Both in atomic and in mesoscopic physics it is interesting to consider the energy time dependence of a parametrically driven chaotic system. We assume an Hamiltonian $H(Q,P;x(t))$ where $x(t)\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}Vt$. The velocity $V$ is slow in the classical sense but not necessarily in the quantum-mechanical sense. The crossover (in time) from ballistic to diffusive energy spreading is studied. Dissipation is the associated irreversible growth of the average energy. It is found that a dimensionless velocity ${v}_{\mathrm{PR}}$ determines the nature of the dynamics, and controls the route towards quantal-classical correspondence. A perturbative regime and a nonperturbative semiclassical regime are distinguished.
Both in atomic and in mesoscopic physics it is interesting to consider the energy time dependence of a parametrically driven chaotic system. We assume an Hamiltonian H(Q,P;x(t)) where x(t)=Vt. The velocity V is slow in the classical sense but not necessarily in the quantum-mechanical sense. The crossover (in time) from ballistic to diffusive energy spreading is studied. Dissipation is the associated irreversible growth of the average energy. It is found that a dimensionless velocity vPR determines the nature of the dynamics, and controls the route towards quantal-classical correspondence. A perturbative regime and a nonperturbative semiclassical regime are distinguished.
The effect of dissipation on the recovery of the classical chaotic dynamics is discussed in the semiclassical regime of the quantum nonintegrable systems. In the classically chaotic Hamiltonian systems, the semiclassical approach breaks down within rather short time scale due to the interference of the wave function. It is shown that the dissipation suppresses the interference and confines the quantum fluctuation around the semiclassical trajectory which is described by the semiclassical Langevin equation. We also show that the semiclassical condition obtained by Schmid is equivalent to the condition for the suppression of interference. The discussion is valid regardless whether the classical motion is chaotic or not and therefore dissipation helps the recovery of the classical dynamics in the chaotic systems.
Two approaches to quantum-classical correspondence are distinguished according to the classical dynamical theory with which quantum theory is compared. The first of these, Ehrenfest correspondence, defines a dynamical regime in which the quantum expectation values follow approximately a classical trajectory. The second of these, Liouville correspondence, applies when the quantum probability distributions remain well approximated by a density in the classical phase space. The former applies only for narrow states, whereas the latter may remain valid even for quantum states that have spread to the system size. A spin model is adopted for this correspondence study because the quantum state is discrete and finite- dimensional, and thus no artificial truncation of the Hilbert space is required. The quantum time-evolution is given by a discrete unitary mapping. The corresponding classical model is volume-preserving (non-dissipative) and the time-evolution is given by a symplectic map. In classically chaotic regimes, the widths of initially narrow quantum states grow, on average, exponentially with time, until saturation at the system size. This initial spreading rate is well approximated by the classical Lyapunov exponent when the accessible classical phase space is predominantly chaotic. Because of the exponential growth rate of the quantum variance, the Ehrenfest regime is delimited by a break-time that grows logarithmically with increasing quantum numbers. The small differences between quantum expectation values and corresponding Liouville averages of dynamical variables also grow exponentially, initially, if the classical behaviour is chaotic. This exponential rate is independent of h and consistently larger than the classical Lyapunov exponent by at least a factor of two. Interestingly, this exponential growth rate does not continue until these differences approach magnitudes of order the system size, but crosses over to power-law growth (or simply saturates), when the differences have reached magnitudes no larger than order ħ. Because of the exponential growth of these quantum-classical differences, the Liouville regime is also delimited by a break-time that grows logarithmically with increasing quantum numbers. However, due to the early saturation of the exponential growth of the differences, the Liouville log break-time applies only in a restricted domain. After spreading to the system size, the quantum states relax to an approximately time-independent state. This is characterised by equilibrium distributions that are subject to rapidly oscillating fluctuations about the coarse-grained classical steady-state. If the accessible classical phase space is predominantly chaotic, then the equilibrium quantum probability distributions are nearly microcanonical. Otherwise the steady-state quantum probability distributions accurately reflect the details of the KAM surfaces in the classical phase space. The equilibrium regime quantum fluctuations, which are quantum-classical differences, are shown to approach zero as a negative power of the quantum numbers. Since the differences between quantum expectation values and individual classical trajectories grow exponentially in time to a magnitude that scales with the system size, the Ehrenfest regime is delimited by an unrestricted log break-time. The log break-time may be short compared to experimental time-scales for some chaotic macroscopic bodies, and, as a result, an observable breakdown of Ehrenfest correspondence may arise for some macroscopic systems. Although the Liouville regime is also delimited by a log break-time, an observable breakdown of correspondence between quantum mechanics and Liouville mechanics for macroscopic bodies is not likely since their differences, expressed relative to the dimensions of the observable, approach zero in the limit of large quantum numbers.
Making use of appropriate quantum-classical correspondence we have examined the differential behavior of regular and chaotic trajectories in terms of quantum decoherence, the evolution of average quantities, and variances and entropy in a driven double-well oscillator in contact with the surroundings. Our numerical analysis has shown that even on a short time scale the decay of the average quantum coherence is multiexponential. While the onset of decoherence is much faster for a regular trajectory, the decay is faster asymptotically for a chaotic trajectory compared to a regular one. The coupling of the system to the environment turns chaotic in a regular evolution. The environment also affects the evolution of quantum variances for a regular trajectory almost from the beginning, while it has an insignificant effect on chaotic evolution up to a time after which, for both trajectories, noise in the quantum variances exhibits remarkable suppression, although fluctuations of the reservoir modes, in general, tend to increase the level of variances. We identify three stages of quantum evolution; a short decoherence regime followed by a Liouville flow, the latter regime being dominated by the classical curvature of the potential. This is the regime at which growth of entropy or quantum variances is exponential, the rate being determined by the classical largest Lyapunov exponent. The last stage is the irreversible flow dominated by diffusion which suppresses noises in the quantum variances. \textcopyright{} 1996 The American Physical Society.
The main idea of "Quantum Chaos" studies is that Quantum Mechanics introduces two energy scales into the study of chaotic systems: One is obviously the mean level spacing Δ∝ℏd, where d is the dimensionality; The other is Δb∝ℏ, which is known as the non-universal energy scale, or as the bandwidth, or as the Thouless energy. Associated with these two energy scales are two special quantum-mechanical (QM) regimes in the theory of driven system. These are the QM adiabatic regime, and the QM non-perturbative regime respectively. Otherwise Fermi golden rule applies, and linear response theory can be trusted. Demonstrations of this general idea, that had been published in 1999, have appeared in studies of wavepacket dynamics, survival probability, dissipation, quantum irreversibility, fidelity and dephasing.
The quantum-classical correspondence stands out as one of the most intriguing challenges in the realm of quantum mechanics. Within the context of quantum chaos, the conventional approach, which relies on spectral statistics to infer distinct classical dynamical properties, fails in typical scenarios, such as slow relaxation processes, dynamical localization, or other deviations from ergodicity. To address this difficulty, we propose a novel approach, the essence of which lies in associating a quantum energy shell with classical finite-time trajectories. Our results extend the knowledge of quantum-classical correspondence, offer explanations for previously conflicting findings, and also bear significance for applications in quantum metrology, sensing, and computation when the determination of the ergodicity degree of the quantum states is in need.
Using the Wigner method, it is shown that a classical-like equation of motion for a quasiprobability distribution ${\mathrm{\ensuremath{\rho}}}_{\mathit{W}}$ can be built up, \ensuremath{\partial}${\mathrm{\ensuremath{\rho}}}_{\mathit{W}}$/\ensuremath{\partial}t=(L${\mathrm{^}}_{\mathrm{cl}}$+L${\mathrm{^}}_{\mathrm{QGD}}$)${\mathrm{\ensuremath{\rho}}}_{\mathit{W}}$, which is rigorously equivalent to the quantum von Neumann--Liouville equation. The operator L${\mathrm{^}}_{\mathrm{cl}}$ is equivalent to integrating classical trajectories, which are then averaged over an initial distribution, broadened so as to fulfill the requirements of the quantum uncertainty principle. It is shown that 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${\mathrm{^}}_{\mathrm{cl}}$ and L${\mathrm{^}}_{\mathrm{QGD}}$ is illustrated. It is shown that the latter operator L${\mathrm{^}}_{\mathrm{QGD}}$ (where QGD stands for quantum generating diffusion), makes the 1/2-spin system ``remember'' its quantum nature, and competes with the irreversibility induced by the former operator. Some ambiguous aspects of ``irreversibility'' and ``growth of quantum fluctuations'' as indicators of semiclassical chaos are discussed.