Wigner Distribution Function for Open Quantum Systems

1997 ◽  
Vol 11 (09n10) ◽  
pp. 391-397 ◽  
Author(s):  
S. Baskoutas
Keyword(s):  
Distribution Function ◽  
Degrees Of Freedom ◽  
Wigner Distribution ◽  
Open Quantum Systems ◽  
Equations Of Motion ◽  
Analytical Form ◽  
Dissipative Systems ◽  
Wigner Distribution Function ◽  
Dissipative Tunneling ◽  
Lie Algebraic Structure

Using the modified biorthonormal Heisenberg equations of motion for non-Hermitian (NH) Hamilton operators, in order to imply a consistent Lie-algebraic structure and also the equivalence between the Heisenberg and Schrödinger pictures, we have obtained the analytical form of the Wigner distribution function which is unavoidable complex. Its imaginary part accounts for the influence of additional degrees of freedom, which are always present in the phenomenological representation of dissipative systems through (NH) Hamiltonians. Applications of the above formalism can be found, for instance, in dissipative macroscopic quantum tunneling (MQT) effect for Josephson junctions, and in the dissipative tunneling of trapped atoms in optical crystals.

scholarly journals Open Quantum Dynamics Theory on the Basis of Periodical System-Bath Model for Discrete Wigner Function

2021 ◽  
Author(s):  
Yuki Iwamoto ◽  
Yoshitaka Tanimura
Keyword(s):  
Distribution Function ◽  
Quantum Dynamics ◽  
Degrees Of Freedom ◽  
Wigner Distribution ◽  
Equations Of Motion ◽  
Heat Bath ◽  
Planck Equation ◽  
Mesh Size ◽  
Dynamics Simulation ◽  
Open Quantum

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 two-dimensional (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.

Sign in / Sign up

Export Citation Format

Share Document