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arXiv:2102.01493 · Physical Review A

Measurement of work and heat in the classical and quantum regimes

Paolo Solinas, Mirko Amico, Nino Zanghı̀

Published Jun 30, 2021• 14 citations
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Abstract

Despite the increasing interest, the research field which studies the concepts of work and heat at the quantum level has suffered from two main drawbacks: first, the difficulty to properly define and measure the work, heat, and internal energy variation in a quantum system and, second, the lack of experiments. Here, we report a full characterization of the dissipated heat, work, and internal energy variation in a two-level quantum system interacting with an engineered environment. We use the IBMQ quantum computer to implement the driven system's dynamics in a dissipative environment. The experimental data allow us to construct quasiprobability distribution functions from which we recover the correct averages of work, heat, and internal energy variation in the dissipative processes. Interestingly, by increasing the environment coupling strength, we observe a reduction of the pure quantum features of the energy exchange processes that we interpret as the emergence of the classical limit. This makes the present approach a privileged tool to study, understand, and exploit quantum effects in energy exchanges.

DOI
10.1103/physreva.103.l060202

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At nonzero 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 characterized 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 characterization of the dynamics of quantum systems, including the measurement process.

arXiv:1707.06159•Physical Review A•Jan 30, 2018•36 citations
Despite the increasing interest, the research field which studies the concepts of work and heat at the quantum level has suffered from two main drawbacks: first, the difficulty to properly define and measure the work, heat, and internal energy variation in a quantum system and, second, the lack of experiments. Here, we report a full characterization of the dissipated heat, work, and internal energy variation in a two-level quantum system interacting with an engineered environment. We use the IBMQ quantum computer to implement the driven system's dynamics in a dissipative environment. The experimental data allow us to construct quasiprobability distribution functions from which we recover the correct averages of work, heat, and internal energy variation in the dissipative processes. Interestingly, by increasing the environment coupling strength, we observe a reduction of the pure quantum features of the energy exchange processes that we interpret as the emergence of the classical limit. This makes the present approach a privileged tool to study, understand, and exploit quantum effects in energy exchanges.
We discuss an approach to determine averages of the work, dissipated heat, and variation of internal energy of an open quantum system driven by an external classical field. These quantities are measured by coupling the quantum system to a quantum detector at different times. This approach allows us to preserve the full quantum features of the evolution. From the measured phase, we are able to obtain a quasicharacteristic function and a quasiprobability density function for the corresponding observables. Despite the fact that these quasiprobability density functions are not the results of direct measurements, they reproduce the expected value of the physical quantities. Analogously to the Wigner function, the negative regions of these quasiprobability density functions are directly related to pure quantum processes which are not interpretable in classical terms. We use this feature to show that in the limit of fast dissipation, the quantum features vanish and interpret this as the emergence of the classical limit of the energy exchange process. Our analysis explains and confirms the behavior observed in recent experiments performed on IBMQ devices [P. Solinas et al., Phys. Rev. A 103, L060202 (2021)]. The possibility to discriminate between classical and quantum features makes the proposed approach an excellent tool to determine if, and in which conditions, quantum effects can be exploited to increase the efficiency in an energy exchange process at the quantum level.
We propose an approach to define and measure the statistics of work, internal energy and dissipated heat in a driven quantum system. In our framework the presence of a physical detector arises naturally and work and its statistics can be investigated in the most general case. In particular, we show that the quantum coherence of the initial state can lead to measurable effects on the moments of the work done on the system. At the same time, we recover the known results if the initial state is a statistical mixture of energy eigenstates. Our method can also be applied to measure the dissipated heat in an open quantum system. By sequentially coupling the system to a detector, we can track the energy dissipated in the environment while accessing only the system degrees of freedom.
Technological advances over the past few decades have made it possible to address quantum systems down to individual quanta of energy. Artificial atoms are now routinely fabricated and manipulated using solid-state technology. Exciting new applications are now within grasp, such as quantum heat engines, which have been theorised to achieve energy efficiencies not possible for their classical counterparts. Energy fluctuations are an inherent part of these devices and a detailed understanding of them is required. However, the proper theoretical framework under which such fluctuations are to be understood is still debated. In this thesis, we study energy fluctuations in open quantum systems at the level of individual runs of an experimental protocol. This work has been carried out in two parts. In the first part, we focus on calorimetric measurements – a promising new experimental protocol to continuously measure energy fluctuations of a qubit interacting with a thermal environment. Applying the protocol to more complex systems is not straightforward. Using a simple two-level approximation, we show that for some experimental parameters this protocol can be extended to a driven harmonic oscillator. Another aspect of the calorimetric protocol is that it can induce non-Markovian effects which affect how work and heat are defined. We quantify the degree of non-Markovianity induced by this measurement protocol, when applied to a qubit, based on a previously developed theoretical model. Importantly, we show that non-Markovianity does not necessarily decrease monotonically with the environment size. The attempt to extend the calorimetric protocol to general systems has led to the second part of this thesis. The calorimetric detection is a particular case of a continuously monitored system. However, other schemes exist that lead to different definitions of work and heat. Reconciling these approaches is not straightforward, as evidenced by the many work definitions present in the literature. The difficulty is centred at the fact that, in the orthodox interpretation of quantum mechanics, energy is not always a well-defined quantity. Here, based on the Bohmian interpretation of quantum mechanics, we put forward a general framework to remove this problem in a clear conceptual manner. In particular, we show that the classical definition of work for closed systems can be straightforwardly generalised to the quantum case. For an open system, we used the Bohmian conditional wave function which provides a stochastic state evolution for the open system. We show that this approach can lead to new insights regarding the role of entanglement in energy fluctuations and has the potential to provide a general and observer-independent framework of energy fluctuations in open quantum systems. The results presented in this thesis give a small contribution to the developing field of quantum thermodynamics.
We consider a collision model representation of nonequilibrium dynamics for an externally driven open quantum system. Specifically we investigate the nondemolition quasiprobability distributions (QPDFs) of work and heat in both Markovian and non-Markovian regimes. In this model, work and heat correspond to different evolution processes, and their contributions can be distinguished. For the system state with initial coherence, in both Markovian and non-Markovian regimes, negative quasiprobabilities at half of the energy gap of the system for work and the variation of internal energy appear, which is considered to be a witness of the quantumness of the system. We also find that when the external driving speed slows down, the quantumness of the system is weakened, and when the driving speed is slow enough, the negative quasiprobability will disappear and the results of the QPDF coincide with those of the two-measurement protocol. Most importantly, we find that non-Markovianity can enhance the quantumness of the system.
Assigning the variations of internal energy into heat or work contributions is a challenging task due to the fact that these properties are trajectory dependent. A number of proposals have been put forward for open quantum systems following an arbitrary dynamics. We here focus on nonequilibrium thermodynamics of a two-level system and explore in addition to the conventional approach, two definitions motivated by either classical work or heat in which the driving Hamiltonian or the trajectory itself, respectively, are used to set up a reference basis. We first give the thermodynamic properties for an arbitrary dynamics and illustrate the results on the Bloch sphere. Then, we solve the particular example of a periodically driven qubit interacting with a dissipative and decoherence bath. Our results illustrate the trajectory-dependent character of heat and work and how contributions originally assigned to dissipation in the Lindblad equation can become a coherent part assigned to work.
At nonzero 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 characterized 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 characterization of the dynamics of quantum systems, including the measurement process.