PHYSICAL REVIEW A VOLUME 45, NUMBER 7

Dynamic Stark effect for the Jaynes-Cummings system

1 APRIL 1992

P. Alsing, D.-S. Guo, * and H. J. CarmichaelDepartment ofPhysics, Chemicai Physics Institute, University of Oregon, Eugene, Oregon 97403

and Institute of TheoreticaL Science, University of Oregon, Eugene, Oregon 97403(Received 26 August 1991)

We calculate the quasienergies and steady states of a coupled two-level atom and quantized elec-tromagnetic cavity mode with the cavity mode driven by a periodic classical field. The atom, the cavitymode, and the classical field are all on resonance. The quasienergies give shifted Jaynes-Cummings levelsplittings. These splittings are reduced by the interaction with the driving field and vanish at a thresholdvalue of the driving field strength. Above the threshold, discrete quasienergies and normalizable steadystates do not exist. Below the threshold, for weak driving fields, the steady states are squeezed and dis-placed Jaynes-Cummings eigenstates. We discuss the relevance of these results to work in cavity quan-tum electrodynamics.

PACS number(s): 42.50.Dv, 32.60.+i, 36.90.+f

I. INTRODUCTION

The Jaynes-Cummings Hamiltonian describes the in-teraction of one mode of the electromagnetic field with atwo-state atom in the electric dipole and rotating-waveapproximations. This Hamiltonian is of fundamental im-portance to the field of quantum optics; it is a central in-gredient in the quantized description of any optical sys-tem involving the interaction between light and atoms.

The problem of a two-state atom interacting with anelectric field is mathematically equivalent to the problemof a spin-

, particle in a magnetic field. The early historyof the Jaynes-Cummings model is therefore found inwork on magnetic resonance [1,2]. Jaynes and Cum-mings presented their analysis in 1963 [3]. Since thattime, a large number of theoretical papers have appeareddealing with various aspects of the Jaynes-CummingsHamiltonian and the evolution in time that it generates[4,5]. For example, the Jaynes-Cummings Hamiltonian isused widely in quantized theories of the laser [6,7]; thisalone accounts for a vast literature on the subject. How-ever, in spite of the extensive theoretical attention it hasreceived, the full, quantized Jaynes-Cummings Hamil-tonian has had little relevance to experiments in opticsuntil recently. This is because most interactions betweenlight and atoms involve highly populated modes of theelectromagnetic field where a semiclassical treatment isall that is required. In the semiclassical limit the electricfield enters the Hamiltonian as a c number. The evolu-tion of the atomic state is represented by a precession onthe Bloch sphere at a frequency determined by thestrength of the electric field. The precessionfrequency the Rabi frequency is produced by a split-ting of the atom's energy levels due to its interaction withthe time-periodic field dynamic Stark splitting [8].

To obtain conditions where a semiclassical treatment isinadequate, it is necessary either that the fundamentalcoupling strength (the dipole coupling strength) be largeor that many weakly coupled modes contribute to thephysics in an important way. In both situations, single-

or few-photon excitations, requiring a quantized treat-ment, can produce observable effects. Spontaneous emis-sion is an example of a phenomenon involving manyweakly coupled modes where a quantized treatment iscalled for [9]. Photon antibunching in resonance fluores-cence is another [10,11]. For interactions with one modeof the electromagnetic field, the size of the dipole cou-pling strength is important; rates of coherent evolution(Rabi frequencies) must be comparable to dissipationrates for the single-mode interaction to be significant.Traditionally, coherent effects are observed in the limit ofvery large photon numbers and small dipole couplingconstant. In contrast, work in cavity quantum electro-dynamics attempts to make the dipole coupling constantitself large [12]. When this is achieved, dynamical effectsassociated with the energy spectrum of the full quantizedJaynes-Cummings Hamiltonian are observable [1216].

Usually, the Jaynes-Cummings Hamiltonian will not,on its own, provide a complete description of an experi-ment, no matter how closely it approaches the two-state,one-mode idealization of the Jaynes-Cummings model.The Jaynes-Cummings Hamiltonian defines a "mole-cule, " a composite system formed from the coupling of atwo-state system and a quantized harmonic oscillator.To interrogate the "molecule, "we must probe it in somemanner. The probe is a perturbation; we must analyzethe problem of "molecule" plus probe to fully understandthe experiment. In this paper, we analyze the problem ofthe Jaynes-Cummings "molecule" probed by a classical(external) coherent field. Recent experiments designed toobserve the so-called "vacuum" Rabi splitting are of thistype [17,18]. Although the splitting for a single atomcoupled to an electromagnetic cavity mode has not yetbeen observed, it appears that this observation is not veryfar off. (Note that vacuum Rabi splitting refers to thesplitting of the first excited state of the Jaynes-CummingsHamiltonian. )

The complete Hamiltonian we consider consists of theJaynes-Cummings Hamiltonian for a two-state atom in-teracting on resonance with one mode of an electromag-

45 5135 1992 The American Physical Society

5136 P. ALSING, D.-S. GUO, AND H. J. CARMICHAEL 45

netic cavity the free Hamiltonian Ho plus an interac-tion Hamiltonian Ht(t) describing the coupling betweenthis system and the external field. The interaction Ham-iltonian has an explicit periodic time dependence due tothe oscillation of the external field at the common reso-nance frequency of the atom and cavity mode. We aretherefore interested in steady states (in the sense ofperiodic states) and quasienergies rather than in energyeigenstates and eigenvalues [19,20]. The external fieldcan either couple to the atom, by illumination throughthe open sides of the cavity, or to the cavity mode by il-lumination through one of the mirrors. The most in-teresting results are found in the latter case. For this casewe find that the resonance frequencies of the Jaynes-Cummings "molecule" undergo a dynamic Stark shift; inplace of the Rabi splittings +&n g, we obtain quasiener-gies

+v n g[1(2@/g) ], n =0, 1,2, . . .where 8 is the amplitude of the external field and g is thedipole coupling constant [21]. The associated steadystates are generally quite complicated. For weak excita-tion they are squeezed and displaced Jaynes-Cummingseigenstates. For an external field amplitude larger than6 =g/2, no normalizable steady states exist. (Note that8=g /2 is the threshold condition for spontaneousdressed-state polarization [16].)

We have used the word "molecule" by analogy and itbecomes awkward with repetition. We therefore drop itin the rest of the paper. We will refer to the Jaynes-Cummings system, and the driven Jaynes-Cummings sys-tem, when the external driving field is added.

To our knowledge, the method we use for finding thequasienergies and steady states of the driven Jaynes-Cummings system has never been applied before. Wetherefore introduce the method in stages. In Sec. II wederive the familiar eigenstates the so-called dressedstates [8]and eigenenergies of the Jaynes-CummingsHamiltonian, in the absence of a driving field. In Sec. IIIwe treat the driven Jaynes-Cummings system with theexternal field driving the atom. Here the quasienergiesare not shifted and the steady states are displaced dressedstates. We calculate the quasienergies and steady statesfor the driven Jaynes-Cummings system with the externalfield driving the cavity mode in Sec. IV. In Sec. V wediscuss the relevance of our results to work in cavityquantum electrodynamics. A concluding summary isprovided in Sec. VI.

II. EIGENSTATES AND EIGENENERGIESOF THE JAYNES-CUMMINGS SYSTEM

In this section we solve the eigenvalue problem for thestandard Jaynes-Cummings system. We formulate theproblem in the language used to treat the driven Jaynes-Cummings system in Secs. III and IV. We therefore be-gin with the definition of the driven Jaynes-CummingsHamiltonian:

dard Jaynes-Cummings system,

Ho= H +Hg =%coo(a a+ ,'cr,)+ifig(a cr ao ~),0(2)

o + 1Q)pta

e0 ENpt

ea

where we allow the driving field to couple either to theatom (upper row inside the large parentheses) or to thecavity mode (lower row inside the large parentheses). aand a are creation and annihilation operators for the cav-ity mode, satisfying the commutation relation

[a,at]=1; (4)o.+, o. , and o, are atomic pseudospin operators, satisfy-ing the commutation relations

[cr+,o ]=2o[oo+]=+o+;coo is the frequency of the driving field, in resonance withthe atom and the cavity mode; 8 is the amplitude of thedriving field; and g is the dipole coupling constant. Arbi-trary phases for the driving field and the dipole couplingconstant may be absorbed into the definition of theoperators; thus, there is no loss of generality in using realquantities 8 and g.

We will denote solutions to the Schrodinger equationobtained from Hamiltonian (1) by

~

g'(t) ):[H, +H, (t)] ~ 1(')

=. [H. +H, +H, (t)]iq'),

where the superscript t on g refers to the Schrodingerpicture, the picture in which the Hamiltonian has an ex-plicit time dependence. The Hamiltonian 0 generates

p

rotations that remove this time dependence. We there-fore also define states ~g) in the "interaction picture, "with

where the Schrodinger equation in the "interaction pic-ture" is

=.(H,+H, )~q),dt i% (8)

with

o+ 0H +H@ =iong(a cr ao+)+ifi6'a a

and Ht(t) is the Hamiltonian for the periodic interactionwith the driving field,

H, (t):H~(t)

H =Ho+Ht(t),where Ho is the (on-resonance) Hamiltonian for the stan-

We use quotes to remind us that the "interaction picture"is defined here by separating H and H +H@(t), rather

p

45 DYNAMIC STARK EFFECT FOR THE JAYNES-CUMMINGS. . . 5137

than Ho and Hl(t). Our objective is to solve the eigenval-ue problem

EItP+)+ifigalg ) =0,Elyz & ivigatlitz+ & =o .

(15a)(15b)

lfz&=eand solutions to the Schrodinger equation (1) in the form

lyt (r) ) e i(zlzz)tlat ) (12)The states Ilitz ) are periodic in time with period 2m/coo;we follow Sambe I 19] in referring to these states as steadystates. The quantities E are the quasienergies. Eachquasienergy defines a frequency shift E/A that is added(in the Schrodinger picture) to every harmonic meio,m =0,1,2,..., present in the periodic oscillation of thesteady state If'z).

From the definition of Hg+H@ in Eq. (9), we write theeigenvalue problem (10) in the form

0i AC

a a+E lyz &

(Hg +Hz ) I yz & =E lgz & (10)The eigenstates lgz ) in the "interaction picture" definethe Schrodinger picture states

i (H /A)t (11} atalqz&= E/fi I~ )

This is the decoupled eigenvalue problem, and in thiscase there is no need for further transformations to con-vert it into an harmonic-oscillator problem. The quasien-ergies are determined by the requirement

E/fi=n, n =0, 1,2, . . .

which gives

ED =0and

(18a)

E=+&n gEi = A&ng

(18b)

(n =1,2, . . .). The states If@ ) are proportional to theFock states

Using Eq. (15a) to substitute for It/iz ) in Eq. (15b), we ob-tain

'2

+ isa +0

+E lli;)I@o &=c, lo&

~/i ) =I11(&=c(n &, n =1,2, . . . (19)

iR+ isa t+ 0 I1lz+ ) =(}, (13b)

where we have expanded the eigenstate as

I 0 & = I 0' & I + &+ I g & I &; (14)I+ ) and I ) are the upper and lower states of the two-

state atom, and lgz ) and lgz ) are field states normal-ized so that (/zlzz ) =1. We will use the following gen-eral approach to solve Eqs. (13). We first decouple theequations to obtain an eigenvalue problem for Igz ) orIfz ) alone in which an effective Hamiltonian appearsthat is quadratic in the operators a and a. We then usedisplacement and squeezing transformations (when neces-sary) to convert the effective Hamiltonian into the Hamil-tonian for an harmonic oscillator. From solutions to theharmonic-oscillator problem, the solutions to Eqs. (13)are---constructed by inverting the transformations thathave been used.

Carrying out this program gets progressively morediScult as we move from the standard Jaynes-Cummingssystem to the driven Jaynes-Cummings system with theexternal field driving the atom, and then to the drivenJaynes-Cummings system with the external field drivingthe cavity mode. We begin with the standard Jaynes-Cuminings system where the conversion of Eqs. (13) intoan harmonic-oscillator problem is almost trivial.

For the standard Jaynes-Cummings system 8=0, andEqs. (13}reduce to

and corresponding to the quasienergies (18b},we obtainIg)=(1W'2)( n 1& I+ &+i In & I &),I q, & =(1W'2)( In 1)I+ ) i ln ) I )) (21b)

(n =1,2, ...) where we have chosen the arbitrary phase forthese states by taking c&to be pure imaginary.

Using Eq. (11), the eigenstates (21) in the interactionpicture give steady states

(22a)and

i [n (1/2) ]captIf'&)=e lg~), n =1,2, . . . , g=u, l .(22b)

Here the time dependence is contained in an overallphase factor, and in the full solution to the Schrodinger

where g=u or l, and the constants co and c& will bedetermined by the normalization. The states lgz+) areobtained from Eqs. (15), (18), and (19) in the form

Iq+g&= etc&In1), n =1,2, . . .where e&=+1,1 for g=u, l. From Eqs. (14), (19), and(20), we can now construct the normalized eigenstatesI fz ) . Corresponding to the quasienergy (18a), we obtain

(21a)

5138 P. ALSING, D.-S. GUO, AND H. J. CARMICHAEL 45

equation [Eq. (12)] the phases (n ,' )coot and (E/A)t addto give (E,/A)t = ,' coot,n =0, and(E,/A)t = [(n ,'ko0+ep'ng]t,

n =1,2, , g=u, l (e~=+1) .This rejects the fact that Hand H commute; there-

0fore, the states (21) are simultaneous eigenstates of bothHamiltonians. Of course, the "quasienergy" and"steady-state'* language is not really necessary for treat-ing this case. But it is needed to handle the drivenJaynes-Cummings system. When the driving field is notzero, H does not commute with H +H@ and we do not0find simultaneous eigenstates; the time dependence of thestates I 1ltz ) will then be nontrivial.

IP&=(1/&2)(I@/g;n 1)I+ &+il@/g;n &I &),(29b)

I y, & =(1/&2)( I @/g; n 1 & I+ & il @/g; n & I & ),(n=1,2,...) where Ib/g;n 1) and Ih/g;n ) are dis-placed Fock states:

la;n ) =D(a)ln & . (30)

Ip')= 'I '"'@/ o&l & (31a)and

From Eqs. (11) and (29), the steady states of the drivenJaynes-Cummings system with the external field drivingthe atom are

III. THE DRIVEN JAYNES-CUMMINGSSYSTEM: COUPLING TO THE ATOM

We first consider the driven Jaynes-Cummings systemwith the external field driving the atom. This problem issolved by a minor extension of what we have just seen.The eigenvalue problem in the "interaction picture"takes the form [from the upper row in Eq. (13)]

Elq;)+tA(ga @)lq, ) =O,EI q & t A(ga 4') I y+ &=0 .

(23a)(23b)

(a @/g)(a @/g) If' ) = E/A I ) (24)

The only change, in comparison with Eq. (16), is thathere we have the eigenvalue problem for a displaced har-monic oscillator. We can remove the displacement bymultiplying on the left by D ( 8/g), where

D (a ) =exp(aa a'a ), (25)

We solve Eq. (23a) for IlltE ) and substitute the solutioninto Eq. (23b) to obtain

2

X(le ' ' @/g;n 1)I+ )

+ie(Ie '6'/g;n &I &),n =1,2, . . . , g=u, l (e&=+1) . (31b)

These states illustrate the features of the steady states fora time-periodic Hamiltonian in a nontrivial way. Theycarry the same time-dependent phases as the states (22);but they also carry the periodic time dependence thatenters through the displaced Fock states. The full solu-tions I/0(t)) and IP'&(t)) to the Schrodinger equation[Eq. (12)] do not involve single frequencies ,'coo and(n ,')coo+e&&ng; they involve infinite sets of frequen-cies: (m

,')coo, m=0, 1,2, ..., for the state Igo(t)) and(m 2)t00+e&&ng, m=0, 1,2,..., for the states IP' t(t)),n=l, 2,..., g=u, l (e&=+1). Thus, the quasienergiesED=0 and E&=e&A&ng characterize the steady statesby defining frequency shifts that are applied to the wholeseries of harmonically spaced frequencies (m

,

' )~o,m =0, 1,2, . . . . Each steady state is also distributed in acharacteristic way across the Fock states. This distribu-tion determines the relative strengths of the different fre-quency components in the state.

and using

D (a)aD(a)=a+a .Then,

a alp~ &E/A

l~ )g

where

I g, & =D'(@/g) I y; & .

(26)

(27)

(28)

IV. THE DRIVEN JAYNES-CUMMINGSSYSTEM: COUPLING TO THE CAVITY MODE

We now turn to our main interest calculation of thequasienergies and steady states of the driven Jaynes-Cummings system with the external field driving the cavi-ty mode. This calculation follows the same general stepsas the calculations in Secs. II and III. However, the de-tails are considerably more complicated. In the "interac-tion picture" we have the eigenvalue problem [from thelower row in Eqs. (13)]

ly, & = I@/g;0& I & (29a)and

Equation (27) is solved by Fock states as before. Thequasienergies are given by Eqs. (18), and, after invertingthe displacement, the eigenstates in the "interaction pic-ture" are

[ iA6(at a)+E]IQE ) +i Agalg~ ) =0,[ iAC(at a )+E)IPE ) i Aga I fE ) =0.

(32a)

Our first task is to decouple these equations and obtain anequation for IQE ) (or IQE ) ) alone. To this end, we mul-tiply Eqs. (32a) and (32b) on the left by at and a, respec-tively, which gi~es

45 DYNAMIC STARK EFFECT FOR THE JAYNES-CUMMINGS. . . 5139

[ ifi@(a a)+E]a lPz )+isa alga ) iAlgz ) =0,(33a)

[ iM(a a)+E]alga ) iRgaa lPz & ikelgz &=0 .

0 (E)0 (E)lyz & =0, (40)where O~(E) and 0 (E) commute. The general solutionwill take the form

(33b) Ip;)=c, lp; , )+c.lg; ..), (41)Then, from Eqs. (32) we have

alpz ) = (6/g)(a a)+i E/R l&z (34a)(42a}(42b)

where lPz.~

) and l gz. ) are solutions to the equationso,(E)ly;., ) =0,o (E)lit;. &=0.

a lpz ) = (@/g)(a a)+i+ E/A (34b)

(v jg)(at a)+i E jfig

+ata

Using Eq. (34b) to substitute for a tl Pz ) in Eq. (33a), andEq. (34a) to substitute for al fz ) in Eq. (33b), we obtain

It may happen that there are quasienergies E for whichEq. (42a) has a solution and Eq. (42b} does not, or viceversa. For these quasienergies one of the constants, c orc, will vanish. But, in general, we allow for the possibil-ity that both of Eqs. (42) have a solution for the samevalue of E. When this is the case, the constants c and cwill be determined by the requirement that Eqs. (32) aresatisfied [this is not guaranteed for arbitrary solutions toEqs. (40)] and by the normalization

( 8 lg)(a a)+ i E/Rg

(@jg) I yz+ & =0, (35a)+aat lgz+ )

2

+(@jg)lyz &=0. (35b)

& qz I gz & &&z l&z&+ & qz I gz & 1 .

Before we can determine the constants, we must find thestate l gz ) that corresponds to the state l fz ) given byEqs. (41) and (42). For this purpose we use Eq. (35a) andEqs. (39) to write

We can now use Eq. (35a) to eliminate llitz ) from Eq.(35b). This gives lyz &=(ejg) '[0 ,'[1++1 (2e jg)']]lyz &,

[[0(E)+,'][0(E),']+(8jg) ] lPz ) =0, (36) (43)where

0(E)= (Bjg)(a a )+i Ejlg

a a+aa (37)2

where we indicate two alternative forms for the operatoron the right-hand side; the + sign goes with the subscriptp, and the sign goes with the subscript m. From Eqs.(42) and (43) we have

+ +,'+1(26/ ) (39a)

Equation (36) is the desired equation for lpga ) alone.But it is not related in an obvious way to the eigenvalueproblem for an harmonic oscillator. In particular, Eq.(36) is quartic, rather than quadratic in the creation andannihilation operators. We must therefore take an addi-tional step before we can proceed as we did in the previ-ous calculations. We observe that the operator on theleft-hand side of Eq. (36) factorizes in the form

j [0(E)+,'][0(E),']+(8jg) ] =0 (E)0 (E), (38)where

2

0 (E)= (8/g)(a a)+iEjlP

lgz ) = (g/26)[c [1++1(2C/g)2]lgz. )+c.[1&1(2@jg)']]lq;..& .

(44)There are two steps left in our calculation: We must

solve Eqs. (42a) and (42b} to determine the allowedquasienergies E and the states

l gz ) and lPz ), and wemust substitute the solutions for the states into Eqs. (32)and apply the normalization condition to determine theconstants c and c . We accomplish the first task by us-ing displacement and squeezing transformations to con-vert Eqs. (42a) and (42b) into eigenvalue equations for anharmonic oscillator. We multiply Eqs. (42a) and (42b) onthe left by S (ri)D (a), where D(a) is defined in Eq. (25)and

0 (E)= (C jg)(a a)+iE/fi+ '+1(2A'/g)

2 2 (39b}

Now we are looking for solutions to the factorized equa-tion

S(ri) =exp[ ,'(riat ri'a )];we transform the operators 0 (E}and 0 (E) usingS (ri)D (a)aD(a}S(7})

=(a +a)coshri+(at+a*)sinhri .Then, if we choose

(45)

(46)

5140 P. ALSING, D.-S. GUO, AND H. J. CARMICHAEL 45

a=P(E): i E/fi 2@/g1 (28/g)

ata Iy ) [I (2g/ )2) 3/2

ri=r, e "=+I(2@/g)Eqs. (42) are replaced by the equations

2

(47a)

(47b)

1 ly, , ),

[ itic'(a a)+E]IPE ) ih'ga lg~ ) = Io) . (52b)[Note that the vacuum state IO) appears on the right-hand side of Eq. (52b), where 0 appears in Eq. (32b).] Itis for this reason that the arbitrary constants c and cappear in Eqs. (41) and (44). The ratio of these constantsis determined by requiring that Eq. (32b) is satisfied. Theseparate constants are then fixed by the normalization

(48a)ata IPF. )=.[1(2@/g) ] ' ' IQE ),. (48b)where

(q, lq, ) =(q,+l1i,+ &+(@;I@;& =I .The calculation is straightforward, but tedious, and wetherefore just quote the results. Corresponding to thequasienergy (soa), we obtain

ly, ., &=S'(r)D (P(E)) P . ),Iy, ..&=S (r)D (P(E))lg . ) .

(49a) Ix, &= Ir, o;o& IM &, (53a)

and corresponding to the quasienergies (sob), we obtain

Equations (48) are satisfied by Fock states when the con-stants on the right-hand sides are the non-negative in-tergers. Thus, the quasienergies are

IX...&

=(I/&2) [lr,p(E...);n I& IP &

+ilr, P(E);n& IM & ],Eo =0, (soa)

ly, ( & =(1/&2)[lr, p(E,();n 1& IP &(53b)

which is permitted by Eq. (48b) but not by Eq. (48a), andE=+R&ng[1(2@/g) ] ~EI= fi&ng[1 (2C/g) ] ~

(50b)(n=1,2, ...) where

ilr, P(E&);n ) IM ) ],(n=1,2,...) which are permitted by both Eqs. (48a) and(48b). The corresponding states are

I&F , &:If&~&.=ln 1&, n =1,2, . . . , g=u, l(Sla)

and

IP) =(I/&2)[[1++1(2@/g) ]'~ I+ )[1&I (26/g)']' 'I ) j,

IM ) =( I/&2) [ [1++I (28/g) ]' I )(S4a)

Ix.,.&=Io&)=' 0

IX.. .

&= In &, n =1,2, . . . , g=u, l .

(5 lb)

[1+I(2@/g)']' 'I+ ) j;(54b)

the states I r, p(E&);n 1 ) and I r, p(E&);n ), g =u, I,are squeezed and displaced Fock states:

(52a)

Equations (41), (44), (49), and (51) define states lyo )and ly&), n=1,2,..., g=u, l, that satisfy Eqs. (33). Weseek solutions to Eqs. (32), and while these must satisfyEqs. (33), the converse is not true. This follows becausewe multiplied Eq. (32b) by a to obtain Eq. (33b); there-fore, in addition to the states that satisfy Eqs. (32), thesolutions to Eqs. (33) also include states that satisfy[ i'd@(a a )+E]I fE ) +i figa I fE ) =0,

Ig, a;n ) =D(a)S(g)ln ) . (55)From Eqs. (11), (53), and (54), the steady states of thedriven Jaynes-Cummings system with the external fielddriving the cavity mode are

lyo) =e ' Ie 'r, o;0) IM, )and

where+exile r, e P(E&);n ) IM, ) ], n =1,2, . . . , (=u, l (e&= 1) (56b)

IP, &=(IW'2)[[1++1(26/g) ]'~ I+) e '[1+I (2e/g) )' I &j, (57a)

45 DYNAMIC STARK EFFECT FOR THE JAYNES-CUMMINGS. . . 5141

~M, ) =(1/&2)[[1++1(2C/g) ]'~

) e [1+1(2A/g) ]' ~+ )], (57b)

V. DISCUSSION

The quasienergies (50) and steady states (56) are thecentral results of this paper. These results have variousapplications to problems in cavity quantum electro-dynamics which we will explore in future work. In thissection we point out some of the more obvious connec-tions with cavity quantum electrodynamics.

The quasienergies (50}define shifted Jaynes-Cummingslevel splittings. The shifts are relevant to proposed spec-troscopic measurements on the Jaynes-Cummings sys-tem. For example, there is interest in making directfrequency-space measurements of the Jaynes-Cummingsspectrum, with particular emphasis on the splitting of thefirst excited state the so-called "vacuum" Rabi split-ting. Comparing Eqs. (18b) and (50b), we see that the sizeof this splitting depends on the way it is observed; prob-ing the atom or the cavity mode gives different results.There are no observations yet of vacuum" Rabi splittingfor a single atom. But there are two observations inmany-atom systems [17,18]. Both of these experimentscoherently excite the cavity mode. In a single-atom ex-periment the modulation technique used by Raizen et al.[17] would observe the splitting 2g [1(2g/g)2] ~,where 8 is the amplitude of the carrier field. For a directtransmission measurement like the one performed by Zhuet al. [18] it is not possible to make a quantitative predic-tion from the present results because this measurementinvolves detunings that are not included in our calcula-tion. We can say, however, that the quasienergies shift inresponse to the driving field and that this shift willchange as the frequency of the driving field is swept. Ofcourse, in either of these methods the frequency shifts in-duced by the driving Geld can be made small by reducingthe driving field amplitude; indeed, if 2C/g is not verysmall ( &0.1), we observe numerically that spectra mea-sured by coherent excitation include contributions frommultiphoton resonances involving states above the firstexcited state.

It is interesting to compare our result for the shift ofthe single-atom "vacuum" Rabi peaks with what wewould expect for many-atom "vacuum" Rabi splitting.There has been some discussion of the fact that, for weakexcitation, the spectroscopic features for the single-atomsystem and the many-atom system are the same and canbe understood in terms of a classical coupled harmonic-oscillator model (linear dispersion theory) [17,18,22].This equivalence does not extend to the frequency shiftsinduced by the driving field. For definiteness we comparesingle-atom and many-atom systems with a resonantcoherent field driving the cavity mode; to probe the fre-quency structure, this field will carry a small modulationin the manner of the experiment of Raizen et al. [17].We identify the quasienergies (single atom} and eigenval-ues of linearized Bloch equations (many atoms) thatdefine the "vacuum" Rabi peaks in the limit of weakdriving fields and ask how these quantities change as a

function of driving field strength. Of course, if the driv-ing field is too strong, major differences between the spec-tra will arise because of excitations beyond the first excit-ed state. For a meaningful comparison, we therefore con-sider only weak-field perturbations of the "vacuum" Rabipeaks (2C/g 1).

For the single-atom system, the vacuum" Rabi split-ting changes according to the expression

(E) E,(}lfi=2g [1(2@/g) ]=2g[13(6'/g) ] . (58)

For the many-atom system, we obtain the "vacuum"Rabi peaks from eigenvalues of the linearized opticalBloch equations. In the limit of weak driving fields, theseequations take the form of coupled oscillator equations:

a= ca+&NgP,P= (y /2)13 2&N gmssa,

(59a)(59b)

where a and P are the amplitudes of the field and polar-ization oscillators, v and y/2 are the half-widths of thecavity and atomic resonances, N is the number of atoms,and mss is the steady-state inversion in the presence ofthe coherent field driving the cavity. Normally, for weakdriving fields we set m ss = ,',' then, when&Ng a, y/2, the eigenvalues of the 2X2 matrixdefined by the right-hand sides of Eqs. (59) give themany-atom "vacuum" Rabi splitting 2&Ng. The first-order correction to this result is obtained using

mss= ,'[12N '(@/v'Ng) ]; (60)this is the saturated inversion calculated taking the intra-cavity absorption into account (8 describes the externalfield, not the field inside the cavity). The many-atom"vacuum" Rabi splitting is then

2&Ng+2~mss~

=2v'Ng [1 N'(v /v'Ng)~] . (61)Here the shift induced by the driving field is negligible inthe large-Nlimit. The N dependence is consistent withthe fact that the many-atom system behaves as a pair ofcoupled harmonic oscillators up to corrections of orderN

Another connection between our results and previouswork in cavity quantum electrodynamics comes from thesqueezing involved in constructing the steady states (56).Carmichael showed that the field transmitted by acoherently driven atom containing a single atom issqueezed [23). This squeezing is related to the squeezingin absorptive optical bistability [24] which has been ob-served in a many-atom system [25]. Rice and Carmichaelshowed that the presence of squeezing induces a narrow-ing of the "vacuum" Rabi peaks in incoherent spectra,replacing Lorentzians by squared Lorentzians [26]. The

5142 P. ALSING, D.-S. GUO, AND H. J. CARMICHAEL 45

steady states (56) provide a new view of squeezing-relatedeffects in the driven Jaynes-Cummings system. For ex-ample, the "ground state" of the driven Jaynes-Cummings system (the steady state corresponding to thequasienergy Eo =0) is the product of a squeezed state forthe field and the state

~M, ) for the atom. With spontane-ous emission and cavity loss included, we can show that,for weak driving fields, the driven Jaynes-Cummings sys-tem settles (to lowest order) in this "ground state. " Thus,our analysis of the Hamiltonian (1) identifies the basicorigin of the squeezing, and the steady states (56) providea natural basis for calculating effects such as squeezing-induced linewidth narrowing [27], which is difficult tocalculate using bare-energy eigenstates (eigenstates ofH ) or standard dressed states (eigenstates of0Ho =H+Hs ).

Our results are also related to the phenomenon ofspontaneous dressed-state polarization [16]. In thisphenomenon the asymptotic quantum state of a drivencavity mode interacting with a two-state atom undergoesa novel symmetry-breaking transition as the strength ofthe driving field is increased. Above the transitionthreshold, the intracavity field shows a bimodality inphase; the two-phase states result from a spontaneous po-larization of the atom in one or the other of the two semi-classical dressed states produced by the intracavity field.To see the connection between this phenomenon and ourpresent work, we observe that the quasienergies (50) andsteady states (56) are only valid for 2C/g (1. For largervalues of 28/g, discrete quasienergies and normalizablesteady states do not exist, although a continuum of statessatisfying a 5-function normalization will exist [28]. Theboundary 28/g= 1 is the threshold found in our work onspontaneous dressed-state polarization [16]. Moreover,the atomic states

~M, ) and ~P, ) given by Eqs. (57) areprecisely the steady states found, below threshold, fromour semiclassical analysis of this phenomenon [Eqs. (24b)and (24c) in Ref. [16]]. In fact, when continued for26 /g & 1 ~M, ) and

~P, ) also reproduce the steady states

given by our semiclassical analysis above threshold[Eqs.(25b) and (25c) in Ref. [16]]. Thus, the basic physics un-derlying spontaneous dressed-state polarization is con-tained in Hamiltonian (1) and can be understood withoutconsidering the dissipative terms that are present in a fulltreatment of this phenomenon.

Finally, we should say something about detunings.There are two detunings that might be added to thedriven Jaynes-Cummings Hamiltonian: a detuning A~between the external field and the atom, and a detuning4C between the external field and the cavity mode. It isstraightforward to generalize our results to include a de-tuning b, .This changes E to E+A'b, in Eq. (13a) andto E fih z in Eq. (13b). After these changes, themethods used to solve Eqs. (13) carry through with minormodifications. As an example of the results, the quasien-ergies

E&=e&fi&ng[1(26/g) ], g'=u, l (p&=+I)are replaced by

Eg=EgN 1 (2@/g)2X [(b,/2)2+ng2V 1 (2g/g)~]'~2 (62)

A detuning between the driving field and the cavity modeis more diScult to handle. A detuning A~ changes E toE+A'b caa in Eqs. (13). The term fihca a does notcommute with the other field operators in Eqs. (13) and,as a result, the methods we used to solve Eqs. (13) forhe =0 cannot be used when b, c%0. One property of thesolution in this case does seem fairly clear, however. Weexpect that a detuning b, c%0 will remove the singularbehavior at 2C/g= 1 and allow discrete quasienergies forall driving field strengths. This follows because the bosonterm ifr8(a a)+irihca a that dominates the Hamil-tonian in the strong-driving-field limit and does not havea discrete spectrum for b,z =0 does have a discrete spec-trum for b, CIAO.

VI. SUMMARY

We have calculated the quasienergies and steady statesof the driven Jaynes-Cummings system a single atomcoupled to a single electromagnetic cavity mode with ei-ther the atom or the cavity mode driven by a coherentexternal field. When the atom is driven by the externalfield, we find that the quasienergies give the usualdressed-state level splittings +fiV n g, n =0,1,2, , where...gis the dipole coupling constant and the steady states aredisplaced dressed states. When the cavity mode is drivenby the external field, we find that the quasienergies giveshifted level splittings

+&ng[1 (2A'/g) ), n =0, 1, , . . . ,where 6' is the amplitude of the driving field. In this casethe steady states are formed from superpositions of atom-ic states multiplied by squeezed and displaced Fockstates; for weak driving fields, they are just squeezed anddisplaced dressed states. Above the threshold 28/g= 1,discrete quasienergies and normalizable steady states donot exist.

We have given explicit results with the atom, the cavi-ty mode, and the driving field all on resonance. The alge-braic method used to obtain these results is easily gen-eralized to include a detuning between the driving fieldand the atom. We have not found an analytically tract-able solution when there is a detuning between the driv-ing field and the cavity mode.

Our results are relevant to work in cavity quantumelectrodynamics. The quasienergies have obviousrelevance to spectroscopic measurements on the Jaynes-Cummings system [17,18] and the steady states provide auseful basis for analytical calculations of spectra. A par-ticular attraction of the steady states as a basis is that

45 DYNAMIC STARK EwvaCT FOR THE JAYNES-CUMMINGS. . . 5143

they automatically incorporate the squeezing e6'ectsfound in earlier work on the driven Jaynes-Cummingssystem [23,26]. Our results also provide insight into thethreshold behavior found in recent numerical calcula-tions for the driven Jaynes-Cummings system withdissipation spontaneous dressed-state polarization [16].There appears to be ample scope for the application ofour results.

ACKNOWLEDGMENTS

This work was supported by the National ScienceFoundation under Grant No. PHY-9096137. We aregrateful to Dr. Marcus Lindberg for sending us hisderivation of the quasienergies (50b) and to Dr. Craig Sa-vage for discussions on the subject of frequency shifts inthe driven Jaynes-Cummings system.

'Present address: Department of Physics, University ofWindsor, Windsor, Ontario, Canada N9B 3P4.

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