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REAL TIME QUANTUM TUNNELING

GEORGE J. PAPADOPOULOS

Published Jan 1, 2015• 1 citation
Publisher

Abstract

The time evolution of the probability and current densities stemming from an initial state of a particle in the form of a wave packet on the left hand side of a square barrier is studied. In particular, the space distribution at a given time of the above densities is given. Furthermore, the evolution in time at the entrance and exit of the barrier is given. The numerical results show repeated reversal in the current density at the barrier entrance, while being unidirectional at the exit. However, the probability entering the barrier over a long time equals the corresponding probability exiting the barrier. Owing to the fact that the wave packet expands on both of its sides, as time goes by, it is possible to have transmission even if the particle’s initial momentum points away from the barrier. The effect, in question, becomes evident in a diagram for the transmission coefficient in terms of the particle initial momentum, over a relatively small region.

DOI
10.17466/tq2015/19.1/n

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The time evolution of the probability and current densities stemming from an initial state of a particle in the form of a wave packet on the left hand side of a square barrier is studied. In particular, the space distribution at a given time of the above densities is given. Furthermore, the evolution in time at the entrance and exit of the barrier is given. The numerical results show repeated reversal in the current density at the barrier entrance, while being unidirectional at the exit. However, the probability entering the barrier over a long time equals the corresponding probability exiting the barrier. Owing to the fact that the wave packet expands on both of its sides, as time goes by, it is possible to have transmission even if the particle’s initial momentum points away from the barrier. The effect, in question, becomes evident in a diagram for the transmission coefficient in terms of the particle initial momentum, over a relatively small region.
Localized wave-packet tunneling through a one-dimensional potential barrier is investigated with respect to the distribution of tunneling times. Specifically, we construct a quantity--- termed the ``tunneled flux''---which has properties of a tunneling-time probability distribution. The tunneled flux is determined completely in terms of the time evolution of an initially localized wave packet. The tunneled flux and the corresponding tunneling-time distributions are investigated analytically via a semiclassical approximation. Additionally, numerical studies are performed with both the semiclassical and quantal versions of the tunneled flux. The semiclassical calculation qualitatively reproduces numerical tunneled fluxes, converging quantitatively as \ensuremath{\Elzxh}\ensuremath{\rightarrow}0. It also provides a wealth of insight into the nature of tunneling processes. Our principle conclusions are as follows: (i) There is an essential disparity between the time- and energy-domain pictures of wave-packet tunneling; (ii) the tunneling-time distribution can be modeled as an exponential of a skewed Lorentzian function, with a width governed primarily by the phase-space asymmetry of the initial wave packet; and (iii) tunneling is faster than simple classical motion from one side of the barrier to another, even assuming an instantaneous transit between the turning points. Effectively, the barrier acts as a filter for the high-momentum components of the initial wave packet.
In this paper, we discuss the tunneling time of a quantum particle through a rectangular barrier. The reflection and transmission times associated with the wave packets representing the particle are discussed. By using an initial Gaussian momentum distribution, we carry out a comparative analysis of the stationary phases of the incident, reflected, and transmitted wave packets leading to the reflection and transmission times $\ensuremath{\Delta}{t}_{R}$ and $\ensuremath{\Delta}{t}_{T},$ respectively. In the present treatment of this old and very known problem we take into account the deformations of the reflected and transmitted momentum distributions. These deformations produce a dependence of the reflection and transmission times on the location of the initial wave packet. In a parallel calculation, by numerically monitoring the time evolution of the system, we characterize a reflection and a transmission time. Such times agree with the ones obtained via the stationary phase method.
We consider the quantum scattering off a time dependent barrier in one dimension. Our initial state is a right going eigenstate of the Hamiltonian at time t=0. It consists of a plane wave incoming from the left, a reflected plane wave on the left of the barrier and a transmitted wave on its right. We find that at later times, the evolving wave function has a finite overlap with left going eigenstates of the Hamiltonian at time t=0. For simplicity we present an exact result for a time dependent delta function potential. Then we show that our result is not an artifact of that specific choice of the potential. This surprising result does not agree with our interpretation of the eigenstates of the Hamiltonian at time t=0. A numerical study of evolving wave packets, does not find any corresponding real effect. Namely, we do not see on the right hand side of the barrier any evidence for a left going packet. Our conclusion is thus that the intriguing result mentioned above is intriguing only due to the semantics of the interpretation.
Analytic expressions are given for the time spent by a particle tunneling through a potential barrier. The expressions are derived for an incident wave packet which is initially Gaussian, centered about a point an arbitrary distance away from a rectangular potential barrier and moving toward the barrier with constant average velocity. Upon collision with the barrier, the packet splits into a transmitted and a reflected packet. The resultant transmission time is positive, nonzero and in principle measurable. Although the transmission time becomes quite large as the incident kinetic energy becomes very small, in general, for nonzero incident momentum and finite potential barriers which are neither very thick nor very thin, the transmission times are less than the time that would be required for the incident particle to travel a distance equal to the barrier thickness. The transmission times for metal-insulator-metal thin film sandwiches, given approximately by δt3=ℏ/(Efφ)12, where Ef is the Fermi energy of the metal, and φ the vacuum work function, are of the order of 10−16 sec, compared to RC time constants of about 10−13 sec.
An interesting analytical expression is obtained for the coordinate probability distribution of a wave packet transmitted through a potential barrier. In the large-time asymptotics the distribution is given in terms of the transmission amplitude and the phase-space quasidistribution of the initial (not necessarily pure) state. The distortion of a (broad) wave packet due to the barrier penetration is determined by two timelike parameters, given by the energy derivative of the complex transmission amplitude. The resulting change of the packet shape reveals a dependence on the detector position and incorporates a possible speed-up effect and a negative dispersion.
The time behavior of a wave packet, transmitted through a potential barrier, is investigated for all times assuming it to be initially located outside the asymmetric barrier. The survival probability of the initial state, which characterizes the decay of the unstable quantum system in three time domains, is discussed. The presence of initial, exponential, and long-time regimes is a general feature of the decay process. We analytically found the three crucial points of time that separate these regimes. These characteristic times are set by simple formulae, including momentum distribution. It is then shown that the radius of convergence of the cumulant expansion for the survival probability is always small. A long-time behavior of survival probability is described by the asymptotic series whose coefficients decrease with increasing particle energy but at different rates. To be specific, we have examined the exactly solvable model of tunneling through a finite range triangular barrier.