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arXiv:1405.0619

Quantum correlation between a particle and potential well or barrier

Frank V. Kowalski, R. S. Browne

Published May 3, 2014• 0 citations
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Abstract

A two-body quantum correlation is calculated for a particle and an infinite potential well in which it is trapped or either a barrier or finite well over which it traverses. Correlated interference results when the incident and reflected particle substates and their associated well or barrier substates overlap. Measurement of the particle in this region causes a splitting of the well or barrier substate at subsequent times. The joint probability density, which is a function both of the different positions and different times at which the particle and well or barrier are measured, is derived assuming that no interaction occurs between the time each is measured.

DOI
10.48550/arxiv.1405.0619

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A two-body quantum correlation is calculated for a particle and an infinite potential well in which it is trapped or either a barrier or finite well over which it traverses. Correlated interference results when the incident and reflected particle substates and their associated well or barrier substates overlap. Measurement of the particle in this region causes a splitting of the well or barrier substate at subsequent times. The joint probability density, which is a function both of the different positions and different times at which the particle and well or barrier are measured, is derived assuming that no interaction occurs between the time each is measured.
Correlations between the behavior of pairs of particles are generated in the following experimental situation. An on-line source emits a stream of two-particle systems, where each system is in one and the same quantum state. After emission, the particles – call then (I) and (II) – move off in opposite directions. Each particle then encounters one of several possible barriers that either it passes or doesn't. A short distance behind each barrier is a detector set to register the presence of the particle, should it get that far. Finally, the detectors are connected by a timed relay and counter that registers a “coincidence count” should the two detectors fire within a set, brief time interval. When the experiment is run with various different barriers, detection rates accumulate for each barrier singly, and coincidence rates (the correlations) for the various pairs of barriers.
Reflection of a microscopic particle from a mesoscopic/macroscopic `mirror' generates two-body correlated interference from the incident and reflected particle substates and their associated mirror substates. The microscopic momentum exchanged generates two mirror substates which interfere to produce fringes which do not vanish as the mirror mass increases. The small displacement between these mirror states can yield negligible environmental decoherence times. Mirror coherence lengths impose constraints on the extent of this interference, which are mitigated using interference of the two-body states associated with the particle reflecting from both of the two surfaces of a slab of matter in a manner analogous to the classical interference of a pulse of light reflecting from a `thin film'. This two-body correlated interference is modeled as a particle traversing a finite well with both the particle and well treated quantum mechanically. Such a treatment predicts the expected `thin-film' interference but only as a special case of a more general result. It is also shown that measurements on only the reflected particle (yielding a marginal probability density function) can act as a probe to reveal the quantum state of the macroscopic reflector. For equal masses, coherence of the particle substate is transferred to the mirror substate, a quantum manifestation of a familiar classical result.
The time-dependent Schr\"odinger equation is solved for a one-dimensional rectangular potential whose height changes sinusoidally in time. By using standard wave-matching techniques, dynamic transmission and reflection probabilities are found. It is shown that the flux transmitted through the barrier region is modulated in time at integer multiples of ${\ensuremath{\omega}}_{b}$, the angular frequency at which the potential oscillates. The results obtained for a delocalized plane wave incident on the barrier are extended by considering the interaction of a Gaussian wave packet with the oscillating potential barrier. The modulation of the transmitted and reflected probability density by the oscillating barrier is clearly demonstrated by a series of calculations performed to illustrate this interaction. These calculations also show that the probability density leaving the barrier region is composed of individual wave packets having different average velocities, a result which is a direct consequence of the transitions induced by the time-dependent potential barrier under consideration. Calculations are also performed which show that the flux transmitted across the potential barrier depends on the phase of the oscillating barrier at the time the incident Gaussian wave packet begins to interact with it. The results of these model calculations suggest that experiments properly designed to measure the transmission probability of a quantum particle as it escapes from a metal surface are likely to provide information about the time formation of the dynamic image potential.
The lack of translational invariance perpendicular to the plane of a single quantum well causes equal probability for spontaneous emission to the left or right. Combining one emission path from the left and one from the right into a common detector leads to interference fringes for fundamentally indistinguishable paths corresponding to geometries where the same in-plane momentum is transferred to the quantum well. For all other paths, no interference is observed because of the entanglement between the photon and extended Bloch states of the many-body system. In multiple-quantum-well structures the interference can be controlled via the spacing between the wells.
Baz' and Rybachenko have proposed the use of the Larmor precession as a clock to measure the time it takes a particle to traverse a barrier. An applied magnetic field is confined to the barrier. The spin of the incident particles is polarized perpendicular to this field. The extent of the Larmor precession occurring during transmission is used as a measurement of the time spent traversing the barrier. However, the particles tunneling through an opaque barrier also acquire a spin component parallel to the field since particles with spin parallel to the field have a higher transmission probability than particles with spin antiparallel to the field. Similar effects are actually used to polarize electrons and neutrons. An interpretation of this experiment compares the results with an approach which determines the traversal time by studying transmission of particles through a time-modulated barrier. This leads to three characteristic times describing the interaction of particles with a barrier. A dwell time measures the average time interval during which a particle interacts with the barrier whether it is reflected or transmitted at the end of its stay, a traversal time measures the time interval during which a particle interacts with the barrier if it is finally transmitted, and a reflection time measures the interaction time of a reflected particle.