- curacies of yyip/m^ in these experiments are

± 2.9 ppm, ± 0.60 ppm, and ± 0.14 ppm. Here we

demonstrate an alternative technique for precision mass spectroscopy in a Penning trap and apply it to an indirect measurement of nip /nig. In our experiment, we measure the axial (i^^), magnetron (i^m). and electric -field- shifted cyclotron (t^c) frequencies of a small cloud of atomic

ions stored in a Penning trap by observing the

changes in ion fluorescence scattering from a laser beam which is focused onto the ion cloud as shown schematically in Fig. 1. That is, when the

ion motional frequencies are excited by an externally applied oscillating electric field, the ion

orbits increase in size causing a decrease in laser fluorescence. To a good approximation,

the electric field excites only the collective center-of-mass modes, whose frequencies are equal

to those of a single isolated ion in the trap.^ The

three measured frequencies can then be combined

to yield the free-space cyclotron frequency (i^c) from the expression®

qB^/2n>n =u, =[iu,')Uu ,^ + u^^V'^SI umts), (1) where B^ is the (uniform) applied magnetic field, q is the ion charge, and m is the ion mass.

Therefore, if f c is measured for different ions, mass comparisons can be made.

As a demonstration of the technique, we have

compared the cyclotron frequency to magneticfield-dependent nuclear-spin-flip hyperfine

?5^Gii^SERBEAM.

FIG. 1. Basic technique. A small sample of ions is confined by static magnetic and electric fields (not shown). A laser beam is focused onto the sample and

the fluorescence scattering is observed. When the cyclotron motion is excited by an externally applied

electric field (E^f) the ion orbits increase in size

which results in a decrease in fluorescence.

628

TN-120
Volume 50, Number 9 PHYSICAL REVIEW LETTERS 28 February 1983

I AMj\ =1 transition frequencies in the ^Be"^ 2s^S^/2

ground state. Approximately twenty ®Be^ ions

(spherical cloud of diam =100 Mm) were stored

in a Penning trap"* with 1.64^o =

>^o =0.417 cm.

Typical conditions were 5^ = 1.134 T and applied

trap voltage V^-l V, giving i^^.' - 1-922 MHz, u-^ = 198 kHz, and i^„ =10.2 kHz. The trap was made

of gold mesh end caps and a molybdenum- mesh

ring electrode. The center of the trap was at one

focus of an ellipsoidal mirror; the second focus was outside the vacuum system. A lens was used

to collimate the fluorescence light into a photomultiplier tube. The ions were laser cooled,

compressed, and pumped into the (Mi,Mj)^{-^,

- j) ground state by a laser tuned to the 2sSi/zi-i, - 2)- ^P^P3/2(- 2,- i) (x. = 313 nm) transition.'' The size of the cloud was determined by

using a second probe laser. The density was determined by measuring the space-charge-shifted

ExB cloud-rotation frequency via the change in

Doppler shift across the cloud by means of the

probe laser. ^ We note that this cloud-rotation

frequency is > i^„; only when space charge is

negligible are these frequencies equal. From

these measurements of cloud size and density,

ion number was determined. The magnetic-fielddependent (- I, - y)- (- T,- i) ground-state spin

flip hyperfine transition frequency (i^^) was measured by an rf-optical double-resonance technique

described elsewhere.^ In some cases, the (|-,

•j) — (t,?) transition was measured. The spinflip frequency and cyclotron frequency were

measured nearly simultaneously on the same ion

cloud by stepping rf oscillators in frequency across y^ 3-nd i^c' • That is, the cyclotron resonance was probed at one frequency near u^' and

then the spin resonance probed at a frequency

near i^^. Then the cyclotron and spin oscillators were incremented (by typically 0.25 and 2 Hz,

respectively) and the cycle repeated. For each

step of the spin and cyclotron resonances, the

laser was shut off while the resonance was driven

to avoid light shifts. Resonance curves taken in this fashion are shown in Fig. 2. This simultaneous sweeping reduced the effect of slow fluctuations in 5q (ca.± 1.5 ppm/min). Axial and magnetron resonances could be taken separately by using the same frequency- stepping technique; however, much better axial resonance curves were

observed by leaving the laser on continuously.

Potentially this causes a light shift as discussed

below. From the measured values of i^j and the

values for the ^Be"^ hyperfine constant (A) and

nuclear-to-electron ^'-factor ratio (gj/gj), the

Breit-Rabi formula was used to determine

i^A'Be')^gj{'Be')eB^/{4n»iJ, (2) where ^^(^Be*) is the electron^ factor in ®Be"

and e is the electron charge. Since ^'^(^Be'') and

I'ei^Be*) are measured under essentially the same

conditions we have

-gjeBel>n{^Be')/m, (3) where /»(^Be*) is the ^Be^ mass. To determine

A and gj/gJ, we have measured the {- ^,+\) ~ (- 5-»+ 7) and (|,- t)- (7,- i) ground-state

transition frequencies at field-independent points.''

We obtain the preliminary values >! =- 625 008-

837.048(10) Hz and ^;/av = 2. 134 779 853(2) x 10" ^

The only significant deviation from the BreitRabi formula is a shift in the effective value of A , proportional to B^^ and approximately equal to - 0.017 Hz at iBg = 1 T.^ This has a negligible ef- fect on the present experiment.

In our determination of R we have considered

the following systematic effects:

(1) Deviations of the electric potential from a pure quadratic (r^ - 2^^), The only clearly observable systematic effect was an axial anharmonicity of approximately {du^/dE Ju/^ =-0.1/eV

(Fg = 1 V). This is about 10 times larger than

that observed with a trap of solid uncompensated

electrodes.^" However, because of the small excitation required to observe the axial resonance,

typical linewidths were < 4 Hz. We have also ob3,000

Counts/s

Background ——*

3,000

Counts/s

Background •

^2 Hz

•

• a • . • •

• •

. • • .

• • *'s—

t 310286 528.0Hz 1

1 . 1 1

^1 1^0.25 Hz

. • • • . •

e • •

1 925 082.75 Hz

FIG. 2, Example cyclotron (v^') and spin (p^) resonances. These curves were taken nearly simultaneously by the technique described in the text. Sq = 1-134 T,

1^,^215 kHz.

629

TN-121
Volume 50, Number 9 PHYSICAL REVIEW LETTERS 28 February 1983

served a slight (<0.1 ppm) negative cyclotron an- harmonicity which agrees in sign and approximate

magnitude with the axial anharmonicity. For

fixed Kg, an error in l'/ or v ^ due to a fourthorder anharmonic term in the electric potential causes an error in R proportional to -Sg'-. An

error in our measurement of v ^ which is independent of B^ causes an error in ft ^B^"".

(2) Induced-charge frequency shifts.^ From the

measured numbers of ions this shift should be

<0.1 ppm and causes an error in ft '-^B^'~ for constant ion number.

(3) Optical-well shift. Since a focused laser

beam creates an optical potential well/^ we might

expect a shift in t^^. For the conditions realized

here, we estimate Ai^^ ^^z10''' and measure

At', 'u,< 10"''. For fixed v^ this shift causes an error in ft ocSg "^.

(4) Magnetic bottle shift. ^ The trap could contribute to the magnetic field inhomogeneity through

its susceptibility. Because the amount of material constituting the electrodes is small, we estimate this shift to be sm.all. It causes an error

inft ^B^-'.

(5) Light shift on spin transitions. Although the

laser is off while the ^^ power is applied we

typically repeated the i' s cycle two or four times

on each point to increase the signal-to-noise

ratio. This might cause a spin coherence between spin pulses which could then be light

shifted. Measurements indicated that such a

shift was <0.5x 10"^. This shift is independent

of^o-

(6) Space charge. Other kinds of ions were

driven from the trap with strong cyclotron excitation; therefore no space-charge shifts should

occur.^

Values of ft vs B^'^ and B^'^ were extrapolated

to the limit -Bq- «=. Axial frequencies of 292 and

215 kHz at magnetic fields of 0.673 T [(|,^) - i^,^) transition], 0.764 T, and 1.134 T were

used. The total spread of unextrapolated values was 0.4 ppm and the total spread of extrapolated

values was approximately 0.15 ppm. From these

extrapolations and our estimates of systematic

effects we find

^^(^Be')*«('Be^)/m,

= 32891.5710(49) (0.15 ppm). (4) The numbers in parentheses represent estimates

of one standard deviation. We note that we have

checked this result to 2 ppm accuracy by comparing i^e(^Be'^) to the cyclotron frequency of electrons which are alternately stored in the trap.

This result, with a theoretical value^^'" of ,^v(^Be") and the value^^ of r,i {^Be")/m p, can be

used to give an indirect determination of nip /nig.

Using Veseth's value^' for ^ji^Be'), we obtain

^>/;,/0><\, =1836.152 38(62) (0.34 ppm). (5) This value agrees with but is (0.34 ± 0.37) ppm

lower than the most precise direct determination.'* If we assume the value of lUp/ni^ from Ref.

4, we obtain

^^('Be') =2.002 262 06(42) (0.21 ppm). (6) The potential accui'acy for direct mass comparisons with use of this method is extremely

high. Assume that cyclotron resonance is performed on a single Be' ion in a trap with z^ = V^^=1 cm, fio = 6 T, f, = 150 kHz. If maximum laser cooling^^ is achieved, then ^max — 1 Mm» ''^(max),r^(niax); this implies that the ef- fects of anharmonicities are greatly suppressed.

The third- and fourth-order axial corrections to the electric potential can be nulled by proper

biasing of the end cap and guard electrodes on a compensated trap.'* The largest uncompensated

terms in the electric potential appear to be due

to residual third- and fourth- order terms which

violate axial symmetry. Using perturbation theory ,^^ we estimate these terms to be at the level

of a few parts in 10^^ (and proportional to B^^).

The induced- charge frequency shift would be at the level of a few parts in 10^^ (proportional to

-Sg"^). Magnetic bottle effects could be nulled^'' and relativistic shifts would be negligible.

Eventually, precision mass comparisons could

be made between isotopes of the same species

or. more interestingly, between different nuclear

isomers. In the case of nuclear isomers, if the

energy difference could also be determined in terms of > -ray wavelengths, a conversion factor

from wavelength to atomic mass unit would be

obtained.^'' Such techniques could also be employed to study radiation pressure forces. ^^ We gratefully acknovvledge the support of the

U. S. Office of Naval Research and the U. S. Air

Force Office of Scientific Research. We thank

R. D. Deslattes, B. N. Taylor, and R. S. Van

Dyck, Jr., for helpful comments. Special thanks

go to L. Veseth, R. Hegstrom, and S. J. Lipson

for sending us their unpublished results.

-D. J. Wineland, Bull. Am. Phys. Soc. _18, 1573

(1973).

630 TN-122
Volume 50, Number 9 PHYSICAL REVIEW LETTERS 28 February 1983

^G. Gartner and E. Klempt., Z. Phys. A 287, 1 (1978).

•'G. Graff, H. Kalinowsky, and J. Traut, Z. Phys. A

297 . 35 (1980).

*R. S. Van Dyck, Jr., and P. B. Schwinberg, Phys.

Rev. Lett. 47, 395 (1981).

^L. S. Brown and G. Gabrielse, Phys. Rev. A 25,

2423 (1982).

«W. M. Itano and D. J. Wineland, Phys. Rev. A 24,

1364 (1981).

^J. J. Bollinger and D. J. Wineland, to be published.

'W. M. Itano, uipublished calculations; S. J. Lipson,

unpublished calculations; see also N. P. Economou,

S. J. Lipson, and D. J. Larson, Phys. Rev. Lett. 38,

1394 (1977).

^D. J. Wineland, P. A. Ekstrom, and H. G. Dehmelt,

Phys. Rev. Lett. 31, 1279 (1973).

'"d. J. Wineland and H. G. Dehmelt, J. Appl. Phys.

46, 919 (1975). "A. Ashkin, Science 210. 1081 (1980).

^^gj~ 2.002 262 73(60). L. Veseth, private communication.

'Vj- = 2.002 262 84(80). R. Hegstrom, to be published.

'Vrt(^Be")/wp= 8.946 534 34(43) (0.048 ppm), B. N. Taylor, private communication (based on the January 1982

midstream atomic mass adjustment of A. H. Wapstra

and K. Bos). '^W. M. Itano and D. J. Wineland, Phys. Rev. A 25,

35 (1982).

'^L. D. Landau and E. M. Lifshitz, Mechanics (Pergamon, New York, 1960), p. 86.

'^G. Gabrielse and H. Dehmelt, Bull. Am. Phys. Soc.

26, 598 (1981).

'^R. D. Deslattes, private communication; see also

R. D. Deslattes, E. G. Kessler, W. C. Sauder, and

A. Henins, Ann. Phys. (N.Y.) 129, 378 (1980).

TN-123 631
Volume 50, Number 17 PHYSICAL REVIEW LETTERS 25 April 1983

ERRATUM

LASER-FLUORESCENCE MASS SPECTROSCOPY.

D, J. Wineland, J. J. Bollinger, and Wayne M.

Itano [Phys. Rev. Lett. 50, 628 (1983)].

The second sentence under the heading "Optical-well shift" on p. 630 should read as follows:

"For the conditions realized here, we estimate an error A/?//? = 10"'' and measure A/?//? < lO"'

due to this effect." The second to the last sentence under the heading "Light shift on spin transitions" should read as follows: "Measurements

indicated that a shift in R due to this effect was

<0.5x 10'"."

Some of the references listed at the end of the

paper are out of order. References 5, 6, 7, 8,

9, and 10 should be renumbered 6, 7, 8, 9, 10,

and 5, respectively.

TN-124

1333
IEEE Transactions on Nuclear Science, Vol. NS-30, No. 2, April 1983 1521

TIME AND FREQUENCY STANDARDS BASED ON CHARGED PARTICLE TRAPPING

Wayne M. Itano, D. J. Wineland, H. Hemmati , J. C. Bergquist, and J. J. Bollinger

Time and Frequency Division

National Bureau of Standards

Boulder, Colorado 80303

Summary

Microwave or optical frequency standards based

on internal resonance transitions of ions confined

in electromagnetic traps have the fundamental advantages of long observation times and small perturbations. These advantages are somewhat offset by low

signal to noise ratios. Work at NBS has concentrated on microwave hyperfine transitions of atomic

ions stored in Penning-type ion traps. The use of narrowband, tunable light sources for state selection and detection and for reducing the average

kinetic energy of the ions (laser cooling) is an important feature of this work. Results to -date include the fluorescence detection and cooling to about 50 mK of a single Mg ion and the observation

of a 0.012 Hz linewidth on a 300 MHz Mg hyccrfijie

transition. A frequency standard based on Hg

ions is under development. Related work, mostly

based on RF-type ion traps, is underway at several other labs.

Introduction

At present, the most accurate (reproducible)

frequency standards are based on microwave transitions of atoms or molecules. The stability of a frequency standard increases with increased Q (transition frequency divided by linewidth) and increased

signal to noise ratio. The reproducibility depends

upon control of environmental factors. Standards

based on narrow optical transitions have the advantage of higher Q for a given interaction time, for cases where the linewidth is limited by interaction

time. However, the use of such a frequency standard

to generate precise time, one of the chief applications of frequency standards, is very difficult with

current technology. The main difficulty is dividing

an optical frequency down to the RF region. Also,

high-stability optical sources are not easy to produce.

An atomic frequency standard can be either

active or passive in nature. In an active device,

such as a self-oscillating hydrogen maser, excited

atoms decay, emitting radiation with a stable fre- quency. In a passive device, such as a cesium

atomic beam, the atomic resonance frequency is probed by radiation derived from an oscillator whose

frequency is controlled in a feedback loop so that

the frequency of the radiation matches that of the

atoms. All of the proposed frequency standards

based on stored ions to be discussed in this paper

are passive devices.

Ions can be confined for long periods (as long

as days) under ultrahigh vacuum conditions in ion

traps by electric and magnetic fields. For frequency standard applications, stored ions have the combined advantages of long interaction times (hence narrow resonance lines), because both the storage

and relaxation times can be long, and small perturbations to transition frequencies. Atoms in atomic

beams also have small perturbations, but the interaction time is limited to the flight time through

the apparatus (<_ 0.01 s). Atoms can be stored

without relaxation in buffer gases or coated cells

for times up to about 1 s, but the transition fre- quencies are significantly perturbed by collisions.

The fundamental disadvantage of ion traps is the low signal to noise ratio, due to the snpll number of ions that can be stored (typically 10 or less in a trap of cm dimensions).

Several laboratories have worked on developing

a frequency standard based ar^gth^ 40.5 GHz ground

state hyperfine splitting of Hg ions sjtjV'ed in a trap of the RF quadrupole (Paul) type. State selection and detection is by optical2 pymping. Resonance light from a lamp containing Hg pumps

ions from the F=l level to the F=0 level. Resonant microwave radiation repopulates the F=l level and is detected by an increase in the resonance fluores- cence intensity. R^onance linewidths of about 1 Hz have been observed. At present, the main accuracy

limitation is the second-order Doppler (time dilation) shift, which is relatively high (about 10 ) because the average ion kinetic energy ,i^ a ^few eV. In a similar experiment on trapped Yb ions,

Blatt et al_j_ have observed a 0.06 Hz linewidth on a 12.6 GHz hyperfine transition, corresponding to a Q

t'

B = B_ 2 (required for Penning trap)

"end caps'

4>{r. 2) = A(r2 - Zz^)

^o^^2z,2

electrode surfaces generated

by 0(r, z) = const.

Figure 1. Electrode configuration for RF or Penning trap.

U. S. Government work not protected by U. S. copyright. TN-125
1522

of 2 X 10 , the highest yet obtained in microwave

spectroscopy. Various proposals have been made for optical frequency standards based on narrow transitions in atomic ions, but none of them have yet

been experimentally realized.

Experimental Methods and Results

The stored ion exper,ijnents at Bureau of Standards (NBS)"'-""-'-^ have

the National

all used Penning

traps. In a Penning trap, an outline of which is shown in Fig. 1, ions are confined by a quadrupole

electric potential and a uniform magnetic field. Since only static fields are used, cooling the- ions

to reduce the second-order Doppler shift is much

easier than in an RF trap, which uses oscillating

inhomogeneous electric fields. Confinement of an ion in a Penning trap is unstable, because collisions

with residual gas molecules increase the radial extent of the orbit. In practice, however, ions can

be confined for days even without laser cooling (to be explained below), and indefinitely with laser

cooling. Laser cooling (also called optical sideband

cooling or radiation pressure cooling) is a method

by which a beam of light can be used to damp the velocity of an atom or ion. The basic mechanism for cooling of a trapped ion by a laser beam tuned

slightly lower in frequency than a strongly allowed

resonance transition is as follows: when the velocity of the ion is directed against the laser beam, the light frequency in the ion's frame is Doppler

shifted closer to resonance so that the light scat- tering takes place at a higher rate than when the velocity is along the laser beam. Since the photons

are reemitted in random directions, the net effect, over a motional cycle, is to damp the ion's velocity, due to absorption of photon momentum. If the laser frequency is tuned above resonance, it causes

heating. The effects of frequency detuning, orientation, and intensity profile of the laser beam on laser cooling of an ion in a Penning trap have been

calculated. ^,q ,, ,^ ^ ,^ Laser cooling of Mg ' ' and Be ions has been achieved, using the strongly allowed first resonance lines. For both ions, the light sources were the second harmonics, generated in nonlinear

crystals, of CW dye lasers. It is easier to reach very low temperatures with low ion densities, be- cause of space charge induced motion. Single ions can be detected by fluorescence, as shown in Fig. 2. The four plateaus are due to the^are^ence in the trap of three, two, one, and zero ^rMg ions, which

were neutraHz£d, one by one, by Mg atoms coming

from an oven. Optical Doppler broadening measurements on a single ion indicated that the effective

temperature for , motion parallel to the laser beam was about 50 mK. Laser cooling and detection of a single trapped ion has also been reported by Neu- hauser et al."^''-^" Long relaxation times for hyperfine and Zeeman sublevels make possible a very sensitive double resonance technique. In some cases, the laser polarization and frequency can be adjusted so that most of the ions are transferred to the sublevel which is coupled most strongly to the excited state and which scatters photons at a high rate. A transition from this level induced, for example, by micro- waves, results in an interruption of the photon

scattering until the ion is pumped back to the original sublevel by weak, off-resonance transitions. The number of photons not scattered during

this period can be very large, in fact greater than

10 , so that the microwave transition can be detected with nearly 100% quantum efficiency, even though

only a small fraction of scattered photons are counted. This is important for maximizing the signal to noise ratio in a frequency standard. Figure 3 shows a hyperfine uesorj^ance obtained

on a small cloud of laser cooled Mg ions. The oscillatory lineshape results from the use of the Ramsey separated oscillatory field technique, ap- plied in the time domain. Two coherent 1.02 s RF pulses separated by 41.4 s were used to drive the transition. This resonance demonstrates the long

relaxation times possible with stored ions. Line

broadening due to magnetic field variations was eliminated by operating the trap near a magnetic

field at which the derivative of the transition

frequency with respect to field is zero.q Figure 4 shows a similar hyperfine resonance of -.^ . Two

2 s RF pulses separated by 4 s were applied.

Future Work

Details of a specific proposal for a microwave

fre(^ianci^ standard based on a hyperfine transition

of Hg ions stored^in a Penning trap have been

published previously. The main advantage of Hg is the high frequency of the transition (26 GHz). The potent-Ul accuracy is estimated to be about 1 part

in 10 . (At present, the most accurate frequency

standards are laboratory C^ atomic , beams , with an accuracy of 1 part in 10 to Ig .) For laser cooling and optical pumping of Hg , a narrowband,

tunable, CW 194 nm source is required, ^uch a source has recently been developed at NBS. The second harmonic of an argon ion 515 nm laser, gener- ated in an ADP (ammonium di hydrogen phosphate)

crystal, is sum frequency mixed with a 792 nm dye

^

t/0 100 s z

LiJ 1—z Oj-itW^

1

LlJ o n z p,_rri J-^'^l'n^.nr-Lj L^ n

UJ u ^ 1 1 u LTLn o >/ ~

C/)

LlJ ' a:o

3 JT—T'^

_1

1

1

^^^^^

ao Z^Mg^ +"Mg-^ "Mg^ +"Mg -/-^ Wl/\xTrflj^^ BACKC

» TIME

24 + Figure 2. Fluorescence from a small cloud of Mg ions. The three large steps are due to the loss of individual ions. TN-126
1523

RF Frequency - 291 996 250 ( Hz

)

25 + Figure 3. Mg hyperfine resonance.

laser in a KB5 (potassium pentaborate) crystal. The

output at 194 nm is about 2 pW, which should be enough for preliminary experiments.

Acknowledgments

We wish to thank the Air Force Office of Scientific Research and the Office of Naval Research for their support of this work.

References

1. F. G. Major and G. Werth, "High-Resolution

Magnetic Hyperfine R.^nance in Harmonically

Bound Ground-State Hg Ions," Phys. Rev. Lett. 30, 1155 (1973).

2. M. D. McGuire, R. Petsch, and G. Werth, "Precision Determinati-nuj of|_ the Gound-State Hyperfine

Separation in Hg Using the Ion-Storage

Technique," Phys. Rev. A 17, 1999 (1978).

3. M. Jardino, M. Desaintfuscien, R. Barillet, J. Viennet, P. Petit, and C. Audoin, "Frequency

Stability of a Mercury Ion Frequency Standard,"

Appl. Phys. 24, 107 (1981).

4. L. S. Cutler, R. P. Gifford, and M. D. McGuire,

"A Trapped Mercury 199 Ion Frequency Standard,"

Proc. 13th Annual Precise Time and Time Interval Applications and Planning Meeting, Washington, D. C, 1981 (NASA Conf. Publ. 2220, NASA Scientific and Tech. Info. Branch, 1982) pp. 563-578.

5. L. Maleki, "Recent Developments and Proposed

Schemes for Trapped Ion Frequency Standards,"

Proc. 13th Annual Precise Time and Time Interval Applications and Planning Meeting, Washington, D. C, 1981 (NASA Conf. Publ. 2220, NASA Scientific and Tech. Info. Branch, 1982) pp. 593-607.

6. R. Blatt, H. Schnatz, and G. Werth, "UltrahighBasolution Microwave Spectroscopy on Trapped

^'•^Yb Ions," Phys. Rev. Lett. 48, 1601 (1982).

7. H. G. Dehmelt, "Mono-Ion OsciTlator as Potential Ultimate Laser Frequency Standard," IEEE

Trans. Instrum. Meas. IM-31 , 83 (1982).

8. F. Strumia, "Analysis of New Microwave and

Optical Frequency Standards Based on Ion Storage," Proc. 32nd Annual Symposium on Frequency

Control, U. S. Army Electronics Command, Fort Monmouth, N. J., 1978 (Electronic Industries

Assoc, 2001 Eye St., N. W. , Washington, D. C. 20006, 1978) pp. 444-452.

9. P. L. Bender, J. L. Hall, R. H. Garstang, F. M.

J. Pichanick, W. W. Smith, R. L. Barger, and J.

B. West, "Candidates for Two-Photon Absorption

Optical Frequency Standards," Bull. Am. Phys.

Sec. 21, 599 (1976).

1 1 1 1 1 Q075 Hz <I0^

AAAT"^

1

y \j V

t 303 016 377.267 Hz

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

0.75 1.00 1.25 1.50 1,75 rf frequency -303 016 376 (Hz)

9 + Figure 4. Be hyperfine resonance.

D. J. Wineland, R. E. Drullinger, and F. L. Walls, "Radiation-Pressure Cooling of Bound Resonant Absorbers," Phys. Rev. Lett. 40, 1639

(1978).

R. E. Drullinger, D. J. Wineland, and J. C. Bergquist, "High-Resolution Optical Spectra of Laser Cooled Ions," Appl. Phys. 22, 365 (1980).

D. J. Wineland, J. C. Bergquist, W. M. Itano,

and R. E. Drullinger, "Double-Resonance and Optical-Pumping Experiments on Electromagnetically Confined, Laser-Cooled Ions," Opt. Lett.

5, 245 1980).

D. J. . neland and W. M. Itano, "Spectroscopy

of a Single Mg Ion," Phys. Lett. 82A, 75 (1981).

W. M. Itano and D. J. Wineland, "Precision Measurement afr the Ground-State Hyperfine Constant of "^^Mg , Phys. Rev. A 24, 1364

(1981).

D. J. Wineland anc^ W. M. Itano, ^ "Spectroscopy

of Laser Cooled Be Ions," Bull. 'Am. Phys. Soc. 27, 471 (1982); J. J. Bollinger and D. J. Wineland, unpublished.

W. M. Itano and D. J. Wineland, "Laser Cooling

of Ions Stored in Harmonic and Penning Traps,"

Phys. Rev. A 25, 35 (1982).

W. Neuhauser, M. Hohenstatt, and P. Toschek,

"Optical-Sideband Cooling of Visible Atom Cloud

Confined in Parabolic Well," Phys. Rev. Lett. 41, 233 (1978).

W. Neuhauser, M. Hohenstatt, and P. Toschek,

"Localized Visible Ba Mono-Ion Oscillator,"

Phys. Rev. A 22, 1137 (1980).

D. J. Wineland, W. M. Itano, and J. J. Bollinger, unpublished.

D. J. Wineland, W. M. Itano, J. C. Bjergquist,

and F. L. Walls, "Proposed Stored ^"^Hg Ion Frequency Standards," Proc. 35th Annual Symposium on Frequency Control, U. S. Army Electronics Command, Fort Monmouth, N. J. , 1981 (Electronic Industries Assoc. , 2001 Eye St. , N. W. , Washington, D. C. , 20006, 1981) pp. 602-611.

H. Hemmati , J. C. Bergquist, and W. M. Itano,

"Generation of CW 194 nm Radiation by Sum Frequency Mixing in an External Ring

Cavity," unpublished.

TN-127
Prog. Quaiu. Electr. 1984. Vol. 8. pp. 139-142 0079-6727 84 SO.OO + .50 Printed in Greal Britain.

FREQUENCY STANDARD RESEARCH USING STORED

IONS

D. J. WiNELAND, Wayne M. Itano. J. C. Bergquist, J. J. Bollinger and

H. Hemmati

National Bureau of Standards. Boulder. Colorado 80303. U.S.A.

Abstract -We summarize research undertaken to de\elop time and frequency standards based on

stored ions. The ion storage method for high resolution spectroscopy is also briefly compared to the methods for stored neutrals and slow atomic beams.

Key words—Atomic clocks, atomic spectroscopy, frequency standards, high resolution spectroscopy,

laser spectroscopy.

1. INTRODUCTION

The possibility of producing very cold atomic beams and stored neutrals is excitmg. Certainly

these experiments are interesting by themselves but they also point the way to some new

experiments in high resolution spectroscopy and may result in new schemes for atomic

frequency standards. The technique of stored ions also has great promise in these applications

and therefore it is useful to compare these techniques so that potential problems can be

addressed in the future. In the case of trapped ions, practical frequency standards techniques

have already been demonstrated. Stabilities comparable to or better than some commercial

standards have been achieved and the potential is clear to obtain long term stability and

accuracy of 10"^^ in a practical device. This paper is biased toward the stored ion technique but

certainly this method is not without difficult problems; moreover the potential problems with

slow atomic beams and stored neutrals might be made small in the future.

1.1. Stored Ions vs Stored Neutrals or Atomic Beams

Probably the main attraction of the stored ion techniques for high resolution spectroscopy is that the ideal of an unperturbed species at rest in space is approached to a high degree.

Specifically, charged particles such as electrons and atomic ions can be stored for long periods of

time (essentially indefinitely) without the usual perturbations associated with confinement (for example the perturbations due to collisions with walls or buffer gasses in a traditional gas cell optical pumping experiment). This advantage would also appear to carry over when the stored

ion technique is compared to the present schemes for stored neutrals. The current proposals for neutral trapping rely on strongly perturbing the internal structure of the atom (or molecule) to provide binding. This is especially clear for magnetostatic' ^

' and electrostatic'"' trapping where

there is a direct trade off between the atom's kinetic energy and internal energy. In the case of

laser traps this trade off is not quite as clear, but in traps using either induced dipole forces'^' or

radiation pressure forces.''^' rather strong ac Stark shifts accompany the trapping process. In all cases, these internal energy shifts associated with trapping may cause unwanted systematic

frequency shifts in high resolution spectroscopy. This problem can be overcome if the trap is turned offwhile the "clock"" transitions are driven, but even if maximum laser cooling is achieved on strongly allowed transitions, minimum \elocities are on the order of lOcm/s which would

limit the time the trap could be off (therefore limiting Q) and would cause an additional heating

mechanism. Of course, the confining fields of electromagnetic traps for ions can cause shifts but

these appear to be controllable down to below 10" ^^'^"^'

In principle, atoms in an atomic beam can be totally free of external fields; this is perhaps the

principal advantage of the method. We note however, that typical residual fields due to contact

potentials etc. in an atomic beam apparatus can be several mV cm. These fields might cause

139 TN-128
140 D. J. WiNELAND et al.

observable systematic frequency shifts in certain experiments. A trapped ion on the other hand

seeks the region of zero electric field; for cooled ions the average magnitude of electric field can be below ImV/cm.'*'* Perhaps the chief limitation of the atomic beam method is caused by

the net linear motion of atoms; even if Doppler cancellation schemes are used, residual first order Doppler shifts can be the limiting systematic effect. This is, for example, true for the cesium

beam frequency standard, the most accurate frequency and time standard available. For the case of trapped ions, the same ions can be confined nearly indefinitely ; therefore the net velocity averages to zero and first order Doppler shifts are nearly absent. The primary disadvantage of the stored ion technique is that the number of trapped ions is typically small—approximately 10^ ions or less for a trap of centimeter dimensions. Therefore one is usually restricted to use simple atomic systems where signal to noise can be maximized.

1.2. Scope of Paper

The remainder of the paper discusses stored ion experiments whose goal is very high

resolution spectroscopy—i.e. stored ion frequency standards: present status and future goals.

2. STORED IONS AND FREQUENCY STANDARDS

2.1. Microwave Frequency Standards

At present, the most accurate (reproducible) frequency standards are based on microwave

transitions of atoms or molecules. The stability of a frequency standard increases with increased

Q (transition frequency divided by linewidth) and increased signal to noise ratio. The

reproducibility depends upon control of environmental factors. Standards based on narrow

optical transitions have the advantage of higher Q for a given interaction time, for cases where

the linewidth is limited by interaction time. However, the use of such a frequency standard to generate precise time, one of the chief applications of frequency standards, is very difficult with

current technology. The main difficulty is dividing an optical frequency down to the RF region.

Also, high-stability optical sources are not easy to produce.

An atomic frequency standard can be either active or passive in nature. In an active device,

such as a self-oscillating hydrogen maser, excited atoms decay, emitting radiation with a stable frequency. In a passive device, such as a cesium atomic beam, the atomic resonance frequency is probed by radiation derived from an oscillator whose frequency is controlled in a feedback loop

so that the frequency of the radiation matches that of the atoms. All of the proposed frequency

standards based on stored ions to be discussed in this paper are passive devices. Ions can be confined for long periods (as long as days) under ultrahigh vacuum conditions in ion traps by electric and magnetic fields. For frequency standard applications, stored ions have

the combined advantages of long interaction times (hence narrow resonance lines), because

both the storage and relaxation times can be long, and small perturbations to transition

frequencies. Atoms in atomic beams also have small perturbations, but the interaction time is limited to the ffight time through the apparatus ( <0.01 s for room temperature atoms). Atoms

can be stored without relaxation in buffer gases or coated cells for times up to about 1 s, but the

transition frequencies are significantly perturbed by collisions. A possible disadvantage of ion

traps is the low signal to noise ratio (S/N), due to the small number of ions that can be stored. However this disadvantage can be offset by obtaining very high Q, since the stability crj,(t) of an

oscillator locked to an atomic transition scales as {S/N Q)'^. For example, if an oscillator is locked to the 26 GHz hyperfine transition in ^°

' Hg^ and a hnewidth of 0.01 Hz is obtained, then

100 stored ions can potentially give a stability crj,(T) = 6 x 10" ''^t"

^'^ "^' This would exceed any

existing passive atomic frequency standard.

Several laboratories have worked on developing a frequency standard based on the 40.5 GHz

ground state hyperfine splitting of ^^^Hg"^ ions stored in a trap of the rf quadrupole (Paul)

type.'^"^^' State selection and detection is by optical pumping. Resonance light from a lamp

TN-129
Frequency standard research using stored ion's 141

containing ^°^Hg* pumps ions from the F = 1 level to the F = level. Resonant microwave

radiation repopulates the F = 1 level and is detected by an increase in the resonance fluorescence

intensity. Resonance linewidths of about 1 Hz have been observed."^' At present, the main

accuracy limitation is the second-order Doppler (time dilation) shift, which is relatively high

(about 10"^^) because the average ion kinetic energy is a few eV. In a similar experiment on

trapped ^ ^^ Yb'*^ ions, Blatt et a/.'^"^' have observed a 0.06 Hz linewidth on a 12.6 GHz hyperfine

transition, corresponding toa2of2 x 10^\ the highest yet obtained in microwave spectroscopy.

2.2. Laser Cooling and Microwave Frequency Standards

Perhaps the chief advantage of applying laser cooling to stored ions is the suppression of

Doppler effects. Even without laser cooling, first order Doppler effects are highly suppressed

because of the long term storage. However, without laser cooling, second order Doppler shifts can be relatively large (approximately 2 x 10" '^ for room temperature Hg"^ ions) and since the

velocity distributions are non-Maxwellian—these shifts are difficult to precisely characterize. With laser cooling, temperatures < 1 K are easily obtained. The lowest temperatures

(approximately 0.01 K) have been obtained for single ions.''^~' ''' However for a microwave

frequency standard, many ions are required in order to keep the signal to noise ratio high

enough to maintain desired stability."'' ^* So far the only reported experiments with the goal of a

laser cooled microwave frequency standard are those of NBS.*'^-'^' These experiments are based on the storage of many ions (10^-10^) in a Penning trap where residual heating

mechanisms*^"' are apparently much less than in the rf trap. The Penning trap'^°' requires static electric and magnetic fields for trapping. Since the required magnetic fields are rather large

( ~ 1 T), one uses extremum points in the clock transition frequency vs magnetic field; for these conditions, stabilities below 10"^^ should be obtained.**^'^**-'^' Linewidths of approximately

0.0 1 Hz have been obtained on 300 MHz nuclear spin flip hyperfine transitions in'^ '

* ^^Mg"^ and

an oscillator has been locked to a similar transition in ^Be^, giving a stability approaching

10"'^. Operation on a 26 GHz transition in ^"'Hg"*^ is anticipated; inaccuracy of < 10~'^ and

stabilities better than 10"^^ appear possible.*^'

2.3 Optical Frequency Standards

The potential accuracy and stability of optical frequency standards, where a single isolated ion can be used to advantage (i.e. signal to noise ratio is sacrificed in favor of very high Q), are extremely high. Anticipated accuracies of 10"'** do not seem unreasonable provided that

sufficiently narrow band lasers (<10Hz) are eventually obtained.

Optical frequency standards have the basic advantage of higher Q for fixed coherent

interaction time. Dehmelt has proposed optical frequency standards based on forbidden

transitions of single, laser cooled group III A ions (Tl "^, In ^ , Ga "^, Al ^ , or B

"^ ) stored in small rf traps.'^' Penning traps or Penning/rf trap combinations might also be used.'^^' The

6^Si/2~6'^Pi/2-5^D3/2 Raman transition in Ba"^ could be used as a reference to generate a

stable infrared diff"erence frequency in a nonlinear crystal.*^-

~^' Also in Ba *, the 5^D5/2 to 5^D3/2

12//m transition*^"*' and the quadrupole-allowed 6'Si/2 to 5'D5y2 1.8 /im transition*^^' have

been proposed as standards. Other high Q optical transitions in Sr"*",*^^' Pb"^, I^, and Bi"^*^^' have been suggested for stored ion frequency standards. The two-photon'~^'^' or single photon

quadrupole"^' 5d'°6s^Si/2 to 5d^6s^^D5/2 transition in Hg"^ has also been suggested. Twophoton transitions have the advantage of being first-order Doppler free even for a cloud of many

ions, where it is impossible to satisfy the Dicke criterion at optical wavelengths. They have the disadvantage that the large fields required to drive the transition cause ac Stark shifts.

3. REFERENCES

1. W. Paul and U. Trinks, Inst. Phys. Conf. Ser. 42, 18 (1978).

2. W. H. Wing, Phys. Rev. Lett. 45, 631 (1980).

3. See for example: A. Ashkin, Science 210, 1081 (1980); V. S. Letokhov and V.G. Minogin, Phys. Rep. 73, 1 (1981), and references therein.

TN-130
142 D. J. WlNELAND et al.

4. V. G. MiNOGiN and J. Javanainen, Optics Commun. 43, 119 (1982).

5. Wayne M. Itano, L. L. Lewis and D. J. Wineland, Phys. Rev. A25, 1233 (1982).

6. D. J. Wineland, in Precision Measurements and Fundamental Constants II, edited by B. N. Taylor and W. D.

Phillips, National Bur. Stds. (U.S.), Spec. Publ. 617, (in press).

7. H. Dehmelt, I.E.E.E. Trans. Instrum. Meas. IM31, 83 (1982).

8. D. J. Wineland, W. M. Itano, J. C. Bergquist and F. L. Walls, Proc. 35f/i Annual Symposium on Frequency

Control, U.S. Army Electronics Command, Fort Monmouth, NJ, 1981 (Electronic Industries Assoc, 2001 Eye St., N. W., Washington, D. C, 20006, 1981) pp. 602-61 1. 9. F. G. Major and G. Werth, Phys. Rev. Lett. 30, 1 155 (1973).

10. M. D. McGuiRE, R. Petsch and G. Werth, Phys. Rev. A17. 1999 (1978).

1 1. M. Jardino, M. Desaintflscien, R. Barillet, J. Viennet, P. Petit and C. Audoin, Appl. Phys. 24, 107 (1981).

12. L. S. Cutler, R. P. Giffard and M. D. McGuire, Proc. 13f/i Annual Precise Time and Time Interval Applications and Planning Meeting, Washington, D. C, 1981 (NASA Conf. Publ. 2220, NASA Scientific and Tech. Info. Branch,

1982) pp. 563-578

1 3. L. Maleki, Proc. 1 3th Annual Precise Time and Time Interval Applications and Planning Meeting, Washington, D.C., 1981 (NASA Conf. Publ. 2220, NASA Scientific and Tech. Info. Branch, 1982) pp. 593-607.

14. R. Blatt, H. Schnatz and G. Werth, Phys. Rev. Lett. 48, 1601 (1982).

15. W. Neuhauser. M. Hohenstatt, P. Toschek and H. Dehmelt, Phys. Rev. All, 1137 (1980).

16. D. J. Wineland and Wayne M. Itano, Phys. Lett. 82A, 75 (1981).

17. W. Nagourney, G. Janik, and H. Dehmelt, Proc. Natl. Acad. Sci. U.S.A., 80, 643 (1983).

18. D.J. Wineland, J. C. Bergquist, R. E. Drullinger, H. Hemmati, W. M. Itano and F. L. Walls, J. de Physique

42, C8-307 (1981).

19. Wayne M. Itano, D. J. Wineland, H. Hemmati, J. C. Bergquist and J. J. Bollinger. I.E.E.E., Trans. Nuc. Sci. NS30, 1521 (1983).

20. D. J. Wineland, Wayne M. Itano and R. S. Van Dyck, Jr., in Advances in Atomic and Molecular Physics, Vol. 19,

p. 135, edited by Bederson and Bates. .Academic Press, New York.

21. Wayne M. Itano and D. J. Wineland, Phys. Rev. A24, 1364 (1981).

22. D. J. Wineland and W. M. Itano, Bull. Am. Phys. Soc. 27, 864 (1982).

23. W. Neuhauser, M. Hohenstatt, P. Toschek. and H. Dehmelt, Proc. 5th Int. ConJ. on Spectral Lineshapes, West

Berlin (1980).

24. H. Dehmelt, W. Nagourney and G. Janik, Bull. Am. Phys. Soc. 11, 402 (1982).

25. H. G. Dehmelt, J. de Physique 42, C8-299 (1981).

26. H. G. Dehmelt and H. Walther. Bull. Am. Phys. Soc. 20, 61 (1975).

27. F. Strumia, Proc. 32nd Ann. Symp. Freq. Control, Fort Monmouth, New Jersey, Electronic Industries Assoc, 2001 Eye St., N. W. Washington, D.C., p. 444 (1978).

28. P. L. Bender, J. L. Hall, R. H. Garstang, F. M. J. Pichanick, W. W. Smith, R. L. Barger and J. B. West, Bull. Am. Phys. Soc. 21, 599 (1976).

TN-131
Precision Measurements of Laser Cooled 'Be^ Ions

J.J. Bollinqer, D.J. Wineland, W.M. Itano, and J.S. Wells Time and Frequency Division, National Bureau of Standards

Boulder, CO 80303, USA

1. Introduction

The long confinement times with minimal perturbations of ion storage

techniques provide a basis for high resolution spectroscopy [1,2]. Line

Q's greater than 10^° and linewidths smaller than a few Hz have been

obtained on ground-state hyperfine transitions in atomic ions stored in

rf quadrupole traps [3-7]. The accuracy of these measurements has been

limited, to a large extent, by the second-order Doppler shift. Radiation pressure from lasers has been used to reduce the second-^rder Doppler

shift by cooling ion temperatures below 1 K for single Ba and Mg ^ions

in an rf trap [8,9] and for singl^e Mg ions and small clouds of Mg ions

in a Penning _|_trap [10-12]. ^Be ions have an electronic structure similar to Mg ions and are consequently easy to cool with a frequencydoubled dye laser (x=313 nm). This paper discusses measurements of cyclotron frequencies, g-factors, hyperfine constants, and ^on cloud

parameters which have been made on clouds of laser-cooled -Be ions.

The ^Be ions are confined by the static magnetic and electric fields

of a Penning trap and stored for hours. The trap is made of gold mesh

endcaps and a molybdenum mesh ring electrode. The center of the trap is at one focus of an ellipsoidal mirror; the second focus is outside the vacuum system. A lens is used to collimate the fluorescence light into a photomul tipl ier tube. The ions are cooled and compressed by a 313 nm narrowband source tuned to the 2s2Sj^ (M. ,M,) = (-3/2,-l/2) -< 2p2P- „ (-3/2,-3/2) transition. The 313 nm^ light source is obtained by

generating the second harmonic of the output of a single mode cw dye

laser in a 90° phase-matched crystal of rubidium dihydrogen phosphate

(RDP). The resulting power is typically 20 uW. In addition to cooling,

the 313 nm light also optically pumps the ions into the (Mj,Mj)=(-3/2,-l/2) ground state [10,11].

2. Laser-Fluorescence Mass Spectroscopy

The axial (v ), magnetron (v ), and |lectric-field-shifted cyclotron (v')

frequencies ^f a small clouifof "^Be ions stored in a Penning trap are measured by observing the changes in ion fluorescence scattering from the laser beam which is focused onto the ion cloud [13]. When the ion motional frequencies are excited by an externally applied oscillating

electric field, the ion orbits increase in size, causing a decrease in laser fluorescence due to a decrease in overlap between the ion cloud and laser beam. To a good approximation, the electric field excites only the collective center-of-mass modes, whose frequencies are equal to those of

a single, isolated ion in the trap [14]. The three measured frequencies

can then be combined to yield the free-space cyclotron frequency (v ) from the expression [15]

168

Laser Spectroscopy VI ^ ed. by H.P.

Weber and W. Luthy, TSpringer Verlag)

1983

TN-132
qB /2Tim =v = [(v'i ^0 c

'-^ c + V r (1)

where B is the applied magnetic field, q is the ion charge, and m is the

ion mass. Mass comparisons can be made by measuring v for different

ions.

This technique was demonstrated by comparing the cyclotron frequency

to magnetic-field-dependent nuclea^r-spin-f 1 ip hyperfine IamJ = 1 transition frequencies in the ^Be ground state. This, along with the Breit-Rabi formula, yielded the ratio [13]

R= g,(5Be"^)m(9Be'^)/m (2)

to 0.15 ppm. This result, with a theoretical value of g,(^Be ) [16] and

the known value [17] of m(^Be"^)/m , can be used to give an indirect

determination of m /m , p e'

m /m = 1836.152 38(62) (0.34 ppm) (3)

This value agrees with the most precise direct determination [18]. If the recent value of m /m from Ref. 18 is used, an indirect determination,

"

gj(^Be^ 2.002 262 63 (33) (0.16 ppm). (4)

is obtained which agrees with the theoretical calculations [16]. Because

of the small cloud sizes and small excitation required to observe the motional resonance, the potential accuracy of the laser fluorescence

method for mass spectroscopy is extremely high due to suppression of field inhomogeneity and trap anharmonicity effects. It is estimated that

ion cyclotron resonance accuracies near 1 part in 10^^ may ultimately be possible [13].

3. Cloud Temperature, Density, and Size

The cloud temperature, density, and size can be determined by using a second focused, frequency-doubled dye laser as a probe laser. If the probe laser is tuned from the optically pumped (-3/2,-1/2) ground state

to the 2p2p (-3/2, +1/2) state, some of the ion population is removed

from the (-377,-1/2) ground state. This results in a decrease in the fluorescence light intensity. The size of this signal depends on the overlap of the probe beam with the cloud. The spatial extent of the cloud can be determined by measuring where the depopulation transition

signal disappears as the probe laser is moved across the cloud. In this way the shape of the clouds is measured to be approximately ellipsoidal

with typical dimensions ranging from 100 to 300 um.

The ion cloud undergoes a slow t x 5 drift rotation about the z axis.

The cloud rotation frequency differs from the single ion magnetron

frequency due to the space charge of the other ions. It can be determined by measuring the change in the Doppler shift of the depopulation transition as the probe laser is moved in the radial

direction. The ion number density is then obtained from the measured

cloud rotation frequency. Measured densities are 1-2x10^ ions/cm^ and are relatively independent of the number of ions in the cloud, the trap

voltage, and other trap parameters. For the small and large cloud sizes,

this gives total ion numbers ranging from a few ions to nearly 1000 ions.

169

TN-133
The temperature of the cyclotron motion can be determined from the

full width at half maximum (fwhm) of the depopulation transition.

Cyclotron temperatures of 20 to 100 mK were obtained for almost all of

the clouds. From the measurements of the cloud size and rotation

frequency, the magnetron kinetic energy averaged over the cloud can be determined and an effective magnetron temperature can be calculated.

This temperature increases with the size of the cloud, but even for the larger clouds, it is less than 200 mK. Because the probe laser is directed perpendicular to the magnetic field, a direct measurement of the

axial temperature cannot be made. The axial motion is indirectly cooled

by collisional coupling to the cyclotron motion but is directly heated by the recoil of the scattered photons [19]. The equilibration time between

the axial and cyclotron motions is determined to be less than 100 ms for

the clouds in this experiment. The axial temperature is measured by

turning off the cooling laser, waiting a variable length of time, and

then measuring the temperature of the cyclotron motion. In this way

axial temperatures are measured to be hotter than typical cyclotron

temperatures, but not more than 200 to 300 mK for most of the clouds. A typical average temperature of 200 mK gives a second-order Doppler shift

of 3 parts in lO^^^

In a frame of reference rotating with the cloud, the ion cloud behaves

like a one-component plasma; that is, the positive charged ions behave as if they were moving in a uniform density background of negative charge

[20]. The properties of such a plasma are determined by the coupling

constant r. r equals the potential energy of nearest neighbors divided

by the thermal energy of the ions. For r's approaching 1, the plasma is called strongly coupled. Theoretical calculations predict that for r>2, the plasma should have characteristics associated with those of a liquid,

and at r sl55, a liquid-solid phase transition should take place [21]. We have measured r's on the order of 3 or 4 for many clouds and as high

as 10-15 for a few clouds. It may eventually be possible to obtain r's where an ordering of the cloud into a lattice structure may take place

[22].

4. Hyperfine Structure Measurements

By measuring the frequency difference between depopulation transitions

when the probe laser is tuned to different 2p2p states, a determination

of the 2p2p hyperfine structure and the 2p2P fine structure separation

is made [23]. The 2p2Pj^ A value is determined to be -114.4(6.0) MHz. This is the first experi!nental measurement of the 2p2p^ A value and it is

in agreement with the theoretical calculation of -116^8(2.4) MHz [24].

The zero field 2p2Pj^ - 2p2P,,p fine structure interval is determined to be 197.151(75) GHz.' In addiction, the zero field 2p2P,,2 ^ ^^^^i and 2p2p * 2s2S^ optical transitions are determined to be TL 935.3198(45)

cm"i*and 31 928.7435(40) cm "^ respectively.

The ground state hyperfine structure is determined by measuring the

(-3/2,1/2) - (-1/2,1/2) and (3/2,-1/2) - (1/2,-1/2) ground state transition frequencies at magnetic field independent points [11] (see

Fig. 1). Microwaves are used to transfer population from the optically

pumped ground state to one of the states of a field independent

transition. The transition is detecteri by a decrease in the fluorescence

light intensity when the field-independent transition is probed. Figure

2 shows the signal obtained for the (-3/2, 1/2) * (-1/2,1/2) transition.

The oscillatory lineshape results from the use of the Ramsey interference

method, which is implemented by driving the transition with two coherent

rf pulses of 0.5 s duration separated by 19 s. The performance of an

170

TN-134
(Ml, Mj)

F=1

F=2

(3/2, -1/2)

(1/2, -1/2)

(-1/2, -1/2)

(-3/2, -1/2)

Figure 1. Hyperfine structure of the ^Be 2s%!, ground state as a function of magnetic ffeld. Two f ield-indeoendent transitions

at 0.68 and 0.82 T are shown

oscillator locked to this transition is measured [25] to be comparable to the performance of a commercial Cs standard. In addition, the (-3/2,1/2)

* (-1/2, +1/2) field independent transition frequency is determined to 4x10-13 accuracy. Work on the (3/2,-1/2)^(1/2,-1/2) field-independent

transition is not completed, but preliminary measurements have determined

its frequency to 4x10 "^^ accuracy. From these two measurements,

preliminary ground-state values of A= -625 008 837.048(4) Hz (6x10-12)

and gi/gj=2.134 779 853(l)xl0 "^ (5x10-1°) g^e obtained.

19500

14500 Frequency

Figure 2. Signal obtained on the (-3/2,l/2)^(-l/2,l/2) fieldindependent transition. The sweep width is 100 mHz and the frequency

interval between points is 5 mHz. The dots are experimental and are

the average of ten sweeps; the curve is a least-squares fit

Acknowledgments

This work was supported in part by the Air Force Office of Scientific

Research and the Office of Naval Research.

References :

1. H. G. Dehmelt: Advan. Atomic and Mol . Phys. 3, 53 (1967) and 5^, 109 (1969)-

2. D. J. Wineland, W. M. Itano, and R. S. Van Dyck, Jr.: Advan. Atomic

and Mol. Phys. _19, to be published

3. H. A. Schuessler, E. N. Fortson, and H. G. Dehmelt: Phys. Rev. 187 , 5 (1969)

171

TN-135
4. F. G. Major and G. Werth: Phys. Rev. Lett. 30, 1155 (1973)

5. R. Blatt, H. Schnatz, and G. Werth: Phys. Rev. Lett. 48, 1601

(1982)

6. M. Jardino, M. Desaintfuscien, R. BariHet, J. Viennet, P. Petit,

and C. Audoin: Appi . Phys. 24, 107 (1981)

7. L. S. Cutler, R. P. Giffard, and M. D. McGuire: "Mercury -199 Trapped Ion Frequency Standard: Recent Theoretical Progress

and Experimental Results," in Proc. 37th Ann. Symp. on Freq.

Control, 1983 (Systematics General Corporation, RDl Box 352 Rt

38 Brinley Plaza, Wall Township, NJ 07-719) to be published

8. W. Neuhauser, M. Hohenstatt, P. Toschek, and H. Dehmelt, Phys. Rev. Lett. 4^, 233 (1978) and Phys. Rev. A22, 1137 (1980)

9. W. Nagourney, G. Janik, and H. Dehmelt: Proc. Nat. Acad. Sci. USA

80, 643 (1983)

10. R. E. Drullinger, D. J. Wineland, and J. C. Bergquist: Appl . Phys.

22, 365 (1980)

11. W. M. Itano and D. J. Wineland: Phys. Rev. A 24, 1364 (1981)

12. D. J. Wineland and W. M. Itano: Phys. Lett. 82A, 75 (1981)

13. D. J. Wineland, J. J. Bollinger, and W. M. Itano: Phys. Rev. Lett.

50, 628 (1983) and erratum 50, 1333 (1983)

14. D. J. Wineland and H. G. Dehmelt: J. Appl. Phys. 46, 919 (1975)

15. L. S. Brown and G. Gabrielse: Phys. Rev. A 25, 2473 (1982)

16. g,= 2.002 262 73(60), L. Veseth, private communication; and g,=

"^ 2.002 262 84(80), R. Hegstrom, to be published "^

17. m(5Be )/mp= 8.946 534 34(43) (0.048 ppm), B. N. Taylor, private

communication (based on the January 1982 midstream atomic mass adjustment of A. H. Wapstra and K. Bos)

18. R. S. Van Dyck, Jr. and P. B. Schwinberg: Phys. Rev. Lett. 47_, 395 (1981) and R. S. Van Dyck, F. L. Moore, and P. B. Schwinberg:

Bull. Am. Phys. Soc. 28, 791 (1983)

19. D. J. Wineland and W. M. Itano: Phys. Rev. A 20, 1521 (1979) and

W. M. Itano and D. J. Wineland: Phys. Rev. A 25, 35 (1982)

20. J. H. Malmberg and T. M. O'Neil: Phys. Rev. Lett. 3|, 1333 (1977)

21. S. Ichimaru: Rev. Mod. Phys. 54, 1017 (1982)

22. The behavior of the small ion cTouds in this experiment may not be exactly the same as that calculated in the thermodynamic limit.

Specifically, the liquid-solid phase transition may take place

at a value different from r = 155

23. J. J. Bollinger, J. S. Wells, and D. J. Wineland: to be published

24. S. Garpman, I. Lindgren, J. Lindgren, and J. Morrison: Z. Phys. A 276 , 167 (1976)

25. J. J. Bojlinger, W. M. Itano, and D. J. Wineland: "Laser Cooled

^Be Accurate Clock," in Proc. 37th Ann. Symp. on Freq.

Control, 1983 (Systematics General Corporation, RDl Box 352 Rt

38 Brinley Plaza, Wall Township, NJ 07719) to be published

172

TN-136
PHYSICAL REVIEW A VOLUME 31, NUMBER 4 APRIL 1985

Hyperfine structure of the 2p ^Pyi state in ^Be

•"

J. J. Bollinger, J. S. Wells, D. J. Wineland, and Wayne M. Itano

Time and Frequency Division, National Bureau of Standards,

Boulder. Colorado 80303

(Received 17 October 1984) An optical-optical double resonance technique has been used on beryllium ions stored in a Penning trap

to measure the magnetic dipole hyperfine interaction constant A^/j of the 2p ^P\/2 level in 'Be"*". The measured value of -4 1/2= - 118.6(3.6) MHz is in good agreement with theoretical calculations. The 2p^P

fine-structure splitting and the 2s^Sy2~' ^P^^i/i.ui optical transition frequencies have also been measured.

The 2p ^P hyperfine structure of the Li i electronic se- quence has attracted much theoretical interest over the past two decades (see, for example, Refs. 1-4 and references contained in Ref. 5). This is because of the computational

feasibility in a three-electron system, the importance of and

interest in polarization and correlation effects, and the availability of good experimental measurements-^"^ in ^Li. The

^1/25 ^3/2> and ^4 3/2, 1/2 hypefine constants of the 2;? ^P state

in ^Li have been measured; these constants determine the

individual spin-dipolar, orbital, and contact contributions to the 2p ^P hyperfine structure. Agreement with theoretical calculations is on the order of a few percent.' Measure- ments of the A 2/2 hyperfine constant of the 2p ^f state have

been made in 'Be+ (Ref. 8) and "8^+ (Ref. 9). In

'Be''",U3/2l has been measured to be less than 0.6 MHz.'- A Hartree-Fock central field calculation yields the value

i43/2 = — 18.7 MHz, but polarization and correlation effects contribute -1-17.2 MHz and produce a final value of

A-ii2= —1.5(1.2) MHz in agreement with the experimental measurements.' Measurements of the A\i2 and ^3/2,1/2 hyperfine constants of the 2p ^P state in 'Be"^ are useful for further tests of the theoretical calculations and are needed

to determine the individual spin-dipolar, orbital, and contact contributions to the 2p ^P hyperfine structure. This paper

reports the first measurement of the A 1/2 hyperfine constant of the 2p'^P state in 'Be"^. The measured value of ^1/2= — 118.6(3.6) MHz is in agreement with the theoretical calculation of Ref. 1 of -116.8(2.4) MHz.

Much of the experimental apparatus used in this measurement has been described in other reports. '"•" 'Be"*" ions were confined by the static magnetic and electric fields of a Penning trap in ultrahigh vacuum ( < 10~^ Pa). The uni- form magnetic field of approximately LIT nearly decoupled

the nuclear spin from the total electronic angular momentum. The confining electric field ( < 1 V/cm) had a negligi- ble perturbation ( < 1 Hz) on the 2s'^Sy2 and 2p ^Pj energy

levels of the ions. The benign environment and long con- finement times of electromagnetic traps have been used to make numerous high-precision rf and microwave measure- ments of the hyperfine structure in the ground state of

ions.'^ Here, however, an optical measurement of the first excited state hyperfine structure was accomplished by using

laser cooling to reduce the ion temperature below a few K

so that the hyperfine components of the optical transition were resolved. A narrow-band ( < 4 MHz) radiation source (power =20 piW) tuned to the low-frequency side of the

2s ^SipiMi.Mj) = ( - f, - |) - 2/^ ^P3/2( - f, - y)

(\ = 313 nm)

transition of 'Be"*" was used to cool and spatially compress

the ions and optically pump them into the (-y, — y)

ground state." The 313-nm source was obtained by fre- quency doubling the output of a single-mode cw dye laser. The resonance fiuorescence induced by this "cooling" laser was used to detect the ions." Typical ion "clouds" consist- ed of at most several hundred ions with cloud densities of 2x 10' ions/cm^ and cloud diameters of 100-300 ixm. The 2/7 ^Pi/2 hyperfine structure was measured by an

optical-optical double resonance technique with a second

frequency-doubled dye laser, denoted the probe laser (power « 1 )LtW).'^ When the probe laser was tuned in resonance with the

2s'Sy2i i)- 2p'Py2i f +|)

transition (transition A), some of the ion population was re- moved from the ( — y. — y ) ground state. This resulted in a decrease in the observed fluorescence induced by the cool- ing laser. The 25^5i/2( - y, - y)— 2/7 2Pi/2( - y, - |)

transition (transition B) was detected in a similar way. Radio-frequency (rf) radiation (frequency approximately

equal to 300 MHz) tuned to the ground-state

( — y,— y)—' ( — y, -y) transition was used to mix the (-y, -y) and (-y, -y) ground states. Owing to this

rf mixing, a decrease in the fluorescent light intensity was

also observed when the probe laser was tuned from the (-y, -y) ground state to the 2;; ^fi/2( - y. + y) state (transition C) . Figure 1 shows the transitions driven by the probe laser. Figure 2(a) shows the depopulation signals obtained when the probe laser was frequency swept through

transitions A and C.

Part of the probe dye laser beam, before doubling, was

picked off and sent into an '"I2 cell. '"I2 saturated absorption spectra were simultaneously recorded as the probe laser was frequency swept through the optical transitions to the 2p^P\/2 state (see Fig. 2). At the magnetic field used in

this experiment, line number 955 of the '^'l2 atlas''* and the next higher-frequency line (uncataloged) approximately

coincided with the 'Be"^ transitions to the A/y=— y and

Mj= +Y states of the 2p^Pi/2 manifold, respectively. The

frequency differences between the I2 hyperfine components

were measured by stabilizing two dye lasers to different I2 hyperfine components and making heterodyne difference frequency measurements on a p-i-n diode. This provided

the frequency scale from which measurements of the fre- quency differences between the transitions detected by the

31 2711 ©1985 The American Physical Society

TN-137
2712 BRIEF REPORTS 31

A j Transitions g I driven by -—C I probe laser

FIG. 1. Sketch of the energy levels of the 2s^Si/2 and Ip^Pj

states of Be''". The cooling/optical pumping transition is indicated by the dotted line. The transitions detected by the probe laser are marked by solid lines and are labeled A, 5, and C. The rf mixing

radiation serves to populate both the (A//,A/y) = ( — y, — y) and

( — y, — j) ground states.

probe laser were made. From data like that shown in Fig.

2, the frequency difference between transitions A and C was measured to be 360.4(2.4) MHz. (Error estimates in this paper are one standard deviation uncertainties.) The measured frequency difference between transitions A and B was

10787.7(5.4) MHz. In addition to random error, the error estimates include uncertainties in systematic shifts due to background slopes, overlap of neighboring transitions, and

calibration of the frequency scale. By alternately chopping

the probe and cooling laser beams (at about 30 Hz), potential light shifts were eliminated. Any ac Stark or ac Zeeman

shifts produced by the ground-state rf mixing were estimat- ed to be negligible.

The 360.4(2.4)-MHz difference between transitions A and

C is the sum of 2p ^Pi/2 hyperfme-structure contributions and the ( - y, — y)—' ( — y, - y) ground-state splitting. The ground-state contribution was separately measured by

an rf-optical double resonance technique:o,ii to be 310.286 815(30) MHz. The spHtting between the ^P^P\/2(-T'+T^ and 2/?^Pi/2(-y. +t) states is then

50.1(2.4) MHz. In addition, the measured value of the

-V--

FIG. 2. (a) Depopulation signals obtained when the probe laser was frequency swept through transitions A and C. The data are the average of eight sweeps, (b) ^^''^ saturated absorption spectra

('^^l2 scattered light intensity vs frequency) simultaneously record- ed with the depopulation signals. The frequency scale of the '^^l2 saturated absorption spectra is half the frequency scale of the 'Be"*" depopulation signals. These '^^l2 hyperfme components belong to the next higher frequency line (uncataloged) above line number 955

in the ^^\ atlas (see Ref. 14).

(-y,— y)— ( — y, — y) ground-state transition was

used, along with measured values of the ground-state hyperfme constant and ^ factors,'"'^ to determine the magnetic field, 5 = 1.136 827 7(22) T. The absolute frequencies of the optical 25^Si/2'" 2p^P\/2

transitions were determined (to less precision than the fre- quency differences) from the simultaneous recordings of the

^Be"'' signals and the '^'l2 saturated absorption signals along

with the tabulated frequencies''' of the '^^l2 lines. Transitions A, B, and C were measured relative to line number

954 [R91(3-8)] of the '"I2 atlas because it was the nearest unblended line. Similarly, the 2s^Si/2( — y, — y) —* 2/7 ^P3/2( ~ T' ~ T^ cooling transition was also measured

relative to line number 960 [P89(4-10)]. These measurements, along with the measured magnetic field, the ground-state hyperfine constant and g factors, '"'^^ and the Lande g factors for the 2p ^Pi/2 and 2p ^Py2 states, deter- mine the 2/7 ^P zero-field fine-structure interval A£, and the 2s^Si/2^ 2p^Pi/2 and 2s^Si/2-^ 2p^Py2 zero-field optical transition frequencies. We obtained

A£//2 = 197 150(64) MHz ,

v(25^5i/2—2;72/'i/2) = 31 928.7436(40) cm"' . vi2s^Sy2--2p^Py2) = ^'^925.3\9S{45) cm-^ .

These values agree with the previous, less precise measure- ments of Ref. 16. A many-body perturbation theory calculation'^'^* of A£//7 = 6.412 cm"' = 192 227 MHz is within 2.5% of the experimental measurement. The Hamiltonian for the 2p ^P manifold is given by

H = //fc -I- Hzf + Hjn + H'nh + Hz,IZe

//fs= (y)A£L- S is the fine-structure interaction. Hze — (gii-z'^ gsSz)fJ.B^ is the Zeeman interaction of the exterTN-138
31 BRIEF REPORTS 2713

nal magnetic field with the electron orbital and spin magnetic moments. We take ^5 = 2.002 319 3 (free-electron value)

and gL = l (reduced mass corrections are included in Hze)-

Hzn'^ —SifJ-Nh^ is the Zeeman interaction of the external magnetic field with -the nuclear magnetic moment, /xg and

fiN are the Bohr and nuclear magnetons, gi is the nuclear g

factor in the 2p ^P state of 'Be"^. For the precision of these measurements, gi can be taken equal to the g factor of the bare nucleus,'' g;= -0.784 955(2). Hyj^ is the hyperfinestructure interaction, //hfs = -^hfsm + -^0 consists of a magnetic dipole interaction //hfsm and an electric quadrupole in- teraction Hq. The electric quadrupole interaction does not

couple the nuclear quadrupole moment with the 2p ^P\ii states and, for the purposes of this experiment, can be

neglected, //hfsm can be parametrized by the three constants'

for /= y. Hze is the relativistic, diamagnetic, and finite nuclear mass correction to the Zeeman interaction Hze-^^'^^

If //z„, //hfs, and Hze are at first neglected, the energies

and eigenstates of //fs -I- Hze can be determined analytically. Let \J,Mj) denote an eigenstate of Jf^ and J^. The

Ip'^Pili Mj= -\-Y and My=-y eigenstates of H[s+ Hze

are, respectively.

(1 !)l/2

and

il-p') 1/2 T' -y) ply. ftp -y)• \

.1,5

Aj =

{J,Mj = J,I,M,='l\H^(sJJMj = J,I.Mi=I)

hIJ _ {J,Mj = J.IMi = l\hAjl-i\J,Mj=J.lMi = I) hIJ

for J= y,y, and the off-diagonal element

AlJ-\='

With the experimental values 5=1.1368277(22) T and

(A£'//!) = 197 150(64) MHz, the theoretically determined

y=y amplitude admixtures are a = 0.037 06(1) and

/3 = 0.039 10(1), and the states are theoretically split by a frequency f/= 10610.57(1) MHz. The effects of Hz„ + //hfsm + //ze were treated by perturbation theory.

Terms with estimated contributions of much less than 0.1 MHz were neglected. The following two equations were

obtained for the measured frequency differences,

^^m,.Mj~^m' M'y^^ in the 2p^Pi/2 manifold:

(£_3/2,i/2 — £'-1/2,1/2)/^ = 50.1(2.4) MHz (experiment)

^-i\~a^)A,f2/2-Al,2/4U + a2^Ay2.U2 + gitJ-NB/h ,

(£'_3/2,i/2-£'-3/2,-i/2)//J = 10787.7(5.4) MHz (experiment)

^U+8gy2t^BB/h-3[\-{a^ + l3^)/2]Ay2/2 + iAl2/4U+ia-fi)3^Ay2.i/2

(1)

(2)

bgy2 is the sum of the corrections to the 2p '^Py2 gj factor due to Hze- Its value was calculated by many-body pertur- bation theory to be - 10.95(3) x 10-^'*

The quantity .^1/2= - 118.6(3.6) MHz was determined

from Eqs. (1) and (2) and is in good agreement with the

theoretical calculation of ^1/2= - 116.8(2.4) MHz.' The

uncertainty is due mainly to the experimental uncertainty in the measured frequencies. A rather poor value for /1 3/2, 1/2 of /i3/2,i/2= — 19.2(28.6) MHz was also determined from

Eqs. (1) and (2). The calculation of Ref. 1 predicts

•^3/2,1/2= ~ 23.2(1.0) MHz. The precision of the ^3/2,1/2 measurement was limited by the small 0.04 amplitude ad- mixture of the 2p ^^3/2 states in the 2p ^P\i2 manifold. Increased accuracy in the hyperfine constants and an in- dependent determination of the 2p ^P\/2 gj factor can be obtained by measuring more transitions at higher magnetic

fields. At approximately 6.26 T, the 2p ^Pii2, Mj= — y and

2/7 ^Pi/2, Mj= +Y levels cross and the 2p^Py2 amplitude

admixture in the 2p ^f1/2 manifold is approximately 4.5 times larger than at 1.1 T. At a magnetic field of approximately 9.39 T, the 2p^Py2, Mj^^-j and 2p''Py2,Mj = — y levels cross and the hyperfine interaction strongly

couples states with AMi= —^.Mj. These antilevel crossings

are particularly sensitive to the off diagonal hyperfine constant ^3/2, l/2-^'*

We gratefully acknowledge the support of the U.S. Air Force Office of Scientific Research and the U.S. Office of Naval Research. We thank J. D. Prestage and R. Blatt for carefully reading the manuscript and for helpful comments.

'S. Garpman, I. Lindgren, J. Lindgren, and J. Morrison, Z. Phys. A

276, 167 (1976); Phys. Rev. A 11, 758 (1975). ^D. R. Beck and C. A. Nicolaides, in The Fourth International Confer- ence on Beam-Foil Spectroscopy, September 15-19, 1975, Gatlinburg, Tennessee, edited by I. A. Sellin and D. J. Pegg (Plenum, New

York, 1976), Vol. 2, p. 105. 3R. K. Nesbet, Phys. Rev. A 2, 661 (1970).

"J. D. Lyons, R. T. Pu, and T. P. Das, Phys. Rev. 178, 103 (1969). ^H. Orth, H. Ackermann, and E. W. Otten, Z. Phys. A 273, 221

(1975).

(•K. C. Brog, T. G. Eck, and H. Wieder, Phys. Rev. 153, 91 (1967); H. Wieder and T. G. Eck, ibid. 153, 103 (1967). 'G. J. Ritter, Can. J. Phys. 43, 770 (1965).

80. Poulsen, T. Andersen, and N. Skouboe, J. Phys. B 8, 1393

(1975). 'O. Poulsen, P. S. Ramanujam, and D. B. Iversen, J. Phys. B 8, L450 (1975).

'"D. J. Wineland, J. J. Bollinger, and W. M. Itano, Phys. Rev. Lett. 50, 628 (1983); 50, 1333(E) (1983). "W. M. Itano and D. J. Wineland, Phys. Rev. A 24, 1364 (1981).

TN-139
2714 BRIEF REPORTS 31

'2d. J. Wineland, W. M. Itano, and R. S. Van Dyck, Jr., Adv. At. Soc. 28, 780 (1983). Mol. Phys. 19, 135 (1983). '^L. Johansson, Ark. Fys. 20, 489 (1961).

13d. J. Wineland, J. C. Bergquist, W. M. Itano, and R. E. Drul- '^L. Veseth, J. Phys. B 16, 2913 (1983).

linger. Opt. Lett. 5, 245 (1980). '*L. Veseth (private communication).

'•S. Gerstenkorn and P. Luc, Atlas du spectre d'absorption de la "W. M. It&no, Phys. Rev. B 27, 1906 (1983). molecule de I'iode entre 14800-20000 cm'^ (Editions du CNRS, ^o^ Abragam and J. H. Van Vieck, Phys. Rev. 92, 1448 (1953).

Paris, 1978); Rev, Phys. Appl. 14, 791 (1979). ^'L. Veseth, Phys. Rev. A 22, 803 (1980); J. Phys. B 16, 2891

15W. M. Itano, D. J. Wineland, and J. J Bollinger, Bull. Am. Phys. (1983).

TN-140
26 October 1984, Volume 226, Number 4673

Trapped Ions, Laser

Cooling, and Better Clocks

In a recent experiment (7) at the National Bureau of Standards (NBS) in Boulder, Colorado, the frequency of a

particular hyperfine transition in the ground state of beryllium atomic ions was measured with an inaccuracy of only

about one part in 10'^ In this experiment, the ions were confined or "trapped"' in a small region of space by

D. J. Wineland

tenths of a percent. On the experimental

side, about 8 years ago a measurement

(3) of beryllium ion hyperfine structure

to an accuracy of 3 ppm (parts per million) was made. Since the accuracy of

this previous measurement is presumably good enough to satisfy the theorists for quite some time in the future, one can

logically ask why anyone would want to

Summary. Ions that are stored in electromagnetic "traps" provide the basis for extremely high resolution spectroscopy. By using lasers, the kinetic energy of the ions can be cooled to millikelvin temperatures, thereby suppressing Doppler frequency

shifts. Potential accuracies of frequency standards and clocks based on such

experiments are anticipated to be better than one part in 10^^.

using static electromagnetic fields and

their kinetic temperature was lowered to

less than 1 K by a process sometimes

called "laser cooling." In all of physics,

only a few measurements can boast a higher accuracy; those experiments measure similar transitions in neutral cesium

atoms. Experimental spectroscopy (the study

of the interaction between radiation and

matter) has traditionally provided a means of checking the theory of quan- tum mechanics which predicts the inter- nal energy structure of atoms and molecules. In the case of hyperfine structure, which represents the magnetic coupling

between the nucleus and atomic elec- trons, the theory quickly becomes very comphcated. For alkali-like ions, the most sophisticated calculations (2) agree with experiment at a level of only a few

The author is a research physicist in the Time and Frequency Division of the National Bureau of Stan- dards. Boulder, Colorado 80303.

make a better measurement. Briefly, I will give three of the reasons.

1) The primary driver in several laboratories (including NBS) is to provide

better clocks and frequency standards. The principal use of atomic clocks is in navigation and communications, where

requirements have continued to press

the state of the art. The way an atomic

clock works is perhaps apparent in the

internationally agreed on definition of

the second: "The second is the duration

of 9,192,631,770 periods of the radiation corresponding to the transition between

the two hyperfine levels of the ground

state of the cesium-133 atom." A simplified model for the practical realization of

the second involves making a device (an atomic beam apparatus) which allows one to induce and detect transitions be- tween the two ground-state hyperfine energy levels. When the frequency of the

radiation for maximum transition probability is attained, the cycles are electronically counted; when 9,192,631,770 cycles have occurred, 1 second has passed.

Several laboratories are trying to apply

this same idea to the internal energy

levels of ions; there is reason to believe

that the inaccuracy of a time standard based on stored ions can eventually be

much smaller than that of the cesium

clock, which can have an accuracy of about one part in lO'^ or less (4). 2) With the extreme accuracy attained with stored ion techniques, it may be

possible to measure various small effects which would otherwise be masked by measurement imprecision. As an example, it should be possible to measure

nuclear magnetic susceptibility as a small perturbation to atomic hyperfine

structure (5). Nuclear magnetic susceptibility, which has not been measured previously, could give a new kind of infor- mation about nuclear structure. Trapped

ions may provide a way to make the required precise measurements.

3) The system itself, laser-cooled

stored ions, is intrinsically interesting

and may provide the basis for other experiments which are only peripherally

related to spectroscopy. An example is the study of strongly coupled three-di- mensional plasmas (6). For brevity, this article will only touch on some aspects of stored ion spectros- copy where laser cooling is employed

and how they are related to better clocks and frequency standards. A recent re- view article (7) includes many interesting

trapped ion experiments which are not discussed here.

Trapped Ions

The principal attraction of the stored

ion technique is that charged particles,

including electrons and atomic ions, can

be stored for long periods of time (days

are not uncommon) without the usual

perturbations associated with confine- ment [for example, the frequency shifts associated with the collisions of ions with buffer gases in a more traditional optical pumping experiment (8)]. Storage has principally been accomplished in four types of "traps": the RF

(radio frequency) or Paul trap, the Pen26 OCTOBER 1984 TN-141

395
6 = Bo z (required for Penning trap)

•End

</.(r,z) = A(r^-2z'')

u = Uq + Vq cos nt

(Vq = for Penning trap)

Electrode surfaces generated by 0(r, z) = const.

Fig. 1. Schematic representation of the electrode configuration for the "ideal" Paul (RF) or Penning trap. Electrode surfaces are figures of revolution about the ; axis and are equipotentials of (|>(r,z) = A(r - 2z'). (Cylindrical coordinates are used with the origin at the center of the

trap.) Typical dimensions are VT Zo = 'o - ' cm. Typical operating parameters are: for the Paul trap, Vq = 300 V/cm, illl-u = 1 MHz; for the Penning trap, t/o = 1 V, B = 1 T.

ning trap, the Kingdon (electrostatic)

trap, and the magnetostatic trap ("magne'ic bottle"). Magnetic bottles have had

limited use in high-resolution work be- cause the trapping relies on spatially

inhomogeneous magnetic fields, which

can cause shifts and broadening of magnetic field-dependent lines. A notable exception is the electron g factor measurements at the University of Michigan

by H. R. Crane, A. Rich, and their colleagues (9). The Kingdon trap (10) is perhaps the simplest, using only static electric fields for trapping. Since an electrostatic potential minimum cannot exist

in a charge-free region, the Kingdon trap

relies on a dynamical equilibrium for trapping (ions orbit around an attractive

wire). Kingdon traps have been used in spectroscopic experiments by Prior and

his colleagues (//), but so far neither Kingdon traps nor magnetic bottles have

been used in laser cooling experiments,

and therefore they will not be discussed

further here. The Paul (12) or RF trap uses inhomogeneous RF electric fields to provide

confinement in a pseudopotential well (7, 13). It is the three-dimensional analog of

the Paul quadrupole mass filter. To see how it works, we first note that in a (homogeneous) sinusoidal RF electric

field, ion motion is sinusoidal but is 180° out of phase with respect to the electric force. If the field is somewhat inhomogeneous, it is easy to show that the force on the ion averaged over one cycle of the driven motion is toward the region of weaker field. Since an electric field mini- mum can exist in a charge-free region,

stable trapping can be accomplished.

Such a trap is shown schematically in Fig. 1, where the three trap electrodes

are shaped to provide an electric poten396

tial of the form (/^ - 2z^) inside the trap. For this "ideal" trap shape, an ion is bound in a nearly harmonic well; detection of the ion well frequencies can then be used to perform mass analysis.

The "ideal" Penning (14) trap uses the same electrode configuration as in Fig. 1 but uses static electric and magnetic

fields. A harmonic potential well is pro- vided along the z axis by static electric

fields. This, however, results in a radial

electric field, which forces the ions to- ward the "ring" electrode. This eflFect can be overcome if a static magnetic field B is superimposed along the c axis. In

this case the x-y motion of the ions is a composite of circular cyclotron orbits (primarily due to the B field) and a circular E x B drift magnetron motion about

the trap axis. Both the Paul and Penning traps can

provide long-term confinement. Storage

times of days are not uncommon; in the

first single-electron (Penning trap) ex- periments (15) the same electron was

used in experiments for several weeks!

This long-term storage is important in spectroscopy because (i) "transit time"

broadening (the broadening of transi- tions due to the time-energy uncertainty

relation associated with the time the ion stays in the trap) can be made negligible

and (ii) the average velocity <v> of the

ions approaches zero. The latter is important because it can make averaged

first-order Doppler frequency shifts negligible. Suppression of such first-order Doppler shifts is perhaps the chief ad- vantage over the atomic beam method;

for example, residual first-order Doppler

shifts (due to the net velocity of the atomic beam) are the main limitation to accuracy for the cesium beam frequency

standard. (Actually, even though

TN-142

<v> -^ for the traps, the first-order Doppler effect is proportional to <k-v>,

where k is the U'ave vector of the radiation. For spatially inhomogeneous radiation fields, shifts can occur. If the life- time of the ion's internal transition is long compared to the periods of ion motion, these effects result in asymmetric sidebands at the ion motional fre- quencies; however, the "pulling"

caused by these sidebands can be ex- tremely small.) The perturbation of the ions' internal structure due to trapping can be ex- tremely small. Perhaps the most trouble- some frequency shifts are caused by

electric fields. Shifts which are linearly

dependent on the electric field are absent because the average electric field <E> = 0. (We know that because if <E> f the ions would leave the trap.) Second-order shifts can be quite small. As an example, the shift of the groundstate hyperfine frequency of Hg^ ions, vq (Hg^), has been calculated (16) as 8v/vo (Hg^) s -1.4 X 10 '^£•^ wherefisin

volts per centimeter. Radio-frequency

electric fields in an RF trap may be as high as 300 V/cm; this could give a

fractional shift of 10" '-\ However, for small samples of laser-cooled ions electric fields can be smaller than 1 mV/cm

(77), yielding negligible shifts. (The re- storing electric forces become less as the

ion's kinetic energy is reduced.) Of

course, there are also shifts associated with electric fields due to ion-ion colli- sions, but these are expected to be smaller than the trapping fields (17). In most cases the magnetic field (B) of

the Penning trap (typically about 1 T or

10" gauss) causes large frequency shifts to the ions' internal structure. The ener- gy separation of the beryllium hyperfine

transition mentioned in the introduction goes to zero at zero magnetic field (/); therefore, in some sense, the energy

separation is entirely due to the external magnetic field. This might be regarded as a severe disadvantage, except that this transition and some others become inde- pendent of magnetic field to first order at certain magnetic fields. Second-order

field dependence can be small; for the beryllium example Av/vq = -0.017 (AS/

B)^. so if the field is held to 10"* of the nominal field (this can be done with a superconducting magnet), the fractional

shifts are only 1.7 x 10" '^ Thus, magnetic fields can strongly perturb the in- ternal structure but these perturbations

may only be a philosophical disadvantage. For the clock application we do not care about their existence if they can be

made reproducible and stable; moreSCIENCE, VOL. 226
over, these perturbations are physically

interesting since they can sometimes be

calculated with high precision.

A third feature of the traps may be

regarded as either a disadvantage or ad- vantage. Typically, tne number of ions

that can be trapped is rather small. Densities in the range of lO'' to 10^ cm~^ are

typical; therefore the total number of

ions may be quite small. Fortunately,

atomic ions can be sensitively detected;

this is apparent in the single-ion experi- ments discussed below. In many spectroscopic experiments it is anticipated

that the signal-to-noise ratio can ap- proach the theoretical limit (18)—that is, that it can be limited only by the statisti- cal fluctuations in the number of ions

that make the transition. We note that if we could obtain much higher densities by using larger confining fields then we

would lose one of the advantages of the technique because electric field frequen- cy shifts would become troublesome in very high resolution work.

The small sample sizes can actually be

regarded as an advantage in a couple of ways. First, the small numbers imply

that the ions can be confined to a small region of space—down to dimensions on

the order of 1 ixm or less for single ions

(19, 20). This means that field imperfections—deviations from the quadratic

electric potential or deviation from uni- formity of the magnetic field in Penning

traps—can be quite small over the ion sample. Therefore accuracy in magnetic

field-dependent studies (mass spectros- copy, g factor measurements) can be extremely high. Second, in single-photon

absorption spectra it is desirable to satis- fy the Lamb-Dicke criterion (21)—that

is, confinement to dimensions sX/2Tr,

where \ is the wavelength of the radiation. When the Lamb-Dicke criterion is satisfied, first-order Doppler effects (broadening or sideband generation) are suppressed. For optical wavelengths this condition can be met only for single confined ions. For high-resolution spectroscopy,

Paul and Penning traps have many desir- able features in common, but they differ

in some important respects. The magnetic field of the Penning trap may be a disadvantage in some experiments but it may also be the clear choice for magnetic

field-dependent studies. The RF trap is able to provide tighter spatial confine- ment and may be the best choice for

optical frequency standards (where it is desirable to satisfy the Lamb-Dicke cri- terion), although the Penning trap can

nearly satisfy the Lamb-Dicke criterion on optical transitions in certain cases.

26 OCTOBER 1984

Heating mechanisms in the Paul trap (due to the large RF fields) are typically more severe than in the Penning trap. This can be an important problem for

large numbers of stored ions.

Laser Cooling

The experiments of Dehmelt and col- laborators (13) in the 1960's showed the

ability of the stored ion technique to obtain very high resolution in atomic

spectra. Prior to 1970, the ground-state

hyperfine transition in the ^He"^

(vq s 8.7 GHz) ion was measured in an RF trap with a line width of only 10 Hz

(22). but the accuracy was limited by the second-order Doppler shift (Av/vq =

-l/2<v->/f-) to 10 Hz (c = speed of

light). The second-order Doppler effect

is due to relativistic time dilation. Be- cause the atoms are moving, their time proceeds slower than a laboratory observer; this effect must be accounted for. The relatively large second-order Doppler frequency shifts imposed by the high

velocities of the stored ions and the

difficulty of measuring their velocity dis- tribution have historically been the main

limitation to achieving high accuracy in stored ion spectra.

In 1975 proposals were made (23. 24)

to get around this general problem of the second-order Doppler shift by a process commonly called laser (or sideband)

cooling. The idea is outlined for the case

of an atom with internal (optical) transi- tion frequency Va having natural (radiative) line width Ava- Assume that the atom is constrained to move in a har- monic well along the z axis (one-dimensional model of the ion trap). Therefore

its velocity is given by i'- = vq cos l-nv^t. where Vv is its oscillation frequency in the well and we assume Vv>>Ava. When

observed along the direction of the motion, the spectrum in the laboratory contains the central resonance line (at fre- quency Va) with sidebands generated by

the first-order Doppler effect at frequencies Va + rtVv having intensity y„'^(voVa/ cvv) (with n a positive or negative inte- ger); here, /„ is the Bessel function of order n. This spectrum is a simple fre- quency modulation (FM) spectrum

where the frequency modulation is supplied by the first-order Doppler effect

(27). If we irradiate the atom with photons of frequency vl = Va + nv^. the fre- quencies of the resonantly scattered photons occur at Va and nearly symmetrically around Va at the sideband frequencies

Va ± Vv, Va ± 2vv Therefore, al- though photons of energy h(v^ + nvj

TN-1A3

are absorbed, on the average photons of energy /iVa are reqmitted; when n is negative, this energy difference causes

the kinetic energy of the atom to de- crease by «/zvv per scattering event. This explanation in terms of sidebands

(24) is easily visualized for Vv>>AvaFor all experiments done so far Ava>>Vv; however, the above conclusion is still valid. An alternative explanation (23) for this limit (note that when

Vv ^ the atoms are free) is that when

VL<Va> the atom predominantly interacts with the incident radiation when it moves toward the source of radiation and Doppler shifts the frequency into resonance—that is, when vl(1 + v./c) = Va. In the absorption process, the photon momentum is first transferred to the atom, causing its momentum to change

by h/k, where h is Planck's constant. Since the remission occurs nearly symmetrically in the ±z directions, the net

effect is to change the velocity of the atom by A\'; = h/Mk (M = mass of the atom). If Av,<<v-, then the kinetic ener- gy of the absorber decreases by an amount Mv-Av- = nhv.j. The cooling process is weak in that it takes about 10"* scattering events to do substantial cooling below room temperature; it is strong

in that, for allowed transitions, the reso- nant scattering rate can approach 10* sec '. Clearly, however, we require a

situation where a two-level system is approached in order to have 10"* scatter- ing cycles; hence we usually think in terms of simple atomic systems. Laser cooling was first observed in 1978 in experiments at both NBS (Boulder) and Heidelberg University. In the NBS experiments (25). the temperature

of Mg^ ions was monitored by observing

the induced currents (26) in the trap electrodes of a Penning trap. At magnetic fields of about 1 T a two-level system

is formed in '''Mg^ (the most abundant

isotope) by driving the 35^5i/2 (Mj =

-1/2) ^ ip^PiP. (Mj = -3/2) transition with linearly polarized light (see inset in Fig. 2). From the selection rule AMJ = 0, ±1, the ion must fall back to the original ground state. In addition,

because other allowed transitions are driven weakly, about 16/17 of the ions are pumped into the Mj = -Ml ground

state (27). In these experiments, cooling

to s40 K was observed and was limited by the noise in the induced current detection. In the first Heidelberg experiments

(28). cooling was observed through the increased storage time of Ba^ ions in a miniature Paul trap. (Separation between

the end caps 2;o - 0.5 mm.) The line

397
+ + +

+

^ + + + +

+ +

+ +

27,412 27,413

Microwave frequency (MHz)

Fig. 2. Microwave/optical double resonance spectrum of -"'Mg*. Inset shows relevant energy

levels of '"Mg* in a magnetic field. With the laser tuned to the transition shown, the ions are pumped into the ^Sy2 (Mj - -1/2) ground state and a two-level system is formed with this ground state and the excited ~P}.i2 (Mj - -3/2) state. When incident microwaves are tuned to the (Mj = - 1/2) «-^ {Mj = + 1/2) ground-sale Zeeman transition, these levels are nearly equally populated, which causes a decrease in fluorescence scattering from the ions. Transitions in other ions are detected in a similar way to this example. [From {27)]

Fig. 3. Photographic image of a single Ba* ion

(indicated by arrow) localized at the center of

a miniature RF trap izn = 0.25 mm). The

lower part of the figure is a drawing of the trap electrodes in the same orientation as in the photograph. [From (19)]

398 TN-144

that was used for cooling was the 6s^S\/2 - 6p'P]/2 transition (X. = 493 nm). A

second laser (\ = 650 nm) was required

to empty the metastable '^Dy2 state to which atoms could decay from the '^P\n state. More recently, laser cooling ex- periments on trapped ions have been

carried out at Seattle (20) and Orsay (29). In addition, laser cooling on neutral atomic beams has also been observed

(30). In both kinds of traps, it has become

customary to describe the resulting ion

kinetic energy in terms of temperature;

however, this must be interpreted with

caution. For a cloud of ions in a Penning

trap it is theoretically possible (31) to cool the cyclotron and axial energy to approximately hAvJl. Equating this en- ergy to k^T where k^ is Boltzmann's

constant, we get T = 1 mK for Avg = 43 MHz (Mg^). However, the kinetic ener- gy in the magnetron motion depends on

the space charge density and size of the cloud (6) and can be much larger than

this for more than a few ions in the trap. In a similar vein, for a cloud of ions in an RF trap, the energy of motion in the pseudopotential well can be cooled to the same limit as the cyclotron or axial motion in a Penning trap (28). However

space charge repulsion tends to push the

ions toward the edge of the cloud where

the energy in the driven motion can be

much larger than this. These problems

(17) which can cause undesirable sec- ond-order Doppler shifts can be sup- pressed in both traps by going to very

small numbers of ions—down to one.

Single Ions

In subsequent Heidelberg experiments

(19) single ions were observed in an RF

trap by laser fluorescence scattering.

Figure 3 is a photograph of a single Ba"^

ion. The size of the image determined the extent of the ion motion; therefore the temperature in the pseudopotential well was measured to be between 10 and 36 mK. [The driven motion "temperature"

will be equal to or larger than this (13,

17).] Laser cooling of single Mg^ ions in Penning (32) and RF (20) traps has also been accomplished. The lowest temperatures attained are those of the Seattle group (20), where the temperature in at least two directions of the pseudopotential motion was determined to be less than 20 mK. [Cooling in all directions

will be straightforward (19).] In both of

the magnesium experiments, the temperature was determined from the Doppler broadening on the optical cooling

SCIENCE, VOL. 226
transition. For temperatures below

about 0.1 K this Doppler broadening

contributes only a small part of the total

line width, which is now primarily due to radiative decay. Therefore, very low

temperature becomes difficult to measure. In the future this problem may be circumvented by probing narrow optical transitions as described below. In any

case, the amount of cooling that has

already been achieved gives a significant reduction in the second-order Doppler

shift correction. If we can assume, for example, that magnesium ions have been

cooled to 10 mK, then the second-order

Doppler shift correction is about one

part in 10'^

Spectra

Strongly allowed transitions, which

are desirable for the laser cooling, are perhaps not so interesting for high-resolution spectroscopy since the resolutions

are limited by the radiative line width (as

in the case of Mg* above). For high- resolution spectroscopy we usually think

in terms of optical pumping/double-reso- nance detection schemes. A simple ex- ample which is characteristic of the method is shown in Fig. 2. In this case,

the object was to detect the {Mj = - 1/2)

«-* (My = -t-1/2) ground-state Zeeman

transition. The ions are both laser-cooled and pumped into the 'S\i2 {Mj = -1/2)

ground state as discussed above. The

fluorescence (scattered) light intensity is monitored while a microwave generator

whose output is directed at the ions is frequency swept through the Zeeman

transition. When the resonance condition is satisfied the ground-state populations are nearly equalized; this causes a decrease in the scattered light, which is then the signature of the microwave res- onance. This example is illustrative of

the various detection schemes used but it

is not so interesting for high-resolution

spectroscopy since here the line width of

the transition was limited by magnetic

field fluctuations. A more interesting example is given by

the ground state (A// = -3/2, My = -H 1/2)

<-» (M/ = - 1/2, Mj = -1-1/2) nuclear spin

flip hyperfine transition of "Mg"^. At

a field of about 1.24 T, the first derivative of this transition frequency with

respect to magnetic field goes to zero; therefore the transition frequency be- comes highly insensitive to magnetic

field fluctuations. At this field, the reso- nance shown in Fig. 4 was measured (33) with a line width of only 0.012 Hz. The

oscillatory line shape results from the

26 OCTOBER 1984

291 996 251.899 Hz

1.88 1.90 1.92 RF freguency-291 996 250 (Hz)

Fig. 4. Graph of a hyperfine resonance of trapped "'Mg^ ions. The oscillatory line shape results from the use of the Ramsey

resonance method, implemented by applying two coherent RF pulses 1.02 seconds long, separated by 41.4 seconds. The solid curve is a theoretical line shape [From (32)]

use of the Ramsey method (34) in the time domain: two phase coherent RF

pulses 1.02 seconds long separated in time by 41.4 seconds were used to drive

the transition. In order to avoid light

shifts, the laser was shut oif while the RF

transition was driven. A similar transition was used for the beryllium "clock" mentioned in the in- troduction. For the beryllium case, the most important limitation to accuracy was caused by a second-order Doppler

shift. This resulted because when the

laser was off during the RF transition period (about 20 seconds), the ions were

heated slightly due to background gas

collisions. This problem can be sup- pressed in the future by using cryogenic

pumping.

Frequency Standards and Clocks

In a frequency standard or clock, measurement imprecision (SverroAo) is ap- proximately equal to (Q 5/yV)"', where

Q = vq/Avo and S/N is the signal-to-noise

ratio for detecting the number of ions

that have made the transition. If the

radiative line width is small enough, then

the experimentally observed line width

(Avo) need only be limited by the length of time taken to induce the transition. Because of this, Avo is probably independent of the species of trapped ion used. Therefore, we would like to use as high a frequency (vy) as possible in order to increase Q and reduce measurement imprecision. This is the single disadvantage

of either Mg' or Be^ ions, since the interesting "clock" transitions are only

around 300 MHz (Q > 10'"). A better

ion for a laser-cooled microwave clock is perhaps Hg^ (18) (v„ s 40 GHz for

''^'Hg*). Very important frequency stan- dard work has already been accomTN-145

plished with this ion (35), but laser cooling is much harder to achieve than for Be^ or Mg'" (partly because the 194-nm

cooling radiation is difficult to produce),

and has not been done yet. A logical extension of this idea is to go

to much higher frequency—for example,

to use a narrow optical transition. The

anticipated Q in this case can be ex- tremely high, 10'^ or more. A number of

transitions in various ions have been

proposed (7); Dehmelt (36) was the first to suggest that such extremely high resolution spectroscopy could be carried out by using single-photon transitions in, for example, single group IIIA ions. For

instance (36), the '5o •^ ^Pq transition in Tl^ (\ = 202 nm) has a ^ s 5 x 10'^ For such single-photon optical transi- tions, it is desirable to approximately

satisfy the Lamb-Dicke criterion; this is most easily accomplished with siftgle trapped ions. Others (37) have proposed

using two-photon Doppler-free transi- tions, for example the ^Sx/i —» ^D^n transitions in Hg^ (\ = 563 nm, Q =

7 X lO'''). Two-photon optical transi- tions with equal frequency photons have

the advantage of eliminating the first- order Doppler efl'ect for a cloud of many

ions, where it is impossible to satisfy the Lamb-Dicke criterion. They ultimately

have the disadvantage that the rather

large optical fields necessary to drive the

transition cause undesirable a-c Stark

shifts. Already, in experiments at Heidelberg

and Washington (38), the two-photon

"51/2 —> ^P\i2 ~* 'Di/2 stimulated Raman

transition in Ba^ has been observed. In these transitions, the first-order Doppler

effect is present; its magnitude (for co- propagating beams) is the same as that of

a single laser beam at the difference frequency. Present results (38) are limit- ed by laser line width broadening, but such transitions should also be extremely narrow; the lifetime of the "D3/2 state

in Ba''^ is 17.5 seconds (39), which would

give an intrinsic 2 of 1.6 x 10'^. For single ions, optical spectra should

give precise temperature information

through the intensity of the motional

sidebands generated by the Doppler ef- fect (35, 40). In the near future, the

resolution of the sideband structure in the transitions noted above will probably

be limited by the laser line widths. This

problem might be alleviated by driving a

stimulated Raman transition between two nearly degenerate levels in the elec- tronic ground state of the ion. In "''Mg"'^, for example, the ~S\i2 (Mj = -1/2)—

^P,I2 {Mj = -1/2) -* -5,/2 (Mj = +1/2)

transition could be driven by using two

399
laser beams separated in frequency by

the Mj = -1/2, and Mj = +1/2 groundstate frequency difference. The effects of

the laser line width would be suppressed

by generating the two laser lines with a phase modulator (4/); the line width of

the overall transition would be limited by

the ground-state lifetime. The intensity of the motional sidebands would depend

on V • (k| - ki), where ki and k: are the wave vectors of the two laser beams and

V is the ion velocity. Thus the angle

between the beams could be chosen to optimize the temperature information.

Conclusions

The projected accuracy for optical fre- quency standards is extremely high. As

an example, in In^, the line width of the

'^0 -^ ^P\ "cooling" transition is about

1.3 MHz: this implies a second-order Doppler shift of 10" " or lower. Other

systematic shifts can occur (7, 7, 13. 16,

18, 33, 35-38. 42) but it is not unreasonable to think they will be controllable to

this level. These extreme accuracies make important the problem of measure- ment imprecision, since the signai-to- noise ratio on a single ion will be about one for each measurement cycle. Practi- cally speaking, this means that a long

averaging time will be required to reach a measurement precision equal to these

accuracies. In fact, for a while, the accu- racy and resolution may be limited by

laser line width characteristics (line width and line width symmetry); however, the potential for extremely narrow

lasers also exists (43). With the great potential for the optical

frequency standards, one can logically ask why we bother thinking about RF or microwave frequency standards, where

the desired large numbers of ions (to increase signal-to-noise and measure- ment precision) causes unwanted sec- ond-order Doppler effects (17). At present, the answer concerns the utility of

optical frequency standards within current technological limitations. To use such devices as clocks as in communications and navigation, one must count

cycles of the radiation. At microwave

frequencies this is straightforward. At

optical frequencies it is technically feasi- ble but very hard (44); it has not been

done yet. To illustrate further, one might

also have asked why we do not push the frequency even higher, that is, make a clock based on narrow Mossbauer transitions in bound nuclei. Here, however,

the technological problems become even more apparent. In spite of the technical problems of making an optical "clock,"

optical frequency standards will, of

course, find many immediate uses. An

obvious class of experiments are cosmological in nature; for example, more precise measurements of the gravitational red shift will arise. In any case, the

potential accuracy for stored ion spec- troscopy in all spectral regions seems

extremely high. Frequency standards and clocks with inaccuracy of one part in 10'^ appear very reasonable; eventually

they could be orders of magnitude better than this.

References and Notes

1. J. J. Bollinger, W. M. Itano, D. J. Wineland,

Proc. 37lh Annii. Symp. Freq. Control (\9H). p. 37 (copies available from Systematics General

Corporation. Brinley Plaza, Route 38, Wall Township, N.J. 07719).

2. S. Garpman, 1. Lindgren, J. Lindgren, J. Morrison, Z. P/ivi. -4 276, 167 (1976); S. Ahmad etal., Phys. Rev. A 27, 2790(1983).

3. J. Vetter, H. Ackermann, G. ZuPutlitz. E. W,

Weber. Z. Phys. A 276. 161 (1976).

4. A. G. Mungall and C. C. Costain, IEEE Trans. Inslrum. Meas. IM-32, 224 (1983); G. Becker,

Proc. lOlh Int. Corygr. Chronomelry 2, 33

(1979); D. J. Wineland. D. W. Allan. U. J. Glaze, H. Hellwig. S. Jarvis, Jr., IEEE Trans. Instrum. Meas. IM-25, 453 (1976).

5. D. J. Larson, Bull. Am. Phys. Soc. 19, 1193

(1974).

6. J. J. Bollinger and D. J. Wineland. Phys. Rev.

Lett. 53. 348 (1984).

7. D. J. Wineland. W. M. Itano, R. S. Van Dyck,

Jr., Adv. At. Mot. Phys. 19, 135 (1983).

8. E. W. Weber, Phys. Rep. 32. 123 (1977).

9. A. Rich and J. C. Wesley, Rev. Mod. Phys. 44, 250(1972).

10. K. H. Kingdon, Phvs. Rev. 21, 408 (1923).

11. M. H. Prior and E. t. Wang, Phys. Rev. A 16, 6

(1977).

12. E. Fisher. Z. Phys. 156, 1 (1959); R. F. Wuerker, H. Shelton, R. V. Langmuir, 7. Appl. Phys. 30. 342 (1959).

13. H. G. Dehmelt, Adv. At. Mol. Phys. 3, 53

(1967); ibid. 5, 109 (1969).

14. F. M. Penning, Phvsica 3, 873 (1936).

15. P. Ekstrom and D. Wineland. 5(7. Am. 243 (No.

2), 104 (1980); D. J. Wineland, P. Ekstrom, H. Dehmelt. Phys. Rev. Lett. 31, 1279 (1973).

16. W. M. Itano, L. L. Lewis, D. J. Wineland,

Phys. Rev. A 25, 1233 (1982).

17. D. J. Wineland, Natl. Bur. Stand. (U.S.) Publ.

617. in press.

18. D. J. Wineland. W. M. Itano, J. C. Bergquist, F. L. Walls. Proc. 35th Annii. Symp. Freq. Control

(1981). p. 602 (copies available from Electronic

Industries Association, 2001 Eye Street, NW,

Washington, D.C. 20006).

19. W. Neuhauser, M. Hohenstatt, P. Toschek, H. Dehmelt. Phys. Rev. A 22, 1137 (1980).

20. W. Nagourney. G. Janik, H. Dehmelt, Proc.

Natl. Acad. Sci. U.S.A. 80, 643 (1983).

21. R. H. Dicke, Phys. Rev. 89. 472 (1953).

22. H. A. Schuessler, E. N. Fortson, H. G. Dehmelt, ibid. 187, 5 (1969).

23. T. W. Hansch and A. L. Schawlow, Opt. Commun. 13, 68 (1975).

24. D. J. Wineland and H. G. Dehmelt, Bull. Am.

Phys. Soc. 20, 637 (1975).

25. D. J. Wineland. R. E. Drullinger, F. L. Walls, Phys. Rev. Lett. 40. 1639 (1978).

26. H. G. Dehmelt and F. L. Walls, ibid. 21, 127

(1968); D. A. Church and H. G. Dehmelt, J. Appl. Phys. 40. 3421 (1969).

27. D. J. Wineland. J. C. Bergquist, W. M. Itano, R.

E. Drullinger, Opt. Lett. 5, 245 (1980).

28. W. Neuhauser, M. Hohenstatt, P. Toschek. H. Dehmelt, Phys. Rev. Lett. 41, 233 (1978).

29. F. Plumelle. Abstracts for the 15th European Group on Atomic Spectroscopy Conference,

Madrid. 5 to 8 July 1983.

30. S. V. Andreev, V. I. Balykin, V. S. Letokhov,

V. G. M\nogin. JFTP Lett. 34, 442 (1981); J. V. Prodan. W. D. Phillips. H. J. Metcalf, Phys. Rev. Lett. 49. 1149 (1982); J. L. Hall. W.

Ertmer, R. Blatt, in preparation; W. D. Phillips, Ed., "Laser-cooled and trapped atoms," Natl. Bur. Stand. (U.S.) Spec. Publ. 653 (1983).

31. W. M. Itano and D. J. Wineland. Phys. Rev. A

25,35(1982).

32. D. J. Wineland and W. M. Itano, Phys. Lett. A

82,75(1981).

33. W. M. Itano and D. J. Wineland, Phys. Rev. A

24, 1364 (1981).

34. N. F. Ramsey, Molecular Beams (Oxford Univ.

Press. London. 1956).

35. F. G. Major and G. Werth. Phys. Rev. Lett. 30, 1155 (1973); M. D. McGuire, R. Petsch, G. Wmh.Phys.Rev.A 17. 1999 (1978); M. Jardino

et al.. Appl. Phys. 24, 107 (1981); L. S. Cutler, R. P. Giflfard, M. D. McGuire. Proc. 37th Annu. Symp. Freq. Control (1983), p. 32 (copies avail- able from Systematics General Corporation,

Brinley Plaza, Route 38, Wall Township, N.J.

07719); M. Jardino, F. Plumelle, M. Desaintfuscien, J. L. Duchene. Proc. 38th Annu. Symp.

Freq. Control (Philadelphia. 1984), in press.

36. H. Dehmelt. IEEE Trans. Inslrum. Meas, IM31, 83 (1982).

37. P. L. Bender et al.. Btdl. Am. Phys. Soc. 21, 599

(1976).

38. W. Neuhauser, M. Hohenstatt. P. E. Toschek.

H. Dehmelt. in Spectral Line Shapes. B. Wende. Ed, (deGruyter. Berlin. 1981); W. Nagourney. Bull. Am. Phys. Soc. 29, 815 (1984).

39. R. Schneider and G. Werth, Z. Phys. A 293, 103

(1979).

40. H. A. Schuessler, Appl. Phys. Lett. 18, 117

(1971); F. G. Major and J. L. Duchene, 7. Phys.

(Orsay) 36, 953 (1975); H. S. Lakkaraju and H.

A. Schuessler, J. Appl. Phys. 53, 3967 (1982); M. Jardino, F. Plumelle. M. Desaintfuscien, in Laser Spectroscopy VI. H. P. Weber and W.

Luthy, Eds. (Springer-Verlag. New York.

1983), p. 173. 41. J. E. Thomas et al., Phvs. Rev. Lett. 48, 867

(1982).

42. D. J. Wineland and W. M. Itano, Bull. Am.

Phvs. Soc. 27, 864 (1982).

43. J. L. Hall, L. Hollberg, Ma Long-Shen. T. Baer, H. G. Robinson. J. Phvs. (Orsay) 42, C8-59

(1981); A. Yarivand K. Vahala, /£££y. Quant.

Electron. QE-19. 889 (1983); R. W. P. Drever et aL.Appl. Phys. B 31, 97 ( 1983); J. Hough etal..

ibid. 33, 179 (1984).

44. See Proceedings of the Third Symposium on

Frequency Standards and Metrology, J. Phys.

42, Colloque C-8, December 1981; V. P. Chebotayev et al., Appl. Phys. B 29, 63 (1982); D. A. Jennings el al.. Opt. Lett. 8. 136 (1983); K. M.

Baird, Phys. Today 36 (No. 1). 52 (1983).

45. I gratefully acknowledge the support of the

Office of Naval Research and the Air Force

Office of Scientific Research. I also thank J. J. Bollinger, J. C. Bergquist. J. Cooper, W. M.

Itano. J. D. Prestage, and S. R. Stein for helpful comments on the manuscript.

TN-146

400 SCIENCE, VOL. 226
SPECTROSCOPY OF STORED ATOMIC IONS

D. J. Wineland, Wayne M. Itano, J. C. Bergquist,

J. J. Bollinger and J. D. Prestage

Time and Frequency Division

National Bureau of Standards

325 Broadway

Boulder, Colorado 80303

I. INTRODUCTION

In this paper, we briefly review measurements of atomic ion

spectra made with the stored ion technique in the last few years. By

design, its scope is limited; for example, in these proceedings G. Werth will review experiments specifically relating to high resolution microwave spectra of stored ions and R. S. Van Dyck Jr. and D.

A. Church will discuss applications of the stored ion technique to g-2 measurements and collision studies respectively. We will also

omit the interesting excited state lifetime measurements made using

traps ^ and measurements of molecular spectra^. More comprehensive

reviews are given elsewhere-*" ; we also refer the reader to the

abstracts for this conference.

The ion storage technique for studies of atomic spectra is actually quite general and could in principle be used on all atomic

and molecular ions; in practice, however, we find that its applicability is more limited. The number of ions that c^n be stored is

typically rather small (densities -10°/cm^, trap volume -10 cm^) and

therefore the signal to noise ratio in many experiments is rather

small. This problem is compounded if the ion population is distributed over many states as in the case of molecules, but in spite of

these difficulties, trapped molecular ion spectra have been obtained

by laser induced fluorescence^. The low densities of course yield

the main advantage of the technique - that is, the perturbations on

the ions' internal structure (due to ion-ion collisions for example) are extremely small. This coupled with long storage times and methods for obtaining cold samples can lead to very high resolution and

accuracy.

II. TRAPPING

The basic methods of trapping have been discussed in the various

reviews referred to in the introduction. The Paul (or rf) trap is the most popular method; both it and the Penning trap can provide

very long trapping times (> hours). The Kingdon trap is finding

increased use (partially because of its simplicity) but the

Atomic Physics 9 , R.S. Vandyck Jr.

and E.N. Fortson,eds. , (World

Scientific, Singapore, 1985)

TN-147
4 SPECTROSCOPY OF STORED ATOMIC IONS

magnetostatic trap or magnetic bottle has not been used very much in spectroscopy because of the inhomogeneous fields required for trapping. Combinations of the above traps have been studied but not extensively used. A summary of trap methods has been given in a recent review^ and in other papers in these proceedings.

III. STATE PREPARATION AND DETECTION

Optical Pumping/Double Resonance

In the past several years the dominant method of state selection

and detection of stored atomic ions has been via optical means. More conventional optical pumping and detection with lamps ' and lasers

has been used for example to measure hyperfine spectra '. In addition, some novel optical pumping and detection effects have been

realized using lasers with trapped ions; we illustrate this with a few examples.

24

In Fig. 1, we show the 3 S Mg ions in a magnetic field.

^/2 ^"^ ^3/2 1®^®! structure of

f a laser is tuned to the (-1/2) -* (-3/2) transition (numbers 2c in parentheses refer to the ground 3 S^ ^p

and excited 3 Po/? values of Mj respectively) then we note that ions must decay to the M, rule AM, 0,±1

•ij - -1/2 ground level because of the selection

At first, one might think that after many photon

scattering events the ions are gradually pumped from the Mj = -1/2 to Mt = +1/2 ground state level because of excitation in the wings of

SCATTBffDUGHT

WTB<SnY(5s/pt)

BACKGROUND -^

+ + +

+

+ + +

+ +

27412 27413

MICROWAVE mEQUENCY IMH2) 24, Fig. 1. Microwave/optical double resonance spectrum of '^''Mg"'. Inset

shows relevant energy levels of Mg'^ in a magnetic field. With the

laser tuned to the transition shown, 15/17 of the ions are pumped

into the S^/2 C^j = -1/2) ground state and a quasi two level system

is formed with this ground state and the excited Po/p (^j = ~3/2)

state. When incident microwaves are tuned to the (Mj = -1/2) O (Mj = +1/2) ground state Zeeman transition, these levels are nearly equally

populated which causes a decrease in fluorescence scattering from the

ions. Transitions in other ions can be detected in a similar way. (From ref . 20)

.

TN-148
SPECTROSCOPY OF STORED ATOMIC IONS

Other allowed transitions (Assume the laser is linearly polarized

perpendicular to the magnetic field so that only AMj = +1 transitions

are allowed). In fact just the opposite occurs. Although excitation

in the wings of the (-1/2) - ( + 1/2) transition pumps ions out of the

(-1/2) ground state, excitation in the wings of the (+1/2) + (-1/2)

transition also occurs which tends to pump ions from the (+1/2) to (-1/2) ground state. Since this latter transition is four times

closer in frequency to the laser than the former one, the net result

is that about 16/17 of the population is pumped into the (-1/2)

ground state. This pumping is very weak, but because of the very

long relaxation times of the ions in the trap it can be very

efficient. Similar optical pumping can also be observed when

hyperfine structure is present; for example , in the case of "Mg"^

and "Be*. Novel hyperfine pumping effects have also been observed

in the excited -'S^ states of Li*. This kind of pumping is far from

universal. For example "depopulation" pumping occurs if one excites

any of the 3 S^/2 "* 3 P-i/p transitions in Mg*. In this case, which

is perhaps more typical, one must redistribute or mix the ground

state population in order to see additional scattering from the laser. In this instance one of the advantages of the traps for high

resolution spectroscopy (weak relaxation) becomes a disadvantage in

terras of observation. Solutions to this problem are: (1) provide a buffer gas for relaxation. For example, Ruster et al.^, observed

fluorescence from single Ba ions in an rf trap by relaxing the ions

against an H2 buffer gas. (2) If the number of ground (and

metastable) states is not too large, artificial relaxation can be

provided by auxiliary microwave or laser radiation.

This kind of pumping described above is important for laser

cooling and also establishes a population imbalance necessary to observe other internal transitions in the ions. In the Mg* example

above, if the scattered fluorescence light from the ions is monitored, the (Mj = -1/2) to (Mj = + 1/2) ground state Zeeman transition

(induced by microwave radiation) can be detected by observing the

decrease in fluorescence as the microwave oscillator is swept through

resonance. (Fig. 1 ) An interesting feature of this optical pumping,

double resonance detection scheme is that each microwave photon absorbed causes a change of about AN* = 2iJB(-1 /2,-3/2)/1 7B(-1/2, + 1/2)

in the number of scattered photons where B(Mt,Mt') is the transition

rate from the Mj ground state to the Mj' excited state. For Mg

ions in a magnetic field of about 1 T, AN* can be as high as 2 x 10 . This technique has sometimes been referred to as "electron shelving"

because the ion's electron is temporarily "shelved" in a state from

which the laser scattering is essentially absent.' It has been used

in all the optical pumping double resonance experiments on Mg* and Be ions stored in Penning traps. Not only is the detection sensitivity increased by this photon number amplification effect but also

by the photon energy upconversion. Perhaps noteworthy is the case of

detecting absorbed 303 MHz photons in Be* hyperfine transitions (see

below) where an energy erthancement factor of AN*X(303 MHz) /X( laser) > 10 is obtained. These impressive numbers are of course not realized

in practice since fluorescence collection efficiencies are typically

significantly less than 100$. However, a more important statement is that if (the absence of) enough scattered photons per ion (typically

2. 2) are observed before repumping takes place, then the signal to noise ratio in such experiments need only be limited by the statistiTN-149
6 SPECTROSCOPY OF STORED ATOMIC IONS

?U cal fluctuations in the number of ions that make the transition . This is the maximum signal to noise ratio possible.

Because of the extremely high detection sensitivity in such

quasi-two level systems, the ability to detect very small numbers of

ions exists; in fact single ions have been detected by several

groups ^*"~. This is perhaps not so surprising if we consider for

example a Mg ion which is cooled down (see below) to where the

Doppler broadening is smaller than the natural width and is localized

to say less than a few ym in the trap. In this case, a 0.05 wW laser

beam (X - 280 nm) focused to a beam waist Uq =• 5 um can scatter about

10' photons/s - an easily detectable signal even with modest

collection efficiency.

Recently, another interesting optical pumping scheme has been

developed for trapped negative ions^"^ ; here a population imbalance

in Zeeman substates has been created using polarization dependent

photodetachment. In these experiments, changes in trapped ion number

were detected by driving the ion motions and detecting the induced currents. This method of state selection and detection is similar to that used in past experiments on Ho* where polarization dependent

photodissociation was employed^.

Cooling

The technique of radiation pressure-^ or sideband-^^ cooling was

first demonstrated^ '^^ in 1978 (see also refs. 27 and 36). It has

become a key element in experiments whose goal is high resolution

spectroscopy because both first and second order Doppler effects are suppressed. Frequency shifts in spectra due to the second order

Doppler effect (time dilation of the ions which are moving with

respect to the lab) have historically been the main limitation to obtaining high accuracy because of the relatively high temperatures

of the ions in the trap - up to about 1 eV for ions in an rf trap. A kinetic energy of 1 eV for ions with mass M = 50 u (atomic mass

units) yields a fractional second order Doppler frequency shift of 2

x 10~ . The kinetic energy of the ions can be determined by

observing first order Doppler effect generated sidebands in microwave

spectral lines^ . It can be reduced by cooling the ions with a light buffer gas such as He •^•'^•^^•^"^^ . With these, techniques,

1 ? hn u? accuracies near one part in 1

-^ should be obtained ' but if one hopes to obtain an accuracy significantly better than the best

existing frequency standards ( a few parts in 10 for cesium beams)

laser cooling may be necessary.

The best experimental results on laser cooling (and those most easily compared to theory) have been obtained using single ions.

This is because for many trapped ions, rf heating in Paul traps (due

to the driven motion) and the kinetic energy in the magnetron motion

for Penning traps gives rise to higher effective temperatures. In the first single ion experiments, done at Heidelberg on Ba ions,

photographs were made of the trapped ion -'. The size of the image

determined the extent of the ion motion; therefore the temperature in the pseudopotential well was measured to be between 10 and 36 niK. The kinetic energy in the driven motion will be approximately the same '^ . Laser cooling of single Mg* ions in Penning and rf ' traps has also been accomplished. The lowest temperatures measured

are those of the Seattle group ' where the temperature for at least

TN-150
SPECTROSCOPY OF STORED ATOMIC IONS

two directions of the pseudopotential motion was determined to be

less than 20 mK (Fig. 5). (Cooling in all directions will be

straightforward). In both of the magnesium experiments, the temperature was determined from the observed Doppler broadening on optical

transitions. For temperatures below about 0.1 K this Doppler

broadening contributes only a small part of the total linewidth which

is now primarily due to radiative decay. Therefore, very low

temperature becomes difficult to measure. In the future this problem

can be circumvented by measuring the first order Doppler effect

generated sidebands in optical spectra as was done for microwave

spectra above. In any case, for Mg* ions cooled to 20 mK, the second

order Doppler effect is only about 2 parts in 10 ! The theory of laser cooling of trapped ions has been elaborated

for a large number of cases, starting with the original proposal-^^,

which introduced the "optical sideband" picture. Calculations based

on this picture are included in refs. 25, 35, and 53- The fundamental process is the absorption of a photon with frequency less than

that of an optical transition, followed by the emission of a photon

whose frequency, on the average, is about equal to the transition

frequency, the deficit coming out of the kinetic energy of the ion. Some of the more recent general discussions of the state of the

theory are given by Javanainen"^ and Stenholm-^'^. For reasons of

theoretical simplicity, the ions are usually considered to be

two-level systems confined by harmonic potentials, and interactions

between ions are generally ignored. This situation corresponds

fairly closely to the experiments carried out with a single ion in an

rf trap.

Of particular interest from an experimental point of view is the

Lamb-Dicke regime or limit, which corresponds to the ion being

confined to dimensions much less than an optical wavelength. In this

limit, the broadening of the optical transition due to the

first-order Doppler effect disappears.

Wineland and Itano^^ calculated cooling rates and limiting tem- peratures using perturbation theory and rate equations. They

considered the limiting cases of the trap oscillation frequency being

much greater than or much less than the natural linewidth (Y) of the

optical transition, calling them the "strong binding" and "weak

binding" cases, respectively. Experiments, so far, correspond to the

weak binding case. In this limit, the minimum terrperature is on the

order of •nY/2kg. They also treated laser cooling in a Penning trap

for the weak binding case and compared the results with experiment^ . All of these calculations were limited to low laser intensities.

Andr^ et al.^' calculated numerically the equilibrium energy

distributions for the weak binding case. In later work^ , the

kinetic energy due to the micromotion in an rf trap was included,

using a semiclassical method.

In a series of papers, Javanainen and Stenholm^^" and

Javanainen have used quantum statistical methods to study

various limiting cases. The Fokker-Planck equation for the Wigner

function was solved in the "heavy particle" limit, which corresponds

roughly to the weak binding case, for one-dimensional^' and

three-dimensional traps. The "fast particle" limit, which

corresponds to high excitation of the ionic motion, was treated by

solving a Fokker-Planck equation in the n (harmonic oscillator

quantum number) representation, for both the strong binding and weak

TN-151
8 SPECTROSCOPY OF STORED ATOMIC IONS

binding cases . The Lamb-Dicke limit was treated by solving the

coupled rate equations for the populations, for both the strong

binding and weak binding cases . Recently, results have been

obtained for the steady state^ and the dynamics , which are exact

in the Lamb-Dicke limit. These results extend and correct the

earlier work.

Laser Sources

A troublesome obstacle that is generally encountered in optical

studies of atomic ions is the radiation source. This is because the

wavelength of the requisite radiation frequently lies in the

ultraviolet (UV). While reliable continuous-wave (cw) lasers are readily available throughout the visible, there are but a few cw lasers in the U.V. region. Presently, most of the narrowband cw UV sources are generated by some nonlinear process that requires high

powers at the fundamental wavelength(s) in order to reach useful

levels of UV power. And so it is only with the development of high

power tunable dye lasers and power enhancement cavities that

narrowband and tunable UV sources with useable powers are possible '. Fortunately, very little power is required to radiatively cool ions

that are confined in electromagnetic traps at high vacuum; usually no more than several microwatts.

At present the most prominent method of generating UV radiation

is by the interaction of high power laser radiation with an optically

transparent material presenting a non-linear response. The two dominant methods are second harmonic generation (SHG), which is the

doubling of laser radiation, and sum frequency mixing (SFM), which is the generation of a higher frequency source by the mixing of the

outputs of two lower frequency lasers' . The phase matching

condition

n^ajT = n, ojh + n^ioo

must be met in order to efficiently produce radiation at too by mixing

lasers at frequencies to, and up (ooo = ui-i+ajp). n^ is the refractive

index of the nonlinear medium for the frequency in^. For second

harmonic generation, u^ = 0^2 snd the phase matching condition reduces

to ndu) = n(2u). Since all optical materials have spectral dispersion, properly oriented uniaxial or biaxial crystals are required to satisfy the phase matched condition to give efficient mixing. In experiments on the ions Mg and Be , tunable UV radiation has been

generated near the first resonance Lines (280 nm & 313 nm respectively) by SHG in KDP isomorphs (AD»P & RDP respectively). These

crystals were temperature tuned to phase match the UV and fundamental

indices (90° phase matched). In both cases the conversion efficiency

exceeded 5 x 10~ W~ so that tens of microwatts of UV radiation

could be obtained by single-passing the light from the tunable dye

laser through the crystal. For radiation pressure cooling and

optical pumping of trapped mercury ions, narrowband and tunable

radiation near the 6s S./p ~ ^P ''1/2 ^^'^^^ resonance line at 19** nm

is required. One method for producing cw radiation at ^9^ nm is by

SFM in a potassium pentaborate (KB5) crystal, the 257 nm second

harmonic of the output of a cw 515 nm argon-ion laser with the output

of a tunable cw dye laser in the 792 nm region°° (Fig. 2). The 257

TN-152
SPECTROSCOPY OF STORED ATOMIC IONS

nm second harmonic is generated in a ADP crystal that is placed in an

.11 external Cing cavity which is held in resonance at the fundamental

frequency"^. To amplify the 19^* nm power, the KB5 crystal is placed

into two intersecting power enhancement cavities. One of these

cavities increases by 15 the circulating 792 nm power and the second

cavity increases by 7 the 257 nm power. By this method 8-10

microwatts of tunable cw 19t nm power has been generated. Thermal

lensing caused by heating in the KB5 crystal due to absorption of

light at 792 nm is the present limit to achieving higher powers at ^9^ nm"^"^.

Sympathetic Cooling

So far, optical pumping and laser cooling have been achieved on only a few different ions. Laser cooling could be extended to certain other ions but in practice this may be difficult to acconqjlish because of the required laser wavelengths or "trapping" in metastable states. Ions which are difficult to cool directly can be

cooled by collisional coupling with other stored ions which are easy

to cool^ . This has been demonstrated' in experiments on Mg* where

Mg* was laser cooled and by collisions "sympathetically" cooled

^Mg* and Mg*. (We note that in subsequent experiments at NBS. it 'mpathetically cool Mg* by laser cooling "Mg ; lefinitive because the laser was tuned to the was possible to sympc this result is more heating side of the Mg* transition.) For ions in a Penning trap,

lighter ions should be held near the center of the trap by heavier

ions which are laser cooled '^. Qualitatively, this should occur because the magnetron frequency is slightly higher for heavy mass

194 nm

u'2A*

^-2 41*

SRKU FlEOUEia

IIIC OTE LASER

792nm 4 Plote

' a

SERVO CONTROL

SINLE nEOUENa

AR-KM LASER

515 nm

DOUBLIK

IIIC UVITT

257 nm

Fig. 194 nm laser source. (From ref. 68)

TN-153
10 SPECTROSCOPY OF STORED ATOMIC IONS

ions; this causes a friction force on the lighter Ions which should

push them towards the center of the trap^ ''^. At low temperatures,

the separation between species should be nearly complete'^. As an example, spectroscopy could be done on 'Be* Ions which are held at

the center of the trap and sympathetically cooled by an outer annulus

of ^ Mg* Ions which are laser cooled. The cooling laser could be

applied so that It would not spatially overlap the 'Be* ions and

therefore light shifts on the 'Be* energy levels cQuld be avoided.

In this way, extremely narrow llnewldths (<< 1 mHz) and high accuracy

spectra on Be* (or other Ions) might be obtained. For Ions In an rf

trap, sympathetic cooling may be limited by rf heating.

IV. rf AND OPTICAL SPECTRA

Radlofrequency Spectra

The ion storage method is capable in principle of achieving

extremely narrow resonance llnewldths on either rf or optical

transitions. It is much easier experimentally to observe narrow

lines on rf transitions because the natural llnewldths are negligible

and because stable, tunable oscillators are readily available.

Sub-hertz llnewldths on hyperfine transitions have been observed on several different atomic ions.

Early work on the hyperfine structures of H2* and the Is and 23

states of ^He* has been reviewed previously^'". Recent work on the

determination of ground state hyperfine structure separations by

microwave-optical double resonance is reviewed elsewhere in these

Proceedings '. We restrict our discussion to studies of negative

ions and to the high-accuracy work at the National Bureau of

Standards (NBS) on laser cooled ions.

Microwave spectra of negative atomic ions have been obtained in

a series of experiments by Larson and coworkers. These experiments

were carried out with Penning traps in magnetic fields of about 1 T. State preparation and detection were carried out by utilizing the

polarization dependence of the photodetachraent cross section.

Microwave transitions between Zeeman components of the ground P3/2 states of ^^S" and ^°0~ were observed^^'^"^ (Fig. 3). The observable

transitions were from M, - +1/2 to Mj - +3/2 and from Mj - -1/2 to Mj

- -3/2. The two transition frequencies are separated due to the

perturbation by the P1/2 state. The average of the two transition

frequencies yields the atomic gj factor, while the frequency

splitting yields an indirect value for the fine structure separation.

The magnetic field was calibrated by detecting the cyclotron

resonance of electrons in the same trap. The accuracy obtained for

the gj factors was sufficient to show the deviations from the Landd

value after correction for the anomalous moment of the electron. In further studies, different Mj conponents of the Mj - -3/2 to Mj - -1/2 transition in the P3/2 state of ^^S" were observed^^ . The

dipole and quadrupole hyperfine parameters were determined from the

frequency splittings.

TN-154
SPECTROSCOPY OF STORED ATOMIC IONS 11

17930 17,933 17936 Frequency (MHz)

Fig. 3. Microwave resonances between magnetic sublevels of the P ground state of ^^S" at about 0.96 T. (From ref. 29).

3/2

to

The ground state hyperfine constants (A) and

nuclear-to-electronic g factor ratios (gj/gj) of Mg* and ^Be"^ have

been measured with high accuracy in a series of experiments at NBS.

In these experiments, the ions were stored for long periods (hours)

in Penning traps. The ions were optically pumped and laser cooled by

light from a frequency-doubled dye laser. Radiofrequency transitions

between ground-state hyperfine Zeeman sublevels were detected by a change in the fluorescence intensity. In most cases, the resonances

are broadened by magnetic field instabilities and inhomogeneities.

However, for certain transitions, and at certain values of the

magnetic field, the first derivatives of the transition frequencies

with respect to the field are zero. If the magnetic field is sufficiently close to one of these values, a resonance can be

observed with a linewidth limited only by the finite observation

time.

In 25Mg^

(Mj = -1/2

magnetic field'near 1.2i4 T. Near this field, a resonance with a width of 0.0T2 Hz and a center frequency of 291.996 251 899(3) MHz was observed (Fig. 4). A and gj/gj were determined with much less

accuracy, because the other transitions observed were field-dependent. The results are A = -596 25^ 376(5^) Hz and gj/gj = 9.299 if84(75) x 10"^.

Similar spectroscopy has been performed with 'Be ions with an even higher degree of accuracy'^. Two field-insensitive transitions

have been observed. They are (Mj - -3/2, Mj = 1/2) to (Mj - -1/2, Mj

- 1/2) at about 0.82 T and (Mj - 3/2, Mj —1/2) to (Mj - 1/2, Mj = -1/2) at about 0.68 T. The first of these has been used as a reference for a frequency standard. The results for the constants

are A = -625 008 837.048(4) Hz and gj/gj - 2.134 779 853(1) x 10"^.

The accuracy of these constants is currently limited by the

theoretical uncertainty of the diamagnetic shift of the hyperfine

structure. The determination of the actual values (as opposed to the

, the first derivative of the (Mj = -3/2, Mj = 1/2)

Ml 1/2) transition goes to zero at a value of the

TN-155
12 SPECTROSCOPY OF STORED ATOMIC IONS

ratios) of the g factors by measuring the cyclptron frequency of the

ions is discussed in the section on mass spectroscopy.

T- — r— —1- --!"•

O

1

<rt .^*^w .^^k. 1

8 J/^

VoOEHz-//\^ c ^ y \ yo \

8 -0 / \ / \ _

tfj y Ky «M,

il

1 ..

t 291 996 251.899 Hz

1 1 1 L

1.8 1.89 1.90 1.91 1.92 RF Frequency - 291 996 250 ( Hz

)

Fig 4 ....

...J in '^^Mg at about ^ .2H T. The (Mt = -3/2, Mj - 1/2) to (Mj - -1/2, Mj = 1/2) resonance

The oscillatory lineshape results from the use of the Ramsey separated oscillatory field method, implemented by

applying two phase-coherent rf pulses 1.02 s long, separated by 41.4

s. (From ref. 21 ).

The repeatability from run to run of the frequency standard

based on the 'Be"^ hyperfine transition is less than 1 part in 10^.

Possible sources of systematic errors have been carefully investigated. At present, the largest source of error is the second-order

(relativistic time dilation) shift, equal to -( 1/2)<v^>/c^. While

the cooling laser is on, this shift is only about -3 x 10"^.

However, the light and the state-preparation microwaves must be shut

off while the rf resonance is driven, in order to prevent resonance

shifts and broadenings. The ions heat up during this period, which

is typically 20 s. The average shift is about -3 x 10"-^ and can be

calibrated by optical Doppler width measurements of the ion

temperature. The magnetic field instability of a few parts in 10 leads to a random error of about 3 x 10~ . Shifts due to thermal

radiation, electric fields, and microwave switch leakage are estimated to be below 10~ . Shifts due to collisions with residual

-1 s gas molecules are estimated to be below 10 -^. A light shift can

exist, even though the light is shut off during the rf resonance

period, if coherence survives the optical pumping period. In order

to eliminate a shift of this type, the rf phase is randomized before

the beginning of each rf resonance period. Even better frequency standard performance might be expected if

a hyperfine transition in Hg* were used, because of the higher

transition frequency . As a preliminary step toward this goal, the

ground-state Zeeman resonance of ' Hg* ions stored in a Penning trap

has been observed at NBS by microwave-optical double resonance. The

value obtained for the g factor is 2.003 17'<3(7'<), which agrees with

a theoretical calculation -^ .

Optical Spectra

In this section we review optical spectroscopy that has been

performed with atomic ions in traps. Very few such studies have been

TN-156
SPECTROSCOPY OF STORED ATOMIC IONS 13

liuorescence oi crappea ions was maae oy i observed the 1493 nm 6 ^S^/p ^o 6 ^Pi/2 (D1

trap, using a pulsed tunable dye laser as

performed, if spectroscopy is narrowly defined as the determination

of energy level separations. The first observation of laser-induced

fluorescence of trapped ions was njade by IfflSnder and Werth , They

)1 ) line of Ba"^ ions in an rf

a source. In later work,

they observed a saturation dip in the D1 fluorescence when the ions were illuminated by counterpropagating laser beams' . This feature

is not broadened nor shifted by the first-order Doppler effect. The

1 57 + hyperfine splitting of the D1 line of the -" Ba isotope was resolved

by Blatt et al.''. The average kinetic energy of the trapped ions in these experiments was a few eV, so the first-order Doppler broadening

was large (typically about 5 GHz).

Experiments at several laboratories have now demonstrated the

Doppler narrowing that can be achieved by laser cooling ,2/ ,30 ,72^ We note again that laser cooling, unlike some other Doppler reduction

techniques, can be used to eliminate second-order as well as first-order Doppler effects. Optical transitions have been observed

for which the widths are dominated by the natural linewidths, due to the reduction of the first-order Doppler broadening by laser cooling.

This degree of line narrowing has been observed on single Mg ions in a Penning trap at NBS and on single Mg* and Ba* ions in rf traps at the University of Washington (U. W. )^'^ '^°. The U.W. group obtained a value for the natural linewidth of the 280 nm 3 S.^p to 3 line, which agreed with previously published Hanle-effect measure- ments, by fitting the observed resonance profiles. (Fig, 5) A narrow spectral feature in a single, trapped Ba two-photon excitation of the 6 S^/2 " 6 P-]/2 ~ 5 D0/2 Ranian resonance was first observed in experiments at the University of

Heidelberg . This feature is potentially very narrow, since its

natural linewidth is on the order of the inverse of

-P3/2 (D2;

ion due to

lifetime, which has been measured to be 17.5 C^.O) s' 3/2

100 200 300

frequency (MHz)

400

2M, Fig. 5. Fluorescence of a single laser-cooled '^"'Mg" ion in an rf

trap as a function of the relative optical frequency. The sharp

decrease in signal above the line center is due to laser-induced

hieating. The fitted linewidth is equal to the natural linewidth to within the measurement error. (From ref. 27)

In recent work at U.W. J greatly increased resolution'

lis feature has been observed with

In these experiments, the blue (493

TN-157
1

4

SPECTROSCOPY OF STORED ATOMIC IONS

nm) laser is tuned to the low side of the 6 ^\/2 ^° ^ 1/2o

transition and the red (650 nm) laser is swept across the 5 ^-^/p ^°

6 P-|/2 transition while fluorescence at the blue wavelength is observed. What is observed is a broad resonance due to the

p p one-photon transition centered at the 5 ^,^2 to 6 ''y/2 transition

with a narrow dip which occurs at the two-photon resonance (i.e.,

when the frequency difference between the blue and red lasers is equal to the 6 S^/2 to 5 D0/2 transition frequency). The

two-photon resonance is broadened, but to a high degree not shifted

by the laser intensities^ . The observed linewidth of the dip is about 5 MHz and is due to the laser frequency widths. This is less

than the natural linewidth of the 6 ^i/p state, which is about 21 MHz. This type of two-photon resonance has been observed previously

in other atomic systems''.

The isotope and hyperfine shifts of the D2 line of Mg* were measured in experiments at NBS' . In these experiments, which were carried out in a Penning trap, one laser was tuned to the low side of

the Mg"*" component, to continuously cool the ion cloud, while the

fluorescence induced by a lower-power laser was observed as its

frequency was swept. Laser cooling was particularly useful for this

measurement because it allowed full resolution of the optical isotope

structure which is normally obscured by the room temperature Doppler

width. These measurements are in agreement with those made by other

methods.

The first measurement of the magnetic dipole hyperfine constant

of the 2 ^Pi/p excited state of ^Be* was made recently at NBS^^'°°.

l^

1/2

M| = -1/2 M|=-3/2

•• •• ^ • • •• «• ^

•• • - ^

•^;

-•— 110 MHz

•t Fig. 6. Optical-optical double resonance signals corresponding to the 2 ^S^/2 (Mj = -3/2, Mj = -1/2) to 2 ^P^/a (Mj » -3/2, Mj - +1/2)

and the 2 S^^^ (Mj = -1/2, Mj = -1/2) to 2 ?-^/2 (Mj - -1/2, Mj - +1/2) transitions at about 1.14 T in a cloud of laser-cooled 'Be*

ions. The signal observed is the fluorescence due to a laser tuned

to the 2 ^S^/2 (^I = "3/2, Mj - -1/2) to 2 ^P3/2 (Mj - -3/2, Mj - -3/2) transition as a function of thfe frequency of another,

lower-power laser. Note that the room temperature Doppler width

would be about 4 GHz. (From ref. 80)

TN-158
SPECTROSCOPY OF STORED ATOMIC IONS 1

5

The Ions were confined In a Penning trap at a magnetic field of about

1.14 T and laser-cooled to about 0.5 K by excitation of a hyperfine-Zeeman conponent of the 313 nm 2 ^S^/g ^° 2 ^^3/2 (D2) line

with a frequency-doubled dye laser. Several hyperfine-Zeeman

components of the 2 S^/g to 2 P1/2 (D1) line were probed with a second frequency-doubled dye laser (Fig. 6). The resonance

frequencies of these components were measured by comparing the dye

laser frequencies to the frequencies of_l 'Ig hyperflne components. The relative frequencies of the 'l^ components were determined by laser heterodyne measurements. The laser cooling was required in order to resolve the hyperflne components of the D1 line.

The value obtained for A(2 P-i/g) was -118.3(3.6) MHz, which is in good agreement with theoretical calculations.

Certainly, the few experiments described here have only Just

begun to exploit the possibilities for high resolution optical

spectroscopy with trapped ions. The fundamental advantages are the same as for microwave spectroscopy (long observation times with small

perturbations and elimination of Doppler effects by laser cooling).

There is the additional advantage of higher fractional resolution for

a given measurement period, due to the higher transition frequencies.

V MASS SPECTROSCOPY

In addition to the study of spectra due to the internal energy

levels in atomic Ions, the possibility also exists to perform mass spectrum analysis using the traps. The Paul (or rf) trap is the

three dimensional analogue of the Paul rf quadrupole mass spectrometer which is commonly used for residual gas analysis. For high

resolution studies, measuring mass ratios by comparing the cyclotron

frequencies of different ions in the same magnetic field has been

more successful.

For several years, trapped ion cyclotron resonance (ICR)

spectrometers have been used by chemists to yield mass spectra"^ "°^. Trapped ICR spectrometers are basically Penning traps with rectangular shaped electrodes. If seraiempirical fits are made to the

cyclotron resonances in order to account for ion number dependent

effects such as space charge, mass determinations near 1 ppm accuracy

are possible"'. However, Penning traps with hyperbolic electrodes

would seem to be better for very high resolution work for the

following reason: To the extent that the electric potential inside

the trap is quadratic and the magnetic field (B) homogeneous, the

(modified) cyclotron, axial, and magnetron motions are harmonic

(neglecting relatlvistic effects) and their frequencies (v^,', v^ v_ respectively) are related by : and

respectively) are related by

^c ^c ^z ^m • ^ ' where v^ is the "unmodified" cyclotron frequency in a magnetic field

B. In principle, a quadratic electric potential is guaranteed if the electrodes are equipotentials of the function 41 » A(r - 2z )

This case is more nearly satisfied for the Penning traps with

hyperbolic electrodes than for the typical ICR cells. Therefore, we could expect higher resolutions and accuracy In the Penning traps

because the higher order terms in the potential (which give rise to anharmonlc frequency shifts and corresponding uncertainties) would be

TN-159
1

6

SPECTROSCOPY OF STORED ATOMIC IONS

less. Additionally, sangjle sizes for ions and electrons in Penning

traps can be quite small (< nm^), therefore suppressing the effects

of magnetic field inhomogeneitles and electric field inper feet ions.

Five recent experiments have demonstrated the usefulness of the

Penning trap for mass spectroscopy. In the three experiments of

refs. 87-89, direct measurements of the electron/proton mass ratio

(m /m ) were made by comparing the cyclotron frequencies of electrons

and protons in the same Penning trap apparatus; this direct

measurement of such widely different masses would be rrearly

impossible in a more conventional mass spectrometer' . These three experiments are very similar in principle; they

differ in the method of detection. For example GSrtner and Klempt

(2.9 ppm accuracy) detect electron/ion cyclotron resonance by

measuring electron/ion loss from the trap after resonant

excitation '. In the work of Graff et al. (0.6 ppm accuracy),

resonant excitation of electron/proton cyclotron motion is detected

by the increase of the electron/proton orbital magnetic moment; this

appears as a change in the time of flight spectrum when the

electrons/protons are ejected out one endcap into an axially

symmetric inhoraogeneous magnetic field.

The most accurate measurements (O.OM ppm) are those of Van Dyck

et al.^'^ . This accuracy comes about primarily because the

experiment is more sensitive to cyclotron excitation, therefore

anharmonic and relativistic effects in the spectra are less. In this

experiment, v^', v^ and v^^ are separately determined yielding v^ via

Eq. 1. ^2 ^^ measured by observing the spectrum of induced currents

in the endcap electrodes. The proton cyclotron resonance is observed

by splitting the ring electrode into quadrants, exciting the

cyclotron motion by applying ac voltage across two of the quadrants

and detecting the induced currents in the other two quadrants in a bridge arrangement. A resulting 76.4 MHz synchronously detected

signal is shown in Fig. 7 where the line is only 0.2 Hz (2.5 ppb)

wide. Recently linewidths on the order of 30 mHz have been

Si observed-^ . Electron cyclotron resonance is detected using the

magnetic bottle technique, and v„ is measured via sideband structure ^ Qq m on the V and v ' spectra . The present uncertainty in "ig/m^ (0.04

ppm) is limited by the uncertainty in the respective positions of the

electrons and protons (and therefore the respective average magnetic

field) when they are alternately stored in the trap.

In the experiments of Wineland et al.'-*, Vp ' , v and v for 'Be

ions in a Penning trap were determined by observing the changes in ion fluorescence scattering from a laser beam which is focused onto

the cooled ion cloud. That is, when the ion motional frequencies are excited by an externally applied oscillating electric field, the ion

orbits increase in size causing a decrease in ion cloud/laser beam

overlap which results in a decrease in laser fluorescence. In these

experiments, the resulting value of v (from Eq. 1) was compared to the 'Be* electron spin flip frequency to 0.15 ppm accuracy, limited

by a rather large anharmonic term in the electric potential. This

result could be viewed as giving an indirect measurement of fOg/m^ (to

0.2 ppm) if a theoretical value of gj('Be*) is assumed or it could be

viewed as yielding a measurement of gj('Be*) (to 0.15 ppm) by using

the Van Dyck et al., measurement of m /m . c p

TN-160
SPECTROSCOPY OF STORED ATOMIC IONS 17

76,365,472

Fig. 7. Graph of proton cyclotron resonance in a Penning trap. This narrow dispersion-shaped curve is the result of direct synchronous

detection of the resonance at v '(p*) = 76 ,365 ,'<76.9 Hz using the

split quadring design in a well-compensated Penning trap (for V = 5^.^ V). The linewidth, limited primarily by observation time,

represents fewer than 40 protons. (From ref. 89)

In the experiments of Schwinberg et al.' the electron/positron

mass ratio was measured to a precision of about 0.1 ppm. The

cyclotron frequency detection method is the same as for the g-2

experiments.'

Discussion

It appears that several orders of magnitude improvement can be

expected if the effects of magnetic and electric field imperfections

can be reduced. (Relativistic effects can also be very important;

see Van Dyck, these proceedings). Two ways this can be done are (1)

directly reduce the field imperfections. Higher order terms in the

electric potential can be reduced by using compensation electrodes

and magnetic bottles could be eliminated' . One must worry about

distortions of the magnetic field by the electrodes; these could be

suppressed by appropriate machining'^. (2) Increase the sensitivity

to ion motion. If much smaller ion motions can be detected, the ions

sample field imperfections to a lesser extent. These effects usually

scale as some high power (> 2) of the ion amplitude so that

improvement here could be substantial. (Frequency shifting effects of

:th order anharmonic terms have been discussed by Landau

these arguments can be extended to higher order

terms). Ideally, one-would like to use single ions at very low

temperature since the extent of ion orbits can be extremely small.

Detection sensitivity to induced currents can be increased by using

SQUID amplifiers'' and therefore can be extended to heavier ions

third and fourtl

and Lifshitz'°;

TN-161
1

8

SPECTROSCOPY OF STORED ATOMIC IONS

(including molecular ions) which are more difficult to detect.

Accuracies near 1 part in 10 -^ are not unrealistic for the laser

fluorescence method'-^; this Is primarily a statement that if the ions can be laser cooled, they can be confined (and detected) in extremely

small regions of space where field imperfection effects are minimized.

For more than one ion in the trap, space charge effects can show

up in several ways; for sinplicity we only discuss the effect on the

axial resonance here but similar arguments could be "made for the

other degrees of freedom. For a single ion species near the center

of the trap where the coupling to the endcap electrodes for all ions

is the same (i.e. the electric field from a voltage applied to one endcap is uniform over the cloud) no space charge shift in v is observed since only the center of mass motion couples to the

electrodes' . This was demonstrated in the work of Ref. 93 v*iere for

example, observed magnetron frequencies were consistent with the free

space value (accuracy - 0.5J) but the magnetron frequency of

individual Ions was shifted by about a factor of 3 because of space

charge^^. If the cloud is spread out radially (e.g. shaped like a pancake whose diameter approaches the trap diameter) then the

Internal modes of oscillation in addition to the center of mass mode

are observable because the z coupling to the endcaps depends on the

ion's radius in the trap . Internal modes also become observable if the ions are only weakly coupled together and the trap is imperfect

so that for example v depends on r. If two species of ions are present in the trap, different effects come into play. For dilute

clouds the center of mass oscillation of one species is space charge

shifted by the other ions . This property was used in ref. 74 to measure Ion density. In the experiment of ref. 93. the presence of

single BeH* ions would broaden the Be* cyclotron resonances by more

than 100 ppm. To further illustrate the above remarks, we present a simple

example: that of two ions In a Penning trap. If the axial excursions

of the Ions (z^ and Zj) are small conqsared to their separation r in the radial direction, then the equations of motion in the z direction

can be approximately written

m^ 2^ + k^ z^ - k-5(z^-Z2)

mpZp + k2 Z2 - k-3(z2-z^)

where u^i - (kj^/m^^) (i - 1,2) are the respective oscillation frequencies of the ions (with the other ion removed from the trap) and

ko » q^q2/r3. These equations can be solved exactly . Some

limiting cases are interesting to examine.

Case 1: k^ - k2, m^ - m2. The center of mass oscillates at the

unshifted value u^. = ui-^^ • 0122' If the coupling to the endcaps is different for the two tons (eg. suppose one ion is at the center of

the trap and the other at a nominal radius r) then some current is induced at the internal mode frequency u^' • ((u^ ~ 2k,/m) . The

strength of the induced current due to the internal mode will scale

with the difference in endcap coupling between the two ions.

TN-162
SPECTROSCOPY OF STORED ATOMIC IONS 19

Case 2: m^ ' ^2, k^ ^ ^2' ^o < Ik^-k2l (weak coupling limit)

Observed induced currents are at u..' = ((d_< ~ k^/m,) =id, ,, ions are nearly independent.

Case 3: m^ « m2, k^ * k2, k^ > Ik^ - k2l (strong coupling

limit). Dominant part of the center of mass oscillation is at frequency oj^' » (u^i + iii^2^^^ > particles appear locked together.

Case ^4: m^ - m2, k^ ^ V.2, k, = lk.| - V.^ (intermediate

coupling). Oscillations near oj^^ and ^^2'' Qualitatively not significantly different than case 2.

The most straightforward solution to space charge problems is to use single ions. Short of this, different ion species can be

eliminated from the trap by: (1) driving the unwanted ions out of the

trap using strong motional excitation"' •"^•'^^. (2) Operate the trap

in a mass selective mode. For rf traps both high and low masses can

be excluded. For the Penning trap, particles with lower charge to mass ratio can be ejected by exceeding the critical voltage^ for

these ions. This was used in ref. 93 to eject ions with M2. 15 u. (3) Selective ionization. In ref. 93, Be ions were created but V12 and He formation was prevented by using an electron beam energy just

slightly above the ionization potential of neutral Be. The problem of space charge frequency shifts will depend on the

particular experiment and can of course be quite complicated. In general, it will not be enough to consider the space charge frequency

shift of one ion due to the other ions. For example, in the method

of observing induced currents in the electrodes, the observable is the sum of the induced currents due to all ions. In the simple case

of a small cloud of identical ions near the center of the trap, space

charge frequency shifts in the spectrum of the total current are

absent.

VI APPLICATIONS

The ion storage technique will continue to find varied

applications. We illustrate here with a few examples; other

applications are mentioned in the reviews referred to in the

introduction and in the acconpanying papers ' ' .

Frequency Standards and Clocks

In several laboratories, the primary motivation for doing high

resolution spectroscopy is to use such spectra as references for

frequency standards and clocks. Clocks are frequency standards where

it is possible to count cycles of the radiation so that time

intervals can be generated. This distinction is an important one in practice; for example, it will probably be much more difficult to obtain a laser "clock" than a microwave "clock" even though the

performance of laser frequency standards should eventually be

superior ^. In a frequency standard" or clock operating at frequency v^,

measurement imprecision ("S^error'^'^o^ ^^ approximately limited to (Q • S/N)~^ where Q r v^/Av. and S/N is the signal to noise ratio for

detecting the number of ions that have made the transition. If the

TN-163
20 SPECTROSCOPY OF STORED ATOMIC IONS

radiative linewidth is small enough then the experimental linewldth

(AVq) is probably independent of the trapped ion that is used (eg.

detennined by interrogation time) . Therefore we would like to use as high a frequency (v^) as possible in order to increase Q. This is the single disadvantage of either Mg* or Be* ions since the

interesting "clock" transitions are only around 300 MHz (Q i, 10^*^). In spite of this limitation, the accuracy achieved with a beryllium

clock "^ approaches that of the best cesium standards and significant

improvement could be expected. A better ion for a Jgser cooled

microwave clock is perhaps Hg* (v = 40 GHz for ^^Hg"^.) Very

important frequency standard work has already been accomplished using

this ion '' ' ^~ ', but laser cooling is much harder to achieve

than for Be or Mg . A logical extension of this idea is to go to much higher frequency; for example, to use a narrow optical

transition. A number of transitions in various ions have been

proposed^; Dehmelt was the first to suggest that such extremely

high resolution spectroscopy could be carried out using single photon

transitions in for example single group IIIA ions. For instancej

the 6^Sq0 63Pq transition in T8,* (A = 202 nm) has a Q s 5 x 10^^. For such single photon optical transitions, it is desirable to approximately satisfy the Lamb-Dicke criterion; this is most easily

accomplished with single trapped ions. In experiments at Heidelberg

and Seattle, the present resolution of the two photon »^-\/2 * ^ \/2

* 6 Dt/p stimulated Raman transition in Ba* is limited by laser

linewidth broadening, but this transition could be extremely

narrow^ ; the lifetime of the ^j/2 state in Ba* is 17.5 s which

would give an intrinsic Q of 1.6 x 10 . The accuracy for optical frequency standards could be extremely

high. As an example, in In"*", the linewidth of the 5 S - 5 P-) "cooling" transition is about 1.3 MHz, this implies a second order

Doppler shift of 10~

^ or lower. Other systematic shifts can occur^"^'^^'^^'^^ '^^'^^'^^^"^^^ but it is not unreasonable to think

they will be smaller than or controllable to this level. These extreme accuracies make important the problem of measurement

imprecision since the signal to noise ratio on a single ion will be

about one for each measurement cycle. Practically speaking, this means that a long averaging time will be required to reach a measurement precision equal to these accuracies. In fact, for a while, it may be that the accuracy and resolution will be limited by

laser linewidth characteristics (linewidth and linewidth symmetry). However the potential for extremely narrow tunable lasers also

exist3l03Jl5,113.

Search for Spatial Anisotropy

Frequency standards, including those based on atomic or nuclear

(Mossbauer) transitions, have traditionally played an important role

in testing gravitational theories. One example is measurements of

the gravitational red shift. In addition, the very high resolution

attained in trapped ion spectroscopy enables other sensitive tests of

the Einstein Equivalence Principle (EEP). All metric theories of

gravity (including General Relativity) are founded on the EEP,

according to which, any nongravitational physics experiment done in a local freely falling frame near a strongly gravitating mass will have

the same outcome when done in a freely falling frame far away from

TN-164
SPECTROSCOPY OF STORED ATOMIC IONS 21

all such masses. Also included in the EEP is local Lorentz Invariance which states that the outcome of the experiment is independent of the velocity of the freely falling frame '. In particular, two different atomic clocks (i.e., clocks based on transitions in two different kinds of atoms) located at the same point in space-time will have relative rates which are independent of

(1) the velocity of the freely falling lab, and (2) the position and mass of strongly gravitating bodies. As a test of the EEP the

frequency of the ^Be"^ "clock" transition (Mj = 1/2, Mj = -3/2) * (Mj

= 1/2, My « -1/2) has been compared to the frequency of a passive

hydrogen maser^-^ to see if a correlation can be found with

orientation in space. Two gravitational interactions have been proposed which violate

the EEP and shift the 'Be"*" "clock" transition relative to the

hydrogen transition. The first is,a direct coupling of a nucleon's

spin to the gravitational field, .

Ug = U(r) I • r = U(r) l2P^(cos B),

where U(r) is the strength of the coupling, r is the unit vector pointing from the particle to the source of the field and B is the

angle between the magnetic field used to confine the 'Be"^ ions (the

quantization axis) and the direction r. In the second model ' the

inertial mass of a nucleon in "Be* depends upon the orientation of

its orbit relative to the direction toward nearby massive bodies in the universe (e.g., the Milky Way Galaxy or the Virgo Supercluster

of galaxies) . This mass anisotropy is a quadrupolar effect and thus

produces a shift of the Be* transition proportional to P2(cos 6).

These experiments are also sensitive to a quadrupolar coupling

between a nucleon's velocity (in the laboratory frame)and the

velocity of the earth through the cosmic microwave background

' . Thus it is possible to test local Lorentz invariance by

investigating the extent to which the mean rest frame of the universe

acts as a preferred frame.

These experiments have searched for a sidereal time variation in the ^Be* "clock" transition which is proportional to P£(cos 6) (X. = 1,2,3) for three directions of the unit vector r: the direction of

the galactic center, the direction of the Virgo supercluster center

and the direction of motion through the apparent mean rest frame of

the universe. Preliminary experiments have achieved resolution better

than 1 mHz and see no such variation. These first results have

decreased the limits set by Hughes ' and Drever^^ on a P2(cos 6)

variation by a factor of about 50. Ultimately, one could expect an increase in resolution of an additional factor of 100 or more.

Non Neutral Plasmas

At sufficiently high ion densities and sufficiently low ion

temperatures such that the Debye length is small compared to the ion

cloud dimensions, the ion cloud in a Penning or rf trap can be

described as a non-neutral plasma . Experiments on ions and

electrons stored In Penning-type traps have studied a variety of

plasma and cooperative effects. Examples include the spectra of

plasma and diocotron waves in three dimensional plasmas -^ and

the detection of waves similar to the drumhead modes of a vibrating

TN-165
22 SPECTROSCOPY OF STORED ATOMIC IONS

membrane, in a nearly two dimensional cloud of ions stored in a hyperbolic Penning trap . Radiation pressure from lasers can be

used to cool and compress stored ions and study non-neutral plasmas

in a state where the thermal energy per ion is less than the Couloni)

energy per ion. Such a plasma is called strongly coupled . A strongly coupled plasma is characterized by the Coulomb coupling

constant r = q /akgT where a = (4Trn/3)~ . q is the ion charge, T is

the ion tenperature, and n is the ion number density. Extensive

theoretical calculations exist for a strongly coupled one component

plasma (OCP) . An OCR consists of a single charge species embedded

in a uniform density background of opposite charge. These calculations predict that at r s 2, the pair correlation function should

begin to show oscillations characteristic of a liquid, and at much

larger values^"* of r (r s ''70). crystallization may take place.

Crystallization has been observed ' in a two dimensional OCP (r = 137) and in a system of charged aluminum particles (several microns

in size) stored in an rf trap . In a frame of reference rotating

with the (Sx8) rotation frequency of an ion cloud in a Penning trap,

the ions can be viewed as being embedded in a uniform charge

distribution of opposite sign. Specifically, the spatial correlations and the values of r for the onset of liquid and solid behavior

are the same for the OCP and the non-neutral plasma in a Penning

1 ?Q trap ^. A value of r on the order of 2 has been estimated for a pure electron plasma stored in a long cylindrical Penning trap and

cooled to near the IK temperature of its surroundings.^ In a small

plasma of laser cooled ^Be* ions stored in a Penning trap, a value of

r as large as 10 (indicating liquid behavior) has been measured''.

In this latter experiment, a second laser was used to probe the ion

plasma and measure the temperature of the ions from the Doppler

broadening of the optical probe transition. The ion number density

was determined by measuring the (Sx§) cloud rotation frequency.

Because the trap electric field and magnetic field were known, the

space charge electric field was extracted from the cloud rotation

frequency and used to determine the ion number density. Ion number

densities of » 2 x ^0'/c^P and temperatures of ^ 75 mK produced

values of r = 10. Values of r large enough to observe a liquid-solid

phase • transition should be accessible in future versions of this

experiment. If the theoretical cooling and density limits can be

obtained, values of r as large as 15,000 are perhaps possible for Be

ions. Because experimental information on three dimensional,

strongly coupled plasmas is almost non-existent, these experiments

can provide some useful tests of the theoretical calculations.

Acknowledgments

The NBS experiments and the preparation of this manuscript were supported by the Office of Naval Research and the Air Force Office of

Scientific Research. We thank E. C. Beaty, G. Leuchs and H. U. Daniel for suggestions and comments on the manuscript.

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TN-166
SPECTROSCOPY OF STORED ATOMIC IONS 23

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121. J. H. Malmberg and J. S. deGrassie, Phys. Rev. Lett. ^, 577

(1975).

122. J. S. deGrassie and J. H. Malmberg, Phys. Fluids 23, 63 (1980).

123. G. Dimonte, Phys. Rev. Lett. _46, 26 (1981).

124. S. Ichimaru, Rev. Mod. Phys. _54, 1017 (1982).

125. W. L. Slatterly, A. D. Doolen, and H. E. DeWitt, Phys. Rev. A

21_, 2087 (1980).

126. J. P. Hansen, Phys. Rev. A_8, 3096 (1973).

127. C. C. Grimes and G. Adams, Phya. Rev. Lett. _42, 795 (1979).

128. R. F. Wuerker, H. Shelton, and R. V. Langmuir, J. Appl. Phys.

30, 342 (1959).

129. J. H. Malmberg and T. M. O'Neil, Phys. Rev, Lett. ^, 1333

(1977).

TN-170
SPECTROSCOPY OF STORED ATOMIC IONS 27

130. J. H. Malmberg, T. M. O'Neil, A. W. Hyatt, and C. F. Driscoll,

Bull. Am. Phys. Soc. _28, 1155 (1983).

131. V. A. Dzuba, V. V. Flambaum, P. G. Silvestrov, and 0. P. Sushkov, Phys. Scripta, to be published.

132. P. M. Radmore and P. L. Knight, J. Phys. B15 , 561 (1982) and

references therein.

133. D. Howe and F. L. Walls, IEEE Trans. Instrum. Meas. IM-32 , 218

(1983).

TN-171
Volume 53, Number 4 PHYSICAL REVIEW LETTERS 23 July 1984

Strongly Coupled Nonneutral Ion Plasma

J. J. Bollinger and D. J. Wineland

Time and Frequency Division, National Bureau ofStandards, Boulder, Colorado 80303

(Received 7 May 1984)

Radiation pressure from a laser has been used to cool and compress small nonneutral plas- mas of 'Be"*" ions confined by static electric and magnetic fields. A second laser has been

used as a probe to measure ion densities of 2x 10^ cm~^ and ion temperatures below 100

mK. A Coulomb coupling constant, Y , as large as 10 has been measured indicating that the plasma is strongly coupled. In the future, values of F large enough to observe a liquid-solid

phase transition should be accessible.

PACS numbers: 52.25.Lp, 52.25.Ps, 52.70.Kz, 52.90.+Z

This Letter reports measurements of the density

and temperature of a small nonneutral plasma composed of 'Be''' ions. The thermodynamic state of

such a classical Coulomb system is determined by

the coupling constant r (defined below) which is a measure of the ratio of the Coulomb energy to kinetic energy per particle.' For values of Y on the

order of 1 or larger, correlation effects become important and the plasma is called strongly coupled.

Recently, values of F on the order of 2 have been

estimated for a pure electron plasma cooled to near

the 4 K temperature of its surroundings.^ By use of

electrodynamic storage, very large values of F have

been obtained with highly charged aluminum parti- cles several microns in size.^ Classical, twodimensional, strongly coupled plasmas have been

realized with the system of electrons on the surface

of liquid helium. In this system, values of F as large as 200 are experimentally accessible, and a

liquid-solid phase transition has been observed"* at Fsl37. To the authors' knowledge, the experiments reported in this paper give the largest values

of F reported in a steady-state, magnetically confined, three-dimensional plasma.

The experimental arrangement used here^'^ is similar to that used in other nonneutral-plasma

studies. ^"'^

It is different in that the spatial extent

of the plasmas could be nondestructively mapped

and in that it was possible to control the shape, density, internal temperature, and angular momentum

(add or subtract) of the plasmas with radiation pres^ sure from lasers. A uniform magnetic field B = Bqz (5o =0.8 T) provided confinement of the

'Be''' ions perpendicular to the magnetic field. Static electric fields produced by three electrodes were

used to provide confinement of the ions along the

magnetic field. The only qualitative difference

between this confinement geometry and that of

Refs. 7-12 is that the electric potential produced by

the electrodes varied continuously over the confinement region and had the form <t)j{r,z) = miojilz^- r^)/4q, where oj^ is the single-particle

"axial" or z oscillation frequency, q is the ion

charge, m is the ion mass, and {r,z) are cylindrical

coordinates. This system has usually been called a Penning trap.'^-''' A pressure below about 10"^ Pa

was maintained in the confinement region. Radiation pressure from a laser was used to cool and

compress the ion plasma.'^' '^ This was accomplished with a focused, frequency-doubled dye laser beam tuned to the low-frequency side of the

transition in 'Be''' (A. = 313 nm) and directed at the

ion plasma perpendicular to the magnetic field. The

beam had a waist of about 60 ^tm and a power of

about 10 /xW. The ions were optically pumped into

the 2s^5i/2(- y, - y) ground state and detected

by observing the ion fluorescence scattering in- duced by this cooling laser beam. The plasma ra- dius was decreased by directing the cooling laser beam to the side of the plasma which receded from

the laser beam because of the ExB rotation. '^"'^ On the average, each photon-scattering event decreased the canonical angular momentum [in the

gauge where A(r) = yBxr] of the system by

dh/\, where d is the distance between the laser beam and the trap axis and h/k is the photon

momentum. Since the scattering rate was large

( =10^ scattered photons per second per ion) this effect could dominate other external torques on the

ions such as collisions of the ions with the background gas. In practice, relatively high-density,

stable plasmas were maintained for many hours.

The ions were expected to be in near thermal

equilibrium with each other. The equilibrium state of a nonneutral plasma confined by static electric

fields and a uniform magnetic field has been dis- cussed by O'Neil and co-workers.''''^ The ion dis348 © 1984 The American Physical Society TN-172
Volume 53, Number 4 PHYSICAL REVIEW LETTERS 23 July 1984

tribution function is

f(r,z.V) = nQ{m/2TTkQTy^^exp[-[]-m{V-ojre)^+q4>ir,z) + ]-mwin-(o)r^/kQT}, (1)

where n^ is the ion density at the center of the trap,

n = qBq/ mc is the cyclotron frequency, w is determined by the total canonical angular momentum of

the system, and 4) = (f) / + 4> j- is the sum of the potential due to the ions (including the potential due

to the induced charges on the trap electrodes) and

the applied trap potential. The distribution function

is a Maxwellian velocity distribution superimposed on a rigid rotation of frequency w. This rotation is due to the Ex B force on the ions and is often called the magnetron rotation. 0(r,z) and the spatial

shape of the plasma are determined from Poisson's

equation (Eq. 3, Ref. 11), V^= -A-rrqnoe''',

where i})= -[q(t>+ Yma>iH-(x))r^]/kQT. In the

limit T-* 0, this is satisfied when the plasma is a uniformly charged figure of revolution with

</>/(r,z)= -27r<7Wo(ar^-f /8z^)/3, where nQ= mw

x{n-cj)/2'rrq^, /3 = 3tu^V2a>(n -co), and 2a + (i = 3. In the limit that the ratio of the plasma

dimensions to the trap dimensions approaches zero (about 10"^ here) the plasma becomes a uniformly

charged spheroid.

In a frame of reference rotating about the trap

axis with frequency co, the ion plasma behaves like a neutral one-component plasma. '^ That is, the

positively charged ions behave as if they are moving

in a uniform, negatively charged background. The

properties of a one-component plasma are ex- pressed in terms of the Coulomb coupling constant' r^qVak^T. (2)

where a is defined by 47rnoa-^/3 = \, and T is the

temperature of the ions from the distribution of Eq.

(1). Theoretical calculations' predict that at r =2,

the pair-correlation function should begin to show

oscillations characteristic of a liquid, and at much

larger values'^-'^ of T (r=170), crystallization

may take place. (For the small plasmas of this ex- periment, surface effects may be important; however, this is not expected to greatly alter the theoretical prediction.)

Values of F in this experiment were determined

by separately measuring the density, wq, and temperature, T, of the ions. A second laser beam

(beam waist =60 /xm) was used to probe the plas- ma. When the probe laser frequency was swept

through the 2s^Sif2(- t- - j")— ^P^Pyii- t> + -) transition, some of the ion population was re- moved from the optically pumped ground state, resulting in a decrease in the resonance fluorescence induced by the cooling laser as shown in Fig.

1 . The size of this depopulation signal depended on

the local density of the plasma at the intersection of

the probe beam and the plasma. With the probe

beam perpendicular to the magnetic field, spatial

maps of the plasmas were taken by moving the

probe beam across the plasma in both the radial and

^^Au rWV^

106 MHz

Background

FIG. 1. Sketch of the plasma with the cooling (C) and

probe iP) laser beams. The cooling beam was directed at the side of the plasma which receded from this beam.

For much of the data, the probe beam was directed ap- proximately parallel to the cooling beam. The rotation frequency, w, was measured by translating the probe

beam in the y direction (radially). Spatial maps of the plasma were obtained by translating the probe beam in both the y (radial and z (axial) directions. The inset shows the depopulation signal obtained when the fre- quency (vp) of the probe laser was slowly swept through

the 25^5,/2(-y, -y)-^ 2p^Py2(-Y, + j) transition.

/ is the 'Be'^ fluorescence intensity from the cooling

laser.

TN-173 349
Volume 53, Number 4 PHYSICAL REVIEW LETTERS 23 July 1984

TABLE I. Summary of measurements on four nonneutral plasmas taken at three values of w^. Ail data were taken at a magnetic fieid of fio = 0.819 T. Tlie coupling F and tlie Debye length \d were calculated with use of the axial temperature T^. Nis the number of ions and r^. is the rms radius of the cyclotron orbit.

(uJItt . 10-^/70 a 7"c t; Ad /c (kHz) (cm-^) (pim) N (mK) (mK) r {^lm) (pim)

243 2.1(6) 23(2) 142(51) 45(12) 144(34) 5.1(1.3) 6(1) 1.1(1)

243 2.0(5) 23(2) 172(62) 56(13) 151(36) 4.9(1.2) 6(1) 1.2(1) 198 2.0(5) 23(2) 126(45) 10(12) 75(30) 9.7(3.9) 4(1) 0.5(3) 175 1.8(5) 24(2) 95(34) 19(12) 82(30) 8.6(3.2) 5(1) 0.7(2)

axial directions. The spatial maps were consistent

with a uniform-density, spheroidal plasma (typical

dimensions 100 to 300 ^m) with a sharp drop in density (in less than a few Debye lengths) at the

plasma edge. The rotation frequency oj was determined by measuring the change in the Doppler shift of the depopulation transition as the probe beam

was moved radially across the plasma (see Fig. 1). «o was then determined from nQ=mu){0. — (x))llTTq^ with use of the measured value of w. (n was determined by auxiliary measurements.^) T

was determined from the width of the depopulation

signals. The 19.4-MHz natural width and the plasma rotation convoluted with the Gaussian spatial

intensity profile of the probe beam contributed to the width of the depopulation signals. In addition

there was a saturation broadening of the depopulation signals when the signal size approached the total fluorescence count rate. These broadening

mechanisms were calculated and the Doppler

broadening due to the nonzero temperature of the

ions was extracted. With the probe laser beam per- pendicular to the magnetic field, the temperature,

Tc, of the cyclotron motion was measured.

The cooling laser beam, directed perpendicular to the magnetic field, only indirectly cooled the axial motion of the ions by collisional coupling of the axial motion with the cooled cyclotron motion. The

axial motion was directly heated by the recoil of the

scattered photons'^; therefore, the axial temperature, Tj, was possibly higher than 7^. T^ was measured by passing the probe beam along a diagonal

between the three electrodes. In this case, the

depopulation signals contained a contribution due

to the Doppler broadening by the axial (z) motion

of the ions. Table I summarizes the measurements

on four separate plasmas in this experiment. T was

calculated by use of T^, because it was the larger of

the two temperatures. In future experiments, by

directing the cooling beam along the diagonal and

cooling the axial motion directly, an axial temperature equal to the cyclotron temperature should be

obtained.

The theoretical cooling limit^^ depends on the

linewidth, Ai^, of the cooling transition and predicts

a temperature equal to h^v/lk^, where h is Planck's constant. For 'Be"^ , Av = 19.4 MHz which

gives a limiting temperature of 0.5 mK. At a magnetic field of 10 T, the density, limited by the Bril- louin density, ''*'^° could be as high as «o=3

X 10'°/cm^ for 'Be''" ions. Therefore, values of F as

large as 15 000 are perhaps accessible and crystallization in a nonneutral ion plasma might be obtained. We note that fairly direct measurements of

the pair-correlation function could be made by observing interferences in the low-angle laser scatter- ing from the ions, similar to x-ray crystallographic

techniques. Ion diffusion (or lack thereof) could be

measured by spatially separating the cooling and

probe beams (along z) and pulsing the probe beam,

similar to that described by Stern, Hill, and Rynn.^^

Classical mechanics has been assumed to give an

adequate description in the preceding discussions.

At temperatures below ^n/Zcg, the discreteness of

the Landau (quantized cyclotron) levels becomes

important. At slightly lower temperatures, T

^Kwp/k^, where Wp=[2cj(Q-w)V^^ is the plasma frequency of the ions, the collective motion of

the ions should be treated quantum mechanically. '° Both of these interesting regions may be accessible

with 'Be"*" or perhaps other ions which have a much

smaller radiative linewidth Ai' where even lower

temperatures could be expected.

We gratefully acknowledge the support of the

U. S. Office of Naval Research and the U. S. Air

Force Office of Scientific Research. We thank T. M. O'Neil, W. M. Itano, and S. Barlow for reading

the manuscript and for helpful comments. Special

thanks go to J. D. Prestage for his help with the fi- nal data acquisition.

350 TN-174
Volume 53, Number 4 PHYSICAL REVIEW LETTERS 23 July 1984

'S. Ichimaru, Rev. Mod. Phys. 54, 1017 (1982), and

references therein.

2J. H. Malmberg, T. M. O'Neil, A. W. Hyail, and C. F. Driscoll, Bull. Am. Phys. Soc. 28, 1155 (1983).

3R. F. Wuerker, H. Shelton, and R. V. Langmuir, J. Appi. Phys. 30, 342 (1959).

"C. C. Grimes and G. Adams, Phys. Rev. Lett. 42, 795

(1979). 5D. J. Wineland, J. J. Bollinger, and W. M. Itano, Phys. Rev. Lett. 50, 628, 1333(E) (1983).

6J. J. Bollinger, D. J. Wineland, W. M. llano, and J. S. Wells, in Laser Spectroscopy VI, edited by H. P. Weber

and W. Luthy, Springer Series in Optical Sciences Vol. 40

(Springer, Berlin, 1983), p. 168.

''J. H. Malmberg and J. S. deGrassie, Phys. Rev. Lett 35,577(1975).

8C. F. Driscoll and J. H. Malmberg, Phys. Rev. Lett. 50, 167 (1983), and references therein. 9G. Dimonte, Phys. Rev. Lett. 46, 26 (1981).

lOj. H. Malmberg and T. M. O'Neil, Phys. "kev. Lett. 39, 1333 (1977).

>1S. A. Prasad and T. M. O'Neil, Phys. Fluids 22, 278

(1979).

12T. M. O'Neil and C. F. Driscoll, Phys. Fluids 22, 266

(1979).

•3f. M. Penning, Physica (Utrecht) 3, 873 (1936).

'"H. G. Dehmelt, Adv. At. Mol. Phys. 3, 53 (1967),

and 5, 109 (1969); D. J. Wineland, W. M. Itano, and

R. S. Van Dyck, Jr., Adv. At. Mol. Phys. 19, 135 (1983). _ '5R. E. Drullinger, D. J. Wineland, and J. C. Bergquist,

Appl.Phys. 22,365 (1980).

'6W. M. itano and D. J. Wineland, Phys. Rev. A 24,

1364 (1981).

'^W. M. Itano and D. J. Wineland, Phys. Rev. A 25, 35

(1982); D. J. Wineland and W. M. Itano, Phys. Rev. A

20, 1521 (1979).

18W. L. Slatterly, G. D. Doolen, and H. E. DeWitt,

Phys. Rev. A 21, 2087 (1980).

19J. P. Hansen, Phys. Rev. A 8, 3096 (1973).

20R. C. Davidson, Theory of Nonneutral Plasmas (Benjamin, Reading Mass., 1974), p. 4. 21R. A. Stern, D. N. Hill, and N. Rynn, Phys. Lett. 93A, 127 (1983).

TN-175

351
Volume 54, Number 10 PHYSICAL REVIEW LETTERS 11 March 1985

Laser-Cooled-Atomic Frequency Standard

J. J. Bollinger, J. D. Prestage, Wayne M. Itano, and D. J. Wineland

Time and Frequency Division, National Bureau ofStandards, Boulder, Colorado 80303

(Received 5 December 1984) The first frequency standard based on iaser-cooleu atoms is reported. Beryllium atomic ions were

stored in a Penning trap and cooled by radiation pressure from a laser. The frequency of the ^Be"*" (M,,Mj) (-f. + y)'~(-y, + y) ground-state hyperfine transition at its magneticfield-independent point was determined to be 303 016 377.265 070(57) Hz. The accuracy of a fre- quency standard referenced to this transition was comparable to the best frequency standards,

which are based on cesium atomic beams.

PACS numbers: 32.30.Bv, 06.30.Ft, 35.70.Fk The principle of laser cooling was introduced''^ as a way of suppressing both first- and second-order

Doppler shifts in high-resolution spectroscopy. Aside

from the interest in the physics of the cooling process

itself, two of the primary goals of this technique are more accurate atomic spectroscopy and improved fre- quency standards. This has been noted in the experimental and theoretical reports on trapped ions,-'"^

cooled atomic beams,''"' and trapped neutrals.'"'^ As a step toward these goals we report an accurate measure- ment of 'Be"*" ground-state hyperfine structure and

present results on the first frequency (and time) standard based on laser-cooled atoms. Such experiments

provide improved measurements of hyperfine constants and ^-factor ratios.^ In addition, they could lead

to measurements of previously undetected physical

quantities and improvements in previous experimental

tests. Examples include the measurement of deviations of atomic hyperfine structure from that predicted

by the Breit-Rabi formula due, for example, to nuclear

diamagnetism''' and various tests of gravitational in- teractions such as tests of spatial anisotropy (HughesDrever-type experiments).'^

In the experiment reported here, 'Be"*" ions were

confined in high vacuum ( < 10~^ Pa) by the static magnetic and electric fields of a Penning trap.^ The

long confinement times and benign environment of

the Penning trap are beneficial for high-resolution

spectroscopy.^ The three axially symmetric electrodes

of this trap provide an electrostatic potential of the

form 4>T= A{2z^— r^), where r and z are cylindrical

coordinates. For /J > 0, positive ions experience a harmonic restoring force along z. A uniform magnetic

field parallel to z provides confinement in the radial direction. Ion radial motion is a superposition of cir- cular cyclotron motion and a circular Ex B drift ''magnetron" motion about the axis of the trap. An essential feature of this experiment was the

reduction of the source of the largest systematic uncertainty, the second-order Doppler sh'ft, by reduction of

the ion kinetic energy with radiation pressure from a

laser. '"^ Cooling of the axial and cyclotron motions of

a trapped ion by a laser beam tuned slightly lower in

frequency than a strongly allowed resonance transition can be understood by considering the following one- dimensional model. Suppose that the ions are constrained to move along the z axis in a harmonic well and assume that a laser is directed at the ions along z. An ion absorbs and reemits photons predominantly

when its velocity is directed against the laser beam, be- cause the light frequency in the ion's frame is Doppler

shifted into resonance. When averaged over all angles

of reemission, the ion's momentum is reduced by hl\

per scattered photon, where A is the laser wavelength.

The "cooling" laser also couples with the magnetron

motion and can be used to reduce the radius of the ion

magnetron orbit and therefore compress the radius of

the ion "cloud. "^''^

Specifically, in this experiment a narrow-band ( < 4

MHz) radiation source (power = 20 /xW) tuned to the

low-frequency side of the

25 25i/2(A/,= -|,A/y=-|)

(X = 313 nm) transition of 'Be"*" was used to cool and

spatially compress the ions and optically pump them

into the (-y.-y) ground state. '^ The 313-nm

source was obtained by frequency doubling of the out- put of a single-mode cw dye laser. The resonance

fluorescence induced by this cooling laser was used to detect the ions.'^-'^ The size, density, and temperature of the ion clouds were determined by use of a similar radiation source as a probe. '^ Typical clouds

ranged from a few hundred to 2000 ions with cloud di- ameters from 300 to 500 /Am and densities of about

3x 10^ ions/cm^. Cyclotron and axial temperatures of

less than 100 mK and effective magnetron temperatures of less than 2 K were obtained with the cooling

laser applied continuously. Ion storage times were many hours without the laser applied; with the laser applied, ion loss is negligible.

At a magnetic field of about 0.8194 T [ground state (-y,-y)—'(-y,y) electron spin-fiip frequency

of 23914.01 MHz], the ( - y.y)^ ( - y.y)

ground-state hyperfine transition, v^ (see Fig. 1),

1000

Work of the U. S. Government

Not subject to U. S. copyright

TN-176
Volume 54, Numbe-.r 10 PHYSICAL REVIEW LETTERS March 1985

F=1

(Ml, Mj)

(-3/2, 1/2) (-1/2, 1/2)

(1/2, 1/2)

(3/2, 1/2)

F=2 _

(3/2, -1/2)

(1/2, -1/2)

(-1/2, -1/2)

(-3/2, -1/2)

FIG. 1. Hyperfine structure (not drawn to scale) of the

'Be"*" 2s ^5i/2 ground state as a function of magnetic field. i>]

is independent of magnetic field to first order at fl = 0.8194

T.

depends only quadratically on the magnetic field deviation, bB, as 8i^,/i/,= -Omi(hB/B)\ The u^ transi- tion was detected by optical-microwave-rf triple reso- nance. Microwave radiation tuned to the electron

spin-flip resonance transferred half of the ion population from the optically pumped (-y, -y) state to the ( - y , + y ) state. Some of the ( - y , + y ) state population was transferred to the ( - y, + y) state by

application of rf near the 303-MHz ^'l transition fre- quency. As a result of the microwave mixing this resulted in an additional decrease in the ( — y, — y)

state population and therefore a decrease in the observed fluorescence.

Ramsey's method of separated oscillatory fields'* was used to interrogate V]. An rf pulse of duration / was applied, followed by a free precession interval of

duration Tand a second rf pulse of duration /coherent

with the first pulse. Data were taken with r= 10 s and

r = 0.1, 0.5, and 2.0 s, and with T=\9 sand / = 0.5 s. Typically the cooling laser and mixing microwaves were on for a period of 3 s, during which the 'Be"'' ions were prepared in the ( — y, — y) and ( — y, + y

)

states. The cooling laser and mixing microwaves were

then turned off in order to avoid light and ac Zeeman

shifts during the interrogation period. After the inter- rogation period, the laser and microwaves were turned

back on, and the signal was obtained from the fluores- cence count rate during the first 0.3 s of this time in- terval. Figure 2 shows the signal obtained for the

(/, r) = (0.5 s, 19 s) interrogation. The linewidth Ai^i = 25 mHz gives a Q = Vl/^v^ of 1.2x10'° on the

303-MHz f 1 transition frequency.

A synthesized rf source near 303 MHz was used to probe V]. A passive hydrogen maser [fractional fre- quency stability'^ cr_^(7) = 1.5x 10~'^r~'^^ for measurement time T in seconds; frequency drift < 3 X 10~'Vd] was used as the external reference for this source. A computer alternately stepped the rf fre- quency by ±^v\/2 about a calculated frequency /,. After each rf frequency step, a measurement of the

>>

-t->

CD

(~> / \ f\

C

QJO

(/) 0)o

Ll_ q/o

\ 25 mHz F \

O ^ o 303 016 377.265 Hz

Frequency

FIG. 2. Signal obtained on the 1*1 field-independent transition for (/,r) = (0.5 s, 19 s). The sweep width was 100 mHz and the frequency interval between points was 5 mHz.

The dots are experimental and are the average of ten sweeps; the curve is a least-squares fit.

signal was made and a new frequency y, + i was obtained about which the next rf step was made. /]+] was obtained through a digital servo system which

steered the frequencies i/j to the i>\ transition fre- quency in a way that was independent of a linear drift

in the total count rate.^°"^^ Data were taken without interruption for a period of

approximately 2 h. After this, the run was stopped

and the magnetic field measured and reset if necessary. The average frequency and o-yiT) were calculat- ed from the frequencies l/j. For each run, a-yiT) was

fitted to a r"'/^ dependence for r greater than the ser- vo attack time ( — 100 s). A total of sixty runs were

used in the final determination of the v^ transition fre- quency. For the 29 runs with a (0.5 s, 19 s) Ramsey

-iu-1/2 interrogation, (Ty(T) ranged from 1.3xlO~"r to 4x 10~"t~'''^. The statistical uncertainty of the aver- age frequency of each run was estimated from the fit- ted (Tyir) at T equal to the total measurement time of

the run. A weighted-average frequency for each group

of identical ( /, T) interrogations was calculated.

In order to determine the frequency offset due to the second-order Doppler or time dilation effect, the

cyclotron and axial temperatures of the ions were measured (via optical Doppler broadening) as a function of

the time that the cooling laser was turned off by a technique similar to that discussed in Ref. 17. Figure

3 shows the results of these measurements on typical

ion clouds. The heating shown in Fig. 3 produced

a second-order Doppler shift, — {v'^/2c'^)v\, of

-107(27) ixHz for the (0.5 s, 19 s) interrogation. The

contribution of the magnetron rotation of the cloud to the second-order Doppler shift was almost an order of

magnitude less. The weighted-average frequency of

each ( /, T) interrogation was corrected for the total second-order Doppler shift. The weighted average of

the resulting frequencies provided (if further sys- TN-177

1001
Volume 54, Number 10 PHYSICAL REVIEW LETTERS 11 March 1985

35 -

30

1

25

2- 20

1— 15

-

._

10 ._

5 I ..

I

TABLE I. Estimated systematic errors. The size of the

effect is the fraction of the transition frequency vi.

Systematic effect Size of effect

10 20

Cooling Laser Off Time (s)

FIG. 3. Temperature, f, of the cyclotron and axial motions as a function of the length of time that the cooling

laser is off.

tematic corrections are neglected) a 33-/LtHz (1.1

xlO~'^) one-standard-deviation determination of the

vi transition frequency relative to the passive hydrogen maser reference.

Table I lists the estimated systematic errors. The 3 parts in 10^ magnetic field fluctuations produced a small offset in the measured v\ frequency through the

quadratic magnetic field dependence of v\. Because vj

is a Am/r = ±1 transition, there was a small shift due

to the Earth's rotation. This shift arises because only one rotating component of the applied (linearly polarized) rf field was responsible for inducing the V] transition. The ions may be viewed as being in an inertial frame with the laboratory (including the trap and applied rf field) rotating around the ions once per sidereal day. This produced a shift equal to //? cos/8, where

fn is the rotation frequency of the Earth and ft is the

angle between the magnetic field and the Earth's rota- tion axis. Microwave leakage during the interrogation

period could have produced a frequency shift in I'l as

large as 1 part in 10''*. By stepping the rf synthesizer

so that the sides of the ninth side lobes (in the (0.5 s, 19 s) interrogation], rather than the central Ramsey

lobe, were sampled, a frequency shift in V] due to a background slope was measured to be less than

5x10" '^ We estimated the frequency shift due to the

blackbody electromagnetic field at 300 K to be

-3(3)x 10"'^.^^ Any frequency shift due to a coher- ence in the hyperfine states that survived the repumping period was eliminated by randomization of the

phase of the rf before each interrogation period. Fractional shifts of the ground-state hyperfine frequency of

'^^Ba+ (Vetter, Stuke, and Weber^") and '^^Hg^

(Cutler, Giffard, and McGuire^^) due to He collisions have been measured to be about 5x10" "/Pa and

4x 10" "/Pa, respectively. On the basis of this, a fre- quency shift in 1^1 due to background neutral collisions with a vacuum better than 10"^ Pa was estimated to be

much less than 1 part in 10'^ We have not considered

Second-order Doppler

Magnetic field fluctuations

Rotation of Earth Microwave leakage

Background slopes

Servo offsets

Blackbody radiation shift Coherence between cycles

Background gas collisions

Stark shifts

First-order Doppler

ac Zeeman shifts

Total systematic offset

-38(9)xlO-'*

-1.3(lij)xl0-'*

-1.8(0.2)xlO-'*

^1x10-'*

^5xlO-'5

«5xlO-'5

-3(3)xlO-'' < 1x10-'^ < 1x10-'^ < lxlO-'5 < lxlO-'5 < 1x10-'^

-41.1(9.4)xl0-'*

any possible frequency shifts due to the presence of

nearby conducting surfaces. ^^ The systematic uncertainty of 9.4x 10"'" is comparable to that of the U.S. cesium standard (8.5x10"''*).^' The combined uncertainty due to the random error and the systematic

errors of Table I was 34 /nHz (l.lx 10"'^). The final measured frequency of the v^ transition was

1^1 = 303016377.265070(57) Hz,

where the uncertainty includes the 1.5 parts in 10'^ uncertainty of the passive hydrogen maser frequency re- lative to the SI second.

The accuracy of the present measurement was limit- ed by the second-order Doppler shift due to the ion

heating shown in Fig. 3. The slow heating near the

end of the interrogation period was consistent with collisional heating by the room-temperature background

gas. The rapid heating observed immediately after the

cooling laser was turned off may be caused by

asymmetry-induced transport. ^^ With the cooling laser

off, axial asymmetries of the trap can increase the total canonical angular momentum of the ions, resulting in an increase in the ion-cloud radius. As the ion cloud

expands, electrostatic potential energy of the ions due

to the space-charge and trap electric fields is converted

into thermal energy of the ions. The observed heating can possibly be reduced by improvement of the trap

axial symmetry, by use of ion clouds of lower density,

and by improvement of the vacuum. The use of a second ion (e.g., ^''Mg"'') to "sympathetically" cool^'^' the 'Be''' ions could help prevent the heating, whatev1002 TN-178
Volume 54, Numher 10 PHYSICAL REVIEW LETTERS 11 March 1985

er its cause. The ^"^Mg"*" cooling-laser beam could be

applied continuously throughout the interrogation

period and could keep the ^Be"*" ions cold by Coulomb

coupling with the cold ^''Mg''" ions. Because of the

centrifugal separation of the ^Be"^ and ^''Mg''' ions,^°

the overlap of the ^Be"^ ions with the ^''Mg"'" coolinglaser beam could be made very small and result in an

ac Stark shift on the i^i transition of less than 1 part in 10'^ Use of these techniques might result in more

than an order of magnitude improvement in the per- formance of the present 'Be"*" frequency standard or one based on other ions.^ We gratefully acknowledge the support of the U.S. Air Force Office of Scientific Research and the U.S. Office of Naval Research. We thank F. L. Walls for information on the passive hydrogen maser, S. R.

Stein for many helpful discussions, and J. C.

Bergquist, H.-U. Daniel, and G. Leuchs for carefully

reading the manuscript.

'T. W. Hansch and A. L. Schawlow, Opt. Commun. 13, 68

(1975). 2D. J. Wineland and H. G. Dehmelt, Bull. Am. Phys. Soc. 20,637 (1975). 3D. J. Wineland, R. E. Drullinger, and P. L. Walls, Phys. Rev. Lett. 40, 1639 (1978).

^W. Neuhauser, M. Hohenslatt, P. Toschek, and H. G. Dehmelt, Phys. Rev. Lett. 4L 233 (1978); H. G. Dehmelt,

IEEE Trans. Instrum. Meas. 31, 83 (1982). 5D. J. Wineland, W. M. Itano, and R. S. Van Dyck, Jr., Adv. At. Mol. Phys. 19, 135 (1983). 6R. Blatt, H. Schnalz, and G. Werth, Phys. Rev. Lett. 48, 1601 (1982).

''v. 1. Balykin, V. S. Letokhov, and V. I. Mushin, Pis'ma

Zh. Eksp. Teor. Fiz. 29, 614 (1979) [JETP Lett. 29, 560

(1979)1.

8W. D. Phillips and H. Metcalf, Phys. Rev. Leu. 48, 596

(1982).

'^ Laser Cooled and Trapped Atoms, edited by W. D. Phillips, U.S. National Bureau of Standards Special Publication No.

653 (U.S. GPO, Washington, D.C., 1983), and Prog. Quantum Electron. 8, 115 (1984).

>0A. Ashkin and J. P. Gordon, Opt. Lett. 4, 161 (1979). •>W. H. Wing, Phys. Rev. Lett. 45, 631 (1980). 12D. E. Pritchard, Phys. Rev. Lett. 51, 1336 (1983).

'^J. Daiibard, S. Raynaud, and C. Cohen-Tannoudji, Opt.

Commun. 47, 395 (1983).

'''D. J. Larson, Hyper/lne Interactions (North-Holland, Amsterdam, 1978), Vol.4, p. 73. '^C. M. Will, Theory and Experiment in Gravitational Physics

(Cambridge Univ Press, Cambridge, 1981), p. 30; J. D.

Prestage, J. J. Bollinger, W M. Itano, and D. J. Wineland, to be published. '6W. M, Itano and D. J. Wineland, Phys. Rev. A 24, 1364

(1981).

''J. J. Bollinger and D. J. Wineland, Phys. Rev. Lett. 53,

348 (1984).

'*N. F. Ramsey, Molecular Beams (Oxford Univ. Press, London, 1956).

'W(t) =((,/, + , (7)-//r)]V2/2), where /,(r) is the

/th measurement of the average frequency over a measure- ment time intervair, the brackets denote an average over all measurements /, f= (./;), and zero dead time is assumed

between measurements. See J. A. Barnes et al., IEEE Trans. Instrum. Meas. 20, 105 (1971).

20m. Jardino, M. Desaintfuscien, R. Barillet, J. Viennet,

P. Petit, and C. Audoin, Appl. Phys. 24, 107 (1981).

2lL. S. Cutler, R. P. Giffard, and M. D. McGuire, in Proceedings of the Thirteenth Annual PTTI Applications and

Planninf; Meeting;. Washington, D.C., 1981 (NASA,

Washington, DC, 1982), NASA Publication No. 2220, p. 563.

22j. J. Bollinger, W. M. Itano, and D. J. Wineland, in Proceedings of the Thirty-Seventh Symposium on Frequency

Control, Philadelphia, 1983 (Systematics General Corp., Wall

Township, N.J., 1983), p. 37. 23W. M. Itano, L. L. Lewis, and D. J. Wineland, Phys. Rev. A 25, 1233 (1982).

24J. Vetter, M. Stuke, and E. W. Weber, Z. Phys. A 273, 129 (1975).

25L. S. Cutler, R. P. Giffard, and M. D. McGuire, in Proceedings of the Thirty-Seventh Annual Symposium on Fre- quency Control, Philadelphia, 1983 (Systematics General

Corp., Wall Township, N.J., 1983), p. 32. 26m. Babiker and G. Barton, Proc. Roy. Soc. London, Ser. A 326, 255 (1972); G. L. Guttrich and K. W. Billman, Phys.

Lett. 24A, 53 (1967). 27D. J. Wineland, D. W. Allan, D. J. Glaze, H. W. Hellwig,

and S. Jarvis, IEEE Trans. Instrum. Meas. 25, 453 (1976).

28C. P. Driscoll and J. H. Malmberg, Phys. Rev. Lett. 50, 167 (1983); D. L. Eggleston, T. M. O'Neil, and J. H. Malmberg, Phys. Rev. LeU. 53, 982 (1984).

29R. E. Drullinger, D. J. Wineland, and J. C. Bergquist,

Appl. Phys. 22, 365 (1980).

30T. M. O'Neil, Phys. Fluids 24, 1447 (1981).

1003

TN-179
Volume 54, Number 22 PHYSICAL REVIEW LETTERS 3 June 1985

Limits for Spatial Anisotropy by Use of Nuclear-Spin-Polarized ^Be"*^ Ions

J. D. Prestage, J. J. Bollinger, Wayne M. Itano, and D. J. Wineland

Time and Frequency Division, National Bureau ofStandards, Boulder, Colorado 80303

(Received 28 February 1985) The frequency of a nuclear spin-flip (\Am,\ = \) transition in 'Be"^ has been compared to the fre- quency of a hydrogen maser transition i\AF\ = \, Amf = 0) to see if the relative frequencies

depend on the orientation of the 'Be"^ ions in space. The present null result represents a decrease

in the limits set by Hughes and Drever on a spatial anisotropy by a factor of about 300.

PACS numbers: 04.80. -I- z, 35.1 0.-d

In metric theories of gravity the influence of external gravitational fields on atomic structure is governed

by the Einstein equivalence principle (EEP).' All metric theories (including general relativity) are based on the EEP which states that (i) all bodies fall in a given gravitational field with the same acceleration

(weak equivalence principle), (ii) the outcome of any

local nongravitational experiment is independent of

the velocity and orientation of the freely falling ap- paratus [local Lorentz invariance, (LLI)], and (iii) the

outcome of any local nongravitational experiment is independent of where and when in the universe it is performed. It follows from LLI that two different

atomic clocks located at the same point in space-time

will have relative rates that are independent of the

velocity and orientation of the freely falling lab. As a test of LLI the frequency of the l^SxiiiMj.Mj) = (- y, +y)-"(-y, +y) (see Fig. 1) ground-state

hyperfine transition in 'Be"'" (hereafter called the

"clock" transition) has been compared to the frequency of a passive hydrogen maser^ to see if a correlation

between the relative rates of the two clocks and the

orientation of the 'Be"*" nuclear spin in space can be

found. Such clock-comparison experiments are inter- preted as the most precise tests of LLI.'-^ Various proposals for violations of LLI have been

made. One of these is a coupling of a particle's spin to the gravitational field, "* Ug= U„\t-\- t4S-f, where U„

and Ug are the strengths of the coupling for a nucleon

(Ml, Mj)

(-3/2, 1/2) (-1/2, 1/2) ^^ (1/2, 1/2) F=1-^=—-— ^Jr""^ ^ (3/2, 1/2)

F=2 __

(3/2, -1/2)

(1/2, -1/2)

(-1/2, -1/2) (-3/2, -1/2) FIG. 1. Hyperfine structure (not drawn to scale) of the

'Be''' 25 ^5i/2 ground state as a function of magnetic field, v

is a first-order magnetic field-independent transition at 0.819

T.

and electron spin, respectively, and f is the unit vector from the atom to the source of the field. This interaction violates parity iP) and time-reversal (D invari- ance. Similarly, and LLI-violating and P-nonconserving coupling of a particle's spin to its motion with

velocity V relative to some preferred frame of refer- ence of the form V^ = K„\ • V + A^^S • V has been pro- posed.^ These couplings produce a shift of the 'Be''' clock transition proportional to cos/3 where /8 is the angle between the quantization axis for the ions and the

direction of f or V. The hydrogen maser transition

(/"= l,A/f = 0) — (/" = 0,A/f = 0) is not sensitive to these perturbations in first order because, in the low

operating magnetic field of the maser ( — 100 /xG) the

proton and electron spins couple together so that

(I -f) = (S -f) =0 for any direction r, where the

averaging is done in either of the two maser states. Another LLI-violating interaction which shifts the

relative rates of the 'Be"*" and maser clocks was originally proposed by Cocconi and Salpeter as a model for

inertial mass.^ The model is motivated by Mach's principle which states that a body's inertial mass is determined by the total distribution of matter in the

universe. Since matter in the nearby universe is dis- tributed anisotropically, inertial mass could show a cor- responding anisotropy. Indeed, in this model the mass

of an orbiting nucleon (orbital angular momentum

/^ 1) will depend upon the orientation of its orbit re- lative to the direction of the matter anisotropy. This

interaction leads to a shift of the 'Be''' clock transition

proportional to P2^cos/3) = (3cos^^- l)/2, where /3 is now the angle between the quantization axis for the

'Be"*' ions and the direction of matter anisotropy in the

nearby universe (e.g., the direction toward the galactic center or Virgo supercluster center). Searches for fre- quency shifts of NMR transitions in ^Li which were

correlated with the direction toward the galactic center were made by Hughes and co-workers^ and Drever*

and their null results ruled out the Cocconi-Salpeter

model. It should be stated that in their model of inertial mass, Cocconi and Salpeter computed the change

in kinetic energy of an orbiting nucleon induced by a

spatial anistropy. Others'' '° have pointed out that when anisotropic effects on both kinetic and potential

Work of the U. S. Government

Not subject to U. S. copyright TN-180 2387
Volume 54, Number 22 PHYSICAL REVIEW LETTERS 3 June 1985

energies are considered, the null results are to be ex- pected provided that the anisotropy couples in the same way to both forms of energy. These experiments

can thus be regarded as a test of the universal coupling

of gravity to all forms of mass-energy. ^'^ Models of electrodynamics in a gravitational field which include the possibility of nonuniversal couplings

are the Dicke-Peebles-Ni formalism^^-'^ and the

THefx formalism.'-^' ^^ In the TH^n formalism the

parameters Tq and //q describe the coupling of gravity

to material particles while eo and /iq describe gravity's

coupling to electromagnetic fields. In this formalism

the limiting speed for material particles is Cq while the

speed of light is c,igh„ where ca/c,igh,( ToeoAto/Z/o)'^^-

If co?^ c,jgh, an electromagnetically bound system of

charged particles experiences a coupling between its internal structure and center-of-mass motion. This

produces a contribution hE=^hmfV'VJ to the elec- tromagnetic part of the nuclear binding energy. Bm/' is the anomalous inertial mass tensor'-^ which depends

upon the electromagnetic structure of the nucleus and

V is the velocity of the nucleus through some pre- ferred frame. This could be the rest frame of the sun

or the mean rest frame of the universe (defined as the

frame in which the cosmic 3-K background radiation is isotropic).'^ In the THeyL formalism^'* —

1- ^o^oMo

//n V ^2 -^ ,3

(r,-,-V)^

'ij (1)"

I.M, where t,j is the coordinate vector from the /th to yth nuclear proton, e is the proton charge, c is the speed of light,

and the summation is over the Z protons. Use of the Wigner-Eckart theorem gives

8£=- 1 1- T'o^oMo

//n

ZM?- ni + l) e'/^iTijXT,j)^^^

nii-i) i^J

l = M,

(VxV)^2)^ (2)

where (. . .)!/' denotes the y,A/ spherical component

of the tensor in parenthesis. The resulting variation in the 'Be"*" clock transition frequency is

hv = 1- T'o^oMo

//n

(Z-l)e^Q V'^ 6hR'

PjC cos/3). (3)

where Q and R are the electric quadrupole moment

and charge radius of the ^Be nucleus. We have used

the following estimate for the nuclear matrix ele-

.15. ment'

i^J

2) ^^ (Z-l)e^(?

^ = ^/ 4lR^ ' (4)

These quadrupolar perturbations do not shift the

hydrogen-maser transition frequency. Finally because

the 'Be''" nuclear spin is /=y, we are sensitive to

shifts of the clock transition up to order P^icos/S) but no higher.

A discussion of the Be"*" clock is given by Bollinger

et al. ^^ A few hundred to 2000 'Be''" ions are stored in a Penning trap.'^ The ions are cooled and optically

pumped (95%) into the 22Si/2(A//,My) = (- y, - y)

ground state by radiation from a frequency-doubled

dye laser (\ = 313 nm, power =20 /xW) which is tuned slightly below the 225i/2( -y, -y)

-^l^Pyii-^, -y) transition frequency.'^' '^ At

this frequency the ions absorb radiation more strongly

when their motion is toward the laser than when they

move away from it. When averaged over all angles of

reemission, the ion's momentum is reduced by h/k

per scattered photon where \ is the laser wavelength

and h is Planck's constant. This results in a cooling of

the Be"*" cloud to about 1 K. The observed intensity of

the scattered fluorescent light from the

2^'5i/2(-y. -y)-" 22F3/2(-y. -y) transition is proportional to the population of the (-y, — i")

ground state. At a magnetic field of about 0.8194 T [ground-state

(-y, -y)— (-y,y) electron spin-fiip frequency

of 23914.01 MHz], the clock transition frequency, v, depends only quadratically on the magnetic field deviation, 85, as hvlv= -O.OnibB/sy. The clock transi- tion is detected by optical-microwave-rf triple reso- nance. Microwave radiation tuned to the electron

spin-fiip resonance transfers half of the ion population

from the optically pumped (-y. — y) state to the

( - y, -f- y) state. Some of the ( - y, + y) state population is transferred to the ( - y, -I- y) state by application of rf near the 303-MHz clock transition fre- quency. Because of the microwave mixing this results

in an additional decrease in the ( — -f. — y) state population and therefore a decrease in the observed

fluorescence.

The clock transition is driven by a synthesizer using

Ramsey's separated oscillatory fields technique^* with

0.5-s coherent Rabi pulses separated by a 19-s free- precession period. A computer adjusts the frequency

of the synthesizer so that it remains centered on the

clock transition. The time base for this synthesizer is provided by a passive hydrogen maser. Any shift of

the 'Be"*" tfansition frequency relative to the maser

transition will appear as a variation of the synthesizer's

frequency. In a 2-h measurement period the signal2388 TN-181
Volume 54, Number 22 PHYSICAL REVIEW LETTERS 3 June 1985

to-noise ratio was sufficient to locate the center of the

25-mHz-wide clock resonance to better than 0.5%.

The results of 29 such measurements taken between

7 May and 15 June, 1984 are shown plotted against

sidereal time in Fig. 2.'^ Because the Earth rotates, the angle (3 changes

throughout the day. That is, /3 = /3(t) where t is the

sidereal time. For example, consider the direction of

motion through the mean rest frame of the universe.

Our laboratory is at 40° north latitude and our magnetic field is horizontal and directed 232° from geographic

north (i.e., almost southeast). Thus, at 3*^ sidereal

time /3= 158° while at 15^/3 = 34°. Since the magnetic field direction is not changed with respect to the

vertical we are insensitive to any shifts of the ^Be"*" transition caused by the Earth's gravitational field. We have searched for a variation in the clock transi- tion frequency of the form

v = vq + A,^P^icos/3iT)) (5)

(/c = 1, 2, or 3) where A/^ measures the magnitude of

any LLI violation. The results of a least-squares fit of

the function of Eq. (5) to the data of Fig. 2 is shown in Table I. Also shown in that table is the direction in space which determines the angle /3(t). The errors

are the quadrature sum of the statistical error (one sigma) and a 21-ixHz uncertainty in the variation of the

second-order Doppler shift of the clock transition over

the 29 runs. This shift arises because, to avoid light shifts of the clock transition, during the 20-s Ramsey

interrogation period the ions are not laser cooled and

consequently heat up from about 1 K to about 30 K.

Also, magnetic field measurements at the beginning

and the end of each run indicate a field deviation of

bB/B= ±1.5x10"^ This leads to a peak-to-peak

frequency fluctuation of the clock transition of about

12 /xHz over the 29 runs. These were the two largest

systematic errors in this measurement, although

several other effects were considered.'^ Fluctuations

in any of the systematic errors could potentially mimic

a /*^(cos/3(t)) variation but the amplitude of such

variations would be below our quoted uncertainty. In addition to the directions listed in Table 1 we have

also fitted the data of Fig. 2 for an arbitrary direction

(excluding those directions within 10° of the Earth's

rotational axis where our sensitivity is reduced). We

find that all resulting values of Ai^ are consistent with zero at the 100-/:zHz level. In the direction of the galactic center, our present

limit on M2I (in ^Be^) of 50 + 42 = 92 /xHz is about a factor of 200 improvement over the limit of Hughes

and co-workers^ and Drever^ on \A2\ (in ^Li) for the same direction. For the direction of the Earth's

motion through the mean rest frame of the universe, we take 81 /xHz as the limit on lAil. Using Eq. (3) and V^/c^=\0-^ (Ref. 13) we place a limit on the

THeiJL preferred-frame parameter''^ of \\-TQ€QfXQ

x/Zq-'I^IO-'^ This is about 300 times smaller

than the Hughes-Drever limit on this parameter if we

use their limit on A 2 in the direction of the galactic center and the same method for estimating the nuclear

matrix element [Eq (4)]. The limits on A] are comparable with limits set by other workers^° while the

limits on A 2 are new. From the limit on /I j we find that t/„,A:„lV| < 100 mHz and because (S,) differs for the two levels of the clock transition by about

2xlO-\ we find Ue,KjW\^0.50 Hz. Although the

limits established here can be regarded as a test of

LLI, we wish to emphasize that they set an upper limit on any dependence of the relative clock rates on their orientation with respect to the sun or the fixed stars, whether from a breakdown of LLI or some other cause, such as a new interaction.

It should be possible to improve the sensitivity of

the present measurement by more than an order of

magnitude.'^ A search for a P^icosfi) interaction due

to the Earth's gravitational field would be facilitated by

orienting the experimental magnetic field along the

>- X

a

7'LU X

P III IT IXI U. >H + <

0) _l CD UJ

CO

1 1 1 1 1 1

1 i 1 1 1

1

1 11 1 1 I1 1 1

i 1

500 - -

i*'

1 |l'l 1 h \

' + -

-500

~

; 1 1 1 1 1 1 1 i 1 1 1 1 1 1 1 1 1 gh i2h 18'^ 24'^ SIDEREAL TIME

FIG. 2. Variation of the 'Be''" clock transition frequency

referenced to a passive hydrogen maser plotted against sidereal time. Tick marks on the vertical scale are 100 /xHz

apart.

TABLE I. Variation of the 'Be"^ clock transition frequency, V, measured relative to a passive hydrogen maser. Ak is a measure of any LLI violation, v = vq + Ai^Pkicos^ir)).

where yS is the angle between the 'Be'^ quantization axis and

the direction in space given in the table, r is sidereal time.

Direction

Ak (/xHz)

A2

Motion through mean

rest frame of the universe

Motion through

solar rest frame

Galactic center Vir^ supercluster

Sun

35 ±40

36 ±42

35 ±46 13 ±54

30 ±51 9 ±63

49 ±38 -50 ±42 -15 ±46

20 ±40 0±49 9±57

40 ±39 -58 ±43 16±47

TN-182 2389
Volume 54, Number 22 PHYSICAL REVIEW LETTERS 3 June 1985

vertical. It should also be possible to make such measurements similar to the present one using nuclear

magnetic resonance techniques on neutral atoms.^°'^^ In fact, relevant data may already exist from NMR

gyroscope experiments."

We gratefully acknowledge the support of the U. S. Air Force Office of Scientific Research and the U. S. Office of Naval Research. We thank F. L. Walls for information on the passive hydrogen maser and gratefully acknowledge helpful discussions with M. P. Haugan. We thank C. M. Will, S. R. Stein, and D. W. AlIan for carefully reading the manuscript.

^C. M. Will, Theory and Experiment in Gravitational Physics

(Cambridge Univ. Press, Cambridge, 1981). 2d. Howe and F. Walls, IEEE Trans. Instrum. Meas. 32,

218(1983).

3M. P. Haugan, Ann. Phys. (N.Y.) 118, 156 (1979), and

doctoral dissertation, Stanford University, 1978 (unpublished).

*J. Leitner and S. Okubo, Phys. Rev. 136, B1542 (1964); N. D. Hari Dass, Phys. Rev. Lett. 36, 393 (1976).

5p. R. Phillips and D. Woolum, Nuovo Cimento 64B, 28

(1969).

^G. Cocconi and E. Salpeter, Nuovo Cimento 10, 646

(1958). ^V. W. Hughes, H. G. Robinson, and V. Beltran-Lopez,

Phys. Rev. Lett. 4, 342 (1960); S. A. Lewis, W. L. Williams,

and V. W. Hughes, Bull. Am. Phys. Soc. 11, 121 (1966);

W. L. Williams, doctoral dissertation, Yale University, 1966

(unpublished).

8R. W. p. Drever, Philos. Mag. 6, 683 (1961).

'S. T. Epstein, Nuovo Cimento 16, 587 (1960); R. H.

Dicke, Phys. Rev. Lett. 7, 359 (1961).

lOW.-T. Ni, in Proceedings of the 1983 International School

Symposium on Precision Measurement and Gravitational Experiments, edited by W.-T. Ni (National Tsing Hua University,

Taiwan, China, 1983), p. 519.

lip.

J. Peebles and R. H. Dicke, Phys. Rev. 127, 629

(1962).

12a. p. Lightman and D. L. Lee, Phys. Rev. D 8, 364

(1973).

I3p. M. Lubin, G. L. Epstein, and G. F. Smoot, Phys. Rev.

Lett. 50, 616 (1983); D. J. Fixsen, E. S. Cheng, and D. J. Wilkinson, Phys. Rev. Lett. 50, 620 (1983).

i^The interaction between two charged particles in the

77/6/1 formalism is given in Eq. (4.13) of Ref. 3. Equation

(1) here is the total interaction for a collection of such particles. i^M. P. Haugan is currently making a more detailed calculation of this matrix element.

16J. J. Bollinger, J. D. Prestage, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett. 54, 1000 (1985).

i^H. G. Dehmelt, Adv. At. Mol. Phys. 3, 53 (1967), and 5, 109 (1969); D. J. Wineland, W. M. Itano, and R. S. Van

Dyck, Jr., Adv. At. Mol. Phys. 19, 135 (1983).

18n. F. Ramsey, Molecular Beams (Oxford Univ. Press, London, 1956).

•'Preliminary results of this experiment were presented at the Ninth International Conference on Atomic Physics Satellite Workshop on Ion Trapping, Seattle, Washington, July

1984 (unpublished).

20D. J. Wineland and N. F. Ramsey, Phys. Rev. A 5, 821

(1972); J. C. Gallop and B. W. Petley, Nature (London)

303, 53 (1983).

21B. C. Grover, Phys. Rev. A 28, 2521 (1983); F. J. Raab,

in Proceedings of the Ninth International Conference on Atomic

Physics. Seattle. Washington. 1984. edited by R. S. Van Dyck,

Jr., etat. (World Scientific, Singapore, 1985); T. E. Chupp,

private communication.

22E. Kanegsberg, Proc. SPIE-Int. Soc. Opt. Eng. 157, 73

(1978).

TN-183

2390
NBS-n4A (REV. 2.8C) U.5. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO. NBS TN-1086

2. Performing Organ. Report No 3. Publ ication Date

July 1985

4. TITLE AND SUBTITLE

Trapped Ions and Laser Cooling

Selected publications of the Ion Storage Group of the Time and Frequency

Division, NBS, Boulder, Colorado

5. AUTHOR(S)

Editors: D. J. Wineland, W. M. Itano, J. C. Bergquist. and J. J. Bollinger

6. PERFORMING ORGANIZATION (If joint or other ilian NBS. see instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City, State, ZIP)

Supported in part by

U.S. Office of Naval Research U.S. Air Force

800 North Quincy and Office of Scientific Research

Arlington, VA 22217 Boiling Air Force Base, DC 20332

10. SUPPLEMENTARY NOTES

I I Document describes a computer program; SF-185, FlPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here)

Selected publications of the Ion Storage Group of the Time and Frequency

Division, NBS, Boulder, Colorado.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

atomic spectroscopy: ion storage* laser cooling: laser spectroscopy; Paul trap:

Penning trap: trapped atomic particles

13. AVAILABILITY

jV] Unlimited

I I For Official Distribution. Do Not Release to NTIS

K~\ Order From Superintendent of Documents, U.S. Government Printing Office, Wachington, D.C. 20402.

I

] Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

19 2

15. Price

USCOMM-DC 6043-P80

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NBS_ Technical Publications

Periodical

Joumai of Research—The Journal of Research of the National Bureau of Standards reports NBS research and development in those disciplines of the physical and engineering sciences in which the Bureau is active. These include physics, chemistry, engineering, mathematics, and computer sciences. Papers cover a broad

range of subjects, with major emphasis on measurement methodology and the basic technology underlying

standardization. Also included from time to time are survey articles on topics closely related to the Bureau's

technical and scientific programs. Issued six times a year.

Nonperiodicals

Monographs—Major contributions to the technical literature on various subjects related to the Bureau's scientific and technical activities. Handbooks—Recommended codes of engineering and industrial practice (including safety codes) developed in cooperation with interested industries, professional organizations, and regulatory bodies. Special Publications—Include proceedings of conferences sponsored by NBS, NBS annual reports, and other

special publications appropriate to this grouping such as wall charts, pocket cards, and bibliographies.

Applied Mathematics Series—Mathematical tables, manuals, and studies of special interest to physicists,

engineers, chemists, biologists, mathematicians, computer programmers, and others engaged in scientific and

technical work.

National Standard Reference Data Series—Provides quantitative data on the physical and chemical properties

of materials, compiled from the world's literature and critically evaluated. Developed under a worldwide pro- gram coordinated by NBS under the authority of the National Standard Data Act (Public Law 90-396).

NOTE: The Journal of Physical and Chemical Reference Data (JPCRD) is published quarterly for NBS by

the American Chemical Society (ACS) and the American Institute of Physics (AIP). Subscriptions, reprints,

and supplements are available from ACS, 1155 Sixteenth St., NW, Washington, DC 20056.

Building Science Series—Disseminates technical information developed at the Bureau on building materials,

components, systems, and whole structures. The series presents research results, test methods, and perfor- mance criteria related to the structural and environmental functions and the durability and safety

characteristics of building elements and systems.

Technical Notes—Studies or reports which are complete in themselves but restrictive in their treatment of a

subject. Analogous to monographs but not so comprehensive in scope or definitive in treatment of the subject

area. Often serve as a vehicle for final reports of work performed at NBS under the sponsorship of other government agencies.

Voluntary Product Standards—Developed under procedures published by the Department of Commerce in Part 10, Title 15, of the Code of Federal Regulations. The standards establish nationally recognized re- quirements for products, and provide all concerned interests with a basis for common understanding of the

characteristics of the products. NBS administers this program as a supplement to the activities of the private sector standardizing organizations.

Consumer Information Series—Practical information, based on NBS research and experience, covering areas of interest to the consumer. Easily understandable language and illustrations provide useful background

knowledge for shopping in today's technological marketplace.

Order the above NBS publications from: Superintendent of Documents, Government Printing Office,

Washington, DC 20402. Order the following NBS publications—FIPS and NBSIR 's—from the National Technical Information Service, Springfield, VA 22161.

Federal Information Processing Standards Publications (FIPS PUB)—Publications in this series collectively constitute the Federal Information Processing Standards Register. The Register serves as the official source of

information in the Federal Government regarding standards issued by NBS pursuant to the Federal Property

and Administrative Services Act of 1949 as amended. Public Law 89-306 (79 Stat. 1127), and as implemented

by Executive Order 11717 (38 FR 12315, dated May 11, 1973) and Part 6 of Title 15 CFR (Code of Federal

Regulations).

NBS Interagency Reports (NBSIR)—A special series of interim or final reports on work performed by NBS

for outside sponsors (both government and non-government). In general, initial distribution is handled by the sponsor; public distribution is by the National Technical Information Service, Springfield, VA 22161, in paf)er

copy or microfiche form.
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