MODULATION EFFECTS IN PASSIVE Rb

FREQUENCY STANDARDS

P. Thomann, G. Busca

To cite this version:

P. Thomann, G. Busca. MODULATION EFFECTS IN PASSIVE Rb FREQUENCYSTANDARDS. Journal de Physique Colloques, 1981, 42 (C8), pp.C8-189-C8-197.<10.1051/jphyscol:1981822>. <jpa-00221717>

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Submitted on 1 Jan 1981

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CoZZoque C8, suppZe'ment au n012, Tome 42, de'cembre 1981 page C8-189

MODULATION EFFECTS I N P A S S I V E Rb FREQUENCY STANDARDS

P. Thomann and G. Busca

ASULAB S. A., 6, passage M a x Meuron, CH-2001 NeucMteZ, Switzer Zand

Abstract.- The lineshape of a microwave resonance is studied theore t i - c a l l y and experimentally i n the case of a phase-modulated RF-field and o p t i c a l detect ion of the resonance. The ca lcu la t ion allows t o optimize t h e e r r o r s i g n a l i n a Rb frequency s tandard with respec t t o the modula- t i o n parameters. Sine-wave and square-wave modulation a r e examined i n some d e t a i l .

1. Introduct ion.- The s h o r t term s t a b i l i t y of a passive Rb standard is mainly li- mited by t h e s igna l t o noise r a t i o of t h e e r r o r s i g n a l needed t o lock t h e quar tz o s c i l l a t o r t o t h e atomic frequency. The e r r o r s i g n a l i s usual ly obtained by modu- l a t i n g t h e phase of the RF f i e l d and de tec t ing t h e corresponding modulation i n the l i g h t i n t e n s i t y t ransmit ted by t h e resonance c e l l . I n t h i s paper we derive ana ly t i - c a l r e s u l t s fo r t h e dependence of t h e e r r o r s i g n a l on modulation frequency, modula- t i o n depth and re laxa t ion r a t e s , with p a r t i c u l a r a t t e n t i o n t o the s p e c i f i c cases of sine-wave and square-wave modulation.

Modulation e f f e c t s have been extensively s tudied i n t h e p a s t ([I] - [8]). A l l au- thors we a r e aware o f , however, concentrate on the lineshape of t h e radio-frequency absorption curve. Here we a r e concerned with o p t i c a l detect ion of an RF resonance s igna l , which means t h a t the population inversion between t h e two hyperfine l e v e l s , not t h e i r coherence, is t h e re levan t parameter. Furthermore, most treatments a r e r e s t r i c t e d a s t o the range of t h e modulation parameters (modulation frequency low [ 6 - 8 o r high 1 3 , 43 compared t o t h e re laxa t ion r a t e s , low modulation amplitu- des [2. $, . In t h i s ca lcu la t ion we pu t no l imi ta t ion on t h e modulation amplitude and frequency, so t h a t t h e e r r o r s i g n a l cannot be r e l a t e d i n general t o der iva t ives of t h e s t a t i c lineshape.

In order t o s implify t h e mathematical treatment and t o obtain meaningful a n a l y t i c a l r e s u l t s , we make the two following assumptions:

1) Although t h e populations of a l l Zeeman subs ta tes a r e coupled through t h e o p t i c a l pumping process , we assume t h a t t h e (small) population changes induced by t h e RF f i e l d i n the two field-independent s t a t e s have a neg l ig ib le e f f e c t on t h e o ther populations. Thus we replace t h e two Zeeman mul t ip le t s by two s i n g l e l e v e l s and assume t h a t t h e e f f e c t of the coupling within m u l t i p l i c i t i e s can be contained i n the re laxa t ion r a t e s of t h e two-level system.

2) We assume t h a t the RF power is low enough t o neg lec t s a t u r a t i o n e f f e c t s ; t h i s allows a per tu rba t ive treatment of t h e equations of motion. I n t h i s respec t our ca lcu la t ion i s s imi la r t o the one by Karplus [I], bu t second-order, i n s t e a d of f i r s t o r d e r , r e s u l t s a r e necessary t o account f o r t h e o p t i c a l detect ion of t h e RF-resonance.

2. Derivation of the resonance lineshape.- The evolut ion of t h e densi ty matrix is s p l i t i n t o t h r e e p a r t s corresponding t o o p t i c a l pumping, re laxa t ion and i n t e r a c t i o n

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981822

JOURNAL DE PHYSIQUE

with t h e RF f i e l d : O P Re l R F

a , ~ = a , ~ + a , P + ~ , P

We take a simple model f o r o p t i c a l pumping where l e v e l s 1 and 2 a r e depleted by t h e pumping l i g h t a t r a t e s rl and r2. The pumping l i g h t i s assumed t o be weak and t o have a broad spectrum, i n which case t h e pumping cycle i s adequately described by r a t e equat ions (no o p t i c a l coherences). The l i g h t i s reemit ted spontaneously from a s i n g l e exc i ted l e v e l which decays t o l e v e l s 1 and 2 with t h e same r a t e 4y, ( ~ > > r ) . We then have the following equations f o r t h e hyperfine 0-sublevels (we neg lec t t h e l i g h t - s h i f t f o r s i m p l i c i t y ) .

We assume t h a t a l l r e laxa t ion mechanisms ( c o l l i s i o n s with o t h e r atoms o r with the walls , magnetic f i e l d inhomogeneities) can be described by a " longi tudinal" and a " t ransverse" re laxa t ion time y i , and y;; t h e s teady s t a t e densi ty matrix elements without o p t i c a l pumping a re

011 = 022 = 4 and p l 2 = 0 , so t h a t

a Rel , Pn = - Y; (Pit - 1/21

a Rel , p22 = - Y; (P22- 1 4 )

Rel a , p 1 2 = - yiplz Fina l ly , the i n t e r a c t i o n with t h e RF-field ( l i n e a r l y po la r ized p a r a l l e l t o t h e C-

f i e l d ) is given by

where p2 / ~ W O pe icp(t)

H - . ( Be-icp(t) -% w o

Uo includes the quadra t ic Zeeman e f f e c t and t h e l i g h t s h i f t

p = .-1 PBBRF i s t h e coupling s t reng th o f the 0-0 t r a n s i t i o n i n Rb-87 with t h e RF f i e l d

Radiofrequency f i e l d , with phase $ ( t ) t o be BRF ( t ) = B R F COS cp(t)

s p e c i f i e d l a t e r

Defining A = pl l - p22, we obtain t h e following equations f o r A and p12

A = - Yl (A-AO) + 4 p Jm ( p12e - i V ( t )

biz=(-y2 + i u o ) p,, - ,ae icp(tIA

where = y; + 1 3 ( r1 + r2 )

y , = Yi + % ( r l + r 2 )

0 - Yz(r - r A -&+b;r~+;:)

In obtaining equations (4) we have a l s o made use of t h e rotating-wave approximation, which i s j u s t i f i e d s ince i n p r a c t i c e both 6 and t h e frequency w, of phase modulation a r e much smaller than W o .

Although a numerical so lu t ion of eqs (4) can read i ly be obtained f o r any value of t h e parameters, we r e s t r i c t t h e range of RF amplitude i n order t o ob ta in an approxi- mate a n a l y t i c a l so lu t ion . I f 6 < yl, yz we can expand A and i n a per tu rba t ion s e r i e s of powers of 8. This is n o t a very r e s t r i c t i v e condition: i n p r a c t i c e one avoids t o s a t u r a t e t h e t r a n s i t i o n s ince s a t u r a t i o n a l s o means broadening of t h e re- sonance curve and decrease of t h e discr iminat ing slope.

We there fore w r i t e

with d") - on

~2)- b n

Inser t ing t h e expressions above i n t o eqs ( 4 ) and s o r t i n g ou t terms of equal powers of B y i e l d s the following orders of approximation f o r t h e permanent solut ions:

Order 0 : A(') = A0 : = o

Order 1: A(') = 0

- i p a0 +iwt7 +iB@(t') .(-Y2 + iw,) ( t - t') d t'

where we have used t h e following d e f i n i t i o n

64I(t) i s the phase modulation of t h e RF f i e l d ; we make use of i ts p e r i o d i c i t y t o w r i t e it a s a Fourier s e r i e s

- we suppose f o r s impl ic i ty t h a t ak is r e a l , which is t h e case i f 6 $ 1 ( t ) = - 6@(- t ) .

Theref ore

( l) iwt a k e ikwmt

p I l ( t ) = - iPAoe 2 -a Y2+i(U +kW,)

where Cf = W - U o is t h e RF-atom detuning.

Order 2: p$) =

I n t e g r a t i n g eq. (9) we obta in , f o r the lowest (second) o rder i n B, the following expression f o r t h e permanent so lu t ion A ( ' ) (t) :

Replacing PI(:) ( t ' ) by i ts e x p l i c i t expression (eq. 8) and performing t h e integra- t i o n , we g e t , a f t e r some manipulations

A (2)(*) = - 2' Cp COs pw, t + sp sin pw,t Y1 Y 2 { P=o

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where A ( x ) = ( i + xZ)- ' x k = (a + k w , ) / y, and

D ( x ) = X . ( 1 + x 2 ) - I Y p = p W m / Y,

The inversion A ( ' ) shows o s c i l l a t i o n s a t a l l harmonics of t h e modulation frequency. A s a funct ion of t h e detuning a , t h e amplitude of these o s c i l l a t i o n s undergoes re- sonances centered a t a = - kq,,. When % << y l , y~ t h e add i t iona l resonances (k # 0) merely broaden the unmodulated l i n e shape whereas i f q,, >>yl , y2 they appear a s resolved sidebands. Since t h e f i r s t harmonic (p = 1) is of most i n t e r e s t here we w i l l concentrate on t h i s component and, i n o rder t o ob ta in s p e c i f i c a n a l y t i c a l re- s u l t s , we w i l l consider two s p e c i a l cases of phase modulation, namely sine-wave and square-wave modulation.

3. F i r s t harmonic of t h e population inversion, sine- and square-wave phase modulation.-

The f i r s t harmonic component of t h e inversion is given by e q . . l l :

where C1 and S 1 a r e given by eqs ( l l b , c )

a ) sine-wave modulation

The phase excursion is then

8g,( t )= m s i n w m t ; t h e Fourier development of ei6'(t) reads

i m s in%t +o = J~ ( m ) e i k w m t

k=-w and we have ak = Jk(m), where Jk(m) is t h e s tandard Bessel funct ion of order k.

b ) square-wave modulation -

i f k = O

= [ i s I i n m i f i f k k is i s even odd ,.,,

Table 1 gives a summary of a n a l y t i c a l r e s u l t s which a r e v a l i d i n some l i m i t i n g cases of i n t e r e s t . For small modulation frequencies t h e l i n e shape i s , a s expected, equal t o t h e der iva t ive of t h e s t a t i c Lorentzian l i n e shape (eq. 15) . A t high modu- l a t i o n frequencies , however, t h e l i n e shape becomes a s tandard dispersion curve (eq. 1 3 ) .

Comparison between these two l i m i t i n g cases (eqs 13 and 15, t a b l e 1) i n d i c a t e s a simple method t o determine Y2 fromthe width of t h e experimental curves and y l from t h e i r amplitude. A t high modulation frequencies, f o r example,the dispersion curve peaks a t a/yZ = I, which gives an immediate measurement of '(2. Once y2 is known, y l can be determined i n t h e following way. Keeping the phase excursion m and the de- tuning a f ixed, t h e amplitude C1 of t h e e r r o r s i g n a l is measured a t two frequencies: (&, y2 and Um2 >> y2. Inser t ing the experimental values C l ( % l ) and C2((&2) i n t o eqs 13 and 15 y i e l d s

ffl ffl " w .rl 0 rn

h 4

0

g 8 I" rn

8 ..i 0

3 " B 8

A

r- =!

0 - " z 5 2

~ I L II e V V s s CI

0 I

-NY- I II II II a - 4J - s 8 -

Y , 8

;

A-

- $ h U .rl ::

2 2 + A " " V

; 31; G v 2 g I 31;

'I ?I

0 - "I U rn

-

S - N

N

* II

A A A A

N m ln W

=! - 5 " - 5

T' 2

al

m

" ffl

B u

A " 8 I

C .rl rn

N A

N * $2 +

.rl rn - rn +

g 8 - U

I

3 2 4 3 I b - II II E 4 - 0 - II V

2? # 2: e

a m I al

.r( rn

4 4

* :: 819 rn 2 + N

u rn - N

,I ,I

0 - 4-1 4

b

II - +J - -

N

4

u

9

A I 8 2 2:

I

u (I

8 G + rn

N - .. ";,

N -

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Once y2 and yl a r e de'termined, t h e modulation frequency and t h e modulation index can be chosen so a s t o optimize t h e s lope of A1 (x) = (c: + s:)% near o = 0, using t h e curves no 3 , 4. I t should be noted t h a t Al, t h e maximum amplitude o f t h e s i g n a l , r a t h e r than C1 o r S1, is t h e re levan t parameter f o r determining t h e discr iminat ing s lope s ince C1 and S 1 a r e both equal t o zero a t o = 0 and a r e proport ional t o each o ther near a = 0. I n o t h e r words t h e phase s e t t i n g of t h e phase s e n s i t i v e de tec tor i n a servo-loop using A ( 2 ) ( t ) a s an e r r o r s i g n a l serves only t o maximize t h e s i g n a l bu t introduces no s h i f t i n t h e s t a b i l i z e d frequency. That is t r u e f o r pure s ine- wave and.pure square-wave modulation, bu t it may not hold f o r an a r b i t r a r y modula- t i o n s igna l .

4. Experimental.- In o rder t o check t h e r e s u l t s es tab l i shed i n t h e preceding sec- t ion , we have used a se tup very s i m i l a r t o the one used normally i n Rb frequency standards: a Rb-87 lamp, followed by a Rb-85 f i l t e r , provides t h e pumping l i g h t f o r a Rb-87 coated c e l l without buffer-gas. The RF f i e l d is produced from the 5 MHz output of a Cs-clock, phase modulated, mul t fp l ied by 1368 and mixed with t h e out- pu t of a swept frequency syn the t ize r . In order t o avoid any coupling of the RF sidebands with t h e magnetic-field dependent Zeeman subs ta tes a t t h e h ighes t modu- l a t i o n frequencies ( - 10 kHz), a r a t h e r l a rge DC magnetic h i e l d of 0.5 Gauss was used t o separate t h e Zeeman l e v e l s by - 350 kHz.

Fig. 1 Amplitude A1 of t h e f i r s t harmonic o f t h e inversion (t) versus RF-atom detuning (theory and experiment). Y P = 2050 s-l, h+,, = 1 4 . 4 . 1 0 ~ s-l, sine-wave phase modulation, m = 1.4

Fig. 1 shows t h e shape of t h e resonance (amplitude A1 of t h e f i r s t harmonic of A ( ' ) ( t ) a s a funct ion of the RE'-atom detuning a ) . In t h i s exemple t h e modulation frequency i s much l a r g e r than t h e re laxa t ion r a t e s , s o t h a t wel l resolved sidebands appear a t i n t e g e r values of t h e r a t i o o / ~ . The quant i ty of i n t e r e s t f o r a frequen- cy s tandard i s t h e s lope of t h e curve near a = 0 , which we define a s

PI = aA_ a c a / r ~ )

(18)

Fig. 2a Slope of t h e discr iminat ing s i g n a l versus modulation frequency f o r y1/Y2 = 0.3; 0.7; 1.5; 3.0, sine-wave phase modulation ( m = 2 ) s o l i d l i n e s : eq. 11; o: experiment

Fig. 2 b Slope of t h e discr iminat ing s i g n a l versus modulation frequency for Y1/Y2 = 0-3; 0-7; 1.5; 3.0, square-wave modulation m = ~ / 4 ; s o l i d l ines : eq. 17 ( t a b l e 1) ; o: experiment

C8- 196 JOURNAL DE PHYSIQUE

The discr iminat ing s lope [ ~ o l t s / ~ z ] is then given by

where V i s t h e DC vol tage change generated a t the photo-detector output when t h e RF power l e v e l is switched from B = 0 t o B >> yly2 ( s a t u r a t i o n ) , thus inducing an atomic inversion v a r i a t i o n equal t o A' (eq. 5 ) .

Figures 2a and 2b show t h e dependence of P l on the modulation frequency f o r seve- r a l values of t h e r a t i o y1/y2. For low values of %, P1 increases with modulation frequency bu t i s independent of Y1. However, t h e modulation frequency a t which P1 reaches a maximum and the maximum value i t s e l f both depend on y l . A measurement of P I a s a funct ion of y, allows, a s described before, a simple experimental determi- na t ion of y1/Y2. In t h e condit ions of our experiment, t h e da ta f i t t h e t h e o r e t i c a l curve corresponding t o Y1/y2 = 0.7 ( f i g . g: square-wave modulation with m = T/4; f i g . &: sine-wave modulation with m = 2 ) . The re laxa t ion r a t e s y; and y; (see eqs 3, 5) could be measured by repeat ing these measurements a t various pumping r a t e s and ex t rapola t ing t o zero l i g h t i n t e n s i t y .

The dependence of PI on (sine-wave) modulation frequency i s shown i n f i g . 3 f o r var ious modulation depths. For low values of t h e modulation index m (m 2) , t h e optimum modulation frequency s t a y s constant bu t the s lope PI increases with m. For m 5 2, t h e maximum slope s t a y s constant b u t t h e optimum modulation frequency de- creases with increasing m: t h e l a r g e s t s i g n a l i s obtained when the frequency excur- s ion mm is roughly equal t o t h e width y2 of t h e resonance.

Fig. 3 Slope o f the discr iminat ing s i g n a l versus modulation frequency (sine-wave modulation) f o r var ious modulation depths ~ 1 / Y 2 = 0.7. S o l i d l ines : eq. 11; experimental data: + (m = 0 .5) ; 0 (m = 1); o ( m = 2 ) ; A ( m = 5 )

Fig. 4 shows how t h e s lope P1 o f t h e e r r o r s i g n a l depends on t h e phase excursion m f o r both sine-wave and square-wave modulation. In the l a t t e r case, P I is pro- por t iona l t o sin2m. The maximum slope i s thus obtained when m = T/4, regard less of a l l o ther parameters. In sine-wave modulation PI depends i n a more complicated way on both t h e modulation index and t h e modulation frequency bu t the optimum s lope

can be up t o 50% l a r g e r than i n square-wave modulation.

Fig. 4 Slope of t h e discr iminat ing s i g n a l versus phase excursion m f o r sine-wave phase modulation and square-wave phase modulation (wm/y2 = 0.77). So l id l i n e s : sine-wave (eq. 11); square-wave (eq. 17) experiment: o sine-wave phase modulation; + square-wave phase modulation, (s lope i n a r b i t r a r y u n i t s )

5. Conclusions.- A dynamical ca lcu la t ion of t h e double-resonance lineshapes i n the presence of phase-modulation o f t h e RF f i e l d has been performed. This calcu- l a t i o n is v a l i d f o r any combination of modulation frequency and depth b u t i s res- t r i c t e d t o unsaturated resonances. I t allows t o p r e d i c t t h e modulation parameters t h a t w i l l maximize t h e discr iminator s lope once t h e re laxa t ion r a t e s of t h e system a r e known. The re laxa t ion r a t e s can both be measured, using t h e same t h e o r e t i c a l r e s u l t s , by comparing the amplitude of t h e s i g n a l a t low and high modulation f re - quencies. Experimental r e s u l t s a r e i n good agreement with t h e o r e t i c a l p red ic t ions and show t h a t t h e assumptions underlying t h e ca lcu la t ions a r e j u s t i f i e d , par t i cu- l a r l y concerning t h e neg lec t of Zeeman pumping.

Acknowledgements

We thank D r . H. Brandenberger f o r having pointed ou t t h a t problem t o us and f o r he lpfu l discussions.

References

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