Model for calorimetric measurements in an open quantum system

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 qubitcalorimeter 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.

Thermodynamic Geometry of Microscopic Heat Engines

“…Here, σ x and σ z are the usual Pauli matrices, ̵ hΩ denotes the overall energy scale and the dimensionless parameters ε ≥ 0 and λ ≥ ε correspond to the tunneling energy and the flux-tunable level-splitting of the qubit [52,53]. The role the environment is played by a normalmetal island, whose temperature can be accurately controlled with established techniques [54] and monitored by means of sensitive electron thermometers, a technology that could soon enable calorimetric work measurements [55][56][57][58][59][60]. This reservoir can be described in terms of two jump operators, V + Λ and V − Λ , defined by the conditions…”

Hybrid master equation for calorimetric measurements

“…We aim to compare temperature predictions by the weak-drive modelling of the qubit dynamics to those of the Floquet modelling studied in [2]. For the numerical integration of the dynamics, we take the similar parameters as [2]. The level spacing of the qubit is ω = 0.5k B × 1 K, the volume of the calorimeter is V = 10 −21 m 3 , Σ = 2 × 10 −9 W K -5 m -3 and γ = 1500k B /(1K).…”

“…Our motivation is an experiment proposed by [1], that aims to measure thermodynamic indicators of a driven qubit system in contact with a thermal environment. For a detailed discussion of the experiment we refer to [1,2]. In essence, the setup by [1] is a nanoscale electric circuit, containing a superconducting qubit and a resistor element.…”

“…In what follows, we refer to this equation as the Floquet master equation. One of the main results of [2] was to derive from the Floquet master equation a Fokker-Planck equation for the probability distribution of the electron temperature on long time scales, by eliminating the underlying qubit dynamics. The disadvantage of using the Floquet stochastic equation is that it requires, additionally to the usual set of assumptions required for the Born-Markov approximation [5,6], that the strength of the drive is much larger than the qubit-calorimeter coupling squared [4].…”

Josephson-Threshold Calorimeter

“…Thermal detectors rely on the conversion of a temperature rise in a measurable variation of an electric signal. In * claudio.guarcello@nano.cnr.it particular, "quantum" calorimetry permits to detect single photons and measure their energy [4][5][6]. These detectors are characterized by shorter photon-induced energy deposition with fast internal equilibration times as compared to the characteristic thermal relaxation time.…”