Quantum tunnelling, real-time dynamics and Picard-Lefschetz thimbles

Abstract

We follow up the work, where in light of the Picard-Lefschetz thimble approach, we split up the real-time path integral into two parts: the initial density matrix part which can be represented via an ensemble of initial conditions, and the dynamic part of the path integral which corresponds to the integration over field variables at all later times. This turns the path integral into a two-stage problem where, for each initial condition, there exits one and only one critical point and hence a single thimble in the complex space, whose existence and uniqueness are guaranteed by the characteristics of the initial value problem. In this paper, we test the method for a fully quantum mechanical phenomenon, quantum tunnelling in quantum mechanics. We compare the method to solving the Schrödinger equation numerically, and to the classical-statistical approximation, which emerges naturally in a well-defined limit. We find that the Picard-Lefschetz result matches the expectation from quantum mechanics and that, for this application, the classical-statistical approximation does not.

19 pages, 6 figures; more references added

References in corpus (6)

• The Sphaleron Rate in the Minimal Standard Model
• Lattice simulations of real-time quantum fields
• Stochastic quantization at finite chemical potential
• Real-time Feynman path integral with Picard--Lefschetz theory and its applications to quantum tunneling
• Some remarks on Lefschetz thimbles and complex Langevin dynamics
• Gradient flows without blow-up for Lefschetz thimbles

Cited by in corpus (17)

• Inflationary Butterfly Effect: Non-perturbative Dynamics From Small-Scale Features
• Path integral contour deformations for observables in $SU(N)$ gauge theory
• Path integral contour deformations for noisy observables
• Quantitative Analysis of the Stochastic Approach to Quantum Tunneling
• Backpropagating Hybrid Monte Carlo algorithm for fast Lefschetz thimble calculations
• Lorentzian path integral for quantum tunneling and WKB approximation for wave-function
• Structure Factors of Neutron Matter at Finite Temperature
• False vacuum decay rates, more precisely
• Mass Renormalization in Lattice Simulations of False Vacuum Decay
• Non-Hermitian BCS-BEC crossover of Dirac fermions
• Resurgence in Lorentzian quantum cosmology: No-boundary saddles and resummation of quantum gravity corrections around tunneling saddle points
• Finite-Temperature Instantons from First Principles
• Parametrized Path Approach to Vacuum Decay
• Toward quantum tunneling from excited states: Recovering imaginary-time instantons from a real-time analysis
• Quantum cosmology of the flat universe via closed real-time path integral
• Jackiw-Teitelboim Gravity and Lorentzian Quantum Cosmology
• Bubble nucleation and quantum initial conditions in classical statistical simulations