One of the most important goals in quantum thermodynamics is to demonstrate advantages of thermodynamic protocols over their classical counterparts. For that, it is necessary to (i) develop theoretical tools and experimental set-ups to deal with quantum coherence in thermodynamic contexts, and to (ii) elucidate which properties are genuinely quantum in a thermodynamic process. In this short review, we discuss proposals to define and measure work fluctuations that allow to capture quantum interference phenomena. We also discuss fundamental limitations arising due to measurement back-action, as well as connections between work distributions and quantum contextuality. We hope the different results summarised here motivate further research on the role of quantum phenomena in thermodynamics.The aim of this short review is to discuss different proposals and definitions of fluctuating work in quantum systems, particularly in the presence of a superposition of energies in the initial state. This is an exciting topic that has recently received a lot of attention in the community of quantum thermodynamics [1,2]. One can identify different motivations behind this research line:

At non-zero temperature classical systems exhibit statistical fluctuations of thermodynamic quantities arising from the variation of the system's initial conditions and its interaction with the environment. The fluctuating work, for example, is characterised by the ensemble of system trajectories in phase space and, by including the probabilities for various trajectories to occur, a work distribution can be constructed. However, without phase space trajectories, the task of constructing a work probability distribution in the quantum regime has proven elusive. Here we use quantum trajectories in phase space and define fluctuating work as power integrated along the trajectories, in complete analogy to classical statistical physics. The resulting work probability distribution is valid for any quantum evolution, including cases with coherences in the energy basis. We demonstrate the quantum work probability distribution and its properties with an exactly solvable example of a driven quantum harmonic oscillator. An important feature of the work distribution is its dependence on the initial statistical mixture of pure states, which is reflected in higher moments of the work. The proposed approach introduces a fundamentally different perspective on quantum thermodynamics, allowing full thermodynamic characterisation of the dynamics of quantum systems, including the measurement process.

We study non-Markovian effects present in a driven qubit coupled to a finite environment using a recently proposed model developed in the context of calorimetric measurements of open quantum systems. To quantify the degree of non-Markovianity we use the Breuer-Laine-Piilo (BLP) measure [H.-P. Breuer, Phys. Rev. Lett. 103, 210401 (2009)]. We show that information backflow only occurs in the case of driving, in which case we investigate the dependence of memory effects on the environment size, driving amplitude, and coupling to the environment. We show that the degree of non-Markovianity strongly depends on the ratio between the driving amplitude and the coupling strength. We also show that the degree of non-Markovianity does not decrease monotonically as a function of the environment size.

We study the equivalence between the recently proposed finite environment quantum jump model and a master equation approach. We derive microscopically the master equation for a qubit coupled to a finite bosonic environment and show that the master equation is equivalent to the finite environment quantum jump model. We analytically show that both the methods produce the same moments of work when the work is defined through the two-measurement protocol excluding the interaction energy. However, when compared to the work moments computed using the power operator approach, we find a difference in the form of the work moments. To numerically verify our results, we study a qubit coupled to an environment consisting of ten two-level systems.

Citation: SAMPAIO, R., SUOMELA, S. and ALA-NISSILA, T., 2016. Calorimetric measurement of work for a driven harmonic oscillator. Physical Review E, 94: 062122.Additional Information:• This paper was accepted for publication in the journal Physi- 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.