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arXiv:1611.06780 · Physical Review A

Path of a tunneling particle

Charis Anastopoulos, Ntina Savvidou

Published May 23, 2017• 5 citations
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Abstract

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.

DOI
10.1103/physreva.95.052120

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We formulate quantum tunneling as a time-of-arrival problem: we determine the detection probability for particles passing through a barrier at a detector located a distance L from the tunneling region. For this purpose, we use a positive-operator-valued measure (POVM) for the time of arrival determined in C. Anastopoulos and N. Savvidou, J. Math. Phys. 47, 122106 (2006). This only depends on the initial state, the Hamiltonian, and the location of the detector. The POVM above provides a well-defined probability density and an unambiguous interpretation of all quantities involved. We demonstrate that for a class of localized initial states, the detection probability allows for an identification of tunneling time with the classic phase time. We also establish limits to the definability of tunneling time. We then generalize these results to a sequential measurement setup: the phase-space properties of the particles are determined by an unsharp sampling before their attempt to cross the barrier. For such measurements the tunneling time is defined as a genuine observable. This allows us to construct a probability distribution for its values that is definable for all initial states and potentials. We also identify a regime in which these probabilities correspond to a tunneling-time operator.
Classical mechanics is a singular theory in that real-energy classical particles can never enter classically forbidden regions. However, if one regulates classical mechanics by allowing the energy E of a particle to be complex, the particle exhibits quantum-like behavior: complex-energy classical particles can travel between classically allowed regions separated by potential barriers. When Im E → 0, the classical tunneling probabilities persist. Hence, one can interpret quantum tunneling as an anomaly. A numerical comparison of complex classical tunneling probabilities with quantum tunneling probabilities leads to the conjecture that as Re E increases, complex classical tunneling probabilities approach the corresponding quantum probabilities. Thus, this work attempts to generalize the Bohr correspondence principle from classically allowed to classically forbidden regions.
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We apply the quasilinearization method of Bellman and Kalaba [Quasilinearization and Nonlinear Boundary-Value Problems (Elsevier, New York, 1965)] to find approximate solutions for the multidimensional quantum tunneling for separable as well as nonseparable wave equations. By introducing the idea of the complex ``semiclassical trajectory'' which is valid for the motion over and under the barrier, and which, in the proper limit, reduces to the real classical trajectory in the allowed region, we obtain an eigenvalue equation for the characteristic wave numbers. This eigenvalue equation is similar to the corresponding equation obtained from the WKB approximation and yields complex eigenvalues with negative imaginary parts. When the barrier changes very rapidly as a function of the radial distance, we can replace the concept of the semiclassical trajectory, which may not be applicable in this case, by the concept of a complex ``quantum trajectory.'' The trajectory defined either way depends on a constant of integration, and by minimizing the action with respect to this constant we can obtain the minimum escape path. The case of two-dimensional tunneling is discussed as an example of this method.
We study the decay of unstable states by formulating quantum tunneling as a time-of-arrival problem: we determine the detection probability for particles at a detector located a distance L from the tunneling region. For this purpose, we use a positive-operator-valued measure (POVM) for the time-of-arrival determined by Anastopoulos and Savvidou [J. Math. Phys. 47, 122106 (2006)]. This only depends on the initial state, the Hamiltonian, and the location of the detector. The POVM above provides a well-defined probability density and an unambiguous interpretation of all quantities involved. We demonstrate that the exponential decay only arises if three specific mathematical conditions are met. Their physical content is the following: (i) the decay time is much larger than any microscopic timescale, so that the fine details of the initial state can be ignored, (ii) there is no quantum coherence between the different “attempts” of the particle to traverse the barrier, and (iii) the transmission probability varies little within the momentum spread of the initial state. We also determine the long time limits of the decay probability and we identify regimes, in which the decays have no exponential phase.
We derive the exact distribution of times spent in the classically forbidden region by a particle as it tunnels through a square barrier potential. We find that the integrated distribution has a simple interpretation. If one restricts the time spent in the barrier to be \ensuremath{\leqslant}\ensuremath{\tau}, then an effective uncertainty \ensuremath{\Delta}V is introduced in the barrier height, such that\ensuremath{\Delta}V\ensuremath{\tau}\ensuremath{\simeq}1. This uncertainty leads to unusual interference effects in the distribution. These interference effects explain why the average speed of a particle traversing a very opaque barrier may exceed the speed of light. \textcopyright{} 1990 The American Physical Society