We obtain the Crooks and the Jarzynski non-equilibrium fluctuation relations using a direct quantum-mechanical approach for a finite system that is either isolated or coupled not too strongly to a heat bath. These results were hitherto derived mostly in the classical limit. The two main ingredients in the picture are the time-reversal symmetry and the application of the first law to the case where an agent performs work on the system. No further assumptions regarding stochastic or Markovian behavior are necessary, neither a master equation or a classical phase-space picture are required. The simplicity and the generality of these non-equilibrium relations are demonstrated, giving very simple insights into the Physics.
By taking full advantage of the dynamic property imposed by the detailed balance condition, we derive a new refined unified fluctuation theorem (FT) for general stochastic thermodynamic systems. This FT involves the joint probability distribution functions of the final phase space point and a thermodynamic variable. Jarzynski equality, Crooks fluctuation theorem, and the FTs of heat as well as the trajectory entropy production can be regarded as special cases of this refined unified FT, and all of them are generalized to arbitrary initial distributions. We also find that the refined unified FT…
The probability densities of work that can be exerted on a quantum system initially staying in thermal equilibrium are constrained by the fluctuation relations of Jarzynski and Crooks, when the work is determined by two projective energy measurements. We investigate the question whether these fluctuation relations may still hold if one employs generalized energy measurements rather than projective ones. Restricting ourselves to a class of universal measurements which are independent of several details of the system on which the work is done, we find sets of necessary and sufficient conditions …
The past two decades witnessed important developments in the field of non-equilibrium statistical mechanics. Among these developments, the Jarzynski equality, being a milestone following the landmark work of Clausius and Kelvin, stands out. The Jarzynski equality relates the free energy difference between two equilibrium states and the work done on the system through far from equilibrium processes. While experimental tests of the equality have been performed in classical regime, the verification of the quantum Jarzynski equality has not yet been fully demonstrated due to experimental challenge…
We show that the quantum Jarzynski equality generalizes to $\mathcal{PT}$-symmetric quantum mechanics with unbroken $\mathcal{PT}$-symmetry. In the regime of broken $\mathcal{PT}$-symmetry the Jarzynski equality does not hold as also the $\mathcal{CPT}$-norm is not preserved during the dynamics. These findings are illustrated for an experimentally relevant system -- two coupled optical waveguides. It turns out that for these systems the phase transition between the regimes of unbroken and broken $\mathcal{PT}$-symmetry is thermodynamically inhibited as the irreversible work diverges at the cri…
The well-known Jarzynski equality, often written in the form $e^{-βΔF}=\langle e^{-βW}\rangle$, provides a non-equilibrium means to measure the free energy difference $ΔF$ of a system at the same inverse temperature $β$ based on an ensemble average of non-equilibrium work $W$. The accuracy of Jarzynski's measurement scheme was known to be determined by the variance of exponential work, denoted as ${\rm var}\left(e^{-βW}\right)$. However, it was recently found that ${\rm var}\left(e^{-βW}\right)$ can systematically diverge in both classical and quantum cases. Such divergence will necessarily po…
A superconducting cavity model was proposed as a way to experimentally investigate the work performed in a quantum system. We found a simple mathematical relationship between the free energy variation and visibility measurement in quantum cavity context. If we consider the difference of Hamiltonian at time $t_0$ and $t_λ$ (protocol time) as a quantum work, then the Jarzynski equality is valid and the visibility can be used to determine the work done on the cavity.
The Jarzynski equality (JE) is a remarkable statement relating transient irreversible processes to infinite-time free energy differences. Although twenty years old, the JE remains unfamiliar to many; nevertheless it is a robust and powerful law. We examine two of Einstein's most simple and well-known discoveries, one classical and one quantum, and show how each of these follows from the JE. Our first example is Einstein's relation between the drag and diffusion coefficients of a particle in Brownian motion. In this context we encounter a paradox in the macroscopic limit of the JE which is fasc…
The target paper presents an experimental verification of a "Jarzynski-related" equality. We show that the latter equality is in fact not related to the Jarzynski equality.
The fluctuation theorems, and in particular, the Jarzynski equality, are the most important pillars of modern non-equilibrium statistical mechanics. We extend the quantum Jarzynski equality together with the Two-Time Measurement Formalism to their ultimate range of validity -- to quantum field theories. To this end, we focus on a time-dependent version of scalar phi-four. We find closed form expressions for the resulting work distribution function, and we find that they are proper physical observables of the quantum field theory. Also, we show explicitly that the Jarzynski equality and Crooks …
A result of great theoretical and experimental interest, Jarzynski equality predicts a free energy change $ΔF$ of a system at inverse temperature $β$ from an ensemble average of non-equilibrium exponential work, i.e., $\langle e^{-βW}\rangle =e^{-βΔF}$. The number of experimental work values needed to reach a given accuracy of $ΔF$ is determined by the variance of $e^{-βW}$, denoted ${\rm var}(e^{-βW})$. We discover in this work that ${\rm var}(e^{-βW})$ in both harmonic and an-harmonic Hamiltonian systems can systematically diverge in non-adiabatic work protocols, even when the adiabatic prot…
Many studies of quantum-size heat engines assume that the dynamics of an internal system is unitary and that the extracted work is equal to the energy loss of the internal system. Both assumptions, however, should be under scrutiny. In the present paper, we analyze quantum-scale heat engines, employing the measurement-based formulation of the work extraction recently introduced by Hayashi and Tajima~[arXiv:1504.06150]. We first demonstrate the inappropriateness of the unitary time evolution of the internal system (namely the first assumption above) using a simple two-level system; we show that…
In this paper, we address an important question of the relationship between fluctuation theorems for the dissipated work $W_{d} = W-ΔF$ with general finite-time (like Jarzynski equality and Crooks relation) and infinite-time (like Gallavotti-Cohen theorem) drive protocols and their time-reversal symmetric versions. The relations between these kinds of fluctuation relations are uncovered based on the examples of a classical Markovian $N$-level system. Further consequences of these relations are discussed with respect to the possible experimental verifications.
Symmetry-Enforced Non-Hermitian Jarzynski Equality in an SU(2)-Rotated Family of Hybrid $\mathcal{PT}$--$\mathcal{APT}$ Systems
Sun Yat-sen University（中山大学）；Shenzhen Research Institute of Sun Yat-Sen University（中山大学深圳研究院）；Quantum Science Center of Guangdong-HongKong-Macao Greater Bay Area（粤港澳大湾区量子科学中心）；Guangdong Provincial Key Laboratory of Quantum Metrology and Sensing（广东省量子计量与传感重点实验室）；State Key Laboratory of Optoelectronic Materials and Technologies（光电信息材料与技术的国家重点实验室）
It has recently been shown that the Jarzynski equality gets modified, when there are experimental errors in computing work. This modified result also holds good in presence of feedback. In this work, we use a simple toy model, that of a Szilard engine, to prove these results both in the presence and in absence of feedback.
Crook's Fluctuation Theorem and Jarzynski equality are immensely powerful tools in obtaining equilibrium properties through non-equilibrium transition between two equilibrium states. In this letter, we propose an extension to the Crook's fluctuation theorem for transition between two non-equilibrium steady states (NESS). Using the proposed theorem, we show that it is possible to obtain free energy differences of multiple equilibrium states from a single set of data obtained from the transition between two NESS. The results are verified using numerical simulations by employing Nose-Hoover dynam…
We give a perturbative analysis of Crooks relation and Jarzynski equality in an arbitrary driven quantum system weakly coupled to a heat bath. Invoking no efficient Hamiltonian nor any restriction on the form of the coupling, we derive the first-order correction to Crooks relation and Jarzynski equality in terms of the interacting Hamiltonian. A Crooks type of relation about energy increment in the system and nonvanishing heat expenditure to the heat bath up to the second order of interaction would be given by the way. Our results tell us that deviation of the quantum fluctuation relations is …
In the global framework of finding an axiomatic derivation of nonequilibrium Statistical Mechanics from fundamental principles, such as the maximum path entropy -- also known as Maximum Caliber principle -- , this work proposes an alternative derivation of the well-known Jarzynski equality, a nonequilibrium identity of great importance today due to its applications to irreversible processes: biological systems (protein folding), mechanical systems, among others. This equality relates the free energy differences between two equilibrium thermodynamic states with the work performed when going bet…
The out-of-time-ordered correlator (OTOC) diagnoses quantum chaos and the scrambling of quantum information via the spread of entanglement. The OTOC encodes forward and reverse evolutions and has deep connections with the flow of time. So do fluctuation relations such as Jarzynski's Equality, derived in nonequilibrium statistical mechanics. I unite these two powerful, seemingly disparate tools by deriving a Jarzynski-like equality for the OTOC. The equality's left-hand side equals the OTOC. The right-hand side suggests a protocol for measuring the OTOC indirectly. The protocol is platform-nons…
In open quantum systems, a clear distinction between work and heat is often challenging, and extending the quantum Jarzynski equality to systems evolving under general quantum channels beyond unitality remains an open problem in quantum thermodynamics. In this Letter, we introduce well-defined notions of guessed quantum heat and guessed quantum work, by exploiting the one-time measurement scheme, which only requires an initial energy measurement on the system alone. We derive a modified quantum Jarzynski equality and the principle of maximum work with respect to the guessed quantum work, which…
The quantum Jarzynski equality is an important theorem of modern quantum thermodynamics. We show that the Jarzynski equality readily generalizes to relativistic quantum mechanics described by the Dirac equation. After establishing the conceptual framework we solve a pedagogical, yet experimentally relevant, system analytically. As a main result we obtain the exact quantum work distributions for charged particles traveling through a time-dependent vector potential evolving under Schrödinger as well as under Dirac dynamics, and for which the Jarzynski equality is verified. Special emphasis is pu…
It has been established that the inclusive work for classical, Hamiltonian dynamics is equivalent to the two-time energy measurement paradigm in isolated quantum systems. However, a plethora of other notions of quantum work has emerged, and thus the natural question arises whether any other quantum notion can provide motivation for purely classical considerations. In the present analysis, we propose the conditional stochastic work for classical, Hamiltonian dynamics, which is inspired by the one-time measurement approach. This novel notion is built upon the change of expectation value of the e…
The free energy of glasses cannot be estimated using thermodynamic integration, as glasses are intrinsically not in equilibrium. We present numerical simulations showing that, in contrast, plausible free-energy estimates of a Kob-Andersen glass can be obtained using the Jarzynski relation. Using the Jarzynski relation, we also compute the chemical potential difference of the two components of this system, and find that, in the glassy regime, the Jarzynski estimate matches well with the extrapolated value of the supercooled liquid. Our findings are of broader interest as they show that the Jarz…
