Cooperative many-body enhancement of quantum thermal machine power

We study the impact of cooperative many-body effects on the operation of periodically-driven quantum thermal machines, particularly heat engines and refrigerators. In suitable geometries, $N$ two-level atoms can exchange energy with the driving field and the (hot and cold) thermal baths at a faster rate than a single atom due to their SU(2) symmetry that causes the atoms to behave as a collective spin-$N/2$ particle. This cooperative effect boosts the power output of heat engines compared to the power output of $N$ independent, incoherent, heat engines. In the refrigeration regime, similar cooling-power boost takes place.


I. INTRODUCTION
One of the central questions in the emerging field of quantum thermodynamics [1][2][3][4][5] pertains to possible quantum advantages ("supremacy") in the operation of thermal machines such as heat engines or refrigerators compared to their classical counterparts [6]. In that context, starting with the seminal work by Scully et al. [7], extensive investigations have focused on the question whether quantum coherence in either the machine's working medium [8][9][10][11][12][13] or the energizing (hot) bath (the "fuel") [14][15][16][17][18][19][20][21][22][23][24] could either boost the power output or the efficiency of quantum engines. Whilst these investigations have been mainly theoretical, impressive experimental progress has also been made such as the first realization of a heat engine based on a single atom [25], the demonstration of quantumthermodynamic effects in the operation of a heat engine implemented by an ensemble of nitrogen-vacancy (NV) centers in diamond [26] and the implementation of a quantum engine fueled by a squeezed-thermal bath [27].
Here we explore the possibility of exploiting collective (cooperative) many-body effects in quantum heat engines and refrigerators [28][29][30][31]. These generic quantum effects have a common origin with Dicke superradiance [32], whereby light emission is collectively enhanced by the interaction of N atoms with a common environment (bath) such that its intensity scales with N 2 . Investigations of this effect for a cloud of closely-packed emitters [38] face a severe problem: The dipole-dipole interaction (DDI) among emitters may diverge and cause an uncontrollable inhomogeneous broadening that may destroy superradiance. By contrast, DDI may be either suppressed [43] or collectively enhanced [54] in appropriate one-dimensional setups [55] such that in either case superradiance persists.
Beyond the need to control DDI, the feasibility of quantum cooperative effects in heat machines depends on the resolution of another principal issue: Is the timing of * Wolfgang.Niedenzu@uibk.ac.at atom-bath interactions important for their cooperativity [56][57][58]? Since the early studies of superradiance and superfluorescence [59] the crucial role of proper initiation of the atomic ensemble has been stressed. If this is the case, is continuous, steady-state interaction of the atoms with heat baths compatible with cooperativity? Here we show that, rather surprisingly, quantum cooperative enhancement of N -atom energy exchange with thermal baths persists under steady-state conditions. This fundamental result is an outcome of our analysis that extends to N atoms the concept of the minimal quantum heat machine [60,61]: A two-level atom that is simultaneously coupled to two (hot and cold) thermal baths and is periodically modulated in energy (transition frequency).
This paper is organized as follows: We introduce the model for an N -partite quantum thermal machine and discuss the collective-spin basis (Dicke basis) in Sec. II. The thermodynamic analysis is based on the Markovian master equation presented in Sec. III. The collective energy currents and the power boost compared to N independent engines are analyzed in Secs. IV and V. A concrete example for a heat engine and a refrigerator is discussed in Secs. VI and VII. In Sec. VIII we address possible experimental realizations before concluding in Sec. IX.

II. MANY-BODY THERMAL MACHINES: MODEL AND COLLECTIVE BASIS
We consider an ensemble of N two-level atoms subject to a periodic driving field that modulates the atomic transition frequency [2,60,61], such that H S (t + τ ) = H S (t). The modulation fulfills the periodicity condition τ −1 τ 0 ω(t)dt = ω 0 , where τ = 2π/Ω is the cycle time and ω 0 the atoms' "bare" (unperturbed) transition frequency. The atoms are all identically coupled to two (cold and hot) thermal baths arXiv:1806.10810v1 [quant-ph] 28 Jun 2018 Figure 1. Energy currents in (a) a heat engine and (b) a refrigerator. A heat engine converts the heat current J h from the hot bath into power P whereas a refrigerator consumes power to establish a heat flow from the cold to the hot bath. Regardless of the operation mode, we here show that all three energy currents Jc, J h and P can be equally enhanced by cooperative effects between the atoms. We use the convention that energy currents directed towards the atoms have positive sign. Heat-engine operation then corresponds to P < 0.
(i ∈ {c, h}), where B i is the coupling operator of the ith bath [53,62,63]. Note that the atoms are indistinguishable to both of the baths. The thermodynamic properties of a setup governed by Eqs. (1) have been studied for N = 1, i.e., a single twolevel atom, in Refs. [60,61]. In particular, it has been shown that this setup constitutes a universal thermal machine that can be operated on-demand as either a heat engine (thereby converting heat obtained from the hot bath into power) or a refrigerator (consuming power to cool the cold bath), see Fig. 1. The classical field that induces the periodic modulation in the Hamiltonian (1a) acts as a piston that either extracts (supplies) power from (to) the machine [1].
We now introduce the collective spin operator J := (J x , J y , J z ) T with components [53] and the transition operators In terms of these collective operators, the Hamiltonians (1) adopt the simple forms The pertinent feature of the collective operators (2) is that they can be cast into the block-diagonal form where the S k j are spin operators of lower dimension. The block-diagonal form (4) is adopted in the basis spanned by the Dicke states, which are entangled many-body states [52,53,64].
The decomposition (4) follows from the theory of spin addition [53,65]: The 2 N -dimensional joint Hilbert space of N two-level atoms can be decomposed into n irreducible subspaces of dimension 2j + 1, each corresponding to an eigenvalue j(j + 1) of J 2 [53]. The possible values of j are j = 0, 1, . . . , N 2 for N even and j = 1 2 , 3 2 , . . . , N 2 for N odd, respectively. We will make use of the notation j k to denote the j belonging to the kth subspace (k = 1, . . . , n) when necessary.
As an example, two spin-1/2 particles couple to a triplet (j = 1) and a singlet (j = 0), whereas three spin-1/2 couple to a quadruplet (j = 3 2 ) and two doublets (j = 1 2 ). These decompositions [66] are commonly denoted as 2⊗2 = 3⊕1 and 2 ⊗ 2 ⊗ 2 = 4 ⊕ 2 ⊕ 2, respectively, where the numbers indicate the dimensionality 2j + 1 of the Hilbert space of the respective spin. The latter example shows that different irreducible subspaces may share the same j (called multiplicity of j). The block-diagonal structure of the collective operators is nevertheless still guaranteed by the eigenstates' different symmetry [52]. The largest possible spin j = N/2, however, is unique and formed by the totally-symmetric N -atom states (see Appendix A) [53].
In this paper we wish to explore the thermodynamic implications of the existence of irreducible subspaces on the operation of thermal machines (see Fig. 1). Phrased differently, we raise the question: Can the symmetry of the Hamiltonian (3) be exploited as a genuine quantumthermodynamic resource that boosts the performance of heat engines or refrigerators? The answer to this question is positive (Fig. 2).

III. MASTER EQUATION
A Lindblad master equation for the reduced state of the atoms for periodic Hamiltonians can be derived using Floquet theory [1,2,53,61,[67][68][69][70]]. For our system described by the Hamiltonian (1), the master equation for the single-atom case N = 1 is given in Refs. [2,60,61]; for a detailed derivation of the master equation we refer the reader to Sec. 4.1 of Ref. [61]. Intriguingly, this derivation also holds for N > 1 upon replacing the single-atom operators by their collective counterparts according to Eqs. (2); the reason being that σ j and J j fulfill the same Lie algebra, i.e., the same commutation relations [65]. Hence, the multi-atom master equation in the interaction picture readsρ = i∈{c,h} q∈Z with the sub-Liouvillians [53]. Physically, the master equation (5) describes the incoherent emission (first line in Eq. (6) with jump operator J − ) and absorption (second line in Eq. (6) with jump operator J + ) of photons carrying energy (ω 0 + qΩ) to (from) the two baths at inverse temperatures β c and β h , respectively. The frequencies ω 0 + qΩ (q ∈ Z) are called Floquet sideband frequencies. The rates associated to these sideband channels are determined by the bath spectra G i (ω) and the weights P (q) which depend on the form of the modulation ω(t) in the Hamiltonian (1a) (see Appendix B).
The merit of the collective operators J ± [Eq. (2b)] is now clear: The master equation (5) conserves the symmetry, meaning that it does not couple the individual irreducible subspaces that correspond to the blocks in Eq. (4). In other words, since [J ± , J 2 ] = 0 these subspaces are invariant under the master equation (5) such that every (2j k + 1)-dimensional subspin evolves individually. Hence, the steady-state solution of the master equation (5) is a weighted direct sum of Gibbs-like states of these individual spin constituents, where (8) and Z k := Tr k exp −β eff ω 0 S k z . Here Π k denotes the projector onto the kth invariant subspace and ρ 0 is the initial condition of the atoms such that n k=1 Π k ρ0 = 1. The inverse effective temperature β eff is implicitly defined via the "global" detailed-balance condition in the master equation (5) [61]: The nominator in Eq. (9) is the total absorption rate in the master equation (5) whereas the denominator corresponds to the total emission rate therein.
The steady state (7) thus reflects the fact that the initial populations Π k ρ0 of the invariant subspaces (i.e., the initial weights of the individual subspins) cannot change dynamically under the master equation (5) since the corresponding subspins do not interact with each other. Each subspin, however, relaxes to the Gibbs-like steady state (8).
We note that the static steady state (7) is attained in the interaction picture w.r.t. the Hamiltonian (1a). In the original Schrödinger picture this state corresponds to a limit cycle with periodicity τ = 2π/Ω [1,69].

IV. COLLECTIVE ENERGY CURRENTS
The steady state (7) of the atoms is an out-ofequilibrium state that is maintained by the interplay of three energy currents: (i) the heat current J c from the cold bath, (ii) the heat current J h from the hot bath and (iii) the power P = −[J c + J h ] originating from the driving field (see Fig. 1). Following the theory of the thermodynamics of periodically-driven open quantum systems [1,2,60,61,69], the heat currents from the two baths to the atoms are (i ∈ {c, h}) This expression accounts for the fact that under Ω-periodic driving photons are not only exchanged at the bare transition frequency ω 0 but also at the Floquet sidebands ω 0 + qΩ and thus carry different energies (ω 0 + qΩ).
Inserting the steady state (7) into Eq. (10), the heat currents and the power evaluate to where is the heat flow induced by the spin-j k associated with the is the corresponding power. The prefactor in Eq. (12) is defined as where p ss,k j is the (thermal) population of the jth level of the spin-j k particle at inverse temperature β eff . The explicit form of Eq. (13) is given in Appendix C.
Consider the case N = 3 as an illustrative example. As mentioned above, 2 ⊗ 2 ⊗ 2 = 4 ⊕ 2 ⊕ 2, such that j 1 = 3 2 , j 2 = 1 2 and j 3 = 1 2 , respectively. There are thus n = 3 invariant subspaces and the power (11b) evaluates to As a consequence, for an initial condition having its support only in the two irreducible doublet spaces such that Π 2 ρ0 + Π 3 ρ0 = 1, the power P induced by the three two-level atoms equals the power generated by a single two-level atom, P = P( 1 2 ). Hence, owing to destructive interference, for such an initial condition there is no benefit to, e.g., heat-engine operation, in adding two additional atoms.
Introducing the heat current J i ( 1 2 ) induced by a single two-level atom, the collective heat currents (12) can be cast into the simple form Since the ratio in Eq. (15) does not explicitly depend on the bath index i, both heat currents J h (j k ) and J c (j k ) are equally amplified [11] such that also the power originating from the spin-j k subspace is equally enhanced with respect to the power generated by a single two-level atom, The results (10)- (16) are general in that they do not only apply to heat engines (that convert heat from the hot bath into power) but also to refrigerators (that consume power to cool the cold bath) or any other kind of setups, e.g., heat distributors where both baths are heated up on the expense of the invested power.

V. COOPERATIVE POWER ENHANCEMENT
Example (14) demonstrates the crucial role of the atomic initial condition ρ 0 in cooperative many-body thermal machines. Perfect constructive interference of the fields scattered by the atoms is achieved under the condition of full atomic cooperativity where all N twolevel atoms assemble to the largest-possible spin N/2. In order to achieve the maximal benefit it is therefore important to start in a favorable initial condition where only the subspace associated with this spin-N/2 is populated, i.e., Π K ρ0 = 1 (j K = N 2 ). This subspace is unique and comprises, for example, all N atoms being initially excited or all N atoms being initially in the ground state (see Appendix A) [53].
The collective power P coll := P N 2 and its counterpart P ind := N P 1 2 established by N individual atoms (cf. Fig. 2) then fulfill the relation The ratio (17) in the low-temperature regime and lim β eff ω0→0 in the high-temperature regime. The superradiant scaling behavior P coll ∼ N P ind = N 2 P( 1 2 ) is thus established for sufficiently high effective temperatures such that the spin-N/2 particle is considerably excited [cf. Eq. (8)].
For β eff ω 0 → ∞ the respective energy currents are equal [Eq. (18)] since for such low effective temperatures both the two-level atoms (in the individual case) and the spin-N/2 (in the collective case) are mostly in their respective ground state [cf. Eq. (8)]. Since, however, these states coincide [53] there is no difference between collective or individual atoms and both give rise to the same heat current, J coll i = J ind i , and thus to the same power, P coll = P ind .
By contrast, in the high effective-temperature limit β eff ω 0 → 0 more and more levels of the spin-N/2 become excited. Since the transition probabilities between its individual levels |j (carrying j excitations; see Appendix A) are enhanced by the Clebsch-Gordan coefficients [53], the coupling of the individual levels to the bath is increased compared to the single two-level case σ − = |g e|. These enhanced transition probabilities play a role once the corresponding levels are populated, which requires a sufficiently high effective temperature in the state (8). This behavior is also reflected in the amplification function (13) where the squared coefficients from Eq. (20) are weighted by the respective thermal populations of the corresponding levels of the spin-N/2 particle at inverse temperature β eff . Conversely, for a given value of β eff (which depends on the modulation ω(t) and the bath properties) we find the saturation relation Again, as expected from Fig. 4, in the low effectivetemperature regime β eff ω 0 1 the r.h.s. of Eq. (21) tends to unity, such that even for large particle numbers no power enhancement compared to the individual-atom case can be achieved. By contrast, in the high-temperature regime β eff ω 0 → 0 the r.h.s. of Eq. (21) diverges as 2(β eff ω 0 ) −1 , such that collective effects significantly enhance the power in this parameter regime (see Fig. 5). The saturation value (21) is then extremely sensitive to the value of β eff ω 0 . Hence, there is always a maximum (saturation) atom number above which the power is not amplified further. Consequently, the ideal superradiant scaling behavior (19) for all N is, strictly speaking, only achieved in the strict limit β eff ω 0 → 0.
Dicke superradiance is commonly related to intense short light pulses emitted by a collection of inverted twolevel atoms [38,52,53]. By contrast, here we rather have "continuous" or "persistent" superradiance (see also Ref. [71]). Under steady-state operation photons are continuously exchanged between the baths, the power source and the atoms, which gives rise to collectivelyenhanced steady-state energy flows (cf. Fig. 2).

VI. EXAMPLE: COOPERATIVE ENERGY CURRENTS IN A HEAT ENGINE
As an example, we consider a sinusoidal frequency modulation of the form ω(t) = ω 0 + g sin(Ωt) (22) in the Hamiltonian (1a) under the condition 0 ≤ g Ω ≤ ω 0 [60,61]. For this modulation the weights P (q) in the master equation (5) evaluate to [61] such that the higher Floquet sidebands |q| > 1 do not contribute significantly and can therefore be neglected.   (23) and (24). The maximum power boost of P coll /P ind ≈ 4 is attained for β h ω0 → 0, i.e., for a very high temperature of the hot bath (cf. Fig. 4 for β eff ω0 ≈ 0.51). This maximum boost cannot be increased by adding more atoms since the saturation value (21) also evaluates to 4 (cf. Heat-engine or refrigeration operation (Fig. 1), respectively, require the photons that are exchanged between the atoms and the hot bath to carry more energy than the photons exchanged between the atoms and the cold bath. This requirement can be realized by spectrally-separated baths, e.g., by imposing [60,61] For the sinusoidal modulation (22) and the spectral conditions (24), the thermal machine described by the master equation (5) can on-demand act either as a heat engine or a refrigerator [60,61]. Figures 6 and 7 show the energy currents generated by 100 individual atoms compared to their collective counterparts (the explicit expressions for the currents are given in Appendix D).
Here the minimal value of β eff ω0 is 0.036, which results in a power boost of P coll /P ind ≈ 28 (cf. Fig. 4). Contrary to the situation in Fig. 6 there is a benefit in adding more atoms since here the saturation value (21) evaluates to roughly 56 (cf. Fig. 5). Owing to the smaller gradient between the two bath temperatures, the power is less than in Fig. 6. The cooperative power boost, however, is larger. Note the different scaling of the axes compared to Fig. 6. energy currents may, however, be small despite the superradiant boost (see Fig. 7).
If, however, the cold-bath temperature is predetermined and the task is to produce large power, irrespective of how large the collective boost is, then it is more favorable to simply increase the hot-bath temperature as much as possible so as to generate the largest possible temperature gradient, as expected for any heat engine [6]. Even though the collective effect may then be small, it will still be present (see Fig. 6).
In general, for predetermined finite bath temperatures it is favorable to strive for a collective behavior of the atoms by imposing the symmetry that leads to the establishment of invariant subspaces, provided that the initial condition is carefully chosen. The SU(2) symmetry that manifests itself in the establishment of invariant subspaces is thus a thermodynamic resources that can be exploited to increase the output power of heat engines (or the cooling current of refrigerators) compared to an incoherent additive (and hence, in a sense, classical) ensemble of individual heat engines. Superradiance is established by entanglement between the atoms and is thus a genuine quantum effect [52].
In summary, the maximum boost (19) is extremely sensitive to the bath temperatures (see Figs. 3-5 where for small β eff ω 0 and large atom numbers the boost increases very steeply). Consequently, even though the parameters in Fig. 7 allow for very small values of β eff ω 0 , the maximum enhancement (21) still saturates to roughly 56 (cf. Fig. 5), which, depending on the atom number N , may be far away from the maximum value (N + 2)/3 that is attained in the limit β eff ω 0 → 0 [Eq. (19)]. In general, the latter limit requires both baths to have comparable temperatures much higher than the bare atomic transition frequency ω 0 such that the absolute value of the power decreases significantly compared to the case of distinct temperatures shown in Fig. 6.
Hence, there is a severe tradeoff between the realization of a large quantum boost and the generation of a large power output. Depending on what feature is to be achieved in an experiment, the operation point of the machine must be chosen accordingly.

VII. COLLECTIVE REFRIGERATION
The ratio (17) of the collective energy currents to their individual counterparts is universally valid, independent of whether the thermal machine is operated as an engine or a refrigerator (cf. Fig. 1). Hence, collective effects are not only favorable to engine operation but can also enhance the performance of a refrigerator (i.e., the cold current J c ). This enhancement, however, comes to the price of an equally-scaled larger external power supply.
As seen from Figs. 6 and 7, the heat flow J c increases the colder the hot bath becomes. The largest boost, however, occurs for the smallest possible β eff ω 0 (see Fig. 3), which is the regime where the energy currents vanish (see Appendix D). Just like for the engine case we can conclude that for given bath temperatures there is always a benefit of collective operation (cf. Fig. 4). However, whereas for the engine the boosted power follows from "free" enhanced heat currents, the enhanced refrigerator requires the cost of an equally-scaled larger power investment compared to the individual-atoms case.

VIII. REALIZATION CONSIDERATIONS
The collective behavior that results in the enhancement of the steady-state energy currents is a consequence of the indistinguishably of the atoms with respect to the two baths. This implies an SU(2) symmetry and the existence of invariant irreducible subspaces. The coupling Hamiltonian (1b) thus describes the optimal (ideal) situation. Fortunately, its strict symmetry is not necessarily required to obtain a symmetry-preserving Markovian master equation of the form (5).
Consider, e.g., the interaction of N two-level atoms with the surrounding electric field in dipole approximation [34], (25) where x j is the position of the jth atom, f k the atomfield-mode coupling and b k annihilates a photon in mode k of the electric field. Owing to the position-dependent phases, the operator (25) cannot be cast in the form (3b). Under the Markovian approximation that leads to the master equation (5), however, the atoms effectively only interact with the resonant bath mode (at frequency ω 0 without modulation [53] and at frequencies ω 0 + qΩ with modulation [70], respectively). In one-dimensional geometries, the atoms can hence effectively appear indistinguishable to the resonant bath mode if they are placed at interatomic distances of integer multiples of the resonance wavelength λ 0 = 2πc/ω 0 , such that all atoms scatter in phase [43,55,72]. Since, however, the atoms not only exchange photons with the bath at the bare transition frequency ω 0 but also at the Floquet sidebands [Eq. (5)], one must additionally require 2πc/(ω 0 +qΩ) ≈ 2πc/ω 0 for the relevant q (in the example of Sec. VI these are q = ±1). Hence, under the Markov approximation the dipole Hamiltonian (25) may effectively be replaced by the fully-symmetric Hamiltonian (1b).
The feasibility of superradiance under the interaction (25) in three dimensions has been experimentally demonstrated for atoms confined within a volume much smaller than the cubed wavelength [38,53]. Then, however, the bath-induced dipole-dipole interaction diverges [34]. By contrast, it has been theoretically derived [55] and experimentally demonstrated [43] that in one dimension maximal collective coupling to the bath can be realized while entirely suppressing the bath-induced dipole-dipole interaction by placing the atoms in a chain at distances d = λ 0 or integer multiples thereof.
The above general requirement of the atoms appearing identical to the bath at all Floquet sidebands can be lifted by imposing a spectral separation of the two baths as introduced in Sec. VI for machine operation. For the modulation (22) and the spectral separation (24) the two baths may be realized by two crossed bad cavities (similar to the experiment [73]) or waveguides at different temperatures T c and T h and different resonance frequencies ω 0 ±Ω. Another possibility may be to consider two commensurable modes of a bad cavity [74] at different temperatures where the free spectral range of the cavity is 2Ω, similar to the two-mode Tavis-Cummings model [75,76]. The feasibility of collective strong coupling in multimode cavities has been demonstrated experimentally [44]. Dicke superradiance has also been experimentally realized for superconducting qubits in microwave cavities [47] and for atoms placed along a photonic crystal waveguide [49].

IX. CONCLUSIONS
We have investigated the impact of cooperative manybody effects on the operation of quantum thermal machines. In suitable geometries and for a carefully-chosen initial condition, a collection of N two-level atoms may behave as a giant collective spin-N/2 particle whose energy levels are entangled many-body states. The underlying SU(2) symmetry is manifested in the conservation of the collective spin. This behavior leads to significantlyenhanced energy currents and to a non-extensive scaling of the power output. Namely, the power generated by a quantum heat engine that involves N collective atoms in its working medium surpasses the power generated by N individual engines that operate with a single-atom working medium. We have mapped the thermal machine with N two-level atoms on a machine with a single, fictitious, spin N/2-particle. Consequently, for a given N , an alternative machine with a single, physical, spin-N/2 particle would exhibit the same performance as the discussed collective machine.
Whilst we have here considered the case of a modulated transition frequency [Eq. (1a)], our results also apply to different kinds of driving Hamiltonians since our study is based on a general thermodynamic framework [1,2,60,61,69] for periodically-driven open quantum systems. Notably, the Floquet-Bloch master equation for stronglydriven two-level atoms evaluates to a form similar to Eq. (5) in the dressed states [67,69], such that similar enhancement features are to be expected in that case too. Our approach is general in that it may be applied to heat engines, refrigerators or any other scenarios like heat distributors that, e.g., occur for the laser-driven cooling of a dephasing bath [69,77,78].
The boosted energy currents are due to "continuous" or "persistent" superradiance (see also Ref. [71]). Whereas the power P ind generated by a heat engine that involves N individual two-level atoms as its working medium is, in a sense, classical as it is additive and does not involve quantum-interference effects, the collective power P coll generated under perfect constructive-interference conditions is a genuine quantum many-body effect that involves entanglement between the two-level atoms.