Dynamical Origin of Decoherence in Clasically Chaotic Systems

The decay of the overlap between a wave packet evolved with a Hamiltonian H and the same state evolved with H}+$\Sigma$ serves as a measure of the decoherence time $\tau_{\phi}$. Recent experimental and analytical evidence on classically chaotic systems suggest that, under certain conditions, $\tau_{\phi}$ depends on H but not on $\Sigma$. By solving numerically a Hamiltonian model we find evidence of that property provided that the system shows a Wigner-Dyson spectrum (which defines quantum chaos) and the perturbation exceeds a crytical value defined by the parametric correlations of the spectra.Comment: Typos corrected, published versio

Sensitivity to perturbations in a quantum chaotic billiard

The Loschmidt echo (LE) measures the ability of a system to return to the initial state after a forward quantum evolution followed by a backward perturbed one. It has been conjectured that the echo of a classically chaotic system decays exponentially, with a decay rate given by the minimum between the width $\Gamma$ of the local density of states and the Lyapunov exponent. As the perturbation strength is increased one obtains a cross-over between both regimes. These predictions are based on situations where the Fermi Golden Rule (FGR) is valid. By considering a paradigmatic fully chaotic system, the Bunimovich stadium billiard, with a perturbation in a regime for which the FGR manifestly does not work, we find a cross over from $\Gamma$ to Lyapunov decay. We find that, challenging the analytic interpretation, these conjetures are valid even beyond the expected range.Comment: Significantly revised version. To appear in Physical Review E Rapid Communication

Almost all quantum states have nonclassical correlations

Quantum discord quantifies non-classical correlations in a quantum system including those not captured by entanglement. Thus, only states with zero discord exhibit strictly classical correlations. We prove that these states are negligible in the whole Hilbert space: typically a state picked out at random has positive discord; and, given a state with zero discord, a generic arbitrarily small perturbation drives it to a positive-discord state. These results hold for any Hilbert-space dimension, and have direct implications on quantum computation and on the foundations of the theory of open systems. In addition, we provide a simple necessary criterion for zero quantum discord. Finally, we show that, for almost all positive-discord states, an arbitrary Markovian evolution cannot lead to a sudden, permanent vanishing of discord

Stability of Quantum Motion: Beyond Fermi-golden-rule and Lyapunov decay

We study, analytically and numerically, the stability of quantum motion for a classically chaotic system. We show the existence of different regimes of fidelity decay which deviate from Fermi Golden rule and Lyapunov decay.Comment: 5 pages, 5 figure

Hypersensitivity to perturbations of quantum-chaotic wave-packet dynamics

We re-examine the problem of the "Loschmidt echo", which measures the sensitivity to perturbation of quantum chaotic dynamics. The overlap squared $M(t)$ of two wave packets evolving under slightly different Hamiltonians is shown to have the double-exponential initial decay $\propto \exp(-{\rm constant}\times e^{2\lambda_0 t})$ in the main part of phase space. The coefficient $\lambda_0$ is the self-averaging Lyapunov exponent. The average decay $\bar{M}\propto e^{-\lambda_1 t}$ is single exponential with a different coefficient $\lambda_1$. The volume of phase space that contributes to $\bar{M}$ vanishes in the classical limit $\hbar\to 0$ for times less than the Ehrenfest time $\tau_E=\frac{1}{2}\lambda_0^{-1}|\ln \hbar|$. It is only after the Ehrenfest time that the average decay is representative for a typical initial condition.Comment: 4 pages, 4 figures, [2017: fixed broken postscript figures

Short time decay of the Loschmidt echo

The Loschmidt echo measures the sensitivity to perturbations of quantum evolutions. We study its short time decay in classically chaotic systems. Using perturbation theory and throwing out all correlation imposed by the initial state and the perturbation, we show that the characteristic time of this regime is well described by the inverse of the width of the local density of states. This result is illustrated and discussed in a numerical study in a 2-dimensional chaotic billiard system perturbed by various contour deformations and using different types of initial conditions. Moreover, the influence to the short time decay of sub-Planck structures developed by time evolution is also investigated.Comment: 7 pages, 7 figures, published versio

Universality of the Lyapunov regime for the Loschmidt echo

The Loschmidt echo (LE) is a magnitude that measures the sensitivity of quantum dynamics to perturbations in the Hamiltonian. For a certain regime of the parameters, the LE decays exponentially with a rate given by the Lyapunov exponent of the underlying classically chaotic system. We develop a semiclassical theory, supported by numerical results in a Lorentz gas model, which allows us to establish and characterize the universality of this Lyapunov regime. In particular, the universality is evidenced by the semiclassical limit of the Fermi wavelength going to zero, the behavior for times longer than Ehrenfest time, the insensitivity with respect to the form of the perturbation and the behavior of individual (non-averaged) initial conditions. Finally, by elaborating a semiclassical approximation to the Wigner function, we are able to distinguish between classical and quantum origin for the different terms of the LE. This approach renders an understanding for the persistence of the Lyapunov regime after the Ehrenfest time, as well as a reinterpretation of our results in terms of the quantum--classical transition.Comment: 33 pages, 17 figures, uses Revtex

Loschmidt Echo and Lyapunov Exponent in a Quantum Disordered System

We investigate the sensitivity of a disordered system with diffractive scatterers to a weak external perturbation. Specifically, we calculate the fidelity M(t) (also called the Loschmidt echo) characterizing a return probability after a propagation for a time $t$ followed by a backward propagation governed by a slightly perturbed Hamiltonian. For short-range scatterers we perform a diagrammatic calculation showing that the fidelity decays first exponentially according to the golden rule, and then follows a power law governed by the diffusive dynamics. For long-range disorder (when the diffractive scattering is of small-angle character) an intermediate regime emerges where the diagrammatics is not applicable. Using the path integral technique, we derive a kinetic equation and show that M(t) decays exponentially with a rate governed by the classical Lyapunov exponent.Comment: 9 pages, 7 figure

Study of Loschmidt Echo for a qubit coupled to an XY-spin chain environment

We study the temporal evolution of a central spin-1/2 (qubit) coupled to the environment which is chosen to be a spin-1/2 transverse XY spin chain. We explore the entire phase diagram of the spin-Hamiltonian and investigate the behavior of Loschmidt echo(LE) close to critical and multicritical point(MCP). To achieve this, the qubit is coupled to the spin chain through the anisotropy term as well as one of the interaction terms. Our study reveals that the echo has a faster decay with the system size (in the short time limit) close to a MCP and also the scaling obeyed by the quasiperiod of the collapse and revival of the LE is different in comparison to that close to a QCP. We also show that even when approached along the gapless critical line, the scaling of the LE is determined by the MCP where the energy gap shows a faster decay with the system size. This claim is verified by studying the short-time and also the collapse and revival behavior of the LE at a quasicritical point on the ferromagnetic side of the MCP. We also connect our observation to the decoherence of the central spin.Comment: Accepted for publication in EPJ

Statistics of finite-time Lyapunov exponents in the Ulam map

The statistical properties of finite-time Lyapunov exponents at the Ulam point of the logistic map are investigated. The exact analytical expression for the autocorrelation function of one-step Lyapunov exponents is obtained, allowing the calculation of the variance of exponents computed over time intervals of length $n$. The variance anomalously decays as $1/n^2$. The probability density of finite-time exponents noticeably deviates from the Gaussian shape, decaying with exponential tails and presenting $2^{n-1}$ spikes that narrow and accumulate close to the mean value with increasing $n$. The asymptotic expression for this probability distribution function is derived. It provides an adequate smooth approximation to describe numerical histograms built for not too small $n$, where the finiteness of bin size trimmes the sharp peaks.Comment: 6 pages, 4 figures, to appear in Phys. Rev.