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Physical review. B, Condensed matter

Quantum turbulence and resonant tunneling

M. Ya. Azbel

Published Mar 15, 1999• 8 citations
Publisher

Abstract

Tunneling of the Hartree liquid through opaque potential barriers separated by a well is considered. It is always unstable and is related to three distinctly different time scales. The shortest one is the period of classical oscillations in the well. It determines the period of tunneling current oscillations. Instability and specifically quantum turbulence develop on the scale of the quantum dwell time in the well. Dissipation yields the largest time scale. Analytical and numerical solutions are presented, and possible experiments are discussed.

DOI
10.1103/physrevb.59.8049

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arXiv:quant-ph/0505099•Journal of Physics A Mathematical and General•Jun 15, 2005•9 citations
Tunneling of the Hartree liquid through opaque potential barriers separated by a well is considered. It is always unstable and is related to three distinctly different time scales. The shortest one is the period of classical oscillations in the well. It determines the period of tunneling current oscillations. Instability and specifically quantum turbulence develop on the scale of the quantum dwell time in the well. Dissipation yields the largest time scale. Analytical and numerical solutions are presented, and possible experiments are discussed.
In the discussion of temporary behaviors of quantum tunneling, people usually like to focus their attention on rectangular barrier with steep edges, or to deal with smooth barrier with semi-classical or even numerical calculations. Very few discussions on analytic solutions of tunneling through smooth barrier appear in the literature. In this paper, we provide two such examples, a semi-infinite long barrier V(x)=A2[1+tanh(x/a)] and a finite barrier V(x) = A sech2(x/a). To each barrier, we calculate the associated phase time and dwell time after obtaining the analytic solution. The results show that, different from rectangular barrier, phase time or dwell time does increase with the length parameter a controlling the effective extension of the barrier. More interestingly, for the finite barrier, phase time or dwell time exhibits a peak in k-space. A detailed analysis shows that this interesting behavior can be attributed to the strange tunneling probability Ts(k), i.e., Ts(k) displays a unit step function-like profile Θ(k − k0), especially when a is large, say, a ≫ 1/κ, 1/k. And k0≡mA/ħ is exactly where the peak appears in phase or dwell time k-spectrum. Thus only those particles with k in a very narrow interval around k0 are capable to dwell in the central region of the barrier sufficiently long.
The physics of dynamic resonant tunneling is investigated.First, the resonant tunneling effect through an opaque barrier via a delta-function well is illustrated.Then, it is shown that, even in the adiabatic regime, where the dynamics can be governed by an analytic solution, the particle can be activated to higher energies.If the well varies quickly enough that the particle cannot escape from the well during the energetic elevation, the activation can be enhanced, as was anticipated by Azbel.However, and this is the main result of this work, the quasi-bound state of the well can even "reduce" the activation.In fact, because the resonant energy of the well matches twice the incoming particle's energy, and if the contribution to the wave function from both parts destructively interferes, then the particle cannot dwell in the well and activation is suppressed.This effect can be utilized in frequency-controlled transistors, and it is even speculated that it may explain the reason that humans can distinguish between tens of thousands of different odors with merely few hundreds of odor receptors.Lastly, the short time dynamics of a very fast perturbative well is also discussed.
The dynamics of particle tunneling is characterized by the concept of a tunneling time. The various times suggested in the literature are considered in order to characterize the complementary features of the tunneling process. Explicit calculations show how to find the particle and the wave aspects of different tunneling times. We also expose the physics behind the various definitions and their interrelations. We show that in the classical limit the particle properties dominate and in the opaque limit the wave aspects do. The complementary analysis helps in the interpretation of the tunneling process and eliminates some of the controversies found in the literature. All tunneling times are found to be special cases of a universal tunneling time definition relating to parameter derivatives of the phases of the wave functions. We propose that these relations can be used to extract the tunneling times for any general potential by integrating the wave packet through the tunneling process and extracting the derivatives numerically from these results. Furthermore, we investigate the particle properties of resonant tunneling. They are shown to dominate the dynamics inside the well regions.
The tunneling time of a particle through a given barrier is commonly defined in terms of “internal clocks” that effectively measure the interaction time with internal degrees of freedom of the barrier. It is known that this definition of the time scale for tunneling is not unique in the sense that it depends on the clock used to define it. For the case of resonance tunneling, a particular choice that in the limit of a high/broad square barrier yields the original result of Büttiker and Landauer ( Phys. Rev. Lett. 1982, 49, 1739) is correlated to the lifetime of the resonance state. This is illustrated for analytically solvable one-dimensional double barrier models and for a realistic model of electron tunneling through a static water barrier. The latter calculation constitutes a novel application of this concept to a 3-dimensional model, and the observed structure in the energy dependence of the computed traversal time reflects the existence of transient tunneling resonances associated with instantaneous water structures. These models, characterized by the existence of shape resonances in the barrier, make it possible to examine different internal clocks that were proposed for measuring tunneling times in situations where a “clock independent” intrinsic time scale (the resonance lifetime) for the tunneling time exists. It is argued that this time may be used in order to estimate the relative importance of dynamical barrier processes that affect the tunneling probability.
Solutions to explicit time-dependent problems in quantum mechanics are rare. In fact, all known solutions are coupled to specific properties of the Hamiltonian and may be divided into two categories: one class consists of time-dependent Hamiltonians which are not higher than quadratic in the position operator, like e.g. the driven harmonic oscillator with time-dependent frequency. The second class is related to the existence of additional invariants in the Hamiltonian, which can be used to map the solution of the time-dependent problem to that of a related time-independent one. In this paper we discuss and develop analytic methods for solving time-dependent tunnelling problems, which cannot be addressed by using quadratic Hamiltonians. Specifically, we give an analytic solution to the problem of tunnelling from an attractive time-dependent potential which is embedded in a long-range repulsive potential. Recent progress in atomic physics makes it possible to observe experimentally time-dependent phenomena and record the probability distribution over a long range of time. Of special interest is the observation of macroscopical quantum-tunnelling phenomena in Bose–Einstein condensates with time-dependent trapping potentials. We apply our model to such a case in the last section.