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Europhysics Letters (EPL)

Frequency response of the Loschmidt echo decay in an open driven nonlinear oscillator

Shihui Zhang, Zhan-Yuan Yan

Published Nov 1, 2015• 0 citations
Publisher

Abstract

The decay of the Loschmidt echo and its relation to the frequency response of the underlying classical dynamics are investigated in an open Duffing system by means of the Wigner function. The initial Wigner function of the system is Gaussian and centered at a phase point ( x 0 , p 0 ). For different ( x 0 , p 0 ), significant peaks are observed in the frequency response curves of the Loschmidt echo decay during the evolution of the Wigner function. Furthermore, there is good correspondence between the frequency response curves of the Loschmidt echo decay and the underlying classical dynamics. This can be attributed to the increase of the fringes of the Wigner function by the external driving force, which can be revealed by the frequency response of the underlying classical dynamics.

DOI
10.1209/0295-5075/112/30006

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The decay of the Loschmidt echo and its relation to the frequency response of the underlying classical dynamics are investigated in an open Duffing system by means of the Wigner function. The initial Wigner function of the system is Gaussian and centered at a phase point ( x 0 , p 0 ). For different ( x 0 , p 0 ), significant peaks are observed in the frequency response curves of the Loschmidt echo decay during the evolution of the Wigner function. Furthermore, there is good correspondence between the frequency response curves of the Loschmidt echo decay and the underlying classical dynamics. This can be attributed to the increase of the fringes of the Wigner function by the external driving force, which can be revealed by the frequency response of the underlying classical dynamics.
The time evolution of the nonclassicality is found to have strong dependence on the driving frequency in a Duffing-type driven nonlinear system with the Gaussian state as the initial Wigner function. Its frequency response curve has a remarkable peak, near which the nonclassicality of the state is enhanced. Furthermore, good correspondences are found between the frequency response curves of the time-average of the nonclassicality indicator and those of the classical mean square amplitude. Such correspondences can also be observed, when using a superposition of two coherent states as the initial Wigner function.
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 de Broglie 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 (nonaveraged) 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.
The Loschmidt echo (LE) is a measure of the sensitivity of quantum mechanics to perturbations in the evolution operator. It is defined as the overlap of two wave functions evolved from the same initial state but with slightly different Hamiltonians. Thus, it also serves as a quantification of irreversibility in quantum mechanics. In this thesis the LE is studied in systems that have a classical counterpart with dynamical instability, that is, classically chaotic. An analytical treatment that makes use of the semiclassical approximation is presented. It is shown that, under certain regime of the parameters, the LE decays exponentially. Furthermore, for strong enough perturbations, the decay rate is given by the Lyapunov exponent of the classical system. Some particularly interesting examples are given. The analytical results are supported by thorough numerical studies. In addition, some regimes not accessible to the theory are explored, showing that the LE and its Lyapunov regime present the same form of universality ascribed to classical chaos. In a sense, this is evidence that the LE is a robust temporal signature of chaos in the quantum realm. Finally, the relation between the LE and the quantum to classical transition is explored, in particular with the theory of decoherence. Using two different approaches, a semiclassical approximation to Wigner functions and a master equation for the LE, it is shown that the decoherence rate and the decay rate of the LE are equal. The relationship between these quantities results mutually beneficial, in terms of the broader resources of decoherence theory and of the possible experimental realization of the LE.
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.
Quantum states extended over a large volume in phase space have oscillations from quantum interferences in their Wigner distribution on scales smaller than variant Planck's over 2pi [W. H. Zurek, Nature (London) 412, 712 (2001)]]. We investigate the influence of those sub-Planck-scale structures on the sensitivity to an external perturbation of the state's time evolution. While we do find an accelerated decay of the Loschmidt Echo for an extended state in comparison to a localized wave packet, the acceleration is described entirely by the classical Lyapunov exponent and hence cannot originate from quantum interference.