# Heat transfer statistics in mixed quantum-classical systems

## TL;DR

This paper develops a mixed quantum-classical framework to analyze heat transfer at the nanoscale, deriving general statistical expressions and demonstrating their application to a spin boson model.

## Contribution

It introduces a novel mixed quantum-classical approach using QCLE and full counting statistics to study heat transfer without simplifying bath assumptions.

## Key findings

- Derived a general expression for the heat moment generating function.
- Showed the steady state fluctuation symmetry holds up to order ℏ.
- Simulated heat transfer in the nonequilibrium spin boson model.

## Abstract

The modelling of quantum heat transfer processes at the nanoscale is crucial for the development of energy harvesting and molecular electronics devices. Herein, we adopt a mixed quantum-classical description of a device, in which the open subsystem of interest is treated quantum mechanically and the surrounding heat baths are treated in a classical-like fashion. By introducing such a mixed quantum-classical description of the composite system, one is able to study the heat transfer between the subsystem and bath from a closed system point of view, thereby avoiding simplifying assumptions related to the bath time scale and subsystem-bath coupling strength. In particular, we adopt the full counting statistics approach to derive a general expression for the moment generating function of heat in systems whose dynamics are described by the quantum-classical Liouville equation (QCLE). From this expression, one can deduce expressions for the dynamics of the average heat and heat current, which may be evaluated using numerical simulations. Due to the approximate nature of the QCLE, we also find that the steady state fluctuation symmetry holds up to order $\hbar$ for systems whose subsystembath couplings and baths go beyond bilinear and harmonic, respectively. To demonstrate the approach, we consider the nonequilibrium spin boson model and simulate its time-dependent average heat and heat current under various conditions.