Path integral approach to the Wigner representation of canonical density operators for discrete systems coupled to harmonic baths

Abstract

We derive a semi-analytical form for the Wigner transform for the canonical density operator of a discrete system coupled to a harmonic bath based on the path integral expansion of the Boltzmann factor. The introduction of this simple and controllable approach allows for the exact rendering of the canonical distribution and permits systematic convergence of static properties with respect to the number of path integral steps. In additions, the expressions derived here provide an exact and facile interface with quasi-and semi-classical dynamical methods, which enables the direct calculation of equilibrium time correlation functions within a wide array of approaches. We demonstrate that the present method represents a practical path for the calculation of thermodynamic data for the spin-boson and related systems. We illustrate the power of the present approach by detailing the improvement of the quality of Ehrenfest theory for the correlation function C zz (t) = Re σ z (0)σ z (t) for the spin-boson model with systematic convergence to the exact sampling function. Importantly, the numerically exact nature of the scheme presented here and its compatibility with semiclassical methods allows for the systematic testing of commonly used approximations for the Wigner-transformed canonical density.

A partially linearized spin-mapping approach for simulating nonlinear optical spectra

“…Due to the presence of excitonnuclear coupling, the Wigner transform of the Boltzmann operator in the single-exciton subspace is challenging to obtain exactly. 60 For simplicity, we approximate this quantity as follows:…”

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“…20 We recently introduced 21,22 a simple, trajectorybased approximate method that makes use of the classical adiabatic theorem to slowly convert the Wigner density of a harmonic reference system to that of the target Hamiltonian. Other recent work 23 has used the quasi-adiabatic propagator path integral methodology 24 to obtain the Wigner distribution of the bath in the case of a system interacting with a bath of independent harmonic oscillators.…”

Wigner Distribution by Adiabatic Switching in Normal Mode or Cartesian Coordinates and Molecular Applications

“…Extensions of the thermal Gaussian approximation which capture quantum corrections have also been proposed. In the special case of a system coupled to a harmonic bath, the quasi-adiabatic propagator path integral has been employed to develop a numerically exact treatment of the bath Wigner density …”

“…In the special case of a system coupled to a harmonic bath, the quasi-adiabatic propagator path integral 10 has been employed to develop a numerically exact treatment of the bath Wigner density. 11 We have recently described 12 a very simple, approximate method for obtaining the Wigner transform of the density operator corresponding to a thermal Boltzmann density using the classical adiabatic theorem. 13 Starting from a suitable zeroth-order Hamiltonian for which the Wigner density is either analytically or numerically available, the phase space distribution is propagated in time via classical trajectories, while the potential is slowly transformed to that of the target Hamiltonian.…”