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

Abstract

Discretizing distribution function in a phase space for an efficient quantum dynamics simulation is 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 find that a twodimensional (2D) periodically invariant system-bath (PISB) model with two heat baths is an ideal platform not only for a periodical system but also for a 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. The stability of the present scheme is demonstrated in a high-temperature Markovian case by numerically integrating the discrete QFPE with by a coarse mesh for a 2D free rotor and harmonic potential systems for an initial condition that involves singularity.

Discretized hierarchical equations of motion in mixed Liouville--Wigner space for two-dimensional vibrational spectroscopies of liquid water

“…65 Such features allow us to dramatically reduce the computational time required to integrate the HEOM, in particular as the size of a system increases. Note that for an unbounded system and a rotationally invariant system, a Lagrange-Fourier mesh method 70 and a discrete Wigner function method 71 are respectively more efficient.…”

Computational perspective on recent advances in quantum electronics: from electron quantum optics to nanoelectronic devices and systems

“…For non-trivial interaction mechanisms a closure problem exists in the Wigner formalism [78, p 98]. Other very recent advances of Wigner function methods for electronics (and photonics, but not further discussed here) can be found in a recent special issue [103] and cover, among others, handling of boundary conditions [104][105][106] and modelling and solution approaches [107][108][109][110][111][112][113].…”

Autobiography of Yoshitaka Tanimura

“…My goal is to simulate a superconducting state in real time under a time-dependent external force. At the moment, however, Kiyoto Nakamura has only been able to solve the HEOM for a four-site Holstein-Hubbard system (137). I think we will need to wait another 15 years for the necessary computational developments.…”