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New Journal of Physics

Calorimetric measurement of work in a quantum system

Jukka P Pekola, Paolo Solinas, Alexander Shnirman, D. V. Averin

Published Nov 12, 2013• 103 citations
Publisher

Abstract

We propose a calorimetric measurement of work in a quantum system. As a physical realization, we consider a superconducting two-level system, a Cooper-pair box, driven by a gate voltage past an avoided level crossing at charge degeneracy. We demonstrate that, with realistic experimental parameters, the temperature measurement of a resistor (environment) can detect single microwave photons emitted or absorbed by the two-level system. This method would thus be a way to measure the full distribution of work in repeated measurements, and to assess the quantum fluctuation relations.

DOI
10.1088/1367-2630/15/11/115006

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We propose a calorimetric measurement of work in a quantum system. As a physical realization, we consider a superconducting two-level system, a Cooper-pair box, driven by a gate voltage past an avoided level crossing at charge degeneracy. We demonstrate that, with realistic experimental parameters, the temperature measurement of a resistor (environment) can detect single microwave photons emitted or absorbed by the two-level system. This method would thus be a way to measure the full distribution of work in repeated measurements, and to assess the quantum fluctuation relations.
To define the work performed on a driven quantum system in a physically sound way has turned out to be a truly non-trivial task, except in some special cases of limited applicability. This topic has been in a focus of intense research recently in the attempts to generalize the classical fluctuation relations into the quantum regime. Here we propose and demonstrate that a calorimetric measurement gives both a theoretical and experimental tool to test the Jarzynski equality (JE) and other fluctuation relations in a quantum system, and to analyze the distribution of dissipation in them, based on the very principle of conservation of energy. We focus on an experimentally feasible two-level system, a superconducting Cooper pair box subject to Landau-Zener interband transitions. Because of the small heat capacity and weak relaxation to the phonon bath, the calorimetric measurement on the electron gas (resistor) turns out to be a very feasible experimental method.
A calorimetric measurement has recently been proposed as a promising technique to measure thermodynamic quantities in a dissipative superconducting qubit. These measurements rely on the fact that the system is projected into energy eigenstates whenever energy is exchanged with the environment. This requirement imposes a restriction on the class of systems that can be measured in this way. Here we extend the calorimetric protocol to the measurement of work in a driven quantum harmonic oscillator. We employ a scheme based on a two-level approximation that makes use of an experimentally accessible quantity and show how it relates to the work obtained through the standard two-measurement protocol. We find that the average work is well approximated in the underdamped regime for short driving times and, in the overdamped regime, for any driving time. However, this approximation fails for the variance and higher moments of work at finite temperatures. Furthermore, we show how to relate the work statistics obtained through this scheme to the work statistics given by the two-measurement protocol.
We describe a qubit linearly coupled to a heat bath, either directly or via a cavity.The main focus of the paper is on calorimetric detection in a realistic circuit, specifically a solid-state qubit coupled to a resistor as an absorber.The bath in the model is formed of oscillators initially in the ground state with a distribution of energies and coupling strengths.A direct numerical solution of the Schrödinger equation for the full system including up to 10 6 oscillators in the bath verifies the expected decay process.We address quantitatively the question of separation of the qubit and bath by adding a cavity in between which by detuning allows one to adjust the decay rate into a convenient regime for detection purposes.Most importantly, we propose splitting a quantum to two uncoupled baths and performing a cross-correlation measurement of their temperatures.This technique enhances significantly the signal-to-noise ratio of the calorimeter.
We discuss work performed on a quantum two-level system coupled to multiple thermal baths. To evaluate the work, a measurement of photon exchange between the system and the baths is envisioned. In a realistic scenario, some photons remain unrecorded as they are exchanged with baths that are not accessible to the measurement, and thus only partial information on work and heat is available. The incompleteness of the measurement leads to substantial deviations from standard fluctuation relations. We propose a recovery of these relations, based on including the mutual information given by the counting efficiency of the partial measurement. We further present the experimental status of a possible implementation of the proposed scheme, i.e. a calorimetric measurement of work, currently with nearly single-photon sensitivity.
A calorimetric measurement has recently been proposed as a promising technique to measure thermodynamic quantities in a dissipative superconducting qubit. These measurements rely on the fact that the system is projected into energy eigenstates whenever energy is exchanged with the environment. This provides a realization of the two measurement protocol in open quantum systems. However, the measurement scheme assumes that only the two lowest levels of the system can be occupied. The possibility of occupying higher levels becomes relevant when the system is driven for a sufficiently long time or the temperature is high enough. This causes a systematic error in the calorimetric measurement. We quantify this by considering a driven quantum harmonic oscillator in contact with an ideal heat bath for which we derive analytic expressions for the average work and its variance. We find that for the average work the two level approximation holds in the limit of small dissipation for short driving time and low temperature. In the limit of strong dissipation it will hold for any driving time, at low temperature. However the approximation is not valid for the variance and the higher moments of work at finite temperature.
Motivated by the recent development of fast and ultrasensitive thermometry in nanoscale systems, we investigate quantum calorimetric detection of individual heat pulses in the sub-meV energy range. We propose a hybrid superconducting injector-calorimeter setup, with the energy of injected pulses carried by tunneling electrons. It is shown that the superconductor constitutes a versatile injector, with tunable tunnel rates and energies. Treating all heat transfer events microscopically, we analyze the statistics of the calorimeter temperature fluctuations and derive conditions for an accurate measurement of the heat pulse energies. Our results pave the way for fundamental quantum thermodynamics experiments, including calorimetric detection of single microwave photons.