Wave-packet Rabi oscillations from phase-space flow in mesoscopic cavity QED

Omri GatRacah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel

�Received 20 February 2008; published 22 May 2008�

The dynamics of the Jaynes-Cummings Hamiltonian is studied in the semiclassical limit by reducing it toquantum normal form and deriving the phase-space dynamics of classically smooth observables. The resultingsystematic semiclassical approximation is used to analyze the Rabi oscillations of a photon wave packetrevealing subtle consequences of the uncertainty principle.

DOI: 10.1103/PhysRevA.77.050102 PACS number�s�: 03.65.Sq, 42.50.Pq

I. INTRODUCTION

The Jaynes-Cummings �JC� Hamiltonian of a two-statesystem coupled to a harmonic oscillator is a basic model ofquantum optics and the theory of atom-light interactions.This simple Hamiltonian describes quite accurately the quan-tum electrodynamics of strongly coupled Rydberg atoms inultrahigh-Q microwave cavities �1�. Because of the universalnature of the JC Hamiltonian, it models many other quantumsystems with interactions between discrete and continuousdegrees of freedom, such as hyperfine levels in trapped ions�2� and spin-orbit interactions in semiconductors �3�. It isalso a building block for more complicated systems such asmicromasers �4� and electromagnetically induced transpar-ency cavities �5�.

In the rotating-wave approximation �RWA�, where non-resonant terms in the Hamiltonian are neglected, the JCmodel becomes integrable, with a conserved quantity in ad-dition to energy—the polariton number that gives the totalnumber of excitations, field and atom. Polariton number con-servation makes it easy to diagonalize the Hamiltonian, sincethe polariton eigenspaces are only two-dimensional, andmakes the dynamics of states with a small photon numbersimple. Nevertheless, the dynamics of states that have manysignificant energy-state components, including the importantexample of phase-space localized wave packets, can showconsiderable complexity �6�, including squeezing �7�, wave-packet Rabi oscillation, collapse-revival �8�, and entangle-ment �9�.

Recently, the interest in wave-packet dynamics in the JCsystem has increased, as it has become a subject of directexperimental investigation in cavities excited to mesoscopiccoherent states with a moderate number of photons �10�. Theexperimental systems are of course also subject to dissipa-tion and decoherence, but the coherence lifetimes that havebeen achieved ensure that the actual dynamics can be ap-proximated by that of an ideal system during the initial timeinterval of wave-packet evolution, where the complex dy-namical phenomena occur.

Complex wave-packet dynamics can be exhibited by theintegrable JC system because its energy spectrum is a non-linear function of the number of polaritons. Many of thecomplex dynamical phenomena occur also in the simplercase of nonlinear oscillators �11,12�, where they can be ana-lyzed using semiclassical nonlinear dynamics. Here we showthat an analogous semiclassical analysis is applicable also tothe JC dynamics.

As always, the semiclassical approximation is valid whenthe typical action scales are significantly larger than Planck’sconstant. In the JC system, this condition holds either whenthe intrinsic action scale BR �defined below�, which is a mea-sure of the detuning, is much larger than �, or when theinitial state is sufficiently excited energetically. Then the fielddegree of freedom becomes semiclassical but, unlike nonlin-ear oscillators, the JC system also has the atomic polarizationinternal degree of freedom that remains fully quantum-mechanical. This contrast is reflected by the much faster dy-namics of the polarization versus the field: The ratio of thenatural frequencies of the polarization and the field dynamicsdiverges in the semiclassical limit. Our semiclassical analy-sis, which retains an exact quantum-mechanical treatment ofthe internal degree of freedom, is in this way different fromthat of cavity dynamics of an ensemble of atoms �13�.

Wave-packet propagation studies �8,9� have provided asemiclassical analysis of the JC system maintaining a quan-tum internal degree of freedom, but their utilization of theclassical dynamics is limited because Schrödinger picturewave-packet propagation does not have a direct classicallimit. Here, in contrast, the dynamics is obtained by evolvingclassically smooth observables in the Heisenberg picture.The dynamics of the observables is cast in terms of theirsemiclassical phase-space quasiflow �11,14�, defined as theirtime-dependent Weyl representation. As a consequence ofthe presence of internal degrees of freedom, the quasiflow ofthe JC system is matrix-valued, giving the components of anoperator in polarization subspace at each phase-space point.

The focus of this work is the Rabi oscillations of wavepackets. The semiclassical quasiflow leads to the explicit cal-culation of the Rabi dynamics beyond the leading-order clas-sical substitution, demonstrating subtle quantum effects inthe interaction between the slow and fast degrees of freedom:The entanglement of eigenstates results in the polarizationoscillations shown in Eq. �6� and Fig. 1, while the quantumfield uncertainty causes the field Rabi oscillations shown inEq. �7� and Fig. 2.

II. THE POLARITON QUASIFLOW

The JC model describes a single electromagnetic mode offrequency �, modeled by a harmonic oscillator with a low-ering operator a, interacting with a two-state system of en-ergy gap �� with a lowering operator �. Under the RWA,

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where the nonresonant terms a� and a†�† are neglected, theJC Hamiltonian is

H = ��a†a + aa†

2+ ��

�†� − ��†

2+ �g�a†� + a�†� . �1�

When the two-level system models an atom, g is an effectivedipole interaction. The commuting a and � operators obeyBose and Fermi algebra rules, respectively, i.e., �a ,a†�=1,�2=0, �� ,�†�+=1.

The RWA JC Hamiltonian is integrable because it com-mutes with the polariton number operator npol=a†a−�†�. Inother words, the system excitations are polaritons �15� withan internal two-state polarization degree of freedom. For thepurpose of studying dynamics, therefore, we define the�single-mode� bosonic polariton annihilation operator b

=����†�+��†�a†aaa† �a, and the fermionic polarization lower-

ing operator �=�a† 1�aa† that commutes with it. That is, b† and

b create and annihilate �respectively� polariton quanta, and �†

and � raise and lower �respectively� the polarization. We de-fine also standard SU�2� generators for the internal degree offreedom of the polaritons, �1=�+�†, �2= i��−�†�, �3=�†�−��†.

In terms of b and �, the Hamiltonian takes the form H

=�B̂+��3+��B̂�1, where �= �2 ��−��, �=��g, B̂=bb†

=��npol+1�, and �B̂ is the unique positive operator whose

square is B̂. The advantage of the polariton representation isthat the Hamiltonian depends on the polariton field only

through the combination B̂; that is, the Hamiltonian has been

reduced to its quantum normal form �16,18�. At this stage, B̂

can be handled as if it were a c number, letting the Hamil-tonian be further reduced by invariant polarization space de-

composition into H= P+�B̂��+�B̂�+ P−�B̂��−�B̂�, introducing

the definitions ��B̂�=�B̂��B̂�, ��B̂�=��B̂+BR, where

BR= �� /��2 is the natural action scale of the system. ��B̂�are scalars in the internal degree of freedom space, i.e., they

commute with � and �†. P�B̂�= 12 �1 �s�B̂��1+c�B̂��3�� are

orthogonal projections such that P++ P−=1, P+P−= P−P+=0,

with c�B̂�=�BR / �B̂+BR�, and s=�1−c2. The P�B̂� are pro-jections on invariant dressed polarization subspaces that, un-like the bare polarization projections 1

2 �1�3�, do not com-mute with the field b.

The reduction of the Hamiltonian to its normal form sim-plifies the solution of the Heisenberg equations of motion.The dynamics of the polarization operator �3 is given by

�3,t = �s�B̂��1 + c�B̂��3�c�B̂� − �s�B̂�c�B̂��1 − �3�

cos���B̂�t� + s�B̂�sin���B̂�t� , �2�

where �= 2�� is the Rabi frequency. The dynamics of the

polariton field b is derived using the identity bf�B̂�= f�B̂+��b that holds for an arbitrary function f ,

bt = ��P+�B̂� + P−�B̂�e−i��B̂�t�P+�B̂ + ��e−i�̃��B̂�t

+ �P+�B̂�ei��B̂�t + P−�B̂��P−�B̂ + ��ei�̃��B̂�t�e−i�tb , �3�

defining the �field-dependent� phase frequency �̃�

= 1� ���·+��−��·��. �̃� is semiclassically slow and has a well-

defined value �̃=�� in the classical limit �→0.

4 Π 8 Π 12 Π�t0.55

0.6

0.65

0.7

0.75�Σ3 t �

4 Π 8 Π 12 Π 16 Π�t

0.30.320.340.360.38�Σ3 t �

FIG. 1. �Color online� The mean polarization of an initiallycoherent photon state, and atomic polarization optimally localizedon the “+” invariant subspace, with BR=6.25�. The continuous blueline is derived from the semiclassical approximation Eq. �6� and theblack dots from a direct numerical solution. Good agreement isobserved already in the top panel where the mean polariton numberis 8 while the bottom panel with 50 polaritons shows agreement toless than 3%. The relative error term in the semiclassical approxi-mation is O��1/2�.

2 Π 4 Π 6 Π 8 Π�t

1.05

1.1

1.15

1.2Re�at �

2 Π 4 Π 6 Π 8 Π�t

1.5

1.55

1.6

1.65

Re�at �

FIG. 2. �Color online� The expectation of the real quadrature ofthe field amplitude of an initially coherent photon and excitedatomic state with BR=6.25�. The continuous blue line and the blackdots have analogous meanings to those of Fig. 1, while the dashedred line shows only the adiabatic terms in the semiclassical fielddynamics Eq. �7�. Good agreement is observed for the lower polar-iton numbers, 4 �top� and 8 �bottom�, since the relative semiclassi-cal error is O���.

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The dynamical analysis presented so far is exact, but theconversion of Eqs. �2� and �3� to physically measurablequantities requires taking matrix elements with respect to theinitial state, and for this purpose the semiclassical approxi-mation will be assumed, letting �→0 keeping other systemparameters fixed. The analysis will rely on the classicalsmoothness of �3,t and bt, and this requires the time t to be ofo�1� in the semiclassical limit. As shown below, the time ofvalidity encompasses many Rabi oscillations and their col-lapse but not their revival.

The semiclassical approximation will be made in theWigner-Weyl representation of the JC system that is a specialcase of the standard Wigner-Weyl representation of many-component systems �17�, in itself a straightforward generali-zation of the scalar Wigner-Weyl representation. Becausethe polarization degree of freedom remains quantum-mechanical, the symbol range consists of the operators in thetwo-dimensional polarization Hilbert space. In the samemanner, the Wigner “function” W is also operator-valued andthe expectation value of an observable is

�L̂ = d2�

2 tr L��,�̄�W��,�̄� , �4�

where L is the Weyl representation of the operator L̂, � is acomplex phase-space coordinate, and tr stands for trace onthe two states of the internal degree of freedom.

The dynamical expectation values presented below werecalculated using Eq. �4� after deriving the quasiflow, i.e., thetime-dependent Weyl representation, of �3 �=�3� and of afrom Eqs. �2� and �3�. Because these operators are classicallysmooth, the leading term quasiflow is obtained simply bysubstituting the classical field amplitude � and its complexconjugate �respectively� for the operators b and b† �note thatthe difference between the a and b operators is of higherorder in ��. The higher-order terms were obtained using thestandard rules of the Moyal calculus �17�. The straightfor-ward but technical details of this calculation are omitted.

III. SEMICLASSICAL RABI OSCILLATIONS

We proceed to a concrete demonstration of the results bythe Rabi oscillations of phase-space localized pure initialstates. Specifically, the initial states are taken to be separablein the bare field-atom representation with a coherent fieldstate. This means that the initial Wigner “function” is of theform 2w exp�−2 ��−�0�2�, where w is a projection operatorin the atomic two-dimensional Hilbert space. The Wignerfunction is combined with the quasiflows to calculate physi-cal expectation values, by carrying out the Gaussian phase-space integral and the two-dimensional trace operation inEq. �4�.

Consider first the case in which the atom is prepared inthe excited state, w= 1

2 �1+�3�, and the observable to be mea-sured is the mean polarization ��3 that gives the differencebetween the probabilities that the atom is in the excited stateand in the ground state. In a classical field of intensity equalto the mean intensity of the quantum field, the two-levelsystem would undergo periodic Rabi oscillations of fre-

quency ��A0�, where A0=����0�2+ 12 � is the expectation

value of the action in the coherent initial state ��0. However,because of the field uncertainty, the Rabi oscillations have afinite lifetime �8� after which they collapse. The semiclassi-cal calculation agrees with this result:

��3t = c�A0�2 + s�A0�2 cos���A0�t�e−�1/2��A0����A0�t�2�5�

exhibits Rabi oscillations of frequency ��A0�, superexponen-tially damped with the rate ��A0���A0�. The collapse time istherefore asymptotic to �−1/2; in particular, it diverges in theclassical limit as expected. It should be mentioned that Eq.�5� is obtained from Eq. �2� using only the leading classicalsubstitution term in the quasiflow; therefore, the same col-lapse dynamics would be observed with a random classicalelectromagnetic field with a Gaussian distribution.

A more subtle consequence of the quantum uncertainty inthe electromagnetic field is the impossibility of concentratinga state that is separable in the bare atom-field decompositionon one of the dressed polarization subspaces, because theenergy eigenstates are entangled. As a result, the dynamics ofany separable initial state exhibits Rabi oscillations betweenthe two internal polariton states. The separable state will bemaximally localized on the + subspace if we choose w= P+�A0�. The mean polarization obtained with this initialstate is

��3t = c�A0� − ����A0�t��A0�

s�A0�� �BR A0

2�BR + A0�+

1

2s�A0�

sin���A0�t� exp�−1

2�A0����A0�t�2� . �6�

Beside the constant term, which is the only one to survive inthe classical limit, the polarization displays Rabi oscillationswith an amplitude proportional to � /A0, which is growing intime. The amplitude does not grow without bound, however,because it is also subject to collapse. Since the collapse timegrows as �−1/2 in the semiclassical limit, the peak amplitudeis of O��1/2�. A comparison of the semiclassical calculationwith results obtained by direct numerical solution of theSchrödinger equation for a particular parameter choice isshown in Fig. 1, showing good agreement for a moderatenumber of photons.

A different manifestation of the quantization of the elec-tromagnetic field is the oscillations of the field amplitude �awith the fast Rabi frequency. One cause for these oscillationsis the periodic energy exchange between the field and atom.More surprising, though, are the Rabi oscillations in the po-lariton field, caused by the nontrivial commutator �b ,b†� thatgenerates the off-diagonal terms in Eq. �3�.

As an example, consider again the case in which the atomis initially in the excited state, w= 1

2 �1+�3�. In the framerotating with the optical frequency �, the expectation valueof the field amplitude is

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�atei�t = �cos��̃�A0t�� − ic�A0�sin��̃�A0�t��

−�

4A0s�A0�2�1 − e−i��A0�t exp�−

1

2�A0���t�2 ��0.

�7�

The terms in the second line express the Rabi oscillations ofamplitude proportional to � /A0. The terms on the first line,on the other hand, result from the weighted average of thetwo diagonal elements in Eq. �3� that describe adiabaticphase oscillations in the dressed polarization invariantsubspaces frequency �̃ �respectively�. In the classical sub-stitution approximation, these phase oscillations lead to thesplitting of the diagonal components of the Wigner function�8–10�.

The time evolution of the field amplitude as obtainedfrom Eq. �7� is shown in Fig. 2 for a particular parameterchoice, and compared with the values obtained from the nu-merical solution. The “baseline” of the Rabi oscillations isprovided by the adiabatic terms, whose contribution is shownin the figure with a dashed line. The adiabatic motion is soslow that in the time interval before the collapse of the Rabioscillations, only a fraction of a phase oscillation is achieved.However, because of the much larger amplitude of the adia-batic motion, its effect is comparable to that of the fast Rabioscillations, illustrating the singular nature of the semiclassi-cal limit, where phenomena occurring on widely separatedtime scales have to be considered simultaneously.

IV. CONCLUSIONS

The dynamics of quantum systems, including integrableones, becomes computationally complex for phase-space lo-calized wave packets, or other initial states having manynonzero energy components. However, this difficulty can be

bypassed when expectation values of classically smooth ob-servables are sought, and the utility of this approach for mul-ticomponent systems was demonstrated with the semiclassi-cal analysis of the Rabi oscillations in the Jaynes-Cummingssystem, a phenomenon that is a direct manifestation of itsquantum multicomponent nature.

Computability is not a practical concern for the JC sys-tem, but the semiclassical analysis provides a thorough andsystematic theory of the dynamics that is lacking in purenumerical analysis. The results presented in Eqs. �5�–�7� il-lustrate the fact that the Rabi oscillations of a phase-spacelocalized wave packet are approximated in the leading termby oscillations in a classical field with parameters deter-mined by the center of the wave packet. This result is thestarting point for the analysis of quantum electromagneticeffects as an expansion in powers of �. The concept wasdemonstrated using photon coherent states, but clearly holdsfor any phase-space localized initial state. Furthermore, theidea is not limited to the analysis of Rabi oscillations: It isapplicable to most other quantities of interest in the JC sys-tem such as squeezing and entanglement, as well as to othermulticomponent quantum system including some cited in theIntroduction.

The approximation of quantum electrodynamic phenom-ena by classical ones is, like all semiclassical dynamical ap-proximations, nonuniform in time. In our case, the time scaleof validity is set by the Heisenberg time tH= �����−1, wherethe discreteness of energy levels becomes manifest with therevival of oscillations �8�, which is not perturbative in thequasiflow. Neither is the semiclassical quasiflow well-suitedto study quantum interference phenomena. These phenomenaare of course amenable to semiclassical analysis, but theyrequire the use of more elaborate tools and considerablyheavier analysis, in contrast with the purpose of this work.

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�1991�.�18� The normal form Hamiltonian is equal to the original Hamil-

tonian �1� except for its action on the ground state. Rather thanadd an encumbering correction term to the Hamiltonian, it willbe assumed that there is a negligible probability amplitude tobe in the ground state, a natural assumption in the semiclassi-cal analysis.

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