QQuantum thermal machines and batteries
Sourav Bhattacharjee ∗ and Amit Dutta Department of Physics, IIT Kanpur, Kanpur-208016, India (Dated: October 28, 2020)The seminal work by Sadi Carnot in the early nineteenth century provided the blueprint ofa reversible heat engine and the celebrated second law of thermodynamics eventually followed.Almost two centuries later, the quest to formulate a quantum theory of the thermodynamic lawshas thus unsurprisingly motivated physicists to visualise what are known as ‘quantum thermalmachines’ (QTMs). In this article, we review the prominent developments achieved in the theoreticalconstruction as well as understanding of QTMs, beginning from the formulation of their earliestprototypes to recent models. We also present a detailed introduction and highlight recent progressin the rapidly developing field of ‘quantum batteries’.
CONTENTS
I. Introduction 2II. Preliminary models 3A. Three level maser 4B. Particle in a one-dimensional box 5III. Open Quantum Systems 7A. Dynamical maps 7B. The GKSL equation for static Hamiltonians 8C. The GKSL equation for periodic Hamiltonians 10IV. Continuous thermal machines 12V. Reciprocating thermal machines 14A. Four stroke devices 161. The Otto Cycle 162. Efficiency at maximum power 183. Quantum friction and shortcuts to adiabaticity 184. Non-thermal baths 195. Four-stroke QTM based on many-body systems 206. Non-Markovian QTMs 20B. Two stroke devices 21VI. Equivalence of thermal machines 21VII. Quantum Szilard Engine 22VIII. Applications in quantum metrology 22A. Quantum Fisher Information 22B. Quantum thermometry 24C. Quantum Magnetometry 25IX. Quantum batteries 28A. Passive states and maximal work extraction 28B. Entangling vs non-entangling protocols 29C. Quantum speed limits 30D. Quantum advantage 311. From speed limit considerations in Hilbert space 322. From speed limit considerations in energy space 33E. Role of inherent entanglement 34F. Usability of stored energy 35 a r X i v : . [ qu a n t - ph ] O c t G. Models 351. Spin models 352. Cavity assisted charging 363. Disordered chains 37H. Quantum batteries as open systems 37X. Outlook 38Acknowledgments 38References 38
I. INTRODUCTION
Thermal machines refer to a broad class of devices whose operation is associated with some form of exchange and/orconversion of heat energy. They usually consist of two or more ‘heat reservoirs’ and a ‘working fluid’ (WF) whichfacilitates the intended process. Commonly known examples are the classical heat engines and refrigerators whichform the backbone of almost all mechanical and industrial machines that utilize thermal energy, ranging from thehousehold air conditioners and refrigerators to the fuel based vehicles and the propeller of a spaceship blasting offinto space. Such devices, by construction, operate irreversibly in far from equilibrium settings. The dynamics of thequantities of interest, such as the heat transferred or the work extracted, are governed by the commonly known laws ofthermodynamics. However, the tremendous technological advancement achieved in the past few decades has bothnecessitated as well as facilitated a rapid miniaturization of such thermal machines over the years. The frontiers ofsuch miniaturization has been pushed down to astonishingly small scales, where quantum effects are prominent andcan therefore no longer be ignored. This has resulted in the resurgence of a few age-old questions that have persistedthroughout the last century − how do thermodynamics, usually associated with macroscopic phenomena, reconcile withquantum mechanics, which describes equations of motion at the microscopic level that are inherently time-reversalsymmetric?It is worth noting that quantum mechanics and thermodynamics belong to a set of two immensely successful, albeitindependent theoretical frameworks that have withstood the test of rigorous experimental verification. Yet, theircompatibility remains an open question, particularly, at scales where quantum effects are expected to dominate thedynamics of a physical system . A priori, one may argue that the well-known laws of thermodynamics (with anexception to the first law) are defined for macroscopic systems described by statistical averages, and hence the questionof their validity for microscopic systems consisting of a few particles or qubits may appear meaningless. However, in1959, Scovil et. al. demonstrated that the working of a quantum three-level maser coupled to two thermal reservoirsresembles that of a heat engine; with an efficiency upper bounded by the Carnot limit . This work provided the firsthint that the laws of thermodynamics, particularly the second law, may have a more fundamental and even quantumorigin. In other words, it might be possible to naturally arrive at the thermodynamic laws starting from a microscopicquantum framework. However, the only known model of a ‘quantum heat engine’ at the time, i.e. the three-levelmaser, relied on a quasi-static description based on the equilibrium population of the energy-levels and hence couldnot provide any further insight into the dynamical processes involved.It was not until the 1990’s when researchers, motivated by the developing field of open quantum systems, beganto look for new toy models of ‘quantum thermal machines’ (QTMs). The aim was to design simplistic models thathave the same functional behavior as classical thermal machines, i.e. conversion of heat energy into useful work andvice-versa, yet at the same time that could be analyzed within the dynamical framework of open quantum systems. Theadvent of the Lindbladian framework made plausible the construction of physically meaningful models of QTMsthat could operate at far from equilibrium settings. The challenge was however to demarcate the dynamical energyexchanges into parts that could be associated with quantum analogues of ‘heat’ and ‘work’ as well as to formulate asecond law in terms of the entropic changes involved in a cycle of operation. The formulation of minimal workingmodels of reciprocating or stroke engines soon followed; these works paved the way for an explosion of research thatextensively analyzed the performance of such models in diverse scenarios, from exploring the role of entanglementand coherences to the consequences of using non-thermal baths and many more. However, to this day, the debateregarding a unique definition of quantities such as quantum work and heat as well as a universal formulation of thesecond law is far from being fully settled.Over the years, a plethora of such simple models of QTMs have been proposed to probe the thermodynamiclaws at the quantum domain, a few of which have also been realized experimentally . Extensive analysis of thesequantum models have strongly pointed to the presence of an upper bound to the efficiency and performance of suchquantum heat engines and refrigerators. The existence of the Carnot bound, which is a manifestation of the celebratedsecond law of thermodynamics, at such small scales re-established a strong case for the validity of thermodynamicsprinciples down to microscopic scales, thereby necessitating further scrutiny of the emergence of the thermodynamiclaws at the fundamental level.The exhaustive analysis of QTMs, nevertheless, has led to an unprecedented understanding of how simple few-levelquantum systems exchange energy as well as information with other such systems or with an external environment.Such understanding has in turn opened up the possibility of engineering microscopic devices in a way that mayrevolutionize nano-scale engineering. As for example, the application of ‘quantum probes’, which are essentially simplequantum systems such as a qubit or a harmonic oscillator, to quantum metrology have only been recently realized. Particularly, they have been shown to be suitable candidates for high precision measurements in thermometry (i.e., temperature measurements of nanoscale devices) as well as magnetometry (magnetic field measurements).As much as the focus has been on thermal energy conversion in the quantum regime, the subtleties of quantumphenomena affecting the process of energy storage and its subsequent extraction had not received much attention untilrecently. Alicki and Fannes, in their pioneering work , showed that a quantum battery composed of many identicalcopies of a single quantum system can, in principle, facilitate a higher energy extraction per cell through cyclic unitaryprocess when compared to a single cell. In addition, they concluded that the maximal work extraction is possibleonly if the battery is driven through intermediate entangled states while discharging the battery. However, it waslater proved that although entangling’ or non-local operations are required for maximal work extraction, it is notnecessary to generate entanglement, per se, in the battery during the discharging process. Further, the use of non-localoperations was also shown to result in a faster scaling of the speed of discharging or charging (depositing energy) thebattery with the battery size, as compared to local driving protocols. The above results have also been verified in anumber of models. In spite of rapid developments, a robust mechanism to identify and utilize quantum affects foroptimizing the usage of quantum batteries has not been established yet.In this article, we present a brief review of the basic design of some of the broad class of QTMs that are widelystudied in literature. As already mentioned, the concepts of quantum heat and quantum work as such are not yetuniquely defined and several definitions of these can be found in literature (see Ref. [6] for a review). However, inthis review article, we limit ourselves to a handful of these definitions relevant for understanding the working ofthe thermal machines discussed. Two parameters of paramount importance which characterize the performance ofQTMs and which we will repeatedly encounter in the course of discussion, are the efficiency ( η ) and the coefficient ofperformance (COP). The efficiency is defined in as the ratio of work output to the heat supplied from the hot bath,when the QTM operates as an engine. Similarly, the COP is defined as the ratio of heat extracted from the cold bathto the work performed on the working fluid, when the QTM acts as a refrigerator. In this regard, it is useful to recallthe second law which states that the maximum efficiency η c = 1 − T c /T h and COP c = T c / ( T h − T c ) is attained ina Carnot cycle which is a reversible cycle operating between baths with temperatures T h and T c . We also review acouple of applications of QTMs in the field of quantum metrology , particularly in thermometry and magnetometry . Finally, we also outline recent developments in theoretical modeling of quantum batteries,with majority of the discussion focused on cyclic unitary protocols.We would like to mention here that this review article is in no way exhaustive. In particular, given the vast amountof literature available as far as QTMs are concerned, this review aims for a brisk introduction to the basics of QTMsoutlining the essential underlying principles. On the other hand, given that the theoretical concept of quantumbatteries is relatively new and still in its nascent stages, we have thus taken care to provide a more in-depth discussionson its fundamentals as well as recent developments. II. PRELIMINARY MODELS
In this section, we outline the working of two very simple yet insightful models of QTMs, which operate quasi-statically and are capable of working as quantum heat engines or quantum refrigerators. The first model we introduceis the three-level maser which, as already mentioned, is the earliest prototype of a quantum heat engine. The othermodel we discuss, is the realisation of a four-stroke Carnot engine where the working substance is comprised of thetext-book system of a single particle in a one-dimensional box potential as working fluid . This model is uniquein the way that it first identifies the analogue of classical ‘force’ and uses the same to calculate the quantum workperformed. Although numerous other models were also proposed in the early days of QTMs, we however, beginby discussing the two models mentioned above. We highlight these particular models to make the reader appreciatethe fact that thermodynamic signatures, as these toy models demonstrate, can manifest in the working of QTMs evenwhen one does not explicitly resort to open system dynamics to analyze them. (a) L F AB C D (b)
Figure 1. (a)Schematic of the three-level maser which can act as a quantum heat engine. A hot bath with temperature T h induces excitations between energy levels E and E , while a cold bath with temperature T c induces excitations between E and E . Work is extracted by an external field resonant with the energy gap E − E . (b) F − L diagram of the quantum Carnotcycle discussed in Sec. II B, depicting the four strokes of the cycle. We have set π (cid:126) /m = 1 while calculating the force F ( L ).Note the striking similarity of the quantum Carnot cycle with the classical version in which the working medium is composed ofan ideal gas enclosed in a three-dimensional volume; a similar cycle is then manifested in the P − V diagram where P and V arethe pressure and volume of the gas enclosed, respectively. A. Three level maser
In 1959, Scovil et.al. realized that the steady-state operation of a three-level maser (see Fig. 1(a)) connected totwo thermal baths through appropriately chosen frequency filters resemble that of a heat engine or refrigerator. The‘quantum’ working fluid is a three-level system with energy levels E , E and E ( E > E > E ) and populations n , n and n , respectively. A hot bath with temperature T h is coupled to the system through a frequency filter so that itcan induce excitations only between E and E with energy (cid:126) ν h ∼ | E − E | . Similarly, a cold bath with temperature T c is allowed to induce transitions between E and E with energy (cid:126) ν c = | E − E | in the system. Further, the systemis also coupled resonantly with a radiation field with frequency ν p = | E − E | / (cid:126) .After the system attains (dynamical) equilibrium, the populations of the energy-levels satisfy, n n = e − (cid:126) νhkBTh , (1a) n n = e − (cid:126) νckBTc . (1b)Within this equilibrium regime, for each quanta of excitation (cid:126) ν h induced by the hot bath, the system loses energy hν c to the cold bath and hν p to the radiation field, so that the populations are held steady. The energy exchangedwith the baths can be thought of as ‘heat’ transferred while the energy supplied to the radiation field is identified asthe work extracted from the system with the radiation field playing the role of the classical ‘piston’. Note that thepreceding characterization of heat and work trivially satisfies the first law, (cid:126) ν h = (cid:126) ν c + (cid:126) ν p . (2)The most remarkable result however appears when the efficiency of the system is considered. An engine-like operationis possible when a ‘population inversion’ is achieved between the levels E and E , i.e. n ≥ n , which leads to thecondition, n n = n n n n = exp (cid:18) (cid:126) ν c k B T c − (cid:126) ν h k B T h (cid:19) ≥ ν c ν h ≥ T c T h . (4)The efficiency, i..e., the ratio of the work extracted to the energy supplied by the hot reservoir, is obtained as, η = (cid:126) ν p (cid:126) ν h = 1 − ν c ν h ≤ − T c T h , (5)where we made use of the first law (see Eq. (2)) for obtaining the second equality.Note that the dynamical equilibrium assumed at all instants of time means that the maser operation is reversible. Inother words, the engine-like operation discussed above can be reversed to obtain refrigerator-like operation, in which aquanta of excitation (cid:126) ν p induced by the radiation field leads to the extraction of an energy quanta (cid:126) ν c from the coldbath and the simultaneous loss of an energy quanta (cid:126) ν h to the hotter bath. The coefficient of performance is easilycalculated as, COP = (cid:126) ν c (cid:126) ν p = ν c ν h − ν c ≤ T c T h − T c . (6)The efficiency as well as the coefficient of performance of the three-level maser when working as a heat engineand refrigerator, respectively, therefore appears to be identical to those known for the classical Carnot cycle. Thisobservation hence provided the astonishing result that the second law can hold true even for few-level quantum systems. B. Particle in a one-dimensional box
Another simple model of a quantum Carnot engine was provided by Bender et.al. which, much like its classicalcounter-part, is operated in a cycle comprising of discrete strokes. The working fluid is made up of a particle confinedin a one-dimensional hard-walled box, in which one of the two walls is movable. The length of the box L is therefore ancontinuous variable which can be controlled externally to perform work on the system. To elucidate, let us consider theparticle wave function | ψ i = P n a n | φ n i , where | φ n i are the energy eigen states of the system. The energy expectationvalue of the system is h E i = X n | a n | E n ( L ) , (7)with E n ( L ) = n π (cid:126) / mL , where m is the mass of the particle. Invoking upon the notion of classical mechanics, theinstantaneous ‘force’ which performs the work is defined as F ( L ) = − d h E i /dL = − X n | a n | dE n ( L ) dL = X n | a n | n π (cid:126) mL . (8)It is important to note that in defining the force we have used h E i . In general, the derivative with respect to theenergy expectation value in the above expression should also contain terms such as E ( L ) d | a n | /dL . However, we recallthat any change in the populations | a n | is necessarily associated with changes in the Von-Neumann entropy of thesystem. For quasi-static processes, which are the only processess involved in the working of the quantum Carnot cycle,an entropy change can result only from heat transfer with external baths or environment and not from any externalforce acting on the system. Hence, it is justified to neglect derivatives with respect to | a n | while defining the force.Having defined the force, the quantum analogues of adiabats and isotherms are identified as follows. The quantumadiabats correspond to the processes of slowly tuning the length L of the box such that the populations a n are heldconstant. On the other hand, the quantum isotherms are defined as operations in which the coefficients a n changealong with E n on tuning L , but in such a way that the energy expectation value defined in Eq. (7) remain constant.One may draw a parallel here with classical isothermal processes in which the temperature (also the internal energy incase of ideal gas) remains constant.For purpose of simplicity, consider the situation in which only the two lowest lying energy eigen states contribute tothe wave function, so that a + a = 1. Let us assume that the system in initialized in the ground state, with the boxlength L = L A , so that a ( L A ) = 1. The energy expectation h E A i is, h E A i = E ( L A ) = π (cid:126) mL A , (9)where the subscript A marks the initial point of the cycle. The start of the subsequent strokes will likewise be labeledas B, C and D . The quantum Carnot cycle is then constructed as follows (see Fig. 1(b)):1. Adiabatic compression ( A → B ) – Length of the box is quasi-statically compressed to L = L B with L B < L A sothat the system remains in the instantaneous ground state. The force acting on the system during this stroke( A → B ) is given by F AB ( L ) = π (cid:126) mL , (10)which performs work and increases the energy of the system to h E B i = π (cid:126) mL B , (11)where h E B i is the energy expectation value at the end of this stroke. The populations do not change and hencewe have a ( L B ) = a ( L A ) = 1.2. Isothermal expansion ( B → C ) – Length of the box is expanded to L = L C = 2 L B such that the energyexpectation value remains constant throughout, i.e. | a ( L ) | π (cid:126) mL + (1 − | a ( L ) | ) 2 π (cid:126) mL = π (cid:126) mL B . (12)Note that the populations depend on L during the process B → C due to the constraint on the energy expectation.At the end of the expansion, the system therefore reaches the first excited state with a ( L C = 2 L B ) = 0. Hence,we have, h E C i = h E B i = π (cid:126) mL B (13)The force acting on the system during this step is, F BC ( L ) = | a ( L ) | π (cid:126) mL + (1 − | a ( L ) | ) 4 π (cid:126) mL , (14)where a ( L ) is constrained by Eq. (12).3. Adiabatic expansion ( C → D ) – As in step 1, the length is quasi-statically changed to L = L D = 2 L A with L D > L C such that the system remains in the first excited state, i.e. a ( L D ) = a ( L C ) = 0. The energyexpectation at the end of this stroke is therefore, h E D i = 4 π (cid:126) mL D = π (cid:126) mL A , (15)and the instantaneous force acting on the system during the expansion is F CD ( L ) = 4 π (cid:126) mL (16)4. Isothermal compression ( D → A ) – Finally, the length is tuned back to L = L A in a way that the energyexpectation value remains constant, | a ( L ) | π (cid:126) mL + (1 − | a ( L ) | ) 2 π (cid:126) mL = π (cid:126) mL A , (17)where the population functions are primed to distinguish them from the functions in the isothermal stroke B → C . This also ensures that the system returns to the instantaneous ground state, a ( L A ) = a ( L A ) = 1,hence closing the cycle. The force acting on the system during this final stroke is given by, F DA ( L ) = | a ( L ) | π (cid:126) mL + (1 − | a ( L ) | ) 4 π (cid:126) mL . (18)The total work performed during the cycle is now calculated as W = Z L B L A F AB ( L ) dL + Z L C =2 L B L B F BC ( L ) dL + Z L D =2 L A L C F CD ( L ) dL + Z L A L D F DA ( L ) dL = π (cid:126) m (cid:18) L B − L A (cid:19) log 2 (19)Heat is absorbed by the system only during the isothermal expansion and equals the work done during the expansion,i.e., Q H = Z L B L B F BC ( L ) dL = π (cid:126) mL B log 2 . (20)The efficiency of this quantum engine cycle is therefore η = WQ H = 1 − L B L A = 1 − h E A ih E B i (21)Note that, by construction, h E A i and h E B i play the role of constant temperatures during the isothermal compressionand expansion steps, respectively. The efficiency therefore turns out to be identical to that of the classical Carnot cycle.As already mentioned, the preliminary models discussed in this section are based on quasi-static processes whichcan be readily analyzed without any need for explicit equations of motion that govern the dynamics of the workingfluid. On the down side, such simplistic description do not provide any fundamental insights regarding the dynamicalprocesses involved. In the following sections, we will take a look at models that operate at far from equilibriumsettings and importantly whose operation bear the hallmarks of classical thermodynamic principles such as a maximumefficiency bounded by the Carnot efficiency. III. OPEN QUANTUM SYSTEMS
An open quantum system refers to a quantum system S which obeys the quantum mechanical laws of motion andis coupled to an external environment E . The environment, in principle, can either be quantum in nature with adiscreet energy spectrum, or classical in the continuum limit of vanishing energy gaps. Although, the dynamics of thecomposite system (i.e., the system S and the environment E taken together) can be described by the Schrodinger or theVon-Neumann equation, the exponentially large degrees of freedom of the environment renders it practically impossibleto analytically solve these equations. One of the ways to resolve this issue is to formulate equations in terms of thereduced state of the system by tracing out the relevant degrees of freedom of the environment from the dynamics of thecomposite system. In this section, we discuss one such equation, namely the Gorini–Kossakowski–Sudarshan–Lindblad(GKSL) equation , which is relevant for the rest of this review. A. Dynamical maps
To begin with, let us assume that the system and the environment are initially decoupled, ρ (0) = ρ s (0) ⊗ ρ E (0), where ρ s (0), ρ E (0) and ρ (0) represent the initial states of the system, environment and the composite system, respectively.The temporal evolution of the system is described by a dynamical map V ( t, ρ s ( t ) = Tr E (cid:2) U ( t, ρ s (0) ⊗ ρ E (0) U † ( t, (cid:3) = V ( t, ρ s (0) , (22)where ρ s ( t ) = Tr E ρ ( t ) is the reduced state of the system obtained by tracing over the environmental degrees of freedomand U ( t,
0) is the unitary evolution operator acting on the composite system.A few remarks about the dynamical map V ( t,
0) are in order. Firstly, V ( t,
0) is a self-mapping on the density matrixspace which implies that it must be completely positive (maps only to positive eigen-values) and trace preserving(so that all density matrices have unit trace), as can be verified from Eq. (22). Importantly, one can also check that V ( t,
0) only comprises of operators defined on the Hilbert space of S .Secondly, if the dynamics is Markovian (memory-less) in nature, the family of maps V ( t, ∀ t > quantum dynamical semi-group , which satisfies V ( t,
0) = V ( t, t ) V ( t , V ( t,
0) to be of the form V ( t,
0) = e L t , where L is referred to as the Linbladian super-operator. Substituting inEq. (22), we therefore find that the master equation governing the evolution of the system is of the general form , dρ s ( t ) dt = L ρ s ( t ) . (23)Note that in the case of isolated system, the evolution of the system is governed by the Von-Neumann equation, dρ s ( t ) dt = − i [ H s , ρ s ( t )] , (24)from which one can immediately identify the super-operator L U for unitary evolutions as, L U = − i (cid:2) H s , ρ isos ( t ) (cid:3) . (25)In what follows, we will explicitly see these emergence of the features discussed above as we outline a short derivationof the GKSL master equation. We shall keep ourselves restricted to the cases where the Hamiltonian of the systemis either independent or periodically modulated in time, as these are the ones relevant in the contextof QTMs. Although periodically modulated Hamiltonians are strictly not time-independent, the invariance of theFloquet Hamiltonian at stroboscopic instants, as we shall elaborate below, permits a dynamical equation of theform in Eq. (23).
B. The GKSL equation for static Hamiltonians
Consider a system-environment composite represented by a time-independent Hamiltonian, H = H s + H E + H I , (26)where H s and H E correspond to the Hamiltonians of the system and the environment, respectively, while H I encapsulates the coupling between them and is of the form, H I = X i S i ⊗ B i . (27)The operators S i and B i are local Hermitian operators pertaining to the Hilbert spaces of the system and theenvironment, respectively. As an illustration, the interaction between a two-level system and a photonic cavity (bath)can be of the form, H I = σ x ⊗ ( a † + a ), where σ x is a Pauli matrix and a ( a † ) is the photonic annihilation (creation)operator. We start with the von-Neumann equation governing the evolution of the composite system in the interactionpicture, d ˜ ρ ( t ) dt = − i (cid:2) ˜ H I ( t ) , ˜ ρ (0) (cid:3) − Z t (cid:2) ˜ H I ( t ) , (cid:2) ˜ H I ( t ) , ˜ ρ ( t ) (cid:3)(cid:3) dt . (28)In the above equation, for any observable O , ˜ O represents the same in the interaction picture.We now make an important approximation, namely the Born or the weak-coupling approximation, which assumesthat (i) the system does not influence the environment so that ˜ ρ E ( t ) = ˜ ρ E and (ii) the composite system exists in atensor-product state at all times, ˜ ρ ( t ) = ˜ ρ s ( t ) ⊗ ˜ ρ E . The Born approximation is valid for fast decaying environmentalcorrelations and we shall return to it later. Under the above approximations, the equation of motion for the reducedstate of the system can be obtained as, d ˜ ρ s ( t ) dt = − i Tr E (cid:2) ˜ H I ( t ) , ˜ ρ s (0) ⊗ ˜ ρ E (cid:3) − Z t Tr E (cid:2) ˜ H I ( t ) , (cid:2) ˜ H I ( t ) , ˜ ρ s ( t ) ⊗ ˜ ρ E (cid:3)(cid:3) dt (29)Next, we decompose the system operators S i into projections on the eigen-space of the Hamiltonian H s as, S i = X ε,ε h ε | S i | ε i | ε i h ε | = X ω = ε − ε S i ( ω ) | ε i h ε | , (30)where | ε i are the energy eigen states of the system and ω = ε − ε are the possible excitation energies. The abovedecomposition leads to a particularly simple form of the Hamiltonian H I in the interaction picture,˜ H I ( t ) = X i e iH s t S i e − iH s t ⊗ X i e iH E t B i e − iH E t = X i,ω e − iωt S i ( ω ) ⊗ B i ( t ) , (31)where, B i ( t ) = X i e iH E t B i e − iH E t (32)One can now derive the following relations,Tr E (cid:2) ˜ H I ( t ) , ˜ ρ s (0) ⊗ ˜ ρ E (cid:3) = X i,ω (cid:2) e − iωt S i ( ω ) , ˜ ρ s (0) (cid:3) h B i ( t ) i E , (33a)Tr E (cid:2) ˜ H I ( t ) , (cid:2) ˜ H I ( t ) , ˜ ρ s ( t ) ⊗ ˜ ρ E (cid:3)(cid:3) = X i,j,ω,ω e i ( ω t − ωt ) (cid:16) S † j ( ω ) S i ( ω )˜ ρ s ( t ) − S i ( ω )˜ ρ s ( t ) S † j ( ω ) (cid:17) h B † j ( t ) B i ( t ) i E + h.c., (33b)where h·i E denotes averaging over the state ˜ ρ E . In most cases, the average of the environmental operators h B i ( t ) i E vanish. However, even if they are finite, one can re-scale the system Hamiltonian appropriately to set h B i ( t ) i E = 0 and thus we shall ignore these averages in all the scenarios which we will consider in the rest of the paper. Substitutingthe above relations in Eq. (29) and shifting the time-coordinate t = t − t , we obtain, d ˜ ρ s ( t ) dt = X i,j,ω,ω Z t dt e i ( ω − ω ) t (cid:16) S i ( ω )˜ ρ s ( t − t ) S † j ( ω ) − S † j ( ω ) S i ( ω )˜ ρ s ( t − t ) (cid:17) e iωt h B † j ( t ) B i ( t − t ) i E + h.c. (34)The next step is to invoke upon the Markovian approximation, which assumes that the two-time environmentcorrelation functions decay rapidly within increasing time separation t − t , so much so that the system does notchange appreciably within their decay time. In other words, if the environment correlations decay within a time τ B and the system relaxes over a time-scale τ R , then τ R (cid:29) τ B . Thus, if we consider a coarse-grained evolution of thesystem with a time-scale ∼ τ R , we can replace ˜ ρ s ( t − t ) with ρ ( t ) in Eq. (34) and extend the integral to infinity, as allcontributions from time t > τ B can be neglected. We therefore arrive at, d ˜ ρ s ( t ) dt = X i,j,ω,ω e i ( ω − ω ) t (cid:16) S i ( ω )˜ ρ s ( t ) S † j ( ω ) − S † j ( ω ) S i ( ω )˜ ρ s ( t ) (cid:17) Γ i,j ( ω ) + h.c. (35)where Γ i,j ( ω ) = Z ∞ dt e iωt h B † j ( t ) B i ( t − t ) i E = Z ∞ dt e iωt h B † j ( t ) B i (0) i E . (36)We have used the fact that ˜ ρ E is stationary, [˜ ρ E , H E ] = 0 in deriving the second equality. Note that in Eq. (35), theevolution at a particular time t is only determined by the state of the system at time t ; hence there are no ‘memory’effects.Finally, we make the rotating wave or secular approximation which allows us to ignore all fast oscillating terms, i.e.we retain only terms with ω = ω . Note that this approximation, like to Markovian approximation, is also valid undera coarse-grained picture of the time evolution. The secular approximation allows to cast Eq. (35) in the Linbladianform (see Eq. (23)), d ˜ ρ s ( t ) dt = ˜ L ˜ ρ s ( t ) , (37a)0where, ˜ L ˜ ρ s ( t ) = − i [ H LS , ˜ ρ s ( t )] + X ω,i,j γ i,j ( ω ) (cid:18) S i ( ω )˜ ρ s ( t ) S † j ( ω ) − { S † j ( ω ) S i ( ω ) , ˜ ρ s ( t ) } (cid:19) . (37b)The renormalized or Lamb shifted
Hamiltonian H LS commutes with H s and is given by, H LS = X ω,i,j η i,j ( ω ) S † j ( ω ) S i ( ω ) , (37c)where γ i,j ( ω ) = 2Re [Γ i,j ( ω )] and η i,j ( ω ) = Im [Γ i,j ( ω )]. he equation derived in Eq. (37) is famously known as theGKSL master equation. Note that the first term in Eq. (37b) captures the unitary part of the evolution while thesecond term encapsulates the dissipative part of the evolution.Let us now return to the Born approximation. The assumption ρ E ( t ) = ρ E is strictly not valid because there alwaysexist a finite relaxation time which the environment requires to equilibriate, even if this relaxation time is very smallin comparison with the relaxation time of the system. However, if one considers a ‘coarse-graining’ of the time-scaleas we have done above in the case of the Markovian and secular approximations, the above assumption is perfectlyvalid as . Similarly, let us consider that a finite correlation χ is built up between the system and the environment, sothat at a given time t , ρ ( t ) = ρ s ( t ) ⊗ ρ E + χ ( t ). If one explicitly calculates the contribution of these correlation in theevolution after a small time ∆ t by integrating Eq. (28), one gets ,∆˜ ρ cors ( t ) = − Z t +∆ tt dt Z t Tr E (cid:2) ˜ H I ( t ) , (cid:2) ˜ H I ( t ) , χ ( t ) (cid:3)(cid:3) dt ∝ X i,j Z t +∆ tt dt Z t h B † j ( t ) B i ( t ) i E dt , (38)where ∆˜ ρ cors quantifies the extra contribution arising from the correlations χ ( t ). One can see that a finite contributionwill only arise for ∆ t < τ B as the environmental correlations h B † j ( t ) B i ( t ) i E are negligible for | t − t | > τ B . Hence,once again, the coarse-grained picture of the time evolution permits us to neglect this extra contribution arising onlyfor a very short duration.To obtain the asymptotic steady-state attained by the system, we revert back to the Schrodinger picture. In thestatic case, the GKSL equation in Eq. (37a) assumes the form, dρ s ( t ) dt = L ρ s ( t ) = − i [ H s , ρ s ( t )] + ˜ L ρ s ( t ) , (39)in the Schrodinger picture. The steady state ρ ss is therefore determined by solving the characteristic equation L ρ ss = 0.We note in passing that the set of reasoning in the derivation above can also be generalized for slowly varyingHamiltonians H s ( t ). The rate of change should be slow enough so that the quantum adiabaticity is maintained. Inother words, the time-scale over which H s ( t ) changes appreciably is much greater than the time-scale of relaxation ofthe system as well as the baths. Under such conditions, the GKSL equation assumes the form, d ˜ ρ s ( t ) dt = L ( t )˜ ρ s ( t ) , (40)where all the quantities in the time-dependent super-operator is derived in terms of the instantaneous Hamiltonianof the system. Likewise, the steady state ρ s s ( t ) is also slowly-varying and is dependent on the instantaneous energyeigen-values of the system. C. The GKSL equation for periodic Hamiltonians
We now consider the case of a periodically driven system coupled to an external environment , H ( t ) = H s ( t ) + H E + H I , (41)where H s ( t + T ) = H s ( t ). A dynamical equation in Linbladian form can be derived in a way similar to the case of staticHamiltonian discussed above, albeit with some alterations. The essential ingredient in the case of periodic Hamiltonians1is the so called Floquet Hamiltonian H F , which is defined as follows. Consider the unitary evolution operatorover a single time period T , in the absence of any coupling to the environment, U s ( T,
0) = T e − i R T H s ( t ) dt = e − iH F T , (42)where T is the time-ordering operator and the Floquet Hamiltonian H F is defined as, H F = iT ln U s ( T, . (43)The Floquet Hamiltonian H F thus acts as an effective static Hamiltonian which drives the evolution of the system whenobserved at stroboscopic instants of time, i.e., U s ( mT,
0) = exp ( − imH F T ). It possesses a set of time-independent quasi-energy eigen-states | φ n i , which satisfy H F | φ n i = ε n | φ n i , where ε n are refereed to as the quasi-energies.Now, let us consider the time evolution operator generating the evolution up to an arbitrary time, U s ( t,
0) = U s ( t, e iH F t e − iH F t = R ( t, e − iH F t , (44)where R ( t ) = U s ( t, e iH F t . Using the relations U s ( mT,
0) = exp( − imH F T ) and U s ( mT + t , mT ) = U s ( t ,
0) where0 < t < T , one can easily verify that R ( t + T ) = R ( t ). Thus, R ( t ) has a discrete time-translational invariance, whichpermits its Fourier decomposition as, U s ( t ) = X q R ( q ) e − iq Ω t e − iH F t , (45)where Ω = 2 π/T is the frequency of the periodic modulation and R ( q ) = 1 T Z T R ( t ) e iq Ω t dt. (46)The operators S i are transformed in the interaction picture as,˜ S i = U † s ( t, S i U s ( t, e iH F t X q ,q (cid:16) R † ( q ) S i R ( q ) e − i ( q − q )Ω t (cid:17) e − iH F t = X m e iH F t S ( m ) e − im Ω t e − iH F t = X m X ω = ε n − ε n h φ n | S i ( m ) | φ n i e − im Ω t e i ( ε n − ε n ) t = X m,ω S i ( m, ω ) e − i ( ω + m Ω) t (47)where ω = ε n − ε n now denotes the difference in quasi-energies or eigen-values of the Floquet Hamiltonian H F . Thisis unlike the static case where ω refereed to the difference in energy eigen-values of the time independent H s . TheHamiltonian H I hence assumes a similar form in the interaction picture as in Eq. (31),˜ H I = X i,ω,m e − i ( ω + m Ω) t S i ( m, ω ) ⊗ B i ( t ) , (48)where B i ( t ) is given by Eq. (32). Assuming the same set of approximations as in Sec. III B, one arrives at the followingequation of the super-operator L ,˜ L ˜ ρ s ( t ) = − i [ H LS , ˜ ρ s ( t )] + X m,ω,i,j γ i,j ( ω + m Ω) (cid:16) S i ( m, ω )˜ ρ s ( t ) S † j ( m, ω ) − { S † j ( m, ω ) S i ( m, ω ) , ˜ ρ s ( t ) } (cid:17) , (49)where H LS commutes with the Floquet Hamiltonian H F and is given by, H LS = X m,ω,i,j η i,j ( ω + m Ω) S † j ( m, ω ) S i ( m, ω ) . (50)2Unlike the static case, transforming the GKSL equation to the Schrodinger picture is not trivial in general. Instead,we first obtain the steady state in the interaction picture itself by solving ˜ L ˜ ρ ss = 0. The corresponding state in theSchrodinger picture is given by, ρ ss ( t ) = U s ( t, ρ ss U † s ( t, , (51)and satisfies ρ ss ( t + T ) = ρ ss ( t ) as U s ( t + T,
0) = U s ( t, IV. CONTINUOUS THERMAL MACHINES
The three-level maser discussed in Sec. II A, apart from being the earliest prototype of quantum thermal machines,can also be identified as the simplest form of continuous thermal machines (CTMs). This class of thermal machinesare characterized by their perpetual (continuous) coupling with both the heat source as well as the sink, unlike thereciprocating class of machines discussed in the next section. CTMs have a greater experimental relevance as they,unlike their reciprocating counter-parts, do not require intermittent couplings and decouplings between the workingfluid and the baths that are particularly difficult to implement at microscopic scales. Moreover, such intermittentcoupling-decoupling mechanisms are bound to generate some finite energetic-costs on the performance of reciprocatingmachines which are mostly ignored in theoretical calculations.Work extraction or refrigeration in CTMs is usually enforced by a periodic modulation of the system Hamiltonian,which in general, drives the system to a periodic steady state. The dynamical exchanges of energy between the systemand the baths as well as the work reservoir (energy source of the external agent which modulates the system) in thesteady state are, in general, out-of-equilibrium processes. In this regard, the three-level maser can be considered as aspecial case of QTMs in which a dynamical equilibrium is maintained throughout, thereby rendering the operationperfectly reversible. On the other hand, for a generic out-of-equilibrium process, the dynamics is irreversible in natureand the performance is found to be worse than the reversible Carnot engine or refrigerator. In what follows, we shalladopt the master equation approach, detailed in Sec. III C, to analyze the performance of a simple CTM operating inthe steady state.For purpose of simplicity, let us consider a periodically modulated two-level system (TLS) in contact with twothermal baths having temperatures T h and T c ( T h > T c ) . The Hamiltonian of the composite system reads, H ( t ) = H s ( t ) + H h + H c + H I , (52)where H s ( t ) and H h ( H c ) are the Hamiltonians of the system and the hot (cold) bath, respectively. The modulation isperformed on the energy-gap of the TLS so that H s ( t ) is of the form, H s ( t ) = 12 ω s ( t ) σ z , (53)with ω s ( t + T ) = ω s ( t ) and σ z being a Pauli matrix. The interaction between the system and the baths is chosen to be, H I = σ x ⊗ ( B h ⊗ I c + I h ⊗ B c ) , (54)where σ x is a Pauli matrix, while B h and B c are Hermitian operators which act locally on the Hilbert spaces of thehot and cold baths, respectively. Note that the modulation imposed is such that [ H s ( t ) , H s ( t )] = 0, which ensuresthat the external driving only modulates the energy levels and do not generate any excitations in the system. On theother hand, the interaction between the system and the baths is chosen such that [ H s ( t ) , H I ] = 0 and thus the bathscan induce excitations and affect the population of the energy levels.On plugging in Eqs. (53) and (54) in the GKSL equation derived Eq. (49) and simplifying, we obtain, ∂ρ ( t ) ∂t = L ρ ( t ) = X j,m L jm ρ ( t ) , (55a)where, L jm ρ = P m (cid:16) γ j ( ω + m Ω)( σ − ρσ + − { σ + σ − , ρ } ) + γ j ( − ω − m Ω)( σ + ρσ − − { σ − σ + , ρ } ) (cid:17) , (55b)where we have ignored the Lamb-shift corrections to the energy levels. The superscript j = h, c label the operatorsor correlation functions defined on the hot and cold baths, respectively. Likewise, ω denotes the mean gap of the3two-level system averaged over T , Ω = 2 π/T is the frequency of modulation and m = 0 , ± , ± . . . corresponds to thedifferent photon sectors or side-bands created as a result of the modulation. The coefficient P m assigns a weight to thecontribution from the m th side-band and is given by, P m = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T Z T e − i R t ( ω s ( t ) − ω ) dt e − im Ω t dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (56)Note that, unlike in the previous subsections, we have not used ˜ O to denote an operator O in the interaction picturefor simplicity in notation.A closer look at Eq. (55) suggests that the effective action of the periodic modulation in conjugation with thecoupling to thermal baths can be interpreted as a dissipative evolution driven by infinite copies of each of the thermalbaths. Each of the copies, which we henceforth refer to as sub-baths, of a particular thermal bath couples to differentside-bands of the Floquet spectrum. The super-operator L jm encodes the dissipative action arising due to the couplingof one of the copies of the j th bath to the m th side-band. The sub-baths therefore induce excitations in the systemhaving energies equal to the energy gaps of the different side-bands. As discussed in Sec. III C, the steady state ρ ss inthe interaction picture is obtained by solving the eigenvalue equation L ρ ss = 0, which transforms to a periodic steadystate in the Schrodinger picture.A crucial assumption in determing the steady state is that the thermal baths satisfy the Kubo-Martin-Schwingercondition γ j ( − ω ) = e − β j ω γ j ( ω ), where β j = 1 /T j is the inverse temperature of j th bath. The steady state is thenfound to be, ρ ss = 11 + r (cid:18) r
00 1 (cid:19) , (57a)where, r = P m,j P m γ j ( ω + m Ω) e − ω m Ω Tj P m,j P m γ j ( ω + m Ω) , (57b)To determine the heat currents and work, we first note that each sub-bath, when acting independently on the system,can in principle drive the system to a Gibbs-like steady state determined by the eigenvalue equation L jm ρ jm,ss = 0.These steady states are of the form , ρ jm,ss = 1 Z exp (cid:18) ω + m Ω ω β j H F (cid:19) , (58)where H F = ω σ z / Z = Tr (cid:16) exp (cid:16) ω + m Ω ω β j H F (cid:17)(cid:17) . Next, we calculate the rate of change of von-Neumann entropy S ( t ) = − Tr ( ρ ( t ) ln ρ ( t )), dS ( t ) dt = − Tr ( ˙ ρ ( t ) ln ρ ( t )) = − X j,m Tr (cid:0) L jm ρ ( t ) ln ρ ( t ) (cid:1) . (59)Further, it follows from from Spohn’s inequality for Markovian dynamics that Tr (cid:0) L jm ρ (ln ρ − ln ρ jm,ss (cid:1) ≤ dS ( t ) dt ≥ − X j,m Tr (cid:0) L jm ρ ( t ) ln ρ jm,ss (cid:1) = X j J j ( t ) T j . (60)The above inequality can be considered to be a dynamical version of the second law. In the steady state, theVon-Neumann entropy maximizes, and we obtain, X j J j T j = − X j,m Tr (cid:0) L jm ρ ss ln ρ jm,ss (cid:1) ≤ . (61)Substituting Eq.(58) in the first equality above, the steady state heat currents can be identified as, J j = X m (cid:18) ω + m Ω ω (cid:19) Tr (cid:16) L jm ρ ss H F (cid:17) , (62)4while the power is calculated from the principle of energy conservation (first law), P = ˙ W = − X j J j . (63)The working and the mode of operation of the thermal machine discussed above depend crucially on the form ofmodulation as well as the bath spectral functions γ j ( ω ). To act as a heat engine or refrigerator, the two baths needto be spectrally separated . As for example, consider the case of a sinusoidal modulation of the system with thebath spectra separated as γ h ( ω ) = 0 ∀ ω < ω , γ c ( ω ) = 0 ∀ ω > ω . One can then show that the QTM operates as arefrigerator, i.e. J h < J c , P >
0, if the modulation frequency Ω > Ω cr , where the critical frequency Ω cr givenby , Ω cr = ω T h − T c T h + T c . (64)For Ω < Ω cr , the QTM operates as a heat engine, characterized by reversal of the signs of the heat currents and power,i.e. J h > J c , P <
0. In this regime, the efficiency is found to be η = 2Ω ω + Ω , (65)The maximum efficiency is achieved as Ω → Ω cr ; at Ω = Ω cr the machine achieves the Carnot efficiency η = 1 − T c /T h ,but the power as well as the heat current vanishes. Similarly, in the refrigerator regime, the COP is also found tobe limited by the Carnot bound. Note that, although the efficiency and COP in general depends upon the choiceof bath spectral function, they are nevertheless, always restricted by the Carnot bounds. Recently, it has also beendemonstrated that using an asymmetric pulse , the switching between different modes of operation can alsobe achieved by tuning the up (or equivalently down) time duration of the pulse. Additionally, when modulated atresonance Ω = ω , tuning the up time duration also allows to QTM to function as a heater. In this mode of operation,the power supplied to the system P > J h < J c < . When compared with the performance of TLSs, it isfound that the presence of degeneracy in the case of multi-level systems can boost the heat currents and power of thethermal machine; however, the efficiency or the coefficient of performance remains identical to that of the case of TLSs.Similarly, it has been shown that using N two-level atoms as working fluid in place of the TLS enhances the poweroutput (as well as cooling capacity in refrigerator mode) of the thermal machine when compared with the net poweroutput from N independent machines . Further, it has also been demonstrated that the hot bath can be cooled tovery low temperatures if one considers a modified version of the continuous thermal machine discussed above, withthe system and interaction Hamiltonians chosen as, H s ( t ) = 12 ω ( t ) σ z + 12 g (cid:0) σ + e − iνt + σ − e iνt (cid:1) (66a)and H I = σ z ⊗ B h ⊗ I c + σ x ⊗ I h ⊗ B c , (66b)respectively. The first equation models a laser driven TLS with the coupling strength between the TLS and thelaser as g , while the second equation describes the coupling of the TLS to a dephasing (does not induce transitionsbetween energy levels of σ z ) hot bath and a cold bath. The particular advantage of this model is that, unlike themodel governed by Eqs. (53) and (54), no spectral separation of the two baths is required for the cooling operation.Finally, we note that the working of CTMs in the regime of non-Markovian dynamics has aslo been explored . V. RECIPROCATING THERMAL MACHINES
A reciprocating thermal machine operates in a cycle which is composed of discrete strokes. Notably, the workingfluid is coupled to the baths only in some of the strokes of the cycle. This class of thermal machines are much simplerto analyze in comparison to continuous ones, although they are in general trickier to implement experimentally. Theparticle in box system, discussed in Sec. II B is an example of reciprocating thermal machine. In spite of it’s tantalizingsimilarity to the classical Carnot cycle , it does not offer much insight into the dynamical exchange of energy between5the working fluid and the environment. In particular, the heat exchanged with the baths is indirectly found from thework performed in the isothermal strokes where the net energy change is zero. In this section, we will first discuss thecommon notions of quantum heat and work widely used in literature, which will be subsequently used to review someof the commonly studied reciprocating thermal machines.Although there are no universally accepted definitions of ‘quantum work’ or ‘quantum heat’ till date, we brieflyunderline here a set of definitions that are particularly useful in the context of reciprocating devices, especially, inthe limit of weak system-bath coupling. Consider a system S with reduced density matrix ρ s ( t ) evolving under atime-dependent Hamiltonian H s ( t ) and coupled to an external environment. A change in energy expectation value of S after duration τ can be expressed as ,∆ E = Z τ ddt [Tr ( ρ s ( t ) H s ( t ))] dt = Z τ Tr (cid:18) dρ s ( t ) dt H s ( t ) (cid:19) dt + Z τ Tr (cid:18) ρ s ( t ) dH s ( t ) dt (cid:19) dt = W + Q, (67)where the work is identified as, W = Z τ Tr (cid:18) ρ s ( t ) dH s ( t ) dt (cid:19) dt, (68a)and the heat exchanged with the environment as, Q = Z τ Tr (cid:18) dρ s ( t ) dt H s ( t ) (cid:19) dt. (68b)The above definitions of quantum work and heat can be intuitively justified as follows: If the system S was isolated,any change in energy expectation can only be associated with a work performed, as there is no environment withwhich S can exchange heat. This is consistent with the definition of heat in Eq. (68b) as for an isolated system, onecan easily check that, Q iso = Z τ Tr (cid:18) dρ s ( t ) dt H s ( t ) (cid:19) dt = − i Z τ Tr ([ H s ( t ) , ρ s ( t )] H s ( t )) dt = 0 , (69)where we have used the Liouville’s equation to arrive at the second equality. Further, both the quantum work andheat, as defined above, depend on the evolution process and are therefore not state functions, similar to their classicalcounterparts. One can therefore, consider Eq. (67) as the quantum equivalent of the first law.Further, the definitions in Eq. (67) also permits a natural formulation of a dynamical version of the second law.Within the framework of GKSL equations derived for time-independent and slowly varying system Hamiltonians H s ( t )in Sec. III, the rate of heat exchanged can be calculated using Eq. (68b) as, J ( t ) = dQdt = Tr (cid:18) dρ s ( t ) dt H s ( t ) (cid:19) = Tr ( L ( t ) ρ s ( t ) H s ( t )) (70)Next, we invoke the Spohn’s inequality ,Tr [ L ( t ) ρ s ( t ) (ln ρ s ( t ) − ln ρ ss )] ≤ , (71)where ρ ss is the steady state which satisfies L ρ ss = 0. For baths obeying the Kubo-Martin-Schwinger (KMS)condition , ρ ss ( t ) corresponds to the thermal Gibbs state ρ ss ( t ) = e − βH s ( t ) / Tr (cid:0) e − βH s ( t ) (cid:1) , where β = 1 /T is theinverse bath temperature. Substituting in the above equation, we arrive at, dS ( t ) dt − J ( t ) T ≥ , (72)where S ( t ) = − Tr ( ρ s ( t ) ln ρ s ( t )) is the von Neumann entropy of the system. The above equation is the dynamicalversion of the second law which we had also obtained in Sec. IV for time-periodic Hamiltonians. Finally, we would liketo mention here that the above definitions of work and heat need to be modified in some cases, such as in the case ofautonomous or self-contained quantum thermal machines .6 Figure 2. Schematic representation of the Otto cycle in the T − S plane, where T denotes the temperature and S is theVon-Neumann entropy. The strokes AB and CD are isentropic strokes in which the energetic changes occur only in the form ofwork. On the other hand, the thermalization with the hot and cold baths occur in the strokes DA and BC, respectively, with nowork performed during these strokes whatsoever. A. Four stroke devices
Carnot, Otto and Stirling engines are prime examples of four stroke thermal machines as their operation are basedon cycles that are made up of four sequential strokes. The quantum analogues of these four stroke machines have beenextensively studied, particularly the Otto engine has received the most attention followed by the Carnot engine. This isbecause the Otto cycle is the simplest to analyze owing to the fact that the heat and work exchanges occur separatelyin different strokes, unlike the Carnot and Stirling cycles, in which both the processes occur simultaneously in theisothermal strokes. In particular, note that the isothermal stroke requires a dissipative evolution with a time varyingHamiltonian. However, exact formulations in terms of the Lindbladian framework exist only for the cases of staticand infinitesimally slowly varying Hamiltonians. Thus, the Lindbladian framework becomes inadequate when one isinterested in studying finite-time performances of the thermal machines. On the contrary, the Otto cycle provides nosuch hindrance, as the natural separation of the work producing and heat exchanging strokes allows the former to berephrased in terms of unitary evolution in isolated conditions and the latter in terms of dissipative dynamics withtime-independent Hamiltonians.In the following, we therefore restrict ourselves to reviewing the working principles of the quantum Otto cycle using the definitions of quantum heat and work discussed previously. We note at the outset that a vast amount ofresearch have gone into analyzing numerous aspects of the quantum Otto cycle and as such, a full-fledged discussion ofall such work is not feasible. We therefore, resort to highlighting only fundamental aspects of its working and outlinesome of the interesting results that have been reported over the past few years.
1. The Otto Cycle
For the Otto cycle (see Fig. 2), we consider a quantum harmonic oscillator (QHO) as the working fluid withHamiltonian, H = ˆ p m + 12 mω ( t )ˆ x = ω ( t ) (cid:0) a † a + 1 / (cid:1) , (73)where ˆ x and ˆ p are the position and conjugate momentum operators, respectively. In addition, a denotes the annihilationoperator and ω is the natural frequency of the oscillator which can be controlled externally. The QHO can be coupledto a hot or a cold thermal bath, having temperatures T h and T c ( T h > T c ), respectively. Further, we assume that theinitial frequency is ω (0) = ω A , i.e., H (0) = H A = ω A (cid:0) a † a + 1 / (cid:1) , and the QHO is initialized in thermal equilibriumwith bath T h so that, ρ A = e − H A /T h Z ( ω A , T h ) , (74)7where Z ( ω, T ) = Tr( e − H A /T ) and we have set k B = (cid:126) = 1. The energy expectation value is found to be, h E A i = Tr( ρ A H A ) = ω A (cid:18) h n i + 12 (cid:19) = ω A (cid:18) ω A T h (cid:19) , (75)where h n i is the mean occupation number. The quantum Otto cycle is now constructed as follows:1. Isentropic compression – The QHO is decoupled from the hot bath and its Hamiltonian is tuned from H A → H B ,where H B = ω B (cid:0) a † a + 1 / (cid:1) and ω B ≤ ω A . The unitary evolution in isolated conditions does not affect theVon-Neumann entropy of the TLS; hence it is an isentropic process. In the ideal cycle, the tuning occursadiabatically, which ensures that the occupation probabilities of the instantaneous eigen-energy levels remainunchanged. The energy exchange, which importantly occurs only in the form of work, is given by W AB = ω B − ω A (cid:18) ω A T h (cid:19) . (76)2. Cold isochore – In this stroke, the QHO is coupled to the cold bath and allowed to thermalize with its Hamiltonianheld constant at H = H B . By virtue of Eq. (68a), this ensures that no work is performed. The only energyexchange occurs in the form of heat transfer and is given by, Q c = ω B (cid:18) coth (cid:18) ω B T c (cid:19) − coth (cid:18) ω A T h (cid:19)(cid:19) (77)3. Isentropic expansion – As in the compression stroke, the QHO is decoupled from bath and the Hamiltonian istuned as H B → H A . Once again, in the ideal case, the populations remain invariant, and the work performed isfound to be, W BA = ω A − ω B (cid:18) ω B T c (cid:19) . (78)4. Hot isochore – The cycle is completed by coupling the QHO to the hot bath so that it thermalises back to itsinitial state as given in Eq. (74). The heat exchanged in the process is, Q h = ω A (cid:18) coth (cid:18) ω A T h (cid:19) − coth (cid:18) ω B T c (cid:19)(cid:19) (79)We reemphasize that the work is performed only during the isentropes while heat is exchanged with the bathsonly during the isochores. As the TLS returns to its intial state after each cycle, the total energy is conserved, i.e. Q c + W AB + Q h + W BA = 0. The efficiency can be calculated as, η o = − W AB + W BA Q h = Q h + Q c Q h = 1 − ω B ω A , (80)where we have used the convention that W is positive if work is done on the system. When operating as a heatengine, the net work output is positive, i.e. W AB + W BA ≤
0. Using Eqs. (76) and (78), this inequality is simplified to ω B /ω A ≥ T c /T h . Hence the efficiency is limited by, η o ≤ − T c T h . (81)Thus, we find that the efficiency of the quantum Otto engine is bounded by the Carnot limit. As in the case of CTMs,one can check that at the Carnot point, ω B /ω A = T c /T h , the net work done as well as the heat exchanges vanish. For ω B /ω A ≤ T c /T h , the machine operates as a quantum refrigerator, where the COP is also found to be upper-boundedby the COP of the Carnot refrigerator.8
2. Efficiency at maximum power
In practice, the power output of an ideal Otto engine, as in the case of Carnot engine, is zero. This is because of thefact that each of the adiabats as well as the isochores ideally requires an infinite time to achieve perfect adiabaticevolution and thermalization, respectively. The efficiency at maximum power is therefore an importantfigure of merit to analyze the engine’s performance. To evaluate the same, we expand the net work output in the hightemperature limit as, W = − ( W AB + W BA ) = ( ω B − ω A ) (cid:18) T c ω B − T h ω A (cid:19) = (cid:18) ω B ω A − (cid:19) T c (cid:16) ω B ω A (cid:17) − T h (82) W is maximum when the ratio ω B /ω A satisfies ω B /ω A = √ T c / √ T h . The efficiency at maximum power ¯ η is thus,¯ η = 1 − r T c T h , (83)which is surprisingly identical to the Curzon-Ahlborn (CA) efficiency of an ideal engine operating with finite power.To achieve finite power, it is necessary to devise ways to achieve both perfect adiabaticity and thermalization withinfinite times. The time required in the former is in general much longer than the latter; a greater effort has thus beendevoted to engineer methods to achieve adiabatic evolutions in finite time, which we shall be briefly discussing below.However, we note that protocols to achieve fast thermalization have also been explored recently .
3. Quantum friction and shortcuts to adiabaticity
The fact that the strokes associated with work extraction are required to be adiabatic is better explained in terms ofquantum coherence. Diabatic excitations, associated with the build up of coherence in between the energy levels of thesystem, are generated when the time allocated to the isentropic processes is finite and [ H ( t ) , H ( t )] = 0. The build upof coherence costs additional work, which effectively reduces the net useful work extracted from the heat reservoirs,thus undermining the efficiency of the engine. Coherence therefore plays the role of ‘quantum friction’ which hampersthe engine’s ability to extract useful work.A powerful technique which can mitigate work losses due to non-adiabatic driving is to utilize certain shortcuts toadiabaticity . One way in which this can be realized is by driving the working fluid along a certain path whichensures that the final state reached at the end of the isentropic stroke does not have any coherence although they mayexist at intermediate times and the populations of the energy eigen-states are thus the same at the beginning and atthe end of the stroke. To illustrate, let us consider an isentropic stroke in which the frequency of the QHO is tunedfrom ω (0) = ω i to ω ( τ ) = ω f in a finite duration of time τ . Now, consider the operator, I ( t ) = 12 (cid:20) mω b ˆ x + 1 m (cid:0) b ˆ p − m ˙ b ˆ x (cid:1) (cid:21) , (84)where b is a time-dependent parameter and ω = const >
0. This operator becomes an invariant of evolution if theparameter b satisfies the Ermakov equation, ¨ b + ω ( t ) b = ω b . (85)Let us now impose the boundary conditions b (0) = 1, ˙ b (0) = ¨ b (0) = 0 and b ( τ ) = p ω /ω f , ˙ b ( τ ) = ¨ b ( τ ) = 0. Thesechoices of boundary conditions lead to ω = ω i , I (0) = H (0) and I ( τ ) = ω H ( τ ) /ω f . The set of these six boundaryconditions allows one to calculate a polynomial form of b ( t ) from which the required time-dependent tuning of ω ( t )can be determined.The above results imply that an initial set of eigen-states of the Hamiltonian H ( t ) are also the eigen-states of theoperator I ( t ) at t = 0. Let us consider a generic initial state | ψ (0) i = P c n (0) | φ n i , where | φ n i are the eigen-states ofboth I (0) and H (0). If ω ( t ) is subsequently tuned following Eq. (85), I ( t ) remains invariant and the state evolves as | ψ ( t ) i = P c n (0) e iθ n ( t ) | φ n ( t ) i , where | φ n ( t ) i are instantaneous eigen-states of I ( t ) but not H ( t ) (0 < t < τ ) and θ ( t )9is some time-dependent phase. Finally, since | φ n ( τ ) i are also the eigen-states of H ( τ ), we conclude that the populationof each of the eigen-states are restored at t = τ .An alternative but similar approach to adiabatic shortcuts is provided by counter-diabatic (CD) protocols .These protocols involve adding additional interactions to the system Hamiltonian; the dynamics resulting from theinclusion of these additional CD interactions suppresses the diabatic excitations which would have been otherwisegenerated in the system due to fast driving of the bare Hamiltonian. A general form of the CD interactions is obtainedas follows. Consider a system Hamiltonian of the form, H ( t ) = X n ε n ( t ) | n ( t ) i h n ( t ) | . (86)Under adiabatic driving, the system initialized in a given energy eigen-state at t = 0 follows the same instantaneousstate throughout the evolution. The time evolution operator is therefore required to be of the form, U ( t ) = X n e iφ n ( t ) | n ( t ) i h n (0) | . (87)where φ n ( t ) is the phase acquired by the n th eigen-state during the unitary evolution and is given by, φ n ( t ) = − (cid:126) Z t dt ( ε n ( t ) − i h n ( t ) | ∂ t n ( t ) i ) . (88)We wish to find the Hamiltonian which mimics the above time evolution without any adiabatic approximations. Thisis easily done by substituting U ( t ) in the (Schrodinger) equation, H ( t ) = i (cid:126) ˙ U ( t ) U † ( t ), which leads to, H ( t ) = H ( t ) + H CD ( t ) = X n ε n ( t ) | n ( t ) i h n ( t ) | + i (cid:126) X n (cid:16) | ∂ t n ( t ) i h n ( t ) | − h n ( t ) | ∂ t n ( t ) i | n ( t ) i h n ( t ) | (cid:17) , (89)where H CD encodes the CD interactions that are to be added to ensure an adiabatic evolution with reference to thebare Hamiltonian H . Of course, one must be careful to take into account the extra work done by the additionalcounter-diabatic terms while calculating the net work output and efficiency of the thermal machine.
4. Non-thermal baths
It is important to realize that coherence is not always detrimental to the performance of QTMs. In fact, coherenceplays the central role in boosting the work output of QTMs which utilize non-thermal baths as heat sources. In 2003,Scully et. al. demonstrated that it is possible to extract work effectively from a single ‘phaseonium’ bath, whichconsist of three-level atoms that have small amount of coherence between almost- degenerate lower energy levels, .Dubbed as the photo-Carnot engine, the working fluid here is the radiation field generated by the atoms operatingbetween a phaseonium bath with temperature T h and a bath T c with T h > T c . The working fluid relaxes to athermal steady state with a temperature T φ = T h (1 − ¯ n(cid:15) cos φ ), where T h is the temperature of the hot bath, ¯ n isthe average photon number in the absence of coherence, (cid:15) is a measure of the magnitude of the coherence and φ is the associated phase. By appropriately tuning φ , it is therefore possible for the working fluid to attain a highertemperature than the hot bath. This in turn allows work extraction even when T h = T c , thus effectively permittingwork extraction from a single reservoir. Similar results were also reported in the context of quantum Otto engines,where the use of such ‘quantum coherent fuels’ was shown to enhance performance .A slightly different mechanism by which the use of non-thermal baths can boost engine performance is when theworking fluid itself is rendered non-thermal after interaction with the bath. This can be done, for example, by usinga squeezed thermal bath as the hot bath with squeezing parameter r . A QHO coupled to such a baththermalizes to a squeezed thermal state with mean phonon number h n i = h n i + (2 h n i + 1) sinh ( r ). Returning tothe analysis of the Otto cycle in Sec. V A 1, we note that the initial energy expectation E A is modified as, h E A i = ω A (cid:18) ω A T h (cid:19) ∆ H r , (90)where ∆ H r = 1 + (2 + 1 / h n i ) sinh ( r ). Proceeding as in Sec. V A 1, we calculate the efficiency which turns out to bethe same as the Otto engine efficiency η o . However, the efficiency at maximum power is found to be,¯ η = 1 − s T c T h (cid:0) ( r ) (cid:1) (91)0For r = 0, ¯ η reduces to the CA efficiency obtained in Eq. (83) for the case of thermal reservoirs. On the other hand,for r → ∞ , we find ¯ η →
1, which appears to surpass the Carnot limit. Nevertheless, this does not violate the secondlaw as the non-thermal hot bath is found to have an effective higher temperature T ∗ h > T h . On properly accounting forthis effective temperature, the upper bound of the efficiency is found to be bounded by the generalized Carnot limit, η gen = 1 − T c T h (cid:0) ( r ) (cid:1) . (92)Using squeezed thermal baths, it was also shown that work extraction is possible even from a single squeezed bathwithout violating the laws of thermodynamics . More recently, the efficiency bound on such quantum thermalmachines has been quantitatively estimated using the notion of ergotropy, which quantifies the maximum amount ofwork that can be extracted from non-passive states through unitary protocols .
5. Four-stroke QTM based on many-body systems
An increasing amount of research in recent times is focusing on deploying quantum many-body quantum systems asworking fluid in quantum stroke engines. This is because many-body systems, besides having thepotential to naturally scale up the work output and efficiency per cycle of QTMs, are also capable of hosting certainnovel phenomena which have no single-particle counterparts and can serve as thermodynamic resources. As for example,finite size scaling theory predicts that if the working fluid is operated closed to its critical point, the efficiency of theOtto engine can approach the Carnot limit at finite power. Similarly, using results known from energy-level statisticsand localization properties of many-body localized phases, it has been shown that a quantum Otto engine operatedwith the working fluid ramped between a localized and a thermal phase has significant advantages . In particular,this engine exhibits lesser fluctuations in work output and can be easily scaled up in size as the localization ensures thatdifferent ‘sub-engines’ work independently of each other. At the same time, it is to be noted that ensuring adiabaticdriving protocols is a more challenging task in many-body systems as compared to single-particle systems. Recentworks have therefore focused on exploring viable shortcuts to adiabaticity protocols for many body quantumheat engines.A remarkable feature which emerges in many-body quantum engines is that non-adiabatic affects in some cases mayeven lead to enhancement of the engine’s performance. Such an enhancement has been demonstrated in the case ofan Otto engine where the working fluid is an interacting Bose gas confined in a time-dependent harmonic trap .The efficiency achieved using the many-particle system is greater than the efficiency of an ensemble of single particleheat engines that have the same amount of thermodynamic resources at their disposal. Another mechanism thoughwhich non-adiabatic affects can be exploited to tap into the cooperative resources of many-body systems is tied to thenotion of passive states . Technically, passive states are characterized by density matrices which are diagonal inenergy basis and the population decrease with increase in energy. If a system is initially prepared in a passive state,then no work can be extracted out of it through cyclic unitary protocols. In a single-particle Otto engine, the finalstates after the completion of the isentropic strokes are passive states and the efficiency, as we have seen, is maximizedwhen the strokes are adiabatic. However, it can be shown that the direct product state of multiple identical copies of apassive state, which is not additionally thermal, need not be a passive . This opens up the possibility of extractingextra amount of work in many-body systems. For maximizing efficiency, the direct product state at the end of theisentropic strokes is required to be passive, which in turn necessitates non-adiabatic excitations so that the populationsamong different copies can be interchanged. We will return to passive states in more detail when we address quantumbatteries in Sec. IX.
6. Non-Markovian QTMs
In all the examples of four-stroke thermal machines considered thus far, the underlying assumption is that thedynamics of the QTM is strictly Markovian in nature. However, significant new results have also been reported recentlyfor QTMs operating in the non-Markovian regime . In general, non-Markovian dynamics can result in anumber of scenarios; such as in the limit of strong system-bath couplings and long decay times of bath correlationfunctions which implies a bath with memory. In Ref [178], it was shown that frequent quantum nondemolitionmeasurements can lead to extraction of useful work from the system-bath correlation energy if the cycle is operatedwithin the bath memory time. Similarly, it was pointed out in Ref [133] that in an Otto cycle with a TLS as theworking fluid, thermalizing with a non-Markovian bath is not necessarily accompanied by a monotonous increase ordecrease in the effective temperature of the TLS. This in turn allows the Otto engine to attain an efficiency exceeding1the Carnot limit when operated in finite time. It has also been found that even if the system shares correlations withonly some degrees of freedom with the baths with the overall evolution remaining Markovian, the power output canstill be boosted . It is fair to say that the impact of non-Markovian dynamics on the operation of QTMs is far frombeing fully explored and more research in this direction is expected in near future.
B. Two stroke devices
Apart from four-stroke thermal machines, it is also possible to construct reciprocating thermal machines basedon two-stroke cycles . The reduction in number of strokes per cycle is compensated by increasing the numberof systems or working fluids to two. As for example, consider two qubits S and S with energy gaps ω and ω ,respectively. These qubits can be individually coupled to two thermal baths with temperatures T and T , such that T > T . The two-stroke cycle consists of the following sequential strokes – (1) a thermalization stroke in which thequbits S and S are coupled with the baths having temperatures T and T , respectively, and allowed to thermalize;(2) a unitary stroke in which the two qubits interact with each other, with no contact whatsoever with the baths, andwork is performed on the composite system. The thermal machine based on this two-stroke cycle can work in threedifferent modes depending on the ratio of the energy gaps ω and ω . The different modes are characterized by therelative sign of the heat gained from the hot (cold) bath Q h ( Q c ) and the work performed W . These modes and theirregime of operation are listed below:• Refrigerator ( Q h < Q c > W > ω /ω > T /T .• Engine ( Q h > Q c < W < < ω /ω < T /T .• Accelerator ( Q h > Q c < W > ω /ω < η two − stroke = 1 − ω /ω < − T /T . In Sec. VIII C, we will discuss the working of the two-strokecycle in more detail with an application of the same to quantum magnetometry . VI. EQUIVALENCE OF THERMAL MACHINES
Naively, the two broad class of quantum thermal machines we have discussed, namely the continuous and reciprocatingthermal machines, may appear to be vastly different in terms of their construction and operation. However, it waspointed out that in the limit of small bath action that, they are indeed thermodynamically equivalent . Theequivalence is valid within the Markovian and rotating wave approximations. To elaborate, consider the mapping ρ N × N → | ρ i × N of the density matrix of a N -level system. The GKLS master equation, governing the evolution ofthe density matrix can therefore be represented as, i (cid:126) d | ρ i dt = H ( t ) | ρ i , (93)where H N × N is the super-operator which consists of terms arising from the system Hamiltonian as well as Lindbladoperators. The bath action is then defined as , s = Z τ cyc k ˜ H ( t ) k dt, (94)where τ cyc is the duration of one full cycle of operation, ˜ H denotes the super-operator H in the interaction pictureand k · k is the operator norm defined as k · k = max p eig ( · † · ). In the regime of small bath action with respect to thePlanck’s constant, s (cid:28) (cid:126) , it can be shown that the state of the system in the continuous and reciprocating thermalmachines differs by order O ( s/ (cid:126) ) before completion of the cycle and by O (( s/ (cid:126) ) ) at the end of the cycle. In fact, thework and heat transferred also differ by the same order of magnitude. Physically, the emergence of the equivalence isexplained as follows. In general, work can be extracted through a coherent mechanism which involves alteration ofthe off-diagonal terms of the density matrix in energy eigen basis, as well as a stochastic mechanism which involvesalteration of populations. In continuous machines, only the coherent mechanism is present while in the reciprocatingmachines, the stochastic mechanism is also present. In the limit of small bath action, the coherent mechanism stronglydominates and hence the thermal machines types become thermodynamically equivalent. Later, this equivalence wasalso extended to non-Markovian systems .2 VII. QUANTUM SZILARD ENGINE
Before concluding our discussion on QTMs, let us briefly mention the Szilard engine which, though not astechnologically relevant, is tremendously important to gain a better understanding of the rapport between informationand thermodynamics . When Maxwell proposed using ‘information’ as a resource to conceive what came to befamously known as the Maxwell’s demon , the second law appeared to be under some serious challenge. Buildingon his work, Szilard proposed an engine in which a feedback assisted cyclic process appeared to allow conversion ofall the heat extracted from a single reservoir into work. This conceptual and ideal engine, operating with a singlemolecule gas as the working fluid, works as follows. Consider a single molecule gas confined in a cylinder which is incontact with a heat reservoir. A piston, having a wide opening in its centre that can be closed with a friction lessshutter, is placed somewhere inside the cylinder. When the shutter is closed, the piston divides the cylinder volumeinto two parts. An intelligent demon then performs a measurement to determine on which side of the piston themolecule is located. Depending on the measurement outcome, the demon attaches a string to the right or left sideof the piston such that the isothermal expansion of the single molecule gas pulls the string, thus lifting any weightattached to the other end of the string. Once the required work is completed, the shutter is opened so that the singlemolecule gas can once again occupy the whole volume of the cylinder. Therefore, the expansion of the gas driven bythe heat extracted from the reservoir is completely converted into work.Classically, the widely accepted solution to this paradox is given by the Landauer’s erasure principle , whichassociates an energetic cost with any logically irreversible manipulation of information . As for example, resettingthe information stored in a memory bit amounts to an increase of entropy ∼ k B log 2 and hence, an amount of heat ∼ k B T log 2 is dissipated to the environment at temperature T . A truly cyclic process demands that the memory ofthe demon is also restored at the end of the complete cycle. This requires an erasure of the information acquired bythe demon during the measurement which results in heat dissipation, thus accounting for a second heat reservoir orsink. Thus it became apparent that information should be visualized as a physical entity that has a direct bearing onthermodynamic processes .Note that the Szilard engine, by construction, is a microscopic engine working with a single molecule and hence, arigorous analysis must also incorporate the quantum effects in play. With this motivation, numerous quantum modelsof the Szilard engine has been investigated over the years . While early works had explored the consequences ofendowing a quantum nature to the measurement and information erasure processes , many recent works have alsoinvestigated exotic variations of the Szilard engine. Similarly, it was pointed out in Ref [202], that an energetic costshould be associated with the insertion or removal of piston in the quantum case, resulting from changes of boundaryconditions. Recently, it has also been shown that it is possible to operate a Szilard engine working without any thermalsource by drawing energy from projective measurements , although it may lose some of its characteristics inthe process . Nevertheless, there is no universal consensus regarding the significance and implications of a fullyquantized Szilard engine and much remains to be understood. VIII. APPLICATIONS IN QUANTUM METROLOGY
In this section, we review a couple of recently proposed applications of quantum thermal machines in the field ofquantum metrology . The basic idea behind the protocols is to use quantum thermal machines as sensitiveprobes, which can be coupled to a given system and some parameter of the system is then estimated through an indirectmeasurement on the probe. The precision attained by using a particular protocol is assessed through a comparisonwith the minimum bound on the relative error dictated by the quantum Cramer-Rao bound . Thisbound is quantified by the so called quantum Fisher information, which unlike the classical Fisher information, has ageometrical origin and is therefore uniquely determined by the state of the system on which the measurement is to beperformed. It is therefore imperative that we present a brief summary of the QFI before moving on to discuss theprotocols in detail.
A. Quantum Fisher Information
Let us consider that we wish to estimate a parameter θ through an indirect measurement on a random variable X .The probability that the outcome of a measurement on X is x i ∈ X , is determined by the conditional probability( x | θ ). The parameter θ is read off from the measurement on X through the estimator ˆ θ ( X ). Let us assume that thetrue value of the parameter is θ = θ . Further, we assume an unbiased estimator, i.e. h ˆ θ ( x ) − θ i = 0 . (95)3Taking a partial derivative w.r.t. θ , we get, ∂∂θ h ˆ θ ( x ) − θ i = Z (cid:16) ˆ θ ( x ) − θ (cid:17) ∂p ( x | θ ) ∂θ dx − Z p ( x | θ ) dx = 0 . (96)or, Z (cid:16) ˆ θ ( x ) − θ (cid:17) p ( x | θ ) ∂∂θ log p ( x | θ ) dx = 1 , (97)where we have used the equalities ∂ θ p ( x | θ ) = p ( x | θ ) ∂ θ log p ( x | θ ) and R p ( x | θ ) dx = 1. Next, consider the followingrelation by virtue of the Cauchy-Schwarz inequality, Z h (cid:16) ˆ θ ( x ) − θ (cid:17) p p ( x | θ ) ihp p ( x | θ ) ∂∂θ log p ( x | θ ) i dx ≤ sh Z (cid:16) ˆ θ ( x ) − θ (cid:17) p ( x | θ ) dx ih Z p ( x | θ ) (cid:18) ∂∂θ log p ( x | θ ) (cid:19) dx i . (98)Recognizing, Z (cid:16) ˆ θ ( x ) − θ (cid:17) p ( x | θ ) dx = Var( θ ) , (99)and rearranging Eq. (97) appropriately, we arrive at the (classical) Cramer-rao bound,Var( θ ) ≥ I ( θ ) , (100)where I ( θ ) is the classical Fisher information (CFI) given by, I ( θ ) = Z p ( x | θ ) (cid:18) ∂∂θ log p ( x | θ ) (cid:19) dx = Z p ( x | θ ) (cid:18) ∂p ( x | θ ) ∂θ (cid:19) dx. (101)The CFI, as defined above, depends on the probability distribution p ( x | θ ). In other words, the CFI is influenced bythe choice of the random variable X used for estimating the parameter θ .From a quantum mechanical viewpoint, the probability distribution p ( x | θ ) is obtained as p ( x | θ ) = Tr [ ρ θ Π x ], where { Π x } refers to the set of elements of a positive-operator-value-measure (POVM) and ρ θ is the state of the quantumsystem which naturally depends on the parameter θ . The CFI can therefore be written as, I ( θ ) = Z Tr h ∂ θ ρ θ Π x i Tr h ρ θ Π x i dx = Z Re (cid:16) Tr h ρ θ Π x L θ i(cid:17) Tr h ρ θ Π x i dx, (102)where we have introduced the symmetric logarithmic derivative (SLD) L θ which is defined as, ∂ρ θ ∂θ = 12 ( L θ ρ θ + ρ θ L θ ) . (103)Using the inequality Re( z ) ≤ | z | , we therefore obtain, I θ ≤ Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Tr h ρ θ Π x L θ ir Tr h ρ θ Π x i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx = Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Tr √ ρ θ √ Π x r Tr h ρ θ Π x i p Π x L θ √ ρ θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx, (104)4Using the Cauchy-Scwarz inequality again, we finally arrive at, I θ ≤ Z Tr [Π x L θ ρ θ L θ ] dx = Tr (cid:2) ρ θ L θ (cid:3) = H θ , (105)where we have used the completeness of the POVM R Π x dx = I in obtaining the second equality. The quantity θ is the quantum Fisher information, which is purely a geometrical quantity and does not depend upon the choice ofmeasurement parameter X . The QFI therefore sets a stricter bound on the relative error bound,Var( θ ) ≥ H θ . (106)To derive a more explicit form of the QFI, we write the solution pf the SLD obtained from Eq. (103) as, L θ = 2 Z ∞ (cid:18) e − ρ θ t ∂ρ θ ∂θ e − ρ θ t (cid:19) dt. (107)Substituting ρ θ = P i p i | φ i i h φ i | , the above equation simplifies to, L θ = X i p i ∂p i ∂θ | φ i i h φ i | + X i = j p i − p j p i + p j h φ j | ∂∂θ | φ i i | φ j i h φ i | . (108)The QFI obtained in Eq. (105) hence assumes the form, H θ = X i p i (cid:18) ∂p i ∂θ (cid:19) + 2 X i = j ( p i − p j ) p i + p j (cid:12)(cid:12)(cid:12)(cid:12) h φ j | ∂∂θ | φ i i (cid:12)(cid:12)(cid:12)(cid:12) . (109)The geometrical origin of the QFI is now explicitly seen as the above expression derived for the QFI is identical tothat of fidelity susceptibility F I ( θ ) = − (cid:15) → ∂ F ( ρ ( θ + (cid:15) ) , ρ ( θ )) /∂(cid:15) , where F ( ρ , ρ ) = Tr (cid:2)p √ ρ ρ √ ρ (cid:3) isthe fidelity between the states ρ and ρ . We note in passing that for a pair of pure states, F ( ρ , ρ ) reduces to thepure state fidelity F ( | ψ i , | ψ i ) = |h ψ | ψ i| . Further, the (ground-state) fidelity in itself exhibits many interestingbehavior, particularly those associated with quantum critical phenomena . B. Quantum thermometry
A QTM operating at Otto efficiency was shown to be capable of measuring very low temperatures with high precisionin Ref. [62], mimicking the role of a nano-scale thermometer. The thermal machine is constructed using a circuit-QEDsetup where the working fluid consists of two LC oscillators with characteristic frequencies Ω h and Ω c , coupled to eachother through a Josephson junction. In addition, each of the oscillator is coupled to distinct thermal baths havingtemperatures T h and T c , respectively, where the temperature T c is unknown and is to be measured. A bias voltage V supplies external energy or power to the set of oscillators. Under the resonance condition 2 eV = Ω h − Ω c theHamiltonian reads , H = E J (cid:16) a † h A h A c a c + h.c. (cid:17) , (110)where E J is the Josephson energy and a h ( c ) is the annihilation operator acting on the Fock space of the oscillator withfrequency Ω h ( c ) and the operators. The operators A α are of the form , A α = 2 λ α e − λ α X n α L (1) n α (4 λ α ) n α + 1 | n α i h n α | , (111)where λ α are the amplitude of oscillator zero-point phase fluctuations, L kn ( x ) are the generalized Laguerre polynomialsand | n α i are Fock-states associated with the oscillator with frequency Ω α . Similarly, the current operator assumes theform, I = − eE J i (cid:16) a † h A h A c a c − h.c. (cid:17) . (112)5The dynamical evolution of the system is assumed to be governed by the Lindblad master equation. In the steadystate operation, the heat currents are given by, J h ( c ) = Ω h ( c ) κ h ( c ) (cid:16) h n h ( c ) i − n h ( c ) B (cid:17) (113a)and the power is calculated as, P = V h I i , (113b)where κ h ( c ) is the decay rate associated with the bath with temperature T h ( c ) , n h ( c ) B is the mean occupation number ofthe corresponding baths and h·i denotes expectation values in the steady state.Using the above equations, one can show that the steady state operation resembles that of a quantum heat engineor refrigerator. Further, the engine like operation is characterized by an efficiency equal to the Otto efficiency η = 1 − Ω c / Ω h , which approaches the Carnot efficiency in the limit of vanishing heat currents and power. The unknowntemperature T c is determined as follows. For a fixed set of Ω c and Ω h , whose values are assumed to be known withhigh precision, one can vary T h until the steady state power vanishes. As the power and heat currents vanish onlywhen the device attains Carnot efficiency, the following condition must be satisfied, T c T ∗ h = Ω c Ω h , (114)where T ∗ h is the temperature of the hot bath for which the power vanishes. The temperature T c can therefore easily becalculated as T c = T ∗ h Ω c / Ω h .The error in measurement can be estimated using the error propagation formula,∆ T c = s(cid:18) ∂ h I i ∂T c (cid:19) − (∆ I ) + (cid:18) Ω c Ω h (cid:19) (∆ T h ) , (115)where ∆ X denotes the root mean square error in the measurement of parameter X . By choosing Ω h (cid:29) Ω c , thesecond term (under the root) on the r.h.s of the above equation can be neglected. To evaluate the remainingterm, an approximate model is considered by substituting E J A h A c / g . In this case, the error evaluates to∆ T c = αT c sinh (Ω c / T c ) / Ω c . At the same time, the minimum possible error in measuring the temperature asdetermined from the Cramer-Rao bound (see Eq. (106)) is found to be ∆ T crc = 1 / H T c = β ∆ T c /α , where β and α depend on the coupling constants. For any choice of the coupling constants, one always finds α ≥ β which is expectedas the minimum error can not be lower than that of the Cramer-Rao bound. The optimal value is found to be α/β ≈ .
55 which shows that the proposed thermometer is capable of measuring temperatures with a precision closeto the maximum theoretically possible precision.
C. Quantum Magnetometry
As in thermometry, it has also been proposed that it is possible to boost precision of weak field magnetometrymeasurements by utilizing a two-stroke QTM acting as a quantum probe . To elaborate, let us consider two qubitslabeled as K and U , which are coupled to two separate magnetic fields of known and unknown intensities, respectively.Further, we assume that the known magnetic field is relatively stronger than the unknown field. This setting thereforegives rise to two TLSs, whose energy levels are denoted as ± ω k and ± ω un , with ω k > ω un . In addition, we also havetwo thermal baths with temperatures T h and T c , such that T h > T c . Initially, K and U are assumed to be in thermalequilibrium with the hot bath T h and cold bath T c , respectively, with no interactions existing between the two. Thedensity matrix of the total system (excluding the baths) is therefore given by, ρ tot (0) = ρ k (0) ⊗ ρ un (0) , (116)where ρ un (0) = (cid:18) n un
00 1 − n un (cid:19) = 1 Z un e − ωunTc e ωunTc ! , (117a) ρ k (0) = (cid:18) n k
00 1 − n k (cid:19) = 1 Z k e − ωkTh e ωkTh ! . (117b)6 T h Hot bathCold bath T c KU H I Unitary stroke Thermaliza � on stroke (a) (b) Figure 3. (a) Schematic representation of the two stroke quantum thermal machine. (b) The heat exchanged with the bathsand work done as a function of ω k . At the Carnot point ω k = 5, all of the quantities reverse their sign signaling the transitionfrom engine-like to refrigerator-like operation or vice-versa. For these particular numerical results, the interaction Hamiltonian isassumed to induce a swap of populations between the two TLSs at the end of the unitary stroke which corresponds to choosing θ = π/ Here, n un(k) denotes the excited state population of U ( K ) and Z un(k) are the respective partition functions. Thedevice is then operated in a two-step cycle as per the following strokes (see Fig. 3(a)):1. Unitary Stroke : The TLSs are decoupled from their respective baths and allowed to interact with each other;this interaction lasts for a duration of time τ U . We specifically consider an interaction of the form, H I ( t ) = 2 ω I ( t ) ( |↑ k i |↓ un i h↓ k | h↑ un | + h . c . ) (118)where { |↑ k(un) i , |↓ k(un) i } is the energy eigenbasis of the TLS K ( U ) and ω I ( t ) is a time-dependent modulation.This choice of H I ( t ) results in a unitary evolution in which only the projection of ρ tot (0) on the subspace spannedby the states |↑ k i |↓ un i and |↓ k i |↑ un i is rotated. We denote this rotation by an angle θ .2. Thermalization Stroke : At the end of the unitary stroke, the interaction H I ( t ) is switched off and each TLSis again coupled to its respective thermal bath, i.e., the bath with which it was initially in equilibrium beforethe unitary stroke. This thermalization stroke last for a duration of time τ T with the assumption that τ T issufficiently long so that each TLS returns to its initial configuration given by Eq. (117). Note that heat exchangesbetween the TLSs and the baths only occur during this second stroke.The work done and the heat exchanged during the cycle are calculated as follows. Since all heat exchanges occurduring the thermalization stroke, one can calculate the same using (68b) as, Q h(c) = Tr[ ρ k(un) ( τ U + τ T ) H k(un) ] − Tr[ ρ k(un) ( τ U ) H k(un) ] (119)where, ρ k(un) ( t ) = Tr un(k) [ ρ tot ( t )]. On simplification, this evaluates to, Q c = 2 ω un ( n un − n k ) sin θ, (120a) Q h = 2 ω k ( n k − n un ) sin θ. (120b)Using the first law, the work done evaluates to W = − ω k − ω un )( n k − n un ) sin θ. (120c)7 Figure 4. The Wheatstone bridge setup is used to measure electrical resistance with high precision. The variable resistance R with known values is tuned until the voltage (measured by a voltmeter V ) across the two arms of the bridge are balanced. Atthis balanced position, the unknown resistance is calculated using the formula R u = R R /R . The thermometry (Sec. VIII B)and the magnetometry (Sec. VIII C) protocol discussed in the text are also based on the Wheatstone bridge principle as can beseen by comparing the above equation with Eqs. (114) and (122). It is easy to see from the above expressions that if n k > n un , the machine acts as an engine while it acts as a refrigeratorif n k < n un (see Fig. 3(b)). In the regime of engine-like operation, the efficiency is found to be equal to the Ottoefficiency, η = − WQ h = 1 − ω un ω k , (121)Importantly, the transition from engine-like to refrigerator-like operation occurs at n k = n un . It follows from Eq. (117)that this is possible when, ω un ω k = T c T h . (122)At this transition point, the efficiency equals the Carnot efficiency accompanied by the vanishing of heat exchangedand work performed during the cycle.The intensity of the unknown magnetic field to which the qubit U was coupled can now be determined usingEq. (122). Experimentally, the Carnot point can be identified by carefully tuning ω k (the known magnetic field) andobserving when the heat exchanged and the work done vanishes. Indeed, one only needs to look for a sign reversalin the magnitude of these quantities, which in general, can be done much more accurately than determining exactvalues of the same. Provided that the temperatures of the baths are accurately known, the error in determining ω un is therefore ∆ ω un = ∆ ω k T c /T h . Further, the error ∆ ω k has contributions arising from two sources – (i) frommeasurement errors in determining the null of the heat exchanges or the work and (ii) the error in directly measuring ω k through standard methods. The former can be shown to be negligibly small under certain feasible conditions .Consequently, the error ∆ ω k arises solely from its direct measurement and this is equal to the error (∆ ω un ) dir thatwould have been present if ω un was measured directly. Hence, we can write,∆ ω un = (∆ ω un ) dir T c T h . (123)Therefore, the indirect measurement by using the two-stroke thermal machine helps in reducing the error in measuringthe weaker magnetic field by a magnitude of T c /T h as compared to a direct measurement.It is important to note that the protocols discussed above are based on the Wheatstone bridge (see Fig. 4) principle,where an unknown parameter is measured by tuning another auxiliary parameter and searching for the zero-pointwhere some observable such as the currents vanish. At this point, the values of the known and the unknown parametersmust satisfy a certain preset ratio which therefore enables one to estimate the unknown parameter. The practical8advantage of these protocols resides in the fact that such zero-point measurements in general can be carried out withvery high precision as compared to absolute value measurements. Further, knowledge of other microscopic parametersin the set up is not required which eliminates possible sources of error arising from the uncertainty in the known valueof such microscopic parameters. IX. QUANTUM BATTERIES
As much as it is important to engineer devices that extract energy or useful work from heat sources, it is alsoequally important to realize ways of efficiently storing this energy. As with thermal machines, it is natural to ponderwhether quantum effects can be utilized to our advantage so as to facilitate a better storage of useful energy, orfaster charging as well as discharging of batteries operating in the quantum regime. The study of so called quantumbatteries deal precisely with these questions and is currently one of the most active fields of research. Inthis section, we introduce the fundamentals of quantum batteries and outline some of the recent advances in thisrapidly developing field. We shall first review the concept of passive states, which are the target states of any unitaryprotocol that aims to facilitate maximal energy extraction from a quantum system. Subsequently, we shall also see howthe presence of entanglement may speed up the process of energy deposition or energy extraction from the battery,commonly referred to as charging or discharging the quantum battery, respectively. Before proceeding with the detail,we would like to remark here that majority of the work in this area so far has focused on unitary charging anddischarging protocols. These include case studies where the ‘charging’ corresponds to a unitary evolution of the batteryunder a time-dependent driving Hamiltonian as well as other cases in which energy is transferred from an externalsource (charger) to the battery under a global unitary evolution. We shall therefore limit the discussions mostly tounitary protocols and only highlight a couple of works dealing with dissipative protocols at the end of this section. A. Passive states and maximal work extraction
In a unitary evolution of a quantum system, the von-Neumann entropy remains invariant, which follows directlyfrom the fact that the eigen-values of the density matrix is not altered through a unitary transformation. However, theconverse is not true – two states having the same von-Neumann entropy are not necessarily connected by a unitarytransformation, except in two-dimensional Hilbert spaces. This simple observation suggested that the maximum work(per unit cell) that can be extracted from a ‘battery’ of cells can be higher than that possible from a single cell . Toelaborate further, we first recall the notion of passive states which we had briefly touched upon previously inSec. (V A).Let us consider a system with Hilbert space dimension d and Hamiltonian H such that h = P dj =1 ε j | j i h j | with ε j ≤ ε j +1 . We are interested in the unitary evolution of the system when an arbitrary local interaction is switched on,so that the total Hamiltonian reads, h ( t ) = h + h ( t ) . (124)Note that this can also be interpreted as quenching the system. The Hamiltonian h ( t ) remains finite for 0 ≤ t ≤ τ andvanishes otherwise; the operation is thus cyclic in nature. Denoting the initial state of the system as ρ , the maximumamount of work which can be extracted from it during this cyclic process is known as ergotropy , defined as, E = W U,max = Tr [ ρh ] − min U ( τ ) ∈ SU ( d ) { Tr (cid:2) U ( τ ) ρU ( τ ) † h (cid:3) } = Tr [ ρh ] − Tr [ σ ρ h ] , (125)where the superscript ‘1’ denotes that we are working with a single cell or single copy of the system, W U,max denotesthe maximum work that can be extracted using unitary operations and U ( τ ) are unitary time-evolution operatorsacting for the duration of time τ . Note that the minimization in the first equality above is over all d -dimensionalunitary matrices (belonging to the SU ( d ) group) and hence an implicit minimization over τ is also implied. In thesecond equality, σ ρ is the passive state corresponding to ρ , defined as the state having zero ergotropy. Hence, no energycan be extracted from σ ρ through cyclic unitary processes, i.e.∆ E = − W U = Tr (cid:2) U σ ρ U † h (cid:3) − Tr [ σ ρ h ] ≥ , ∀ U ∈ SU ( d ) . (126)where a positive value of ∆ E corresponds to a negative work extraction, or equivalently, a work deposition on thesystem. In general, it can be shown that a state is passive if it is diagonal in the energy eigen-basis of the system9with non-decreasing diagonal elements (populations), when arranged in the order of non-increasing energies, i.e. σ = P dj =1 s j | j i h j | with s j ≥ s j +1 for ε j ≤ ε j +1174,175 . Consequently, the passive state σ ρ which can be attained bymeans of local unitary operations on ρ is unique in nature. Unless otherwise mentioned, we shall use the notation σ ρ to denote the unique passive state corresponding to ρ in the rest of the article.Since ρ and σ ρ are connected by a unitary transformation, they have the same Von-Neumann entropy, S ( ρ ) = S ( σ ρ ).However, there may exist other states that have the same entropy as ρ but with energies even lower than σ ρ , althoughnot accessible by unitary transformations alone. Using the fact that the thermal state ζ ρ = e − βh / Tr (cid:2) e − βh (cid:3) minimizesthe free energy, Tr( ρh ) − β S ( ρ ) ≥ Tr( ζ ρ h ) − β S ( ζ ρ ) , (127)where the inverse temperature β is determined by the entropy constraint S ( ζ ρ ) = S ( ρ ), we have Tr( ρh ) ≥ Tr( ζ ρ h ).Hence, the maximum work that can be extracted from ρ is , W max = Tr [ ρh ] − Tr[ ζ ρ h ] (128)Hence, there remains an unattainable work of magnitude W max − W U,max within a constraint of constant entropy,which can not be extracted by local unitary protocols acting on the system.However, the above scenario completely changes when one considers a battery of cells where each cell corresponds toan identical copy of ρ . In this case, it becomes possible to attain a higher work extraction capacity per system thanthat possible from a single copy of the system. The possibility arises because in general, for n copies of the system, ⊗ n σ ρ may not necessarily be the same as σ ⊗ n ρ . In fact, the equality holds true only for thermal states. In other words,if the passive state corresponding to ρ is a thermal state, σ ρ = ζ ρ , then it can be easily shown that ⊗ n ζ ρ = σ ⊗ n ρ .Hence, thermal states are also known as completely passive states. Denoting the battery Hamiltonian as H = P ni h ,i ,where h ,i is the Hamiltonian of the i th cell . Let us now define the maximum extractable work per unit cell, w max = W nmax n = 1 n (cid:16) Tr [ ⊗ n ρH ] − Tr [ σ ⊗ n ρ H ] (cid:17) , (129)In the limit n → ∞ , it can be shown that , lim n →∞ w max = W max , (130)In other words, the maximum work extracted per system or ‘cell’ in a quantum battery can saturate the maximumavailable thermodynamic energy within the constraint of a constant entropy. Another way to look at the higher energyextraction per cell is that, although the battery as a whole is driven unitarily, the dynamics of the individual cells areno longer necessarily unitary. This allows them to attain final states which have lesser energy than their correspondingpassive states. B. Entangling vs non-entangling protocols
Let us suppose that we wish to extract the maximum work available from a battery through a unitary evolution U such that U ( ⊗ n ρ ) U † = σ ⊗ n ρ . The initial state of the battery ⊗ n ρ , by construction, is separable as it exists as adirect product of the states of the individual cells. The final state σ ⊗ n ρ is diagonal in the eigen-basis of the batteryHamiltonian, H = P ni =1 h ,i and hence is also separable. We can now categorize the unitary operators U facilitatingthe required work extraction into two groups – one in which the battery remains separable at all intermediate timesand another in which the individual cells are allowed to get entangled at intermediate times.As an illustration , consider the simple case in which the initial state ⊗ n ρ is also diagonal in the energy basisbut not passive, ⊗ n ρ = diag ( p , p , . . . , p d n ). The required unitary protocol for maximum work extraction thereforecorresponds to a set of permutation operations which rearrange the populations of the density matrix. Note thatpermutation operations are non-local in nature and is capable of generating entanglement. Let us consider one suchoperation h α | ⊗ n ρ | α i ≡ p α (cid:28) p β ≡ h β | ⊗ n ρ | β i , where | α ( β ) i = | i α ( β )1 , i α ( β )2 , . . . , i α ( β ) n i with | i α ( β ) j i denoting the stateof the j th cell. One possible way to carry out this transposition while preserving the separability of the state at alltimes is to carry out a sequence of 2 n − | i α , i α , . . . , i αn i (cid:28) | i β , i α , . . . , i αn i (cid:28) | i β , i β , . . . , i αn i (cid:28) | i β , i β , . . . , i βn i . (131)0On the other hand, if entanglement generation is permitted, than the above operation can be completed in a single stepwith a unitary operation of the form U = P µ = α,β | µ i h µ | + | α i h β | + | β i h α | . The second approach can be consideredequivalent to taking a shortcut through the subspace of entangled states in the Hilbert space, as opposed to restrictingto the subspace of separable states.The above illustration provides two important results . Firstly, maximal work can be extracted from a batterywithout requiring any generation of entanglement between the individual cells. Secondly, a large number of operationsare required to extract the work if the battery is to remain in a separable state at all instants of time, thus requiring along time in the process. These observations naturally give rise to the question – can quantum entanglement providean advantage in terms of the speed of the charging or discharging of quantum batteries. C. Quantum speed limits
It is important to realize that for a set of fixed initial and final states (and hence a fixed energy difference), the speedof charging is intricately tied to the concept of quantum speed limits (QSL) . QSLs put a fundamental lowerbound on the minimum time of evolution between two given states; these bounds are consequences of the energy timeuncertainty relation ∆ E ∆ t ≥ (cid:126) . Various such bounds have been proposed in literature depending on the evolutionprocess under consideration. For our purpose, consider the evolution of the density matrix ρ ( t ) = P i p i | φ i ( t ) i h φ i ( t ) | ,driven by a time-dependent Hamiltonian H ( t ). The Bures angular distance defines the distance between two states ρ and ρ in the density matrix space as, D ( ρ , ρ ) = arccos [ F ( ρ , ρ )] , (132)where F ( ρ , ρ ) = Tr (cid:2)p √ ρ ρ √ ρ (cid:3) is the Uhlmann’s fidelity . For two states separated by an infinitesimal evolution,it can be shown from Eq. (132) that the distance is given by ,lim ∆ t → ds ( ρ ( t ) , ρ ( t + ∆ t )) = 14 H t ∆ t , (133)where H t is the QFI, previously discussed in Sec. VIII A. Using the general expression for QFI derived in Eq. (109)and the fact that the eigen values of ρ remain invariant in unitary evolution, we evaluate the QFI with respect to timeas follows, H t = 2 X i = j ( p i − p j ) p i + p j (cid:12)(cid:12)(cid:12)(cid:12) h φ j | ∂∂t | φ i i (cid:12)(cid:12)(cid:12)(cid:12) = 2 X i = j ( p i − p j ) p i + p j (cid:12)(cid:12)(cid:12)(cid:12) h φ j | H ( t ) − h H ( t ) i (cid:126) | φ i i (cid:12)(cid:12)(cid:12)(cid:12) ≤ X i = j ( p i + p j ) (cid:12)(cid:12)(cid:12)(cid:12) h φ j | H ( t ) − h H ( t ) i (cid:126) | φ i i (cid:12)(cid:12)(cid:12)(cid:12) = 4 (∆ H ( t )) (cid:126) , (134)where ∆ H ( t ) = p h H ( t ) i − h H ( t ) i and we have used the inequality ( p i + p j ) ≥ ( p i − p j ) for p i , p j ≥
0. The equalityis satisfied in the above equation for p i p j = δ i,j ; or equivalently, for pure states. The speed of evolution is thereforefound to be, dsdt = 12 p H t ≤ ∆ H ( t ) (cid:126) . (135)Thus, given a pair of initial and final states ρ (0) = ρ i and ρ ( τ ) = ρ f , respectively, the time required to traverse thedistance between them is calculated by integrating over the Bures distance between them and the time taken, Z D ( ρ i ,ρ f )0 ds ≤ (cid:126) Z τ ∆ H ( t ) dt (136)Therefore, we obtain a lower bound on the minimum time required in any unitary evolution as, τ ≥ τ QSL ≥ (cid:126) D ( ρ ( t ) , ρ ( t + τ ))∆ E τ , (137)1where, ∆ E τ = 1 τ Z τ ∆ H ( t ) dt. (138)As already mentioned, the second inequality in Eq. (137) saturates for pure states ρ ( t ) = | ψ ( t ) i h ψ ( t ) | and we obtain, τ pureQSL = (cid:126) arccos |h ψ ( t ) | ψ ( t + τ ) i| ∆ E τ (139)The bound derived in Eq. (137) is known as the Mandelstam-Tamm bound for arbitrary mixed states. Sometimes,a unified bound is used which combines the Mandelstam-Tamm bound with the Margolus-Levitin bound and isgiven as, τ ≥ τ QSL ≥ (cid:126) D ( ρ ( t ) , ρ ( t + τ )min { ∆ E τ , E τ } , (140)where, E τ = 1 τ Z τ H ( t ) dt. (141)Note that the Bures angular distance considered above measures the physical distinguishability of the initial andfinal states. However, from the perspective of quantum batteries, it is also important to consider how much the statesare distinguishable in terms of their average energy. With this motivation, a different speed limit was recently analyzedin Ref. [226], where the distance between a pair of states is measured in the energy space as follows. Let us considerthe probability distribution of a pair of density matrices ρ and ρ in the energy eigen basis as, p k = Tr [ ρ Π k ] and q k = Tr [ ρ Π k ], respectively, where Π k = {| k i h k |} is the projection operator on the energy eigen-state | k i . The distanceis then defined as the relative entropy distance or the Kullback-Liebler divergence between these two distributions , D KL ( p, q ) = X k p k log p k q k . (142)For states separated by infinitesimal time ∆ t , the distance can be expanded upto second order in ∆ t as,lim ∆ t → D KL ( p ( t ) , p ( t + ∆ t )) = X k p k (cid:18) dp k dt (cid:19) ∆ t . (143)The speed limit in the energy space is therefore defined as , v E ( t ) = lim ∆ t → p D KL ( ρ ( t ) , ρ ( t + ∆ t ))∆ t = r I E ( t )2 , (144)where I E ( t ) = X k p k (cid:18) dp k dt (cid:19) , (145)is the classical Fisher information (see Eq. (101)) defined on the energy eigen-space. D. Quantum advantage
The power of a quantum battery is defined as the speed at which energy can be deposited on it (charged) or therate at which useful work can be extracted from it (discharged). Any enhancement in the power of a quantum batteryresulting from quantum effects such as entanglement is dubbed as quantum advantage . As we shall see below, QSLsenforce certain bounds on any possible quantum advantage in the charging or discharging process of quantum batteries.We discuss these bounds by analyzing QSL restrictions on both the Hilbert space as well as the energy space.2
1. From speed limit considerations in Hilbert space
The comparison of the charging power of quantum batteries using local (non-entangling) and non-local (entangling)operations was first considered in Refs. [68,227]. It was argued that a quantum advantage can be meaningful only whenthe thermodynamic resources available to the entangling protocols is same as that of non-entangling protocols. In otherwords, any trivial enhancement of the charging power simply resulting from availability of more energy in the drivingHamiltonian must be properly accounted for. This necessitates appropriate re-scaling of the driving Hamiltonian inthe entangling protocol, which ultimately results in an extensive scaling of the quantum advantage.To elaborate, let us consider a quantum battery composed of n cells. We wish to compare two charging protocols asgiven below:1. Parallel charging : Each of the battery cell is driven simultaneously under the action of an identical Hamiltonian, H k ( t ) = h + V k ( t ) , (146)were h is the un-driven Hamiltonian of a single cell.2. Collective charging : The battery is collectively driven with the Hamiltonian, H ( t ) = H + V ( t ) , (147)where H = P ni =1 h ,i , with h ,i = h ∀ i .The charging takes place for a duration τ and V k ( t ) = V ( t ) = 0 for t < , t > τ . We also assume that the same finalstate is achieved using both the protocols starting from a given initial state. The quantum advantage for collectivecharging is then defined as the ratio of the power for collective and parallel charging ,Γ = P P k = τ k τ . (148)We now make use of the QSL results discussed in Sec. IX C; specifically in context of the inequality presented inEq. (140). As already mentioned, one must rule out any trivial enhancements in power due to transfer of more energy.For this purpose, let us first put a constraint (C1) on the driving Hamiltonian of the form E τ ≤ nE k τ , where E , k τ isthe time-averaged mean energy defined in Eq. (141). From Eq. (140), we therefore obtain, τ ≥ τ QSL ≥ D n /E τ ≥ D n nE k τ , (149)and hence the quantum advantage is upper bounded as,Γ C ≤ nE k τ D n τ k = nβ D D n , (150)where β = τ k /τ k QSL ≥ D ( D n ) is the Bures angular distance between the initial and final density matrices of thecell (battery). Similarly, if the variance is constrained (C2) in stead of the mean energy, one can show that,Γ C ≤ √ nβ D D n . (151)The inequalities derived above shows that collective charging protocols provide a quantum advantage over parallelones that scales extensively. The exact scaling depends on the particular choice of the constraint applied on the drivingHamiltonian of the collective charging protocol. However, if the initial state lies within the separable ball , defined as the‘ball’ of region centered on the maximally mixed state containing only separable states, the same quantum advantagecan be achieved even though the cells remain separable at all instants of time during the charging process . In otherwords, although non-local ‘entangling’ operations are necessary for a quantum advantage, generation of entanglementbetween the battery cells is not required.In practical situations, it is difficult to engineer very long-range non-local interactions. This brings into question thescaling of the quantum advantage for arbitrarily large battery size. In fact, it turns out that for a driving Hamiltoniancontaining k -local terms (terms which act over k cells at any given time) and where each individual cell interacts withonly m other cells, the quantum advantage scales as Γ ≤ γ (cid:2) k ( m −
1) + k (cid:3) . (152)3The constant γ is independent of battery size and hence there is no extensive scaling of the quantum advantage with n . We note though that, unlike the constraints C C
2, this bound is derived with a constraint imposed on theoperator norm of the driving Hamiltonian , E τ ≤ n E k τ , with, E k ( τ = 1 τ k ( Z k ( || H k ( ( t ) || op dt, (153)where || H ( t ) || op denotes the largest singular value of H ( t ).
2. From speed limit considerations in energy space
An alternate bound on the instantaneous power of a quantum battery can be derived using the QSL restrictionsin the energy eigen space (see Eq. (144)). Considering a driving Hamiltonian of the form H ( t ) = H + V ( t ), theinstantaneous power is defined as, P ( t ) = ddt h H i = ddt Tr ( ρ ( t ) H ) . (154)Expanding in the eigen-basis of H = P k ε k | k i h k | , we can write, P ( t ) = X k ε k dp k ( t ) dt = X k ( ε k − c ( t )) dp k ( t ) dt , (155)where p k ( t ) = Tr ( ρ ( t ) | k i h k | ) and c ( t ) ∈ R . Note that in deriving the second equality, we have used P k dp k /dt = d ( P k p k ) /dt = 0. Next, we square and rearrange the above equation as, P ( t ) = X k ( ε k − c ( t )) √ p k √ p k dp k ( t ) dt ! . (156)Applying the Cauchy-Schwarz inequality, we arrive at, P ( t ) ≤ X k p k ( ε k − c ( t )) ! X k p k (cid:18) dp k dt (cid:19) ! (157)It is easy to check that the r.h.s is minimized for c ( t ) = h H i . We therefore arrive at the inequality , P ( t ) ≤ ∆ H ( t ) I E ( t ) , (158)where ∆ H ( t ) is the variance of the un-driven or bare Hamiltonian of the battery and I E ( t ) is the Fisher informationdefined on the energy space (see Eq. (145)). The above inequality implies that the power of a quantum battery ata particular instant is crucially dependent on two aspects of the charging/discharging protocol. Firstly, it dependson how much non-local the driving protocol is in the energy space as quantified by the instantaneous variance. Inother words, it depends on how many of the energy eigen-states have a significant population at that instant of time.Secondly, it also depends on the instantaneous speed of the driving protocol in the energy space, where the distancebetween the states is measured in terms of their energetic difference.Let us now have a closer look at the variance ∆ H ( t ) . Assuming a battery of n identical cells, the bare Hamiltonianreads H = P ni =1 h i and the variance can be expanded as ,∆ H ( t ) = Tr n X i =1 h i ! ρ ( t ) − Tr " n X i =1 h i ! ρ ( t ) = n X i =1 (cid:16) Tr (cid:2) h i ρ ( t ) (cid:3) − Tr [ h i ρ ( t )] (cid:17) + n X i = j (cid:16) Tr [ h i h j ρ ( t )] − Tr [ h i ρ ( t )] Tr [ h j ρ ( t )] (cid:17) (159)4The first term in the second equality above captures the total contribution of the local variance from each cell andtherefore scales linearly with the battery size n . For the parallel charging protocol, the second term vanishes and hencethe maximum power follows the same linear scaling. Thus, a quantum advantage can be achieved for protocols forwhich the second term remains finite and as well as scales faster than n . However, the second term is non-zero onlywhen the cells are entangled and hence the protocol must generate entanglement between the battery cells to achieveany quantum advantage. The variance therefore bridges a connection between the scaling of the quantum advantageand entanglement generating protocols.It is important to note that the upper bound of the power in Eq. (158) and the resulting extensive scaling of thequantum advantage has been derived without incorporating any constraint or re-scaling of the driving Hamiltonian.This is unlike the bounds (of the quantum advantage) derived in Eqs. (150) and (151) where the constraints imposedon the driving Hamiltonians resulted in the extensive scaling of the quantum advantage.Once again, it is also necessary to analyze the scaling of the bound with the range of entanglement generated in thesystem. For a battery consisting of n qubits where at most k qubits are entangled at a given time t , it can beshown that the variance satisfies , 4∆ H ( t ) ≤ rk + ( n − rk ) , where r is the integer part of n/k . The power istherefore bounded as , P ( t ) ≤ (cid:0) rk + ( n − rk ) (cid:1) I E ( t ) . (160)The inequality derived above thus provides a deeper insight into how the instantaneous entanglement generated duringthe charging or discharging protocol affects the power of a quantum battery. E. Role of inherent entanglement
So far, we have only considered entangling protocols in charging batteries composed of independent cells. How doesthe situation change if some amount of entanglement pre-exists among the battery cells and is not generated by thecharging Hamiltonian? This question was addressed by Le et.al. , who considered a many-body spin chain withtwo-body interactions as a model of quantum battery and analyzed the power while charging with a local externaldriving field. In particular, they considered the XXZ Hamiltonian as the bare or un-driven Hamiltonian of the battery,which is given by, H = H B + H g , (161a)where, H B = B n X i =1 σ zi , (161b) H g = − X i 0. For p > n . Note that these results are in agreement with the bounds in quantum advantage derived for k = 2order interactions with m number of participating cells derived in Eq. (152). An extensive scaling of the quantumadvantage is possible only in the limit of p → m ∼ n − 1, while it scales as ∼ O (2 m ) = O ( const ) for p > m is finite. However, the crucial difference is that the advantage is facilitated by the entanglement generated bythe internal Hamiltonian H g and not by the charging field V . F. Usability of stored energy The quality of a quantum battery is not only determined by its energy storage capacity and rate of charging ordischarging, but also on the ‘usability’ of the stored energy, which may depend on a number of factors. As for example,a battery having a high uncertainty of the stored energy, characterized by the ratio ∆ H ( τ ) /E ( τ ) where E ( τ ) is themean energy stored, can be potentially hazardous to use for work extraction . This is because a high variancetypically means that the energy which the battery can deliver has a large fluctuation about its mean energy. Similarly,large temporal fluctuations during the charging process implies that the amount of stored energy is highly sensitive tothe charging duration, thereby making it difficult to estimate the stored energy.Another important figure of merit of the quality of the stored energy is the fraction of the mean energy available forextraction after the charging process. Mathematically, it is defined as , f ( τ ) = E ( τ ) E ( τ ) , (163)where E ( τ ) and E ( τ ) = h H i are the ergotropy (see Eq. (125)) and mean energy of the battery, respectively, aftercharging for a duration of time τ . This fraction is unity if the ground state energy is zero and the battery is in apure state. Therefore, the quantity f becomes relevant when the battery exist in a mixed state. This may happenfor charger-battery systems (see Sec. IX G 2 below) where energy is transferred from another system which acts as a’charger’ to the battery. While the composite system evolves unitarily and hence remains pure if initialized in thesame, the reduced state of the battery may be mixed in case of entanglement generation between the charger and thebattery. Similarly, in some cases, one may need to consider energy extraction from only a subset of M number of cellsfrom the battery as the full system may not be accessible . The reduced system of the M cells ρ M is likely toexist in a mixed state and hence one needs to consider the fraction of usable energy from this subset.It has been argued (and numerically demonstrated for some models) that, whereas entanglement provides aboost in the optimal charging power of batteries, the same may be responsible for lesser availability of usable energyfrom the battery. This is because of the fact that the amount of entanglement within the battery cells directly influencesthe mixed nature of ρ M . Similarly, highly entangled states of the battery and charger also renders the battery statemixed, thereby leading to f ( τ ) (cid:28) 1. However, for systems having certain integrals of motion in which the dynamicalevolution is constrained to a small part of the full Hilbert space, it is expected that in the thermodynamic limit ,lim n →∞ f ( τ ) = 1 . (164)This can be understood from the fact that the entanglement entropy of subsystems in integrable systems (a measure ofbipartite entanglement), does not scale extensively with the system size. On the other hand, the amount of energystored in the battery scales linearly with the size of the battery. Hence, in the thermodynamic limit, the energy lockedaway due to entanglement is expected to be negligible in comparison with the energy stored. G. Models 1. Spin models A plethora of physically realizable models of quantum batteries have been proposed over the past few years. Acommon starting point in a large chunk of these models is to consider an array of non-interacting spins or qubits,6placed in a local external field. The Hamiltonian of the battery thus reads, H = B n X i σ zi , (165)while a charging Hamiltonian V ( t ) is used for the charging process so that the battery evolves under the action of theHamiltonian, H ( t ) = H + V ( t ) = B n X i σ zi + V ( t ) (166)The exact form of charging Hamiltonian V ( t ) characterizes the model under consideration and the capacity and powercan then be analyzed. We discuss some of these models below.Firstly, one can rule out all integrable models of spin chains, such as the Ising, XY or extended Ising models, aspotential candidates for attaining any quantum advantage . This is because the Hilbert space of such models have alocal decoupled structure in the quasi-momentum basis. As the size of the quasi-momentum basis increases linearlywith the system size, the power can also be shown to scale linearly, at best. Also, the different quasi-momentum modesdo not maximize their local expectation energy simultaneously, which severely limits the maximum storage capacity ofthese models.The Lipkin-Meshkov-Glick (LMG) Hamiltonian , H LMG = λn X i 2. Cavity assisted charging A collection of two-level atoms (TLAs) charged collectively through cavity induced excitations also forms a promisingmodel of quantum battery, as the collective charging may naturally lead to a quantum advantage. The Dicke model,described by the Hamiltonian , H DK = ωJ z + ω c a † a + 2 ω c λ ( t ) J x (cid:0) a + a † (cid:1) , (168)was explored in Ref. [232] as the charging Hamiltonian of a quantum battery composed of n TLAs. In the aboveequation, J α P ni =1 σ αi and the operator a ( a † ) annihilates (creates) a cavity photon. The model is initialized in a directproduct state of the ground state of the TLAs and a Fock state, | ψ (0) i = | G i ⊗ | N i , where | G i = ⊗ n | g i is the groundstate of n TLAs and | N i is the Fock state in the Hilbert space of the cavity with N number of photons. The chargingoccurs when a finite λ is switched on. Note that this setup is different from the ones previously discussed in the waythat the state of both the TLAs which form the battery as well as the cavity which acts like a charger , is trackedduring the evolution. However, only the energy deposited on the TLAs is considered relevant in the context of energystorage and hence the battery Hamiltonian is considered to be H = ωJ z . Such battery-charger composites have alsobeen explored in a number of other works .Under resonant conditions ( ω = ω c ), the charging power of the above model was compared to that of a parallel model where the n TLAs interact with n distinct cavities. Through numerical analysis, it was found that the modelindeed exhibits a quantum advantage which scales as ∼ √ n . However, if one re-scales the coupling between the TLAsand the cavity as λ → λ/ √ n so as to have a well defined thermodynamic limit, the quantum advantage has beenshown to disappear with the power scaling only linearly with n . This is despite the fact that the variance of thebare Hamiltonian ∆ H with H = ωJ z scales quadratically with n in the strong-coupling regime and the Fisherinformation I E scales linearly in both the strong and weak coupling regimes. The absence of super extensive scaling ofthe quantum advantage has been attributed to a poor saturation of the bound in Eq. (158).In addition, the super-extensive scaling of the variance in the strong-coupling regime is also found to be present atthe end of the charging protocol, which means the ratio δH ( τ ) /E ( τ ) does not vanish even in the thermodynamiclimit, thereby undermining the quality of the energy stored (see Sec. IX F). In addition, a significant amount ofentanglement is also found to be present between the TLAs and the cavity in the final state which implies that a largefraction of the energy stored is likely to be unavailable for extraction.7 3. Disordered chains Disordered quantum many-body systems are known to host certain ‘localized’ phases which may have potentialapplications in constructing quantum batteries as hinted in recent results. Consider once again, a battery composedof an array of spins with the bare and the driving Hamiltonians given as in Eqs. (165) and (166), respectively. Thecharging Hamiltonian is of the form V ( t ) = λ ( t ) H , where H = − λ ( t ) n X i =1 (cid:0) J i σ xi σ xi +1 − J σ xi σ xi +2 (cid:1) . (169)As before, we have λ ( t ) = 1 during the charging time 0 ≤ t ≤ τ and is zero otherwise. The nearest-neighbor couplingis of the form J j = J + δJ j , where δJ j is random chosen from a uniform distribution satisfying − δJ ≤ δJ j ≤ δJ . For J = 0 the model described by the Hamiltonian H + H exist in the Anderson-localized (AL) phase. Similarly, for J = 0 and δJ > δJ c > 0, the model exhibits a many-body localized (MBL) phase . On the contrary for J = 0and δJ < δJ c , the spectrum of the model is characterized by a mobility edge , i.e. all energy eigen-states below anenergy threshold are localized while states with energy above the threshold are thermal and show ergodic behavior.Numerical analysis have shown that the fraction of the stored energy in the battery which can be subsequentlyextracted as work, i.e. the quantity f ( τ ) discussed in Sec. IX F, drastically improves in the localized phases of the ALand the MBL as compared to ergodic phases . In the AL phase, this enhancement arises due to the integrability ofthe model as discussed in Sec. IX F. Similarly, the MBL phase is also characterized by the presence of an extensivenumber of localized integrals of motion, thus restricting the dynamics in the Hilbert space. Further, the presence ofinteractions in the MBL phase suppresses temporal fluctuations in the stored energy which are otherwise prominent inthe AL phase.Similarly, it has also been demonstrated that for a battery prepared in the ground state of the quantum XYZchain and charged through a local magnetic field (in x − direction), the power is enhanced when the nearest-neighborcouplings are chosen randomly from a Gaussian distribution . Note that in this case, the disorder is introduced inthe battery Hamiltonian H ; unlike the previous cases where disorder was present in the couplings of the chargingHamiltonian.Finally, we would like to mention that the exploration of disordered or random interactions in models of quantumbattery has only just begun to pick up pace and is still in its nascent stages. As for example, recent results have shownthat using the fermionic Sachdev-Ye-Kitaev (SYK) Hamiltonian for charging a battery of spins, it is possible to achieveexplicit quantum advantage in the power of quantum batteries as well as improve the quality of the energy stored bysuppressing unwanted fluctuations . We also note that a general approach to analyze battery models based ondisordered systems has been recently introduced in Ref. [242]. H. Quantum batteries as open systems Finally, we would like to mention that the working of quantum batteries has also been explored in the context ofopen system dynamics. The charger-battery setting, discussed in Sec. IX G 2, has been extended to the case where thecharger extends as a mediator between an external energy supply and the battery with the energy transfer facilitatedby thermalization with the bath or coherent pumping by classical fields coupled to the charger . The evolution of thecharger-battery composite is thus dissipative in nature and is consequently analyzed using the GKSL master equation.Using this approach, different implementations of charger-battery setup has been considered using harmonic oscillatorand qubit systems to analyze the interplay between coherent pumping and thermalization mechanisms in the batteryoperation. We refer the reader to Ref. [234] for detail.An interesting protocol of charging a quantum battery was considered in Ref. [243], where the system to be chargedis coupled sequentially to a series of auxiliary systems prepared in Gibbs state. The coupling with each of the auxiliarysystem lasts for a time τ during which the systems undergoes dissipative dynamics. Using this method, it is possibleto drive the system to an active equilibrium state, in which no external work is required to sustain the state once it isreached. Importantly, the equilibrium state is not passive thus allowing work extraction and thus provides advantageover regular thermalization processes in which the steady state is generally thermal and passive in nature. Once again,we refer to Ref. [243] for detail. Recently, other aspects of dissipative charging of quantum batteries are also beingexplored .8 X. OUTLOOK It is fair to say that quantum thermal machines have come a long way since the inception of the early prototypes.A rigorous analysis of their operating mechanism over the years has provided a much deeper understanding of howthermodynamic signatures are manifested at the scale of a few-level systems. The nature of the heat reservoirs hasbeen found to significantly impact the performance of the QTMs; which at times have been shown to outperform theirclassical counterparts. Nevertheless, all QTMs proposed thus far have been found to operate within the premises ofthe well established classical thermodynamic laws. While QTMs employing simple systems such as qubits or harmonicoscillators as working fluids and operating within the framework of Markovian dynamics are well understood by now,the trend is shifting more towards incorporating various sources and aspects of non-Markovian dynamics into theoperation of QTMs. Likewise, the pros and cons of using quantum many-body systems as working fluids are beingactively explored. Newer variants of QTMs, aimed at utilizing entanglement as a resource, are being envisaged .It is also to be noted that there has also been a spurt in the number of experimental implementations of the QTMsand the verification of their predicted performance.On the other hand, there remains much to be understood as of how the quantumness of batteries exactly influencestheir utility as energy storage devices. It is yet to be established if the advantages reported in the context of quantumbatteries have a pure quantum origin or are merely the artefacts of intelligent and efficient charging/dischargingprotocols. Moreover, the stability of the stored energy, particularly with respect to environmental exposure, also needsto be carefully examined. Nevertheless, given the current pace of development and the amount of research beingdevoted, these questions are expected to be answered in the near future. ACKNOWLEDGMENTS We are grateful to Utso Bhattacharya, Victor Mukherjee, Wolfgang Niedenzu and Saikat Mondal for collaborationand useful discussions on related works. We also thank Arnab Ghosh, Souvik Bandyopadhyay and Somnath Maity forcritical comments and suggestions. S.B. acknowledges CSIR, India for financial support. A.D. acknowledges financialsupport from SPARC program, MHRD, India. ∗ [email protected] M. Zemansky, R. Dittman, Heat and thermodynamics 7th ed. , McGraw Hill Education (1997). J. Gemmer, M. Michel, G. Mahler, G., Quantum Thermodynamics. Springer-Verlag, Berlin and Heidelberg (2010). T. Jahnke, G. Mahler, Europhys. Lett. 90 (5), 50008 (2010). R. Kosloff, Entropy , 2100 (2013). M. Horodecki, J. Oppenheim, Nat. Commun. 4, 2059 (2013). S. Vinjanampathy, J. Anders, Contemporary Physics , 545 (2016). J. Anders and M. Esposito, New J. Phys. F. Campaioli, F. A. Pollock, and S. Vinjanampathy, Thermodynamics in the Quantum Regime, Fundamental Aspects andNew Directions (Springer, Cham, Switzerland, 2018). S. Deffner and S. Campbell, Quantum Thermodynamics, An introduction to the thermodynamics of quantum information ,IOP Publishing, (2019). R. Kosloff, J. Chem. Phys. , 204105 (2019). A. Tuncer, O. E. Mustecaplioglu, arXiv:2009.04387 (2020). H. E. D. Scovil, E. O. Schulz-DuBois Phys. Rev. Lett. , 262 (1959). S. Carnot, Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance (Bachelier,Paris, 1824). G. Lindblad, Commun.Math. Phys. 48, 119 (1976). V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, J. Math. Phys. 17, 821 (1976). E. Geva, R. Kosloff, J. Chem. Phys. 104 (19), 7681 (1996). C. M. Bender, D. C. Brody, B. K. Meister, J. Phys. A: Math. Gen. , 4427 (2000). M.O. Scully, Phys. Rev. Lett. 87, 220601 (2001). J. He, J. Chen, and B. Hua, Phys. Rev. E , 036145 (2002). E. Geva, J. Mod. Opt. 49, 635 (2002). B. Lin, J. Chen, Phys. Rev. E 67, 046105 (2003). D. Segal, A. Nitzan, Phys. Rev. E , 026109 (2006). H.T. Quan, Y.D. Wang, Y.X. Liu, C.P. Sun, F. Nori, Phys. Rev. Lett. , 180402 (2006). D. Segal, Phys. Rev. Lett. 101, 260601 (2008). N. Linden, S. Popescu, P. Skrzypczyk, Phys. Rev. Lett. 105, 130401 (2010). Y.X. Chen, S.W. Li, Europhys. Lett. 97 (4), 40003 (2012). A. Levy, R. Kosloff, Phys. Rev. Lett. 108, 070604 (2012). A.M. Zagoskin, S. Savel’ev, F. Nori, F.V. Kusmarsev, Phys. Rev. B , 014501 (2012). A.E. Allahverdyan, K.V. Hovhannisyan, A.V. Melkikh, S.G. Gevorkian, Phys. Rev. Lett. , 050601 (2013). G. S. Agarwal and S. Chaturvedi, Phys. Rev. E , 012130 (2013). L.A. Correa, J.P. Palao, G. Adesso, D. Alonso, Phys. Rev. E 87, 042131 (2013). H. P. Goswami, U. Harbola, Phys. Rev. A 88 013842, (2013). D. Venturelli, R. Fazio, V. Giovannetti, Phys. Rev. Lett. 110, 256801 (2013). D. Gelbwaser-Klimovsky, R. Alicki, G. Kurizki Phys. Rev. E , 012140 (2013). R. Alicki, Open Syst. Inf. Dyn. (01n02), 1440002 (2014). N. Brunner, M. Huber, N. Linden, S. Popescu, R. Silva, P. Skrzypczyk, Phys. Rev. E , 032115 (2014). L.A. Correa, J.P. Palao, D. Alonso, G. Adesso, Sci. Rep. 4, 3949 (2014). R. Gallego, A. Riera, J. Eisert, New J. Phys. 16 (12), 125009 (2014). R. Alicki, D. Gelbwaser-Klimovsky, K. Szczygielski1, J. Phys. A: Math. Theor. P. A. Erdman, F. Mazza, R. Bosisio, G. Benenti, R. Fazio, and F. Taddei, Phys. Rev. B , 245432 (2017). P. Chattopadhyay, G. Paul, Sci Rep , 16967 (2019). P. Chattopadhyay, Eur. Phys. J. Plus , 302 (2020). K. Ono, S. N. Shevchenko, T. Mori, S. Moriyama, F. Nori, arXiv:2008.10181 (2020). A. Usui, W. Niedenzu, M. Huber, arXiv:2009.03832 (2020). G. Manzano, R. Sánchez, R. Silva, G. Haack, J. B. Brask, N. Brunner, P. P. Potts, arXiv:2009.03830(2020). R. Dann, R. Kosloff, P. Salamon, arXiv:2009.02801(2020). Y. Huangfu, S. Qi, J. Jing, arXiv:2008.12486(2020). T. R. de Oliveira, D. Jonathan, arXiv:2008.11694(2020). K. Kaur, V. Singh, J. Ghai, S. Jena, O. E. Mustecaplioglu, arXiv:2008.10258(2020). Y. Fang, Y. Zheng, J. Chang, arXiv:2010.04856(2020). Q. Bouton, J. Nettersheim, S. Burgardt, D. Adam, E. Lutz, A. Widera, arXiv:2009.10946(2020). V. Blickle, C. Bechinger, Nat. Phys. 7 (12), 1–4 (2011). O. Abah, J. Roßnagel, G. Jacob, S. Deffner, F. Schmidt-Kaler, K. Singer, E. Lutz, Phys.Rev.Lett. , 203006 (2012). J. Roßnagel, S. T. Dawkins, K. N. Tolazzi, O. Abah, E. Lutz, F. Schmidt-Kaler, and K. Singer, Science , 325 (2016). D. von Lindenfels, O. Gräb, C. T. Schmiegelow, V. Kaushal, J. Schulz, M. T. Mitchison, J. Goold, F. Schmidt-Kaler, and U.G. Poschinger, Phys. Rev. Lett. , 080602 (2019). J. P. S. Peterson, T. B. Batalhão, M. Herrera, A. M. Souza, R. S. Sarthour, I. S. Oliveira, and R. M. Serra, Phys. Rev. Lett. , 240601 (2019). J. Klatzow, J. N. Becker, P. M. Ledingham, C. Weinzetl, K. T. Kaczmarek, D. J. Saunders, J. Nunn, I. A. Walmsley, R.Uzdin, and E. Poem, Phys. Rev. Lett. , 110601 (2019). V. Giovannetti, S. Lloyd, and L. Maccone, Nature Photonics , 222 (2011). C. L. Degen, F. Reinhard, and P. Cappellaro, Rev. Mod. Phys. , 035002 (2017). G. Kurizki, G. A. Alvarez, and A. Zwick, Technologies , 1 (2017). L. Pezzè, A. Smerzi, M. K. Oberthaler, R. Schmied, and P. Treutlein, Rev. Mod. Phys. , 035005 (2018). P. P. Hofer, J. B. Brask, M. Perarnau-Llobet, and N. Brunner, Phys. Rev. Lett. , 090603 (2017). V. Mukherjee, A. Zwick, A. Ghosh, X. Chen, G. Kurizki Communications Physics , 162 (2019). S. Mondal, S. Bhattacharjee, A. Dutta, Phys. Rev. E 102, 022140 (2020). S. Bhattacharjee, U. Bhattacharya, W. Niedenzu, V. Mukherjee, A. Dutta, New J. Phys. , 013024 (2020). R. Alicki and M. Fannes, Phys. Rev. E 87, 042123 (2013). K. V. Hovhannisyan, M. Perarnau-Llobet, M. Huber, and A. Acin, Phys. Rev. Lett. 111, 240401 (2013). F. Campaioli, F. A. Pollock, F. C. Binder, L. Céleri, J. Goold, S. Vinjanampathy, and K. Modi, Phys. Rev. Lett. 118, 150601(2017). M. G. A. Paris, Int. J. Quantum Inf. 07, 125 (2009). J. Ma, X. Wang, C. P. Sun, F. Nori, Physics Reports , 89 (2011). L. A. Correa, M. Mehboudi, G. Adesso, and A. Sanpera, Phys. Rev. Lett. 114, 220405 (2015). S. Zhou, M. Zhang, J. Preskill, and L. Jiang, Nat. Comm. 9, 78 (2018). A. De Pasquale, D. Rossini, R. Fazio, and V. Giovannetti, Nat. Commun. 7, 12782 (2016). G. De Palma, A. De Pasquale, and V. Giovannetti, Phys. Rev. A , 052115 (2017). K. V. Hovhannisyan, and L. A. Correa, Phys. Rev. B , 045101 (2018). S. N. Shevchenko and D. S. Karpov, Phys. Rev. Applied , 014013 (2018). M. Mehboudi, A. Sanpera, L. A. Correa, J. Phys. A: Math. Theor. 52, 303001 (2019). P. P. Potts, J. B. Brask, and N. Brunner, Quantum 3, 161 (2019). W. Muessel, H. Strobel, D. Linnemann, D. B. Hume, and M. K. Oberthaler, Phys. Rev. Lett. , 103004 (2014). J. B. Brask, R. Chaves, and J. Kolodynski, Phys. Rev. X , 031010 (2015). F. Albarelli, M. A. C. Rossi, M. G. A. Paris, and M. G. Genoni, New J. Phys. , 123011 (2017). E. Polino, M. Valeri, N. Spagnolo, and F. Sciarrino, AVS Quantum Sci. 2, 024703 (2020). C. M. Bender, D. C. Brody, B. K. Meister, Proc. R. Soc. Lond. A. R. Alicki, J. Phys. A , L103 (1979). R. Kosloff, J. Chem. Phys. 80, 1625 (1984). E. Geva, R. Kosloff, J. Chem. Phys. , 3054 (1992). T. Feldmann, E. Geva, R. Kosloff, P. Salamon, 1996, Am. J. Phys. 64 (4), 485–492 (1996). R. Kosloff, E. Geva, J.M. Gordon, J. Appl. Phys. , 8093 (2000). T. Feldmann, R. Kosloff, Phys. Rev. E 61, 4774 (2000). J.P. Palao, R. Kosloff, J.M. Gordon, Phys. Rev. E , 056130 (2001). T. Feldmann, R. Kosloff, Phys. Rev. E 70, 046110 (2004). H.P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2002). S. Kryszewski, J. Czechowska-Kryszk,arXiv:0801.1757 (2008). R. Alicki, D. Gelbwaser-Klimovsky, G. Kurizki, arXiv:1205.4552. A. Levy, R. Alicki, and R. Kosloff, Phys. Rev. E. , 061126 (2012). M. Kolar, D. Gelbwaser-Klimovsky, R. Alicki, G. Kurizki, Phys. Rev. Lett. 109, 090601 (2012). R. Kosloff and A. Levy, Annu. Rev. Phys. Chem. , 365 (2014). D. Gelbwaser-Klimovsky, W. Niedenzu, and G. Kurizki, Adv. At. Mol. Opt. Phys. , 329 (2015). V. Mukherjee, W. Niedenzu, A. G. Kofman, and G. Kurizki, Phys. Rev. E , 062109 (2016). G. Floquet,Sci. Ecole Norm. Sup. 12, 47–88 (1883). M. Bukov, L. D’Alessio, A. Polkovnikov, Advances in Physics, 64:2, 139-226 (2015). A. Eckardt, Rev. Mod. Phys. 89, 011004 (2017). H. Spohn, J. Math. Phys. , 1227 (1978). K. Ono, S. N. Shevchenko, T. Mori, S. Moriyama, and F. Nori, Phys. Rev. Lett. , 207703 (2019). D. Gelbwaser-Klimovsky, W. Niedenzu, P. Brumer, G. Kurizki, Sci. Rep. 5, 14413 (2015) W. Niedenzu, D. Gelbwaser-Klimovsky, G. Kurizki, Phys. Rev. E 92, 042123 (2015). W. Niedenzu, G. Kurizki, New J. Phys. 20, 113038 (2018). D. Gelbwaser-Klimovsky, K. Szczygielski, U. Vogl, A. Saß, R. Alicki, G. Kurizki, M. Weitz, Phys. Rev. A 91, 023431 (2015). K. Szczygielski, D. Gelbwaser-Klimovsky, R. Alicki, Phys. Rev. E 87, 012120 (2013). V. Mukherjee, A.G. Kofman, G. Kurizki, Commun Phys 3, 8 (2020). R. Kubo, J. Phys. Soc. Jpn. 12, 570 (1957). P. C. Martin, J. Schwinger, Phys. Rev. 115, 1342 (1959). R. Haag, N.M. Hugenholtz, M. Winnink, Commun. Math. Phys. 5, 215 (1967). D. Gelbwaser-Klimovsky, G. Kurizki, Phys. Rev. E 90, 022102 (2014). D. Gelbwaser-Klimovsky, G. Kurizki, Sci. Rep. 5, 7809 (2015). W. Niedenzu, M. Huber and E. Boukobza, Quantum , 195 (2019). T.D Kieu, Phys. Rev. Lett. , 140403 (2004). Y. Rezek, R. Kosloff, New J. Phys. 8 (5), 83 (2006). H.T. Quan, Y.X. Liu, C. P. Sun, F. Nori, Phys. Rev. E , 031105 (2007). M.J. Henrich, F. Rempp, G. Mahler, Eur. Phys. J. 151 (1), 157 (2007). H. Wang, S. Liu, J. He, Phys. Rev. E , 041113 (2009). L. He, X. He, W. Tang, Sci. China Ser. G-Phys. Mech. Astron. , 1317 (2009). G. Thomas, R. Johal, Phys.Rev.E 83, 031135 (2011). X. He, J. He, Sci. China Phys. Mech. Astron. , 1751 (2012). X. L. Huang, T. Wang, X. X. Yi, Phys. Rev. E , 051105 (2012). J. Deng, Q.H. Wang, Z. Liu, P. Hanggi, J. Gong, Phys. Rev. E 88, 062122 (2013). K. Zhang, F. Bariani, and P. Meystre, Phys. Rev. Lett. , 150602 (2014). J. Roßnagel, O. Abah, F. Schmidt-Kaler, K. Singer, E. Lutz, Phys. Rev. Lett. , 030602 (2014). A. del Campo, J. Goold, M. Paternostro, Sci. Rep. , 6208 (2015). M. Campisi and R. Fazio, Nat. Comm. , 11895 (2016). G. Manzano, F. Galve, R. Zambrini, and J. M. R. Parrondo, Phys. Rev. E , 052120 (2016). R. Kosloff and Y. Rezek, Entropy 19, 136 (2017). G. Thomas, N. Siddharth, S. Banerjee, and S. Ghosh, Phys. Rev. E , 062108 (2018). M. Pezzutto, M. Paternostro, Y. Omar, Quantum Sci. Technol. , 025002 (2019). N. Y. Halpern, C. D. White, S. Gopalakrishnan, and G. Refael, Phys. Rev. B , 024203 (2019). A. Hartmann, V. Mukherjee, W. Niedenzu, W. Lechner, Phys. Rev. Research 2, 023145 (2020). C. Van den Broeck, Phys. Rev. Lett. 95, 190602 (2005). M. Esposito, K. Lindenberg, C. Van den Broeck, 2009, Phys. Rev. Lett. 102, 130602 (2009). M. Esposito, R. Kawai, K. Lindenberg, C. Van den Broeck, Phys. Rev. Lett. 105, 150603 (2010). M. Esposito, N. Kumar, K. Lindenberg, Phys. Rev. E 85, 031117 (2012). C. Van den Broeck, N. Kumar, K. Lindenberg, Phys. Rev. Lett. 108, 210602 (2012). R. Wang, J. Wang, J. He, Y. Ma, 2013, Phys. Rev. E 87, 042119 (2013). P. A. Erdman, V. Cavina, R. Fazio, F. Taddei and V. Giovannetti, New J. Phys. , 103049 (2019). F.L. Curzon, B. Ahlborn, Am. J. Phys. 43 (1), 22–24 (1975). R. Dann, A. Tobalina, and R. Kosloff, Phys. Rev. Lett. 122, 250402 (2019). R. Dann, A. Tobalina, and R. Kosloff, Phys. Rev. A 101, 052102 (2020). X. Chen,A. Ruschhaupt, S. Schmidt, A. Del Campo, D. Guéry-Odelin, J.G. Muga, Phys. Rev. Lett. 104, 063002 (2010). X. Chen, J.G. Muga, Phys. Rev. A, 82, 053403 (2010). E. Torrontegui, S. Ibánez, S. Martínez-Garaot, M. Modugno, A. del Campo, D. Guéry-Odelin, A. Ruschhaupt, X. Chen, J.G.Muga, Adv. At. Mol. Opt. Phys., 62, 117 (2013). Y. Y Cui, X. Chen, J. G. Muga, J. Phys. Chem. A, 120, 2962 (2016). K. Funo, N. Lambert, F. Nori, C. Flindt, Phys. Rev. Lett. , 150603 (2020). L. Dupays, I. L. Egusquiza, A. del Campo, and A. Chenu, Phys. Rev. Research 2, 033178 (2020). O. Abah, E. Lutz, Phys. Rev. E 98, 032121 (2018). O. Abah, M. Paternostro, Phys. Rev. E 99, 022110 (2019). B. Çakmak and Ö. E. Müstecaplıoğlu, Phys. Rev. E 99, 032108 (2019). K. Funo, N. Lambert, B. Karimi, J. P. Pekola, Y. Masuyama, and F. Nori, Phys. Rev. B 100, 035407 (2019). O. Abah, M. Paternostro, E. Lutz, Phys. Rev. Research 2, 023120 (2020). M. O. Scully, M. S. Zubairy, G. S. Agarwal, and H. Walther, Science , 862 (2003). M. O. Scully, M. S. Zubairy, Quantum Optics (Cambridge Press, London, 1997). A. Hardal, O. Müstecaplıoğlu, Sci Rep 5, 12953 (2015). O. Abah, E. Lutz, Europhys. Lett. , 20001 (2014). J. Rossnagel, O. Abah, F. Schmidt-Kaler, K. Singer, and E. Lutz, Phys. Rev. Lett. 112, 030602 (2014). W. Niedenzu, D. Gelbwaser-Klimovsky, A. G. Kofman, and G. Kurizki, New J. Phys. 18, 083012 (2016). J. Klaers, S. Faelt, A. Imamoglu, and E. Togan, Phys. Rev. X , 031044 (2017). W. Niedenzu, V. Mukherjee, A. Ghosh, A.G. Kofman, G. Kurizki, Nat Commun 9, 165 (2018). J. M. Diaz de la Cruz, M. A. Martin-Delgado, Phys. Rev. A 89, 032327 (2014). M. Beau, J. Jaramillo, A. Del Campo, Entropy 18, 168 (2016). J. Jaramillo, M. Beau, A. Del Campo, New J. Phys. 18, 075019 (2016). Yu-Han Ma, Shan-He Su, and Chang-Pu Sun, Phys. Rev. E 96, 022143 (2017). J. Lekscha, H. Wilming, J. Eisert, and R. Gallego, Phys. Rev. E 97, 022142 (2018). Y. Chen, G. Watanabe, Y. Yu, X. Guan. A. del Campo, npj Quantum Inf. , 88 (2019). D. Gelbwaser-Klimovsky, W. Kopylov, and G. Schaller, Phys. Rev. A 99, 022129 (2019). A. Hartmann, V. Mukherjee, G. B. Mbeng, W. Niedenzu, and W. Lechner, arXiv:2008.09327 (2020). W. Pusz, S. L. Woronowicz, Commun. Math. Phys. , 273–290 (1978). A. Lenard, J. Stat. Phys. 19, 575 (1978). A. E. Allahverdyan, R. Balian, and T. M. Nieuwenhuizen, Europhys. Lett. 67, 565 (2004). N. Erez, G. Gordon, M. Nest, G. Kurizki, Nature 452 (7188), 724–727 (2008). D. Gelbwaser-Klimovsky, N. Erez, R. Alicki, G. Kurizki, Phys. Rev. A 88, 022112 (2013). A. Pozas-Kerstjens, E. G. Brown, and K. V. Hovhannisyan, New J. Phys. , 043034 (2018). P. Abiuso, V. Giovannetti, Phys. Rev. A 99, 052106 (2019). A. Das, V. Mukherjee, Phys. Rev. Research 2, 033083 (2020). A.E. Allahverdyan, R.S Johal, G. Mahler, Phys. Rev. E , 041118 (2008). A. E. Allahverdyan, K. Hovhannisyan, and G. Mahler, Phys. Rev. E , 051129 (2010). M. Campisi, J. Pekola, and R. Fazio, New J. Phys. , 035012 (2015). R. Uzdin, A. Levy, and R. Kosloff, Phys. Rev. X , 031044 (2015). M. Campisi and R. Fazio, J. Phys. A: Math. Theor. , 345002 (2016). L. Buffoni, A. Solfanelli, P. Verrucchi, A. Cuccoli, and M. Campisi, Phys. Rev. Lett. , 070603 (2019). R. Uzdin, A. Levy, R. Kosloff, Entropy 18, 124 (2016). J. Goold, M. Huber, A. Riera, L. del Rio, P. Skrzypczyk, J. Phys. A Math. Theor. 49, 143001 (2016). J. C. Maxwell, Theory of Heat (London: Longmans, 1871). K. Maruyama, F. Nori, V. Vedral, Rev. Mod. Phys. , 1 (2009). L. Szilard, Z. Phys. 53, 840 (1929). R. Landauer, IBM J. Res. Dev. 5, 183 (1961). R. Landauer, Phys. Lett. A 217, 188 (1996). O. Penrose, Foundations of Statistical Mechanics: A Deductive Treatment (Oxford: Pergamon, 1970) C. H. Bennett, Int. J. Theor. Phys. 21, 905 (1982). C. H. Bennett, Stud. Hist. Phil. Mod. Phys. 34, 501 (2003). W. H. Zurek, Maxwell’s demon, Szilard’s engine and quantum measurements Frontiers of Nonequilibrium Statistical Physics ed G T Moore and M O Scully (Boston, MA: Springer) pp 151–61 (1986). L. C. Biedenharn, J. C. Solem, Found Phys 25, 1221(1995). S. Lloyd, Phy. Rev. A 56, 3374 (1997). J. M. R. Parrondo, Chaos 11, 725 (2001). S. W. Kim, T. Sagawa, S. De Liberato and M. Ueda, Phys. Rev. Lett. 106, 070401 (2011). H. Li, J. Zou, J. G. Li, B. Shao, L. A. Wu, Ann. Phys. 327, 2955 (2012). C. Y. Cai, H. Dong, C. P. Sun, Phys. Rev. E 85, 031114 (2012). J. J. Park, K-H Kim, T. Sagawa, S. W. Kim, Phys. Rev. Lett. 111, 230402 (2013). M. Plesch, O. Dahlsten, J. Goold and V. Vedral, Sci. Rep. 4, 6995 (2014). W. H. Zurek, Phys. Rep. 755, 21 (2018). J. Bengtsson, M. N. Tengstrand, A. Wacker, P. Samuelsson, M. Ueda, H. Linke, S. M. Reimann, Phys. Rev. Lett. 120,100601 (2018). A. Aydin, A, A. Sisman, R. Kosloff, Entropy 22, 294 (2020).v C. Elouard, D. A. Herrera-Martí, M. Clusel, A. Auffèveset , npj Quantum Inf 3, 9 (2017). C. Elouard, D. Herrera-Martí, B. Huard, and A. Auffèves, Phys. Rev. Lett. 118, 260603 (2017). M. H. Mohammady and Janet Anders, New J. Phys. 19, 113026 (2017). S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72, 3439 (1994). V. Giovannetti, S. Lloyd, L. Maccone, Phys. Rev. Lett. 96, 010401 (2006). Y. M. Zhang, X. W. Li, W. Yang, and G. R. Jin, Phys. Rev. A 88, 043832 (2013). T. Gefen, F. Jelezko, and A. Retzker, Phys. Rev. A 96, 032310 (2017). A. Dutta, G. Aeppli, B. K. Chakrabarti, U. Divakaran, T. F. Rosenbaum, D. Sen, Quantum phase transitions in transversefield spin models: from statistical physics to quantum information (Cambridge University Press , 2015) P. P. Hofer, J.R. Souquet, and A. A. Clerk, Phys. Rev. B 93, 041418(R) (2016). S. Deffner and E. Lutz, J. Phys. A Math. Theor. 46, 335302 (2013). S. Deffner and S. Campbell, J. Phys. A Math. Theor. 50, 453001 (2017). I. Bengtsson and K. Zyczkowski, Geometry of Quantum States: An Introduction to Quantum Entanglement, 2nd ed. (Cambridge University Press, Cambridge, 2017). M. L. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge,2000). L. Mandelstam, I. Tamm, J. Phys. 9, 249 (1945). L. B. Levitin, Y. Toffoli, Phys. Rev. Lett. 103, 160502 (2009). N. Margolus, L. B. Levitin, Physica D 120, 188 (1998). S. Julia-Farre, T. Salamon, A. Riera, M. N. Bera, and M. Lewenstein, Phys. Rev. Research 2, 023113 (2020). F. C. Binder, S. Vinjanampathy, K. Modi, and J. Goold, New J. Phys. 17, 075015 (2015). T. P. Le, J. Levinsen, K. Modi, M. M. Parish, and F. A. Pollock, Phys. Rev. A 97, 022106 (2018). N. Friis and M. Huber, Quantum 2, 61 (2017). G. M. Andolina, M. Keck, A. Mari, M. Campisi, V. Giovannetti,nand M. Polini, Phys. Rev. Lett. 122, 047702 (2019). D. Rossini, G. M. Andolina, and M. Polini, Phys. Rev. B 100, 115142 (2019). D. Ferraro, M. Campisi, G. M. Andolina, V. Pellegrini, and M. Polini, Phys. Rev. Lett. 120, 117702 (2018). G. M. Andolina, D. Farina, A. Mari, V. Pellegrini, V. Giovannetti, and M. Polini, Phys. Rev. B 98, 205423 (2018). D. Farina, G. M. Andolina, A. Mari, M. Polini, and V. Giovannetti, Phys. Rev. B 99, 035421 (2019). Y. Y. Zhang, T. R. Yang, L. Fu, and X. Wang, Phys. Rev. E 99, 052106 (2019). G. M. Andolina, M. Keck, A. Mari, V. Giovannetti, and M. Polini, Phys. Rev. B 99, 205437 (2019). A. Crescente, M. Carrega, M. Sassetti, D. Ferraro, arXiv:2009.09791(2020). S. Ghosh, T. Chanda, and A. Sen(De) Phys. Rev. A 101, 032115 (2020). D. Rossini, G. M. Andolina, D. Rosa, M. Carrega, M. Polini, arXiv:1912.07234. D. Rosa, D. Rossini, G. M. Andolina, M. Polini, M. Carrega, arXiv:1912.07247. M. Carrega, J. Kim, D. Rosa, arXiv:2007.03551 (2020). F. Caravelli, G. Coulter-De Wit, L. P. García-Pintos, and A. Hamma, Phys. Rev. Research 2, 023095 (2020). F. Barra, Phys. Rev. Lett. 122, 210601 (2019). K. V. Hovhannisyan and A. Imparato, New J. Phys. , 052001 (2019). J. Liu, D. Segal, G. Hanna, J. Phys. Chem. C 123, 30, 18303 (2019). K. V. Hovhannisyan, F. Barra, A. Imparato, Phys. Rev. Research 2, 033413 (2020). J. Q. Quach, W. J. Munro, arXiv:2002.10044(2020). M. Carrega, A. Crescente, D. Ferraro, M. Sassetti, New J. Phys. 22, 083085 (2020). S. Zakavati, F.T.Tabesh, S.Salimi, arXiv:2003.09814(2020). S. Ghosh, T. Chanda, S. Mal, A. Sen De, arXiv:2005.12859(2020). S. Bai, J. An, arXiv:2009.06982(2020). A. Crescente, M. Carrega, M. Sassetti, D. Ferraro, New J. Phys. 22 063057 (2020). T. Purves, T, Short, arXiv:2008.09065(2020). Z. Chen, Phys. Rev. A 71, 052302 (2005). O. Guhne, G. Toth, and H. J. Briegel, New J. Phys. 7, 229 (2005). G. Toth, Phys. Rev. A 85, 022322 (2012). P. Hyllus, W. Laskowski, R. Krischek, C. Schwemmer, W. Wieczorek, H. Weinfurter, L. Pezzé, and A. Smerzi, Phys. Rev. A85, 022321 (2012). H. J. Lipkin, N. Meshkov, and A. J. Glick, Nucl. Phys. A 62, 188 (1965). R. H. Dicke, Phys. Rev. 93, 99 (1954). P. W. Anderson, Phys. Rev. 109, 1492 (1958). R. Nandkishore and D. A. Huse, Annu. Rev. Condens. Matter Phys. 6, 15 (2015). F. Alet and N. Laflorencie, C. R. Phys. 19, 498 (2018). D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Rev. Mod. Phys. 91, 021001 (2019). A. Tavakoli, G. Haack, N. Brunner, and J. B. Brask, Phys. Rev. A 101, 012315 (2020). S. Khandelwal, N. Palazzo, N. Brunner and G. Haack, New J. Phys. 22, 073039 (2020). M. Josefsson and M. Leijnse, Phys. Rev. B 101, 081408(R) (2020).267