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arXiv:1803.11015 · Physical Review A

Model for calorimetric measurements in an open quantum system

Brecht Donvil, Paolo Muratore-Ginanneschi, Jukka P Pekola, Kay Schwieger

Published May 8, 2018• 13 citations
arXivHTMLPDF

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 a 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.

DOI
10.1103/physreva.97.052107

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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 a 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.
Ongoing experimental activity aims at calorimetric measurements of thermodynamic indicators of quantum integrated systems. We study a model of a driven qubit in contact with a finite-size thermal electron reservoir. The temperature of the reservoir changes due to energy exchanges with the qubit and an infinite-size phonon bath. Under the assumption of weak coupling and weak driving, we model the evolution of the qubit-electron temperature as a hybrid master equation for the density matrix of the qubit at different temperatures of the calorimeter. We compare the temperature evolution with an earlier treatment of the qubit-electron model, where the dynamics were modeled by a Floquet master equation under the assumption of drive intensity much larger than the qubit-electron coupling squared. We numerically and analytically investigate the predictions of the two mathematical models of dynamics in the weak-drive parametric region. We numerically determine the parametric regions where the two models of dynamics give distinct temperature predictions and those where their predictions match.
Recent developments in experimental methods allow for the study of thermodynamic properties of quantum systems. In quantum integrated circuits, quantum systems are elements in an electric circuit that can straightforwardly be coupled to other elements. This manipulability allows one to construct quantum heat engines, Maxwell demons etc. Quantum integrated circuits also are one of the main potential settings to realise a working quantum computer. Calorimetric measurements in integrated circuits serve as a promising technique to probe thermodynamic laws of the quantum regime and to study the inner workings of quantum devices. Due to this experimental accessibility, the theoretical study of open quantum systems in the context of quantum integrated circuits is highly relevant. Open quantum systems are typically small systems, e.g. qubits or oscillators, in contact with one or more reservoirs. The research on which this thesis is based can roughly be divided into two parts. The first part is concerned with the thermodynamics of a driven qubit in contact with a thermal bath. This system is the archetype of a quantum out-ofequilibrium system. In one case the qubit is strongly driven by a semiclassical driving field. Building on earlier works, the thermodynamic relations of the system are found by proving the equivalence with an easier to study qubit-oscillator system. In the other, case the qubit is driven by being in contact with two baths with a temperature gradient. The full generating function is derived in a proper approximative scheme and a fluctuation-dissipation relation is found. The second part focusses on a specific experimental scheme to perform calorimetric measurements. The scheme relies on coupling a quantum system to a finite reservoir and performing fast temperature measurements on the reservoir. Doing so allows one to infer energy changes in the reservoir and therefore to obtain the heat exchanged with the system. The dynamics of this system are modelled for weak system-reservoir coupling and concrete experimental predictions are made. In new work, the dynamics for a toy model of a system interacting with a finite reservoir are derived from first principles for a specific model. The first principle derivation matches with the earlier modelled dynamics in the weak coupling and allows to consider strong system-reservoir coupling as well.
Abstract Performing imperfect or noisy measurements on a quantum mechanical system both impacts the measurement outcome and the state of the system after the measurement. In this paper we are concerned with imperfect calorimetric measurements. In calorimetric measurements one typically measures the energy of a thermal environment to extract information about the system. In our setting the measurement is imperfect due to noise directly acting on the detector. Concretely, we consider an additional noise bath such that its energy is measured simultaneously with the calorimeter energy. Under weak coupling assumptions, we derive a hybrid master equation for the state of the system and the detected energy and find that the presence of the noise bath manifests itself by modifying the jump rates of the reduced system dynamics. We study an example of a driven qubit interacting with a resonant boson calorimeter and demonstrate that increasing the additional noise leads to an apparent reduction in the power flowing from qubit to calorimeter and thus to a seemingly detected heating up of the calorimeter.
Quantum calorimetry, the thermal measurement of quanta, is a method of choice for ultrasensitive radiation detection ranging from microwaves to gamma rays. The fundamental temperature fluctuations of the calorimeter, dictated by the coupling of it to the heat bath, set the ultimate lower bound of its energy resolution. Here we reach this limit of fundamental equilibrium fluctuations of temperature in a nanoscale electron calorimeter, exchanging energy with the phonon bath at very low temperatures. The approach allows noninvasive measurement of energy transport in superconducting quantum circuits in the microwave regime with high efficiency, opening the way, for instance, to observe quantum jumps, detecting their energy to tackle central questions in quantum thermodynamics.
The large-deviation method can be used to study the measurement trajectories of open quantum systems. For optical arrangements this formalism allows to describe the long time properties of the (nonequilibrium) photon counting statistics in the context of a (equilibrium) thermodynamic approach defined in terms of dynamical phases and transitions between them in the trajectory space [J. P. Garrahan and I. Lesanovsky, Phys. Rev. Lett. 104, 160601 (2010)]. In this paper, we study the thermodynamic approach for fluorescent systems coupled to complex reservoirs that induce stochastic fluctuations in their dynamical parameters. In a fast modulation limit the thermodynamics corresponds to that of a Markovian two-level system. In a slow modulation limit, the thermodynamic properties are equivalent to those of a finite system that in an infinite-size limit is characterized by a first-order transition. The dynamical phases correspond to different intensity regimes, while the size of the system is measured by the transition rate of the bath fluctuations. As a function of a dimensionless intensive variable, the first and second derivatives of the thermodynamic potential develop an abrupt change and a narrow peak, respectively. Their scaling properties are consistent with a double-Gaussian probability distribution of the associated extensive variable.