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Physical Review E

Irreversibility with quantum trajectories

Diego A. Wisniacki, F. Borondo, R. M. Benito

Published Oct 27, 2005• 1 citation
Publisher

Abstract

Irreversibility is an important issue for many quantum processes. Loschmidt echoes, originally introduced as a way to gauge sensitivity to perturbations in quantum mechanics, have turned out to be a useful tool for its investigation. Following the philosophy supporting this idea, and using quantum trajectories as defined in the causal interpretation of quantum mechanics due to Bohm, we introduce in this paper a more informative alternative measure for irreversibility. The method is applied to the Bunimovich stadium billiard, a paradigmatic example of chaotic system, that constitutes an excellent model for mesoscopic devices.

DOI
10.1103/physreve.72.046219

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Irreversibility is an important issue for many quantum processes. Loschmidt echoes, originally introduced as a way to gauge sensitivity to perturbations in quantum mechanics, have turned out to be a useful tool for its investigation. Following the philosophy supporting this idea, and using quantum trajectories as defined in the causal interpretation of quantum mechanics due to Bohm, we introduce in this paper a more informative alternative measure for irreversibility. The method is applied to the Bunimovich stadium billiard, a paradigmatic example of chaotic system, that constitutes an excellent model for mesoscopic devices.
Irreversibility in quantum processes in the Bunimovich stadium and rectangular billiard in the presence of noise is studied. For this purpose, a novel method based on Loschmidt echo and quantum trajectories, as defined in the de Broglie-Bohm formulation, is used. Our results indicate that the dynamics along the diagonal of the billiard is most sensitive to noise when the wave packet (or alternatively the quantum trajectories) collide with the corners of the billiard. © 2005 American Institute of Physics.
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 carried out. This dependence can be removed by averaging the fidelity over the Haar measure, and the Lyapunov regime is recovered, as has been 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.
The Loschmidt echo (LE) measures the ability of a system to return to the initial state after a forward quantum evolution followed by a backward perturbed one. It has been conjectured that the echo of a classically chaotic system decays exponentially, with a decay rate given by the minimum between the width Gamma of the local density of states and the Lyapunov exponent. As the perturbation strength is increased one obtains a crossover between both regimes. These predictions are based on situations where the Fermi golden rule (FGR) is valid. By considering a paradigmatic fully chaotic system, the Bunimovich stadium billiard, with a perturbation in a regime for which the FGR manifestly does not work, we find a crossover from Gamma to Lyapunov decay. We find that, challenging the analytic interpretation, these conjectures are valid even beyond the expected range.
Loschmidt's paradox asks why macroscopic irreversibility is universal despite the time-reversal symmetry of microscopic dynamics. We argue that irreversibility is not a property of the dynamics but of accessibility: chaotic evolution drives phase-space structure below the quantum resolution scale ℓℏ\ell_\hbarℓℏ​, at a critical time tc=λ−1ln⁡(δ0/ℓℏ)t_c = λ^{-1}\ln(δ_0/\ell_\hbar)tc​=λ−1ln(δ0​/ℓℏ​), after which the time-reversed microstate exists as a valid solution of Hamilton's equations but cannot be selected by any physically admissible operation. The mechanism operates entirely within the semiclassical regime tc≤tEt_c \leq t_Etc​≤tE​, where classical geometry is exact. This provides a dynamical resolution of the Loschmidt paradox. The quantum foundation is established using a Krylov-complexity framework: we prove that for any H(t)=H(−t)H(t)=H(-t)H(t)=H(−t), the quantum Lyapunov exponent satisfies λLforward=λLbackwardλ_L^{\rm forward} = λ_L^{\rm backward}λLforward​=λLbackward​. The arrow of time is not in the dynamics. The mechanism predicts sigmoid fidelity decay, logarithmic scaling of tct_ctc​ with λ−1λ^{-1}λ−1, and ensemble-size independence of the inaccessibility threshold -- all consistent with three decades of Loschmidt echo experiments and confirmed in a stadium-billiard simulation reported here. Underlying everything: quantum mechanics conserves information exactly. Entropy, defined as the logarithm of the multiplicity ΩΩΩ -- the number of possibilities consistent with the available information -- can only increase when information becomes operationally inaccessible. The second law reflects not a breakdown of microscopic reversibility, but the dynamical inaccessibility of the information required to reverse it.
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.
Non-Markovianity can be characterized by performing a series of successive measurements and analyzing departures of the corresponding outcome statistics from a Markovian probabilistic structure. Considering a system coupled with its environment via a dephasing interaction, we show that joint outcome probabilities can be written in terms of a set of two-time environment correlations. Their definition involves forward and backward propagators with different Hamiltonians, associated with a recently introduced generalization of standard Loschmidt echoes [Cormick and Budini, Phys. Lett. A 593, 132011 (2026)]. This result establishes a solid connection between quantum non-Markovianity defined in an operational (measurement-based) way and complex quantum dynamics studied through their sensitivity to dynamical perturbations. We find conditions that guarantee a Markovian (system) behavior and also determine how the generalized echoes can identify information exchanges between the system and its environment. We illustrate our ideas considering examples of spin environments that realize these different dynamical regimes.