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Vacuum-field Rabi oscillations in a bimodal cavity

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1992 Quantum Opt. 4 85

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Quantum Opt. 4 (1992) 85-91. Printed in the U K

Vacuum-field Rabi oscillations in a bimodal cavityHo Trung Dung and A S ShumovskyLaboratory of Theoretical Physics, Joint Institute for Nuclear Research, Head PostOffice, PO Bo x 19, Moscow 1 01w 0, RussiaReceived 2 December 1991Abstract. Examining the spontaneous emission of a two-level atom interacting at thesame time with two modes of the cavity radiation field we show that such multimodecoupling may lead to irregular patterns in the vacuum -field Rabi oscillations. It also leadsto the appearance of a new line in the spontaneo us emission spectra in com parison withthe case of single-mode single-atom coupling.

1. IntroductionWhen an excited atom is placed inside a high-Q resonant cavity, after spontaneousemission the atom can reabsorb the pho ton and the n emit it again, and so on. T heatomic inversion undergoes Ra bi oscillations induced by th e vacuum of the radiationfield. Such vacuum-field R ab i oscillations have b een sho wn to lead t o a splitting in thespon taneo us emission spectra [ l ,21 as well as the absorption spectra of a system of Natoms contained in a cavity [3]. In the latter case, the magnitude of the cavityresonance splitting increases with the square root of the number of atom s inserted.This fact has been successfully employed to observe vacuum Rabi splitting inexperiments on absorption spectra [4]. The effects of transition to a neighbouringlevel, cooperativity, multiphoton transitions [2] and cavity damping [5] on vacuum-field Rabi oscillations have also been treated. It is interesting to investigate anothersituation where under resonance or near-resonance conditions two modes interacteffectively with the atom.Th e spontaneous emission of a single atom which is initially excited in the presenceof N- 1 initially unexcited atoms, interacting with M modes of the field has beenexactly solved under the condition that the atoms are at random space positions [ 6 ] .Our treatment, however, is closer in spirit to a recent work by Papadopoulos [7]where t he a utho r has examined the interaction of a two-level ato m with M degeneratemodes of radiation. It has been shown that this model is analytically tractable byintroducing new B ose ope rato rs which reduce th e task of finding th e wavefunctions a ttime f to a routine exercise. Th e results were applied to obtaining the evolution of theatom ic inversion and th e Row of energy between th e modes and the atom for the caseof spo nta neo us emission and the case where only on e m ode is initially energetic andthe atom is excited.In this pap er, we relax the degenerate-mode condition but confine ourselves to thecase of spontaneous emission of a two-level atom into two modes of the cavity field.We focus on analysing the influence of such a two-mod e coupling on the vacuum-field0954-8998/92/020085+07 $4.500 992 IOP Publishing Ltd 85

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86Ra bi oscillations and spontaneous emission spectra. W e show that in the case of non-deg ene rate modes, the system exhibits noticeably diffe ren t features as compared withthose in the case where the modes are of the same frequency.

Ho Trung Dung a n d A S Shumousky

2. Vacuum-field Rabi oscillationsW e conside r a system consisting of a single two-level atom and two cavity field modes.Th e model Hamiltonian is a straightforward generalization of the Jaynes-CummingsHamiltonian [8] or the two-mode case. In the rotating wave and electric dipoleapproximations it is given by ( h=1)

2 2

H= oR'+ op: a,+ g,( R+ a,+ R-a:). (1),=I , = 1Here R and R' are pseudospin ope rato rs describing th e atom of frequency o, anda ,, i# 1 , 2 are the creation and annihilation operators for the radiation mode offrequency o,,nd gl and g2ar e the atom-field coupling constants.It is convenient to separate the Hamiltonian (1) into two mutually commutingparts 191

H =Hoi H I [Ha,Hi]=0where

Ho=o(a:al +a l a 2+R'),H I =- E p f a i + ~ g i ( R + a , i R - a f )2 2

!=I i = l

with th e detuning param eters A; being defined byA . = o -0,> i = l , 2 . (4)

Since we assume a lossless cavity and consider the case of spontaneous emissionwhere the ato m is initially excited an d the cavity field is initially em pty , only thre estates of the com bined atom-field system11)- le; 0,O) P)=Ig;1,0) 1 3 ) ~g;0,1) ( 5 )

are to be dealt with. Using these states as a basis, one can write the matrixrepresentations of Ha and HI asH,=qoI ( 6 )

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Vacuum Rabi oscillations in a bimodal cavity 87where I is a 3 x 3 unit matrix. The eigenvalues of HI in (7) are to be found from thethird-order characteristic equation

They readfor the particular case of two dege nerate modes ( A , = A 2 = A ) , and

X 3 + A1 +A2)X2- (g? +& -A lAz )X- (g:AZ+&Al)=O. (8 )XI, ,= fA&(&+&+:A2)1'2 X3= -A (9)xl=: ~ I + : ( P ? - 3 ~ 2 ) 1 n ~ ~ ~ ( e )x2=~ r , + : ( ~ U : - 3 ~ 2 ) ' " C O S ( e + ~ ~ ) (10)x3 -+PI ++(pi - /,L2)"2 cos(@+$E)P I = A I+A2 P Z = -(d +&- A I & ) ~ 3 = - ( g : & + & A i )

where

in general. Now by writing the wavefunction fo r the system in the course of time as

and solving the Sch rodinge r equation with the initial conditionI&)) =e x p ( - h t ) [ C,( f) ll )+ C df )l 2)+c3(t)13)1IW))= e; O,O)

(12)

(C,(O)=1,C,(O) =C,(O)=0)on e gets

where the abbreviationsP I = (XI-XZ)(XI-&) 8 2 =(&-xI)(x2X3) 83=W3-X1)(& -X2)(14)have been used.

Once the state vector of the system at time f is known, on e can easily calculatevarious quan tities relating to atomic coherence and statistical properties of the field.In particular, the probability of finding the atom in its excited state and the photonnumbers in modes 1 and 2 are given byP $ ) = lC I(t)l2 ( d a l ) , =c2(f)12 (a;aJ ,= IC3(Oi2 (15)

respectively. If the two m odes ar e of th e same frequ ency, by substituting (9 ) into (13),

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88 Ho Trung Dung and A S Shumousky

1.0

Figorel. Evolution of (a) the probability of finding the atom in its excited state[ P c ( l )+11. (b ) the photon number in mode 1 [ (a :aJ , ] , (c) the photon number n mode 2[ (a ;aJ, ] for A I = A 2 = g 1 n d g2 = 0. 5g ,

(14) and making use of (15) we obtainPe(t)=cosft+-sin2ftA4f

d .fa;a,) ,=- infr

.(aa )-- mft I-fwheref = (d+&+$A2).

T he above results ar e in agreement with those in [7]and show clearly tha t in the caseof spontaneous emission into degenerate modes, the total energy is shared betweenthe atom and t he field as if there w ere only one interacting m ode with the effectivecoupling constant equal t o Th e energy of th e radiation field, in tur n, is sharedbetween the modes in accordance with the ratio (&If). An example of numericalcomputations for (16) is presented in figure 1, from which we see that in th e case oftwo dege nera te modes th e time evolution of the atomic excited-state population andthe photon num bers of modes a re of simple sine or cosine form .Th e situation is changed if the mode frequencies are different. Then, the beatingof th re e non-com mensu rate eigenvalues may lead to a chaotic-like behaviour of P.( t ) ,(a :a l ) ,and (a:a2), as can be seen in figure 2. Th e interacting modes no longer can beeffectively treated as one mode and the field energy is now distributed between the

F i g . Thesame asin figure 1 but for A , = O , A 2 = g , andg2=0.5g ,

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Vacuum Ra bi oscillations in a bimodal cavity 89modes in such a way depending not only on the strength of the atomic coupling butalso on the detuning parameters. H ere we should note that even when th e modes arenon-degenerate, there exist particular values of the mode frequencies (for exampleA l = -A , , gl=gz) or which the eigenvalues of (7) are simplified and the regularevolution patterns of P c ( t ) , a;al), an d @:a2), ap pea r aga in.3. The spontaneous emission spectraIn our analysis of the spontaneous emission spectra we use the definition of thephysical transient spectrum introduced by Wodkiewicz and Eberly [lo]S(v,T)=2r73 R e d r ex p[ (r- iv)r] dr exp[ -2T (T- t)](R +(t+ r ) R - ( t ) ) (17)where T is the time of the measurement, r is the bandwidth of the detectingmechanism and B is a measure of fluorescence int o othe r modes. F rom the definitionof the Heisenberg operators, we obtain for the two-time correlation function D(t,r)the expression

I , ' ID ( t , r ) = ( R + ( f + t ) R - ( t ) )

= v(U)lexp[iH(t+s)]R'exp(-iHt)R- exp(-iHt)(q(O))=exp(iwt)C,(t)C:(t+ r) .

(18)After inserting the result (18) into (17), the integral over f and r can be easilyevaluated and o ne gets

1 {exp[iT(Xl- X,,,)] - xp[-rT- iT(v - J -XI)])x (r+ (v - w - ,){exp[-TT- iT(v -w -X I)] - xp(-2TT)}1r - (v - - , )-

If we consider th e long-time limit 1 and if we ignore the terms corresponding toexp[iT(X,-X,)] with I # m [5], retaining only those terms such tha t X,-X,,,=O, i.e.l = m , t hen

where P I are given by (14). From (20) one can see that in this limit the spontaneousemission spectra in general consist of thr ee peaks in contrast with two peaks as in th ecase of th e single-mode single-atom interaction [l,21. Th e positions of these peaks aredetermined by OJ +XI, l = 1, 2, 3, an d since no source of relaxation has been assumedin the model, the widths of these peaks are determined by the bandwidth of thedetecting mechanism r. From (20) it also follows that in the special case of two

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90 Ho Trung Dung and A S Shumovsky

0

- L - 3 -2 -1 0 1 2 3 I- J - wl -

Figure3. Spontaneous emission spectra for g,=g,, A , = O and various values of A 2 / g , .Other parameters are detector width r =O.lg, and the counting time T = lWp;'.

degenerate modes the height of the peak connected with X,= -A'(see equation (9))vanishes and the total number of peaks reduces to two.In figures 3 and 4 we have used exact formula ( 1 9 ) to plot the spontaneousemission spectra in the long-time limit with T = lOOg;', T=O.lg, for various values ofthe detuning parameters and coupling constants. In figure 3 we have taken g, and g2 asequal; A , = O while A2 was varied in the unit of g , . The value A 2 = 0 impliesdegeneracy between the modes and, as mentioned above, the peak associated with X ,is quenched resulting in a doublet structure of the spontaneous emission spectra. Formoderate values of A2 we observe spectra with triplet structure which is graduallyreplaced by the doublet structure when the value of A 2 ncreases, since then mode 2 isdecoupled from interaction with the atom. In figure 4 we have fixed the values of thedetuning parameters as A , = -Az=g, and changed g in the unit of g,. For g,= 0 onenaturally has two-peak spontaneous emission spectra characterizing the one-atomone-mode interaction. The triplet structure appears when the value of g , increases. Ifg, becomes large as compared with g , . the atom effectively interacts only with mode 2and we again observe spectra with the two-peak structure.In conclusion, w e have studied the vacuum-field Rabi oscillations and the splittingof the spontaneous emission spectra of an atom in a bimodal cavity. We have shown

0

-1 - 3 -2 -1 0 1 2 3 L- J - W I -Figure4. The same as in figure 3 except for A , = - A 2 = g , and g,/g, being changed.

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Vacu um Rabi oscillations in a bimodal cavity 91that due to the beating between three non-equal eigenvalues of the system, irregularpatterns in the evolution of the atomic excited-state population and the photonnumbers of the modes occur. The spontaneous emission spectra in the general case ofdetuning parameters exhibit a three-peak structure. It has also been shown that theeffects of multimode coupling reveal themselves better when the modes are non-degenerate. Finally, we notice that the problem treated here is mathematically verysimilar to that concerning the influence of a transition to a neighbouring level on thevacuum-field Rabi oscillations [2].

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[lo] Eberly J Hand Wodkiewicz K 1977 1. Opt. Soc. Am. 67 1252