Continuous-time random walk for open systems: fluctuation theorems and counting statistics

Semi-Markov processes in open quantum systems. II. Counting statistics with resetting

A semi-Markov process method for obtaining general counting statistics for open quantum systems is extended to the scenario of resetting. The simultaneous presence of random resets and wave function collapses means that the quantum jump trajectories are no longer semi-Markov. However, focusing on trajectories and using simple probability formulas, general counting statistics can still be constructed from reset-free statistics. An exact tilted matrix equation is also obtained. The inputs of these methods are the survival distributions and waiting-time density distributions instead of quantum operators. In addition, a continuous-time cloning algorithm is introduced to simulate the large-deviation properties of open quantum systems. Several quantum optics systems are used to demonstrate these results.

Asymptotic large deviations of counting statistics in open quantum systems

We use a semi-Markov process method to calculate large deviations of counting statistics for three open quantum systems, including a resonant two-level system and resonant three-level systems in the Λ and V configurations. In the first two systems, radical solutions to the scaled cumulant generating functions are obtained. Although this is impossible in the third system, since a general sixth-degree polynomial equation is present, we still obtain asymptotically large deviations of the complex system. Our results show that, in these open quantum systems, the large deviation rate functions at zero current are equal to two times the largest nonzero real parts of the eigenvalues of operator -iH[over ̂], where H[over ̂] is a non-Hermitian Hamiltonian, while at a large current these functions possess a unified formula.

Operationally accessible uncertainty relations for thermodynamically consistent semi-Markov processes

Semi-Markov processes generalize Markov processes by adding temporal memory effects as expressed by a semi-Markov kernel. We recall the path weight for a semi-Markov trajectory and the fact that thermodynamic consistency in equilibrium imposes a crucial condition called direction-time independence for which we present an alternative derivation. We prove a thermodynamic uncertainty relation that formally resembles the one for a discrete-time Markov process. The result relates the entropy production of the semi-Markov process to mean and variance of steady-state currents. We prove a further thermodynamic uncertainty relation valid for semi-Markov descriptions of coarse-grained Markov processes that emerge by grouping states together. A violation of this inequality can be used as an inference tool to conclude that a given semi-Markov process cannot result from coarse graining an underlying Markov one. We illustrate these results with representative examples.

On the hybrid Davies like generator for quantum dissipation

We provide a class of quantum evolution beyond Markovian semigroup. This class is governed by a hybrid Davies like generator such that dissipation is controlled by a suitable memory kernel and decoherence by standard Gorini-Kossakowski-Lindblad-Sudarshan generator. These two processes commute and both of them commute with the unitary evolution controlled by the systems Hamiltonian. The corresponding memory kernel gives rise to semi-Markov evolution of the diagonal elements of the density matrix. However, the corresponding evolution needs not be completely positive. The role of decoherence generator is to restore complete positivity. Hence, to pose the dynamical problem, one needs two processes generated by classical semi-Markov memory kernel and purely quantum decoherence generator. This scheme is illustrated for a qubit evolution.

Dissipation-Time Uncertainty Relation

We show that the entropy production rate bounds the rate at which physical processes can be performed in stochastic systems far from equilibrium. In particular, we prove the fundamental tradeoff ⟨S[over ˙]_{e}⟩T≥k_{B} between the entropy flow ⟨S[over ˙]_{e}⟩ into the reservoirs and the mean time T to complete any process whose time-reversed is exponentially rarer. This dissipation-time uncertainty relation is a novel form of speed limit: the smaller the dissipation, the larger the time to perform a process.

Exact Coarse Graining Preserves Entropy Production out of Equilibrium

The entropy production rate associated with broken time-reversal symmetry provides an essential characterization of nanosystems out of equilibrium, from driven colloidal particles to molecular motors. Limited access to the dynamical states is generally expected to hinder the correct estimation of this observable. Here we show how memoryless jump processes can be coarse grained, exactly preserving its average and fluctuations at stationarity. This supports univocal applicability of fluctuation theorems for entropy and allows inference of the genuine thermodynamics together with inaccessible process details.

Inferring broken detailed balance in the absence of observable currents

Identifying dissipation is essential for understanding the physical mechanisms underlying nonequilibrium processes. In living systems, for example, the dissipation is directly related to the hydrolysis of fuel molecules such as adenosine triphosphate (ATP). Nevertheless, detecting broken time-reversal symmetry, which is the hallmark of dissipative processes, remains a challenge in the absence of observable directed motion, flows, or fluxes. Furthermore, quantifying the entropy production in a complex system requires detailed information about its dynamics and internal degrees of freedom. Here we introduce a novel approach to detect time irreversibility and estimate the entropy production from time-series measurements, even in the absence of observable currents. We apply our technique to two different physical systems, namely, a partially hidden network and a molecular motor. Our method does not require complete information about the system dynamics and thus provides a new tool for studying nonequilibrium phenomena.

Non-equilibrium systems with hidden states are relevant for biological systems such as molecular motors. Here the authors introduce a method for quantifying irreversibility in such a system by exploiting the fluctuations in the waiting times of time series data.

Non-Markovianity and negative entropy production rates

Entropy production plays a fundamental role in nonequilibrium thermodynamics to quantify the irreversibility of open systems. Its positivity can be ensured for a wide class of setups, but the entropy production rate can become negative sometimes. This is often taken as an indicator of non-Markovianity. We make this link precise by showing under which conditions a negative entropy production rate implies non-Markovianity and when it does not. For a system coupled to a single heat bath, this can be established within a unified language for two setups: (i) the dynamics resulting from a coarse-grained description of a Markovian master equation and (ii) the classical Hamiltonian dynamics of a system coupled to a bath. The quantum version of the latter result is shown not to hold despite the fact that the integrated thermodynamic description is formally equivalent to the classical case. The instantaneous fixed point of a non-Markovian dynamics plays an important role in our study. Our key contribution is to provide a consistent theoretical framework to study the finite-time thermodynamics of a large class of dynamics with a precise link to its non-Markovianity.

Heat transfer statistics in mixed quantum-classical systems

The modelling of quantum heat transfer processes at the nanoscale is crucial for the development of energy harvesting and molecular electronic devices. Herein, we adopt a mixed quantum-classical description of a device, in which the open subsystem of interest is treated quantum mechanically and the surrounding heat baths are treated in a classical-like fashion. By introducing such a mixed quantum-classical description of the composite system, one is able to study the heat transfer between the subsystem and bath from a closed system point of view, thereby avoiding simplifying assumptions related to the bath time scale and subsystem-bath coupling strength. In particular, we adopt the full counting statistics approach to derive a general expression for the moment generating function of heat in systems whose dynamics are described by the quantum-classical Liouville equation (QCLE). From this expression, one can deduce expressions for the dynamics of the average heat and heat current, which may be evaluated using numerical simulations. Due to the approximate nature of the QCLE, we also find that the steady state fluctuation symmetry holds up to order ℏ for systems whose subsystem-bath couplings and baths go beyond bilinear and harmonic, respectively. To demonstrate the approach, we consider the nonequilibrium spin boson model and simulate its time-dependent average heat and heat current under various conditions.

Motion of kinesin in a viscoelastic medium

Kinesin is a molecular motor that transports cargo along microtubules. The results of many in vitro experiments on kinesin-1 are described by kinetic models in which one transition corresponds to the forward motion and subsequent binding of the tethered motor head. We argue that in a viscoelastic medium like the cytosol of a cell this step is not Markov and has to be described by a nonexponential waiting time distribution. We introduce a semi-Markov kinetic model for kinesin that takes this effect into account. We calculate, for arbitrary waiting time distributions, the moment generating function of the number of steps made, and determine from this the average velocity and the diffusion constant of the motor. We illustrate our results for the case of a waiting time distribution that is Weibull. We find that for realistic parameter values, viscoelasticity decreases the velocity and the diffusion constant, but increases the randomness (or Fano factor).

The stretch to stray on time: Resonant length of random walks in a transient

First-passage times in random walks have a vast number of diverse applications in physics, chemistry, biology, and finance. In general, environmental conditions for a stochastic process are not constant on the time scale of the average first-passage time or control might be applied to reduce noise. We investigate moments of the first-passage time distribution under an exponential transient describing relaxation of environmental conditions. We solve the Laplace-transformed (generalized) master equation analytically using a novel method that is applicable to general state schemes. The first-passage time from one end to the other of a linear chain of states is our application for the solutions. The dependence of its average on the relaxation rate obeys a power law for slow transients. The exponent ν depends on the chain length N like ν=-N/(N+1) to leading order. Slow transients substantially reduce the noise of first-passage times expressed as the coefficient of variation (CV), even if the average first-passage time is much longer than the transient. The CV has a pronounced minimum for some lengths, which we call resonant lengths. These results also suggest a simple and efficient noise control strategy and are closely related to the timing of repetitive excitations, coherence resonance, and information transmission by noisy excitable systems. A resonant number of steps from the inhibited state to the excitation threshold and slow recovery from negative feedback provide optimal timing noise reduction and information transmission.

Generalized Master Equations Leading to Completely Positive Dynamics

We provide a general construction of quantum generalized master equations with a memory kernel leading to well-defined, that is, completely positive and trace-preserving, time evolutions. The approach builds on an operator generalization of memory kernels appearing in the description of non-Markovian classical processes and puts into evidence the nonuniqueness of the relationship arising due to the typical quantum issue of operator ordering. The approach provides a physical interpretation of the structure of the kernels, and its connection with the classical viewpoint allows for a trajectory description of the dynamics. Previous apparently unrelated results are now connected in a unified framework, which further allows us to phenomenologically construct a large class of non-Markovian evolutions taking as the starting point collections of time-dependent maps and instantaneous transformations describing the microscopic interaction dynamics.

Feedback control of waiting times

Feedback loops are known as a versatile tool for controlling transport in small systems, which usually have large intrinsic fluctuations. Here we investigate the control of a temporal correlation function, the waiting-time distribution, under active and passive feedback conditions. We develop a general formalism and then specify to the simple unidirectional transport model, where we compare costs of open-loop and feedback control and use methods from optimal control theory to optimize waiting-time distributions.

Stochastic thermodynamics of Langevin systems under time-delayed feedback control: Second-law-like inequalities

Response lags are generic to almost any physical system and often play a crucial role in the feedback loops present in artificial nanodevices and biological molecular machines. In this paper, we perform a comprehensive study of small stochastic systems governed by an underdamped Langevin equation and driven out of equilibrium by a time-delayed continuous feedback control. In their normal operating regime, these systems settle in a nonequilibrium steady state in which work is permanently extracted from the surrounding heat bath. By using the Fokker-Planck representation of the dynamics, we derive a set of second-law-like inequalities that provide bounds to the rate of extracted work. These inequalities involve additional contributions characterizing the reduction of entropy production due to the continuous measurement process. We also show that the non-Markovian nature of the dynamics requires a modification of the basic relation linking dissipation to the breaking of time-reversal symmetry at the level of trajectories. The modified relation includes a contribution arising from the acausal character of the reverse process. This, in turn, leads to another second-law-like inequality. We illustrate the general formalism with a detailed analytical and numerical study of a harmonic oscillator driven by a linear feedback, which describes actual experimental setups.

Thermodynamics of quantum-jump-conditioned feedback control

We consider open quantum systems weakly coupled to thermal reservoirs and subjected to quantum feedback operations triggered with or without delay by monitored quantum jumps. We establish a thermodynamic description of such systems and analyze how the first and second law of thermodynamics are modified by the feedback. We apply our formalism to study the efficiency of a qubit subjected to a quantum feedback control and operating as a heat pump between two reservoirs. We also demonstrate that quantum feedbacks can be used to stabilize coherences in nonequilibrium stationary states which in some cases may even become pure quantum states.

Generalized fluctuation-dissipation and reciprocal relations for Brownian sieves and molecular machines

A Brownian sieve is a spatially periodic microstructured device that combines the effects of thermal noise, local spatial asymmetry, and external forces to separate particles based on their transport properties. By treating the motion of an individual particle as a cyclical process in which the particle fluctuates away from and returns to the origin of some unit cell, I derive generalized fluctuation-dissipation and reciprocal relations for the averages (and for all moments) of the number of periodic displacements that are exact and valid for arbitrary values of the external forces. These relations hold not only for Brownian sieves, but for all molecular machines in which a nanoscale system couples two chemical, mechanical, or transport processes by a cycle in which the molecular machine itself fluctuates away from and then returns to some arbitrary reference state, in the process doing or receiving work on or from the environment.

Quantum semi-Markov processes

We construct a large class of non-Markovian master equations that describe the dynamics of open quantum systems featuring strong memory effects, which relies on a quantum generalization of the concept of classical semi-Markov processes. General conditions for the complete positivity of the corresponding quantum dynamical maps are formulated. The resulting non-Markovian quantum processes allow the treatment of a variety of physical systems, as is illustrated by means of various examples and applications, including quantum optical systems and models of quantum transport.