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Journal of Physics A Mathematical and General

Complex time path for the tunnelling at intermediate energy

E.‐M. Ilgenfritz, H. Perlt

Published Nov 7, 1992• 9 citations
Publisher

Abstract

The authors investigate tunnelling through a potential barrier in quantum mechanics extending Feynman's path integral to complex time paths. This problem has gained new interest in the context of baryon number violating processes in the standard electroweak model, where the transitions between topological sectors must be described at high energies and temperatures. The authors reduce the number of degrees of freedom to one in order to test their method. It is found that a path in the complex time plane consisting of purely real or imaginary steps minimizes the real and imaginary parts of the action simultaneously. Jumps from real to imaginary time increments are situated at the turning points of the classical problem. Possible extensions to field theory are discussed.

DOI
10.1088/0305-4470/25/21/026

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The authors investigate tunnelling through a potential barrier in quantum mechanics extending Feynman's path integral to complex time paths. This problem has gained new interest in the context of baryon number violating processes in the standard electroweak model, where the transitions between topological sectors must be described at high energies and temperatures. The authors reduce the number of degrees of freedom to one in order to test their method. It is found that a path in the complex time plane consisting of purely real or imaginary steps minimizes the real and imaginary parts of the action simultaneously. Jumps from real to imaginary time increments are situated at the turning points of the classical problem. Possible extensions to field theory are discussed.
A detailed real time description of quantum tunneling in the semiclassical limit is given, using complex classical trajectories. This picture connects naturally with the ideas of post-selection and weak measurement introduced by Aharonov and collaborators. I show that one can precisely identify the complex classical trajectory which a post-selected tunneling particle has followed, and which dominates the path integral in the limit as Planck's constant tends to zero. Detailed analytical calculations are presented for tunneling in cubic and quartic potentials. For a long post-selected tunneling time, the imaginary part of the tunneling coordinate is found to achieve very large values just before the particle tunnels. I discuss how the real and imaginary parts of the particle's coordinate may, in principle, be independently measured using weak measurements. It would be very interesting to observe this effect, which would demonstrate the essential role of complex numbers in our closest possible classical description of reality. Extensions to quantum field theory and general relativity are briefly discussed.
We study tunneling in one-dimensional quantum mechanics using the path integral in real time, where solutions of the classical equation of motion live in the complex plane. Analyzing solutions with small (complex) energy, relevant for constructing the wave function after a long time, we unravel the analytic structure of the action, and show explicitly how the imaginary time bounce arises as a parameterization of the lowest order term in the energy expansion. The real time calculation naturally extends to describe the wave function in the free region of the potential, reproducing the usual WKB approximation. The extension of our analysis to the semiclassical correction due to fluctuations on the saddle is left for future work.
We adapt the semiclassical technique, as used in the context of instanton transitions in quantum field theory, to the description of tunneling transmissions at finite energies through potential barriers by complex quantum mechanical systems. Even for systems initially in their ground state, not generally describable in semiclassical terms, the transmission probability has a semiclassical (exponential) form. The calculation of the tunneling exponent uses analytic continuation of degrees of freedom into a complex phase space as well as analytic continuation of the classical equations of motion into the complex time plane. We test this semiclassical technique by comparing its results with those of a computational investigation of the full quantum mechanical system, finding excellent agreement.
It is widely believed that the WKB approximation to quantum barrier penetration is described in the path-integral formalism by a saddle-point approximation based on an expansion about an imaginary-time semiclassical path. This article examines the relationship between tunneling amplitudes and imaginary-time paths. It is shown, using a simple example, that for some tunneling amplitudes the dominant semiclassical path cannot be chosen to have purely imaginary time if all other variables are chosen to be real.
Abstract It is well known that quantum tunneling can be described by instantons in the imaginary-time path integral formalism. However, its description in the real-time path integral formalism has been elusive. Here we establish a statement that quantum tunneling can be characterized in general by the contribution of complex saddle points, which can be identified by using the Picard-Lefschetz theory. We demonstrate this explicitly by performing Monte Carlo simulations of simple quantum mechanical systems, overcoming the sign problem by the generalized Lefschetz thimble method. We confirm numerically that the contribution of complex saddle points manifests itself in a complex “weak value” of the Hermitian coordinate operator x^ \hat{x} x^ x ̂ evaluated at time t, which is a physical quantity that can be measured by experiments in principle. We also discuss the transition to classical dynamics based on our picture.