tandards

F£

Ti""i^""\ I I MB i\

346

On The Natural Shift of A

Resonance Frequency

ROBERT J. HARRACH

Q

i

.** >3i

\ £,^tAU 0*

o

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards
THE NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards1 provides measurement and technical information services

essential to the efficiency and effectiveness of the work of the Nation's scientists and engineers. The

Bureau serves also as a focal point in the Federal Government for assuring maximum application of the physical and engineering sciences to the advancement of technology in industry and commerce. To

accomplish this mission, the Bureau is organized into three institutes covering broad program areas of research and services: THE INSTITUTE FOR BASIC STANDARDS . . . provides the central basis within the United

States for a complete and consistent system of physical measurements, coordinates that system with the measurement systems of other nations, and furnishes essential services leading to accurate and uniform

physical measurements throughout the Nation's scientific community, industry, and commerce. This

Institute comprises a series of divisions, each serving a classical subject matter area: —Applied Mathematics—Electricity—Metrology-—Mechanics—Heat—Atomic Physics—Physical

Chemistry—Radiation Physics—Laboratory Astrophysics2—Radio Standards Laboratory,2 which

includes Radio Standards Physics and Radio Standards Engineering—Office of Standard Refer- ence Data. THE INSTITUTE FOR MATERIALS RESEARCH . . . conducts materials research and provides

associated materials services including mainly reference materials and data on the properties of materials. Beyond its direct interest to the Nation's scientists and engineers, this Institute yields services which are essential to the advancement of technology in industry and commerce. This Institute is or- ganized primarily by technical fields: —Analytical Chemistry—Metallurgy—Reactor Radiations—Polymers—Inorganic Materials—Cryogenics2—Materials Evaluation Laboratory—Office of Standard Reference Materials.

THE INSTITUTE FOR APPLIED TECHNOLOGY . . . provides technical services to promote the use of available technology and to facilitate technological innovation in industry and government. The

principal elements of this Institute are: —Building Research—Electronic Instrumentation—Textile and Apparel Technology Center

Technical Analysis—Center for Computer Sciences and Technology—Office of Weights and Measures—Office of Engineering Standards Services—Office of Invention and Innovation—Clearinghouse for Federal Scientific and Technical Information.3

1 Headquarters and Laboratories at Gaithersburg, Maryland, unless otherwise noted; mailing address Washington, D. C,

20234.

2 Located at Boulder, Colorado, 80302.

3 Located at 5285 Port Royal Road, Springfield, Virginia, 22151.
UNITED STATES DEPARTMENT OF COMMERCE • John T. Connor, Secretary

NATIONAL BUREAU OF STANDARDS • A. V. Astin, Director

NBS TECHNICAL NOTE 346

ISSUED SEPTEMBER 29, 1966

ON THE NATURAL SHIFT OF A

RESONANCE FREQUENCY

ROBERT J. HARRACH

Radio Standards Laboratory

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

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

Price 25 cents.

TABLE OF CONTENTS

Page

1. ABSTRACT.

2. INTRODUCTION.

3. THEORY.

4. EXPERIMENT. 10

5. APPENDIX. 17

6. REFERENCES. 19

7. FIGURES. 21

iii

ON THE NATURAL SHIFT OF A RESONANCE FREQUENCY

Robert J„ Harrach

National Bureau of Standards, Boulder, Colorado

ABSTRACT

The natural resonance frequency shift, caused

by the transition-inducing radiation field, is examined

for a magnetic dipole transition between hyperfine

structure levels in the ground state of a thallium atom.

A calculation predicts, for an atomic beam experiment,

n 10 a natural shift magnitude of 1 . 4 parts in 10 of the thallium

resonance frequency, per mW/Oe. In the experiment,

frequency shifts caused by overlap of neighboring resonances were observed, but the natural shift was unresolved, indicating that its size is more than an order of

magnitude below the calculated value. Subsequently it has been shown that the natural shift was inhibited by the

particular radiation field mode used in the experiment.

When the theory correctly takes this mode into account,

the calculated natural frequency shift is consistent with

the experimental results.

Key Words: Atomic Beam, Frequency Shift, Radiation Field,

Resonance, Thallium

INTRODUCTION

The magnetic and electric components of a radiation field that

induces transitions between a pair of stationary state energy levels,

perturb the levels in a manner that depends on the radiation field intensity,

Consequently the resonance frequency for the transition is a function of

this intensity, exhibiting a "natural" shift away from the Bohr separation

frequency of the two levels in the unperturbed atom.

In a calculation the natural shift arises from two contributions.

One depends on the radiation field coupling neighboring energy levels to

the pair involved in a given transition. A second contribution is due to

the "anti-rotating" or non-resonant time component of the oscillating

radiation field. The latter effect has as its leading term the familiar

Bloch-Siegert expression, and the former has been discussed by

balwen, Ramsey, Mizushima, Shirley, and others.

In this paper a derivation is given of the natural shift for a fourlevel quantum system of the type displayed by the ground state hyperfine

structure levels of a hydrogen or thallium atom in a weak external

magnetic field. An attempt to measure this shift with an atomic beam

experiment using thallium atoms and Rabi-type (single radiation field)

excitation is then described. In studying a magnetic dipole V -transition

between the levels (F,M ) = (1,1) and (F,M ) = (0,0), it was found that

r F

the natural shift was not resolved, indicating that its size is more than

an order of magnitude below the calculated value. Subsequently, it

[ 17l

has been shown by J. Shirley that the natural shift was inhibited

by the particular radiation field mode used in the experiment. Appreciable radiation field intensity-dependent shifts were observed for very

small external magnetic field magnitudes (extreme proximity of neighboring energy levels), with the shifts becoming larger than 2 parts in
9

10 of the resonance frequency value, per milliwatt variation of input

intensity to the radiation field. These are attributed to ordinary overlap

of the (1, 1) *-* (0,0) transition with the weakly excited (1,0) «-> (0,0) transition,

In the Appendix consideration is given to the natural frequency shift

of an atomic beam resonance when the excitation is made by a pair of

separated radiation fields „ Shift expressions are given for two transitions

between hyperfine structure levels in the ground state of cesium- 133.

THEORY

Interest will be confined to magnetic dipole interactions in quantum

systems for which the Stark effect is negligible. The radiation field is

then specified as an oscillating magnetic field, H (t) = H cos(cot), and the

interaction between the quantum system and field is given by the semiA classical Hamiltonian V(t) = -

jj *H cos(cot), where juis the magnetic dipole

a ~ ~ r ~

moment. Let K be the Hamiltonian of the system in a weak external

o

' magnetic field, H , with nondegenerate eignevalues W corresponding to ~ c k

a complete orthonormal set of eigenfunctions ) U (x) {*&) AK U (x) = W U (x), W >W , (1)

O lC /%_• JK xC /-^ iC"rl xC /

U. (x)U. (x)dx = 6 . (2)

j ~ k _ _ jk

(3)

The problem is then to solve the Schro'dinger equation:

ih^- Y(x,t) = \k +V(t)l Y(x,t).

ot ~ L ° J ~

If a solution is sought as an expansion in the eigenfunctions of

AK in the form t o -it, t Y(x,t) = V C. (t)e U, (x), (4)

then the Schro'dinger equation is replaced by a set of coupled differential

equations in the time -dependent amplitudes C (t)
r t

i(()J + ^W " 1?v

t i± c .(t)= U. - ?. C.(t) + e

J >,b(j,k)e C (t), all j, (5)

dt j L j jj j frg. *

J

where u). = W./fi, and

J J

2b(j,k) s _ 1 Aj*(x)u-H U (s)dx. (6)

To obtain Eq. (5) it has been assumed that b(j, j) = 0, for all j, and the

rotating field approximation has been used, replacing cos(uot) = (e + e )/2

iuut / by the single exponential function e /2. This neglect of the anti-rotating

component of the radiation field precludes a derivation of the Bloch-Siegert

shift in what follows . The time -dependent part of the expansion (4) has been written in

[7] the form of J. Shirley's "phase factoring" transformation. For many

quantum systems it is possible, by making judicious choices for the

values of the phase factors, < 5, > , to satisfy the set of equations

U>+ ?.

-

'?. =0, (7)

J k

[8]

for all j and k for which b(j, k) ^ 0. In this case, Eqs. (5) simplify to

i ~C.(t) = [ttlj - Sjc.ft) +Vb(j,k)C

k

(t). all j (8)

Eqs. (8) are amenable to a perturbation treatment, similar to that of

[2]

Salwen, by which resonance frequency shifts may be calculated.
A well known special case is that where the quantum system maybe represented to a very good approximation as a two-level system, so

that only two of the C (t) in (l|), say C (t) and C (t), are nonzero. Then

[7]

P q

the choice of phase factors can be

£ = (tw + uu - uu)/2 , p p q

S = (in + uu + «))/2 . (9)

q p q

The pair of coupled differential equations, written as a single matrix

equation, are then

d /

c

p

w \ /"tV"' b(pq)

dt

\

c

q

(t)

/ ^

b<pq) + T

(u,qp- ,,,) ' \c

q

(t)

Eq. (10) can be solved exactly, and lead to the Rabi transition

[91 probability. The resonance frequency for the transition in this case

equals the Bohr frequency, id = uu - uu . qp q P

The result of bringing in energy levels other than the pair (p, q)

involved in a given transition is to give, in a way to be described, an

"effective" two-level problem, but with the elements of the 2X2 matrix

Hamiltonian in Eq. (10) shifted to slightly different values. Thus

consider the generalized two-level problem:

(ii)

d p \ / w p
The general solution is

C

P

() \ -v*-^ / V<+> = a(t)e P

+ a_(t )< C (t, / \^ X < +)

P

(12)

where the eigenvalues are (using the upper or lower sign throughout)

h-

(o Mi± 4

^(a- e)

2 + l4 |b|

2

. (13)

and the corresponding orthonormal eigenvectors are

/ K&

xq

(±)

A b

\* 7?

(a - e)

^

V(a-3)

2 + l^|b|2

,

1 (a- 8)

1 2

/ 2 2

' V(a- B) + l/ |b|

\

/

(H)

The constants a (t ) are determined by the given initial state at t = t . The probability, P, that the quantum system makes a transition

from its initial state after being exposed to the radiation field for a duration T is defined in terms of the absolute value squared of the

"overlap" of the initial state (at t = t ) onto the state vector at time

t + T: o P = 1 - C (t )C (t + t) + C (t )C (t + t)

p o p o q o q o

(15)

k |ajt )|

2

|a (t )|

2 sin2

| + o ' ' - o [ +

]
If the initial state is either p or q, we find that the transition probability

for an effective two-level system is

P(p ~q) = fr-«»!''ibii

sin

t

V

<a " B,z+ " b| 1' (16)

For the special case of no neighboring energy levels, the matrix elements

a, $, and b are given as in Eq. (10), and the Rabi transition probability

results.

In treating an ensemble or beam of atoms interacting with the

field, the sine-squared term in Eq. (16) is averaged over the ensemble.

The condition that defines the resonance frequency is that the denominator

in P (p —>q) take a minimum value. When b is independent of frequency,

this implies that the resonance condition is

a- 3 = 0. (17)

Evaluating a and p for a given stationary state energy spectrum thereby

determines the resonance frequency. This evaluation is now carried out

for a specific example.

The ground state of a thallium (or hydrogen) atom in a weak

external magnetic field, H , has four hyperfine structure levels, shown

in Fig. 1, corresponding to the possible values of the total angular

momentum quantum number, F, and the projection quantum number,

M„, in the direction of H . Our interest will be in transitions F ^jc

from the single lower level, (F, M ) = (0, 0), to one of the three F

relatively closely spaced upper levels. The frequencies of these

transitions are in the microwave region.

In considering the effect on the lj.<—»1 transition (ie, the (1,1)

«—*(0, 0) transition) due to the levels 2 and 3, we encounter four
simultaneous differential equations of the form (5). The most significant radiation field couplings are represented (see Eq. (6)) by the matrix

elements b(l, 4), b(l, 3), and b(l, 2). The element b(2, 4) is identicallyzero by the magnetic dipole selection rule, A M = 0, =•= 1. The elements F

b(3, 4) and b(2, 3) can be neglected as regards their influence on the

[12]

4 «—>1 transition.

Retaining only the three matrix elements, Eqs. (7) are satisfied

by taking

?

! = (

{"

l + W

4

- U) )/2 J (18)

? 4

= (^ + ^

4

+ u>)/2 = 3

=§

2

This produces four coupled equations of the form (8), that can be written

as a single matrix equation containing the following 4x4 matrix

Hamiltonian:

/

\

(to- «)41 )/2 b(l,2) b(l,3) b(l,4)

b*(l,2) (aJ 2r%2 _UJ)//Z ° b*(l,3) (0J 31

-0J 43

-0))/2

b*(l,4) («>41

-w)/2

/

An iteration method is used to reduce this to a 2 x 2 matrix

with diagonal elements cx

} 3and off-diagonal elements b, b*. The procedure — iXt

is to make the transformation C. ft) = \ e , k = 1, 2, 3, 4, in (8), thereby k k

producing a 4 x 4 eigenvalue problem. The component equations are

U^a^+b^r (w- tu41

)X

1 /2 + b(l,2)X

2 + b(l, 3) \

3 + b(l,4)x4

, (19)

X\

4

= b*Xj + 3X 4

- b*(l,.4) X

x + (U)41

- w) \

4/2 , (20)

W

2

= b*(l,2) X

i + (a> 2i

- uj42- oj) \

2/ 2 , (21)

U

3 = b*(l, 3) ^ + (^ - w 43

-uj) \

3

/2 . - (22)
The first two equations define ex, g, and b for the effective two -level

problem.

The zero -order approximation consists of taking X = X =

(i.e. , completely neglecting the neighboring energy levels). In this case,

a= (uo - UJ )/2 = - S, and the resonance condition (17) gives (JO = uj . The eigenvalue, by Eq. (13), is

* = ± 1^ (

(^-%1

) 2 + 4|b(l,4)| 2,

In first-order and higher approximations, X and X in Eqs. (21)

and (22) are solved in terms of X , X X, and Ul). In these solutions X

and uo are iteration parameters, denoted by X' and (JO

1 . We find that

X

2 = - 2X^(1, 2)/(«J"

21

- «) 42

-W' - 2X-), (23)

X

3 = - 2X

1 b*(l,3)/(U)31

- UJ 43- «)» - 2X»).

Substituting these relations into (19) and (20) gives a pair of equations in

X and X alone, and identifies ex, p, and b for the problem as:

CL= (U)- U0 41

)/2 -

2[b(l,2)l

2

Z]b(l, 3>1 2

- ^21 " ^42 "

(tJ°' + 2X,) ^31 ~ %3 " ^ + 2V)

S = (U) 41

- (i))/2,

b = b(l,4). (24)

In n th - order, (A)' is taken to be the resonance frequency determined in

r 13] order (n-1), and X' is the eigenvalue, Eq. (13), evaluated at U)1

. Thus, for first-order results, U) 1 = uu and X' = |b(l,4)| .The resonance

8
condition (17) then gives the frequency shift:

2lb(l,2)l

2 2|b(l,3)| 2

Vs " m

41 " +

(uj 21-uu42

- uu 41

- 2|b(l,4)|)

+

(uj 31

-iu

43

-u)

41

- 2|b(l,4)|)

(25)

Quantities like |b(j, k)| ' are proportional to the square of the

radiation field magnitude, and are thereby proportional to the radiation

[14]

field intensity. Using the expressions of Torrey . it is found for

thallium that

and

|b(l, 4)|

2

as u

2 H2

sin

2

(rfl/72h

2 = |b(l, 2)|2

, (26) o r

|b(l,3)|

2 ^u 2 H

2

cos

2

(Ti)/36h

2

) (27) o r

where u is the Bohr magneton, and H sin(r)) and H cos(ri) are,

o r r respectively, the components of the radiation field magnitude perpendicular to and parallel v to the static magnetic field, H . & ~c

The Bohr separation frequencies are evaluated using the BreitRabi formula . For thallium,

W, 3! UU , U) 2S (JO - KH , (A) S: (JU + XH , 31 o 21 o c 41 o c

«)._ 2= + 2KH , (JU = +KH , 42 c 43 c

(28)

where U) /2TT s v ^ 2. 13108 X10 cps, and x/2n^ 4. 65 X 10

5

(cps/Oe)
Collecting these results, the first-order frequency shift expression

for the (1, 1) «^»(0, 0) transition in thallium is

6uo = uu _ m =

res 41

|i H sin (r\) o r (144)h KH

1 4ctn (T]) / U

oHr sin(n)\ / U

oH

r

8in(Ti)^

L^1 + 12^hHHc ) \

+ GJT^KH /

J

2 TT2 . 2, . U H sin (ti) r o r

(144) ft KH

1 + 4ctn (r\) (29)

The approximate form applies when the static field magnitude is large

compared to the radiation field. The shift then varies linearly with

radiation field intensity and inversely with the magnitude of the static

magnetic field.

EXPERIMENT

An atomic beam experiment of the Rabi type, in which atoms

traverse a single radiation field region, was performed to measure the

natural frequency shift (29). A circular cylindrical, variable length

[15] cavity operated in a TE mode provides the radiation field. In this

mode the important magnetic component oscillates parallel to the cavity

axis, with a maximum amplitude directly on axis given by

H (t) = H sin(2TTx/\ )cos(uot), 0^x^\ g/2, (30) r r g

&

where X is the radiation wavelength in the cavity, and x measures distance

along the axis. Also present in this mode are a radial magnetic component

and circumferential electric component, each of which has zero magnitude

directly on axis. The radiation source is a phase -locked klystron, and

power transmitted to the cavity through standard K-band waveguide is

measured by a bolometer.

10
A beam of thallium-205 atoms effuses from an oven through a narrow rectangular slit and passes along the axis of the resonant cavity.

The oven is operated at 900 K, so that the most probable velocity

for atoms in the beam is /3/2 v = /3k-nT/m = 3. 4 X 10 cm/sec„ The

-7

resonant length of the cavity in vacuum (4x10 torr) is X - 16 cm,

giving a theoretical resonance line width of (1. 07) v /t= 1.9 kcps

at optimum excitation intensity.

A constant magnetic field (c-field) is applied transverse to the

beam by means of a current supplied to four rods in a rectangular array.

The cavity is suspended into the center of the c-field, and the entire

region is magnetically shielded. The polarizations of the applied c-field

and oscillating magnetic field are mutually perpendicular, and magnetic

dipole TT-transitions are excited.

Deflection of the thallium atoms is made by two water-cooled

inhomogeneous electromagnets at the entrance and exit to the shielded

region. Each magnet has a 4-in. gap length, and gives a field

3 4 magnitude and gradient at the beam of 6x10 Oe and 2 3 X 10 Oe/cm,

respectively. Beyond the second inhomogeneous magnet, atoms which

have made a transition converge on an oxidized tungsten wire detector,

operated at a temperature of 1Z80 K.

Measurement of the resonance frequency is made by a manual

(as opposed to servo) technique. A single experiment consists of

adjusting the c-field current to a certain value and measuring the

resonance frequency of the (1,1) <—*(0, 0) transition for various levels of

input power to the cavity.

Statistical uncertainties of the measured frequencies were typically

20 to 80 cps (i.e., several parts in 1 of the resonance frequency value).

However, since the transition is strongly dependent on magnetic field, the

measurement precision was limited by small, sporadic variations in the

c-field. From the expression for oi^j in (28), it is seen that a change in

11
-4

c-field magnitude of only 1X10 Oe results in a resonance frequencyshift of almost 50 cps. Such a change can arise from a variation of about

0.1% in the c-field current, or from imperfect shielding of variations in

the magnetic environment of the laboratory.

Within the uncertainty of the results, no radiation field intensitydependent frequency shift was observed for the (1, 1) •*-* (0,0) resonance

until the c-field magnitude was reduced to a value slightly less than 0.0 50 Oe

Some representative data are shown in Figs. 2-4, where the input

microwave power to the cavity is denoted by p, and the experimentally

determined optimum input power is p =0.1 mW . The frequency shifts o are normalized to be zero at p = 0, and data are shown for values of

p/p up to 240 . The observed frequency shifts are fairly linear with p, but show

a definite "saturation" effect for small c-field magnitudes and high input

powers. This tendency is very pronounced for the lowest c-field value

shown (0.0108 Oe). For the other c-field magnitudes it is useful to find,

by least squares analysis, the straight lines which best approximate the

data. These appear as the solid lines in Figs. 2 and 3.

To interpret these results, we first evaluate the natural shift

expression (29) for the atomic beam experiment. When the radiation

field intensity is optimum for exciting an atomic beam resonance between

any pair of levels (j,k), the matrix element coupling these levels is

. [10] given by

b (j,k) | = 1.885 v /I. (31) o o

Using also the relations

2 „2 /„2 iMj.k) r/i b a,k) r =h7*t = P/p

,

o r r o o

12
and Eqs. (26) and (27), we can express Eq. (29) as

6 V = 5^ ( 1.885v

oA)p

2n ' 4nKH pCO

1 + 4 ctn (r\

)

(32)

With the appropriate values of v , t, and H, this becomes

o

6 V -

0. 30p

H p

c o

1 + 4 ctn (ri) (33)

where Sv is in cps and H in Oe

c The applied c -field is perpendicular to the oscillating radiation

field, but since the magnetic shielding is imperfect a residual static

magnetic component exists in the apparatus, making the angle r\ deviate

somewhat from tt/2 radians. A lower limit for the magnitude of the

natural shift is found by setting T] = tt/2 in (33), and this lower limit is

much larger than any shift observed. For example, at H = 0.0469 Oe,

the predicted shift is about -60 cps/mW, while that observed (Fig. 2 ) is

about -2 cps/mW. At very low c -field magnitudes, the expression for

the natural shift has to be carried to higher than first-order, and the

shift is no longer linear but levels off at high intensity. This Is in

qualitative agreement with the data, but quantitatively the situation is

not improved: the predicted natural shifts are more than an order of

magnitude larger than any observed.

Part of this disagreement is accounted for in an "intuitive" way

by considering the expression (30) for the transition -inducing magnetic

field component. The field magnitude is not constant throughout the

transition region, but varies as a half sine wave. This suggests that the

Z H term in (29) should be replaced by

H

2

<sin

2

(2TT x A )> = H

2

A- r g r'

13
This would reduce the natural shift by a factor of two. But, strictlyspeaking, the variation of the oscillating field magnitude makes the

natural shift a function of atom velocity, and invalidates the simple

[17]

resonance condition (17). Dr. J. Shirley has made a detailed analysis

of the natural frequency shift which should result when the radiation field

is of the form (30), and his results completely explain the depression of

the natural shift magnitude below the level that could be clearly resolved

in this experiment.

A study of other effects that could be responsible for the observed

frequency shifts leads to the conclusion that overlap of the (1 , 1) — (0, 0)

transition with the (1 , 0) «—» (0, 0) transition could be significant. The

latter is a magnetic dipole cr-transition, excited by a component of the

radiation field which is parallel to the static c-field. Because of

imperfect magnetic shielding, such a component was present, and the

(1 , 0) «—» (0, 0) transition was weak but observable. The other thallium

transition that could be important, the (1,-1) <—+ (0, 0) transition, was not

observed, since atoms which make this transition are deflected away

from the detector by the inhomogeneous magnets.

To calculate an expression for the overlap frequency shift, we

can represent the line shapes by the Lorentzian function

2 2 2

q(v) = qQw / 0(v - v Q

) + w ] , (31+)

which gives a line centered at v Q) with maximum height q , and line

width at half-maximum given by w. The variation of the line shape with

radiation field intensity is fixed by determining experimentally the

dependence of w and qQ on the microwave input power, p.

For two neighboring resonance lines of the form (31+), it is easily

found that the shift of the line center of one due to overlap with the other

is given by

6 V - (qyqo

)(w')

2

w

2

Av/[l|(Av)

2 + (W) 2

] 2

, (35)

1U
where Av is the separation of the unshifted line centers, and the prime

refers to the neighboring line. From Eqs. (28), we have Av = - xH I 2-ti . The experimental determination of the input power dependence of the line

width, shown in Fig. 5, gives

w= (2.2)10

3

(p/p

o

) 1/4

C ps, (36)

and w' is given by the same relation. Thus the overlap frequency shift

of the (1, 1) «—» (0, 0) resonance is expected to be

-(l.l)10

7

(q^/qo )(p/p

o )Hc 6v = p =—

1/712 CpS " (37)

[(8.6)10 V + 4.8(p/p

o

) i/Z

J

When the line separation is large, this simplifies to

6v = -(1.5)10"

5

( qyqo )(p/p

o )/H

3

cps. (38)

The factor (q"/q ) was found to be approximately 1/4, though it varied

slightly with both p and H . Comparing the expressions (37) and (38) with the observed

frequency shifts, it is found that they agree within an order of magnitude.

Also the observed rates of frequency shift display the inverse cube

dependence on c-field magnitude given in Eq. (38), as shown by the

least squares determined line in Fig. 6. The observed shifts are

therefore consistent with an interpretation in terms of overlap.

Despite the shortcomings of the experiment that has been described,

two improvements would make it very well suited to an observation of the

natural resonance frequency shift. First, one should use a radiation field

mode in which the oscillating magnetic field component has constant amplitude

15
over the length of the transition region (for a given instant in time),

rather than a variable amplitude as given by Eq. (30). The natural

shift would then have its full magnitude as given by Eqs. (29) and (32).

Second, by using improved magnetic shielding and improved c-field

geometry in the apparatus, the neighboring transition (F,M ) = (1,0) «—

(F,M-r-,) - (0,0) could be almost eliminated, and one would expect no

sizeable frequency shifts due to overlap. The most significant effect

that would compete with the natural shift would then probably be shifts

[3] due to unbalanced sidebands in the excitation spectrum. The presence

of such shifts could be ascertained by intentionally altering the sideband

intensities „

16
APPENDIX

[10] When an atomic beam experiment is of the Ramsey type,

employing excitation by a pair of radiation fields separated by a "drift"

region, the natural frequency shift calculation is more complicated.

The transition probability for the effective two-level problem in this

case is found to be

P(p~q>

H(a

4 b

2 i .2

B) +4 b )

. iV iA t

r_ i /

- iX +

T

~ iX T X I cos (w - uu)T/2 ( e + e )

(39)

i(a - B)

V(a- B) Z +4|b| Z sm

1/ - iX +

T - iX

_ T \

(Ou - uj)T/2 I e - e )

where uu is the space average Bohr frequency in the drift region of the

apparatus, T = L/v is the time required by an atom with velocity v to

traverse the drift region of length L, T = t/v is the time required to

traverse each of the radiation field regions of length I, X is given by

Eq. (13), and a, B, b are defined by Eqs. (11). As before, a, (3, and b

are evaluated for a given stationary state energy level spectrum. A

frequency increment 6ui) is defined, in a given order of the iteration

calculation, by

a - B - w - uu qp

•0)

, (40)

where ll) is the Bohr frequency corresponding to the c-field magnitudes

(assumed equal) in the two transition regions. But the resonance condition is no longer (X - 8 = 0. Rather,

17
u, ~ m + M (u, + 6uo - uu ) (41)

res qp L qp qp

The coefficient k depends on the value of (id - li) ) and on the level of

[ 18 ] qp qp

radiation field intensity. For the "ideal" case where U) = oj , k

qp qp

is independent of intensity and approximately equals unity. The natural

resonance frequency shift is then given by — t

id _ a) = — 8uo . (42)

res qp Ij

A special case of interest is the frequency standard transition

between the F = 4, M = and F = 3, M = hyperfine structure levels F F

of a cesium atom. Taking into account the nearest neighboring energy

levels, shown in Fig. 7 , and retaining only the radiation field couplings

indicated in the Figure, the natural frequency shift of the (4, 0)*—»(3, 0)

resonance is derived to be

6uu = (uu - eu ) res res 52

L

[

lb(2,6|

2

|b(2,4)|

2

|b(l,5)|

2

|b(3,5)| 21

L

(W62- m

52

+m

65 )

" (m 52- m

42

+a,

54) { \l~ V"zJ (V W

53 + Vj

^ + 31 |b | 2

tan

2

(ri)p/Lp

o

, (43)

where T) is the angle between the c -field and radiation field, and in this

case, I bQ I = (0. 942) Vq /^ . The latter form results when the matrix elements

are evaluated for cesium through terms linear in c-field, and the Bohr

frequencies are evaluated through the quadratic c-field terms.

A similar calculation for the (4, l)«-»(3, 1) transition in cesium

gives

6u) = I |b | tan (T|)p/l5Lp 2n (7. 00)10 H . (44)

res o r' o c

18
REFERENCES

[1] F„ Bloch and A. Siegert, Phys. Rev. 57, 522 (1940). See also

references l 3] , [7], andflO].

[2] H. Salwen, Phys. Rev. 99, 1274 (1955).

[3l N. Ramsey, Phys. Rev. 100 , 1191 (1955), and in Recent Research

in Molecular Beams , I. Estermann, Ed. (Academic Press, New

York, 1959).

[4] M. Mizushima, Phys. Rev. 133 , A414 (1964). In the last section

of this paper, the frequency shifts of some resonances in cesium

that are interpreted by Mizushima as natural shifts are instead

most likely caused by c-field nonuniformity. See reference [ 18] .

[5] J. Shirley, Thesis (California Institute of Technology, 1963), and

Phys. Rev. L38, B979 (1965).

[61 J. Winter, Ann. Phys. (Paris) 4, 745 (1959), S. Autler and

C. Townes, Phys. Rev. 100 , 703 (1955).

[7] J. Shirley, J. Appl. Phys. 34, 783 (1963). See also, K. Freed,

J. Chem. Phys. 43, 1113 (1965).

[ 8j In a given problem few of the b(j,k) will be zero or even small.

However, by the same reasoning that justifies the two-level

approximation we can neglect those matrix elements that would

induce a transition with resonance frequency greatly different from

the frequency of the resonance under consideration.

[9] I. Rabi, Phys. Rev. 51_, 648 (1937). See also references [10]

and [14].

L 10] N. Ramsey, Molecular Beams (Oxford University Press, New

York, 1956).

19
[11] R. Mockler, Advan. Electron. Electron Phys . 15, 1 (1961) „

[12] The "Zeeman couplings", b(2,3) and b(3,4), induce transitions with

5 resonance frequencies smaller than go by a factor of 10 or more

for the c-field magnitudes of interest (see reference [8]).

[13] Of the two choices for. the eigenvalue, \ or \ , one is a "proper"

choice (corresponding to Salwen's proper root in [2j)» The other

is non-physical, leading to the possibility of an infinitely large

frequency shift,,

[ 14] H. Torrey, Phys. Rev. 59, 293 (1941).

[15] F. Horner, T. Taylor, R. Dunsmuir, J. Lamb, andW. Jackson,

J. Instn. ElecJ. Engrs. 93 III, 53 (1946).

[16] R. Beehler, W. Atkinson, L. Heim and C. Snider, IRE Trans, on

Instr. 1-11 , 231 (1962).

L17] J. Shirley, "Effect of a sinusoidal excitation amplitude on the

performance of an atomic beam spectrometer" (to be published).

[18] R. Harrach, "Radiation field dependent frequency shifts of atomic

beam resonances" (to be published).

20
M

LEVEL

NUMBER

F=1

F =

Fig. 1. Hyperfine structure levels of the ground state of a thallium-205

atom in a weak external magnetic field.

21
Si/

(cps)

Sv

(cps)

-300 -

-200 -

-100 -

P/Pr

Fig. 2. Measured frequency shifts of the (1, 1) *— (0, 0) resonance for the

c-field magnitudes H = 0. 0469 Oe (upper) and 0. 0217 Oe (lower).

22
8^

(cps)

P/P,

(cps)

P/P

Fig. 3. Measured frequency shifts of the (1, 1) <—>(0, 0) resonance for

H = 0. 0141 Oe (upper) and 0. 015 Oe (lower).

23
-1000

-800

-600-

(cps)

-400 -

- 200 -.

P/ P,

Fig. 4. Measured frequency shift of the (1, 1) --(0, 0) resonance for

H =0. 0108 Oe.

c

24
8 —

7 -

Cl

o

* 5 -

X

ILU

4 -

3

2

I —

(P/Po)

1/4

Fig. 5. Measured variation of the resonance line width as a function of

excitation intensity.

25
o

Q.

01

LU

CL

</> CLO

X

in

>-o

UJ

Z)

o

LU

or

U_

U_O

LU

f-

<

-4 /^~-3 Hp x 10"* (0e~ 3

)

Fig. 6. Rate of frequency shift as a function of c-field magnitude.

26
MF

I

F = 4

-I

-I

F = 3

I

> >

43 ^53

< <

^42

>

52

>

62

>

51 >61

LEVEL

NUMBER — 6

5

4

3

2

Fig. 7. Some hyperfine structure levels of the ground state of a cesium- 133

atom in a weak external magnetic field.

27 G PO 830-230


U.S. DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20230

POSTAGE AND FEES PAID

U.S. DEPARTMENT OF COMMERCE

OFFICIAL BUSINESS