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arXiv:1010.2115 · Physical Review E

Lyapunov decoherence rate in classically chaotic systems

Marcus V. S. Bonança

Published Apr 20, 2011• 14 citations
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Abstract

We provide a path integral treatment of the decoherence process induced by a heat bath on a single particle whose dynamics is classically chaotic and show that the decoherence rate is given by the Lyapunov exponent. The loss of coherence is charaterized by the purity, which is calculated semiclassically within diagonal approximation, when the particle initial state is a single Gaussian wave packet. The calculation is performed for weak dissipation and in the high-temperature limit. This situation allows us to simplify the heat bath description to a single random potential. Although the dissipative term is neglected in such approach, the fluctuating one can be treated phenomenologically to fit with the above regime. Our results are therefore valid for times shorter than the inverse of the dissipation rate.

DOI
10.1103/physreve.83.046214

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We provide a path integral treatment of the decoherence process induced by a heat bath on a single particle whose dynamics is classically chaotic and show that the decoherence rate is given by the Lyapunov exponent. The loss of coherence is charaterized by the purity, which is calculated semiclassically within diagonal approximation, when the particle initial state is a single Gaussian wave packet. The calculation is performed for weak dissipation and in the high-temperature limit. This situation allows us to simplify the heat bath description to a single random potential. Although the dissipative term is neglected in such approach, the fluctuating one can be treated phenomenologically to fit with the above regime. Our results are therefore valid for times shorter than the inverse of the dissipation rate.
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 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.
We discuss a deterministic model of detector coupled to a two-level system (a qubit). The detector is a quasiclassical object whose dynamics is described by the kicked rotator Hamiltonian. We show that in the regime of quantum chaos the detector acts as a chaotic bath and induces decoherence of the qubit. We discuss the dephasing and relaxation rates and demonstrate that the main features of single-qubit decoherence due to a heat bath can be reproduced by our fully deterministic dynamical model. Moreover, we show that, for strong enough qubit-detector coupling, the dephasing rate is given by the rate of exponential instability of the detector's dynamics, that is, by the Lyapunov exponent of classical motion. Finally, we discuss the measurement in the regimes of strong and weak qubit-detector coupling. For the case of strong coupling the detector performs a measurement of the up∕down state of the qubit. In the case of weak coupling, due to chaos, the dynamical evolution of the detector is strongly sensitive to the state of the qubit. However, in this case it is unclear how to extract a signal from any measurement with a coarse-graining in the phase space on a size much larger than the Planck cell.
We study a finite-resolution route to decoherence-like behavior in quantum systems whose classical limit is chaotic. The mechanism does not require tracing over an external environment, adding phenomenological noise, or changing unitary quantum mechanics. Instead, chaotic phase-space transport drives Wigner-function interference fringes to progressively shorter wavelengths, while any observation, reconstruction, or deliberately coarse description has finite phase-space resolution. Modeling that resolution by Gaussian coarse-graining, we show that interference components transported below the resolution scale are exponentially attenuated in the coarse-grained phase-space description. The exact quantum state remains pure under unitary evolution; the suppression is, therefore, operational rather than objective. In the semiclassical pre-Ehrenfest window, the instability scale is set by finite-time Lyapunov exponents, while the dimensionless suppression factor is determined by the initial fringe geometry and the coarse-graining covariance. The same geometry gives a robustness criterion: interference components with strong projection onto covector-expanding Oseledets directions are suppressed first, while components avoiding those directions remain visible longer. A hyperbolic cat-map example illustrates the mechanism explicitly. We state the regime of validity, distinguish the mechanism from environment-induced decoherence, and give a practical extraction procedure for numerical or experimental tests.
The time evolution of purity for an initially localized state of a symmetric two-level system coupled to a dissipative bath is investigated using numerically exact real-time path integral methods. With strong system-bath coupling and high temperature, the purity decays monotonically to its fully mixed value, with a short-time Gaussian behavior, which is subsequently followed by exponential evolution. However, under low-temperature and weak coupling conditions, a substantial recovery of purity is observed. A simple theoretical analysis reveals three contributions that correspond to a completely incoherent, eigenstate population difference and rate terms. The last two of these terms can counter the early drop of purity and are responsible for its rebound. These findings caution against using purity as a measure of decoherence in the dynamics of quantum dissipative systems.
We describe the quantum-mechanical spreading of a Gaussian wave packet by means of the semiclassical WKB approximation of Berry and Balazs [J. Phys. A 2,2,2, 625 (1979)]. We find that the time scale $\ensuremath{\tau}$ on which this approximation breaks down in a chaotic system is larger than the Ehrenfest times considered previously. In one dimension $\ensuremath{\tau}=\frac{7}{6}{\ensuremath{\lambda}}^{\ensuremath{-}1}\mathrm{ln}(A/\ensuremath{\Elzxh}),$ with $\ensuremath{\lambda}$ the Lyapunov exponent and A a typical classical action.