We formulate the general approach based on the Lindblad equation to calculate the full counting statistics of work and heat produced by driven quantum systems weakly coupled with a Markovian thermal bath. The approach can be applied to a wide class of dissipative quantum systems driven by an arbitrary force protocol. We show the validity of general fluctuation relations and consider several generic examples. The possibilities of using calorimetric measurements to test the presence of coherence and entanglement in the open quantum systems are discussed.
We formulate the general approach based on the Lindblad equation to calculate the full counting statistics of work and heat produced by driven quantum systems weakly coupled with a Markovian thermal bath. The approach can be applied to a wide class of dissipative quantum systems driven by an arbitrary force protocol. We show the validity of general fluctuation relations and consider several generic examples. The possibilities of using calorimetric measurements to test the presence of coherence and entanglement in the open quantum systems are discussed.
We develop a comprehensive framework for characterizing fluctuations in quantum transport and nonequilibrium thermodynamics using two complementary approaches: full counting statistics and first-passage times. Focusing on open quantum systems governed by Markovian Lindblad dynamics, we derive general ensemble relations that connect the two approaches at all times, and we clarify how the steady states reached at long times relate to those reached at large jump counts. In regimes of metastability, long-lived intermediate states cause violations of experimentally testable cumulant relations, as we discuss. We also formulate a fluctuation theorem governing the probability of rare fluctuations in the first-passage time distributions based on results from full counting statistics. Our results apply to general integer-valued trajectory observables that do not necessarily increase monotonically in time. Three illustrative applications, a two-state emitter, a driven qubit, and a variant of the Su-Schrieffer-Heeger model, highlight the physical implications of our results and provide guidelines for practical calculations. Our framework provides a complete picture of first-passage time statistics in Markovian quantum systems, encompassing multiple earlier results, and it has direct implications for current experiments in quantum optics, superconducting circuits, and nanoscale heat engines.
We employ the generalized master equation to calculate the statistics of work produced by driven quantum systems interacting with a heat bath. We consider several generic systems such as a harmonic oscillator, single and two coupled qubits to demonstrate that quantum fluctuations of work can be either sub- or super-Poissonian. The latter regime indicates a steady state entanglement between coupled qubits. We show that quantum work fluctuations can be studied experimentally by measuring the statistics of photons emitted by a quantum system coupled to an overdamped electromagnetic cavity mode.
Recently, a "unified" quantum master equation was derived and shown to be of the Gorini-Kossakowski-Lindblad-Sudarshan form. This equation describes the dynamics of open quantum systems in a manner that forgoes the full secular approximation and retains the impact of coherences between eigenstates close in energy. We implement full counting statistics with the unified quantum master equation to investigate the statistics of energy currents through open quantum systems with nearly degenerate levels. We show that, in general, this equation gives rise to dynamics that satisfy fluctuation symmetry, a sufficient condition for the Second Law of Thermodynamics at the level of average fluxes. For systems with nearly degenerate energy levels, such that coherences build up, the unified equation is simultaneously thermodynamically consistent and more accurate than the fully secular master equation. We exemplify our results for a "V" system facilitating energy transport between two thermal baths at different temperatures. We compare the statistics of steady-state heat currents through this system as predicted by the unified equation to those given by the Redfield equation, which is less approximate but, in general, not thermodynamically consistent. We also compare results to the secular equation, where coherences are entirely abandoned. We find that maintaining coherences between nearly degenerate levels is essential to properly capture the current and its cumulants. On the other hand, the relative fluctuations of the heat current, which embody the thermodynamic uncertainty relation, display inconsequential dependence on quantum coherences.
In this work, we consider a general feedback protocol based on quantum-jump detections, where the last detected jump channel is stored in a memory and subsequently used to implement a feedback action, such as modifying the system Hamiltonian conditioned on the last jump. We show that the time evolution of this general protocol can be described by a Lindblad master equation defined in a hybrid classical-quantum space, where the classical part encodes the stored measurement record (memory) and the quantum part represents the monitored system. Moreover, we show that this new representation can be used to fully characterize the counting statistics of a system subject to a general jump-based feedback protocol. We apply the formalism to a three-level system coupled to two thermal baths operating as a thermal machine, and we show that jump-based feedback can be used to convert the information obtained from the jump detections into work. Our framework provides analytical tools that enable the characterization of key statistical properties of any counting observable under jump-based feedback, such as the average current, noise, correlation functions, and power spectrum.
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.