Thermodynamics of trajectories promises to make possible the thorough analysis of the dynamical properties of an open quantum system, a sought-after goal in modern physics. Unfortunately, calculation of the relevant quantities presents severe challenges. Determining the large-deviation function that gives access to the full counting statistics associated with a dynamical order parameter is challenging, if not impossible, even for systems evolving in a restricted Liouville space. Acting on the realisation that the salient features of most dynamical systems are encoded in the first few moments of the counting statistics, in this article we present a method that gives sequential access to these moments. Our method allows for obtaining analytical result in several cases, as we illustrate, and allows using large deviation theory to reinterpret certain well-known results.
We apply the large-deviation method to study trajectories in dissipative quantum systems. We show that in the long time limit the statistics of quantum jumps can be understood from thermodynamic arguments by exploiting the analogy between large-deviation and free-energy functions. This approach is particularly useful for uncovering properties of rare dissipative trajectories. We also prove, via an explicit quantum mapping, that rare trajectories of one system can be realized as typical trajectories of an alternative system.
This article sets up a new formalism to investigate stochastic thermodynamics of out-of-equilibrium quantum systems, where stochasticity primarily comes from quantum measurement. In the absence of any bath, we define a purely quantum component to heat exchange, that corresponds to energy fluctuations induced by measurement back-action. By providing a central position to quantum measurement, our formalism bridges the gap between quantum optics and quantum thermodynamics: Thermal signatures of measurement induced irreversibility are identified in a simple implementation of Jarzynski equality, wh…
We consider Markovian open quantum systems subject to stochastic resetting, which means that the dissipative time evolution is reset at randomly distributed times to the initial state. We show that the ensuing dynamics is non-Markovian and has the form of a generalized Lindblad equation. Interestingly, the statistics of quantum-jumps can be exactly derived. This is achieved by combining techniques from the thermodynamics of quantum-jump trajectories with the renewal structure of the resetting dynamics. We consider as an application of our analysis a driven two-level and an intermittent three-l…
We present a general method to undertake a thorough analysis of the thermodynamics of the quantum jump trajectories followed by an arbitrary quantum harmonic network undergoing linear and bilinear dynamics. The approach is based on the phase-space representation of the state of a harmonic network. The large deviation function associated with this system encodes the full counting statistics of exchange and also allows one to deduce for fluctuation theorems obeyed by the dynamics. We illustrate the method showing the validity of a local fluctuation theorem about the exchange of excitations betwe…
We apply a large-deviation method to study the diffusive trajectories of the quadrature operators of light within a reservoir connected to dissipative quantum systems. We formulate the study of quadrature trajectories in terms of characteristic operators and show that in the long time limit the statistics of such trajectories obey a large-deviation principle. We take our motivation from homodyne detection schemes which allow the statistics of quadrature operator of the light field to be measured. We illustrate our approach with four examples of increasing complexity: a driven two-level system,…
We undertake a thorough analysis of the thermodynamics of the trajectories followed by a quantum harmonic oscillator coupled to $N$ dissipative baths by using a new approach to large-deviation theory inspired by phase-space quantum optics. As an illustrative example, we study the archetypal case of a harmonic oscillator coupled to two thermal baths, allowing for a comparison with the analogous classical result. In the low-temperature limit, we find a significant quantum suppression in the rate of work exchanged between the system and each bath. We further show how the presented method is capab…
For classical nonequilibrium systems, the separation of the total entropy production into the adiabatic and nonadiabatic contributions is useful for understanding irreversibility in nonequilibrium thermodynamics. In this article, we formulate quantum analogs for driven open quantum systems describable by quantum jump trajectories by applying a quantum stochastic thermodynamics. Our main conclusions are based on a quantum formulation of the local detailed balance condition.
I formulate a quantum stochastic thermodynamics for the quantum trajectories of a continuously-monitored forced harmonic oscillator coupled to a thermal reservoir. Consistent trajectory-dependent definitions are introduced for work, heat, and entropy, through engineering the thermal reservoir from a sequence of two-level systems. Within this formalism the connection between irreversibility and entropy production is analyzed and confirmed by proving a detailed fluctuation theorem for quantum trajectories. Finally, possible experimental verifications are discussed.
A recent experiment by Minev et. al [arXiv:1803.00545] demonstrated that in a dissipative (artificial) 3-level atom with strongly intermittent dynamics it is possible to "catch and reverse" a quantum jump "mid-flight": by the conditional application of a unitary perturbation after a fixed time with no jumps, the system was prevented from getting shelved in the dark state, thus removing the intermittency from the dynamics. Here we offer an interpretation of this phenomenon in terms of the dynamical large deviation formalism for open quantum dynamics. In this approach, intermittency is seen as t…
The equivalence of thermodynamic ensembles is at the heart of statistical mechanics and central to our understanding of equilibrium states of matter. Recently, it has been shown that there is a formal connection between the dynamics of open quantum systems and the statistical mechanics in an extra dimension. This is established through the fact that an open system dynamics generates a Matrix Product state (MPS) which encodes the set of all possible quantum jump trajectories and permits the construction of generating functions in the spirit of thermodynamic partition functions. In the case of c…
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 (non-equilibrium) 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…
Quantum open systems evolve according to completely positive, trace preserving maps acting on the density operator, which can equivalently be unraveled in term of so-called quantum trajectories. These stochastic sequences of pure states correspond to the actual dynamics of the quantum system during single realizations of an experiment in which the system's environment is monitored. In this chapter, we present an extension of stochastic thermodynamics to the case of open quantum systems, which builds on the analogy between the quantum trajectories and the trajectories in phase space of classica…
On the basis of information theory, a new formalism of classical non-relativistic mechanics of a mass point is proposed. The particle trajectories of a general dynamical system defined on an (1+n)-dimensional smooth manifold are treated geometrically as dynamical variables. Statistical mechanics of particle trajectories are constructed in a classical manner. Thermodynamic variables are introduced through a partition function based on a canonical ensemble of trajectories. Within this theoretical framework, classical mechanics can be interpreted as an equilibrium state of statistical mechanics. …
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 i…
We propose a piecewise deterministic Markovian jump process in Hilbert space such that the covariance matrix of this stochastic process solves the thermodynamic quantum master equation. The proposed stochastic process is particularly simple because the normalization of the vectors in Hilbert space is preserved only on average. As a consequence of the nonlinearity of the thermodynamic master equation, the construction of stochastic trajectories involves the density matrix as a running ensemble average. We identify a principle of detailed balance and a fluctuation-dissipation relation for our Ma…
Conditional fluctuation theorems and entropy production for monitored quantum systems under imperfect detection
Institute for Cross-Disciplinary Physics and Complex Systems IFISC (UIB-CSIC)（交叉物理与复杂系统研究所）；Departamento de Física Aplicada, Universidad de Alicante（阿利坎特大学应用物理系）
The Dirac equation has been applied to fermions scattering from the downward potential step. The results show some particles do not fall off the edge of the step and reflect. Also, based on de Broglie-Bohm interpretation of quantum mechanics (Bohmian mechanics) and Bohmian trajectories we have resolved the problem. Lastly, a phenomenological study of the Bohmian trajectory of the Klein paradox has been discussed.
The dissipation generated during a quasistatic thermodynamic process can be characterised by introducing a metric on the space of Gibbs states, in such a way that minimally-dissipating protocols correspond to geodesic trajectories. Here, we show how to generalize this approach to open quantum systems by finding the thermodynamic metric associated to a given Lindblad master equation. The obtained metric can be understood as a perturbation over the background geometry of equilibrium Gibbs states, which is induced by the Kubo-Mori-Bogoliubov (KMB) inner product. We illustrate this construction on…
This article sets up a formalism to describe stochastic thermodynamics for driven out-of-equilibrium open quantum systems. A stochastic Schrödinger equation allows to construct quantum trajectories describing the dynamics of the system state vector in presence of an eventually monitored environment. Thermodynamic quantities are defined at the single quantum trajectory level, independently of any energy measurement, at any time of the protocol. We thereby identify coherent contributions, without classical counterparts, leading to quantum fluctuations of thermodynamic quantities. This formalism …
We use a near quantum limited detector to experimentally track individual quantum state trajectories of a driven qubit formed by the hybridization of a waveguide cavity and a transmon circuit. For each measured quantum coherent trajectory, we separately identify energy changes of the qubit as heat and work, and verify the first law of thermodynamics for an open quantum system. We further establish the consistency of these results by comparison with the master equation approach and the two-projective-measurement scheme, both for open and closed dynamics, with the help of a quantum feedback loop…
We introduce and realize demons that follow a customary gambling strategy to stop a nonequilibrium process at stochastic times. We derive second-law-like inequalities for the average work done in the presence of gambling, and universal stopping-time fluctuation relations for classical and quantum non-stationary stochastic processes. We test experimentally our results in a single-electron box, where an electrostatic potential drives the dynamics of individual electrons tunneling into a metallic island. We also discuss the role of coherence in gambling demons measuring quantum jump trajectories.
We present a new trajectory formulation of high harmonic generation that treats classically allowed and classically forbidden processes within a single dynamical framework. Complex trajectories orbit the nucleus, producing the stationary Coulomb ground state. When the field is turned on, these complex trajectories continue their motion in the field-dressed Coulomb potential and therefore tunnel ionization, unbound evolution and recollision are described within a single, seamless framework. The new formulation can bring mechanistic understanding to a broad range of strong field physics effects.
The thermodynamics of quantum systems driven out of equilibrium has attracted increasing attention in last the decade, in connection with quantum information and statistical physics, and with a focus on non-classical signatures. While a first approach can deal with average thermodynamics quantities over ensembles, in order to establish the impact of quantum and environmental fluctuations during the evolution, a continuous quantum measurement of the open system is required. Here we provide an introduction to the general theoretical framework to establish and interpret thermodynamics for quantum…
The result that closed systems evolve toward equilibrium is derived entirely on the basis of quantum field theory for a model system, without invoking any of the common extra-mathematical notions of particle trajectories, collapse of the wave function, measurement, or intrinsically stochastic processes. The equivalent of the Stosszahlansatz of classical statistical mechanics is found, and has important differences from the classical version. A novel result of the calculation is that interacting many-body systems in the infinite volume limit evolve toward diagonal states (states with loss of al…
A microscopic definition of the thermodynamic entropy in an isolated quantum system must satisfy (i) additivity, (ii) extensivity and (iii) the second law of thermodynamics. We show that the diagonal entropy, which is the Shannon entropy in the energy eigenbasis at each instant of time, meets the first two requirements and that the third requirement is satisfied if an arbitrary external operation is performed at typical times. In terms of the diagonal entropy, thermodynamic irreversibility follows from the facts that the Hamiltonian dynamics restricts quantum trajectories under unitary evoluti…
We propose a fully operational framework to study the non-equilibrium thermodynamics of a quantum system $S$ that is coupled to a detector $D$ whose state is continuously monitored, allowing to single out individual quantum trajectories of $S$. We focus on detailed fluctuation theorems and characterize the entropy production of the system. We establish fundamental differences with respect to the thermodynamic of unmonitored, unitarily evolved systems. We consider the paradigmatic example of circuit-QED, where superconducting qubits can be coupled to a continuously monitored resonator and show …
We introduce a pedagogical discussion on Bohmian mechanics and its physical implications in connection with the important role played by the quantum phase in the dynamics of quantum processes. In particular, we focus on phenomena such as quantum coherence, diffraction, and interference, due to their historical relevance in the development of the quantum theory and their key role in a myriad of fundamental and applied problems of current interest.
We consider continuously monitored quantum systems and introduce definitions of work and heat along individual quantum trajectories that are valid for coherent superpositions of energy eigenstates. We use these quantities to extend the first and second laws of stochastic thermodynamics to the quantum domain. We illustrate our results with the case of a weakly measured driven two-level system and show how to distinguish between quantum work and heat contributions. We finally employ quantum feedback control to suppress detector backaction and determine the work statistics.
A central topic in the emerging field of quantum thermodynamics is the definition of thermodynamic work in the quantum regime. One widely used solution is to define work for a closed system undergoing non-equilibrium dynamics according to the two-point energy measurement scheme. However, due to the invasive nature of measurement the two-point quantum work probability distribution leads to inconsistencies with two pillars of thermodynamics: it breaks the first law and the time-reversal symmetry expected for closed dynamics. We here introduce the quantum histories framework as a method to charac…
The first step in quantum information theory is the identification of entanglement as a valuable resource. The next step is learning how to exploit this resource efficiently. We learn how to exploit entanglement efficiently by applying analogues of thermodynamical concepts. These concepts include reversibility, entropy, and the distinction between intensive and extensive quantities. We discuss some of these analogues and show how they lead to a measure of entanglement for pure states. We also ask whether these analogues are more than analogues, and note that, l o c a l l y, entropy of entangle…
We develop a microscopic theory for biasing the quantum trajectories of an open quantum system, which renders rare trajectories typical. To this end we consider a discrete-time quantum dynamics, where the open system collides sequentially with qubit probes which are then measured. A theoretical framework is built in terms of thermodynamic functionals in order to characterize its quantum trajectories (each embodied by a sequence of measurement outcomes). We show that the desired biasing is achieved by suitably modifying the Kraus operators describing the discrete open dynamics. From a microscop…
Recent progress in the study of dynamical phase transitions has been made with a large-deviation approach to study trajectories of stochastic jumps using a thermodynamic formalism. We study this method applied to an open quantum system consisting of a superconducting single-electron transistor, near the Josephson quasiparticle resonance, coupled to a resonator. We find that the dynamical behavior shown in rare trajectories can be rich even when the mean dynamical activity is small and thus the formalism gives insights into the form of fluctuations. The structure of the dynamical phase diagram …
A damped and driven collective spin system is analyzed by using quantum state diffusion. This approach allows for a mostly analytical treatment of the investigated non-equilibrium quantum many body dynamics, which features a phase transition in the thermodynamical limit. The exact results obtained in this work, which are free of any finite size defects, provide a complete understanding of the model. Moreover, the trajectory framework gives an intuitive picture of the two phases occurring, revealing a spontaneously broken symmetry and allowing for a qualitative and quantitative characterization…
We discuss the thermodynamic aspects of a single qubit based device, powered by weak quantum measurements, and feedback controlled by a quantum Maxwell's demon. We discuss both discrete and time-continuous operation of the measurement based device at finite temperature of the reservoir. In the discrete example where a demon acquires information via discrete weak measurements, we find that the thermodynamic variables including the heat exchanged, extractable work, and the entropy produced are completely determined by an information theoretic measure of the demon's perceived arrow of time. We al…
