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arXiv:1907.04865 · Physical review. D/Physical review. D.

Generalized escape paths for dynamical tunneling in QFT

Luc Jean-Marie Darmé, Joerg Jaeckel, Marek Lewicki

Published Nov 19, 2019• 9 citations
arXivHTMLPDF

Abstract

We present a formalism based on the functional Schr\"odinger equation to analyze time-dependent tunneling in quantum field theory at the semiclassical level. The full problem is reduced step by step to a finite dimensional quantum mechanical setup and solved using the Wentzel-Kramers-Brillouin approximation. As an example, we consider tunneling from a homogeneous oscillating initial state in scalar quantum field theory.

DOI
10.1103/physrevd.100.096012

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We present a formalism based on the functional Schr"odinger equation to analyze time-dependent tunneling in quantum field theory at the semiclassical level. The full problem is reduced step by step to a finite dimensional quantum mechanical setup and solved using the Wentzel-Kramers-Brillouin approximation. As an example, we consider tunneling from a homogeneous oscillating initial state in scalar quantum field theory.
Starting with the exact variational formulation of quantum mechanics (or quantum field theory) in the Schroedinger picture, and using Gaussian trail wave functions, an exact functional Schroedinger equation is obtained. The time-dependent Hartree-Fock approximation is obtained and extended to many oscillators and to field theory. The problem of quantum roll, in which an initial Gaussian wave packet starting at an unstable classical equilibrium point evolves in time, is treated using this formalism. Quantum tunnelling is also studied. (LEW)
Despite the fact that quantum tunneling has been studied since the advent of quantum mechanics, the literature appears to contain no simple (textbook) formula for tunneling in generic asymmetric double-well potentials. In the regime of strong localization, I derive a succinct analytical formula based on the Wentzel-Kramers-Brillouin semiclassical approach. Two different examples of asymmetric potentials are discussed: the cases when the two localized levels are degenerate and when they are not degenerate. For the first case, I also discuss a time-dependent problem showing the quantum Zeno effect.
A Brief History of Quantum Tunneling Some Basic Questions Concerning Quantum Tunneling Simple Solvable Problems Time-Dependence of the Wave Function in One-Dimensional Tunneling Semiclassical Approximations Generalization of the Bohr - Sommerfeld Quantization Rule and its Application to Quantum Tunneling Gamow's Theory, Complex Eigenvalues, and the Wave Function of a Decaying State Tunneling in Symmetric and Asymmetric Local Potentials and Tunneling in Nonlocal and Quasi-Solvable Barriers Classical Descriptions of Tunneling Tunneling in Time-Dependent Barriers Decay Width and Scattering Theory The Method of Variable Reflection Amplitude Applied to Solve Multichannel Tunneling Problems Path Integral and Its Semi-Classical Approximation in Quantum Tunneling Heisenberg's Equations of Motion for Tunneling Wigner Distribution Function in Quantum Tunneling Decay Widths of Siegert States, Complex Scaling and Dilatation Transformation Multidimensional Quantum Tunneling Group and Signal Velocities Time-Delay, Reflection Time Operator and Minimum Tunneling Time More about Tunneling Time Tunneling of a System with Internal Degrees of Freedom Motion of a Particle in a Waveguide with Variable Cross Section and in a Space Bounded by a Dumbbell-Shaped Object Relativistic Formulation of Quantum Tunneling Inverse Problems of Quantum Tunneling Some Examples of Quantum Tunneling in Atomic and Molecular Physics Some Examples from Condensed Matter Physics Alpha Decay.
We study the decay of general initial states out of a metastable potential well in quantum mechanics. We provide a closed-form expression for the probability current that tunnels through the barrier in terms of the resonant states into which the initial state can be decomposed. All ingredients are computed analytically to first subleading order in the semiclassical limit. Specializing to a coherently-oscillating state, we derive an equivalent semiclassical approximation to the current via the path integral, which circumvents the decomposition into resonant states. We test the accuracy of the approximation by comparing it to a numerical solution of the Schrödinger equation.
Abstract This chapter the reviews essential concepts and methods of time-dependent quantum mechanics. (1) Formal solutions of the time-dependent Schrödinger equation. (2) The two-level system. (3) Interacting systems: The time-dependent self-consistent field approximation and the Ehrenfest approximation. (4) The Born–Oppenheimer approximation and nuclear potential surfaces. (5) Time-dependent Hamiltonians: A two-level system in a time-dependent field. (6) Non-adiabatic dynamics: The surface-hopping method. (7) Expressing the time evolution in terms of the Green operator. (8) Representations (the Schrödinger and Heisenberg representations, the interaction representation). (9) Time-dependent perturbation theory. (10) Quantum dynamics of the free particles. (11) Quantum dynamics of the harmonic oscillator. (12) The raising/lowering operators formalism. (13) Tunneling.
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.