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Journal of Experimental and Theoretical Physics

Relativistic version of the imaginary-time formalism

V. D. Mur, Boris Mikhailovich Karnakov, Vladimir S Popov

Published Sep 1, 1998• 51 citations
Publisher

Abstract

A relativistic version of the quasiclassical imaginary-time formalism is developed. It permits calculation of the tunneling probability of relativistic particles through potential barriers, including barriers lacking spherical symmetry. Application of the imaginary-time formalism to concrete problems calls for finding subbarrier trajectories which are solutions of the classical equations of motion, but with an imaginary time (and thus cannot be realized in classical mechanics). The ionization probability of an s level, whose binding energy can be of the order of the rest energy, under the action of electric and magnetic fields of different configuration is calculated using the imaginary-time formalism. Besides the exponential factor, the Coulomb and pre-exponential factors in the ionization probability are calculated. The Hamiltonian approach to the tunneling of relativistic particles is described briefly. Scrutiny of the ionization of heavy atoms by an electric field provides an additional argument against the existence of the “Unruh effect.”

DOI
10.1134/1.558679

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A relativistic version of the quasiclassical imaginary-time formalism is developed. It permits calculation of the tunneling probability of relativistic particles through potential barriers, including barriers lacking spherical symmetry. Application of the imaginary-time formalism to concrete problems calls for finding subbarrier trajectories which are solutions of the classical equations of motion, but with an imaginary time (and thus cannot be realized in classical mechanics). The ionization probability of an s level, whose binding energy can be of the order of the rest energy, under the action of electric and magnetic fields of different configuration is calculated using the imaginary-time formalism. Besides the exponential factor, the Coulomb and pre-exponential factors in the ionization probability are calculated. The Hamiltonian approach to the tunneling of relativistic particles is described briefly. Scrutiny of the ionization of heavy atoms by an electric field provides an additional argument against the existence of the “Unruh effect.”
Tunnel-ionization is investigated in the framework of relativistic quantum mechanics. \nFor an arbitrary constant electromagnetic field a gauge invariant energy \noperator is introduced in order to identify the classically forbidden region for \ntunnel-ionization. Furthermore, relativistic features of tunnel-ionization are explored. \nA one-dimensional intuitive picture predicts that the ionized electron wave \npacket in the relativistic regime experiences a momentum shift along the laser’s \npropagation direction. This is shown to be consistent with the well-known strong \nfield approximation. Furthermore, spin dynamics in tunnel-ionization process \nis discussed in the standard as well as in the dressed strong field approximation. \nNext, the tunneling time delay is investigated for tunnel-ionization by extending \nthe definition of the Wigner time delay. Later, this concept is redefined in terms of \nthe phase of the fixed energy propagator. The developed formalism is applied to \nthe deep-tunneling and the near-threshold-tunneling regimes. It is shown that in \nthe latter case signatures of the tunneling time delay can be measurable at remote \ndistance. Finally, the path-dependent formulation of gauge theory is discussed. It \nis demonstrated that this equivalent formulation of gauge theory leads to a canonical \ngauge fixing, in which the Feynman path integral becomes more intuitive and \nthe calculation of the quasiclassical propagator is considerably simplified
A relativistically invariant representation of the generalized momentum of a particle in an external field is proposed. In this representation, the dependence of the potentials of the interaction of the particle with the field on the particle velocity is taken into account. The exact correspondence of the expressions of energy and potential energy for the classical Hamiltonian is established, which makes identical the solutions to the problems of mechanics with relativistic and nonrelativistic approaches. The invariance of the proposed representation of the generalized momentum makes it possible to equivalently describe a physical system in geometrically conjugate spaces of kinematic and dynamic variables. Relativistic invariant equations are proposed for the action function and the wave function based on the invariance of the representation of the generalized momentum. The equations have solutions for any values of the constant interaction of the particle with the field, for example, in the problem of a hydrogen-like atom, when the atomic number of the nucleus is Z > 137. Based on the parametric representation of the action, the expression for the canonical Lagrangian, the equations of motion, and the expression for the force acting on the charge are derived when moving in an external electromagnetic field. The Dirac equation with the correct inclusion of the interaction for a particle in an external field is presented. In this form, the solutions of the equations are not limited by the value of the interaction constant. The solutions of the problem of charge motion in a constant electric field, the problems for a particle in a potential well and the passage of a particle through a potential barrier, the problems of motion in an exponential field (Morse), and also the problems of a hydrogen atom are given.
Abstract Tunneling ionization in static or slowly varying electric fields is a cornerstone of strong-field physics and provides the entry point for semiclassical descriptions of above-threshold ionization and high-harmonic generation. In conventional quantum mechanics, the Perelomov–Popov–Terent’ev theory and its Ammosov–Delone–Krainov (ADK) form yield an ionization rate whose defining feature is an exponential dependence governed by an under-barrier (imaginary-time) action. Here we develop an analytical ADK-like tunneling model within space-fractional quantum mechanics, where the quadratic kinetic energy is replaced by the Riesz fractional Laplacian of order 1 < α ⩽ 2 . Working in a static electric field in the length gauge, we derive a closed-form tunneling exponent for a triangular exit barrier. The fractional kinetic operator deforms the conventional I p 3 / 2 scaling to I p 1 + 1 / α and introduces a characteristic sin ⁡ ( π / α ) factor encoding the complex-phase structure associated with nonlocal dispersion. We position this benchmark relative to prior tunneling studies in fractional quantum mechanics (primarily scattering through model barriers and fractal potentials) and provide a validation protocol for testing the exponent in time-dependent simulations of the fractional Schrödinger equation under a constant field. The result establishes a transparent reference for static-field ionization in nonlocal quantum dynamics and a baseline for strong-field approaches extensions.
Tunneling ionization in static or slowly varying electric fields is a cornerstone of strong-field physics and provides the entry point for semiclassical descriptions of above-threshold ionization and high-harmonic generation. In conventional quantum mechanics, the Perelomov--Popov--Terent'ev (PPT) theory and its Ammosov--Delone--Krainov (ADK) form yield an ionization rate whose defining feature is an exponential dependence governed by an under-barrier (imaginary-time) action. Here we develop an analytical ADK-like tunneling model within \emph{space-fractional} quantum mechanics, where the quadratic kinetic energy is replaced by the Riesz fractional Laplacian of order 1<α≤21<α\le21<α≤2. Working in a static electric field in the length gauge, we derive a closed-form tunneling exponent for a triangular exit barrier. The fractional kinetic operator deforms the conventional Ip3/2I_p^{3/2}Ip3/2​ scaling to Ip1+1/αI_p^{1+1/α}Ip1+1/α​ and introduces a characteristic sin⁡(π/α)\sin(π/α)sin(π/α) factor encoding the complex-phase structure associated with nonlocal dispersion. We position this benchmark relative to prior tunneling studies in fractional quantum mechanics (primarily scattering through model barriers and fractal potentials) and provide a validation protocol for testing the exponent in time-dependent simulations of the fractional Schrödinger equation under a constant field. The result establishes a transparent reference for static-field ionization in nonlocal quantum dynamics and a baseline for strong-field approaches extensions.
The quantum tunneling rate of a macroscopic particle in one-dimensional potentials subject to oscillating external fields is considered using the complex time path integral method and the time-dependent Schrödinger equation. A simple calculation scheme for obtaining the tunneling rate within quasi-classical accuracy is described and applied to two systems. The one is the tunneling motion through an inverted parabola potential and the other is the escaping motion from a cubic potential well. Both systems are subject to small oscillating fields. It is found that both of the rates are exponentially enhanced due to the imaginary time motion of the tunneling particle under oscillating potential barriers, and that time scale is identical to the Büttiker-Landauer traversal time.
The analytical continuation of classical equations of motion to complex times suggests that a tunneling particle spends in the barrier an imaginary duration i|T| . Does this mean that it takes a finite time to tunnel, or should tunneling be seen as an instantaneous process? It is well known that examination of the adiabatic limit in a small additional AC field points towards |T| being the time it takes to traverse the barrier. However, this is only half the story. We probe the transmitted particle's history, and find that it very little of the field's past behavior, as if the transit time were close to zero. The ensuing contradiction suggests that the question is ill posed, and we explain why. Published by the American Physical Society 2024