Open quantum dynamics theory on the basis of periodical system-bath model for discrete Wigner function

• Published in: Journal of computational electronics Vol. 20; no. 6; pp. 2091 - 2103
• Main Authors: Iwamoto, Yuki, Tanimura, Yoshitaka
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-12-2021; Springer Nature B.V
• Subjects: Accuracy; Boundary conditions; Distribution functions; Eigenvectors; Electrical Engineering; Electrons; Engineering; Equations of motion; Equilibrium; Finite element method; Fokker-Planck equation; Heat; High temperature; Hilbert space; Initial conditions; Invariants; Mathematical analysis; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematical models; Mechanical Engineering; Optical and Electronic Materials; Plating baths; Quantum physics; Quantum theory; Theoretical; Wigner distribution; Quantum Fokker–Planck equation; Hierarchical equations of motion; Discrete Wigner distribution function; Open quantum dynamics theory
• ISSN: 1569-8025; 1572-8137
• DOI: 10.1007/s10825-021-01754-z

Discretizing a distribution function in a phase space for an efficient quantum dynamics simulation is a non-trivial challenge, in particular for a case in which 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.