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Physical Review A

Transmission through a potential barrier in an arbitrary quantum state

Georg Sulyok, Johann Summhammer, H. Rauch

Published Jul 30, 2012• 5 citations
Publisher

Abstract

We investigate the scattering of a quantum-mechanical particle on an oscillating rectangular potential barrier. In the first step, we show the calculation for a classical potential revealing coherent, quantized energy exchange between the particle and potential, respectively, the likewise classical field that generates the latter. In order to explore the quantized energy transfer in more detail, the transition to a quantum field is performed and the Schr\"odinger equation for the composite quantum system consisting of particle and field is solved analytically as well. The result allows a physically very intuitive explanation of the underlying energy exchange processes and explicitly displays the occurring entanglement between particle and field due to their interaction, thus offering the possibility to use this combined system for quantum information experiments. Depending on the initial field state, incoherent or coherent energy exchange is observed. In the limit of high mean photon numbers the classical results can be fully retrieved.

DOI
10.1103/physreva.86.012124

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We investigate the scattering of a quantum-mechanical particle on an oscillating rectangular potential barrier. In the first step, we show the calculation for a classical potential revealing coherent, quantized energy exchange between the particle and potential, respectively, the likewise classical field that generates the latter. In order to explore the quantized energy transfer in more detail, the transition to a quantum field is performed and the Schr"odinger equation for the composite quantum system consisting of particle and field is solved analytically as well. The result allows a physically very intuitive explanation of the underlying energy exchange processes and explicitly displays the occurring entanglement between particle and field due to their interaction, thus offering the possibility to use this combined system for quantum information experiments. Depending on the initial field state, incoherent or coherent energy exchange is observed. In the limit of high mean photon numbers the classical results can be fully retrieved.
When a quantum particle traverses a rectangular potential created by a quantum field both photon exchange and entanglement between particle and field take place. We present the full analytic solution of the Schrödinger equation of the composite particle-field system allowing investigation of these phenomena in detail and comparison to the results of a classical field treatment. Besides entanglement formation, remarkable differences also appear with respect to the symmetry between energy emission and absorption, resonance effects and if the field initially occupies the vacuum state.
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.
We investigate the time-dependent Schr\"odinger equation for the case of a plane matter wave incident on an oscillating potential step. Approximate solutions are given for different regimes of the problem. The resulting energy spectra are quantized. This may either be interpreted as a phase modulation of the wave or as a coherent multiphonon exchange. A comparison to the results that apply for a beam of classical particles is made. In particular, the transition from the quantum-mechanical to the classical case is examined. We have performed an experiment with very cold neutrons that clearly revealed the discussed effects. Energy splittings of 2.8 to 9.1 neV of the reflected waves were analyzed with an energy resolution of \ensuremath{\sim}1 neV, thus demonstrating the feasibility of such high-resolution experiments with neutron reflectometry. \textcopyright{} 1996 The American Physical Society.
Quantum tunneling is conventionally described using the stationary Schrödinger equation with an externally imposed potential barrier. In this work, we present a dynamical formulation of quantum tunneling in which the barrier is modeled as a structured medium possessing quantized internal modes that interact coherently with an incident electron. Using the Heisenberg operator formalism and a second-quantized representation of the barrier medium, we derive coupled dynamical equations governing the electron–barrier interaction. Under a continuum-mode and mean-field approximation, the collective response of the barrier modes generates an effective potential that reproduces the conventional rectangular barrier model and the standard tunneling transmission probability obtained from the Schrödinger equation. Within this framework, a tunneling traversal time is naturally defined from the dynamical evolution of the electron and is shown to depend on the barrier width, barrier height, and incident electron energy. Numerical simulations illustrating the transmission probability and tunneling-time behavior are presented. The results provide a complementary microscopic interpretation of tunneling processes in structured quantum media and may be relevant to nanoscale transport, photonic barriers, and coherent quantum devices.
Quantum tunneling is conventionally described by the Schrödinger wave equation with a prescribed static potential barrier, providing accurate transmission probabilities but offering limited insight into the microscopic dynamics of particle–barrier interactions. In this work, we develop a dynamical second-quantization framework in which the barrier is modeled as an ensemble of quantized internal modes rather than as an externally imposed classical potential. The tunneling particle interacts directly with these microscopic barrier excitations through a coupled particle–barrier Hamiltonian, from which the Heisenberg equations of motion are derived. Collective coherent excitations of the barrier modes give rise to an emergent effective barrier that naturally recovers the conventional rectangular barrier and the WKB transmission limit under appropriate conditions. Unlike standard treatments, the present formulation explicitly incorporates microscopic barrier dynamics and provides a unified description of particle–barrier coupling within a second-quantized formalism. The framework further suggests that repeated tunneling events may experience different microscopic interaction histories, motivating a statistical interpretation of tunneling times. Because both the particle and barrier are treated within the same operator formalism, the theory provides a natural foundation for extension to relativistic quantum transport, photonic barriers, cavity quantum electrodynamics, and other structured quantum media.
Scattering of a quantum particle from an oscillating barrier or well does not generally conserve the particle energy owing to energy exchange with the photon field, and an incoming particle-free state is scattered into a set of outgoing (transmitted and reflected) free states according to Floquet scattering theory. Here we introduce two families of oscillating non-Hermitian potential wells in which Floquet scattering is fully suppressed for any energy of the incident particle. The scattering-free oscillating potentials are synthesized by application of the Darboux transformation to the time-dependent Schr\"odinger equation. For one of the two families of scattering-free potentials, the oscillating potential turns out to be fully invisible.