We investigate the fundamental dimensional limits to thermodynamic machines. In particular we show that it is possible to construct self-contained refrigerators (i.e. not requiring external sources of work) consisting of only a small number of qubits and/or qutrits. We present three different models, consisting of two qubits, a qubit and a qutrit with nearest-neighbour interactions, and a single qutrit respectively. We then investigate fundamental limits to their performance; in particular we show that it is possible to cool towards absolute zero.
We investigate whether size imposes a fundamental constraint on the efficiency of small thermal machines. We analyse in detail a model of a small self-contained refrigerator consisting of three qubits. We show analytically that this system can reach the Carnot efficiency, thus demonstrating that there exists no complementarity between size and efficiency.
We study the thermodynamics of a quantum system interacting with different baths in the repeated interaction framework. In an appropriate limit, the evolution takes the Lindblad form and the corresponding thermodynamic quantities are determined by the state of the full system plus baths. We identify conditions under which the thermodynamics of the open system can be described only by system properties and find a quantum local detailed balance condition with respect to an equilibrium state that may not be a Gibbs state. The three-qubit refrigerator introduced in [N. Linden, S. Popescu and P. Sk…
We consider a quantum Otto cycle operating between two squeezed thermal reservoirs. The influences of the squeezing degree on the optimization performance of quantum Otto heat engines and refrigerators are investigated. We demonstrate that under symmetric condition, the efficiency at maximum power (EMP) of heat engines and the coefficient of performance (COP) at maximum \c{hi} criterion of refrigerators are equal to Curzon-Ahlborn (CA) efficiency and CA COP, respectively. We also found that under asymmetric condition, the EMP of heat engines can be improved when the squeezing degree of hot the…
In this paper, we study the quantum self-contained refrigerator [N. Linden, S. Popescu and P. Skrzypczyk, Phys. Rev. Lett. \textbf{105}, 130401 (2010)] in the strong internal coupling regime with engineered reservoirs. We find that if some modes of the three thermal reservoirs can be properly filtered out, the efficiency and the working domain of the refrigerator can be improved in contrast to the those in the weak internal coupling regime, which indicates one advantage of the strong internal coupling. In addition, we find that the background natural vacuum reservoir could cause the filtered r…
We analyze the performance of a quantum Otto cycle, employing time-dependent harmonic oscillator as the working fluid undergoing sudden expansion and compression strokes during the adiabatic stages, coupled to a squeezed reservoir. First, we show that the maximum efficiency that our engine can achieve is 1/2 only, which is in contrast with the earlier studies claiming unit efficiency under the effect of squeezed reservoir. Then, we obtain analytic expressions for the upper bound on the efficiency as well as on the coefficient of performance of the Otto cycle. The obtained bounds are independen…
A small quantum absorption refrigerator, consisting of three qubits, is discussed in the transient regime. We discuss time scales for coherent dynamics, damping, and approach to the steady state, and we study cooling and entanglement. We observe that cooling can be enhanced in the transient regime, in the sense that lower temperatures can be achieved compared to the steady-state regime. This is a consequence of coherent dynamics, but can occur even when this dynamics is strongly damped by the dissipative thermal environment, and we note that precise control over couplings or timing is not need…
The inexorable miniaturisation of technologies, the relentless drive to improve efficiency and the enticing prospect of boosting performance through quantum effects are all compelling reasons to investigate microscopic machines. Thermal absorption machines are a particularly interesting class of device that operate autonomously and use only heat flows to perform a useful task. In the quantum regime, this provides a natural setting in which to quantify the thermodynamic cost of various operations such as cooling, timekeeping or entanglement generation. This article presents a pedagogical introd…
We discuss a quantum refrigerator to increase the ground state probability of a target qubit whose energy difference between the ground and excited states is less than the thermal energy of the environment. We consider two types of quantum refrigerators: (1) one extra qubit with frequent pulse operations and (2) two extra qubits without them. These two types of refrigerators are evaluated from the viewpoint of quantum thermodynamics. More specifically, we calculate the heat removed from the target qubit, the work done for the system, and the coefficient of performance (COP), the ratio between …
A single qutrit with transitions selectively driven by weakly-coupled reservoirs can implement one of the world's smallest refrigerators. We analyze the performance of $N$ such fridges that are collectively coupled to the reservoirs. We observe a quantum boost, manifest in a quadratic scaling of the steady-state cooling current with $N$. As $N$ grows further, the scaling reduces to linear, since the transitions responsible for the quantum boost become energetically unfavorable. Fine-tuned inter-qutrit interactions may be used to maintain the quantum boost for all $N$ and also for not-perfectly…
We show that one can construct a quantum absorption refrigerator that provides refrigeration only in the transient regime, by using three interacting qubits, each of which is also interacting with local heat-bath. The machine either does not provide cooling in the steady state, or the steady state is achieved after a long time. We propose a canonical form of qubit-bath interaction parameters that generates transient cooling without steady-state cooling, and claim that such a phenomenon is generic to small-scale quantum absorption refrigerators. We also show that it is generically possible to h…
We propose to use a few-qubit system as a compact quantum refrigerator for cooling an interacting multi-qubit system. We specifically consider a central qubit coupled to $N$ ancilla qubits in a so-called spin-star model as our quantum refrigerator. We first show that if the interaction between the qubits is of the longitudinal and ferromagnetic Ising model form, the central qubit is colder than the environment. The colder central qubit is then proposed to be used as the refrigerant interface of the quantum refrigerator to cool down general quantum many-qubit systems. We discuss a simple refrig…
We show that quantum absorption refrigerators, which have traditionally been studied as of three qubits, each of which is connected to a thermal reservoir, can also be constructed by using three qubits and two thermal baths, where two of the qubits, including the qubit to be locally cooled, are connected to a common bath. With a careful choice of the system, bath, and qubit-bath interaction parameters within the Born-Markov and rotating-wave approximations, one of the qubits attached to the common bath achieves a cooling in the steady state. We observe that the proposed refrigerator may also o…
We study the effect of Kerr nonlinearity in quantum thermal machines having a Kerr-nonlinear oscillator as working substance and operating under the ideal quantum Otto cycle. We first investigate the efficiency of a Kerr-nonlinear heat engine and show that by varying the Kerr-nonlinear strength the efficiency surpasses in up to 2.5 times the efficiency of a quantum harmonic oscillator Otto engine. Moreover, the Kerr-nonlinearity makes the coefficient of performance of the Kerr-nonlinear refrigerator to be as large as 3 times the performance of quantum harmonic oscillator Otto refrigerators. Th…
Small quantum absorption refrigerators have recently attracted renewed attention. Here we present a missing design of a two-qubit fridge, the main feature of which is that one of the two machine qubits is itself maintained at a temperature colder than the cold bath. This is achieved by 'reversing' the couplings to the baths compared to previous designs, where only a transition is maintained cold. We characterize the working regime and the efficiency of the fridge. We demonstrate the soundness of the model by deriving and solving a master equation. Finally, we discuss the performance of the fri…
Multilevel autonomous quantum thermal machines are discussed. In particular, we explore the relation between the size of the machine (captured by Hilbert space dimension), and the performance of the machine. Using the concepts of virtual qubits and virtual temperatures, we show that higher dimensional machines can outperform smaller ones. For instance, by considering refrigerators with more levels, lower temperatures can be achieved, as well as higher power. We discuss the optimal design for refrigerators of a given dimension. As a consequence we obtain a statement of the third law in terms of…
Clarifying the impact of quantumness in the operation and properties of thermal machines represents a major challenge. Here we envisage a toy model acting either as an information-driven fridge or as heat-powered information eraser in which coherences can be naturally introduced in by means of squeezed thermal reservoirs. We study the validity of the transient entropy production fluctuation theorem in the model with and without squeezing as well as its decomposition into adiabatic and non-adiabatic contributions. Squeezing modifies fluctuations and introduces extra mechanisms of entropy exchan…
We explore the possibility of enhancing the performance of small thermal machines by the presence of common noise sources. In particular, we study a prototypical model for an autonomous quantum refrigerator comprised by three qubits coupled to thermal reservoirs at different temperatures. Our results show that engineering the coupling to the reservoirs to act as common environments lead to relevant improvements in the performance. The enhancements arrive to almost double the cooling power of the original fridge without compromising its efficiency. The greater enhancements are obtained when the…
An autonomous quantum thermal machine comprising a trapped atom or ion placed inside an optical cavity is proposed and analysed. Such a machine can operate as a heat engine whose working medium is the quantised atomic motion, or as an absorption refrigerator which cools without any work input. Focusing on the refrigerator mode, we predict that it is possible with state-of-the-art technology to cool a trapped ion almost to its motional ground state using a thermal light source such as sunlight. We nonetheless find that a laser or similar reference system is necessary to stabilise the cavity fre…
We propose a general protocol for low-control refrigeration and thermometry of thermal qubits, which can be implemented using electronic spins in diamond. The refrigeration is implemented by a probe, consisting of a network of interacting spins. The protocol involves two operations: (i) free evolution of the probe; and (ii) a swap gate between one spin in the probe and the thermal qubit we wish to cool. We show that if the initial state of the probe falls within a suitable range, and the free evolution of the probe is both unital and conserves the excitation in the $z$-direction, then the cool…
We study how to achieve the ultimate power in the simplest, yet non trivial, model of a thermal machine, namely a two-level quantum system coupled to two thermal baths. Without making any prior assumption on the protocol, via optimal control we show that, regardless of the microscopic details and of the operating mode of the thermal machine, the maximum power is universally achieved by a fast Otto-cycle like structure in which the controls are rapidly switched between two extremal values. A closed formula for the maximum power is derived, and finite-speed effects are discussed. We also analyse…
Small self-contained quantum thermal machines function without external source of work or control, but using only incoherent interactions with thermal baths. Here we investigate the role of entanglement in a small self-contained quantum refrigerator. We first show that entanglement is detrimental as far as efficiency is concerned---fridges operating at efficiencies close to the Carnot limit do not feature any entanglement. Moving away from the Carnot regime, we show that entanglement can enhance cooling and energy transport. Hence a truly quantum refrigerator can outperform a classical one. Fu…
Absorption refrigerators are autonomous thermal machines that harness the spontaneous flow of heat from a hot bath into the environment in order to perform cooling. Here we discuss quantum realizations of absorption refrigerators in two different settings: namely, cavity and circuit quantum electrodynamics. We first provide a unified description of these machines in terms of the concept of virtual temperature. Next, we describe the two different physical setups in detail and compare their properties and performance. We conclude with an outlook on future work and open questions in this field of…
The extension of thermodynamics into the quantum regime has received much attention in recent years. A primary objective of current research is to find thermodynamic tasks which can be enhanced by quantum mechanical effects. With this goal in mind, we explore the finite-time dynamics of absorption refrigerators composed of three qubits. The aim of this finite-time cooling is to reach low temperatures as fast as possible and subsequently extract the cold particle to exploit it for information processing purposes. We show that the coherent oscillations inherent to quantum dynamics can be harness…
We investigate the performance of a quantum Otto refrigerator operating in finite time and exploiting local counterdiabatic techniques. We evaluate its coefficient of performance and cooling power when the working medium consists a quantum harmonic oscillator with a time-dependent frequency. We find that the quantum refrigerator outperforms its conventional counterpart, except for very short cycle times, even when the driving cost of the local counterdiabatic driving is included. We moreover derive upper bounds on the performance of the thermal machine based on quantum speed limits and show th…
An implementation of a small quantum absorption refrigerator in a circuit QED architecture is proposed. The setup consists of three harmonic oscillators coupled to a Josephson unction. The refrigerator is autonomous in the sense that it does not require any external control for cooling, but only thermal contact between the oscillators and heat baths at different temperatures. In addition, the setup features a built-in switch, which allows the cooling to be turned on and off. If timing control is available, this enables the possibility for coherence-enhanced cooling. Finally, we show that signi…
The thermodynamic uncertainty relation, originally derived for classical Markov-jump processes, provides a trade-off relation between precision and dissipation, deepening our understanding of the performance of quantum thermal machines. Here, we examine the interplay of quantum system coherences and heat current fluctuations on the validity of the thermodynamics uncertainty relation in the quantum regime. To achieve the current statistics, we perform a full counting statistics simulation of the Redfield quantum master equation. We focus on steady-state quantum absorption refrigerators where no…
We propose a three-qubit setup for the implementation of a variety of quantum thermal machines where all heat fluxes and work production can be controlled. An important configuration that can be designed is that of an absorption refrigerator, extracting heat from the coldest reservoir without the need of external work supply. Remarkably, we achieve this regime by using only two-body interactions instead of the widely employed three-body interactions. This configuration could be more easily realised in current experimental setups. We model the open-system dynamics with both a global and a local…
We study the impact of cooperative many-body effects on the operation of periodically-driven quantum thermal machines, particularly heat engines and refrigerators. In suitable geometries, $N$ two-level atoms can exchange energy with the driving field and the (hot and cold) thermal baths at a faster rate than a single atom due to their SU(2) symmetry that causes the atoms to behave as a collective spin-$N/2$ particle. This cooperative effect boosts the power output of heat engines compared to the power output of $N$ independent, incoherent, heat engines. In the refrigeration regime, similar coo…
We present a unified approach to study continuous measurement based quantum thermal machines in static as well as adiabatically driven systems. We investigate both steady state and transient dynamics for the time-independent case. In the adiabatically driven case, we show how measurement based thermodynamic quantities can be attributed geometric characteristics. We also provide the appropriate definition for heat transfer and dissipation owing to continous measurement in the presence and absence of adiabatic driving. We illustrate the aforementioned ideas and study the phenomena of refrigerati…
We analyze the performance of slowly driven meso- and micro-scale refrigerators and heat engines that operate between two thermal baths with small temperature difference. Using a general scaling argument, we show that such devices can work arbitrarily close to their Carnot limit only if heat-leaks between the baths are fully suppressed. Their power output is then subject to a universal geometric bound that decays quadratically to zero at the Carnot limit. This bound can be asymptotically saturated in the quasi-static limit if the driving protocols are suitably optimized and the temperature dif…
Diverse models of engines energised by quantum-coherent, hence non-thermal, baths allow the engine efficiency to transgress the standard thermodynamic Carnot bound. These transgressions call for an elucidation of the underlying mechanisms. Here we show that non-thermal baths may impart not only heat, but also mechanical work to a machine. The Carnot bound is inapplicable to such a hybrid machine. Intriguingly, it may exhibit dual action, concurrently as engine and refrigerator, with up to 100% efficiency. We conclude that even though a machine powered by a quantum bath may exhibit an unconvent…
Effects of Magnetic Anisotropy on 3-Qubit Antiferromagnetic Thermal Machines
Pontificia Universidad Católica de Valparaíso（天主教瓦爾帕萊索大學）；Universidad Técnica Federico Santa María（費德里科·聖塔瑪麗亞理工大學）；Millennium Nucleus in NanoBioPhysics (NNBP)（納米生物物理千年核）；Universidad Mayor（馬約爾大學）；Ohio University（俄亥俄大學）
Quantum speed limit, furnishing a lower bound on the required time for the evolution of a quantum system through the state space, imposes an ultimate natural limitation to the dynamics of physical devices. Quantum absorption refrigerators, on the other hand, have attracted a great deal of attention in the last few years. In this article, we discuss the effects of quantum speed limit on the performance of a quantum absorption refrigerator. In particular, we show that there exists a trade-off relation between the steady cooling rate of the refrigerator and the minimum time taken to reach the ste…
We present a theoretical study of an electronic quantum refrigerator based on four quantum dots arranged in a square configuration, in contact with as many thermal reservoirs. We show that the system implements the basic minimal mechanism for acting as a self-contained quantum refrigerator, by demonstrating heat extraction from the coldest reservoir and the cooling of the nearby quantum-dot.
The aim of this book chapter is to indicate how quantum phenomena are affecting the operation of microscopic thermal machines, such as engines and refrigerators. As converting heat to work is one of the fundamental concerns in thermodynamics, the platform of quantum-thermal machines sheds light on thermodynamics in the quantum regime. This chapter focuses on the basic features of quantum mechanics, such as energy quantization, the uncertainty principle, quantum coherence and correlations, and their manifestation in microscopic thermal devices. In addition to indicating the peculiar behaviors o…
We present a systematic study of the first excited-state population in a 3D transmon qubit mounted in a dilution refrigerator with a variable temperature. Using a modified version of the protocol developed by Geerlings et al. [1], we observe the excited-state population to be consistent with a Maxwell-Boltzmann distribution, i.e., a qubit in thermal equilibrium with the refrigerator, over the temperature range 35-150 mK. Below 35 mK, the excited-state population saturates to 0.1%, near the resolution of our measurement. We verified this result using a flux qubit with ten-times stronger couplin…
We combine the formalisms of Floquet theory and full counting statistics with a Markovian embedding strategy to access the dynamics and thermodynamics of a periodically driven thermal machine beyond the conventional Born-Markov approximation. The working medium is a two-level system and we drive the tunneling as well as the coupling to one bath with the same period. We identify four different operating regimes of our machine which include a heat engine and a refrigerator. As the coupling strength with one bath is increased, the refrigerator regime disappears, the heat engine regime narrows and…
Freie Universität Berlin（柏林自由大学）；Rice University（莱斯大学）；Microsoft Research（微软研究院）
The optimal control of open quantum systems is a challenging task but has a key role in improving existing quantum information processing technologies. We introduce a general framework based on Reinforcement Learning to discover optimal thermodynamic cycles that maximize the power of out-of-equilibrium quantum heat engines and refrigerators. We apply our method, based on the soft actor-critic algorithm, to three systems: a benchmark two-level system heat engine, where we find the optimal known cycle; an experimentally realistic refrigerator based on a superconducting qubit that generates coher…
The introduction of the quantum analogue of a Carnot engine based on a bath comprising of particles with a small amount of coherence initiated an active line of research on the harnessing of different quantum resources for the enhancement of thermal machines beyond the standard reversible limit, with an emphasis on non-thermal baths containing quantum coherence. In our work, we investigate the impact of coherence on the thermodynamic tasks of a collision model which is composed of a system interacting, in the continuous time limit, with a series of coherent ancillas of two baths at different t…
Qubit error bursts in superconducting quantum processors of Quantum Inspire: quasiparticle pumping and anomalous time dependence
QuTech and Kavli Institute of Nanoscience, Delft University of Technology（代尔夫特理工大学量子纳米科学卡弗里研究所和QuTech）；Netherlands Organisation for Applied Scientific Research (TNO)（荷兰应用科学研究组织（TNO））；Faculty of Electrical Engineering, Mathematics, and Computer Science, Delft University of Technology（代尔夫特理工大学电气、数学与计算机科学学院）
We present a study of the performance of endoreversible thermal machines optimized with respect to the thermodynamic force associated with the cold bath in the regime of small thermodynamic forces. These thermal machines can work either as an engine or as a refrigerator. We analyze how the optimal performances are determined by the dependence of the thermodynamic flux on the forces. The results are motivated and illustrated with a quantum model, the three level maser, and explicit analytical expressions of the engine efficiency as a function of the system parameters are given.
The characterization and control of quantum effects in the performance of thermodynamic tasks may open new avenues for small thermal machines working in the nanoscale. We study the impact of coherence in the energy basis in the operation of a small thermal machine which can act either as a heat engine or as a refrigerator. We show that input coherence may enhance the machine performance and allow it to operate in otherwise forbidden regimes. Moreover, our results also indicate that, in some cases, coherence may also be detrimental, rendering optimization of particular models a crucial task for…
