PHYSICAL REVIEW A VOLUME 7, NUMBER 6

Parametric Frequency Conversion of Resonance Radiation in Optically Pumped Rb"Vapor"

Henry TangColumbia Radiation Laboratory, Department of Physics, Columbia University, ¹w York, ¹w York 10027

(Received 26 December 1972)

This paper reports an investigation of parametric frequency conversion in a vapor of Rb atoms jn

which nearly resonant optical waves are coupled with microwaves at the ground-state hyperfine-structure

{hfs) frequency of Rb" by the nonlinear susceptibilities of Rb" vapor. These nonlinear susceptibilities

are greatly enhanced by optical pumping of the ground-state hfs levels of Rb" and by using nearly

resonant light. The frequency-conversion process generates optical sideband waves which are separated

in frequency from the carrier light waves by the ground-state hfs frequency of Rb (6.835 GHz). These

sidebands have been directly observed in the spectral profiles of Rb"' resonance light which had passed

through Rb" vapor. The conversion efficiency of power from the carrier waves to the sideband waves

in a 45-cm length of vapor was greater than 1%. The observed behavior of the sidebands is in

excellent qualitative agreement with the predictions of a semiclassical theory.

I. INTRODUCTION

The index of refraction of an optically pumpedvapor undergoing coherent magnetic resonancetransitions can have components which oscillate intime at the magnetic resonance frequency. The ef-fects of these oscillating indices of refraction onthe propagation of light through the vapor werefirst observed in certain light-modulation experi-ments in optically pumped vapors. Following asuggestion by Dehmelt, ' Bell and Bloom observedthat Na resonance light, which has passed througha vapor of optically pumped Na atoms, undergoingground-state Zeeman transitions, was intensitymodulated at the Zeeman frequency. A similartype of light-modulation experiment using Kresonance light in a vapor of optically pumped Katoms undergoing ground-state hyperfine structuretransitions (at 458 MHz) was performed by Firesterand Carver. Subsequently, Mathur et al. useda newly developed crossed-field photomultipliertube to detect intensity modulation of Rb reso-nance light at the Rb'7 ground-state hfs frequency(6. 835 GHz). This latter experiment settled a con-troversy over whether, in this type of experiment,the light can be modulated at such high freqeuncies.The controversy was due in part to poor theoreticalunderstanding of light-modulation experiments andin part to the lack of sensitive photodetectors withmicrowave bandwidths.

The basic difference between light-modulationexperiments at microwave frequencies and thoseat Lower radiofrequencies is that in the former, thetransit time of the light through the vapor is com-parable to or exceeds the period of oscillation ofthe applied microwave field. In this case, the ex-tent of the modulation will depend on not only thesize of the coupling susceptibility but also on how

well the optical field, the microwave field, and the

optical sideband field, which is generated, arephasematched. %e have recently reported on thedirect observation of optical sidebands generatedonto the spectral profile of a beam of Hb D&-

resonance light in a vapor of optically pumped Rb 8$

atoms undergoing ground-state hfs transitions.The sidebands could be easily observed becausethe spectral width of the Rb" light (- l. 5 GHz) wassmaller than the ground-state hfs transition fre-quency of Rb', so that the sidebands were well re-solved. Also, because a long length of Rb vaporwas used, and the optical and microwave fieldswere kept well phasematched, the sidebands couldbe made quite intense. An additional feature ofthis experiment was that the long length of vaporwhich was used entirely absorbed the resonantportions of the probing light so that it was only pos-sible to observe the sidebands generated from theoff-resonance parts of the probing light.

In this paper, we present the results of a detailedexperimental study of these parametric processesin optically pumped Rb vapor. %e also presentin Sec. II a semiclassical theory which predictedthis effect.

II. THEORY

In this section, we first calculate the dielectricsusceptibility of optically pumped Rb'~ vapor withground-state hfs coherence. Then we incorporatethis susceptibility into Maxwell's equations andsolve for wave-propagation solutions in the vapor.In the calculation for the susceptibility we makeuse of the formulas for the polarizability operatorfor Hb vapor derived in a paper by Mathur, Tang,and Happer, which will hereafter be referred toas MTH. These formulas were derived using theeffective operator formalism of Happer and Mathur,whose paper we will hereafter refer to as HM.

2010

FREQUENCY CONVE

A. Polarizability Operator

VERSION 0VE OF RESONANC E . .

'g the theory of HAccordin to.d-l-t-. p.lar bar1zability of an po

b s e expectation valu

p pr1ate polarizabil t1y operator n,

(n)= Tr [™np],

where p is the

(l)

e ground-state densit

fined ascep ibilityc b' '

(X) of the vapor is de-

&x&=~& &,

where N is t" ' i 1n

(2)

he atomic densit in po.

'y operator for Rb 7

MTH (39). Ac dcor ing to that forvapor is given b

the polarizabil'ti1yasasumof formula we can w 'trl eour terms

(n)= n~((u)+n h&.(~) &f »+i&~ n.~(F F '

O

C3CED

2Ol

LL

(a)

500I

looo I 500Magnetic field ( gauss)

m=+t

m=-t

m=-2I

2000

07

m=-1

0 m=+p

x(J(F, F'))x+ 5~ n, (F F .x JF ', ~ b, , F;~) &Q(F, F')&.

All the terms in (3) except for the(3

e coefficients e inre unctions of the o t

re

~

op

q po1c 1s 1nde en

e second term in 3f'ne polar ab'l't wh'chhich is proportional to the

asure of the ground-

tai th ot o 'iza ion. The third tropic polarizabiliti

proportional to thi 1 ies, which are

tronic angulao e expectation value of the ele-ec-

1n 1s term indica es a vector crosse ourth term in 3 pen polarizabilities of the vapor, wh1ch

o the expectaquadrupole e operator Q(F F'

xp tat1on values of th e

The eneenergy levels of the rh'"n '" F

are

angular -momentum uanesignated by total

er', a static magneticie to remove th

of the magnetic sublhe degeneracy

i (

i

profile of Rb D ls ows a representat'a 1on of the spectral

po, light superimes. The hfs splitting of Rb '

6Hz). Neverthelessz is smaller than th at of Rb {6385

r e ess, the splittin ith6 spectral pr fil~ ..fab"D-g

is well resolved

components of Bb, light and the hfs

5, which has a nu p

e y E=3 and E=2pp er broadening and th ' e'ng, and the isotope shift, the

m =+1—

-RO-N

C3

.-2.0—CS

I

~ -l.O—

Ld

LU

CLcA )0

C3

~~2.0—CXLLI

LI

3.0—

s„~IRb )

{e)P/

(

J

4.0—

5.0—

6.0—

FIG gy levels of the 5 S(a) Ener &/2 g o" ransition is indi

87t db pa an b re res

So

5 P& trans' 'ion 1ne of

db nt a typical spectral profile of D

amp. 4 represents the E= 3 hf w 1 ets th 2 hf s component of th Bb8~e, light.

2012 HENRV TANG

E= 3 component of Rb light overlaps the E= 2component of the Rb ~ absorption. We make useof this accidental overlap in our experiment byilluminating the Rb vapor with Rb resonancelight. Thus, we mere able to deplete the E= 2 hfslevel and produce an excess population into theE= 1 hfs level. Having established a populationimbalance in the hfs levels of Rb by optical pump-ing, a resonant microwave magnetic field parallelto the static field was applied to the vapor to excitethe IF =1; ml, = 0&-to- IF = 2; ml, =0& transition asindicated in Fig. 1(a). This transition will be,hex eaftex, x efex red to as the "0-0"tx'ansltlon. Thecombined effect of the optical pumping and theresonant microwave magnetic field is that theatoms in the vapor go into a coherent superposi-tion of the tmo hfs levels, or me say that the vaporhas acquired hfs coherence. The effect of theoptical pumping on the polarizability of the vaporis to give it a nonvanishing hfs polarizability. Thevalue of &I J)= —+q when all the atoms are in theE= 1 hfs level. One cannot usually pump all theatoms in the vapor into one hfs level so that in ourcase &I. J& lies somewhere between 0 and -f.Since me have not affected the population distri-bution among the Zeeman levels in each hfs level,no static components of the gyrotropic or bire-fringent polarizability have been generated. How-ever, because the vapor was undergoing coherent0-0 transitions, the ground-state density matrixacquired nonvanishing oscillatory off-diagonal ele-ments. This gives rise to two nonvanishing termsin the gyrotropic susceptibility (see Appendix A).One can also show (see Appendix B) that 0-0 hfscoherence gives no nonvanishing terms to the bire-fringent polarizability. Therefore, under the con-ditions we have stated, the polarizability of theRb vapor can be written as

&& &= c' (ol)+ o'hf. (ol) &1. J&+(i/2) o.l(12;ol)

x (20 I P I 10)e, x + (i/2) n„(21;ol ) &10 I P I 20& e,x,(4)

where (1, 0 I p I 2, 0) and &2, 0 I p I1, 0) are off-diagonalelements of the ground-state density matrix, e, isa unit vector in the direction of the static magneticfield which we will call the z direction, and we havesubstituted for the matrix elements of J; (2, 0 I J I 1, 0)= &1, 0I JI2, 0)= ——,'e, (see Appendix A).

B. Ground-State Density Matrix

The sublevels of the Rb ground state are num-bered according to Fig. 1(a). Since the microwavefield excites transitions only between levels 3 and7, me will assume that the only nonvanishing off-diagonal elements of the density matrix are p37 and

p». The density matrix of the eight sublevels canbe written as

pat 0 0 0 0 0 0 0

0 p» 0 0 0 0 0 0

0 0 p33 0 0 0 ps' 0

0 0 0 p44 0 0 0 0

0 0 0 0 p5q 0 0 0

0 0 0 0 0 p66 0 0

0 0 p» 0 0 0 p„ 0

0 0 0 0 0 0 pss

The time rate of change of any element of (5) isgoverned by the equation

[(~+X„),p] + —.„[MCP —pMC']

1,.q [&O P]33=- i~OPOV (8)

and

1[+or pjv3 3olop73& (9)

where Qpo = (Eo —Eq)/8,If we let the frequency of the applied micromave

field be 0, then me can write the microwave fieldH in the form

H(A) = Hl cosQt. (10)

Then K~ can be written as

Kl=ggpo J' HlcosAf.

The term (1/N) [X~,pj can be written in componentform as

where J(o is the unperturbed Hamiltonian of theatoms, 3C~ describes the effect of the applied mi-crowave magnetic field, 5X is a non-Hermitianeffective Hamiltonian' which describes the effectof the optical-pumping light. The term (BP/Bt), ~~accounts for the rate of repopulation of the ground-state sublevels owing to spontaneous decay fromthe excited states, and (BP/Bt)„„„accountsfor thereturn to thermal equilibrium of the populationsof the sublevels and of the coherences betmeenthese levels.

Writing the first term on the left side of (6) incomponent form as

1 1.„-[&op] "=@(p E p..E.), -$8

we find that under the conditions we have assumedfor the experiment, the only nonvanishing com-ponents of (7) are

PARAMETRIC FREQUENCY CONVERSION OF RESONANCE ~ . . 2013

1 Z

[5C~ p] ~ = 5~ gg peHg cossetwrite"

rln&=0 for n=6, 7, 8, (17a)

[p @IJ In' &- &nl~. l»ps. 1.

(12)By substituting (7I Jgl 3&= &3I I

I

7)= and droppingthe rapidly oscillating terms, we write the nonvan-ishing components of (12) as

1@

[5C~ ~ p] ss = 2P impvs e (13a)

1[5C~, p]„=—2p Impvs e '"',

sI

1P]vs = —iP (Pvv Pss) e

(13b)

(13c)

and

1zh

[K~ P]sv = il3(pvv Pss) e

where p= psIIq/2h.The effective Hamiltonian gK according to HM

(2. 37) is expressible in terms of two Hermitianoperators, &8 and &I',

5+=58-N(51'/2) = —g+ n (14)

where 5$ is the light-shift operator defined by HM

(2. 38) and 51' is the light absorption operator de-fined by HM (2. 39). The term (I/itf) [5Ãp —PMCv]

in (6) may be written in component form as

[5xp —P5x'], , = ~~ —(p, ()'3I58

In ')

ZS k

( I5hl»„., ) —;(& 15I'

+p., &ul5rln &'. (1.5)

The diagonal matrix elements of &8 account forshifts of the ground-state energy levels owing tothe light, while the diagonal matrix elements of5~ account for the excitation of atoms" from theground state to the excited state. The off-diagonalmatrix elements of 58 and pI' contribute only atzero magnetic field or when the light is intensitymodulated. ' Otherwise, those terms may beneglected, and (15) becomes

1.@

[5Ãp —pÃC ]

&nl5rln&=r for n=1, 2, 3, 4, 5, (17b)

where I" is defined as the optical-pumping rate ex-pressed in photons absorbed per atom per second.Since we shall be exciting magnetic resonancetransitions between levels 3 and 7 we need only beconcerned with the light shifts of these two levels.We write

(I/g) [(3I5h

I3) —(7

I58

I7&] = b(d g, (18)

where 4~, is defined as the light shift in radiansper second of the transition frequency betweenlevels 3 and 7. We now write for the componentsof (16)

(1/iK) [NCP —p53C']„,= —I' p,for n=1, 2, 3, 4, 5, (19a)

(I/ih) [5Ãp —p6Ã']vs = [-—,' I'+i n~, ]p„, (19b)

(I/iK) [5KP —p63C ] =[--', I' —ib&u, ] ps . (19c)

We shall assume that repopulation of the eightground-state sublevels occurs with equal probabili-ty; so we write

(dp .I (»&+P—22+P33+P44+ pss)k df)„„y

for n = 1-8. (20)

This last assumption will be valid if a buffer gasat a sufficiently high pressure is present in theresonance cell to collisionally depolarize the ex-cited states of the Rb atoms.

Ground-state relaxation of alkali atoms due tospin exchange, ' buffer-gas collisions, anddiffusion to the walls' is exceedingly complicated,although a great deal is known about it. However,to account for all presently known aspects of ground-state relaxation in detail would introduce so many

mathematical complexities that it would obscurethe basically simple analysis of the light propaga-tion. We therefore represent the ground-staterelaxation with the following simplified equations,which, while not strictly correct, nevertheless re-produce the important qualitative properties of the

relaxation:0

— -'&nl»ln&+-(nl«ln& p.." (16) = —y, (p ——,') for n =1-8, (2»)rei~

Since the optical-pumping light (Rb resonancelight) excites Rb v atoms out of only the E = 2 hfslevel of the ground state [see Fig. 1(b)] we can

= —'Yp p73p (21b)

HENRY TANG

and

!dP87 &

rel ax(21c)

On combining (6), (9), (13), (17)-(21), and rec-ognizing that p» = pa& = p44= p55 and p«= p«, we ob-tain the following set of equations for the elementsof (5):

dpgqP22+ F( P22+P38) yi(P22 ) (22 )

1

dpssdf

P83+ F(4P22+ pss) yl( P83 )1

+ 2p Impvse '"', (22b)

dp« 1

dtF(4P22 +P33) yl( P66 ) (22c)

„f"= 8 I'(4P22+Pss) -yi(pvv- 8) —2P ImP73e *"',

(22d)

= i&uopvs —ip( pvv—P33) e'"' —[-,' I'+ys] pvs

(22e)+ &~g PVsy

dPsg = z(doP3v+iP(Pv7 p38) eI

F +y2] Psv~ jQg 1

(22f)2+QPg Psy s

On comparing (22e) and (22f) we can see that p, v

= p73. To obtain the steady-state value of psv we let

p e-i&t

and substitute in (22f) to give

P( P77 P33)(0 —

ohio

—aoi, ) +i [(1j2)I"+y, ](24)

By equating the left side of (22a)-(22d) to zero, wecan solve for the steady-state value of (pvv —pss)to give

(Pvv Pss)

—,'Iy (I+y )[—,' I' y, +~6 I'yi yy, ] + [-,' I' ++4 I'y, + 2y', ] 2P'6

fl I'+ys1(0 —010 —aoi, ) +[-,' I'+ys] (26)

From (23) and (24) we note that the off-diagonal den-

sity matrix elements of "coherences" exist whenthere is a population imbalance between levels 3and 7 and when a resonant or nearly resonantmicrowave field induces transitions between thesetwo levels. Furthermore, the coherences areoscillating in time at the frequency of the appliedmicrowaves. Consequently, the gyrotropic com-ponents of the vapor polarizability in (4), whichare proportional to the coherences, are oscillating

Xo=&[n (&)+nsv (&) (I' J&] ~ (27)

The explicit expressions for the complex coeffi-cients n„(&o)and n„„(01)for Rb vapor are foundin MTH (41) and (42). A plot of the real part of Xo

versus the optical frequency & in the region nearthe D, transition of Rb is shown in Fig. 4. Fig-ure 4(a) shows the case when the vapor has no hfspolarization ((I ~ J)= 0), while Fig. 4(b) shows thecase when there is a sizable excess population inthe lower (E = 1) ground-state hfs level. In general,the degree of hfs polarization depends on the pump-ing-light intensity, the atomic density, and thebuffer-gas pressure. Therefore, the static suscep-tibility of the vapor will depend on theseparameters.

The oscillating part of the Rb ~-vapor susceptibil-ity may be written as

i n„(12;01)H,e ' 'eg(X )=

4 (Pvv —Pss)&ps fl ~ ) [ I y ]

n.,(21;07) P,e'"'e.x( )(0 —Mo —boi, ) —i f ,' I'+ys]-

The explicit expressions for the coefficients n„(12;01) and n«(21;&u) for Rb are found in MTH (53).A plot of n«(12;&o) and n«(21; 01) versus optical

at the microwave frequency. A plot of ~p3~~ versusthe magnitude of the microwave field

~ Hl ~is shown

in Fig. 2. In this figure the microwave frequencywas taken to be on resonance (0 =010+h&u, ), and thevapor parameters were chosen to be representativeof actual experimental conditions. We note in Fig.2 that for small values of ) H, I, j p» j varies linear-ly with (H, ~. Ip37~ attains a maximum for somevalue of the field, and for large values of the field,the transition 3- 7 saturates and ~ psv~ goes to zero.A plot of I psv l versus the microwave frequency 0is shown in Fig. 3. Figure 3(a) shows the case oflow microwave field [4p «(-,' I'+ys)]. We note that)p») shows a normal resonance line shape, where-as, in Fig. 3(b) which shows the case when themicrowave field is large [4p» (—,

' I'+ys)], I psv I

decreases at the center of the resonance owing tosaturation of the transition 3- 7. Therefore, wewould expect the optical sideband amplitudes, whichare roughly proportional to ~ps~~, to have the samebehavior with microwave field strength and micro-wave frequency as is illustrated in Figs. 2 and 3.

87C. Steady-State Susceptibility of Rb Vapor

Using the expression for the off-diagonal densitymatrix elements derived in (23) and (24) in (4) andusing the relation given by (2) we obtain the com-plete susceptibility for Rb vapor with 0-0 hfs co-herence. This susceptibility can be separated intoa static part and a part which oscillates at the driv-ing microwave frequency. The static part yo canbe written as

PARAMETRIC FREQUENCY CONVERSION OF RESONANCE ~ ~ ~ 2015

0.0'5—

0.02— FIG. 2. I p37I vs the magnitudeof the microwave field H~. Thevapor parameters were assumed tobe F = &0 sec; y&

= 360 sec"', y2=230 sec ~.

O,OI -—

00

H) {10 4 gc»s)

frequency in the region near the Dj transition ofRb is shown in Fig. 5.

.0. V4ve Propagation

I et us represent the electric field of a quasi-monochromatic -light wave by

E(r, f, ~) =h(g, f, |d)e'"' ""+c.c., (29)

where c.c. denotes complex conjugate, and ( is thedistance along the direction of propagation,

(32)

where (d, is the upper-sideband frequency and

y, &(up, = &+&@) is a component of the up-couplingsusceptibility.

We will keep in mind that both the static andmicrowave magnetic fields are oriented along thez direction. On comparing (32) with the first termin (28) we find that the only nonvanishing componentsof the up-coupling susceptibility are

K = (1/k)~k. r. (30) (Q7 = 0 +(0)= x &(Q) = A +Q))& (33)

The amplitude 8(f, f, &o) is a slowly varying functionof and t. One can think of an incoherent light beamas a superposition of many monochromatic waves,such as (29) with various frequencies and randomphases.

The wave (29) will induce dielectric polarizationP in the Rb vapor,

X~~(Q) = —Q + Qp) = —g~y(Q7 = —0 +Q) )p (36)

where

( )i N&o(Pm —Pss) ng, (12;&u)

(fl —~o —~(u, )+i [,'I' V+]-s(34)

Similarly, we can show that the only nonvanishingcomponents of the down-coupling susceptibility are

P (Q~ g(g f ) ef(k~1' Ale) + c (31)where

The static part of the susceptibility will generatea component of polarization which oscillates at thesame frequency & as the incident light wave. How-ever, the two terms of the oscillating susceptibilitywill generate components of polarization whichoscillate at frequencies +0 and & —Q. We willadopt terminology commonly used by radio en-gineers and call the light wave at frequency + thecarrier and the light waves at ~+0 and ~ —0 theupper and lower sidebands, respectively. We canwrite the Cartesian components of the upper side-band polarization P, (&g,) as

i Nyo(P~~ Pss) n~t(21;(o)y„~=-0+~j=—4 (0 —(oo —4(ug) -i [,'&+vs]-

(36)From the forms of the nonvanishing components ofthe coupling susceptibilities of (34) and (36) we notethat when the carrier light is linearly polarized,the sidebands will also be linearly polarized but inan orthogonal direction. If, however, the carrierlight is circularly polarized, then the sidebandsare also circularly polarized and in the same senseas the carrier. In the latter case, the carrierwaves and the sideband waves mix, causing intensi-

HENR+2016

t beats have beeThese light jn the light.f Firester an

tybea s i „.me~~~ orved in the exPobservethur et a „-,f;elds «

'"20 and Ma

. f the electricCarver

h behavior o. rporate the

redice mustthe carri

d plarizationsnd sldeban Pocarrier an

VC

t the amplitudestions t»tQy ma the assump

t'on for theve eq,ell's eguatioin Maxwbe written as

4 aPl ctric field can

3

2Q

xgxE—

(aI

p x(o g«"Hi=

20—

l6—

l2

20000I

-Io 000I

0( rad/ sa~ ~

I

(0000I

20 000

28~b~

4H =8

20—

l6—

l2—

I

2000oI

i000oI

0( rad/sec j

I- (0 000I

200000

y for h&gmicrowave fr &1

la. g~ I»sec- »=

I vs the mac& 23p sec

ave fie -3 g + —36p

low m].cp ] P sec

~ owave

gregale

&re assumed

I vs the microameters we

a~ I~3~ . The vapo" ~ rmacrZ&G . f eld.

PARAMETRIC FREQUENCY CONVERSION OF RESONANCE. . .

(a) N = lA I x 10 cm

&I J&= 0Y= 0.5

0

-300 -220 -I40 -60 0 60 I40 220 500 380

Frequency displacement (v-v, e )/c (mK )

4 — (b) N= I.OI ~ lO cm

& I ~ J & = -0.600Y=0.5

Qx1

-300 -220 -140 -60 0 60 I40 220 300 380

Frequency displacement (v-v, )/c (mK)

FIG. 4. (a) gp'(~) vs co for D& light for the case of no hfs polarization. gp (ct)) =ReXp(u). (b) gp'(co) vs v for D~ light forthe case with hfs polarization. N is the atomic density which was taken to be that of a saturated rubidium vapor at 70 C.Jt" is the broadening parameter to account for the buffer-gas-collision broadening of the optical-absorption line. SeeRef. 7 for details of this calculation.

l5-

IO

5ECP

CQ

I 0

-15—

lI

I\II

(z", ( l2)

/ y an't(2&)

I

~ ~~ ~ 4 ~~ e ~~ ~

l

~~2

—a'g, ( 2 I )4

~ ~~ ~~ ~

~ ~

-20-150

!

l50(mK}

—&00 0 50 IOO 200FREQUENCY OISPLACEMENT ( v - "c.g.)

"=ImnRef. 7 fox details of this calculatio&. @gg=He™gt'» gt=jF'yG 5 ~ &O2; ~) a~d Q&&(21; (d) vs fo& Df ig

of the electric fields are slowly varying functionsof time a, nd distance one can show that Eq. (3V) canbe reduced to

8$(r, t, (u) 1 Bh(f, t, (o)8$ c Bt

2vi 8 P(t~ t~ &) i(R "r-~t-)(33)8 tMc ~t

where 8(~) is the a,mplitude of the electric field atfrequency ~ as defined by (29). The polarizationcan be written in the form

P(x, t, (o) =6'(K, t, ~)e""'""+c.c. (39)

Substituting (39) into (38), we obtain

~~

—+ ——! $(f, t, (u)= 2vika (K, t, (o). (40)

9& c 8t)In general, (40) represent an infinite set of coupledequations. But if we refer to Fig. 5, which showsthe dependence of the coupling susceptibilities onoptical frequency, we see that n«(12;co), which isthe frequency-response function for the up-cou-pling susceptibility, is large only in the vicinity ofthe I'= 2 component of the Rb B~ transition, andit goes to zero for higher and lower frequencies.S'milarly n (21 &), which is the frequency re-

litsponse function for the down-coupling susceptibi i y,is large only in the vicinity of the I" = 1 componentof the D& transition, and it also goes to zero forhigher and lower frequency. It should also be notedthat the imaginary parts of n«(12;u) and n«(2lpu)are nonzero only on resonance. Therefore, theimaginary parts of the coupling susceptibilities arenot useful in generating sidebands because the por-tions of the carrier-light spectral profile whichare on resonance suffer strong absorption by the

vapor. The real parts of the coupling susceptibili-ties, however, remain sizable even off resonance.Therefore, because the coupling susceptibilitiesare strongly frequency dependent we can assumethat a monochromatic carrier wave couples effi-ciently only to an upper and a lower sideband. Thisassumption reduces the number of equations givenby (40) to three. The expression for the micro-wave field given by (10) is for a field which isstationary in space. In our experiment, the inter-action of light and microwaves occur over an ex-tended region. The microwave field is applied bymeans of a resonant cavity whose axis is parallelto the light beam. The TE modes of this cavityhave components of magnetic field which are par-allel to the cavity axis. In the cavity, the fieMsare standing-wave fields which can be thought of asbeing composed of two oppositely directed travel-ing-wave fields. We will only consider the case ofa traveling microwave field which propagates inthe same direction as the light, and we will assumethat the microwave field propagating in the oppositedirection is badly phase matched so that it can beneglected. Hence, for the above case we can writethe longitudinal component of the microwave mag-netic field in the cavity in the form

H(z, t, 0) = H, (A) e" ' ""+c.c., (41)

where K is the guide wave vector of the microwavesin the cavity. There will also be radial componentsof the microwave magnetic field in the cavity whichwe will ignore. We have also assumed in (41) thatthe amplitude of the microwave field remains con-stant. One can show that when the microwave fieldis of the form of (41) the coupling susceptibilitiesneed only be modified by multiplying (34) by e'»",and (36) by e' '~ We can now write the com-

PARAMETRIC FREQUENCY CONVERSION OF RESONANCE. . .

( fl )ff (g) -e &&r-sot g&( )&(s s rabat&-

g )fi (il) & (&s-o g ( )& s s-a&-&

+c.c. , (42b)

p ( ) &p (+ )e&&ih s kl &) ~c

)g (. )e&&s s ~ s&

( fl )ff (fl)e-&&Es ot&g ( -)e&&ss rat)-

+c.c., (42c)

where 0, =p). /c and I) =p& /c. By substituting (42)into the wave equation of (40) we arrive at the fol-lowing set of coupled equations:

g, (u), ) = 2&&ik [Xp(p), )g, (&d, )

+ y,„((o.= 0+(0)H,(Q)g„((0)e'~r'], (43a)

dg „(&d)= 2&&ik [Xp(p) )g „((o) + X„,(p) = 0+p) )H, (Q)

d8

xg, (p) ) e' "'+li„((p= —0 y&d, )

x H, (Q)g„(p).)e-"r'], (43b)

dg,((p ) = 2&&ik [yp(p) )g, ((o.)

+ X,„(p).= —fl+&d) H, (A)g„((o)e ' '], (43c)

where ~=K+k —k, =K —k+k, and because we

seek steady-state solutions, we have set the timederivatives to zero.

E. Phase Matching

One can show from (43) that for optimum cou-pling between the carrier wave and the upper side-band wave, it is necessary to fulfill the followingphase-matching condition:

np((o, )k, = nt(p) )k+ K, (44)

where np(&u, ) and np(&d) are the real parts of theisotropic indices of refraction of the vapor at fre-quencies , and , respectively. We can write

ponents of the polarization at the upper-sideband,the carrier, and the lower-sideband frequencies,assuming that the carrier wave is linearly po-larized in the x direction and propagates in the g-direction as

p ( ) ( )&&s - s&

y (~ )g (~ )ei&s s «-)

(~ fl +/)H (g) ei(rs ot&g -( ) ei&ss (lit&-

+c.c., (42a)

P ( ) + ( )e&&ssa& )&+c

the real part of the isotropic index of refractionas

no(p)) = I +2vX&)((p), (46)

5A = 2&&k [yp(&d, ) —y p(p) )], (48)

and to satisfy (46), the value of 5K required can bewritten as

5K = 2&(k [Xp((p) —Xt(&p )] .From (48), (49), and the expression for the iso-tropic susceptibility given by (2'7), we note that thephase-matching conditions depend on the atomicdensity N, and the quantity (I J). The behaviorXt&(p)) as a function of optical frequency in the vi-cinity of the D, transition for the case without opti-cal pumping and for the case with optical pumping

is shown in Figs. 4(a) and 4(b), respectively. Theatomic density assumed in Fig. 4 was estimatedfrom published rubidium-vapor -pressure datafor a saturated vapor at 70 'C. The value of (I ~ J)used in Fig. 4(b) was estimated to represent actualexperimental conditions. Gne can see in Fig. 4that because gp(&d) varies so rapidly with opticalfrequency, it would be difficult to select a com-bination of a carrier frequency, an upper-side-band frequency, and a lower-sideband frequencywhich would satisfy both (48) and (49) with the samevalue of gK.

The propagation equation (43) can be solved nu-

merically by computer. In Fig. 6 we show thecalculated modification of the spectral profile ofRb' D& light which has passed through a 45-cmlength of Rb vapor. The transmitted spectralprofile was obtained by solving (43) for a range of

carrier frequencies which spanned the spectralprofile of Rb D~ light. The vapor parameters(N and (I ~ J)) used in the calculation were assumedto represent the typical experimental conditions.The values of the frequency-response functions for

where yp(&d) is the real part of the complex isotrop-ic susceptibility defined by (27).

Simili. arly, one can show that for optimum cou-pling between the carrier wave and the lower-side-band wave, it is necessary to fulfill the followingphase- matching condition:

Np(p) )k =np'((o)k —K. (46)

The phase-matching conditions of (44) and (46) canbe met by adjusting the guide wave vector K of themicrowaves. The value of K required for properphase matching can be written as

K= Kp+ 6K, (47)

where Kp is the free-space wave vector of themicrowaves (Ep= 0/c). Therefore, to satisfy con-dition (44), the value of 6K required can be writtenas

2020 HENRY TANG

IO

IO

TI

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

g =1.41 x 10 cm

&1 J&= -0.600+6 = 4.51 cm

)o + 6 l, I, I

"MO -200 -loo 0 IOO 200 500Frequency displacement (I -I &)&c (mK)

400

FIG. 6. Calculated spectral profile of carrier side-bands after carrier light has traversed 45 cm of Rbvapor. The initial spectral profile of the carrier light isrepresented by the broken line.

the Rb susceptibilities yo(ur), o.«(12;~), and

n«(21; &u) were taken from the curves shown inFigs. 4(b) and 5. The microwave frequency wastaken to be on resonance (Q = &go+ 6+,) and the mag-nitude of the coherence, )o ~, was estimated fromFig. 2 to be (o ) =0.0325. In Fig. 6, the initialprofile of the Rb" light (indicated by the brokenline) was assumed for simplicity to be two rectan-gular-shaped lines designated as components e and

f to represent the E= 8 and E= 2 hfs components,respectively. In the calculation me have taken thevalue of the guide mavelength of the microwavesto be A~=4. 51 cm such that the phase-matchingcondition of (49) is satisfied to give perfect phasematching between a carrier at —40 mK and anupper sideband as 188 mK (the microwave fre-quency being 288 mK). The spectral profile of thetransmitted light is represented by the solid curvein Fig. 6. %e note that the components of thecarrier light have been partially attenuated be-cause of the overlap with the Rb absorption line.But in addition, sideband components designatedby a, 0, c, and d which were not present in theinitial spectral profile have appeared. The strong-est sideband component is the one which peaks at188 mK, because we had chosen to precisely phasernatch the optical mave at 188 mK with an optical

wave at —40 mK. Other sideband components whichare not so well phase matched are also present butare much weaker. Needless to say, the spectralprofile of the sidebands depends on which wavesone has phase matched. In designing an experi-ment to observe the sidebands, one needs to firstdecide which waves one wishes to phase match andthen to adjust the guide wave vector of the micro-waves to the proper value as determined by the con-ditions (48)-(50). However, to apply these condi-tions, one needs to know precisely the values ofcertain vapor parameters such as the atomic den-sity N and the hfs poLarization (1 ~ J). Unfortunate-ly, these quantities are very difficult to determineaccurately; furthermore, they tend to be ratherinhomogeneous throughout the resonance cell.Therefore, in practice one is not always able topredict precisely the spectral profile of the side-bands.

III. EXPERIMENT

A. Experimental Procedure

A schematic diagram of the experimental ap-paratus is shown in Fig. 7. A 45-cm-long cylin-drical resonance cell filled with a small amount ofRb metal and 12 torr of nitrogen buffer gas wasplaced inside of a microwave cavity which was tunedto resonate at 6. 835 GHz in the TE»„mode. Thecavity and cell were situated within three pairs ofmutually orthogonal Helmholtz coils which servedto cancel out the earth's magnetic field and to pro-vide a static 2-6 magnetic field along the z direc-tion. The resonance cell was optically pumped byresonance light from a long Rb" lamp, which waslocated parallel to the cavity axis. The Rb' atomsin the resonance cell are pumped from the I" = 2hfs level into the I' =1 hfs level of the ground state.Microwaves from a stabilized klystron were intro-duced in the cavity to drive the 0-0 hfs transition inthe Rb atoms, thus producing coherence in thevapor. The microwaves were square-wave modu-lated to permit the accompanying changes in thesideband intensity to be detected by phase-sensi-tive detection.

The optical carrier light was provided by a sec-ond Rb lamp which mas situated at one end of thecavity. A linear polarizer P& polarized the carrierlight before it enters the cavity. The transmittedlight emerging from the cavity passed through an-other polarizer P&, which was orthogonal to P, .The tmo linear polarizers P& and P~ served to fil-ter out the carrier light while passing the sidebandlight. Finally, the transmitted light beam was fre-quency analyzed with a scanning Fabry-Perot in-terferometer for which a Rb D&-interference filterwas used as a preselector. The optical detectorused with the interferometer mas a refrigerated

PARAMETRIC FREQUENCY CONVERSION OF RESONANCE. . . 2021

0

Rb LAMP85

D, FI~TERCANNING ETALON

LAMP

I

REFRIGERATEDPHOTO MULTIPLIE R

TUBEI IL

LINEAR R'b87 ~ L I NEARPOLARIZE R CELL CAVIT Y POL ARIZ ER

PIll Pp

MODULATOR—

8TAB I L I Z EDKLYSTRON

REFERENCE

SEN SIT I YEDETECTOR

RECORDER

FIG. 7. Experimentalapparatus.

photomultiplier tube. The photomultiplier signalwas detected with a phase-sensitive detector insynchronism with the modulation of the microwaves

B. Apparatus

1. Microwave Cavity

The microwave cavity (see Fig. 8) is made froma number of 5&&5&&&'-in. brass plates, each havinga circular hole in its center. The plates were ar-ranged in a structure which kept them 1 cm apartwith the holes aligned. One end of the cavity wasterminated by a plate with a smaller hole while theother end had a 10-cm-long section of ordinarycircular waveguide terminated by a perforatedshorting plug. This type of cavity will only containaxially symmetric TE modes and the cavity usedin this experiment was designed to operate in theTEpg„mode. The unloaded Q of the cavity without

the resonance cell was on the order of 10 .The resonance cell, which was situated coaxially

inside the cavity was supported by annular spacersmade from Ecco foam. Placing the quartz cellin the cavity reduced the Q substantially. This mayhave been due to either imperfect alignment of theaxis of the cell with the axis of the cavity or due toan aximuthal nonuniformity in the wall thickness ofthe quartz tube. A nonaxially symmetric distribu-tion of quartz in this cavity would result in mixingof the TE»„modes with nonaxially symmetricmodes which are very lossy. The Q of the cavitywith the resonance cell in place was on the orderof 100.

In order to satisfy the phase-matching conditionsgiven by (44) or (46), the guide wave vector in thecavity must be adjusted to the proper value deter-mined by (48) and (49): The guide wave vector inthe cavity can be adjusted by varying the transverse

46.4cm

FIG. 8. Microwave cavity.

l2.7crn

HENRY TANG

dimensions of the cavity; however, one must takeinto account the loading effects of the quartz reso-nance cell. The quartz, which has a relativelyhigh dielectric constant will tend to lengthen theguide wave vector. Given the inner and outerradii of the quartz resonance cell, one can usuallysolve for the cavity radius which mould give aparticular value for the guide wave vector. In acavity like the one shown in Fig. 8, the fieldspenetrate a small distance between the plates sothat the effective radius of the cavity is somewhatlarger than the radius of the holes in the plates.Therefore, the holes of the plates must be cut withtheir radii smaller than the required cavity radiusby an amount equal to the penetration depth of thefields. The penetration depth was determined bymeasuring the guide wave vector of the cavity with-out the cell, and relating the measured value to aneffective radius of the cavity.

The advantages of the cavity shomn in Fig. 8were the folloming: Large areas of the resonancecell inside the cavity were made accessible topumping light from a lamp which mas outside thecavity. The resonance cell could be illuminateduniformaly by the pumping light along its entirelength. This latter advantage was particularlyimportant in that it made phase-matching possibleand also reduced the inhomogeneous broadening ofthe microwave resonance due to light shifts.

2. Resonance Cell

The resonance cell mas made from fused-quartztubing (see Fig. 9). The quartz tubing was firstetched in a 20/p solution of HF to remove the graph-ite film that is usually found on the mails of com-mercial quartz tubing. The cell was then con-structed and sealed onto a vacuum system whereit was baked at 500 C for 24 hours while a pres-sure of 2x10 torr was maintained. After the cellwas allowed to cool, 99. 2%%uq isotopically pure Rbmetal was distilled into one of the reservoirs.Care mas taken not to allow the rubidium metal tocondense in the cell proper as this would increasethe microwave loss when the ceQ is placed insidethe cavity. Nitrogen gas at a pressure of 12 torr

was also introduced into the cell to serve both asa buffer gas and also as a quenching gas for fluo-reseently scattered radiation. ' After sealing offthe cell, we found that there was a sharp gradientin the rubidium-vapor density along the length ofthe cell away from the reservior. This was proba-bly due to some sort of chemical reaction takingplace between the rubidium vapor and the quartzwalls. Such reactions have been reported by otherworkers. This problem was particularly severein our case because of the long cylindrical geometryof our cells. However, the gradient was found todiminish with time and disappeared entirely afterabout two months. This "aging" period for the cellswas later reduced considerably by first coating thecell walls with Paraf lint, which seemed to be lessreactive with rubidium. Then rubidium was dis-tilled into the cell reservoir and the cell was sealedoff without buffer gas. %'ith no buffer gas to hinderthe diffusion of the rubidium atoms, the cell wallssaturated in about one week. Buffer gas was addedto the cell via a break-off seal after the gradientshad disappeared.

3. Resonance Lamps

The Rb" lamp which was used for optical pumpingof the Rb'7 vapor is shown in Fig. 10. The lampconsisted of a Pyrex dc discharge tube containingRb separated isotope and 2 torr of Krypton gas.The lamp electrodes were made from self-heatingoxide-coated filaments which were capable of sus-taining the discharge at currents of as high as 2 A.An evacuated jacket surrounded most of the activeregion of the discharge to keep that region hot.The Rb ' resonance light from this lamp was fo-cused onto the resonance cell by an elliptical re-flector.

The lamp which provided the carrier light was acommercially available Varian X49-609 spectrallamp containing Rb" separated isotope and about 2 torrof krypton gas.

4. Interferometer

The Fabry-Perot interferometer mas a C. D.Grant model FP-1. The instrument was scanned

4.56cm 5.06ce FIG. 9. Resonance cell.

45cm

7 PARAMETRIC FREQUENCY CONVERSION OF RESONANCE. . .

57|:e

2.2t:+

FIG. 10. Hb resonance lamp used foroptical pumping.

—I.P. cm

by applying a ramp voltage to three stacks of pi-ezoelectric disks which translated one of the twoplane parallel plates. The 2-in. -diam-plates wereobtained from the Perkin-Elmer Corp. The manu-facturer claimed the flatness of the plates to beX/200. The reflectance of the dielectric coatingson the plates was 97%%uq at the D, and D~ lines of ru-bidium. The central fringe of the interferencepattern was imaged onto a 500- p, m pinhole by a300-mm-focal-length lens. As the plate spacingwas scanned, the change in intensity of the centralfringe was recorded by a photomultiplier tube lo-cated directly after the pinhole.

A plate spacing of 0. 5 cm was used which gavea Q'ee spectral range of 1000 mK. A narrow-band

interference filter (from Baird Atomic) which hada passband of 90 A centered at V948 A and a peaktransmittance of 52%%u~ was used as a preselector forthe interferometer. The finesse of the interferom-eter, which is the ratio of the free spectral rangeto the instrumental width, was estimated to be 35at V94V A.

5. Micro~aves

A schematic of the apparatus used to providethe microwaves is shown in Fig. 11. The micro-waves were generated by a Varian X-268 reflexklystron which was stabilized by a Frequency En-gineering Laboratories A130K frequency synchro-nizer. The synchronizer phase locks the klystron

OSC ILL OSC OPE

KLYSTRONSYNCHRONIZFR

lllllllillllllllllllllilllllllllllillllll

llillllllllllllllllliillllllllllllliillll

CAV ITY

SLIDE-SCREW

TUNER

1DC

KLYSTRON

REF

P;4IXER

PREC IS I ONATTENUATOR—

Ml cR OwAvESWITCH

l3 Hz FROMLOCK-INAMP I IF IER

6.835GHz

F ILTER

VARACT OR 60 MHzHARMONIC TOGENERATOR 6.835 G Hz

1 TO60MHz

MULT IP LIER1 MHz

FREQUENCYSTANDARD

FIG. 11. Schematic diagram of circuit used to provide stabilized microvraves.

2024 H ENR Y TANG

frequency to that of an oven-controlled quartz crys-tal oscillator; thus the short term stability of themicrowave frequency was kept stable to better thanone part in 10 . Frequency stability of this orderwas necessary as the width of the microwave reso-nance was on the order of 1 kHz. As a check onthe long-term stability of the klystron, the micro-wave frequency was compared with 6835th har-monic of a 1-MHz NBS quartz frequency standard.The microwaves from the klystron first passedthrough an FXR C164A precision attenuator andwere then chopped by a Hewlett-Packard 33053-Amicrowave switch which was actuated by a 13-Hzreference signal from a lock-in amplifier. Themicrowaves were fed to the cavity through a slide-screw tuner. Despite the low Q of the cavity whenthe resonance cell was inside, the klystron hadmore than enough power to drive the microwavetransition. The power required to saturate themicrowave transition is only on the order of 10 ' W.

ponents a and b of Fig. 12(a) are displaced 228 mK(228 mK= 6. 8 6Hz) to the low-frequency side ofcomponents e and f of Fig. 12(c), respectively.Also, components c and d of Fig. 12(a) are dis-placed 228 mK to the high-frequency side of com-ponents e and f of Fig. 12(c), respectively. Fur-thermore, the spectral components of Fig. 12(a)were not due to fluorescently scattered pumpinglight as they do not coincide with the spectral com-ponents of Rb87 resonance light I Fig. 12(d)] and theyalso disappeared when the carrier light was re-moved as shown in Fig. 12(e). The componentsremaining in Fig. 12(e) were due to the smallamount of carrier light which is transmitted evenwhen P& and P~ were crossed. Since the absorptionof the overlapping component of Rb light by thevapor is increased when the microwaves are on,the transmitted carrier light is modulated at thechopping frequency of the microwaves and was,

6. Detection

The photomultiplier was an EMI 9658HA whichhad an extended-red-response version of the S-20photosurface. The quantum efficiency was claimedby the manufacturer to be 5% at V94V A. To re-duce the dark current, the tube was cooled to—30 C in a thermoelectric refrigerator housing,and was used with a magnetic lense before the pho-tocathode. Operated in this way, the tube's darkcurrent was less than 0. 1 nA. The output of thephotomultiplier went directly to a Princeton Ap-plied Research HR-8 lock-in amplifier.

The linear polarizers P, and P2 were PolaroidHN-38 sheet polarizers. The transmission forV94V A light was 89% when P, and P2 were parallelbut only 0. 2% when they were crossed.

1.0-

0 400CAI—

~ I.O-CL

CC)CL

(Q)

800

(b)

I.O-

0 400 800

C. Experimental Results

Observations with the Fabry-I'erot Interferometer

0-I—

00 400 800

I.O- (e)

The data recorded with the apparatus of Fig. 7is shown in Fig. 12. Figure 12(a) shows the spec-tral profile of the transmitted light from the cavityobserved with the polarizers P, and Pz crossed;hence the carrier-light components have been most-ly removed. This signal was recorded with 30-msec time constant on the lock-in amplifier, andthe duration of the interferometer sean was 1 min.Figure 12(b) shows the spectral profile of the trans-mitted Rb carrier light (observed with P, and Pzparallel). The initial spectral profile of the car-rier light before it enters the cavity is shown inFig. 12(c). For comparison, the spectral profileof Rb D, light is shown in Fig. 12(d). We note inFig. 12(a) that four components (designated by a,b, c, and d) are clearly resolved. On comparingFig. 12(a) with Fig. 12(c) we also not that com-

(c)

I.O—

0 400 800

0 400 800FREQUENCY DISPLACEMENT(mK) (ORIGIN ARBITRARY)

FIG. 12. (a) Spectral profile observed with the lock-in detection when the polarizers P& and P~ were crossed.(b) Spectral profile observed when the polarizers P~ and

P2 were parallel. (c) Spectral profile of Rb D~ light(carrier light). (d) Spectral profile of Rb D& light. (e)Spectral profile observed with lock-in detection withpolarizers P& and P2 crossed when the carrier light wasremoved.

PARAMETRIC FREQUENCY CONVERSION OF RESONANCE. . . 2025

therefore, detected by the lock-in amplifier.%e therefore conclude that the spectral com-

ponents seen in Fig. 12(a) are upper and lower side-bands which have been generated onto the spectralprofile of Rb ' resonance light by the microwavecoherence in the Rb' vapor. The intensity of themost intense sideband component was about 1.3%of the intensity of its corresponding carrier com-ponent.

The sideband signals were detected with suffi-ciently high signal-to-noise ratio to allow observa-tion of the sideband without lock-in detection. Fig-ure 13 shows the result when the output of the pho-tomultiplier was connected directly to an oscillo-scope. Here, the microwaves were not chopped,and the horizontal sweep of the oscilloscope wassynchronized with the interferometer scan. Thefull horizontal sweep corresponds to 1000 mK.

Figure 13(a) shows the spectral profile observedwhen the polarizers P, and P& were parallel; i.e. ,

only the transmitted carrier light passed Pz. Thespectral profile observed when P& and P2 werecrossed is shown in Fig. 13(b). The sideband com-ponents a, b, c, and d are indicated in this figure.Also present in Fig. 13(b), were the componentse and f or Rb light from the pumping lamp whichwas scattered by the cavity plates and the resonancecell walls. Figure 13(c) shows the spectral profileobserved when the carrier lamp was removed. Onecan see in this figure that in addition to componentse and f, there were also present the small compo-nents g and h which were due to the fluorescentlyscattered Rb ~ light from the vapor. The Rb ~ flu-orescence had been greatly reduced by the pres-ence of the nitrogen gas in the cell. Figure 13(d)shows the spectral profile which was observedwhen the carrier 'lamp was replaced but the micro-wave frequency was tuned off resonance. We seethat the sideband components were present onlywhen the microwave frequency was on resonance.

IOm V/cm .5mV/cm

I

4) f

(e)

.5mV/cm .5mV/cm

tlC V

(w)(e)

(e)(f)

(c)FIG. 13. (a) Oscillogram of the photomultiplier output when the polarizers P& and P2 were parallel. (b) Oscillogram

of the photomultiplier output when the polarizers P& and P2 were crossed. (c) Oscillogram of the photomultiplier outputwhen the polarizers P~ and P2 were crossed but the carrier light was removed. (d) Oscillogram of the photomultiplieroutput when the polarizers P~ and P2 were crossed but the microwave frequency was off resonance.

2. Observations without the Interferometer

The following data were recorded with the inter-ferometer removed; i.e. , the photomultiplier wassituated immediately after the polarizer P2. Thepolarizers P, and P2 were crossed so that the sig-nals represent the integrated intensity of all thespectral components of Fig. 12{a). Figure 14 showsthe integrated intensity of the sidebands as a func-tion of microwave frequency. The microwave fre-quency was swept over a range of 20 kHz about thehfs resonance frequency. The sideband intensityversus microwave-frequency signal obtained forlow microwave power is shown in Fig. 14(a) whileFig. 14(b) shows the same signal obtained for highmicrowave power. We observed in Fig. 14(a) thatfor low microwave power, the sideband intensity asa function of microwave frequency obeyed a normalresonance shape. However, in Fig. 14(b), we ob-serve that for high microwave power, the intensityof the sidebands at the center of the resonance haddecreased. This is what one would expect sincethe sideband intensity is approximately proportionalto the square of the coherence (ips~l ) for smallvalues of coherence. We have already shown IseeFig. 3(b)] that when the hfs transition is saturated,the magnitude of the coherence decreases at thecenter of the hfs resonance.

In Fig. 15 we have plotted the integrated intensity

-20dB

FIG. 14. (a) Integrated sideband intensity vs micro-wave frequency for low microwave power. (b) Integratedsideband intensity vs microwave frequency for highmicrowave power.

of the sidebands versus the magnitude of the micro-wave magnetic field, H„while the microwave fre-quency was kept on the center of the resonance.%'e see that the sideband intensity attained a maxi-mum for a field strength of about 1.Ox10 4 6, anddecreased for greater or lesser field strengths.Upon comparing Fig. l5 with Fig. 2, we note thatthe sidebands of Fig. 12(a) have the same qualita-tive dependence on the microwave field strength asdoes the coherence I03~ ). The functional depen-dence of the sideband intensities on the coherenceis quite complicated for long lengths of vapor butone can obtain it by solving the propagation equa-tions of (43).

3. Studies ofPhase Pi'utehing

One can show by solving (43) that the intensity ofa sideband wave which is perfectly phase matchedwith its carrier wave will increase monotonicallywith the length of the cell until the power in thecarrier wave is depleted. On the other hand, theintensity of an imperfectly phase-matched side-band wave will reach a maximum before the carrierpower is depleted and then will decrease. The in-teraction length I for a particular monochromaticsideband wave is that distance of propagation atwhich the sideband intensity reaches its first maxi-mum before the carrier is depleted.

A study was made of the growth of the sidebandsas a function of the active length of the cell. Thelatter quantity was varied by blocking off segmentsof the resonance cell length from the pumping light,beginning from the end away from the carrier Lamp.Then in the unilluminated regions of the cell, thecoherence vanished and no conversion takes place.Since the sidebands do not overlap the Rb absorp-tion lines, they propagate through the inactive re-gion of the cell unattenuated. The effects of vary-ing the active length of the cell on the sidebandsare shown in Fig. 16. We can see that all the re-solvable sideband components grew monotonicallywith increasing cell length, although they do notall grow at the same rate. It appeared that theupper sideband components grew faster initiallythan the lower sideband components, but the lattereventually caught up with and exceeded the former.

Figure 1V(a) shows the dependence of the inte-grated sideband intensity on the active cell length.In this case, the average guide wavelength of themicrowaves was 4. 5 cm. We note that for each ofthe cell temperatures shown in Fig. 1V(a), the in-tegrated sideband intensity grew monotonically withactive cell length with the rate of the growth in-creasing with the cell temperature. However, forcell temperatures higher than VO C, the rate ofgrowth decreased with temperature, but in addition,a small oscillation appeared superimposed on theotherwise monotonic growth of the integrated side-

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t t bactions. saccount for the bac

netlc lei the mlc' rowave magobtained by taking

l.O-

CA

~ l.0

CL

CDCL

0I—

LLI

I ~

0 400 800

(cj

I.O- L=22cm

0I I

0 400 800

l.of

L=28cm

l.o .(e)

L= 30cm

0

0 400 800

L=36cm

l.o.

0

0 400 800

be of the form

(5o)i( Es-Qg)= —H, (Q) e*

it into the polarizationand incorporating it into

0 400 8000 400 800NT(mK) (OR lGIN ARBlTRARY)FREQUENCY DlSDl SPL ACEME

cell length.al r01 ef'1 vs active ce eFIG. 16. Sz ed band spectr p

2028 H ENR Y TANG

The polarizations are then substituted, as before,into the first-order wave equation of (40). Thetotal sideband field then is the superposition of thesolutions of (43) and those of the counterpart of (43)for backward wave interactions. Suppose a side-band wave of frequency , is composed of compo-nents due to both forward and backward wave in-teractions, but neither component is well phasematched. Figure 18(a) shows the wave-vector dia-gram representing the generation of an upper side-band of frequency &o, and wave vector n'(~.)k. dueto the forward wave interaction of a carrier waveof wave vector n'(&o)k and a microwave field ofwave vector K. Figure 18(b) shows the wave-vectordiagram representing the generation of an upper-sideband wave of the same frequency but due to thebackward wave interaction of the same carrierwave and an oppositely directed microwave field.The quantities ~, and 4' in Fig. 18 give the ex-tent of the phase mismatch for the forward and the

backward wave interactions. One can show by ob-taining the first-order solutions to Eg. (43) and itscounterpart for backward wave interactions that theamplitudes of the sideband components which areimperfectly phase matched will vary sinusoidallywith the distance of propagation. We can, there-fore, write the amplitude of the sideband wave atfrequency ~, to first order as

S((g,) =A sin(~, z) —Bein(~zz), (51)

where A is the maximum amplitude of the compo-nent arising from the forward wave interactionand B is the maximum amplitude of the componentarising from the backward wave interaction. Equa-tion (51) can be rewritten to give

8(&u,) = (A —B)sin(~, z)

+2Bcos [n'(a, )k. - n'(&o)k] z sinKz, (52)

where we have used the relations taken from Fig.18 that

IOO—I-CA

80—

~ CA

~ ~60—CG

C) &-

ca ~n ~40—~ m

lK ~Ld pp

{a) &g =4.58cm ,69'CDI -CARRIER LIGHT o

'000

00

00

00

00

0 40 bbb

4

.59'C

0 49'G0 b 0 00 4 00 0

0 4 0 00 0 4 4 ~a

o D 00 OO~0 0gd 4 oo

40 000do 0f'o

O 0bq &5OQ

0 i~NY0

I

Io l5ACTIVE CELL LENGTH (cm)

I

20

20—I-CA

LaJ

~ CA

X DfxI ~ IO—

lK

—lK

O ~UJ ~I—~lK

UJ

5—0

ooooooo4

044040 . m~rsochoo. I

0 5

A. g= 4.58 cm

D,—CARRIER LIGHT

IP l5

ACTIVE CELL LENGTH(cm}

0 4 4404

oo o 4 4 4o oo ooo 0 75 C

OO0

4004 0

004

000o oo

o'o 40

00 00

FIG. 17. (a) Integrated sideband intensity vs active cell length at three different cell temperatures each showing amonotomic increase of the sideband intensity with cell length. (b) Integrated sideband intensity versus active cell lengthshowing oscillatory behavior of the sideband intensity with cell length.

PARAMETRIC FREQUENCY CONVERSION OF RESONANCE. . . 2029

p'(e) k

n'(@i )k+

n'(e) k

n'(rug) k+

FIG. 18. (a) Wave-vector diagram representing im-perfectly phase-matched forward-wave interaction (not

drawn to scale). (b) Wave-vector diagram representingimperfectly phase-matched backward wave interaction(not drawn to scale}.

and

~I = n'((o, )k, —n'((o)k —K

~g = 0'(Id, )k, —n'(Id)k+ K.

We can see from (52) that the sideband-wave ampli-tude will have a minimum when sinKz is zero.Furthermore, these minima occur at the samepositions along the length of the cavity as do thenodes of the microwave field. One can show thatthe same effect takes place for lower-sidebandwaves.

ff (fl) ei(Ks-Dt&

and a microwave field traveling to the left

(53)

—H (0)e" "' ""+c c. (54)

Assuming that only forward wave interaction takes

In order to further demonstrate the relationshipbetween the phases of the sideband waves and thephase of the microwave fields in the cavity, werearranged our apparatus as shown in Fig. 19.Here we reflect the transmitted light from the cavi-ty back into the cavity by means of a sphericalmirror. The mirror holder was mounted on tracksso that it could be translated parallel to the axisof the cavity. The reflected light, after it hadpassed through the cavity, was detected at the endnear the carrier lamp. A dielectric coated beamsplitter was used to direct the light through a linearpolarizer P& and a D, filter, and into a photomulti-plier tube. The polarizers P, and P~ were crossedto eliminate the carrier light. The integrated side-band intensity was recorded as the mirror wasscanned in one direction. %hat was observed wasa set of interference fringes (see bottom of Fig.19) with the fringe spacing being equal to half thefree-space wavelength of the microwaves.

These fringes can be understood quantitativelywith the aid of the propagation equations (43). How-ever, a qualitative understanding of the effectshown in Fig. 19 can be gotten by considering thefollowing. Suppose the wave front of the opticalcarrier wave enters the cavity from the left attime t. Inside the cavity there is a microwave-field traveling to the right,

BEAMSPLITTFR Rb PUMPING LAMP

Rb LAMPMIRROR

P,'

Pq

P HOTOMULTIP LIERTUBE

FIG. 19. Apparatus used todemonstrate interference of side-bands. The lower portion of thisfigure shows the intensity de-tected by the photomultiplier tubeas the mirror spacing wasscanned.

I—CA

ILL

1

I I I I I I I I I

4 8 I2 t6 20MIRROR DISPLACEMENT (cm)

2030 HENRY TANG

place, the carrier wave as it propagates to theright interacts with the right-running microwavefield, generating sideband waves. The time ofarrival of the carrier wave front at the right endof the cavity is t+L/n'(~)c, where L is the lengthof the cavity. Thus, as the carrier wave frontleaves the cavity, the phase of the right runningmicrowave field is KL —A[t+L/n'(&u)c]. Afterleaving the cavity, the carrier wave, along withthe sideband waves which were generated in thefirst pass, propagates a distance d through freespace towards the mirror which reflects thesewaves back toward the right end of the cavity. Thetime of arrival of the wave front back at the rightend of the cavity is t +L/n'(&u) c+ 2d/c. The carrierwave which now propagates to the left will interactonly with the left-running microwave field. Thephase of the left-running microwave field as thecarrier wave front reenters the cavity from theright is

v+ (-KL —II [t+L/n'(~) c+ 2d/c] }. (55)

Solving for d in (55) we obtain

d=a nAo/2 —(const ), (57)

where Ao is the free-space wavelength of the micro-waves. Therefore, from (57) we see that the sep-aration of successive maxima of the sideband in-tensity corresponds to a translation of the mirrorthrough a distance of Ao/2.

IV. CONCLUSIONS

We have directly observed the sideba, nds whichwere generated onto the spectral profile of an in-coherent optical carrier by an alkali-metal-atomvapor with ground-state hfs coherence. This ef-fect had been predicted, by the semiclassical theorywhich we presented in Sec. H. We have not yet at-tempted to make any quantitative comparisons be-tween our theory and our experimental results,because to make such comparisons, we would needto have accurate values for the various parametersof the alkali vapors. In particular, we would needto know the atomic density N and the degree of

In order for the sideband waves which are generatedduring the second pass of the carrier wave to be inphase with the sideband waves which were generatedduring the first pass, the phase of the left-runningmicrowave field, as the carrier wave front reen-ters the cavity, must be the same as that of theright-running microwave field when the carrierwave left the cavity. This condition may be writtenas

(KL —Q [t L+/n'((g ) c]}= v+(- EL —0 [t+L/n'(&o) c+ 2d/c] }en(2v),

n = integer. (56)

ground-state hfs polarization &I J), to an accuracyof a few percent. This is because the spectralprofile of the sidebands is very sensitive to how

one has phase matched the interacting waves, andthe phase-matching conditions are primarily de-termined by N and &I J) [see (47)-(49)]. Al-though several methods are known for measuringN, and for measuring &I ~ J&, none of these meth-ods wouM give measurements of the required ac-curacy.

However, we were able to make several qualita-tive comparisons between the theory and the ex-periment. The dependence of the sideband inten-sities on the microwave field shown in Figs. 14and 15 clearly demonstrates the relationship be-tween the sidebands and the microwave coherencein the vapor. The data shown in Figs. 16 and 17demonstrate the sensitive dependence of the side-band conversion efficiencyon the, phase matching.Finally, the interference fringes shown in Fig. 19demonstrate that the phases of the sidebands isdetermined by either the sum or difference of thephases of the carrier wave and the phase of themicrowave field. Therefore, our observationsseem to be in excellent qualitative agreement withour theory.

ACKNOWLEDGMENT

The author is indebted to Professor W. Happerfor suggesting this experiment, and for advice andencouragement throughout the course of this work.

APPENDIX A: EVALUATION OF & f(F, F').I FOR RbGROUND STATE

The ground-state density matrix p can be written

&&FJp, r', m, m'

The expectation value of the operator J(11) is

&J(11)&=Tr [pJ(11)]=Q &Fm~

pJ(11) Em&Fm

= 5~ ( lm~p

~

lm'& ( lm'~

J(ll)~

lm). (A2)mm'

We assume there are no coherences or populationimbalances between the Zeeman sublevels withinthe E=1 hfs level so that

lm )=0 for mmmm',

and all the diagonal density matrix elements&lm

~p

~

lm) are equal. Hence (A2) can be writtenas

&J(11)&= &»~p~»& Z &lm~J, ~lm&n„(A4)

where we have expanded the angular-momentum

PARAMETRIC FREQUENCY CONVERSION OF RESONANCE. . . 2031

operator J into its components

(A5)

C(ill;m 0 m) = —C(111; —m 0 —m) (AVa)

We can evaluate the matrix elements of t, usingthe Wigner-Eckart theorem to give

(IJlmlJ~lIJlm&= (-1) C(ill;m 0 m)[(2J+1) 3] i

x W(J1J2;Il) [(Z(J+ 1)] i~ (A6)

Substituting (A6) in (A4) and using the properties ofthe Clebsch-Gordan coefficients that

Q (FF') = Q (- 1) Q2„T2e(FF'), (81)

where Q2„is an irreducible basis dyadic definedby HM (Alo) and Tz „(FF')is an irreducible basisoperator defined by HM (Al). The expectation valueof Q(ll) is

(Q(11))=Tr [pQ(ll)]=5~ (Em pQ(11) Fm)

APPENDIX 8: EVALUATION OF &Q(FF ))'FOR Rbsv GROUNDSTATE

The definition of the quadrupole operator Q (EE')is given in MTH (56) as

C (11lg 000) = 0,

we conclude that

(J(ll))= 0.

(A7b)

(As)

Using the same reasoning as above, we can alsoshow that

(J(21))= Tr [ p J(21)]= E ( lmlp l

2m') ( 2m' J lm).(A'10)

If we assume that coherence exists only betweenlevels !E=1;m=0) and IF=2;m=0) we can writefor (J(21)),

&J(21)&= &lolpl2o&&20 J.110)e,. (All)

Similarly, one can also write for (J(12)),

(J(12))= (20 pl10) (lol J', 20) e, . (A12)

The matrix element (20l J', 10) can be evaulatedusing the Wigner-Eckart theorem

&IJ20lJ, lIJ10& = (-1)' c(112;000)[(2J+1)3]'

x W(JiJ2; Il) x [J(J+ 1)]' . (A13)

The Clebsch-Gordan and Racah coefficients in

(A13) can be evaluated to give C(112;000)= K-,' and

W(~ 1 2 2; 2 1)= I//12.Thus we obtain

1O&= ——,'.In a similar way, we can evaluate (loiZ, i 20) togive

(A14)

&lol J,l2o&= --,'. (A15)

Substituting (A14) in (All), and (A15) in (A12), weobtain

&J(21)&=--, &lol pl2o&e, (A16a)

(J(22)) = 0. (Ao)

The expectation values of the operator J(21) can bewritten as

= 0~ (1mlp lm')(lm lQ(ll) lm).mm (82)

We assume there are no coherences or populationdifference between the Zeeman sublevels within theE= 1 hfs level so we can write (82) as

(Q(11))= (lllpl 11) F~ (lmlQ(11) Im&. (83)

From the definition in (81) we can write the ma-trix element (lm [Q(ll) [ lm) as

&lmlQ(») I1m) =Q20 &lmIT20(») llm&

According to HM (Al) the matrix element of thebasis operator may be written as

(lmlT20(ll)

llm) = (- 1)" C(112;m 0 m). (85)

Substituting (84) and (85) in (83) we write for&Q(11)),

(Q(11))= (lllpl ll) Q&0%~ (-1)" C(112;m 0 m).(86)

The sum in (86) can be shown to vanish. Therefore, we conclude

(Q(ll)) = 0.

Using the same reasoning as above, we can alsoshow that

(Q™(22))= 0. (86)

The expectation value of the quadrupole operatorQ(12) is

(Q (12))= Tr [ p Q™(12)]

= 5 (2ml p l

lm ') ( lm 'lQ(12)

l2m).

mme

Since coherence exists only between the )lE= 1;m = 0) and i F = 2;m = 0) levels, the only nonvanish-

ing off-diagonal density matrix elements are(20 i p i 10) and (10 i p i 20) . Hence, we can write for

(Q(12)&,

(J'(l2)).= ——,' (20

lp

l10)e, . &Q(12)&= &lolQ(12) l2o& &2ol pl lo). (810)

H ENR Y TANG

Similarly, we can show that the expectation value ofQ(21) can be written as

(Q(21))= &201Q(21) 10) (101p120). (811)

Using the definition given in (81) we can write

&2oIQ(»)I10)=Qao (201 T»(») 110). (812)

From the definition of T2„given in HM (Al), wefind that

(201T»(12) 110)= —C(212; 000) = 0. (813)

Therefore, substituting (813) and (812) in (810) weconclude that

(814)(Q(12))= 0.

Similarly, the matrix element (201Q(21)110) in(Bll) can be written as

&201 Q(»)I») =Q» & 101T»(»)120). (815)

&101T»(21) 20) = —C(122; 000) = 0. (816)

Substituting (816) and (81V) in (811), we also con-clude that

(Q(21))= 0. (81V)

Once again from the definition of T» we can write

'Work supported in part by the Joint Services Electronics Pro-gram (U. S. Army, U. S. Navy, and U. S. Air Force) under Con-tract No. DAAB07-69-C-0383, and in part by the U. S. Air ForceOffice of Scientific Research under Contract No. AFOSR-68-14548.

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