coupled-pendulum model of the stimulated resonance...

Vol. 5, No. 8/August 1988/J. Opt. Soc. Am. B 1613

Coupled-pendulum model of the stimulated resonanceRaman effect

P. R. Hemmer

Rome Air Development Center, Hanscom Air Force Base, Bedford, Massachusetts 01731

M. G. Prentiss*

Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Received October 16, 1987; accepted April 5, 1988

A set of three classical coupled pendulums is used to model the stimulated resonance Raman interaction. Thismodel provides a simple, intuitive, physical description of the resonance Raman process and can also be used tointerpret experimental observations, including the dynamics of Raman-induced transparency, the physical natureof Ramsey fringes in separated-field excitation, and the effects of off-resonant laser excitation. The model has alsobeen extended to suggest what might be observed for strong laser fields.

INTRODUCTION

Recently the stimulated resonance Raman effect has foundnumerous potential applications in such diverse areas asspectroscopy, collisional studies, and Raman lasers.1-3 Inaddition, in our own experiments we are exploring the possi-bility of using the stimulated resonance Raman interactionfor development of a portable clock.4 Despite this largenumber of experimental studies, a simple physical model hasnot been available to describe this interaction. Theoreticaltreatments using either perturbation theory or dressedstates have been devised.5 The dressed-state approach hasbeen more successful because it describes the existence of atrapped state that is transparent to resonant excitationfields. This accounts for the nonabsorption resonances thathave been observed experimentally by a number of research-ers.6 However, these models do not offer a simple physicalexplanation of either the formation of the trapped state orthe influence of experimental conditions on observables.

In this paper we show that an analogy between the stimu-lated resonance resonance Raman interaction and the well-known system of three classical coupled pendulums can pro-vide considerable physical- insight. Before studying thethree-level analogy in detail, it will be useful first to showhow a coupled-pendulum system can be used to describe themore familiar two-level system. In the case of a two-levelatomic system, an atom in only one state will remain in thatstate forever. Similarly, in an uncoupled two-pendulumsystem, if only one pendulum is initially oscillating then onlythat pendulum will be oscillating at all later times. In thepresence of a laser field the two atomic states are coupled sothat an atom initially in only one state will no longer remainin that state but will oscillate back and forth between statesat the Rabi frequency, which is proportional to the strengthof the coupling between states.7 Analogously, in the pres-ence of a weak coupling spring the energy can be transferredfrom one pendulum to another at a rate proportional to thecoupling-spring strength.8 In both cases it is possible to

find new eigenstates of the interaction. For the atom-fieldcase, these are the dressed states derived by Cohen-Tan-noudji and co-workers9 and Berman and co-workers,10

whereas for the coupled-pendulum system they are the well-known normal modes.

Applying a similar analogy to the three-level case allows usto explain, in a simple way, how the resonance Ramantrapped state is formed and to predict the influence of vari-ous experimental parameters on the observables. Amongthe experimental observations that can be interpreted bythis model are the generation of line-shape asymmetries insingle-zone Raman excitation, the nature of Ramsey fringesin separated-field excitation, and the effects of correlateddetuning and laser intensity on Ramsey fringes. The modeleven suggests what might be observed in the case of stronglaser intensities.

In this paper we will show the correspondence betweenvariables in the three-level atomic system and variables inthe three-pendulum system. Table 1 summarizes such cor-respondences along with the notation to be used later in thepaper. It should be noted that the analogies in Table 1make use of the usual approximations, such as the assump-tion of weak interaction, electric dipole, classical laser field,and rotating-wave approximation.7"10 In addition, the ap-proximation of weak coupling (compared with the damping)is also employed initially, which is reasonable in many appli-cations, for example, the Raman clock studies in our labora-tory.

The coupled-pendulum analogy to the three-level reso-nance Raman interaction will now be considered.

COUPLED-PENDULUM ANALOGY FORRESONANCE RAMAN INTERACTION

Figure la shows schematically a stimulated resonance Ra-man interaction involving long-lived atomic states, 11) and13), which are coupled to a short-lived intermediate state 12)by two near-resonance laser fields at frequencies xl and W2-

0740-3224/88/081613-11$02.00, C 1988 Optical Society of America

P. R. Hemmer and M. G. Prentiss

1614 J. Opt. Soc. Am. B/Vol. 5, No. 8/August 1988

Table 1. Summary of Correspondences between theStimulated Resonance Raman Interaction (Three-Level Atom) and a Set of Three Classical Coupled

Pendulumsa

Coupled-Pendulum System Three-Level Atom

Pendulum oscillations Atom-field composite statesXi1, fl2, 73 11)161), 12), 13)12)

Coupling springs Laser-field coupling

k, k 2 JOKE, U23E2fQ1 = M-, 2 = MU Q = H2 , Q2 = 22WI, MW2' ~ h '2 h

Frictional damping Spontaneous emission72' 72

Pendulum oscillation amplitudes Composite-state amplitudesN 1 , N2 , N3 Al, A2, A3

Pendulum energy ( /2Mwk'2) Composite-state probabilityIN111, IN212, IN312 IA112 IA2 12, IA3 12

Pendulum natural frequency Composite-state energy (+ h)

W1', WY, 3E 1,2 ( 3h h hNormal modes Dressed states

n-, (n+, n12) I-, +)

aRelevant notations are also introduced.

As shown, state 12) is also permitted to decay at a rate Y2through spontaneous emission. Experimentally, if isheld fixed at the 1 - 2 transition frequency and 2 isscanned over the 3 - 2 transition, fluorescence line shapessuch as the one shown in Fig. lb result. Here, the Ramaninteraction appears as a dip in fluorescence (trapped-stateformation), which, in this case, occurs at the center of the Y2-

wide 3 - 2 transition.The coupled-pendulum analogy to this resonance Raman

system is shown in Fig. 2. Here, undamped pendulums 1and 3 represent the long-lived atom-field composite statesIl)1c 2) and 13) 1w2), respectively, and the damped pendulum2 represents the short-lived intermediate state 12). Theanalogy of pendulums to composite states rather than pureatomic states is necessary because the composite states aredegenerate (on resonance) so that identical pendulums canbe used. As shown, pendulums 1 and 3 are both coupled topendulum 2 by weak springs, k and k2. These two springscouple the pendulum oscillations in analogy to the way thatthe two laser fields couple atom-field composite states in theRaman system. Finally, the frictional damping of pendu-lum 2, 7Y2', serves to model spontaneous emission in theRaman system.

In terms of atom-field composite states, the wave functiondescribing the resonance Raman interaction is as follows:

T = [al(t)ll) 1w)exp(-i6t)exp(-'/2 iAt) + a2(t)12)

+ a 3(t)13)1W2 )exp(-ibt)exp('/2iAt)]exp[-i(e2 /h)t], (1)

where 6 and A are the laser-correlated and difference-fre-quency detunings, respectively, which are defined as

6 = /2 (W + 2) - -/ 2 (C + 3)J/h,

(A1 - /2 iA l/2 j~1

*

A2 = 1/2igl -/2(2-2i6)

\A3 ° 1/2i22*

0 /A

/2~2 f A 2 )'

1/2iA A 3

(3)

where Al, A2, and A3 are the rotating-wave amplitudes ofI1)cwl), 12), and 13)Io,2), respectively, and are defined as

Al = a, exp(-/ 2iAt),

A 2 = a 2 exp(i6t),

A 3 = a 3 exp('/ 2iAt). (4)

Similarly, the equations of motion for the coupled-pendu-lum system are easily obtained from Newtonian mechanicsand are as follows:

Xl + [(wo + 2j A') + M] - ?7 = 0,

712 [(@-6 )+ M2 M 2M Al- M 3 -12(W , k, k21 k, k2~~~2'+MJ72~ M "I1M 1 3 y2%,

3 2 ) + ]3 - p 12 = 0. (5)

Here, the pendulum natural frequency w0 = [(g/l)1 2 + 6'] ischosen in analogy to the reference frequency of the Ramansystem, [(e2/h) + 6]. Making an analog to the rotating-waveapproximation permits the following substitution:

(6)

Using this substitution in the coupled-pendulum equationsgives the following pendulum amplitude equations:

a

11>

0

< E~~~~~~~D-2Y2 0 2y,

(x2((A)I fixed)

Fig. 1. a, Schematic diagram of stimulated resonance Raman in-teraction. b, Experimental data showing the Raman interaction asa decrease in fluorescence.

A = (1 - W2) - ( 3 - CO)/

Substituting this wave function into the Schr6dinger equa-tion gives the following well-known amplitude equation:

Fig. 2. Coupled-pendulum model for the stimulated resonanceRaman interaction.

7 (t) = [N(t)exp(-iw 0t) + c.c].

(2)



-'/2i(A' + Sl)J2 i = l2iol

N,3 0

/2172' - 2i(6' -/2S22)]

0

122 f N 2 ' (7)

/2(A'- s33) J N3

The details are in Appendix A, and the new quantities aredefined as

kKM

1 k2S21 =-

Si = Al1,

S22 = 21 + i22,

s33 = £2'. (8)

The strong mathematical similarities between the pendu-lum amplitude equations [Eq. (7)] and the resonance Ramanamplitude equations [Eq. (3)] are evident by inspection. Infact, the only difference is that the pendulum equationsinclude the nonzero terms sil, s2 2, and s33 , which describe ashift or pulling of the pendulum natural frequencies becauseof the presence of the coupling springs and are analogous toself-energy terms in the Raman system, (2/h)(11erI1), (2/h)(21erI2), and (2/h)(31er13), respectively. These self-ener-gy terms are of course zero because of electric-dipole selec-tion rules.'" In the pendulum analogy to the two-level sys-tem this frequency pulling is unimportant because both pen-dulum frequencies are shifted by the same amount.However, in the three-level pendulum model it is necessaryto compensate for these frequency shifts by introducing acorrection to the pendulum natural frequency, which de-pends on the coupling-spring strengths. Although this iseasy to do with a computer-generated pendulum model, it isless elegant and would complicate the construction of anyreal model. Fortunately, there is one cause of special inter-est for which no such compensation is required. This is thecase of equal coupling springs that are also weak comparedwith the pendulum 2 damping. This case is of special inter-est because it corresponds to most of our experimental stud-ies and, therefore, will now be treated in more detail.

WEAK-COUPLING APPROXIMATION

The resonance Raman system amplitude equations are easi-ly solved by using the approximation of small Rabi frequen-cies (compared with the state 12) decay rate). Briefly, this isaccomplished by neglecting A2 compared with -Y2A2. Thesolution is plotted in Fig. 3 for the simplest case of resonantlaser fields and equal Rabi frequencies, Q1 = 2 = 0. 2

7r2. Asshown, if the Raman system starts in composite state 11) 1w),with the initial values A, = 1, A2 - i[/y 2]A, and A3 = 0,

then it smoothly decays to the well-known trapped state5 Al= +1/2, A2 = 0, and A3 =-1/2. Here, it should be noted thatthe nonzero state 12) amplitude at t = 0 is a consequence ofthe weak-coupling approximation.

Similarly, the motions of the coupled-pendulum systemcan also be solved by assuming weak coupling springs (com-

pared with pendulum 2 damping forces). However, unlikethe Raman system, the coupled-pendulum dynamics havethe advantage that they can be illustrated pictorially. Thisis done in Fig 4a for the case of equal coupling-springstrengths. In this figure the dashed pendulums illustratethe average motions of each pendulum, whereas the solidpendulums and arrows show relative phases through instan-taneous position and velocity, respectively. As seen, if pen-dulum 1 is initially oscillating, then the system smoothlydecays to a normal mode, consisting of pendulums 1 and 3oscillating 1800 out of phase, with no pendulum 2 oscillation.In analogy to the Raman system, this normal mode will becalled the trapped mode. A similar result (not shown) isalso obtained in the case when pendulum 3 is initially oscil-lating. Here, as in the Raman system, the nonzero initialoscillations of pendulum 2 are a consequence of the weakcoupling approximation.

It is customary in coupled-pendulum systems to separatecomplex motions into normal modes that have relativelysimple time dependencies. This is done in Figs. 4b and 4c,which show the normal-mode decomposition of the motionin Fig. 4a. There are only two normal modes, rather thanthree, because the weak-coupling approximation has simpli-fied the interaction. As can be seen, the trapped mode (Fig.4b) is constant in time. This is as expected for a normalmode. However, the other mode decays with time because itincludes damped motions of pendulum 2. This mode will becalled the damped mode. Within the approximations de-scribed, the motion of the pendulum system in Fig. 4a can beregenerated by linearly combining only the trapped anddamped modes. More-careful examination of the dampedmode (first few rows of Fig. 4c) also shows that pendulum 2oscillates 900 out of phase with both pendulums 1 and 3.This is as expected in the approximation of weak coupling,for in this case pendulum 2 essentially behaves as a drivendamped oscillator, 8 in which the resonant driving forcecomes from pendulums 1 and 3 through the springs.

A similar analysis can be applied to the resonance Ramansystem and results in the following two dressed states:

n -002Y2

22A1 -A

1n~ A

0 2Fig. 3. Plot of Raman-system composite-state amplitudes versustime in the case of equal Rabi frequencies, 2 = 0. 2 y2.



a C

o,2t7 t2

0

1/5 !n)

21(/5 (4(D\,

31(/5 Q(

41(/5 uQ

1((!,Q

_QQ).10Q _ CQ111 Q`

-Q-Q_ -QQ0 _0, QQ

_Q_ Q"'2) "CTQQQ0,2) Q_ZFig. 4. a, Motion of the coupled-pendulum model in the case of equal coupling-spring strengths, Q' = 0.2

-Y2'. b, Trapped-mode contribution tomotion in a. c, Damped-mode contribution to motion in a.

I-) = '/2[11)lwol) - 13)"2)],

1+) =2 [11)61) + 3)IC2) + 2i ( -2i6) 12)J. (9)

Here, as for the pendulum system, only two dressed statesare needed to describe the Raman interaction when theweak-coupling approximation is made. The dressed stateI-) is of course the familiar trapped state5 and, as alreadymentioned, corresponds to the trapped-pendulum mode.The second dressed state +) decays with time because itincludes a contribution from the spontaneously decayingstate 12). This state is reminiscent of the damped-pendu-lum mode and will therefore be called the damped state. Tobe more precise, in the damped-pendulum mode the pendu-lum 2 amplitude is a fixed fraction, 2Q'/-y2, of the pendulum 1and 3 amplitudes and is 900 out of phase with both. Simi-larly, in the damped dressed state the amplitude of state 12)is a fraction, 2Q/y 2, of the state 11)1wj) and 13)IW2) ampli-tudes and also possesses a 900 phase shift. Moreover, theamplitude of the damped-pendulum mode decays asexp[-(Q'2/ y2 ')t], and the damped dressed-state amplitudedecays as exp[-(Q2/72)t].

Thus the coupled-pendulum model clearly illustrates thedynamics of trapped-state formation and, therefore, of Ra-man-induced transparency. Moreover, the observable fluo-rescence signal can be inferred directly from the amplitudeof the damped pendulum 2 oscillation because this is propor-tional to the amplitude of the spontaneously decaying state12). Here it should be mentioned that experiments that relyon fluorescence detection can directly observe only thedamped-state population. The presence of the trappedstate must be inferred from a lack of fluorescence.

Now we will show how the Ramsey fringes observed inseparated-field excitation can be interpreted by using pen-dulums.

SEPARATED-FIELD RAMAN EXCITATION

Separated-field excitation for the resonance Raman processis illustrated schematically in Fig. 5a. Here, atom-fieldsuperposition states are first created by the laser fields ininteraction zone A and are then permitted to propagate in

the laser-field-free dark zone before being probed by similarlaser fields in zone B. This two-zone excitation schemeresults in Ramsey12 interference fringes that can be observedin the zone B fluorescence. Experimentally, these fringeslook like those shown in Fig. 5b, where the rapid damping ofthe fringes for large difference-frequency detunings, A, is aconsequence of velocity-averaging effects. As can be seen,these fringes have widths that are characteristic of the tran-sit time between interaction zones, T = L/v, and, as will beshown, contain both relative amplitude and phase informa-

a

41(/T 0 41(/T

( TL/vFig. 5. a, Separated-field excitation of the Raman interaction. b,Experimental data showing Ramsey fringes for a 15-cm interactionzone separation in a sodium atomic beam.

Fig. 6. Uncoupled-pendulum model for the dark-zone Raman in-teraction.



tion for the atom-field superposition states created in zone

In terms of pendulums, the dark-zone Raman system canbe modeled by a set of three pendulums with no couplingsprings, as illustrated in Fig. 6. Here again, pendulums 1, 2,and 3 correspond to resonance Raman composite statesI1)Ilc), 12), and 13)Ic'2), respectively. Now, however, unlikefor the previous pendulum model, we will no longer assumethat all the pendulums have exactly equal oscillation fre-quencies. This is because typical dark-zone transit timesare much longer than single-zone interaction times; thuseven small frequency differences can no longer be ignored.

The solution to the resonance Raman system in the darkzone is especially simple because the composite states

11)jcwl), 12), and 13)1W2) decouple and once again becomestationary eigenstates of the three-level system. Similarly,the oscillations of the individual pendulums 1, 2, and 3 inFig. 6 are normal modes of the uncoupled-pendulum system.In fact, the resonance Raman and the coupled-pendulumamplitude equations become identical in the limit of zerocoupling.

The coupled-pendulum analogy to two-zone or separated-field Raman excitation therefore consists of first permittinga coupled-pendulum system to interact for a time r', inanalogy to the Raman zone A interaction time r. The cou-pling springs are then suddenly removed, and the pendu-lums are permitted to oscillate without coupling until amuch later time T', analogous to the Raman dark-zone tran-sit time T. At this later time the springs are replaced, andthe resulting normal-mode amplitudes in this second cou-pled-pendulum system correspond to the Raman superposi-tion state amplitudes in zone B.

To see how two-zone Raman excitation produces Ramseyfringes, two cases will be considered. These two cases ap-pear in Figs. 7a and 7b and correspond to a short zone ARaman interaction time and a long zone A interaction time,respectively. In both cases the Raman system is assumed tostart out in composite state Il)kj) prior to zone A.

As can be seen, in the limit of a short zone A interactiontime (Fig. 7a) the pendulum system is essentially still in itsinitial configuration (only pendulum 1 is oscillating). Inthis case it is seen that only pendulum 1 continues to oscil-late at all later times. This, of course, is as expected becausethe oscillations of the individual pendulums are normalmodes of the uncoupled-pendulum system. Here, it shouldbe noted that the time-dependent phase shift that appearsin this figure results from the fact that the pendulum 1oscillation frequency is A'/2 lower than wo.

Conversely, in the case of a long zone A interaction time(Fig. 7b) the pendulum system has decayed to a puretrapped mode by the time the coupling springs are removed.Here also, both pendulums 1 and 3 continue to oscillate withconstant magnitude, as expected. Now, however, the sys-tem as a whole physically resembles a damped mode (with a900 phase shift) after every odd quarter cycle, A'T'/2 = 7r/2or A'T/2 = 37r/2. Although this has no consequence in theuncoupled-pendulum system, it would lead to much differ-ent dynamics if the coupling springs were suddenly reintro-duced, as in the analogy to Raman two-zone excitation. Forexample, if the springs were restored after either one half orone exchange cycle, A'T'/2 = r or A'T'/2 = 27r, then a puretrapped mode would result and no decay would occur. In

a0ZT AT' \

0

1(/4 r > al

TK / 4 ~ ~ ¶~ /

SIC/4 ; o t3 t /2,I' O 'Q i71(/4 Kr)

21K ,lln- --4aFig. 7. Motion of the uncoupled-pendulum system correspondingto a, short zone A interaction times; b, long zone A interaction times.

contrast, if the springs were restored after one quarter orthree quarters of a cycle, A'T'/2 = 7'/2 or AT/2 = 37r/2, thena pure damped mode would be produced and the systemwould decay to zero-oscillation amplitude.

The exchange oscillations between trapped and dampedmodes in the uncoupled-pendulum system suggest that itwould be instructive to rewrite the dark-zone Raman solu-tion [Al(T) = Al(r)exp(-iAT/2) and AA(T) = A3(,)exp(iAT/2)] in terms of (zero-field) trapped I-) and damped I+ ) stateamplitudes, B- and B+, respectively:

B_(T) = B.(T)cos(/ 2 AT) - iB+(r)sin('/2 AT),

B+(T) = -iB_(T)sin(1/2 AT) + B+(T)cos(1/2 AT)- (10)

From these expressions and the dark-zone pendulummodel it is easy to see that laser difference-frequency detun-ings A (for fixed T) produce Ramsey fringes because theymodulate the (observable) zone B damped-state populationin the same way that the damped-pendulum-mode contribu-tion was modulated in the above uncoupled-pendulum sys-tem. In fact, the pendulum drawings suggest that the Ram-sey-fringe minima and maxima in the zone B fluorescencecan be interpreted as a direct measure of the final zone Atrapped- and damped-state populations, respectively.

Finally, as suggested by the earlier pendulum model ofFig. 4, Raman-Ramsey fringes are expected to increasesmoothly in amplitude with increasing zone A interactiontime (UT), approaching a constant maximum amplitude.This is in contrast to the two-level system, in which Rabioscillations cause the fringes to appear and disappear alter-natively for increasingly long interaction times.' 2

CORRELATED LASER DETUNINGS

Of particular interest in our experimental investigations arethe effects of correlated laser detunings, 6. As already men-tioned, correlated detunings arise when both laser fields are



a b

12>

I1 I1

Fig. 8. Schematic diagrams illustrating a, correlated laser detuning( and b, laser difference-frequency detuning A.

frequency shifted by the same amount so that the differencefrequency is unaffected. This situation is illustrated sche-matically in Fig. 8a. In addition, the complementary case oflaser difference-frequency detuning, A, is illustrated in Fig.8b for comparison. We will now consider the effects ofcorrelated laser detunings in a single interaction zone and inseparated fields, with the single-zone case considered first.

The effects of correlated laser detunings in a single zoneare illustrated in Fig. 9 by using coupling pendulums. Thecase shown corresponds to a positive detuning of twice thestate 12) decay rate, = 2 2. Here, correlated detuning ismodeled by an equal increase in the natural frequencies ofpendulums 1 and 3, represented in the figure by shorterpendulum lengths. As shown in Fig. 9a, if only pendulum 1is initially oscillating, then the system eventually decays to apure trapped mode as before. Now, however, pendulums 1and 3 experience amplitude exchange oscillations as thedecay proceeds. For clarity, these exchange oscillations areplotted as a function of time in Fig. 10, where each tick markon the time axis corresponds to one row of the pendulumdrawing of Fig. 9a.

Figures 9b and 9c illustrate the trapped- and damped-mode contributions, respectively, to the pendulum motionsof Fig. 9a. As shown, the trapped mode is unaffected by thechange in pendulum 1 and 3 oscillation frequencies. This isbecause both pendulums 1 and 3 still have equal naturalfrequencies. However, the pendulum-damped mode nowexperiences a time-dependent phase shift that was not ob-served for the on-resonance case. This phase shift arises

a

because the oscillation frequency of the damped mode ispulled slightly lower, relative to the trapped-mode frequen-cy, by pendulum 2. As before, the original pendulum mo-tions of Fig. 9a can be regenerated by linearly combiningonly these trapped and damped modes. When this is done itbecomes clear that the exchange oscillations involving pen-dulums 1 and 3 are a consequence of the time-dependentphase difference between trapped and damped modes.

We will now show that these results can be used to inter-pret the fluorescence line-shape asymmetries that are ob-served experimentally for non-zero-correlated laser detun-ings.4 Typical asymmetric single-zone line shapes are pre-sented in Figs. 11a and 11b for correlated detunings of 6 =,y2/

2 and = 2, respectively. To model such asymmetrieswith pendulums, two systems will be considered, corre-sponding to opposite difference-frequency detunings of A ±0.65 (Q2/7y2) but identical correlated detunings of 6 = 2/2 .These pendulum systems are illustrated in Figs. 12a and 12cfor the positive and negative A cases, respectively. Forclarity, the pendulum motions shown in these two figures arealso plotted as a function of time in Figs. 13a and 13b,respectively. Again, each tick mark on the time axis corre-sponds to one row of the pendulum drawing.

As can be seen, for positive difference-frequency detun-ing, A' > 0 in Fig. 12a, the pendulum oscillations quicklydecay with a relatively large damped-mode component.But for negative detuning, A' < 0 in Fig. 12c, they decaymore slowly with almost no damped component, This canbe seen more clearly in Figs. 12b and 12d, which show onlythe instantaneous damped components of the motions inFigs. 12a and 12c, respectively. For brevity, the trappedcomponents are not shown.

As these and previous pendulum drawings (Figs. 4 and 7)suggest, both correlated and difference-frequency detuningstend to produce Raman damped-state components that are900 out of phase with the trapped state. If both thesedamped-state contributions are in the same direction, theywill be reinforced; otherwise, they will tend to cancel. Thisleads to either a larger or a smaller overall damped-statecontribution and, therefore, a faster or slower decay, respec-tively. Of course, in the Raman-system analogy faster decaymeans more fluorescence, thereby accounting for the ob-served fluorescence line-shape asymmetries.

b c

2S-7t t0

'K/2

31(/2 1',l

~Q Q)- I'I111 C1,

_QQ72 d Uu' ciQ- Qi")&"t

,1.

,Q

041

9,C'11

2zK CQ7s _Q (@" of) r7C Q51(/2 d,'i C ) : Cj DI

Fig. 9. a, Coupled-pendulum model to an off-resonance Raman interaction, 6' = 2 Y2': coupling-spring strength, ' = 0.2 y2'. b, Trapped-

mode contribution to motion in a. c, Damped-mode contribution to motion in a.



s = 2Y2nQ= 0 0 2Y>

TX

IcH-

(-5

1 Q/2I c 4-;

,K 2,tW _ C _

Fig. 10. Plot of average pendulum oscillation amplitudes versustime for the off-resonance model of Fig. 9a.

a S Y2/2Q0 3 Y2

row). In contrast to the previous dark-zone model, the in-stantaneous damped-pendulum-mode contribution now be-comes a minimum for A'T' 5, 0, specifically A'T/2 = 7r/5.This is easier to see when the motion is separated into in-stantaneous trapped and damped components, as shown inFigs. 15b and 15c, respectively. In the Raman-system anal-

a b

0

'1(/8 ((0-D1(/4 (p

3T/8 Q%/2 Q51(/8 Q

S Y2Q+- 0.3Y2

-Q-QC ..

-Q Q)

C

ip4gI.Q . -Q-

Q-Q-

Q)

Q1

Q.0

d

-Y2/4 0 Y2 /4 -Y2/4 0 Y2 14

Fig. 11. Experimental data showing single-zone line-shape asym-metries obtained for correlated laser detunings of a, 6 = 0.

5Y2 and b,a = Y2. Rabi frequencies in both cases are Q1 = 0.

4Y2 and Q2 =

0.1372 Q+ = (1/2)(Q12 + Q22)1/21.

Our experimental studies also showed that the Ramanline-shape asymmetries reversed sense either for oppositecorrelated laser detuning, (, or for opposite initial popula-tions in the composite states 1)w1i) and 13)1W2). The cou-pled-pendulum model likewise exhibits asymmetry rever-sals, as is shown in Fig. 14. Here, Fig. 14a shows the pendu-lum motions corresponding to a positive difference detuning(as in Fig. 12a) but a negative correlated detuning (oppositeto Fig. 12a). The slow decay in this case indicates that theasymmetry has reversed sense for opposite Y', in analogywith the Raman system. Similarly, Fig. 14b shows the caseof A' > 0 and (' < 0, which is the same as in Fig. 12a, only nowpendulum 3 is oscillating initially instead of pendulum 1.Here also, slow decay results, thereby verifying the depen-dence of asymmetry on initial conditions as well.

The pendulum analogy for correlated laser detunings willnow be extended to separated-field excitation to show howsuch laser detunings produce phase shifts in Ramsey fringes.

To this end, Fig. 15a shows the motions of the uncoupled-pendulum system (dark-zone Raman model) for a represen-tative case corresponding to a non-zero-correlated detuningin zone A. In this example the initial conditions were ob-tained from the correlated detuning model of Fig. 9a, for 6' =2 y2', after an interaction time of (Q22/26')r' = -r/2 (second

A'0

0

-K/4 0 a r -3/8 V Q Q Q

-51(/8 QQQ Q QQFig. 12. Coupled-pendulum models showing the physical basis ofthe line-shape asymmetries in Fig. 11: a, pendulum model corre-sponding to 6' = 0.572' and A' = +0.65(UY2/Y2'); b, instantaneousdamped-mode contribution to motion in a; c, pendulum model cor-responding to (' - 0. 572' and A' = -0.65(Q' 2/Y2'); d, instantaneousdamped-mode contribution to motion in c.

a bA' >0S'= 0.5Y 2 'Q'= 0.242'

LL

;_

C-

A'< 0S'= 0.5Q'= 0.2

1 1w 2UiM-

Y2 '-Y2

Fig. 13. a, Plot of average pendulum oscillation amplitudes for theline-shape asymmetry model of Fig. 12a. b, The same as a but forthe model of Fig. 12d.



aIA't

0

1(/8 N')

1(/4 ratio

31(/8 (|vQTQ

1(/25t/8 TV,

_Q Q)11 Q11,)1

_QE c ) Q_;11

AQ2 Q7:,_QQ~ Q711 U)1_C~)_ Q)11 U11

Fig. 14. Pendulum models showing asymmetry reversals: a, oppo-site correlated detuning case, = -

0.572 and A = +0.65(Q 2/Y2) b,

pendulum 3 oscillating initially instead of pendulum 1 for -+ 0.572 and A = +0.65(Q2/Y2)-

ogy this translates into a frequency shift of the central Ram-sey-fringe minimum by an amount A = 2r/5T (assumingfixed T). Such central-fringe frequency shifts (also calledRamsey-fringe phase shifts) are in fact observed experimen-tally for non-zero-correlated laser detunings, as will be seenbelow. From the pendulum drawings it is easy to show thatthese phase shifts arise whenever the trapped and dampeddressed states are not exactly in phase at the start of the darkzone.

At this point it is possible to interpret some additionalexperimental observations in Raman separated-field excita-tion, such as that illustrated by the data in Figs. 16a and 16b,which were recently recorded in our laboratory. These data

a

show Ramsey-fringe phase shifts, labeled A, as a function ofcorrelated laser detuning, 6, for two different zone A interac-tion strengths, corresponding to (Q2/-Y2)r = 7r/8 and ( 2/Y2)T= r/2, respectively. As can be seen, the observed phaseshifts vary linearly with 6 for small values of ( 2 /y2)T, where-as for large ( 2 /Y2)r the phase shifts nearly vanish, at leastfor 6 < 2. These observations can be explained by recallingthat for a long enough zone A interaction time, the dampeddressed state decays to zero, leaving only the trapped state.However, if no damped state remains, then its relative phasebecomes unimportant, and no Ramsey-fringe phase shiftsare expected to result.

Experiments also showed that correlated laser detuningsproduced only in zone B (for example by Doppler shifts)caused much smaller Ramsey-fringe phase shifts than simi-lar laser detunings produced only in zone A. The pendulumdrawings show that this occurs primarily because thetrapped and damped states mix only in the dark zone (forsmall A). Thus, although phase differences betweendressed states do arise in zone B for correlated detunings,they do not cause Ramsey-fringe phase shifts.

STRONG LASER FIELDS

We will now consider the limit of strong coupling, for whichthe Rabi frequency is much greater than the state 12) decayrate. As mentioned earlier, first it is necessary to correct forthe effects of the coupling springs on the pendulum oscilla-tion frequencies. When this is done the resulting modifiedpendulum amplitude equations (not shown) are mathemati-cally identical to the resonance Raman amplitude equations.

The motions of this modified coupled-pendulum modelare shown in Fig. 17a in the limit of strong laser fields(compared with the state 12) damping). As can be seen,when only pendulum 1 is oscillating initially, somewhat

b c

0,N'T'

0

1(/10

1(/5 {v

31(/10 1Q'k.9

21(/5 sfC

1(/2 '\)

31C/5

71t/10

41/5

(DJII"

(Q:11)

0~11_11)

'I a.IQ(7 )ZQ- _QQ ''

II'IQ Q 1

Fig. 15. Effects of correlated detuning on the uncoupled (dark-zone) pendulum model fortwo of Fig. 9a, ('/26') = r/4; b, instantaneous trapped-mode contribution to motion inmotion in a.

-Q a -Q-Q-Q-J Q Q

HeQ u-Q ¢

(5I' 2y2': a, initial conditions obtained from row

a; c, instantaneous damped-mode contribution to


"IO

"I111'0

11Z'01

"II-P.111

I


a(2 Y2)t -t/8

/ 10 T

It iT0 -A

21( ]10OT

I I I

-2Y 2 0 2aY2

S - 1

b(2 Y2)t=1/2

still eventually expected to decay to a pure trapped mode forlong enough interaction times, in agreement with previoustheoretical treatments.5 This case is yet to be studied indetail experimentally.

/P 10 T1 0

A. _10 T

-2Y2 0 2Y2

Cs

Fig. 16. Experimental data showing Ramsey-fringe phase shifts Aversus correlated laser detuning 6 for zone A laser intensities suchthat a, ( 2 /72) T = r/8 and b, Q2 2 2r = r/2.

complex amplitude exchange oscillations result. These os-cillations can of course be simplified by dividing the motioninto trapped and damped components, as shown in Figs. 17band 17c, respectively. Once again, the trapped componentis independent of time, just as in the weak-coupling limit.Now, however, because the system is underdamped, thedamped component exhibits amplitude exchange oscilla-tions involving pendulum 2 and a superposition oscillationcomposed of pendulums 1 and 3. The damped mode istherefore no longer a normal mode of the system; thus itsdynamics can be broken down further into two additionalnormal modes, which appear in Figs. 17d and 17e. Here, itshould be noted that the pendulum normal modes describedin Figs. 17d and 17e are not the same as the normal modesusually derived for a coupled three-pendulum system.8 Inparticular, pendulum 2 oscillates with a different magnitudethan expected. This difference results from the modifica-tions made to the pendulum natural frequencies to correctfor the effects of the coupling springs.

The above analysis shows that in the large Rabi-frequencylimit Rabi-like oscillations are expected to be observed inthe Raman-system fluorescence. Nonetheless, the system is

a b

UNEQUAL RABI FREQUENCIES

Finally, we would like to consider the case of resonanceRaman excitation with unequal Rabi frequencies. In termsof pendulums this can be modeled by using unequal couplingsprings and the modified pendulum model.

Figure 18a shows the time evolution of a coupled-pendu-lum system when the spring attached to pendulum 1 is threetimes as strong as the one attached to pendulum 3. Asshown, if the system starts out with only pendulum 1 oscil-lating, then it smoothly decays to a trapped mode over timeslong compared with 1/Q'+, where Q+'2 = (2 + Q2

12 )/2.

However, this trapped mode, unlike the previous trappedmode, has unequal oscillation amplitudes for pendulums 1and 3; the pendulum with the weak-coupling spring has thelargest amplitude. This is required so that pendulums 1 and3 will tend to induce equal but opposite motions in pendu-lum 2, thereby causing the damped pendulum 2 to remainstationary.

From this discussion it is evident that the trapped state, inspite of its name, is transparent not to arbitrary resonantlaser fields but only to those having the same Rabi-frequen-cy ratio (and the same relative phase) as those used to createit. Thus a pure, trapped superposition state created withone combination of Rabi frequencies would decay to a differ-ent trapped state if the ratio of the Rabi frequencies weresuddenly changed, for example, in a two-zone atomic-beamexcitation scheme. This situation is illustrated by using thependulums in Fig. 18b for the case when the pendulumsystem initially resembles a pure trapped mode from theequal coupling-spring system. As expected, this system de-cays to the new trapped mode but with a smaller amplitudethan in Fig. 18a.

Once again, the coupled-pendulum analogy provides thephysical insight needed to interpret the resonance-Raman

C d

not ,'p' I

1(/4

1(/2 QD:Q1 Q~D31(/4 Q o

1( Q Q (],D

51/4 Q G

31(/2 "s 9_ Q71/4 (D - Q

at.1( \t8D Q C

X)Q-Q Q Cab A) 9 Q (Q • (Q cC94 A 6X;

' QQ-Q n Q 2 Q Qd2nQ} Q SAn g (A X' ' Q'

h Q-0- , Q(XQ •C Q a C;Q ) C;Q) Q (Q-,)ah Q UP (- P _- (x3;) an~ 4:1 Go? (O X:'

D _Q Con (ID fl ah 14"Cl : V G V -03 4 D

Fig. 17. a, Motion of modified coupled-pendulum model in the limit of strong coupling springs (compared with pendulum-2 damping). b,Trapped-mode contribution to motion in a. c, Damped-mode contribution to motion in a. d and e, Normal-mode contributions to dampedmode of c.

e



a012MtY2

0

1(/5 1"'Q a

21(/5 ;O a-Q-

31(/5 ',0 _Q

4K/ 5 ,,, _Q

Q 111-•1 Q4QC11-Q)1~ -roQ Q )Q411 I 'iQ Ci)(1 I 11""1 aQ )

where the complex-conjugate parts have been dropped forsimplicity. These equations are simplified by making thefollowing assumptions, which are consistent with the pendu-lum rotating-wave approximation:

V << CON,

R << coNA,

72 << 0

1' << co,

Al << wo. (A2)

With these approximations, the pendulum amplitude equa-tions reduce to the simple form of Eq. (7).

t~~L Q an CFig. 18. a, Motion of modified coupled-pendulum model for un-equal coupling springs. b, Motion for initial conditions resemblinga trapped mode from the equal spring model.

system dynamics, this time for the case of unequal Rabifrequencies.

SUMMARY

In this paper we have shown that the analogy between thestimulated resonance Raman system and a system of threecoupled pendulums provides a simple, intuitive, physicaldescription of the Raman process and can also be used toelucidate various experimental observations. These includethe dynamics of the formation of the trapped dressed state,the physical nature of Ramsey fringes in separated-fieldRaman excitation, and both single-zone line-shape asymme-tries and Ramsey-fringe phase shifts that are observed in thepresence of non-zero-correlated laser detunings.

APPENDIX A: DERIVATION OF SIMPLIFIEDPENDULUM AMPLITUDE EQUATIONSStarting with the coupled-pendulum equations of motion[Eqs. (5)] and making the pendulum rotating-wave substitu-tion of Eq. (6) gives the following pendulum amplitude equa-tions:

N1 - 2iwoN, + (oA' + I A/2 + k,)N1-k4N2 =-

N 2 - (2iwo -72'A

-(2wob - *2 _-k _k 2 .M- - - IW 1V 2

-k, N, -k2 N3 =0,

3 - 2iwoN 3 + (-C0oA + A 2 + k2 k2

(Al)

ACKNOWLEDGMENTS

Our special thanks to S. Ezekiel of the Massachusetts Insti-tute of Technology for a critical reading of the manuscriptand for numerous discussions and suggestions. This re-search was supported by the Rome Air Development Center,the U.S. Air Force Office of Scientific Research, the NationalScience Foundation, and the Joint Services Electronics Pro-gram.

P. R. Hemmer is also a research affiliate with the ResearchLaboratory of Electronics, Massachusetts Institute of Tech-nology, Cambridge, Massachusetts 02139.

* Permanent address, AT&T Bell Laboratories, Holmdel,New Jersey 07733.

REFERENCES

1. F. Shimizu, K. Shimizu, and H. Takuma, "Selective vibrationalpumping of a molecular beam by a stimulated Raman process,"Phys. Rev. A 31, 3132 (1985); A. Sharma, W. Happar, and Y. Q.Lu, "Sub-Doppler-broadened magnetic field resonances in theresonant stimulated electronic Raman scattering of multimodelaser light," Phys. Rev. A 29, 749 (1984); R. E. Tench and S.Ezekiel, "Precision measurements of hyperfine predissociationin 2 vapor using a two-photon resonant scattering technique,"Chem. Phys. Lett. 96, 253 (1983); K. Takagi, R. F. Curl, and R.T. M. Su, "Spectroscopy with modulation sidebands," Appl.Phys. 7, 181 (1975).

2. E. Buhr and J. Mlynek, "Collison-induced Ramsey resonancesin Sm vapor," Phys. Rev. Lett. 57, 1300 (1986); A. A. Dabagyan,M. E. Movsesyan, T. 0. Ovakimyan, and S. V. Shmavonyan,"Stimulated processes in potassium vapor in the presence of abuffer gas," Sov. Phys. JETP 58, 700 (1983).

3. D. Krokel, K. Ludewigt, and H. Welling, "Frequency up-conver-sion by stimulated hyper-Raman scattering," IEEE J. QuantumElectron. QE-22, 489 (1986); R. S. F. Chang, M. T. Duignan, R.H. Lehmberg, and N. Djeu, "Use of stimulated Raman scatter-ing for reducing the divergence of severely aberrated laserbeams," in Excimer Lasers: Their Applications and NewFrontiers in Lasers, R. W. Waynank, ed., Proc. Soc. Photo-Opt.Eng. 476, 81 (1984); J. C. White, "Up-conversion of excimerlasers via stimulated anti-Stokes Raman scattering," IEEE J.Quantum Electron. QE-20,185 (1984); N. V. Znamenskii and V.I. Odintsov, "Infrared stimulated Raman scattering in rubidiumvapor with a tunable pump frequency," Opt. Spectrosc. (USSR)54, 55 (1983); R. Wyatt, N. P. Ernsting, and W. G. Wrobel,"Tunable electronic Raman laser at 16 microns," Appl. Phys. B27, 175 (1982); M. L. Steyn-Ross and D. F. Walls, "Quantumtheory of a Raman laser," Opt. Acta 28, 201 (1981).

4. P. R. Hemmer, G. P. Ontai, and S. Ezekiel, "Precision studies ofFtimulated resonance Raman interactions in an atomic beam,"



J. Opt. Soc. Am. B 3, 219 (1986); P. R. Hemmer, S. Ezekiel, andC. C. Leiby, Jr., "Stabilization of a microwave oscillator using aresonance Raman transition in a sodium beam," Opt. Lett. 8,440 (1983); P. Knight, "New frequency standards from ultra-narrow Raman resonances," Nature 297, 16 (1982); J. E. Thom-as, P. R. Hemmer, S. Ezekiel, C. C. Leiby, Jr., R. H. Picard, andC. R. Willis, "Observation of Ramsey fringes using a stimulatedresonance Raman transition in a sodium atomic beam," Phys.Rev. Lett. 48, 867 (1982); J. E. Thomas, S. Ezekiel, C. C. Leiby,Jr., R. H. Picard, and C. R. Willis, "Ultrahigh resolution spec-troscopy and frequency standards in the microwave and far-infrared region using optical lasers," Opt. Lett. 6, 298 (1981).

5. B. J. Dalton, T. D. Kieu, and P. L. Knight, "Theory of ultra-high-resolution optical Raman Ramsey spectroscopy," Opt.Acta 33, 459 (1986); D. Pegg, "Interaction of three-level atomswith modulated lasers," Opt. Acta 33,363, (1986); P. F. Fracassi,L. Angeloni, and R. G. DellaValle, "Effects of dampings anddephasings in resonance Raman spectroscopy," Chem. Phys.Lett. 115, 428 (1985); P. L. Knight, M. A. Lauder, P. M. Rad-more, and B. J. Dalton, "Making atoms transparent: trappedsuperpositions," Acta Phys. Austriaca 56, 103 (1984); V. Zad-kov, N. Koroteev, M. Rychev, and A. Fedorov, "Saturated co-herent Raman scattering spectroscopy for molecular gasses,"Moscow Univ. Phys. Bull. 39, 30 (1984); N. I. Shamrov, "In-duced transparency in resonant induced Raman scattering," Zh.Prikl. Spektrosk. 40, 346 (1984); F. H. Mies, "Resonance fluo-rescence and Raman line shapes produced by monochromaticlaser fields: effects of branching ratio and homogeneous broa-dening," J. Quant. Spectrosc. Radiat. Transfer 29, 237 (1983); P.M. Radmore, "Population trapping in a multilevel system,"Phys. Rev. A 26, 2252 (1982); S. Swain, "Conditions for popula-tion trapping in a three-level system," J. Phys. B 15, 3405(1982); P. M. Radmore and P. L. Knight, "Population trappingand dispersion in a three-level system," J. Phys. B 15, 561(1982); B. Halperin and J. A. Koninfstein, "Conditions for excit-ed-state Raman and absorption processes during optical pum-ping," Can. J. Chem. 59, 2792 (1981); F. H. Miew and Y. B.Aryeh, "Kinetics and spectroscopy of near-resonant opticalpumping in intense fields," J. Chem. Phys. 74, 53 (1981); J. E.Thomas and W. W. Quivers, Jr., "Transit-time effects in opti-cally pumped coupled three-level systems," Phys. Rev. A 22,2115, (1980); G. Orriols, "Nonabsorption resonances by nonlin-ear coherent effects in a three-level system," Nuovo Cimento53B, 1 (1979); H. R. Gray, R. M. Whitley, and C. R. Stroud, Jr.,"Coherent trapping of atomic populations," Opt. Lett. 3, 218

(1978); C. Cohen-Tannoudji and S. Reynaud, "Modification ofresonance Raman scattering in very intense laser fields," J.Phys. B 10, 365 (1977); E. Courtens and S. Szoke, "Time andspectral resolution in resonance scattering and resonancefluorescence," Phys. Rev. A 15, 1588 (1977); R. M. Whitley andC. R. Stroud, Jr., "Double optical resonance," Phys. Rev. A 14,1498 (1976); M. Sargent III and P. Horwitz, "Three-level Rabiflopping," Phys. Rev. A 13, 1962 (1976); R. G. Brewer and E. L.Hahn, "Coherent two-photon processes: Transient and steadystate cases," Phys. Rev. A 11, 1641 (1975j; A. Szoke and E.Courtens, "Time-resolved resonance fluorescence and reso-nance Raman scattering," Phys. Rev. Lett. 34, 1053 (1975).

6. P. Kumar and J. H. Shapiro, "Observation of Raman-shiftedoscillation near the sodium D lines," Opt. Lett. 10, 226 (1985);M. Kaivola, P. Thorsen, and 0. Poulsen, "Dispersive line shapesand optical pumping in a three-level system," Phys. Rev. A 32,207 (1985); M. S. Feld, M. M. Burns, T. U. Kuhl, P. G, Pappas,and D. E. Murnick, "Laser-saturation spectroscopy with opticalpumping," Opt. Lett. 5, 79 (1980); G. Alzetta, L. Moi, and G.Orriols, "Nonabsorption hyperfine resonances in a sodium va-por irradiated by a multimode dye-laser," Nuovo Cimento 52B,209 (1979); R. P. Hackel and S. Ezekiel, "Observation of sub-natural linewidths by two-step resonant scattering in I2 vapor,"Phys. Rev. Lett. 42, 1736 (1979); R. L. Shoemaker and R. G.Brewer, "Two-photon superradiance," Phys. Rev. Lett. 28,1430(1972).

7. A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York,1975), Chap. 15.

8. R. A. Becker, Introduction to Theoretical Mechanics(McGraw-Hill, New York, 1954), Chaps. 7 and 14.

9. C. Cohen-Tannoudji, "Atoms ((Habilles)) per des photons op-tiques ou de radiofrequence," J. Phys. C 5a, 11 (1971); C. Cohen-Tannoudji and S. Haroche, "Absorption et diffusion de photonsoptiques par un atome en interaction avec des photons de radio-frequence," J. Phys. (Paris) 30, 153 (1969).

10. P. R. Berman and R. Salomaa, "Comparison between dressed-atom and bare-atom pictures in laser spectroscopy," Phys. Rev.A 25, 2667 (1982); P. R. Berman and J. Ziegler, "Generalizeddressed-atom approach to atom-strong-field interactions-ap-plication to the theory of lasers and Bloch-Siegert shifts," Phys.Rev. A 15, 2042 (1977).

11. A. S. Davydov, Quantum Mechanics (Pergamon, Oxford, 1965),Chap. 9.

12. N. F. Ramsey, Molecular Beams (Oxford U. Press, London,1963), Chap. 5.


coupled-pendulum model of the stimulated resonance...

Documents