Physica D 155 (2001) 311–323

Structural Hamiltonian chaos in the coherent parametricatom–field interaction

V.I. Ioussoupov, L.E. Kon’kov, S.V. Prants∗Laboratory of Nonlinear Dynamical Systems, Pacific Oceanological Institute of the Russian Academy of Sciences, 43 Baltiiskaya St.,

Vladivostok 690041, Russia

Received 2 October 2000; received in revised form 25 February 2001; accepted 7 March 2001Communicated by E. Ott

Abstract

We consider one of the simplest semiclassical models in laser and atomic physics, a collection of two-level atoms interactingcoherently with an electromagnetic standing wave in an ideal single-mode cavity within the rotating-wave approximation(RWA). In the strong-coupling limit, atoms and a cavity field constitute a strongly coupled atom–field dynamical systemwhose atomic and field variables oscillate in a self-consistent way with the Rabi frequency. It is proven analytically by theMelnikov method and numerically by computing maximal Lyapunov exponents and Poincaré sections that the parametricRabi oscillations in a vibrating cavity may be chaotic in a sense of exponential sensitivity to initial conditions. Wavelet spectracomputed with typical signals of the oscillations demonstrate clearly that Hamiltonian chaos in the coherent atom–fieldinteraction with modulated coupling is structural. Structural chaos is characterized by positive values of the maximal Lyapunovexponents and regular structures which coexist in the same signal. Duration of a train of regular oscillations is defined by theperiod of modulation, but oscillations between successive trains are chaotic. Results of numerical experiments and estimationof the control parameters and approximations involved show that a Rydberg atom maser operating with a collection oftwo-level atoms inside a high-quality superconducting microwave cavity is a promising device for observing manifestationsof structural Hamiltonian chaos in real experiments. © 2001 Elsevier Science B.V. All rights reserved.

PACS: 05.45.−a; 42.65.Sf

Keywords: Hamiltonian chaos; Two-level atoms; Rabi oscillations; Wavelets

1. Introduction

The fundamental model for the interaction of radi-ation and matter, comprising a collection of two-levelquantum systems coupled with a single-mode elec-tromagnetic field, provides the basis for laser physicsand describes a rich variety of nonlinear dynamical

∗ Corresponding author. Tel.: +7-4232-313081;fax: +7-4232-312573.E-mail address: prants@mail.ru (S.V. Prants).

behavior. The discovery that a single-mode laser, asymbol of coherence and stability, may exhibit deter-ministic instabilities and chaos is especially impor-tant since lasers provide nearly ideal systems to testgeneral ideas in statistical physics. From the standpoint of nonlinear dynamics, laser is an open dissipa-tive system which transforms an external excitationinto a coherent output in the presence of loss. It iswell known [1] that a single-mode, homogeneouslybroadened laser, operating on resonance with thegain center, can be described in the semiclassical

0167-2789/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved.PII: S0 1 6 7 -2 7 89 (01 )00264 -0

312 V.I. Ioussoupov et al. / Physica D 155 (2001) 311–323

rotating-wave approximation (RWA) by three realMaxwell–Bloch equations which have been shown tobe equivalent to the Lorenz model for fluid convection[2]. Some manifestations of a Lorenz-type strangeattractor and dissipative chaos have been observedwith different types of lasers (for a review, see [3]).

Physics of the radiation–matter interaction can besimplified even more by considering the interactionof two-level atoms with their own radiation field ina perfect single-mode cavity without any externalexcitation. Fully quantum models of Dicke [4] andJaynes and Cummings [5] provide the basis of cav-ity quantum electrodynamics [6]. Recent excitingachievements in this rapidly growing field, especially,creating micromasers and microlasers, are now allow-ing direct tests of foundations of quantum mechanics.These devices can serve as a testing ground for theproblem of correspondence between classical andquantum dynamics since micromasers and micro-lasers can be easily extended to many-atom operationproviding one to step from regimes which ask fora quantum description to regimes which ask for asemiclassical description and vice versa.

Here we study extremely nonlinear dynamics ofa strongly coupled atom–field system in a losslesssingle-mode cavity, assuming the number of atoms tobe sufficiently large in order to adopt the semiclassi-cal approximation. Semiclassical equations of motionfor this system may be reduced to the Maxwell–Blochequations for three real independent variables which,in difference from the laser theory, do not includelosses and pump. These equations are, in general, non-integrable, but they become integrable immediately af-ter adopting the RWA [5] that implies the existence ofan additional integral of motion, conservation of theso-called number of excitations. In early studies of theMaxwell–Bloch equations, it has been shown theoret-ically and numerically that they may demonstrate dy-namical instabilities and chaos of Hamiltonian type ifone goes beyond the RWA [7] or if one stays within theRWA, but atoms move through a cavity in a directionalong which a cavity sustains a field that is periodicin space [8]. Numerical experiments have shown thatin the first case, prominent chaos arises when the den-sity of atoms is very large (approximately 1020 cm3 in

the optical range [7]). In the second case, the speed ofatoms should be very high (approximately a few per-cent of the speed of light in vacuum [8]) in order toobserve a transition to chaos numerically.

In this paper, we propose another physical mech-anism that may lead within the RWA to Hamiltonianchaos with atoms at rest in a single-mode cavity.From the theoretical point of view, the idea is tointroduce a time-dependent parameter to make theMaxwell–Bloch set of equations nonautonomous andnonintegrable. Practically, it may be done by mod-ulating cavity length by means of an electro-opticalmodulator. Modulation of positions of the nodes of acavity standing wave gives rise to modulation of thecoupling strength between atoms and the field. Theresulting parametric Rabi oscillations of the atomicpopulation inversion will be studied by differentmethods in the context of nonlinear dynamics.

2. The nonlinear atom–field oscillator

2.1. The equations of motion

We start with the time-dependent extension of thestandard quantum–optical Hamiltonian

H = 1

2ωa

N∑j=1

σjz + ωf

(a†a + 1

2

)

+Ω0(t)

N∑j=1

(aσj+ + a†σ j−), (1)

which describes the interaction of N identicaltwo-level atoms with a single mode of a quantizedelectromagnetic field in the RWA and the point-likeapproximation. The energy separation of all the atomsis supposed to be the same and equal to ωa. ThePauli matrices σz and σ± = 1

2 (σx ± iσy) describethe internal atomic dynamics, and the Boson oper-ators a and a† characterize the field mode with thefrequency ωf . In difference from the standard Dickeand Jaynes–Cummings models [4,5], we incorporateinto the Hamiltonian a time-dependent atom–fieldcoupling Ω0(t) which may describe the effect of

V.I. Ioussoupov et al. / Physica D 155 (2001) 311–323 313

a parametric modulation imposed on the system.We assume a harmonic modulation of cavity lengthwhich causes the respective variance of the nodesof a standing light wave which, in turn, results inthe harmonic modulation of the atom–field couplingΩ0(1 + α sinωmt), where α and ωm are the depthand frequency of the modulation, respectively. Thepoint-like approximation means that a collection ofatoms is so confined that the amplitude value of thecoupling Ω0 = dE0/ may be put to be the same forall the atoms in the ensemble (here d is the magnitudeof the dipole matrix element, E0 = √

ωf/2εV , V isthe effective cavity mode volume).

The next step is to derive in the Heisenberg repre-sentation the equations of motion from the Hamilto-nian (1). In the semiclassical approximation which isvalid with the accuracy of the order of 1/N [9], onecan replace all the operators by their expectation val-ues over an arbitrary initial quantum state of the atomsand the field. Choosing the following expectation val-ues as dynamical variables:

x = 1

N

N∑j=1

〈σ jx 〉, y = 1

N

N∑j=1

〈σ jy 〉,

z = 1

N

N∑j=1

〈σ jz 〉, e = 1√N

〈a + a†〉,

p = i√N

〈a† − a〉, (2)

we close the respective set in the form of the dimen-sionless Maxwell–Bloch equations:

x = −y −ΩN(1 + α sin δτ)zp,

y = x −ΩN(1 + α sin δτ)ze,

z = ΩN(1 + α sin δτ)(xp + ye),

e = ωp −ΩN(1 + α sin δτ)y,

p = −ωe −ΩN(1 + α sin δτ)x, (3)

where dot denotes the derivative with respect to thedimensionless time τ = ωat . Since we do not includelosses and pump into consideration, this set possessestwo integrals of motion:

R = x2 + y2 + z2 = 1, W = e2 + p2 + 2z, (4)

reflecting a conservation of the length of the Blochvector and of the number of excitations, respectively.With the help of semiclassical factorization made, wereduce the infinite-dimensional state space of the fullyquantum atom–field system to the five-dimensionalphase space, which is a direct product space of theBloch sphere and the oscillator plane. In fact, we havethree independent real-valued variables, one for a fieldcomponent (e or p), another one for the density ofthe atomic population inversion z, and the third onefor an atomic polarization component (x or y). Thenormalized collective Rabi frequency

ΩN = Ω0√N

ωa, (5)

the normalized detuning from resonance

ω = ωf

ωa, (6)

and the normalized modulation frequency

δ = ωm

ωa(7)

play the role of the control parameters of the nonau-tonomous nonlinear dynamical system (3). In con-clusion of the characterization of the model, we notethat an equivalent set of semiclassical equations maybe obtained in the Schrodinger representation by cou-pling the Maxwell equations with the Schrodingerequations for atomic transition amplitudes (see, forexample, [11]).

2.2. Regular free Rabi oscillations

Without modulation, the set (3) has an additionalintegral of motion

C = ΩN(xe − yp)+ (1 − ω)z, (8)

reflecting a conservation of the interaction energywhich is valid within the RWA. By differentiating thethird equation in the set (3) with α = 0 and makinguse of the conservation laws (4) and (8), we derivethe following closed equation for the density of theatomic population inversion:

z = f (z), z(0) = ΩN(x0p0 + y0e0),

z(0) = z0, (9)

314 V.I. Ioussoupov et al. / Physica D 155 (2001) 311–323

where

f (z)= 3Ω2Nz

2 − [(ω − 1)2 +Ω2NW ]z

+(ω − 1)C +Ω2N. (10)

Thus, the autonomous Maxwell–Bloch equationswithout losses and pump are reduced to a single or-dinary differential equation of the second order de-scribing a free classical oscillator (9) driven by thenonlinear restoring force f (z). The energy integral ofEq. (9):

E = 1

2z2 +Π(z), Π(z) = −

∫f (z) dz (11)

helps us to integrate it in the form of the elliptic inte-gral of the first order:

√2 dτ =

∫ z(τ )

z0

dz√E −Π(z)

. (12)

By inverting this integral, it is easy to find the solu-tion for the atomic inversion in terms of the ellipticJacobian sine:

z(τ ) = z1 + (z2 − z1) sn2(√z1 − z3ΩNτ + ϕ, k),

(13)

where

ϕ = sn−1(

z1√z1 − z2

, k

), k =

√z1 − z2

z1− z3,

(14)

and z1, z2 and z3 are the roots of the algebraic cubicequation E −Π(z) = 0.

At exact resonance between the atoms and the cavitymode, i.e. at ω = 1, and under the initial conditionscorresponding to fully inverted atoms, z0 = 1, thegeneral solution (13) takes the form of the specialsolution:

zJC(τ ) = −1 + 2 sn2(√n0 + 1ΩNτ + ϕJC, kJC),

ϕJC = sn−1(

1,1√

n0 − 1

), kJC = 1√

n0 + 1, (15)

first found by Jaynes and Cummings [5]. Here n0 =14 (e

20 +p2

0) is the initial density of photons in a cavity.For a large number of photons, n 1, the elliptic

functions are well approximated by the trigonometricones. In this limit, the solution

zh(τ ) sin(√

2n0ΩNτ + const) (16)

describes harmonic free oscillations of the atomic pop-ulation inversion. In the limit of initial vacuum field,n0 = 0, they are approximated by the hyperbolic func-tions yielding the solution on the separatrix

zs(τ ) 1 − 2 sech2ΩNτ (17)

that describes the locus of states in which atoms radiateand reabsorb their own radiation field in infinite time.

Projections of the phase portrait of the atomic oscil-lator (9) onto the plane z–z consist of a collection ofperiodic orbits and a special orbit, a separatrix loop,connecting the stable fixed point S−: (z = 0, z = −1)which corresponds to initially depopulated atoms, z =−1, x = y = 0, and the vacuum field, e = p = 0,and the unstable fixed point S+: (z = 0, z = 1) corre-sponding to initially fully inverted atoms, z = 1, x =y = 0, and the vacuum field, e = p = 0.

General exact solutions of the autonomousMaxwell–Bloch equations for the field, e and p, andthe atomic polarization components, x and y, havebeen found in our paper [10]. It is important that thenonautonomous Maxwell–Bloch equations (3) withα = 0 are integrable at exact resonance for any kindof modulation of the coupling ΩN(τ). It follows fromthe existence of the third integral of motion xe− yp =const that is valid at ω = 1 for any differentiable func-tion of time ΩN(τ). General exact solutions of theresonant nonautonomous Maxwell–Bloch equationsare easily found from the respective solutions in theautonomous limit by putting in the latter ω = 1 andtransforming to the new “time” τ → ∫ τ

0 ΩN(τ′) dτ ′.

2.3. Chaotic parametric Rabi oscillations

The integrable limits of the Maxwell–Bloch equa-tions (3) and the respective solutions, discussed in thepreceding section, describe the regular Rabi oscilla-tions of the internal energy of two-level atoms inter-acting with a single-mode cavity field which occur,at least, in the two following cases: if modulation of

V.I. Ioussoupov et al. / Physica D 155 (2001) 311–323 315

Fig. 1. Poincare sections on the field plane e–p with ΩN = 1, ω = 0.9, δ = 0.01 and the different values of the depth of modulation: (a)α = 0.25; (b) α = 0.5; (c) α = 0.75; (d) α = 1.

the cavity length is absent at all (α = 0), and if theeigenfrequency of a vibrating cavity is tuned to exactresonance with atomic working transition (ω = 1). Togive an idea about increasing the complexity of motionwith α = 0 and ω = 1, we represent in Fig. 1 Poincarésections of the Hamiltonian flow (3) on the field planee–p at different values of the depth of modulation α.These sections are computed with the following ini-

tial conditions: e0 = p0 = 1, x0 = 0, y0i =√

1 − z20i

where the initial atomic inversion z0i (i = 1, . . . , 8)takes eight values in the range from −1 to 1. The

values of the control parameters, the collective Rabifrequency ΩN , detuning ω and the modulation fre-quency δ are given in the figure caption. With increas-ing values of α, the invariant sets are broken, and theaccessible part of the phase plane is filled with dots,but a kind of regular structure is still visible in thesections even with large values of the depth of modu-lation.

In Fig. 2 we show the long-time evolution of theatomic population inversion z(τ ) with ΩN = 1, ω =0.9 and α = 1 at different values of the modulation

316 V.I. Ioussoupov et al. / Physica D 155 (2001) 311–323

Fig. 2. Parametric Rabi oscillations of the atomic population inversion with ΩN = 1, ω = 0.9, α = 1 and the different values of themodulation frequency: (a) δ = 0.001 (Tm 6280, λ 0); (b) δ = 0.01 (Tm 628, λ 0.006); (c) δ = 0.05 (Tm 125.6, λ 0.022).

frequency δ for the atom–field system (3) prepared ini-tially in the following state: x0 = y0 = 0, z0 = 1, e0 =p0 = 1, that corresponds to initially fully invertedatoms and initial density of photons to be equal to n0 =12 . A characteristic periodical structure of the paramet-ric Rabi oscillations with trains of high-amplitude os-cillations intermitted by their collapses, that are dueto the modulation, is seen in this figure. At timesτm = (3 + 4m)π/2δ (m = 0, 1, 2, . . . ), when themodulation function takes its minimal values, the ef-fective atom–field coupling goes to zero (at α = 1).The rate of change of z(τ ) decreases with decreas-ing the atom–field coupling (see the third equation inthe set (3)), and the parametric oscillations begin toslowdown near the time moments τm (that is seen inall the three fragments of Fig. 2) resulting eventually

in the collapses of the Rabi oscillations. With a givenvalue of δ, all the trains have the same duration equalto the modulation period Tm = 2π/δ, but the ampli-tudes of the oscillations and their behavior betweensuccessive trains look more and more chaotic with in-creasing δ. In order to check, it we compute the max-imal Lyapunov exponent λ for each of these signals.The computation confirms that the signal in Fig. 2awith Tm 6280 is quasiperiodic with λ 0, the sig-nal in Fig. 2b with Tm 628 is weakly chaotic withλ 0.006, and the signal in Fig. 2c with Tm 125.6is chaotic with λ 0.022.

To understand a mechanism of arising the chaoticparametric Rabi oscillations shown in Fig. 2 and man-ifested in Fig. 1, we make use the Melnikov methodof analyzing solutions of nonautonomous dynamical

V.I. Ioussoupov et al. / Physica D 155 (2001) 311–323 317

systems in a neighborhood of their unperturbed invari-ant sets [12]. Without modulation, the evolution of theatom–field system is quasiperiodic. The main effect ofmodulation is to produce, out of resonance (ω = 1), ahomoclinic structure in the vicinity of the separatrix ofthe unperturbed Maxwell–Bloch set of equations. Thebasic idea of the Melnikov analysis is to make use ofexact solutions of the unperturbed integrable system(α = 0) in the computation of the perturbed system(3) if the perturbation may be considered as small, i.e.the depth of modulation is supposed to be small ascompared with the Rabi frequency, α ΩN . With thehelp of the integrals of motion, one can represent thehomoclinic manifold implicitly by the equations R =1, W = 2, C = 1 − ω. The signed distance betweenthe stable and unstable manifolds of the equilibriumhyperbolic point S+ at the moment τ0 along the nor-mal to the unperturbed homoclinic manifold is propor-tional to the expression αM(τ0) + O(α2), where theMelnikov function M(τ0) is evaluated along the sepa-ratrix of the Maxwell–Bloch equations without mod-ulation which has been found in the paper [10] (seeEq. (33) in [10]). Details of calculating the Melnikovfunction for equations of the Maxwell–Bloch type can

Fig. 3. The maximal Lyapunov exponent λ versus the collective Rabi frequency ΩN = Ω0√N/ωa with ω = 0.9, α = 1 and the following

values of the modulation frequency: (a) δ = 0.001; (b) δ = 0.01.

be found in [10,13]. In our case, the final result is thefollowing:

M(τ0) = 2π(1 − ω)δ2

Ω3N sh(δπ/βΩN)

cos(δτ0), (18)

where β2 = 4 − [(ω − 1)/ΩN ]2.It is evident from (18) that out of resonance,

ω = 1, the Melnikov function has simple zeroes asa function of τ0, where τ0 parameterizes time of themotion of a representative point on a trajectory. IfM(τ0) has simple zeroes, then the stable and unstablemanifolds of the hyperbolic point intersect transver-sally resulting in Smale horseshoe chaos. This inter-section occurs with the frequency of modulation δ

manifesting itself in Fig. 2 as irregular oscillationsbetween successive trains of the parametric Rabioscillations. A splitting of the stable and unstablehyperbolic manifolds, coinciding in the integrablelimit, takes place under an arbitrary small depth ofmodulation α.

We have performed some computer simulation onthe atom–field system (3) assuming the depth of mod-ulation of the Rabi frequency to be large. In closeddynamical system, chaos has its origin in extremal

318 V.I. Ioussoupov et al. / Physica D 155 (2001) 311–323

Fig. 4. Topographical λ-map on the plane ω–ΩN with α = 1 and δ = 0.01.

sensitivity to initial conditions which is characterizedby positive values of the maximal Lyapunov expo-nent. The local exponential divergence of trajectoriesproduces a local stretching, but because of the global

Fig. 5. Topographical λ-map on the plane ω–ΩN with α = 1 and δ = 0.1.

confinement in the phase space of our conservativeHamiltonian system with two degrees of freedom,this stretching is accompanied by folding. Repeatedstretching and folding produces very complicated

V.I. Ioussoupov et al. / Physica D 155 (2001) 311–323 319

Fig. 6. Topographical λ-map on the plane log10δ–ΩN with α = 1 and ω = 0.9.

motion that is known as chaotic. The dependence ofthe maximal Lyapunov exponent on the values of thecollective Rabi frequency ΩN is shown in Fig. 3 withtwo different values of the modulation frequency δ.Since the number of atoms may be written with thehelp of Eq. (5) as follows: N = (ΩNωa/Ω0)

2, thesefigures may be considered as a dependence of λ onthe number of atoms N . In the regime of very weakchaos, the curve (a) demonstrates a curious periodicstructure.

When a dynamical system possesses more thana single control parameter, it is useful to computetopographical λ-maps [8] that give a representative“portrait” of chaos. Such a map shows the valuesof the maximal Lyapunov exponent λ of a systemwith given initial conditions as a function of two ofthe control parameters with fixed values of the otherones. We have computed the topographical λ-mapswith all the three control parameters ΩN , ω, and δ

varied. The values of λ are encoded by the color us-ing a linear scale which is shown on the right sidesof the maps. Figs. 4 and 5 show the ω–ΩN maps

calculated at the fixed values of the normalized detun-ing δ = 0.01 and δ = 0.1, respectively. It is clear fromthe figures that chaos disappears when the atomictransition frequency is closed to the field frequency,and the chaotic “sea” enlarges with increasing δ. Fig.6 shows the topographical λ-map on the frequencyplane log10δ–ΩN at α = 1 and ω = 0.9. The mapsgive us, in fact, the values of λ as a function of thenumber of atoms N which may be considered as anadjustable control parameter of the atom–field systemtogether with δ and ω.

3. Wavelet analysis of structural Hamiltonianchaos

The signals of the parametric Rabi oscillationscomputed in the preceding section demonstrate avisible structure even if they are proven to be chaoticin a sense of extremal sensitivity to initial condi-tions which is characterized by a positive value ofthe maximal Lyapunov exponent. Homoclinic nature

320 V.I. Ioussoupov et al. / Physica D 155 (2001) 311–323

Fig. 7. High-frequency range of the wavelet spectra of the paramet-ric Rabi oscillations shown in Fig. 2: (a) δ = 0.001; (b) δ = 0.01;(c) δ = 0.05. Reciprocals of the numbers on the axis T correspondto the respective values of the frequency components of a signal.

of Hamiltonian dynamical chaos in the atom–fieldsystem has been proven above by the Melnikovmethod. In order to know a temporal kind of thisstructural chaos we compute in this section a wavelettransformation of the time series z(τ ) asfollows:

F(T , τ) =∫ ∞

−∞z(t)φ∗

T τ (t) dt, (19)

where φT τ (t) = (π√

2τmax/4T )φ[(t − τ)/T ], φ =exp(ik0t) exp(−t2/2) is a basic Morlet wavelet, τmax

the maximal duration of a signal z(τ ), k0 the fittingparameter, t the integration time and T −1 the fre-quency in dimensionless units. A two-dimensionalmatrix F(T , τ) computed can be represented as atwo-dimensional chart with absolute values of thewavelet transformation (19) encoded by the colorusing a linear scale and increasing from white toblack color. The frequency and temporal scales ofthe signals z(τ ) are indicated in Figs. 7 and 8 alongthe axes T and τ , respectively. Each point (Ti, τi) inthe wavelet chart is a convolution of the signal z(τ )with the Morlet wavelet φ shifted to the value τi andstretched to the value Ti . In such a way, the transfor-mation F(T , τ) gives an information about temporaland frequency peculiarities of a signal simultaneously.In fact, wavelet spectra show in which way frequencycomponents of a signal under consideration changein time.

In the fragments (a), (b) and (c) of Figs. 7 and8, we demonstrate results of the wavelet transforma-tion (19) of the Rabi oscillation signals z(τ ) shownin the respective fragments of Fig. 2. Fig. 7 showshigh-frequency parts of the respective wavelet spectracomputed with the fitting parameter k0 = 96. The val-ues of the maximal Lyapunov exponent computed inthe previous section have indicated that the signal ofthe parametric Rabi oscillations with the modulationfrequency δ = 0.001 (Fig. 2a) is regular with λ 0,the signal with δ = 0.01 (Fig. 2b) is weakly chaoticwith λ 0.006, and the signal with δ = 0.05 (Fig. 2c)is a chaotic one with λ 0.022. This notwithstanding,the high-frequency components of all of these signalsdemonstrate regular behavior characterized by two pe-riodic functions. One of them describes variations in

V.I. Ioussoupov et al. / Physica D 155 (2001) 311–323 321

time of much more intensive frequency components(the black lines in the fragments) than the other one.These regular variations in time reflect, mainly, a vis-ible structure of the parametric Rabi oscillations thatis due to modulation with the period Tm and charac-teristical slowdown of these oscillations in time mo-ments τm = (3 + 4m)π/2δ. Fig. 7a shows that thehigh-frequency components of the regular signal withthe comparatively long period of modulation Tm 6280 change in time regularly with the periods Tm and2Tm in a rather wide frequency ranging from T −1 0.01 to 0.3. With the decrease in the modulationperiod Tm in n times, extremal values of this rangedecreases approximately in n times extending fromT −1 0.1 to 3 (Fig. 7b with Tm 628) and fromT −1 0.5 to 12 (Fig. 7c with Tm 125.6). Thehigh-frequency domains of the wavelet spectra of thechaotic signals (b) and (c) demonstrate in their tops ir-regular appearing and disappearing components withcomparatively low frequencies. One may conclude thatthe very high-frequency components of the paramet-ric Rabi oscillations change in time regularly both forperiodic and for chaotic signals.

In Fig. 8, we demonstrate low-frequency parts ofthe wavelet spectra of the same signals shown in Fig. 2and computed with the formula (19) but with the fittingparameter k0 = 2. In contrast to the high-frequencywavelets, the low-frequency spectra are very differentfor regular and chaotic signals. Low-frequency com-ponents of the regular signal (Tm 6280) up to thevalues of the order of 10−4 corresponding to the lengthof the signal integrated are again changed in time peri-odically (see Fig. 8a). The components of the weaklychaotic signal (Tm 628) with λ 0.006 demon-strate in the range from 10−1 to 10−3 chaotic be-havior (please note the black spots in Fig. 8b), whereasthe components with T −1 > 10−3 are rather regular.The components of the chaotic signal (Tm 125.6)with λ 0.022 appear and disappear chaotically upto the values of the order T −1 5 × 10−3 (Fig. 8c).Thus, the wavelet spectra computed with typical sig-nals of the parametric Rabi oscillations of the atomicpopulation inversion elucidate a structural temporalkind of Hamiltonian chaos in the simple atom–fieldsystem with modulation.

Fig. 8. Low-frequency range of the wavelet spectra of the paramet-ric Rabi oscillations shown in Fig. 2: (a) δ = 0.001; (b) δ = 0.01;(c) δ = 0.05. Reciprocals of the numbers on the axis T correspondto the respective values of the frequency components of a signal.

322 V.I. Ioussoupov et al. / Physica D 155 (2001) 311–323

4. Summary and conclusion

We have analyzed the RWA nonlinear dynamics inone of the simplest models of laser and atomic physicsthat comprises two-level atoms in an ideal high-Qsingle-mode cavity. The main problem in which wehave been interested in this paper was a transition toHamiltonian chaos in the coherent atom–field inter-action with the atom–field coupling to be modulated.The Heisenberg equations for expectation values of acomplete set of the atomic and field observables havebeen shown to be integrable in the autonomous limitwithout any modulation and in the resonance limitwith an arbitrary modulation. They possess special or-bits that are homoclinic to the state with fully invertedatoms and vacuum cavity field which is an equilib-rium one in the semiclassical approximation. With thehelp of the Melnikov method, we have proved analyt-ically transverse intersections of stable and unstablemanifolds of this equilibrium point under a smallmodulation of the Rabi frequency. These transverseintersections are believed to provide Smale horseshoemechanism of chaos. To confirm numerically thechaotic dynamics in the semiclassical RWA model,we have computed Poincaré sections and maximalLyapunov exponents under strong modulation of theRabi frequency. The Lyapunov topographical mapsshowing the regions of regular and chaotic motion inthe space of the control parameters provide represen-tative numerical “portraits” of the system’s dynamicsin different ranges of its control parameters.

We could show that Hamiltonian chaos in thecoherent parametric atom–field interaction is of aspecial kind that is characterized both by a sensitivedependence on initial conditions and by long-livedstructures visible in the parametric Rabi oscillationsof the atomic population inversion and in the re-spective Poincaré sections. An intermittent route tothis structural Hamiltonian chaos has been clearlydemonstrated in the wavelet spectra of the typical sig-nals of the parametric Rabi oscillations. It has beenshown that with increasing modulation frequency, thelow-frequency components of the respective waveletspectrum changed irregularly in time appearing anddisappearing in a chaotic way. At the same time,

the high-frequency components of the same waveletspectrum change periodically in time even for thesignals with positive values of the maximal Lya-punov exponents. Coexistence of regular and irregularcomponents in the same signal is a visiting card ofstructural Hamiltonian chaos.

In conclusion, we would like to present some spec-ulations on a possibility to observe manifestations ofstructural Hamiltonian chaos with real devices. TheHamiltonian approach we have adopted throughoutthe paper is valid over time intervals shorter than allthe characteristic relaxation times. This approach isbased on the strong-coupling limit in the atom–fieldinteraction which is characterized by the following in-equality: (Ω0

√N)−1 Ta, Tf , where Ta and Tf are

the atomic and field relaxation times, respectively. Innumerical experiments computing maximal Lyapunovexponents, Hamiltonian chaos can be diagnosed overtime interval of the order of the correlation decouplingtime Tcor = 2π/ωaλ. In terms of the collective Rabifrequency ΩN and the maximal Lyapunov exponentλ, the condition for observing Hamiltonian chaos nu-merically can be rewritten as ΩN λ.

A Rydberg maser, operating in the strong-couplingregime with two-level Rydberg atoms inside a high-Qcavity, seems to be a promising device for observingsemiclassical Hamiltonian chaos in real experiment.The Rydberg atoms have the transition frequency ofthe order of ωa 1011–1012 rad/s, the magnitude ofthe electric dipole matrix element d 103 atomicunits, the single-photon vacuum Rabi frequency Ω0 105–106 rad/s, and the lifetime of the circular Ryd-berg states, Ta 10−2 s [14]. Parameters of a typicalsuperconducting microwave maser cavity are the fol-lowing: the quality factor Q 1010, the cavity lengthLc 1 cm, and the lifetime of intracavity photonsTf 10−1–10−2 s [14]. As it follows from our numer-ical results, one can reach the regime of the chaoticparametric Rabi oscillations with the strength of chaosof the order of λ 0.01 operating with a droplet con-sisting approximately of 1010 atoms and the frequencyof modulation of the order of δ 0.01.

We used the semiclassical approximation through-out the paper ignoring subtle questions of quantum–classical correspondence in the coherent parametric

V.I. Ioussoupov et al. / Physica D 155 (2001) 311–323 323

atom–field interaction. It should be mentioned thatsome manifestations of quantum chaos have been stud-ied with a system of N two-level atoms in a perfectcavity. In the model [15], atoms were assumed to in-teract with a resonant cavity field taking into accountvirtual atom–field transitions violating the RWA. InRef. [16], atoms were considered to interact with a res-onant cavity field and with an external coherent fieldin the framework of the RWA. It was shown with boththe models that in the range of parameters for devel-oped semiclassical chaos (when neglecting quantumcorrelations), the semiclassical approximation was vi-olated by quantum effects at the time scale τ ∼ lnNknown as the breaktime of the quantum–classical cor-respondence [17]. We plan to study in future this effectin parametric Rabi oscillations of N two-level atomsin a cavity with modulated atom–field coupling.

Acknowledgements

This work was supported by the grant from the Rus-sian Foundation for Basic Research No. 02-17269.

References

[1] H. Haken, Phys. Lett. A 53 (1975) 77.[2] E.N. Lorenz, J. Atmos. Sci. 20 (1963) 130.[3] R.G. Harrison, Contemp. Phys. 29 (1988) 341.[4] R.M. Dicke, Phys. Rev. 93 (1954) 493.[5] E.T. Jaynes, F.W. Cummings, Proc. IEEE 51 (1963) 89.[6] P.R. Berman (Ed.), Cavity Quantum Electrodynamics,

Academic Press, New York, 1994.[7] P.I. Belobrov, G.M. Zaslavskii, G.Kh. Tartakovskii, Zh.

Eksp. Teor. Fiz. 71 (1976) 1799 [Sov. Phys. JETP 44(1976) 945].

[8] S.V. Prants, L.E. Kon’kov, Phys. Lett. A 225 (1997) 33.[9] S.V. Prants, L.E. Kon’kov, Zh. Esksp. Teor. Fiz. 115 (1999)

740 [JETP 88 (1999) 406].[10] S.V. Prants, L.E. Kon’kov, I.L. Kirilyuk, Phys. Rev. E 60

(1999) 335.[11] L.E. Kon’kov, S.V. Prants, J. Math. Phys. 37 (1996)

1204.[12] V.K. Melnikov, Trans. Moscow Math. Soc. 12 (1963) 3.[13] D.D. Holm, G. Kovacic, T.A. Wettergen, Phys. Lett. A 200

(1995) 299.[14] J.M. Raimond, S. Haroche, Confined Electrons and Photons,

Plenum Press, New York, 1995, p. 383.[15] G.P. Berman, E.N. Bulgakov, G.M. Zaslavsky, Chaos 2 (1992)

257.[16] G.P. Berman, E.N. Bulgakov, D.D. Holm, Phys. Rev. A 49

(1994) 4943.[17] G.P. Berman, G.M. Zaslavsky, Physica A 91 (1978) 450.