We study how the interaction with an external incoherent environment induces a crossover from quantum to classical behavior for a particle whose classical motion is chaotic. Posing the problem in the semiclassical regime for weak bath coupling, we find that noise rather than dissipation is primarily responsible for the dephasing that leads to the quantum inhibition of mixing. However, the noise is also shown to inhibit the mixing by enhancing paths that interfere coherently.
We study how the interaction with an external incoherent environment induces a crossover from quantum to classical behavior for a particle whose classical motion is chaotic. Posing the problem in the semiclassical regime for weak bath coupling, we find that noise rather than dissipation is primarily responsible for the dephasing that leads to the quantum inhibition of mixing. However, the noise is also shown to inhibit the mixing by enhancing paths that interfere coherently.
I compare the quantum and classical dynamics of a particle moving in a cosine potential while subject to a time-dependent force. I concentrate here on the behavior of an initially well-localized wave packet at times before the classically chaotic motion is fully developed. I find that the quantum and classical dynamics are indistinguishable well beyond the Ehrenfest time where the wave packet delocalizes. The quantum and classical descriptions first differ precisely when the classical probability density is folded in the vicinity of a hyperbolic fixed point. At this point, the wave function acquires a nodal structure which I show to be the result of a simple beating phenomenon between paths in the semiclassical propagator. When the interaction of the classical manifold with the hyperbolic fixed point leads to escape from the remnant separatrix, representing the onset of classical chaos, the interference associated with the tendril leads to a long-lived accumulation of quantum amplitude on top of the cosine barrier. This effect also has a semiclassical interpretation, meaning that it arises from the interference between classical paths, but one must expand to second order in Planck's constant to describe the behavior correctly. Both the nodal structure and this pinning of amplitude on the barrier are dynamic mechanisms for the quantum inhibition of mixing. I then couple the system to a bath of harmonic oscillators in order to study the effect of the environment on these mechanisms. Although an oscillator bath environment brings dissipation as well as noise to the problem, the noise effect dominates for the high temperature, weak coupling regime I study. When there is sufficient noise to render the interfering classical paths indistinguishable, I find that the quantum interference gets erased. This dephasing occurs at very early times, long before there is appreciable dissipation of energy to the environment. Consequently, one can argue that the presence of an environment, even if its effect would be negligible in a nonchaotic setting, allows for the possibility of quantum mixing.
We investigate the effect of dephasing (decoherence) on quantum transport through open chaotic ballistic conductors in the semiclassical limit of small Fermi wavelength to system size ratio, ${\ensuremath{\lambda}}_{F}∕L⪡1$. We use the trajectory-based semiclassical theory to study a two-terminal chaotic dot with decoherence originating from (i) an external closed quantum chaotic environment, (ii) a classical source of noise, and (iii) a voltage probe, i.e., an additional current-conserving terminal. We focus on the pure dephasing regime, where the coupling to the external source of dephasing is so weak that it does not induce energy relaxation. In addition to the universal algebraic suppression of weak localization, we find an exponential suppression of weak localization $\ensuremath{\propto}\mathrm{exp}[\ensuremath{-}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}∕{\ensuremath{\tau}}_{\ensuremath{\phi}}]$, with the dephasing rate ${\ensuremath{\tau}}_{\ensuremath{\phi}}^{\ensuremath{-}1}$. The parameter $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}$ depends strongly on the source of dephasing. For a voltage probe, $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}$ is of order the Ehrenfest time $\ensuremath{\propto}\mathrm{ln}[L∕{\ensuremath{\lambda}}_{F}]$. In contrast, for a chaotic environment or a classical source of noise, it has the correlation length $\ensuremath{\xi}$ of the coupling or noise potential replacing the Fermi wavelength ${\ensuremath{\lambda}}_{F}$. We explicitly show that the Fano factor for shot noise is unaffected by decoherence. We connect these results to earlier works on dephasing due to electron-electron interactions and numerically confirm our findings.
Fidelity is a convenient tool to study the sensitivity of quantum motion under Hamiltonian perturbations. In this paper we first show that classical chaos can produce the dephasing necessary to suppress quantum interference, even in the absence of any environment. To this end we consider the fidelity of mixed states, which takes into account interference amplitudes, and directly relate its decay to the decay of an appropriate classical correlation function, which is totally unrelated to quantum phases. We then discuss the dephasing in a two-qubit system, induced by the coupling to a single-particle, deterministic chaotic environment. The latter is shown to behave as a pure dephasing many-body object which induces decoherence in the system; memory effects are also taken into account.
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.
The population decay due to a small opening in an otherwise closed cavity supporting chaotic classical dynamics displays a quantum correction on top of the classical exponential form, a pure manifestation of quantum coherence that acquires a universal form and can be explained by path interference. Being coherent, such enhancement is prone to decoherence effects due to the coupling of the system to an external environment. We study this interplay between incoherent and coherent quantum corrections to decay by evaluating, within a Caldeira-Leggett scenario, off-diagonal contributions to the decoherence functional coming from pairs of correlated classical paths in the time regime where dissipative effects are neglected and decoherence does not affect the classical dynamics, but quantum interference must be accounted for. We find that the competing effects of interference and decoherence lead to a universal nonmonotonic form for the survival probability depending only on the coupling strength and macroscopic parameters of the cavity.
Environmental noise leads to dephasing and relaxation in a quantum system. Often, a rigorous treatment of multiple noise sources within a system-bath approach is not possible. We discuss the influence of environmental fluctuations on a quantum system whose dynamics is dephasing already due to a phenomenologically treated additional noise source. For this situation, we develop a path-integral approach, which allows us to treat the system-environment coupling in a numerically exact way, and additionally we extend standard perturbative approaches. We observe strong deviations between the numerically exact and the perturbative results even for weak system-bath coupling. This shows that standard perturbative approaches fail for additional, even weak, system-bath couplings if the system dynamics is already dissipative.