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Register a new userUniform semiclassical approach to fidelity decay in the deep Lyapunov regime
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We use the uniform semiclassical approximation in order to derive the fidelity decay in the regime of large perturbations. Numerical computations are presented which agree with our theoretical predictions. Moreover, our theory allows to explain previous findings, such as the deviation from the Lyapunov decay rate in cases where the classical finite-time instability is non-uniform in phase space.
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We derive fidelity decay and parametric energy correlations for random matrix ensembles where time--reversal invariance of the original Hamiltonian is broken by the perturbation. Like in the case of a symmetry conserving perturbation a simple relation between both quantities can be established.
We study the fidelity decay of the $k$-body embedded ensembles of random matrices for bosons distributed over two single-particle states. Fidelity is defined in terms of a reference Hamiltonian, which is a purely diagonal matrix consisting of a fixed one-body term and includes the diagonal of the perturbing $k$-body embedded ensemble matrix, and the perturbed Hamiltonian which includes the residual off-diagonal elements of the $k$-body interaction. This choice mimics the typical mean-field basis used in many calculations. We study separately the cases $k=2$ and $3$. We compute the ensemble-averaged fidelity decay as well as the fidelity of typical members with respect to an initial random state. Average fidelity displays a revival at the Heisenberg time, $t=t_H=1$, and a freeze in the fidelity decay, during which periodic revivals of period $t_H$ are observed. We obtain the relevant scaling properties with respect to the number of bosons and the strength of the perturbation. For certain members of the ensemble, we find that the period of the revivals during the freeze of fidelity occurs at fractional times of $t_H$. These fractional periodic revivals are related to the dominance of specific $k$-body terms in the perturbation.
Work in closed quantum systems is usually defined by a two-point measurement. This definition of work is compatible with quantum fluctuation theorems but it fundamentally differs from its classical counterpart. In this paper, we study the correspondence principle in quantum chaotic systems. We derive a semiclassical expression of the work distribution for chaotic systems undergoing a general, finite time, process. This semiclassical distribution converges to the classical distribution in the usual classical limit. We show numerically that, for a particle inside a chaotic cavity, the semiclassical distribution provides a good approximation to quantum distribution.
Mean fidelity amplitude and parametric energy--energy correlations are calculated exactly for a regular system, which is subject to a chaotic random perturbation. It turns out that in this particular case under the average both quantities are identical. The result is compared with the susceptibility of chaotic systems against random perturbations. Regular systems are more susceptible against random perturbations than chaotic ones.
Chaotic dynamics with sensitive dependence on initial conditions may result in exponential decay of correlation functions. We show that for one-dimensional interval maps the corresponding quantities, that is, Lyapunov exponents and exponential decay rates are related. For piecewise linear expanding Markov maps observed via piecewise analytic functions we provide explicit bounds of the decay rate in terms of the Lyapunov exponent. In addition, we comment on similar relations for general piecewise smooth expanding maps.
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