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Effects of Phase Fluctuations in theAtom-field Coupling Coefficient ofthe Jaynes-cummings ModelAmitabh Joshi aa Department of Mathematics, University of ManchesterInstitute of Science and Technology, PO Box 88, ManchesterM60 1QD, EnglandPublished online: 18 Jun 2010.

To cite this article: Amitabh Joshi (1995) Effects of Phase Fluctuations in the Atom-fieldCoupling Coefficient of the Jaynes-cummings Model, Journal of Modern Optics, 42:12,2561-2569, DOI: 10.1080/713824334

To link to this article: http://dx.doi.org/10.1080/713824334

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JOURNAL OF MODERN OPTICS, 1995, VOL . 42, NO . 12, 256 1 -2569

Effects of phase fluctuations in the atom-fieldcoupling coefficient of the Jaynes-Cummings model

AMITABH JOSHItDepartment of Mathematics, University of Manchester Instituteof Science and Technology, PO Box 88, Manchester M601QD, England

(Received 1 March 1995 ; revision received 12 May 1995)

Abstract . The Jaynes-Cummings model of a two-level atom interacting witha single quantized mode of radiation field is generalized to include the effects ofstochastic phase fluctuations in the atom-field coupling coefficient . Thefluctuations are modelled by random telegraph process and an equation for thedensity operator averaged over the distribution of fluctuations has been obtained .The solution of this equation is used to study the resulting decoherence effectsin the dynamical behaviour of the atom and the statistical properties of the field .

1 . IntroductionThe Jaynes-Cummings model (JCM) is the simplest quantum electrodynamic

model of an atom-field interaction describing the interaction of an undampedtwo-level atom with a single non-decaying electromagnetic field mode [1] (also seefor example the recent extensive topical review by Shore and Knight [2]) . Not onlyis this model mathematically tractable but also its predictions are physically realistic .With the state-of-the-art high-Q cavity technology the experimental realization ofsome of the predictions of JCM has become possible both in the optical as wellas in the microwave regime. For instance, experimental observations of Rabioscillations [3] and the quantum phenomenon of collapse revivals [4] have beenreported recently .

In the recent past, the JCM has been generalized and extended in manydirections . It has been extended to include the effects of cavity mode decay [5] aswell as black-body photons [6] . It has been extended further for multiatom systems[7] as well as for multilevel systems [8] . The effects of spatial mode structure of thecavity [9] and the transient effects arising from a time-dependent atom-fieldcoupling coefficient [10] in JCM evolution have also been reported .

In this paper we further extend the JCM in yet another direction to incorporatethe effects of stochastic fluctuations in the atom-field coupling parameter of theJCM. For this purpose we consider the interaction of a single-mode coherent fieldwith a two-level atom undergoing one-photon transition in an ideal cavity. Thestochastic fluctuations in the cavity could be inherited from the source of theprivileged single-mode coherent field coupled to the cavity . Another mechanism forthe fluctuations in the atom-field coupling parameter could be due to the followingreasons. In the present experiments on cavity quantum electrodynamics a stable

-Permanent address : Laser and Plasma Technology Division, Bhabha Atomic ResearchCentre, Trombay, Bombay 400 085, India .

0950-0340/95 $10 . 00 © 1995 Taylor & Francis Ltd .

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beam of Rydberg atoms enters a superconducting cavity from one end, interacts witha single privileged mode of the cavity and comes out from the other end . The flowrate of the atomic beam is well controlled so that the single atom-field interactionis a dominant process . Any variations in the mechanism of the production of Rydbergatoms either due to instability in the atomic vapour production or in the dye lasersystem responsible for the transitions of atoms to the Rydberg state could give riseto a kind of stochastic fluctuation . Hence the process of interaction of electromag-netic field with the atomic beam in the cavity will also acquire these fluctuations .Recently, it has also been reported that the fluctuations of vacuum Rabi frequencyor the atom-field coupling coefficient wash out the trapping states in the micromasersystem (for example) . The most probable source of such fluctuations is a strayelectric field generated by rubidium deposits at the cavity coupling holes or theelectric field between adjacent crystal domain in the cavity walls fabricated ofniobium. Experiments confirm that the atom-field coupling coefficient fluctuatesabout 20% and the fluctuations are quite random . Hence we are motivated to themodel such fluctuations in the atom-field coupling parameter [11] .

Another motivation for considering the JCM with stochastic fluctuations isbecause of its applications to study the motion of an ion in a harmonic trap interactingwith a standing or/and travelling wave . This is because under a certain approxi-mation [12] the equations governing the motion of the ion in the trap may be reducedto a form that is similar to the JCM with field variables replaced by the vibrationalmodes of the quantized centre-of-mass motion of the ion in the trap . The couplingcoefficient now involves the amplitude and the phase of the standing wave, andtherefore, it becomes natural to consider the fluctuations in these quantities .

2. The model and resultsThe model under consideration in this work involves the interaction of a single

two-level atom with a single mode of radiation field . The atom is characterized byspin-2 angular momentum operators S ± and Sz while the field is described by theannihilation operator a and creation operator a+ . For simplicity it is assumed thatthe field is in resonance with the atomic transition frequency w o . In the usualrotating-wave approximation, the Hamiltonian of the system takes the form

H=cwoSz +woa+a+[g*(t)S + a+g(t)S-a+],

(1)

in which the interaction part is

Hint(t) =g*(t)S+a+g(t)S-a ,

(2)

where the coupling g(t) between the atom and the field is assumed to be timedependent. Transient effects arising from various forms of deterministic modulationof coupling coefficient g(t) have been already studied in a recent paper [10] .Moreover, it is easy to introduce stochasticity in the problem through the couplingcoefficient g(t) irrespective of its origin . This has been illustrated in recent work[13, 14], where the stochastic fluctuations in the atom-field coupling were treated .In these papers the coupling coefficient was assumed to fluctuate in phase and/or inamplitude. The phase fluctuations were described by a Wiener-Levy (phasediffusion) process and the amplitude fluctuations by a coloured Gaussian noise .An alternative model which represents noise by means of discrete jump processeswas first introduced into quantum optics by Burshtein and Oseledchik [15] . A simpleexample of such a jump process is the two-state random telegraph . These models

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Atom field coupling coefficient

2563

are very convenient and elegant to study the noisy electromagnetic field-atominteraction in a non-perturbative manner. While the models based on Gaussian noisehave been generally popular in quantum optics, random telegraph models have alsobeen studied considerably by several workers (for example [16,17]) . Models ofGaussian noise can be handled analytically using the so-called cumulant approxi-mation only when very short coherence times are involved . In contrast, the randomtelegraph models, whether associated with phase, frequency or amplitudefluctuations lead to an equation for the average response that has an exact algebraicsolutions [16] . The model of random telegraph (jump-type) noise is physically verysound to describe the noise arising from electromagnetic field fluctuations or fromcollisions of various kinds or from other external forces . The inclusion of jump-likenoise to the traditional setting of atom-field interaction in the cavity quantumelectrodynamics causes some important changes in the dynamical evolution of thesystem. This as we shall show in the present paper, leads to interesting features suchas the decoherence effect in the collapse and revival phenomenon of the Rabioscillations .

In order to illustrate our model we assume that

g(t) = go exp [ + io(t)], (3)

where the non-stochastic amplitude go is a positive real quantity while the phase ¢(t)is treated as a stochastic variable . There are now two ways in which the randomtelegraph model can be used in the system . The first is the so-called random phasetelegraph where 0(t) itself fluctuates in the manner of a jump . The second caseinvolves writing

¢(t) =J

p(t') dt',

(4)

where ~ = p(t) is a random telegraph . We may appropriately call this case the randomfrequency telegraph . The correlation functions (g(t)g(t')) are different in the twocases. However, we restrict ourselves only to the random phase telegraph noise inthe present work ; so the interaction Hamiltonian Hint(t) depends parametrically on0 . In other words the element of randomness is introduced in the time dependenceof the Hamiltonian H = Ho + Hint(t) only by a change in ¢(t) . We assume further thatthe change in O(t) occurs instantaneously jumpwise, and the jumps are separated bytime intervals of the order To, in which 4(t) = constant. Let O(t) follow thedistribution

dQ(i) = exp ( -ti)di

TO To(5)

The above distribution specifies the probability of duration of each such jumpinterval . We consider only that case in which 4(t)s in the neighbouring intervals arenot correlated . So, at any instant, the probability of finding a given q5 remains thesame and equal to dQ(r), and there is no limitation on the form of this distribution .In other words, 0(t) is undergoing random continuous change of Markov type .

To determine the dynamical evolution of system we need to know the unitarytransformation U(t, t') such that

p(t) = U* t,t') p(t')U - ' (¢, t, t') .

(6)

At the end of each (ith) interval we find the density matrix p(t) which is the initial

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condition for the next matrix ; so, if in the interval (0, t) there are k changes in 0, then

p(t, tk) = U(4)(tk) ; t, tk) . . . U(4)(t1) ; t2, t1)U(4)(0) ; t1, 0)P(0)

X U 1 (x(0) ; t1, 0)U 1 (4)(t1) ; t2, t1) . . . U 1 O(tk) ; t, tk) .

( 7 )The expression presented in equation (7) is of multiplicative nature and hence it isquite easy to average it over. The probability (in the interval (0, t)) that k changesof 0 have actually occurred at successive instants t1, t2, . . ., tk and that a certainsequence 01, 02 . . . , Wk (where )i = 4(t i )) was realized between them is obviouslyequal to

k

kdQ(t1, t2, . . ., tk ; y)1, 4)2, . . . , 4k, t) = exp ( - t ) 11 (d) 11 dQ(4)i) .

To i=1 To i=oThe average density operator can be thus written as

t

- 1

`

`kP(t) eXp (-

)=I k f dtk f dtk- 1 . . .

To

k=o To 0

0

(8)

X f dt1 JdQ(4)k)JdQ(4)l) . . .JdQ(4)o)p(t,tk) .k- (9)0

Note that the term with k = 0 (when 0 does not change at all in the interval (0, t))will not contain integrals with respect to the time and is equal simply to

f dQ(o o) p(t, 0) .

By using the recurrence relation (6), we can multiply both sides of equation (9) fromthe left and the right by U- 1 ((a, T, t) and U(4), T, t) respectively and also bydQ(4)) dt/To, then integrate with respect to time from 0 to T and eliminate the entireseries using equations (9) . After some simplifications it is easy to show that [15]

P(T) exp(TO) = f U(4) ; T, 0)P(0)U-1(4) ; T, 0) dQ(4))

+ o f exp (o) f U(4) ; T, t)p(t)U

t) dQ(O) dt,

(10)

where we have dropped the horizontal bar sign over p for convenience .Next, we define

Glk = f Uik(4' ;T,t)Ulm 1 (4;T,t)dQ()),

Gilk-m - Gk1m :

so we getT

Pim(T) = exp -i0 Tr [G'-(r, 0)P(0)]

+ o fo dt exp ( - ( T To t) ) Tr [G`m (T, t)p(t)] .

(12)

This is the statistical average over the random variable 4)(t) . In order to determinedynamical evolution of the system one has to determine G . The problem is now

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U = exp ( - iH;, Tt) =

Cos (B„t)- i exp ( + io) sin (B„t)

Atomfield coupling coefficient

256 5

simplified because we have to deal with an interval in which 0 (or alternatively g)is constant and the change in p is perfectly regular . Thus, by knowing G, we can findthe average variation in the system, unbounded with respect to time, during therelaxation process [15] . We obtain without difficulty the unitary transformation

c-i exp ( - i4) sin (B„t)1

os (B„t)

j '

in which On =go(n+ 1) 1 / 2 . The elements to be averaged in the calculations of G arethose containing the factors exp ( ± i4,) . Since the phases are equally probable, thatis

dQ = 2o,hence most of the terms vanish after averaging. The remaining (relevant for ourpurpose) non-vanishing elements of G are

G 111 = G 22 = cost [B„(i - t)], G iz = G > > = sin e [B„(i - t)], etc .

(15)

Here GZZ (G2112 ) represents the probability of finding the system at the instant 'C in thestate 12) if it was completely located in the state 11) at the instant t .

We need not calculate all the elements of the density matrix elements using (11) .For a two-level atom we have p11 = 1 - P22 and P12 = pz1 ; so it is perfectly sufficientto know only p11 and P12 to determine the complete dynamics of the system . We havethus

The quantity of experimental interest called inversion is defined as W„ =P11 - P22

and can be evaluated using equations (17) and (18) :

Wn exp ( +i)

= W„(0)[1 - 2 sin e (B„z)]To

+ o Jtoexp ( +

o){1 - 2 sin e [B„(t - T)]} W„(t) dt.

(19)

This equation describes the relaxation of the population and can be easily solved by

'rcos 2 [B„(i - t)]

0G11 = (16)L 0

sine [B„(z - t)]

and using equations (12) and (16) we obtain

P11 exp ( + io) = P11(0) + [P22(0) - pi (0)] sin e (B„i)

f T

+o Jo

exp ( + o) {P11(t) + [P22(t) - P11(t)] sine [B„(t - i)] } . (17)

Similarly,

P22 exp ( + io) = P22(0) + [P11(0) - P22(0)] sine (B„z)

+ o foexp ( + o) {P22(t) + [ P11(t) - P22(01 sin e [0,,(t - T)]} . (18)

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the known procedure of differentiation . Differentiating (19) three times and carryingout necessary simplification gives us

d' W,T 3

2

+ 2o d 22 Wn+ ( o + ( 20n)2)

ddWt n + (200)2 Wn = 0,

(20)

where the required solution should satisfy the following initial conditions :

d Wn = 0, d2Wndt

dt2-- -(20n) ZWn(0) .

(21)

We make use of the Laplace transform to solve equation (20) together with the initialconditions mentioned in equation (21), and obtain the following expression for thetime evolution of inversion Wn(t) :

Wn(t) = (cos [2go(n + 1) 112Zn t]

1

t+112 sin [2go(n + 1) 112Zn t] exp - - ,

(22)2goto(n + 1) Zn

toin which

1

1/2Zn = 1 - 2 24got o(n + 1)

If the initial field is in a coherent state described by

( -

1 2) «nIa) _

CHI n), Cn = expn-o

2 (n !)1/21

then the inversion W(t) can be written as

W(t) = E I CnI 2 Wn(t)n=oa

_

ICnI 2 (cos [2go(n + 1)' 12Znt]

n=0

(23)

(24)

+ 2goto(n + 1)112Zn sin [2go(n + 1) 1 f 2Znt]) exp ( - o) .

(25)

This result allows us to study the phenomenon of collapses and revivals in the JCMin the presence of phase fluctuations . The phase fluctuations affects only the dipoleor the transverse relaxation mechanism of the system, . Hence there is no dissipationof energy and the system finally reaches an equilibrium state consisting of an equalmixture of excited state and ground state . Note that, as goto --> X , the expression ofW(t) reduces to the usual JCM result without fluctuations as expected . The effectsof phase fluctuations on the inversion W(t) is pictorially shown in figure 1 for thefield which is initially in a coherent state with mean photon number n = 10. It is clearfrom figure 1 that the damping of Rabi oscillations is more pronounced when themean time interval to between phase jumps become shorter and shorter . In otherwords the decoherence mechanism is faster for the shorter-phase jump intervals . Onthe other hand, we have verified that, for got o % 500, W(t) is same as that for astandard JCM without any fluctuations .

At this stage it would be interesting to compare this dephasing mechanism withthe usual dissipation mechanism such as cavity-field damping and spontaneousemission decay or radiative damping of the system . In these cases, one finds that the

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7

6

5

4

3

2

1

0

Atom field coupling coefficient

2567

D

V

C

1

A, .

B

A NI , }lo- Nl4A A If10 20 30

T40 50 60

Figure 1 . Inversion W(t) as a function of time T = got for the initial coherent field with meanphoton number n = 10: curve A, W(t), gt0 = 500 ; curve B W(t) + 2, gto = 50; curve C,W(t) + 4, gT o = 5 ; and curve D, W(t) + 6, gTo = 0 .5 .

total energy of the atom-field system is no longer a constant of motion [18] . Thiskind of dissipation mechanism affects both the energy as well as the coherence butthe latter decays very rapidly and thus the collapse-revival phenomenon vanishesquickly much before any significant change is made in the energy of the system [18] .

The field statistical properties of the cavity field can be conveniently studied interms of the second-order intensity-intensity correlation function defined by

(2)

(a +2 a2)g (t) = (a+a)2 .

(26)

The function g(2) (t) is a measure to find out the statistical nature of the cavity field .g(2) (t) < 1 means sub-Poissonian statistics . By making use of the solution of p(t) wecan calculate both (a + a) and (a +2a2) . For the field initially in the coherent state(equation (24)), these quantities are given by the following expressions :

(a + a) = 1 +

1CnI 2n - (1)

1Cnl 2 (cos [2go(n + 1) 112Zn t]2 n =0

2 n =0

1

t

+ 2goio(n + 1)1/2Znsin [2go(n + 1) 1 J2Zn t] exp -

To

(27)

(a +2 a2) _ iIC-1 2n2 - E njCn l 2 (cos [2go(n + 1) 112Zn t]

n=O

n=0

+

1

1/2 sin [2go(n + 1)1/2Znt]) exp ( - t .

(28)

2goTO(n + 1) Zn

f

To)

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A . Joshi

0 .14

0.12

0 .1

0 .08

0 .06

0.04

0 .02

D

C

B

Al W I

-0 .02

'0

10

20

30

40

50

60T

Figure 2 . Intensity-intensity correlation function G(t) =-g(2)(t) as a function of time T =gotfor the initial coherent field with mean photon number n = 10: curve A, G(t), gT o = 500 ;curve B, G(t) + 0 .04, gTo = 50 ; curve C, G(t) + 0 . 08, gTo = 5; curve D, G(t) + 0- 12,gTo = 0 . 5 .

Again it is straightforward to verify the fact that, as goto -* -, the expression of g(2) (t)reduces to the usual JCM result without fluctuations . Figure 2 shows the behaviourof the intensity-intensity correlation function g(2) (t) with time for some values offluctuation parameter goto . Here also, like figure 1, the oscillations damp out morerapidly for the shorter-phase jump intervals .

The decoherence mechanism presented here has an origin in the stochastic phasefluctuations of the atom-field coupling parameter . However, recently Milburn [19]has proposed a model for intrinsic decoherence, based on a simple modification ofunitary Schrodinger evolution or modified von Neumann equation for a densityoperator . According to Milburn's model [19] the off-diagonal terms of the densityoperator are intrinsically suppressed because the system evolves under a stochasticsequence of identical unitary transformation leading to the destruction of quantumcoherence . In fact, using the Milburn equation, Moya-Cessa et al . [20] have analysedthe dynamical evolution of the JCM and predicted that phase diffusion leads to thecomplete deterioration of the collapse-revival phenomenon .

In conclusion, we have introduced here the phase fluctuations in the usual JCMand obtained an exact master equation for the density operator averaged over thefluctuations . The fluctuating parameter need not necessarily be the phase and canalso be the amplitude or the frequency or the energy of the stationary states .However, the fluctuations introduce a kind of damping (different from the radiativedamping of atom) in the dynamical evolution of the JCM . This damping affects only

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Atom field coupling coefficient 2569

the transverse relaxation mechanism related to the dipole phase decay of the system .Hence it would introduce a finite width in the cavity transmission spectrum even fora lossless cavity .

AcknowledgmentsThe author is grateful to Professor R . K. Bullough, University of Manchester

Institute of Science and Technology, for providing hospitality for carrying out thiswork .

References[1] JAYNES, E. T ., and CUMMINGS, F. W ., 1963, Proc. Inst . electric . electron . Engrs, 51, 89 .[2] SHORE, B . W., and KNIGHT, P . L ., 1993, J. mod. Optics, 40, 1195 .[3] GENTILE, T. R., HUGHEY, B. J ., and KLEPPNER, D ., 1989, Phys . Rev ., 40, 5103 .[4] REMPE, G., WALTHER, H., and KLEIN, N ., 1987, Phys . Rev. Lett ., 58, 353 .[5] BARNETT, S. M ., and KNIGHT, P . L ., 1986, Phys. Rev., A, 33, 3610 ; PURI, R. R ., and

AGARWAL, G . S., 1986, Phys . Rev . A, 33, 3610 .[6] KNIGHT, P . L ., and RADMORE, P. M ., 1982, Phys . Lett ., A, 90, 342; PURI, R. R ., and

JOSHI, A., 1989, Opt . Comm ., 69, 369 .[7] TAVIS, M ., and CUMMINGS, F . W., 1966, Phys . Rev ., 170, 379 ; DENG, Z ., 1983, Optics

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N. N ., JR., LE KIEN, F., and SHUMOVSKI, A. S ., 1984, Phys . Lett. A, 101, 201 .[9] SCHLICHER,R. R., 1989, Optics Commun ., 70,97 ; JOSHI, A., and LAWANDE, S. V ., 1990,

Phys. Rev. A, 42, 1752 .[10] JosHI, A ., and LAWANDE, S. V ., 1993, Phys . Rev. A, 48, 2276 .[11] RAITHEL, G., WAGNER, C., WALTHER, H ., NARDUCCI, L . M ., and SCULLY, M . 0., 1994,

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BLOCKEY, C. A., WALLS, D. F ., and RISKEN, H., 1992, Europhys . Lett., 17, 509 .[13] JosHI, A., and LAWANDE, S. V ., 1994, Phys. Lett ., A, 184, 390 .[14] LAWANDE, S. V ., and JosHI, A., 1994, Phys . Rev., A, 50, 1692 .[15] BURSHTEIN, A. S., and OSELEDCHIK, Y. S., 1966, Zh. eksp . teor . Fiz., 51, 1071

(Engl . Transl ., Soviet Phys. ,'ETP, 24, 716) .[16] ZOLLER, P ., and EHLOTZKY, F., 1977, J. Phys . B, 10, 3023 ; WODKIEWICZ, K., SHORE,

B. W., and EBERLY, J .H., 1984, J. opt . Soc . Am . B, 1, 398 ; EBERLY, J . H., WODKIEWICZ,K., and SHORE, B. W., 1984, Phys . Rev. A, 30,2381 ; WODKIEWicz, K ., SHORE, B . W .,and EBERLY, J. H ., 1984, Phys . Rev . A, 30, 2390 ; LAUDER, M . A., KNIGHT, P . L ., andGREENLAND, P. T ., 1986, Optica Acta, 33, 1231 .

[17] SHORE, B. W., 1990, The Theory of Coherent Atomic Excitation, Vol 2 (New York : Wiley),chapter 23, and references therein .

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