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Physical Review Letters

Transition Path Time Distribution, Tunneling Times, Friction, and Uncertainty

Eli Pollak

Published Feb 15, 2017• 28 citations
Publisher

Abstract

A quantum mechanical transition path time probability distribution is formulated and its properties are studied using a parabolic barrier potential model. The average transit time is well defined and readily calculated. It is smaller than the analogous classical mechanical average transit time, vanishing at the crossover temperature. It provides a direct route for determining tunneling times. The average time may be also used to define a coarse grained momentum of the system for the passage from one side of the barrier to the other. The product of the uncertainty in this coarse grained momentum with the uncertainty in the location of the particle is shown under certain conditions to be smaller than the ℏ/2 formal uncertainty limit. The model is generalized to include friction in the form of a bilinear interaction with a harmonic bath. Using an Ohmic friction model one finds that increasing the friction, increases the transition time. Only moderate values of the reduced friction coefficient are needed for the quantum transition time and coarse grained uncertainty to approach the classical limit which is smaller than ℏ/2 when the friction is not too small. These results show how one obtains classical dynamics from a pure quantum system without invoking any further assumptions, approximations, or postulates.

DOI
10.1103/physrevlett.118.070401

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•Jul 1, 2017•1 citation
A quantum mechanical transition path time probability distribution is formulated and its properties are studied using a parabolic barrier potential model. The average transit time is well defined and readily calculated. It is smaller than the analogous classical mechanical average transit time, vanishing at the crossover temperature. It provides a direct route for determining tunneling times. The average time may be also used to define a coarse grained momentum of the system for the passage from one side of the barrier to the other. The product of the uncertainty in this coarse grained momentum with the uncertainty in the location of the particle is shown under certain conditions to be smaller than the ℏ/2 formal uncertainty limit. The model is generalized to include friction in the form of a bilinear interaction with a harmonic bath. Using an Ohmic friction model one finds that increasing the friction, increases the transition time. Only moderate values of the reduced friction coefficient are needed for the quantum transition time and coarse grained uncertainty to approach the classical limit which is smaller than ℏ/2 when the friction is not too small. These results show how one obtains classical dynamics from a pure quantum system without invoking any further assumptions, approximations, or postulates.
The standard approaches to tunneling times are replaced by considering time correlation functions. A class of correlation functions that is always positive is identified and used to define quantum mechanical transition time probability distributions. The formalism is used to study the quantum dynamics of a thermal position correlation function of a parabolic barrier Hamiltonian. The transition time probability distribution between two points distributed symmetrically about the barrier top shifts to shorter times as the temperature is reduced and tunneling is increased. A study of the mean transition time as a function of the distance between the center of the initial and final densities shows that when the temperature is sufficiently low and tunneling dominates the dynamics, increasing the length of the path traversed decreases the mean transition time. The introduction of friction to the dynamics does not "destroy" this phenomenon, except when the friction coefficient is very large.
We define, using a path-integral method, an amplitude distribution for the traversal time of a particle tunneling through a barrier. We calculate this without approximation for the case of a square barrier potential. The form of the integrated distribution is shown to be determined by the uncertainty principle: if the particle is known to spend a time \ensuremath{\le}\ensuremath{\tau} inside the barrier, then an effective uncertainty \ensuremath{\Delta}V is introduced in the barrier height, such that \ensuremath{\Delta}V\ensuremath{\tau}\ensuremath{\approxeq}\ensuremath{\Elzxh}. The distribution has a number of interesting properties: it is noisy at very short times, oscillates periodically at intermediate times, and saturates to a constant at long times. The first two properties arise as a result of resonances in the transmission amplitude, which play an important role in the distribution. The characteristic (median) time scale for the distribution is shown to be ${\mathrm{\ensuremath{\tau}}}_{\mathit{Q}}$=2md2{\mathit{md}}^{2}md2/\ensuremath{\Elzxh}. We show that this represents the average traversal time for particle energies that are close to the energy of the barrier height. For very opaque barriers, the oscillations have a very short period. We show that this can lead to anomalously short average traversal times, and resolves the outstanding problem that the average velocity of a tunneling particle may exceed the speed of light. For particles with energies above the height of the barrier (classical traversal), we find that the real part of the distribution is dominated by the classical path, and has a sharp peak around the classical traversal time. It is argued that the form of the distribution found in this work may be applied to general barrier shapes.
The transition-path-time distribution is formalized for quantum systems and applied to a number of examples. Using a symmetrized thermal density, transition times are studied for the free particle, a $\ensuremath{\delta}$-function potential, a square-barrier potential, and symmetric-double-well dynamics at very low temperature. These studies exemplify extreme nonlocality for motion in $\ensuremath{\delta}$-function potentials, vanishing tunneling times for the square-barrier potential, and varying transit times in the symmetric-double-well potential. In all cases, there are regions where the longer the distance traversed, the shorter the mean transit time is. For the thermal density correlation functions studied here, the Hartman effect exemplifies itself through the independence of the transit time on the barrier height. However, due to the thermal distribution, the transit time does depend on the barrier width, initially decreasing with increasing width but then increasing again.
A variety of different theoretical procedures have been used to determine tunneling times. These include (1) phase time; (2) the Stevens procedure; (3) Larmor times; (4) a complex “time”; (5) dwell time; (6) the Büttikker-Landauer time; (7) Feynman path-integrals; (8) scattering theory; (9) a stochastic formulation; (10) oscillatory perturbation of the barrier; (11) kinetic time; and (12) the semiclassical solution. Interest in tunneling times is not purely pedagogical, since correction for changes in image potential during transit is required in determining the conductance of several new high-speed semiconductor devices. A new procedure to evaluate transit time is presented, which assumes that energy fluctuations keep a tunneling particle above the barrier while transiting the classically forbidden region. For large rectangular barriers, the most probable fluctuations minimize the product of their magnitude and the transit time. This results in the semiclassical solution. For the case of very small barriers, there is a separate solution in which the transit time is indeterminate but bounded. The new solution is compared with 12 different results by others. Numerical values, limiting forms, and interpretations are presented for these various tunneling times. Difficulties in using time domain numerical solutions to determine the transit time for a wave packet will also be described. © 1992 John Wiley & Sons, Inc.
We have presented a simple approach to quantum theory of Brownian motion and barrier crossing dynamics. Based on an initial coherent state representation of bath oscillators and an equilibrium canonical distribution of quantum-mechanical mean values of their co-ordinates and momenta we have derived a c number generalized quantum Langevin equation. The approach allows us to implement the method of classical non-Markovian Brownian motion to realize an exact generalized non-Markovian quantum Kramers' equation. The equation is valid for arbitrary temperature and friction. We have solved this equation in the spatial diffusion-limited regime to derive quantum Kramers' rate of barrier crossing and analyze its variation as a function of the temperature and friction. While almost all the earlier theories rest on quasiprobability distribution functions (e.g., Wigner function) and path integral methods, the present work is based on true probability distribution functions and is independent of path integral techniques. The theory is a natural extension of the classical theory to quantum domain and provides a unified description of thermally activated processes and tunneling.
This work is about a quantum-mechanical generalisation of transition rate theory that accounts for temperature-dependent quantum-dynamical effects. The starting point of this investigation is a formulation of the classical transmission coefficient where the activated dynamics is needed to obtain the time evolution of the classical state function and the rate is expressed as an average flux in momentum space. Traditional approaches to quantum transition rate theory are briefly discussed, with emphasis on the fact that these do not generally require knowledge of the open quantum dynamics. The latter can be studied by means of the Lindblad equation for mixed states, which is shown to be analogous to an average over realisations of a certain type of stochastic Schrödinger dynamics. The stochastic evolution in Hilbert space is used in combination with an ansatz for the wave function as a Gaussian wave packet to show that under appropriate assumptions the stochastic Schrödinger equation converges to the classical Langevin equation. A novel formulation of the quantum transmission coefficient is presented which features a quantum-mechanical version of the state function obtained via the density operator, so that the activated quantum dynamics can be taken into account by solving the Lindblad equation. A numerical investigation of the quantum transmission coefficient thus obtained for different temperatures is carried out, along with a comparison to the classical results. The quantum-mechanical transmission coefficient is found to display some expected features, such as vanishing in the limit as t → 0, displaying a “plateau” value for certain values of the temperature, and decaying to zero exponentially fast when the fundamental assumptions of TRT do not hold. Additionally the plateau value of the quantum transmission coefficient is found to be smaller than the classical counterpart in all regimes, and to approach the corresponding classical value in the limit of large temperatures.