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Journal of Physics A Mathematical and Theoretical

Initial or final values for semiclassical evolutions in the Weyl–Wigner representation

Alfredo M. Ozorio de Almeida, Raúl O. Vallejos, Eduardo Zambrano

Published Mar 13, 2013• 17 citations
Publisher

Abstract

Initial value representations are constructed to avoid the search for trajectories that are only defined in semiclassical approximations by their boundary conditions. We show how to incorporate these procedures within the full Weyl representation, so that quantum expectation values are given by phase space integrals over the evolving Wigner function. Spurious semiclassical singularities at caustics are cancelled, even though there is no increase in the number of trajectories, as compared to usual semiclassical formulae. The whole construction remains exact in the case of quadratic Hamiltonians. The evolution of (density) operators depends on either a forward and backward trajectory, given by an initial value, or else on a pair of trajectories, propagating backwards from their final value. The latter option reduces numerical errors in the computation of trajectories. The general scheme also leads to analogous algorithms for evolving the quantum fidelity, which can be approximated perturbatively with a single trajectory, reducing to the 'dephasing representation' for small times. The theory is developed within a generalized 'Maslov method', based on semiclassical Fourier transforms, in order to avoid singularities in the limit of small times.

DOI
10.1088/1751-8113/46/13/135304

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Initial value representations are constructed to avoid the search for trajectories that are only defined in semiclassical approximations by their boundary conditions. We show how to incorporate these procedures within the full Weyl representation, so that quantum expectation values are given by phase space integrals over the evolving Wigner function. Spurious semiclassical singularities at caustics are cancelled, even though there is no increase in the number of trajectories, as compared to usual semiclassical formulae. The whole construction remains exact in the case of quadratic Hamiltonians. The evolution of (density) operators depends on either a forward and backward trajectory, given by an initial value, or else on a pair of trajectories, propagating backwards from their final value. The latter option reduces numerical errors in the computation of trajectories. The general scheme also leads to analogous algorithms for evolving the quantum fidelity, which can be approximated perturbatively with a single trajectory, reducing to the 'dephasing representation' for small times. The theory is developed within a generalized 'Maslov method', based on semiclassical Fourier transforms, in order to avoid singularities in the limit of small times.
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.
Abstract The trace of an arbitrary product of quantum operators with the density operator is rendered as a multiple phase space integral of the product of their Weyl symbols with the Wigner function. Interspersing the factors with various evolution operators, one obtains an evolving correlation. The kernel for the matching multiple integral that evolves within the Weyl representation is identified with the trace of a single compound unitary operator. Its evaluation within a semiclassical approximation then becomes a sum over the periodic trajectories of the corresponding classical compound canonical transformation. The search for periodic trajectories can be bypassed by an exactly equivalent initial value scheme, which involves a change of integration variable and a reduced compound unitary operator. Restriction of all the operators to observables with smooth non-oscillatory Weyl symbols leads naturally to classical evolution within a single phase space integral, if each observable undergoes independent Heisenberg evolution. The linear response function for infrared spectroscopy provides an example depending on a double phase space integral. This is compared to an alternative scheme that employs the widely used Herman-Kluk propagator.
The trace of an arbitrary product of quantum operators with the density operator is rendered as a multiple phase space integral of the product of their Weyl symbolswith the Wigner function. Interspersing the factors with various evolution operators, one obtains an evolving correlation. The kernel for the matching multiple integral thatevolves within the Weyl representation is identified with the trace of a single compound unitary operator. Its evaluation within a semiclassical approximation then becomes asum over the periodic trajectories of the corresponding classical compound canonical transformation. The search for periodic trajectories can be bypassed by an exactly equivalent initialvalue scheme, which involves a change of integration variable and a reduced compound unitary operator. Restriction of all the operators to observables with smooth nonoscillatoryWeyl symbols reduces the evolving correlation to a single phase space integral. If each observable undergoes independent Heisenberg evolution, the overallcorrelation evolves classically. Otherwise, the kernel acquires a nonclassical phase factor, though it still depends on a purely classical compound trajectory: e.g. the fasefor a double return of the quantum Loschmidt echo does not coincide with twice the phase for a single echo.
The quantum Hamiltonian generates in time a family of evolution operators. Continuity of this family holds within any choice of representation and, in particular, for the Weyl propagator, even though its simplest semiclassical approximation may develop caustic singularities. The phase jumps of the Weyl propagator across caustics have not been previously determined. The semiclassical appproximation relies on individual classical trajectories together with their neighbouring tangent map. Based on the latter, one defines a continuous family of unitary tangent propagators, with an exact Weyl representation that is close to the full semiclassical approximation in an appropriate neighbourhood. The phase increment of the semiclassical Weyl propagator, as a caustic is crossed, is derived from the facts that the corresponding family of tangent operators belong to the metaplectic group and that the products of the tangent propagators are obtained from Gaussian integrals. The Weyl representation of the metaplectic group is here presented, with the correct phases determined within an intrinsic ambiguity for the overall sign. The elements that fully determine the phase increment across a particular caustic are then analysed.
We present a comprehensive study of semiclassical phase-space propagation in the Wigner representation, emphasizing numerical applications, in particular as an initial-value representation. Two semiclassical approximation schemes are discussed. The propagator of the Wigner function based on van Vleck's approximation replaces the Liouville propagator by a quantum spot with an oscillatory pattern reflecting the interference between pairs of classical trajectories. Employing phase-space path integration instead, caustics in the quantum spot are resolved in terms of Airy functions. We apply both to two benchmark models of nonlinear molecular potentials, the Morse oscillator and the quartic double well, to test them in standard tasks such as computing autocorrelation functions and propagating coherent states. The performance of semiclassical Wigner propagation is very good even in the presence of marked quantum effects, e.g., in coherent tunneling and in propagating Schrodinger cat states, and of classical chaos in four-dimensional phase space. We suggest options for an effective numerical implementation of our method and for integrating it in Monte-Carlo-Metropolis algorithms suitable for high-dimensional systems.
The Moyal description of quantum mechanics, based on the Wigner-Weyl isomorphism between operators and symbols, provides a comprehensive phase space representation of dynamics. The Weyl symbol image of the Heisenberg picture evolution operator is regular in ħ and so presents a preferred foundation for semiclassical analysis. Its semiclassical expansion "coefficients," acting on symbols that represent observables, are simple, globally defined (phase space) differential operators constructed in terms of the classical flow. The first of two presented methods introduces a cluster-graph expansion for the symbol of an exponentiated operator, which extends Groenewold′s formula for the Weyl product of two symbols and has ħ as its natural small parameter. This Poisson bracket based cluster expansion determines the Jacobi equations for the semiclassical expansion of "quantum trajectories." Their Green function solutions construct the regular ħ ⇓ 0 asymptotic series for the Heisenberg-Weyl evolution map. The second method directly substitutes such a series into the Moyal equation of motion and determines the ħ coefficients recursively. In contrast to the WKB approximation for propagators, the Heisenberg-Weyl description of evolution involves no essential singularity in ħ, no Hamilton-Jacobi equation to solve for the action, and no multiple trajectories, caustics, or Maslov indices.