Abstract
Discretizing a distribution function in a phase space for an efficient quantum dynamics simulation is a non-trivial challenge, in particular for a case that a system is further coupled to environmental degrees of freedom. Such open quantum dynamics is described by a reduced equation of motion (REOM) most notably by a quantum Fokker-Planck equation (QFPE) for a Wigner distribution function (WDF). To develop a discretization scheme that is stable for numerical simulations from the REOM approach, we employ a two-dimensional (2D) periodically invariant system-bath (PISB) model with two heat baths. This model is an ideal platform not only for a periodic system but also for a non-periodic system confined by a potential. We then derive the numerically ''exact'' hierarchical equations of motion (HEOM) for a discrete WDF in terms of periodically invariant operators in both coordinate and momentum spaces. The obtained equations can treat non-Markovian heat-bath in a non-perturbative manner at finite temperatures regardless of the mesh size. As demonstrations, we numerically integrate the discrete QFPE for a 2D free rotor and harmonic potential systems in a high-temperature Markovian case using a coarse mesh with initial conditions that involve singularity.
Highlights
A central issue in the development of a computational simulation for a quantum system described in a phase space distribution is the instability of the numerical integration of a kinetic equation in time, which depends upon a discretization scheme of the coordinate and momentum[1,2,3,4,5]
We introduce a new approach to construct a Wigner distribution function (WDF) for an open quantum dynamics system on the basis of a finite dimensional quantum mechanics developed by Schwinger [6]
To apply this formalism to an open quantum dynamics system, we found that a rotationally invariant system-bath (RISB) Hamiltonian developed for the investigation of a quantum dissipative rotor system is ideal [62]
Summary
A central issue in the development of a computational simulation for a quantum system described in a phase space distribution is the instability of the numerical integration of a kinetic equation in time, which depends upon a discretization scheme of the coordinate and momentum[1,2,3,4,5]. Our approach is an extension of a discrete WDF formalism introduced by Wootters [61] that is constructed on the basis of a finite dimensional quantum mechanics introduced by Schwinger [6] To apply this formalism to an open quantum dynamics system, we found that a rotationally invariant system-bath (RISB) Hamiltonian developed for the investigation of a quantum dissipative rotor system is ideal [62]. We employ a 2D periodically invariant system-bath (PISB) model to derive a discrete reduced equation of motion that is numerically stable regard less of the mesh size.
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