Weak and ultrastrong coupling limits of the quantum mean force Gibbs state

TL;DR

This paper derives general expressions for the mean force Gibbs state of a system interacting with a bosonic reservoir, exploring both weak and ultrastrong coupling regimes, and reveals how the state’s properties change with interaction strength.

Contribution

It introduces a new expansion method for the ultrastrong coupling regime and characterizes the mean force Gibbs state in this limit, highlighting the basis in which it becomes diagonal.

Findings

In the weak coupling limit, the state retains coherences with respect to the system Hamiltonian.

In the ultrastrong coupling regime, the state is diagonal in the interaction basis.

The methods are demonstrated on systems including a qubit, a V-system, and coupled qubits.

Abstract

The Gibbs state is widely taken to be the equilibrium state of a system in contact with an environment at temperature $T$. However, non-negligible interactions between system and environment can give rise to an altered state. Here we derive general expressions for this mean force Gibbs state, valid for any system that interacts with a bosonic reservoir. First, we derive the state in the weak coupling limit and find that, in general, it maintains coherences with respect to the bare system Hamiltonian. Second, we develop a new expansion method suited to investigate the ultrastrong coupling regime. This allows us to derive the explicit form for the mean force Gibbs state, and we find that it becomes diagonal in the basis set by the system-reservoir interaction instead of the system Hamiltonian. Several examples are discussed including a single qubit, a three-level V-system and two coupled qubits all interacting with bosonic reservoirs. The results shed light on the presence of coherences in the strong coupling regime, and provide key tools for nanoscale thermodynamics investigations.