Abstract
We address the sensitivity of quantum mechanical time evolution by considering the time decay of the Loschmidt echo (LE) (or fidelity) for local perturbations of the Hamiltonian. Within a semiclassical approach, we derive analytical expressions for the LE decay for chaotic systems for the whole range from weak to strong local perturbations and identify different decay regimes which complement those known for the case of global perturbations. For weak perturbations, a Fermi-golden-rule (FGR)-type behavior is recovered. For strong perturbations, the escape-rate regime is reached, where the LE decays exponentially with a rate independent of the perturbation strength. The transition between the FGR regime and the escape-rate regime is non-monotonic, i.e. the rate of the exponential time-decay of the LE oscillates as a function of the perturbation strength. We further perform extensive quantum mechanical calculations of the LE based on numerical wave packet evolution, which strongly support our semiclassical theory. Finally, we discuss in some detail possible experimental realizations for observing the predicted behavior of the LE.
Highlights
One of the most prominent manifestations of chaos in classical physics is the hypersensitivity of the dynamics to perturbations in the initial conditions or Hamiltonian
We address the sensitivity of quantum mechanical time evolution by considering the time decay of the Loschmidt echo (LE) for local perturbations of the Hamiltonian
In the present work we extend the semiclassical theory of the LE by lifting the second of the two above-mentioned assumptions, i.e. we allow for a local perturbation in coordinate space
Summary
The discovery [5] of the Lyapunov regime for the decay of the averaged LE in classically chaotic systems, M(t) ∼ exp(−λt) with λ being the average Lyapunov exponent, provided a strong and appealing connection between classical and quantum chaos: it related a measure of instability of the quantum dynamics, such as the LE, to a quantity characterizing the instability of the corresponding classical dynamics, i.e. the Lyapunov exponent. In the present work we extend the semiclassical theory of the LE by lifting the second of the two above-mentioned assumptions, i.e. we allow for a local perturbation in coordinate space In this context the LE decay was previously addressed in the case of a strong local perturbation [23], i.e. for a billiard exposed to a local boundary deformation much larger than the de Broglie wavelength.
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