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NBS SPECIAL PUBLICATION 653

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

Laser-Cooled and

Trapped Atoms
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Laser-Cooled and Trapped Atoms

Proceedings of the Workshop on Spectroscopic

Applications of Slow Atomic Beams

NBS Gaithersburg, MD, 14-15 April 1983

William D. Phillips, Editor

Electrical Measurements and Standards Division

Center for Absolute Physical Quantities

National Measurement Laboratory

National Bureau of Standards

Washington, DC 20234

NATIONAL BUREAU. OF STANDARDS

LI3RARY

JUL 5 1983

Sponsored by:

Office of Naval Research

Arlington, VA 22217

and

National Bureau of Standards

Washington, DC 20234

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

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued June 1983
Library of Congress Catalog Card Number: 83-600541

National Bureau of Standards Special Publication 653

Natl. Bur. Stand. (U.S.), Spec. Publ. 653, 172 pages (June 1983)

CODEN: XNBSAV

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1983

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

Price

(Add 25 percent for other than U.S. mailing)
PREFACE

My personal interest in the field of laser-cooled and trapped atoms dates from Ashkin's 1978 paper on laser trapping and cooling of atoms [1]. While atom traps had been proposed as early as 1968 by Letokhov

[2] and 1970 by Ashkin [3] and a good bit of theoretical work had been done, the 1978 proposal caused a renewed interest in the field both because the proposal seemed so promising and because of the beautiful

demonstration of laser cooling in ion traps by Wineland et al. [4] and Neuhauser et al. [5].

Since 1978 there has been extensive theoretical work on the motion of atoms in laser fields as well as discussions of other kinds of traps for neutral atoms. Bjorkholm et al. demonstrated the dipole forces

which could make optical trapping possible fc]o An experimental realization of an optical trap, however,

proved to be more elusive. A basic problem was that the atoms had to be cooled considerably before they

could be loaded into the very shallow optical traps.

The earliest attempts to cool an atomic beam by Letokhov and col leagues at the Institute of Spectroscopy

in Moscow, were hampered by optical pumping [7]. The Moscow group overcame the optical pumping problem

and reported a clear demonstration of laser deceleration in 1981 [8]. In 1982 the group at the National

Bureau of Standards in Gaithersburg reported a different method which eliminated optical pumping and in

addition compensated for the changing Doppler shift of the decelerated atoms [9]. More recently the NBS

group succeeded in cooling a beam of atoms to energies comparable to the depth of proposed optical traps

[10]. By this time it had become obvious that the production of slow atomic beams was an interesting

development in itself, apart from the possibility that such atoms might eventually find their way into

an optical trap.

In the light of all these developments, Bob Junker of the Office of Naval Research suggested that the

time was ripe for an assessment of where the field of slow neutral atoms stood and where it was headed.

From these discussions came the idea for a small gathering of scientists in a workshop atmosphere to discuss slow atoms and particularly to speculate on the possibilities for future work, especially work

which might lead to applications in high resolution spectroscopy or time and frequency standards.

The resulting workshop, under the joint sponsorship of the Office of Naval Research and the National

Bureau of Standards, did not stray too far from the original goal. It included a review of the present

status and direction of frequency standards work, the theory of atom-field interaction and traps, and

experiments on laser cooling of beams. A number of proposals for traps for neutral atoms were presented

as well as new ideas for cooling atoms in traps, beams, and gases.

Thirty-four invited scientists attended the workshop, and contributed 21 papers to these proceedings [11].

They represented universities, government and private laboratories across the country, and included some foreign visitors at U.S. institutions. While this attendance exceeded the "small gathering" initially

envisioned, the discussions were as lively and informal as we had hoped they would be. Discussions often

lasted as long as presentations, and the Workshop was enriched by a number of impromptu presentations.

It is to be hoped that the May 2 deadline for final submission of manuscripts has allowed authors to consider the discussions and ideas which grew from the Workshop. I would like to mention two instances

of this: G. L. Greene has kindly included his impromptu remarks concerning magnetic trapping of ultra-cold

neutrons in these proceedings, and J. P. Gordon has presented a new general proof concerning the

possibility of radiative confinement of atoms in traps which rely only on the scattering force.

I would like to thank some of the many people who helped to make this Workshop a success: Kathy Stang of

the NBS National Measurement Laboratory offices who guided the coordination of the Workshop from the

beginning and who, with Naomi Crockett, handled the registration; Sara Torrence and Kathleen Kilmer of

the NBS Public Information Division who made so many of the arrangements needed for a smooth operation;

Brenda Main and Lesa Childress of the NBS Electrical Measurements and Standards Division Office, who

handled the considerable secretarial effort which goes into such a meeting; Barry Taylor and John Prodan

who, in addition to participating in the meeting provided invaluable assistance in planning and arranging;

Sam Stein who coordinated the efforts of the NBS Time and Frequency Division in providing the overviews

of the status of time and frequency standards; Bob Junker and the Office of Naval Research for proposing

this Workshop and providing important guidance as well as the major financial support for the Workshop and

for this publication; finally, all the participants in the workshop who through a very long day and a half

of talksand discussions maintained the high level of interest and enthusiasm which made the Workshop such

a rewarding experience.

[l] A. Ashkin, Phys. Rev. Lett. 40, 729 (1978).

[2] V. S. Letokhov, Pis'ma Zh. Eksp. Teor. Fiz. 7, 348 (1968) [JETP Lett. 7, 272 (1968)] .

i i i
[3] A. Ashkin, Phys. Rev. Lett. 25, 1321 (1970).

[4] D. J. Wineland et al. , Phys. Rev. Lett. 40, 1639 (1978).

[5] W. Neuhauser et al ., Phys. Rev. Lett., 41, 233 (1978).

[6] J. E. Bjorkholm et al. , Phys. Rev. Lett. 41, 1361 (1978).

[7] V. I. Balykin, V. S. Letokhov, and V. I. Mishin, JETP Lett. 29, 561 (1979). [Pis'ma Zh. Eksp. Teor.

Fiz. 29, 614 (1979)]; Sov. Phys. JETP, 51, 692 (1980) [Zh. Eksp. Teor. Fiz. 87, 1376 (1980)];

V. I. Balykin, Opt. Commun. 33, 31 (19807.

[8] S. V. Andreev et al ., Pis'ma Zh. Eksp. Theor. Fiz. 34, 463 (1981) [JETP Lett. 34, 442 (1981)],

[9] W. D. Phillips and H. Metcalf, Phys. Rev. Lett. 48, 596 (1982).

[10] J. V. Prodan, W. D. Phillips, and H. Metcalf, Phys. Rev. Lett. 49, 1149 (1982).

[11] The opinions expressed by non-NBS authors in these proceedings are their own and their appearance

in this publication does not represent an endorsement by the Bureau of Standards or the U.S. Government. Any mention of commercial products in this publication does not imply a recommendation

or endorsement by the National Bureau of Standards.

William D. Phillips

Gaithersburg, Maryland

May 1983

iv
TABLE OF CONTENTS

PAGE

PREFACE iii

W. D. PHILLIPS, J. V. PRODAN, and H. J. METCALF -- Neutral atomic beam cooling experiments

at NBS 1

S. R. STEIN -- The design of atomic frequency standards and their performance in specific

applications 9

D. J. WINELAND, W. M. ITANO, J. C. BERQUIST, J. J. BOLLINGER, and H. HEMMATI -- Frequency

standard research using stored ions 19

K. M. EVENSON, D. A. JENNINGS, F. R. PETERSON, J. S. WELLS, and R. E. DRULLINGER -- Optical

frequency synthesis spectroscopy 27

L. LEWIS -- Limitations of atomic beam frequency standards 38

P. R. HEMMER, S. EZEKIEL, and C. C. LEI BY -- Stabilization of a microwave oscillator using

a resonance Raman transition in a sodium beam 47

M. H. MITTLEMAN The force on an atom in a laser and D.C. field 53

H. J. METCALF -- Magnetic trapping of decelerated neutral atoms 59

J. P. GORDON -- The motion of atoms in a radiative trap 68

W. H. WING -- Some problems and possibilities for quasistatic neutral particle trapping 74

W. H. WING -- Gravitational effects in particle traps 94

W. C. STWALLEY -- A hybrid laser-magnet trap for spin-polarized atoms 95

D. E. PRITCHARD -- Good and bad aspects of traps for neutral particles 103

E. GIACOBINO and P. R. BERMAN -- Cooling of vapors using col 1 isional ly aided radiative

excitation 112

K. RUBIN and M. S. LUBELL -- A proposed study of photon statistics in fluorescence through

high-resolution measurements of the transverse deflection of an atomic beam 119

M. S. LUBELL and K. RUBIN -- Velocity compression and cooling of a sodium atomic beam using a frequency modulated ring laser 125

J. V. PRODAN and W. D. PHILLIPS -- Chirping the light — fantastic? Recent NBS atom cooling

experiments 137

R. BLATT, W. ERTMER, and J. L. HALL -- Cooling of an atomic beam with frequency- sweep techniques 142

W. ERTMER, R. BLATT, and J. L. HALL -- Some candidate atoms and ions for frequency standards

research using laser radiative cooling 154

G. L. GREENE -- Trapping of low energy neutrons 162

LIST OF WORKSHOP PARTICIPANTS 166

v

Laser-Cooled and Trapped Atoms, (Proceedings of the Workshop on Spectroscopic Applications of Slow Atomic

Beams, held at NBS, Gaithersburg , MD, 14-15 April 1983) Ed. by W. D. Phillips, Natl. Bur. Stand. (U.S.)

Spec. Publ. 653 (1983).

NEUTRAL ATOMIC BEAM COOLING EXPERIMENTS AT NBS

William D. Phillips, John V. Prodan* and Harold J. Metcalf**

Electrical Measurements and Standards Division

National Bureau of Standards

Washington, DC 20234

We have decelerated and cooled a neutral atomic sodium beam using a near resonant,

councerpropagating laser beam. A spatially varying magnetic field compensates for the

changing Doppler shift as the atoms decelerate, keeping the atoms in resonance with the

laser. We have observed final velocities as low as 40 m/s, or about 25 times slower than the

initial thermal velocity. By compressing the atomic velocity distribution we have increased

the atomic density per unit velocity by as much as a factor of 30 over that of the thermal

distribution.

Key words: frequency standards, high resolution spectroscopy, laser cooling, laser trapping,

slow atomic beam.

1. Introduction

The techniques and results for resonant deceleration of neutral atoms have been presented in some

detail elsewhere [1,2]. In this paper we review those techniques and results, discuss more recent

results, describe the problems associated with neutral atom cooling, and discuss some possibilities for

appl ications.

Laser deceleration of an atomic beam is accomplished by directing a resonant laser beam against the

propagation direction of an atomic beam. The atoms absorb photons from the laser beam, receiving

repeated momentum kicks in the direction of the laser beam. Since the reradiation pattern is symmetric

with respect to reflection in any plane, no net average momentum transfer results from the reradiation,

and the atoms slow down. (Of course, the reradiation contributes to a random walk of the atomic

momentum about its average value.) The velocity change per photon is Av=hv/Mc, where v is the photon

frequency and M is the atom's mass. For Na atoms making transitions at 589 nm (3S-*3P) this is a velocity change of 3 cm/s per photon.

The maximum acceleration can be calculated by noting that when the transition is heavily saturated, the

atom spends half its time in the excited state and while excited it radiates at a rate determined by

the natural lifetime : b 1 ,i ,2 /•, \ a max

= m 27^ 10 m/s (1)

for the case of Na where x = 16 ns. Na atoms with a typical thermal velocity of 1000 m/s can, with

this acceleration, be stopped in 1 ms over a 0.5 m distance, after scattering over 30,000 photons.

The situation for atom beam deceleration should be contrasted to the case of cooling ions in a trap

f3]. In that case, the laser must be tuned on the red side of the transition frequency to cool. A

blue tuning will heat the ions. For the atomic beam, counterpropagation of the cooling beam guarantees

deceleration, regardless of the tuning. The tuning only affects the rate of deceleration. This rate,

however, should be maximized since in contrast to the case of an ion trap the atoms are in the

apparatus for only a short time.

Because of the large number of photons needed for deceleration the relative size of the transverse

spread in momentum, caused by the random reradiation, is small. This spread is given approximately by

the square root of the number of photons radiated, and for Na is about 1/2% of the initial velocity.

*NBS-NRC Postdoctoral Fellow

**Permanent Address: Dept. of Physics, SUNY, Stony Brook, NY

1
The disadvantage of this large number is that during the many optical cycles involved, undesired

optical pumping effects are likely to occur. Hyperfine interaction splits the Na 3S]/2 state by 1772 MHz into F=l and F=2 components. Since the natural linewidth is only 10 MHz, if the laser induces a transition from 3S]/2> F

= 2 t0 a 3P state which decays to 3Sj/2, F=l, that atom will not be reexcited at

a reasonable rate, because it is so far out of resonance with the laser. This kind optical pumping is

likely to occur after about 100 optical cycles, so the amount of deceleration would ordinarily be

severely limited. Another process limiting deceleration is the changing Doppler shift as the atoms

decelerate. 200 photons will shift the velocity by 6 m/s, which corresponds to a Doppler shift of 10 MHz, the natural linewidth. This process would soon take the atoms out of resonance, effectively

ending the deceleration process.

We have solved the optical pumping problem by using a magnetic field applied along the direction of the

laser-atomic beam axis and a circularly polarized laser. The details are given in Ref. [1], but in essence this creates an effective two-level system between the F=2, Mp=2 component of the ground state

and the F=3, Mp=3 component of the excited 3P 3/2 state. With a field greater than 0.05 T the

probability of transitions outside these two levels is small enough to be negligible.

The magnetic field also allows us to solve the Doppler shift problem. By winding a solenoid with more

turns at one end than the other, we can create a spatially varying magnetic field. The Zeeman shift of

the transition frequency (14 GHz/T) is then also spatially varying and can be chosen to compensate for

the varying Doppler shift. If we assume that the atoms will decelerate at a constant rate a, the

velocity as a function of distance traveled will be

v(z) = v/v0 *-2az . (2)

Equating the Doppler shift corresponding to this velocity to the Zeeman shift induced by the field

yields a field whose spatial dependence is

B(z) = B

b + B

0 /l-2az/v

0

2

. (3)

The bias field B^ is chosen to keep the field high enough to avoid optical pumping. B 0is the field

which produces a Zeeman shift equal to the Doppler shift of an atom having velocity v 0

.

A field with this shape will allow the deceleration to zero velocity of any atom with a velocity less

than or equal to v 0

. Lower velocities simply begin to decelerate at larger distances, when they first

come into resonsnce.

2. Apparatus

We have constructed a multi-layer solenoid to reproduce the field given by Eq. (3). The length of the

solenoid and strength of the field are chosen so that Na atoms at 1000 m/s can be stopped in about 1 m with an acceleration about half the theoretical maximum given by Eq. (1).

The solenoid is incorporated into an atomic beam apparatus shown schematically in Fig. 1. The atomic

beam, from a 950 K Na source, collimated to 15 mrad full angle, enters the high field end of the

solenoid, where the peak field is 0.16 T. When a given atom reaches a point in the solenoid where the

combined Zeeman and Doppler shifts place it in resonance with the fixed frequency cooling laser, it

absorbs photons and begins to decelerate. As long as the field does not change so rapidly that the

rate of Zeeman shift change exceeds the possible rate of Doppler shift change, the atom will stay in resonance as it travels down the varying field of the solenoid.

The cooling laser beam is converging so as to approximately match the divergence of the atomic beam.

In this way, the small component of transverse velocity which results from the imperfect collimation of

the atomic beam is reduced along with the much larger longitudinal velocity.

The atomic velocity distribution is determined by observing o fl uorescence induced by a second, very weak

probe laser which crosses the atomic beam at an angle of 11 . As the probe frequency is scanned, the

induced fluorescence gives the atomic density as a function of velocity component along the probe. At

11° this corresponds to 98% of the longitudinal velocity of the atoms. Calibration of the probe laser

scan is provided by directing a small portion of the probe beam perpendicular to the atomic beam.

Since the atoms have nearly zero velocity component along this beam, the fluorescence it induces can be

2
used as a frequency marker for zero Doppler shift. The known hyperfine structure of Na, which is well

resolved for the perpendicular excitation, provides a scale calibration for the probe scan.

COLLECTION

OPTICS

Na

Source

PROBE

LASER BEAM MECHANICAL

CHOPPER

10

cm

iO cm" -40 cm—

1

COOLING

LASER

BEAM

Fig. 1. Block diagram of experimental set-up.

In order to avoid confusion from the fluorescence induced by the much stronger cooling laser, we use a mechanical chopper to shut off the cooling beam just before observing fluorescence from the probe. A boxcar integrator is used to restrict the observation time to a short period after the cooling laser

shuts off. The time for which the cooling laser is on is long compared with the atomic transit time

through the apparatus so the detection technique gives the steady-state velocity distribution for

continuous laser cooling.

3. Results

Figure 2 shows the effect of the cooling laser on the velocity distribution when only a uniform field

is applied along the atomic beam. In this case, the amount of deceleration is limited by the

uncompensated Doppler shift. Atoms are removed, from that velocity group which is resonant with the

laser and "pushed" to a lower velocity until they are too far out of resonance to absorb many more

photons. The result of Fig. 2 is quite similar to the results reported by Andreev et a]_. [4]. In the latter experiment, two cooling laser frequencies, separated by the ground hyperfine splitting, are

used to overcome optical pumping. In both cases, there is only about a 15% reduction from the velocity

which is resonant with the laser.

Figure 3 shows the effect when the spatially varying field is added to the uniform field. In this

case, where the Doppler shift is compensated by the Zeeman shift, a factor of two reduction in velocity

is achieved. The density of atoms at the lower velocity is 30 times higher than at the peak of the

thermal distribution, and the full width of the cooled velocity distribution is only 5% of the central

velocity.

Since the total deceleration is limited by the total change in the magnetic field, to achieve lower

velocities one simply tunes the laser to be initially resonant with slower atoms. Figure 4 shows the

velocity distributions which result from a sequence of different laser tunings. As expected, when the

tuning is to lower initial velocity, a lower final velocity is achieved. The final velocity actually

changes more than the initial velocity, i.e., the total deceleration increases as the starting velocity

3
I—I—t—I—I—t- t2

H

3 DOPPLER SHIFT (GHz)

HI 1-

I 2 VELOCITY (x I 0 M/S)

Fig. 2 Atomic velocity profile produced with the cooling laser at a frequency resonant with atoms

(in a uniform magnetic field) whose initial Doppler shift is v-j

.

Vo ~ 1130 M/S

V = 520 M/S

Vo

t DOPPLER SHIFT

Fig. 3 Atomic velocity profile produced using the spatially varying magnetic field. The cooling

laser is tuned to be in resonance (in the highest field) with atoms whose initial Doppler

shift is v]. The dotted line represents the unmodified velocity profile.

4
Fig. 4 Series of atomic velocity distributions

using the spatially varying magnetic

field and different tunings of the cooling laser. In each profile the position

in the velocity distribution where the

laser is tuned is marked by an arrow.

The profile labeled (g) is the unmodified distribution produced by blocking

the cooling laser.

-f

1 1—t—i—i 1—H y

0 4 8 12 VELOCITY (x I O

l M/S)

is lowered. This is because the faster atoms see a more rapidly changing field near the end of the

solenoid and they cannot decelerate fast enough to keep pace with the changing Zeeman shift. As a result, they "drop out" of the cooling process and do not experience the total deceleration implied by

the available field change.

At a laser tuning such that the final atomic velocity is 200 m/s, about five times slower than the

initial velocity, the amplitude of the slow atom peak decreases sharply and disappears entirely if the

laser is tuned to a still slower velocity. This occurs because the deceleration of the atoms does not

completely stop when the atoms leave the solenoid. Even though the magnetic field falls rapidly as the

atoms exit the solenoid, when the velocity is below 200 m/s the atoms can still stay in resonance, or close enough to resonance that the cooling laser will actually stop them. This is not surprising

considering that the stopping distance for saturated atoms at 200 m/s is only 2 cm.

5
To circumvent this problem, as well as to confirm the hypothesis given to explain it, we devised a modified detection scheme. Instead of observing the fluorescence immediately after the cooling laser

shuts off (typical delay is 50 ys) we now wait a significant time before beginning the observation—on

the order of several milliseconds. This allows the atoms which reach low velocities near the end of

the magnet to drift in the dark, unaffected by any more deceleration, until they reach the observation

region 40 cm away.

Figure 5a shows a sequence of velocity distributions obtained with various delay times. For longer

delays, lower velocity atoms are observed. In fact, because the atoms are decelerated to near zero velocity over such a short distance near the end of the solenoid, the observed velocity is nearly

inversely proportional to the delay time. Using the known delay times and the measured velocities, we

can determine the location in the solenoid where the atoms originated. We find that the entire range

VELOCITY (I0

2 m/s)

0 1.6 3.2

Fig. 5

(a) The velocity distribution of cooled

atoms with various observational delay times.

The broad background is from nonbeam Na atoms, (b) Atoms observed at a velocity of

40 m/s and a delay of 9.0 ms. The leftmost two peaks in all these distributions

are frequency markers from which the residual Doppler shift (and hence atomic

velocity) is measured.

-0.4-0.2 0 0.2 0.4

V-v0 (GHz)

-O.I 0 O.I

0 40 80

VELOCITY, m/s

6
of velocities in Fig. 5a come from a few centimeters near the end of the solenoid. Slower atoms come

from regions of lower field, nearer the observation region, as expected.

The narrowness of the velocity distribution of slow atoms in Fig. 5a is in some sense artifical, since

it is related to the narrowness of the gate time used for the observation. This is essentially a time-of-fl ight velocity selection, and longer gate time will usually result in a broader observed

velocity distribution. Nevertheless, Fig. 3 shows a dramatic compression of the velocity, under

steady-state conditions. Our working assumption, which is supported by the data, is that at any

position in the solenoid, there exists in steady-state a narrow distribution of velocity whose central

velocity corresponds to the magnetic field at that position. If we could extract atoms from a specific

position we could obtain a continuous atomic beam with a narrow velocity distribution. Our delayed

observation technique is a substitute for such an extraction.

Figure 5b shows the slowest velocity distribution we have obtained. Using a 9 ms delay we observed a central velocity of 40 m/s with a width of about 10 m/s. This is the true velocity width, since we found that it did not change as the gate time was increased. This is because at this very low

velocity, so little atomic density exists at still lower velocity that admitting slower atoms, by

extending the observation time, does not affect the results. After accounting for the width due to the

natural, radiative width inherent in the detection (6 m/s) we infer an equivalent "temperature" of

relative motion of 0.07 K. We estimate the atomic density within the natural linewidth to be on the

order of 10 5 cm~3.

One of the most striking features of Fig. 5 is the rapid decline in observed atomic density as v^.

For different tunings of the cooling laser we have observed dependences as strong as v3»5. Simple

considerations predict that the density at the observation point should vary as v^ due to the angular

divergence of the beam which increases as 1/v, and as v due to the increase in density which comes as the velocity decreases but the flux stays the same. These effects combine to predict a density varying

as v. This assumes that the gate time for observation is long enough to include the range of arrival

times implied by the 6 m/s velocity width corresponding to the natural lifetime.

While we originally ascribed the rapid dependence we observe to the scattering of very slow atoms by

background gas [2], improvements in the vacuum systems which allowed us to reduce the pressure by two

orders of magnitude have shown that this is not the case. We do not now have a satisfactory

explanation of the velocity dependence of the density. Work is continuing on this problem: we are doing a Monte Carlo simulation of the cooling process, analyzing the detection system in more detail

and planning more experiments. We do believe that we understand certain aspects of the problem such as the dependence on cooling laser tuning (some tunings make it unlikely that any atoms will reach the

very low velocities) and the dependence on observation time (too short observation windows restrict the

observable velocity to a smaller range than the 6 m/s corresponding to the natural width), but we do

not understand the persistence of a dependence on velocity which is always stronger than our simple model

predicts.

4. Future Plans

Because we now believe that atoms are not being lost due to scattering, it seems reasonable to assume

that whatever mechanism causes the density to decrease, it can be overcome by collecting the atoms

efficiently and conducting them to the observation region. We intend to use a hexapole focussing

magnet to achieve this. The hexapole acts as a positive lens to atoms. Its collection efficiency

varies as v"^, so we expect to be able to achieve significant improvements in the density of slow

atoms.

An important unsolved problem for our cooling process is seen in Fig. 4. Curve (d) shows a large peak

at about 300 m/s, with some smaller peaks at higher velocity. Under certain conditions we have

observed four or more narrow, strong peaks of about equal intensity. They do not seem to be related in

a simple way to known irregularities in the solenoid field, nor do they change significantly as those

irregularities are varied. They depend strongly on tuning and power of the cooling laser. More

cooling power always strengthens the slowest peak and weakens the faster ones. We do not have a suitable explanation for these peaks, although they would appear to be related to some process that causes atoms to drop out of the cooling process at specific velocities. Experiments in progress on cooling by scanning the cooling laser rather than using a varying field may help to shed some light on this question [5].

Extraction of atoms from the solenoid at a particular position corresponding to a particular velocity

would be an important advance. We are considering using a mirror-coated micro-channel -pi ate as a

7
device which will pass the atomic beam into a dark region and still allow the cooling laser beam to

overlap the atomic beam up to that point. If successful, such a scheme would allow production of a continuous beam of slow atoms, as opposed to the pulsed production which is imposed by our detection

technique.

The ability to produce high intensity, nearly monoenergetic atomic beams such as in Fig. 3 raises the

possibility of some interesting atomic beam experiments. One of these involves the use of transverse

deflection of an atomic beam by light to study quantum statistics [6]. An experiment along these lines

is discussed by Rubin and Lubell elsewhere in these proceedings [7].

An important application of slow atom techniques is in ultra-high resolution spectroscopy. This is discussed by W. Ertmer et al . elsewhere in these proceedings [8], We simply note that if Ca atoms were decelerated even a very modest amount compared to what we have observed for Na, then the 2nd

order Doppler effects which now limit the resolution of the Ca intercombi nation line at 657 nm [9]

would be virtually eliminated.

One of the primary original motivations for this and other similar work has been the possibility of

trapping neutral atoms in electromagnetic fields. Traps are discussed elsewhere [10,11,12,13], but a basic problem is that most such traps are so shallow in energy that very slow atoms are required. We have now produced atoms close to the energy of proposed traps.

A recent proposal for laser trapping by Minogin [14] is an extension to three dimensions of a one-dimensional radiation pressure trap proposed by Ashkin [10] Since the trap depends only on radiation pressure rather than the dipole force [10,11] it can be much larger than the tightly focussed

dipole traps —on the order of 1 mm rather than several microns. This has two distinct advantages:

first, with densities as high as 10 5 cm~3 which we have achieved for 40 m/s atoms, about 100 atoms

could be expected to be in the trap if it were turned on at the proper time after the cooling laser

shut off. This sort of operation would just be an extension of our delayed observation technique.

Second, 1 mm is longer than the distance needed to bring saturated atoms at 40 m/s to rest. This gives

a reasonable chance that some atoms will lose enough energy to remain in the trap. Two trapping

frequencies would have to be used to avoid optical pumping, and with proper tuning of each, the trap

would further cool the atoms. In principle, temperatures in the order of 10" 3 K are possible.

This work was supported in part by the Office of Naval Research.

[I] W. D. Phillips and H. Metcalf, Phys. Rev. Lett. 48 596(1982).

[2] J. V. Prodan, W. D. Phillips and H. Metcalf, Phys. Rev. Lett 49 1149 (1982).

[3] D. J. Wineland et al_. Phys. Rev. Lett 40, 1639 (1978).

[4] S. V. Andreev et al_. JETP Lett 34, 442 (1981) [Pis'ma Zh. Eksp. Teor. Fiz. 34, 463 (1981)].

[b] J. V. Prodan and W. D. Phillips, these proceedings

[6] R. J. Cook, Opt. Commun. 35, 347 (1980).

[7] K. Rubin and M. S. Lubell, these proceedings.

[8] W. Ertmer et aj_. , these proceedings.

[9] Ch. Salomon etaU, J. Phys. (Paris) 42, C8-3 (1981).

[10] A. Ashkin, Phys. Rev. Lett. 40, 729 (1978).

[II] J. P. Gordon and A. Ashkin, Phys. Rev. A 21, 1606 (1980).

[12] W. H. Wing, Phys. Rev. Lett. 45, 631 (1980).

[13] See also papers by W. H. Wing, W. C. Stwalley, D. E. Pritchard, J. P. Gordon, H. Metcalf

and G. L. Greene in these proceedings.

[14] V. G. Minogin, Sov. J. Quantum Electron 12 299 (1982) .C^art. Elektron. (Moscow) 9, 505

(1982)], but see also J. Gordon, these proceedings for a cntidsm of such traps.
Laser-Cooled and Trapped Atoms , (Proceedings of the Workshop on Spectroscopic

Applications of Slow Atomic Beams, held at NBS, Gaithersburg, MD, 14-15 April

1983) Ed. by W. D. Phillips, Natl. Bur. Stand. (U.S.) Spec. Publ. 653 (1983).

THE DESIGN OF ATOMIC FREQUENCY STANDARDS AND THEIR

PERFORMANCE IN SPECIFIC APPLICATIONS

S. R. Stein

National Bureau of Standards

Boulder, Colorado 80303

ABSTRACT

The reduction of timing errors in atomic clocks is shown to be

important for secure communications and navigation. An approach based

upon control of all systematic frequency shifts and a reduction in both

the first order Doppler shift and confinement effects is recommended.

Ion storage is a promising technique because of its ability to achieve

extremely long observation times, negligible confinement perturbations,

and laser cooling. The applicability of atom cooling should be

evaluated.

Key Words: Atomic Clocks, Frequency Standards, Navigation,

Secure Communications

Introduction

The development of frequency standards and clocks has been one of the

most visible applications of atomic and molecular spectroscopy. For this

reason, it is natural for a workshop on slow beams to consider some of the

requirements for improved frequency standards and what conclusions may be

drawn about the scientific research necessary for their development. At least

half the applications of atomic frequency standards are in high technology

defense systems whose development now requires approximately a decade from

planning to implementation. Although the role of atomic clocks is often

critical, the clock is such a small part of such systems that plans rarely

take into account required improvements in clock performance. The subject of

this workshop directs our attention towards issues of timekeeping over long

periods of time since much simpler techniques are adequate for short times.

There are three general types of atomic frequency standards which are commercially available today: the rubidium gas cell frequency standard, the

cesium beam frequency standard, and the hydrogen maser. These three are only

a small fraction of the instruments which have been demonstrated in the

laboratory. Devices not developed commercially include active rubidium

masers, cesium gas cell frequency standards, thallium atomic beam frequency

standards, superconducting cavity stabilized oscillators, ammonia absorption

frequency standards, calcium atomic beam frequency standards, methane

stabilized lasers and many others. This paper will examine the ultimate

performance limitations of today's state-of-the-art atomic frequency standards

and the projected requirements for improvement. It will attempt to clarify

9
the most likely direction for research if we are to achieve these ends.

Through analysis of these questions we hope to maximize the likelihood of

producing additional important atomic frequency standards. Other performance

characteristics such as acceleration sensitivity, size, weight and power are not considered.

The Role of Clocks in Navigation, Communications and Data Acquisition

For the sake of non-practitioners in the clock field, it may be

worthwhile to start from the beginning and ask what essential function clocks

serve. The answer is that they provide a uniform scale of the time coordinate

between synchronization experiments. Without periodic ^synchronizations , two

independent clocks would necessarily differ from one and other by an amount

which grows without any bound. Thus, for purposes of science and general

commerce, the world would require only a single clock in a large

communications network to disseminate the time of that clock. In fact, we are

all quite used to operating in this fashion. Nearly every wall clock in the

entire country is loosely locked to the time scale of the National Bureau of

Standards by the actions of electrical utilities. Our wall clocks are only

displays that register the number of cycles of the voltage delivered by the

local power company.

But practically everyone wears a wristwatch in order to provide a memory

of the last synchronization to Universal Time. Thus it is possible to have

time information in locations not served by electric power or other means of

resynchronization. Of course, depending upon his requirements, one will

resynchronize his watch with the wall clock at appropriate intervals.

Similarly, behind every application of an atomic frequency standard, one can discern some benefit for extending the interval of independent operation

before resynchronization is required.

At first, this example may seem facetious, but it illustrates important

aspects of the utilization of atomic frequency standards. In fact, the best

wristwatches which are sold today are comparable in quality to the national

time standards of only 150 years ago. Atomic frequency standards are employed

when the application cannot or chooses not to provide frequent

^synchronizations. Military systems often utilize atomic clocks to prevent

jamming or increase the security. VLBI systems use remote clocks for time

tagging the data acquisition process since the timing requirements are more stringent than any known means of remote time synchronization. NASA uses atomic clocks for deep space navigation for the same reasons. Commercial

broadcasts and communications systems use atomic frequency standards to

enhance system reliability. The benefits of using clocks - improved security,

freedom from interference and redundancy - are intangibles but have great

importance to modern society.

Performance of Atomic Frequency Standards

Since clocks are used to mark off uniform increments of time (phase),

their performance is analyzed in terms of frequency variability. The

generally accepted measure of this variability is the Allan variance,

CK (T) y defined in reference [1]. This variance is a measure of the

changes between successive frequency determinations. Figure 1 is the

performance of the clocks in the NBS atomic time scale. It shows that there

are two regions of averaging time for frequency measurements in which the

performance of atomic frequency standard is qualitatively very different. The

10
~ 15 2 3 4 5 6 7 LOG TAU (seconds)

Figure 1: The estimated stability of the clocks in the NBS atomic

time scale. NBS-4 is a laboratory cesium standard.

PHM4 is a laboratory passive hydrogen standard.

STD. COMM. Cs refers to the standard performance

cesium standards from commercial manufacturers.

OPT. 004 Cs is a high performance commercial option.

first performance region, called the short-term, is characterized by improving

stability. In the second, long-term region, the performance ceases to improve

and begins to get worse. Examination of the performance of passive and

active frequency standards will illustrate the features of these two regions.

A passive frequency standard is one in which an external oscillator is

used to probe an atomic absorption. Appropriate signal processing allows the

formation of a discriminator signal and permits the probe oscillator to be

locked to the atomic feature. In the short-term region, the frequency

deviations are well understood and may be calculated quite accurately. The

error between the frequency of the probe oscillator and the atomic resonance

results from noise. For example, in the case of the cesium beam frequency

standard the noise is due to a combination of the statistical fluctuations in

the number of atoms that make the transition and shot noise of the detected

signal. The result is white frequency noise which is described by [2]

aa

h

where Q is the atomic quality factor, (S/N) is the voltage signal to noise

ratio in a 1Hz BW and T is the averaging time.

The short-term performance of a hydrogen maser is qualitatively different

because the feedback to the atoms is high enough to permit power at the atomic

transition frequency to be obtained from the ensemble of atoms. There is

coherence between the individual atomic emitters so that white noise within

the atomic linewidth perturbs the phase but not the frequency of the

oscillation. Since the phase fluctuations are bounded, the frequency

stability improves inversely with the averaging time [3].

11
This feature of the hydrogen maser makes it the most stable atomic frequency

standard in the region from approximately ten-seconds to a few hours. In the

short-term, the frequency stability improves with the averaging time.

Although such behavior should continue indefinitely, the standard eventually

enters the long-term region in which the frequency stability begins to degrade

due to imperfections in the control electronics. The atomic resonance frequency itself is subject to change resulting from variations in parameters

such as magnetic field. In addition, there is always a difference between the

frequency of the standard and the atomic line center and this differential is

subject to change. As a result, the frequency of the standard displays an approximate random walk behavior and the frequency stability is given by the

approximate formula

where the value of k. cannot be predicted from the design, but must be obtained

emperical ly.

The frequency stability of an atomic clock will not deteriorate

indefinitely as *3* because frequency excursions beyond a certain size are exceedingly unlikely. For example, it would be unreasonable for the frequency

to change by as much as a linewidth. In actuality, one knows with a fair

degree of confidence that the frequency will change by much less than that

over the life of the standard. Today, a manufacturer of cesium beam standards

may specify a maximum lifetime frequency change of 3 x 10 . Thus the Allan

variance is bounded by the limit

< 5 */o~

XL

(or 3- > \oo s.

Although there is no reason to expect that OyC?) would remain constant at

lower levels than implied by the above argument, such a model (flicker noise)

has been improperly used for the long-term frequency stability of atomic

frequency standards. New techniques of statistical analysis indicate that a random walk of frequency model is more conservative. Primary standards

utilize techniques which quantitatively relate the frequency to the

unperturbed atomic frequency. They are therefore said to have the property

called accuracy^ which is the ability to reproduce a specified frequency.

NBSt6, the NBS primary frequency standard is estimated to be accurate to 8 x 10 . Thus NBS-6 could be used to assure that

-IH r

Time Dispersion: The Bottom Line

To compare the timekeeping capability of various clocks one must compute

the rms error x(T) which a clock accumulates in time T- In order to simplify

this very complicated problem we assume that the behavior of the clock is

known and the frequency variations are described by

ol(t) ~ L t r +c+dr]

12
where b, c and d are the 1 second intercepts of the three dominant noise types

- white frequency, "flicker" frequency, and random walk frequency. In

practice, one must also take into account the uncertainties in clock

parameters such as drift so this approach will yield slightly optimistic

results. The time error of the clock has been shown to be [4]

X(t)~ C fe*r + i.Htr*+ J

V

s J*.

Table 1 shows the time errors of two clocks. The performa/ice ofclock A is

dominated by random walk frequency noise such that oy (lo s ) ° 10 . Clock B_/( has an accuracy of 10 and the noise is conservatively estimated as slit)* 19

TABLE 1: Time Error in Nanoseconds

Clock A Clock B

? x&2 x(T)

10^s 1 1

lO^s 32 12

10 's 1000 120

Some Applications of Atomic Frequency Standards

Let us consider two applications which now or in the near future need

improved frequency standards: secure diplomatic and military communications

and military navigational systems. First, we consider the direct benefits of

decreased time dispersion and extended update intervals. Later we will

examine other important benefits of improving clock technology.

Secure communications are generally provided through the use of data

encryption and steganography. (Encryption is the generation of a cypher text

from a plain text message while steganography consists of techniques which

hide the transmission of a message.) The encryption process is accomplished

by a device called a crypto which uses an algorithm similar to the Data

Encryption Standard published by the National Bureau of Standards and an encryption key. Decryption is accomplished by a second crypto using a decryption algorithm and key. Such systems require that the receiving crypto

be synchronized to the transmitting crypto and this function is usually

provided by the communication system itself.

• The level of security provided by such a system is generally sufficient

for even the most sensitive commercial information. However, further levels of

security are required for diplomatic and military communications. For these

applications, it may also be desirable to employ some technique to hide the

signal in order to protect against jamming or so that the signal may not be

detected and decryption attempted. In fact, in some cases knowledge that a signal has been sent at all is more than one is willing to reveal. The most

common methods of hiding communications signals are frequency hopping and

pseudorandom noise spread spectrum modulation. Spread spectrum techniques

hide the signal beneath a high level of simulated noise. For example, an audio voice channel may be mixed with an rf pseudorandom noise signal before

transmission from the secure facility. Alternatively information at a rate of

a few bits per minute may be phase modulated on a carrier in a band near 76

13
Hz [5]. In order to receive the signal it is necessary to integrate for

substantial periods of time to synchronize the receiving crypto with the

transmitting crypto and to extract the signal from the background noise. To

make signal interception as difficult as possible, information required to

synchronize the receiving crypto to the transmitting crypto is generally not

transmitted with the message. The receiver contains a clock which maintains

an imperfect memory of the original synchronization. After a period of time

the synchronization degrades due to time dispersion between the two cryptos

and the receiver must search over a larger and larger time interval for

indication of successful signal decryption. A 20 MHz communications channel

must be searched with tens of nanoseconds resolution. A 75 Hz channel

requires only milliseconds resolution but long integration intervals are required to separate signal from noise. Once decryption commences, the crypto

machines may be locked together. Thus the most difficult operational

requirements for the frequency standard is the need to permit receivers which

have been out of communication for substantial periods of time to quickly

regain entry into the communications net. Communications systems now in the

planning and development phases need frequency stability in the range of oj('!r)*io over long time intervals [6],

Satellite based military navigation systems place comparably stringent

requirements on the satellite atomic frequency standards. For example, the

desired performance of GPS satellite clocks is o$(T)i lc"

>s for time

intervals between 1 and 10 days. This performance would result in a clock

contribution to the navigation error of approximately 3 meters at one-day and

30-meters at 10-days. Today, it appears to be acceptable for the GPS system

to operate for 10 days without updates from the ground. This policy may be

based more on what is practical given today's technology than what is

desirable for system operation. Longer periods of autonomous operation would

serve to increase the security and survivability of the GPS system. It is

important to bear in mind that the clock performance in the GPS system cannot

be treated as an independent issue. The performance of the system during

periods of autonomous operation is also degraded by the absence of updated

ephemeris information. Improvements in both areas are necessary to extend the

autonomous operation period at a given performance level. The availability of

improved clocks would allow the study of the ephemeris in greater detail, thus

speeding the solution of this overall problem. Improvement of clocks to the

0^(T)$ lo""*" level would extend the autonomous operation period for the GPS

system to 100 days with no degradation in current performance levels.

In addition to the actual timekeeping performance of atomic clocks, there

are other aspects of performance which could be improved at the same time.

There is considerable interest in reducing the turn-on time, and the

sensitivity to radiation effects. The reliability and the lifetime of clocks

need to be increased. Many or all of these are amenable to improvement.

Factors Limiting the Long-term Timekeeping Performance of Atomic Clocks

We have seen that for times of a day and longer, the performance of

atomic frequency standards is limited by systematic offsets of the frequency

from the ideal. By far the largest of the systematic offsets result from

Doppler effects. In general, we may write the absorption frequency of an atom

initially moving at velocity v with respect to a source at rest as

14
The measured frequency £j is equal to the nominal frequency U)0 with three

corrections. The second term in the equation is an offset proportional to the

velocity difference,^, between an atom and observer called the first order

Doppler effect; k is the radiation wave vector. The third term is due to time

dilation. Because of its quadratic behavior, this term is usually called

second order Doppler effect. & is the speed of light. The fourth term takes

into account the recoil of the atom. It will be neglected here since it is

generally small except in the optical case.

The first order Doppler effect is generally the largest of all the

systematic errors in the atomic frequency standard. A cesium atom from the

vapor at 350K has a most probable velocity of approximately 200 meters per

second. Thus the first order Doppler shift is approximately one part per

million. Although at the present time there is little agreement on what

limits the long-term performance of commercial atomic frequency standards, the

situation is more clear in the case of laboratory standards which perform

roughly a factor of 10 better than their commercial counterparts. It is fair

to say that these devices are limited in performance either by residual first

order Doppler shift or by whatever technique is used to cancel this shift.

There are several different Doppler cancellation approaches in common

use. In the case of the hydrogen maser and the rubidium gas cell frequency

standard, the atom is confined to a region of space smaller than one-half

wavelength. Under these conditions the Doppler 1 i neshape consists of a discrete, narrow line well resolved from the broad Doppler pedestal. The

linewidth of this feature is determined by other mechanisms such as finite

observation time or relaxation processes. In a hydrogen maser, the atoms are confined within a teflon coated vessel, whereas, in a rubidium frequency

standard they are confined by collisions with a buffer gas. In either case,

the atoms experience frequency perturbations which are extremely difficult to

measure and control.

In a cesium beam frequency standard, collision effects are avoided by

using a low density atomic beam which passes through a microwave cavity

in a region where there are no reversals of the microwave phase. In a perfect, lossless microwave cavity, the first order Doppler shift would be

perfectly nulled. However, the finite conductivity of the walls results in

asymmetry of the counter propagating microwave fields and a residual first

order Doppler shift which is proportional to the misalignment of the atomic

beam from the orthogonality with microwave Poynting vector.

-12

The second order Doppler effect is only on the order of 10 for room temperature atomic species, but is even more difficult to characterize.

Furthermore, aging effects have been observed to change the velocity

distribution by a significant amount. Figure 2 shows the velocity

distribution of a cesium beam tube before and after a modification. The

broken line was

2

data taken after the standard had drifted in frequency

w/tJ ^ 2 x 10 over the course of one year. Note the non-Maxwell ian nature

of the density of atoms at high velocity. Since detector aging was the

suspected cause, changes were made in the collector potentials and a subsequent velocity distribution measurement, indicated by the solid line, was more nearly normal. The frequency returned to its original value. Data such

as this indicates that if all the more serious problems in atomic beam frequency standards were eliminated, variations in the second order Doppler

effect would probably limit performance at the 10 level for a 350K beam

temperature.

15
1.0

.SO

25

—NORMAL —ABNORMAL

-J 1 J 1 i i_

75" too US' iso /7F zoo Velocity )

Figure 2: The change in detected atom velocity distribution

due to aging of the detector in a commercial cesium

beam tube. The accompanying frequency change was approximately ty/iS =: 2 x 10

The electronic control circuitry that determines the center of the resonance is subject to a very long list of problems that produce substantial

frequency offsets. For analog control systems, the problems include harmonic

distortion in the phase modulation, spurious lines in the microwave signal,

variations in the mean microwave power and pulling by the microwave cavity.

Digital control circuits reduce the problem associated with modulation and

spurious signals but add additional difficulties related to maintaining

symmetry of the squarewave modulation cycle. As a result of all of these

problems, the best existing eJectronics has difficulty maintaining consistent

operation even down to the 10 level.

Probable Direction for Breakthroughs in the Long-term Performance of

Atomic Frequency Standards

The current generation of atomic frequency standards is extremely mature.

Laboratory devices reach a performance floor of approximately 10_^3 ^or ^ imes

longer than one day. Performance deteriorates to worse than 10 for times

of several months to one-year. Engineering improvements to existing

commercial standards and some new commercial devices such as room temperature

ion standards may very well achieve performance comparable to today's

laboratory standards but significant improvements beyond what is achieved

today in the laboratory are unl i kely without some fundamental changes.

Major advances are likely to come through research and development

focused in three directions. First, low velocity ions or atoms should be used

in order to reduce the magnitude of the Doppler effects. Second,

significantly higher atomic line-Q and longer observation times must be

obtained. This would improve the ability to cancel the first order Doppler

shift. Frequency offsets due to the electronic control circuitry would also

be reduced at least in proportion to the increase in the Q. Finally, all

variables which affect the operating frequency of the standard should be

referenced directly back to the atomic frequency itself including such things

as the Zeeman shift due to the finite magnetic field.

16
A reasonable objective for the next generatjxw of advanced atomic

frequency standards will be to achieve routine 10 performance for very

long times. To accomplish this goal, the first order Doppler effect either

needs to be reduced in absolute magnitude by a factor of 100 or nulling

techniques must be improved by this same factor. The second order Doppler

effect needs to be reduced to below the 10 level. For a trapped ion

device, the first order Doppler effect is reduced to negligible proportions by

confinement. Reduction of the second order effect requires a temperature of

300 mK. As explained earlier, traditional attempts to exploit this technique

have resulted in significant frequency shifts due to the confinement

mechanism. But the dc electric and_a\agnetic fields used to confine ions

result in uncertainties less than 10 , three orders of magnitude smaller

than the best prior state-of-the-art. In the case of a neutral atomic beam,

cooling to approximately 30 mK is needed to null the first order Doppler

effect.

The ability to confine ions with negligible perturbations allows enormous

enhancement of the atomic line-Q. Experiments performed on magnesium have

demonstrated 10 millihertz.linewidths at a transition frequency of 300 MHz or

an atomic line-O^of 3 x 10 [7]. Other experiments have demonstrated Q's as high as 2 x 10 [8]. This is a factor of 10 to 100 improvement over the

hydrogen maser, and a factor of 100 to 1000 improvement over the highest Q

c,g^ium atomic beams. Assuming that the same linewidth can be obtained -.u^ing

Hg which has a hyperfine frequency of 26 GHz, the Q would be 10 . Improvement in Q may also be obtained in a cold atomic beam. However, the

effect of gravity on the beam inhibits obtaining very long interaction times

and trapping is indicated. It is not known whether confinement effects can be

made sufficiently small.

Frequency standards have reached the point where many of the parameters

must be controlled to extremely high levels of precision. State-of-the-art

control is required for temperature, the influences of external magnetic

fields, and certain parameters of the electronic circuitry. In the past, this

has been accomplished using a variety of sensors and references. In the

future, it will be increasingly necessary to reference all critical variables

directly to invariant atomic properties. For example, Zeeman frequency shifts

can be measured by observing the magnetic field sensitive transitions. This

philosophy may sound very much like the design principles of a primary

frequency standard such as NBS-6. This is because the ability to guarantee

from first principles that the frequency of a standard is within 10 of some arb|£rary value assures that long term frequency stability must be better than

10 forever. Once such a standard is constructed, examination of the

control signals over a long period of time will make it possible to design

simpler devices which, although they have lost the property of frequency

accuracy, retain adequate long term frequency stability for many applications.

This design philosophy is exactly the one which led to the cesium beam

standard and can probably be employed again to achieve dramatic results.

Building an accurate standard in order to achieve long term frequency

stability can have significant additional advantages. Such a device should

have greatly increased reliability due to its self-monitoring and correction

capabilities. It would also have enhanced capability to withstand adverse

environments such as extremes of temperature, radiation or EMP, since the

atomic resonance itself is relatively unperturbed by these effects. The

17
accuracy approach would also improve the speed with which the standard could

be brought into operation. Present day frequency standards suffer from retrace

error. That is, the frequency after start up differs from the previous

operating frequency. Sometimes the operating environment prevents the rapid

evaluation of this new frequency. For example, when a cesium standard, is

started up on a GPS satellite, its frequency can be predicted to only 10

The perturbations duetto the atmospheric delays, permit the new frequency to be

measured to ~5 x 10 in 1 day and 1 x 10 after 5 days. An accurate

frequency standard would evaluate its own frequency without the need to_ 13

reference a standard on the ground. Its frequency would be known to 1 x 10

in hours.

In conclusion, we believe that it is now possible to make major improvements in the timekeeping ability of atomic clocks. Three approaches are recommended: The use of narrower atomic features, the reduction of Doppler

effects and the evaluation of all systematic frequency offsets in terms of

atomic properties. Laser cooling of stored ions is an extremely promising

technique which should make it possible to achieve frequency accuracy and

stability at the 10 level. Secure diplomatic and military communications,

space and military navigation, and scientific investigations will be the prime

beneficiaries.

REFERENCES

[1] J. A. Barnes, A. R. Chi, L. S. Cutler, D. J. Healey, D. B. Leeson, T. E. McGunigal, J. A. Mullen, W. I. Smith, R. L. Sydnor, R. F. C. Vessot and

G. M. R. Winkler, IEEE Trans. Instrum. Meas. IM-20, 105 (1971).

[2] J. Vanier and L. G. Bernier, IEEE Trans. Instrum. Meas. IM-28, 188

(1979).

[3] L. S. Cutler and C. L. Searle, Proc. IEEE 54, 136 (1966).

[4] D. W. Allan and H. Hellwig, in Proceedings of the Position Location and

Navigation Symposium, 29 (1978).

[5] Defense Electronics 14, 59 (1982).

[6] Private Communication, Department of the Navy (unclassified).

[7] W. M. Itano and D. J. Wineland, Phys. Rev. A24, 1364 (1981).

[8] R. Blatt, H. Schmatz and G. Werth, Phys, Rev. Lett. 48, 160 (1982).

18
Laser-Cooled and Trapped Atoms , (Proceedings of the Workshop on Spectroscopic

Applications of Slow Atomic Beams, held at NBS, Gai thersburg, MD, 14-15 April

1983) Ed. by W. D. Phillips, Natl. Bur. Stand. (U.S.) Spec. Publ. 653 (1983).

\

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

ABSTRACT

We summarize research undertaken to develop 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 exciting. 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 poten-

-15

tial 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 certain19
ly 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 [1] and electrostatic [2] 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 [3] or radiation pressure forces [4], 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 off while

the "clock" transitions are driven, but even if maximum laser cooling is

achieved on strongly allowed transitions, minimum velocities are on the order

of 10 cm/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 [5-7]

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 observable

systematic frequency shifts in certain experiments. A trapped ion on the

20
other hand seeks the region of zero electric field; for cooled ions the aver- age magnitude of electric field can be below lmV/cm [6] . 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--!. 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.

21
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 flight 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 a (x) of an oscillator locked to an atomic

-1 ^

transition scales as (S/N • Q) . For example, if an oscillator is locked to

the 26 GHz hyperfine transition in ^*Hg+ and a linewidth of 0.01 Hz is obtained, then 100 stored ions can potentially give a stability a (t) = 6 x -14 -h ^

10 X [8] . This would exceed any existing passive atomic frequency standard.

Several laboratories have worked on developing a frequency standard based

199 + on the 40.5 GHz ground state hyperfine splitting of Hg ions stored in a trap of the rf quadrupole (Paul) type [9-13] . State selection and detection

202 +

is by optical pumping. 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. Resonance linewidths of about 1 Hz have been observed [12].

At present, the main accuracy limitation is the second-order Doppler (time

22
fine transition, corresponding to a Q of 2 x 10^, the highest yet obtained in

dilation) shift, which is relatively high (about 10

11

) because the average

ion kinetic energy is a few eV. In a similar experiment on trapped *^Yb+

ions, Blatt et al. [14] have observed a 0,06 Hz linewidth on a 12.6 GHz hyperfine transition, corres 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

-13 + 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 < IK are easily obtained. The lowest

temperatures (approximately 0.01 K) have been obtained for single ions [15-17].

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

[6,8]. So far the only reported experiments with the goal of a laser cooled

microwave frequency standard are those of NBS [18,19]. These experiments are

2 5 based on the storage of many ions (10 -10 ) in a Penning trap where residual

heating mechanisms [20] are apparently much less than in the rf trap. The

Penning trap [20] requires static electric and magnetic fields for trapping.

Since the required magnetic fields are rather large (y IT) , one uses extremum

points in the clock transition frequency vs. magnetic field; for these condi-

-15 tions, stabilities below 10 should be obtained [8,18,19]. Linewidths of

approximately 0.01 Hz have been obtained on 300 MHz nuclear spin flip hyper25 + fine transitions in Mg [21] and an oscillator has been locked to a similar

9 + -13 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 [8]

23
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. Antici-

~ 18 pated accuracies of 10 do not seem unreasonable provided that sufficiently

narrow band lasers (< 10 Hz) 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 [7]. Penning traps

2 2 or Penning/ rf trap combinations might also be used [22] . The 6 S1 - 6 Pj - 2 + % 1

5 D 3 y

2 Raman transition in Ba could be used as a reference to generate a stable infrared difference frequency in a nonlinear crystal [7,23]. Also in

+ 2 2

Ba , the 5 D5/2 to 5 D 3/2 12 pm transition [24] and the quadrupole-allowed

2 2

6 Sx to 5 D 5. 2 1.8 [im transition [25] have been proposed as standards. Other

* + + + + high Q optical transitions in Sr [26] Pb , I , and Bi [27] have been suggested for stored ion frequency standards. The two-photon [28,8] or single

10 2 9 2 2 + photon quadrupole [18] 5d 6s Si to 5d 6s D 5/2 transition in Hg has also

been suggested. Two-photon transitions have the advantage of being firstorder 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. No. 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.

[4] V. G. Minogin and J. Javanainen, Optics Commun. 43, 119 (1982).

24
[5] Wayne M. Itano, L. L. Lewis, and D. J. Wineland, Phys. Rev. A 25, 1233

(1982).

[6] D. J. Wineland, in "Precision Measurements and Fundamental Constants II"

(B. N. Taylor and W. D. Phillips eds.) National Bur. Stds. (U. S.), Spec.

Publ. 617, in press.

[7] H. Dehmelt, I.E.E.E. Trans. Instrum. Meas. JM31, 83 (1982).

[8] D. J. Wineland, W. M. Itano, J. C. Bergquist, and F. L. Walls, Proc. 35th

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. 602-611.

[9] F. G. Major and G. Werth, Phys. Rev. Lett. 30, 1155 (1973).

[10] M. D. McGuire, R. Petsch, and G. Werth, Phys. Rev. A 17, 1999 (1978).

[11] M. Jardino, M. Desaintfuscien, R. Barillet, J. Viennet, P. Petit, and C.

Audoin, Appl. Phys. 24, 107 (1981).

[12] L. S. Cutler, R. P. Gifford, and M. D. McGuire, 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.

[13] L. Maleki, 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.

J [14] R. Blatt, H. Schnatz, and G. Werth, Phys. Rev. Lett. 48, 1601 (1982).

[15] W. Neuhauser, M. Hohenstatt, P. Toschek, H. Dehmelt, Phys. Rev. A 22,

1137 (1980).

[16] D. J. Wineland and Wayne M. Itano, Phys. Lett. 82A, 75 (1981).

25
[17] W. Nagourney, G. Janik, and H. Dehmelt, Proc. Nat. Acad. Sci., USA, 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. To be published in I.E.E.E., Trans. Nuc. Sci., NS30 , No. 2,

Apr. 1983.

[20] D. J. Wineland, Wayne M. Itano, and R. S. Van Dyck, Jr., in Advances in

Atomic and Molecular Physics , Eds. Bederson and Bates (Academic, New

York) Vol. 19. To be published.

[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, H. Dehmelt, Proc. 5th Int. Conf. on Spectral Lineshapes, West Berlin (1980).

[24] H. Dehmelt, W. Nagourney, G. Janik, Bull. Am. Phys. Soc. 27 402 (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).

26
Laser-Cooled and Trapped Atoms , (Proceedings of the Workshop on Spectroscopic

Applications of Slow Atomic Beams, held at NBS, Gaithersburg, MD, 14-15 April

1983) Ed. by W. D. Phillips, Natl. Bur. Stand. (U. S.) Spec. Publ. 653 (1983).

Optical Frequency Synthesis Spectroscopy

K. M. Evenson, D. A. Jennings, F. R. Petersen, J. S. Wells, and R. E. Drullinger

Time and Frequency Division

National Bureau of Standards

ABSTRACT

In order to measure the super narrow spectral features of cooled atoms and ions, in the

optical region, optical frequency synthesis (OFS) techniques rather than wavelength techniques must be used. It is anticipated that many of these resonances will be in the optical

region of the spectrum, and this paper will address the state-of-the-art of the measurements of frequencies in that region. Two recent optical frequency measurements of iodine

transitions in the visible will be described as well as recent improvements in fabricating

the point-contact diode used in these measurements.

Keywords: optical spectroscopy, optical frequency synthesis (OFS), iodine

Introduction

One of man's most accurate and precise endeavors is that of frequency measurement; therefore, it

is not surprising that a significant part of his understanding of the physical world is through the

application of these measurement techniques. Specifically, the extension of both frequency measurement and other radio frequency techniques to the optical region has made a significant impact on the

fields of metrology and spectroscopy.

Optical frequency synthesis (OFS) [1] is an extension of radio and microwave frequency measurement

techniques into the infrared and visible region of the electromagnetic spectrum. This advance has

been made possible by the development of a source of radiation (the laser), sub-Doppler frequency

stabilization techniques, and the broadband harmonic generating capability of metal-insulator-metal

(MIM) diodes. OFS is a "bootstrap" method whereby harmonic combinations of spectrally pure and mea- surable lower frequency laser oscillators are used to generate higher laser frequencies. This technique has now been used to extend "direct" or "absolute" frequency measurements from the far infrared

into the visible spectrum. Most importantly, direct frequency measurements are now being made in this

region of the spectrum for the first time. It is in this region that the most accurate wavelength

measurements traditionally have been made and the meter has been defined. Now we have two different

methods for making excellent comparisons of radiations in the visible, one utilizing wavelength metrologic techniques and the other utilizing frequency metrology (OFS). At the current state of the art,

wavelength comparisons are subject to inaccuracies of the order of a few parts in 10*^ because of

diffraction, imperfections in the optical components, etc. However, frequency comparisons can be made

which are limited only by the quality of the oscillators and can be referred to a primary standard

that is four orders of magnitude better than the primary length standard!

27
The metrologic significance of laser frequency measurement was first most clearly demonstrated in

1972 when both the frequency and wavelength of the methane stabilized He-Ne laser at 3.39 |Jm (88 THz) were accurately measured. The product of these is a value of the speed of light [2] 100 times more accurate than the value accepted at that time, a value limited only by the krypton primary length

standard itself. The possibility of improving length metrology with the use of laser radiation, and

also fixing the speed of light gave impetus to the Consultative Committee for the Definition of the

Meter (CCDM) to propose a redefinition of the meter based on the second. In 1982 the CCDM proposed:

"The meter is the length of the path travelled by light in vacuum during a time interval 1/299 792 458

of a second" [3] . With this definition, the meter could be realized using the wavelength of any laser

which is stabilized to a narrow atomic or molecular absorption feature for which the frequency is

known. The wavelength, A, would be determined from the relation k - c/v, where c is the fixed value

of the speed of light, and V is the measured frequency of the stabilized laser. To realize this

definition, it is necessary to accurately measure the frequency of certain absorption lines in the

visible spectral region.

The first such measurements were performed at NBS in the spring of 1982. Two iodine lines which

are suitable for realizing the new meter were accurately measured with optical frequency synthesis

techniques. The first of these lines [4], a visible yellow iodine absorption line at 520 THz (576

nm) , was measured using nonlinear crystals as frequency doublers above 130 THz (2.3 |Jm) and the MIM

diode for harmonic generation at lower frequencies. This accomplishment was quickly followed by the

measurement [5] of the red He-Ne iodine stabilized laser at 473 THz (633 nm) , where the final frequency was synthesized by resonant enhanced four wave mixing of three known frequencies in a helium-neon

plasma

Additionally, because of the importance of the MIM diode to this technology, we shall describe

some experiments which revealed improved coupling techniques and some surprising results concerning

the sharpness of the diode tip which resulted in more stable diodes.

The OFS measurements of iodine

127

We describe the measurement of the frequency of the o hyperfine component of the visible I

? 127

17-1 P(62) transition at 520 THz (576 nm) and the i hyperfine component of the \^ 11-5 R(127) - stabilized He-Ne laser [6] at 473 THz (633nm) to 1.6 parts in 10 10

. The frequencies were measured in

three steps. First, a CC^ laser line was compared with the CH^-stabilized He-Ne laser at 88 THz (3.39

|Jm) [7], which presently is the most accurately known of all laser frequencies. Then, various CC^

lasers referred to this CH^ referenced laser were used in separate measurements of the 520 THz and 473 THz I 2 transitions. Since the 520 THz iodine line was used to measure the 473 THz frequency, the 520

THz frequency measurement will be discussed first.

The OFS technique here consists of a "chain" of lasers linking lower frequency lasers to higher

ones with harmonic generation and heterodyning techniques as is shown in Fig. 1. In a MIM diode,

three harmonics of one 26 THz CO^ laser radiation plus two harmonics of another are heterodyned with

the 130 THz color center laser radiation to produce a rf beat which is measured. Part of the color

center laser radiation is subsequently frequency doubled and locked to the 260 THz He-Ne laser radia23
tion. The He-Ne laser radiation is in turn frequency doubled and frequency locked to a 520 THz cw dye

laser. The 520 THz cw dye ring laser at the top of the chain was servo-locked to the o hyperfine

127 ' J,. , component of the I^, 17-1 P(62) transition observed in saturated fluorescence. The line had a full

width at half maximum of 1 MHz for an iodine pressure of 4 Pa. The center of the transition could be

determined to within 40 kHz.

The He-Ne 260 THz laser employed an 8 m discharge tube and generated 100 mW of single mode output

power. The He and Ne gas mixture was adjusted to optimize single mode output power. A resonant

reflector with a thin lossy metallic film positioned 9 cm from the high reflectivity end mirror provided tuning and mode selection. Approximately 90 mW of the output power was focused into a temperature tuned lithium niobate frequency-doubling crystal (T = 185°C) to generate 50 (jw of power at 520

THz. This visible radiation was collinearly mixed with a portion of the cw dye laser output on a Si

photodiode to produce a rf beat used to lock the two lasers together.

The cw color center laser operating at 2.3 |Jm was used as a frequency transfer oscillator for

connecting the He-Ne laser frequency to the accurately known CO. laser frequency. The color center

+ laser used (F^ )^ centers in lithium-doped KC1 [8] , and provided broadly tunable laser output from 2 to 2.5 pm. When pumped with 4 W from a cw 1.33 (Jm Nd:YAG laser, the color center laser produced over

100 mW of output power at 2.3 (Jm. The laser was operated in a ring configuration to insure efficient

single mode operation and was actively stabilized to a stable optical resonator with a resulting

linewidth of less than 20 kHz.

The 2.3 |Jm radiation was focused into a temperature phase matched lithium niobate frequency-doubling crystal (T = 530°C) and generated approximately 10 (jW of 1.15 |Jm radiation. This second harmonic output was combined collinearly on a beamsplitter with 10 mW from the He-Ne laser, and both beams were focused onto a high speed Ge photodiode. The resulting beat frequency was displayed on a rf

spectrum analyzer and measured with an adjustable marker oscillator which was frequency counted.

The remainder of the non-doubled 2.3 pm radiation was focused onto a W-Ni point contact MIM

13 16 diode. Coincident on the diode were the beams from two C 0^ lasers operating on the adjacent lines

P

T (50) and P_(52). The P_(50) laser was stabilized to the saturated fluorescence signal from an

13 16 external 5.3 Pa C 0^ absorption cell. The P^(52) laser was phase locked to the Pj(50) laser via a stabilized 62 GHz klystron. These two CO^ lines were selected such that twice the frequency of one CO^ laser plus three times the frequency of the other CO^ laser nearly equalled the necessary color

center laser frequency (130 THz). The free running CO^ lasers each had approximately a 3 kHz linewidth; however, the average frequency difference of the two lasers was held stable to within 1 Hz in a

1 second averaging period.

The frequency of the Pj(50) laser was determined by comparing it against a CH^-stabilized He-Ne

laser operating at 88 THz. International intercomparisons have shown frequency reproducibility of

this device to be about i 3 x 10 ^, and a number of absolute measurements of its frequency have been

reported [3] . We took the average value of the 4 most recent measurements of this frequency as the

basis for measurements described in this paper: the CH^-stabilized He-Ne laser frequency was taken to be 88 376 181.609 MHz ± 0.009 MHz.

29
The result of the frequency chain measurement of the o hyperfine component of the

P(62) transition was

17-1

v = 520 206 808.547 MHz ± 0.081 MHz

.

This value is in agreement with previous, less accurate measurements [9]. The estimated frequency error represents a line center uncertainty of 1.6 parts in 10*^. Most of this uncertainty arises from

the dispersion of the measured frequencies of the CH^ standard, with the rest of the error appearing

from effects such as reproducibility of the lock point, uncompensated pressure shifts, and counting

and interpolation errors at each stage of the chain. Table 1 shows the error budget for this measurement.

The 633 nm He-Ne laser frequency was measured in a significantly different manner. Several years

ago Klementyev et al. proposed a method of synthesizing 473 THz radiation using sum frequency mixing

in Ne [10]. In this scheme, shown in Fig. 2, 473 THz radiation is generated by the nonlinear resonant

mixing of radiation at the three measurable transition frequencies: 88 THz (3.39 pm) , 125 THz (2.39

[Jm) , and 260 THz (1.15 nm) . The sum frequency at 473 THz is radiated by coherent polarization on the

3s2~2p^ transition in Ne which results from the nonlinear interaction of the three fields resonant

with three cascade transitions connecting the same two atomic levels. In our experiment, we used a separate laser for each of the frequencies, and mixed them in a low pressure He-Ne discharge tube.

The experimental arrangement is shown in Fig. 3. Summed in the tube were radiations at 88 THz from a CH^-stabilized He-Ne laser (amplified to 8 mW) , 125 THz from a 2.39 |Jm color center laser (50 mW)

frequency-referenced to C0„ , and 260 THz from a 1.15 [im He-Ne laser (200 mW) referenced to I. at 520

-8 THz. Approximately 5x10 W was generated at 473 THz. Mixing could be verified by blocking any one of the three input beams, which immediately caused the generated 473 THz red light to disappear or by

frequency shifting any of the three source lasers which caused a corresponding shift in the synthesized frequency.

The 473 THz radiation from the summing tube was combined on a beamsplitter with 40 pW of radia3 20 127

tion from a He- Ne 473 THz laser stabilized on either the g or i hyperfine component of the I^,

11-5, R(127) transition. From this beat and knowledge of the three infrared laser frequencies, the

frequency of the i-component adjusted to the operating conditions recommended by the CCDM was determined to be

The estimated frequency error represents a line center uncertainty of 1.6 parts in 10 . Again, most

of this uncertainty arises from lack of knowledge of the CH, standard frequency, with the rest due to uncertainty in the reproducibility of the lock points of both the I_-stabilized He-Ne laser (1 x

10

10

), and of the ^-stabilized second harmonic of the 1.15 \im laser. The error budget for the

measurement is shown in Table II.

Both visible frequency measurements described above represent an improvement in accuracy by

almost 3 orders of magnitude over previous frequency measurements to the visible [9]. However, the

= 473 612 214.830 MHz ± 0.070 MHz

10

30
uncertainty reported here does not yet approach the limits of this measurement technique, but is

rather limited by the quality of the laser sources [11]. Our measurements were limited mainly by the

frequency reproducibility of the lasers. Using the currently proposed redefinition of the meter,

either of these lines could serve as a length standard with an improvement of about a factor of 10 86 compared to the present official length standard based on the Kr discharge lamp. However, we have

the techniques at this time to improve these or similar measurements by many orders of magnitude.

The speed of visible light

In order to check the accuracy of the value of the speed of light used in the new definition of

the meter, a value of c using visible radiation is obtained by multiplying the value of the frequency

of the red iodine absorption by the most accurately available wavelength value. The wavelength for

this transition is obtained from four published values which are 632 991 399.0 ±0.8 fm [12], 632 991

399.8 ± 0.9 fm [13], 632 991 400.0 ±12. fm [14], and 632 993 398.0 ±3 fm [15]. The weighted average

of these measurements is 632 991 399.4 ±0.6 fm. The value for the speed of light is, of course, the

product of the frequency and wavelength and is c = 299 792 458.6 ±0.3 m/ s , with a one sigma uncertainty.

This value of c, the most accurate ever measured for visible light, is in good agreement with the

defined value of c proposed by the CCDM [14] and will, thus, maintain "continuity" in the meter with

the new definition.

The MIM Diode

Metal-Insulator-Metal diodes in a point contact configuration, used at room temperature, are the

most successful and versatile devices for absolute measurements of laser frequencies from the far

infrared to the visible [16]. Field induced electron tunneling is the most widely accepted explanation for their extremely broadband response [17-19]. However, there is also some evidence for more

than one phenomenon occurring in the diodes which contribute to the video detection at visible wavelengths [20] . Most experiments and theories have indicated that the contact area should be as small

as possible in order to decrease the capacitance and hence minimize the time of response. Consequently, tip radii in the 40-90 nm range have been generally used. This feature seriously affects the

mechanical stability of the diodes, sometimes preventing their use as routine laboratory devices.

Also, delicate and short lived contacts can make quantitative and systematic investigation of the

physical mechanisms involved in detection and harmonic generation difficult.

Encouraged by a chance measurement at infrared frequencies in which video signals were almost

unaffected with the use of a dull point, we performed more quantitative tests on the diode. Most of

the interest in point contact MIM diodes lies in their high speed of response used in generating laser

harmonics. In this experiment, that property was tested by the generation of third harmonic signals.

Radiations from a 3.39 |Jm He-Ne laser at 88 THz and from a 1^(30) CC> 2 laser emission at 29 THz were focused on the diode with conical antenna coupling. Microwave power at 48 GHz from a phase-locked

klystron also radiated the diode to produce a beat signal of a few tens of megahertz. After amplification in a broadband amplifier, the beat note with a signal-to-noise (S/N) ratio up to 25-30 dB in a

31
30 kHz bandwidth couid be observed on a spectrum analyzer. The power focused on the diode was 30 mW

from each of the lasers. The microwave power was in general adjusted to be close to the saturation

power for the diode. The S/N ratio of the heterodyne signal was measured as a function of the resistance (adjusted mechanically) for each diode. The video sensitivity at 10 pm, 3.39 urn, and 48 GHz was

also recorded. Typically the rectified signals increased almost linearly with the resistance up to about 1 kQ and then showed some saturation (the signal at 2 kQ about 507o higher than the one at 1 kQ)

The shape of the video response vs the impedance was the same at 3.39 [Jm and 10 |Jm, but with a de- crease of about two orders of magnitude in the detected signal strength at the shorter wavelength.

With a resistance of 500 Q, typical dc signals were 20 mV at 10 |Jm and 200 |JV at 3.39 |Jm. The parabolic mirror used for focusing was machined for operation at wavelengths greater than 10 \im; therefore, a good share of the decrease was probably due to the optics. The measured dc signals were almost the same for all the tip radii between 50 and 760 nm with resistances below 1 kQ. A small

decrease (about 10% of the sensitivity at higher resistance could be observed for the less sharp

diodes. The beat S/N ratio was optimum for resistance values between 300 and 500 Q with a gradual

decrease of the signal at increasing resistance and a rapid decrease at lower values. Typical results

are shown in Fig. 4. Diodes with four significantly different tip radii were used. No significant

decreases in the S/N were observed from "sharp" to "dull" points. This is very surprising for the

capacitance should be determined by the area of the contact, i.e., the square of the tip radius. For

a radius of some tens of nanometers, a frequency cut-off can be calculated of the order of 100 THz,

and a radius of a few hundred nanometers should lead to a frequency cut-off two orders of magnitude

lower. This is in complete disagreement with our experimental results. The "spreading resistance" in

the base, proportional to the inverse of the contact radius, increases the RC time of response proportional to the radius [21], but it, by itself, does not explain the high speed operation of these

blunter diodes. Our measurements were performed to help understand the operation of the diode; however, they seem to have had the opposite effect. For instance, the "effective contact area" may not

simply be determined by the square of the radius of the tip. It is, moreover, not clear how the

dielectric constant of the oxide in the contact changes and affects the capacitance under the high

mechanical and thermal stresses present when the contact is made.

These results, besides stimulating questions about the theory of operation of the diode, have

pointed out a practical design improvement. In particular, with the use of less sharp tip radii, the

stability of the diode has been dramatically increased without a significant decrease in the sensitivity or speed of response. The same contact can be stable for hours to days in normal laboratory

environmental conditions without readjustment. Also, the diode resistance can be more precisely

adjusted and reset with a resolution of a few tens of ohms. This is particularly important for the

maximization of the beat S/N ratio in harmonic generation and mixing experiments. This experimental

result opens a new theoretical puzzle and new possibilities for practical applications of the point

contact MIM diodes. For instance, in the far infrared, the sensitivity is comparable to that of a Golay cell, but the speed of response is twelve orders of magnitude faster!

Conclusion

The application of OFS techniques to the measurement of the super narrow resonances from cooled

atoms and ions will introduce a new era in spectroscopy. Optical frequency synthesis spectroscopy has

32
already improved the accuracy of the spectral measurements of conventional spectra by two orders of

magnitude compared with the use of wavelength techniques and promises further increases in accuracy.

One possible application of the OFS technique could be the measurement of the Rydberg, which should be

the first in a series of measurements promising unprecedented accuracy in our measurements of the

fundamental constants. In this new era, measurements will be limited by ones ability to characterize

the spectral features rather than the measurement techniques!

References

1. A frequency synthesizer in the radio or microwave region of the spectrum has phase coherence

between the synthesized frequency and a reference or standard. Optical frequency synthesis

implies phase coherence in a given mixing step but not necessarily between steps, i.e., there is

not necessarily phase coherence between the optical frequency and the reference frequency.

2. K. M. Evenson, J. S. Wells, F. R. Petersen, B. L. Danielson, G. W. Day, R. L. Barger, and J. L. Hall, Phys. Rev. Lett. 29, 1346 (1972).

3. Comite Consultatif pour la Definition Du Metre, 7th Session, June 2-5, 1982 (Paris).

4. C. R. Pollock, D. A. Jennings, F. R. Petersen, J. S. Wells, R. E. Drullinger, E. C. Beaty, and K. M. Evenson, Opt. Lett. 8, 133 (1983).

5. D. A. Jennings, C. R. Pollock, F. R. Petersen, R. E. Drullinger, K. M. Evenson, J. S. Wells, J.

L. Hall, and H. P. Layer, Opt. Lett. 8, 136 (1983).

6. H.P. Layer, IEEE Trans, on Inst, and Meas., Vol, IM-29, No. 4 (1980).

7. F.R. Petersen, (private communication)

8. I. Schneider and C.L. Marquardt, Opt. Lett. 5, pp. 214-15 (1980).

9. K.M. Baird, K.M. Evenson, G.R. Hanes, D.A. Jennings, and F.R. Petersen, Opt. Lett. 4, pp. 263-4

(1979).

10. V.M. Klementyev, Yu. A. Matyugin, V.P. Chebotayev, ZhETF Pisma 24, 8 (1976).

11. See reference 7, and H. P. Layer, W. R. C. Rowley, and B. R. Marx, Opt. Lett.

6, pp. 188 (1981).

12. W. R. C. Rowley and A. J. Wallard, J. Phys. E6, 647 (1973).

13. W. G. Schweitzer, Jr., E. G. Kessler, R. D. Deslattes, H. P. Layer, and J. R. Whetstone, Appl.

Opt. 12, 2829 (1973).

14. Rapport: Comite Consultatif pour la Definition du Metre, 5th Session, page M54 (1973) Bureau

International des Poids et Measures, Sevres, France.

15. G. R. Hanes, K. M. Baird, and J. DeRemigis , Appl. Opt. 12, 1600 (1973).

16. K. M. Evenson, D. A. Jennings, and F. R. Petersen, J. Phys. (Paris) 42, C8-473 (1981).

17. S. M. Faris, T. K. Gustafson, and J. C. Wiesner, IEEE J. Quantum Electron. QE-9 737 (1973)

18. G. M. Elchinger, A. Sanchez, C. F. Davis, Jr., and A. Javan, J. Appl. Phys. 47, 591 (1976)

19. A. Sanchez, C. F. Davis, K. C. Liu, and A. Javan, J. Appl. Phys. 49, 5270 (1978).

20. H. U. Daniel, M. Steiner, and H. Walther, Appl. Phys. 25, 7 (1981).

21. D. J. E. Knight and P. T. Woods, J. Phys. E. Scient. Instr. 9, 898 (1976).

33
Table I. Error Budget for the 520 THz Visible Frequency Measurement

Component Error

1. CO^ laser uncertainty (3.4 kHz times

effectively 20 harmonics)" 68 kHz

2. Electronic resettability of entire

servo system 40 kHz

3. Uncertainty in first derivative

offsets and pressure shifts 15 kHz

4. Statistical fluctuation in data 14 kHz

Total Uncertainty (summed Quadratically) 81 kHz

•10

" This 3.4 kHz is due to an estimated 1 x 10 uncertainty in the

CH^-stabilized He-Ne frequency added in quadrature to the measurement

uncertainties in the determination of the Pj(50) frequency, (ref. 7)

Table II. Error Budget for the 473 THz Visible Frequency Measurement

Laser Frequency in MHz Measurement Correlated

Uncertainty in kHz Uncertainty (due

(independent of CH^ to CH^ in kHz)

uncertainty)

CH. 88 THz

4 88 376 181 609 9

Color Center

125 THz 125 132 754 610 11 12

I 2 /2 260 THz 260 103 404 273 25 30

27.4 51

1 01

I

2 11-5 R(127)g 473 612 340 492 47

1 ?7

'l 11-5 R(127)i 473 612 214 789

a 47

Total Uncertainty [(27. 4)

2 + (51)

2 + (47)2 ]^ = 74 kHz

The Comite Consultatif pour la Definition du Metre [3] has recommended that the following conditions

127 be realized when the \^ molecule, transition 11-5, R(127), component i is used for intracavity

stabilization of the 473-THz He-Ne laser: (;) I cold-finger temperature of 15°C, (2) cavity standing127 c wave power of 40 mW, and (3) modulation peak-to-peak amplitude of 6 MHz. Therefore, the 11-5

R(127)i frequency adjusted for the recommended operating conditions is 473 612 214.830 ±0.074 MHz [5].

34
SERVO

SERVO

l/B

= 1539

SPECTRUM

ANALYZER

l/5

« 520 206 808. 547

DYE, I 2 (576nm)

1

LiNbO, x2

*4

He-Ne (1.15 /Am)

LiNb03 in

x2

COLOR CENTER (2.3 p)

<X) 3f, + 2l/2

ty„= 61721

H KLYSTRON

SERVO

3C02 , P,<52), (11.5/i.m)

= 2 1/4 + A;

1/4 = 2z/3 + A;

v%

= 3 1/, + 2z/2 + i/B

V{ -26035339.978"

,SC02 , P,(50), (ll.5^m)

l/2

s I/, + tyw +A

LASER FREQUENCY SYNTHESIS CHAIN

(11.5/i.mTO 576 nm, ALL FREQUENCIES IN MHz)

(X) MIM Diode 0Photodiode

Fig. 1. The OFS chain for generating 520 THz radiation

35
3.39 /J.m

2.39 urn —

125 THz

260 THz

3S:

0.633 urn ——— 1 473 THz

2 P.

Fig. 2. Energy level diagram of Ne showing the pertinent cascade transitions used to synthesize 473

THz radiation.

Fig. 3. Experimental apparatus for measuring the 473 THz laser frequency.

36
CDO

q:

LjlJ coo

z

o

COo

z

o

q:

<

"D

30

20

10

TIP RADIUS

A 500 A

AA

- ^ A A *

A

I® O O

+ A

8

0

+

900 A

1710 A

7600 A

A

A

I

i>,,

1000 2000

RESISTANCE, a

3000

Fig. 4. Signal-to-noise ratios in a 30 kHz bandwidth for the observed beat signals in the third

harmonic experiment as a function of the diode resistance. The beat was generated by the

synthesis scheme v_ = v„ 0_ - 3v_ -48.7 GHz. Results for diodes with different tip

B o . jy (Jm K (.oUj

radii of curvature are reported. I

37
Laser-Cooled and Trapped Atoms, (Proceedings of the Workshop on Spectroscopic Applications of Slow Atomic

Beams, held at NBS, Gaithersburg, MD, 14-15 April 1983) Ed. by W. D. Phillips, Natl. Bur. Stand (U S ) Spec. Publ. 653 (1983).

v

'

Limitations of Atomic Beam Frequency Standards

Lindon Lewis

Time and Frequency Division

National Bureau of Standards

Boulder, Colorado 80303

ABSTRACT

Atomic beam frequency standards may be placed into two categories: field standards and

laboratory standards. While this distinction is somewhat artificial, because the two types

of standards are interdependent, each category does have different requirements of accuracy,

size, and cost. Despite this separation, generally the developments which produce the best

laboratory standards eventually give rise to improved field standards. Existing field

-13

standards are limited in long term fractional frequency stability to a (t) 3 x 10 , for

X 6 months. A laboratory standard such as NBS-6, the U. S. primary cesium standard, is

-14

limited in inaccuracy to Ay ^ 8 x 10 . Proposed new cesium field standards are expected

-14

to yield long term stabilities of a (t) * 1 x 10 (l = 6 months). Stored ion standards,

prime candidates for new laboratory frequency standards, are expected to have better than Ay

: 1 x 10 ^ inaccuracy. As other approaches to atomic beam frequency standards are considered, they should attempt to compete favorably with these emerging technologies.

Key Words: atomic beams, atomic frequency standard, cooled atoms, optical pumping.

General Characteristics of Atomic Beam Standards

Atomic frequency standards may be divided into two categories--field standards and laboratory

standards. The distinction is useful because the most highly desired characteristics in each cate- gory often preclude the realization of the properties most sought after in the other category. For

example, a field standard must be durable, portable, reliable, and easily operated, with unsupervised

operating periods of months to years. A laboratory standard, on the other hand, must attain the

highest possible accuracy, even if the device requires considerable space, power, money, and the

patience of five Ph.D. physicists. In discussing the limitations of existing standards, or in projecting the usefulness of proposed new standards, it is helpful to keep in mind the category of interest, as the criteria vary with application. At the same time, it is important to realize that this

distinction is somewhat artificial, as field standards have historically descended from the laboratory standards. Generally, the knowledge and technologies which have been developed in order to adequately describe the frequency offsets and uncertainties in high accuracy laboratory standards

have been sucessfully applied to the development of field standards. A good example of this relationship between research and application is the production of the commercial cesium beam standard, which

followed the creation of the laboratory cesium standard. Because of this widespread use of cesium

beams, this paper will discuss the limitations of atomic beam frequency standards in terms of the

38
cesium microwave resonance, although most of the comments may be applied to other devices as well.

All commercially available atomic beam frequency standards today use the 9.19263177 GHz hyper133 fine splitting of ground state Cs measured by the separated cavity Ramsey technique [1]. In addition, most national time metrology laboratories realize the definition of the second using a laboratory version of this same device. However, the reasons that the cesium microwave resonances

has become the most popular timekeeping transition are partly technical [the atom is easily detected

using an ionizing hot wire] , and partly historical [once the second was defined in terms of the

cesium HFS, other methods received less attention].

The main advantage of atomic beam standards over other types of atomic standards, such as rubidium cells and hydrogen masers, is that perturbations of the atoms are reduced by using only atoms

which travel relatively freely through a vacuum chamber. In the rubidium and hydrogen standards,

buffer gases and coated walls interact with the atoms to produce frequency shifts. These shifts

cannot in general be calculated, and must instead be measured, which places limitations upon the

accuracy and long term stability of the devices. On the other hand, confining the atoms with buffer

gases or walls increases the interaction time of the atoms with the resonant microwave energy, producing narrower linewidths. In general, narrower linewidths result in some smaller systematic effects

and better short term stability of the clock. The interaction time in an atomic beam, however, can only be increased either by lengthening the standard, or by reducing the atomic velocity.

All atomic beam frequency standards have the same generic features. There is a source of atoms,

usually a multichannel collimator in the case of cesium clocks. The atoms are then prepared in a special energy state, generally by deflection in a strong inhomogeneous magnetic field. After state

selection, the atoms pass through an interaction region which produces the clock transition, such as

a Ramsey type of microwave cavity. The interaction time in this region determines the resonance

linewidth. Finally, atoms which have made the clock transition are detected in a third region,

traditionally another set of magnets which deflect atoms into a hot wire detector.

Table I summarizes several of the characteristics of such traditional cesium standards, both for

commercially produced field standards, and for one laboratory cesium standard, NBS-6.

1982 Evaluation of NBS-6

Limitations to the stability and accuracy of both field and laboratory cesium standards can be

described by referring to the results of the accuracy evaluation of NBS-6 completed in April 1982. A

summary of this evaluation is given in Table II. Detailed discussions of the systematics in NBS-6

are given in references 2 and 3.

-14 While several of the uncertainties given in this table appear at the 1 to 3 x 10 level, this

is partly due to the fact that not much effort has been spent trying to reduce these uncertainties,

due to the dominance of the cavity phase shift uncertainty (Ay *v 8 x 10 . The latter systematic

is usually separated into two parts [4] an end-to-end microwave cavity phase shift, and a distributed

cavity phase shift. In a primary standard, such as NBS-6, the frequency shift due to the former

effect is cancelled to considerable accuracy (depending upon the clock stability and reproducibility

39
of velocity distributions) by reversal of the atomic beam direction. Roughly speaking, the average

of the clock frequencies for the two directions is the correct value. This frequency offset is about

-13

3.6 x 10 in NBS-6. To the degree that the atomic beam path is retraced upon beam reversal, the

frequency shift due to distributed cavity phase shift (a consequence of residual 1st order Doppler)

-13

is also cancelled. In NBS-6, the size of this effect is about 1 x 10 /mm of spatial offset of the

atomic beam [2] . Fractional frequency offsets Ay^ due to microwave cavity phase shift A<|) are given

by

p o

where L is the interaction length, is the effective atomic beam velocity, and V

q is the resonance

frequency. This expression shows explicitly that the effect of cavity phase shift can be reduced by

increasing the interrogation time T = (L/V^).

If cavity phase shifts in commercial standards are comparable in size to those in NBS-6, then

the expected frequency shifts due to distributed cavity phase shifts should be

QUaboratory std) -13

= -12 % Q(field std)

X iU '

This would suggest that cavity phase shifts may be a limiting systematic in field standards as well

as in laboratory cesium standards.

Of the remaining systematics, those of most concern for improving standards' performance are second order Doppler and the various electronics servo limitations. For an atomic beam of effective

velocity V^, the clock fractional frequency is shifted by an amount

Ay

d

= -% 4> (2)

C

Evaluation of this systematic requires knowledge of the atomic beam velocity distribution, as seen by

the complete microwave resonance spectrometer. In the past this distribution has been determined by

using pulsed interrogation techniques [5] and by unfolding the microwave spectrum at different micro-

-14 wave power levels [6]. While the uncertainty in this shift is quoted at the 1 x 10 level, it is

anticipated that an order of magnitude reduction of the uncertainty may be had with exercise of con- siderable care in the measurement. This process is greatly helped if a narrower velocity distribution is used in the atomic beam.

The evaluation of NBS-6 requires that the electronic servo find the center of the microwave

5 resonance to better than about one part in 10 . Implicit is the assumption that the physical reson- ance itself is symmetric to this level. This assumption is supported by the fact that no significant

frequency shift is observed when the amplitude of microwave phase modulation is changed. Further

checks on frequency shifts due to imperfect electronics include measurement of integrator input offsets and frequency shifts with microwave power change. Again, the demands placed upon the servo are reduced with increase in microwave interaction time, since a narrower microwave resonance need not be

split as finely.

40
One Approach to Improved Field Standards

One approach to considerably improved long term stability in field usable frequency standards

involves the use of laser optical pumping in an "introspective" cesium beam standard. In an optically

pumped cesium standard [7,8,9] the state selection region consists of an optical interaction region

where atoms are converted into the desired energy level instead of rejected by inhomogeneous magnets.

The laser pumping may be performed with either one or two lasers. In the first case, A laser beam

2 2 tuned to the Sj , F=4 «-> P3/2J F'=3 transition, (Figure I) crosses the upstream atomic beam. This

laser pumps atoms into the F=3 ground state. In order to detect atoms in the F=4 ground state, a 2 2 second laser is tuned downstream to the S 1 , F=4 <-» P3/2> F'=5 transition, which can produce many

fluorescence photons for each atom in the F=4 ground state. With complete pumping upstream, no signal is observed in the detection region. As the microwave signal is tuned through the ground

state HFS resonances, however, downstream fluorescence is increased, as shown in Figure II. (The

apparent asymmetry between the (4,-1) <-> (3,-1) and (4,1) «-» (3,1) transitions in this curve is due

to a synthesizer malfunction, and is not a true asymmetry). A preliminary version of such a standard

is under construction now at NBS/Boulder. It is expected to have short term stability comparable to or better than that of conventional commerical cesium standards; laser diodes are used as the light sources [ 10]

MF = -4 -2 0 2 4

253 MHz 5-i

203 MHz

1 1 1

i 1 11 152 MHz

852 nm

MF = -4 -2 0 2 4

9.19263177 GHz

i i i i i i i

4

3

F'=2

F

133

6

2 P3/2

4

3

6

2

Si/2

Figure I. Level diagram for Cs , showing hyperfine structure frequencies (not to scale)

o

c

d)o >

<n *z

0) CO O a) rr c

_sJ\j\J VsJ VnJ W \yJ \_AJ

-3,-3 -2,-2 -1,-1 0,0 1,1 2,2 3.3

Microwave

Frequency

Figure II. One-laser optical pumping

F=4 <-» F'=4, a-polarization pump laser, and

a F=4 •<-> F'=5 detection laser.

41
A further refinement of this method is to use two lasers for optical pumping [11]. One laser

2 2 may be tuned to the Sj , F=3 F 3/2 , F'=3 transition using n polarization. This pumps all magnetic

•5 / midlevels of the F=3 ground state HFS into the F=4 HFS, with the exception of the m=0 sublevel. The

2 2 second laser, tuned to the S, , F=4 <-» P 3/2 , F'=3 transition pumps atoms back to the F=3 HFS. After

many cycles of this process, most atoms should reside in the F=3, m=0 sublevel. An example of this

process is given in Figure III, where the pumping into m=0 is incomplete because of an insufficient

number of pumping cycles. Further work on this technique is in progress.

u

c

a)o >

<n as

a> co O Q) rr c

-3,-3 -2,-2 1.-1w VaJ

0,0 1.1 2,2 3,3 Microwave

Frequency

Figure III. Two-laser optical pumping, using both F=4 <-» F'=4, a-polarization and F=3

rt-polarization pump lasers, and a F=4 <-* F'=5 detection laser.

F'=3,

Optical pumping is expected to improve the performance of field standards by both reducing the

size of systematics and by allowing more precise evaluation of their uncertainties. Since nearly all

atoms are selected and detected, regardless of their spatial position or velocity, the effect of the

distributed cavity place shift is simplified. A smaller cesium beam diameter is possible, resulting

in a smaller microwave cavity window, which should give a smaller frequency offset [4]. In addition,

since atoms which have been pumped into the F=3 HFS are invisible to the F=4 > F'=5 detection laser,

it may be possible to operate simultaneously counterpropagating cesium beams, which would permit the

evaluation of the cavity phase shift in a field standard.

Other benefits of optical pumping and fluorescence detection in a field standard include increased S/N, more symmetric microwave spectra (of the Zeeman split lines), and reduced liklihood of

Majorana transitions, which may produce fairly substantial frequency shifts [12,13].

Further improvement in the performance of the proposed standard comes from use of on-board systematics evaluation. This "introspective" approach involves the measurement of sytematic effects by

monitoring the atoms themselves. For example, the Zeeman splitting can be measured to give a value of

the magnetic field which will shift the clock resonance a known amount. Other systematics to be

measured in similar ways include second order Doppler and microwave power dependence.

42
It is expected that a thorough development of these principles will result in a cesium field

-14

standard with long term (six months) stability of 1 x 10 . Such a standard would find application

in precision navigation and in highly secure communications sytems [14] .

An Approach to Improved Laboratory Standards

Many of the systematics discussed above are reduced as the interaction time is increased. One

approach to achieving longer interaction times is the use of stored ions [15]. Stored ions have

allowed observation times of tens of seconds, with measured microwave Q's of more than 2 x 10^ [16],

12 and projected microwave Q's of 3 x 10 . These long observation times are obtained without the perturbations experienced in other confinement methods, such as buffer gases and coated walls. The

systematic uncertainties are generally less than 1 x 10 ^. This suggests that a new type of primary

standard with inaccuracy less than 1 x 10 ^ may be constructed using stored ions.

Cooled Atomic Beams

The preceding description of existing and projected frequency standards may be used as a framework for discussion of the possible application of cooled beams to atomic frequency standards. In particular, the expected performance of new cesium beam and stored ions frequency standards places a lower limit on performance goals of any new technology, including cooled beams. In addition, many of

the systematics discussed earlier will also be present in a cooled beam clock, as well as new sources

of frequency offset.

While reduction of atomic beam velocity can increase the microwave Q in a cooled beam device,

resulting in a reduction of certain systematics, larger Q in itself will not result in a more accurate

frequency standard. The standard must be treated as a system, with a complete treatment of the interrelating parts in order to obtain an estimate of the system performance. For the purposes of this

discussion, an abbreviated list of considerations will be given. • Although the list is necessarily

incomplete, it should serve as an example of the type of problems which must be solved if cooled

neutral atoms are to be useful in frequency standards.

1. System goals. Inaccuracy of less than 1 x 10 ^, for a primary standard type of laboratory

standard

-14

Long term stability of < 1 x 10 , for a field usable clock.

2. Lower atomic velocities. Considerable cooling of a sodium beam has been reported [17], with mean velocity of about 40 m/s and a velocity spread of about 10 m/ s (down from a mean velocity of 1000

m/s), but this absolute value is not very different than that obtained in more traditional cesium

beam standards. Becker [18] has reported a mean velocity of 93 m/s and a velocity width of 7 m/s

for CS-1, the PTB primary cesium standard, which uses magnetic state selection, and no laser

cooling. While this is not seen as a fundamental difficulty, very low atomic velocities have not

yet been observed in a cooled beam.

3. Beam flux. As the technology is improved, and lower velocities are achieved, it is important

that the reduction in atomic velocity is not achieved at an unacceptable cost in beam flux. The

short term stability [19] of a clock is generally given by

a

y

(x) = K[Q(sm h ]' l x~ h

(3)

43
where S/N is the power signal-to-noise in a one hertz bandwidth, and K is a constant which depends

upon the resonance lineshape. Generally, K lies between 0.1 and 1.0. For traditional cesium

standards, K ~ 0.2. The quantity S/N is generally limited by the "shot noise" on the number of

atoms in the atomic beam, which decreases relative to the signal as the beam flux is increased.

The short term stability should be sufficient to permit measurement of clock performance within a -11 -h reasonable period of time. In a field standard, this means a (t) < 1 x 10 t

2

. In a primary

.i) .i y standard, CTy

(t) < 1 x 10 T .

4. Transverse heating. As an atomic beam is cooled, increased energy is associated with the atoms' transverse motion, resulting in a spreading of the beam. This effect might be reduced with the use of additional lasers or some other cooling mechanism. However, whatever method is chosen to prevent loss of beam flux through transverse heating, it must not introduce additional systematics, such as frequency shifts associated with the microwave cavity.

5. Cavity phase shift. In itself, this frequency shift is expected to be reduced in a slow atomic

clock. (See Eq. 1). However, slow atoms are perturbed for a longer time by gravity and acceleration, resulting in larger deflection angles of the atomic beam. This can produce larger clock

frequency shifts associated with the distributed cavity phase shift as discussed above.

6. Gravity. As longer and longer observation times are employed with free neutral atoms, gravity

plays a more and more important role. Some effects are interactive, such as the cavity phase

shift mentioned above. A more direct consequence is size and geometry, as a clock which uses very slow atoms must accomodate the falling atoms.

7. Light shifts. The use of lasers to perform the cooling of the atoms, and the concommitant fluor- escence light can introduce substantial light shifts [20] . Methods would need to be developed to greatly reduce or eliminate this effect on the standard's performance.

References

[1] N. F. Ramsey, "Molecular Beams" (Oxford Univ. Press, 1956).

[2] D. J. Wineland et al., IEEEE Trans Instr. Meas. IM-25 (1976) 453.

[3] L. L. Lewis, F. L. Walls, and D. J. Glaze, J. de Phys (Paris) 42 (1981) C8-241.

[4] R. F. Lacey, 22nd Ann SFC (1968) 545.

[5] D. A. Howe, 30th Ann. SFC (1976) 451, H. Hellwig et al., Proc. 27th Ann. SFC (1973) 357.

[6] S. Jarvis, Jr., Metrologia 10 (1974) 87.

[7] L. L. Lewis, M. Feldman, and J. C. Bergquist, J. de Phys (Paris) 42 (1981) C8-271.

[8] G. Singh, P. DiLavore, and C. 0. Alley, IEEE J. Quan. Elect. QE-7 (1971) 196.

[9] M. Arditi and J. L. Picque, J. Phys (Paris) 41 (1980) L-379.

[10] L. L. Lewis and M. Feldman, 35th Ann SFC (1981) 612.

[11] L. Cutler, private communication (1979).

[12] G. Becker, IEEE Trans . Instr. Meas. IM-27 (1978) 319.

[13] H. Hellwig, Proc. 5th Conf. on Atomic Measses and Fund. Constants, ed. by J. Sanders and A

Wapstra, Plenum Press (1976) 330, D. W. Allan et al., Proc. 31st SFC (1977) 555.

[14] S. R. Stein, paper included in this proceedings.

[15] D. J. Wineland, paper included in this proceedings.

[16] R. Blatt, H. Schnatz, and G. Werth, Phys. Rev. Lett. 48 (1982) 1601.

[17] J. V. Prodan, W. D. Phillips, and H. Metcalf, Phys. Rev. Lett. 49 (1982) 1149.

[18] G. Becker, Proc. of the 11th Ann PTTI (1979) 113.

44
[19] J. Vanier and L.-G. Bernier, IEEE Trans. Inst, and Meas. IM-30 (1981) 277.

[20] A. Brillet, Metrologia 17 (1981) 147.

Table I. Characteristics of Cesium Atomic Beam Frequency Standards

Commercial Stds NBS-6

Short term

Stability 1.5xl0" U x' h 5X10" 13

X h

Long term

Stability

Accuracy

-13

3x10 (6 months)

3x10 ^ (5 years)

7xl0"

12

lxlO

-14

(1 week)

8x10

-14

Volume ~ .03 nf ^ 5 nf

Interaction

Length

7.5 cm 375 cm

Microwave Q 1x10 2.8x10

8

Mean atomic velocity 100 m/s 190 m/s

45
Table II. NBS-6 Evaluation, April 1982.

Systematic Value (xlO ) Uncertainty (xlO )

C-field 5335 3

Mag. field Inhomo. 0.2

Majorana 0.3

tail pulling 2

cavity pulling 0.1

RF spectrum 1.0

2nd order doppler 26 1.0

cavity phase shift 36 8.0

amplifier offset 1.0

2nd harmonic distortion 2.0

uncertainty in (j)^ due to 6[p(v)] 1.0

blackbody shift 1.7 0.0

RSS 9.2

random 1.0

total (RSS)

46
Laser-Cooled and Trapped Atoms , (Proceedings of the Workshop on Spectroscopic

Applications of Slow Atomic Beams, held at NBS, Gai thersburg, MD, 14-15 April

1983) Ed. by W. D. Phillips, Natl. Bur. Stand. (U. S.) Spec. Publ. 653 (1983).

Stabilization of a Microwave Oscillator

Using a Resonance Raman Transition in a Sodium Beam

P.R. Hemmer and S. Ezekiel

Research Laboratory of Electronics

Massachusetts Institute of Technology, Cambridge, MA 02139

and C.C. Leiby, Jr.

Rome Air Development Center

Hansom Air Force Base, MA 02139

Preliminary results of stabilization of a 1772 MHz oscillator

to a resonance Raman transition in an atomic beam of sodium are -11

presented. Short term stability of 5.6 x 10 (x = 100s) for a 15 cm interaction region separation has been achieved. Sources

of frequency drift are briefly discussed.

Earlier experimental work [1] has shown that extremely narrow optical

features can be obtained using a stimulated resonance Raman transition between two long-lived hyperfine ground sublevels in a sodium atomic beam.

Such studies are of interest because they have possible applications both in

spectroscopy and in the development of new time and frequency standards,

especially in the microwave to millimeter regions of the spectrum. In this

paper we present preliminary results of attempts to stabilize the frequency

of a microwave oscillator using these narrow optical features.

Figure 1(a) shows schematically a stimulated resonance Raman transition

between two long-lived states, 1 and 3, induced by two laser fields co^ and

032» In a well collimated atomic beam, the Raman transition linewidth for

weak copropagating laser fields is determined by the decay rates of states

1 and 3, with a negligible contribution from state 2, i.e., the linewidth

is set by the transit time, since states 1 and 3 are long lived [2]. To

obtain a small transit time linewidth we use separated oscillatory field

excitation [3,1] as illustrated in Figure 1(b), where L is the separation

between the interaction regions. In the present experiment, states 1 and 3

correspond to the F = 1, m^, = 0 and F = 2, m^, = 0 hyperfine sublevels of the

sodium 3

2 $-^/2 Srounc* state, separated by 1772 MHZ, and state 2 is the

3 2p

-L/2 = 2» +1) excited state, having a 16 nsec. lifetime.

47
The laser field at frequency oo^ is obtained from a single mode dye

laser which is locked to the 1 -<-*- 2 transition. The laser field at frequency u)^ is generated from that at by an acoustooptic frequency shifter

[1] driven with a microwave oscillator near 1772 MHZ. This greatly reduces

the effects of laser jitter by correlating the jitters in the two laser

frequencies so as to obtain a stable difference frequency oo^ - u .

Figure 2(a) shows a typical fringe lineshape corresponding to an interaction region separation of L = 15 cm. To obtain this trace the dc

fluorescence collected from the second interaction region is monitored with

a photomultiplier while the difference frequency to^ - co is scanned over

the Raman transition.

To stabilize the frequency of the microwave oscillator to the central

fringe in Figure 2(a) a discriminant is needed. This discriminant, which

is shown in Figure 2(b), is obtained by frequency modulating the microwave

source and demodulating the second region fluorescence signal with a lockin amplifier. The output of the lock-in amplifier is then used in a feedback loop to hold the microwave oscillator frequency at the central zero of

the discriminant [4]. The stability of this oscillator is then measured by

comparing it with a commercial cesium clock.

Figure 3 shows a plot of the measured fractional frequency deviation

of the microwave oscillator a (t) , as a function of averaging time, x. For

y -11

x = 100 seconds, the stability is 5.6 x 10 . The predicted [5] fractional

frequency deviation, based on shot noise, appears as the dashed line superimposed on the data in Figure 3. As can be seen, the data is shot noise

limited up to about T = 500 seconds.

For x greater than 500 seconds, the fractional frequency deviation no

-1/2 longer decreases with T , indicating the presence of long term frequency

drifts. Work is presently being performed to identify and study in detail

the important sources of frequency error. These include:

(a) The relative misalignment of the laser beam at CO.. and

away from cop ropagating.

(b) Phase shifts due to relative path length differences measured

from the beam splitter in Figure 1(b) to each of the two interaction regions.

48
(c) Phase shifts which arise when the polarizations of the laser

beams at co^ and differ and there are birefringence optics

present.

(d) Detunings of the pump laser, at u)^, away from resonance with

the 1 2 transition

(e) Laser intensity variations

(f) Optically induced atomic recoil

(g) Fluorescence from the interaction regions that propagates

along the atomic beam [6].

(h) Effects of the sodium 3 2p

1/ 2

^ F = ^ state which is 189 MHz

below the 3 ^ F = 2 ^ state *

(i) Laser and atomic beam misalignments

(j) External magnetic fields

(k) Second order Doppler

We have already incorporated in our setup means for minimizing some of

the effects mentioned above in order to obtain the data in Figure 3. A

detailed discussion of error sources for the stimulated resonance Raman

scheme will be the subject of a future publication [4].

Although the preliminary results presented here are for a sodium atomic

beam, the techniques discussed are readily applicable to cesium. In fact,

if cesium were substituted for sodium in our apparatus and everything else

being equal, we would expect an improvement in stability by a factor of 16,

-12

i.e., a (t) = 3.5x10 for x = 100 sec. This increase takes into con- sideration the larger hyperfine transition frequency of cesium, the greater

atomic mass of cesium and the lower oven temperature needed to obtain the

same atomic flux used in our present sodium apparatus. Moreover, if the

interaction region separation were increased from our present 15 cm to

-12

3.75 m, the separation of NBS-6, [7] we would obtain a a (t) 2.5 x 10

-13 ^

(T = 100s) for a Na beam and O^(t) * 1.5 x 10 (x = 100s) for a C

g

beam.

Finally we are extremely interested in developing techniques for improving stability by slowing down the atomic beam. The recent radiation

cooling experiments by Prodan, Phillips and Metcalf [8] at NBS are of much

interest to us and are very applicable since their experiments were also

conducted on sodium.

49
This research was supported by the Rome Air Development Center,

Contract No. F19628-80-C-0077 and by the National Science Foundation.

References

[1] Thomas, J.E., Hemmer, P.R., Ezekiel, S., Leiby, Jr., C.C., Picard, R.H.

,

and Willis, C.R, "Observation of Ramsey Fringes Using a Stimulated

Resonance Raman Transition in a Sodium Atomic Beam." Phys. Rev. Lett.

48, 867, 1982.

[2] Thomas, J.E., Ezekiel, S. , Leiby, Jr., C.C., Picard, R.H., and Willis,

C.R. "Ultrahigh - resolutions spectroscopy and frequency standards in

the microwave and far-infrared regions using optical lasers." Optics

Lett. 6, 298, 1981.

[3] Ramsey, N.F. Molecular Beams. Oxford University Press, London, 1963.

[4] Hemmer, P.R. , Ezekiel, S., Leiby, Jr., C.C. To be published.

[5] Lacey, R.F., Helgesson, A.L. and Holoway, J.H. "Short-term Stability

of Passive Atomic Frequency Standards." Proc. IEEE 54, 170, 1966.

[6] Brillet, A. "Evaluation of the Light Shifts in an Optically Pumped

Cesium Beam Frequency Standard." Metrologia 17, 147, 1981.

[7] Glaze, D.J. , Hellwig, H., Allan, D.W. , Jarvis, Jr., S. "NBS-4 and

NBS-6: The NBS Primary Frequency Standards." Metrologia 13, 17, 1977.

[8] Prodan, J.V., Phillips, W.D., and Metcalf, H. "Laser Production of a Very Slow Monoenergetic Atomic Beam." Phys. Rev. Lett. 49, 1149, 1982.

50
Figure 1. (a) Stimulated resonance Raman system. (b) schematic of setup for separated

oscillatory field excitation using a stimulated resonance Raman transition.

Figure 2. (a) Ramsey fringes corresponding to L = 15 cm. Scan rate 330 Hz/sec;

T = 10 msec; peak-to-valley amplitude ~ lpA; central fringe width FWHM approx.

2.16 kHz. (b) Discriminant obtained by frequency modulation. Scan rate

330 Hz/sec; T = 100 msec.

51
Or

y

(T)

10"3 10"2 0-1 1 10 102 103 104

T(seconds)

Figure 3. Fractional frequency stability vs. averaging time.

52

I
Laser- Cooled and Trapped Atoms , (Proceedings of the Workshop on Spectroscopic

Applications of Slow Atomic Beams, held at NBS, Gaithersburg, MD, 14-15 April

1983) Ed. by W. D. Phillips, Natl. Bur. Stand. (U. S.) Spec. Publ . 653 (1983).

The Force on an Atom in a Laser and D.C. Field*

Marvin H. Mittleman

The City College of The City University of New York

The deflection of an atom depends upon the coupling of the field to

the internal motion of the atom. This internal motion can be profoundly

altered by a laser field so the deflection can depend upon the laser

intensity and frequency. The force that an atom experiences in a combined

D.C. and laser field is reviewed. The fluorescence of the atom is shown to

play an important role.

1. Introduction

This note is a brief review of the theory of the force on an atom due to D.C.

electric and/or magnetic fields (the external field) and a resonant laser field. It

1 2

contains nothing new. The laser is described here as a classical single mode

electromagnetic field. Quantum corrections are not significant and multimode

corrections can be included at the end of the calculation by a suitable averaging

procedure. The laser-atom interaction is treated in dipole approximation as far as

the internal motion of the atom is concerned. Corrections to this are usually

negligible. However the center of mass motion of the atom will extend over many

laser wavelengths so no such approximation is possible there.

The theory of the deflection of atoms by external fields is ancient. Only

energy conservation is necessary to understand the effect: an atom in an external

field will have its energy levels shifted by the field and it is usually sufficient

to use perturbation theory, in one form or another, to obtain these shiftsMoreover, in the usual case, each of the levels will be shifted by a different

amount. The total energy of the atom must be conserved since the external field is

conservative. Therefore the energy shift of the internal state of the atom must be

compensated by the change in the center of mass kinetic energy. Another way of

saying this is that the energy level shift will act as a potential energy for the

*This work was supported by a contract from the U.S. Office of Naval Research, No.

N00014-76C-0014.

53
center of mass motion. (This is analogous to the atom-atom scattering problem in

which the electronic eigenvalue acts as an internuclear potential in the

Born-Oppenheimer approximation.) The magnitude of the level shift will differ from

level to level so the center of mass potential will also depend upon the level. In

the absence of the laser, the change of the level shift with center of mass position

will occur very slowly on the internal time scale of the atom so the atomic state

will deform adiabatically and only a single level shift (usually that of the ground

state) will enter. The laser will change this since it can strongly couple a pair

of atomic states so that the atom will be in a superposition of these two states

rather than in either one. Therefore the level shifts of both states will be

relevant to the center of mass motion. The precise way that they enter will depend

upon the way that the atom is resonant with the transition between the two states.

This is governed by the ratio of two. numbers. The first is the detuning frequency

(* = 1)

Aw = wL

-W

10

(R), (1)

where is the laser frequency and W^q(^) = W^(^) - W

Q (1b where w

n

(^0 is

the energy of the n-th electronic state of the atom which depends upon ]£, the center

of mass coordinate of the atom. The atomic states coupled by the laser have been

labeled "1" and "0". The second number is the coupling matrix element of the two

states

A<R) = (1 |f 0) = |A|ei

f (2)

1j

where E is the amplitude of the electric field of the laser. A convenient ratio is,

the detuning parameter, u

sinhy(R) = .-y- • ( 3 )

For small values of y the two states are significantly coupled whereas large values

of y give essentially one state or the other.

54
We have so far ignored the motion of the atom. The considerations described

above are relevant in the rest frame of the atom. They can be transformed to the

lab frame and the only significant change (for non-relativistic atoms) is that the

laser frequency is Doppler shifted so that w is replaced by uL where

BL- ML- R-k

L

(4)

and where v = R. Then the center of mass potential will depend upon the position

and velocity of the atom. (In fact the dependence is much more complex in that this

"potential" will depend upon the temporal history of the center of mass motion.

This correction is negligible unless the atoms undergo accelerations comparable to

the electronic accelerations in the atom. This is not realizable now.)

An objection can be raised to the reasoning just presented. It is that energy

conservation was the underpinning of the logic but the laser is a time dependent

field so energy conservation fails. The solution of the dilemma lies in Shirley's

3 theorem which states that the energy transferred to the particles must be an

integer multiple of the laser frequency. (Energy is transferred in units of

photons.) Then the energy of the particles is conserved modulo (not w^).

But the potentials discussed above are continuous functions of their variables so

energy conservation modulo ojt is equivalent to absolute energy conservation and so

XJ the validity of the logic is restored.

At this point we must introduce the fluorescence of the atom. If the laser-atom

coupling is effective (|y| < 1) the atom will spend a significant fraction of its

time in the excited state. (The fraction will be 1/2 for y = 0. ) In an experiment

—8

in which the atom is in the fields for many natural decay times ( ^ 10 sec),

which is the usual case, the atom will fluoresce many times during its passage

through the fields. This will modify the internal state of the atom and therefore

the potential experienced by the center of mass. Fluorescence is a stochastic

process and so the internal state of the atom will become an incoherent

superposition of the atomic states after a few fluorescent emissions. We can

restate this: in the absence of fluorescence, the atom starts in its ground state

and moves, adiabatically , into the external and laser fields. The two internal

atomic states in these fields can be written

e~

1

( ^-t + Mv«R) , i . „ N 2 r ± u/2 -S (w_ t - k. *R)

$ = 1 ie M u_e L L

(2coshy) 1 U

(5)

± e±v/,2 - 16 u e -*5 (

<*> Lt - k * R>} < TM0 +w"l ± e ) t

55
where u

Q

and are the atomic states in the external field and the Rabi

frequency is

e = / | A | 2 + Aco 2 (6)

These states have been obtained by using the rotating wave approximation which is

correct to terms of order Aw/oj^ which is usually very small. The atomic ground

state will deform adiabatically into either $+

or depending upon the sign of

Aw, but it will remain in this state. Fluorescence will however couple these states

so the atom will go into an incoherent superposition of these two states after a few

fluorescent times. The center of mass potential will then depend upon the details

of the fluorescence.

It is easy to show that there can be many hundreds of fluorescent emissions

during a typical experiment so that the treatment of the emission process by

straightforward perturbation theory is out of the question. There are a variety of

alternate treatments which can be used to deal with the problem. Perhaps the

4 simplest is that due to Mollow who showed that an approximate "constraint" can be

written for the wave function of the fluorescent state. The method is sufficient to

calculate some simple properties of the state which are all that is needed to

describe the force acting on the center of mass of the atom. Since the atom is

thrown into an incoherent superposition of the state $+and $_ it is clear that

the force must depnd upon the probability of finding each of these states, P

+

and

P_. The Mollow technique gives these parameters. For times which are large

compared to the natural decay time they are

where we have also assumed that the Rabi frequency is large compared to the natural

decay rate. (The assumption is easily removed.)

There is also a recoil force due to fluorescence. This may be understood in a simple way. The fluorescent process is one in which a laser photon is absorbed and

then emitted as a fluorescent photon in (for example) a dipole pattern. In the rest

frame of the atom there is no net momentum emitted into the f luorscent field so the

process converts the momentum of the laser photon into momentum of the atom. (In

the lab frame the fluorescent photons do carry away some momentum but this is a

/ 2 cos h 2y , (7)

56
small relatlvistic effect which is actually cancelled by the relativistic effect

associated with the change of mass of the atom"*). The force resulting from this

recoil is

*\N, (8)

where N is the rate of emission of fluorscent photons. Again the Mollow technique

yields

N = ^y(cosh 2y)

-1

, (9)

where y is the natural decay rate of the excited state.

Finally there is one more contribution to the force on the atom which arises

from the spatial inhomogeneit y of the laser beam, the ponderomotive force. The

laser enters the atomic Hamiltonian through the vector potential which we write as

X = 4r^- cos(wT t - k_ •R) . (10)

CO- Li Li

Li

The Hamiltonian contains the kinetic energy term

^(P + *A) 2

=. |i + £ p

. A + |l A2 . (11)

The last term may be written

e

2 E2 (R) , e

2 E

2 (R) ,/-«. , „ n /-.ox \ + -, _ 0 cos 2((oT t-k_»R). (12) 4m w

1 4m w * L L

Li Li

The second term of (12) has been assumed to be non-resonant with any atomic

transition and so contributes terms similar to the counter rotating terms which have

been dropped in obtaining (5) . The first term of (12)^which we call U , the

57
ponderomotive potential, does not couple to the electronic motion within the atom,

it only acts as a potential for the center of mass motion.

We can now assemble the results to give the force

W +W +e W

0

+Wl- £ * - F = -P

+

V( U

2

L

) - P_V( U

2 L

) -VUp + h^N. (13)

If we interpret ^(Wq + W^ + e ) as the "energy" of the states $+

(5), the

result (13) is readily understood.

This result applies for low intensity-near-resonant lasers since the shift of

the energy levels of the atom due to the dynamic Stark effect has been neglected.

This is usually a small effect but if it is large enough to affect the resonance

condition it is significant. That is, if the dynamic Stark shift, AE^, is

comparable to |a| it must be included. This can be done by using a complete set of

atomic states which contain this shift, instead of the "bare" atomic states implied

in the discussion here. The main change in the force due to this effect is the

inclusion of the dynamic Stark shift terms in Wq and W^. (Care must be taken to

prevent counting intermediate states more than once.) These terms are the same

2 order of magnitude as the ponderomotive potential and must be included with it.

The force can now be used in a classical equation of motion to describe the

motion of the atoms but all the details will not be presented here. It should

however be pointed out that the resonance condition is velocity dependent so that

the laser can force the atom to change its velocity and thereby decouple itself from

the laser. A judicious choice of the external field can compensate for this.

Finally, it should also be noted that the force (13) is the expectation value of

an operator which has a non-vanishing variance. This will cause the velocity of the

atom to spread. We shall not pursue this here.

2. References

[1] The material discussed here is from M. H. Mittleman, K. Rubin, R. H. Callender,

and J. I. Gersten, Phys. Rev. A _16 , 583(197 7).

[2] The details of the mathematics of this discussion can be seen in "Introduction

to the Theory of Laser-Atom Interactions," M. H. Mittleman, Plenum, New York,

1982.

[3] J. H. Shirley, Phys. Rev. B 138 , 979(1965).

[4] B. R. Mollow, Phys. Rev. A JL2 , 1919(1975),

[5] This fact was pointed out to me by Dr. Richard Cook, private communication.

58
Laser-Cooled and Trapped Atoms , (Proceedings of the Workshop on Spectroscopic

Applications of Slow Atomic Beams, held at NBS, Gai thersburg, MD, 14-15 April

1983) Ed. by W. D. Phillips, Natl. Bur. Stand. (U.S.) Spec. Publ. 653 (1983).

Magnetic Trapping of Decelerated Neutral Atoms

Harold J. Metcalf

State University of New York

Stony Brook, N.Y. 11794

A scheme is proposed for trapping atoms having finite magnetic

moments using inhomogeneous magnetostatic fields. Fields of ^ 0.1 T can be used to contain atoms decelerated to v ^ few m/s. Application

to the N.B.S. decelerated atom project is discussed.

Key words: cooling, trapping, precision spectroscopy

1. Introduction

The use of electromagnetic forces to influence the motion of free,

neutral atoms has been a subject of interest for many years. On the

practical side, there are many spectroscopic and collision experiments

that would benefit from the use of slow and/or cold atomic samples. On

the theoretical side, the various limits to the control of atomic motion,

as well as the approaches to them, present interesting and challenging

problems. In this paper, we discuss some of the methods of confinement of

neutral particles, with an emphasis on the use of inhomogeneous magnetic

fields for those particles having magnetic moments. Then we describe the

magnetic trap for slow sodium atoms proposed for use at the N.B.S. Since

all of the trapping schemes described below share the common feature of

being rather shallow compared with thermal energy at room temperature,

deceleration and cooling are often inseparable from confinement.

The most popular mechanism for producing a force on free neutral

atoms uses the electric field associated with light, particularly light at or near the resonant frequencies of atomic transitions. The influence of

radiation force on free atoms using light from a resonance lamp was first

observed in 1933 by Frisch [1] and in 1972 by Picque and Vialle [2]. In

that same year, Walther [3] and coworkers detected atomic deflection using

a tunable dye laser beam. At least two applications for this phenomenon

had been proposed two years earlier by Ashkin [4,5]. Although the

phenomenon was regarded as a curiosity for many years, the advent of

modern dye lasers has stimulated new interest in both influencing and

controlling the motion of free atoms.

Interest in cooling (and trapping) grew rapidly as a result of work at

Bell Laboratories [6] and at the U.S.S.R. Academy of Sciences [7]; the

first experimental results were reported in 1978 [81. Additional experimental results were reported by both the Russian group [9] and by us at

the N.B.S. [10], but all this work was related to deceleration and cooling.

59
Trapping of atoms using resonance radiation has been proposed and

discussed extensively by Letokhov and coworkers [11] and by Ashkin and

coworkers [12]. These schemes exploit both the dipole force associated

with the inhomogeneous electric field of both a non-uniform light beam and

of the standing wave pattern from counter propagating beams, as well as the

recoil associated with the transfer of momentum from the optical field to

the atom (scattering). In the first case the dipole moments are produced

by the light itself by mixing with or excitation to a state of opposite

parity, resulting in an atomic superposition state having a non-zero

dipole moment [13,14]. There have been a number of interesting geometries

proposed for these optical traps, including a rather novel tetrahedral

configuration [15] and an electromagnetic mirror [16]. The combined use

of optical radiation for state preparation by excitation to Rydberg states,

and electrostatic forces on the resulting highly polarizable atoms has been

proposed for both trapping [14] and deceleration [17].

There has been considerably less work on the use of magnetic fields

for steering and trapping. The deflection of particles having magnetic

moments was first demonstrated by Stern and Gerlach [18] in a landmark

experiment more than 60 years ago. It was not until the early 1950 's that

a generalized Stern-Gerlach approach to deflection appeared in the form of

Paul's hexapole lens [19]. This idea has been exploited by both the atomic

beam community [20] and the slow neutron people [21]

2. Magnetic Trap for Slow Neutral Atoms

The principle advantage to magnetic forces for slow atoms arises from

the long time these atoms spend in any particular part of the field

gradient. Even though the force is limited by the product of the magnetic

moment (typically a Bohr magneton) and the field gradient (typically one or two Tesla/meter) , the impulse delivered to the atoms is enhanced by the long

interaction time. For example, the deflection in the Stern-Gerlach experiment is proportional to l/v2.

In this section we describe the magnetic trap proposed for use at

the N.B.S. In brief, the cylindrically symmetrical trap consists of two

end mirrors separated by L composed of regions of sharply increasing

magnetic fields for longitudinal confinement, and hexapole lens of focal

length L/4 placed axially and halfway between them for transverse confinement. The optical cavity analog of this trap is shown below.

60
An end mirror can be made using a several cm diameter coil of a few

hundred turns carrying a few amperes that produces a maximum field of

about B = 0.1 Tesla and gradient of about 1 Tesla/meter. Since the

magnetic moment yg

of a ground state Na atom is yg= yR

, this coil can reflect an axially moving atom having kinetic energy E = h Mv2 < yfi B = 10 J

corresponding to a kinetic temperature of about 0.06°K. These atoms move

at about 6 m/s.

It is very curious to note that there is an intimate relation between

the depth of a magnetic trap and the ultimate final velocity that can be

obtained by laser deceleration of the kind we have achieved at the

N.B.S. [10]. This relationship arises because of the heating associated

with the random fluorescence of the light absorbed by the atoms from the

decelerating laser beam. Although the average impulse transferred to

atoms moving with initial velocity v0 by the fluorescence of n = MvD X/h quanta is expected to be zero, the rms fluctuations are not

zero because of the quantization of momentum in units of h/A. The rms momentum is thus Ap = /n h/X and the resulting kinetic energy is

E = Ap2 /2M = nh2/2MA2 = hvQ /2X = hvD

/2 where Vj) = vD

/A is the Doppler

shift required to bring the laser into resonance with the atom moving at

its initial velocity v

q

[22], Therefore hvD

/2 is the minimum trap depth.

In our experiment this Doppler shift is just compensated by a Zeeman

shift produced by an applied magnetic field B, and because the excited

state has magnetic moment y

p

= 2y

fi

, the optical frequency is shifted

by hvg = ygB. Therefore = and the applied Zeeman field B is just

twice as large as it needs to be: the main magnet used in the deceleration

process in our experiments can easily serve as one end of the trap [23].

Construction of an appropriate magnet to serve as a mirror for the other

end of the trap is straightforward.

Lateral confinement in our trap will be provided by a hexapole electromagnet. These have been described many years ago [19,20] and permanent

magnet devices are available commercially [24]. The principle idea is to

produce a field that varies quadratically with distance from the axis so that the field gradient (thus the force) varies linearly and produces

harmonic motion. Solutions to Laplace's equation for the magnetic potential

$(r,6) under the boundary condition of six-fold symmetry results in a series with leading term r^. The field is thus quadratic as desired. An

atom moving at some small angle away from the axis will be deflected back

towards it in such a way as to produce focussing; these devices have the

properties of a lens.

61
The basic configuration of the magnet is shown perpendicular to the

axis [25] below. Although the pole pieces shown here are hyperbolae, the

more ideal case of cylindrical boundary conditions at alternating potential

0 and V was solved in Ref . 20 (1959) and the potential is

n=l

sin(n6) . na

n" sin ~2

N

I (-D^sint ( i £=1 2N

where N is the number of pole pairs, rQ is the distance to the boundary,

and a is the angle subtended by each pole. Then the magnetic field for

N = 3 and a = fr/6 is

2 r 6

B(r,6) =Bo (|) 1 - 2 ( | ) cos69 + ^ ( | ) o L o o

The dominant term is the desired quadratic, and the next term is down by

an order of magnitude for about 3/5 of the aperture area.

We have designed and built the device shown adjacent to the ideal

case. The three independent coils of about 500 turns each of #20 AWG

wire, formvar insulated, have a resistance of about 2 ohms each, and stay

cool carrying 3 amperes with only ambient air cooling. Careful measurements on a milling machine bed confirmed that the resulting field has the

required geometrical symmetry, reaches the design maximum of B0

= 0.2 Tesla

in the gaps, and is very nearly parabolic in the 2 cm diam. central aperture.

62
In order to calculate the optical properties of this device, we first

note that the oscillation frequency in the harmonic potential is given by

= 2y sB

0 /Mr( 2 and the total phase change of the oscillation for an atom that traverses a magnet of length Z at velocity v is Af = w^Z/v.

Then straightforward geometrical considerations show that atoms diverging

from a point on the axis a distance s away from the center of the magnet

are bent back to cross the axis again at a distance s' such that

(s + s')/ss' = 1/f where f = v

2 /wT?Z = (Ekin/Emag) (rQ

2 /Z) . For thermal atoms

it takes a 10 cm long magnet with aperture only 2 mm diam to get f = 20 cm,

resulting in a lens of speed f/100. But for atoms decelerated to 40 m/s as we have observed [10], a lens as short as 2.5 cm and as wide as 2 cm diam

has a focal length only 13 cm and a speed f/6.5. The collecting power for

slow atoms is enhanced by a factor v

2

, and hexapole lenses can be tailored

for particular needs.

In addition to strongly enhanced collecting power, the use of hexapole

lenses for decelerated atoms has another strong advantage over their

traditional use with thermal beams. Since the focal length depends on v > there are severe aberrations associated with the Maxwellian spread of

thermal velocities. This problem is sharply reduced for the compressed

velocity distribution of decelerated atoms reported in Ref . 10.

Although we would also like to further cool the atoms in this trap,

there is another curious coincidence in Ref. 10 that would seem to make

further cooling impossible. The rms velocity Av - Ap/M = v

Q

/i/n = 6 m/s

results in an rms Doppler shift Avp = Av/X = 10 MHz which is almost

exactly the natural width of the transition. It would seem that the natural

width would present a limit to further deceleration because atoms could

also be accelerated in the laser beam. Fortunately this view is too naive,

and further cooling can be achieved by weaker, slightly off-resonant light.

Of course, the hexapole lens of the trap would have the right focal length

only for atoms of a particular speed, but we can compensate for this

difficulty by properly tapering the field of the end mirrors. If their field

increased linearly with distance along the axis, the position of the turning

point would move away from the center of the lens quadratically with velocity

and would just compensate the changing focal length of the lens [26].

Loading atoms into this kind of trap presents the usual problems:

there must be some way to extract energy from the atoms while they are

inside. Our deceleration process does this by its very nature, and

therefore provides an ideal damping or velocity-dependent force. Some

other methods have been described in the literature cited, and still

others appear in these proceedings.

In order for this proposed scheme to work at all, the atomic magnetic

moments must be oriented so that they are repelled from regions of strong

field. This is certainly the orientation produced by the optical pumping

process that occurs during cooling, but it must be preserved while the atom

is in the trap, even though the trap fields change directions in a very com- plicated way. As long as the atom is moving slowly enough, the magnetic

63
field changes sufficiently slowly that the magnetic moment precesses about

the field and follows it adiabatically . The condition is violated, however,

if the atom sees a field change with strong enough Fourier component at

the frequency of magnetic resonance that would cause a transition to a state

of different orientation. That frequency is simply w

L

= %UgB/n. The transition moment is approximately ^SUg, and therefore transitions occur at the

rate Wj^UgAB/n where AB = vVBt is the field change seen by an atom during

the time t - tt/2o)l

. Then wR

= (VB/B) (uv/2)=Trv/r=iTw

T

for motion in the

lens, and it appears that an atom cannot survive against Majorana transitions for more than a few oscillations in the trap.

Fortunately, this is not the case. An atom oscillating perpendicular

to the trap axis sees an oscillating magnetic field having dB/dt parallel to

B, the equivalent of z-polarization, which does not produce the Am = ± 1

transitions required to re-orient the magnetic moment. An atom moving parallel

the lens' axis sees dB/dt = 0 and no transitions occur. Only orbital motion

produces the necessary rotating magnetic field, but the atomic moment can precess about the field resulting in a substantially decreased transition

rate. More careful considerations of the problem of a magnetic moment in a hexapole trap have been worked out by Matora [27], and suggest that the

trap is indeed very nearly stable against this problem. Similar considerations hold for the end mirrors. In fact, the effect of the anisotropy of

the gravitational force on the atoms may provide the ultimate limit of the

trapping time.

Another consideration for traps of this kind is the vacuum required.

Although a trapped ion at thermal energies can survive a few collisions,

and even a cooled trapped ion that is struck by residual gas atom having

thermal energy can still stay in a typical ion trap, a cooled neutral can

not. Since the neutral's trap is so shallow, and since these atoms are 'sitting ducks' for thermal energy background gas, the vacuum requirements

are more stringent. The mean free time between collisions must exceed the

trapping time. Estimating the cross section for destructive collisions is

very difficult because even a very gentle collision (i.e., large impact

parameter b) can impart enough energy to eject the atom from the trap.

Nevertheless, a mean free time of 1 sec demands P = 10"^ Torr for b = few

Bohr radii, and the required pressure decreases as 1/b^.

3, Other Magnetic Traps

There are two other kinds of magnetic traps for sufficiently slow,

neutral objects. To illustrate one of these, consider a hexapole field

made from six parallel, straight conductors arranged so that they cross a perpendicular plane at the vertices of a hexagon [19,28]. If they each

carry constant currents of the same magnitude but alternating directions,

then the field produced will be the same hexapole field shown earlier.

Adequately strong fields can be made by using very high currents in superconducting wires. A very long, thin structure of this type can be used as a guide to transport slow neutral particles much like a graded-index optical

fiber. We can imagine a long structure of this type bent around into a torus

as shown below. The particles are guided around the circle in a very

similar way to those in very high energy storage rings. As recently as 1977 [29] it was not clear whether or not such a storage ring would work, but one has been built for ultracold neutrons by Paul and coworkers [30]

.

64
The other kind of magnetic trap, also a hexapole field and closely

related to this one, is shown below. The field configuration is imagined

by rotating the hexapole field shown earlier about a vertical line in the

plane of the paper. The magnet is constructed from three superconducting

rings of wire carrying appropriate currents. Such a trap has also been

built and is being tested in Paul's laboratory [31] with sodium atoms.

Although this trap has the desired hexapole field configuration in any

vertical plane, it does not have the appropriate symmetry in the horizontal

plane and it is not known if the trap will be stable.

4. Future Plans for the N.B.S. Experiment

The atom deceleration experiment of Ref . 10 has not reported atoms

slower than about 40 m/s because the signal disappears rapidly at low

velocities. We believe that this is an observation problem and not any

limit to the particular deceleration process. Most of the difficulties are believed to arise from the unavoidable space between the end of the solenoid

and the observation region. This drift region cannot be eliminated because

the stray magnetic fields from the solenoid will necessarily cause difficulties in the observation region. We have been concerned with collisions

with background gas, with beam divergence from both transverse heating and

residual imperfect collimation [32], and with the effect of the laser light

in this drift space. Various incomplete solutions to these problems have

been partially successful [10]

Much of the signal loss is believed to originate from beam divergence

produced by various effects and our first experiment with the new hexapole

will therefore be to focus the diverging beam into the interaction region.

We hope to produce much higher atomic densities at velocities near 40 m/s,

and to observe much slower atoms.

65
One of the principal characteristics of our hexapole is the relative

magnetic independence of the pole pairs. By using different currents in the

separate yoke coils, we will be able to move the magnetic center off the axis,

and therefore deflect as well as focus our atomic beam. This combination of

capabilities should result in a sample of high density, very slow atoms off

the optical axis (out of the laser beam)

.

The next step is to place a small aperture in the off-axis focal plane

that will admit the focussed atoms to a differentially pumped, ultra-high

vacuum region where we will try to trap them. Many schemes for optical

traps have been mentioned above, and others are described in this volume. If we build an optical trap, it will surely be closely related to one of these.

If we build a magnetic trap of the kind described above, we will have to

deal with two important problems. First, we do not know if it is stable

against magnetic dipole transitions that would re-orient the atom and cause ejection from the trap. Although the toroidal hexapole has worked for

neutrons, our proposed trap is complicated by the end mirrors and by the

additional possible transitions associated with the hfs of the sodium atom.

The second problem we face is detection. Although detection of trapped atoms

by fluorescence of light from a weak probe beam is a possibility, we would

have to be very careful not to eject the atoms by momentum transfer from the

probe and not to heat them by too much recoil momentum. We do not know the

solution to either of these problems.

This work was supported by the N.B.S. and the O.N.R.

References

1. R. Frisch, Z. Phys. 86, 42 (1933).

2. J.L. Picque*and J.L. Vialle, Opt. Commun. _5, 402, (1972).

3. R. Schieder, H. Walther, and L. Woste, Opt. Commun. _5, 337 (1972).

4. A. Ashkin, Phys. Rev. Lett. 24_, 156 (1970).

5. A. Ashkin, Phys. Rev. Lett. 25, 1321 (1970).

6. A. Ashkin, Phys. Rev. Lett. 40, 729 (1978); A. Ashkin and J. P. Gordon,

Opt. Lett. 4, 161 (1979).

7. V. Letokhov, V. Minogin, and B. Pavlik, Soviet Physics J.E.T.P. 45_, 698

(1977); V. Letokhov and V. Minogin, J. Opt. Soc. Am. 69, 413 (1979).

8. J.E. Bjorkholm et al., Phys. Rev. Lett. 41_, 1361 (1978).

9. S.V. Andreev et al., J.E.T.P. Lett. 34, 442 (1981).

10. W. Phillips and H. Metcalf, Phys. Rev. Lett. 48, 596 (1982); J. Prodan,

W. Phillips, and H. Metcalf, Phys. Rev. Lett. 4J9, 1149 (1982).

11. V. Letokhov and V. Minogin, Physics Repts. 73» 3 (1981). This extensive

review contains very many excellent references.

12. J. P. Gordon and A. Ashkin, Phys. Rev. A21, 1606 (1980); also see Ref's. 4 and 6.

66
13. M. Mittleman et al., Phys. Rev. A16, 583 (1977).

14. W.H. Wing, Phys. Rev. Lett. 45, 631 (1980).

15. V. Minogin and J. Javanainen, Opt. Commun. 43_, 119 (1982).

16. R.J. Cook and R.K. Hill, Opt. Commun. 42, 258 (1982); also see several

other papers by Cook referenced therein.

17. T. Breeden and H. Metcalf, Phys. Rev. Lett. 47_, 1726 (1981).

18. W. Stern and 0. Gerlach, Z. Phys. 110 (1921).

19. H. Friedberg and W. Paul, Naturwiss. 38, 159 (1951).

20. A. Lemonick, F. Pipkin, and D.R. Hamilton, Rev. Sci. Inst. 26^, 1112

(1955); R. Christensen and D.R. Hamilton, Rev. Sci. Inst. 30, 356 (1959).

21. K.J. Kugler, W. Paul, and U. Trinks, Phys. Lett. 72B, 422 (1978).

22. In our experiments v

q

= 10 m/s, n = 3x10 , X = 590 nm, M = 23 a.m.u.,

and B = 0.1 Tesla.

23. For our experiments there are two other considerations. First, this

calculation assumes that the atoms have been completely decelerated

to vQ //a . Although we haven't observed these yet, we believe this

has been accomplished. Second, the bias field may adequately compensate for any residual velocity.

24. Frequency and Time Systems, Danvers, Massachusetts.

25. N.F. Ramsey, Molecular Beams , Oxford Univ. Press, 1956.

26. Of course we would have to be careful of the effect of the 'mirror' on the lens and vice versa. This scheme may only work in a limited

region of space and velocities.

27. I.M. Matora, Sov. J. Nucl. Phys. 16, 349 (1973).

28. See illustration in R. Golub et al., Scientific American 240,

134, June 1979.

29. V.I. Luschikov, Physics Today, 30, 42, June 1977.

30. K.-J. Kugler, W. Paul, and U. Trinks, Phys. Lett. 72B, 422 (1978).

31. N. Niehues, Thesis, Univ. Bonn Physikalishes Institut, A Study of

Loading Sodium Atoms Into a Magnetic Bottle (German)

.

32. Because the transverse velocity component of the diverging beam

is not reduced in the deceleration, a beam originally collimated

to 0.01 at thermal velocity will diverge at an angle 0.3 when

decelerated to vo /30. Focussing the laser light helps reduce this

effect.

67
Laser-Cooled and Trapped Atoms , (Proceedings of the Workshop on Spectroscopic

Applications of Slow Atomic Beams, held at NBS, Gaithersburg, MD, 14-15 April

1983) Ed. by W. D. Phillips, Natl. Bur. Stand. (U. S.) Spec. Publ . 653 (1983).

The Motion of Neutral Atoms in a Radiative Trap

J. P. Gordon

Bell Laboratories

Holmdel, New Jersey 07733

A number of proposals have been made in recent years for the cooling and/or trapping of neutral atoms in a near-resonance optical radiation field [1-6]. As we have heard in this workshop,

substantial slowing and cooling of an atomic beam has already been achieved. Making a trap is a more difficult problem; so far the only neutral atom traps are on paper. The possibility of success of

such a trap depends on two factors; first, the creation of the necessary force field, and second, the

ability to cool the atoms sufficiently that they stay contained. Our overall conclusions [7] are that a harmonic oscillator model atom can be trapped successfully, whereas a two-level atom cannot. In the case of a real atom such as sodium, a successful trap must involve at least three levels and three

radiation fields of different frequencies.

Let / be the ponderomotive force exerted by the radiation field on an atom. We assume that the

trapping radiation field is stationary, and the the atom-field interaction is electric dipole. The force

field for the trap is the mean value of / considering the atom to be motionless (note that the atom

can have appreciable momentum without much motion if it is heavy enough). The first order

dependence of J on the atom's velocity provides the damping mechanism, while the quantum

fluctuations of / add an inescapable heating mechanism.

If the_electricJicld of the radiation and the electric dipole of the atom are expressed respectively as E + E

f and n + //, then the jth Cartesian component of the force on the atom is

fj-n f

- (BE/dxj) + m/Bxj)u (1)

Here E and \i are the "positive frequency" (a exp(-i&>t» parts of the field and dipole; E and n

7

are their Hermetian conjugates. Classically (1) is the Lorentz force (including both the electric and

magnetic components). Quantum mechanically it is the Heisenberg equation for the time derivative

of the momentum if one uses the -n - E form of the interaction Hamiltonian. We have also used

the rotating wave approximation.

Consider first the trapping force field. For this we need the mean value of (1) for a stational atom.

If the trapping field incoherent and monochromatic, its mean is a classical field E. If we express

the mean dipole by <p> * (x + ix )£* then the mean of (1) becomes

68
<fj> - (x-*x")£ * • 9E/Bxj + complex conjugate (2)

If we now define the real vectors a and $ by

E

* • d£/dx,. - (a, +10,) |£ | 2 (3)

then (2) becomes

</> -2(X a + x>)|£|

2 (4)

The two terms of (4) have been called the dipole force and the scattering force. The real part of (3) is

2a\E\2 -gradi\E\2

) (5)

The dipole force can evidently be used to trap atoms at either a field intensity maximum or minimum, depending on the sign of x- In some common cases the dipole force trap can be

described in terms of a trapping potential (J , such that the dipole is the negative gradient of \J . In particular, if x saturates with increasing field strength according to x(£)TMx(0)/(l+p), where

p— |Ep/£f2

, then the trap potential is

U —x'(0)£

f

2 /n(l+p) (6)

69
Here %(0) is the linear susceptibility of the atom at low field strength, Es is the saturation field strength, and p is the saturation parameter. Deeper traps can be made by going farther off resonance, so increasing x (0)£,

2

.

From the imaginary part of (3) and the wave equation V

2£+*2£-0 » one can easily show that

divifi |£|

2)-0 (7)

It is therefore clear that the scattering force alone cannot be used to form a trap, since this force cannot point inward everywhere on a closed surface. For a simple harmonic oscillator or a two level atom the mean force can be expressed as

<f> - h<n>[-2ila+T0) (8)

Here ft is the detuning w<i>0 of the field frequency from the atomic resonance, T is the radiative

decay rate of the atom, and <n> is either the mean excitation of the oscillator in units of ha>0

, or the mean upper level population of the atom. The difference between the two is that <n> for the

atom is p/2(l + p) and so cannot be larger than 1/2. The dipole force potential in the two cases is JJ-hft<n> for the SHO or (hft/2)ln(l+p) for the two level atom.^The next problem is

damping. This is determined by the first order velocity dependence of <f>. If we ignore the

polarization anisotropy caused by the atom's motion, then the relation

E

* dE/dt-E * (-i«£+(vV)£)-[-i«+v(a+ij8)]|£|2 (9)

(where v is the atom's velocity, and we have used (3)) jmplies that the motion of the atom results

in an_ effective change in frequency detuning to ft-v/3, and an effective change in damping to r+2v a. These changes alter the atom's susceptibility and so alter </>. Because the trapping

field must be tuned far from resonance to increase the trap depth, it follows that the change in

70
<f> due to these perturbations is quite small, and in fact is too small to be useful. The

implication is that one must use a separate radiation field tuned closer to resonance for damping,

but here the SHO and the two level atom react quite differently. In the SHO case, the

susceptibility x remains linear, and the mean forces due to two separate beams of rather different frequencies are independent and simply add together. In the case of the two level atom, the

dynamic level shifts caused by the intense trapping beam make it impossible to keep the damping

beam close to resonance throughout the trap, and its effectiveness is lost. It is fruitful to evaluate

the damping for the SHO case, however. _If the damping beam is effectively a plane wave in the

region of the trap, then it has a— 0, and its force is therefore <fd ><-T<nd >hk, where

<nd > is the oscillator excitation due to the damping beam alone. _Since <nd > a [£l d+ (T/2)

2

]-1

,

the first order velocity change in <nd > is B<nd >/<nd >"2v kCl d / [ftJ+(T/2)

2

], so if fij--

172 (for maximum effect), then

« <fd >-iB<nd >/<nd >]<fd >—2<nd >h(vk)k (10)

This velocity dependent force properly opposes the velocity component along the direction of the

damping beam, and as we shall see it is large enough to keep the atoms in the trap cold. Three

separate damping beams could be used to provide damping for all velocity components.

The final and most difficult problem is finding the momentum diffusion constant. The contribution

from spontaneous emission recoil is relatively easy to understand, but the contribution from induced

emission is more complicated. Finding the momentum diffusion constant involves the calculation of

the two time autocorrelation function </(0 /(0) > . This can be found exactly [7,8], if velocity

effects are ignored, by methods similar to those used to find the spectrum of resonance

fluorescence[9,10]. In fact these two problems are interrelated. For the SHO the result is (here P

is the momentum)

2D. - —[<P2 >-<P>2] - h

2T<n>(k2+a+0l

) (11)

The k term can alternatively be regarded as the cumulative effect of spontaneous emission recoils, or as the effect of the mean dipole interacting with the zero-point fluctuations of the field gradient.

The a+ff1 term can be regarded as due to fluctuations of induced absorption and emission

processes, or as the effect of the zero-point fluctuations of the dipole interacting with the mean field gradient. For a two level atom the k

2 term is unaltered, but the a and p terms become more

71
complicated functions of the saturation parameter, and an a fi term appears. There is one

particularly important term in the two level atom diffusion constant which effectively limits the trap

to small values of the saturation parameter. It is an a term of the form

^a2p

4

(4fl

2+r

2)

2r(i+p)3

It is the only term that does not saturate with increasing field intensity. It is also the only term

which increases with the detuning fl at fixed p. For high saturation it can be explained nicely as the result of the random walk the atom undergoes via spontaneous emission relaxation into one or the other of its two quasi-stationary "dressed states". One can see by comparison with (12), which is otherwise applicable for small <n>TMp/2(l+p), that this term can become dominant even for small values of p because ifi/Tj> > 1 for the trapping beam.

Finally we will show that the SHO can be trapped and cooled to a very low temperature, and

comment on the way this might be accomplished for a real atom. The mean translational energy W

of the atom in the trap is increased by momentum diffusion and decreased by the average damping b according to

dW/dt - DplM - hW (13)

where M is the mass of the atom. Jn equilibrium then W — Dp /Mh. For the SHO, according to

(11) we obtain 5W-, 2<nrf >h(v it) 2. Roughly then because the damping works only on one

velocity component, and the kinetic energy is about half of the total energy, we obtain the relation -vM = <nrf >h&2/3 and therefore

fF

efl =K3Ar<nl >/<n4,> (14)

where we have assumed that for the trapping beam a+fP^k , and <nt > and <n4 > are

respectively the mean oscillator excitations by the trapping beam and damping beam as though they

acted alone. There seems to be no obstacle in this case to cooling to an energy of the order of hT or

72
less, corresponding to a temperature of M7k, or about one millidegree K if FTM 10 sec .

For a real atom; e.g. sodium, we have calculated [7] that even without an effective damping

mechanism the atom can remain in the trap for a time of the order of 1 sec. But of course it would

be preferable to have a stable trap. For this one must make the atom look more like an SHO. In

particular, by using a double resonance, with three levels and two superimposed trapping beams, one

can envisage canceling out the dynamic "light shift" of the lower resonance so that a damping beam

can stay near resonance with it. I see no reason why such a scheme should not work in principle,

although it yet lacks proof.

References

[1] G. A. Askar'yan, Zh. Eksp. Teor. Fiz. 42. 1567 (1962) [Sov. Phys. JETP 15. 1088

(1962)].

121 A. Ashkin Phys. Rev. Lett. 24, 156; 25. 1321 (1970); 40, 729 (1978).

[3] T. W. Hansen and A. L. Schawlow, Opt. Commun. 13, 68 (1975).

[4] D. J. Wineland and H. Dehmelt, Bull. Am. Phys. Soc. 20. 637 (1975).

15] V. S. Letokhov, V. G. Mingolin, and B. D. Pavlik, Zh. Eksp. Teor. Fiz 72. 1328 (1977)

[Sov. Phys. JETP 45, 698 (1977); V. S. Letokhov and V. G. Mingolin, Appl. Phys. 17. 99 (1978).

[6] A. Ashkin and J. P. Gordon, Opt. Lett. 4. 161 (1979).

[7] J. P. Gordon and A. Ashkin, Phys. Rev. A21, 1606 (1980). The arguments above are

essentially a resume of this paper, which gives the complete expression for the diffusion constant of the two level atom.

[8] R. J. Cook, Phys. Rev. A22 1078 (1980).

[9] M. Lax, Phys. Rev. 129, 2343 (1963).

[10] B. R. MoUow, Phys. Rev. 188, 1969 (1969).

73
Laser-Cooled and Trapped Atoms , (Proceedings of the Workshop on Spectroscopic Applications of Slow Atomic

Beams, held at NBS, Gaithersburg, MD, 14-15 April 1983) Ed. by W. D. Phillips, Natl. Bur. Stand. (U.S.)

Spec. Publ. 653 (1983). #

Some Problems and Possibilities for Quasistatic Neutral Particle Trapping

William H. Wing

Physics Department, Optical Sciences Center, and

Arizona Research Laboratories, University of Arizona

Tucson, Arizona 85721

Some basic considerations regarding the confinement of bound neutral systems of

particles in limited regions of space by slowly varying electromagnetic fields are

reviewed. Electrostatic traps are described, spectroscopic linewidth is discussed, and

several specific cases are described briefly. Use of a light field to lengthen the

trapping time of decaying states is proposed. A universal refrigeration system utilizing

trapped ions is presented, and its utility for cooling neutrals as well as other charged

particles is discussed.

Keywords: atomic hydrogen, cooling, electric and magnetic fields, high-resolution

spectroscopy, neutral atoms, polar molecules, positronium, Rydberg atoms, trapping.

1. Introduction

This paper will present and elaborate on several ideas regarding the trapping and confinement of

small neutral objects by slowly varying electromagnetic fields, with the particular aim of producing

circumstances amenable to high resolution spectroscopy: the approximation of Professor Dehmelt's

"single atomic particle at rest in space" and isolated from disturbances. In keeping with the nature of

this forum and with the fact that the subject has enjoyed only limited experimental success up to the

present, no attempt at completeness will be made, some of the ideas will be both prospective and

speculative.

I shall first describe the general principles quasistatic traps employ and discuss the questions

of stability and of spectroscopic resolution. Much of this discussion will be applicable to static

magnetic field traps as well. Next I shall discuss the specific cases of trapping bulk dielectrics,

Rydberg atoms, metastable atoms such as atomic hydrogen and positronium in the 2S state, and polar

molecules. Following that, I shall discuss the idea of lengthening trapping times by coherent

reexcitation of the trapping internal state and shall present a possible universal means of

refrigerating trapped particles by collisions with laser-cooled trapped ions or conceivably other

neutrals. A conclusion which may be drawn is that neutral particle traps seem unlikely to replace

ion traps as high precision atomic clocks unless a neutral atomic system is found which possesses

overwhelming advantages for this purpose compared to trappable atomic or molecular ions.

The effects of gravity on particle traps are discussed in a brief separate note elsewhere in this

volume [1].

2. Principles

2.1. The Quasistatic Regime

In quasistatic traps the characteristic frequencies of field oscillation and of particle motion are much smaller than the internal resonance frequencies of the trapped particles. Consequently,

transitions between stationary states are unlikely, except for Majorana transitions which may be

induced if an accidental degeneracy or near-degeneracy occurs at some point in the particle's orbit. By

contrast, in the high-frequency or "optical" traps discussed elsewhere in this volume, the field

oscillates at frequencies close to the particle's resonance frequencies and transitions are frequent.

Direct momentum transfer between particle and field (as contrasted with momentum transfer between

74
particle and trap structure mediated by the field) is negligible in the quasistatic regime. This

eliminates the radiative recoil method of applying trapping force to the particle and damping (cooling)

its motion. At the same time, it eliminates the momentum fluctuations inherent in that force.

Therefore the minimum temperature of kinetic motion should be less in a quasistatic trap than in an optical trap, provided that a powerful non-optical method of cooling can be found. Another point of

contrast between quasistatic and optical traps is the field strength that can be applied: It is much

higher in the quasistatic case, if equipment of modest cost and complexity is to be used; and in

addition, the high fields achievable at the focus of powerful laser beams are likely to induce

ionization or dissociation of the particle to be trapped. Thus the potential wells of some proposed

quasistatic traps [2-4] are deeper than those of optical traps proposed thus far [5,6]. The differing

characteristics, advantages, and disadvantages of the two types suggest that both should be

investigated. Ultimately, mixed-regime traps using, for example, quasistatic fields for trapping and

optical fields for cooling, as is done in ion traps, may prove to be the best choice.

2.2. Basic Ideas and Constraints

2.2.1. Interaction potential energy

The interaction energy U of a stationary charge or current distribution with an electric or magnetic field can be expanded in multiple moments:

3E (R) 0(H) = e<J)(1) - p.1(H) -1/6 I -T3— + «. , (1)

where, of course, e=0 for a current distribution. H is the coordinate of the particle and E* denotes,

for the present, either an electric or magnetic field, as appropriate. The dipole moment can further

be expanded as

p

= h

+ "iV^ = *i + I

p2^ ' (2)

the second expression applying when the particle is spherically symmetric and the field is not too

strong. In the electric case, the orders of magnitude of the trapping energies given by the three

terms in Eq. (1) are

monopole: e<t>Q

dipole: e<|> P (a /r ) (3) o o o o

2 quadrupole: ed> Q (a /r ) , o o o o

'

where e is now the electron charge, <j> 0 is the range of variation of <|>(R), aQ

is the Bohr radius, rQ

is

the range of motion of the trapped particle, and P

Q and QQ are the dipole and quadrupole moments in

atomic units. Since ao=5x10"9 cm and rQ

is at least 10~2 to 10"3 cm, and since PQ and QQ are of order

unity for atomic-scale particles, the successive interactions decrease rapidly in size, and the first one not identically zero will dominate. Usually the permanent dipole moment, if not identically zero, will also dominate the induced moment. Ion traps can be rather strong on the atomic scale (a few eV

in depth), whereas neutral dipole-moment traps are rather weak (a few yeV to a few meV in depth).

Neutral quadrupole-moment traps are so weak that none has, to my knowledge, been considered seriously.

It is instructive to compare the depths of magnetic and electric dipole traps. Writing an atomic

magnetic moment u as (en/2mc) MQ where m, c, and n are the electron mass, speed of light, and angular

Planck's constant, and M

Q

is the moment in atomic units, we find

*No such scheme appears to exist at the present.

*0r at most 10 2 to 10

1*, in Rydberg states.

75
yB

pE

(eti/2mc) ea 2P,

where a = e

2 /nc = 1/137 and a

Q

= ftVmca. Relation (4) suggests that when the magnetic and electric

moments are comparable, for example, an atom bearing an unpaired electron or a polar molecule, the

magnetic interaction energy should be about 10 2 times smaller than the electric. This would be true

except for the fact that readily achievable laboratory magnetic fields are somewhat over 10 2 times

larger than readily achievable laboratory electric fields (=140,000 G vs. 500 sV/cm).* Thus, in fact,

the maximum magnetic and electric interactions are about equal in this example. In Rydberg states, PQ

= n

2 whereas MQ

~ g^l £ n. Thus the electric interaction grows with increasing n more rapidly than

the magnetic as long as Emax and Bmax are both limited by field technology. Ultimately, however, an n

is reached at which a level crossing occurs before the technological field limit is encountered. Now

the maximum interaction energy is approximately a

2mc 2 /2n3, half the distance between Rydberg levels

[2]. The limit applies in both the electric and magnetic cases, albeit at different n. Figure 1 is a sketch of the maximum interaction strength Wmax versus n based on these qualitative considerations.

1000

500 1000

n

2.2.2. Static stability

The starting point in discussing the stability of a particle in a quasistatic electromagnetic field

is Earnshaw's Theorem, which states:

A charged body placed in an electric field of force cannot rest in stable equilibrium

under the influence of the electric forces alone [7].

The proof follows immediately from Laplace's Equation

2 2 2 3x 9y 3z

Considering the complex nature of superconductors and of dielectric breakdown phenomena, this fact

may be coincidental, but it is interesting to speculate that its origin is the existence of a mechanismindependent limit on the interaction energy of a material's valence electrons with the field it can stably support, which, again considering Eq. (4), would yield B_ ax/Emax

= 1/a.

76
and the relation U = e<|>, which show that in at least one direction of excursion away from a suspected

minimum in U, U cannot increase, so that the equilibrium is unstable or at best neutral. Any

combination of potential energies mathematically similar to <j> will obey the same rule, so that, in

particular, no combination of electric charges, permanent magnetic multipoles (regarded as assemblages

of magnetic charges) and masses located in a region of constant electric, magnetic, and gravitational

fields can rest in stable equilibrium, since each field is derivable from a potential obeying Laplace's

equation.

The observed stability of ordinary matter despite the above . argument "suggests, with a probability

approximating to certainty, that the electrons in an atom or molecule must be in rapid orbital motion.

Thus the problem of the structure of the molecule is removed from the province of Earnshaw's Theorem,"

as Sir James Jeans put it so elegantly [7]. Nonetheless, if we wish to approach the ideal of a particle

at rest in an electromagnetic trap, we must try not to invade Earnshaw's territory in a serious way.

Thus some form of motion or time variation of the trapping force must contribute significantly to the

particle's positional stability.

In the simplest case - that the fields are constant in time and the particle is stationary - we

must employ the latter 's internal motion. The dipole energy term in Eq. (1), when incorporated into the

particle's Hamiltonian, yields field-dependent energy levels W which will also be pg^sitio^i-dep^endent,

and hence will lead to external forces, when the field is nonuniform. The force F = -VRWa

(R) on a particle in state a will push it toward the region of lowest W. For ground -state particles, increasing

the field strength will always reduce the energy. For a permanent dipole p"-| this follows from the

first-order perturbation energy

M E «, --4,-h.- <?,.}> (6)

and the fact that Mj ranges from -J to J. Here nJ is the particle's total angular momentum operator.

For an induced dipole ^(E) the same result follows from the second-order perturbation expression

<5W.

Lt T.I W -W, _TJ p 2

o i

(7)

where since state o is the ground state, all ^-W^ are negative.

Now, unfortunately for static trapping, the following theorem regarding field strength maxima and

minima is also true:

In a region devoid of charges and currents, the strength of a quasistatic electric or magnetic field can have local minima but not local maxima.

The theorem holds for quasistatic gravitational fields as well. The proof, which requires a few

lines of vector algebra, is given in Appendix A. Consequently, particles attracted toward strong field

regions, such as those in their ground states, will eventually encounter the source of the field.

Although they might orbiljj about it if given sufficient kinetic energy, they will ultimately stick to it

if cooled toward stasis. Places isolated from the field source can exist, however, at which field

strength minima occur. The regions of minima can be isolated points, lines, surfaces, and conceivably

volumes, and the minimum field strength can be either zero [2] or nonzero [3]. Thus stable static

trapping is possible only in excited states, which is in a sense self-contradictory, since excited

states eventually decay; one might call this "Earnshaw's Revenge." Nonetheless, certain situations can

be found in which the excited-state lifetime is interestingly long. Electrostatic examples include the

upper inversion level of a polar molecule such as NHg, the metastable 2

2Sf/2 state of the H atom, the

metastable 2^s -j state of the Ps (positronium) atom, and certain upper Rydberg levels of any atom.

Some particulars of hypothetical trapping experiments on these substances are discussed later in this

article.

*Since the above theorem does not apply to light fields, points of stability are possible at light field

intensity maxima devoid of sources, e.g., on the symmetry axis of a Gaussian laser beam.
Magnetic induced-moment trapping should also be possible. An example might be the ground state

of the H atom, whose Zeeman diagram is shown in Fig. 2. Here a difficulty arises, however, which seems

to occur inevitably with particles containing permanent magnetic dipoles: at B=0 a degeneracy occurs between a trapping state, such as |F=1, Mp=l> or |F=1, Mp= o>, and an antitrapping state, such as |F=1,

Mp=-1>. This seems to be unavoidable since the magnetic moment is an intrinsic particle property:

Thus the field interaction energy is orientation-dependent, and the loss of a preferred quantization

axis at B=0 will inevitably lead to degeneracy. The likelihood of Majorana transitions at B=0 then

arises, the cure for which will be either to make B^O at the trap center or to erect a centrifugal

barrier (hence add motion) to keep the particle away from the trap origin. The latter will prevent

cooling of the motion to the ultimate extent, at which the particle's motional wave function would be a

E state, with nonzero amplitude at the origin; the former introduces a perturbation.

Figure 2. Hyperfine states

of H(1 2

S-|/2^ a magnetic

field.

Given that we are willing to add a nonzero field at the trap center, we should not neglect the

following possibility. Avoided crossings between energy levels may occur at fields other than zero; this happens, for example, between states of the same nonzero Mp when the Breit-Rabi energy level

diagram includes F>1, which occurs in alkali atom ground states when the nuclear spin I is >1/2. These

avoided crossings, unlike the one between the Mp=o levels in Fig. 2, are usually nondegenerate, and so the upper one will provide a (meta)stable potential well in a magnetic field.

A polar molecule need not have a Stark level degeneracy at E=0, because its permanent electric

dipole moment is not an intrinsic particle property but rather a structural property. For example, in

NHg the inversion oscillation splits orientation states which would otherwise be degenerate in the

absence of an external field.

2.2.3. Dynamical traps

a. Stability induced by motion . Many more possibilities for stability arise when motion or timevarying fields are introduced. Motion, for example, can permit stable trapping near field maxima if the

resulting centrifugal effective potential barrier prevents collisions with the source of the field, as

is the case for a charged particle orbiting a charged wire (Kingdon trap). However, if damping

mechanisms are present, particles trapped this way will ultimately be lost, and if our goal is to cool

the particle motion as much as possible, we must look elsewhere for stability. Ultimately, we would

like to reduce the kinetic energy to a value approaching the quantum-mechanical minimum, as in an atomic ground state. However, in a trap of dimensions much larger than atomic, this minimum kinetic

78
energy will be so small that if it alone provides stability, the system will likely be extremely

susceptible to upset by external perturbations, such as vibration, stray background gas collisions, or

agitation by thermal radiation, which may, for example, induce Majorana transitions to an antitrapping

state. Thus, although ingenious exceptions to this rule may well be found, it seems more profitable to seek trapping situations that exhibit stability in the classical regime, where energies are large and

forces are strong, and then ascertain that the stability persists at the quantum limit. Accordingly, I will not consider trapping near static-field-strength maxima further in this paper.

b. Stability induced by time-varying potential energy. I will assume at the outset that the

potential energy is quadratic in the coordinates and that its time variation is sinusoidal. Although

this is by no means the general case, it lends itself to analytic treatment, and the analytic results

exhibit qualitative features which no doubt will persist in at least some of the other useful

arrangements. Furthermore, many more complicated potentials and time dependences reduce to the above

case when the amplitude of motion is small enough.

We may usefully define an intermediate regime of operation, the "Goldilocks Regime" [2]: the

motion is neither too hot nor too cold. At lower temperatures quantum effects in the motion become

important; at higher temperatures nonlinearities set in and, ultimately, classical chaos may result.

The Goldilocks Regime, however, is nearly classical and is susceptible to analytic treatment. The

motion of each particle coordinate obeys Mathieu's differential equation (see Appendix B and Section 4). For stable trapping, the stability regions of each of the coordinate equations must overlap; this

generally leads to a lower limit on the field oscillation frequency.

2.3. Spectroscopic Linewidth

If the particle is in a well-defined quantum state of trap motion in both its initial and final

internal states, the absorption spectrum of the trap + particle system will consist of a set of

extremely sharp lines, which can be resolved using a spectrometer of sufficiently high resolution, such

as an extremely high-resolution tunable laser. However, if the resolution is not high enough, or if the

coherence time of a trap state is shorter than the reciprocal frequency difference between states, the

individual trap resonances will be obscured and the spectral line will be broadened considerably.

Defining a "bound" state as one in which the particle and trap momenta are coupled and a "free" state

as one in which the momenta are decoupled, we may consider two limiting cases:

(1) initial state free, final state free or bound, uniform external potential, and

(2) both states free, different nonuniform external potentials.

2.3.1. Uniform external potential

I will treat only the energy and momentum conservation aspects, but will do so r^lativistically.

In the initial state we have a free particle (m,p*) of internal energy 0, a trap (M,P), and several

photons l^. In the final state the particle and trap have associated to form a state of internal

energy WQ and the photons have all been devoured. We may take the potential energy to be zero. Conservation rules state:

(momentum) Enk,^ + p + P* =

p"' (8a)

2 2 2 U 1/2 p p p 11 1/2 p p ? ? U 1/2

(energy) Zhv± + (p c +m c ) +(P o +M e ) =(P c +(m+M+W0/c ) c ) (8b)

To make the situation as favorable as possible, let us have the field, take care of its own momentum

conservation, rather than giving some away to the masses, by se_t£ing Eki=C^ for Doppler-free multiphoton

absorption. Let us also simplify the arithmetic by setting P=0. Then P'=p, and when we expand both

sides of Eq. (8b) to second order and solve for EhV;, we find

If the source of the field is small enough and microscopically smooth and if the potential energy

increases more slowly than 1/R2

, a stable lowest motional quantum state may, in principle, exist near

it [8].

79
2 2 Div = (M+m+W /c

2

)c 2 +

E + 0(p

U

) - me

2 - ^ + 0(p4

) - Mc

2

1 ° 2(M+m+W0/c2

) 2m

2 r W + vo 2

1 M+m+W /c

2 m

o

(9)

2 For atomic systems, W /c <<m. For a free-free transition, M=0 and Eq. (9) reduces to

o

Zhv = W

o[1-(1/2)(p/mc)

2

] , (10)

exhibiting the "second-order Doppler shift," whose origin can be seen from this perspective to be the

mass-energy shift accompanying the change of internal state. For a free-bound trap transition, M>>m

and Eq. (9) reduces to

Ehv

i

= W

q

- p

2 /2m . (11)

The p-dependent term is larger by the factor mc2 /W0

in the free-bound case than in the free-free case.

If p is statistically distributed, the free-bound case will yield much broader lines than the freefree case and thus will be unsuitable for high-resolution spectroscopy. The range of p

2 which will

yield broadening in Eq. (11) will be the lesser of its thermal spread and the mean-square momentum

distribution in the bound state; thus a free-bound transition may offer a means of determining the

latter.

2.3.2. Different nonuniform potentials

I will treat this case nonrelativistically, assuming for simplicity that we have used the condition

£^=5 to eliminate first-order Doppler broadening. The particle is considered "free" in both the

initial and the final states. If the transition occurs over a time At and the particle speed is v, the

inhomogeneous frequency spread due to variation of the transition frequency (W2

-Wi)/fi = [WQ

+ U2

(R) - cyi)]/* is

(5w)inhom = 'H[U2

(1l) " V*)] At/*' (12)

and the uncertainty-principle spread is (6u))hom r 1/At. If we assume these add in quadrature, we find

the total spread to be

Aw =((1/At)

2 + [v'vXuyiyAt/ft]2

}

172

. (13)

Minimizing with respect to At yields the optimum interaction time

5 opt

(At) = Iri/v^OJg-uyl

172

0*0

and minimum transition width

1/2

(Am) . = /2/(At) . = l.4l|v»%U_-U 1 )/A • (15)

min opt 1 2 1

The value of (At)opt can be set by, for example, adjusting the width of a laser beam illuminating the

portion of the trap on which spectroscopy is being done.

80
Evidently it is desirable to do the trap spectroscopy near an extremum of ^"^i* (Aco) mjjj is not

zero here, however. Expanded to second order, the inhomogeneous frequency spread now becomes

(&>) inhom

3x

(16)

where the x direction is the direction of v. Combining with (5w)hom as bef>ore and minimizing, we find

(At)

Qpt

= 2(1/2)

1/6 In/^'-U^Vf

and

(Aa»m

. n

= 1.54/(At)o= 0.87|(U2 "-upv2

/*|

1/3

.

(17)

(18)

These formulae become more transparent physically when applied to a one-dimensional harmonic trap

where

1 2 2

1,2 2 1,2

(19)

Then the linewidth expressions, Eqs. (15) and (18), become

(in general) (6w) . mm

(at extrema) (&>) . = 0.87|Mv2(ft 2-fi 2)/ri| min 1 d \ 1

1/2

(15')

(18')

Using the relation x=v/fl for

2a "typical" value of x in Eq. (15'), averaging inside the absolute value

brackets, and substituting <Mv >=n&Kn>>1), we find

I d 2l (in general) (6a>) . = 1.42 niS2 -n.ft J mm d d 1 I (at extrema) (6u) . = 0.87|n.nLn,fi^| min ' 2 2 11'

2,1/2

(15")

(18")

where n-| 2 are the approximate quantum states of the trap motion. Since the Franck-Condon principle

will be applicable, trap motional energy will be approximately conserved:

n^1 ~ n^l

2 = n n ~ kT/n . (20)

If we take and Q2 quite different, say ft 2

= (3/2)ft and ^-, = (1/2)!^, we finally obtain

1/2

(in general) (&o) . = n

1/2 B = (J^kT/n) mm

1/3

(at extrema) (&)) . = n

1/3 Q = (J2 2kT/n) min

(15"')

(18"')

2.4. Should One Do High-Resolution Neutral-Particle Spectroscopy in a Trap?

Let us note first of all that the inhomogeneous broadening discussed in the previous section is

absent in ion traps, since for monopole interactions fi 2 always equals &<|. We could try to achieve this

condition in special cases of neutral trapping as well. When we cannot, it will be useful to compare

the trap inhomogeneity broadening with the second-order Doppler broadening in a free-particle

spectroscopic experiment at the same kinetic temperature. The second-order Doppler broadening becomes,

81
from Eq. (10) with 6(p 2 /2m)=kT,

kT

(&>)_. = ~ u> . (21) dl 2 o mc

For a hydrogen atom at 300 K, using kT/h=6.2x10 12 Hz, Mc 2 /h=2.2x10 23 Hz, and wo (1S-2S)=2.2x10 15 Hz, we

find

(5u)) OJ ~ 6x10

4 Hz (22)

in the same conditions, with a trap of radius r=1 cm, so that f2=v/r=(105 cm/s)/(1 cm)=10^ rad/srl^xlO 1

* Hz, Eqs. (15"') and (18"') yield inhomogeneous broadenings of at least

g

(in general) (&o) . = 3x10 Hz

7 (23)

(at minima) (6w) . = 1x10 Hz. mm

Thus H-atom spectroscopy can be done much better in a beam than in a trap!

How can we bias the situation in the trap's favor? Lowering the temperature would seem to make

its case comparatively worse rather than better, since the Doppler broadening will fall off more

quickly than will either inhomogeneous broadening. Reducing Q will eventually eliminate the trap

altogether. The only reasonable approach will be to raise Q until coherent interaction of light with

particle can be maintained throughout an entire orbit or, what amounts to the same thing, until we can resolve the inhomogeneously broadened spectral line into discrete transitions between trap + particle

quantum states.

This can certainly be_done by setting f^-kT/n so that only the lowest one or two quantum states

are occupied. For lower ft, as the motional quantum_number n=kT/nft grows, the spectrum will become

dense and ultimately confused._ The limiting value of ft will be reasonable only if kT is extremely low;

for example, if T^10"3k an(j n *100, ft can be =2x1

0

5 Hz. Now advantage could be made of the good

features a trap offers, such as elimination of residual first-order Doppler broadening due to laser

beam misalignment.

3. Electrostatic Trapping Examples

3.1. Bulk Dielectrics

It is often shown in elementary physics demonstrations that a dielectric slab is always drawn

into a parallel-plate capacitor [Fig. 3(a)]. From the atomic point of view, the explanation is that the

slab consists of ground-state atoms, whose Stark shifts are inevitably negative, and which therefore

are falling downhill into the region of strongest electric field. The polarizability of these atoms is

positive, which results in a bulk dielectric constant e>1. However, as we have seen, the neutral point at the center of the capacitor cannot be a field strength extremum and so cannot give stable

equilibrium. We might therefore wish to make a dielectric out of excited atoms having positive Stark

energies, and e<1, so that it would be expelled from the capacitor and could be suspended at a field

strength minimum [Fig. 3(b)], Such a dielectric would not last long; however, we can find an equivalent more permanent arrangement by noting that the energetics of such an object would be similar to those

of an object of dielectric constant e immersed in another medium of e'>e. A nice realization is a small air bubble floating in oil [Fig. 3(c)].

An electrostatic trap for air bubbles in oil is shown in Fig. 4. The quadrupole end caps are made

from two tiny (#0-80) stainless-steel machine screw heads, and the ring electrode is a small (#1-72)

stainless-steel machine washer. The end caps are connected electrically. The assembly is installed in

a glass envelope which is filled with transparent high-voltage transformer oil, with an airspace left,

and sealed off. To excite the "atoms" in the trap, the device is shaken vigorously. A potential of

about 5 kV is then applied between the end caps and the ring, and a bubble is almost always trapped at the center (one can be seen in the photograph). Once charged, the trap can be disconnected from the

82
a) n = 1 atoms Us <0

a >0

Drawn in

Figure 3. Dielectric

particles in electric fields

(see text).

b) n > 1 atoms Us >0

a <0

Expelled

C) € = 1

voltage source and put on an overhead projector for demonstration before an audience. A small highvoltage capacitor is usually clipped across the trap leads to increase the RC leak-down time of the

trap charge, and a short-focal-length lens is placed on top to compensate for the focusing effect of

the round oil column.

Figure 4. Electrostatic

quadrupole trap containing

an air bubble in oil.

If the oil viscosity is low enough, a bubble once set in motion will vibrate back and forth across

the trap center a few times before damping to rest. This is sufficient to demonstrate the inherent

anisotropy in the trap's restoring force: since the electrodes have roughly cylindrical symmetry, the

quadrupole potential is of the form

p (x +y -2z^)/2r

o o

(24)

which satisfies V

2 =0. The bubble's potential energy is

83
12 1 ± 2 a9o 2 2 2 2 U = -^ole" = ^Ct(-7<|>) = ~f (x +y +4z^)/R^ , (25)

where a(>0) is the polarizability of the oil which has been displaced. Writing this in standard

harmonic-oscillator form as U=M(^x^+^|y^+^!z ^) shows that

a :ft = 1:1:2 . (26) x y z

Here M is the effective mass of the oil the bubble carries along as it moves.

The coup de grace in this demonstration is to short the capacitor with, for example, the metallic

pocket clip of a plastic writing pen. At the crack of the spark, the bubble seems to bestir itself and

lazily drifts upward, away from the trap and out of the field of view. The possibility that the

lecturer may administer himself a healthy shock in the process (although he never does) heightens the

drama and, hence, pedagogical impact of the demonstration.

3.2. Rydberg Atoms

As shown earlier in this article, a Rydberg atom appears to exhibit the deepest trap potential

well of any system proposed to date, because its polarizability increases so rapidly with principal

quantum number n. The feasibility of trapping sodium atoms in high np states has been discussed

elsewhere [2]. To summarize briefly, it was found that a I6p Na atom could be trapped in an electrostatic well of depth 2.1x10~3 eV (17 cm" 1 ) and roughly parabolic shape (Fig. 5). The dimensions

can be expressed by stating the oscillation frequency, which then determines the restoring force

constant. Choosing fix

=2iTx10' rad/s gave a maximum amplitude of oscillation of 2.2 um, and the end cap

separation could be as large as 300 um before danger of surface breakdown arose. The limit to trapping time is the lifetime of the excited state, which, if it is predominately the zero-field I6p

state, is t=24 usee. Thus, in the absence of perturbations, x-axis trap motion would evolve through

fi xx=1500 radians, enough to form a well-defined orbit, before 1/e decay.

440

Figure 5. Some excited

states of Na (M = 0) L

=0) about

500 cm" 1 below the

ionization limit; black dots

indicate trapping states. (Adapted from M. L. Zimmerman et al., (Ref. 9). Uc and E

c correspond to wmax and Emax of FiS- 1 -

540

I 2 E

c 3 ELECTRIC FIELD (kV/cm)

To try out this idea, Dr. Craig Jensen and I built an experiment (Fig. 6) at JILA, Boulder, Colorado

in the spring of 1980. The idea was to excite Na atoms on the 2441 -A 3s-l6p transition in a trap

consisting of three aligned meshes with the outer two biased relative to the center one. A quadrupole

trap well would appear in the center of each hole in the middle mesh. Trap oscillations following the

84
laser pulse would frequency-modulate the 4167-8 fluorescence. A narrowband etalon would convert this to an amplitude modulation, which would be digitized, averaged, and displayed. An estimated 10^ to 10

fluorescent photons from Na(l6p) would be detected with each laser shot. Unfortunately, we found that even the highest-purity quartz obtainable fluoresced much more strongly than this, and although the

l6p fluorescence could just be detected, no oscillations were seen in the short time our other

commitments allowed us to spend on this idea. Nonetheless, the principle appears sound, and

accordingly the experiment seems worth describing even though a failure. A different excitation

method, and perhaps a different trap design, would enhance the chance of success.

Ex periment for Observing Trop Osci Motions

Oven

Quartz!

Mesh

"ytrap

Figure 6. Experiment to detect electrostaticallytrapped I6p Na atoms.

. Filters Etalon

Bluesensitive P.M.

Quanta -Ray

Nd! Yag+dye+mixers

Pulsed laser system

Trigger

Transient

digitizer

Trap period »t0"^ec

oiCT'sec

Electric field Time

Loser pulse

3.3. Atomic Hydrogen and Positronium in Their 2S States

Both the 2

2S 1/f2 state of atomic hydrogen and the 2^S

1 state of positronium exhibit positive Stark

energies, the former because it is repelled upward more strongly by the nearby 2

2P 1/2 state than

downward by the more distant 2

2Po/ 2 state, and the latter because it lies above all the other n=2

states. The weak-field Stark levels of H are shown in Fig. 7 and the fine structure of Ps(n=2) in Fig.

8. Since the Stark shift depends on |Mj|, atomic hydrogen has only one 2S Stark sublevel (|Mj| = 1/2),

whereas Ps will have two (|Mj|=0,1). I will discuss the H case in detail because it is simpler (if

hyperfine structure is neglected).

Figure 7. 2

2

S-|/2 and 2

2

P -j /2

states of H in an electric

field, neglecting hyperfine

structure (from Ref. 10).

I 1

S

~2

-0.9

-12

-1.8 -

_ 2

—

1 F 0.1

| 0.2 0.3 6 WV/an.

The immediate difficulty would seem to be quenching of the 2S state via coupling to 2P by the

trap electric field. However, this is not inevitably severe. Let us neglect the 2^3/2 state, and

define a dimensionless field parameter % = (n 2-1) 1/2 njn E/S, where E and S are the electric field and

Lamb shift W(nS 1/2 )-W(nP 1 / 2 ), respectively, in atomic units, n=2, and m=|Mj| = l/2. The field mixes 2S and

2P, producing a long-lived and a short-lived state. In weak fields, the energy W and decay rate y of

85
singlet S P

triplet S P

Figure 8. Fine structure of

positronium, n=2 (from Ref.

10).

the long-lived state are

1

<->0 S

3

p,

'p

w = w

2S

+ 5 s

Y = Y2S K\?

(27a)

(27b)

Numerically, Y2S = 1 Hz» Y2P ~ Hz» and S = 10$ Hz. Equation (27a) shows that the trapping potential

energy U=W-W2S is harmonic. Eliminating yields

Y = Y2S + (U/S)y

2P

(27c)

Thus, when averaged over the trap motion, y will be small for trap states having small average U. If we consider a definite motional quantum state in a quadrupole trap having quantum numbers n

x, n

y

, n z and classical oscillation frequencies 8X, 8

y, 8Z

, the eigenenergy is

W(n ,n ,n ) = Woc, + (n +1/2)lfa + (n +1/2)ftl + (n + 1/2)ii0 , x' y' z 2S x x y y z z

' (28)

and since <U>=<p 2 /2m>=(1/2)[W(nx

,ny,n z )-W2S] for a harmonic oscillator,

y(n ,n ,n ) = Yoc + ~ f(n +1/2)8 +(n +1/2)8 +(n +1/2)8 1(y„/S) .

' x y z '2S ?l x x y y z zJ '2P

(29)

We can simplify the spectrum further by using an axially symmetric trap; then 8x =8

y =8z /2 [Eq. (26)]

and the trap energies and decay rates become

W

N

= W

2S + CN+2) tox YN

= Y2S

+ (N+2)(y

2p/S)(8x

/2) ,

(30)

(3D

where N=nx+n

y+2n z

. The energy spectrum consists of a series of equally-spaced levels separated by 1t8 x that grow progressively broader (Fig. 9) and become confused at N~23. The resolution of adjacent trap

levels, if accessed spectroscopically from another bound state of the trap + particle system, is

86
v/6v 2P

(32)

Figure 9. Qualitative sketch

of the H(22S 1/2 ) state N=0 1 2 3 4 5 energy in an electrostatic

trap. vQ

is the free-atom

energy level.

The last form becomes accurate when f2x>>Y2S> a^ which the maximum resolution of 8 is achieved for the

N=0 to 1 splitting. It should be noted that if the spectrometer responds only to changes in n

z , the

resolution will be doubled.

The advantage of spectroscopy in an electrostatic trap rather than in a freely expanding beam is

that stray electric fields and their first derivatives become part of the trap field in the former

case, and therefore can be accounted for if the individual trap levels can be resolved. However, it

should be noted that if the spectroscopic line is a bound-free transition, it will be broadened by the

kinetic energy of the absorbers, as shown earlier.

3.4. Polar Molecules

In polar molecules such as NH^, the Stark levels of the upper inversion state are trapping

(positive-energy) levels, whereas those of the lower inversion level are antitrapping levels [11,12].

This fact offers the possibility of stable electrostatic trapping of NH^, the upper inversion level,

with a trap well depth of a few kelvin [cf. Fig. (1)]. We could contemplate using, for example, a quadrupole or hexapole racetrack to store them. The long lifetime against radiative decay of the lowlying rotational states [131 should permit trapping for extended periods of time. Because many polar

molecules can be obtained in bulk, a large trap population should be achievable.

Laser cooling would seem difficult for polar molecules because of the large number of low-lying

states: after a few cycles of pumping to an excited state and radiative decay, the population would be

spread among many Stark sublevels, some of which would be antitrapping and most of which would be out

of resonance with the light field. Thus we must consider alternate methods.

If a racetrack trap were filled at room temperature, only those molecules having velocities

predominantly along the track, and transverse kinetic energies of at most a few kelvin, would stay in

initially. Collisions would result in further loss of most of the population until the total kinetic

energies dropped to a few kelvin, at which point further evaporative cooling would be quite slow. The

density would presumably also be low enough that condensation would be unlikely. To cool further, we

could store simultaneously a population of ions such as Mg+ in the trap, and cool them using the

optical sideband method. This would require exciting the trap with a radiofrequency voltage rather

than a constant voltage, but that change should not affect the stability of the polar molecules' orbits

greatly. Ion-molecule collisions would then maintain thermal equilibrium between the two populations,

allowing the polar molecules to be laser-cooled indirectly.

A potential problem with this scheme is the reactions or inversion-state flips which might be

induced by collisions with both the ions and other polar molecules. The solution for this problem, if

there is one, would stem from the temperature dependence of the collision cross sections. At high

thermal velocities, the cross sections should fall off as (1/v) 2 ' (s-D, where s=2 to 4 is the inverse

power law of the collisional interaction, because of the decreasing collision time. At low thermal

velocities the cross sections should also fall off because the frequencies of the main Fourier

components of the collisional interaction will be too low to induce a transition at the inversion

frequency. In between there will be a "thermal hump" in the collisional loss rate. Whether or not it

If the apparatus were contained in a high-Q cavity whose dimensions were small compared to the

wavelength of the inversion-line radiation, the lifetime might, in principle, be extended beyond the

natural lifetime, because there would be no radiation field modes available to receive the emitted

photon.

87
will be possible to oool through the hump without losing all the molecules will depend on whether or not the maximum loss cross section is smaller than the thermalization (momentum transfer) cross

section. If so, sufficiently rapid cooling will carry the sample through the loss hump with an acceptable survival rate.

Once the minimum temperature has been reached, the ions can be dumped from the trap by switching

back to constant electrode voltages. Now, if the sample is cold enough, the very slow moleculemolecule collisions will be ineffective in causing inversion state flips, and we may anticipate a storage lifetime of at least several seconds, and perhaps much longer. The nature of the collisions in

this situation should be quite interesting.

It should be noted finally that a racetrack geometry leads to the possibility of cyclotron

acceleration using appropriate alternating field gradients, and that the coherence properties of a beam

of molecules emitted by such a device in rather well-defined transverse quantum states should prove

quite interesting as well.

4. The Motion of an Atom in a Quasistatic Harmonic Potential Well When

Excited by a Light Field

To maintain an excited atom in a trap well beyond its natural lifetime, we may try re-exciting

with a light field. This is a complex problem because there are no stationary states. In the full

quantum treatment, we must include the wave motion of both the internal coordinates and the center-of- mass coordinates, and no doubt should include momentum fluctuations resulting from the quantum nature

of the radiation field as well. Such a treatment is beyond the scope of this paper. It is useful,

however, to examine the much simpler case of classical center-of-mass motion in a harmonic potential

well under the influence of a classical light field. It seems plausible that when the potential varies

little over the deBroglie wavelength of the particle, and also over the distance the particle moves

during a light-induced transition, this model will represent the motion of an ensemble of real

particles reasonably well, at least for times not too long.

If the particle has two internal states a and b, with external potentials Ua and Ub

, which may be

different, the wave function for the internal coordinates is

liKt)> = C (t)la> + C. (t)|b> . (33) a b

The average force on the particle is

<?> = <-VnU> = - VD

<^)\U\\\)> n n

= vLCIC (t)l 2 u + lcK

(t)l 2 uJ . (34) R a a b b

If the light field is intense enough and close enough to resonance that saturation occurs [|Ca

(t)| 2 =

|Cb

(t)| 2 = 1/2], then whenever the mean curvature of the two external potentials is positive, the force

is a restoring force and the particle motion is bounded for small enough kinetic energy. Since the

Stark effect generally increases with principal quantum number, the stabilizing propensity of a weakfield-seeking excited-state external potential will tend to outweigh the destabilizing propensity of

the strong-field-seeking ground state. Similar cases can be found with the Eeeman effect. The above

analysis then offers, in principle, a mechanism for confining a two-level atom at rest in an electrostatic or magnetic trap for times longer than the upper-state decay time.

If the light field is monochromatic, and the initial condition is |\|Ko)> = lb>, the probability of

finding the particle in state a subsequently is

|C (t)| 2

= (V/ft) 2 sin

2

(fit/2) , (35) a

and that of b is |Cb

(t)| 2 = 1-|Ca(t)| 2

, where V is the light-atom interaction matrix element and

88
a = [(v-v ) 2

+ v

2

]

1 (36)

is the Rabi frequency. Here v is the light frequency and vQ

is the atomic transition frequency. Now

the average potential energy <\|)|U|\|J> will be time dependent, as will the force given by Eq. (34). If,

for example, Ub -0 and Ua ~Ma|(x^+y2+4z2

)/2, which is the case for a high np sodium atom or a 2S

hydrogen atom confined in an axially symmetric quadrupole trap, the force will be

I = -(Mft2 /2) (V/fi) 2 (1-cosnt) (x+y+4z) . (37)

If the amplitude of motion is small enough, we can neglect the variation of £2 with position and

velocity. Appendix B then shows that the motion obeys Mathieu's differential equation with the

parameters

a = a

x y

a A = 2q

z x

2q = q /2

y z

2(V/ft) 2

(ft /Q)

2 x

(38)

The first region of stability (0<B<1) is shown in Fig. 10. As the Rabi frequency increases, the

system moves toward the origin along the operating curve a=2q. Stability occurs when fl> 1.743(V/Q)^fix

.

1.0

Figure 10. Mathieu stability

diagram for an atom in an axially symmetric staticfield trap and a light

field.

0.2

-0.2

r > i i—i—i—i—i—i

X

°

1 1 1 1—

ft" 1.0 t

• \ /V

: //\ \\ - / \

/

/ \\ i \

_. 1 , 1 . 1 . r J_„ 1 1 1 0.2 0.4 0.6 1x

= 4*1

Ofl 1.0 1.2

Effects such as spontaneous emission which reduce the coherence between states a and b will

reduce the Mathieu q's without altering the a's. The operating curve of the particle will become more

and more nearly vertical. A substantial stable region will always exist; in fact, when q=0 [the

vertical axis in Fig. 10], all values will give stability, since another stable region extends from 6=1

to 8=2. However, the stability at (a=1, q=0) is marginal since both stable and unstable regions touch

this point. We may anticipate that near here there will be large fluctuations, and it will be wise to turn up the light intensity further.

5. A Universal Trapped-Particle Refrigerator

The optical-sideband cooling method [14-16] has proved very effective for certain kinds of ions,

such as Ba+ or Mg+

, which can be laser-excited readily. For many interesting cases, however, this

Including these effects will introduce nonlinearities and (positive or negative) damping.

89
method is impractical. The cases include ions with no convenient energy levels, such as He+ or high-z

ions; ions with too many low-lying levels to allow a simple closed cooling cycle, such as molecular

ions; ions with no energy levels, such as elementary paricles; and many neutral atoms and molecules

without convenient optically excitable levels. A useful extension of the optical-sideband method would

be to trap two species together, A being laser-coolable and B being the subject of interest. Collisions

could then, in principle, be used to link A and B together thermally so that both could be cooled. The

characteristic optical sideband cooling time is of the order of

where y is the excitation-fluorescence cycle rate, hv is the photon energy, v is the ion mean velocity,

and K is the ion mean kinetic energy. The ion-ion thermalization time is of the order of

where b is the ion cloud radius, e is the ion charge, and x = eV2Kb is the "coupling parameter," the

ratio of coulomb interaction energy to relative kinetic energy. In typical ion-trapping conditions,

requiring T th <Tc demands x*,10-2 to 10~3, or b£l0"3 to 10~2

, for a hot (500 K) plasma; now

thermalization occurs in 10 2 to 1 03 collisions. For a cold (1 K) plasma, x>1 is easy to achieve, so that thermalization occurs in one collision. Thus it is easy to keep the ion cloud constituents in

thermal equilibrium while they are being cooled. Evidence for the success of this process has been

seen in spectroscopy of trapped Mg+ isotopes [11], but it does not appear to have been widely exploited.

Will it work also for ion-neutral or even neutral-neutral collisions? The interaction is shorter

range now, and so the verdict is less certain. However, the coupling between an ion and a polarizable

neutral is still quite strong, and my guess is that at least this combination will also work, offering

a universal means of trapped-particle refrigeration of considerable usefulness.

In the ion-neutral or neutral-neutral case there is, in principle, a possibility that the

collidants will stick together. However, this seems unlikely unless one of them is complex enough, and

has a long enough Poincare recurrent time, that an internal vibration can absorb the collision energy

and pass it from one internal mode to another until infrared radiation or another collision carries it

away.

This work has been supported by the National Bureau of Standards Precision Measurement Grants

Program and by the National Science Foundation. The author is also grateful for the hospitality of the

Joint Institute for Laboratory Astrophyics, Boulder, the Research Laboratory for Electronics,

Massachusetts Institute of Technology, Cambridge, and the Laboratoire de Spectroscopie Hertzienne, Ecole

Normal Superieure, Paris; and for fellowship support from the John Simon Guggenheim Memorial

Foundation.

6. References

[1] W. H. Wing, this volume.

[2] W. H. Wing, Phys. Rev. Lett. 45, 631 (1980).

[31 D. E. Pritchard, this volume.

m C. V. Heer, Rev. Sci. Instrum. 34, 532 (1963).

[5] A. Ashkin, Phys. Rev. Lett. 40, 729 (1978); A. Ashkin and J. P. Gordon, Opt. Lett. 4, 161 (1979).

[6] V. G. Minogin, Sov. J. Quantum Electron. 12(3), 299 (1982).

(39)

(40)

90
[7] Sir James Jeans, The Mathematical Theory of Electricity and Magnetism (Cambridge University

Press, Cambridge, U.K., 1925), 5th Ed.

[8] L. D. Landau and E. M. Lifshitz, Quantum Mechanics, Non-Relativistic Theory, Second English

Edition (Addison-Wesley, Reading, Mass., 1965), § 35.

[9] M. L. Zimmerman, M. G. Littman, M. M. Kash, and D. Kleppner, Phys. Rev. A 20, 2251 (1979).

[10] H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms (Academic

Press, New York, 1957).

[11] J. P. Gordon, H. J. Zeiger, and C. H. Townes, Phys. Rev. 95, 282L (1954).

[12] J. M. Jauch, Phys. Rev. 72, 715 (1947).

[13] J. P. Gordon, personal communication.

[14] D. J. Wineland, R. E. Drullinger, and F. L. Walls, Phys. Rev. Lett. 40, 1639 (1978).

[15] W. Neuhauser, M. Hohenstatt, P. Toschek, and H. G. Dehmelt, Phys. Rev. Lett. 41, 233 (1978).

[16] D. J. Wineland and W. M. Itano, Phys. Rev. A 20, 1521 (1979).

[17] D. J. Wineland, J. C. Bergquist, R. E. Drullinger, H. Hemmati, W. M. Itano, and F. L. Walls, J. de

Physique 42, (suppl. au n°12), C8-307 (1981).

[18] J. Green, personal communication.

[19] J. Meixner and F. W. Schafke, Mathieusche Funktionen und Spharoidfunctionen, mit Anwerdungen auf

physikalische und technische Probleme (Springer, Berlin, 1954).

[20] J. Meixner, F. W. Schafke, and G. Wolf, Mathieu functions and spheroidal functions and their

mathematical foundations, further studies (Springer, New York, 1980).

[21] M. J. 0. Strutt, Lamesche-Mathieusche-und verwandte Funktionen in Physik und Technik (Chelsea,

New York, 1967).

APPENDIX A: A Theorem on Field Maxima and Minima

The theorem states:

In a region devoid of charges and currents, the strength of a quasistatlc electric or magnetic field can have local minima but not local maxima.

a) Maxima (nonexistence: proof by contradiction)

E may be either an electric or a magnetic field, and it will be e_vident that the theorem also

holds for gravitational fields. Put the suspected local maximum in |E| at the coordinate origin

Then at any point r,

E(r) = E(o) + 6E and E

2

(r) = E

2

(o) + 2E(o>6E + (6E) 2

. (A.1)

T_he f^rst and last terms of the rightmost expression are nonnegative. Thus if we have a maximum,

E(o)*5E mus£ _be <0 fqr^arbj-trary r" near o". (Note: r" need not be an infinitesimal.) If we take the z axis along E(o), then E(o)*<$E = Ez (o)6Ez and since Ez (o)>0, the maximum condition reads

6E

z

(r) < 0 for arbitrary r near o. (A.2)

Is this possible? Recall the vector identity

91
$ x ($cE) = V(^'E) - V

2! . (A.3)

For a quasistatic field, with p=0 and j=0, to a sufficiently good approximation

t'E = 0 and v'xl = 0 . (A.4)

The first relation is exact and the second^ becomes exact as_£he time derivative of the electromagnetic

field goes to zero. Thus (A.3) becomes V

2E=0, and thus V2Ez

=0, and since 5E is a possible field, V 26ez =0. If we now apply Green's Theorem

J (<t>V 2

ib - xpV 2

d),dT = / - i|>V<J>)«ds , (A.6) V S

with \p= 5EZ and <|)=1/r [V 2(|)=- 1»TT6(r)] to a sphere of radius reentered at o, we find easily that the

average of <SE Z over the surface equals 6E zCo")=0. Thus if 6E z (r)<0 in some direction, it must be >0 in some other direction, and condition (A.2) above cannot be true. QED.

b) Minima (existence: proof by demonstration)

If a minimum in |e! occurs at 1$, then except for the trivial case ? = const, Eq. (A.1) requires

2F/o) •6E + (<5E) 2 > 0 for arbitrary r near o. (A.7)

This certainly can be true. The possibilities are:

a) ?(o) = t). Example: A quadrupole field E = + c y

y + c

z

z formed from the potential <}> = -(cx

x

2 + c yy2 + c z

z

2

)/2, where since V«E = 0, c x

+ c y+ c

z

= 0.

b) %o>4 but Example: A linear quadrupole field [c z =0 in a)] plus a uniform field in the z E(o)*6E=0. direction. To make a true minimum in the middle, the ends of the quadrupole electrodes

should be bent together somewhat.

c) F/o)»5E>0 Now if 5E

z (z)>0 for both positive and negative z near o, it must vary^ as z 2 or a higher

even power law. Assuming the power is 2, <))(z)-<j>(o) varies as z^ near o [3].

It is clear that several different minimum geometries exist, a), for example, gives a point minimum

unless one c^ is zero, in which case it gives a line, b) also gives a line. A combination of a hexapole

field and a uniform field gives a surface [18]. Bending a linear multipole field into a closed loop

produces a racetrack which has a closed line minimum on the axis. The field can be nonzero there if some current or time-varying magnetic flux links the loop. No doubt ingenuity will reveal a number of

other interesting geometries as well.

APPENDIX B: Motion in a Quasistatic Time-Varying Potential

This regime of operation is the "Goldilocks Regime" [2]: the motion is neither too hot nor too

cold. If the particle is cooled sufficiently further, quantum wave effects in the motion will become

important; if it is heated sufficiently further, classical chaotic motion may result.

We may with sufficient generality write the potential energy as

U(xyz) = -M(£}

2x2 + Q2y2 + ft 2

z

2

) . (B.1) 2 x y z

If the fl's were constants, the motion of a particle of mass M would be 3-dimensional simple harmonic

oscillation at frequencies fix, fi y, Q, z . Now, however, each Q,^ may include a constant and a time-varying

92
term:

k=x,y, or z . (B.2)

Newton's second law then yields

k=x,y, or z . (B.3)

This is Mathieu's differential equation, which in standard form reads

2 2

d n/d£ + (a-2qcos2^)n = 0 . (B.M)

Identifying corresponding variables between (B.3) and (B.M) results in

n = xk, $ = (w k

t+<()k

)/2, a = ^Qk/\, and q = -2^

1k/^ . (B.5)

The solutions of Eq. (B.U) may be classified by an order index 8, whose interpretation is that as fi^O,

the 8th limiting solution becomes a sinusoidal oscillation at the frequency 8^0

, where 8 is now an

integer. The solutions may be bounded (stable) or unbounded (unstable) as the first region of

stability lies between 8=0 and 8=1, corresponding to the region in the a-q plane bounded by the curves

The region of three-dimensional stability is then the region of overlap of the three individual

stability diagrams. If the six coefficients (ax

... qz

) are related to a smaller number of independent

physical variables, the stability region can be represented in a space of less than six dimensions;

frequently only two dimensions are needed.

References [19-21] are extensive treatments of Mathieu's differential equation.

and

(8=0) a = -q

2 /2 + q4 /128 - 29 q

6 /2304 + 68687 q

8 /l887U368 + ...

(8=1) a = 1-q - q

2 /8 + q

3 /64 - q

4 /1536 - 11 q

5 /3586H + ... .

(B.6)

93
Laser-Cooled and Trapped Atoms , ( Proceedings of the Workshop on Spectroscopic Applications of Slow Atomic

Beams, held at NBS, Gaithersburg, MD, 14-15 April 1983) Ed. by W. D. Phillips, Natl. Bur. Stand. (U.S.)

Spec. Publ. 653 (1983).

Gravitational Effects in Particle Traps

William H. Wing

Physics Department, Optical Sciences Center, and

Arizona Research Laboratories, Tucson, Arizona 85721

Schiff and Barnhill 1 have shown that an electron (mass m, charge -e) in a cavity within a metal at rest in a gravitational field -gz experiences an electric field whose average value -mg/e is just

sufficient to cancel the pull of gravity. Similarly, a charged particle (mass M, charge q) in an electromagnetic trap fixed to the earth will feel an average field E0

= Mg/q, which amounts to 10~? V/m

for a proton. If the trap's vertical restoring force obeys Hooke's Law (Fz =-kz z=-MJ2§z, where Q. z is the

vertical oscillation frequency), the gravitational pull will shift the particle's equilibrium position and

total energy. The latter shift is equal to the equilibrium-point potential-energy shift

6U0=-M2g2/2icz =-Mg2 /2f2|. This shift is not detectable spectroscopically, however, since it is the same

for all states of the trap + particle system. The only detectable effect will be the second-order

Stark shift induced by E0, which is of order (M/m)(f2|/(jO a

a)) 5uo

, where (i) a

is the Bohr frequency = 4x10 1 ^ rad/s and w is an atomic transition frequency > 10TM rad/s. For any reasonable value of fi z the Stark

energy is tiny compared to 6uQ and probably is too small to measure.

A neutral particle suspended electromagnetically will also experience a nonzero-average

gravitationally induced electromagnetic field. The Schiff-Barnhill field EQ becomes infinite, implying

that a higher order treatment is needed, the result of which will depend on the details of the

suspension method. However, the case of a harmonic-oscillator effective suspension potential energy

(averaged over the micromotion, if any) covers quite a wide variety of situations, including certain

permanent 2 and induced^ dipole traps and laser traps,1

* and so the formula given above for 6uc will still

apply. Since the trap stiffness in such traps generally depends on the internal or orientational

coordinates of the particle, Q z and hence 6U0 may change by a large percentage during a spectroscopic

transition, and the change may be large enough to measure in a delicate experiment. For example, for a weakly trapped heavy atom for which Mr200 amu and fi z =103 rad/s, 6uo

/'fi=1.6x105 rad/s. Considering only

vertical motion, the spectroscopic transition frequency is w 0-(Mg2 /2fi)A(1 A2|)+A[(nz +1/2)£2 Z

], where 0) o

is

the transition frequency of the free particle at rest, n z is its vertical motional quantum number in the

trap, and A implies the final-initial state difference. To achieve the necessary resolution, the Doppler

effect must be eliminated, i.e., the photon momentum must be delivered either to the trap structure or back to the radiation field. Several tricks can be used to separate out the gravitational term despite our inability to alter g. We may vary the applied trap fields and analyze the nonlinear change in the

transition frequency. We may rotate the trap 90° in the yz plane, or equivalently rotate the

polarization of the spectrometer, and analyze the frequency shift, most simply when ky=kz

. We may

accelerate the trap and make use of the Equivalence Principle to alter the effective g. In all of these

it will be desirable to avoid gravitational stress distortions of the trap structure. In certain kinds

of traps (e.g., electrical) it may in principle be possible to assure this by employing a charged test

particle in the same trap to measure k

y

and kz via its oscillation frequencies, and servoing them to constancy or equality.

Additional signatures of the gravitational effect may be discerned. A shift in the equilibrium

point during the transition will result in altered transition probabilities between quantized trap motion

states, compared to the g=0 situation. The quenching of a metastable state via a close non-metastable

one, e.g., the 2S state of atomic hydrogen via the 2P, will be affected by a gravitationally induced

electric field. Finally, if the trap is anharmonic (i.e., the potential energy is quartic or of higher

order in the particle height), the analysis will be more complicated, but the observable energy shifts

will be of the same order of magnitude as computed here, and still negligible for a charged particle.

A conversation with D. Pritchard valuably clarified these ideas.

1 L. I. Schiff and M. V. Barnhill, Phys. Rev. 151, 1067 (1966).

2C. V. Heer, Rev. Sci. Instrum. 3j», 532 (1963).

3w. H. Wing, Phys. Rev. Lett. 45, 631 (1980).

4A. Ashkin, Phys. Rev. Lett. 40, 729 (1978).

94
Laser-Cooled and Trapped Atoms , (Proceedings of the Workshop on Spectroscopic

Applications of Slow Atomic Beams, held at NBS, Gaithersburg, MD, 14-15 April

1983) Ed. by W. D. Phillips, Natl. Bur. Stand. (U. S.) Spec. Publ. 653 (1983).

A Hybrid Laser-Magnet Trap for Spin-Polarized Atoms

W. C. Stwalley*

University of Iowa

Iowa City, Iowa 52242

This paper presents a brief survey of theoretical issues related to a hybrid laser-magnet trap for neutral spin-polarized atoms. At low densities,

such a trap might be used to address a number of fundamental questions, e.g.

the interaction of an individual atom with an electromagnetic field, while at

high densities it might be used for "containerless" preparation of bulk

amounts of a new metastable form of matter, spin-polarized atoms. In particular, a discussion is presented of the feasibility and limitations of a trap

based on a near-resonant CW TEMJ X ("doughnut mode") laser beam, which provides trapping in two dimensions, and on a strong homogeneous solenoidal magnetic field (the axis of which is also the laser axis), which provides trapping in the third dimension.

Key words: atom trap; laser; low temperature; magnetic field; neutral atom;

spin-polarized atom; trapping

1. Introduction

Among the most exciting developments in atomic and molecular science in the past

decade have been the prediction and preparation of bulk quantities of an entirely new

(and metastable) form of matter, spin-polarized atoms (discussed in Section 3), and the

preparation and detection of very small numbers of various species (even individual

ions or atoms) (discussed in Section 2). A new hybrid magnet-laser concept for a low

temperature trap for gaseous neutral spin-polarized atoms -is proposed here (in Section

4). Such a trap could be used at low density for unique studies of small numbers of

atoms and at high density for unique studies of spin-polarized atoms, in each case without confining walls.

2. Preparation and Detection of Very Few Atoms

The detection of very few atoms by laser fluorescence or ionization is now well

established [1]. Much effort recently has been devoted to laser cooling [2-4] and

laser trapping [3-8] of neutral atoms, inspired by the early work of Ashkin on macroscopic particles [9] and the successful RF quadrupole and Penning traps and laser

cooling of atomic ions [10-15]. Related theoretical and experimental studies have

also been carried out on atomic beams: deflection [16-19], focusing [20,21] and deceleration (to 70 mK!) [22]. Other 3 dimensional neutral atom traps which have been

proposed include a toroidal hexapole magnet [23] for atoms with spin and an electrostatic trap [24] for highly polarizable Rydberg atoms. A new hybrid laser-magnet trap

for spin-polarized neutral atoms based initially on the desire for "containerless"

confinement of spin-polarized atoms is proposed here.

One of the main motivations for atom and ion traps is to be able to study these

species as the temperature approaches zero and as the transition frequency becomes

* Iowa Laser Facility and Depts. of Chemistry and Physics. Supported by NASA and NSF.

95
increasingly well defined. Also, because a single atom or ion can absorb and emit

millions of photons per second (assuming optical pumping effects are avoided), laser

fluorescence is easily detected and laser cooling easily achieved, assuming only the

resonant or near resonant CW lasers are available.

3. Spin-Pol ari zed Atoms

In the 1950s, it was proposed that a spin-selected sample of H atoms might be

stabilized [25,26] and, if stabilized, would be a unique quantum (Bose) gas [27].

Interest was revived in the 1970s by greatly improved calculations of the properties

of spin-polarized hydrogen [28-34] and greatly improved cryogenic capabilities, leading to the first successful preparation of ~10 llt spin-polarized H atoms per cm

3 by

Silvera and Walraven [35]. Since then, densities exceeding 10 17 /cm 3 have been

achieved, but nuclear spin relaxation followed by recombination on the relatively

inert helium-coated walls seems to be a fundamental limit preventing long time storage of higher densities (see e.g. Reference 36). Such higher densities are of great

interest, e.g. a Bose-Einstein condensation (and gaseous superfluidity) is predicted

[30,37] in the 10 1

9

- 10 2 °/cm 3 range (depending on temperature) and solidification [38]

is predicted at ~10 22 /cm 3

.

Other spin-polarized atoms are also of considerable interest, e.g. D, Li, N, Na,

although only D has yet been prepared. The first accurate calculations of the properties of spin-polarized nitrogen have recently been completed [39]. Although this substance should be a solid at low temperature, it has the highest energy content and

also the promise of greater stability than spin-polarized hydrogen or deuterium. The

properties of spin-polarized alkali metal atoms are also being examined; they should

be non-metallic ( rare -gas-1 ike) solids at low temperature. Since the hybrid magnetlaser trap discussed below involves a CW laser near a resonance line of the atom to be

trapped, this manuscript will emphasize the Li and Na atoms since appropriate CW la- sers are readily available. Vacuum ultraviolet CW lasers for analagous traps for H, D or N atoms are clearly many years away (although pulsed lasers are now available for H and D [40]). Indeed the shortest wavelength CW laser we are aware of is 194 nm [41].

However, the use of magnetic and laser fields for confinement in place of walls is a very intriguing and attractive alternative.

It should be noted that spin-polarized atoms are of interest primarily for fundamental reasons: as a new form of matter, highly metastable, with extreme quantum behavior for H and D. However, possible long-term applications include energy storage,

production of polarized beams, and even polarized controlled thermonuclear fusion [42,

43]. The remarkable persistence of spin-polarization in samples at room temperature , and above should also be noted in this connection [44].

4. A Hybrid Magnet-Laser Trap for Neutral Spin-Polarized Atoms

As noted above, there are a variety of schemes for "contai nerless" (i.e. without

contact with walls [45]) confinement of neutral atoms, although none have yet been implemented. Of particular interest is "containeHess" confinement of spin-polarized

atoms [37], especially now that the limit on spin-polarized hydrogen atom density is

apparently being determined by the helium-coated walls. The minimal changes (Figure

1) to convert the existing University of Iowa low temperature -high magnetic field (T

£ 0.012 K, B < 110 kilogauss) system [46,47] to a hybrid laser-magnet spin-polarized

atom trap would be to introduce a cylindrical laser beam on the magnet axis (defined

as z). Some theoretical issues for this trap currently under study are discussed below and given in Table I. It should also be noted that the Iowa Laser Facility, con- taining both state-of-the-art laser and dilution refrigerator systems, is one of

very few labs where such traps could in fact be built with existing equipment once these theoretical issues are resolved.

96
MIRROR

DILUTION REFRIGERATOR INSERT

SUPERCONDUCTING MAGNET

LIQUID HELIUM DEWAR

TEM*

1 LASER

Na SOURCE

Figure 1. Schematic diagram of the hybrid laser-magnet trap apparatus includes

a liquid helium cooled Na beam and coaxial laser directed into an opening in a dilution refrigerator operating below 1 K. The solid line at the center represents the trap region.

Table I. Theoretical Issues for the Hybrid Laser-Magnet Trap

A. The Trap Itself

1. Determination of optimal laser field

a. single -mode versus multimode

b. detuning Av (versus laser intensity, magnetic field strength)

c. extracavity versus intracavity (also ring cavity versus linear cavity)

d. upper limits on power (multiphoton ionization, four-wave mixing, etc.)

e. optical pumping concerns

f. beam parameters (size, focused or not, polarized linear or circular,

etc.), cryogenic mirrors (reflectivity, absorption)

g. "noisy" feedback stabilized high power continuously tunable dye laser versus highly stable medium power line tunable optically pumped dimer lasers

(e.g. work in this lab at 5889 A in 6 Li

2

) 2. Determination of optimal magnetic field

a. magnetic field strength (presumably maximum)

b. field i nhomogenei ty effects

B. Injection into the Trap

1. Laser cooling techniques

a. Doppler profile cooling

b. cooling by anti stokes scattering using Zeeman structure

2. Colli si onal cooling

a. magnitude of low temperature helium cross sections

b. magnitude of inelastic collisions involving Zeeman levels

3. Heating sources (black-body absorption, recoil, background gas)

C. Low Density Trap Effects

1. Radiative destruction of atoms (e.g. multiphoton ionization)

97
Table I (continued).

2. Atomic manipulations (escape through laser field minima (for single-mode

standing wave), steering with a focused laser, periodic array generation,

frustrated spontaneous emission, coherent motions, etc.)

3. Tunnelling and other quantum behavior as DeBroglie wavelength approaches trap

size, laser wavelength

D. High Density Trap Effects

1. Three-body atomic recombination

a. spin-allowed to triplet ( 3 £j)

b. spin-forbidden to singlet ( Eg)

2. Nucleation on diatomic or polyatomic molecules

3. Possibility of supersaturation by continual photodi ssociation of diatomic

molecules with a different laser

4. Pressure broadening of spin-polarized atoms

5. Light scattering and cavity losses from atoms

6. Quantum fluid effects (e.g. possibility of Bose condensation in super-saturated spin-polarized 7 Li or

23 Na)

7. Limiting pressure (or density) at a given temperature that can be supported by

a given potential well depth

8. Laser-induced radiative collisions

E. Other Issues

1. Implementation of other cooling or trapping ideas given the hybrid laser-magnet trap as an initial condition

2. Frequency standards applications

3. Experimental tests of gauge-invariant and alternative formulations of atom/

field interactions

4. Hidden variable tests

5. Ultrahigh sensitivity tunable detector

6. Fundamental temperature limits

7. Multi stable devices

In particular, if the laser frequency is slightly to the blue of the atomic resonance frequency, the atom will experience a relatively strong "transverse dipole"

force pushing it into the central region of weaker light intensity. This force has

been dramatically demonstrated in the Na atom focusing experiments of Bjorkholm and

coworkers [20,21,48]. If one employs a TEM* : ("doughnut mode") laser beam (or, alternatively, a ring of ordinary overlapping TEM 00 laser beams), one confines the atom in

two dimensions (x and y, X to the laser and magnetic axis z). They are, of course,

already confined along the z direction within the magnet by their spin-polarization

[49]; the potential well depth (energy to remove the atom from the magnet) in our system is -7 K. The depth of the two dimensional laser trap has 0 been estimated to be ~1

K by Ashkin [4] for a typical 1 watt dye laser near the 5890 A Na resonance line.

Commercial dye lasers already exceed this power at 5890 A and, if an intracavity dye

laser is used, much higher powers and much deeper trap potential wells are possible.

In such a case, one need only consider injection of atoms roughly described by a temperature T « 7 K in order to have an effective trap, i.e. one has a very "robust"

trap compared to the others discussed in Section 2. Such cold atoms could possibly be

formed by mixing a Na beam with 4 K He. This beam would then be cooled by laser cooling techniques discussed in Section 2 or more simply by cooling with added helium gas

(even at 1 K, the vapor pressure of 3 He is 8.8 torr [50]). Note the helium gas is

very nearly unaffected by the laser and magnetic fields, and the condensed species

such as NaHe or Na 2 should also condense on the walls or pass through the system.

Even if only a small fraction of the initial Na atoms was trapped, this should be adequate for low density experiments at least. Note also that other impurities (e.g. Li or K) would not be effectively trapped in a Na trap.

98
In addition to these questions of optimal laser field (and also optimal magnetic

field) and of atomic injection, there are many other issues of high interest. For

example, considering low densities first, there are obvious problems such as what

power of near-resonant photons will produce an unacceptable rate of multiphoton ionization. Recall that gain or loss from the trap is readily monitored by laser fluor- escence. The interaction of individual atoms with a single-mode standing wave field

or with a "steering" laser field and other sorts of manipulations are of interest.

For small numbers of atoms, possibilities such as generation of periodic arrays ("gratings") or of coherent motion of atoms arise. Quantum effects such as tunnelling and

diffraction arise as the temperature drops and the de Broglie wavelength approaches

first the laser wavelength and second the characteristic trap dimensions.

One interesting issue is the lowest kinetic temperature to which an atom can be

cooled in such a trap. Pobell [51], in discussing condensed matter, has alluded to 10

uK as a "brick wall" for refrigeration, although 50 nK has been achieved for nuclear

spins (not in full thermal equilibrium) [52]. There are, of course, a great many reasons for being interested in "near zero" [51,53]. In discussing laser cooling, Ashkin [4] has estimated 10~ 3 - 10

_lt K can be achieved in his 3D laser trap; in ion

traps, optical cooling to <36 mK has been achieved in two labs [14,15] and in the

later case, a "magnetron temperature" of 1.3 yK has been achieved. Neuhauser et al

[14] feel their technique is capable of achieving a kinetic temperature of 10 nK!

At high densities, additional issues need to be addressed. It is clear that an important loss process will be three -body atomic recombination. Except for recent

studies involving low temperature hydrogen atoms [54], atomic recombination has not

been studied below 77 K. Even at -500-620 K, Na atoms still recombine primarily on

the walls, not in the gas phase [55]. The dominant recombination of Na or Li should

be to form 3 eJ dimers by a spin-allowed process; it has been estimated for H that the

spin-forbidden process to form : Zg molecules will be ~5 x 10" 8 times slower [56]. The

3 eJ molecules (in any vi bra ti onal -rotational level) could be efficiently photodissociated (to the repulsive 3 n

g

state) at -5515 A for Na [57] or at -5875 A for Li 2 [58].

Thus, even though spin-polarized Li or Na might be expected to be a solid at low temperature, it is possible that a supersatura ted spin-polarized atomic vapor can be

maintained! Depending on the ultimate temperature in fact achievable with this supersaturated vapor, it might even undergo Bose condensation ( 7 Li or

23 Na) and other extreme quantum phenomena!

There are a number of other intriguing questions listed in Table I, e.g. questions of experimental tests of predicted line shapes and other observables for one or

two photon transitions using gauge-invariant, mul tipolar-gauge or conventional formalisms [59]. Suffice it to say that the hybrid laser-magnet trap for neutral spin-polarized atoms is an interesting alternative low density trap concept and a quite

speculative but promising possibility for a high density spin-polarized atom trap to

overcome the wall recombination problem; clearly a great deal of interesting science

lies ahead.

5. References

1. V. I. Balykin, G. I. Bekov, V. S. Letokhov and V. I. Mishin, Sov . Phys. Usp. 23,

651 (1980).

2. V. I. Balykin, V. S. Letokhov and V. I. Mishin, JETP Letters 29, 560 (1979).

3. A. Ashkin and J. P. Gordon, Optics Letters 4, 161 (1979).

4. A. Ashkin, Science 210, 1081 (1980).

5. A. Ashkin, Phys. Rev. Letters 40, 721 (1978).

99
6. V. S. Letokhov and V. G. Minogin, J. Opt. Soc. Am. 69, 413 (1979).

7. J. P. Gordon and A. Ashkin, Phys. Rev. A 21_, 1606 (1980).

8. V. G. Minogin, Sov. J. Quantum Electron. U, 299 (1982).

9. A. Ashkin, Phys. Rev. Letters 24, 156 (1970).

10. H. G. Dehmelt, Advan. At. Mol . Phys. 3, 53 (1967).

11. H. G. Dehmelt, Advan. At. Mol. Phys. 5_, 109 (1969).

12. D. J. Wineland, R. E. Drullinger and F. L. Walls, Phys. Rev. Letters 40, 1639

(1978).

13. W. Neuhauser, M. Hohenstatt, P. Toschek and H. Dehmelt, Phys. Rev. Letters 42,

233 (1978).

14. W. Neuhauser, M. Hohenstatt, P. E. Toschek and H. Dehmelt, Phys. Rev. A 22, 1137

(1980).

15. W. M. Itano and D. J. Wineland, Phys. Rev. A 25, 35 (1982)

16. A. Ashkin, Phys. Rev. Letters 25, 1321 (1970).

17. E. Arimondo, H. Lew and T. Oka, Phys. Rev. Letters 43, 753 (1979).

18. A. F. Bernhardt and B. W. Shore, Phys. Rev. A 23 , 1290 (1981).

19. J. E. Bjorkholm, R. R. Freeman and D. B. Pearson, Phys. Rev. A 23, 491 (1981).

20. J. E. Bjorkholm, R. R. Freeman, A. Ashkin and D. B. Pearson, Phys. Rev. Letters

41_, 1361 (1978).

21. D. B. Pearson, R. R. Freeman, J. E. Bjorkholm and A. Ashkin, Appl . Phys. Letters

36, 99 (1980).

22. J. V. Prodan, W. D. Phillips and H. Metcalf, Phys. Rev. Letters 49, 1149 (1982).

23. V. S. Letokhov and V. G. Minogin, Opt. Commun. 35, 199 (1980).

24. W. H. Wing, Phys. Rev. Letters 45, 631 (1980).

25. J. T. Jones, M. H. Johnson, H. L. Mayer, S. Katz and R. S. Wright, Aeroneutronics

Systems, Inc. Publication No. U-218 (1958).

26. M. W. Windsor, in Formation and Trapping of Free Radicals , edited by A. M. Bass

and H. P. Broida (Academic Press, New York, 1960), page 400f.

27. C. E. Hecht, Physica 25, 1159 (1959).

28. J. V. Dugan and R. D. Etters, J. Chem. Phys. 59, 6171 (1973).

29. R. D. Etters, J. V. Dugan and R. W. Palmer, J. Chem. Phys. 62, 313 (1975).

30. W. C. Stwalley and L. H. Nosanow, Phys. Rev. Letters 36, 910 (1976).

31. M. D. Miller and L. H. Nosanow, Phys. Rev. B 15, 4376 (1977).

100
32. R. D. Etters, R. L. Danilowicz and R. W. Palmer, J. Low Temp. Phys. 33_, 305

(1978).

33. L. J. Lantto and R. M. Nieminen, J. Low Temp. Phys. 3J_, 1 (1979).

34. Y. H. Uang and W. C. Stwalley, J. de Physique 5J_, C7-33 (1980).

35. I. F. Silvera and J. T. M. Walraven, Phys. Rev. Letters 44, 164 (1980).

36. I. F. Silvera, Physica 109 and HOB , 1499 (1982).

37. W. C. Stwalley, Y. H. Uang, R. F. Ferrante and R. W. H. Webeler, J. de Physique

41_, C7-27 (1980).

38. L. H. Nosanow, J. de Physique 41_, C7-1 (1980).

39. R. F. Ferrante and W. C. Stwalley, J. Chem. Phys. 78, 3107 (1983).

40. K. H. Welge and H. Zacharias, Proc. Lasers '81 Conference (1980), p. 1129

41. H. Hemmati , J. C. Bergquist and W. M. Itano, Proc. Topical Meeting on Laser

Techniques for Extreme Ultraviolet Spectroscopy (Boulder, Colorado, March 1982).

42. R. M. Kulsrud, H. P. Furth, E. J. Valeo and M. Goldhaber, Phys. Rev. Letters 49,

1248 (1982).

43. D. Kleppner and T. J. Greytak, Proc. 5th Internatl . High Energy Spin Symposium

(Brookhaven National Laboratory, September 1982).

44. N. D. Bhaskar, W. Happer and T. McClelland, Phys. Rev. Letters 49, 25 (1982).

45. Containerless processing, in Materials Processing in Space (National Academy of

Sciences, Washington, D. C, 1978), p. 27.

46. W. C. Stwalley, J. de Physique 39, C6-108 (1978).

47. R. W. H. Webeler, R. F. Ferrante and W. C. Stwalley, J. de Physique 41_, C7-161

(1980).

48. J. E. Bjorkholm, R. R. Freeman, A. Ashkin and D. B. Pearson, Phys. Rev. Letters

5, 111 (1980).

49. R. W. Cline, T. J. Greytak, D. Kleppner and D. A. Smith, J. de Physique 41_,

C7-151 (1980).

50. J. Wilks, Properties of Liquid and Solid Helium (Clarendon Press, Oxford, 1967).

51. F. Pobell, Physica 109 and HOB , 1485 (1982).

52. 0. V. Lounasmaa, Physica 109 and HOB , 1880 (1982).

53. W. M. Fairbank, Physica 109 and HOB , 1404 (1982).

54. W. N. Hardy, M. Morrow, R. Jochemsen, B. W. Statt, P. R. Kubik, R. M. Marsolais,

A. J. Berlinsky and A. Landesman, J. de Physique 4T_, C7-157 (1980).

55. J. P. Woerdman, S. S. Eskildsen and W. J. J. Rey, ACS Symp. Series 179, 61

(1982).

101
56. J. M. Yuan and T. K. Lim, J. de Physique 4]_, C7-39 (1980).

57. J. P. Woerdman and J. J. deGroot, ACS Symp. Series 179 , 33 (1982).

58. M. E. Koch, W. C. Stwalley and C. B. Collins, Phys. Rev. Letters 42, 1052

(1979).

59. K.-H. Yang, Phys. Lett. A 94, 259 (1983).

102
Laser-Cooled and Trapped Atoms , (Proceedings of the Workshop on Spectroscopic

Applications of Slow Atomic Beams, held at NBS, Gai thersburg, MD, 14-15 April

1983) Ed. by W. D. Phillips, Natl. Bur. Stand. (U. S.) Spec. Publ . 653 (1983).

Good and Bad Aspects Of Traps for Neutral Particles

David E. Pritchard

Department of Physics and

Research Laboratory of Electronics

Massachusetts Institute of Technology

Cambridge, Massachusetts 02139

Ve exhibit a configuration Of magnetic fields which has a field

minimum at its center and is therefore suitable for trapping paramagnetic neutral atoms or molecules. The trap has a large field component

along the z axis and is therefore conducive to high resolution spectroscopy. We propose an RF-laser cooling cycle which appears to be capable

-6

of attaining ultimate temperatures of ^10 K. Limits to the attainable

resolution due to this temperature are discussed.

In order to label the properties of traps for neutral particles as

"good" or "bad" it is necessary to have some value system. Mine is that

good properties enhance the utility of these traps for studies of properties of the trapped atoms: precise spectroscopy of their levels,

collision studies using a dense cold target, and possibly the "solid

state" physics of their mutual interaction at high density. (From the

contrasting perspective of interesting and/or challenging physics relating to slow atoms, the labels good and bad may well be reversed!)

The Trap Itself

The main requirement for achieving the objectives listed above is

the ability to trap atoms in a particular quantum state with a long

103
(minutes) lifetime; in most cases this means atoms in their ground electronic state. We could build the trap of electric and/or magnetic

fields, so it is natural to look at the strength of the relevant mag1 p netic and electric interactions: u_B and -r- otE . For my favorite atom,

Na, these have strength 5 cm ^ and 0.02 cm

: respectively at large but

readily achievable laboratory fields (10 gauss and 10 V/cm respectively). From this it seems that magnetic fields are preferable, but if

you consider polar diatomic molecules (eg. NaCl with dipole moment *

ea

Q

) the energy of interaction will be ^4 cm ^ for the lowest rotational

levels (where j is no longer a good quantum number at this field).

These numbers are good news: they are considerably larger than the

depths of proposed optical traps (ASH78), moreover, recent experiments

show that atoms whose kinetic energy is less than the above limit can be

produced either by laser slowing in a beam (PPM82) or cryogenic techniques (CSG80) (note that 1 cm

-1 = 1.4 Kelvin = 3x10

10

Hz).

The second piece of good news is that traps for neutral particles

can be made from static E and/or B fields. The most obvious such trap

for neutrals is a long hexapole or quadrupole magnet which is bent into

a circle; it would confine weak field seekers to a ring on its axis.

The addition of a solenoidal field around the quadrupole would not alter

the property that the weakest field was at the center: the solenoidal

field would have no components in the transverse direction and the multipole field would have components only in this direction so there would

104
be no cross terms in |B| = |Bmp

+ B

g

| . If the solenoidal field were

not quite uniform, then there would be confinement at any point(s) along

the ring where the field was minimum - i.e. the solenoidal field would

become a weak magnetic bottle, B [(l + 7r(kz) - -r(kp) J z ~

s 4

l/2(kz)(kp)p where z is along the ring of confinement, confining the

weak field seekers along z while expelling them along p. However, if

the quadrupole field (whose magnitude also increases linearly in p) is

sufficiently strong, it will overcome the outward instability of the

bottle. Clearly a ring-shaped region of stability is not necessary: a

straight (and relatively weak) bottle plus a co-axial quadrupole field

should create a trap for weak field seekers with non-zero field at the

center. If the uniform B-field along z is large, the angular variation

of the quantization axis will be small and Majorana flops will not

occur.

Even in a trap which has zero field at its center the trapped particles will be displaced from this center by gravi ty (WIN83) . For Na

(again), the gravitational potential is ^5 MHz per cm, which can obviously be quite significant in a precision spectroscopy experiment where

a fairly uniform field is desired.

Trapping Potential Depends on the Quantum Level

In thinking about particle traps for neutrals, one must realize

that the interaction of the trap fields with the particle, and hence the

particle's motion, depends strongly on the internal quantum numbers of

105
the particle. Confining the particle requires a spatial variation of

the potential energy of the particle; this results from the spatial

variation of the field strength. If the particle makes a transition to

a different quantum state with a different relationship between potential energy and field, the force and the potential energy will be different. This introduces new possibilities for .cooling or heating the

atoms (good), but it also introduces inhomogeneous line broadening (bad)

as well as difficulties with coherent superpositions of two different

states (bad). This is not true of experiments on trapped ions where, to

an excellent approximation, the motion depends on only the charge and is

therefore independent of the quantum state.

Let us first consider the effects of transitions between two quantum levels 1 & 2 which have different field-independent magnetic

moments, \i. and \i 0

.

z

*

Fig. 1

106
Assume that the field strength varies as a function of z between

and as shown in the lower part of Fig. 1 ; then the potentials for

motion in states 1 and 2 are ^

= ^ 2^*

^ ^e particle is initially

in state 1 with total mechanical energy (Kinetic plus Potential) ,

then it will oscillate in z out to where = . Since we took > \i

^ in this example, the separation between and is not constant, leading to a broadening of the transition by the amount hAcj = (B^-B^) (ij^-

\i

^ ) = E^ ( u

—

^ 1 )/m j • This is an inhomogeneous broadening which depends

on the position of the particle. It is noteworthy that a superposition

of states and vp ^ (produced, for example, by a very short pulse of

radiation at t = 0) will decay in a time vH/Agj. If \i

^ or ^ s ^ 0,

then the particle will be bound in that state, causing an even more

rapid decay of any superposition than the above limit. These considerations make us pessimistic about some proposed spectroscopy experiments

on atoms trapped in E-field traps (WIN80 but see WIN83 for a possible

way out). The above situation bears considerable similarity to molecular spectroscopy; in particular the last case is analogous to transitions from a bound to a repulsive potential curve, and the inhomogeneous

broadening discussed previously results from the population of many

vibrational levels.

Efficient Cooling

The fact that the potential energy depends on the state can be used

to generate very efficient cooling, essentially because the

107
inhomogeneous broadening allows one to make a transition at a selected

point in space. We consider now a three state system in a trap whose

energy levels vary as shown in Fig. 2.

Fig. 2

Imagine that the objective is to produce cold atoms in state 2, and that

initially the particles have total mechanical energy E^. Their maximum

excursion is to z^. If radio frequency radiation is now applied at frequency RF

q it will equalize the populations in levels 1 and 2. On

account of the classical Franck-Condon principle, the kinetic energy at

z^ will be the same in state 1 and 2 (nearly zero as shown) but the

total energy of the particles in state 1 will be reduced to E^ due to

the fact that U,lo(z ) << U_do(z ). If the atoms are now irradiated by

108
laser tuned to u, , atoms in level 1 will undergo transitions to 2' which

may decay spontaneously back to 2 in time T^. Neutral particle traps

frequently operate in the regime where T^ << the period of oscillation

in states 1 and 2 - under these conditions there will be little change

of kinetic energy in the laser-induced cycle except from the momentum

transferred by the light. Thus the combined RF-laser cycle has reduced

the mechanical energy of the atoms in state 2 by a factor of ^2, which

restricts their motion to |z|<z^. If the RF frequency is now lowered

slowly, it will eventually come into resonance with the atoms a second

time, cooling them again. This type of cooling cycle is very efficient,

and has an ultimate lower momentum limit of a few hu /c in contrast to

Jj

1 /2 Doppler schemes for trapped ions where the limit is [mh/T^] which is

typically ^xlO larger. For Na atoms the velocity corresponding to 3

photons is 9 cm/sec, and the corresponding kinetic energy is 8x10 ^ cm

^

(0.2 MHz).

High Resolution Spectroscopy?

Use of neutral particle traps for high resolution spectroscopy

requires a method of eliminating the inhomogeneous broadening which

arises when A straightforward way to do this is to bias the trap

eff eff r so that = at the bias field, B [the effective moment is

u (B)

eff

=

" 3 ^

B) so at this field

3-D the transition frequency field is

field-independent (see Fig. 3).]

Fig.

109
If we now assume that some of the trapped particles have mechanical

energy kT, it is clear that they will make excursions to regions of the

eff

trap where the field exceeds B

q by the amount AB = kT/u , shifting the

1 P d^ transition energy by the amount -^(AB) [ —^(U -U >. ]

, and causing

2 broadening of the transition by the order (kT) /aU. Taking kT to be the

cooling limit obtained previously and AU to be «H GHz (a typical h.f.s.

separation) gives a broadening ^40 Hz. This is obviously bad from the

standpoint of precise hyperfine spectroscopy.

In order not to end our discussion On such a bad note, we observe

2

that an S + S transition in an ^\j2 a ^ om^ c ground state has the property that the two moments \i

^ and are exactly equal at high magnetic

fields where the hyperfine interaction is completely decoupled.

These speculations were supported by NSF, and the main ideas

presented here were conceived and abandoned in 1977 because I could not

see how to get any atoms in the trap! Recent work on both slow (PPM82)

and cold (CSG80) atoms promises to overcome the filling problem, and has

rekindled my interest. However, this workshop provided the real impetus

for me to write them down in a coherent form.

110
References

ASH 78 A. Ashkin, Trapping of Atoms by Resonance Radiation Pressure,

Phys. Rev. Lett. 40, 729 (1978).

C3G 80 R. Cline, D. Smith, T.J. Greytak and D. Kleppner, Confinement

of Spin Polarized Atomic Hydrogen, Phys. Rev. Lett. 45_, 2117 (1980).

PPM 82 John V. Prodan, William D. Phillips, Harold Metcalf, Laser Production of Very Slow Monoenergeti c Atomic Beams, Phys. Rev. Lett. 49,

1 149 (1982).

WIN 80 W. Wing, Electrostatic Trapping of Neutral Atomic Particles,

Phys. Rev. Lett. 4_5, 631 (1980).

WIN 83 W. Wing, see proceedings of this conference.

Ill
Laser-Cooled and Trapped Atoms, Proceedings of the Workshop on Spectroscopic

Applications of Slow Atomic Beams, held at NBS, Gaithersburg, MD, 14-15 April

1983) Ed. by W. D. Phillips, Natl. Bur. Stand. (U.S.) Spec. Publ. 653 (1983).

Cooling of Vapors Using Collisionally Aided Radiative Excitation

E. Giacobino and P.R. Herman

Physics Department, New York University, New York, N.Y. 10003

Collisionally aided radiative excitation (CARE) is proposed as a mechanism for cooling an atomic vapor. With a CW laser power of

2 1.0 W/cm and the resonant dipele-dipole interaction providing the collision mechanism, we estimate that temperature gradients of tens of degrees

16 3

per cm can be achieved at vapor densities of order 10 atoms/cm .

Key words: cooling, collisions, resonant broadening, laser assisted

collisions.

1. Introduction

Over the past several years, there has been considerable interest in collisional processes that occur in the presence of radiation fields such as reactions

of the form

A

x + Bx + fifl •* A

2

+ B

x

(A^, atom A in state i; B^ t atom B in state i), which are commonly referred to as collisionally aided radiative excitation (CARE) [ij. Most experimental studies

to date have concentrated on measurements of either the CARE cross sections or the

frequency distribution of the reemitted light. However, it can easily be seen

that CARE also produces a change in the atoms' translational energy and, as such

may have potential use as a method for heating or cooling an atomic vapor [2].

Supported in part by NSF Grant INT 7921530 and by the U.S. Office of Naval

Research.

112
Let us consider radiation of amplitude E and frequency ft acting on a twolevel atomic system whose Bohr frequency is u » ft-A (|A| » Doppler width). As

a result of the interaction with the field, the atom may scatter a Rayleigh photon

or be excited to its upper state. The latter process, although possible without

collisions in strong fields, is greatly enhanced by collisions [ij. Since the

final energy level of the atom is different from the energy of the absorbed photon,

the energy difference must be compensated by a change in the translational energy

of the colliding atoms. As pointed out previously [2] the net result is a cooling

or a heating of the vapor. Tuning with ft > w produces heating and that with ft < u> produces cooling.

2. Expected Cooling

The magnitude of the effect depends strongly on the detuning A » ft - o>s each

time an atom interacts with a photon the energy "fiA is removed or added to the vapor.

One may choose 4iA to be an appreciable fraction of kT (at 500°K kT corresponds to

4 2-1 a frequency of 10 GHz or a wave number of 3 x 10 cm ) , so that velocity changes

are large compared with those of order 1.0 cm/sec which occur as a result of photonrecoil processes.

Of course, the process considered here involves a non-resonant atom-field

interaction and the rate of excitation is reduced compared to a resonant excitation

2 2 by a factor T /A where T is the collision rate. (For the sake of simplicity we

use expressions calculated using the impact limit of line broadening) . We shall

consider the case of resonant broadening; i.e. the resonant dipole-dlpole interaction between two atoms of the species gives rise to the rate F.

The rate of excitation to level 2 from level 1, is given by

2

"2 - *5r <ni " n

2

) (1)

A

where n^ is the population of state i, x is the Rabi frequency defined by

X-f . (2)

and u is the dipole matrix element for the transition. If n^ » nj, the steadystate upper population of state 2 is

113
(3)

where y is the decay rate of the upper level. For cooling to actually occur, the

atom must decay back to the ground state via spontaneous emission rather than via

stimulated emission (stimulated emission would "heat" the atom if absorption cools

it). This means that it is advisable to restrict the laser power such that

< y • <4 > A

Beyond this limit, any additional laser power is useless since there is as much

stimulated emission as absorption.

For an atomic density N and cylindrical interaction volume of length L and

cross-sectional area A, the power dH/dt removed from the sample is

2

J!

. m Ql. nal, (5)

A

which may be rewritten as

f-T7Y <><V (6 >

where P is the laser power and is the maximum laser power consistent with con- dition (4). This result suggest a potentially large heating or cooling effect.

Taking

-KA/kT » 0.1

P/P * 10"

m

one finds

±— % u 6 *A sec"

1

(7) NAL dt

Under such conditions, the time scale at which the energy is removed or doubled is

(A/2tt - 10 GHz) ; y/2i\ - 10 MHz;

(P -5,0 W/cm2

; N - 10

15

cm"

3

),

114
1.0 sec. In calculating dH/dt, ve assumed the medium to be optically thin.

The above calculation has to be modified when effects of radiation trapping are

taken Into account. Equation (4) must be replaced by

^ < Y' , (8)

where y' Is the effective upper state decay rate In the presence of radiation

trapping. Typically, one finds [3]

Y* - Y/[KL (tt Jin KL) 1/2

], (KL » 1) (9)

where

K - Na (y/<0 (10)

O u

Is the absorption coefficient for resonant radiation in a Doppler broadened medium

((Up Doppler width). The cross section O

q

is given by

o

'

where y is the dipole matrix element and X the wavelength for the transition.

The rate of energy loss for the sample is now given by

f - w f- NAL

- (T<r^> (12)

m

where is the maximum laser power consistent with condition (8). Note that

P' may be considerably less than P . However, since P'/y' " p /Y* Eq. (12) may m m m m

be recast in the form

f -$f §-NAL . [P < p; - FB

(YVY)] (13)

m

Equation (13) is identical to the result (6) which we found in the absence of

radiation trapping, except that P is limited to a smaller value.

115
17

Even though F is limited to a value equal to P^, significant cooling can still be achieved. For example, for the parameters chosen above, one can use

Eqs. (9-11) to estimate P'/P Y'/Y ~ 1°

6

» assuming a cell length on the order

tax m

^ ©f 5cm. Since P/P was taken equal to 10 , one finds P - 10TM P' which does not

m m

violate the condition P < P'. Thus, the expected cooling is the same as that

m

found in Eq. (7).

3. Temperature Gradient

We can now estimate the temperature gradient in a cell created by this

cooling. Considering that the atoms are perfectly thermalized on the wall of

the cell, the amount of energy we must remove per second, to keep a temperature

gradient dT/dr between the laser interaction region and walls is

S dt dr •

where S is the surface of the laser beam [S « 2/rrAL » &L; beam circumference

» I] and (C is the thermal conductivity of the gas approximately given by

tc * ku/a (15)

c

where k is the Boltzmann constant, u a mean speed, and 0

£ a collision cross section.

Combining Eqs. (14) and (13) and dividing by the wall temperature T

w

we obtain

We may also note that the limiting value P determined from condition [ h] is given m

by

o

To optimize the temperature gradient which can be obtained, we proceed as follows:

(1) We choose a density N such that the optical depth for the laser radiation is

116
equal to unity. That Is, we set

Na Ty

X

2/T

N - 2A

2

/(a LTy) (18) o

For L s 5cm, and T determined from theories of resonance broadening [k] (i.e.

3

r 0.023 N X y), X * 600 nm, and the values of A and y used above, one finds

16 3 N a 2 x 10 atoms/cm . By choosing an optical depth of unity, we ensure that

each photon "does its duty" in removing energy from the sample. (2) For this

value of N, Eq. (9) yields a value y'/y - P'/P. - 10" 7 mm (3) Using Eq. (17),

we estimate the maximum allowed value P' for P to be

m

P'

f » 1 x

10~ 7 |£ * 0.2 W/cm2

2

(4) Since this value of P'/A is readily obtainable with a CW laser for A < 5 cm

m

we set P/P equal to its maximum allowed value (P'/P ) in Eq. (16). Moreover,

m ^ mm

we take N<?c * 0.1 T/u to finally obtain

For A « 4.0 cm

2

, T - 0.6 x 10

10

sec"

1

, u

2 * 10

10

cm/sec, (iiA/kT) » 0.1,

t£ " < 20>

The temperature gradients predicted by Eq. (19) (of order 10 degrees/cm)

may be somewhat optimistic. However, this order of magnitude calculation does

seem to indicate that significant cooling can be achieved using collisionally

aided radiative excitation.

One of us (E.G.) would like to thank Dr. F. Laloe for stimulating discussions

on this subject.

*We assume that the excited state population is close enough to saturation to

ensure that it is resonant collisions that provide the major contribution to

the thermal conductivity.

117
References

l] See, for example, the review article by P.R. Bernan and E.J. Robinson in

Photon-Assisted Collisions and Related Topics edited by N.K. Rahman and

C. Guidotti, (Uarwood Academic Pub., Chur, Switz. , 1982) pp. 15-34 and

other articles in this book.

2] P.R. Benaan and S. Stenholm, Opt. Comm. 24, 155 (1978).

33 M.I. Dyakonov and V.I. Perel, Sov. Phys. - JETP 20, 997 (1965).

4] A recent review is given by E.L. Lewis [E.L. Lewis, Phys. Rep. 58, 3 (1980)].

118
Laser-Cooled and Trapped Atoms , (Proceedings of the Workshop on Spectroscopic

Applications of Slow Atomic Beams, held at NBS, Gaithersburg, MD, 14-15 April

1983) Ed. by W. D. Phillips, Natl. Bur. Stand. (U. S.) Spec. Publ. 653 (1983).

A Proposed Study of Photon Statistics in Fluorescence

Through High Resolution Measurements of the Transverse Deflection

of an Atomic Beam

K. Rubin and M. S. Lubell*

Department of Physics, The City College of C.U.N.Y. , New York, NY 10031

Information about the photon statistics in fluorescence can be

obtained from the study of the variance in the angular deflection of

a beam of atoms interacting with a transverse laser beam. The

quantity of interest, Q = (An2 - n)/n, is a measure of the

departure of the photon statistics from a Poisson distribution, where

n is the mean and An2 is the variance of the number of photons

emitted by a "two-level" atom. We demonstrate in this paper that our

existing apparatus has sufficient resolution to make a statistically

significant measurement of Q.

1. Introduction

The photon statistics associated with resonance fluorescence are of considerable

fundamental interest and have been the subject of several recent papers. ^

^ In

particular Mandel''" has shown that it is possible for the statistics to be Poisson,

2 sub-Poisson, or super-Poisson according to whether the variance, An , of the

number of emitted photons is respectively equal to, less than, or greater than n,

the mean of the number of photons emitted. For a two-level system, Mandel defined a parameter Q, given by

Q = An2/

n

(1) n

(which is a measure of the deviation of the statistics from a Poisson distribution)

and showed, for example, that at resonance the value of Q should be -3/4. It is

*Alfred P. Sloan fellow.

119
important that a measurement of Q be made to verify Mandel's predictions. A

possible method of measuring Q, first suggested by Cook, is the study of the

deflection of a beam of atoms interacting with a transverse laser beam. To make a meaningful measurement of Q, however, requires a high resolution atomic beam

apparatus. We propose to use such an apparatus already in existence to perform the

measurement, and we will show in this paper that the apparatus has sufficient

resolution to accomplish the task.

2. Proposed Experiment

The atom-laser beam geometry is shown in Figure 1. The deflection of the beam

is due to absorption and a subsequent spontaneous re-emission of photons. As the

Figure 1. Atom-laser beam geometry.

atom travels through the laser field it absorbs photons and consequently recoils in

the 4x direction. If, in the time it takes to travel through the laser field the

atom absorbs n photons it will acquire a transverse momentum equal to (nhv/c), where

h is Planck's constant , v is the near-resonant laser frequency, and c is the speed of

light. The n photons will be reradiated in a random direction, within the dipole

distribution, adding a random component to the atom's momentum in the +x direction.

The average angular deflection <j) of an atom of mass m initially travelling with a velocity v_ is then given by

LASER n

BEAM_

nhv (2) cmv

0

120
for small The variance In <$> is due to both the statistical fluctuations in n and

the randomness of the spontaneous decays. Thus for a monoenergetic beam of zero initial angular spread, the variance of the beam after interaction with the laser

takes the form

in which case Q can be expressed in terms of Acf) and <f> . However, in unfolding the

experimental data, the initial angular dispersion and velocity distribution of the

atom beam must be taken into account. Both of these quantities produce additional

variances in <f> . Consequently Q can be expressed as

^cmv

Q^

^

Acf, 2 - Acf) 0 2 - S

2

<j> 2

- - ^ • "J"'

(4)

2 2-2

where A<J> 0 is the angular dispersion in the initial beam and S <j> is due to the

velocity distribution in the beam. If the velocity spread is small, S can be

2 written as S ^ 2Av/v_. For an accurate measurement of Q both At)) and

2 2 2

S <j> must be small compared to A(f> . These conditions will be satisfied if

and

Avv 2n 2 (6)

For example, if n = 100, Eqs. (5) and (6) yield the requirements

121
2 1/2 -4

(At}) ) < 3 x 10 rad and v/v

Q

<l/20. Both of these requirements can

be satisfied with our present apparatus.

In addition to satisfying the requirements expressed by Eqs. (5) and (6), the

atom beam must be prepared as a two-state system. For sodium, all the experimental

conditions can be met in a spatially narrow, velocity-selected beam prepared in the

F = 2, mp = 2 hyperfine ground state. If the state-selected sodium beam

intersects a circularly polarized laser beam tuned to the transition 3

2

S

1 ^ 2

(F=2,

mj,=2) to 3 V^.^i^-^* mj,=3), a two-state approximation can be used since the

laser field will cause the atom to oscillate between the (F=2, 1^=2) and (F=3,

mj,=3) states until a spontaneous decay occurs with such a decay returning the atom

to the initial (F=2, 1^=2) ground state.

The velocity and state selection can be accomplished as shown in Figure 2. The

offset source together with the inhomogeneous magnetic field and system of slits

9 serve as a velocity selector. It can be shown that for the given dimensions

and with the magnet operating in the high-field region, the range of velocities in

the beam exiting from slit will be given by (Av/v^) ^(s^/d), where s^ is

the width of the source slit and d is the displacement of the source slit from the

beam axis. A ratio of 1/50 is easily attainable. The state selection is

Q

accomplished by the use of the E-H gradient magnet. This magnet, operated in the

low-field region, is adjusted such that the state of interest (F=2, m =2) passes

r through undeflected. For this state the magnetic force is balanced by the electric

force while for other hyperfine states angular deflections will occur that will

prevent them from passing through slits and S^. The arrangement shown in

Figure 2 therefore will provide a beam in the (F=2, nij,=2) hyperfine state with the

required range of velocities. The angular divergence of the beam at the interaction

region for the given dimensions will be less than 4 x 10 ~* rad, well within the

required range of values.

A scale drawing of the apparatus is shown in Figure 3, with a surface-barrier

(hot-wire) atom detector (labeled I) located 130 cm downstream from the interaction

region (labeled G) . For the given geometry the intensity of the beam arriving at

the detector can be estimated easily. Assuming that the source is operated at a vapor pressure of 10 ^ Torr and that the dimensions of all slits are

8 2 25 ym x 2.5 mm, M.0 atoms/cm s will arrive on the beam axis at the plane of the

detector with the desired velocity distribution and state selection. This

-14

corresponds to ^2 x 10 A leaving the hot-wire detector. With the use of an

electron multiplier having a gain of 10"*, the final detector current will be

-9

^2 x 10 A, about ten times greater than the background noise. Furthermore, if

the laser beam is modulated and phase sensitive detection, employed, recoil currents
ATOMIC

BEAM

INHOMOGENEOUS E-H GRADIENT

DIPOLE MAGNET MAGNET

Figure 2. Schematic diagram of experiment showing the inhomogeneous dipole-magnet velocity selector and

the E-H gradient magnet

8

state selector. The slits, SQ

, S^, and are all taken to have dimensions

25 yum x 2.5 mm.

Top View

Figure 3. Scale drawing of apparatus showing atom source, A; bellows, B; velocity selector, C; adjustable slits, D; E-H gradient magnet, E; beam flag, F; interaction region, G; drift space, H; surface

barrier atom detector, I; dye laser, N; argon ion laser, 0; mirror, P; power metter, Q; and stepping

motor, R. 123
that are less than 1/100 of the initial beam can be measured. Thus the beam

intensity should be quite sufficient for the proposed measurements.

If the atom beam detector has a transverse resolution of 25 urn, the measurements

can be carried out with an angular resolution of 2 x 10 ^ rad. Now if the

distribution is Poisson, the angular variance will be given by

Assuming a laser operating single mode with a power output of ^50 mW (more than

sufficient to saturate the transition) and a beam width of ^2.5 mm, an atom will

scatter 'VLOO photons in passing through the laser beam. Therefore from Eqs. 2, 3

-3 2 1/2 and 6, for a Poisson distribution, <j> will be ^10 rad and (A<j> ) will be

-4

^4 x 10 rad. Both of these quantities should be readily observable. Thus

deviations from the Poisson character should be measurable, and Mandel's formulation

should be eminently amenable to test.

3. References

[1] L. Mandel, Opt. Lett. ^, 205 (1979).

[2] D. Stoler, Phys. Rev. Lett. 33, 1397 (1974).

[3] H. J. Carmichael and D. F. Walls, J. Phys. B £, L43, 1199 (1976).

[4] H. J. Kimble, M. Dagenais, and L. Mandel, Phys. Rev. Lett. 39, 691 (1977).

[5] L. Mandel, J. Optics (Paris) 10, 51 (1979).

[6] R. J. Cook, Opt. Commun. _35, 347 (1981).

[7] D. F. Walls and P. Zoller, Phys. Rev. Lett. 4_7, 709 (1981) and references

therein.

[8] B. Bederson, J. Eisinger, K. Rubin, and A. Salop, Rev. Sci. Instr. 31^, 852

(1960).

[9] K. Rubin, J. Perel, and B. Bederson, Phys. Rev. 117 , 151 (1960).

124
Laser-Cooled and Trapped Atoms , (Proceedings of the Workshop on Spectroscopic

Applications of Slow Atomic Beams, held at NBS, Gaithersburg, MD, 14-15 April

1983) Ed. by W. D. Phillips, Natl. Bur. Stand. (U. S.) Spec. Publ. 653 (1983).

Velocity Compression and Cooling of a Sodium Atomic Beam Using a Frequency Modulated Ring Laser*

M. S. Lubell1" and K. Rubin

Department of Physics, The City College of C.U.N. Y. , New York, NY 10031

A thorough understanding of the effects associated with velocity

compression and cooling of atomic beams through laser-atom interactions

requires a detailed study of the dipole radiation force. Such a study can

be carried out with the use of a high-resolution atomic beam apparatus and

a frequency-swept ring laser employed in a counter-propagating colinear

laser-atom beam geometry. The principles of the proposed experimental

program are discussed as well as directions for future research.

Keywords: Laser-cooling; atomic beams; dipole radiation force.

1. Introduction

The study of resonance radiation pressure on atoms is an old field dating back

to Einstein's theoretical analysis in 1917 of the motion of an atom in an

le

2

1 electromagnetic wave and Frisch's experimental observations in 1933 of the

deflection of an atomic beam by a transverse beam of incoherent resonance light.'

The development of high power tunable lasers during the last two decades has sparked

3 a new interest in the field, with emphasis being placed both on the fundamental

physics of dressed atoms and on the applications of radiation pressure to atom beam

4-7 * , 8 ~12 a 13-25 j -j . . , , 13 cooling, focusing, deflection, and velocity analysis as

24

precursors to the development of isotope separators, neutral atom storage rings,

6 7 2 5~27 28—32

and neutral atom ' and ion traps. Of the various applications there

is no doubt that the potential development of neutral atom traps has generated the

greatest excitement. That such a fascination for atom trapping should exist is no

surprise, given the wide range of applications which atom traps would have. Since

the confined atoms could be cooled continually, their velocities would be maintained

*Research supported in part by the National Science Foundation under Grant No

PHY83028868.

+ Alfred P. Sloan fellow.

125
3

substantially below 10 cm/s. Thus high resolution spectroscopy, free from both

first and second order Doppler effects as well as motional field effects, could be

carried out on a limited ensemble of atoms over long periods of time. The

implications for the precision measurement of fundamental constants and the

maintenance of time standards are obvious.

While the various applications of atom traps and radiation pressure in general

can engender almost endless speculation, the realization of useful devices requires

a fundamental understanding of the problem of an atom in strong inhomogeneous

radiation fields. The understanding must be experimental as well as theoretical, in

the sense that the practical limitations of laser-atom beam configurations must be

determined, particularly since most proposed neutral-atom trapping schemes have

-2 33 well-depths ( < 10 eV) that are quite shallow, judged both absolutely and in

comparison with ion traps.

Radiation forces on atoms fall into two categories—a recoil or scattering force

due to the absorption and spontaneous emission of a photon and a dipole force due to

the interaction of an induced dipole moment with spatial inhomogeneities either in

the impinging radiation field or in externally applied dc electric or magnetic

fields. In a colinear, counter-propagating laser-atom beam geometry the recoil

force can be employed to alter the velocity distribution of a beam. Specifically,

the distribution can be compressed, with the mean velocity of the beam shifted

3 toward extremely low values of the order of 10 cm/s. In a transverse laser-atom

beam geometry, the recoil force can be used to deflect an atomic beam, with

deflection angles of the order of 1 mrad easily obtained in a single interaction

region.

The dipole force, likewise, can be used to induce deflections of atomic beams.

However, in addition to this, it can be employed to produce strong focusing and

defocusing effects which are central to the various trapping schemes. While the

theoretical analysis of the recoil force is relatively simple, needing only a semi-classical treatment, the corresponding analysis of the dipole force is

considerably more complex, requiring a quantized field theory or dressed atom

approach if, as is often the case, the pumping of the excited state of the atom is

fast compared to the natural lifetime of the excited state and the laser line is

narrow compared to the natural linewidth of the atomic transition. The quantum

273433 3 6 A

2

mechanical fluctuations ' ' and photon statistics associated with such

resonance-fluorescence processes are of considerable fundamental interest and have

been the subject of several recent papers. Also of interest are the effective

polarizabilities and magnetic moments which are needed to characterize the dressed

20 states when external dc electric or magnetic fields are present

126
2. Experimental Approach

2.1 Laser Cooling and Velocity Analysis

Recent work at the National Bureau of Standards (NBS) has demonstrated that a colinear counter-propagating laser-atom beam geometry can yield substantial atom

beam cooling.^ In contrast to the NBS approach we propose a frequency sweeping

scheme rather than a Zeeman sweeping scheme to tune the appropriate atomic

transitions. The frequency sweeping procedure, while technically somewhat complex,

can be adapted more easily to the high-resolution atomic beam apparatus required for

detailed studies of the radiation force and moreover with such an apparatus can

produce sizable velocity compression as well as cooling.

For simplicity we consider the case of a sodium atomic beam and a ring dye laser

beam in a counter-streaming configuration. Then, atoms whose Doppler-shif ted

absorption frequencies lie well within the laser line will have a high probability

of absorbing a photon. If the laser intensity is far below the saturation limit,

the lifetime of the excited atom will be determined by the spontaneous decay rate,

which is approximately 6 x 10 s . Since the spontaneous-decay photons are emitted in random directions determined by the dipole radiation law, while the

absorbed photons are always traveling opposite to the atomic beam directions, a net

transfer of momentum occurs from the radiation field to the atoms in the beam, with

the consequence that the beam velocity is decreased.

In the case where n photons have been absorbed and spontaneously reemitted, the

average velocity decrease is nhv/mc where v is the frequency of the photon, mis the

mass of the atom, and h and c are respectively Planck's constant and the speed of

light. The stochastic nature of the spontaneous emission produces, in addition, a random distribution of velocities each with a magnitude /nhv/mc, as expected in a random walk problem with n encounters. Thus a spreading of the beam occurs as its

longitudinal velocity decreases, with the spreading proportional to v^n and the mean

decrease of the beam velocity proportional to n. In effect, the atomic beam is

cooled. For a fixed laser frequency and the colinear counter-streaming geometry

considered, the cooling process ceases when the reduction in the beam velocity

Doppler shifts the resonant absorption frequency out of the laser line.

Consider the case of a sodium atom exposed to 3S-3P resonance radiation from a single moded laser with a linewidth of 2 MHz. The Doppler shifted frequency, v' , is

given in terms of the unshifted frequency, by the expression v' = vy(l - 8), where

2 -1/2

Y = (1 - B ) and g = v/c with v the speed of the atom. For atomic

velocities v 'V' 10^ cm/s , v' can be approximated by v

1 ^ v(l - v/c), so that a
change in velocity Av produces a change in frequency given by

Av1 = -(v/c)Av = -(l/X)Av. If an atom undergoes spontaneous emission n times, Av in

turn is given by Av ^ nhv/mc. Since X = 5890.6 A for the case considered and

hv/mc * 3 cm/s, the spontaneous emission of about 40 photons reduces the velocity by

120 cm/s, an amount sufficient to shift the absorption frequency out of the laser

line. Thus the velocity profile of the atomic beam will have a gap in it of width

120 cm/s centered at a value determined by the mean Doppler-shif ted laser

frequency. If the 3S-3P transition is saturated, spontaneous emission will take

place at the rate of 3 x 10^ photons/s. Therefore a sodium beam travelling at a mean velocity of ^10 cm/s will have a hole of width 120 cm/s burned into its

velocity distribution after it has travelled a distance of only 0.13 cm.

If the laser providing the Doppler shifted resonance radiation is operated

multi-mode rather than single-mode, a series of holes will be burned into the

velocity distribution separated by a distance 5v = XSv, where Sv is the

mode-spacing. Evidence of such holes has been found in exploratory studies at CCNY,

where a multi-mode dye laser with 500 MHz mode-spacing and a 20 MHz linewidth was

used, yielding the expected hole separation of ^3 x 10 cm/s.

From the foregoing description it is clear that for the purpose of slowing down

an atomic beam and compressing its velocity distribution, some means must be devised

to keep the atoms in resonance with the incident radiation as their velocities are

reduced. Added to this problem are the difficulties introduced by the hyperfine

structure which can cause populations to build up in states which are completely out

of resonance with the incident radiation. In order to overcome these complications

we will consider a cooling scheme which utilizes circularly polarized light from a frequency swept single-moded dye laser.

The energy levels, and transitions, pertinent to the cooling scheme are shown in

Figure 1. As can be seen, the ground state hyperfine splitting is 1771 MHz,

considerably larger than an achievable single-mode laser linewidth of ^2 MHz. In

general, the sodium atomic beam will have populations in both the F = 1 and F = 2

2 3 hyperfine levels of the 3

^2_/2 S roun<* state. The v -Maxwellian velocity

profiles for this situation are illustrated in Figure 2(b) for effusive flow of

sodium from an oven at a temperature of 750 K.

Suppose at the outset the laser is tuned to a frequency which drives the T

19

4 transition (shown in Figure 1) for atoms at the lower end (^10 cm/s) of the

velocity distribution. If M = +1 circularly polarized light is employed, only

r the M , = 0, +1, +2 sublevels of the F' = 2 hyperfine state will be excited as

F

indicated in Figure 1. No other transitions will be driven initially because of the

level spacings, the laser linewidth, and the low-velocity Doppler tuning condition.

128
P

3/2 / 59MH2

-l— 2

3P

35 MHz

16 MHz

I r 0

515.5 GHz

JL

\ 189 MHz V_i_

508.93THZ

(5890.6 A)

3Si

1771 MHz

Figure 1. Energy level diagram of

sodium. Shown at the left are the

frequency spacing s of the hyperfine levels of the 3S and 3P states

together with the allowed optical

transitions from the 3S to 3P

manifold. The T^

2

and 1^

transitions (indicated in bold) are

the transitions to be used for the

beam cooling process. Shown at the

right are the relevant transitions

between hyperfine magnetic substates

for the circularly polarized light

used in the cooling process. Single

lines are used for transitions from

2

the 3 Sjy2> F = 1 sublevels and

double lines, for transition from

2 the 3 F = 2 sublevels.

The dashed double line indicates an

allowed transition from a sublevel

which has no available population

during the cooling process.

Consequently, the F = 2 hyperfine level of the ground state will be optically

pumped, with the M = -1, 0, +1, +2 substates all represented.

r If the laser frequency now is swept downward as shown in Figure 2(a), the

depletion of the F = 1 velocity profile will be swept upward in velocity. For a frequency sweep rate of 1.5GHz/ms, a regime which is now technically accessible,

and a laser linewidth of 2 MHz any atom will remain in resonance for 1,3 us, long

enough under saturation conditions for 40 spontaneous decays to occur. Thus

substantial pumping of the F = 2 hyperfine state will take place. This situation is

illustrated in Figure 2(b) at a time t^ = 0.28 ms after the start of the frequency

sweep. As the sweep proceeds, the frequency of the laser will eventually come into

resonance with the low-velocity Doppler-shif ted ^ rans^^on ' This situation is

illustrated in Figure 2(c) at a time - 0,13 ms after the start of the sweep.

The T AM = +1 resonance absorption light optically pumps the M = +2

2. j r F

sublevel of the F = 2 hyperfine state, as shown in Figure 2(d) at a time

t^ =2,0 ms after the start of the sweep.

129
v(cm/s) y (cm/s) v (cm/s)

Figure 2. Illustration of the laser cooling process. Shown in the upper left

figure is a plot of the swept laser frequency as a function of time. Shown in the

remaining five figures are plots of the velocity profile of the sodium atomic beam

at the five times, t^ through t,., indicated in the upper left figure. A

temperature of 750 K is assumed for the sodium oven. The insets in each of the

five velocity profile figures display the relevant transitions induced by the

circularly polarized light. The notation follows that of Figure 1. The locations

of the Doppler tuned resonance for the T^ and transitions are indicated

by arrows on the velocity axis. Profiles for the F = 1 and F = 2 manifolds of the

2

3 8 roun<^ state are indicated respectively by dots and hatches. The cross

hatching in the lower figures, representing the profile of the F = 2, M = +2

r magnetic substate, shows the buildup of the population in this state and, in the

figure at the extreme right, the onset of velocity compression and cooling.

By the time the laser is swept into resonance with the low-velocity

Doppler-shifted T__ transition only the M = +2 sublevel will be populated,

ZZ t thereby precluding any excitation to the F' = 2 hyperfine state with AM^, = +1

circularly polarized radiation. Thus after 3 ms the entire population of the

130
velocity profile will be contained in the M

p = +2 sublevel of the 3 F=2

hyperfine state, and only transitions, upward or downward, between the

M = +2 and M , = +3 sublevels will be possible under the influence of

F F

A Mj, = +1 circularly polarized radiation. This condition, shown in Figure 2(e),

will be achieved within a beam path length of 665 cm. Now suppose the laser frequency is swept upward as shown in Figure 2(a). Both

cooling and velocity compression will take place if the sweep rate is sufficiently

low to stay in step with the cooling rate. Since the atoms are slowed down at the

rate of 3 cm/s per spontaneously emitted photon, then at saturation, where

spontaneous emission takes place at the rate of 3 x 10^ photons/s, the atoms will

8 2

be slowed down at a rate dv/dt = 0*9 x 10 cm/sec . With the frequency sweep

rate given by dv'/dt = -(1/X) (dv/dt )

, a rate of 1#5 GHz/ms will suffice for

X = 5890.6 A. The cooling and compression are illustrated in Figure 2(f) at a time

tr = 1.45 ms after the onset of the upward sweep. Assuming that atoms initially

3 5

travelling at velocities of up to 2 x 10 cm/s are to be brought essentially to

rest, a beam path length of approximately 225 cm will be required. Thus in order to

accommodate both the pumping and the cooling a beam path length of 890 cm must be

provided. A scale drawing of the required apparatus is shown in Figure 3, with a number of beam-forming and beam-diagnostic devices included.

In the foregoing discussion we have assumed perfect circular polarization for

the exciting radiation. In actuality, with the use of a polarizing beam splitter

and a Soleil-Babinet compensator, a circular polarization of 0.995 can be obtained.

This deviation from unity, combined with the finite laser linewidth, will cause some

atoms during the cooling process to undergo transitions (labelled * n ^^8ure 1)

to the F' = 2 excited hyperfine state from which a fraction will be lost by eventual

decay to the F = 1 ground hyperfine state. However, for a laser linewidth of 2 MHz,

a hyperfine separation of 59 MHz between the F ' =3 and F' = 2 excited states, and

the required number of transitions of 10~* during the cooling process, fewer than

10 percent of the atoms should be so lost, assuming a laser power of ^10 mW.

In undergoing the cooling process, the beam will acquire transverse velocities

2 lying in the range of 5 x 10 cm/s, corresponding to a temperature of 0.02 K.

Under reasonable experimental conditions, the longitudinal velocity will probably be

at least an order of magnitude larger. The actual determination of the longitudinal

velocity of the cooled beam can be accomplished by a technique employed in the CCNY

exploratory studies; namely, deflection in an inhomogeneous magnetic field. In

order to avoid pumping effects, however, the laser beam must be blocked during the

velocity measurement. This task could be most easily accomplished with a Pockels

cell shutter.

131
Figure 3. Scale drawing of existing atomic beam apparatus with drift region H

added, components shown are A, sodium source; B, bellows; C, velocity selector;

44

D, adjustable slits; E, E-H gradient magnet for state selection; F, beam flag;

G, test chamber; H, drift region with deflector magnet at downstream end; I, atom

beam detector; J, gate valve; K, LN

2

traps; L, water baffles; M, diffusion

pumps; and N, laser.

2.2 Dipole Force Studies

The use of the atomic beam apparatus illustrated in Figure 3 will also permit

20 the dipole force to be studied in detail. Since all optical trapping methods

rely on the effects of this force, a thorough experimental investigation of its

properties in a realistic situation must be regarded as a prudent course of action.

In order to permit treatment of the atom as a two-level system, the atomic beam must

be prepared with a narrow velocity spread in the F = 2, M

f = +2 hyperf ine level,

characteristics readily achieved with the use of the velocity selector and E-H

44 gradient magnet shown in Figure 3. With collimating aperatures having

dimensions of 25 urn x 2.5 mm and with the alkali oven operated for a source pressure

132
of 0.1 Torr, both standard running conditions, a beam flux of ^10 atoms/cm s with a divergence of <4 x 10 rad can be obtained. If this narrow, well

collimated beam now intersects at right angles in chamber G of Figure 3 a circularly

polarized laser beam chosen for = +1 transitions, an effective two-state

system results. By adjusting the width, w, of the laser beam, the time which an

atom spends in the radiation field can be varied, and by adjusting the laser power,

I, and the detuning factor, Aa>, the rate of spontaneous emission can be varied.

Therefore from measurements of the beam deflection, the magnitude of the centroid

and variance of the dipole force can be examined as functions of w, I, and Auj,

parameters crucial for the design of optical traps.

Without too much effort it can be shown that in the absence of spontaneous

emission, the values of w, I, and Aw can be chosen so that a thermal atom passing

through the laser will suffer an impulse giving rise to an angular displacement of

-4

^10 rad with respect to the incident direction. Such a deflection is just

resolvable with the experimental conditions assumed. Cooling of the beam prior to

the interaction with the transverse laser will obviously enhance the effect. From a 45 dressed-state analysis it can further be shown that for a judicious choice of

laser parameters, the effect of spontaneous emission on the centroid of the dipole

force can be neglected. The variance in the force, however, will be dominated by

recoil effects associated with spontaneous emission, resulting in a beam spreading

-4

angle of M.0 . Thus, any effects associated with predicted fluctuations of

46 dressed atom states induced by spontaneous emission will be masked.

Conceivably, they could be uncovered through the presence of interference phenomena,

but the theoretical understanding of these phenomena is too imperfect to predict the

size of their effect.

3. Visions of the Future

While much attention of future applications of cooled beams has rightfully been

devoted to their use as injectors for neutral atom traps, several other possible

applications should be mentioned. Low atom beam velocities, for example, will make

neutral atom storage rings based upon dipole magnetic deflectors and hexapole

magnetic f ocusers a distinct possibility. With the high atom densities which might

be achieved with such devices, low rate processes particularly those related to rare

isotopic species, could be studied effectively. Applications to efficient isotope

47 48 separation and the study of collective effects * in strong radiation fields

also deserve to be mentioned as prospects for the future. Doubtless, as compressd , cooled beams become a reality many other uses will become self evident.
4. References

[I] A. Einstein, Phys. Z 18_, 121 (1917).

[2] 0. Frisch, Z. Phys. 86, 42 (1933).

[3] See for example review articles by A. Ashkin, Science 210 , 1081 (1980) and

A. P. Kazantsev, Usp . Fiz. Nauk. 124 , 113 (1978) [Sov. Phys. Usp . _21, 58

(1978)] and references therein.

[4] T.W. Hansen and A.L. Schawlow, Opt. Commun. JL3, 68 (1975).

[5] V.I. Balikin, V.S. Letokhov, and V.I. Mishin, Pis'ma Zh. Eksp. Teor. Fiz. 2_9,

614 (1979) [Sov. Phys. JETP Lett. 29, 560 (1979)]; Zh. Eksp. Teor. Fiz. 78,

376 (1980) [Sov. Phys. JETP 51 , 692 (1980)].

[6] V.S. Letokhov, V.G. Minogin, and B.D. Pavlik, Opt. Commun. r9, 72 (1976); V.S.

Letokhov, V.G. Minogin, and B.D. Pavlik, Zh. Eksp. Teor. Fiz. 7_2, 1328 (1977)

[Sov. Phys. JETP 4^5, 698 (1977)] .

[7] W. D. Phillips and H. Metcalf, Phys. Rev. Lett. 4_8, 596 (1982); J.V. Prodan,

W.D. Phillips, and H, Metcalf, Phys. Rev. Lett. 49, 1149 (1982).

[8] J.E. Bjorkholm, R.R. Freeman, A. Ashkin, and S.B. Pearson, Phys. Rev. Lett.

41, 1361 (1978).

[9] D.B. Pearson, R.R. Freeman, J.E. Bjorkholm, and A. Ashkin, Appl. Phys. Lett.

36, 99 (1980).

[10] G.A. Askar f yan, Zh. Eksp. Teor. Fiz. 4_2, 1567 (1962) [Sov. Phys. JETP 15_, 1088

(1962)] .

[II] D. Grischkowsky, Phys. Rev. Lett. 24_, 866 (1970).

[12] J.E. Bjorkholm and A. Ashkin, Phys. Rev. Lett. 32, 129 (1974).

[13] A. Ashkin, Phys. Rev. Lett. 25, 1321 (1970).

[14] R. Schieder, H. Walther, and L. Woste , Opt. Commun. _5, 337 (1972).

[15] J.L. Picque and J.L. Vialle, Opt. Commun. _5, 402 (1972).

[16] J.E. Bjorkholm, R.R. Freeman, and D.B. Pearson, Phys. Rev. A 23, 491 (1981).

[17] R.J. Cook and A.F. Bernhardt, Phys. Rev. A 18, 2533 (1978).

[18] R.M. Hill and T.F. Gallagher, Phys. Rev. A 12, 451 (1975).

[19] B. Jaduszliwer, R. Dang, P. Weiss, and B. Bederson, Phys. Rev. A 21, 808

(1980).

[20] M.H. Mittleman, K. Rubin, R.H. Callender, and J.I. Gersten, Phys. Rev. A 16,

583 (1977).

[21] A.C. Tarn and W. Happer, Phys. Rev. Lett. 38, 278 (1977).
[22] A.F. Bernhardt, Appl. Phys. j>, 19 (1976).

[23] P. Jacquinot, S. Liberman, J.L. Pic que , and J. Pinard , Opt. Coramun. J3, 163

(1973).

[24] A.F. Bernhardt, D.E. Duerre, J.R. Simpson, and L.L. Wood, Appl. Phys. Lett.

^5, 617 (1974).

[25] A. Ashkin, Phys. Rev. Lett. _24, 156 (1970).

[26] A. Ashkin, Phys. Rev. Lett. 40, 729 (1978).

[27] J. P. Gordon and A. Ashkin, Phys. Rev. A 21, 1606 (1980),

[28] D.J. Wineland and H. Dehmelt , Bull. Am. Phys. Soc . .20, 637 (1975).

[29] D.J. Wineland, R.E. Drullinger, and F.L. Walls, Phys. Rev. Lett. 40, 1639

(1978).

[30] W. Neushauser, M. Hahenstatt, P. Toeschek, and H. Dehmelt, Phys. Rev. Lett.

41, 233 (1978).

[31] W. Neushauser, M. Hohenstatt, P. Toeschek, and H. Dehmelt, Appl. Phys. 17, 123

(1978).

^~

[32] D.J. Wineland and W.M. Itano, Phys. Rev. A 20, 1521 (1979).

[33] See for example W.H. Wing, Phys. Rev. Lett. 4_5, 631 (1980).

[34] J.E. B jorkholm, R.R. Freeman, A. Ashkin, and D.B. Pearson, Opt. Lett. 5, 111

(1980). ~

[35] R.J. Cook, Phys. Rev. Lett. 4_4, 976 (1980).

[36] L. Mandel, Opt. Lett. _4, 205 (1979).

[37] D. Stoler, Phys. Rev. Lett. 3J3, 1397 (1974).

[38] H.J. Carmichael and D.F. Walls, J. Phys. B _9, L43, 1199 (1976).

[39] H.J. Kimble, M. Dagenais, and L. Mandel, Phys. Rev. Lett. 3_9, 691 (1977).

[40] L. Mandel, J. Optics (Paris) 10, 51 (1979).

[41] R.J. Cook, Opt. Commun. 35., 347 (1981).

[42] D. F. Walls and P. Zoller, Phys. Rev. Lett. 4_7, 709 (1981) and references

therein.

[43] W. D. Phillips, "Rapid Frequency Scanning of Ring Dye Lasers",

to be published.

[44] B. Bederson, J. Eisinger, K. Rubin, and A. Salop, Rev. Sci. Instrum. 31_, 852

(1960).

[45] C. Cohen-Tannoudji and S. Reynaud, J. Phys. 13 1£, 345 (1977).

135
[46] Actually fluctuations occur even in the absence of spontaneous decays, but

these arise from photo fluctuations in the laser field, and their effect is

small compared to those produced by spontaneous decays.

[47] N.I. Zhokova, A.P. Kazantsev, E.F. Kazentsev, V.P. Sokolov Zh. Eksp. Teor.

Fiz. 7j6, 896 (1979). [Sov. Phys. JETP 49, 452 (1979)].

[48] B. R. Mollow, Phys. Rev. A 12_, 1919 (1975).

136
Laser-Cooled and Trapped Atoms , (Proceedings of the Workshop on Spectroscopic Applications of Slow Atomic

Beams, held at NBS, Gaithersburg, MD, 14-15 April 1983) Ed. by W. D. Phillips, Natl. Bur. Stand. (U.S.)

Spec. Publ. 653 (1983).

Chirping the Light — Fantastic?

Recent NBS Atom Cooling Experiments

Electrical Measurements and Standards Division

John V. Prodan* and William D. Phillips • National Bureau of Standards

Washington, DC 20234

We have successfully decelerated and cooled a neutral atomic sodium beam using a counter-propagating laser beam tuned nearly resonant with the 0^ transition. In order to compensate for the changing Doppler shift as the atoms slow down, the laser frequency was rapidly scanned or "chirped". We have observed final velocities of 6 x lCr m/s, or about 0.5

of the initial velocity. As of yet we have not been able to cool a significant number of

atoms to lower velocities.

Key words: atomic beam velocity modification, frequency scanning, high resolution

spectroscopy, laser cooling.

Many of the difficulties in studying the spectrum of atoms come from the fact that these atoms are generally in motion. This motion causes the spectral lines to be Doppler broadened which, in turn,

limits the information one can get concerning the detailed structure of the atom. In the past several

years some very ingenious techniques, the so-called Doppler free spectroscopies, have been developed to drastically reduce 1st order Doppler broadening. Saturation techniques are sensitive to atoms in a very narrow velocity range and hence significantly reduce the problems due to the overall motion of the

atoms, while two-photon techniques simply cancel the 1st order Doppler shift.

Another way to deal with the problems imposed by Doppler broadening is to modify the velocity of the

atoms themselves. This technique has the advantage of not only reducing 1st order Doppler broadening

but also reduces 2nd order Doppler effects. Cooling atoms using near resonant laser light was proposed

several years ago [1] and recently we have made significant progress at NBS working with this idea

[2,3,7]. A major problem in using laser radiation to slow atoms is that as these atoms slow down, they

Doppler shift out of resonance with the laser. Using a changing Zeeman shift produced by a spatially

varying magnetic field, we were able to compensate for this change that occurs in the Doppler shift.

In this way, the fixed frequency counter-propagating laser remained nearly resonant with the atomic

transition as the atom slowed down to near zero (longitudinal) velocity.

An alternate way of compensating for the changing Doppler shift is to quickly frequency scan or "chirp"

the cooling laser [4]. For atoms moving at 1000 m/s, corresponding to a Doppler shift of 1.70 GHz, and

fully radiatively saturating the D transition (F=2 to F=3 sublevels) in sodium, the deceleration time

would be 1.07 ms. This would correspond to a maximum scanning rate for the laser of about 1.6 GHz/ms.

The capability of producing such a scan rate and scan width was demonstrated previously by Phillips

[5].

One of the motivations for our studying the cooling process using the chirped laser is the ability to look at details of the process not accessible in the spatially varying magnetic field scheme. During

our experiments using the magnetic field [3,7], we found that the observed density of slow atoms did

not have the velocity dependency that we had predicted. As of yet we do not understand the reasons for

this discrepancy. However, with the chirped technique we can investigate this density dependence by

varying the rate at which the laser scans as well as the magnitude of that scan. These two parameters

would correspond to the spatial rate of change of the magnetic field and the magnitude of the field

difference between the high and low field ends of the solenoid, respectively, in our earlier

experiments [3,7]. Neither of these two parameters could be easily varied with the solenoid design

that was used. The results given below represent the status of our experiments using the chirped laser

techniques.

*NBS-NRC Research Associate
One final point which needs to be addressed before proceeding with the results of the present

experiments is the problem of optical pumping that occurs in sodium. Previously [2,3,7], we dealt with

this problem using the same solenoid that provided the changing Zeeman shift. This solenoid was designed so that field created was the sum of a spatially constant "bias" field and spatially changing

"taper" field. The bias field was introduced to provide a means of overcoming optical pumping. This

field produces sufficient Zeeman splitting and shifts to enable us to make use of certain selection

rules that combine to virtually eliminate optical pumping. In the present chirping experiments, this

bias field was still used so that we could avoid most problems which might arise due to optical pumping

effects.

The experimental set-up is shown in Fig. la. The cooling laser is a tunable ring dye laser, modified

to scan rapidly and is described in Ref. [5]. It's frequency is modulated with a sinewave drive, Fig.

lb, whose phase is adjusted so that the most linear part of the modulation is centered on the open time

of the chopper. During the period when the chopper allows the laser to be transmitted, the frequency

is scanned from v-j to The amplitude and frequency of the sinewave modulation as well as the open

time for the laser chopper are all separately adjustable parameters.

ANALYSIS

LASER

SODIUM

SOURCE

COOLING

LASER

CHOPPING

WHEEL

P R E A M P BOXCAR GATED

AVERAGER AMP

COOLING LASER

INTENSITY

COOLING LASER

FREQUENCY

TIME

Fig. 1 (a) Block diagram of experimental set up

(b) Timing diagram

138
The cooling laser is directed in opposition to an atomic sodium beam. The majority of the cooling

takes place in a constant magnetic field of 0.064 T in a region about 0.8 meters long. The initial

frequency of the cooling laser, V] , is such that it is in resonance with atoms moving at a particular

velocity in the magnetic field. The velocity distribution of the atomic beam as modified by the

cooling laser is analyzed by a second very weak dye laser. This second laser intersects the atomic

beam at an angle of 11° from being co-propagating. As this laser is slowly scanned in frequency, the

fluorescence that it induces is directed by the collection optics to a photomultipl ier tube. The

resulting photo-current is processed by a boxcar averager which is set up to observe the current

generated after the chopping wheel has shut the cooling laser off. In this manner the modified

velocity distribution of the atomic beam is determined.

Some preliminary results of our chirping experiments are shown in Figures 2 and 3. Figure 2 shows a

0

I J2 3 DOPPLER SHIFT (GHz)

I—H 1 1 1 1 1 1 1 0 4 8 12 , VELOCITY < x I 0* M/S)

Fig. 2 Velocity distribution of the atomic beam for the case of (a) cooling laser blocked,

(b) cooling laser on but not scanned, (c) cooling laser scanned for 480 MHz, and

(d) cooling laser scanned for 750 MHz.

sequence of velocity distributions which clearly demonstrate the effects of scanning the cooling laser.

An unmodified distribution is presented in Fig. 2a for comparison. The arrow, designated V on the

abscissa, indicates those atoms in the velocity distribution with which the initial laser frequency

would be resonant, within the magnet). This frequency corresponds to an atomic velocity of about

139
1.1 x lO^m/s. To produce the distribution shown in Fig. 2b, the cooling laser was not scanned. A

peak in the distribution was produced at approximately 9.8 x 10^m/s. These atoms represent the ones which were initially in resonance with the cooling laser but Doppler shift themselves out of resonance

fairly quickly and hence were lost to the cooling process. Considering the power, amount of focussing,

the detuning (the difference between the frequency of the laser and the resonant frequency of the

atomic transition in the atom's rest frame) and the length of time that the laser interacted with the

atoms, the achieved velocity was not inconsistent with predictions made by Minogin [6].

In Figure 2c, the laser was scanned 480 MHz at a rate of 0.64 GHz/ms, well below maximum rate at which

atoms should be able to keep in resonance with the laser. We can see that atoms have been compressed

into a narrow velocity spike at 8.3 x 10 2 m/s with a FWHM of 40 m/s. To produce the distribution shown

in Fig. 2d, the scan width was increased to approximately 750 MHz (at a rate of 1 GHz/ms). The peak

representing the cooled atoms is at a velocity of 6.2 x 10 2 m/s with a FWHM of about 40 m/s. For both

of these two conditions, the velocity of the spike is close to what one would calculate assuming the

atom's velocity had resonantly followed the frequency scan from the initial to the final frequency of

the laser. Velocities predicted in this manner would have been 8.5 x 10 ^ m/s and 6.6 x 10 ^ m/s

respectively.

There are several differences between the two schemes, mentioned earlier, for cooling atoms. In addition to the ability to easily change the rate of cooling and the amount of cooling, the chirping

technique should also produce a different spatial distribution of atomic velocities than does the

varying Zeeman shift approach to cooling. For the spatially varying magnetic field situation, there

should be a unique maximum velocity associated with a given position in the solenoid for a given laser

tuning. However, for the chirping scheme, atoms of a given velocity could exist over a rather large

( a >

<b)

<c)

4 1 J

1 1 1 1 O I 2 f 3 DOPPLER SHIFT (GHz)

I

1 1 1 f 1 1 1—-—• 1 —

0 4 6 12 VELOCITY (i I O M/S)

Fig. 3 Velocity distribution of the atomic beam after a delay of (a) 500 us, ( D ) 800 uS and (c) 1200 uS » The cooling laser was scanned .87 GHz in 750 us. 140
distance, depending primarily on the length of the scan and length of the solenoid. In order to see if

such a situation exists, all that we have to do is to change the amount of delay time between when the

laser is turned off by the chopping wheel and when the gate of the boxcar is opened. This action

allowed us to observe slower and slower atoms originating from from a constant point in space when the

spatially varying field was used. However, with the chirping technique we should see atoms at a fixed

velocity but originating at different points in space.

The results of a study of the velocity dependence of the detected signal as a function of gate delay

are presented in Figure 3. Figure 3a shows the distribution obtained after a delay of 500 us and we see that there is a narrow group of atoms at a velocity of about 8.5 x 102 m/s. Figures 3b and 3c show

velocity distributions after a delay of 800 us and 1200 ys respectively. One can see that the cooled

atoms appear at essentially the same velocity for all three delay times. The atoms that produced the

signal shown in Fig. 3a came from a position about 0.5 m from the observation region (5 cm inside the

magnet) while the atoms represent in Fig. 3c came from 1.0 m from the observation region (still well

within the magnet). These results essentially confirm the predictions stated in the preceeding

paragraph.

There are still several items which remain to be done using this scanning technique. As of yet, we have not been able to produce atoms as cold as we were able to achieve using the varying magnetic field

scheme [3]. In fact we have not even been able to produce significant numbers of atoms below about 600

m/s. The reasons for this need to be determined. Although we have looked at the velocity distribution

produced by a few different scan rates and scan widths, we still have insufficient data to make any

statement as to the effects of these parameters. We plan to accumulate more data relating to these two

parameters in order to understand their effects. We would also like to investigate the effects of

focussing or defocussing the cooling laser as we have some data in this area using the varying magnetic

field scheme. In conclusion, while we have made some progress in understanding the details of the

cooling process using a chirped laser, there is still much more work that needs to done.

This work was supported in part by the Office of Naval Research

[1] T. W. Hansch and A. Schawlow, Opt. Comm. 1_3 68 (1975).

[2] W. D. Phillips and H. Metcalf, Phys. Rev. Lett. 48, 596 (1982).

[3] J. V. Prodan, W. D. Phillips and H. Metcalf, Phys. Rev. Lett. 49, 1149 (1982).

[4] V. S. Letokhov, V. G. Minogin and B. D. Pavlik, Opt. Comm. 19, 72 (1976).

[5] W. D. Phillips, Applied Optics 20, 3826 (1981).

[6] V. G. Minogin, Opt. Comm. 34, 265 (1980).

[73 W. D. Phillips, J. V. Prodan and H. Metcalf, these proceedings.

141
Laser-Cooled and Trapped Atoms, (Proceedings of the Workshop on Spectroscopic

Applications of Slow Atomic Beams, held at NBS, Gaithersburg, MD, 14-15 April

1983) Ed. by W. D. Phillips, Natl. Bur. Stand. (U. S.) Spec. Publ. 653 (1983).

Cooling of an Atomic Beam with Frequency-Sweep Techniques

R. Blatt,* W. Ertmer* and J. L. Hall*

Joint Institute for Laboratory Astrophysics, University of Colorado

and National Bureau of Standards, Boulder, Colorado 80309 U.S.A.

We review the ideas of velocity-modifying an atomic beam by resonant

scattering of intense light from a frequency-swept laser source. Some of

the realistic parasitic effects, particularly fluctuations and residual

transverse heating are considered. A computer Monte-Carlo simulation program is described which provides encouragement and insights useful in the

design of a real atomic beam apparatus. We describe a new rf /optical modulation method to generate the desired laser optical frequency and intensity

profiles versus time. This digital method provides the necessary sweep accuracy and program flexibility in a convenient and powerful way.

Key words: radiative cooling, laser spectroscopy, atomic beam velocity

distributions

The possibility of influencing the motion of atoms by resonance laser radiation has

received considerable attention for several years now [1], A particular interest in this

field is naturally the reduction of the atomic translational energy because high resolution spectroscopic experiments are ultimately limited by velocity-related effects. Since

the pioneering proposals by Hansch and Schawlow [2] and Wineland and Dehmelt [3] a large

amount of theoretical and experimental work has been done. The "cooling" of trapped ions was first demonstrated [4,5] in 1978, while the first experiments on neutral atoms were published by a group in Moscow [6] in 1979 and very recently by a group at NBS Gaithersburg

[7,8] in 1982. Trapped ions have been cooled to a residual temperature of a few milliKelvins, single ion detection has been demonstrated, and the general advantages of ion

trapping techniques for precision measurements have been demonstrated on various ions

[4,5,9,10]. All the reported experiments on neutral atoms have been done with Na atoms,

the basic scheme being an atomic beam and a counterpropagating cooling laser beam. In order to cool effectively the laser frequency must match the transition frequency of the

moving atoms. It has been shown, however, that a laser tuned to the maximum of an atomic

beam velocity distribution, but kept fixed in frequency, can change the mean velocity and narrow the width of the distribution [11]. But this scheme is unlikely to be successful

in achieving temperatures in the sub-Kelvin regime.

Two methods have been proposed to keep the resonance condition between the cooling

laser beam and the moving atoms even as they slow down. The first, reported by Letokhov

and Minogin [12], is based on scanning the laser frequency, but no further results using

that method have been published yet.

Permanent address: Johannes Gutenberg University, Mainz, West Germany.

Permanent address: University of Bonn, Bonn, West Germany.

Staff Member, Quantum Physics Division, National Bureau of Standards.

142
The second technique uses the Zeeman shift to keep the resonance condition and led for

the first time to very promising cooling results on neutral atoms [8]. However, the application of a strong magnetic field (several hundred gauss) can bring its own problems.

In the following, exploiting the ideas of Letokhov and Minogin, a proposal is made

to cool the atoms by rapidly sweeping the laser frequency under precise control. We are optimistic that the absence of magnetic fields coupled with precise time-domain control of

the laser frequency will allow clear interpretation of the final velocity distributions.

It could well be that, armed with what we hope to learn from these swept frequency experiments, it would be interesting to return to cw beam cooling using electric or magnetic

fields with optimized spatial gradients. However, the pulsed cooling method has its own advantages in producing a relatively high density "rod" of essentially-stopped atoms, .iust freely falling under gravity. The high resolution interactions with these atoms can thus occur away from their cooling experiences. Further, the "falling rod" geometry is attractive for successive excitations with longitudinal (possibly weakly-focused) laser beams (as

in the multiple-excitation Ramsey interference fringe method), and for output interrogation

using optical heterodyne techniques. Finally, the atomic beam machine may be oriented a bit off the true vertical to produce the "fountain beam" of emotional significance to atomic beamists and of possible utility in filling a weak trap for neutral species.

In order to study the basic properties of the frequency-swept cooling concept and to investigate the features of such a device, a Monte-Carlo simulation of the proposed experiment was performed. The results of that calculation agree with preliminary computations

based on a differential equation.

1. Basic Idea of Atomic Beam Cooling with Chirped Lasers

As mentioned above, the main problem in cooling an atomic beam by means of a counterpropagating laser beam is the changing Doppler shift. Once the laser is on resonance with

atoms of a certain velocity of the atomic beam, those atoms begin to slow down and clearly

walk out of resonance due to their altered velocity. To retain the resonance condition

either the frequency of the atomic absorption has to be adjusted [8] or the laser frequency

has to be tuned to the new absorption maximum.

The resonance frequency of an atom depending on its velocity is given by the Doppler

formula

v L

(t) -v 0

(l

where vq is the unshifted transition frequency of the atom, v(t) is the (now time dependent) velocity and c denotes the speed of light. Or, if the time dependent laser frequency

v^(t) were given, v(t) then reads

v

T (t)

v(t) = (1 -~Z ) • c

with its derivative

v(t) =

V 0

V 0

On the other hand the change in the velocity of an atom due to the momentum transfer

by absorbed photons is given by

h v

T (t)

v(t) =

m« c»VLi provided the laser intensity is high enough to ensure that the transition is well saturated, and assuming a two-level system. Here m is the mass of the atom, h is Planck's

constant and x is the natural lifetime of the excited state. To determine the frequency

143
dependence vj/t) and the velocity dependence v(t) we solve the resulting differential

equation:

h. VQ

v L

(t) - ov L(t) = 0 , a

= mc 2t The solution reads:

v T

( t ) = v L s at

where v

s denotes an initially given starting frequency, corresponding to the initial

velocity of the atoms. For the velocities this yields

v(t) = (1 - ^ eat

).c . v 0 —1 —1 If we take Na atoms as an example, we get the numerical value a

a 3 x 10 s which gives

a time to stop atoms of initially 1000 m/s in t 1 x 10 -3 This clearly indicates

that the exponential behavior of the frequency sweep is almost entirely determined by the

linear approximation of the exponential expression and thus the laser sweep is given by

v T (t) = v (1-hxt) L S which is indicated in Fig. 1. Due to the cooling and the corresponding frequency scan all

velocities less than the starting velocity ultimately come into resonance with the laser,

and will join the group of already-slowed atoms.

Atomic Beam

2(t) = 2 + v(t)

Basic Idea

Time

Fig. 1. Basic geometry and scan scheme for atomic beam cooling by

a frequency-swept laser.

Assuming a starting velocity for Na atoms

of 1000 m/s and a final velocity near zero, we

get a sweep constant of dvj/dt = 1.7 GHz/ms.

An atom requires about 33000 cycles of laser

Laser Beam 2, 2 + v(t) absorption and spontaneous emission to be

stopped.

2. Computational analysis of the chirped

cooling

In spite of its simplicity, this approach

indicates the feasibility of a frequency-swept

("chirped") laser atomic beam cooling. A se- rious calculation, however, has to take into

account many additional effects which have been

neglected in this first approach.

These are, in particular:

(a) The laser beam intensity. It has been assumed above that the transition is very

well saturated. That is not necessarily satisfied because of the gaussian spatial profile

of the laser beam and the intensity variation due to the spatial position of the atoms.

(b) Statistical absorption and emission processes. No statistical processes have been

included in the differential equation above. Due to the random absorption and emission the

position and velocity of the atoms are not determinate, giving some variations in the

intensity the atoms see during the cooling process, and of course in the final position and

velocity as well.

(c) Optical pumping. These effects have been totally neglected in the first approach. For a real atom, the two-level system approach can hold only approximately, so corrections have to be included.

144
(d) Experimental scan scheme. In the basic calculation only a simple frequency

sweep is taken into account; a more elaborate scanning scheme might be better suited for

ultimate cooling and narrowing of the velocity distribution. For example, it might be

useful to stop the laser's sweep and reduce its power at the end of the cooling cycle.

A general possibility to evaluate these effects should be included.

(e) More dimensional aspects. The approach above finds results only for a one- dimensional problem. More dimensions in velocity and position lead to coupled differential

equations and should be properly included.

(f) Dipole forces due to stimulated transitions. Only forces due to spontaneous scattering have up to now been included. Dipole forces that may lead to further cooling are to

be entirely included.

In order to obtain reliable results which will provide reasonable values for the design of any atomic beam cooling device based on laser chirping, and in order to understand

the basic physics of this process, we have to consider all the mentioned influences.

The scattering force is dependent on the saturation parameter P. Therefore it now

reads [13] sp _ ->

_ L J_ P m m» c i 1+2

with F the force induced by the spontaneous scattering processes, P is defined [14] as

P = B »^a(v)/A where B and A denote the Einstein coefficients for the correspondent transition, and o(v) is the spectral energy density. The vector notation accounts for the various dimensions and hence for the angle between the velocity v and the counterpropagating

laser beam (k^). The saturation parameter depends on the intensity I and detuning Av

,

which are in turn functions of position and velocity. P is given by [14]

P(v) = Kr.z) 1 I | i f 2Avi

sat 1 +

n with

=

1 ^n 2%i

2_l_

2P 2 2

T , v 0 w 2 2 . \~ 2 I(r,z) = —j e . » w = w

Q + —2~2 z HW % Wq

4ir^v1 «h»v T"\ I n

sat . 2 X where Pq denotes the applied laser power in watts and wq the laser beam waist radius in the

focal plane of the gaussian beam.

The experimental scanning scheme can be included by proper calculation of the detuning

Av = Vq(v) - v L

(t)

as a function of the atomic velocity and the time dependent laser frequency.

The dipole forces are given by [14,15]

f,. =--aVl dip c where a is the polarizability of the corresponding transition.

A full theoretical treatment of the cooling problem leads to a differential equation

of the Fokker-Planck type [13].

145
In order to calculate the cooling process for laser chirping including all experimental requirements, two main approaches were pursued. The first, a Monte-Carlo simulation

study is based on a step-by-step calculation and hence requires considerable calculation

time. The second approach, based on an integration of a simplified Fokker-Planck equation,

is much more involved in accounting for all the experimental details.

A. Monte-Carlo Simulation

This Monte-Carlo program simulates the cooling process of an atomic beam by a counterpropagating, fast scanned laser beam. The basic idea of this program is to take into account each change in velocity or position of the atoms caused by the subsequent absorption

or emission of photons.

Input: —Waist Configuration —Scanspeed —Laser Power —Scan Time t_,v

Atom Lost

Initial Conditions: —Position s —Velocity v According Maxwell Distribution

Absorption: —Calc. Angle (k, v) New v

Spontaneous Emission: —

t

sp (Time Until State Decays) —P (Emission Angle) —

F

dip (Dipole Force)

No

(Calc. Next Atom)

Fig. 2. Flow chart for MonteCarlo simulation.

According to their nature, all statistical processes are calculated in a random manner, to include all effects which

contribute to the residual velocity distribution or density in a given atomic beam.

Another important aspect is the con- sideration of all experimental details:

e.g., atomic beam properties, laser beam

configurations, and sweeping of the laser

frequency. Thus we are able to learn

easily the main influences of experimental

parameters on the cooling process and can start our experiment with an optimized experimental setup and sweep-scheme. Moreover, in the future, comparison of these

calculations with experimental results will

be a versatile tool in interpreting the

experiment.

The flow chart of the program is given

in Fig. 2. The input parameters are the

laser beam properties (waist and waist

location, laser power) and the sweep scheme

(scan speed, one or more laser frequencies,

timing of the frequency scan).

In the beginning the program generates

randomly a set of, for example, 100 atoms

with the spatial density and velocity distribution of a usual atomic beam. Each

atom is then treated separately during the

cooling process in the following way: first

the atom absorbs a photon and changes its

velocity according to the It-vector at that

position. Then the lifetime of the excited

state is calculated randomly taking into

account the influence of the saturation

parameter P = P(r,v,t) for the" time that

the atom spends in the excited state. After calculating randomly the emissionangle of the spontaneously emitted photon, the resulting new position and

velocity are calculated including the

146
influence of the dlpole forces. The dipole forces are considered to be constant between

the two absorptions. Before the next absorption takes place it is verified that the atom

has not walked out of the atomic beam limits and that the end of the sweep time is not reached. Then the atom absorbs the next photon, and so on. Having reached the end of

sweep time or the atomic beam limits, the cooling process is finished and the next atom is

cooled. The result of each complete cooling process is the end-position and end-velocity

of all atoms, in addition to other interesting parameters, such as the saturation parameter

at the end.

All computations were done for Na atoms, the atomic beam divergence is assumed

(according to the design of the apparatus) to be 3 mrad.

Figure 3 shows the random initial velocities, with the transverse velocity plotted

versus the longitudinal velocity according to a Maxwellian distribution in the atomic

beam. The cooling process transforms these "initial velocities" into the "final velocities" after one cooling cycle corresponding to one laser sweep. A typical final distribution is given in Fig. 4. The run was taken with tmax

= 1.140 ms, dvj^/dt = 1.489 GHz/ms,

starting at a maximum initial velocity of 1000 m/s.

Laser "compression" of the original longitudinal velocity distribution — with a maxi- mum at 760 m/s and a width of several hundred m/s — leads to a final distribution shown in

Fig. 5. The width of each bar is 1 m/s thus indicating a residual width of about 3 m/s.

The choice of the stopping time controls the final longitudinal velocity.

03 Is>

•Ha

o

3.0

0.0

-3.0

x x X

x* x x

X

X XX \ x

X

X\

X

X x x

X X w

x

200. 400. 609. 800. longitudinal velocity (m/s)

1000.

Fig. 3. Initial velocities in the atomic beam according to a beam-Maxwellian distribution (each x represents one atom)

147
COH

>

longitudinal velocity (m/s)

Fig. 4. Final velocities in the atomic beam after one frequency scan (square indicates a lost atom).

CDC

C

cfl X!a

u

CD

.£>E3

C

longitudinal velocity V (m/s)

Fig. 5. Histogram of the final longitudinal velocities corresponding to Fig. 4.

148
c

e

too

M-lO

u

0)e

c

20

16

12

0.0 transverse velocity V^(m/s)

8.

Fig. 6. Histogram of the final transverse velocities corresponding to Fig. 4.

The final transverse velocity distribution also behaves as expected. Initially

the transverse velocity width is given by the atomic beam divergence and amounts to about

3 m/s. From Fig. 6 we find a final width of about 5 m/s, which is due to "transverse

heating" caused by the emission processes. However, this value is already reduced by

the dipole forces which help in focusing the atomic beam. A case was run with the same initial parameters as in Fig. 6 but without dipole forces. The transverse velocity width

came out to be about 7 m/s, indicating in the previous case the well-known transverse

focusing caused by the dipole forces [16].

The program allows us also to change experimental parameters such as laser-power,"

-waist, and -frequency sweep during each cooling process. If it is desired, the program

can also calculate the induced processes randomly in place of using the average of the

population time in the excited state or ground state.

Although quite successful in calculating experimental details, the program is far from

being complete. The known limitations of the present program are the assumption of a pure

two-level system, isotropic emission of the spontaneous photons and a calculation only in

the x, y plane. Therefore, the optical pumping process and the related radiation characteristic for spontaneous emission as well as calculations in three-dimensional space will

be included in future. The latter is especially important when gravity is to be taken into

account

When all these effects are included, the program will be well suited to calculate the

filling process of neutral atoms into an "optical trap," as well as calculating the longand short-term behavior of atoms in those traps. The generality of our calculation allows

detailed evaluation of any trapping scheme.

149
B. Integration of Simplified Equation of Motion

To check this program as well as to learn more about diffusion and related effects

influencing the cooling process, we are also trying a different approach to calculating

this cooling process. It is similar to calculations made by Minogin [13]. In these calculations, kinetic equations are derived which evaluate the distribution functions of twolevel atoms interacting with a laser beam. In addition to Minogin's computations, these

equations of the Fokker-Planck type also take the laser frequency sweep scheme into consideration. As a consequence, computer solution of these equations has encountered several

numerical problems. Due to these difficulties only preliminary results are available at present. However, in cases where the two approaches can be compared, all important features, such as final velocity and velocity width, show good agreement with those provided

by the Monte-Carlo method. We are anxious to confront all these calculational "experiments" with laboratory data!

3. Experimental Design

After first calculations, we designed an apparatus to cool an atomic sodium beam by a periodically fast-scanned counter-propagating laser beam.

2 2 The hyperfine structure splitting of the cooling transition $1/2 *~> P3/2 * s snown

in Fig. 7. As can be seen from this figure, optical pumping Is one of the main problems

in cooling sodium. An elegant way to overcome it is to use two laser frequencies : one laser frequency is tuned to the transition 2 S

1

/ 2 (F=2 ) ++ 2 ?^/2^ F=^ ^ see Fi £* 7 ^ witn

circular polarization, the other one is tuned to the 2

S

1 / 2

( F=1 ) 2p

3/2 ( F=2 ) transition. The first realizes a nearly perfect two-level scheme (F=2, mF =2 <-> F=3, mp=3)

whereas the second pumps the atoms in the beginning from the F=l level into the F=2 level

of ^S]/2. During the cooling the second field also restores atoms lost due to imperfect

polarization etc.

Na I(23Na) = 3/2

F

3

60 MHz

3

2

P-3/2 2 36 MHz

16 MHz

0

589 nm

2

3

2S'1/2 1772 MHz

1

Fig. 7. Level scheme of Na with the cooling transitions.

150
Control of Frequency-Sweep

Laser Q T W Q,Q±v (t) Optional

EOM Filter

Wt)

Q + v(t)

(Polarization) (Frequency)

Swept

Frequency-Output

rf Power

Amplifier

Electronic Attenuator Power

Control

v = v B(t)-vA

Programmer

PROMS

(Replaceable!)

©Frequency —

vco

V : . rale

0.7-2 HS/Step

1024 Steps

12 Bits

Time

VJ

Time

Fig. 8. Control of the frequency sweep using Programmable Read Only Memory.

The necessary second laser frequency can be produced, for example, by an electro-optic

modulator, to convert a part of the carrier intensity into the desired side band.

As mentioned above, the laser frequency must be tuned very rapidly (1-2 ms sweep time) over a relatively large scan range (~1.7 GHz) with high precision. This can be achieved,

too, by an electro-optic modulator. Figure 8 shows an experimental setup for generating

the precise, fast-scanning frequency needed. The main feature is a PROM (Programmable Read

Only Memory) controlled rf source, which allows both very fast and precise scanning of the

rf and an easy and quick change of the scan scheme. The basic voltage-tunable oscillators

shown in Fig. 8 operate for example in the 2.5-3.5 GHz range, tuning this range and

settling within a few microseconds. With this setup, which has already been tested for

frequencies up to 800 MHz, a power of about 20% of the carrier power in the first sideband

should be possible with a 10 W rf power amplifier. Conversion efficiency above 12% has

been observed at 2.5 W.

151
It should be explicitly noted that the cooling radiation will be taken to be the upper

sideband produced by the modulation process. This will allow the "unused" laser carrier

and lower sideband to provide transverse confinement through the dipole force term. Also

the end of the cooling cycle will occur when the modulation frequency is at its maximum

value. The predictable high frequency efficiency loss of the modulator will actually be an advantage in reducing the power broadening of the cooled transition, thus leading at the last moment to a further (although perhaps small) velocity narrowing. Finally, we note that the 1772 MHz needed for repumping the F=l atoms can easily be inserted into the

rf power amplifier along with the frequency-swept rf from the heterodyne source. An alternative configuration would be separate power amplifiers combined with an appropriate

hybrid coupler.

PM

Fig. 9. Scheme of the experimental setup.

The atomic beam experiment is shown schematically in Fig 9. The apparatus consists

of three parts: the sodium oven chamber, the main interaction and detection chamber, and a chamber for controlling the laser frequencies by monitoring the fluorescence light from the

crosspoint of atomic beam and perpendicular laser beam. The whole atomic beam has a length

of about 1.3 m. The apparatus is designed to work equally as well in horizontal as in near vertical positions. The big "middle chamber" has two large side windows for the detection

laser, which monitors the density and velocity distribution over a relatively long part

of the atomic beam. On top of the chamber is a related large window for detection of the

fluorescence light. The chamber is also suited to detect very slow atoms, which "fall out" below the atomic beam. At both ends of this chamber are cold traps, which provide good vacuum and low sodium background.

For the measurement of the momentary velocity distribution, different schemes are

9 possible. One is laser excitation of the P3/2 atoms into higher s-states, which decay In cascades to the ground state. This allows an easy discrimination of the fluorescence light

from the background sources. In order to cool the transverse velocity also, three transverse laser beams can be applied to the atomic beam at several positions.

3. Conclusions

Despite the many proposals and theoretical publications about cooling and trapping of

neutral atoms only a few experimental approaches are known today. The most interesting and

successful approach is described in Ref. 8. It has the advantage of working with a "fixed"

frequency dye laser and of producing a continuous, strong, polarized slow atomic beam in

152
the $\/2 (F=2) state. One disadvantage of this scheme is the still-unsolved problem of

optical pumping (many of the atoms stay in the ^Sj/2( F=1 ) state with their initial velocity

distribution) especially at the entrance and exit regions of the magnetic field. Fast beam

components may also reduce the density of cool atoms collisionally . A second problem concerns detailed interpretation of the experimental results. In that apparatus, as presently

configured, it is not easy to study the cooling process as it evolves inside the magnet.

We are optimistic that the approach described in this paper, which avoids the magnetic

field and its related problems, will turn out to be a rather promising alternative. One

future advantage of the frequency-swept scheme is that it need not depend strongly on the

polarization of the laser light. This may be particularly important for the next step: the

optical trap. It is almost impossible in most of the proposed schemes to work in a pure

two-level system, if isotopes with nonzero nuclear spin are to be investigated.

The Monte-Carlo simulation of the cooling process as well as of the trapping process appears to be a versatile tool in the preparation and understanding of the whole

experiment. This has already been proven by our first calculations of the planned cooling scheme.

Sodium was chosen as the first candidate because of technical reasons and the possibility to compare directly with other experiments. Future candidate atoms of potentially

high intrinsic interest are described in Ref. 17.

Acknowledgments

It is a pleasure to acknowledge stimulating discussions with J. Prodan, W. D. Phillips, and P. Zoller. P. Zoller was particularly involved in the calculations con- cerning the Fokker-Planck equation. Two of the authors (R. B. and W. E. ) gratefully ac- knowledge fellowship support by the Deutsche Forschungsgemeinschaf t. This research at JI1A

is supported in part by the National Science Foundation and the Office of Naval Research,

and in part by the National Bureau of Standards through its program of research on topics

of potential application to fundamental standards of measurement.

References

1. S. Stenholm, Phys. Rep. _4_3, 151 (1978).

2. T. W. Hansch and A. L. Schawlow, Opt. Comm. 13_, 68 (1975).

3. D. J. Wineland and H. G. Dehmelt, Bull. Am. Phys. Soc. 20_, 637 (1975).

4. W. Neuhauser, M. Hohenstatt, P. Toschek, and H. Dehmelt, Phys. Rev. Lett. ^J_, 233

(1978) . 5. D. J. Wineland, R. E. Drullinger, and F. L. Walls, Phys. Rev. Lett. 40, 1639 (1978).

6. V. I. Balykin, V. S. Letokhov, and V. I. Miskin, Pis 'ma Zh. Eksp. Teor. Fiz. 29, 614

(1979) . 7. W. D. Phillips and H. Metcalf, Phys. Rev. Lett. 48, 596 (1982).

8. J. V. Prodan, W. D. Phillips, and H. Metcalf, Phys. Rev. Lett. 49^ 1149 (1982).

9. D. J. Wineland, J. J. Bollinger, and W. M. Itano, Phys. Rev. Lett. j>0, 628 (1983).

10. D. J. Wineland, see contribution to this workshop.

11. S. V. Andreev, V. I. Balykin, V. S. Letokhov, and V. G. Minogin, Sov. Phys. JETP 55,

828 (1982).

12. V. S. Letokhov and V. G. Minogin, Phys. Rep. J73_, 1 (1981).

13. V. G. Minogin, Sov. Phys. JETP 52, 1032 (1980).

14. A. Ashkin, Phys. Rev. Lett. ^0, 729 (1978).

15. J. P. Gordon, Phys. Rev. A8, 14 (1973).

16. J. E. Bjorkholm, R. R. Freeman, A. Ashkin, and D. B. Pearson, Phys. Rev. Lett. 41 , 1361 (1978).

17. W. Ertmer, R. Blatt and J. L. Hall, see contribution to this workshop.

153
Laser-Cooled and Trapped Atoms , (Proceedings of the Workshop on Spectroscopic

Applications of Slow Atomic Beams, held at NBS, Gaithersburg, MD, 14-15 April

1983) Ed. by W. D. Phillips, Natl. Bur. Stand. (U. S.) Spec. Publ. 653 (1983).

Some Candidate Atoms and Ions for Frequency Standards Research

Using Laser Radiative Cooling Techniques

W. Ertmer,* R. Blatt,* and J. L. Hall*

Joint Institute for Laboratory Astrophysics, National Bureau of Standards

and University of Colorado, Boulder, Colorado 80309 U.S.A.

The rapid progress in laser atomic cooling ideas and also in laser

spectral coverage and frequency stabilization make it appropriate to re- examine the periodic table for atoms and ions that may be of special interest for radiatively-cooled frequency standards research. We discuss

Hg+

, Pb+, Ba+ , neutral alkalis, Mg, Ca, and Ag. Optical two-photon and

weak single-photon transitions are represented, along with microwave hfs

and fs transitions.

Key words: radiative cooling, optical frequency standards, laser

stabilization

Several years ago we saw the first spectacular, experimental demonstrations of radiative cooling to sub-Kelvin kinetic temperatures of trapped Mg+

ions by Wine land and his

colleagues [1], and of a few, even single Ba+

ions by Toschek, Dehmelt and their colleagues

[2]. The beauty of these experiments was marred only by the relative scarcity of ions and

matching laser sources appropriate for this kind of application. However, there have been

many improvements of tunable laser performance and tuning range, notably through the use

of new laser dyes and color centers for the active lasers, and through the application of

nonlinear optical mixing and optical harmonic techniques to extend the range of useful output far into the ultraviolet.

In this report, we note a few transitions in candidate ions and atoms made accessible

by these and earlier laser developments. We particularly want to focus attention on those

atomic systems which have the super-narrow transitions that will be of interest in connection with atomic frequency standard research, even though many of these very same candidate

ions and atoms may pose severe problems for our ingenuity and technological possibilities

in achieving effective radiative cooling.

Probably the most exciting ion being considered at present is Hg+. The relevant

energy levels of Hg+ are shown in Fig. 1. For microwave standards work a high transition

frequency is useful to improve the "Q": the 40 GHz hfs transition in 199 Hg+ and 26 GHz in

201flg+ Q ff er welcome increases in this way over Rb and Cs. These Hg+ transitions have long

been appreciated and studied since the early work by Major and Dehmelt, down through con- temporary work at Hewlett Packard, the Laboratoire de l'Horloge Atomique, and now at NBS,

Boulder by Bergquist and colleagues. A really new and decisive breakthrough in this work

likely will come out of the remarkable recent success of Hemmati, Bergquist and Itano in

producing the coherent 194 nm radiation needed to radiatively cool Hg+

. By summing in a

Permanent address: University of Bonn, West Germany.

Permanent address: Johannes Gutenberg University, Mainz, West Germany.

Staff Member, Quantum Physics Division, National Bureau of Standards.

154
Hg +

6

2P3/2

second nonlinear crystal the output of a frequency-doubled Ar+ laser operating at 514 nm

with the output of a dye laser operating at 792 nm, these workers were able to obtain a few |j,W of coherent UV output at 194 nm [3].

The marvelous thing about appropriate ion trapping techniques is that they are so benign relative to reheating the ions' motion that just a few microwatt optical power level

will suffice to achieve good cooling, thus greatly relieving the pressure on coherent UV

generation techniques in the sub-200 nm part of the spectrum where the resonance lines of

ions are typically found.

Surely other interesting ions will be identified and studied as the laser UV techniques evolve. In some respects the Hg+ ion may remain unique however, considering the

remarkable two-photon optical transition [4] at 563 nm connecting the ground state with

the D5 I2 metastable level. See Fig. 1. The calculated [5] E2 decay rate of the upper

level is 9.5 s -1

, leading to an expected potential line Q of this transition of almost

1015 ! After the ion cloud has been radiatively cooled, one will use a repetitive cycle

of excitation of the sharp resonance, followed by checking for population in the excited

level. For example, a Ramsey two-pulse excitation sequence will lead to two coherent am- plitudes for populating the 6s2 2 D5 / 2 metastable level. A subsequent strong pulse laser

at 399 nm can transfer this D-state population to 6p 2p°/2 leading to 165 nm fluorescence

decay. This operation of testing for excited state population will reveal the Ramsey interference term between the two excitation amplitudes and thus lead to supernarrow resonances. With such a sequential excitation and interrogation scheme one should be able to have both narrow lines and high detection efficiency. Alternatively, with fewer ions and

stronger excitation of the "clock" transition, one could test for the decrease in light

scattering on the cooling transition 6s 2 S^/2 + 6p 2 P°/

2 at 194 nm, following the suggestion by Dehmelt [6].

An ion interesting for its clock transition was identified by Strumia [7]: Pb+. The fine-structure splitting between ground 6p 2 P°/2 and excited 6p 2p3/2 states lies at 710 nm and has a calculated lifetime [5] of 38 ms. Such a sharp magnetic dipole transition

might be ideal for studying the final phase of the cooling process and, conceivably, could

be used to drive the system (slowly!) to a new regime of ultralow kinetic temperatures.

155
Unfortunately, direct cooling of Pb+ would require 168 nm radiation for direct P S cooling, so application of Pb+ as an optical atomic clock will have to wait for further

laser advances even farther into the UV. It may not be impossible to cool ions such as Pb+ with long-range Coulomb collisions with a good "refrigerator ion" such as the Mg+

transition used by Wineland et al . [1] in their first experiments.

No discussion of ion trapping candidates would be complete without consideration of Ba+, the other ion for which dramatic cooling has been demonstrated [2], The relevant

energy levels for Ba+ are shown in Fig. 2. One has strong cooling on the 6

2 S1 /2 * 6 2pl/2 resonance line at 493 nm. Unfortunately, there is an important leakage from 6

2 Pi/2 down

to 5

2 D3/2 with a branching ratio of «1:3; The solution adopted by Toschek et al . is to re-excite the ion back to the 2 Pi/2 level with a 650 nm broadband laser. Cooling and stable trapping was only observed in the presence of both laser fields. A variety of interesting three-level effects remain to be studied in this system. In particular, with a narrow laser at 650 nm as well as at 493 nm, one will look for the "population trapping" or "resonance Raman lineshapes" recently discussed by numerous authors [8] and spectacularly

demonstrated by Thomas et al . [9]. Such a Raman coherence may well allow study of a sharp

transition such as the 2.05 |am quadrupole line indicated in Fig. 2 without a primary laser source operating at that wavelength. We note that this clock transition could also be ex- plored by two-photon spectroscopy using a 4.1 ^m source.

Turning now to atomic species of interest for radiative cooling experiments in atomic

beams, without doubt the most popular atom for this work is sodium. Certainly the wavelength match with conventional Rh 6G dye lasers and easy atomic beam formation and detection represent important advantages of using Na in the radiative cooling experiments.

Unfortunately, the relatively large hyperfine splitting and strong optical pumping effects

add difficulty to the design of clear experiments. Since one needs to scatter - 30, 000

photons from each atom to cool it, even a small leakage effect (due to slightly defective

laser polarization for example) will ultimately cause an important effect. As several of

the papers at this workshop deal with these problems in depth, we take this opportunity to consider alternative atoms for atomic beam laser cooling experiments.

Fig. 2. Partial energy level diagram for Ba .

156
One first approach is to consider other alkali atoms. For example, Li could be cooled

with a dye laser operating at 671 nm with the dyes DCM or Rh 101. K (767 nm) could be

cooled with LD 700 as the dye. The advantage of these atoms relative to sodium lies in

their smaller hyperfine splitting interval (804 MHz and 462 MHz respectively) so that it

would be an easier technical task to use electro-optical modulation to produce a frequencyoffset laser field for repumping the atoms lost into the "wrong" hfs state. Rb (780 nm)

also can be cooled with an ID 700 dye laser. The 3 or 6.8 GHz hfs splittings of Rb (isotopes 85 and 87) are probably too large to address with current bulk EO Modulator technology relative to the repumping problem [10]. However, Rb has the interesting advantage that

two-photon transitions to Rydberg states in Rb fall right in the spectral domain ~ 595 nm

where sub-100 Hz dye laser stability has been demonstrated [11]. Certainly a good demonstration of laser-cooled Cs would be of interest to the frequency standards community.

Lewis [12] at NBS and Arditi and Picque [13] in France have explored the use of diode

lasers for optically-pumping Cs beams to increase the population in the useful mp = 0 level

and for tuning purposes. The recent demonstrations of high diode laser power levels [14]

and rapid tuning rates [15] promise an exciting future prospect to strongly slow an ordinary atomic Cs beam with "simple" lasers.

Continuing our discussion of slow beam atomic clock systems based on magnetic dipole

transitions between appropriate states, one of the more attractive systems is provided by

Mg. See Fig. 3. Here the cooling can be effected in a direct way with 285 nm radiation,

probably derived as the doubled frequency output of a Rh 110 dye laser operating at 570 nm. The absence of nuclear spin and an S^" -type ground state means a total absence of problems due to optical pumping effects during the cooling cycle. The penalty for Mg may lie

in the difficulty of producing frequency-doubled dye laser output of sufficient intensity,

Fig. 3. Partial energy level diagram for Mg.

157
but it seems certain that this problem can be addressed with "build-up cavity" techniques

using a ring cavity topology. Two very attractive clock possibilities are to be found in

Mg. In the first, one uses the 601.277 16(5) GHz 3

3 P£ 3

3 1£ Ml transition which has

recently been measured by Bava et al. [16]. Population reaching 3

3 P° could be detected by

magnetic deflection analysis and a hot wire detector. Alternatively, the 3 P° atoms could

be excited upwards with 517 nm laser pulses to the 4

3 S^ level from which fluorescence could

be observed. Pavlik and Letokhov suggested that interesting optical transitions could be

provided by the alkaline earth singlet-triplet intercombination lines [17]. For Mg this

line falls at 457 nm for which the laser dye Stilbene 3 would be suitable, but it requires

an expensive Ar2 + laser for pumping the highly stabilized 457 nm laser. The lifetime of

the triplet level involved, 3

3 P°, has recently been measured to be (2.4±0.2) ms [181.

Similar schemes can also be discussed in Ca. The cooling transition would be

4 ^S

Q - 4

1 P° at 423 nm. Unfortunately, in Ca and the heavier alkaline earths the

lies below 1 P. In Ca the allowed decay to 3* D is at 5.5 um wavelength. The v

3 density

of states factor is more than 2000-fold in favor of the decay to 1 P, but as we need roughly

30,000 scattering events, this leakage problem will have to be addressed. It is perhaps

worth noting that a CO laser line is tabulated to be within 1.2 GHz of the frequency needed

to repump this line. Alternatively, one might be able to recover many of the captive atoms

by pumping upward with 504 nm light to 4s5p *P° which decays rather strongly to ground.

The fine structure transition 3 P^ * 3 Pq in Ca is at 1.565 THz which certainly is beginning

to become unreasonably difficult relative to coherent rf/sub mm source technology. Ultimately one expects optically-pumped far IR lasers to fill this need.

The first real metrological interest in slow Ca would be in probing the intercombination line 4

1 S0 - 43 Pj at 657 nm. Barger has already shown Ramsey interference resonances

on this line with a ~1 kHz "width" almost totally determined by the atomic beam velocity

distributions working through the second-order Doppler shift [19]. The natural width of

this transition is thought to be ~ 300 Hz based on recent lifetime measurements giving

(0.55±0.04) ms [18].

Still another interesting clock transition in Ca would be the two-photon line at 915 nm connecting 4

1 So to 31 D2 . This stable laser could well be based on the color

center in LiF [20]. Presumably this level could also be excited by an E2 transition at 458 nm.

An attractive and technically less -demanding opportunity exists in Ba for which the

primary cooling radiation at 554 nm can be directly obtained from a Rh 101 dye laser.

However, as may be seen in Fig. 4, a serious leakage channel to 5*D will ultimately trap

the population after ~20 cycles in the cooling transitions. Thus this system will require

a "repopulation" laser at 1.5 ^.m, perhaps based on F

2 + centers [20] in NaCJl or the remarkable KCX.:TJt laser [21]. The resonance Raman three-level coupling process [8,9] discussed

previously will be particularly interesting here as well.

We turn now to the last and perhaps most exciting candidate atom in this little list,

neutral Ag [Fig. 5]. We know from the epoch of Stern and Gerlach that Ag forms nice atomic

beams although the speed of their deposition-type detector does not ideally match our laser servo bandwidth needs. The cooling of Ag in a contemporary experiment would use the resonance lines at 328 nm or 338 nm obtained by frequency doubling a DCM dye laser operating

at 656 or 676 nm. Unfortunately, the lower wavelength D2 line falls in a region where no crystal for non-critical phase matching is known.

158
Ba

Fig. 4. Partial energy level diagram for Ba.

Aq I=y2 ( 107Ag, 109Ag)

Fig. 5. Partial energy level diagram for Ag.

159
The Dl line at 338 is very near the limit of cooled RDA. Using the Dl line for cooling will require successful attention to optical pumping effects, but the Ag ground-state

hyperfine frequencies (1.7 and 2.0 GHz) are low enough that we can provide the needed optical sidebands by E0 modulation. The exciting thing about Ag is its two-photon visible

transition [4] at 661 nm, 5

2 Si/2 > 52 Ds/2- The D5 /2 level has an estimated lifetime of 0.25 s, leading to a sub-Hz natural width and a line 0 of over 1 x 10* 5 !

Detection of many of these long-lived excited states have been discussed above in the

context of radiative excitation followed by fluorescence decay. We should also note the

effective detection of metastable atom beam constituents by Auger electron emission from a low work-function surface. Such Auger detectors were pioneered by Lurio and his associates

and used successfully in many atomic beam measurements, including the measurement of analogous weak two-photon transitions in an atomic beam of Bi [22]. As may be expected when a monolayer surface film is the active element, such a detector lives a longer life in an ultrahigh vacuum environment.

Dye lasers of spectral width ~ 1 MHz, available commercially, are completely appropriate for the radiative cooling tasks we have discussed. Lasers to explore these ultranarrow

lines, with widths measured in Hz, are definitely not available commercially. However, recent cross-stimulation between the stable laser and gravity-wave detector communities has

led to some useful progress, basically the reinvention in an optical context of Pound's rf microwave frequency stabilizer. (Interestingly enough, this techique was developed independently and essentially simultaneously by two groups [11,23].) A new paper reports sub100 Hz dye laser line widths [11] and sub-Hz line widths are expected from contemporary

experiments using improved techniques. We cannot claim that the laser technology is really

well in hand and ready to explore these incredibly-narrow resonances, but from the above

discussion of several atomic systems, it does appear that there will be something interesting to measure when ultraslow atomic beams and ultrastable lasers are available together.

Acknowledgments

Two of the authors (R. B. and W. E. ) gratefully acknowledge fellowship support by

the Deutsche Forschungsgemeinschaft. This research at JILA is supported in part by the

National Science Foundation and the Office of Naval Research, and in part by the National

Bureau of Standards through its program of research on topics of potential application to fundamental standards of measurement.

References

1. D. J. Wineland, R. E. Drullinger and F. L. Walls, Phys. Rev. Lett. 40, 1639 (1978).

2. W. Neuhauser, M. Hohenstatt, P. Toschek and H. Dehmelt, Phys. Rev. Lett. _41_, 233

(1978).

3. H. Hemmati, J. C. Bergquist and W. M. Itano, Opt. Lett. _8, 73 (1983).

/ 4. 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. 2±, 599 (1976).

5. R. H. Garstang, J. Res. NBS 68A, 61 (1964).

6. H. Dehmelt, Bull. Am. Phys. Soc. 20_, 60 (1975); D. J. Wineland, J. C. Bergquist, W. M. Itano and R. E. Drullinger, Opt. Lett. 5, 245 (1980).

7. F. Strumia, Proc. 32nd Annual Freq. Control Symposium 444 (1978).

8. See, e.g. B. J. Dalton and P. L. Knight, Opt. Comm. _42, 411 (1982) and references

therein.

\ 9. J. E. Thomas, P. R. Hemmer, S. Ezekiel, C. C. Leiby, Jr. , R. H. Picard and C. R. Willis, Phys. Rev. Lett. 48, 867 (1982).

160
10. However, some remarkable advances In EO Modulator design have recently given 10%

sidebands at 4 GHz. G. C. Bjorklund and N. H. Tran, private communication.

^11. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley and

H. Ward, Appl. Phys. B 31 , in press.

12. L. Lewis, private communication.

13. M. Arditi and J. L. Picque, Opt. Comm. 15, 317 (1975).

-v/14. Remarkable results have recently been obtained with coherently-phased multi-diode

laser arrays. See, e.g. D. R. Scifres, R. D. Burnham and W. Streifer, Appl. Phys.

Lett. 41_, 1030 (1982).

V15. P. Pokrowsky, W. Zapka and G. C. Bjorklund, Opt. Comm. _44_, 175 (1983).

16. E. Bava, A. de Marchi and A. Godone, presented at conference on sub mm lasers, 1983.

17. V. S. Letokhov and B. D. Pavlik, 1975 Vavilov Conference, Novosibirsk.

18. G. Giusfredi, P. Minguzzi, F. Strumia and M. Tonelli, Z. Physik A274, 279 (1975).

19. R. L. Barger, Opt. Lett. 6, 145 (1981).

20. L. F. Mollenauer, Opt. Lett. j_, 164 (1977).

21. W. Gellerman and F. Liity, Conference Proceedings: Lasers '81, 364.

22. J. L. Hall, 0. Poulsen, S. A. Lee and J. C. Bergquist, J. Opt. Soc. Am. 68, 697

(1978), and references therein.

V23. G. C. Bjorklund, Opt. Lett. 5, 15 (1980).

t

161
Laser-Cooled and Trapped Atoms , (Proceedings of the Workshop on Spectroscopic

Applications of Slow Atomic Beams, held at NBS, Gaithersburg, MD, 14-15 April

1983) Ed. by W. D. Phillips, Natl. Bur. Stand. (U.S.) Spec. Publ. 653 (1983).

Trapping of Low Energy Neutrons

G.L. Greene

Quantum Metrology Group

National Bureau of Standards

Washington, DC 20234

A brief review of recent work concerning the trapping

of low energy neutrons is given. Particular emphasis is placed

on magnetic confinement schemes which might also be applicable

to neutral atoms. Important differences between the neutron and

neutral atom cases are mentioned.

Key words: hexapole; trapping; ultra cold neutrons.

In considering the problem of neutral atom trapping, it is highly instructive

to examine the effort which has been devoted to the confinement of another neutral

particle, the neutron. The neutron is often facetiously described as an atom with

Z=0, and indeed atomic physics techniques are often applicable to neutron research.

(There is even a chapter devoted to neutron beams in Ramsey's Molecular Beams [1].)

There are, however, important differences between neutrons and atoms which should

be emphasized. It is therefore appropriate to briefly note the neutron properties

which are relevant to the trapping question.

The neutron is a neutral, spin 1/2 particle with a magnetic dipole moment of

approximately -1.91 nuclear magnetons [2,3], (This is of course three orders of

magnitude less than typical atomic spin and orbital moments.) The dielectric

polarizability of the neutron is conjectured to be very small [4] and may be

considered zero for laboratory fields. The combination of neutrality, small

magnetic moment and zero polarizability implies a very low neutron-neutron

cross section. This in turn implies that a "gas" of neutrons is very nearly

non-interacting

The interaction between other matter and low energy neutrons is usually

dominated by nuclear forces. It can be shown [5,6,7] that the coherent scattering

of low energy neutrons with matter is well described by assigning an index of

refraction to the matter. For most materials [9] this index is less than one,

implying the possibility of reflection below some critical angle at the surface of

such material. For sufficiently low energy neutrons, the critical angle for

reflection can reach 90° and therefore neutrons at all angles will be reflected.

Neutrons with such low energy have come to be called Ultra Cold Neutrons. Ultra

cold neutrons typically have energies of ^10 7 eV; this corresponds to a temperature of ^ 1 mK.

Since reflection occurs at all angles, Ultra Cold Neutrons can be trapped in a "bottle" [10] . Such trapping was first observed by Lushikov [11] and has since

become the basis for an exciting experimental search for a static neutron electric

dipole moment. However as this method of trapping is not appropriate to atoms, the

reader is referred to several excellent review articles for further information

[9, 12-16].

162
An important difference between neutrons and atoms arises from the very small

neutron-neutron scattering cross section. Since a neutron gas is nearly noninteracting, a thermal beam of neutrons will have the predicted population at the

extreme low energy tail of the Maxwellian distribution. Thus, unlike the atomic

case where "cooling" is required to obtain nearly stationary particles, very slow

neutrons are available by velocity selection alone. This method of producing very

low energy neutrons has been the most widely employed to date. Other schemes have

been proposed and show great promise.

Because it is neutral and unpolarizable, a purely electromagnetic trap must

couple to the neutron magnetic dipole moment. A natural candidate for such a trap

is the hexapole field suggested by W. Paul [17, 18]. The possibility of total

confinement of neutrons by such a field was suggested by Vladimirski [19] . In

recent years this notion has led to an active program in magnetic neutron confinement by W. Paul and his colleagues [20, 21].

Magnets having two geometries have been constructed by Paul and his co-workers.

These are shown schematically in Fig. 1. The spherical bottle (la) is a true trap

for neutrons in one spin state. There is a point of zero field at the center of

the trap, but it is not anticipated that the loss of particles due to Majorana

flips near this point will be significant. Though the use of this bottle has not

yet been successfully demonstrated, a program is now in progress in which it will

be filled using the superthermal superfluid He phonon source proposed by Golub and

Pendelbury [22].

Figure 1. (a) Spherical hexapole trap, and (b) Toroidal hexapole trap (taken from

Paul and Trinks, Ref. 20).

163
The suggestion that neutrons could be stored in a hexapole torus, as shown

schematically in Fig. lb, was first made by Heer [23] and others [24]. This

geometry has been successfully used to store neutrons by Paul and his co-workers

using the ultra cold neutron source at the Institut Laue Langevin (ILL) in Grenoble

[20, 21].

The actual geometry employed at the ILL is somewhat different from that

indicated in Fig. lb. The inner two current loops were omitted "opening" the trap

in the direction of the torus center. This effect was compensated for by the

centrifugal barrier arising from the circular motion around the torus. A useful

effect of this geometry is that it "moves" the zero field point where Majorana

transitions may occur out of the volume occupied by the neutrons. As a result,

losses due to Majorana transitions are probably negligible. The magnet was superconducting with a maximum usable field of 3.5T. The acceptance velocity of

the trap in the azimuthal direction (around the torus) was ^ 7-20 m/s. In the

perpendicular direction the acceptance was ^ ± 4 m/s. This corresponds to energies

up to 2 x 10 6 eV. A typical "fill" loaded the ring with ^ 100 neutrons. Neutrons were detected in the trap after times up to 20 minutes. It is interesting to note

that this corresponds to approximately two neutron lifetimes [25]. (It is also

interesting to note that this experimental tour-de-force was accomplished with only

20 hours of beam time.) The dominant loss mechanism in the trap probably arises

from a coupling between azimuthal and perpendicular velocities [13, 20, 21]. This

is a problem which must be considered in any "racetrack" trapping scheme.

The successful confinement of neutrons by purely magnetic fields poses both a hope and a challenge to those seeking to trap neutral atoms. Paul, Trinks and

Kugler have clearly demonstrated that neutral particles can be trapped by electromagnetic fields. The problems associated with trapping neutral atoms are quite

different and will certainly require ingenuity on the same level as that demonstrated

in their elegant experiments.

I am indebted to numerous colleagues at the Institut Laue Langevin for many

educational discussions. In particular I wish to thank R. Golub and J.M. Pendelbury

who have shared with me some of their extensive knowledge of ultra cold neutrons.

References

[1] N.F. Ramsey, Molecular Beams , Oxford U.P. (1956), p. 189.

[2] G.L. Greene et al . , Phys. Rev. D 20, 2139 (1979).

[3] G.L. Greene et al. , Metrologia 18, 93 (1982).

[4] Aleksandrov et al . , JETP Lett. 4-, 134 (1966).

[5] E. Fermi, Nuclear Physics , Univ. of Chicago Press (1949), p. 201.

[6] I.I. Gurevich and L.V. Tarasov, Low Energy Neutron Physics , North Holland

(1968).

[7] V.F. Turchin, Slow Neutrons , Israel Program for Scientific Translations (1965).

164
[8] L. Koester, in Neutron Physics , Springer Verlag (1977).

[9] See also R. Golub and J.M. Pendelbury, Rep. Prog. Phys. 42, 439 (1979), p.

443, for a simple explanation of this phenomenon.

[10] Yu. B. Zeldovich, JETP % 1389 (1959).

[11] Luschikov et al. , JETP Lett. 9, 23 (1969).

[12] A. Steyerl, in Neutron Physics , Springer Verlag (1977).

[13] R. Golub and J.M. Pendelbury, Contemp. Phys. 13, 519 (1972).

[14] R. Golub et al. , Scientific American 240, 134 (1979).

[15] W.B. Dress et al., Phys. Rev. D 15, 9 (1977).

[16] I.S. Altarev et al. , Nuclear Physics A341 , 269 (1980).

[17] H. Friedberg and W. Paul Naturwiss. 38, 159 (1951).

[18] W. Paul, Proc. Intern. Conf . on Nuclear Physics and Physics of Fundamental

Particles , Chicago (1951).

[19] V.V. Vladimirski, JETP 12, 740 (1961).

[20] W. Paul and U. Trinks, in Fundamental Physics with Reactor Neutrons and

Neutrinos , ed. T. von Egidy, Institut of Physics Conference Series 42^ (Bristol) 1977.

[21] K.J. Kugler, W. Paul and U. Trinks, Phys. Lett. 72B , 422 (1978).

[22] R. Golub and J.M. Pendelbury, Phys. Lett. 62A, 337 (1977). See also Ref. 9

for a thorough review.

[23] C.F. Heer, Rev. Sci. Instrum. 34, 532 (1963).

[24] I.M. Matora, Sov. J. Nucl. Phys. 16, 349 (1973).

[25] J. Byrne et al. , Phys. Lett. 92B , 274 (1980), and C.J. Christensen et al.

,

Phys. Rev. D 5_, 1628 (1972).

165
LIST OF WORKSHOP PARTICIPANTS

Lloyd Armstrong

National Science Foundation

1800 G St. , NW

Washington, DC 20550

Arthur Ashkin

Bell Laboratories

Crawford's Corner Road

Holmdel, NJ 07733

Ted G. Berl incourt

Office of Naval Research

800 N. Quincy St. Arlington, VA 22217

Paul R. Berman

Physics Dept.

New York University

4 Washington Place

New York, NY 10003

John E. Bjorkholm

Bell Laboratories

Room 4C-318

Holmdel, NJ 07733

R. Blatt

JILA/Univ. of Colorado

Boulder, CO 80309

Donald A. Emmons

Frequency and Time Systems

34 Tozer Road

Beverly, MA 01915

W. Ertmer

JILA/Uni versity of Colorado

Boulder, CO 80309

Kenneth Evenson

National Bureau of Standards

Boulder, CO 80303

Shaoul Ezekiel

Massachusetts Institute of Technology

33-113

Cambridge, MA 02139

El isabeth Giacobino

New York University

Dept. of Physics

New York, NY 10003

James P. Gordon

Bell Telephone Labs

Rm. 4E 410

Holmdel, NJ 07733

G. L. Greene

National Bureau of Standards

Quantum Metrology Group

Washington, DC 20234

J. L. Hall

National Bureau of Standards

JILA

Boulder, CO 80303

Bobby R. Junker

ONR, Code 412

800 N. Quincy St. Arlington, VA 22217

Karl G. Kessler

National Bureau of Standards

Bldg. 221, Rm. B160

Washington, DC 20234

Daniel J. Larson

University of Virginia

Dept. of Physics

Charlottesville, VA 22901

Howard P. Layer

National Bureau of Standards

Bldg. 221, Rm. B160

Washington, DC 20234

Lindon Lewis

National Bureau of Standards

325 Broadway

Boulder, CO 80303

Michael S. Lubell

The City College of CUNY

Dept. of Physics

Convent Ave. & 138th St. New York, NY 10031

Harold Metcalf

State Univ. of New York

Physics Dept.

Stony Brook, NY 11790

Marvin Mittleman

City College-of CUNY

Physics Department

138th St. at Convent Ave. New York, NY 10031

William D. Phillips

National Bureau of Standards

Bldg. 220, Rm. B258

Washington, DC 20234

David E. Pritchard

Massachusetts Institute of

Technology

Rm. 26-231

Cambridge, MA 02139

John V. Prodan

National Bureau of Standards

Bldg. 220, Rm. B258

Washington, DC 20234

Keeneth Rubin

City College of New York

138th St. & Convent Ave. New York, NY 10031

James J. Snyder

National Bureau of Standards

Quantum Metrology Group

Washington, DC 20234

Duncan G. Steel

Hughes Research Labs

3011 Malibu Cyn. Road

Malibu, CA 90265

Samuel R. Stein

National Bureau of Standards

Div. 524

325 Broadway

Boulder, CO 80303

William C. Stwalley

Univ. of Iowa

Dept. of Chemistry

Iowa City, IA 52242

Barry N. Taylor

National Bureau of Standards

Bldg. 220, Rm. B258

Washington, DC 20234

Harry T. M. Wang

Hughes Research Labs.

3011 Malibu Canyon Road

Malibu, CA 90265

David Wineland

National Bureau of Standards

Div. 524

Boulder, CO 80303

W. H. Wing

Univ. of Arizona

Physics Dept.

Tucson, AZ 85721

166
U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

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1. PUBLICATION OR

REPORT NO. NBS SP 653

2. Performing Organ. Report No. 3. Publication Date June 1983

4. TITLE AND SUBTITLE Laser-Cooled and Trapped Atoms

Proceedings of the Workshop on Spectroscopic Applications of Slow Atomic Beams, held

at the National Bureau of Standards, Gaithersburg, MD, on 14-15 April 1983.

5. AUTHOR(S)

William D„ Phillips, Editor

6. PERFORMING ORGANIZATION (If joint or other than NBS, see instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered Final

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State, ZIP) National Bureau of Standards Qff1 f Nava] Research

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10. SUPPLEMENTARY NOTES

Library of Congress Catalog Card Number: 83-600541

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| Document describes a computer program; SF-185, FIPS 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) The proceedings of the Workshop on Spectroscopic Applications of Slow Atomic Beams

contain papers on laser cooling of atomic beams, gases, and trapped particles and

papers on trapping of neutral atoms in laser, electric, and magnetic fields. These

and other papers discuss possible applications of cooled atoms to spectroscopy and

to time and frequency standards.. The present status and directions of research in

time and frequency standards is also discussed.

12. KEY WORDS (Six to twelve entries; a/phobetico) order; capitalize only proper names; and separate key words by semicolon s) atom traps; clocks; cooled atoms; frequency standards; high resolution spectroscopy;

ion traps; laser cooling; laser deceleration; laser traps; optical frequency standards;

slow atomic beams.

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