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Structural Hamiltonian chaos in the coherent parametric atom–field interaction
Physica D 155 (2001) 311–323
Structural Hamiltonian chaos in the coherent parametric atom–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 2001 Communicated by E. Ott
Abstract We consider one of the simplest semiclassical models in laser and atomic physics, a collection of two-level atoms interacting coherently 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 system whose atomic and field variables oscillate in a self-consistent way with the Rabi frequency. It is proven analytically by the Melnikov method and numerically by computing maximal Lyapunov exponents and Poincaré sections that the parametric Rabi oscillations in a vibrating cavity may be chaotic in a sense of exponential sensitivity to initial conditions. Wavelet spectra computed with typical signals of the oscillations demonstrate clearly that Hamiltonian chaos in the coherent atom–field interaction with modulated coupling is structural. Structural chaos is characterized by positive values of the maximal Lyapunov exponents and regular structures which coexist in the same signal. Duration of a train of regular oscillations is defined by the period of modulation, but oscillations between successive trains are chaotic. Results of numerical experiments and estimation of the control parameters and approximations involved show that a Rydberg atom maser operating with a collection of two-level atoms inside a high-quality superconducting microwave cavity is a promising device for observing manifestations of 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 radiation and matter, comprising a collection of two-level quantum systems coupled with a single-mode electromagnetic field, provides the basis for laser physics and describes a rich variety of nonlinear dynamical ∗ Corresponding author. Tel.: +7-4232-313081; fax: +7-4232-312573. E-mail address: [email protected] (S.V. Prants).
behavior. The discovery that a single-mode laser, a symbol of coherence and stability, may exhibit deterministic instabilities and chaos is especially important since lasers provide nearly ideal systems to test general ideas in statistical physics. From the stand point of nonlinear dynamics, laser is an open dissipative system which transforms an external excitation into a coherent output in the presence of loss. It is well known [1] that a single-mode, homogeneously broadened laser, operating on resonance with the gain center, can be described in the semiclassical
0167-2789/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 7 8 9 ( 0 1 ) 0 0 2 6 4 - 0
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rotating-wave approximation (RWA) by three real Maxwell–Bloch equations which have been shown to be equivalent to the Lorenz model for fluid convection [2]. Some manifestations of a Lorenz-type strange attractor and dissipative chaos have been observed with different types of lasers (for a review, see [3]). Physics of the radiation–matter interaction can be simplified even more by considering the interaction of two-level atoms with their own radiation field in a perfect single-mode cavity without any external excitation. Fully quantum models of Dicke [4] and Jaynes and Cummings [5] provide the basis of cavity quantum electrodynamics [6]. Recent exciting achievements in this rapidly growing field, especially, creating micromasers and microlasers, are now allowing direct tests of foundations of quantum mechanics. These devices can serve as a testing ground for the problem of correspondence between classical and quantum dynamics since micromasers and microlasers can be easily extended to many-atom operation providing one to step from regimes which ask for a quantum description to regimes which ask for a semiclassical description and vice versa. Here we study extremely nonlinear dynamics of a strongly coupled atom–field system in a lossless single-mode cavity, assuming the number of atoms to be sufficiently large in order to adopt the semiclassical approximation. Semiclassical equations of motion for this system may be reduced to the Maxwell–Bloch equations for three real independent variables which, in difference from the laser theory, do not include losses and pump. These equations are, in general, nonintegrable, but they become integrable immediately after adopting the RWA [5] that implies the existence of an additional integral of motion, conservation of the so-called number of excitations. In early studies of the Maxwell–Bloch equations, it has been shown theoretically and numerically that they may demonstrate dynamical instabilities and chaos of Hamiltonian type if one goes beyond the RWA [7] or if one stays within the RWA, but atoms move through a cavity in a direction along which a cavity sustains a field that is periodic in space [8]. Numerical experiments have shown that in the first case, prominent chaos arises when the density of atoms is very large (approximately 1020 cm3 in
the optical range [7]). In the second case, the speed of atoms should be very high (approximately a few percent of the speed of light in vacuum [8]) in order to observe a transition to chaos numerically. In this paper, we propose another physical mechanism that may lead within the RWA to Hamiltonian chaos with atoms at rest in a single-mode cavity. From the theoretical point of view, the idea is to introduce a time-dependent parameter to make the Maxwell–Bloch set of equations nonautonomous and nonintegrable. Practically, it may be done by modulating cavity length by means of an electro-optical modulator. Modulation of positions of the nodes of a cavity standing wave gives rise to modulation of the coupling strength between atoms and the field. The resulting parametric Rabi oscillations of the atomic population inversion will be studied by different methods 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 the standard quantum–optical Hamiltonian H =
N 1 1 j ωa σz + ωf a † a + 2 2 j =1
+Ω0 (t)
N j =1
j
j
(aσ+ + a † σ− ),
(1)
which describes the interaction of N identical two-level atoms with a single mode of a quantized electromagnetic field in the RWA and the point-like approximation. The energy separation of all the atoms is supposed to be the same and equal to ωa . The Pauli matrices σz and σ± = 21 (σx ± iσy ) describe the internal atomic dynamics, and the Boson operators a and a † characterize the field mode with the frequency ωf . In difference from the standard Dicke and Jaynes–Cummings models [4,5], we incorporate into the Hamiltonian a time-dependent atom–field coupling Ω0 (t) which may describe the effect of
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a parametric modulation imposed on the system. We assume a harmonic modulation of cavity length which causes the respective variance of the nodes of a standing light wave which, in turn, results in the harmonic modulation of the atom–field coupling Ω0 (1 + α sin ωm t), where α and ωm are the depth and frequency of the modulation, respectively. The point-like approximation means that a collection of atoms is so confined that the amplitude value of the coupling Ω0 = dE0 / may be put to be the same for all the atoms in the ensemble (here d is the magnitude √ of the dipole matrix element, E0 = ωf /2V , V is the effective cavity mode volume). The next step is to derive in the Heisenberg representation the equations of motion from the Hamiltonian (1). In the semiclassical approximation which is valid with the accuracy of the order of 1/N [9], one can replace all the operators by their expectation values over an arbitrary initial quantum state of the atoms and the field. Choosing the following expectation values as dynamical variables: N
N
1 j x= σx , N
1 j y= σy , N
j =1
z=
1 N
N
j =1
1 e = √ a + a † , N
j
σz ,
j =1
i p = √ a † − a, N
(2)
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reflecting a conservation of the length of the Bloch vector and of the number of excitations, respectively. With the help of semiclassical factorization made, we reduce the infinite-dimensional state space of the fully quantum atom–field system to the five-dimensional phase space, which is a direct product space of the Bloch sphere and the oscillator plane. In fact, we have three independent real-valued variables, one for a field component (e or p), another one for the density of the atomic population inversion z, and the third one for an atomic polarization component (x or y). The normalized collective Rabi frequency √ Ω0 N , (5) ΩN = ωa the normalized detuning from resonance ωf ω= , ωa and the normalized modulation frequency ωm δ= ωa
(6)
(7)
play the role of the control parameters of the nonautonomous nonlinear dynamical system (3). In conclusion of the characterization of the model, we note that an equivalent set of semiclassical equations may be obtained in the Schrodinger representation by coupling the Maxwell equations with the Schrodinger equations for atomic transition amplitudes (see, for example, [11]).
we close the respective set in the form of the dimensionless Maxwell–Bloch equations:
2.2. Regular free Rabi oscillations
x˙ = −y − ΩN (1 + α sin δτ )zp,
Without modulation, the set (3) has an additional integral of motion
y˙ = x − ΩN (1 + α sin δτ )ze,
C = ΩN (xe − yp) + (1 − ω)z,
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 the dimensionless time τ = ωa t. Since we do not include losses and pump into consideration, this set possesses two integrals of motion: R = x + y + z = 1, 2
2
2
W = e + p + 2z, 2
2
(4)
(8)
reflecting a conservation of the interaction energy which is valid within the RWA. By differentiating the third equation in the set (3) with α = 0 and making use of the conservation laws (4) and (8), we derive the following closed equation for the density of the atomic population inversion: z¨ = f (z), z(0) = z0 ,
z˙ (0) = ΩN (x0 p0 + y0 e0 ), (9)
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where f (z) = 3ΩN2 z2 − [(ω − 1)2 + ΩN2 W ]z +(ω − 1)C + ΩN2 .
(10)
Thus, the autonomous Maxwell–Bloch equations without losses and pump are reduced to a single ordinary differential equation of the second order describing a free classical oscillator (9) driven by the nonlinear restoring force f (z). The energy integral of Eq. (9): 1 2 E = z˙ + Π (z), Π (z) = − f (z) dz (11) 2 helps us to integrate it in the form of the elliptic integral of the first order: z(τ ) √ dz 2 dτ = . (12) √ E − Π (z) z0 By inverting this integral, it is easy to find the solution for the atomic inversion in terms of the elliptic Jacobian sine: √ z(τ ) = z1 + (z2 − z1 ) sn2 ( z1 − z3 ΩN τ + ϕ, k), (13) where ϕ = sn
−1
z1 ,k , √ z1 − z 2
z2 k = z1 − − z3 , z1 (14)
and z1 , z2 and z3 are the roots of the algebraic cubic equation E − Π (z) = 0. At exact resonance between the atoms and the cavity mode, i.e. at ω = 1, and under the initial conditions corresponding to fully inverted atoms, z0 = 1, the general solution (13) takes the form of the special solution: zJC (τ ) = −1 + 2 sn2 ( n0 + 1ΩN τ + ϕJC , kJC ), 1 1 −1 ϕJC = sn 1, √ , kJC = √ , (15) n0 − 1 n0 + 1 first found by Jaynes and Cummings [5]. Here n0 = 1 2 2 4 (e0 + p0 ) 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 trigonometric ones. In this limit, the solution (16) zh (τ ) sin( 2n0 ΩN τ + const) describes harmonic free oscillations of the atomic population inversion. In the limit of initial vacuum field, n0 = 0, they are approximated by the hyperbolic functions yielding the solution on the separatrix zs (τ ) 1 − 2 sech2 ΩN τ
(17)
that describes the locus of states in which atoms radiate and reabsorb their own radiation field in infinite time. Projections of the phase portrait of the atomic oscillator (9) onto the plane z˙ –z consist of a collection of periodic 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) corresponding to initially fully inverted atoms, z = 1, x = y = 0, and the vacuum field, e = p = 0. General exact solutions of the autonomous Maxwell–Bloch equations for the field, e and p, and the atomic polarization components, x and y, have been found in our paper [10]. It is important that the nonautonomous Maxwell–Bloch equations (3) with α = 0 are integrable at exact resonance for any kind of modulation of the coupling ΩN (τ ). It follows from the existence of the third integral of motion xe − yp = const that is valid at ω = 1 for any differentiable function of time ΩN (τ ). General exact solutions of the resonant nonautonomous Maxwell–Bloch equations are easily found from the respective solutions in the autonomous limit by putting in the latter τ ω = 1 and transforming to the new “time” τ → 0 ΩN (τ ) dτ . 2.3. Chaotic parametric Rabi oscillations The integrable limits of the Maxwell–Bloch equations (3) and the respective solutions, discussed in the preceding section, describe the regular Rabi oscillations of the internal energy of two-level atoms interacting with a single-mode cavity field which occur, at least, in the two following cases: if modulation of
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Fig. 1. Poincar´e 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 the eigenfrequency of a vibrating cavity is tuned to exact resonance with atomic working transition (ω = 1). To give an idea about increasing the complexity of motion with α = 0 and ω = 1, we represent in Fig. 1 Poincaré sections of the Hamiltonian flow (3) on the field plane e–p at different values of the depth of modulation α. These sections are computed with the following ini-
2 tial conditions: e0 = p0 = 1, x0 = 0, y0i = 1 − z0i 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 Rabi frequency ΩN , detuning ω and the modulation frequency δ are given in the figure caption. With increasing values of α, the invariant sets are broken, and the accessible part of the phase plane is filled with dots, but a kind of regular structure is still visible in the sections even with large values of the depth of modulation. In Fig. 2 we show the long-time evolution of the atomic population inversion z(τ ) with ΩN = 1, ω = 0.9 and α = 1 at different values of the modulation
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Fig. 2. Parametric Rabi oscillations of the atomic population inversion with ΩN = 1, ω = 0.9, α = 1 and the different values of the modulation 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 initially in the following state: x0 = y0 = 0, z0 = 1, e0 = p0 = 1, that corresponds to initially fully inverted atoms and initial density of photons to be equal to n0 = 1 2 . A characteristic periodical structure of the parametric Rabi oscillations with trains of high-amplitude oscillations intermitted by their collapses, that are due to the modulation, is seen in this figure. At times τm = (3 + 4m)π/2δ (m = 0, 1, 2, . . . ), when the modulation function takes its minimal values, the effective atom–field coupling goes to zero (at α = 1). The rate of change of z(τ ) decreases with decreasing the atom–field coupling (see the third equation in the set (3)), and the parametric oscillations begin to slowdown near the time moments τm (that is seen in all the three fragments of Fig. 2) resulting eventually
in the collapses of the Rabi oscillations. With a given value of δ, all the trains have the same duration equal to the modulation period Tm = 2π/δ, but the amplitudes of the oscillations and their behavior between successive trains look more and more chaotic with increasing δ. In order to check, it we compute the maximal Lyapunov exponent λ for each of these signals. The computation confirms that the signal in Fig. 2a with Tm 6280 is quasiperiodic with λ 0, the signal in Fig. 2b with Tm 628 is weakly chaotic with λ 0.006, and the signal in Fig. 2c with Tm 125.6 is chaotic with λ 0.022. To understand a mechanism of arising the chaotic parametric Rabi oscillations shown in Fig. 2 and manifested in Fig. 1, we make use the Melnikov method of analyzing solutions of nonautonomous dynamical
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systems in a neighborhood of their unperturbed invariant sets [12]. Without modulation, the evolution of the atom–field system is quasiperiodic. The main effect of modulation is to produce, out of resonance (ω = 1), a homoclinic structure in the vicinity of the separatrix of the unperturbed Maxwell–Bloch set of equations. The basic idea of the Melnikov analysis is to make use of exact 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 as compared with the Rabi frequency, α ΩN . With the help of the integrals of motion, one can represent the homoclinic manifold implicitly by the equations R = 1, W = 2, C = 1 − ω. The signed distance between the stable and unstable manifolds of the equilibrium hyperbolic point S+ at the moment τ0 along the normal to the unperturbed homoclinic manifold is proportional to the expression αM(τ0 ) + O(α 2 ), where the Melnikov function M(τ0 ) is evaluated along the separatrix of the Maxwell–Bloch equations without modulation which has been found in the paper [10] (see Eq. (33) in [10]). Details of calculating the Melnikov function for equations of the Maxwell–Bloch type can
317
be found in [10,13]. In our case, the final result is the following: M(τ0 ) =
2π(1 − ω)δ 2 ΩN3 sh(δπ/βΩN )
cos(δτ0 ),
(18)
where β 2 = 4 − [(ω − 1)/ΩN ]2 . It is evident from (18) that out of resonance, ω = 1, the Melnikov function has simple zeroes as a function of τ0 , where τ0 parameterizes time of the motion of a representative point on a trajectory. If M(τ0 ) has simple zeroes, then the stable and unstable manifolds of the hyperbolic point intersect transversally resulting in Smale horseshoe chaos. This intersection occurs with the frequency of modulation δ manifesting itself in Fig. 2 as irregular oscillations between successive trains of the parametric Rabi oscillations. A splitting of the stable and unstable hyperbolic manifolds, coinciding in the integrable limit, takes place under an arbitrary small depth of modulation α. We have performed some computer simulation on the atom–field system (3) assuming the depth of modulation of the Rabi frequency to be large. In closed dynamical system, chaos has its origin in extremal
√ 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.
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Fig. 4. Topographical λ-map on the plane ω–ΩN with α = 1 and δ = 0.01.
sensitivity to initial conditions which is characterized by positive values of the maximal Lyapunov exponent. The local exponential divergence of trajectories produces a local stretching, but because of the global
confinement in the phase space of our conservative Hamiltonian system with two degrees of freedom, this stretching is accompanied by folding. Repeated stretching and folding produces very complicated
Fig. 5. Topographical λ-map on the plane ω–ΩN with α = 1 and δ = 0.1.
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Fig. 6. Topographical λ-map on the plane log10 δ–ΩN with α = 1 and ω = 0.9.
motion that is known as chaotic. The dependence of the maximal Lyapunov exponent on the values of the collective Rabi frequency ΩN is shown in Fig. 3 with two different values of the modulation frequency δ. Since the number of atoms may be written with the help of Eq. (5) as follows: N = (ΩN ωa /Ω0 )2 , these figures may be considered as a dependence of λ on the number of atoms N . In the regime of very weak chaos, the curve (a) demonstrates a curious periodic structure. When a dynamical system possesses more than a single control parameter, it is useful to compute topographical λ-maps [8] that give a representative “portrait” of chaos. Such a map shows the values of the maximal Lyapunov exponent λ of a system with given initial conditions as a function of two of the control parameters with fixed values of the other ones. We have computed the topographical λ-maps with all the three control parameters ΩN , ω, and δ varied. The values of λ are encoded by the color using a linear scale which is shown on the right sides of the maps. Figs. 4 and 5 show the ω–ΩN maps
calculated at the fixed values of the normalized detuning δ = 0.01 and δ = 0.1, respectively. It is clear from the figures that chaos disappears when the atomic transition frequency is closed to the field frequency, and the chaotic “sea” enlarges with increasing δ. Fig. 6 shows the topographical λ-map on the frequency plane log10 δ–ΩN at α = 1 and ω = 0.9. The maps give us, in fact, the values of λ as a function of the number of atoms N which may be considered as an adjustable control parameter of the atom–field system together with δ and ω.
3. Wavelet analysis of structural Hamiltonian chaos The signals of the parametric Rabi oscillations computed in the preceding section demonstrate a visible structure even if they are proven to be chaotic in a sense of extremal sensitivity to initial conditions which is characterized by a positive value of the maximal Lyapunov exponent. Homoclinic nature
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of Hamiltonian dynamical chaos in the atom–field system has been proven above by the Melnikov method. In order to know a temporal kind of this structural chaos we compute in this section a wavelet transformation of the time series z(τ ) as follows: ∞ F (T , τ ) = z(t)φT∗ τ (t) dt, (19) −∞
Fig. 7. High-frequency range of the wavelet spectra of the parametric 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 correspond to the respective values of the frequency components of a signal.
√ where φT τ (t) = (π 2τmax /4T )φ[(t − τ )/T ], φ = exp(ik0 t) exp(−t 2 /2) is a basic Morlet wavelet, τmax the maximal duration of a signal z(τ ), k0 the fitting parameter, t the integration time and T −1 the frequency in dimensionless units. A two-dimensional matrix F (T , τ ) computed can be represented as a two-dimensional chart with absolute values of the wavelet transformation (19) encoded by the color using a linear scale and increasing from white to black color. The frequency and temporal scales of the signals z(τ ) are indicated in Figs. 7 and 8 along the axes T and τ , respectively. Each point (Ti , τi ) in the wavelet chart is a convolution of the signal z(τ ) with the Morlet wavelet φ shifted to the value τi and stretched to the value Ti . In such a way, the transformation F (T , τ ) gives an information about temporal and frequency peculiarities of a signal simultaneously. In fact, wavelet spectra show in which way frequency components of a signal under consideration change in time. In the fragments (a), (b) and (c) of Figs. 7 and 8, we demonstrate results of the wavelet transformation (19) of the Rabi oscillation signals z(τ ) shown in the respective fragments of Fig. 2. Fig. 7 shows high-frequency parts of the respective wavelet spectra computed with the fitting parameter k0 = 96. The values of the maximal Lyapunov exponent computed in the previous section have indicated that the signal of the parametric Rabi oscillations with the modulation frequency δ = 0.001 (Fig. 2a) is regular with λ 0, the signal with δ = 0.01 (Fig. 2b) is weakly chaotic with 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 signals demonstrate regular behavior characterized by two periodic functions. One of them describes variations in
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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 visible structure of the parametric Rabi oscillations that is due to modulation with the period Tm and characteristical slowdown of these oscillations in time moments τm = (3 + 4m)π/2δ. Fig. 7a shows that the high-frequency components of the regular signal with the comparatively long period of modulation Tm 6280 change in time regularly with the periods Tm and 2Tm in a rather wide frequency ranging from T −1 0.01 to 0.3. With the decrease in the modulation period Tm in n times, extremal values of this range decreases approximately in n times extending from T −1 0.1 to 3 (Fig. 7b with Tm 628) and from T −1 0.5 to 12 (Fig. 7c with Tm 125.6). The high-frequency domains of the wavelet spectra of the chaotic signals (b) and (c) demonstrate in their tops irregular appearing and disappearing components with comparatively low frequencies. One may conclude that the very high-frequency components of the parametric Rabi oscillations change in time regularly both for periodic and for chaotic signals. In Fig. 8, we demonstrate low-frequency parts of the wavelet spectra of the same signals shown in Fig. 2 and computed with the formula (19) but with the fitting parameter k0 = 2. In contrast to the high-frequency wavelets, the low-frequency spectra are very different for regular and chaotic signals. Low-frequency components of the regular signal (Tm 6280) up to the values of the order of 10−4 corresponding to the length of the signal integrated are again changed in time periodically (see Fig. 8a). The components of the weakly chaotic signal (Tm 628) with λ 0.006 demonstrate in the range from 10−1 to 10−3 chaotic behavior (please note the black spots in Fig. 8b), whereas the 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 up to the values of the order T −1 5 × 10−3 (Fig. 8c). Thus, the wavelet spectra computed with typical signals of the parametric Rabi oscillations of the atomic population inversion elucidate a structural temporal kind of Hamiltonian chaos in the simple atom–field system with modulation.
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Fig. 8. Low-frequency range of the wavelet spectra of the parametric 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 correspond to the respective values of the frequency components of a signal.
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4. Summary and conclusion We have analyzed the RWA nonlinear dynamics in one of the simplest models of laser and atomic physics that comprises two-level atoms in an ideal high-Q single-mode cavity. The main problem in which we have been interested in this paper was a transition to Hamiltonian chaos in the coherent atom–field interaction with the atom–field coupling to be modulated. The Heisenberg equations for expectation values of a complete set of the atomic and field observables have been shown to be integrable in the autonomous limit without any modulation and in the resonance limit with an arbitrary modulation. They possess special orbits that are homoclinic to the state with fully inverted atoms and vacuum cavity field which is an equilibrium one in the semiclassical approximation. With the help of the Melnikov method, we have proved analytically transverse intersections of stable and unstable manifolds of this equilibrium point under a small modulation of the Rabi frequency. These transverse intersections are believed to provide Smale horseshoe mechanism of chaos. To confirm numerically the chaotic dynamics in the semiclassical RWA model, we have computed Poincaré sections and maximal Lyapunov exponents under strong modulation of the Rabi frequency. The Lyapunov topographical maps showing the regions of regular and chaotic motion in the space of the control parameters provide representative numerical “portraits” of the system’s dynamics in different ranges of its control parameters. We could show that Hamiltonian chaos in the coherent parametric atom–field interaction is of a special kind that is characterized both by a sensitive dependence on initial conditions and by long-lived structures visible in the parametric Rabi oscillations of the atomic population inversion and in the respective Poincaré sections. An intermittent route to this structural Hamiltonian chaos has been clearly demonstrated in the wavelet spectra of the typical signals of the parametric Rabi oscillations. It has been shown that with increasing modulation frequency, the low-frequency components of the respective wavelet spectrum changed irregularly in time appearing and disappearing in a chaotic way. At the same time,
the high-frequency components of the same wavelet spectrum change periodically in time even for the signals with positive values of the maximal Lyapunov exponents. Coexistence of regular and irregular components in the same signal is a visiting card of structural Hamiltonian chaos. In conclusion, we would like to present some speculations on a possibility to observe manifestations of structural Hamiltonian chaos with real devices. The Hamiltonian approach we have adopted throughout the paper is valid over time intervals shorter than all the characteristic relaxation times. This approach is based on the strong-coupling limit in the atom–field interaction which √ is characterized by the following inequality: (Ω0 N )−1 Ta , Tf , where Ta and Tf are the atomic and field relaxation times, respectively. In numerical experiments computing maximal Lyapunov exponents, Hamiltonian chaos can be diagnosed over time interval of the order of the correlation decoupling time Tcor = 2π/ωa λ. In terms of the collective Rabi frequency ΩN and the maximal Lyapunov exponent λ, the condition for observing Hamiltonian chaos numerically can be rewritten as ΩN λ. A Rydberg maser, operating in the strong-coupling regime with two-level Rydberg atoms inside a high-Q cavity, seems to be a promising device for observing semiclassical Hamiltonian chaos in real experiment. The Rydberg atoms have the transition frequency of the order of ωa 1011 –1012 rad/s, the magnitude of the electric dipole matrix element d 103 atomic units, the single-photon vacuum Rabi frequency Ω0 105 –106 rad/s, and the lifetime of the circular Rydberg states, Ta 10−2 s [14]. Parameters of a typical superconducting microwave maser cavity are the following: the quality factor Q 1010 , the cavity length Lc 1 cm, and the lifetime of intracavity photons Tf 10−1 –10−2 s [14]. As it follows from our numerical results, one can reach the regime of the chaotic parametric Rabi oscillations with the strength of chaos of the order of λ 0.01 operating with a droplet consisting approximately of 1010 atoms and the frequency of modulation of the order of δ 0.01. We used the semiclassical approximation throughout the paper ignoring subtle questions of quantum– classical correspondence in the coherent parametric
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atom–field interaction. It should be mentioned that some manifestations of quantum chaos have been studied with a system of N two-level atoms in a perfect cavity. In the model [15], atoms were assumed to interact with a resonant cavity field taking into account virtual atom–field transitions violating the RWA. In Ref. [16], atoms were considered to interact with a resonant cavity field and with an external coherent field in the framework of the RWA. It was shown with both the models that in the range of parameters for developed semiclassical chaos (when neglecting quantum correlations), the semiclassical approximation was violated by quantum effects at the time scale τ ∼ ln N known as the breaktime of the quantum–classical correspondence [17]. We plan to study in future this effect in parametric Rabi oscillations of N two-level atoms in a cavity with modulated atom–field coupling.
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Acknowledgements This work was supported by the grant from the Russian Foundation for Basic Research No. 02-17269.
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H. Haken, Phys. Lett. A 53 (1975) 77. E.N. Lorenz, J. Atmos. Sci. 20 (1963) 130. R.G. Harrison, Contemp. Phys. 29 (1988) 341. R.M. Dicke, Phys. Rev. 93 (1954) 493. E.T. Jaynes, F.W. Cummings, Proc. IEEE 51 (1963) 89. P.R. Berman (Ed.), Cavity Quantum Electrodynamics, Academic Press, New York, 1994. P.I. Belobrov, G.M. Zaslavskii, G.Kh. Tartakovskii, Zh. Eksp. Teor. Fiz. 71 (1976) 1799 [Sov. Phys. JETP 44 (1976) 945]. S.V. Prants, L.E. Kon’kov, Phys. Lett. A 225 (1997) 33. S.V. Prants, L.E. Kon’kov, Zh. Esksp. Teor. Fiz. 115 (1999) 740 [JETP 88 (1999) 406]. S.V. Prants, L.E. Kon’kov, I.L. Kirilyuk, Phys. Rev. E 60 (1999) 335. L.E. Kon’kov, S.V. Prants, J. Math. Phys. 37 (1996) 1204. V.K. Melnikov, Trans. Moscow Math. Soc. 12 (1963) 3. D.D. Holm, G. Kovacic, T.A. Wettergen, Phys. Lett. A 200 (1995) 299. J.M. Raimond, S. Haroche, Confined Electrons and Photons, Plenum Press, New York, 1995, p. 383. G.P. Berman, E.N. Bulgakov, G.M. Zaslavsky, Chaos 2 (1992) 257. G.P. Berman, E.N. Bulgakov, D.D. Holm, Phys. Rev. A 49 (1994) 4943. G.P. Berman, G.M. Zaslavsky, Physica A 91 (1978) 450.
Structural Hamiltonian chaos in the coherent parametric atom–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 2001 Communicated by E. Ott
Abstract We consider one of the simplest semiclassical models in laser and atomic physics, a collection of two-level atoms interacting coherently 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 system whose atomic and field variables oscillate in a self-consistent way with the Rabi frequency. It is proven analytically by the Melnikov method and numerically by computing maximal Lyapunov exponents and Poincaré sections that the parametric Rabi oscillations in a vibrating cavity may be chaotic in a sense of exponential sensitivity to initial conditions. Wavelet spectra computed with typical signals of the oscillations demonstrate clearly that Hamiltonian chaos in the coherent atom–field interaction with modulated coupling is structural. Structural chaos is characterized by positive values of the maximal Lyapunov exponents and regular structures which coexist in the same signal. Duration of a train of regular oscillations is defined by the period of modulation, but oscillations between successive trains are chaotic. Results of numerical experiments and estimation of the control parameters and approximations involved show that a Rydberg atom maser operating with a collection of two-level atoms inside a high-quality superconducting microwave cavity is a promising device for observing manifestations of 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 radiation and matter, comprising a collection of two-level quantum systems coupled with a single-mode electromagnetic field, provides the basis for laser physics and describes a rich variety of nonlinear dynamical ∗ Corresponding author. Tel.: +7-4232-313081; fax: +7-4232-312573. E-mail address: [email protected] (S.V. Prants).
behavior. The discovery that a single-mode laser, a symbol of coherence and stability, may exhibit deterministic instabilities and chaos is especially important since lasers provide nearly ideal systems to test general ideas in statistical physics. From the stand point of nonlinear dynamics, laser is an open dissipative system which transforms an external excitation into a coherent output in the presence of loss. It is well known [1] that a single-mode, homogeneously broadened laser, operating on resonance with the gain center, can be described in the semiclassical
0167-2789/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 7 8 9 ( 0 1 ) 0 0 2 6 4 - 0
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rotating-wave approximation (RWA) by three real Maxwell–Bloch equations which have been shown to be equivalent to the Lorenz model for fluid convection [2]. Some manifestations of a Lorenz-type strange attractor and dissipative chaos have been observed with different types of lasers (for a review, see [3]). Physics of the radiation–matter interaction can be simplified even more by considering the interaction of two-level atoms with their own radiation field in a perfect single-mode cavity without any external excitation. Fully quantum models of Dicke [4] and Jaynes and Cummings [5] provide the basis of cavity quantum electrodynamics [6]. Recent exciting achievements in this rapidly growing field, especially, creating micromasers and microlasers, are now allowing direct tests of foundations of quantum mechanics. These devices can serve as a testing ground for the problem of correspondence between classical and quantum dynamics since micromasers and microlasers can be easily extended to many-atom operation providing one to step from regimes which ask for a quantum description to regimes which ask for a semiclassical description and vice versa. Here we study extremely nonlinear dynamics of a strongly coupled atom–field system in a lossless single-mode cavity, assuming the number of atoms to be sufficiently large in order to adopt the semiclassical approximation. Semiclassical equations of motion for this system may be reduced to the Maxwell–Bloch equations for three real independent variables which, in difference from the laser theory, do not include losses and pump. These equations are, in general, nonintegrable, but they become integrable immediately after adopting the RWA [5] that implies the existence of an additional integral of motion, conservation of the so-called number of excitations. In early studies of the Maxwell–Bloch equations, it has been shown theoretically and numerically that they may demonstrate dynamical instabilities and chaos of Hamiltonian type if one goes beyond the RWA [7] or if one stays within the RWA, but atoms move through a cavity in a direction along which a cavity sustains a field that is periodic in space [8]. Numerical experiments have shown that in the first case, prominent chaos arises when the density of atoms is very large (approximately 1020 cm3 in
the optical range [7]). In the second case, the speed of atoms should be very high (approximately a few percent of the speed of light in vacuum [8]) in order to observe a transition to chaos numerically. In this paper, we propose another physical mechanism that may lead within the RWA to Hamiltonian chaos with atoms at rest in a single-mode cavity. From the theoretical point of view, the idea is to introduce a time-dependent parameter to make the Maxwell–Bloch set of equations nonautonomous and nonintegrable. Practically, it may be done by modulating cavity length by means of an electro-optical modulator. Modulation of positions of the nodes of a cavity standing wave gives rise to modulation of the coupling strength between atoms and the field. The resulting parametric Rabi oscillations of the atomic population inversion will be studied by different methods 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 the standard quantum–optical Hamiltonian H =
N 1 1 j ωa σz + ωf a † a + 2 2 j =1
+Ω0 (t)
N j =1
j
j
(aσ+ + a † σ− ),
(1)
which describes the interaction of N identical two-level atoms with a single mode of a quantized electromagnetic field in the RWA and the point-like approximation. The energy separation of all the atoms is supposed to be the same and equal to ωa . The Pauli matrices σz and σ± = 21 (σx ± iσy ) describe the internal atomic dynamics, and the Boson operators a and a † characterize the field mode with the frequency ωf . In difference from the standard Dicke and Jaynes–Cummings models [4,5], we incorporate into the Hamiltonian a time-dependent atom–field coupling Ω0 (t) which may describe the effect of
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a parametric modulation imposed on the system. We assume a harmonic modulation of cavity length which causes the respective variance of the nodes of a standing light wave which, in turn, results in the harmonic modulation of the atom–field coupling Ω0 (1 + α sin ωm t), where α and ωm are the depth and frequency of the modulation, respectively. The point-like approximation means that a collection of atoms is so confined that the amplitude value of the coupling Ω0 = dE0 / may be put to be the same for all the atoms in the ensemble (here d is the magnitude √ of the dipole matrix element, E0 = ωf /2V , V is the effective cavity mode volume). The next step is to derive in the Heisenberg representation the equations of motion from the Hamiltonian (1). In the semiclassical approximation which is valid with the accuracy of the order of 1/N [9], one can replace all the operators by their expectation values over an arbitrary initial quantum state of the atoms and the field. Choosing the following expectation values as dynamical variables: N
N
1 j x= σx , N
1 j y= σy , N
j =1
z=
1 N
N
j =1
1 e = √ a + a † , N
j
σz ,
j =1
i p = √ a † − a, N
(2)
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reflecting a conservation of the length of the Bloch vector and of the number of excitations, respectively. With the help of semiclassical factorization made, we reduce the infinite-dimensional state space of the fully quantum atom–field system to the five-dimensional phase space, which is a direct product space of the Bloch sphere and the oscillator plane. In fact, we have three independent real-valued variables, one for a field component (e or p), another one for the density of the atomic population inversion z, and the third one for an atomic polarization component (x or y). The normalized collective Rabi frequency √ Ω0 N , (5) ΩN = ωa the normalized detuning from resonance ωf ω= , ωa and the normalized modulation frequency ωm δ= ωa
(6)
(7)
play the role of the control parameters of the nonautonomous nonlinear dynamical system (3). In conclusion of the characterization of the model, we note that an equivalent set of semiclassical equations may be obtained in the Schrodinger representation by coupling the Maxwell equations with the Schrodinger equations for atomic transition amplitudes (see, for example, [11]).
we close the respective set in the form of the dimensionless Maxwell–Bloch equations:
2.2. Regular free Rabi oscillations
x˙ = −y − ΩN (1 + α sin δτ )zp,
Without modulation, the set (3) has an additional integral of motion
y˙ = x − ΩN (1 + α sin δτ )ze,
C = ΩN (xe − yp) + (1 − ω)z,
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 the dimensionless time τ = ωa t. Since we do not include losses and pump into consideration, this set possesses two integrals of motion: R = x + y + z = 1, 2
2
2
W = e + p + 2z, 2
2
(4)
(8)
reflecting a conservation of the interaction energy which is valid within the RWA. By differentiating the third equation in the set (3) with α = 0 and making use of the conservation laws (4) and (8), we derive the following closed equation for the density of the atomic population inversion: z¨ = f (z), z(0) = z0 ,
z˙ (0) = ΩN (x0 p0 + y0 e0 ), (9)
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where f (z) = 3ΩN2 z2 − [(ω − 1)2 + ΩN2 W ]z +(ω − 1)C + ΩN2 .
(10)
Thus, the autonomous Maxwell–Bloch equations without losses and pump are reduced to a single ordinary differential equation of the second order describing a free classical oscillator (9) driven by the nonlinear restoring force f (z). The energy integral of Eq. (9): 1 2 E = z˙ + Π (z), Π (z) = − f (z) dz (11) 2 helps us to integrate it in the form of the elliptic integral of the first order: z(τ ) √ dz 2 dτ = . (12) √ E − Π (z) z0 By inverting this integral, it is easy to find the solution for the atomic inversion in terms of the elliptic Jacobian sine: √ z(τ ) = z1 + (z2 − z1 ) sn2 ( z1 − z3 ΩN τ + ϕ, k), (13) where ϕ = sn
−1
z1 ,k , √ z1 − z 2
z2 k = z1 − − z3 , z1 (14)
and z1 , z2 and z3 are the roots of the algebraic cubic equation E − Π (z) = 0. At exact resonance between the atoms and the cavity mode, i.e. at ω = 1, and under the initial conditions corresponding to fully inverted atoms, z0 = 1, the general solution (13) takes the form of the special solution: zJC (τ ) = −1 + 2 sn2 ( n0 + 1ΩN τ + ϕJC , kJC ), 1 1 −1 ϕJC = sn 1, √ , kJC = √ , (15) n0 − 1 n0 + 1 first found by Jaynes and Cummings [5]. Here n0 = 1 2 2 4 (e0 + p0 ) 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 trigonometric ones. In this limit, the solution (16) zh (τ ) sin( 2n0 ΩN τ + const) describes harmonic free oscillations of the atomic population inversion. In the limit of initial vacuum field, n0 = 0, they are approximated by the hyperbolic functions yielding the solution on the separatrix zs (τ ) 1 − 2 sech2 ΩN τ
(17)
that describes the locus of states in which atoms radiate and reabsorb their own radiation field in infinite time. Projections of the phase portrait of the atomic oscillator (9) onto the plane z˙ –z consist of a collection of periodic 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) corresponding to initially fully inverted atoms, z = 1, x = y = 0, and the vacuum field, e = p = 0. General exact solutions of the autonomous Maxwell–Bloch equations for the field, e and p, and the atomic polarization components, x and y, have been found in our paper [10]. It is important that the nonautonomous Maxwell–Bloch equations (3) with α = 0 are integrable at exact resonance for any kind of modulation of the coupling ΩN (τ ). It follows from the existence of the third integral of motion xe − yp = const that is valid at ω = 1 for any differentiable function of time ΩN (τ ). General exact solutions of the resonant nonautonomous Maxwell–Bloch equations are easily found from the respective solutions in the autonomous limit by putting in the latter τ ω = 1 and transforming to the new “time” τ → 0 ΩN (τ ) dτ . 2.3. Chaotic parametric Rabi oscillations The integrable limits of the Maxwell–Bloch equations (3) and the respective solutions, discussed in the preceding section, describe the regular Rabi oscillations of the internal energy of two-level atoms interacting with a single-mode cavity field which occur, at least, in the two following cases: if modulation of
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Fig. 1. Poincar´e 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 the eigenfrequency of a vibrating cavity is tuned to exact resonance with atomic working transition (ω = 1). To give an idea about increasing the complexity of motion with α = 0 and ω = 1, we represent in Fig. 1 Poincaré sections of the Hamiltonian flow (3) on the field plane e–p at different values of the depth of modulation α. These sections are computed with the following ini-
2 tial conditions: e0 = p0 = 1, x0 = 0, y0i = 1 − z0i 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 Rabi frequency ΩN , detuning ω and the modulation frequency δ are given in the figure caption. With increasing values of α, the invariant sets are broken, and the accessible part of the phase plane is filled with dots, but a kind of regular structure is still visible in the sections even with large values of the depth of modulation. In Fig. 2 we show the long-time evolution of the atomic population inversion z(τ ) with ΩN = 1, ω = 0.9 and α = 1 at different values of the modulation
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Fig. 2. Parametric Rabi oscillations of the atomic population inversion with ΩN = 1, ω = 0.9, α = 1 and the different values of the modulation 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 initially in the following state: x0 = y0 = 0, z0 = 1, e0 = p0 = 1, that corresponds to initially fully inverted atoms and initial density of photons to be equal to n0 = 1 2 . A characteristic periodical structure of the parametric Rabi oscillations with trains of high-amplitude oscillations intermitted by their collapses, that are due to the modulation, is seen in this figure. At times τm = (3 + 4m)π/2δ (m = 0, 1, 2, . . . ), when the modulation function takes its minimal values, the effective atom–field coupling goes to zero (at α = 1). The rate of change of z(τ ) decreases with decreasing the atom–field coupling (see the third equation in the set (3)), and the parametric oscillations begin to slowdown near the time moments τm (that is seen in all the three fragments of Fig. 2) resulting eventually
in the collapses of the Rabi oscillations. With a given value of δ, all the trains have the same duration equal to the modulation period Tm = 2π/δ, but the amplitudes of the oscillations and their behavior between successive trains look more and more chaotic with increasing δ. In order to check, it we compute the maximal Lyapunov exponent λ for each of these signals. The computation confirms that the signal in Fig. 2a with Tm 6280 is quasiperiodic with λ 0, the signal in Fig. 2b with Tm 628 is weakly chaotic with λ 0.006, and the signal in Fig. 2c with Tm 125.6 is chaotic with λ 0.022. To understand a mechanism of arising the chaotic parametric Rabi oscillations shown in Fig. 2 and manifested in Fig. 1, we make use the Melnikov method of analyzing solutions of nonautonomous dynamical
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systems in a neighborhood of their unperturbed invariant sets [12]. Without modulation, the evolution of the atom–field system is quasiperiodic. The main effect of modulation is to produce, out of resonance (ω = 1), a homoclinic structure in the vicinity of the separatrix of the unperturbed Maxwell–Bloch set of equations. The basic idea of the Melnikov analysis is to make use of exact 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 as compared with the Rabi frequency, α ΩN . With the help of the integrals of motion, one can represent the homoclinic manifold implicitly by the equations R = 1, W = 2, C = 1 − ω. The signed distance between the stable and unstable manifolds of the equilibrium hyperbolic point S+ at the moment τ0 along the normal to the unperturbed homoclinic manifold is proportional to the expression αM(τ0 ) + O(α 2 ), where the Melnikov function M(τ0 ) is evaluated along the separatrix of the Maxwell–Bloch equations without modulation which has been found in the paper [10] (see Eq. (33) in [10]). Details of calculating the Melnikov function for equations of the Maxwell–Bloch type can
317
be found in [10,13]. In our case, the final result is the following: M(τ0 ) =
2π(1 − ω)δ 2 ΩN3 sh(δπ/βΩN )
cos(δτ0 ),
(18)
where β 2 = 4 − [(ω − 1)/ΩN ]2 . It is evident from (18) that out of resonance, ω = 1, the Melnikov function has simple zeroes as a function of τ0 , where τ0 parameterizes time of the motion of a representative point on a trajectory. If M(τ0 ) has simple zeroes, then the stable and unstable manifolds of the hyperbolic point intersect transversally resulting in Smale horseshoe chaos. This intersection occurs with the frequency of modulation δ manifesting itself in Fig. 2 as irregular oscillations between successive trains of the parametric Rabi oscillations. A splitting of the stable and unstable hyperbolic manifolds, coinciding in the integrable limit, takes place under an arbitrary small depth of modulation α. We have performed some computer simulation on the atom–field system (3) assuming the depth of modulation of the Rabi frequency to be large. In closed dynamical system, chaos has its origin in extremal
√ 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.
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Fig. 4. Topographical λ-map on the plane ω–ΩN with α = 1 and δ = 0.01.
sensitivity to initial conditions which is characterized by positive values of the maximal Lyapunov exponent. The local exponential divergence of trajectories produces a local stretching, but because of the global
confinement in the phase space of our conservative Hamiltonian system with two degrees of freedom, this stretching is accompanied by folding. Repeated stretching and folding produces very complicated
Fig. 5. Topographical λ-map on the plane ω–ΩN with α = 1 and δ = 0.1.
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Fig. 6. Topographical λ-map on the plane log10 δ–ΩN with α = 1 and ω = 0.9.
motion that is known as chaotic. The dependence of the maximal Lyapunov exponent on the values of the collective Rabi frequency ΩN is shown in Fig. 3 with two different values of the modulation frequency δ. Since the number of atoms may be written with the help of Eq. (5) as follows: N = (ΩN ωa /Ω0 )2 , these figures may be considered as a dependence of λ on the number of atoms N . In the regime of very weak chaos, the curve (a) demonstrates a curious periodic structure. When a dynamical system possesses more than a single control parameter, it is useful to compute topographical λ-maps [8] that give a representative “portrait” of chaos. Such a map shows the values of the maximal Lyapunov exponent λ of a system with given initial conditions as a function of two of the control parameters with fixed values of the other ones. We have computed the topographical λ-maps with all the three control parameters ΩN , ω, and δ varied. The values of λ are encoded by the color using a linear scale which is shown on the right sides of the maps. Figs. 4 and 5 show the ω–ΩN maps
calculated at the fixed values of the normalized detuning δ = 0.01 and δ = 0.1, respectively. It is clear from the figures that chaos disappears when the atomic transition frequency is closed to the field frequency, and the chaotic “sea” enlarges with increasing δ. Fig. 6 shows the topographical λ-map on the frequency plane log10 δ–ΩN at α = 1 and ω = 0.9. The maps give us, in fact, the values of λ as a function of the number of atoms N which may be considered as an adjustable control parameter of the atom–field system together with δ and ω.
3. Wavelet analysis of structural Hamiltonian chaos The signals of the parametric Rabi oscillations computed in the preceding section demonstrate a visible structure even if they are proven to be chaotic in a sense of extremal sensitivity to initial conditions which is characterized by a positive value of the maximal Lyapunov exponent. Homoclinic nature
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of Hamiltonian dynamical chaos in the atom–field system has been proven above by the Melnikov method. In order to know a temporal kind of this structural chaos we compute in this section a wavelet transformation of the time series z(τ ) as follows: ∞ F (T , τ ) = z(t)φT∗ τ (t) dt, (19) −∞
Fig. 7. High-frequency range of the wavelet spectra of the parametric 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 correspond to the respective values of the frequency components of a signal.
√ where φT τ (t) = (π 2τmax /4T )φ[(t − τ )/T ], φ = exp(ik0 t) exp(−t 2 /2) is a basic Morlet wavelet, τmax the maximal duration of a signal z(τ ), k0 the fitting parameter, t the integration time and T −1 the frequency in dimensionless units. A two-dimensional matrix F (T , τ ) computed can be represented as a two-dimensional chart with absolute values of the wavelet transformation (19) encoded by the color using a linear scale and increasing from white to black color. The frequency and temporal scales of the signals z(τ ) are indicated in Figs. 7 and 8 along the axes T and τ , respectively. Each point (Ti , τi ) in the wavelet chart is a convolution of the signal z(τ ) with the Morlet wavelet φ shifted to the value τi and stretched to the value Ti . In such a way, the transformation F (T , τ ) gives an information about temporal and frequency peculiarities of a signal simultaneously. In fact, wavelet spectra show in which way frequency components of a signal under consideration change in time. In the fragments (a), (b) and (c) of Figs. 7 and 8, we demonstrate results of the wavelet transformation (19) of the Rabi oscillation signals z(τ ) shown in the respective fragments of Fig. 2. Fig. 7 shows high-frequency parts of the respective wavelet spectra computed with the fitting parameter k0 = 96. The values of the maximal Lyapunov exponent computed in the previous section have indicated that the signal of the parametric Rabi oscillations with the modulation frequency δ = 0.001 (Fig. 2a) is regular with λ 0, the signal with δ = 0.01 (Fig. 2b) is weakly chaotic with 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 signals demonstrate regular behavior characterized by two periodic functions. One of them describes variations in
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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 visible structure of the parametric Rabi oscillations that is due to modulation with the period Tm and characteristical slowdown of these oscillations in time moments τm = (3 + 4m)π/2δ. Fig. 7a shows that the high-frequency components of the regular signal with the comparatively long period of modulation Tm 6280 change in time regularly with the periods Tm and 2Tm in a rather wide frequency ranging from T −1 0.01 to 0.3. With the decrease in the modulation period Tm in n times, extremal values of this range decreases approximately in n times extending from T −1 0.1 to 3 (Fig. 7b with Tm 628) and from T −1 0.5 to 12 (Fig. 7c with Tm 125.6). The high-frequency domains of the wavelet spectra of the chaotic signals (b) and (c) demonstrate in their tops irregular appearing and disappearing components with comparatively low frequencies. One may conclude that the very high-frequency components of the parametric Rabi oscillations change in time regularly both for periodic and for chaotic signals. In Fig. 8, we demonstrate low-frequency parts of the wavelet spectra of the same signals shown in Fig. 2 and computed with the formula (19) but with the fitting parameter k0 = 2. In contrast to the high-frequency wavelets, the low-frequency spectra are very different for regular and chaotic signals. Low-frequency components of the regular signal (Tm 6280) up to the values of the order of 10−4 corresponding to the length of the signal integrated are again changed in time periodically (see Fig. 8a). The components of the weakly chaotic signal (Tm 628) with λ 0.006 demonstrate in the range from 10−1 to 10−3 chaotic behavior (please note the black spots in Fig. 8b), whereas the 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 up to the values of the order T −1 5 × 10−3 (Fig. 8c). Thus, the wavelet spectra computed with typical signals of the parametric Rabi oscillations of the atomic population inversion elucidate a structural temporal kind of Hamiltonian chaos in the simple atom–field system with modulation.
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Fig. 8. Low-frequency range of the wavelet spectra of the parametric 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 correspond to the respective values of the frequency components of a signal.
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4. Summary and conclusion We have analyzed the RWA nonlinear dynamics in one of the simplest models of laser and atomic physics that comprises two-level atoms in an ideal high-Q single-mode cavity. The main problem in which we have been interested in this paper was a transition to Hamiltonian chaos in the coherent atom–field interaction with the atom–field coupling to be modulated. The Heisenberg equations for expectation values of a complete set of the atomic and field observables have been shown to be integrable in the autonomous limit without any modulation and in the resonance limit with an arbitrary modulation. They possess special orbits that are homoclinic to the state with fully inverted atoms and vacuum cavity field which is an equilibrium one in the semiclassical approximation. With the help of the Melnikov method, we have proved analytically transverse intersections of stable and unstable manifolds of this equilibrium point under a small modulation of the Rabi frequency. These transverse intersections are believed to provide Smale horseshoe mechanism of chaos. To confirm numerically the chaotic dynamics in the semiclassical RWA model, we have computed Poincaré sections and maximal Lyapunov exponents under strong modulation of the Rabi frequency. The Lyapunov topographical maps showing the regions of regular and chaotic motion in the space of the control parameters provide representative numerical “portraits” of the system’s dynamics in different ranges of its control parameters. We could show that Hamiltonian chaos in the coherent parametric atom–field interaction is of a special kind that is characterized both by a sensitive dependence on initial conditions and by long-lived structures visible in the parametric Rabi oscillations of the atomic population inversion and in the respective Poincaré sections. An intermittent route to this structural Hamiltonian chaos has been clearly demonstrated in the wavelet spectra of the typical signals of the parametric Rabi oscillations. It has been shown that with increasing modulation frequency, the low-frequency components of the respective wavelet spectrum changed irregularly in time appearing and disappearing in a chaotic way. At the same time,
the high-frequency components of the same wavelet spectrum change periodically in time even for the signals with positive values of the maximal Lyapunov exponents. Coexistence of regular and irregular components in the same signal is a visiting card of structural Hamiltonian chaos. In conclusion, we would like to present some speculations on a possibility to observe manifestations of structural Hamiltonian chaos with real devices. The Hamiltonian approach we have adopted throughout the paper is valid over time intervals shorter than all the characteristic relaxation times. This approach is based on the strong-coupling limit in the atom–field interaction which √ is characterized by the following inequality: (Ω0 N )−1 Ta , Tf , where Ta and Tf are the atomic and field relaxation times, respectively. In numerical experiments computing maximal Lyapunov exponents, Hamiltonian chaos can be diagnosed over time interval of the order of the correlation decoupling time Tcor = 2π/ωa λ. In terms of the collective Rabi frequency ΩN and the maximal Lyapunov exponent λ, the condition for observing Hamiltonian chaos numerically can be rewritten as ΩN λ. A Rydberg maser, operating in the strong-coupling regime with two-level Rydberg atoms inside a high-Q cavity, seems to be a promising device for observing semiclassical Hamiltonian chaos in real experiment. The Rydberg atoms have the transition frequency of the order of ωa 1011 –1012 rad/s, the magnitude of the electric dipole matrix element d 103 atomic units, the single-photon vacuum Rabi frequency Ω0 105 –106 rad/s, and the lifetime of the circular Rydberg states, Ta 10−2 s [14]. Parameters of a typical superconducting microwave maser cavity are the following: the quality factor Q 1010 , the cavity length Lc 1 cm, and the lifetime of intracavity photons Tf 10−1 –10−2 s [14]. As it follows from our numerical results, one can reach the regime of the chaotic parametric Rabi oscillations with the strength of chaos of the order of λ 0.01 operating with a droplet consisting approximately of 1010 atoms and the frequency of modulation of the order of δ 0.01. We used the semiclassical approximation throughout the paper ignoring subtle questions of quantum– classical correspondence in the coherent parametric
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atom–field interaction. It should be mentioned that some manifestations of quantum chaos have been studied with a system of N two-level atoms in a perfect cavity. In the model [15], atoms were assumed to interact with a resonant cavity field taking into account virtual atom–field transitions violating the RWA. In Ref. [16], atoms were considered to interact with a resonant cavity field and with an external coherent field in the framework of the RWA. It was shown with both the models that in the range of parameters for developed semiclassical chaos (when neglecting quantum correlations), the semiclassical approximation was violated by quantum effects at the time scale τ ∼ ln N known as the breaktime of the quantum–classical correspondence [17]. We plan to study in future this effect in parametric Rabi oscillations of N two-level atoms in a cavity with modulated atom–field coupling.
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Acknowledgements This work was supported by the grant from the Russian Foundation for Basic Research No. 02-17269.
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