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arXiv:1510.03140 · Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences

Path integral approach to the quantum fidelity amplitude

Jiří Vaníček, Doron Cohen

Published May 3, 2016• 11 citations
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Abstract

The Loschmidt echo is a measure of quantum irreversibility and is determined by the fidelity amplitude of an imperfect time-reversal protocol. Fidelity amplitude plays an important role both in the foundations of quantum mechanics and in its applications, such as time-resolved electronic spectroscopy. We derive an exact path integral formula for the fidelity amplitude and use it to obtain a series of increasingly accurate semiclassical approximations by truncating an exact expansion of the path integral exponent. While the zeroth-order expansion results in a remarkably simple, yet non-trivial approximation for the fidelity amplitude, the first-order expansion yields an alternative derivation of the so-called 'dephasing representation,' circumventing the use of a semiclassical propagator as in the original derivation. We also obtain an approximate expression for fidelity based on the second-order expansion, which resolves several shortcomings of the dephasing representation. The rigorous derivation from the path integral permits the identification of sufficient conditions under which various approximations obtained become exact.

DOI
10.1098/rsta.2015.0164

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The Loschmidt echo is a measure of quantum irreversibility and is determined by the fidelity amplitude of an imperfect time-reversal protocol. Fidelity amplitude plays an important role both in the foundations of quantum mechanics and in its applications, such as time-resolved electronic spectroscopy. We derive an exact path integral formula for the fidelity amplitude and use it to obtain a series of increasingly accurate semiclassical approximations by truncating an exact expansion of the path integral exponent. While the zeroth-order expansion results in a remarkably simple, yet non-trivial approximation for the fidelity amplitude, the first-order expansion yields an alternative derivation of the so-called 'dephasing representation,' circumventing the use of a semiclassical propagator as in the original derivation. We also obtain an approximate expression for fidelity based on the second-order expansion, which resolves several shortcomings of the dephasing representation. The rigorous derivation from the path integral permits the identification of sufficient conditions under which various approximations obtained become exact.
We obtain an initial-value representation for the quantum Loschmidt echo from the semiclassical theory of Wigner function evolution, together with the classical first-order perturbation theory. In the limit of small actions, the amplitude of each trajectory reduces to unity, just as in the dephasing representation introduced by Vaníček [Phys. Rev. E 70, 055201(R) (2004)], but these trajectories are generated here by the mean Hamiltonian for both the forward and the backward motion. This slight change of action may substantially alter the phase. The amplitude correction depends on the second derivative of the action. This improved dephasing approximation is verified to work even for quadratic Hamiltonians, for which the semiclassical evolution is exact, thus extending the range of application beyond its original scope in quantum chaos.
A 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 is 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. Beyond justifying the approximation, the shadowing theorem makes the dephasing representation practical: without shadowing it would be impossible to find numerically the precise trajectories needed in a semiclassical approximation. It is shown how the general expression reduces to the previously known special forms for localized states. The superiority of the general over the specialized forms is explained and supported by numerical tests for wave packets, nonlocal pure states, and for simple and random mixed states. The tests are done in nonuniversal regimes in mixed phase space where detailed features of fidelity are important. Although semiclassically motivated, the 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, to problems of quantum computation.
Due to the Heisenberg uncertainty principle, various classical systems differing only on the scale smaller than Planck's cell correspond to the same quantum system. We use this fact to find a unique semiclassical representation without the Van Vleck determinant, applicable to a large class of correlation functions expressible as quantum fidelity. As in the Feynman path integral formulation of quantum mechanics, all contributing trajectories have the same amplitude: that is why we denote it the "dephasing representation." By relating our approach to the problem of existence of true trajectories near numerically-computed chaotic trajectories, we make the approximation rigorous for any system in which the shadowing theorem holds. Numerical implementation only requires computing actions along the unperturbed trajectories and not finding the shadowing trajectories. While semiclassical linear-response theory was used before in quasi-integrable and chaotic systems, here its validity is justified in the most generic, mixed systems. Dephasing representation appears to be a rare practical method to calculate quantum correlation functions in nonuniversal regimes in many-dimensional systems where exact quantum computations are impossible.
In order to rigorously derive the amplitude factor of the semiclassical approximation to the quantum propagator, we extend an existing method originally devised to evaluate Gaussian path-integral expressions. Using a result which relates the determinant of symmetric block-tridiagonal matrices to the determinants of their blocks, two difference equations are obtained. The first one allows to establish the connection of the amplitude factor to Jacobi’s accessory equations in the continuous-time limit, while the second one leads to an additional factor which, however, contributes to the final result only in exceptional cases. In order to demonstrate the wide applicability of these difference equations, we treat explicitly the case where the time-sliced Lagrangian is written in generalized coordinates, for which a general derivation has so far been unavailable.
The extremal paths that arise in the stationary phase evaluation of coherent state path integrals do not seem to have a simple physical interpretation, in contrast to the extremal paths that occur in the conventional path integrals. On the other hand, a recently derived semiclassical formula for the coherent state propagator involves a path that is determined in exactly the same manner as the extremal paths of the conventional path integrals. Since both the semiclassical and the stationary phase analyses yield asymptotic (h(cross) to 0) approximations, the stationary phase and the semiclassical expressions for the propagator should be identical. The author presents a simple and direct proof that, in spite of the apparent differences, this is indeed the case. The simplification in the semiclassical formula is due to the utilisation of an appropriate set of canonical variables to describe the classical dynamics. In order to illustrate the usefulness of the semiclassical formula they present an application to the problem of the degenerate parametric amplifier, which had been treated before by operator ordering and path integral methods. The semiclassical approach has a simple classical interpretation that is absent in the alternative treatments.
The Loschmidt echo (LE) is a measure of the stability and reversibility of quantum evolution under perturbations of the Hamiltonian. One of the expected and most relevant characteristics of this quantity for chaotic systems is an exponential decay with a perturbation-independent decay rate given by the classical Lyapunov exponent. However, a non-uniform decay—instead of the Lyapunov regime—has been reported in several systems. In this work, we find an analytical semiclassical expression for the averaged fidelity amplitude that can be directly related to the anomalous—unexpected—behaviour of the LE.