PHYSICAL REVIE%' A VOLUME 49, NUMBER 1 JANUARY 1994

State evolution in the two-photon atom-field interaction with large initial fields

Ho Trung Dung and Nguyen Dinh HuyenLaboratory of Theoretical Physics, Joint Institute for Nuclear Research, Head Post 0+ce P.O. Box 79, Moscow I01000, Russia

(Received 29 April 1993)

The state evolution in the two-photon atom-field interaction with large initial fields is studied. It isshown that under certain conditions, for the initial coherent field, at one-quarter and three-quarters ofthe revival time the atom is converged into unique pure states, and at the revival time both the atom andfield return to their initial states. The possibility for the field macroscopic superposition states to be gen-erated is discussed. The e6'ects of the cavity damping, photon statistics, and dynamic Stark shifts are in-

vestigated. Our results give some further insight into the validity of the effective two-photon Hamiltoni-an which neglects the Stark shifts.

PACS number(s): 42.50.—p, 03.65.—w, 42.52.+x

I. INTRODUCTION

The Jaynes-Cummings model (JCM) [1] describing theinteraction of a single two-level atom with a single-modequantized radiation field is one of the simplest nontrivialsystems in quantum optics. In spite of its mathematicalsimplicity, the model gives rise to an interesting andsurprisingly rich dynamical behavior (for a review see[2]). Much attention has been focused on the collapsesand revivals of the Rabi oscillations which provide evi-dence for the quantum nature of the radiation field [3].Successful realization of the JCM by using highly excitedRydberg atoms enclosed in high-Q superconducting cavi-ties has been reported [4] and the collapse and revivalphenomenon has been experimentally tested.

Recently, Gea-Banacloche has studied the evolution ofthe atomic and field states in the JCM [5]. He showedthat an arbitrary initial pure atomic state evolves into aunique pure state in the middle of the collapse region,provided the field is initially in a coherent state with largeintensity. Moreover, at the half-revival time, the cavityfield represents a coherent superposition of the two ma-croscopically distinct states with opposite phases. Thefact that the atom and field in the JCM most closely re-turn to pure states during the collapse region can betraced out from the behavior of the two quantities: thetrace of the square of the density operator [5] and the en-tropy [6]. Using the entropy concepts, Orszag, Retamal,and Saavedra have pointed out that the pure atomic statecan be generated even from the initially mixed ones [7].

The modification of the JCM in which the atomsmakes two-photon transitions has also attracted consider-able interest due to recent development of the two-photon micromasers [8]. Alsing and Zubairy [9] and Puriand Bullough [10] have shown that the revivals of theRabi oscillations in this model are both compact and reg-ular, in contrast to the one-photon case. The eftects ofthe field statistics [11]and cavity damping [12] have beenexplored. In [13],Sherman and Kurizki have proposed ascheme for preparation and subsequent detection of mac-roscopic quantum superposition states based on the two-photon JCM. Phoenix and Knight have calculated the

entropy in this model and pointed out the periodic recov-ering of the initial atom-field state occurring under aproper choice of the detuning parameter [14].

In the present paper, we investigate the state evolutionof the atom and the field in the two-photon JCM, usingthe effective-Hamiltonian approach. We consider bothcases when the dynamic Stark shifts are not included andwhen they are, and compare the obtained results. Vari-ous initial field states, namely, the coherent, squeezed-vacuum, and chaotic ones are treated. We show that,when the intensity-dependent energy shifts of the two lev-els are equal (the atom is prepared initially in a pure stateand the field in a highly-excited coherent state), the atomand the field most closely return to pure states twice be-fore the revival times and right at the revival times. Theatomic and field states at these times are found. Theeffects of the cavity damping are discussed within thedressed-state approximation. For initial squeezed-vacuum and chaotic states, numerical calculations areperformed, revealing features that are absent in the stan-dard JCM. In the Appendix, the atomic and field entro-pies are calculated with due account taken for the dy-namic Stark shifts.

II. EVOLUTION OF THE FIELDAND ATOMIC STATES

The two-photon JCM under consideration is obtainedwhen a cascade of the atomic transitions

~e & ~ ~i & ~ ~g &

is resonant with twice the field frequency co,g=2',

whereas the intermediate transition frequenciesco„=co+6 and co, =co—4 are strongly detuned from co.

After adiabatically eliminating the intermediate state, onearrives at the effective-interaction-picture Hamiltonian[8,10], in the rotating-wave approximation.

H=&g(a'le & &g I+a "Ig & & e I)

+Pia a [g &(g)+Pz(ata+1)(e &(e[,

where the Stark-shift parameters p, and p2 of the two lev-els and the effective two-photon coupling g are defined in

1050-2947/94/49(1)/473(8)/$06. 00 49 473 1994 The American Physical Society

474 HO TRUNG DUNG AND NGUYEN DINH HUYEN 49

terms of the coupling constants g, (for lg & ~ li & ), g2 (forli & ~ le & ), and b as follows:

2 2g& g2 Ni2

(2)

Notice that in Eq. (1) the spontaneous contribution tothe energy-level shifts has been included [8,10]. Anotherform of the effective two-photon Hamiltonian, wherep2a a le & & e

lstands for pz(a a + 1)le & & e l, is also of fre-

quent use. In the high-field limit we are interested in, thetwo forms obviously lead to identical results.

As has been pointed out by Toor and Zubairy [15], theeffective Hamiltonian (1) is valid for strong fields and fortimes and detunings such that

effective Hamiltonian approach have also been discussedin detail in [16]and [17]. In the case of r = 1, it is knownthat the Stark shifts give rise to an additional overallphase factor, which can only play a role in the off-diagonal elements of the density matrix. Keeping thispoint in mind, we put aside for a while the last two termsin Eq. (1) and work with the effective Hamiltonian

H=irig(a'le&&gl+a 'lg&&el) . (4)

The case with the complete Hamiltonian (1) will be ana-lyzed in the next section.

Using the Hamiltonian (4) to solve the correspondingequations of motion with the initial atomic condition

ll(„(0) & =ale &+Pig &,2

n r+1gt ((—, (3)

and initial field condition

where n is the mean number of photons and r =g, /g2.The conditions and limitations for the validity of the

I

one can easily find the state vector of the system at time t,

lp(t) &= g [[ Ca„cos(gv'n+2t) i pC„—+&sin(g vn +2t)]le &+[ iaC„—&sin(g vn t)+pC„cos(g&n t)]lg &] ln &, (5)n=0

where C„=O for n &0. The exact solution (5) representsa strongly entangled atom-field state. Following [5], weintroduce the semiclassical Hamiltonian corresponding toEq. (4),

H =fig ( u2

Ie & & g I

+ v*

g & & eI ) (6)

obtained by replacing the annihilation operator a by acomplex number v = lv lexp(iy) The e.igenstates of (6)are

[ l e &+exp( 2i q ) l g—& ] .1

2(7)

If the atom enters the cavity in either of the states (7) andthe field is treated classically, the physical observables donot evolve. In a fully quantized theory, though the quan-

I

turn nature of the electromagnetic field implies that thesystem would evolve dynamically, one can expect that thestates (7) still exhibit features distinguishable from theothers. Indeed, let the field be initially in the coherentstate

1/200 —n

le(0)& =exp2 „o n! exp(in'&)ln &

which is the closest quantum counterpart of the stablemonochromatic excitation in the semiclassical theory.For n ))1, by employing the solution (5) and the relationC„2 C„exp-—( 2iy) h—olding for n in the neighborhoodof the mean n, one finds for the semiclassical state ap-proximately

1oo —n

lit,+, &lu &l, o~ —[exp( i2gt)le &—+exp( 2iy)lg —&]expV2 2 „o n!

1/2

exp(in')exp[ igv'n(n —l)t]—ln &,

(9a)

1oo —n

lg,, & lv & l, o~ —[exp(i2gt)le &—exp( 2iy)lg &]—expt=0

2 „o n!

1/2

exp(in y )exp[ig v'n (n —1)t] ln & . (9b)

Equations (9) show that starting from the initial condi-tions (7), the state of the atom-field system can be roughlydecoupled into atomic and field parts; each of themevolves, remaining in a pure state. Even more interestingis that the atomic states appearing in Eqs. (9a) and (9b)exactly coincide at times

and

TRt, =(4k+1)

TRt2 =(4k+3)

(10a)

(lob)

49 STATE EVOLUTION IN THE TWO-PHOTON ATOM-FIELD. . . 475

and

2[i Ie &

—exp( —2iqr)Ig &]

1

2—[i Ie )+exp( 2i—p)Ig )],

(1 la)

(1 lb)

respectively.Since the states (7) are orthonormal and can serve as a

basis, it follows that any pure initial atomic state will

converge into the states (11) at times (10). In otherwords, we observe the so-called crossings of the atomic"trajectories" in the Hilbert space of the atomic states[5]. In contrast to the one-photon transition case, wherethe crossings are reached at precisely half the time of thepeak revivals, in the system at hand they take place twicein each time interval between a collapse and a subsequentrevival, at one-quarter and three-quarters of the revivaltime. If the initial atomic state is a linear superpositionof If,*,), say, the excited state

(12)

then the cavity field at times (10) is a coherent superposi-tion of the macroscopically distinct states

1/200 —n

I4+(t) ) =expo

exp(in qr )

X exp[ Fig +n (n —1)t]In )

(13)

(a "Schrodinger cat"). The field states given in Eq. (13)are difFerent from their one-photon counterparts by thephase factors g&n (n —1), which are nothing else but thefrequencies of the two-photon Rabi notations betweenIe ) and I g ) . Recall that in the one-photon JCM, the

Rabi frequencies are of the form A, &n, with A, being theone-photon coupling constant. By expandingg&n (n —1) in powers of n

T

g&n (n —1)=gn 1—1

2n

1 + e ~ ~

8n(14)

and retaining only terms of the order n we obtain, insteadof

I 4+(t) ), two coherent states]/2

exp —— g exp[in(y+gt)]In ),2 „o n! (15)

which exactly coincide with each other at every time ofrevival of the Rabi oscillations

t3=kT~ (k=1,2, . . . ) . (16}

They are identical to the initial state for even k and un-dergo a global phase change of m from the initial state forodd k, whereas the atomic state is the same as that att =0 for all k.

where k is an integer and Tz is the period of revivals ofthe Rabi oscillations in the two-photon JCM: TJt =(n Ig)[9,10], and are equal to

As a result, there exist three series of the re-creationsof the field and atomic state vectors in the two-photonJCM, provided that the cavity field is initially in acoherent state with large enough intensity [under "series"we mean just the first re-creations for which the condi-tion (3) is fulfilled]. This is clearly visible in Fig. 1(a),where we have plotted the quantity Tr(p„}as a functionof the dimensionless time gt for the atom being initially inthe upper state and the field in a coherent state withn =50. It is not difficult to check that if both the atomicand field subsystems are prepared at t=0 in pure states

Tr(pf ) =Tr(p„).The results (9)—(12) can be generalized to the field

states having a sufficiently well-defined phase; for exam-ple, the squeezed states [18]with a dominant contributionfrom the coherent excitation, in the way similar to thatemployed by Gea-Banacloche for the one-photon transi-tion case [19]. Unfortunately, this asymptotic operatorsolution approach does not apply for such field states asthe squeezed-vacuum state [18]

(where r is the squeezing parameter and n =sinh r),which has a double-peaked phase distribution [20], or thechaotic state

00 —n

pf(0)= g " „,I )(„=0 (n+1)"+' (18)

whose phase is randomly distributed. For these, we haveevaluated the quantity Tr(p„) using a computer. The re-sults are presented in Figs. 1(b) and 1(c} for the initialconditions (17) and (18), respectively, for n =50 and theatom being initially inverted. Figure 1(b) shows thatwhen the cavity initially contains a squeezed-vacuumstate, besides the re-creations of the state vectors occur-ring at ti, t2, and t3, there appears one more series of there-creations at times k(T& j2) (k =1,2, . . . ). As for theinitial chaotic state [Fig. 1(c)], we still observe one seriesof the re-creations at times t3 =kTz. This is a quite non-trivial result: Despite the fact that in our system theatom, which is a two-state system, is coupled to the elec-tromagnetic field, which is a system with an infinite num-ber of degrees of freedom and initially prepared in a com-pletely mixed state, the initial atomic purity is not ab-sorbed forever by the field but periodically returns to theatom. We have also calculated Tr(pf ) (not shown in thefigure), which in this case is not equal to Tr(p„) and haveseen that is undergoes oscillations near zero, with ampli-tudes insignificant compared with those of Tr(p„). Therecoverings of the state vectors under the initialsqueezed-vacuum and chaotic states are entirely due tothe two-photon character of the atomic transitions anddo not appear in the one-photon JCM.

I.et us proceed to consider another important aspect ofthe problem, namely, the effect of the photon leakagefrom the cavity on the state evolution in the two-photonJCM. It is natural to expect that the purity of the atomicand field states should be very sensitive to the factors ofthis kind. With the cavity damping taken into account, if

476 G AND NGUYEN DINH HUYENHO TRUNG DUN 49

] .00

0.5o—

s values of thef ti me for vari«T ( ) as a function o0 000] and (b)ity relaxation p

' 'll repared in the e

ca. Theatomislnltla yp

0g '

nt state with n =and the fi ld in the coherent s

0.500

1.00

0.50.) [& (

84%ure state is a ofirst time ah o- hootr/ =10 . n

l. 8, h h

dis mediate oI po -p

ld "'"' '"d t39P&&2 wi

r' i.e. d three or-=10' and 1,r

ders hig erhat the other factors,

the again acquirehnta may worspurity of the atom and the e

for the atom initia y'' '

ll intheFIG. 1. Time evolu tion of Tr(p„orcoherent state, (b) qs ueezed-eld in the(a co e

n number

excite sd ( ) haotic state.vacuum state, dandcc

n =50.

the density matrixta are ignored, t e e ixt e qis iven ydescri ing'b'

g our system' g'

[12]

]—v(a ap —p

—2a a +pa a), (19)pBt A'

. (4) and 2~ is theis iven in q.eof

(19), we refer t e rethe solution of, warwal. '

) for various. 2th tt T(„b'f

It can be seen that t e e

K DYNAMIC STARK SHIFTSIII. EFFECTS GF THK DY

Hlq.*&=~~;~q;,A.

+ =P, (n+2)+P2 nn

sinO„

+1), A,„=O,cosO„

~n+2, g &,—sinO„n, e &+cosO„

p2(n+1)

P, (n+2)tanO„=

(20)

Hence, we have

our discussion towe restricte ouh 1ff t've Hamiltonianlified e e

T. -" ""':ow the com-h state evolution, b

H miltonian.of th ffct t -pcan be easily diagonahzeThe Hamiltonian (1 can e ea

' 'hze

results [8,10]

exp exp[ —t [p,(n

1) t —1) isn„Ocso„O~n +2gex — P (n+2)+P, (n+1)]t]—1 s» „c+(exp[ i, n—'n 0 +cos 0„)~n,e)n e = +2)+P2(n+1)]t ]sin „„etHt

i

)—'

(21a)

exp expI —i [p, (n

0 +si 0 )~n+2, gn+1) t]cos O„sin+ exp i [P,(n+2)+—P2(n+ O„sin

cos0 ~n, e )+2)+P,(n + 1)]t ]—1)ssnO„c

iHt~

+2 g) =(—fg

(21b)

49 QTON A'rQM-FIELDN IN THE TWO-PHoSTATE EVQLUTI 477

s. They read

Q+A,„+ i u

ei envaluesthe corresponding 'gwhere A,—are e

1 ersion of (1). Ase et the semiclassica v su = v !exp(iy) we geevolution under arge

' '-ion o erator a y

crucial role in eofthi i 1 i abefore, the eigenstates o is'

am

(22

tial fi

A'~,*,=—[p, lv +p, u(lul +1)+Q],SC

2 4Q=V'[P, lul' —P,(lv I+ ]» '+4g lvl (23)

h

lv l exp EQ7

p[ —i2(p, +pi) l &+QQ(A, ,+, —p, lv l )

V —p~ ~

' 'll in the coherentthe fie asld before is initia y ie cavity in states (22), and

1 ebra one a'h h id fEstate (8) wit n'h »1. T en wi

sc V t =p~

X expn

2 n=0 nfexp snap e —'

(n —1)]t$ fn &,('

)exp [ i [pin +—pz n— (24a)

(24b)

d the asymptotic rela-24) we have used t eg q

u'

i the classical limi

fi ld ft rd 1

e atomic states app,24a) andtime, except or

ate that ori-th t thcan easily verify t ae & at times t,evo ves

'gand t2, wit

'h lf & being roug

&—exp( 2i lp ) l g & ],—e —exp— (25)l Psc

account the sharplocation o

n +P2(n —1)] by i 2

24)i to ohfield state in aThis

we transform the'

h the time-depend dent p asee concrete value ofrnd"nd-lyof .--

it'

at timest te coincides, in epenith the initial state awit

—1

1,00

CC5

~0.V5—

I

0 1 2 3

1.00

0.500 1 2 3

1.00

(b)

1t =2k~ r+ —g3 (k=1,2, . . . ) . (26)

als oft the times of revivais nothing else but t e a- hthe Rabi osci

'e w-'.lations in t e w-

. ( )w mgh o i iiill

Psaeations of t e s

es s-t rees

th h hilarl to e~ ~

(1Oa),H il oM (4).lified efFective

(), hIt is clearly seen from a c

0.50 I I

4 5 60 1 2 3gt

resence of Stark'

n of time in the presen. Th o

'1.d,h. a.ld;. . ...h'ft: (a) r =1, (b) r =

in its upper state an t e

478 HO TRUNG DUNG AND NGUYEN DINH HUYEN 49

have used the exact solution (21) to plot Tr(p„) for theatom initially in the excited state and the field in thecoherent state, with Fig. 1(a). However, the two Hamil-tonians (4) and (1) predict different state evolutions: Theattractor state (25) is not equal to the states (11). Thusour results complement the conclusions of Toor and Zu-bairy, stating that the effective Hamiltonian (4) is ade-quate only for describing the quantities in which diagonalelements of the density matrix are involved but is not val-id for the description of such quantities as squeezing.

At t, and t2, when the attractor state (25) is reached,the cavity field is a superposition of the two macroscopi-cally distinct states with the relative phase of m/2. If oneuses the state reduction scheme proposed by Shermanand Kurizki [13], i.e., if one projects the atom-field sys-tern onto one of the atomic energy states, for example,the upper state, one will have a field macroscopic quan-tum superposition state with arbitrary relative phase.

In Figs. 3(b) and 3(c), we have depicted Tr(p„) forr=0. 5 and 0.3, respectively. As is visible from thefigures, the effects of the dynamic Stark shifts are morepronounced when r deviates from unity. On the onehand, the minimal values of Tr(p„) increase, indicatingthat the atomic (and field) state becomes less mixed. Onthe other hand, the convergences of the atomic state intothe unique state is destroyed. These effects resemblethose occurring in the standard JCM when the atom-fielddetuning takes nonzero values [19,21], which is under-standable since the ac Stark shifts can be treated as theintensity-dependent detunings. In contrast to the one-photon JCM, in the two-photon model the atom-field sys-tem always returns to its original state at the revivaltimes, regardless of the chosen value of r.

We have also depicted Tr(p„) for the field being initial-ly in the squeezed-vacuum state (Fig. 4) and chaotic state(Fig. 5), for various values of r By co.mparing Fig. 4(a)with Fig. 1(b) and Fig. 5(a) with Fig. 1(c), we see that,when r =1, the two Hamiltonians (1) and (4) lead to an

1 00

0:r0 i:;~ &)

i 00

0 -&0

() ]

FIG. 5. Same as in Fig. 3 with (a) r =1, {b) r =0.5, but nowthe field is initially in the chaotic state.

identical behavior of Tr(p„), as is expected. For r&1,when the cavity field initially contains a squeezed-vacuum state, only the re-creation of state-vector series atthe revival and half-revival times survive, while when thecavity initially contains a chaotic state, the re-creationsof the atomic state vector are no longer observed.

In the above discussion, we have used the quantityTr(p„) to determine the purity of the atomic state.Another way of looking at this problem is via the entropy[6]. The atomic and field entropies in the two-photonJCM have been calculated and compared with those inthe Raman-coupled model [22] by Phoenix and Knight[14]. However, the authors of [14] have ignored theStark shifts. We present the solution for the case inwhich the dynamic Stark shifts are taken care of in theAppendix.

1.00—

0 1

1.00

0.50—0 1

FIG. 4. Same as in Fig. 3 with (a) r =1, (b) r =0.5, but nowthe field is initially in the squeezed-vacuum state.

IV. CONCLUSIONS

We have considered the state evolution in the two-photon JCM with large fields. We have found that for aninitially coherent state field and equal Stark shifts of thelevels, at —,

' and —,' of the revival time, the atom is con-

verged into a unique pure state, no matter how the initialatomic state is chosen, while the cavity field representscoherent superpositions of the two distinguishable corn-ponents shifted from each other by m/2. Right at the re-vival times, the initial atomic and field states recover.The explicit forms have been obtained for the atomic at-tractor states and the field "Schrodinger cat" states. Tomake the model more realistic, we have included theleakage of photons from the cavity. The sensitivity of thestate purity to the cavity loss has been graphically illus-trated f'or various damping rates. We have also investi-gated the field being initially in the squeezed-vacuum andchaotic states. For both these initial conditions, in con-trast with the one-photon transition situation, one stillobserves the re-creations of the state vectors. The model,in which the atom makes two-photon transitions, is not

49 STATE EVOLUTION IN THE TWO-PHOTON ATOM-FIELD. . . 479

simply a generalization of the one-photon JCM. It pro-vides us with a much richer dynamics.

APPENDIX

By using the relation g =+pip&, one finds the follow-ing identity, after some straightforward algebra,

Pz(a a+1) ga"+'

P2(a a+1)M" ga N"

ga P,a a ga M" P,ataN"

iHtexp

(. )„P2(ata+1)

n=0

ga

p,a'a

In this appendix we derive expressions for the atomicand field entropies in the two-photon JCM in the pres-ence of Stark shifts. In the atomic basis and interactionpicture the time evolution operator can be written as

where

M=Pi(a a+2)+P2(a a+1),N=Pia a+P2(a a —1),

and eventually obtain

(A2)

(A3)

(A4)

iHTexp

P2(ata+1) P +P,(ata +2)

t2 exp(iMt) —1ga

z exp(iNt) —1ga

Pia a +P2(ata —1)—exp(iNt) t 1(AS}

Assume that the atom is initially in the excited stateand the field in the pure state pf(0) = Igf(0) & & ff (0) I.Then, the density operator of the atom-field system attime t is given by

8=sinh —1

2(A12)

p(t) =

where the notation

Cpf (0)C Cpf (0)S

Spf (0)C Spf (0)S

C C C S

s c s s (A6)

The field entropy is found from Eq. (Al 1) to be

Sf = (n+1nsr+—+n 1nm ) . (A13)

The atomic entropy can be determined in a simpler way[6,7,23]. By tracing Eq. (A6) over the field variables, onegets for the reduced atomic density matrix,

T

11 12

(A14)

S= gaexp( iNt ) 1 —t2—

N

Ic&=CIA(0)&, ls&=slqf(0)&

(A8)

(A9)

pf(t)=lc &&el+Is&&sl . (A10)

According to Phoenix and Knight [6], if the density ma-trix is of the form (A10), its eigenvalues are

m+= &c lc &+I &c ls & Iexp(+8),

where

(A 1 1)

has been introduced. Clearly, the field density matrix

pf (t)=Tr«[p( t) ] is equal to

where A,; =Trf [p; (t)] and p; (t) are the matrix elementsof p(t). The eigenvalues of (A14) read

1+V (~i( —~22)'+4l~t2l'

so that the atomic entropy is given by

S« = —(a+ina++a lna ) .

(A15)

(A16)

When both the atom and field are initially in the purestates, it is not diScult to prove that Eq. (A15) is identi-cal to Eq. (All}, i.e., the atomic and field entropies areequal. This fact can also be derived from the Araki-Liebtriangle inequality for the entropies of the two interactingquantum systems [24].

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480 HO TRUNG DUNG AND NGUYEN DINH HUYEN

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