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Physical Review A

Quantum fluctuations in classical chaos

Wei-Min Zhang, Jian‐Min Yuan, D.H. Feng, Qi Wei Pan, John A. Tjon

Published Sep 1, 1990• 11 citations
Publisher

Abstract

The effects of quantum fluctuation on the classical dynamical structure are investigated by using the semiquantal dynamics derived from the effective action. A pulsed-spin system is studied as an example, which shows that global chaos is suppressed by quantum fluctuation. The quantum-classical correspondence is explicitly examined for low-spin cases.

DOI
10.1103/physreva.42.3646

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The effects of quantum fluctuation on the classical dynamical structure are investigated by using the semiquantal dynamics derived from the effective action. A pulsed-spin system is studied as an example, which shows that global chaos is suppressed by quantum fluctuation. The quantum-classical correspondence is explicitly examined for low-spin cases.
The nonlocal effect of quantum mechanics upon the classical chaos around the separatrix in a Hamiltonian system is investigated by extending the definition of the Melnikov function in the semiclassical approximation. It is shown that the quantum correction of the Melnikov function is related to the quantum fluctuation of the energy on the stable and unstable manifolds. This correction is a constant shift of the center of oscillation of the classical Melnikov function from zero. Because of this shift, the effect of quantum mechanics suppresses the classical chaos around the separatrix. Physical estimates are made of the magnitude of the quantum effect for a double-well oscillator system. As examples, we treat the case of the electron for the molecular scale and the proton for the nuclear scale, and also comment on the ammonia molecules NH3{\mathrm{NH}}_{3}NH3​, ND3{\mathrm{ND}}_{3}ND3​, and NT3{\mathrm{NT}}_{3}NT3​.
The time-dependent variational principle using generalized Gaussian trial functions yields a finite dimensional approximation to the full quantum dynamics and is used in many disciplines. It is shown how these ``semiquantum'' dynamics may be derived via the Ehrenfest theorem and recast as an extended classical gradient system with the fluctuation variables coupled to the average variables. An extended potential is constructed for a one-dimensional system. The semiquantal behavior is shown to be chaotic even though the system has regular classical behavior and the quantum behavior had been assumed regular.
The evolution of classically chaotic quantum systems is analyzed within the formalism of Quantum Pseudo-Probability Distributions. Due to the deep connections that a quantum system shows with its classical correspondent in this representation, the Pseudo-Probability formalism appears to be a useful method of investigation in the field of "Quantum Chaos." In the first part of the thesis we generalize this formalism to quantum systems containing spin operators. It is shown that a classical-like equation of motion for the pseudo-probability distribution ρw can be constructed, dρw/dt = (L_CL + L_QGD)ρw, which is rigorously equivalent to the quantum von Neumann-Liouville equation. The operator L_CL is undistinguishable from the classical operator that generates the semiclassical equations of motion. In the case of the spin-boson system this operator produces semiclassical chaos and is responsible for quantum irreversibility and the fast growth of quantum uncertainty. Carrying out explicit calculations for a spin-boson Hamiltonian the joint action of L_CL and L_QGD is illustrated. It is shown that the latter operator, L_QGD makes the spin system 'remember' its quantum nature, and competes with the irreversibility induced by the former operator. In the second part we test the idea of the enhancement of the quantum uncertainty triggered by the classical chaos by investigating the analogous effect of diffusive excitation in periodically kicked quantum systems. The classical correspondents of these quantum systems exhibit, in the chaotic region, diffusive behavior of the unperturbed energy. For the Quantum Kicked Harmonic Oscillator, in the case of quantum resonances, we provide an exact solution of the quantum evolution. This proves the existence of a deterministic drift in the energy increase over time of the system considered. More generally, this "superdiffusive" excitation of the energy is due to coherent quantum mechanical tunnelling between degenerate tori of the classical phase space. In conclusion we find that some of the quantum effects resulting from this fast increase do not have any classical counterpart, they are mainly tunnelling processes. This seems to be the first observation of an effect of this kind.
The effect of a large environment on a finite-size quantum mechanical system is two-fold: It brings dissipation, but also fluctuations of thermal and quantum origin. While dissipation tends to stabilize the dynamics, we question if and how environmental quantum fluctuations can generate chaos in an otherwise classically non-chaotic system. We work out a paradigmatic model of quantum optics: the dissipative Dicke model, where a large spin interacts with a dissipative harmonic mode. We dial in the classical/quantum correspondence by working in the semiclassical regime at large but finite spin. We demonstrate that, starting from a classically regular phase space in the superradiant regime, quantum noise can generate a strange attractor with fractal dimension and a positive Lyapunov exponent. We unveil the deep connection with shear-induced chaos that was recently developed in the mathematical community.
We identify a transition to chaos in the semiclassical spin-boson model that occurs for relatively large boson fields, as one of two periodic orbits becomes unstable. We have studied the quantum dynamics in the vicinity of this transition, as characterized by (i) phase-space trajectories followed by quantum expectation values, (ii) spectra of such trajectories, (iii) subsystem entropies for the spin and boson systems, and (iv) growth of operator variances for the boson system. We find that the transition to chaos in the classical system has no apparent effect on any of these variables, in the spin-1/2 case. This is in disagreement with some claims made in earlier studies of this system. \textcopyright{} 1996 The American Physical Society.
Previous work [Gong and Brumer, Phys. Rev. Lett. 97, 240602 (2006)] motivates this study as to how asymmetry-driven quantum ratchet effects can persist despite a corresponding fully chaotic classical phase space. A simple perspective of ratchet dynamics, based on the Heisenberg picture, is introduced. We show that ratchet effects are in principle of common origin in classical and quantum mechanics, although full chaos suppresses these effects in the former but not necessarily the latter. The relationship between ratchet effects and coherent dynamical control is noted.