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Conference on Lasers and Electro-Optics

Quantum-classical correspondence in dissipative optical systems with limit cycle

K. N. Alekseev, Jan Peřina

Published Aug 29, 2005• 1 citation
Publisher

Abstract

Summary form only given. The situation when nonlinear interactions involve a large number of photons, N, is quite typical of many of the problems of quantum and nonlinear optics. To describe the dynamics of such quantum systems a classical approach is often used. In this theoretical work we compare classical and quantum dynamics of several dissipative quantum optical systems, which have a limit cycle in the classical limit N/spl rarr//spl infin/. In particular, we consider the dynamics of intracavity second harmonic generation (SHG) described by the Hamiltonian.

DOI
10.1109/iqec.2000.908062

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Summary form only given. The situation when nonlinear interactions involve a large number of photons, N, is quite typical of many of the problems of quantum and nonlinear optics. To describe the dynamics of such quantum systems a classical approach is often used. In this theoretical work we compare classical and quantum dynamics of several dissipative quantum optical systems, which have a limit cycle in the classical limit N/spl rarr//spl infin/. In particular, we consider the dynamics of intracavity second harmonic generation (SHG) described by the Hamiltonian.
We develop a semiclassical method to determine the nonlinear dynamics of dissipative quantum optical systems in the limit of large number of photons N; it is based on the 1/N-expansion and the quantum-classical correspondence. The method is used to tackle two problems: the study of the dynamics of nonclassical state generation in higher order anharmonic dissipative oscillators and the establishment of the difference between the quantum and classical dynamics of the second-harmonic generation in a self-pulsing regime. In addressing the first problem, we obtain an explicit time dependence of the squeezing and the Fano factor for an arbitrary degree of anharmonism in the short-time approximation. For the second problem, we analytically find a characteristic time scale at which the quantum dynamics differs insignificantly from the classical one.
We numerically investigate the classical limit of quantum trajectories in optical second-harmonic generation. This is a dissipative system of two nonlinearly coupled harmonic oscillators, corresponding to the fundamental and second-harmonic optical cavity modes. Classically it is described by a system of nonlinear differential equations which have a range of attractors: fixed points, limit cycles, and chaotic attractors. We consider the field-amplitude mean values corresponding to the quantum trajectory wave functions. We find that in the classical limit of large photon number the trajectories of these mean values approach the corresponding classical attractors. This is because the mean values of operator products factorize into products of mean values in the classical limit.
Dynamical chaos in quantum systems (quantum chaos) has been studied by a large number of physicists [1-7]. It has been shown that quantum chaos is possible only in the quasiclassical region, for example in the case of quantum states near the dissociation limit in molecules or near the ionization limit in atoms and molecules. However, the quantum equations of motion are linear and hence they cannot contain dynamical chaos. In the present paper we discuss the solution of quantum problems which are nonlinear in the classical limit. It will be show that in solving the quantum problem in this case it is necessary to use the solution of the nonlinear classical problem forming the classical limit of the quantum problem. i. We note first that here we are not interested in the usual field nonlinearities considered, for example, in quantum optics. This is because field nonlinearities cannot determine the bifurcation properties of the quantum equations of motion since the latter are linear in the wave function ~ or density matrix p. In other words, dynamical chaos, which is intimately connected with the bifurcation properties of the corresponding equations, is not determined by nonlinearities in the fields in quantum equations of motion linear in ~ or 0o
We develop a method for the determination of the dynamics of dissipative quantum systems in the limit of large number of quanta N , based on the 1/ N -expansion of Heidmann et al. (1985 Opt. Commun. 54 189) and the quantum-classical correspondence. Using this method, we find analytically the dynamics of nonclassical states generation in the higher-order anharmonic dissipative oscillators for an arbitrary temperature of a reservoir. We show that the quantum correction to the classical motion increases with time quadratically up to some maximal value, which is dependent on the degree of nonlinearity and a damping constant, and then it decreases. Similarities and differences with the corresponding behavior of the quantum corrections to the classical motion in the Hamiltonian chaotic systems are discussed. We also compare our results obtained for some limiting cases with the results obtained by using other semiclassical tools and discuss the conditions for validity of our approach.
The nonlinear dynamics of dissipative quantum systems in periodic fields is studied in the framework of a Gisin-like nonlinear Schr\"odinger equation with deterministic nonunitary quantum friction terms describing the system-bath couplings. The virtue of this nonunitary evolution is that it is compatible with Dirac's superposition principle and the Hilbert-space structure of quantum kinematics. Floquet theory and the generalized Van Vleck nearly degenerate perturbation method are used to facilitate both analytical and numerical solutions. Closed-form analytic solutions can be obtained in the long-time average approximation or within the rotating-wave approximation. The methods are applied to the study of dissipative quantum dynamics of two-level systems driven by intense periodic fields. It is found that the system asymptotically approaches a limit cycle (whose orientation is subject to the quantum friction constraint), regardless of the strength of the perturbed fields and the nonlinearity constant, indicating quantum suppression of classical chaos. Further, each point of the limit cycle is found to be an attractor and \ensuremath{\psi}(t) exhibits a fractal-like evolution pattern in the course of time. The structure of the limit cycle depends strongly upon field intensity and frequency as well as the order of nonlinear multiphoton transitions. The power spectrum of the Bloch vector trajectory exhibits a dynamical symmetry inherent in the dissipative system and in the asymptotic limit cycle. A theoretical analysis is presented for the understanding of the origin and the role of the dynamical symmetry.