Coherence-assisted single-shot cooling by quantum absorption refrigerators
Mark T. Mitchison, Mischa P. Woods, Javier Prior, Marcus Huber
aa r X i v : . [ qu a n t - ph ] D ec Coherence-assisted single-shot cooling by quantum absorption refrigerators
Mark T. Mitchison,
1, 2, ∗ Mischa P. Woods,
3, 4, † Javier Prior, ‡ and Marcus Huber
6, 7, § Quantum Optics and Laser Science Group, Blackett Laboratory,Imperial College London, London SW7 2BW, United Kingdom Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543 University College of London, Department of Physics & Astronomy, London WC1E 6BT, United Kingdom Universidad Polit´ecnica de Cartagena, C/Dr Fleming S/N 30202 Cartagena, Spain Departament de F´ısica, Universitat Aut`onoma de Barcelona, E-08193 Bellaterra, Spain ICFO-Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain
The extension of thermodynamics into the quantum regime has received much attention in recentyears. A primary objective of current research is to find thermodynamic tasks which can be enhancedby quantum mechanical effects. With this goal in mind, we explore the finite-time dynamics ofabsorption refrigerators composed of three quantum bits (qubits). The aim of this finite-time coolingis to reach low temperatures as fast as possible and subsequently extract the cold particle to exploitit for information processing purposes. We show that the coherent oscillations inherent to quantumdynamics can be harnessed to reach temperatures that are colder than the steady state in orders ofmagnitude less time, thereby providing a fast source of low-entropy qubits. This effect demonstratesthat quantum thermal machines can surpass classical ones, reminiscent of quantum advantages inother fields, and is applicable to a broad range of technologically important scenarios.
I. INTRODUCTION
The development of classical thermodynamics in the19th century underpinned the Industrial Revolution, andthe enormous economic growth and social changes thatfollowed. Now, in the 21st century, the burgeoning quan-tum technological revolution promises unprecedented ad-vances in our computation and communication capabili-ties, enabled by harnessing quantum coherence. As ourmachines are scaled down into the quantum regime, itis of prime importance to understand how quantum me-chanics affects the operation of these devices. This prob-lem has attracted great interest to the field of quantumthermodynamics over the last few years.One useful approach in this regard is to explore simplephysical models which highlight novel aspects of quantumthermal machines. The quantum absorption refrigeratoris the quantum extension of a classical machine devised inthe 19th century (see, for example, Ref. [1] and referencestherein). The smallest possible model with couplings be-tween physical particles and thermal reservoirs was firststudied by Linden et al. [2], who considered a three-qubit refrigerator. While the history of studying thesemachines dates back a long time even in the quantumregime, this three-qubit model was the first where therole of quantum information resources was studied [3],revealing that entanglement in the steady state prohibitsachieving perfect Carnot efficiency, but potentially in-creases cooling efficiency. A wide variety of other quan-tum absorption refrigerator models have also been pro- ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] posed in the recent literature [4–8].Designing thermodynamic processes that can be en-hanced by quantum dynamics is a pivotal challenge in thefield of quantum thermodynamics. One of the paradig-matic thermodynamic tasks concerns work efficiency atthe quantum scale. Here already the very definition ofquantum mechanical work is debated [9–11], yet in differ-ent scenarios quantum mechanical advantages seem pos-sible [12–15]. Another avenue in this endeavour is theengineering of the environment itself to enhance quan-tum processes [16–19]. We circumvent the potential con-troversies regarding the practical value of work and ef-ficiency by concentrating on a different figure of merit,namely the achievable temperature, and by using ther-mal baths to drive the refrigerator, therefore needing nonotion of work.Previous research on the quantum aspects of heat en-gines and refrigerators has focused almost exclusivelyon their operation in the steady state. However, inmany applications one wishes to reach low temperaturesas rapidly as possible, in which case understanding theshort-time behaviour is essential. In particular, the cool-ing may be applied only transiently, after which the coldobject is extracted for use. As a somewhat frivolous yetillustrative example, consider the problem of refrigerat-ing a beverage on a hot day. Maximum enjoyment is ob-tained if the beverage can be cooled and then consumedquickly, at a temperature significantly lower than that ofthe environment. A more serious example could be theinitialisation of a register of qubits for quantum informa-tion processing, where the aim is to produce states withhigh purity (low entropy). Fast cooling is advantageoushere since it may reduce the overall time taken to com-plete the quantum information protocol. Both of thesesituations exemplify what we call single-shot cooling : theone-time application of a refrigeration device in order toconsiderably and rapidly reduce the temperature of theobject in question.In the present work, we study the application of three-qubit absorption refrigerators to single-shot cooling. Byconsidering variations of the basic processes underlyingenergy dissipation and transport, we elucidate the roleof coherence in the operation of such a device. Further-more, we demonstrate that dramatic improvements canbe obtained in both the cooling time and the achievabletemperatures by taking advantage of coherent oscillationsthat appear in the transient dynamics of the refrigerator.Most importantly, the presence of coherence in the deviceallows one to reach lower temperatures than the steadystate. This use of quantum coherence in order to cool be-low the steady-state temperature we refer to as quantumsingle-shot cooling .Note that several physical implementations of quan-tum thermal machines have recently been proposed [20–25], and experimental efforts to construct such devicesin the laboratory are currently under way. Our find-ings could be relevant for future experiments, which maybe practically limited by finite coherence times. Froma more fundamental perspective, our scheme providesone of the first examples in which quantum coherenceplays an active and necessary role in improving the per-formance of machines driven only by thermal noise. II. THEORETICAL FRAMEWORKA. Description of the refrigerator
We consider a quantum absorption refrigerator com-prising three qubits described by standard Pauli opera-tors σ x,y,zi , with the local Hamiltonian H loc = 12 X i =1 E i σ zi . (1)Throughout this article, we employ units of energy, timeand temperature such that E = 1, ~ = 1 and k B = 1.The qubits are coupled together according to a three-body interaction V = g | i h | + h . c ., (2)where the computational basis states | i , | i denote theeigenvectors of σ z . In order for this interaction to beenergy-conserving, in the sense that [ H loc , V ] = 0, we de-mand that the qubit energies satisfy E = E + E . Theinteraction (2) then drives resonant transitions within the transport subspace spanned by the states | i and | i .Each qubit i is in contact with an independent heatbath B i , which drives it towards a thermal equilibriumstate at inverse temperature β i = 1 /T i , where T ≤ T In order to quantitatively analyse the refrigerator wemust specify a thermalisation model. On the other hand,one would like to obtain general results that are indepen-dent of any particular model. In order to avoid being toorestricted by our assumptions, we employ two differentapproaches to modelling the thermal baths. In each case,we assume that the baths are Markovian (memory-less)and weakly coupled to the refrigerator, and we describethe dynamics by a master equation in Lindblad form. Inthe following paragraphs, we describe these models in aqualitative way, deferring the full details to Appendix A. 1. Model I The canonical procedure to model Markovian reser-voirs is to suppose that each qubit is coupled to an infi-nite collection of harmonic oscillators spanning a broadrange of frequencies (Fig. 1(b)). In our case, the bathsare described by identical Ohmic spectral functions of theform J ( ω ) = αω e − ω/ Ω . (5)This function quantifies the strength of the coupling be-tween each qubit and the oscillators near frequency ω ,weighted by the density of states of the reservoir (seeAppendix A). The effect of the baths is therefore char-acterised by two parameters: a dimensionless couplingstrength α , and a frequency cut-off Ω leading to a bathmemory time of order Ω − . Markovian dynamics is ob- tained when Ω is much larger than all other frequencyscales and α ≪ 1, so that the dissipation rates are muchsmaller than the natural frequencies { E i } of the qubits.Some care must be taken when treating the effect ofthe inter-qubit coupling on the thermalisation dynamics.We are able to derive two different master equations de-pending on the magnitude of g . The first equation is validin the strong-coupling limit, where g is much larger thanthe dissipation rates [29]. The second master equationholds in the weak-coupling limit, when g is comparableto or smaller than the dissipation rates [30, 31]. 2. Model II An alternative bath model consists of assigning to eachqubit of the refrigerator an additional fictitious qubit,which is damped by a perfectly Markovian thermal bathat temperature T i , such that the spontaneous emissionrate is γ (Fig. 1(c)). Excitations are then allowed to hopbetween each fictitious qubit and its associated physicalqubit at a rate ηγ . This simulates an effective thermalreservoir with a memory time of order γ − , coupled withstrength η to the physical qubit. Markovian dynamics ofthe physical qubits are therefore obtained when η ≪ γ is larger than all other frequency scales. The en-ergy splitting of each fictitious qubit is chosen to be res-onant with the frequency E i of the physical qubit, whichensures that each physical qubit in isolation is driven to-wards a local thermal state at temperature T i .This model enables us to explore the strong-couplinglimit of large g without invoking the rotating-wave ap-proximation, which involves a time-averaging over theautonomous dynamics of the refrigerator. This time-averaging has been shown to lead to unphysical predic-tions in certain scenarios where the open system is cou-pled to multiple heat baths at different temperatures [30].Thus Model II provides us with an independent check onthe validity of our results. III. RESULTSA. Short-time dynamics of the refrigerator We now present our quantitative results, obtained bynumerical solution of the equations of motion. Our firstobservation is that sufficiently strong coherent couplingbetween the qubits drives damped Rabi oscillations ofthe local qubit populations. However, unlike Rabi oscil-lations due to local driving fields, the three-body inter-action (2) does not induce any local coherences betweenthe qubit populations. The reduced state of each qubitis diagonal at all times and may therefore be assigned aneffective temperature˜ T i ( t ) = − E i − h σ zi ( t ) i . (6)To illustrate this feature of the short-time dynamics,we plot several examples of the evolution of the coldqubit temperature in Fig. 2, which demonstrate that theRabi oscillations allow the cold qubit to reach lower tem-peratures than the steady state. In the strong-couplingregime we find damped temperature oscillations with theapproximate period π/g . Therefore, the optimal quan-tum single-shot cooling procedure consists of extractingthe cold qubit after a time t ≈ π/ (2 g ). So long as thecoupling g is larger than the relaxation rate, the samequalitative behaviour is found in both Model I (solid linein Fig. 2(a)) and Model II (solid line in Fig. 2(c)). Incontrast, when g is much smaller than the relaxationrate, these oscillations are over-damped so that no tem-perature minimum occurs in a finite time (solid line inFig. 2(b)). Nevertheless, we have found that quantumsingle-shot cooling is possible over a very broad rangeof parameters, so long as the coupling g is significantlylarger than the relaxation rate.The physical origin of the temperature oscillations isthe exchange of energy between the refrigerator qubitsdue to the interaction (2). Our second important obser-vation is that this energy transport is driven by coherencein the transport subspace. This can be seen straightfor-wardly by examining the Heisenberg equations of motionfor the local energy expectation values h i = E i h σ zi i / h d t = ˙ Q ( t ) − gE Im C ( t ) . (7)This equation represents an energy balance between therate of heat absorbed from the bath ˙ Q ( t ) and the co-herent flow of energy into the other qubits, which is pro-portional to the imaginary part of the coherence in thetransport subspace: C ( t ) = Tr [ ρ ( t ) | ih | ] , (8)where ρ ( t ) denotes the quantum state. Eqs. (7) & (8) givea direct link between the presence of coherence and theflow of energy across the refrigerator. We expect an anal-ogous relationship to hold for any quantum refrigeratorin which energy transport between distinct sub-systemsoccurs via coherent Hamiltonian evolution.Note that the same mechanism, whereby coherence en-hances the flow of energy, has recently been shown to leadto an operational advantage for heat engines and powerrefrigerators working in the steady state [15]. We alsomention the similarity of our protocol with algorithmiccooling [32, 33]. Indeed, the three-qubit unitary swap op-eration we employ is formally identical to that proposedfor specific algorithmic cooling protocols [34], which mayalso be used in a single-shot operation similar to our pro-posal. The novel feature of our set-up is that no workis required to perform cooling. Rather, the free energysource is provided by the temperature difference betweenthe hot thermal bath at temperature T h and the ambientenvironment at temperature T r . This temperature dif- FIG. 2. Effective temperature dynamics of the cold qubit,with E = 1, E = 2, T r = T = T = 50 and T h = T = 100.(a-c) Blue solid lines indicate the evolution of an initial ther-mal product state, red dotted lines depict the evolution with1% of the maximum coherence added to the initial state.(a) Model I in the strong-coupling regime. (b) Model I inthe weak-coupling regime, plotted with the correspondingstochastic model (black dot-dashed line). (c) Model II in thestrong-coupling regime. (d) Short-time effective temperaturedynamics of an ensemble of cold qubits, where coherence withsome phase noise is added to the initial state. The magnitudeof the initial coherence is 5% of the maximum and the phaseuncertainties are δφ = 0 (black solid line), δφ = π/ δφ = π (red dot-dashed line). The tempera-ture minimum in the absence of phase noise is shown by thegreen vertical dashed line. The data shown in (d) are calcu-lated using Model I with the same parameters as (a). ference establishes a population bias between the states | i and | i , which enables the unitary generated by V to redistribute excitations, thereby cooling the targetqubit. B. Effect of initial coherence To further elucidate the fundamental role of coherencein energy transport, we consider a situation where somecoherence in the transport subspace is added to the initialstate, without modifying the thermally distributed popu-lations. In order to ensure the positivity of the quantumstate, the magnitude of this coherence is upper boundedby |C (0) | < C max , where C max = Y i =1 12 sech (cid:18) β i E i (cid:19) . (9)For each example in Figs. 2(a-c) we have also plottedthe dynamics with a very small amount of initial coher-ence C (0) = i C max / C (0) de-termines the direction of the initial flow of energy into thequbit. Therefore, it is even possible, for example, to heatthe qubit by adding initial coherence with the oppositephase, even when the system behaves as a refrigeratorin the steady state. Likewise, one can transiently cooleven when the steady-state behaviour is that of a heatpump. Similar phase effects have recently been describedin Ref. [41], where the authors show that bath fluctua-tions can revert the detailed balance condition in certainopen quantum systems, creating a net flux of energy fromthe environment into the system.Due to the aforementioned sensitivity of the dynamicsto the phase of the coherence, it is important to con-sider the impact of any phase uncertainty resulting fromimperfect initial state preparation. Specifically, we sup-pose that the initial coherence is given by C (0) = i r e i φ ,where φ is a zero-mean random variable. Assuming thatthe cooling protocol is performed repeatedly on multiplequbits, the relevant quantity to consider is the tempera-ture of the ensemble of cold qubits. For the rest of thissection it should be understood that the term tempera-ture refers to the property of the ensemble. (A charac-terisation of the temperature fluctuations of individualqubits is beyond the scope of this article.)The introduction of phase noise leads to two effects: anincrease of the minimum temperature, and a shift in thetime at which this minimum occurs. In Appendix B weprovide an approximate analytical quantification of theimpact of phase noise, assuming that the phase fluctu-ations are small and approximately Gaussian. We havealso numerically investigated the case of non-Gaussianphase noise, in particular the case where φ is uniformlydistributed in the range φ ∈ [ − δφ/ , + δφ/ C. Stochastic absorption refrigerator In the previous two sections we presented strong evi-dence that coherence is a useful resource that may be har-nessed to produce a significant advantage to single-shotcooling. On the other hand, we can show that coherenceis not a necessary ingredient for quantum absorption re-frigerators operating in the steady state. This is becauseenergy transport can also be described by a stochasticprocess in which excitations are transferred incoherentlybetween the qubits. Such a classical description is ap-propriate in the presence of strong dephasing [42], whichin our case comes directly from the thermal baths (seeRef. [15] for an alternative dephasing model in the con-text of power refrigerators).As a concrete example, in Appendix C we derive aneffective master equation describing the asymptotic re-laxation of the populations in the computational basis,valid when the coupling g is much smaller than the dissi-pation rates. We show that energy transport in this limitcan be effectively modelled by stochastic transitions be-tween the states | i and | i at a rate 2 g / Γ, whereΓ is a characteristic dephasing rate due to the action ofthe thermal baths (see Eq. (C14)).The effective master equation accurately describes thelong-time temperature dynamics of the cold qubit on itsapproach to the steady state (black dot-dashed line inFig. 2(b)). This result demonstrates that three-qubit re-frigerators are capable of cooling even when their dynam-ics is entirely classical. We see from Fig. 2(b) that thereis no difference between the quantum and classical modelin terms of the temperatures which can be achieved in thesteady state.In the transient regime, on the other hand, the temper-ature dynamics are qualitatively different in the quantumand classical cases. Classical refrigeration models de-scribed by stochastic rate equations generically exhibitpure exponential relaxation, precluding the possibility ofsingle-shot cooling below the steady-state temperature.This lies in stark contrast to the generic behaviour wehave found in strongly coupled quantum absorption re- FIG. 3. (a) Example evolution using the same parametersas Fig. 2(a), with (blue solid line) and without (green dot-ted line) a switch-off of the interaction at time t = π/ (2 g ),illustrating the quantum temperature and time advantages∆ T and t Q , respectively. The steady-state temperature T ∞ is shown by the red dashed line. (b) Trade-off between thefractional temperature advantage and the time advantage (inunits of the relaxation time γ − r ), with the same parametersas (a) apart from the temperatures. Each line shows the re-sults for a fixed room temperature T r and a range of hot bathtemperatures T h ∈ [ T r +1 , T r +200]. The value of t Q decreasesas T h is increased in each case. frigerators, i.e. temperature oscillations enabling us torapidly achieve temperatures lower than the steady state. D. Comparison with steady-state cooling Returning now to the fully quantum case, we wouldlike to compare the performance of quantum single-shotcooling and steady-state refrigeration. Let us define thetemperature advantage ∆ T = T ∞ − ˜ T ( t ) as the dif-ference between the steady-state temperature T ∞ of thecold qubit and the quantum single-shot cooling minimum˜ T ( t ). This is an important figure of merit quantifyingthe advantage gained from quantum single-shot coolingcompared to steady-state cooling. Once the cold qubitis decoupled from the refrigerator at time t , its temper-ature increases as it equilibrates with the environment.The advantage gained from quantum single-shot coolinglasts only until the temperature of the cold qubit growslarger than the steady-state temperature, which occursat a time t . This motivates us to define the quantumadvantage time t Q = t − t , which is another impor-tant quantity characterising the performance of the quan-tum single-shot refrigerator compared to its steady-statecounterpart. See Fig. 3(a) for a graphical depiction ofthese quantities.For simplicity, in this section we work in the strong-coupling limit of Model I, where there always exists somequantum single-shot cooling advantage and the optimalswitch-off time is given approximately by t ≈ π/ (2 g ).In this case, the quantum advantage time t Q tends todecrease as either of the bath temperatures T r and T h is increased. Of course, as the room temperature is var-ied, the rate of equilibration of the cold qubit with itsenvironment changes, which in turn affects the value of t Q . In order to remove this trivial dependence on T r ,we measure the quantum advantage time in units of the relaxation time γ − r , where γ r = [ γ ( E ) + γ ( − E )] / γ ( ± E ) are given in Appendix A).For a given set of fridge energies { E i , g } and a fixedroom temperature T r , one would like to optimise the tem-perature of the free energy source T h to obtain the largestfractional temperature advantage ∆ T /T ∞ and also thelongest possible advantage time t Q . We show in Fig. 3(b)that there exists a trade-off between these two quanti-ties, which cannot be simultaneously maximised. Thistrade-off is reminiscent of the competing goals of effi-ciency and power when optimising thermal machines thatoperate in the steady-state. Interestingly, there exists a“sweet spot” where the fractional temperature advan-tage is maximised, while the advantage time remains ad-equately large. This immediately suggests a favorable op-erating regime in which the quantum single-shot coolingprotocol is preferable to steady-state cooling. However,in a practical setting the optimum refrigerator param-eters will of course depend on the specific applicationin question. Finally, it is worth noting that introduc-ing coherence in the transport subspace into the initialquantum state further enhances the fractional tempera-ture advantage for a given t Q , thus coherence is also auseful resource in this context. IV. CONCLUSIONS We have studied the dynamical evolution of three-qubit absorption refrigerators in different regimes, usingtwo different models of thermal dissipation in open quan-tum systems. In both these models we encounter oscil-lations in the temperature of the target qubit below thesteady-state temperature, reinforcing the notion of a uni-versal and robust feature of quantum refrigerators. Weshow that with the right timing these oscillations can beexploited in a quantum single-shot cooling protocol toyield a constant stream of cold qubits. Numerous quan-tum information processing protocols require (approxi-mately) pure input states [44], potentially making quan-tum thermal machines a useful addition as pre-cursors toinformation processing protocols.Experimental implementation of quantum single-shotcooling calls for the strong-coupling limit, in which therate of coherent energy transport exceeds the rate of ther-mal dissipation. This parameter regime may be difficultto achieve in currently proposed set-ups [21–23]. Never-theless, we hope that our results will serve as an encour-agement for further theoretical and experimental worktowards achieving strong coupling in quantum absorp-tion refrigerators. Note that a recent work has foundevidence that quantum single-shot cooling is even possi-ble in the weak-coupling limit [45], providing yet moremotivation to experimentally study transient effects inquantum thermal machines.The advantage of the absorption refrigerator lies in itshigh degree of autonomy: no external energy source isrequired to keep the machine running. Since the initialstates we consider come for free in the context of thermo-dynamic resource theories [46], or more explicitly comedirectly from their respective baths, two thermal reser-voirs at a given temperature are sufficient to let the ma-chine run for as long as their temperatures do not changesignificantly (which for macroscopic baths in contact withquantum systems is sufficiently long for all practical pur-poses).Note that our scheme is not completely autonomous,because we have assumed that the qubit-qubit interactioncan be switched on and off by external control. However,it is also possible to envisage a scenario where the switch-ing is performed by a quantum clock [47]. In this case,the global Hamiltonian would be time-independent andthe protocol becomes fully autonomous. However, a moredetailed study is required in order to understand how aquantum mechanical clock may be used to control themachine: this will form the topic of a future publication.Apart from the prospect of information processing, ourresults help to elucidate the quantum nature of thermo-dynamics. Compared to its classical counterpart, ther-modynamics at the quantum scale often results in ad-ditional constraints and limitations [10, 48] due to thediscretised nature of the fundamental states. The cen-tral question in this context concerns the actual impactof coherence, entanglement and other genuine quantumfeatures on the potential transformations in quantumthermodynamics. Indeed, researchers have only just re-cently started investigating coherences in the context ofresource theories, and a complete understanding is stillelusive [36]. As we show using the stochastic absorption refrigerator, weakly coupled three-qubit fridges operat-ing in the steady state are in some sense equivalent toclassical ones, as one can achieve identical steady-statetemperatures from a classical stochastic model. On theother hand, the coherent transport of energy inducing os-cillations in the population of the cold qubits constitutesa unique quantum feature. This points towards the po-tential of harnessing genuine quantum resources to takethermal machines beyond the classically possible. ACKNOWLEDGEMENTS We acknowledge inspiring discussions with JonatanBohr-Brask, Nicolas Brunner, Karen Hovhannisyan,Marti Perarnau and Martin Plenio. We also thankMartin Plenio for helpful comments on the manuscript.M.T.M. is grateful to the LIQUID institute for their kindhospitality during the completion of this work, and ac-knowledges financial support from the UK EPSRC viathe Controlled Quantum Dynamics CDT. M.W. acknowl-edges funding from the Singaporean Ministry of Edu-cation, Tier 3 Grant Random numbers from quantumprocesses (MOE2012-T3-1-009). J.P. acknowledges fund-ing by the Spanish Ministerio de Econom´ıa y Compet-itividad under Project No. FIS2012-30625. M.H. ac-knowledges funding from the Juan de la Cierva fellowship(JCI 2012-14155), the European Commission (STREP“RAQUEL”) and the Spanish MINECO Project No.FIS2013-40627-P, the Generalitat de Catalunya CIRITProject No. 2014 SGR 966. Data underlying workfunded by EPSRC can be found in a MATLAB file onthe arXiv preprint server (arXiv:1504.01593 [quant-ph]). [1] A. Levy and R. Kosloff. Quantum absorption refrigera-tor. Phys. Rev. Lett. , 108:070604, 2012.[2] N. Linden, S. 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Oppenheim. (Quantumness in thecontext of) resource theories. Int. J. Mod. Phys. B ,(27):1345019, 2013.[47] A. S. L. Malabarba, A. J. Short, and P. Kammerlander.Clock-driven quantum thermal engines. New J. Phys. ,17(4):045027, 2015.[48] F. G. S. L. Brand˜ao, M. Horodecki, N. H. Y. Ng,J. Oppenheim, and S. Wehner. The second lawsof quantum thermodynamics. Proc. Nat. Ac. Sci. ,112(11):201411728, 2015.[49] H. P. Breuer and F. Petruccione. The Theory of OpenQuantum Systems . Oxford University Press, 1st edition,2007. Appendix A: Thermalisation models In this Appendix we describe the two models of ther-malisation used throughout this work, and derive the ap-propriate master equations. 1. Model I In this model, each heat bath is represented as a col-lection of harmonic oscillators, so that the total Hamil-tonian for the three baths is H B = P i =1 H B i , with H B i = X k ν i, k b † i, k b i, k , (A1)where the bosonic mode operators satisfy canonical com-mutation relations [ b i, k , b † j, k ′ ] = δ ij δ kk ′ and [ b i, k , b j, k ′ ] =0. The qubit-bath interaction is given by H AB = X i =1 A i ⊗ X i (A2)where A i = σ xi and the collective bath coordinates aredefined by X i = X k (cid:16) λ i, k b i, k + λ ∗ i, k b † i, k (cid:17) , (A3)with constants λ i, k that control the strength of the cou-pling of qubit i to its associated bath. a. Strong-coupling limit We now sketch the derivation of the strong-couplingmaster equation, valid when g & E i . This master equa-tion describes dissipation as resulting from incoherenttransitions between the eigenstates of the full coupledHamiltonian H A = H loc + V . When g = 0, these eigen-states are simply the computational basis states. When g = 0, the interaction splits the degenerate states span-ning the transport subspace into two new eigenstates,denoted by |±i = ( | i ± | i ) / √ 2, with correspond-ing energy eigenvalues E ± g . The remaining eigenstatesand eigenvalues are left unchanged.Working in an interaction picture with respect to H A + H B , the time evolution of the system coupling operatorsis given by A i ( t ) = e i H A t A i e − i H A t . We decompose this into Fourier components as A i ( t ) = X ω e − i ωt A i ( ω ) , [ H A , A i ( ω )] = − ωA i ( ω ) , (A4) where the Bohr frequencies { ω } denote the set of all pos-sible (positive and negative) energy differences betweenthe eigenvalues of H A .We assume that the initial state of the systemfactorises as ρ (0) = ρ A (0) N i =1 ρ B i , where ρ B i =e − β i H Bi / Tr (cid:0) e − β i H Bi (cid:1) , and that the system-bath cou-pling is sufficiently weak that perturbation theory canbe applied. The master equation is derived by project-ing the interaction-picture von Neumann equation for thedensity operator onto the subspace spanned by states ofthe form ρ A ⊗ ρ B , where ρ A = Tr B ( ρ ), and truncatingthe resulting equation at second order in the system-bathcoupling (Born approximation). The Markov approxima-tion then consists of assuming that the memory time ofthe bath is much shorter than all typical time scales of thereduced qubit evolution, see Ref. [49] for details. Tracingover the baths results in the following equation of mo-tion for the qubit density operator ˜ ρ A in the interactionpicture:d˜ ρ A d t = X ω,ω ′ X i =1 e i( ω ′ − ω ) t Γ i ( ω ) h A i ( ω )˜ ρ A ( t ) A † i ( ω ′ ) − A † i ( ω ′ ) A i ( ω )˜ ρ A ( t ) i + h . c ., (A5)where we defined the self-energyΓ i ( ω ) = Z ∞ d t e i ωt h X † i ( t ) X i (0) i , (A6)with X i ( t ) = e i H Bi t X i e − i H Bi t . In general, the self-energycan be written Γ i ( ω ) = γ i ( ω ) + i S i ( ω ) where the realpart γ i ( ω ) corresponds to an incoherent transition rate,and the imaginary part S i ( ω ) corresponds to an energyshift which we assume to be negligibly small.We now perform the rotating-wave approximation byaveraging over the oscillating terms in Eq. (A5), so thatterms with ω = ω ′ drop out. This approximation isvalid when the typical energy differences are much largerthan the incoherent transition rates, i.e. min { E i , g } ≫ max { γ i ( ω ) } . Transforming back to the Schr¨odinger pic-ture results in a Lindblad equation of the formd ρ A d t = − i[ H A , ρ A ] + X i =1 X ω γ i ( ω ) D [ A i ( ω )] ρ A , (A7)where the dissipators are given by D [ L ] ρ = LρL † − { L † L, ρ } (A8)for a general Lindblad operator L . Explicitly, the Lind-0blad operators are A ( E ) = | i h | + | i h | A ( E + g ) = 1 √ (cid:0) | i h + | − |−i h | (cid:1) A ( E − g ) = 1 √ (cid:0) | + i h | + | i h−| (cid:1) A ( E ) = | i h | + | i h | A ( E + g ) = 1 √ (cid:0) | i h + | + |−i h | (cid:1) A ( E − g ) = 1 √ (cid:0) | + i h | − | i h−| (cid:1) A ( E ) = | i h | + | i h | A ( E + g ) = 1 √ (cid:0) | i h + | − |−i h | (cid:1) A ( E − g ) = 1 √ (cid:0) | + i h | + | i h−| (cid:1) , while the remaining non-zero Lindblad operators, cor-responding to the reverse processes, are found from A i ( − ω ) = A i ( ω ) † .In order to actually evaluate the rates γ i ( ω ), we intro-duce the spectral function of each bath: J i ( ω ) = 2 π X k | λ i, k | δ ( ω − ν i, k ) . (A9)In the limit of an infinite bath with a smooth density ofstates, the sum over the quantum numbers k can be ap-proximated by an integral, and J i ( ω ) becomes a contin-uous function. We assume that the baths have identicalspectral functions of the Ohmic form J ( ω ) = αω e − ω/ Ω , (A10)where α is a dimensionless coupling parameter and Ω isa cut-off frequency of the system-bath interaction, whichmust be much larger than all other energy scales in orderfor the Markov approximation to hold. The incoherentrates are then given by γ i ( ω ) = (cid:26) J ( ω )[1 + n ( ω, β i )] ( ω > J ( | ω | ) n ( | ω | , β i ) ( ω < , (A11)where n ( ω, β ) = (e βω − − denotes the Bose-Einsteindistribution. b. Weak-coupling master equation We now consider the weak-coupling limit where g iscomparable to the dissipation rates. In this case, theprevious derivation is no longer valid since the rotating- wave approximation does not apply to counter-rotatingterms of frequency ( ω − ω ′ ) ∼ g . Instead we should workin an interaction picture generated by H loc + H B andtreat the interaction V between the qubits as a smallperturbation[30, 31]. The time evolution of the systemcoupling operators is now given by A i ( t ) = e i H loc t A i e − i H loc t , with the corresponding Fourier decomposition A i ( t ) = X ω e − i ωt A i ( ω ) , [ H loc , A i ( ω )] = − ωA i ( ω ) , where the frequencies { ω } represent the eigenvalue dif-ferences of H loc only.As before, we assume that the initial state of the sys-tem factorises as ρ (0) = ρ A (0) ⊗ ρ B . Now we project theinteraction-picture von Neumann equation onto states ofthe form ρ A ⊗ ρ B and truncate the resulting equation atsecond order in the qubit-bath interaction and the qubit-qubit interaction. We then perform the Markov approx-imation and trace over the bath variables; see Ref. [31]for full details of the derivation. The resulting equationof motion isd˜ ρ A d t = − i[ V, ˜ ρ A ]+ X ω,ω ′ X i =1 e i( ω ′ − ω ) t Γ i ( ω ) h A i ( ω )˜ ρ A ( t ) A † i ( ω ′ ) − A † i ( ω ′ ) A i ( ω )˜ ρ A ( t ) i + h . c . ! , (A12)where the self-energy Γ i ( ω ) is defined by Eq. (A6). Wethen write Γ i ( ω ) ≈ γ i ( ω ), neglecting the imaginary partcorresponding to small energy shifts of the qubit energysplittings. The rotating-wave approximation now con-sists of crossing off counter-rotating terms with ω = ω ′ ,all of which have frequencies of order E i . In order to beconsistent with our assumptions, we must have E i ≫ g .The resulting master equation in the Schr¨odinger pictureis d ρ A d t = − i[ H A , ρ A ]+ X i =1 (cid:2) γ i ( E i ) D [ σ − i ] ρ A + γ i ( − E i ) D [ σ + i ] ρ A (cid:3) , (A13)where the rates are given by Eq. (A11). 2. Model II In this model, the baths are represented by three ad-ditional fictitious qubits described by Pauli operators1 τ x,y,zi , with Hamiltonian H F = 12 X i =1 E i τ zi . (A14)We have chosen these qubits to have identical energysplittings to their associated physical qubits, in order toavoid renormalising the physical qubit energy splittings.We introduce a Lindblad dissipator for each fictitiousqubit corresponding to damping by a perfectly Marko-vian (delta-correlated in time) reservoir at temperature T i . For simplicity, we assume identical spontaneous emis-sion rates γ for the three fictitious qubits. The couplingto the refrigerator is described by the Hamiltonian H AF = X i =1 ηγ ( σ + i τ − i + σ − i τ + i ) , (A15)where σ ± i = ( σ xi ± i σ yi ) and τ ± i = ( τ xi ± i τ yi ), while η isa small dimensionless parameter. The density operator ρ AF of the six-qubit system is therefore described by themaster equationd ρ AF d t = − i[ H A + H F + H AF , ρ AF ]+ γ X i =1 (cid:2) D [ τ − i ] ρ AF + e − β i E i D [ τ + i ] ρ AF (cid:3) . (A16)The effective spectral density seen by each physical qubitis of the Lorentzian form J i ( ω ) = η γ Γ i Γ i + ( E i − ω ) , (A17)where Γ i = γ (1 + e − β i E i ) / 2. For a fixed η and T i , thebandwidth of the effective bath’s frequency response istherefore controlled by modifying the parameter γ . Appendix B: Phase sensitivity analysis In this Appendix we consider the effect on coolingof coherence in the initial quantum state with a smallamount of phase noise. In order to gain some analyticalinsight, we suppose that the dynamics can be treated asapproximately unitary over the time scales of interest.This is a decent approximation in the strong-couplinglimit, where g is much larger than the relaxation rate.We define the populations a ( t ) = h | ρ A ( t ) | i and b ( t ) = h | ρ A ( t ) | i , and the coherence C ( t ) = h | ρ A ( t ) | i , where ρ A ( t ) denotes the reduced quan-tum state of the refrigerator qubits. We also introducethe convenient parametrisations S = [ a (0) + b (0)] / D = [ b (0) − a (0)] / C (0) = i r e i φ , where r > φ is the phase. The refrigeration con-dition T v < T , where T v is the virtual temperature (6) and T is the initial temperature of the cold qubit, impliesthat D > a ( t ) , b ( t ) is analogous to that of a resonantly driven two-level system. In this approximation, the populations attime t are given by a ( t ) = S − D cos(2 gt ) + r cos( φ ) sin(2 gt ) , (B1)and b ( t ) = 2 S − a ( t ). The population difference of thecold qubit is given by h σ z ( t ) i = h σ z (0) i + 2 [ a (0) − a ( t )] , (B2)which is related to the effective temperature by Eq. (6).From this we find that the first temperature minimumoccurs at time t min ( φ ) = 12 g (cid:18) π − tan − (cid:20) r cos( φ ) D (cid:21)(cid:19) . (B3)with a ( t min ) = S + p D + r cos ( φ ) . (B4)The amplitude of the population oscillations is thereforemaximised by choosing φ = 0, i.e. a purely imaginaryinitial coherence, as expected from Eq. (7). With thechoice of φ = 0, the optimal time to extract the coldqubit is given by t opt = t min (0).We now suppose that each qubit entering the refriger-ator is prepared with a random phase φ of the coherencein the transport subspace, due to experimental noise, forexample. These phase shifts are described by some prob-ability distribution function p ( φ ). In order to find thedynamics of the resulting ensemble of cold qubits, we av-erage Eq. (B1) over the distribution p ( φ ). This procedureleads to the replacement of cos( φ ) in all expressions bycos( φ ) = Z π − π d φ p ( φ ) cos( φ ) . (B5)As a simple but broadly relevant example, we assumethat phase shift has zero mean φ = 0, and that its vari-ance φ is smaller than unity. We also suppose that itscumulants are rapidly decreasing, so that p ( φ ) may beapproximated by a Gaussian distribution. Under theseassumptions, we obtaincos( φ ) = e − φ / . (B6)We find from Eq. (B3) that the minimum temperatureof the ensemble occurs at time t ′ opt = 12 g π − tan − " r e − φ / D , (B7)so that t ′ opt > t opt in this approximation. In the follow-2ing, primed variables indicate quantities in the presenceof noise, while the unprimed variables denote the corre-sponding quantities in the absence of noise.In order to find the shift in the ensemble temperaturedue to the phase noise, we consider two distinct scenar-ios. In the first scenario, we assume that some informa-tion about the phase distribution is known, so that it ispossible to predict in advance the optimal time t ′ opt toextract the qubit. This information could be obtainedby expending some qubits in order to characterise thenoisy preparation by quantum state tomography. In thiscase, the phase noise leads to a change in the populationdifference of the cold qubit given by∆ h σ z i = h σ z ( t ′ opt ) i ′ − h σ z ( t opt ) i = 2 p D + r − q D + r e − φ ≈ r √ D + r (cid:16) − e − φ (cid:17) . (B8)In the second scenario, we assume that no informa-tion about the phase noise distribution is known, so thatthe experimenter extracts the cold qubit at the incor-rect time t opt . For example, this scenario applies if onlya single qubit is available for cooling, so that the noisypreparation cannot be characterised in advance. In thiscase, phase noise changes the population difference bythe amount∆ h σ z i = h σ z ( t opt ) i ′ − h σ z ( t opt ) i = 2 r √ D + r (cid:16) − e − φ / (cid:17) . (B9)Note that it is also possible to consider the case whereonly partial information about the noise distribution isknown, in which case we expect ∆ h σ z i to lie somewherebetween Eqs. (B8) and (B9).In any case, assuming that the effect of phase noiseis small, we find that the expected shift in the ensembletemperature to lowest order in ∆ h σ z i is∆ ˜ T = 2 ˜ T E cosh (cid:18) E T (cid:19) ∆ h σ z i , (B10)where ˜ T denotes the optimal temperature in the absenceof phase noise. This equation indicates that the effect ofphase fluctuations is minimised when ˜ T ∼ E , while themost drastic effects occur for very large or very smalltarget temperatures. In particular, the right-hand sideof Eq. (B10) diverges as ˜ T → 0, which indicates thateven vanishingly small fluctuations in the initial coher-ence phase may prevent one from using this coherence tocool the cold qubit to its ground state.In Fig. 4(a) we numerically calculate and plot an ex-ample of the change in the final temperature of the coldqubit when it is disconnected from the machine, eitherwith or without sufficient knowledge of the phase noisedistribution to predict the optimal extraction time. We x 10 -3 FIG. 4. Comparison of exact numerical and approximate ana-lytical results for the effect of Gaussian zero-mean phase noisecharacterised by its variance φ . The same parameters areused as in Fig. 2(a), with 5% of the maximum initial coher-ence. (a) Shift in the cold qubit temperature when extracted:numerical (blue solid line) and analytical (red dashed line)results when the phase noise distribution is not known, nu-merical (black dot-dashed) and analytical (green dotted line)results when the phase noise distribution is known. (b) Shiftin the time of the temperature minimum ∆ t opt = t ′ opt − t opt :numerical (blue solid line) and analytical (red dashed line)results. see that in fact it makes little difference to the tempera-ture whether or not the optimal extraction time is knownprecisely, since the shift in this time due to phase noiseis rather small (Fig. 4(b)). Our approximate analyticalresults are also shown in Fig. 4 for comparison. Appendix C: Stochastic refrigerator model In this Appendix we derive an effective stochasticmodel of the quantum absorption refrigerator in the deepweak-coupling limit. Our analysis follows Ref. [42], whereit was shown that coherent propagation of a quantumparticle on a lattice under strong local dephasing can beapproximated by a stochastic hopping process. In ourcase, the dephasing is provided directly by the actionof the thermal baths, which destroy the coherences inthe computational basis at a characteristic rate Γ. Inthe limit where g ≪ Γ, we can derive a closed equationof motion for the populations, describing their dynamicsover times coarse-grained on the scale Γ − .Working in the weak-coupling limit of Model I, we be-gin with the master equation (A13). This may be writtenas d ρ d t = L ρ, (C1)where we introduced the Liouvillian superoperator L ,and for brevity we have written ρ = ρ A for the reducedstate of the refrigerator qubits. We decompose the Li-ouvillian as L = L + V , where V ρ = i[ ρ, V ], while thesuperoperator L contains both the local Hamiltonianand dissipative contributions to the Liouvillian.The quantum state can be expanded in the eigenbasis3of L as ρ ( t ) = X λ ρ λ e λt , (C2)where ρ λ is an eigenvector of L with complex eigenvalue λ . Since Re( λ ) ≤ 0, we see that only the eigenvalues withthe largest real part are relevant as t → ∞ . In this basisthe master equation reduces to the eigenvalue equation( L + V ) ρ λ = λρ λ . (C3)The eigenvalues of L are of order Γ, where Γ is a char-acteristic dephasing rate of the thermal dissipation. Onthe other hand, the eigenvalues of V are of order g , whichis much smaller than Γ by assumption. We can thereforetreat the effect of V as a small perturbation.We introduce a projector P defined by P ρ = X n =0 h n | ρ | n i | n i h n | , (C4)where {| n i} are the computational basis states. Thisprojects onto the space of populations (diagonal matrixelements) in the computational basis. We also define itsorthogonal complement by Q = 1 − P , which projectsonto the space of coherences (off-diagonal matrix ele-ments). We refer to the spaces of populations and co-herences as the P -space and the Q -space, respectively.It is readily verified that the following properties hold: PVP = 0 , (C5)[ P , L ] = [ Q , L ] = 0 . (C6)Eq. (C5) states that the interaction V only couples pop-ulations to coherences, while Eq. (C6) reflects the factthat the local Liouvillian L does not couple populationsand coherences, i.e. it is block-diagonal.By introducing the identity 1 = Q + P into both sidesof the eigenvalue equation (C3), we obtain λ P ρ λ = L P ρ λ + PVQ ρ λ (C7) λ Q ρ λ = L Q ρ λ + QVQ ρ λ + VP ρ λ , (C8)where Eqs. (C5) and (C6) have been used. We now solveEq. (C8) for Q ρ λ and substitute the result back into Eq. (C7), finding λ P ρ λ = L P ρ λ + PV ( λ − QV − L ) − VP ρ λ . (C9)Since we are looking for eigenvalues λ which are smallin magnitude, we may neglect the term λ − QV , whichis of order g , in comparison to L , which is of order Γ.Hence we obtain the approximate eigenvalue equation λ P ρ λ = L eff P ρ λ , (C10)where L eff = L − PVL − VP . (C11)As t → ∞ , the master equation (C1) can therefore beapproximated in the P -space byd ρ d t = L eff ρ. (C12)Note that L eff is well defined by Eq. (C11) despite thefact that the unperturbed Liouvillian possesses a zeroeigenvector (the steady-state solution): L ρ ∞ = 0. Thisis because the unique steady-state solution lies in the P -space, i.e. ρ ∞ = P ρ ∞ . Furthermore, one can write VP = QVP by virtue of Eq. (C5). Therefore, the operator L − acts only in the Q -space, where all eigenvalues of L arenon-zero.In order to calculate L eff explicitly, we note that V annihilates all populations apart from | i h | and | i h | . The action of L eff can therefore be foundfrom its action on these two populations. We also makeuse of the fact that L | i h | = − Γ | i h |L | i h | = − Γ | i h | , (C13)with Γ = 12 X i =1 [ γ i ( E i ) + γ i ( − E i )] , (C14)where γ i ( E i ) is defined by Eq. (A11). Using these prop-erties, a straightforward calculation yields the result L eff = L + 2 g Γ D [ B ] + 2 g Γ D [ B † ] , (C15)where B = | i h | , and D is defined by Eq. (A8).This effective Liouvillian describes transport as astochastic process, which transfers population symmetri-cally between the states | i and | i at a rate 2 g //