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Quantum dynamics with the parallel transport gauge. (English) Zbl 1440.65113

Summary: The dynamics of a closed quantum system is often studied with the direct evolution of the Schrödinger equation. In this paper, we propose that the gauge choice (i.e., degrees of freedom irrelevant to physical observables) of the Schrödinger equation can be generally nonoptimal for numerical simulation. This can limit, and in some cases severely limit, the time step size. We find that the optimal gauge choice is given by a parallel transport formulation. This parallel transport dynamics can be simply interpreted as the dynamics driven by the residual vectors, analogous to those defined in eigenvalue problems in the time-independent setup. The parallel transport dynamics can be derived from a Hamiltonian structure and is thus suitable to be solved using a symplectic and implicit time discretization scheme, such as the implicit midpoint rule, which allows the usage of a large time step and ensures the long time numerical stability. We analyze the parallel transport dynamics in the context of the singularly perturbed linear Schrödinger equation and demonstrate its superior performance in the near adiabatic regime. We demonstrate the effectiveness of our method using numerical results for linear and nonlinear Schrödinger equations, as well as the time-dependent density functional theory (TDDFT) calculations for electrons in a benzene molecule driven by an ultrashort laser pulse.

MSC:

65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35Q41 Time-dependent Schrödinger equations and Dirac equations
35Q55 NLS equations (nonlinear Schrödinger equations)
65P10 Numerical methods for Hamiltonian systems including symplectic integrators
81V70 Many-body theory; quantum Hall effect
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35B25 Singular perturbations in context of PDEs
35Q49 Transport equations
82M36 Computational density functional analysis in statistical mechanics
78A60 Lasers, masers, optical bistability, nonlinear optics
81V55 Molecular physics

Software:

RODAS; KELLEY; VMD; DGDFT
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

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