arXiv subjects

Marcin Łobejko

Publications and source records attributed to Marcin Łobejko.

15 recordsLinked to original sources

arXiv2026

Which Otto Engine Is the Fastest?

A general thermal machine is characterized by distinct time or energy scales: temperature, coupling to the heat baths, internal free dynamics and interactions, and external driving. We address the question: what parameters determine the speed of a given engine, and how can they be used to reliably compare different types of engine? Specifically, we consider four realizations of the Otto engine, each operating with the same Otto efficiency, and aim to characterize and compare their power outputs. The main insight of this paper is that, irrespective of their very different dynamical implementations, the power of each engine can be factorized into a common characteristic work and an implementation-specific characteristic time. The characteristic work depends only on the internal frequencies of the machine and the temperatures, whereas the characteristic time depends on the coupling strength to the baths and on the implementation-specific driving. This observation allows us to compare the power of different engines through their characteristic times and thereby reduce the relevant parameter space to parameters that capture the dynamical aspects of engine performance. Despite different dynamical implementations, this framework reveals simple common features. In all cases, the maximum operating speed is limited by a common timescale given by the sum of the characteristic thermalization times of the hot and cold baths. Moreover, the four engines fall into two distinct asymptotic classes: implementations in which driving and thermalization occur simultaneously exhibit quadratic scaling, whereas those in which work extraction and thermalization alternate exhibit linear scaling. These results expose general dynamical features that determine the operational speed of quantum Otto engines.

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arXiv2025

Equivalence of Discrete and Continuous Otto-Like Engines assisted by Catalysts: Mapping Catalytic Advantages from the Discrete to the Continuous Framework

The catalytic extension of a discrete two-stroke engine employs a cyclic auxiliary system - the catalyst - that remains decoupled from the baths and performs no work, yet enhances power and efficiency beyond the corresponding non-catalytic counterpart. Theoretical models of discrete engines are relatively easy to analyze but remain challenging for experimental implementation due to the required control over individual strokes. In contrast, externally driven engines that are simultaneously coupled to both heat baths - the so-called continuous engines - are more experimentally feasible. Here, we establish an equivalence between discrete and continuous machines, both with and without a catalyst, by mapping the discrete unitary processes and thermalization steps onto an interaction Hamiltonian and a Markovian model of dissipation. As a result, by replacing probability flows with probability currents, we construct an analogous continuous machine corresponding to previously demonstrated catalytic schemes that generalize Otto engines. We illustrate this mapping for the simplest catalytic extension of the Otto engine, demonstrating catalytic enhancement in the continuous regime.

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arXiv2025

Open quantum dynamics of Josephson charge pumps

We investigate the macroscopic dynamics of Josephson charge pumps in the light of Alicki et al.'s theoretical description of the Josephson junction as an open quantum system described by a Markovian master equation. Once the electrostatic interaction between the terminals is taken into account via nonlinear capacitive terms in the Hamiltonian, we find that the resulting description of pumping is physically reasonable and in good qualitative agreement with experimental observations. We comment on how this approach relates to other theoretical treatments of quantum pumps based on time-dependent potentials or scattering amplitudes. We also highlight the significance of our results in the broader context of the dynamics of charge pumping by active systems.

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arXiv2025

Corrections to the Hamiltonian induced by finite-strength coupling to the environment

If a quantum system interacts with the environment, then the Hamiltonian acquires a correction known as the Lamb-shift term. There are two other corrections to the Hamiltonian, related to the stationary state. Namely, the stationary state is to first approximation a Gibbs state with respect to original Hamiltonian. However, if we have finite coupling, then the true stationary state will be different, and regarding it as a Gibbs state to some effective Hamiltonian, one can extract a correction, which is called "steady-state" correction. Alternatively, one can take a static point of view, and consider the reduced state of total equilibrium state, i.e., system plus bath Gibbs state. The extracted Hamiltonian correction is called the "mean-force" correction. This paper presents several analytical results on second-order corrections (in coupling strength) of the three types mentioned above. Instead of the steady state, we focus on a state annihilated by the Liouvillian of the master equation, labeling it as the "quasi-steady state." Specifically, we derive a general formula for the mean-force correction as well as the quasi-steady state and Lamb-shift correction for a general class of master equations. Furthermore, specific formulas for corrections are obtained for the Davies, Bloch-Redfield, and cumulant equation (refined weak coupling). In particular, the cumulant equation serves as a case study of the Liouvillian, featuring a nontrivial fourth-order generator. This generator forms the basis for calculating the diagonal quasi-steady-state correction. We consider spin-boson model as an example, and in addition to using our formulas for corrections, we consider mean-force correction from the reaction-coordinate approach.

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arXiv2025

Dynamics of the Non-equilibrium spin Boson Model: A Benchmark of master equations and their validity

In recent years, there has been tremendous focus on identifying whether effective descriptions of open quantum systems such as master equations, can accurately describe the dynamics of open quantum systems. One particular question is whether they provide the correct steady state in the long time limit. Transient regime is also of interest. Description of evolution by various master equations - some of them being not complete positive - is benchmarked against exact solutions (see e.g. Hartmann and Strunz, Phys. Rev. A 101, 012103). An important property of true evolution is its non-Markovian features, which are not captured by the simplest completely positive master equations. In this paper we consider a non-Markovian, yet completely positive evolution (known as refined weak coupling or cumulant equation) for the Spin-Boson model with an Overdamped Drude-Lorentz spectral density and arbitrary coupling. We bench-marked it against numerically exact solution, as well as against other master equations, for different coupling strengths and temperatures. We find the cumulant to be a better description in the weak coupling regime where it is supposed to be valid. For the examples considered it shows superiority at moderate and strong couplings in the low-temperature regime for all examples considered. In the high-temperature regime however its advantage vanishes. This indicates that the cumulant equation is a good candidate for simulations at weak to moderate coupling and low temperature. Our calculations are greatly facilitated due to our concise formulation of the cumulant equation by means of representation of the density matrix in the SU(N) basis.

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arXiv2024

Catalytic enhancement in the performance of the microscopic two-stroke heat engine

We consider a model of heat engine operating in the microscopic regime: the two-stroke engine. It produces work and exchanges heat in two discrete strokes that are separated in time. The working body of the engine consists of two $d$-level systems initialized in thermal states at two distinct temperatures. Additionally, an auxiliary non-equilibrium system called catalyst may be incorporated with the working body of the engine, provided the state of the catalyst remains unchanged after the completion of a thermodynamic cycle. This ensures that the work produced by the engine arises solely from the temperature difference. Upon establishing the rigorous thermodynamic framework, we characterize two-fold improvement stemming from the inclusion of a catalyst. Firstly, we prove that in the non-catalytic scenario, the optimal efficiency of the two-stroke heat engine with a working body composed of two-level systems is given by the Otto efficiency, which can be surpassed by incorporating a catalyst with the working body. Secondly, we show that incorporating a catalyst allows the engine to operate in frequency and temperature regimes that are not accessible for non-catalytic two-stroke engines. We conclude with general conjecture about advantage brought by catalyst: including the catalyst with the working body always allows to improve efficiency over the non-catalytic scenario for any microscopic two-stroke heat engines. We prove the conjecture for two-stroke engines when the working body is composed of two $d$-level systems initialized in thermal states at two distinct temperatures, as long as the final joint state leading to optimal efficiency in the non-catalytic scenario is not product, or at least one of the $d$-level system is not thermal.

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arXiv2024

Catalytic advantage in Otto-like two-stroke quantum engines

We demonstrate how to incorporate a catalyst to enhance the performance of a heat engine. Specifically, we analyze efficiency in one of the simplest engines models, which operates in only two strokes and comprises of a pair of two-level systems, potentially assisted by a $d$-dimensional catalyst. When no catalysis is present, the efficiency of the machine is given by the Otto efficiency. Introducing the catalyst allows for constructing a protocol which overcomes this bound, while new efficiency can be expressed in a simple form as a generalization of Otto's formula: $1 - \frac{1}{d} \frac{ω_c}{ω_h}$. The catalyst also provides a bigger operational range of parameters in which the machine works as an engine. Although an increase in engine efficiency is mostly accompanied by a decrease in work production (approaching zero as the system approaches Carnot efficiency), it can lead to a more favorable trade-off between work and efficiency. The provided example introduces new possibilities for enhancing performance of thermal machines through finite-dimensional ancillary systems.

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arXiv2023

The Josephson junction as a quantum engine

We treat the Cooper pairs in the superconducting electrodes of a Josephson junction (JJ) as an open system, coupled via Andreev scattering to external baths of electrons. The disequilibrium between the baths generates the direct-current bias applied to the JJ. In the weak-coupling limit we obtain a Markovian master equation that provides a simple dynamical description consistent with the main features of the JJ, including the form of the current-voltage characteristic, its hysteresis, and the appearance under periodic voltage driving of discrete Shapiro steps. For small dissipation, our model also exhibits a self-oscillation of the JJ's electrical dipole with frequency $Ω= 2 e V / \hbar$ around mean voltage $V$. This self-oscillation, associated with "hidden attractors" of the nonlinear equations of motion, explains the observed production of monochromatic radiation with frequency $Ω$ and its harmonics. We argue that this picture of the JJ as a quantum engine resolves open questions about the Josephson effect as an irreversible process and could open new perspectives in quantum thermodynamics and in the theory of dynamical systems.

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arXiv2022

Extraction of ergotropy: free energy bound and application to open cycle engines

The second law of thermodynamics uses change in free energy of macroscopic systems to set a bound on performed work. Ergotropy plays a similar role in microscopic scenarios, and is defined as the maximum amount of energy that can be extracted from a system by a unitary operation. In this analysis, we quantify how much ergotropy can be induced on a system as a result of system's interaction with a thermal bath, with a perspective of using it as a source of work performed by microscopic machines. We provide the fundamental bound on the amount of ergotropy which can be extracted from environment in this way. The bound is expressed in terms of the non-equilibrium free energy difference and can be saturated in the limit of infinite dimension of the system's Hamiltonian. The ergotropy extraction process leading to this saturation is numerically analyzed for finite dimensional systems. Furthermore, we apply the idea of extraction of ergotropy from environment in a design of a new class of stroke heat engines, which we label open-cycle engines. Efficiency and work production of these machines can be completely optimized for systems of dimensions 2 and 3, and numerical analysis is provided for higher dimensions.

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arXiv2022

The asymptotic emergence of the Second Law for a repeated charging process

In one of its versions, the Second Law states: "It is impossible to construct an engine which will work in a complete cycle, and produces no effect except the raising of a weight and cooling of a heat reservoir." While the Second Law is considered as one of the most robust laws of Nature, it is still challenging how to interpret it in a fully quantum domain. Here we unpack the true meaning of the "cyclicity" and formulate the Second Law for a generic quantum battery via its asymptotic properties of a charging process rather than in terms of a single cycle. As a paradigm, we propose a machine consisting of a battery that repeatedly interacts with identically prepared systems. We then propose the Second Law in the form: The ergotropy of the battery may increase indefinitely if and only if systems are in a non-passive state. One of the most interesting features of this new formulation is the appearance of the passive states that naturally generalize the notion of the heat bath. In this paper, we provide a handful of results that supports this formulation for diagonal systems. Interestingly, our methodology meets a well-known theory of Markov chains, according to which we classify the general charging processes based on the passivity/non-passivity of charging systems. In particular, the adopted mathematics allows us to distinguish a subtle asymptotic difference between the indefinite increase of the battery's energy (induced by the maximally mixed states) and of ergotropy (induced by the non-passive states) in terms of the so-called null-recurrent versus transient Markov chains.

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arXiv2022

Work and Fluctuations: Coherent vs. Incoherent Ergotropy Extraction

We consider a quasi-probability distribution of work for an isolated quantum system coupled to the energy-storage device given by the ideal weight. Specifically, we analyze a trade-off between changes in average energy and changes in weight's variance, where work is extracted from the coherent and incoherent ergotropy of the system. Primarily, we reveal that the extraction of positive coherent ergotropy can be accompanied by the reduction of work fluctuations (quantified by a variance loss) by utilizing the non-classical states of a work reservoir. On the other hand, we derive a fluctuation-decoherence relation for a quantum weight, defining a lower bound of its energy dispersion via a dumping function of the coherent contribution to the system's ergotropy. Specifically, it reveals that unlocking ergotropy from coherences results in high fluctuations, which diverge when the total coherent energy is unlocked. The proposed autonomous protocol of work extraction shows a significant difference between extracting coherent and incoherent ergotropy: The former can decrease the variance, but its absolute value diverges if more and more energy is extracted, whereas for the latter, the gain is always non-negative, but a total (incoherent) ergotropy can be extracted with finite work fluctuations. Furthermore, we present the framework in terms of the introduced quasi-probability distribution, which has a physical interpretation of its cumulants, is free from the invasive nature of measurements, and reduces to the two-point measurement scheme (TPM) for incoherent states. Finally, we analytically solve the work-variance trade-off for a qubit, explicitly revealing all the above quantum and classical regimes.

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arXiv2020

Thermodynamics of Minimal Coupling Quantum Heat Engines

The minimal-coupling quantum heat engine is a thermal machine consisting of an explicit energy storage system, heat baths, and a working body, which alternatively couples to subsystems through discrete strokes -- energy-conserving two-body quantum operations. Within this paradigm, we present a general framework of quantum thermodynamics, where a work extraction process is fundamentally limited by a flow of non-passive energy (ergotropy), while energy dissipation is expressed through a flow of passive energy. It turns out that small dimensionality of the working body and a restriction only to two-body operations make the engine fundamentally irreversible. Our main result is finding the optimal efficiency and work production per cycle within the whole class of irreversible minimal-coupling engines composed of three strokes and with the two-level working body, where we take into account all possible quantum correlations between the working body and the battery. One of the key new tools is the introduced "control-marginal state" -- one which acts only on a working body Hilbert space, but encapsulates all features regarding work extraction of the total working body-battery system. In addition, we propose a generalization of the many-stroke engine, and we analyze efficiency vs extracted work trade-offs, as well as work fluctuations after many cycles of the running of the engine.

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arXiv2020

The tight Second Law inequality for coherent quantum systems and finite-size heat baths

We propose a new form of the Second Law inequality that defines a tight bound for extractable work from the non-equilibrium quantum state. In classical thermodynamics, the optimal work is given by the difference of free energy, what according to the result of Skrzypczyk \emph{et al.} can be generalized for individual quantum systems. The saturation of this bound, however, requires an infinite bath and an ideal energy storage that is able to extract work from coherences. The new inequality, defined in terms of the ergotropy (rather than free energy), incorporates both of those important microscopic effects. In particular, we derive a formula for the locked energy in coherences, i.e. a quantum contribution that cannot be extracted as a work, and we find out its thermodynamic limit. Furthermore, we establish a general relation between ergotropy and free energy of the arbitrary quantum system coupled to the heat bath, what reveals that the latter is indeed the ultimate thermodynamic bound regarding work extraction, and shows that ergotropy can be interpreted as the generalization of the free energy for the finite-size heat baths.

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arXiv2018

Self--averaging of random quantum dynamics

Stochastic dynamics of a quantum system driven by $N$ statistically independent random sudden quenches in a fixed time interval is studied. We reveal that with growing $N$ the system approaches a deterministic limit indicating self-averaging with respect to its temporal unitary evolution. This phenomenon is quantified by the variance of the unitary matrix governing the time evolution of a finite dimensional quantum system which according to an asymptotic analysis decreases at least as $1/N$. For a special class of protocols (when the averaged Hamiltonian commutes at different times), we prove that for finite $N$ the distance (according to the Frobenius norm) between the averaged unitary evolution operator generated by the Hamiltonian $H$ and the unitary evolution operator generated by the averaged Hamiltonian $\langle H \rangle$ scales as $1/N$. Numerical simulations enlarge this result to a broader class of the non-commuting protocols.

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arXiv2017

Work distributions for random sudden quantum quenches

The statistics of work performed on a system by a sudden random quench is investigated. Considering systems with finite dimensional Hilbert spaces we model a sudden random quench by randomly choosing elements from a Gaussian unitary ensemble (GUE) consisting of hermitean matrices with identically, Gaussian distributed matrix elements. A probability density function (pdf) of work in terms of initial and final energy distributions is derived and evaluated for a two-level system. Explicit results are obtained for quenches with a sharply given initial Hamiltonian, while the work pdfs for quenches between Hamiltonians from two independent GUEs can only be determined in explicit form in the limits of zero and infinite temperature.

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