The strong-coupling quantum thermodynamics of quantum Brownian motion based on the exact solution of its reduced density matrix

TL;DR

This paper derives the exact quantum thermodynamics of quantum Brownian motion using the reduced density matrix, revealing significant renormalizations, a squeezing thermal state, and resolving previous controversies in strong-coupling regimes.

Contribution

It provides an exact analytical solution for the reduced density matrix of quantum Brownian motion, including nonperturbative renormalizations and a detailed analysis of strong coupling effects.

Findings

Reduced Hamiltonian includes frequency shift and squeezing interaction

Exact reduced density matrix is a squeezing thermal state

Resolves controversies on internal energy and negative heat capacity in strong coupling

Abstract

We derive the quantum thermodynamics of quantum Brownian motion from the exact solution of its reduced density matrix. We start from the total equilibrium thermal state between the Brownian particle and its reservoir, and solve analytically and exactly the reduced density matrix of the system by taking the partial trace over all the reservoir states. We find that the reduced Hamiltonian and the reduced partition function of the Brownian motion must be renormalized significantly, as shown in the general nonperturbative renormalization theory of quantum thermodynamics for open quantum systems we developed recently [Phys. Rev. Res. 4, 023141 (2022)]. The reduced Hamiltonian contains not only a frequency shift but also a squeezing pairing interaction, where a momentum-dependent potential is generated naturally from the strong coupling between the Brownian particle and the reservoir, after traced over all the reservoir states. The resulting exact reduced density matrix of the Brownian motion is given by a squeezing thermal state. Moreover, beyond the weak coupling limit, in order to obtain correctly the reduced partition function of the Brownian motion, one must take into account the non-negligible changes of the reservoir state induced by the system-reservoir coupling. Using the exact solutions of the reduced density matrix, the reduced Hamiltonian as well as the reduced partition function of the Brownian motion, we show that the controversial results obtained from the different definitions of internal energy and the issue of the negative heat capacity in the previous studies of strong-coupling quantum thermodynamics are resolved.