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arXiv:chao-dyn/9710017 · Physical Review Letters

Exponentially Rapid Decoherence of Quantum Chaotic Systems

Arjendu K. Pattanayak, Paul W. Brumer

Published Nov 24, 1997• 40 citations
arXivHTMLPDF

Abstract

We use a recent result to show that the rate of loss of coherence of a quantum system increases with increasing system phase-space structure and that a chaotic quantal system in the semiclassical limit decoheres exponentially with rate $2{\ensuremath{\lambda}}_{2}$, where ${\ensuremath{\lambda}}_{2}$ is a generalized Lyapunov exponent. As a result, for example, the dephasing time for classically chaotic systems goes to infinity logarithmically with the temperature, in accord with recent experimental results.

DOI
10.1103/physrevlett.79.4131

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We use a recent result to show that the rate of loss of coherence of a quantum system increases with increasing system phase-space structure and that a chaotic quantal system in the semiclassical limit decoheres exponentially with rate 2\ensuremathλ22{\ensuremath{\lambda}}_{2}2\ensuremathλ2​, where \ensuremathλ2{\ensuremath{\lambda}}_{2}\ensuremathλ2​ is a generalized Lyapunov exponent. As a result, for example, the dephasing time for classically chaotic systems goes to infinity logarithmically with the temperature, in accord with recent experimental results.
The mean-field limit of a bosonic quantum many-body system is described by (mostly) nonlinear equations of motion which may exhibit chaos very much in the spirit of classical particle chaos, i.e., by an exponential separation of trajectories in Hilbert space with a rate given by a positive Lyapunov exponent $\ensuremath{\lambda}$. The question now is whether $\ensuremath{\lambda}$ imprints itself onto measurable observables of the underlying quantum many-body system even at finite particle numbers. Using a Bose-Einstein condensate expanding in a shallow potential landscape as a paradigmatic example for a bosonic quantum many-body system, we show that the system loses its coherence at an exponentially fast rate. Furthermore, we show that the rate is given by the Lyapunov exponent associated with the chaotic mean-field dynamics. Finally, we demonstrate that this chaos-induced loss of coherence imprints itself onto the visibility of interference fringes in the total density after time of flight, thus, opening the possibility to measure $\ensuremath{\lambda}$ and with it the interplay between chaos and nonequilibrium quantum matter in a real experiment.
We compare quantum decoherence in generic regular and chaotic systems that interact with a thermal reservoir via a dipole coupling. Using a time-dependent, self-consistent approximation in the spirit of Hartree, we derive in the high temperature limit an expression for the off-diagonal elements of the system density operator that initially corresponds to a coherent superposition of two adjacent wave packets. We relate the decoherence rate to the Lyapunov exponent in the Ehrenfest regime. In this regime, the greater the instability of the system the faster the loss of coherence occurs.
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.
Classical chaotic dynamics is characterized by exponential sensitivity to initial conditions. Quantum mechanics, however, does not show this feature. We consider instead the sensitivity of quantum evolution to perturbations in the Hamiltonian. This is observed as an attenuation of the Loschmidt echo M(t), i.e., the amount of the original state (wave packet of width sigma) which is recovered after a time reversed evolution, in the presence of a classically weak perturbation. By considering a Lorentz gas of size L, which for large L is a model for an unbounded classically chaotic system, we find numerical evidence that, if the perturbation is within a certain range, M(t) decays exponentially with a rate 1/tau(phi) determined by the Lyapunov exponent lambda of the corresponding classical dynamics. This exponential decay extends much beyond the Eherenfest time t(E) and saturates at a time t(s) approximately equal to lambda(-1)ln[N], where N approximately (L/sigma)(2) is the effective dimensionality of the Hilbert space. Since tau(phi) quantifies the increasing uncontrollability of the quantum phase (decoherence) its characterization and control has fundamental interest.
The environment – external or internal degrees of freedom coupled to the system – can, in effect, monitor some of its observables. As a result, the eigenstates of these observables decohere and behave like classical states: Continuous destruction of superpositions leads to environment-induced superselection (einselection). Here I investigate it in the context of quantum chaos (i.e., quantum dynamics of systems which are classically chaotic). I show that the evolution of a chaotic macroscopic (but, ultimately, quantum) system is not just difficult to predict (requiring an accuracy exponentially increasing with time) but quickly ceases to be deterministic in principle as a result of the Heisenberg indeterminacy (which limits the resolution available in the initial conditions). This happens after a time t ℏ which is only logarithmic in the Planck constant. A definitely macroscopic (if somewhat outrageous) example is afforded by various components of the solar system which are chaotic, with the Lyapunov timescales ranging from a bit more than a month (Hyperion) to millions of years (planetary system as a whole). On the timescale t ℏ the initial minimum uncertainty wavepackets corresponding to celestial bodies would be smeared over distances of the order of radii of their orbits into “Schrödinger cat – like” states, and the concept of a trajectory would cease to apply. In reality, such paradoxical states are eliminated by decoherence which helps restore quantum-classical correspondence. The price for the recovery of classicality is the loss of predictability: In the classical limit (associated with effective decoherence, and not just with the smallness of ℏ) the rate of increase of the von Neumann entropy of the decohering system is independent of the strength of the coupling to the environment, and equal to the sum of the positive Lyapunov exponents. Algorithmic aspects of entropy production are briefly explored to illustrate the effect of decoherence from the point of view of the observer. We show that “decoherence strikes twice”, introducing unpredictability into the system and extracting quantum coherence from the observers memory, where it enters as a price for the classicality of his records.
Chaos in classical systems can be characterized by Lyapunov exponents that measure the exponential divergence of nearby trajectories, but directly extending this framework to quantum mechanics has been a persistent challenge. The wavelike nature of quantum states and the non-commutative geometry of quantum phase space obstruct a straightforward generalization of classical chaos theory. Here we develop a rigorous approach to quantum chaos by leveraging quantum optimal transport theory, which provides the missing geometric foundation for measuring distances between extended quantum distributions. We define quantum Lyapunov exponents that naturally avoid divergences encountered when naïvely generalizing classical exponents, and show that in the semiclassical limit they recover the classical global expansion rate, and hence the usual maximal Lyapunov exponent when they coincide. Our framework provides a tight connection between the divergence of classical trajectories and semiclassical phase space evolution, and additionally clarifies the role of out-of-time-order correlators as diagnostics of quantum chaos. These results establish quantum optimal transport as a unifying mathematical foundation for quantum chaos theory, providing new tools to characterize dynamical behavior across the full range of quantum dynamics from simple few-body models to complex many-body systems.