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arXiv:quant-ph/0406056 · Physical Review E

Classical-quantum correspondence for the scattering dwell time

Caio Lewenkopf, Raúl O. Vallejos

Published Sep 29, 2004• 13 citations
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Abstract

Using results from the theory of dynamical systems, we derive a general expression for the classical average scattering dwell time tau . Remarkably, tau depends only on a ratio of phase space volumes. We further show that, for a wide class of systems, the average classical dwell time is not in correspondence with the energy average of the quantum Wigner time delay.

DOI
10.1103/physreve.70.036214

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Using results from the theory of dynamical systems, we derive a general expression for the classical average scattering dwell time tau . Remarkably, tau depends only on a ratio of phase space volumes. We further show that, for a wide class of systems, the average classical dwell time is not in correspondence with the energy average of the quantum Wigner time delay.
The Wigner delay time is addressed semiclassically using Miller's S -matrix expressed in terms of open orbits. This leads to a very appealing expression, in terms of classical paths, for the energy averaged Wigner time delay in chaotic scattering. The same approach also puts in evidence the semiclassical incapability to correctly assess the time delay higher moments. This limitation suggests that the use of the semiclassical approximation to quantify fluctuations in scattering phenomena, like in mesoscopic physics, has to be considered with great caution.
Using a time-dependent scattering theory for light the authors obtain expressions for the time that light spends in a finite dielectric medium surrounded by vacuum, the so-called dwell time. Given an incoming wavepacket, they focus upon the light scattered in an arbitrary direction. In view of the similarity between Maxwell's equations and the Schrodinger equation, some useful results are derived for the case of Schrodinger potential scattering as well. They show that the dwell time of a Schrodinger particle is a derivative of the phase shifts with respect to the potential, rather than to energy as is the case for the phase-delay time. They indicate the relation to absorption arguments. Because the potential for classical waves is energy dependent, derivatives of the phase shift with respect to potential enter into the phase-delay time for classical waves.
In a recent paper we showed the equivalence, under certain well-characterized assumptions, of Redfield’s equations for the density operator in the energy representation with the Gaussian phase space ansatz for the Wigner function of Yan and Mukamel. The equivalence shows that the solutions of Redfield’s equations respect a striking degree of classical-quantum correspondence. Here we use this equivalence to derive analytic expressions for the density matrix of the harmonic oscillator in the energy representation without making the almost ubiquitous secular approximation. From the elements of the density matrix in the energy representation we derive analytic expressions for Γ1n(1/T1n) and Γ2nm(1/T2nm), i.e., population and phase relaxation rates for individual matrix elements in the energy representation. Our results show that Γ1n(t)=Γ1(t) is independent of n; this is contrary to the widely held belief that Γ1n is proportional to n. We also derive the simple result that Γ2nm(t)=|n−m|Γ1(t)/2, a generalization of the two-level system result Γ2=Γ1/2. We show that Γ1(t) is the classical rate of energy relaxation, which has periodic modulations characteristic of the classical damped oscillator; averaged over a period Γ1(t) is directly proportional to the classical friction, γ. An additional element of classical-quantum correspondence concerns the time rate of change of the phase of the off diagonal elements of the density matrix, ωnm, a quantity which has received little attention previously. We find that ωnm is time-dependent, and equal to |n−m|Ω(t), where Ω(t) is the rate of change of phase space angle in the classical damped harmonic oscillator. Finally, expressions for a collective Γ1(t) and Γ2(t) are derived, and shown to satisfy the relationship Γ2=Γ1/2. This familiar result, when applied to these collective rate constants, is seen to have a simple geometrical interpretation in phase space.
The delay time associated with a scattering process is one of the most important dynamical aspects in quantum mechanics. A common measure of this is the Wigner delay time based on the group velocity description of a wave-packet, which my easily indicate super-luminal or even negative times of interaction that are unacceptable. Many other measures such as dwell times have been proposed, but also suffer from serious deficiencies, particularly for evanescent waves. One important way of realising a timescale that is causally connected to the spatial region of interest has been to utilize the dynamical evolution of extra degrees of freedom called quantum clocks, such as the spin of an electron in an applied magnetic field or coherent decay or growth of light in an absorptive or amplifying medium placed within the region of interest. Here we provide a review of the several approaches developed to answer the basic question - how much time does a quantum particle (or wave) spend in a specified region of space? While a unique answer still evades us, important progress has been made in understanding the timescales and obtaining positive definite times of interaction by noting that all such clocks are affected by spurious scattering concomitant with the very clock potentials, however, weak they be and by eliminating the spurious scattering.