MMany-body quantum technologies
Victor Mukherjee ∗ Department of Physical Sciences, IISER Berhampur, Berhampur 760010, India
Uma Divakaran † Department of Physics, Indian Institute of Technology Palakkad, Palakkad, 678557, India
Thermodynamics of quantum systems and associated quantum technologies are rapidly developingfields, which have already delivered several promising results, as well as raised many intriguingquestions. Many-body quantum technologies present new opportunities stemming from many-bodyeffects. At the same time, they pose new challenges related to many-body physics. In this shortreview we discuss some of the recent developments on technologies based on many-body quantumsystems. We mainly focus on many-body effects in quantum thermal machines. We also brieflyaddress the role played by many-body systems in the development of quantum batteries and quantumprobes.
CONTENTS
I. Introduction 1II. Technical details 2A. Dissipative dynamics of a many-body quantum system collectively coupled to a bath 2B. Scaling theory in critical systems 5C. Dissipative dynamics in free-Fermionic systems 6D. Counterdiabatic driving 7III. Thermal machines with collective coupling 9A. Collective effects in Otto engines 9B. Collective effects in continuous thermal machines 11C. Collective effects due to spin statistics 13IV. Interacting many-body quantum thermal machines 13A. Critical engines 13B. Shortcuts to adiabaticity 16C. Quantum advantage in many-body thermal machines 19D. Quantum engines based on localized states 20E. Quantum Szilard engine with interacting Bosons 21V. Other quantum technologies 22A. Quantum batteries 22B. Quantum probes 22VI. Discussion and outlook 23Acknowledgments 23References 23
I. INTRODUCTION
The recent years have witnessed a plethora of theoretical and experimental studies of technologies at the microscopicscales [1, 2] . The laws of quantum mechanics in general play vital roles in dictating the behaviors of systems at the ∗ [email protected] † [email protected] a r X i v : . [ qu a n t - ph ] F e b atomic scales [3]. At the same time, the operation of thermal machines is guided by the laws of thermodynamics,as has been well known since the last two centurie [4, 5]. Naturally, studying the operation of quantum machinesposes the challenge of understanding the laws of thermodynamics in the quantum regime, which has in turn led tothe burgeoning field of quantum thermodynamics [6–18]. The recent remarkable progress in our understanding of thethermodynamics in the quantum regime, and in realizing technologies based on quantum systems, have been largelydriven by the current expertise in experimentally probing and controlling systems at the microscopic scales [19–23].This advancements in experimental know-how has given us unprecedented ability to devise microscopic machines[24–30], which may be guided by the laws of quantum thermodynamics.One of the major aims of the field of quantum thermodynamics is the development of quantum technologies whichcan prove to be significantly beneficial, when compared to the existing technologies based on classical physics. Mostof the works done till now in this field have addressed technologies based on single or few-body quantum systems.On the other hand, achieving significant advantage through quantum technologies in general demands scaling-up ofsuch technologies to many-body systems. This is a highly non-trivial problem, owing to the exponentially increasingdimension of the Hilbert space with system size. However, this also gives us the opportunity to understand thefundamental physics of thermodynamics of many-body quantum systems. In order to address this challenge, we needto develop our understanding of the dynamics of many-body systems driven out of equilibrium, and in presence ofdissipative thermal and non-thermal baths. Recently several works have addressed this issue. For example, masterequations aimed at studying the dynamics of many-body quantum systems in presence of dissipative environments havebeen developed [31–33]. Such studies can help to answer questions regarding several intriguing many-body effectsin presence of dissipative baths, such as topological properties [34, 35], phase transitions [36–38] and many-bodylocalization [39–41] to name a few.There are already several works which give the readers in-depth reviews of different aspects of quantum ther-modynamics, principles of quantum thermal machines and related quantum technologies; for example, see Refs.[8–10, 12, 13, 16–18]. In contrast, in this short non-exhaustive review, we specifically focus on the role played bymany-body systems in different emerging quantum technologies. We shall mainly discuss quantum engines and re-frigerators based on many-body working mediums (WMs), and also briefly address the important role played bymany-body systems in the studies of quantum probes and quantum batteries. The many-body effects may arisedue to collective coupling between a many-body WM and external dissipative baths [42–44], or due to inter-particleinteractions in the WM [45–50]. Several works have focussed on utilizing these many-body effects to design novelquantum technologies [16]. In light of the recent rapid progress in experimental studies of quantum systems, one canenvisage experimental realizations of such machines in very near future, in several existing platforms, such as thosebased on Rydberg atoms [51, 52], ion traps [25], optical lattices [53] and nitrogen vacancy centers in diamonds [28].The review is organized as follows: in Sec. II, we present some of the technical details that would be helpful for thereaders to follow this review, such as dissipative dynamics in presence of collective coupling in Sec. II A, scaling theoryin quantum critical systems in Sec. II B, dissipative dynamics of free-Fermionic systems in Sec. II C and shortcutto adiabaticity using counter-diabatic driving in Sec. II D. Then we discuss quantum engines based on collectivecoupling in Sec. III; we focus on Otto cycles with collective coupling in Sec. III A, while we consider continuousthermal machines in Sec. III B. We also discuss collective effects arising due to spin statistics in Sec. III C. SectionIV deals with quantum thermal machines based on interacting many-body systems; we discuss the effect of criticalityon the operation of quantum thermal machines in Sec. IV A, shortcut to adiabaticity in many-body quantum enginesin Sec. IV B, quantum advantage in quantum engines in Sec. IV C, quantum engines based on localized states in Sec.IV D and quantum Szilard engines in Sec. IV E. Next we briefly discuss other quantum technologies in Sec. V, viz.quantum batteries in Sec. V A and quantum probes in Sec. V B. Finally, we conclude in Sec. VI. II. TECHNICAL DETAILS
In this section we discuss some of the results and tools which can be useful for studying machines based on many-body quantum systems, which are addressed in this review.
A. Dissipative dynamics of a many-body quantum system collectively coupled to a bath
In order to understand the possible advantages offered by collective phenomenon in many-body quantum thermalmachines, let us first delve deeper into the dynamics and heat capacities of quantum systems collectively coupled tothermal baths. We discuss below the dissipative dynamics of N ≥ H S correspondingto the N spins is given by H S = N (cid:88) r =1 ω r J rz , (1)where J rα is the local angular momentum operator associated with the r -th spin, along α = x, y, z direction. In thisreview we take (cid:126) and k B as unity. Here we focus on the case of indistinguishable spins, brought about by ω r = ω ∀ r ,in which case (1) can be written as H S = ω J z , (2)where we have used the collective operators J α = (cid:80) Nr =1 J rα . We consider the system (Eq. (2)) coupled collectively toa thermal bath through an interaction Hamiltonian given by H int = λB ⊗ J x , (3)where B denotes a Hermitian bath operator and λ is the system-bath interaction strength.The collective operators J α gives rise to Dicke states | j, m (cid:105) , which are simultaneous eigenstates of J = J x + J y + J z and J z : J | j, m (cid:105) i = j ( j + 1) | j, m (cid:105) i ; J z | j, m (cid:105) i = m | j, m (cid:105) i . (4)The collective basis {| j, m (cid:105) i } satisfies the constraints: j ∈ [ j ; N s ] , m ∈ [ − j ; j ] , i ∈ [1; l j ], where j = 0 for s ≥ j = 1 / N odd and s = 1 / s being the dimension of each spin. Here l j is the multiplicity of the eigenspacesassociated with the eigenvalue j [54, 55]. We note that the largest possible spin j = N s is unique, formed by thetotally-symmetric N -atom states. Analogous to single particle operators, one can also define the raising and loweringoperators J + and J − , respectively, through the relations: J ± = J x ± i J y J ± | j, m (cid:105) i = (cid:112) ( j ∓ m ) ( j ± m + 1) | j, m ± (cid:105) i . (5)In the limit of weak system-bath coupling ( | λ | (cid:28) ρ = Γ( ω ) (cid:0) J − ρ J + − J + J − ρ (cid:1) + Γ( − ω ) (cid:0) J + ρ J − − J − J + ρ (cid:1) + h.c. , (6)where Γ( ω ) = λ (cid:82) ∞ exp [ iωu ] Tr ( ρ B B ( u ) B ) du . Here ρ B denotes the state of the bath at temperature T = 1 /β , while B ( u ) = exp[ iH B u ] B exp[ iH B u ] is the bath operator in the interaction picture, with respect to the Bath Hamiltonian H B .As one can see from Eq. (5), the raising and lowering operators J ± acting on a state | j, m (cid:105) i do not change thevalue of j . Consequently, for an initial state devoid of any correlation between different eigenspaces of J , the masterequation (6) keeps the dynamics confined within each j eigenspace. Therefore for each j , the dynamics is same as thatof the thermalization of a system consisting of 2 j + 1 non-degenerate energy levels. One can use the master equation(6) to arrive at dynamics of the populations ρ j,m,i = i (cid:104) j, m | ρ | j, m (cid:105) i :˙ ρ j,m,i = G ( ω ) [( j − m )( j + m + 1) ρ j,m +1 ,i − ( j + m )( j − m + 1) ρ j,m,i ]+ G ( − ω ) [( j + m )( j − m + 1) ρ j,m − ,i − ( j − m )( j + m + 1) ρ j,m,i ] , (7)where G ( ω ) = Γ( ω ) + Γ ∗ ( ω ).The system reaches a steady-state ρ ss ( β ), defined by ˙ ρ = 0 for ρ = ρ ss ( β ), at long times. This steady-state maynot be a thermal state. For an initial state satisfying the constraint i (cid:104) j, m | ρ | j, m (cid:105) i (cid:48) = 0 for i (cid:54) = i (cid:48) , we arrive at thesteady state [44] ρ ss N ( β ) = Ns (cid:88) j = J l j (cid:88) i =1 p j,i ρ th j,i ( β ) . (8)The steady-state (8) depends on the initial state ρ through the probabilities p j,i = j (cid:88) m = − j i (cid:104) j, m | ρ | j, m (cid:105) i (9)and ρ th j,i ( β ) = Z j ( β ) − j (cid:88) m = − j e − mωβ | j, m (cid:105) ii (cid:104) j, m | ; Z j ( β ) = j (cid:88) m = − j e − mωβ . (10)In contrast to the collective coupling scenario discussed above, N such spins interacting independently to the thermalbath reaches the direct-product thermal steady state ρ thind = e − ωβ J z Tr e − ωβ J z = ⊗ Nr =1 e − ωβJ rz (cid:80) sm = − s e − mωβ , (11)which is clearly different from the steady-state (Eqs. (8) - (10)) reached through collective coupling.The heat capacity C = ∂E/∂T of a system in a thermal state ρ at temperature T quantifies the change in itsmean energy E = Tr [ H S ρ ] as a function of the change in its temperature. The difference in the steady-states reachedthrough collective and independent coupling indicate that the corresponding heat capacities should be different aswell. Previous studies have shown the importance of specific heat in the context of critical heat engines [50]. On asimilar note, as we discuss below, the heat capacity plays a central role in the performance of quantum heat machinesin presence of collective coupling with thermal baths as well [44].The collective heat capacity C col ( β ) of the spin ensemble in the steady state (8) is given by C col ( β ) = − β ∂E ss ( β ) ∂β = ns (cid:88) j = j l j (cid:88) i =1 p j,i C j ( β ) , (12)where the steady-state energy E ss ( β ) of the spin ensemble is E ss ( β ) = ω Tr( J z ρ ssN ( β )) = ns (cid:88) j = j l j (cid:88) i =1 p j,i e j ( β ); e j ( β ) = ω j (cid:88) m = − j me − mωβ Z j ( β ) , (13)and C j ( β ) = − β ∂e j ( β ) ∂β = ( ωβ ) (cid:34)(cid:18)
12 sinh( ωβ/ (cid:19) − (cid:18) j + 1 / j + 1 / ωβ (cid:19) (cid:35) . (14)One can use Eqs. (12) and (14) to show that the largest heat capacity C col+ ( β ) = C j = ns ( β ) is obtained for p j = ns = 1.Here we note that the multiplicity l j = 1 for j = ns [55, 56]. In the following we compare the best case scenario, C col+ ( β ) to the independent heat capacity C ind ( β ), given by: C ind ( β ) = − β ∂E th ( β ) ∂β = N C j = s ( β ) , (15)where E th ( β ) = ω Tr( J z ρ th ( β )) = ne J = s ( β ) . (16)One can show thatlim ω | β |(cid:29) C col+ ( β ) C ind ( β ) ∼ N − , (17)while lim ω | β |(cid:28) C col+ ( β ) C ind ( β ) ∼ N s + 1 s + 1 + O (cid:2) N ( N ωβ ) (cid:3) . (18)The collective C col+ ( β cr = 1 /T cr ) becomes equal to the independent heat capacity C ind ( β cr = 1 /T cr ) at a critical bathtemperature T cr , given by T cr ( n, s ) ω (cid:39) (cid:18) N s ( s + 1) + 112 (cid:19) / . (19) B. Scaling theory in critical systems
Till now we have considered many-body effects arising due to the collective coupling between a many-body systemand a thermal bath. However, many-body effects can also arise due to the presence of interactions between thesubsystems of a many-body system. An intriguing phenomenon arising in such interacting many-body systems, isthat of phase transitions. Phase transitions are associated with phases characterized by different symmetries separatedby critical points; classical phase transitions occur due to thermal fluctuations at non-zero temperatures [57], whilequantum phase transitions result from quantum fluctuations at absolute zero temperature [58]. Phase transitionshave generated significant interest in the field of quantum thermodynamics, owing to the divergences of length andtime scales close to critical points [50, 59–63]. These divergences in turn lead to the presence of universal features,through the general scaling relations described below. Such a universality is also reflected in the non-equilibriumdynamics arising due to the dynamics of systems driven through quantum phase transitions, which will be discussedin this section [64, 65].Quantum phase transitions are zero temperature phase transitions where the nature of the ground state changesabruptly at a quantum critical point (QCP), due to change in some parameter g characterizing the Hamiltonian ofthe system [58]. The QCP can also be idenitified by the vanishing of the gap between the ground state and the firstexcited state at the quantum critical point g = g c . It can be shown that for a second order quantum phase transition,the correlation length ξ diverges following a power law ξ ∼ | g − g c | − ν (20)when the critical point g c is approached. Similarly, the correlation time ξ τ also diverges with a power law as follows: ξ τ ∼ | g − g c | − νz . (21)Here, ν and z are the correlation length and dynamical critical exponents associated with the critical point.In the last two decades, several works have addressed the non-equilibrium dynamics across a quantum critical point[64–66]. In particular, researchers started working on understanding the effect of critical point when a system, initallyprepared in the ground state, is driven across a critical point g c by varying g linearly with speed v [67, 68]. When thesystem is far away from the critical point ( g (cid:29) g c ), the correlation time is small compared to the time scale in whichthe Hamiltonian is varied. This enables the system to follow the instantaneous ground state. As soon as the relaxationtime becomes comparable to this scale, the system is no longer able to follow the instantaneous ground state andgets excited. The universal relation connecting density of excitations n to the speed v with which the Hamiltonian isvaried and the critical exponents associated with the quantum critical point crossed is known as Kibble Zurek scaling,and is given by n ∼ v νdνz +1 . (22)where d is the dimensionality of the system. As expected, the defects decreases when the speed decreases. Thenon-adiabatic excitations generated due to the dynamics close to QCP results in non-zero excitation energy E ex aswell, i.e., the energy of the system in excess to its ground state energy. For quenches ending at the QCP, we have[37, 69] E ex ∼ v ν ( d + z ) νz +1 . (23)On the other hand, for quenches across the critical point, in general E ex does not follow universal scaling form.However, for systems in which the excitation energy is proportional to the density of excitations, such as in theone-dimensional transverse Ising model discussed below, we have E ex ∼ v νdνz +1 . (24)The universality seen in the non-equilibrium dynamics attracted lot of attention of the scientists opening a plethoraof papers in the related subject.A prototypical model studied to exemplify critical phenomena in quantum systems is transverse Ising model givenby H ( t ) = − J (cid:88) i σ xi σ xi +1 − h ( t ) (cid:88) i σ zi . (25)Here J denotes the interaction strength between any two nearest-neighbor spins, h ( t ) is a uniform time-dependentmagnetic field along the transverse ( z ) direction and σ αi denotes the Pauli matrix along α = x, y, z axis, correspondingto the spin at site i . The excitation spectrum of this Hamiltonian can be obtained by mapping the Pauli matrices toJordan Wigner (JW) fermions c i [70–72], where the transformation equation is given by σ − i = (cid:0) e iπ (cid:80) j
1= 0 .
01= 0 . Figure 1. The excess energy E ex as a function of τ . The data with solid circles correspond to κ = 0 case (or no bath case)which shows KZ scaling given by τ − / with ν = 1 , z = 1 for transverse Ising model. As κ is increased (Cf, Eq. (31)), moredefects are generated for large τ resulting in a minimum in the E ex − τ graph . Reproduced from Ref. [31]. Continuing the discussion in the previous section, we now consider the dynamics of many-body free-Fermionicsystems similar to that given in Eq. 27, but now in presence of a dissipative bath [31]. Such an analysis is importantfrom the perspective of various many-body quantum technologies, such as adiabatic quantum computation and quan-tum annealing where the interaction of the time dependent Hamiltonian with the environment is unavoidable due tolonger time scales involved, resulting in dissipation and decoherence. We start from the most general time dependentHamiltonian having the following quadratic form in terms of fermions H ( t ) = (cid:88) m,n [ c † m A m,n ( t ) c n + 12 ( c † m B m,n c † n + hermitian conjugate)] (29)where A, B are symmetric and antisymmetric matrices, respectively, with elements depending upon the parametersof the original Hamiltonian. Many spin dependent Hamiltonians like transverse Ising model, Kitaev model, XY modelin presence of transverse field, and others take this form under appropriate transformation.As expected, the general equation for evolution of the density matrix would include the unitary evolution as wellas a term involving system-bath coupling or the dissipative term D [ ρ ]: ∂ρ∂t = − i [ H ( t ) , ρ ] + D [ ρ ] (30)We now discuss the effect of presence of a critical point in the time evolution of systems which are coupled to abath. For simplicity the bath used is Markovian bath where the dissipation term can be written in Lindblad form D [ ρ ] = (cid:88) n κ n ( L n ρL † n − { ρ, L † n L n } ) (31)where L n are local Lindblad operators describing the environment, and κ n are the site dependent coefficients relatedto system-bath coupling strength [54].In Ref. [31], the authors have considered three different types of Lindblad operators, namely, (i) L n = c † n , (ii) L n = c n , (iii) L n = c † n c n . Here, we present the results corresponding to the first type of bath with L n = c † n . Such abath helps in studying the competition between unitary dynamics and baths in an almost exact way. As mentionedbefore, the competition arises because defects due to Kibble Zurek will decrease when the quench time scale τ isincreased. But this will cause the system to interact for more time with the environment resulting in increased defectgeneration due to environment. One must also note that such a bath need not take the system to a thermalized state,but to some steady state. The example Hamiltonian used to demonstrate these competitions is that of transverseIsing model given in Eq. 25, where the transverse field h is varied as t/τ , so that the speed v with which the quantumcritical point h = J is crossed, is given by 1 /τ . The quantity studied in Fig. 1 is the excitation energy E ex . Forsimplicity, κ n in the dissipative term is taken as a constant κ independent of site index n . As seen in the figure andexplained in the text above, there is a non-monotonic behavior in the defects generated with an optimal value of τ for which E ex takes its minimum value. Different scalings have been explained assuming that the defects generatedduring the unitary evolution and while interacting with the environment are unrelated, i.e., the total defects createdare sum of those due to unitary evolution and interaction with the environment. D. Counterdiabatic driving
The performance of quantum thermal machines is quantified by efficiency and output power in case of engines, andrate of refrigeration in case of refrigerators. As has been known for classical as well as quantum thermal machines,the maximum efficiency is bounded by the Carnot limit, through the second law of thermodynamics [4, 5, 74]. Ingeneral, such efficiencies are reached only in the absence of non-adiabatic excitations, i.e., for long cycle time limit.However, in the absence of any control, such high efficiencies are achieved at the price of vanishing output power orrefrigeration rate, which are inversely proportional to the total cycle period. Consequently, application of shortcuts toadiabaticity (STA) to devise control protocols aimed at enhancing the power, refrigeration rate and efficiency in finite-time thermal machines through suppression of non-adiabatic excitations, has gained a lot of attention lately. STAhas been developed and studied thoroughly in the context of closed [75–81] and open [82, 83] quantum and classicalsystems in presence of time-dependent Hamiltonians, and has also been implemented experimentally [20, 84]. In thelast few years, it has also been extended to the field of quantum thermodynamics [85], and proved to be immenselysuccessful in enhancing the performance of a wide class of quantum thermal machines [86, 87]. Application of STAto enhance the performance of quantum thermal machines have been accompanied by several interesting questionregarding the cost of implementation of such control protocols as well [88–90].Scaling-up of quantum technologies demands analysis of many-particle quantum machines. However, the expo-nentially increasing size of the Hilbert space significantly increases the complexity of the problem. This increasedcomplexity is reflected in the application of STA in many-body quantum systems subjected to time-dependent Hamil-tonians as well. For example, finding the exact STA protocol may involve the knowledge of the complete energyspectrum of a system, which in general can be highly non-trivial for interacting many-body systems [77, 91]. Thisissue can be tackled through the method of approximate counterdiabatic driving [92–94], which does not requireexplicit knowledge of the many-body eigenstates. Recently, STA through counterdiabatic driving has proved to behighly beneficial for enhancing the performance of many-body quantum thermal machines as well [49, 95, 96].Here following Refs. [92] and [93], we discuss the derivation of the approximate gauge potential aimed at constructingthe corresponding counderdiabatic Hamiltonian, which can completely eliminate, or significantly reduce, non-adiabaticexcitations arising due to finite rate of change of the original Hamiltonian. To this end, we consider a state | ψ ( t ) (cid:105) evolving under a time-dependent Hamiltonian H ( ϑ ( t )) following the Schr¨odinger equation: i ∂∂t | ψ ( t ) (cid:105) = H ( ϑ ( t )) | ψ ( t ) (cid:105) . (32)Here the parameter ϑ ( t ) introduces time-dependence in the Hamiltonian. In a frame rotating w.r.t. to the unitaryoperator U ( ϑ ( t )) = exp[ − i/ T (cid:82) t H ( ϑ ( t (cid:48) ) dt (cid:48) )], Eq. (32) can be written as i ∂∂t | ˜ ψ ( t ) (cid:105) = ˜ H m | ˜ ψ ( t ) (cid:105) ˜ H m = ˜ H ( ϑ ( t )) − ˙ ϑ ˜ A ϑ , (33)where T denotes the time-ordering operator,˜ H ( ϑ ( t )) = U † H ( ϑ ( t )) U (34)and | ˜ ψ ( t ) (cid:105) = U † | ψ (cid:105) . (35)The adiabatic gauge potential ˜ A ϑ is given by ˜ A ϑ = U † A ϑ U , where A ϑ = i∂ ϑ . According to the construction (Eq.(35)) above, ˜ H ( ϑ ( t )) is diagonal in its instantaneous energy eigenbasis. Consequently, any non-adiabatic excitationarises due to the term ˙ ϑ ˜ A ϑ in Eq. (33). Therefore in order to eliminate the non-adiabatic excitations, it suffices toadd ˙ ϑ A ϑ to the original Hamiltonian H ( ϑ ( t )), such that we finally arrive at the STA Hamiltonian H STA = H + H CD , (36)where the counter-diabatic Hamiltonian is given by H CD = ˙ ϑ A ϑ . (37)One can find the exact counter-diabatic Hamiltonian, in terms of the instantaneous eigenbasis | m (cid:105) and the respectiveeigenvalues E m , through the relation (cid:104) m |A ϑ | n (cid:105) = i (cid:104) m | ∂ ϑ H | n (cid:105) E n − E m ; m (cid:54) = n. (38)Equation (38) requires complete knowledge of the instantaneous eigenstates at all times, and therefore can be imprac-tical to implement, specially in many-body systems. Therefore we aim to find an approximate solution A ∗ ϑ to A ϑ ,which would reduce the non-adiabatic excitations significantly, while being implementable in experimental setups.The choice of the specific form of A ∗ ϑ depends on the constraints involved, such as the range of interactions allowedin the control terms. Accordingly, we define the operator M ϑ through the relation i∂ ϑ H = [ A ϑ , H ] − iM ϑ , (39)such that M ϑ = − (cid:88) n ∂E n ( ϑ ) ∂ϑ | n ( ϑ ) (cid:105)(cid:104) n ( ϑ ) | . (40)Now let us define the Hermitian operator G ϑ ( A ∗ ϑ ) = ∂ ϑ H + i [ A ∗ ϑ , H ] . (41)As one can see from Eqs. (39) and (41), G ϑ ( A ϑ ) = − M ϑ . Next we employ the variational principle method; insteadof solving for A ϑ directly, which would require detailed knowledge of the spectrum (see Eq. (38)), we minimize theoperator distance D ( A ∗ ϑ ) = Tr (cid:104) ( G ϑ ( A ∗ ϑ ) + M ϑ ) (cid:105) (42)between G ϑ ( A ∗ ϑ ) and − M ϑ , with respect to the parameter A ∗ ϑ . Clearly, D ( A ∗ ϑ ) assumes the minimal value (zero)for A ∗ ϑ = A ϑ . One can show that minimizing this operator distance D ( A ∗ ϑ ) is equivalent to minimizing the term S ( A ∗ ϑ ) = Tr (cid:2) G ϑ ( A ∗ ϑ ) (cid:3) , (43)i.e., finding the solution for the equation δ S ( A ∗ ϑ ) δ A ∗ ϑ = 0 , (44)where δ denotes the functional derivative. Therefore to summarize, finding an approximate counter-diabatic Hamil-tonian (see Eq. (37)) boils down to finding the solution to the above equation (44). III. THERMAL MACHINES WITH COLLECTIVE COUPLING
As with classical thermodynamics where studies on thermal machines such as heat engines and refrigerators [6, 7, 97–109] are inherenctly connected with fundamental principles of thermodynamics [11, 14, 15], the corresponding quantumregime is no different. We shall now use the technical details presented above to discuss specific examples of some ofthe many-body quantum machines studied in the literature. In this section we shall focus on many-body quantummachines where non-trivial co-operative effects arise due to collective coupling between the many-body WM and thethermal baths.
A. Collective effects in Otto engines
Quantum engines in general consist of a central system, termed as the working medium (WM), which is subjectedto time-dependent Hamiltonians, and coupled to a cold thermal bath at a temperature T c and a hot thermal bath ata temperature T h > T c . The setup is engineered so as to convert a part of the thermal energy of the hot bath intousable output work, following the laws of thermodynamics [4, 5, 8]. One of the most widely studied quantum thermalengines is the Otto engine, which is described by the following four strokes (see Fig. 2): A BCD
Figure 2. Schematic diagram showing the Otto cycle in the entropy ( S ) - ϑ plane. A → B and C → D denote the unitary strokes,wherein the Hamiltonian parameter ϑ is tuned between ϑ c and ϑ h . Heat Q h ( Q c ) flows between the hot (cold) bath and theWM during the non-unitary stroke B → C (D → A). • Stroke 1, A → B: We start with the WM in thermal equilibrium with a cold bath at temperature T c . The WMHamiltonian H S ( ϑ ) is changed as a function of time from H S ( ϑ c ) to H S ( ϑ h ) in a time τ , where as before, ϑ isa parameter which characterizes the Hamiltonian. The system evolves isentropically during this unitary stroke. • Stroke 2, B → C: The WM is coupled to a hot thermal bath at temperature T h for a time duration τ . TheHamilotnian is kept constant during this non-unitary stroke. For a stroke duration τ of the order of the0thermalization time τ th or longer, the WM reaches the steady state corresponding to the hot bath. The heatflow Q h between the WM and the hot bath is given by Q h = E C − E B , (45)where E j is the energy of the WM at point j = A , B , C , D (see Fig. 2). • Stroke 3, C → D: The WM is decoupled from the thermal bath, following which, the Hamiltonian is changedfrom H S ( ϑ h ) to H S ( ϑ c ) in a time duration τ . As for the stroke 1, the entropy remains constant during thisunitary stroke. • Stroke 4, D → A: The WM is coupled to the cold thermal bath at temperature T c for a time duration τ . Asfor the second stroke, τ (cid:38) τ th allows the WM to reach the steady-state corresponding to the cold bath, thuscompleting the cycle. The heat exchanged Q c during this non-unitary stroke is given by Q c = E A − E D . (46)The output of the thermal machine described above is quantified by the work W = Q h + Q c , (47)power P = Wτ cyc (48)and the efficiency η = − W Q h , (49)where τ cyc = τ + τ + τ + τ is the total cycle duration.Following Ref. [44], we now discuss the operation of an Otto engine consisting of a N-spin WM, collectively coupledto hot and cold thermal baths, during the respective non-unitary strokes. The WM is in presence of the Hamiltonian(see Eq. (2)) H S ( ϑ ( t )) = ϑ ( t ) ω J z . (50)Here ω is a time-independent constant, while the parameter ϑ ( t ) introduces time-dependence in the Hamiltonianduring the unitary strokes, and assumes the constant value ϑ h ( ϑ c ) during stroke 2 (stroke 4). For the spins interactingcollectively with the thermal baths during the non-unitary strokes 2 and 4 (see Eq. (3)), the corresponding steady-states are described in Sec. II A (See Eqs. (8) and (10)). For comparison, we also consider below the case whereeach spin interacts independently with a bath during a non-unitary stroke; in this case the spin ensemble reachesthe usual thermal equilibrium state ρ th ( T x , ϑ x ) = Z − ( β x , ϑ x ) e − β x H S ( ϑ x ) , with β x = 1 /T x , x = h , c and Z ( T x , ϑ x ) =Tr e − β x H S ( ϑ x ) , at the end of a non-unitary stroke.In Ref. [44], the authors considered the Otto cycle described above to show that the work output per cycle can berelated to the specific heat of the WM through the relation W = ∆ ηϑ ( β c − β h ) C ( θ h ) θ h + O (cid:0) ∆ η (cid:1) . (51)Here θ x = ϑ x β x , ∆ η = η c − η = ϑ c /ϑ h − β h /β c is the difference between the Carnot efficiency η c = 1 − β h /β c andthe actual efficiency, which is considered to be small here, and C ( θ h ) denotes the collective or the independent heatcapacity, depending on whether the spins interact collectively or independently with the baths.As discussed in Sec, II A, in case of collective WM-bath coupling, the steady-state ρ ss N ( β ), and hence the cor-responding specific heat C col ( β ), depend non-trivially on the initial state ρ . Therefore in order to focus on themaximum possible advantage obtained through collective coupling, we consider the best case scenario for the col-lective spin machine, which corresponds to ρ belonging to the symmetrical subspace, i.e., j = N s and p j = Ns = 1.Consequently we have C col+ ( β ) = C j = Ns ( β ), so that we need to compare W col+ ≈ ∆ ηλ h ( β c − β h ) C j = Ns ( θ h ) θ h with W ind ≈ ∆ ηλ h ( β c − β h ) N C j = s ( θ h ) θ h (see Eq. (51)). Evaluating C j ( θ h ) /θ h , one can show that keeping ∆ η , β c − β h and1 ϑ h fixed, both W ind and W col assume maximum values for β h →
0, such that one arrives at the following relations: W ind ≤ W ind max := ∆ ηϑ β c ω N (cid:104) (2 s + 1) − (cid:105) (52)and W col+ ≤ W col max := ∆ ηϑ β c ω (cid:104) (2 N s + 1) − (cid:105) = N s + 1 s + 1 W ind max , (53)where we have neglected terms of the order of O (∆ η ) or smaller.Let us now focus on the more practical scenario of finite bath temperature T h ; as one can infer from Eqs. (19) and(51), for a fixed finite temperature T h , increasing the size N of the WM increases the work output W col for small N ,until it reaches the critical number N = N cr (cid:39) T h /ωϑ h ) − / s ( s +1) . Beyond this size of the WM, the independent-spinengine performs better than the collective-spin one, if we focus solely on the work output per cycle. On a similar note,keeping the bath temperatures constant, the collective engine performs better than the independent engine only for ϑ c < ϑ h , cr (cid:39) T h √ / (cid:104) ω (cid:112) N s ( s + 1) + 1 (cid:105) , in terms of output work per cycle.However, in order to understand the advantage offered by collective engines, one needs to focus on the output power,instead of output work. One can use Eq. (7) to show that collective coupling between the WM and a bath leadsto faster dynamics. For a system starting from a thermal state at inverse temperature β with ω | β | (cid:29)
1, collectivesystem-bath interaction shortens the equilibration timescale by at least N times, as compared to that obtained inpresence of independent dissipation. This collective-coupling induced speed-up in equilibration in turn translates toenhancement in power output of collective engines as compared to their independent counterpart, quantified by theratio lim T h (cid:29) ϑ h T cr ( N,s ) P col+ P ind ∼ N ( N s + 1) s + 1 . (54)On the other hand, for fixed T h and N (cid:29) N cr ( T h , λ h ), the advantage offered by faster equilibration for a collectiveheat engine is cancelled out by the less work output per cycle, so that the output powers of the two machines becomeequivalent.The above results are derived for Otto cycles with long durations of thermalizations strokes ( τ , τ (cid:38) τ th ), whichallows the WM to reach steady states at the end of the non-unitary strokes. Interestingly, the advantage due tocollective effects is present in finite-time collective heat engines as well, brought about by τ , τ < τ th [43]. In thiscase, the short durations of the non-unitary strokes do not allow the WM to reach the corresponding steady-states.Nevertheless, the setup operates in a limit cycle, albeit with less output work per cycle. However, the shorter durationof each cycle eventually leads to enhanced power output, for collective, as well as independent engines. For high enoughtemperature of the hot bath, the power scales as N for the collective engine, as opposed to a linear scaling obtained incase of independent engines. In contrast, the shorter duration of the non-unitary strokes reduce the work output andheat input equally, thereby keeping the the efficiency unchanged. Consequently, in terms of power output, collectiveheat engines are more beneficial than independent heat engines for τ , τ (cid:46) τ th . We note that in general the regime τ , τ > τ th is detrimental for operation of heat engines, since the work output per cycle increases with increasing τ , τ only for τ , τ ≤ τ th . B. Collective effects in continuous thermal machines
Till now we have considered dissipation in presence of time-independent Hamiltonians only. However, analysisof several thermal machines may involve time-dependent Hamiltonians as well. For example, continuous thermalmachines operate in presence of periodically modulated WM Hamiltonian, while they are simultaneously coupledto a hot and a cold thermal bath [9, 12, 110]. In contrast to the stroke thermal machines, one does not need torepeatedly couple and decouple the WM with the thermal baths, in order to operate the machine. Instead, herewe consider spectral separation of baths, which eventually leads to non-zero output power [111]. Below we studythe dissipative dynamics of a many-body quantum system subjected to a periodically modulated Hamiltonian, andcollectively coupled to a hot and a cold thermal bath [42].As before (see Eq. (2)), we consider indistinguishable spins in presence of the Hamiltonian H S = ω ( t ) J z . (55)However, in contrast to the Otto cycle discussed in the previous section, we now consider ω ( t ) modulated periodically2in time at a frequency Ω, such that ω ( t + τ cyc ) = ω ( t ), τ cyc = 2 π/ Ω being the time period of modulation. Secularapproximation demands the cycle period τ cyc to be much less than the thermalization time scale [54]. Further, weassume weak system-bath coupling, such that one can apply the Born, Markov and secular approximations to arriveat the master equation governing the dynamics in the interaction picture [9, 12, 110–112]:˙ ρ = (cid:88) v ∈{ c,h } (cid:88) q ∈ Z L v,q ρ, (56)where L v,q ρ := 12 P ( q ) G v ( ω + q Ω) D [ J − ] ρ + 12 P ( q ) G v ( ω + q Ω) e − β v ( ω + q Ω) D [ J + ] ρ. (57)Here ω = τ cyc (cid:82) τ cyc ω ( t ) dt is the bare (unperturbed) frequency of each spin, G v ( ν ) denotes the bath spectral functionat frequency ν , for the v -th ( v = h, c ) bath, and we have considered the KMS condition G v ( − ν ) = G v ( ν ) exp [ − β v ν ][54]. P ( q ) denotes the weight corresponding to the q -th harmonic, and is given by P ( q ) = (cid:12)(cid:12)(cid:12)(cid:12) τ cyc (cid:90) τ cyc exp (cid:20) i (cid:90) t ( ω ( t (cid:48) ) − ω ) dt (cid:48) (cid:21) e − iq Ω t dt (cid:12)(cid:12)(cid:12)(cid:12) . (58)In Ref, [42], the authors considered a WM comprised of N spin-1 / G c ( ν ) ≈ ν ≥ ω ,G h ( ν ) ≈ ν ≤ ω . (59)In such a continuous engine, the collective output power P col and its counterpart P ind := N P (cid:0) (cid:1) , established by N spin-1 /
2s independently coupled to the thermal baths, follow the relationlim β eff ω →∞ P col P ind = 1 (60)in the low-temperature regime, whilelim β eff ω → P col P ind = N + 23 (61)in the high-temperature regime. Here the inverse effective temperature β eff is defined through the relationexp ( − β eff ω ) := (cid:80) v ∈{ c,h } (cid:80) q ∈ Z P ( q ) G v ( ω + q Ω) e − β i ( ω + q Ω) (cid:80) v ∈{ c,h } P ( q ) G v ( ω + q Ω) . (62)As one can see from Eq. (61), a superradiant scaling behaviour P col ∼ N P ind = N P (cid:0) (cid:1) is exhibited at sufficientlyhigh effective temperatures, when the spin- N/ β eff (see Eq. (62)) one arrives at the saturation relationlim N →∞ P col P ind = coth (cid:18) β eff ω (cid:19) . (63)As seen from Eq. (63), P col → P ind in the low effective-temperature regime β eff ω (cid:29)
1, such that collective couplingdoes not provide any advantage as compared to the independent coupling in this case, even for large particle numbers.In contrast, the rhs of Eq. (63) diverges as 2( β eff ω ) − in the high-temperature regime of β eff ω →
0, implying theexistence of significant enhancement in power output in this regime.Let us compare Eq. (61) with the equivalent result in case of stroke engines, viz. Eq. (54). In the case of strokeengines, collective effects lead to an enhancement of the order of N at high temperatures, which is N times higherthan the enhancement obtained in case of continuous engines (see Eq. (61)). The additional enhancement in case ofstroke engines stem from the reduction in thermalization times by a factor of N , due to collective coupling betweenthe WM and the thermal baths. This effect is not present in continuous engines, which lack any thermalization stroke[44].Finally, one can show that the advantage offered by collective coupling is present in the refrigerator regime as well,3where the collective coupling leads to enhanced rate of refrigeration of the cold bath. C. Collective effects due to spin statistics
Spin statistics of a many-body WM can also lead to non-trivial effects of quantum thermal machines [113]. Recentlyit has been shown that collective effects arising due to the Bosonic statistics of indistinguishable particles can also leadto enhancement in the performance of quantum thermal machines [114]. When multiple indistinguishable Bosonicwork resources are coupled to an external system, the output of such a setup, quantified by the internal energy changeof the external system, exhibits an enhancement, as compared to when the setup consists of distinguishable workresources.
IV. INTERACTING MANY-BODY QUANTUM THERMAL MACHINES
In this section we focus on quantum thermal machines with WMs comprised of interacting many-body systems.Different platforms have been used to study such machines, for example, Rydberg atoms [115], multiferroic chain[116], etc. Such WMs allow us to study quantum thermal machines in presence of several many-body effects, in-cluding topological phase transitions [60, 117, 118], superfluid to insulating phase transition [63] and time-translationsymmetry breaking [119], to name a few. Here we discuss a few such quantum engines studied recently, to highlightthe non-trivial role played by inter-particle interactions in the operation of quantum machines.
A. Critical engines
Criticality is a vibrant field of study both in classical as well as quantum condensed matter physics, owing tothe divergence of different parameters close to critical points, and the resultant universality [57, 58]. Therefore itis no surprise that quantum machines with critical WMs have received a significant amount of attention as well[50, 59–63, 120, 121].As mentioned earlier, the maximum efficiency of an engine is bounded by the Carnot limit, which usually occursin engines operating in infinite time, such that the Carnot limit is reached only at the cost of zero power output P . Consequently, one of the major challenges in quantum thermodynamics, is to design quantum engines which candeliver finite power, even as the efficiency approaches the Carnot limit. In Ref. [50], the authors addressed thisproblem by focussing on critical heat engines; they showed that the divergence of specific heat close to criticality canallow us to design quantum engines which can operate infinitesimally close to the Carnot efficiency η C , but withoutsacrificing the output power. The authors quantified the operation of an engine as˙Π = P ∆ η , (64)where ∆ η = η C − η is the deviation of the efficiency η from the Carnot limit η C . For N identical engines working inparallel, the total output power P ∼ N , while η ∼ N . Consequently, we have ˙Π ∼ N . However thislinear scaling with system size N is obtained at the cost of larger resources (larger number of independent engines),and therefore does not signify any advantage obtained due to many-body effects. On the other hand, let us considera quantum engine with a WM comprised of N interacting particles. A super-extensive scaling of ˙Π ∼ N a with thesystem size, signified by a > η ∼ N − a for P ∼ N , thus implying one can reach Carnot efficiencyby increasing N , without compromising on the average power per particle. This can indeed be the case for quantummachines operated close to a second order phase transitions. The total work output W per cycle of an Otto engine isgiven by W = Q + Q , (65)where Q j = C ∆ T j is the heat exchange during the j = 2 , C and the effective change in temperature ∆ T j during the j th stroke. In order to design an engine which can harnessthe advantage provided by second order phase transitions, the authors of Ref. [50] considered an Otto cycle in whichthe temperature of the hot bath coincides with the critical temperature of the WM. A second order phase transitionis accompanied by a diverging specific heat C ∼ N α/νd , and hence diverging work output. On the other hand,critical slowing down close to the phase transitions implies the time scale τ cyc for the cycle varies as τ cyc ∼ N z/d .4 UnitaryStrokeUnitaryStrokeUnitaryStrokeUnitaryStroke
ABC D
Figure 3. Schematic diagram of a many-body Otto cycle close to quantum phase transition. The WM is driven across quantumcritical points during the unitary strokes A → B and C → D. Energy Q in flows from an energizing bath B E to the WM duringthe non-unitary stroke B → C, while energy Q out flows from the WM to a dissipative relaxing bath B D during the non-unitarystroke D → A. (After Ref. [62])
Combining the above results, for ∆ η (cid:28)
1, one arrives at the universal relation˙Π ∼ N a a = α − νzνd , (66)where α, ν and z are respectively, the specific heat, correlation length and dynamical critical exponents, while d denotes the dimensionality of the WM. The universal relation Eq. (66) allows us to choose the WM wisely, such thatone can operate it at the Carnot efficiency, without significantly compromising on the power. For example, ν and d are positive. Therefore α > νz implies one can asymptotically approach the Carnot efficiency at non-zero P , whilethe stronger condition α − νz ≥
1, which can happen for example in Dy Ti O , ensures that one can reach the Carnotefficiency asymptotically, without compromising on the output power per particle [50].Recently, universal behaviors in finite-time quantum engines have also been studied close to quantum phase tran-sitions. Classical phase transitions occur due to thermal fluctuations. In contrast, quantum phase transitions areaccompanied by quantum fluctuations, which arise due to vanishing energy gaps close to quantum critical points [58].Consequently, signatures of quantum phase transitions, such as universality, are very fragile in presence of thermalfluctuations, at non-zero temperatures [36, 122]. However, in Ref. [62] the authors used Kibble-Zurek mechanism[67, 68, 123, 124] to show that universal behaviors may persist in appropriately designed finite-time many-body Ottoengines operated close to criticality, in spite of the presence of thermal fluctuations during the non-unitary strokes (seeSec. II B). The authors considered a many-body Otto engine, which is driven across quantum critical points duringthe unitary stroke 1 (A → B) at a rate τ − for a time interval τ , followed by a non-unitary stroke 2 (B → C), duringwhich input energy Q in is provided by a dissipative, but not necessarily thermal, energizing bath B E ; unitary stroke3 (C → D) during which the WM is driven back across the quantum critical points at a rate τ − for a time interval τ , where τ may not be equal to τ ; and finally, a non-unitary stroke 4 (D → A) to complete the cycle, during whichenergy Q out flows from the WM to a dissipative relaxing bath B D (see Fig. 3). The dynamics during the non-unitarystrokes can be modelled following Sec. II C. The output of such an engine shows universal scaling forms with respectto the duration of the unitary stroke 1 (A → B), provided the engine satisfies the following general conditions: • The relaxing bath B D takes the WM close to its ground state. For example, B D can be a cold thermal bath attemperature T c (cid:28) E , E being the typical energy scale of the WM during the stroke 4 (D → A).5
100 150 300 τ W − W ∞ (a)(b)
200 400 τ η
100 1000 τ | P | Figure 4. Work output follows universal Kibble-Zurek scaling (Eq. (68)) in a quantum engine modelled with a transverse Isingchain WM, driven across quantum critical points. The points are the numerical values and red solid line corresponds to τ − / .For transverse Ising model, d = ν = z = 1. Inset (a): Variation of η with τ . (b) Variation of Power with τ . The green dashedline corresponds to 1 /τ scaling, points represent numerical data and solid line is the analytical expression. The parametersused are: L = 100 , h = 70 , h = − , τ = 0 . , µ (cid:48) E = 1 , µ E = 0 . , µ (cid:48) R = 0 , µ R = 1 with W ∞ =-6481.205. (After Ref. [62]) • The WM is driven at a finite rate τ − across a quantum critical point (or points) during the unitary stroke D → A, i.e., τ is finite. • The energizing bath B E takes the WM to a unique steady state, which does not depend of the state of the WMat the beginning of the stroke 2. • The state of the WM does not change appreciably during the unitary stroke 3 (C → D). This can result forexample if the energizing bath B E takes the WM to a steady state with high entropy. Therefore analogous to B D , B E can be modelled by a hot thermal bath at temperature T h (cid:29) E , where E is the typical energy scaleof the WM during the stroke 2 (B → C). Alternatively we may consider fast quench during the stroke 3 (i.e., τ → τ , given by [37, 62, 69] W − W ∞ ∼ τ − ν ( d + z ) νz +1 , (67)where we have considered the stroke 1 (A → B) ends at a quantum critical point at B. Here W ∞ is the work output ininfinite time Otto cycles operating in the limit τ → ∞ . Alternatively, for unitary strokes which cross critical points,the work output is not universal in general. However, for systems and quench protocols in which the excitation energy E ex during the stroke 1 is proportional to the density of defects, such as in transverse Ising model WM, the scaling(67) is modified as (see Fig. 4) W − W ∞ ∼ τ − νdνz +1 . (68)Furthermore, in the limit where τ is the most dominant time scale of one cycle such that τ cyc ≈ τ , one can use (67)and (68) to derive a scaling relation for the output power PP = Wτ cyc ≈ W ∞ τ + Rτ − νd + xνz +1 νz +1 , (69)where R is a WM-dependent constant. Here x = 1 corresponds to crossing the critical point, while x = 2 is for Bbeing a critical point. The optimal quench rate τ − = τ − delivering the maximum power can be found from the6condition ∂ P ∂τ (cid:12)(cid:12) τ opt = 0 which yields τ opt = (cid:20) R ( νd + xνz + 1) | W ∞ | ( νz + 1) (cid:21) ( νz +1) / [ νd +( x − νz ] , (70)with the corresponding efficiency ˆ η at maximum power beingˆ η = − W ∞ + E ex ( τ opt ) Q ∞ in − E ex ( τ opt ) . (71)Here Q ∞ in is the input energy corresponding to the infinite time Otto engine limit of τ → ∞ .One can use (68)-(71) to choose WMs with appropriate critical exponents and dimensionality, so as to designoptimal many-body quantum critical engines. For example, other factors remaining constant, enhancement in outputpower would demand a WM with large dimension d (see (68)). Furthermore, in case of free-Fermionic WMs operatedin presence of locally thermal baths, one can also arrive at a maximum efficiency bound η max which shows universalscaling with respect to the length L of the WM, given by the relation1 − η max ∝ L − z . (72)Clearly, η max increases with increasing L , thus indicating that many-body quantum engines can be more efficient thanfew-body ones.Recently, many-body quantum machines with ultracold gas WM, driven across superfluid and insulating phase hasalso been studied. The authors have shown that the existence of many-body effects and the critical point can boostthe performance of a N -particle many-body engine, as compared to N single particle engines, through enhancementin efficiency and power. Shortcuts to adiabaticity can further improve the performance of such many-body quantumengines close to criticality [63]. Quantum criticality has also been shown to improve the efficiency in quantum heatengine [59] based on the Lipkin-Meshkov-Glick model [125–127], as well as maximize work in interaction drivenquantum heat engines, with interacting Bose gas as a WM [47]. Recently, effects of topological phase transitions onthe performance of quantum heat engines have been studied [60]. In particular, studies on Otto heat engine witha finite length Kitaev chain as the working medium has shown that topological phase transition can enhance theefficiency, as well as work output of such engines [117]. B. Shortcuts to adiabaticity
In this section, we focus on enhancing the performance of quantum thermal machines through STA. As we discussedin Sec. II D, adiabatic quantum heat engines operate with high efficiency, at the cost of vanishing power. On theother hand, non-adiabatic excitations in finite-time heat engines can enhance the power, but at the cost of reducingthe efficiency. STA provides a solution to this conundrum, through suppression of non-adiabatic excitations in finite-time heat engines, thereby enhancing the power output, while keeping the efficiency close to the value obtained inits adiabatic counterpart. Following Ref. [49] here we discuss STA in many-body quantum heat engines, throughapproximate counterdiabatic driving [92–94].We consider an Otto cycle with a spin chain WM, described by the Hamiltonian H ( t ) = − N (cid:88) i =1 h i ( t ) σ xi − N (cid:88) i =1 b i ( t ) σ zi − N (cid:88) i =1 J i ( t ) σ zi σ zi +1 . (73)Here N is the total number of spins, J i ( t ) denotes the interaction strength between the spins at sites i and i + 1, b i ( t )is the longitudinal field strength along z direction, while h i ( t ) is the transverse field strength along x direction, at site i . We impose periodic boundary conditions, given by σ N +1 = σ . The explicit time-dependence of the Hamiltonianparameters during the unitary strokes 1 and 3 are taken to be h i ( t ) = h i, i + ( h i, f − h i, i ) sin (cid:2) π sin (cid:0) πt τ (cid:1)(cid:3) b i ( t ) = b i, i + ( b i, f − b i, i ) sin (cid:2) π sin (cid:0) πt τ (cid:1)(cid:3) J i ( t ) = J i, i + ( J i, f − J i, i ) sin (cid:2) π sin (cid:0) πt τ (cid:1)(cid:3) , (74)where the index i (f) refers to the initial (final) values of the parameters, and τ = τ = τ is the duration of a unitarystroke. As one can see, in general [ H ( t ) , H ( t (cid:48) )] (cid:54) = 0 for t (cid:54) = t (cid:48) . Consequently, in absence of control, non-adiabaticexcitations are generated during the unitary strokes, which are in general detrimental to the operation of the heat7engine [128, 129]. Therefore we aim to improve the performance of the above described finite-time heat engine, i.e.,enhance the efficiency and power, through the application of counter-diabatic driving, following Sec. II D. We achievethis through the application of the STA Hamiltonian H STA = H + H CD , (75)where the counter-diabatic Hamiltonian H CD is given by H CD = ˙ ϑ ( t ) A ϑ ( t ) . (76)The control function ϑ ( ϑ , t ) = ϑ sin (cid:20) π (cid:18) πt τ (cid:19)(cid:21) (77)is chosen so as to ensure smoothness at the beginning and end of the unitary strokes, quantified by ˙ ϑ ( t = 0) = ˙ ϑ ( t = τ ) = ¨ ϑ ( t = 0) = ¨ ϑ ( t = τ ). Here A ϑ ( t ) is the adiabatic gauge potential, obtained by following the protocol detailed inSec. II D [92–94] and ϑ is a global control parameter strength, introduced to tune the accuracy of the strokes (seebelow).Let us assume H and H CD are supplied by two independent work reservoirs. The counterdiabatic Hamiltonian H CD is designed so as to enhance the performance of the QHE, which in this case is quantified by the total workoutput W STA = (cid:90) τ cyc Tr (cid:104) ρ ˙ H STA (cid:105) dt = W + W CD , (78)where the useful work output is W = (cid:90) τ cyc Tr (cid:104) ρ ˙ H (cid:105) dt, (79)while the unusable output work, arising due to the control mechanism, is W CD = (cid:90) τ cyc Tr (cid:104) ρ ˙ H CD (cid:105) dt. (80)We note that in case of a single work reservoir implementing both H and H CD , the division Eq. (78) is in generalnot operationally relevant any more.The output power is given by P = W τ cyc , (81)while the efficiency is η = − W Q h for W CD ≤ − W Q h + W CD for W CD > . (82)Here Q h = (cid:90) τ cyc Tr [ ˙ ρH ] dt (83)is the heat input during the non-unitary stroke 2. The dependence of the expression of η on the sign of W CD (see Eq.(82)) stems from the fact that for W CD >
0, the setup acts as a hybrid thermomechanical engine, where part of theinput energy comes from the hot bath, and the remaining comes from the control Hamiltonian H CD .We now use the local ansatz H ∗ CD ( t ) = ˙ ϑ ( ϑ , t ) N (cid:88) j =1 ζ j ( t ) σ yj (84)8to approximate the counter-diabatic Hamiltonian Eq. (76). Optimization of the control Hamiltonian following Sec.II D leads to the following form of ζ j ( t ) (see Eqs. (41) - (44)) ζ j ( t ) = 12 ˙ h j ( t ) b j ( t ) − ˙ b j ( t ) h j ( t ) h j ( t ) + b j ( t ) + J j − ( t ) + J j ( t ) . (85)Consequently, the local STA Hamitonian assumes the form H ∗ STA ( t ) = − N (cid:88) i =1 h j ( t ) σ xj − N (cid:88) j =1 b j ( t ) σ zj − N (cid:88) j =1 J j ( t ) σ zj σ zj +1 + H ∗ CD ( t ) , (86)with H ∗ CD ( t ) given by Eqs. (84) and (85). Here the asterisk signifies that the Hamiltonian is inexact. (a)(c) (b)(d) Figure 5. .(a) Power P of the sped-up Otto cycle governed by (i) the original protocol H ( t ) and (ii) the shortcut-to-adiabaticityprotocol H ∗ STA ( t ) as a function of the isentropic-stroke duration τ = τ = τ . The machine acts as an engine if P < W and W CD pertaining to the piston (load) and the external control device, respectively. The green(left) and yellow (middle) shaded areas depict the regions where the machine operates as a heat engine ( W CD <
0) and athermo-mechanical engine ( W CD > W CD < − . (c) Efficiency η [for the heat-engine regime (green-shaded area) and for the hybrid thermo-mechanical regime(yellow-shaded area), respectively]. (d) Success fidelities of the isentropic strokes with and without STA protocol. Inset: Zoom.Parameters: Duration of the isentropic strokes: τ = τ = τ , duration of the thermalization strokes: τ = τ = 0 .
1. The otherparameters are T c = 0 . T h = 22, h j, i = 0 . b j, i = 0, h j, f = 0, b z, f = 1, J j, i = 0 for each spin. Disorderness is introducedin the interaction strengths where the 100 final interaction strengths J j, f are randomly chosen from a Gaussian distributionwith standard deviation σ = 0 . ϑ bounded in[0 , The approximate counter-diabatic Hamiltonian H ∗ CD ( t ) drives the WM to a state ρ (cid:48) B ( ρ (cid:48) D ) at point B (D) (see Fig.2), which is in general different from the state ρ B ( ρ D ) obtained in the adiabatic limit. However, one can increase thereliability of H ∗ CD ( t ) by optimizing the global control parameter strength ϑ , subject to the constraint 0 ≤ ϑ ≤
1, soas to maximize the Fidelity F ( ρ, ρ (cid:48) ) = Tr (cid:113) √ ρρ (cid:48) √ ρ. (87)9We note that restricting the approximate counterdiabatic Hamiltonian H ∗ CD to the single spin form (Cf. Eq. (86))simplifies the analysis and makes it experimentally implementable. At the same time, as shown in Fig. 5, eventhis single body ansatz leads to significant enhancement in the performance of the heat engine. The approximatecounterdiabatic driving allows the setup to act as a heat engine with large output power (see Fig. 5a), work (see Fig.5b) and efficiency (see Fig. 5c), while maintaining a high fidelity F (Cf. Eq. (87)) with the actual cycle (see Fig.5d) even for small cycle period τ ; this is in sharp contrast to the results obtained in absence of control, where thesetup fails to act as a heat engine for small τ . Similar advantages can be obtained in case of many-body quantumrefrigerators as well, in which case STA protocol enhances the refrigeration rate and the efficiency of refrigeration,above their non-adiabatic counterparts. Furthermore, remarkably, it can be shown that the work component W CD vanishes if the additional adiabatic gauge potential and consequently the counterdiabatic Hamiltonian H CD ( t ) is exact,i.e., A ∗ ϑ ( t ) = A ϑ ( t ) and H ∗ CD ( t ) = H CD ( t ), ∀ t [96]. C. Quantum advantage in many-body thermal machines
Designing quantum machines which show quantum advantage, i.e, outperform equivalent classical machines, is oneof the main aims of the field of quantum technology. In Ref. [45], the authors showed that many-particle quantumeffects, coupled with non-adiabatic effects, in a finite-time many-body Otto engine can be beneficial in this respect.The authors considered a WM comprising of N interacting bosons, described by the Hamiltonian H = N (cid:88) i =1 (cid:20) − m ∇ + 12 mϑ ( t ) r i (cid:21) + (cid:88) i 0, or equivalently, infinite durations of theunitary strokes 1 and 3 ( τ , τ → ∞ ). The efficiency of the engine is given by η na = W na (cid:104)Q (cid:105) = 1 − ϑ c ϑ h Q ∗ CD E C − ϑ h ϑ c E A E C − Q ∗ AB ϑ h ϑ c E A . (90)As can be seen from Eq. (90), η na is maximum, and equals the Otto efficiency 1 − ϑ c /ϑ h , in the adiabatic limit, when Q ∗ AB , Q ∗ CD = 1. However, this in turn leads to vanishing output power P na = W na /τ cyc .Interestingly, the authors showed that non-adiabatic effects arising due to finite τ cyc can lead to quantum advantage.In order to study quantum non-adiabatic effects induced enhancement in the performance of many-body engines, theauthors considered the specific example of a one-dimensional WM with the inter-particle interaction given by V ( z i , z j ) = 1 m (cid:88) i The unique features of many-body localization (MBL) can be beneficial for designing quantum machines. Many-body localization prevents systems from thermalizing under their intrinsic dynamics [130–133]. On the other hand,thermalizing (or weakly localized) systems obey the eigenstate thermalization hypothesis [134, 135]. In Ref. [136], theauthors used the difference in energy level statistics of these two kinds of systems to design a quantum heat enginewith a many-body WM exhibiting a MBL phase.In order to understand the operation of such a heat engine, let us look at the energy-level statsistics followed by thesystems in the above two regimes. For systems in the MBL regime, the probability P ( δ ) of an energy gap assuminga size δ , approximately obey the Poisson statistics [137, 138] : P ( E )MBL ( δ ) ≈ (cid:104) δ (cid:105) E e − δ/ (cid:104) δ (cid:105) E . (93)Here (cid:104) δ (cid:105) E is the average gap at the energy E . As can be seen from Eq. (93), an energy gap δ has a finite probabilityof vanishing, given by P ( E )MBL ( δ → ≈ / (cid:104) δ (cid:105) E > 0. On the other hand, energy gaps in systems in the thermalizingregime obey the Gaussian orthogonal ensemble (GOE) statistics [137]: P ( E )GOE ( δ ) ≈ π δ (cid:104) δ (cid:105) E e − π δ / (cid:104) δ (cid:105) E . (94)In contrast to the MBL spectra Eq. (93), small gaps appear with vanishing probability in the thermalizing regime,i.e., P ( E )GOE ( δ → → H meso ( t ) = εκ ( ϑ ( t )) [(1 − ϑ ( t )) H GOE + ϑ ( t ) H MBL ] . (95)Here the average energy density per site ε sets the unit of energy. One can change the qualitative nature of the WMby tuning the parameter ϑ ( t ) ∈ [0 , ϑ ( t ) = 0, the Hamiltonian Eq. (95) reduces to H GOE with a spectrum suchthat the energy gaps δ are distributed according to GOE statistics Eq. (94). On the other hand, ϑ ( t ) = 1 results in H meso (Eq. (95)) taking the form of H MBL , whose gaps follow the Poisson statistics Eq. (93). The renormalizationfactor κ ( ϑ ( t )) is chosen such that the average energy gap (cid:104) δ (cid:105) E is kept constant, thereby emphasizing the effect oflevel-statistics on the operation of the heat engine.One can exploit the difference in level statistics described above to design an Otto cycle which delivers a net outputwork. To this end, we consider a cold bath with bandwidth W b (cid:28) (cid:104) δ (cid:105) , such that that energy flow between the WMand the cold bath occurs only for gaps δ ≤ W b (cid:28) (cid:104) δ (cid:105) . We note that such anomalously small gaps can appear withfinite probabilities in the MBL regime (Cf. Eq. (93)), while this is not the case for the ETH regime (Cf. Eq. (94)).We consider the following Otto cycle using the many-body WM (Eq. (95)): the cycle starts with the WM in theETH phase ( ϑ ( t ) = ϑ h = 0), when the WM is in thermal equilibrium with the hot bath at temperature T h . Duringthe (unitary) stroke 1, ϑ ( t ) is tuned from zero to ϑ = ϑ c = 1, such that H ( t ) changes from H GOE to H MBL . Weassume the tuning is slow enough so as to result in adiabatic dynamics during the unitary strokes, i.e., non-adiabaticexcitations are negligible. The WM is coupled to a cold bath at temperature T c during the second (non-unitary)stroke. As per the MBL level statistics Eq. (93), the gap is small enough with probability ∼ W b (cid:104) δ (cid:105) , so as to allowthermalization with the cold bath. In the third (unitary stroke), ϑ ( t ) is tuned to zero such that we arrive at H GOE .Finally, the WM thermalizes with the hot bath during the fourth stroke, thereby completing the cycle.In the adiabatic limit of long cycle times, T c (cid:28) W b (cid:28) (cid:104) δ (cid:105) , T h = ∞ and the gap distributions Eqs. (93), (94)and the average gap (cid:104) δ (cid:105) E being independent of energy E , one can show that the average work output for the cycledescribed above is W tot ≈ −W b + 2 ln 2 β C , (96)1whereas the efficiency is η ≈ − W b (cid:104) δ (cid:105) . (97)Clearly, a small W b allows one to operate the engine with high efficiency, while producing a non-zero output work W tot , brought about by the different energy gap statistics of H MBL and H GOE . Furthermore, one can robustly scaleup the engine described above to the thermodynamic limit, wherein effectively independent subengines run in parallelwithout affecting each other, owing to the finite localization length of MBL systems. We note that non-adiabaticexcitations arising due to finite rate of driving during the unitary strokes can reduce the work output of the aboveengine [136].On a related note, mobility edges separating localized and delocalized states have also been used to design quantumheat engines. In Ref. [139], the authors considered a central WM coupled to a hot left bath with temperature T h andchemical potential µ h , and a cold right bath with temperature T c < T h and chemical potentials µ h (cid:54) = µ c . The WMwas taken to be a generalized Aubry-Andre-Harper (GAAH) model, given by the Hamiltonian H = N − (cid:88) i =1 ta † i a i +1 + h.c. + N (cid:88) i =1 V i a † i a i , (98)where t is the tunneling constant, a i is the Fermionic annihilation operator at site i and the onsite potential at site i is given by the quasiperiodic function V i = 2 θ (2 πυi + φ )1 − α cos (2 πυi + φ ) . (99)Here θ is the strength of the potential, the phase φ shifts the origin of the potential, υ is an irrational number and α ∈ ( − , E c of the mobility edge, separating the delocalized and localized states, can be tuned throughthe parameter α , and is given by [140] E c = 1 α sign ( θ ) ( | t | − | θ | ) . (100)The WM is coupled at the boundaries to the two non-interacting baths. The authors showed that the presence of amobility edge introduces an energy filter, which results in asymmetry in the dynamics of particles and holes. Thisin turn leads to non-zero steady-state heat current and power output, such that the setup can act as an autonomousthermoelectric heat engine. E. Quantum Szilard engine with interacting Bosons In contrast to the heat engines described above, which convert heat energy into useful output work, Szilard enginesoperate isothermally, and use information to do the same [141–145]. Szilard engines in the microscopic regime havebeen realized experimentally [24, 146–148]. The setup of a Szilard engine comprises of N ≥ L . The operation of such an engine can be described by the following quasistatic steps [141]: (i) insertionof a partition in the box, at position 0 ≤ l ≤ L , (ii) measurement of the particle number n ≤ N on one side (left,say) of the partition (iii) reversibly translating the partition to its final position 0 ≤ l (cid:48) ≤ L and (iv) removal of thepartition at l (cid:48) , thereby completing the cycle. Here we use the information about the particle numbers n and N − n to determine l (cid:48) , which eventually allows us to generate output work. The engine is operated so that the setup isin thermal equilibrium with a bath at temperature T at all times. The quasi-static insertion and removal of thepartition in steps (i) and (iii) respectively, can be assumed not to incur any cost, as long as we restrict ourselves to theclassical regime. However, the situation changes dramatically for a quantum Szilard engine - in this case, the insertionof a partition non-trivially changes the wavefunction of the particles inside the box, thereby making it necessary toensure that the insertion and removal of the partition are done isothermally as well. Interestingly, in Ref. [144], theauthors showed that the performance of a quantum Szilard engine can benefit from many-body effects. A quantumSzilard engine with attractively interacting Bosons was shown to enhance the conversion from information to work,as compared to non-interacting or repulsive Bosons.2 V. OTHER QUANTUM TECHNOLOGIES In addition to quantum engines and refrigerators, the broad field of quantum technologies [1, 2] also encompassesstudies on several related setups, including quantum simulators [149, 150], quantum batteries [151], quantum probes[152], quantum transistors [153], heat rectifiers [154] and quantum clocks [155, 156]. While they are vibrant fields ofresearch, and subjects deserving detailed discussions on their own, in this section we briefly discuss the role playedby many-body systems in the development of a couple of such technologies closely related to the thermodynamics ofquantum systems, viz. quantum batteries [151] and quantum probes [152]. A. Quantum batteries An engine converts energy from a dissipative reservoir into usable output power. On the other hand, batteries canbe used to store energy, which can be extracted and used later, at an opportune time. Consequently the field ofquantum batteries, i.e., batteries based on quantum systems and following the laws of quantum mechanics, is closelyrelated to the field of quantum engines and refrigerators.A discussion regarding quantum batteries necessitates the introduction of the concepts of passive states and er-gotropy; a passive state is a state whose energy cannot be reduced any further through cyclic unitary transformation- such states are diagonal in the energy eigenbasis, with populations of energy levels decreasing with increasing energyeigenvalues. On the other hand, ergotropy is the maximum amount of work that can be extracted from a non-passivestate, through unitary transformations [157, 158]. Naturally, charging a quantum system (battery) involves impartingergotropy to the system, so as to take it to a non-passive state. Thereafter, when needed, one can extract energy,i.e., discharge the battery through cyclic unitary transformations. The performance of a quantum battery can bequantified through the rate of charging / discharging and the maximum amount of work that can be extracted.Previous studies have suggested that entanglement can be beneficial for the performance of such quantum batteries[159]. In case of a quantum battery comprised of a many-body system, global entangling operations can enhancethe performance of such a battery [151, 160]. Collective effects have also been used to charge many-body quantumbatteries through quantum heat engines [161] and through dissipative thermal baths [162], and has been shown toyield quantum advantage [163]. Many-body systems have also been shown to be beneficial for desigining quantumbatteries, for example, through enhanced charging power [164, 165], and ultra-stable charging [166]. B. Quantum probes Studies of quantum systems in presence of dissipative baths, or designing quantum machines, usually involve preciseestimation of system and bath properties, which then leads us to the field of quantum metrology. Quantum metrology[152, 167] deals with probes based on quantum systems, such as quantum thermometers [168–175] and magnetometers[176, 177]. The precision of estimation of a parameter x (such as temperature) of a system is quantified by the relativeerror of measurement e rel = δxx , (101)which is bounded by the quantum Fisher information (QFI) G through the Cramer-Rao bound [167] e rel ≥ x √MG , (102)where δx denotes the absolute error in measurement and M is the number of measurements. As can be seen from Eq.(102), e rel decreases with increasing G . Consequently, one of the major aims of research in this field has been to findways to enhance the QFI, so as to result in high precision measurements. For example, several works have addressedthe issue of high-accuracy estimation of low temperatures, through achievement of large QFI [169, 175, 178]. To thisend, one generally considers a probe interacting with a thermal bath at temperature T , such that the state of theprobe at some optimal time (which is usually taken to be the steady state) is given by ρ ( T ). In this case, the QFI isgiven by G ( T ) = lim δ → ∂ F ( ρ ( T ) ρ ( T + δ )) ∂δ . (103)3Probes based on many-body quantum systems have been shown to advantageous in this context [172, 179]; criticalpoints are associated with divergences in QFI, thereby raising the possibility of high-precision quantum metrologyusing many-body systems [174, 180, 181]. In addition, as for quantum heat engines, collective effects have also beenshown to aid in high-precision quantum thermometry [44]. VI. DISCUSSION AND OUTLOOK Harnessing many-body effects to design high-performing quantum technologies is a rapidly progressing area ofresearch. Many-body effects may offer us the possibility of developing novel quantum machines, which can present uswith significant advantages as compared to equivalent classical machines. At the same time, studies of many-bodysystems can be accompanied by significant challenges, owing to the diverging size of the associated Hilbert space.These challenges have neccetitated the development of several techniques focussed on dealing with the dynamics ofclosed and open many-body quantum systems [31, 33, 39, 92–94]. Such techniques have in turn enabled researchersto design and study different quantum technologies based on many-body systems, in the last few years. In this shortreview, we have discussed some of the recent literature on this fascinating subject.Many-body effects in different forms can aid in the performance of quantum technologies. For example, cooperativeeffects arising due to collective coupling between many-body systems and dissipative baths have been shown tobe beneficial for designing quantum engines [42–44], quantum thermometers [44] and quantum batteries [161]. Inparallel, machines based on interacting many-body systems give rise to rich physics as well, for example in the form ofcriticality [62, 63], and can necessitate the introduction of shortcuts to adiabaticity for enhancing the performance ofsuch machines [49, 96]. Many-body systems have also been shown to aid in fast charging of batteries [151, 165]. Herewe have mainly focussed on quantum engines and briefly addressed quantum batteries and quantum probes. However,studies on many-body systems to develop other quantum technologies, such as quantum clocks [155, 156, 182] andquantum transistors [153], may also lead to interesting results.The recent advances in experimental know-how have made the realization of several quantum technologies a pos-sibility in various platforms. For example, interacting spin-chains models exhibiting phase transitions have beenexperimentally realized using quantum simulators [149, 150, 183], trapped ions [184] and quantum annealer [185]. Inthe recent years single or few-particle engines have already been realized experimentally, for example, using singleions [25], mechanical oscillators [26], nitrogen vacancy centers in diamonds [28], Rydberg atoms [51, 52] and opticallattices [20, 53, 186]. WMs based on strontium [187–189] or rubidium [190] atoms may be used for realizing collectiveeffects in QHEs. With the rapid advancement in development and control of systems in the quantum regime, suchmany-body quantum technologies can be expected to be realized experimentally in the near future. ACKNOWLEDGMENTS It is a pleasure to acknowledge Adolfo del Campo, Andreas Hartmann, Wolfgang Lechner, Glen Bigan Mbeng,Wolfgang Niedenzu and Revathy B. S. for related collaborative works. VM acknowledges SERB, India for Start-upResearch Grant SRG/2019/000411 and IISER Berhampur for Seed grant. UD acknowledges DST, India for INSPIREResearch grant. [1] G. Kurizki, P. Bertet, Y. Kubo, K. Mølmer, D. Petrosyan, P. Rabl, and J. Schmiedmayer, Proceedings of the NationalAcademy of Sciences , 23 (2016).[3] J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics , 2nd ed. (Cambridge University Press, 2017).[4] H. B. Callen, Thermodynamics and An Introduction to Thermostatistics (John Wiley & Sons Inc, New York, 1985, 1985).[5] D. Kondepudi and I. Prigogine, Modern Thermodynamics (John Wiley & Sons Ltd, Chichester, 2nd edn, 2015).[6] H. E. D. Scovil and E. O. Schulz-DuBois, Phys. Rev. Lett. , 262 (1959).[7] M. O. Scully, M. S. Zubairy, G. S. Agarwal, and H. Walther, Science , 862 (2003),https://science.sciencemag.org/content/299/5608/862.full.pdf.[8] J. Gemmer, M. Michel, and G. Mahler, Quantum thermodynamics: Emergence of thermodynamic behavior within com-posite quantum systems , Vol. 784 (Springer, 2009).[9] R. Kosloff, Entropy , 2100 (2013).[10] R. Kosloff and A. Levy, Annual Review of Physical Chemistry , 365 (2014). [11] F. Brand˜ao, M. Horodecki, N. Ng, J. Oppenheim, and S. Wehner, Proceedings of the National Academy of Sciences , 329(2015).[13] S. Vinjanampathy and J. Anders, Contemporary Physics , 545 (2016), https://doi.org/10.1080/00107514.2016.1201896.[14] M. N. Bera, A. Riera, M. Lewenstein, and A. Winter, Nat. Commun. , 2180 (2017).[15] L. Masanes and J. Oppenheim, Nat. Commun. , 14538 (2017).[16] F. Binder, L. A. Correa, C. Gogolin, J. Anders, and G. Adesso, eds., Thermodynamics in the quantum regime (SpringerInternational Publishing, 2018).[17] A. Tuncer and O. E. M¨ustecaplıo¯glu, Turk J Phys , 404 (2020).[18] S. Bhattacharjee and A. Dutta, “Quantum thermal machines and batteries,” (2020), arXiv:2008.07889 [quant-ph].[19] F. Dolde, H. Fedder, M. W. Doherty, T. N¨obauer, F. Rempp, G. Balasubramanian, T. Wolf, F. Reinhard, L. C. L.Hollenberg, F. Jelezko, and J. Wrachtrup, Nature Physics , 459 (2011).[20] M. G. Bason, M. Viteau, N. Malossi, P. Huillery, E. Arimondo, D. Ciampini, R. Fazio, V. Giovannetti, R. Mannella, andO. Morsch, Nature Physics , 147 (2012).[21] G. Kucsko, P. C. Maurer, N. Y. Yao, M. Kubo, H. J. Noh, P. K. Lo, H. Park, and M. D. Lukin, Nature , 54 (2013).[22] A. W. Laskar, P. Adhikary, S. Mondal, P. Katiyar, S. Vinjanampathy, and S. Ghosh, Phys. Rev. Lett. , 013601(2020).[23] S. Pal, S. Saryal, D. Segal, T. S. Mahesh, and B. K. Agarwalla, Phys. Rev. Research , 022044 (2020).[24] J. V. Koski, V. F. Maisi, J. P. Pekola, and D. V. Averin, Proceedings of the National Academy of Sciences , 325 (2016).[26] J. Klaers, S. Faelt, A. Imamoglu, and E. Togan, Phys. Rev. X , 031044 (2017).[27] J. P. S. Peterson, T. B. Batalh˜ao, M. Herrera, A. M. Souza, R. S. Sarthour, I. S. Oliveira, and R. M. Serra, Phys. Rev.Lett. , 240601 (2019).[28] 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).[29] G. Maslennikov, S. Ding, R. Habl¨utzel, J. Gan, A. Roulet, S. Nimmrichter, J. Dai, V. Scarani, and D. Matsukevich,Nature Communications , 202 (2019).[30] J. P. S. Peterson, R. S. Sarthour, and R. Laflamme, “Implementation of a quantum engine fuelled by information,”(2020), arXiv:2006.10136 [quant-ph].[31] M. Keck, S. Montangero, G. E. Santoro, R. Fazio, and D. Rossini, New Journal of Physics , 113029 (2017).[32] X. Xu, J. Thingna, C. Guo, and D. Poletti, Phys. Rev. A , 012106 (2019).[33] F. Nathan and M. S. Rudner, Phys. Rev. B , 115109 (2020).[34] A. Carmele, M. Heyl, C. Kraus, and M. Dalmonte, Phys. Rev. B , 195107 (2015).[35] S. Bandyopadhyay, S. Bhattacharjee, and A. Dutta, Phys. Rev. B , 104307 (2020).[36] J. A. Hoyos, C. Kotabage, and T. Vojta, Phys. Rev. Lett. , 230601 (2007).[37] C. De Grandi, V. Gritsev, and A. Polkovnikov, Phys. Rev. B , 012303 (2010).[38] P. Wang and R. Fazio, “Dissipative phase transitions in the fully-connected ising model with p -spin interaction,” (2020),arXiv:2008.10045 [cond-mat.quant-gas].[39] M. H. Fischer, M. Maksymenko, and E. Altman, Phys. Rev. Lett. , 160401 (2016).[40] Z. Lenarˇciˇc, O. Alberton, A. Rosch, and E. Altman, Phys. Rev. Lett. , 116601 (2020).[41] H. P. L¨uschen, P. Bordia, S. S. Hodgman, M. Schreiber, S. Sarkar, A. J. Daley, M. H. Fischer, E. Altman, I. Bloch, andU. Schneider, Phys. Rev. X , 011034 (2017).[42] W. Niedenzu and G. Kurizki, New Journal of Physics , 113038 (2018).[43] M. Kloc, P. Cejnar, and G. Schaller, Phys. Rev. E , 042126 (2019).[44] C. L. Latune, I. Sinayskiy, and F. Petruccione, New Journal of Physics , 083049 (2020).[45] J. Jaramillo, M. Beau, and A. del Campo, New J. Phys. , 075019 (2016).[46] S. C¸ akmak, F. Altintas, and ¨O. E. M¨ustecaplioglu, The European Physical Journal Plus , 197 (2016).[47] Y.-Y. Chen, G. Watanabe, Y.-C. Yu, X.-W. Guan, and A. del Campo, npj Quantum Information , 88 (2019).[48] E. Yunt, M. Fadaie, and ¨Ozg¨ur E. M¨ustecaplıo˘glu, “Topological and finite size effects in a kitaev chain heat engine,”(2019), arXiv:1908.02643 [cond-mat.stat-mech].[49] A. Hartmann, V. Mukherjee, W. Niedenzu, and W. Lechner, Phys. Rev. Research , 023145 (2020).[50] M. Campisi and R. Fazio, Nat. Commun. , 11895 (2016).[51] H. Kim, Y. Park, K. Kim, H.-S. Sim, and J. Ahn, Phys. Rev. Lett. , 180502 (2018).[52] A. Omran, H. Levine, A. Keesling, G. Semeghini, T. T. Wang, S. Ebadi, H. Bernien, A. S. Zibrov, H. Pichler, S. Choi,J. Cui, M. Rossignolo, P. Rembold, S. Montangero, T. Calarco, M. Endres, M. Greiner, V. Vuleti´c, and M. D. Lukin,Science , 570 (2019), https://science.sciencemag.org/content/365/6453/570.full.pdf.[53] M. Schreiber, S. S. Hodgman, P. Bordia, H. P. L¨uschen, M. H. Fischer, R. Vosk, E. Altman, U. Schneider, and I. Bloch,Science , 842 (2015).[54] H. P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, 2002).[55] L. Mandel and E. Wolf, Optical coherence and quantum optics (Cambridge University Press, 1995).[56] C. L. Latune, I. Sinayskiy, and F. Petruccione, Phys. Rev. Research , 033192 (2019). [57] P. M. Chaikin and T. C. Lubensky, Principles of Condensed Matter Physics (Cambridge University Press, 1995).[58] S. Sachdev, Quantum Phase Transitions (Cambridge University Press, Cambridge, England, 1999).[59] Y.-H. Ma, S.-H. Su, and C.-P. Sun, Phys. Rev. E , 022143 (2017).[60] M. Fadaie, E. Yunt, and O. E. M¨ustecaplıo˘glu, Phys. Rev. E , 052124 (2018).[61] S. Chand and A. Biswas, Phys. Rev. E , 052147 (2018).[62] R. B. S, V. Mukherjee, U. Divakaran, and A. del Campo, Phys. Rev. Research , 043247 (2020).[63] T. Fogarty and T. Busch, Quantum Science and Technology , 015003 (2020).[64] J. Dziarmaga, Advances in Physics , 1063 (2010).[65] A. Dutta, G. Aeppli, B. K. Chakrabarti, U. Divakaran, T. F. Rosenbaum, and D. Sen, Quantum phase transitions intransverse field spin models: from statistical physics to quantum information (Cambridge University Press, Cambridge,2015).[66] A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore, Rev. Mod. Phys. , 863 (2011).[67] W. H. Zurek, U. Dorner, and P. Zoller, Phys. Rev. Lett. , 105701 (2005).[68] A. Polkovnikov, Phys. Rev. B , 161201 (2005).[69] Z. Fei, N. Freitas, V. Cavina, H. T. Quan, and M. Esposito, Phys. Rev. Lett. , 170603 (2020).[70] E. Lieb, T. Schultz, and D. Mattis, Annals of Physics , 407 (1961).[71] P. Pfeuty, Annals of Physics , 79 (1970).[72] J. E. Bunder and R. H. McKenzie, Phys. Rev. B , 344 (1999).[73] J. Dziarmaga, Phys. Rev. Lett. , 245701 (2005).[74] A. del Campo, A. Chenu, S. Deng, and H. Wu, “Friction-free quantum machines,” in Thermodynamics in the QuantumRegime: Fundamental Aspects and New Directions , edited by F. Binder, L. A. Correa, C. Gogolin, J. Anders, andG. Adesso (Springer International Publishing, Cham, 2018) pp. 127–148.[75] M. Demirplak and S. A. Rice, The Journal of Physical Chemistry A , 9937 (2003).[76] M. V. Berry, Journal of Physics A: Mathematical and Theoretical , 365303 (2009).[77] A. del Campo, M. M. Rams, and W. H. Zurek, Phys. Rev. Lett. , 115703 (2012).[78] S. Deffner, C. Jarzynski, and A. del Campo, Phys. Rev. X , 021013 (2014).[79] A. Patra and C. Jarzynski, New Journal of Physics , 125009 (2017).[80] D. Gu´ery-Odelin, A. Ruschhaupt, A. Kiely, E. Torrontegui, S. Mart´ınez-Garaot, and J. G. Muga, Rev. Mod. Phys. ,045001 (2019).[81] A. Patra and C. Jarzynski, “Semiclassical fast-forward shortcuts to adiabaticity,” (2021), arXiv:2101.05901 [quant-ph].[82] G. Vacanti, R. Fazio, S. Montangero, G. M. Palma, M. Paternostro, and V. Vedral, New Journal of Physics , 053017(2014).[83] S. Alipour, A. Chenu, A. T. Rezakhani, and A. del Campo, Quantum , 336 (2020).[84] S. An, D. Lv, A. del Campo, and K. Kim, Nature Communications , 12999 (2016).[85] A. del Campo, J. Goold, and M. Paternostro, Sci. Rep. , 6208 (2014).[86] B. C¸ akmak and O. E. M¨ustecaplıo˘glu, Phys. Rev. E , 032108 (2019).[87] O. Abah, M. Paternostro, and E. Lutz, Phys. Rev. Research , 023120 (2020).[88] S. Campbell and S. Deffner, Phys. Rev. Lett. , 100601 (2017).[89] O. Abah and E. Lutz, Phys. Rev. E , 032121 (2018).[90] O. Abah and M. Paternostro, Phys. Rev. E , 022110 (2019).[91] V. Mukherjee, S. Montangero, and R. Fazio, Phys. Rev. A , 062108 (2016).[92] D. Sels and A. Polkovnikov, Proceedings of the National Academy of Sciences , 1 (2017), geometry and non-adiabaticresponse in quantum and classical systems.[94] P. W. Claeys, M. Pandey, D. Sels, and A. Polkovnikov, Phys. Rev. Lett. , 090602 (2019).[95] M. Beau, J. Jaramillo, and A. del Campo, Entropy , 168 (2016).[96] A. Hartmann, V. Mukherjee, G. B. Mbeng, W. Niedenzu, and W. Lechner, Quantum , 377 (2020).[97] H. T. Quan, Y.-x. Liu, C. P. Sun, and F. Nori, Phys. Rev. E , 031105 (2007).[98] B. Cleuren, B. Rutten, and C. Van den Broeck, Phys. Rev. Lett. , 120603 (2012).[99] M. Kol´aˇr, D. Gelbwaser-Klimovsky, R. Alicki, and G. Kurizki, Phys. Rev. Lett. , 090601 (2012).[100] J. Roßnagel, O. Abah, F. Schmidt-Kaler, K. Singer, and E. Lutz, Phys. Rev. Lett. , 030602 (2014).[101] R. Uzdin, A. Levy, and R. Kosloff, Phys. Rev. X , 031044 (2015).[102] G. Watanabe, B. P. Venkatesh, P. Talkner, and A. del Campo, Phys. Rev. Lett. , 050601 (2017).[103] A. Friedenberger and E. Lutz, EPL (Europhysics Letters) , 10002 (2017).[104] N. Freitas and J. P. Paz, Phys. Rev. E , 012146 (2017).[105] W. Niedenzu, V. Mukherjee, A. Ghosh, A. G. Kofman, and G. Kurizki, Nature Communications , 165 (2018).[106] A. Ghosh, D. Gelbwaser-Klimovsky, W. Niedenzu, A. I. Lvovsky, I. Mazets, M. O. Scully, and G. Kurizki, Proceedingsof the National Academy of Sciences (2018), 10.1073/pnas.1805354115.[107] A. Ghosh, V. Mukherjee, W. Niedenzu, and G. Kurizki, Eur. Phys. J. Special Topics , 2043 (2019).[108] P. A. Erdman, V. Cavina, R. Fazio, F. Taddei, and V. Giovannetti, New Journal of Physics , 103049 (2019).[109] V. Mukherjee, A. G. Kofman, and G. Kurizki, Communications Physics , 8 (2020).[110] R. Alicki, Open Systems And Information Dynamics , 1440002 (2014).[111] D. Gelbwaser-Klimovsky, R. Alicki, and G. Kurizki, Phys. Rev. E , 012140 (2013). [112] R. Alicki, D. Gelbwaser-Klimovsky, and G. Kurizki, arXiv:1205.4552 (2012).[113] Y. Zheng and D. Poletti, Phys. Rev. E , 012110 (2015).[114] G. Watanabe, B. P. Venkatesh, P. Talkner, M.-J. Hwang, and A. del Campo, Phys. Rev. Lett. , 210603 (2020).[115] F. Carollo, F. M. Gambetta, K. Brandner, J. P. Garrahan, and I. Lesanovsky, Phys. Rev. Lett. , 170602 (2020).[116] M. Azimi, L. Chotorlishvili, S. K. Mishra, T. Vekua, W. H¨ubner, and J. Berakdar, New Journal of Physics , 063018(2014).[117] E. Yunt, M. Fadaie, O. E. M¨ustecaplıo˘glu, and C. M. Smith, Phys. Rev. B , 155423 (2020).[118] A. Kumar and C. Benjamin, “A thermodynamic probe of the topological phase transition in a floquet topological insu-lator,” (2020), arXiv:2012.02172 [cond-mat.mes-hall].[119] F. Carollo, K. Brandner, and I. Lesanovsky, “Nonequilibrium many-body quantum engine driven by time-translationsymmetry breaking,” (2020), arXiv:2007.00690 [cond-mat.stat-mech].[120] L. Fusco, M. Paternostro, and G. De Chiara, Phys. Rev. E , 052122 (2016).[121] P. Abiuso and M. Perarnau-Llobet, Phys. Rev. Lett. , 110606 (2020).[122] D. Patan`e, A. Silva, L. Amico, R. Fazio, and G. E. Santoro, Phys. Rev. Lett. , 175701 (2008).[123] T. Kibble, Physics Reports , 183 (1980).[124] B. Damski and W. H. Zurek, Phys. Rev. A , 063405 (2006).[125] H. Lipkin, N. Meshkov, and A. Glick, Nuclear Physics , 188 (1965).[126] N. Meshkov, A. Glick, and H. Lipkin, Nuclear Physics , 199 (1965).[127] A. Glick, H. Lipkin, and N. Meshkov, Nuclear Physics , 211 (1965).[128] T. Feldmann and R. Kosloff, Phys. Rev. E , 025107 (2006).[129] R. Dann, R. Kosloff, and P. Salamon, Entropy (2020), 10.3390/e22111255.[130] I. L. Aleiner, B. L. Altshuler, and G. V. Shlyapnikov, Nature Physics , 900 (2010).[131] R. Nandkishore and D. A. Huse, Annual Review of Condensed Matter Physics , 15 (2015),https://doi.org/10.1146/annurev-conmatphys-031214-014726.[132] F. Alet and N. Laflorencie, Comptes Rendus Physique , 498 (2018), quantum simulation / Simulation quantique.[133] K. S. C. Decker, C. Karrasch, J. Eisert, and D. M. Kennes, Phys. Rev. Lett. , 190601 (2020).[134] M. Rigol, V. Dunjko, V. Yurovsky, and M. Olshanii, Phys. Rev. Lett. , 050405 (2007).[135] M. Rigol, V. Dunjko1, and M. Olshanii, Nature , 854 (2008).[136] N. Yunger Halpern, C. D. White, S. Gopalakrishnan, and G. Refael, Phys. Rev. B , 024203 (2019).[137] V. Oganesyan and D. A. Huse, Phys. Rev. B , 155111 (2007).[138] A. Pal and D. A. Huse, Phys. Rev. B , 174411 (2010).[139] C. Chiaracane, M. T. Mitchison, A. Purkayastha, G. Haack, and J. Goold, Phys. Rev. Research , 013093 (2020).[140] S. Ganeshan, J. H. Pixley, and S. Das Sarma, Phys. Rev. Lett. , 146601 (2015).[141] S. W. Kim, T. Sagawa, S. De Liberato, and M. Ueda, Phys. Rev. Lett. , 070401 (2011).[142] E. Lutz and S. Ciliberto, Physics Today , 30 (2015), https://doi.org/10.1063/PT.3.2912.[143] M. H. Mohammady and J. Anders, New Journal of Physics , 113026 (2017).[144] J. Bengtsson, M. N. Tengstrand, A. Wacker, P. Samuelsson, M. Ueda, H. Linke, and S. M. Reimann, Phys. Rev. Lett. , 100601 (2018).[145] A. B´erut, A. Arakelyan, A. Petrosyan, S. Ciliberto, R. Dillenschneider, and E. Lutz, Nature , 187 (2012).[146] S. Toyabe, T. Sagawa, M. Ueda, E. Muneyuki, and M. Sano, Nature Physics , 988 (2010).[147] ´E. Rold´an, I. A. Mart´ınez, J. M. R. Parrondo, and D. Petrov, Nature Physics , 457 (2014).[148] J. V. Koski, A. Kutvonen, I. M. Khaymovich, T. Ala-Nissila, and J. P. Pekola, Phys. Rev. Lett. , 260602 (2015).[149] H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner,V. Vuleti´c, and M. D. Lukin, Nature , 579 (2017).[150] S. Ebadi, T. T. Wang, H. Levine, A. Keesling, G. Semeghini, A. Omran, D. Bluvstein, R. Samajdar, H. Pichler, W. W. Ho,S. Choi, S. Sachdev, M. Greiner, V. Vuletic, and M. D. Lukin, “Quantum phases of matter on a 256-atom programmablequantum simulator,” (2020), arXiv:2012.12281 [quant-ph].[151] F. Campaioli, F. A. Pollock, and S. Vinjanampathy, “Quantum batteries,” in Thermodynamics in the Quantum Regime:Fundamental Aspects and New Directions , edited by F. Binder, L. A. Correa, C. Gogolin, J. Anders, and G. Adesso(Springer International Publishing, Cham, 2018) pp. 207–225.[152] V. Giovannetti, S. Lloyd, and L. Maccone, Nature Photonics , 222 (2011).[153] K. Joulain, J. Drevillon, Y. Ezzahri, and J. Ordonez-Miranda, Phys. Rev. Lett. , 200601 (2016).[154] S. H. S. Silva, G. T. Landi, R. C. Drumond, and E. Pereira, “Heat rectitication on the xx chain,” (2020), arXiv:2012.04811[quant-ph].[155] V. Buˇzek, R. Derka, and S. Massar, Phys. Rev. Lett. , 2207 (1999).[156] P. Erker, M. T. Mitchison, R. Silva, M. P. Woods, N. Brunner, and M. Huber, Phys. Rev. X , 031022 (2017).[157] W. Pusz and S. L. Woronowicz, Communications in Mathematical Physics , 273 (1978).[158] A. Lenard, Journal of Statistical Physics , 575 (1978).[159] R. Alicki and M. Fannes, Phys. Rev. E , 042123 (2013).[160] F. C. Binder, S. Vinjanampathy, K. Modi, and J. Goold, New Journal of Physics , 075015 (2015).[161] K. Ito and G. Watanabe, “Collectively enhanced high-power and high-capacity charging of quantum batteries via quantumheat engines,” (2020), arXiv:2008.07089 [quant-ph].[162] B. i. e. i. f. m. c. C¸ akmak, Phys. Rev. E , 042111 (2020). [163] F. Campaioli, F. A. Pollock, F. C. Binder, L. C´eleri, J. Goold, S. Vinjanampathy, and K. Modi, Phys. Rev. Lett. ,150601 (2017).[164] T. P. Le, J. Levinsen, K. Modi, M. M. Parish, and F. A. Pollock, Phys. Rev. A , 022106 (2018).[165] D. Rossini, G. M. Andolina, D. Rosa, M. Carrega, and M. Polini, Phys. Rev. Lett. , 236402 (2020).[166] D. Rosa, D. Rossini, G. M. Andolina, M. Polini, and M. Carrega, Journal of High Energy Physics , 67 (2020).[167] M. G. A. PARIS, International Journal of Quantum Information , 125 (2009).[168] M. Brunelli, S. Olivares, and M. G. A. Paris, Phys. Rev. A , 032105 (2011).[169] L. A. Correa, M. Mehboudi, G. Adesso, and A. Sanpera, Phys. Rev. Lett. , 220405 (2015).[170] L. A. Correa, M. Perarnau-Llobet, K. V. Hovhannisyan, S. Hern´andez-Santana, M. Mehboudi, and A. Sanpera, Phys.Rev. A , 062103 (2017).[171] P. P. Hofer, J. B. Brask, M. Perarnau-Llobet, and N. Brunner, Phys. Rev. Lett. , 090603 (2017).[172] K. V. Hovhannisyan and L. A. Correa, Phys. Rev. B , 045101 (2018).[173] A. De Pasquale and T. M. Stace, “Quantum thermometry,” in Thermodynamics in the Quantum Regime: FundamentalAspects and New Directions , edited by F. Binder, L. A. Correa, C. Gogolin, J. Anders, and G. Adesso (SpringerInternational Publishing, Cham, 2018) pp. 503–527.[174] P. P. Potts, J. B. Brask, and N. Brunner, Quantum , 161 (2019).[175] V. Mukherjee, A. Zwick, A. Ghosh, X. Chen, and G. Kurizki, Communications Physics , 162 (2019).[176] S. Bhattacharjee, U. Bhattacharya, W. Niedenzu, V. Mukherjee, and A. Dutta, New Journal of Physics , 013024(2020).[177] A. Levy, M. G¨ob, B. Deng, K. Singer, E. Torrontegui, and D. Wang, New Journal of Physics , 093020 (2020).[178] S. Campbell, M. G. Genoni, and S. Deffner, Quantum Science and Technology , 025002 (2018).[179] W.-K. Mok, K. Bharti, L.-C. Kwek, and A. Bayat, “Optimal probes for global quantum thermometry,” (2020),arXiv:2010.14200 [quant-ph].[180] P. Zanardi, M. G. A. Paris, and L. Campos Venuti, Phys. Rev. A , 042105 (2008).[181] M. M. Rams, P. Sierant, O. Dutta, P. Horodecki, and J. Zakrzewski, Phys. Rev. X , 021022 (2018).[182] A. Peres, American Journal of Physics , 552 (1980), https://doi.org/10.1119/1.12061.[183] J. Zhang, G. Pagano, P. W. Hess, A. Kyprianidis, P. Becker, H. Kaplan, A. V. Gorshkov, Z.-X. Gong, and C. Monroe,Nature , 601 (2017).[184] J.-M. Cui, Y.-F. Huang, Z. Wang, D.-Y. Cao, J. Wang, W.-M. Lv, L. Luo, A. del Campo, Y.-J. Han, C.-F. Li, and G.-C.Guo, Scientific Reports , 33381 (2016).[185] Y. Bando, Y. Susa, H. Oshiyama, N. Shibata, M. Ohzeki, F. J. G´omez-Ruiz, D. A. Lidar, S. Suzuki, A. del Campo, andH. Nishimori, Phys. Rev. Research , 033369 (2020).[186] A. M. Kaufman, M. E. Tai, A. Lukin, M. Rispoli, R. Schittko, P. M. Preiss, and M. Greiner, Science , 794 (2016).[187] M. A. Norcia, R. J. Lewis-Swan, J. R. K. Cline, B. Zhu, A. M. Rey, and J. K. Thompson, Science , 259 (2018),https://science.sciencemag.org/content/361/6399/259.full.pdf.[188] D. Barberena, R. J. Lewis-Swan, J. K. Thompson, and A. M. Rey, Phys. Rev. A , 053411 (2019).[189] K. Tucker, D. Barberena, R. J. Lewis-Swan, J. K. Thompson, J. G. Restrepo, and A. M. Rey, Phys. Rev. A , 051701(2020).[190] T. M. Karg, B. Gouraud, C. T. Ngai, G.-L. Schmid, K. Hammerer, and P. Treutlein, Science369