Equilibrium and nonequilibrium steady states with the repeated interaction protocol: Relaxation dynamics and energetic cost

TL;DR

This paper analyzes the relaxation dynamics and energetic costs of a qubit system interacting repeatedly with thermalized ancilla spins, revealing the emergence of nonequilibrium steady states distinct from thermal equilibrium.

Contribution

It provides an analytical characterization of the nonequilibrium steady states and their dependence on interaction parameters in a repeated interaction protocol.

Findings

System populations and coherences evolve independently towards a steady state.

The steady state generally differs from the ancilla's thermal state.

The number of steps to reach steady state and the associated energetic costs are bounded.

Abstract

We study the dynamics of a qubit system interacting with thermalized bath-ancilla spins via a repeated interaction scheme. Considering generic initial conditions for the system and employing a Heisenberg-type interaction between the system and the ancillas, we analytically prove the following: (i) The population and coherences of the system qubit evolve independently toward a nonequilibrium steady-state solution, which is diagonal in the qubit's energy eigenbasis. The population relaxes to this state geometrically, whereas the coherences decay through a more compound behavior. (ii) In the long time limit, the system approaches a steady state that generally differs from the thermal state of the ancilla. We derive this steady-state solution and show its dependence on the interaction parameters and collision frequency. (iii) We bound the number of interaction steps required to achieve the steady state within a specified error tolerance, and we evaluate the energetic cost associated with the process. Our key finding is that deterministic system-ancilla interactions do not typically result in the system thermalizing to the thermal state of the ancilla. Instead, they generate a distinct nonequilibrium steady state, which we explicitly derive. However, we also identify an operational regime that leads to thermalization with a few long and possibly randomized collisions.