# The heat and work of quantum thermodynamic processes with quantum coherence

## TL;DR

This paper investigates how quantum coherence influences the definitions and distinctions of heat and work in various quantum thermodynamic processes, providing a detailed analysis and exact solutions for specific quantum systems.

## Contribution

It clarifies the role of quantum coherence in thermodynamic energy exchanges and demonstrates that quantum adiabatic processes are sufficient but not necessary for thermodynamic adiabaticity.

## Key findings

- Energy exchanges depend on eigenenergy and probability distribution changes.
- Quantum coherence affects thermodynamic properties in non-adiabatic processes.
- Quantum adiabatic process is sufficient but not necessary for thermodynamic adiabaticity.

## Abstract

Energy is often partitioned into heat and work by two independent paths corresponding to the change in the eigenenergies or the probability distributions of a quantum system. The discrepancies of the heat and work for various quantum thermodynamic processes have not been well characterized in literature. Here we show how the work in quantum machines is differentially related to isochoric, isothermal, and adiabatic processes. We prove that the energy exchanges during the quantum isochoric and isothermal processes are simply depending on the change in the eigenenergies or the probability distributions. However, for a time-dependent system in a non-adiabatic quantum evolution, the transitions between the different quantum states representing the quantum coherence can affect the essential thermodynamic properties, and thus the general definitions of the heat and work should be clarified with respect to the microscopic generic time-dependent system. By integrating the coherence effects in the exactly-solvable dynamics of quantum-spin precession, the internal energy is rigorously transferred as the work in the thermodynamic adiabatic process. The present study demonstrates that quantum adiabatic process is sufficient but not necessary for thermodynamic adiabatic process.