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.
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.
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 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.
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.
The purpose of this paper is to study an effect of an environment on a quantum tunneling. Here, the system we consider is of a symmetric two-level type with “bare” tunneling frequency Δω0, and the correlation time of actions due to the environment, τc, is assumed to be much shorter than Δω0-1. In the investigations, the method of density matrix is used, and a coupling between the system and the environment is included up to lowest nontrivial order. After calculations, we reach the following conclusions: First, the possibility of the quantum tunneling is, in a high temperature range βℏΔω0 ≪1, washed out by the thermal actions from the environment. In the inverse case βℏΔω0 ≪1, where the quantum fluctuation of the environment plays a dominant role, the possibility of the quantum tunneling remains. Secondaly, in the case βℏΔω0 ≪1, the dissipation effect depresses the tunneling probability, while the potential barrier is effectively lowered by the actions due to the environment.
Attempts to find a quantum-to-classical correspondence in a classically forbidden region leads to nonphysical paths involving, for example, complex time or spatial coordinates. Here, we identify genuine quasiclassical paths for tunneling in terms of probabilistic correlations in sequential time-of-arrival measurements. In particular, we construct the postselected probability density ${P}_{\text{p.s.}}(x,\ensuremath{\tau})$ for a particle to be found at time $\ensuremath{\tau}$ in position x inside the forbidden region, provided that it later crossed the barrier. The classical paths follow from the maximization of the probability density with respect to $\ensuremath{\tau}$. For almost monochromatic initial states, the paths correspond to the maxima of the modulus square of the wave function ${|\ensuremath{\psi}(x,\ensuremath{\tau})|}^{2}$ with respect to $\ensuremath{\tau}$ and for constant x inside the barrier region. The derived paths are expressed in terms of classical equations, but they have no analogs in classical mechanics. Finally, we evaluate the paths explicitly for the case of a square potential barrier.
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/\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.