5,994 research outputs found
Semiclassical analysis and sensitivity to initial conditions
We present several recent results concerning the transition between quantum
and classical mechanics, in the situation where the underlying dynamical system
has an hyperbolic behaviour. The special role of invariant manifolds will be
emphasized, and the long time evolution will show how the quantum
non-determinism and the classical chaotic sensitivity to initial conditions can
be compared, and in a certain sense overlap
Origin of the exponential decay of the Loschmidt echo in integrable systems
We address the time decay of the Loschmidt echo, measuring the sensitivity of quantum dynamics to small Hamiltonian perturbations, in one-dimensional integrable systems. Using a semiclassical analysis, we show that the Loschmidt echo may exhibit a well-pronounced regime of exponential decay, similar to the one typically observed in quantum systems whose dynamics is chaotic in the classical limit. We derive an explicit formula for the exponential decay rate in terms of the spectral properties of the unperturbed and perturbed Hamilton operators and the initial state. In particular, we show that the decay rate, unlike in the case of the chaotic dynamics, is directly proportional to the strength of the Hamiltonian perturbation. Finally, we compare our analytical predictions against the results of a numerical computation of the Loschmidt echo for a quantum particle moving inside a one-dimensional box with Dirichlet-Robin boundary conditions, and find the two in good agreement
Dephasing representation of quantum fidelity for general pure and mixed states
General semiclassical expression for quantum fidelity (Loschmidt echo) of
arbitrary pure and mixed states is derived. It expresses fidelity as an
interference sum of dephasing trajectories weighed by the Wigner function of
the initial state, and does not require that the initial state be localized in
position or momentum. This general dephasing representation is special in that,
counterintuitively, all of fidelity decay is due to dephasing and none due to
the decay of classical overlaps. Surprising accuracy of the approximation is
justified by invoking the shadowing theorem: twice--both for physical
perturbations and for numerical errors. It is shown how the general expression
reduces to the special forms for position and momentum states and for wave
packets localized in position or momentum. The superiority of the general over
the specialized forms is explained and supported by numerical tests for wave
packets, non-local pure states, and for simple and random mixed states. The
tests are done in non-universal regimes in mixed phase space where detailed
features of fidelity are important. Although semiclassically motivated, present
approach is valid for abstract systems with a finite Hilbert basis provided
that the discrete Wigner transform is used. This makes the method applicable,
via a phase space approach, e. g., to problems of quantum computation.Comment: 11 pages, 4 figure
Lyapunov decay in quantum irreversibility
The Loschmidt echo -- also known as fidelity -- is a very useful tool to
study irreversibility in quantum mechanics due to perturbations or
imperfections. Many different regimes, as a function of time and strength of
the perturbation, have been identified. For chaotic systems, there is a range
of perturbation strengths where the decay of the Loschmidt echo is perturbation
independent, and given by the classical Lyapunov exponent. But observation of
the Lyapunov decay depends strongly on the type of initial state upon which an
average is done. This dependence can be removed by averaging the fidelity over
the Haar measure, and the Lyapunov regime is recovered, as it was shown for
quantum maps. In this work we introduce an analogous quantity for systems with
infinite dimensional Hilbert space, in particular the quantum stadium billiard,
and we show clearly the universality of the Lyapunov regime.Comment: 8 pages, 6 figures. Accepted in Phil. Trans. R. Soc.
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 for a chaotic oscillator
Chaotic dynamics of a nonlinear oscillator is considered in the semiclassical
approximation. The Loschmidt echo is calculated for a time scale which is of
the power law in semiclassical parameter. It is shown that an exponential decay
of the Loschmidt echo is due to a Lyapunov exponent and it has a pure classical
nature.Comment: Submit to PR
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