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Acta Physica Polonica A

Decoherence and Noise in Loschmidt Echo Experiments

Robert Alicki, W. Miklaszewski, M. Pogorzelska

Published Jan 1, 2006• 1 citation
Publisher

Abstract

We discuss some recent results concerning the decoherence in controlled quantum open systems within the mathematical setting corresponding to motion reversal experiments (the Loschmidt echo).We compare the case of randomly chosen sequence of unitary dynamical maps with the case of a constant dynamics corresponding to a classically chaotic evolution.The interplay between chaos and decoherence is illustrated by the new numerical results on the quantum Arnold cat map perturbed by a measurement process.Open problems related to the simple operational characterization of the decoherence strength are discussed.

DOI
10.12693/aphyspola.109.121

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We discuss some recent results concerning the decoherence in controlled quantum open systems within the mathematical setting corresponding to motion reversal experiments (the Loschmidt echo).We compare the case of randomly chosen sequence of unitary dynamical maps with the case of a constant dynamics corresponding to a classically chaotic evolution.The interplay between chaos and decoherence is illustrated by the new numerical results on the quantum Arnold cat map perturbed by a measurement process.Open problems related to the simple operational characterization of the decoherence strength are discussed.
We discuss the stability of quantum motion under system's perturbations in the light of the corresponding classical behavior. In particular we focus our attention on the so called "fidelity" or Loschmidt echo, its relation with the decay of correlations, and discuss the quantum-classical correspondence. We then report on the numerical simulation of the double-slit experiment, where the initial wave-packet is bounded inside a billiard domain with perfectly reflecting walls. If the shape of the billiard is such that the classical ray dynamics is regular, we obtain interference fringes whose visibility can be controlled by changing the parameters of the initial state. However, if we modify the shape of the billiard thus rendering classical (ray) dynamics fully chaotic, the interference fringes disappear and the intensity on the screen becomes the (classical) sum of intensities for the two corresponding one-slit experiments. Thus we show a clear and fundamental example in which transition to chaotic motion in a deterministic classical system, in absence of any external noise, leads to a profound modification in the quantum behavior.
Decoherence in quantum systems which are classically chaotic is studied. The Arnold cat map and the quantum kicked rotor are chosen as examples of linear and nonlinear chaotic systems. The Feynman-Vernon influence functional formalism is used to study the effect of the environment on the system. It is well known that quantum coherence can obliterate chaotic behavior in the corresponding classical system. But interaction with an environment can under general circumstances quickly diminish quantum coherence and reenact classical chaotic behavior. How effectively decoherence works to sustain chaos, and how the resultant behavior qualitatively differs from the quantum picture, depending on the coupling of the system with the environment and the spectral density and temperature of the environment. We show how recurrence in the quantum cat map is lost and classical ergodicity is recovered due to the effect of the environment. Quantum coherence and diffusion suppression are instrumental to dynamical localization for the kicked rotor. We show how environment-induced effects can destroy this localization. Such effects can also be understood as resulting from external noises driving the system. Peculiar to decohering chaotic systems is the apparent transition from reversible to irreversible dynamics. We show such transitions in the quantum cat map and the kicked rotor and distinguish it from apparent irreversibility originating from dynamical instability and imprecise measurements. By performing a time reversal on and following the quantum kicked rotor dynamics numerically, we show how the otherwise reversible quantum dynamics acquires an arrow of time upon the introduction of noise or interaction with an environment.
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.
The Loschmidt echo and the purity are two quantities that can provide invaluable information about the evolution of a quantum system. While the Loschmidt echo characterizes instability and sensitivity to perturbations, purity measures the loss of coherence produced by an environment coupled to the system. For classically chaotic systems both quantities display a number of —supposedly universal— regimes that can lead to think of them as equivalent quantities. We study the decay of the Loschmidt echo and the purity for systems with finite-dimensional Hilbert space and present numerical evidence of some fundamental differences between them.
The Loschmidt echo and the purity are two quantities that can provide invaluable information about the evolution of a quantum system. While the Loschmidt echo characterizes instability and sensitivity to perturbations, purity measures the loss of coherence produced by an environment coupled to the system. For classically chaotic systems both quantities display a number of —supposedly universal— regimes that can lead to think of them as equivalent quantities. We study the decay of the Loschmidt echo and the purity for systems with finite-dimensional Hilbert space and present numerical evidence of some fundamental differences between them.
Classical chaotic dynamics is characterized by exponential sensitivity to initial conditions. Quantum mechanics, however, does not show this feature. We consider instead the sensitivity of quantum evolution to perturbations in the Hamiltonian. This is observed as an attenuation of the Loschmidt echo M(t), i.e., the amount of the original state (wave packet of width sigma) which is recovered after a time reversed evolution, in the presence of a classically weak perturbation. By considering a Lorentz gas of size L, which for large L is a model for an unbounded classically chaotic system, we find numerical evidence that, if the perturbation is within a certain range, M(t) decays exponentially with a rate 1/tau(phi) determined by the Lyapunov exponent lambda of the corresponding classical dynamics. This exponential decay extends much beyond the Eherenfest time t(E) and saturates at a time t(s) approximately equal to lambda(-1)ln[N], where N approximately (L/sigma)(2) is the effective dimensionality of the Hilbert space. Since tau(phi) quantifies the increasing uncontrollability of the quantum phase (decoherence) its characterization and control has fundamental interest.