Abstract
Thermodynamics is a branch of science blessed by an unparalleled combination of generality of scope and formal simplicity. Based on few natural assumptions together with the four laws, it sets the boundaries between possible and impossible in macroscopic aggregates of matter. This triggered groundbreaking achievements in physics, chemistry and engineering over the last two centuries. Close analogues of those fundamental laws are now being established at the level of individual quantum systems, thus placing limits on the operation of quantummechanical devices. Here we study quantum absorption refrigerators, which are driven by heat rather than external work. We establish thermodynamic performance bounds for these machines and investigate their quantum origin. We also show how those bounds may be pushed beyond what is classically achievable, by suitably tailoring the environmental fluctuations via quantum reservoir engineering techniques. Such superefficient quantumenhanced cooling realises a promising step towards the technological exploitation of autonomous quantum refrigerators.
Introduction
An absorption or heatdriven quantum refrigerator is a system capable of establishing a net steadystate transport of energy from a cold bath (c) to a hot bath (h), assisted only by the residual heat coming from an additional work reservoir (w)^{1,2,3}. In this picture, the cold bath would play the role of the macroscopic or mesoscopic object to be cooled. In addition to their potential technological applications, these autonomous quantumthermal devices are also appealing from the fundamental perspective, as they are naturally well suited for the study of thermodynamics at the level of individual open quantum systems^{1,4,5,6}.
In spite of the increasing interest that quantum absorption cooling has attracted over the last few years^{5,7,8,9,10,11,12}, the field is far from new. A heatdriven quantum fridge is just one specific configuration of the more general quantum heat pump, that can function either as a heater, a chiller or even an engine. The use of threelevel solidstate masers as physical support for heat pumps was already discussed in the late 1950s^{13,14}, when spin refrigeration was also experimentally demonstrated^{15}. The consistent quantumthermodynamic description of these elementary threelevel prototypes was object of further study^{1,4} and, just recently, alternative finitedimensional quantum systems realising autonomous heat pumps have been put forward in the literature^{2,3}.
The different designs of quantum heat pumps share limitations that can be understood from the assumptions on their interactions with the environments. Under the familiar conditions usually met in the quantumoptical regime, the dissipative processes may be assumed purely Markovian^{17,18,19}, which severely restricts the performance of any heatdriven device and confers a distinctive spectral structure to the environmental fluctuations^{16}. In particular, once their steady state builds up, quantum heat pumps are governed by formal analogues of the laws of thermodynamics and, as a consequence, their absolute efficiency ideally saturates to the corresponding Carnot limits ε_{C}, albeit at vanishing ‘cooling power’^{14}, i.e. in the reversible limit, the exchange of any finite amount of energy with the heat baths is performed in infinite time.
For practical purposes, however, one needs to operate at nonvanishing power. In this case, the relevant issue to assess the functionality of these devices demands the optimisation of more practical figures of merit such as the efficiency at maximum cooling power ε_{*}. The natural question arises whether ε_{*} can approach ε_{C} arbitrarily closely even at finite cooling power, or if, on the contrary, it is upper bounded by some fundamental limit. The efficiency at maximum ‘mechanical’ power is extensively used to benchmark the operation of heat engines and a lot of effort has been devoted to establish a universal upper bound therefor^{20,21,22}. Unfortunately, the general arguments used for engines do not provide simple bounds when the cycle is reversed into a refrigerator and consequently, a different approach is needed to arrive to modelindependent performance bounds. Here, we rigorously prove that the ‘smallest’ quantum absorption refrigerators, supported on ideal threelevel masers, are limited in their efficiency at maximum power by a fraction of ε_{C}, only related to the spectral properties of the environmental fluctuations at low frequencies. We show that this general performance bound applies as well to ‘larger’ and nonideal designs^{2,3,8}, as it is independent of the details of the working material of the refrigerator.
Achieving a good understanding of the quantummechanical origin of the limitations of heat pumps can also provide key clues about how to surmount them. We show indeed that, by feeding an absorption fridge with engineered thermal resources, one can push its performance bounds considerably further, allowing for classically impossible superefficient quantum cooling. Namely, at given fixed environmental temperatures, the addition of squeezing to the work bath leads to efficiencies above ε_{C} and, most interestingly, to a systematic enhancement of the output harnessed power. This is achieved strictly within the framework of quantum thermodynamics and thus, in no violation of its laws^{6,23}.
Results
Models of absorption refrigerator
As already advanced, a minimal model of autonomous heat pump^{13} consists of a threelevel system with each of its transitions weakly coupled to one of the three independent heat baths [see Fig. 1(a)]. Essentially, as the steady state builds up, the ‘heat’ collected from the cold bath is dumped into the hot bath with the assistance of the extra energy provided by the work bath, which closes the cooling cycle. Of course, the opposite heating cycle also takes place in the steadystate and it is the imbalance between these two stationary processes which renders the device either a refrigerator or a heater. As we shall see below, refrigeration occurs as long as the frequency of the transition coupled to the cold bath remains below a certain threshold ω_{c} ≤ ω_{c}_{,max}^{1}.
We shall also consider the twoqubit design^{3,6} of Fig. 1(b) in which the cold and the hot bath are each addressed through a twolevel contact, while the work reservoir introduces dissipation in the subspace {0_{c}〉⊗1_{h}〉, 1_{c}〉⊗0_{h}〉}, where 0_{α}〉 and 1_{α}〉 stand for the ground and excited states of contact α ∈ {h, c}. This allows for energy flows between the hot and cold ends and eventually results in net refrigeration. Remarkably, this fridge works within the same cooling window (ω_{c} ≤ ω_{c}_{,max}) as the threelevel prototype.
Finally, we shall comment on another model^{2,5,7} featuring three contact qubits connected via a threebody interaction [see Fig. 1(c)]. Practicallyoriented issues have been recently studied in connection with this design, including its potential experimental realisations^{9,10} and the investigation of its efficiency at maximum power^{8}.
The first and second laws
When the interaction of the working material with the environments is sufficiently weak, one can tackle its effective dynamics via a quantum master equation like
It is an equation of motion for the reduced state of the heat pump , where are the dissipation superoperators associated with each bath. An initial preparation , factorised between system and baths degrees of freedom is assumed. The intrinsic dynamics has been eliminated by taking the interaction picture with respect to the free Hamiltonian of the working material .
If the dissipation is much slower than both the environmental fluctuations and the coherent evolution of the heat pump, the Born, Markov and rotatingwave approximations may be safely applied^{16}. This leads to dissipation superoperators of the well known LindbladGoriniKossakovskiSudarshan (LGKS) type^{17,18}, which are the workhorse of quantum thermodynamics^{23}. We shall specifically denote them by in what follows:
Here, {·, ·}_{+} stands for an anticommutator and , for the jump operator associated with the decay process into channel ω, which occurs at a rate given by the corresponding element of the spectral correlation tensor Γ_{ω}_{,α}. For equilibrium reservoirs, these latter relate via the detailed balance condition^{16}
Individually, each LGKS dissipator generates a completely positive and trace preserving contractive dynamics of the working material, converging towards its local stationary thermal state ^{19}. As time goes to infinity, this contractivity translates into (see Methods for details)
where T_{α} are the equilibrium temperatures of the baths and represents the energy per unit time flowing from bath α into the pump once in the steady state. In particular, we will refer to as the ‘cooling power’ or just power.
Since is timeindependent, the stationarity of the average energy at implies
This balance equation plays the role of a first law in quantum thermodynamics as soon as one identifies the steadystate currents with heat flows. Similarly, Eq. (4) can be regarded as a quantumthermodynamic statement of Clausius theorem, i.e. the second law.
The combination of Eqs. (4) and (5) places the ultimate thermodynamic bounds on the efficiency of a heat pump in its various modes of operation^{14}. For instance, in the chiller configuration (i.e. , and ), the efficiency is defined as the ratio of the cooling power to the input heat (per unit time) provided by the work bath, i.e. . Using Eq. (5) to eliminate from Eq. (4), yields
where ε_{C} is nothing but the Carnot efficiency of a (macroscopic) heat driven quantum absorption fridge operating between baths at temperatures {T_{w}, T_{h}, T_{c}}^{24}.
The cooling window
We now want to delimit the region within the space of parameters of a quantum heat pump where cooling is permitted by the second law. This is of course highly modeldependent but one can always stick to ndimensional working materials with the basic threestroke cooling mechanism of the threelevel maser built in^{13}. If the bath α couples to the heat pump only allowing for transitions with a gap of among the eigenstates of and if the resonance condition ω_{h} = ω_{c} + ω_{w} holds, one should have
This was already acknowledged as a distinctive feature of ideal threelevel heat pumps in the seminal paper by Geusic et al.^{14} and it is easy to see that it remains true for the twoqubit design of Fig. 1(b). Eq. (7) essentially says that in a cooling cycle, every single cold excitation is traded for one hot excitation at the expense of consuming a single work excitation^{7}.
Hence, combining Eq. (4) with (7) yields
This inequality defines the ‘cooling window’. Note that the work temperature T_{w} must be larger than T_{h} (and T_{c}) in order to have a positive ω_{c}_{,max}.
In principle, the efficiency of any ideal heat pump satisfying Eq. (7) saturates to the Carnot bound in the reversible limit of ω_{c} → ω_{c}_{,max}, i.e. when the contact transitions locally equilibrate with their corresponding baths so that the equality in Eq. (4) holds.
On the contrary, the operation of a heat pump might become intrinsically irreversible if additional mechanisms of energy exchange were present. For instance, the threequbit device of Fig. 1(c) only behaves as an ideal heat pump when the effects of the global interaction term on the dissipative dynamics are entirely neglected [see caption of Fig. 1(a)]. Indeed, explicit analytical formulas for its steady state consistent with Eq. (7) may be written down in that limit^{7}. However, since the contact transitions are chosen among the eigenstates of rather than those of , dissipation is always strictly ‘delocalised’ regardless of the interaction strength and neither Eq. (7), nor the equality in Eq. (4) can be satisfied. Consequently, this specific design is an example of nonideal heat pump, as it is unable to reach the Carnot efficiency^{8}.
Efficiency at maximum cooling power: a modelindependent bound
In spite of its fundamental importance, the attainability of the Carnot efficiency in microscopic quantum heat pumps^{7,11} is not the central issue for practical applications. Indeed, when operating at the reversible limit the power exactly vanishes.
One would like instead to run a refrigeration cycle, carefully tuning the design parameters so that efficiency and power are maximised jointly, pretty much in the spirit of finitetime thermodynamics. The practically relevant questions would then be whether there exists a tight upper bound for the efficiency at maximum power ε_{*} other than ε_{C} and whether such bound is modelindependent. Longstanding problems of this sort have been intensively studied in classical macroscopic heat engines and refrigerators^{20,21,22,25,26}, as well as in their quantummechanical counterparts^{27,28,29,30,31,32,33}. In particular, a specific bound for ε_{*} has been recently established for the model of Fig. 1(c) based on a numerical analysis, together with design prescriptions for its saturation^{8}.
In this paper, under natural assumptions on the environmental fluctuations, we prove analytically that the efficiency at maximum cooling power of any ideal fridge made up of elementary threelevel heat pumps is tightly upper bounded by
where d_{c} stands for the spatial dimensionality of the cold bath. This applies directly to the ideal absorption refrigerators of Figs. 1(a) and 1(b), but we verify that Eq. (9) holds as well for the nonideal refrigerator of Fig. 1(c), thus validating and generalising the bound obtained in^{8}. Eq. (9), which is the first main result of this paper, establishes then a quantumthermodynamic limitation which holds for all the models of quantum absorption fridges existing in current literature and is manifestly independent of the details of the working material of the refrigerators. While full details are deferred to the Methods section, we sketch a proof of the bound in what follows.
One may generically characterise the dissipation into a ddimensional free bosonic field α at thermal equilibrium with flat spectral density, by decay rates of the form^{16}
where . Eq. (10) follows just from the application of the Born and Markov approximations to a general microscopic model of relaxation into a bosonic bath. From it, one can see that the cooling power of a threelevel fridge [Fig. 1(a)] writes as
where x ≡ ω_{c}/ω_{c}_{,max}, a_{i} and b_{i} are positive real constants and p(x) is a positive function. The cooling power vanishes at both edges of the cooling window and is a concave function of x. Eq. (11) can always be conveniently recast as
with P(x) again a positive function (see Eq. (41) in Methods). The maximum of is attained at some x_{*} such that
where the prime denotes differentiation with respect to x. Note that the positive and concave function f(x) is maximised precisely at x = d_{c}/(d_{c} + 1).
Previous literature on performance bounds for heat engines has established that the efficiency at maximum power is maximised in the limit of small ε_{C}^{21,34}, as it is also the case for quantum absorption fridges^{8}. We shall therefore expand P(x) around ε_{C} → 0, where in fact P(x) ~ p(x) [see Eq. (41)] and study its analytical properties. We then prove the claim by reductio ad absurdum. Let us first assume that x_{*} > d_{c}/(d_{c} + 1), so that the bound in Eq. (9) would not hold [recall Eqs. (7) and (8)]. P′(x) has at most one root in the interval x ∈ [0, 1] and can be seen to be negative in the neighbourhood of x = d_{c}/(d_{c} + 1), for vanishing ε_{C}. Being both P(x) and f(x) positive in the whole unit interval, the violation of the bound Eq. (9) would contradict Eq. (13). Therefore, one must have instead x_{*} ≤ d_{c}/(d_{c} + 1) and hence,
the bound being saturated if P′(x_{*}) = 0.
Now, we shall consider the twoqubit model [Fig. 1(b)]. We find that its steadystate heat currents can be broken up as (see Methods for details)
where all q_{i} > 0. There are two contributions to the total heat fluxes, namely and , which individually satisfy the first and second law of Eqs. (5) and (4) within the cooling window delimited by Eq. (8). Furthermore, for i = {1, 2} and hence, each threelevel component behaves as an ideal refrigerator on its own. Most importantly, the cold fluxes have the same analytic structure of Eq. (11), so that their efficiency at maximum power is limited by . As a consequence, the performance of the combination of the two is also bounded by ε_{C}d_{c}/(d_{c} + 1), since
The heat currents of the threequbit fridge [Fig. 1(c)] are much more involved analytically. However, extensive numerical evidence confirms that Eq. (9) also applies to this case^{8} [see Fig. 2(c)].
A numerical investigation, presented in Fig. 2, certifies that the bound of Eq. (9) is tight for all the three models. One can further see that choosing is a sufficient condition to approach it closely under large temperature differences . These are to be regarded as analytical design prescriptions for the practical implementation of optimal quantum heat pumps, e.g. following recent proposals involving superconducting qubits or arrays of quantum dots^{9,10}. Note as well that the limit also implies , as should be expected.
Our bound has been analytically established as a constraint on the performance of the fridge operation mode of the threelevel maser and that of any ideal absorption refrigeration cycle reducible to threelevel systems and it has been also shown to hold for fundamentally different nonideal devices. It is in order to remark, however, that Eq. (9) has in fact a general validity transcending specific models. Let us consider a totally generic model of quantum absorption fridge: We can always write the leading contribution to its cooling power as a sum of terms of the form
where stands for excitation/relaxation rates of the contact transitions at ω_{α}. Eq. (17) just formalises the imbalance between the elementary threestroke cooling and heating cycles which has to underly any implementation of quantum absorption cooling and contains no details on the specific working material of the refrigerator, nor about the spectral properties of the reservoirs to which it couples. Given that these three reservoirs are in thermal equilibrium, we can make use of the detailed balance condition of Eq. (3) and thus arrive to
which is formally identical to Eq. (11), with the positive function . Therefore, the whole line of reasoning between Eqs. (11)–(14) is still applicable, provided that the condition p′(x_{*}) ≤ 0 is verified. This weak assumption is sufficient for Eq. (9) to hold, regardless of the physical support of the refrigerator and the specific properties of the baths. Under these premises, we conclude that our bound is modelindependent for quantum absorption refrigerators.
Let us eventually connect this result with the other known bounds from finitetime thermodynamics. In the seminal work by Curzon and Ahlborn^{20}, an upper limit to the efficiency at maximum power was derived for a Carnot engine under the endoreversible approximation^{35}. In spite of its seemingly limited scope, such bound succeeds in capturing the universal behaviour of ε_{*} at small efficiencies^{21,34} and recurrently appears in different models of heat engines^{22,27,31,34,36}, including the threelevel maser in its engine configuration^{34}. However, the obtention of similar results for refrigerators would require a more careful phenomenological modeling of their main sources of irreversibility and can lead to highly modeldependent performance bounds^{37,38,39}.
In analogy with the CurzonAhlborn limit, our result in Eq. (9) accurately represents ε_{*} at low efficiencies for all embodiments of quantum absorption refrigerators, which encompass fundamentally different models. The same limit holds as a strict upper bound to the efficiency at maximum power of ‘classical’ endoreversible absorption chillers^{39}. It must be noted as well, that it relies on a consistent microscopic description of the systembaths interactions rather than on a phenomenological ansatz. From the physical point of view, such a modelindependent bound can be understood from the intuitive notion that the maximisation of the cooling power in the steady state must be governed by the lowfrequency dependence of the corresponding cold decay rate [see Eq. (10)].
Superefficiency: squeezing the second law
Up to now, we have seen how the standard laws of thermodynamics place fundamental constraints on heat pumps, even when they are made up of a single finitedimensional open quantum system. This is perhaps not surprising taking into account that there is nothing really ‘quantum’ about the operation of these devices, other than the discreteness of their energy spectra^{8}. For instance, the ideal fridges of Figs. 1(a) and 1(b) operate in completely classical (diagonal) steady states, while the quantum coherence that builds up asymptotically in the nonideal fridges of Fig. 1(c) does not seem to affect their performance in any crucial way. Even if steadystate bipartite entanglement may exist in this case, it usually appears only in the regime of very low efficiencies and vanishing cooling power^{11}. Likewise, other types of quantum correlations, though widely present, do not have any influence on optimal cooling^{8}.
Is it then possible at all for quantum heat pumps to operate past the ‘classical’ limits? Ideally, one would like to devise tricks to push their performance bounds further, of course remaining always within the standard framework of quantum thermodynamics and not advocating any violation of its laws. We shall devote this section to illustrate how one can indeed go beyond Eqs. (6) and (9), by exploiting nonequilibrium environmental features that can be mimicked with suitable quantum reservoir engineering techniques.
The whole idea would consist in initialising the work reservoir in a squeezedthermal state
Such a state results from the action of the unitary squeezing operator^{40}
on a thermal preparation , where stand for creation and annihilation operators on mode μ of the work reservoir. The parameter ξ ≡ r exp iθ is in general a complex number, although for our purpose we can set the phase θ to zero and consider a real squeezing parameter . The quantumness of a squeezed state resides in the asymmetry of the variances of its field quadratures , as opposed to a (classical) coherent state, for which .
It is well known that squeezed states, even in absence of entanglement, allow for quantum enhancement in several applications of information theory^{41}, including quantum cryptography^{42} and most notably precision measurements^{43} and quantum metrology below shot noise^{44}, with applications e.g. for gravitational wave interferometry^{45,46,47}.
The key property to be exploited here is the nonstationarity of squeezed preparations. This results in a periodic time modulation of the correlation functions of the work reservoir, which is somewhat equivalent to the action of an external driving on the heat pump. In this sense, the work reservoir may now play an active role in the cooling process. As we shall now see, this is achieved without compromising the thermodynamic consistency of the whole setting.
In order to account for the squeezing of the work reservoir, the quantum master equation Eq. (1) must be replaced by
where the modified work dissipator can be cast in the standard LGKS form^{16} (see Methods for details).
Clearly, if the work heat current is suitably redefined as , the asymptotic stationarity of energy implies again the first law as stated in Eq. (5).
However, the dissipator will not generally bring the corresponding contact transition to thermal equilibrium at temperature T_{w}, but to some other steady state . Still, the LGKS form of the overall generator of the dynamical map in Eq. (21) guarantees its full contractivity^{19}, so that the second law generalises to
It is convenient to fit by a thermal state of the form , with a suitable squeezingdependent work temperature . Since the contact with the work reservoir only involves two levels, this is always possible. Eq. (22) may be thus rewritten in the more familiar way
For any r > 0, the effective temperature exceeds T_{w} and it diverges for r → ∞. Therefore, the generalised squeezeddependent Carnot efficiency will always exceed its classical value
Note that, as should be expected, in the limit of r → ∞, Eq. (24) saturates to the maximum efficiency of a powerdriven refrigerator (i.e. a reversed heat engine) ε_{C}(∞) = T_{c}/(T_{h} − T_{c})^{14}.
Hence, we have demonstrated how, for a given set of thermal resources {T_{c}, T_{h}, T_{w}}, classically forbidden superefficient quantum absorption cooling may be realised by just squeezing the work reservoir. This is illustrated in Fig. 3, with a set of performance characteristics corresponding to different environmental squeezing parameters in the threelevel maser heat pump of Fig. 1(a). Perhaps even more remarkable than the seemingly counterintuitive possibility of cooling above the classical efficiency threshold, is the fact that the efficiency at maximum power and the cooling power itself systematically increase jointly with the squeezing parameter. This is a striking consequence of the nonequilibrium environmental fluctuations.
In the macroscopic domain, the practical drawbacks of an absorption chiller^{48}, as compared to an equivalent workdriven refrigeration cycle, are typically its much lower efficiency and output power. Nevertheless, as we can see in Fig. 3, a quantumenhanced heatdriven fridge does not only outperform its purely thermal counterpart, but is also capable of approaching very closely the performance of a Carnot fridge, when provided with a finite amount of environmental squeezing. Thus, as opposed to the classical case, quantum absorption cooling may compete on an equal footing with conventional refrigeration, when combined with reservoir engineering techniques^{49,50,51} to allow for the exploitation of nonequilibrium squeezed environmental fluctuations. This observation could be a promising first step towards a number of applications of heatdriven refrigeration to a new generation of quantum and nanoscale technologies.
Discussion
Let us start by summarising the two main results of this paper. On the one hand, we proved that the efficiency at maximum power of all the known models of quantum absorption refrigerator (both ideal and nonideal) is tightly upper bounded by a fraction of the Carnot efficiency ε_{C}, which is independent of the details of the device and only relates to the spectral properties of the thermal fluctuations of the environment. On the other hand, we showed most remarkably how by squeezing the heat source that drives an absorption cooling cycle, one may boost its performance to the extent of making it comparable to a conventional powerdriven cooling device.
A key path to the optimisation of autonomous quantum heat pumps is thus found to reside in applying suitable reservoir engineering techniques^{49,50,51} rather than in exploiting any resource intrinsic to their quantum mechanical working materials^{8,11}. Indeed, the importance of reservoir manipulation has been very recently acknowledged in the context of quantum thermodynamics: It has been recently speculated that superefficient operation of quantum heat engines, in apparent violation of the second law, may be achieved e.g. by reservoir squeezing^{52} or using more general types of nonequilibrium reservoirs^{53,54} and by connecting the working material to an adiabatically isolated ancilla^{55}. However, to our knowledge, this paper represents the first demonstration of a systematic enhancement in the performance of a quantummechanical thermal device. Furthermore, this unconditional enhancement is, in principle, achievable without manipulating the working material or the given hot and cold baths: It is only the arguably controllable heat source that needs to be tailored.
At this point, one could wonder if generating squeezing in the work reservoir is really worth the effort when one could raise instead its equilibrium temperature to obtain effectively the same results. Back to Fig. 3, we see that a squeezing parameter of r = 1.5 (~13 dB, which is currently at reach^{45}), would already take the heat pump close to its best equivalent powerdriven counterpart. In order to achieve a comparable amplification without squeezing, one should increase the work temperature by at least one order of magnitude, which might just not be possible in engineered or natural environments. Thus, it seems reasonable that in many concrete situations, reservoir squeezing could be indeed the best and most natural way to boost the performance of a heatdriven fridge. However, supporting the claim that a quantumenhanced absorption refrigerator can really compete with a heat pump driven by work is a more delicate issue: To mimic a nonstationary state for the work bath^{49,50,51}, one has pay an extra cost that should be added to the overall efficiency balance for a fair comparison. Again, depending of the specifics of the implementation, it may well be that a quantumenhanced quantum absorption chiller came to cool cheaper than the corresponding quantum ‘compression’ cycle. Furthering this issue demands a study on its own.
We also wish to make clear the realm of validity of our analysis. In all of the above, we have taken the fulfilment of the first and second laws as a guarantee of thermodynamic consistency, but we have neither discussed the third law, nor probed the neighbourhood of the absolute zero^{6}. Since we approach the open system dynamics from a LGKStype master equation, the Markov approximation should better hold, which in turn implies that the thermal fluctuations must be sufficiently fast as compared with the dissipation time scales. Thus, to be always on the safe side, we must limit our environmental temperatures from below.
When it comes to the experimental realisation of quantumenhanced absorption technologies it is worth mentioning the pioneering refrigeration experiments by Geusic et al.^{15} using the energy level structure of Cr^{3+} ions in a ruby crystal as support for a threelevel maser. As already mentioned, detailed proposals exist as well for the nonideal threequbit model of Fig. 1(c) using superconducting qubits^{9} and quantum dots^{10}. The squeezing can be engineered by coherently driving the work transition, potentially involving auxiliary levels and vacua; feasible schemes have been proposed e.g. involving trapped ions or Rydberg atoms^{49,50,51}.
In conclusion, we have demonstrated the possibility of genuine quantumenhanced absorption refrigeration beyond the fundamental bounds imposed by classical thermal environments. The application of reservoir engineering techniques to autonomous quantum heat pumps might render them practically useful and competitive for many applications of quantum technologies, in primis quantum cooling. Moreover, the distinctive simplicity of these devices is ideal to get a clean glimpse of how thermodynamics looks like beyond the standard scenario.
Methods
Quantum master equation
We will now write down explicitly the equation of motion for the working material of a quantum heat pump, introduced in Eqs. (1) and (21). The total Hamiltonian is generically (see caption of Fig. 1)
where the dissipative systemreservoir interactions may be taken as
Here, the ground and excited states of the twolevel contact port with bath α are denoted by 0_{α}〉 and 1_{α}〉 respectively. Let us first assume that all three reservoirs are prepared in thermal equilibrium. Under the further assumptions of vanishing initial correlations between system and environments, weak systemenvironment interaction and separation of time scales of free and dissipative dynamics (Born, Markov and rotatingwave approximations), one arrives to the well known LGKStype quantum master equation^{16} with dissipators as those of Eq. (2).
The elements of the spectral correlation tensor, given explicitly in Eq. (10), follow from the power spectrum of the environmental correlations . On the other hand, result from the decomposition of as eigenoperators of ^{16} and the discrete index ω labels all the open decay channels, i.e. all the energy differences that correspond to nonvanishing jump operators.
In the case of an ideal fridge, for which 0_{α}〉 and 1_{α}〉 are picked among the eigenstates of , Eq. (1) rewrites as
where and and the inner summation runs over the frequencies Ω = {ω_{w}, ω_{h}, ω_{c}}. Eq. (27) accounts for the reduced dynamics of the threelevel and twoqubit heat pumps of Figs. 1(a) and 1(b). The case of the nonideal threequbit model of Fig. 1(c) is more involved. All the details may be found elsewhere^{8}.
Let us now relax the assumption of equilibrium environments to allow for squeezing in the work reservoir. The key difference is that the rates Γ_{α,ω} become explicitly timedependent as a consequence of the nonstationarity of squeezed preparations. After performing the rotatingwave approximation, new terms appear in the work dissipator^{16}, that becomes
where the squeezingdependent coefficients and are given by
with and . A new set of jump operators can always be found so that the work dissipator in Eq. (28) takes the standard LGKS form of Eq. (2), which in turn, guarantees that the effective dynamics of the working material remains that of a dynamical semigroup (i.e. completely positive and tracepreserving) in spite of the squeezing^{17}.
The second law at steady state
It is known that the entropy production of a dynamical semigroup is a strictly positive quantity^{19}. It is defined as
where is the quantum relative entropy and is a steady state of the dissipative dynamics. Each of the dissipators appearing in Eq. (1) individually satisfies the inequality
where is now stationary to the local dissipator alone, i.e. . Now, summing Eq. (31) over α yields
where stands for the von Neumann entropy. We may replace in Eq. (32) with the steady state of the full dynamics and the local steady states , with equilibrium states . Note that indeed ^{19}. We thus obtain
that is, we recover the second law as stated in Eq. (4). When the work reservoir is prepared in a squeezed thermal state, the derivation remains exactly the same, only replacing T_{w} with the squeezingdependent effective temperature^{52}
is such that . This leads to the modified second law of Eq. (22).
Efficiency at maximum power of a threelevel fridge
We will devote this section to prove the ultimate bound ε_{*} ≤ d_{c}/(d_{c} + 1)ε_{C} on the efficiency at maximum power for the threelevel prototype of absorption refrigerator of Fig. 1(a). From Eq. (27), the steady state of the three level system may be readily found to be
Here, the ground, first and second excited states of are labeled 1〉, 2〉 and 3〉 respectively and all coherences asymptotically vanish. is a shorthand for and the denominator Δ is given by
Combining the excitation and relaxation rates , of the cold transition with its steadystate populations , one obtains a stationary heat current given by
which, using the detailed balance relations, becomes (from now on we will set
Given the constraint ω_{h} = ω_{c} + ω_{w}, we may fix the values of {ω_{w}, T_{w}, T_{h}, T_{c}} and take as the only independent variable in Eq. (37). Of course, one would obtain the same results by fixing ω_{h} instead of ω_{w}. This brings us back to Eq. (11)
where , , , and p(x) is positive. As already mentioned, the cold heat current is concave and positive in the unit interval, vanishing at the boundaries .
Let us take a closer look to . Its analytic continuation into the complex plane has, in addition to its zeroes on the real axis, the following complex roots
By applying the Hadamard factorisation theorem^{56}, one may conveniently rewrite g(z) as
with . In particular, c = ω_{w}ε_{C}(T_{c} + T_{h})/2T_{c}T_{h}. Back into the real axis, becomes
that is, we recover Eq. (12) by identifying the first factor in the r.h.s with the positive function P(x). The dimensionless constant is defined as . From there, the performance bound ε_{*} < d_{c}/(d_{c} + 1) follows without difficulties [see Eqs.(12)–(14)]. Note that, the fact that P′(x) has at most one zero in the unit interval follows from the concavity of .
From Eq. (41), one sees that taking ε_{C} → 0 yields P(x) ~ p(x), while p(x) itself tends to the constant value : That is, . To probe this limit, one can just expand P(x) around ε_{C} → 0 and take its derivative with respect to x at x = d_{c}/(d_{c} + 1). This yields
which is a strictly negative function of . Consequently, P′(x) ≤ 0 at x = d_{c}/(d_{c} + 1) for ε_{C} → 0. Indeed, it can be seen that P′(x) ≤ 0 ∀ x ∈ (0, 1) in this limit.
Threelevel breakup of the twoqubit fridge
We shall finally show how the twoqubit fridge of Fig. 1(b) can be broken up into two coupled threelevel masers. As a consequence, following the steps of the preceding section, one can prove that the performance bound of Eq. (9) also applies to the twoqubit refrigerator.
Let us denote the eigenstates of the working material by {0_{h}0_{c}〉, 0_{h}1_{c}〉, 1_{h}0_{c}〉, 1_{h}1_{c}〉}, with energies {0, ω_{c}, ω_{h}, ω_{h} + ω_{c}}. The cold bath couples locally to the cold qubit [see Fig. 1(b)] and thus, it only drives the transitions 0_{h}0_{c}〉 ↔ 0_{h}1_{c}〉 and 1_{h}0_{c}〉 ↔ 1_{h}1_{c}〉. On the other hand, the hot bath is connected to 0_{h}0_{c}〉 ↔ 1_{h}0_{c}〉 and 0_{h}1_{c}〉 ↔ 1_{h}1_{c}〉, while the work reservoir operates in the subspace 0_{h}1_{c}〉 ↔ 1_{h}0_{c}〉 [see Fig. 4].
Intuitively, one can already see that the twoqubit fridge is comprised of two elementary threelevel masers sharing the work transition. To make this statement more precise, let us write down explicitly the stationary heat currents
where ω_{w} = ω_{h} − ω_{c} (ω_{h} > ω_{c}). From here, we can recover Eqs. (15) by just defining
As already pointed out, within the cooling window ω_{c} ≤ ω_{c}_{,max}, the heat currents of each of the two threelevel components behave as those of two ideal fridges, individually satisfying the first and the second laws of Eqs. (5) and (4). Closed formulas for and , analogous to Eq. (37), may be obtained from the steadystate solution of Eq. (27), namely
where
and
The denominator has the same form of Eq. (35d), only replacing with . The remaining constants in Eqs. (44) and (45) are
It is clear that both and can be cast in the generic form of Eq. (38), so that their associated efficiencies at maximum power must be bounded by .
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Acknowledgements
The authors are grateful to K. Hovhannisyan, P. Skrzypczyk, N. Brunner, M. Huber, R. Silva, J. Goold, K. Modi, R. Kosloff, J. Rossnagel, J. G. Coello, A. Acín and S. Lloyd for fruitful discussions and useful comments. This project was funded by the Spanish MICINN (Grant No. FIS201019998) and the European Union (FEDER), by the Canary Islands Government through the ACIISI fellowships (85% co funded by European Social Fund), by the COST Action MP1006, by the University of Nottingham through an Early Career Research and Knowledge Transfer Award and an EPSRC Research Development Fund Grant (PP0313/36) and by the Brazilian funding agency CAPES (Pesquisador Visitante EspecialGrant No. 108/2012).
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L.A.C. and J.P.P. conceived the main idea. L.A.C., J.P.P., D.A. and G.A. participated in the mathematical derivations, the discussion of the results and the preparation of the manuscript.
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Correa, L., Palao, J., Alonso, D. et al. Quantumenhanced absorption refrigerators. Sci Rep 4, 3949 (2014). https://doi.org/10.1038/srep03949
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