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Journal of the Physical Society of Japan

Crossover from Classical to Quantum Behavior in Duffing Oscillator through “Pseudo-Lyapunov Exponent”

Yukihiro Ota, Ichiro Ohba

Published Jan 1, 2003• 0 citations
Publisher

Abstract

We discuss the quantum–classical correspondence in a specific dissipative chaotic system, Duffing oscillator. We quantize it on the basis of quantum state diffusion (QSD) that is a certain formulation for quantum open systems and a useful tool for analyzing nonlinear problems and their classical limit. We define a “pseudo-Lyapunov exponent” and investigate the behavior of this quantity varying Planck constant effectively. We show that in a dissipative system there exists some critical stage in which the crossover from classical to quantum behavior occurs. Furthermore, we show that the role of dissipation changes for quantized Duffing oscillator since the effect of dissipation is important for the occurrence of chaos in classical mechanics but suppresses it in quantum region.

DOI
10.1143/jpsjs.72sc.119

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We discuss the quantum–classical correspondence in a specific dissipative chaotic system, Duffing oscillator. We quantize it on the basis of quantum state diffusion (QSD) that is a certain formulation for quantum open systems and a useful tool for analyzing nonlinear problems and their classical limit. We define a “pseudo-Lyapunov exponent” and investigate the behavior of this quantity varying Planck constant effectively. We show that in a dissipative system there exists some critical stage in which the crossover from classical to quantum behavior occurs. Furthermore, we show that the role of dissipation changes for quantized Duffing oscillator since the effect of dissipation is important for the occurrence of chaos in classical mechanics but suppresses it in quantum region.
We discuss the quantum-classical correspondence (QCC) in a specific dissipative chaotic system, the Duffing oscillator. The quantum version of the Duffing oscillator is treated as an open quantum system and analyzed numerically by the use of quantum state diffusion (QSD). We consider a pseudo-Lyapunov exponent and investigate it in detail, varying the Planck constant effectively. We show that there exists a critical stage in which the crossover from classical to quantum behavior occurs. Furthermore, we find that a dissipation effect suppresses the occurrence of chaos in the quantum region, while it, combined with the periodic external force, plays a crucial role in the chaotic behavior of the classical system.
We apply the QSD to the quantum version of Duffing oscillator and discuss the QCC and dissipative quantum chaos in this system. We calculate the constant phase map and the pseudo-Lyapunov exponent, varying Planck constant effectively and show there is a certain clear critical stage in which a crossover from classical to quantum behavior occurs. Furthermore, we discuss the effect of dissipation in the dissipative quantum chaos by means of a degrees of localization introduced in the QSD formalism[5]. Unfortunately, the pseudo-Lyapunov exponent is not a quantity characterized a dissipative system. Therefore, it will be necessary to discuss the connection between the classical trajectory (strange attractor) and the quantum observable which characterize a dissipative quantum system.
Using the decoherence formalism of Gell-Mann and Hartle, a quantum system is found which is the equivalent of the classical chaotic Duffing oscillator. The similarities and the differences from the classical oscillator are examined; in particular, a new concept of quantum maps is introduced, and alterations in the classical strange attractor due to the presence of scale- dependent quantum effects are studied. Classical quantities such as the Lyapunov exponents and fractal dimension are examined, and quantum analogs are suggested. These results are generalized into a framework for quantum dissipative chaos, and there is a brief discussion of other work in this area.
The classical-quantum transition for chaotic systems is understood to be accompanied by the suppression of chaotic effects as the relative variant Planck's over 2pi is increased. We show evidence to the contrary in the behavior of the quantum trajectory dynamics of a dissipative quantum chaotic system, the double-well Duffing oscillator. The classical limit in the case considered has regular behavior, but as the effective variant Planck's over 2pi is increased we see chaotic behavior. This chaos then disappears deeper into the quantum regime, which means that the quantum-classical transition in this case is nonmonotonic in variant Planck's over 2pi.
Quantum trajectories given by the quantum state diffusion theory are used to study the dynamics of a pair of qubits interacting with the quantized Duffing oscillator. The latter is a nonlinear dissipative and driven quantum system that in the semiclassical limit displays typical bifurcations and chaotic behavior of the classical model. It is shown that the oscillator part of the qubits-oscillator system in the semiclassical limit is well described by the classical model despite the oscillator-qubit interaction. On the other hand, exact quantum dynamics of the qubits pair is completely different from the classical model as long as there is non-negligible qubit-oscillator interaction. Dynamics of the interqubits entanglement and the qubits pair von Neumann entropy is studied in the deeply quantum and various degrees of semiclassical regimes and for different values of the oscillator's bifurcation parameter. It is concluded that the decoherence of the qubits pair by the dissipative nonlinear oscillator is more effective when the oscillator is in the more classical regime, and if the semiclassical oscillator dynamics is chaotic rather than regular.
We study the quantum dissipative Duffing oscillator across a range of system sizes and environmental couplings under varying semiclassical approximations. Using spatial complexity metrics based on Kullback-Leibler distances between phase-space attractors and temporal complexity metrics based on the Lyapunov exponent, we isolate the effect of the environment on quantum-classical differences. Moreover, we quantify the system sizes where quantum dynamics cannot be simulated using semiclassical or noise-added classical approximations. We find that a remarkable parametrically invariant meta-attractor emerges at a specific length scale and noise-added classical models deviate strongly from quantum dynamics below this scale. We also generalize the previous surprising result that classically regular orbits can have the greatest quantum-classical differences in the semiclassical regime and show that the dynamical growth of quantum-classical differences is not determined by the degree of classical chaos.