MICROWAVE RAMSEY PATTERN IN A LASER PUMPED CESIUM BEAM. APPLICATION TO VELOCITY DISTRIBUTION AND 2nd ORDER DOPPLER-SHIFT IN FREQUENCY STANDARD

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MICROWAVE RAMSEY PATTERN IN A LASER PUMPED CESIUM BEAM. APPLICATION TO VELOCITY DISTRIBUTION AND 2nd ORDER DOPPLER-SHIFT IN FREQUENCY STANDARD

M. Arditi

To cite this version:

M. Arditi. MICROWAVE RAMSEY PATTERN IN A LASER PUMPED CESIUM BEAM.

APPLICATION TO VELOCITY DISTRIBUTION AND 2nd ORDER DOPPLER-SHIFT IN FREQUENCY STANDARD. Journal de Physique Colloques, 1981, 42 (C8), pp.C8-261-C8-269.

�10.1051/jphyscol:1981833�. �jpa-00221728�

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CoZZoque C8, suppZ6ment au nO1 2, Tome 42, de'cernbre 1981 page C8-261

MICROWAVE RAMSEY PATTERN I N A LASER PUMPED CESIUM BEAM, A P P L I C A T I O N TO VELOCITY D I S T R I B U T I O N AND 2nd ORDER DOPPLER-SHIFT I N FREQUENCY STANDARD

M. Arditi

I n s t i t u t d 'Etectronique FonondamentaZe, 91405 Orsay, France

Abstract.- Analysis of experimental microwave Xamsey patterns, obtained with a laser pumped cesium beam frequency standard, shows the cesium atoms velocities to follow a broad 14axwellian distribution. As a consequence, the estimate of the second-order D6ppler-shift and cavity 2hase-shift could be more accurately obtained with this device than with the classical cesium beam frequency standard

.

I. Introduction.- An optically pumped cesium beam atomic clock, recently developed, has been already described (I). Phis device has potentialities for high accuracy as a frequency standard because it eliminates the magnets used for state selection and consequently the n~icrowave resonance can be induced in a more uniforrne magnetic field, closer to theoretical considerations (reduction of Majorana transitions, for example). The mechanical construction is also simplified, and with the on-axis alignement of the oven and resonant cavity, the cavity phase-shift no longer has a spatial de2endence across a section of the beart1 and, for the same reason, the fre- quency shift due to variations in microwave power should be greatley reduced.

The system can be made entirely symmetrical by ?lacing permanently a cesium oven at each end, thus facilitating the measurement of cavity phase-shift by the direct method of beam reversal. Also, a precise knowledge of the velocity distri-

bution and average velo-

OUTPUT 5 M H z city of the cesium atoms

is possible in this appa- ratus and should permit an accurate evaluation of the second order ~gppler-shift.

Figure 1 shows a schematic of the device : briefly, it consists of a

AMPLl cesium beam in vacuum with

3 0 H z optical pumping (2) by a laser diode to replace the Stern-Gerlach magnets in the classical Rabi magne- tic resonance apparatus, and optical detection of the microwave resonance

I

I by a change of the beam fluorescence detected by a ohotocell in B. In both UJ ^: PHASE DETECTION F.C.: FREQUENCY CONTROL

Pig.] : Cesium beam frequency standard with optical pumping and optical detection.

cases, a Ramsey-tyge ca- vity (3) is used with the same electronics for exci- tation and detection of the microwave resonance.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981833

C8-262 JOURNAL DE PHYSIQUE

The purpose of this paper is to compare the microwave patterns obtained with the Rabi-Ramsey or Kastler-Ramsey type of apparatus and to deduce some conclusions regarding the estimates for correction of the

znd

order ~appler-shift or cavity phase-shift in both cases.

11. Magnetic resonance methods with separated oscillating fields (Ramsey type micro- wave cavity) : The well known Ramsey's method of separated oscillating fields (4) is

currently used in the classical cesium atomic frequency standard to overcome the effect of field inhomogeneity and to produce a narrower resonance pattern.

For an atom of velocity v, near resonance, the transition probability is given by :

1 L

P = sin'

2 2

cosz

-

^(u-uo);

P 9 q v 2 ⁽¹⁾

with wo, the Bohr frequency, b2 a factor proportional to the exciting microwave power, .p the length of the oscillating regions and L the distance between the oscil- lating regions.

For an atomic beam effusing from an oven, the velocity distribution of the atoms in the beam is such that if I(v)dv is the beam intensity from atoms between velocities v and v+dv :

with a = 2kT/m ( 3 )

(m : atomic mass, and T oven temperature).

This is the so-called Maxwellian distribution.

Assuming then such a velocity distribution in the beam, equation ( 1 ) averaged over the atomic velocities gives (with y = via)

The microwave resonance signal is proportional to <P- _ > and to the excess P ? q

population of atoms in either the p or q states. A population difference is creat- ed in the atomic beam 0-0 states either by magnetic state selection or by the op- tical pumping method. However, both methods can alter the velocity distribution in the atomic beam at the entrance of the Ramsey cavity and thus the averaged tran- sition probability <P > is no longer given by equation (4). In this paper we shall examine the res81eing Ramsey resonance pattern in both cases.

1. Velocity distribution with magnetic state selection (Rabi-Ramsey apparatus) The deflection angle B in a dipole magnet state selector of length Qo is given by :

For an oven which is off the axis of a microwave cavity, 6 is fixed and thus only the atoms of velocity v go through the cavity. However, due to the length of the cavity opening there is a small dispersion of values around B and thus one could considered a high and a low velocities cut-off and a mean velocity in a truncated Maxwellian distribution to best fit the Ramsey resonance pattern (5). However, such a procedure cas be laborious and may not give a unique solution.

Deconvolution methods have been proposed to obtain the velocity distribution from the experimentally recorded resonance pattern (6). These methods require la- borious computer operations.

With a pulse excitation of the atomic beam in a Ramsey cavity, a more direct method of obtaining the velocity distribution is possible (7)(8), and the normal Ramsey pattern calculated from this measured velocity distribution agrees quite well with the measured microwave resonance. The method has shown, in particular, that the velocity distribution could change with the aging of the cesium beam tube.

2. Velocity distribution with optical pumping (Kastler-Ramsey apparatus).

2a. Optical pumping with spectral lamps.

Before lasers, in the near infra-red, became available, optical pumping with spectral lamps has been used in the realization of a rubidium atomic clock (9).

Rubidium 87 atoms were used, in a beam of natural rubidium, because of the possibi- lity of hyperfine pumping with isotopic filtering with rubidium 85 filter-cells.

A linear approximation in a theoretical analysis (10) of the operation of this clock (well justified by the relatively low intensity of the lamps) shows that a factor l/v2 is introduced by the process of optical pumping and optical detection which increases the statistical weight of the slow atoms in the over-all signal.

Consequently the average transition probability is given, near resonance, by :

instead of equation (4).

In this case <P > has a maximum value of 0.68 when

-

4bR = 2.7

.

P14 d<Po-O>

Figure 2 shows a derivative

---

_dF of <Po-O> versus frequency F, obtained experimentally (9), and the cross ~oints are values computed from equation (6).

It can be seen that the agreement between theory and experiment is quite good. By tracing, on the graph, the derivative of <P > as given by equation (4) for a

0,o full Maxwellian distribution of velocities, the line-narrow-

Rb and SPECTRAL LAMPS ing produced by optical pumping

(about 40%) is clearly apparent L . .

"

4bl/0(

=

2.7 2b. Optical pumping with a

I ,I

i

e x p e r i m e n t a l

4 = 300 m / s . laser diode :

I

^\ ----maxwellian

I F

1 !

In more recent ex~eri-

ments (I), a laser diode has been used to optically pumped a beam of cesium atoms. The laser diode was operated under c.w., single mode operation, with its temperature stabiliz-

ed around 25 K in a cryostat.

The output wavelength was coar- sely adjusted in the vicinity of the cesium D p resonance line

(852,l nm) by changing the tem- perature, and finely tuned by varying the injected current.

The atomic beam fluorescence was used to stabilize and lock

the laser frequency to a spe- cific hyperfine component of the D2 line ( 1 1 ) . For pumping the laser diode delivered Fig. 2 : Derivative of Ramsey resonance curve about 1 to 2 milliwatts of

for ~b~~ beam and spectral lamps pumping.

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infra-red radiation focus- ed on the cesium beam in a

Cs and Ga A s LASER spot about 1 cm2, and about

114 of this power was used

L,

21.5 cm. in the detection region.

4 b 1/42 4 Under these conditions, the

0-0 clock transition in the

-experimental cesium beam could be de-

- * * a maxwelllan

tected with a much better signal to noise ratio than with the rubidium beam

clock. However, it should be noted also that in these experiments there was no saturation in the detection of the microwave signal, i.e., the microwave signal decreased by 112 when the laser intensity was reduc-

Figure 3 shows a ty- pical experimental record- ing of the derivative d < Po-O'

dF of the microwave resonance as the frequency Fig. 3 : Derivative of Ramsey resonance curve for

of the exciting microwave csl 33 beam and laser diode pumping. was slowly swept around

the resonance. This curve was obtained with a Ramsey cavity (L = 21.5 cm, .9 = 1 cm) and an oven temperature of 90 "C, corresponding to a value of CY of 215 m. sec-l, and for a condition slightly above optimum microwave power. On this graph, the dotted points are computed from equation (4) and the best fit correspond to a value of 4bRla = 4 (theoretically (4) optimum power cor- respond to 4bRla = 3.77). The agreement is quite good over almost the entire pattern. Numerous such recordings, with variations in the laser light intensity, the oven temperature, the cavity length, the rf modulation, the microwave power, etc. well established experimentally that the atomic velocity distribution was indeed a broad Maxwellian distribution, with no narrowing of the Ramsey pattern as obtained with the rubidium beam pumped with spectral lamps.

In that respect, it should be noted that the spectral density of the mono- chromatic laser radiation is here about l o 3 times higher than for a spectral lamp, so that the efficiency of optical pumping is about the same for all the atoms, in- dependently of their transit time across the light beam. Yet the fact, mentioned above, that there is no saturation of the detected microwave signal for the laser intensities used in these experiments suggests that perhaps some hyperfine relaxa- tion process, in the ground state, decreases the population inversion produced by optical pumping. This relaxation could be due to spin- exchange by collisions between cesium atoms if the vapor density of cesium atoms, in the pumping region near the oven, was not low enough. A cooling trap in this region, to condense cesium vapor, could be helpful1 in reducing the relaxation by collisions. Experi- ments along these lines will be pursued.

However, the fact that the velocity distribution in the beam can be consider- ed as broad Maxwellian has importailt application in the theoretical estimate of 2nd order ~6p~ler-shift and cavity phase-shift as will be discussed now.

111. Second-order Dijppler-shift and cavity phase-shift in optically pumped cesium beam with laser diode.

For an atom with velocity v, moving transversly to the direction of observa- tion, and from the special theory of relativity, the relative shift of the transi- tion frequency is given by :

In an atomic beam, the shift must be averaged over the velocity distribution, and :

With the classical Rabi-Ramsey apparatus, the velocity distribution is not broad Maxwellian and cannot be simply deduced from the oven temperature because the beam optics will modify the distribution to a large extent ( 7 ) ( 8 ) .

Here, with on-axis alignment of the cesium bean and the microwave cavity, and with laser diode pumping, the distribution of atomic velocities across the cavity's openings is isotropic and with a broad Maxwellian distribution of velocities the (D.S) can be easily calculated.

As shown in reference ( 7 ) , near resonance :

where p(v) is the velocity distribution. For a broad Maxwellian distribution

Substituting (10) in (9) and with y = v/a

and

(These integrals are rapidly convergent and an upper limit of 3 for the highest velocity gives sufficient accuracy. Also these integrals can be easily calculated with a programmable pocket computer).

Equation (12) is in agreement with the theoretical results previously report- ed by Harrach (12) where a broad Maxwellian distribution of velocities was consi- dered. For a given oven temperature, i.e. a fixed value of a

,

the microwave

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F i g . 4 : T h e o r e t i c a l (D.S) a s a f u n c t i o n of e x c i t a t i o n power.

power dependence of t h e (D.S) i s g i v e n i n F i g u r e 4. I n t h i s f i g u r e Po, t h e optimum power i s o b t a i n e d f o r a v a l u e of 4bRla = 3.77, where b2 i s p r o p o r t i o n a l t o t h e microwave power. This f i g u r e i s s i m i l a r a l s o t o F i g u r e 5 of Harrach's paper.

4b9.

For

-

= 3.77 e q u a t i o n (12) g i v e s :

f o r B = 21.5 10 cm.sec- 3 ¹

1 4

An accuracy of 10- f o r t h e (D.S) r e q u i r e s t h a t PIPO, around t h e optimum power, should be known t o w i t h i n f 0.5 dB. On t h e o t h e r hand t h e same accuracy f o r (D.S) r e q u i r e s t h a t a should b e known w i t h i n

+

3 m.sec-l, i . e . , according t o e q u a t i o n ( 3 ) , t h a t t h e oven temperature should be known w i t h i n i 12 ^{O C} around a mean v a l u e (94.5" f o r example, corresponding t o a = 215.6 m.sec-I). This should not be too d i f f i c u l t t o o b t a i n .

With t h e v a l u e of a given by e q u a t i o n ( 3 ) , a f i r s t approximation f o r t h e v a l u e of b corresponding t o a given microwave power s e t t i n g ( c l o s e t o optimum power) can be o b t a i n e d i n t h e f o l l o w i n g way: i t i s observed t h a t t h e second z e r o of t h e d e r i v a t i v e <

-

dP dF ^> of t h e Ramsey resonance curve corresponds t o a v a l u e of t h e microwave f r e q u e n c e f such t h a t :

( I n t h a t curve a l s o , from t h e r a t i o of t h e peaks, t h e v a l u e of 4bRla c a n be deduced independently of a ) .

By computing t h e Ramsey resonance p a t t e r n from e q u a t i o n ( 6 ) f o r s l i g h t l y d i f f e r e n t v a l u e s of a and b t h a n t h e i n i t i a l v a l u e s found above, and f o r t h e b e s t f i t w i t h t h e Ramsey experimental curve, one could o b t a i n t h e e x a c t v a l u e s of a and b n e c e s s a r y f o r t h e e s t i m a t e of t h e (D.S).

An o t h e r method t o o b t a i n t h e v a l u e of a with a n o p t i c a l l y pumped cesium beam i s t o use t h e technique of "time of f l i g h t " a s used i n spectroscopy ( 1 3 ) , which

requires no microwave, only a light chopper and a multichannel analyzer, and would give also the velocity distribution in the beam.

2. Cavity phase-shif t (C .S)

If there is a phase difference 6 between the two microwave sections of the Ramsey cavity, the resulting shift in the resonance frequency is given by (7) :

r ^$

p(v) sin'

-

2bR _v ^dv

To obtain a cavity phase-shift smaller than 10-14, equation (1 7) indicates that a synchronization of the microwave field at both ends of the order of 10-l7 second is required !

Actually the measurement of the residual 6 by the method of beam reversal should be made easier in this case by placing permanently a second oven at the end of the tube since the device is symmetrical and the pumping and detection regions can be interchanged without any modification of the apparatus, except for inter- changing the electronics of the photocells A and B.

IV. Conclusions.

-

The optically pumped cesium beam frequency standard has attrac- tive potentialities as a high accuracy primary standard because of its simplified construction, a more uniform magnetic field, a broad Mamellian velocity distri- bution which simplifies the estimate of the 2nd order ~Sppler-shift and the possi- bility of easy beam reversal experiment to measure accurately the cavity phase- shift. However, experiments should be conducted to see if eventual "light-shifts"

produced by optical pumping and fluorescence of the cesium beam do not limit the accuracy of he device. The recent development of stable cw laser diodes operat- ing at room temperature should facilitate the realization of such a frequency standard.

V. Appendix.- One can also compute the ( D . S ) using the momenta method of Audoin et al. ( 1 4 ) . In that reference, the (D.S) of the resonance maximum, without micro- wave modulation, is given by :

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m

where .

k =

\ ^(+lk

£(T) sin2 2 b ~ d~

T is the time of flight of the atoms inside each of the interacting regions and

Substituting (10) in (19) equation (18) becomes identical to (12) with :

2 L~ 4bR

e sin2 dy = 0.304

3

for

-

^{= 4}

a4 aY

Similarly, with the values of these coefficients, the error biases introduced by square-wave or sinusoidal modulation of the microwave power can be easily com- puted with the equations provided in reference (14).

Also, as shown in reference (14), the values of the Tk can be obtained with- out an explicit knowledge of the velocity distribution function by approximating the central peak of the Ramsey pattern with the following polynomial expression:

This was applied to the theoretical value of P given by equation (4) and limited to the central part of the resonance (about full width at half-maximum).

For the case where 4bR/a = 4, the best fit gives the following ratios :

-

^{T2 = 0.716 and}

2

T ⁼ 0.72 10-l2

To

To

whereas equations (21) give :

T2 T

-

_T ^{= 0.8 and} 2 = 0 . 8 1 0 - ' ~

0

To

The discrepancy may come from a lack of accuracy in the approximation of P by (22), and may be significant only at the order of

.

V I

.

R e f e r e n c e s .

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