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arXiv:0811.0789 · Physical Review A

Relation between quantum dwell times and flux-flux correlations

J. Muñoz, D. Seidel, Juan Gonzalo Muga

Published Jan 13, 2009• 11 citations
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Abstract

We examine the connection between the dwell time of a quantum particle in a region of space and flux-flux correlations at the boundaries. It is shown that the first and second moments of a flux-flux correlation function which generalizes a previous proposal by Pollak and Miller [Phys. Rev. Lett. 53, 115 (1984)], agree with the corresponding moments of the dwell-time distribution, whereas the third and higher moments do not. We also discuss operational approaches and approximations to measure the flux-flux correlation function and thus the second moment of the dwell time, which is shown to be characteristically quantum and larger than the corresponding classical moment even for freely moving particles.

DOI
10.1103/physreva.79.012108

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Physical Review A•Nov 23, 2004•44 citations
We examine the connection between the dwell time of a quantum particle in a region of space and flux-flux correlations at the boundaries. It is shown that the first and second moments of a flux-flux correlation function which generalizes a previous proposal by Pollak and Miller [Phys. Rev. Lett. 53, 115 (1984)], agree with the corresponding moments of the dwell-time distribution, whereas the third and higher moments do not. We also discuss operational approaches and approximations to measure the flux-flux correlation function and thus the second moment of the dwell time, which is shown to be characteristically quantum and larger than the corresponding classical moment even for freely moving particles.
The theoretical connections between quantum trajectories and quantum dwell times, previously explored in the context of 1D time-independent stationary scattering applications, are here generalized for multidimensional time-dependent wavepacket applications for particles with spin 1/2. In addition to dwell times, trajectory-based dwell time distributions are also developed, and compared with previous distributions based on the dwell time operator and the flux–flux correlation function. Dwell time distributions are of interest, in part because they may be of experimental relevance. In addition to standard unipolar quantum trajectories, bipolar quantum trajectories are also considered, and found to relate more directly to the dwell time (and other quantum time) quantities of greatest relevance for scattering applications. Detailed calculations are performed for a benchmark 3D spin-1/2 particle application, considered previously in the context of computing quantum arrival times.
Muñoz, Seidel, and Muga [Phys. Rev. A 79, 012108 (2009)10.1103/PhysRevA.79.012108], following an earlier proposal by Pollak and Miller [Phys. Rev. Lett. 53, 115 (1984)10.1103/PhysRevLett.53.115] in the context of a theory of a collinear chemical reaction, showed that suitable moments of a two-flux correlation function could be manipulated to yield expressions for the mean quantum dwell time and mean square quantum dwell time for a structureless particle scattering from a time-independent potential energy field between two parallel lines in a two-dimensional spacetime. The present work proposes a generalization to a charged, nonrelativistic particle scattering from a transient, spatially confined electromagnetic vector potential in four-dimensional spacetime. The geometry of the spacetime domain is that of the slab between a pair of parallel planes, in particular, those defined by constant values of the third (z) spatial coordinate. The mean Nth power, N = 1, 2, 3, …, of the quantum dwell time in the slab is given by an expression involving an N-flux-correlation function. All these means are shown to be nonnegative. The N = 1 formula reduces to an S-matrix result published previously [G. E. Hahne, J. Phys. A 36, 7149 (2003)10.1088/0305-4470/36/25/316]; an explicit formula for N = 2, and of the variance of the dwell time in terms of the S-matrix, is worked out. A formula representing an incommensurability principle between variances of the output-minus-input flux of a pair of dynamical variables (such as the particle's time flux and others) is derived.
We put forward several inherently quantum characteristics of the dwell time, and propose an operational method to detect them. The quantum dwell time is pointed out to be a conserved quantity, totally bypassing Pauli's theorem. Furthermore, the quantum dwell time in a region for one dimensional motion is doubly degenerate. In presence of a potential barrier, the dwell time becomes bounded, unlike the classical quantity. By using off-resonance coupling to a laser we propose an operational method to measure the absorption by a complex potential, and thereby the average time spent by an incoming atom in the laser region.
In a previous paper [G. E. Hahne, J. Phys. A 36, 7149 (2003)] the author studied a nontraditional boundary value problem associated with Schr\"odinger's partial differential equation for the wave function of a structureless particle moving in four-dimensional spacetime: in this boundary value problem, instead of the conventional specification of initial wave-function values on a time=constant\text{time}=\text{constant}time=constant surface, suitable time-dependent boundary and normal-derivative values are given on a three-dimensional space-time surface surrounding a slablike region of interaction in four-dimensional spacetime. The particle's time coordinate plays a natural role as an operator and observable in the modified formalism. In the present paper, the formalism is extended to describe a system of two nonrelativistic particles---each with its own time coordinate---scattering from background potentials and from one another in four-dimensional spacetime. The two-body interaction is taken as a generic noninstantaneous action-at-a-distance, which depends independently on the space-time positions of the two particles. The dynamics is expressed in terms of an integral equation for the wave function, that is, a nonrelativistic version of the Bethe-Salpeter equation. An optical theorem is derived for the transition operator associated with scattering processes; when the theorem holds, the pointwise probability current density derivable from the wave function is conserved globally, that is, in a region covering the space-time domain of significant interparticle interaction. A general formula for the expected dwell-correlation time for the two particles in the space-time region in terms of the scattering matrices is worked out.