A model for calorimetric measurements in an open quantum system

TL;DR

This paper models calorimetric measurements in an open quantum system, specifically a driven qubit coupled to a finite electron reservoir, deriving a stochastic process for the calorimeter temperature and analyzing its steady-state behavior.

Contribution

It introduces a generalized quantum jump process including calorimeter temperature as a dynamical variable, and derives a Fokker-Planck equation for its distribution.

Findings

The qubit-calorimeter system reaches a steady state asymptotically.

The temperature distribution can be described by a Fokker-Planck equation.

Measurable indicators depend on the qubit-calorimeter coupling constant.

Abstract

We investigate the experimental setup proposed in [New J. Phys., 15, 115006 (2013)] for calorimetric measurements of thermodynamic indicators in an open quantum system. As theoretical model we consider a periodically driven qubit coupled with a large yet finite electron reservoir, the calorimeter. The calorimeter is initially at equilibrium with an infinite phonon bath. As time elapses, the temperature of the calorimeter varies in consequence of energy exchanges with the qubit and the phonon bath. We show how under weak coupling assumptions, the evolution of the qubit-calorimeter system can be described by a generalized quantum jump process including as dynamical variable the temperature of the calorimeter. We study the jump process by numeric and analytic methods. Asymptotically with the duration of the drive, the qubit-calorimeter attains a steady state. In this same limit, we use multiscale perturbation theory to derive a Fokker--Planck equation governing the calorimeter temperature distribution. We inquire the properties of the temperature probability distribution close and at the steady state. In particular, we predict the behavior of measurable statistical indicators versus the qubit-calorimeter coupling constant.