Probabilistically Violating the First Law of Thermodynamics in a Quantum Heat Engine
PProbabilistically Violating the First Law of Thermodynamicsin a Quantum Heat Engine
Timo Kerremans, Peter Samuelsson, and Patrick P. Potts ∗ Department of Physics and Nanolund, Lund University, Box 118, 221 00 Lund, Sweden (Dated: February 3, 2021)Fluctuations of thermodynamic observables, such as heat and work, contain relevant informationon the underlying physical process. These fluctuations are however not taken into account in thetraditional laws of thermodynamics. While the second law is extended to fluctuating systems bythe celebrated fluctuation theorems, the first law is generally believed to hold even in the presenceof fluctuations. Here we show that in the presence of quantum fluctuations, also the first law ofthermodynamics may break down. This happens because quantum mechanics imposes constraintson the knowledge of heat and work. To illustrate our results, we provide a detailed case-study ofwork and heat fluctuations in a quantum heat engine based on a circuit QED architecture. Wefind probabilistic violations of the first law and show that they are closely connected to quantumsignatures related to negative quasi-probabilities. Our results imply that in the presence of quantumfluctuations, the first law of thermodynamics may not be applicable to individual experimental runs.
I. INTRODUCTION
The laws of thermodynamics are cornerstones of mod-ern science, providing fundamental constraints on phys-ically allowed processes [1]. The first law of thermody-namics states that a change in energy can be divided intoheat and work (cid:104) ∆ U (cid:105) = (cid:104) Q (cid:105) − (cid:104) W (cid:105) , (1)which is a statement of the fundamental law of energyconservation (the signs being chosen such that heat con-sumed and work produced by a heat engine are positive).The second law of thermodynamics states that entropynever decreases [2] (cid:104) ∆ S (cid:105) ≥ , (2)reflecting a general tendency towards disorder. In addi-tion, the zeroth law of thermodynamics provides a defi-nition for the concept of temperature, and the third lawprovides constraints on the behavior at zero temperature.The theory of thermodynamics as originally devel-oped applies to macroscopic systems, where fluctuationsaround mean values are irrelevant. For small systems,where fluctuations can no longer be neglected, stochas-tic thermodynamics [3, 4] provides an extension to thetraditional theory. In the presence of fluctuations, en-tropy production becomes a stochastic variable describedby a probability distribution P (∆ S ) [5]. Importantly,processes with negative entropy production may be ob-served, probabilistically violating the second law of ther-modynamics. On the level of these probabilities, a gener-alized formulation of the second law is provided by fluc-tuation theorems [3, 6, 7]. As a consequence, Eq. (2)still holds, but the left-hand side now denotes the aver-age entropy change. Similar to entropy, in small systemsalso work and heat fluctuate. In classical systems, energyconservation enforces the first law for every process, thatis, there is no probabilistic violation of the first law ofthermodynamics. In quantum systems the situation is less clear. Re-cently, a considerable effort went into including quantumeffects, such as the superposition principle and entangle-ment, into the theory of thermodynamics [8]. Despiteconsiderable progress, defining the basic thermodynamicobservable work as a fluctuating quantity still presents atopic of debate [9]. At the heart of this debate stands theactive role that the observer takes in quantum theory:while classical fluctuating systems can be described bywell-defined trajectories through phase space, the super-position principle prevents such an observer-independentpicture. Indeed, it has been shown that any definition offluctuating work cannot simultaneously fulfill a numberof desirable properties that are taken for granted in theclassical regime [10]. While work fluctuations are in gen-eral affected by an observer, this is not necessarily thecase for heat fluctuations. In particular, for weak cou-pling between system and reservoir, heat is mediated byenergy quanta that are exchanged with the reservoir. Thenumber of exchanged quanta is well-defined and indepen-dent of any observer. In this case, one may define heatas a fluctuating quantity in complete analogy to classicalstochastic systems [11–14].In this work we show that this qualitative differencebetween heat and work can have profound physical conse-quences: unlike in classical systems, the first law may notbe formulated for individual experimental runs. That is,in quantum systems the first law of themodynamics canbe violated probabilistically. Such violations generallyoccur when heat and work are independently accessedin a situation where quantum superposition prevents anobserver-independent definition for work. Importantly,probabilistic violations of the first law do not imply aviolation of energy conservation. Rather, they reflect anuncertainty on our knowledge of energy changes, oncethey are split into heat and work.To illustrate these effects, we provide a detailed casestudy of heat and work fluctuations in the heat engineproposed in Ref. [15], illustrated in Fig. 1. There, workmay be defined through the time-integral of power, and a r X i v : . [ qu a n t - ph ] F e b we find probabilistic violations of the first law which area consequence of the non-commutativity of the power op-erator with the Hamiltonian. The engine we consider isparticularly suitable for our purposes because of multi-ple reasons: As a thermoelectric device, work fluctua-tions are directly linked to current fluctuations, a topicthat is discussed extensively in the literature [16]. Fur-thermore, heat and work are carried by photons and elec-trons respectively, providing a natural separation of thesequantities and their fluctuations. Finally, the same ar-chitecture was recently used to produce entangled photonbeams [17], illustrating the experimental feasibility of theheat engine. We note that the variances of heat and workwere investigated in a very similar architecture [18]. Aswe show below, the probabilistic violations of the firstlaw are only manifested in higher cumulants.The rest of this article is structured as follows. InSec. II, we discuss probabilistic violations of the first law,how they are defined and when they appear. In Sec. III,we introduce the heat engine that is the subject of ourcase study. Using a local master equation, the laws ofthermodynamics follow from a consistent treatment asdiscussed in Sec. IV. Section V introduces the definitionsfor the fluctuations of heat and work that we employ. Wepresent our quantitative results in Sec. VI. In Sec. VII, wediscuss the implications of measurements on our results.Conclusions and an outlook are provided in Sec. VIII. II. PROBABILISTIC VIOLATIONS OF THEFIRST LAW
Let us consider the joint distribution P ( Q, W, ∆ U ) ofheat entering the system ( Q ), work produced by the sys-tem ( W ), and internal energy changes ( ∆ U ), withoutspecifying how these quantities, or their distribution, aredefined. We assume that the averages of these quantitiesfulfill the first law of thermodynamics given in Eq. (1),where the averages are defined as (cid:104) X (cid:105) = (cid:90) dQ (cid:90) dW (cid:90) d ∆ U X P ( Q, W, ∆ U ) , (3)where X denotes some function of Q , W , and ∆ U . Anydistribution that fulfills P ( Q, W, ∆ U ) (cid:54)∝ δ ( Q − W − ∆ U ) , (4)is then said to exhibit probabilistic violations of the firstlaw.With the definition for probabilistic first law violationsat hand, we turn to the question of when we shouldexpect such violations. To simplify this discussion, wedrop the term ∆ U without loss of generality. Whilethis is physically justified in the scenario discussed be-low, we may always do so by defining a new variable W (cid:48) = W + ∆ U and consider the distribution P ( Q, W (cid:48) ) ,dropping the prime in the following. The first law of ther-modynamics then reduces to (cid:104) W (cid:105) = (cid:104) Q (cid:105) . As discussed FIG. 1. Quantum heat engine based on a voltage biasedsuperconducting circuit. Two single-mode microwave res-onators (LC-circuits) with frequencies Ω h and Ω c are coupledvia a Josephson junction, with an effective coupling strength g . Each resonator is further coupled, with strength κ h and κ c respectively, to a thermal reservoir kept at temperatures T h ≥ T c . above, we are interested in a scenario where heat has anobserver-independent definition while work does not. Inthis case, heat can be measured perfectly and withoutdisturbing the dynamics (at least in principle, see alsoSec. VII). At the same time, due to quantum coherence,any measurement of work is plagued by a fundamentalback-action vs. imprecision trade-off [19, 20]. As we nowdiscuss, in such a scenario probabilistic violations of thefirst law are to be expected.Let us first consider a scenario where only heat isbeing measured. Imposing that the first law holds onthe level of the distribution, we may define P ( Q, W ) = P ( Q ) δ ( Q − W ) [21–23]. In this case, work is inferredfrom the measured heat, circumventing the imprecisionvs. back-action tradeoff. Let us now consider the sce-nario where both heat and work are being measured. Be-cause the work measurement is affected by an imprecisionvs. back-action trade-off, while the heat measurement isnot, we will in general find probabilistic violations of thefirst law, i.e., P ( Q, W ) (cid:54) = P ( Q ) δ ( Q − W ) (as long asthe work measurement does not completely destroy anyquantum coherence that make work observer-dependentin the first place). One may be tempted to salvage thefirst law by assigning the missing energy to the measure-ment. However, any attempt to measure this energy willresult in further disturbances, challenging the first lawanew (see also Sec. VII).The above discussion is particularly relevant for heatengines, where the objective is the production of work.Then a measurement of work is arguably a necessary in-gredient (otherwise, it can hardly be argued that the pro-duced work is of any use). For this reason, we will nowexplore a scenario in detail where P ( Q, W ) is definedsuch that it relates to a direct measurement of work andexhibits probabilistic violations of the first law.We note that there are approaches to quantum ther-modynamics, where heat and work are defined as condi-tional averages, conditioned on the same measurements[24–27]. In these approaches, the first law holds on thelevel of probabilities by construction, as heat and workare not accessed individually. III. THE HEAT ENGINE
To investigate the joint fluctuations of heat and work,as well as the occurence of probabilistic violations ofthe first law, we consider the heat engine proposed inRef. [15], see Fig. 1. The engine consists of a voltagebiased superconducting circuit that defines two single-mode microwave resonators with frequencies Ω h and Ω c .Each resonator is in radiative contact with a thermalreservoir kept at temperatures T h and T c respectively,with T h ≥ T c . A Josephson junction in series mediates acoupling between the resonators by the exchange of pho-tons with tunneling Cooper pairs [28, 29]. Choosing thevoltage bias such that (we set (cid:126) = 1 ) eV = Ω h − Ω c , (5)allows Cooper pairs to tunnel against the voltage bias byabsorbing photons from the resonator with frequency Ω h (henceforth denoted as the hot resonator) and emittingphotons to the resonator with frequency Ω c (the cold res-onator). When operating the system as a thermoelectricheat engine, the temperature gradient drives a net flowof photons from the hot to the cold reservoir. This heatflow drives Cooper pairs, i.e., a supercurrent, against thevoltage bias, hence producing electrical work. As thesupercurrent is dissipationless, it carries no entropy andall the heat is carried by the photons, while the work isprovided by the Cooper pairs. This separation of heatand work is very useful when defining these quantities asfluctuating variables.It is convenient to introduce a simplified pictorial rep-resentation of the heat-to-work conversion process (seethe inset in Fig. 1): First, a photon enters the hot res-onator from the hot reservoir. This photon is then con-verted into a photon in the cold resonator by a Cooperpair tunneling against the voltage bias. Finally, the pho-ton leaves the system into the cold reservoir. In thisprocess the electrical work performed by the Cooper pairis eV , the heat provided by the hot reservoir is Ω h ,and the heat emitted into the cold reservoir is Ω c . It istempting to use this representation to describe the fullstatistics of heat and work. However, we find it to be anoversimplified picture: While it describes the behavior ofmean values well, it fails to capture the behavior of heatand work fluctuations due to the coherent nature of theheat-to-work conversion process.For a quantitative description, we model the systemby the Hamiltonian (for details on the derivation and theinvolved approximations, see Ref. [15]) ˆ H = (cid:88) α = h,c Ω α ˆ a † α ˆ a α + g (cid:104) ˆ a † c ˆ a h e ieV t + ˆ a † h ˆ a c e − ieV t (cid:105) , (6) We note that a Hamiltonian of this form has been in-troduced for a heat engine by Kosloff in 1984 [30]. Thecouplings of the hot and cold resonators to the reservoirsare characterized by κ h and κ c , respectively. Consideringthroughout the paper the regime g, κ α (cid:28) Ω α , (7)the dynamics of the system can be described by a lo-cal Master equation (without any constraint on the ratio g/κ α ) [31] ∂ t ˆ ρ = − i [ ˆ H, ˆ ρ ] + L h ˆ ρ + L c ˆ ρ, (8)where L α ˆ ρ = κ α ( n αB + 1) D [ˆ a α ]ˆ ρ + κ α n αB D [ˆ a † α ]ˆ ρ, (9)with D [ ˆ A ]ˆ ρ = ˆ A ˆ ρ ˆ A † − { ˆ A † ˆ A, ˆ ρ } / , and the Bose-Einsteindistribution n αB = 1 exp (Ω α /k B T α ) − . (10)The first (second) term on the right hand side in Eq. (9)corresponds to the process where one photon is emittedto (absorbed from) the thermal reservoir. IV. THE LAWS OF THERMODYNAMICS
Local master equations have been criticized for not be-ing thermodynamically consistent, which may result inviolations of the second law of thermodynamics [32–34].These violations have been shown to be small (i.e., of theorder of terms that are neglected in deriving a local mas-ter equation) [35, 36]. Here we show how the smallnessof g/ Ω α , a requirement for the validity of Eq. (8), can beexploited to obtain thermodynamic consistency. To thisend, we drop the coupling term in Eq. (6) in the ther-modynamic bookkeeping, introducing a thermodynamic Hamiltonian ˆ H TD = Ω h ˆ a † h ˆ a h + Ω c ˆ a † c ˆ a c . (11)This Hamiltonian will be used to define the internal en-ergy of the system (cid:104) U (cid:105) ≡ Tr { ˆ H TD ˆ ρ } . (12)Importantly, ˆ H TD is only used for the thermodynamicbookkeeping, the dynamics is still described by Eq. (8)with the full Hamiltonian given in Eq. (6). We note thatthis is consistent with a microscopic derivation of thelocal master equation, which implies that the energy ex-changed with the reservoirs cannot be resolved on thescale of g and κ α [31]. We now demonstrate that this ap-proach results in thermodynamic consistency. For a dif-ferent approach to treating the term dropped in Eq. (11),see Ref. [37]. A. The zeroth law
The zeroth law of thermodynamics states that a sys-tem tends to thermal equilibrium when in contact withan equilibrium environment, which here is obtained bysetting the temperatures equal T = T c = T h and re-moving any voltage bias V = 0 . As our model relies onEq. (5), the latter enforces Ω h = Ω c . For these equilib-rium conditions, the steady-state solution of Eq. (8) isfound to be the Gibbs state with respect to ˆ H TD ˆ ρ β ( ˆ H TD ) = e − β ˆ H TD Tr (cid:110) e − β ˆ H TD (cid:111) , (13)where β α = 1 /k B T α . We thus find that the zeroth lawholds, and that thermal equilibrium is characterized by ˆ H TD . We note that this is indeed the expected form ofthe equilibrium state within our approximations ˆ ρ β ( ˆ H TD ) g (cid:28) Ω α (cid:39) ˆ ρ β ( ˆ H ) κα (cid:28) Ω α (cid:39) Tr B { ˆ χ β } , (14)where ˆ χ β denotes the Gibbs state with respect to thetotal Hamiltonian, including the reservoirs, and Tr B de-notes the partial trace over the reservoir degrees of free-dom [38, 39]. B. The first law
To formulate the first law of thermodynamics, we re-quire definitions for heat and work. In thermoelectricdevices work is usually accessed through the electricalcurrent, which is related to the power operator ˆ P ≡ − ∂ t ˆ H = 2 eV ˆ I, (15)where ˆ I = − ig (cid:104) ˆ a † c ˆ a h e ieV t − ˆ a † h ˆ a c e − ieV t (cid:105) , (16)is the particle-current operator for tunneling Cooperpairs. We note that we are only interested in the time-averaged dc-current, and do not consider any ac-currentarising from the Josephson effect. The average power isthe time derivative of the average work and reads (cid:104) ˙ W (cid:105) ≡ Tr { ˆ P ˆ ρ } = − i Tr { [ ˆ H, ˆ H TD ]ˆ ρ } , (17)where the dot denotes a time-derivative. Next, we definethe average heat current from reservoir α to the systemas (cid:104) ˙ Q α (cid:105) ≡ Tr { ˆ H TD L α ˆ ρ } . (18)We note that we used the thermodynamic Hamiltonianto define heat, in consistency with the discussion above.Using Eq. (12), it is then straightforward to show thatthe first law of thermodynamics holds (cid:104) ˙ U (cid:105) = (cid:104) ˙ Q h (cid:105) + (cid:104) ˙ Q c (cid:105) − (cid:104) ˙ W (cid:105) . (19) C. The second law
The second law of thermodynamics states that the en-tropy production cannot be negative. Using Eq. (18) forthe average heat flow, we find ˙ S = − k B ∂ t Tr { ˆ ρ ln ˆ ρ } − (cid:104) ˙ Q h (cid:105) T h − (cid:104) ˙ Q c (cid:105) T c = k B (cid:88) α = h,c Tr { [ L α ˆ ρ ] ( ln ˆ ρ β α − ln ˆ ρ ) } (cid:62) , (20)where we used Spohn’s inequality [40] which relies on L α ˆ ρ β α = 0 [where ˆ ρ β α = ˆ ρ β α ( ˆ H TD ) , cf. Eq. (13)]. Onceagain, the use of ˆ H TD is crucial for obtaining a consistentthermodynamic bookkeeping.We have thus shown that our approach is thermody-namically consistent in the sense that it yields averagevalues for heat and work that satisfy the laws of ther-modynamics. We stress that these laws make statementsabout mean values and may not simply be taken over tofluctuating quantities. V. FLUCTUATIONS: FULL COUNTINGSTATISTICS
We now expand our analysis from average quantitiesto the full statistics of heat and work. To this end, weconsider Q α as the heat exchanged with reservoir α and W as the work produced in the time interval [0 , t ] . Asmentioned above, defining work as a fluctuating quan-tity is a non-trivial issue. In this section, we introducethe definitions of work and heat employed throughout.Connections to explicit measurements are discussed inSec. VII. We do not consider changes in internal energy,as these become negligible in the long-time limit consid-ered below (see App. A). A. Heat fluctuations - counting photons
We first focus on the statistics of heat fluctuations.As mentioned before, the heat flow is exclusively carriedby photons exchanged with the environment. To lowestorder in g/ Ω α and κ α / Ω α , every photon exchanged withreservoir α carries an equal amount of energy, Ω α . Thus,the heat exchanged with reservoir α is fully determinedby the number of photons exchanged with the reservoir,denoted by q α , and we can identify Q α = q α Ω α such that P ( Q h , Q c ) = (cid:88) q h ,q c δ ( Q h − q h Ω h ) δ ( Q c − q c Ω c ) P ( q ) , (21)where q = ( q h , q c ) . We stress that q α denotes the net number of photons exchanged with reservoir α duringthe time interval [0 , t ] . The sign is chosen such that apositive q α denotes photons entering the system.The distribution P ( q ) , known as the full countingstatistics, can be obtained by introducing the photoncounting fields χ = ( χ h , χ c ) in the master equation [41–43] ∂ t ˆ ρ ( χ ) = − i [ ˆ H, ˆ ρ ( χ )] + L χ h h ˆ ρ ( χ ) + L χ c c ˆ ρ ( χ ) , (22)with the superoperators L χ α α ˆ ρ = κ α ( n αB + 1) (cid:20) e iχ α ˆ a α ˆ ρ ˆ a † α − (cid:8) ˆ a † α ˆ a α , ˆ ρ (cid:9)(cid:21) + κ α n αB (cid:20) e − iχ α ˆ a † α ˆ ρ ˆ a α − (cid:8) ˆ a α ˆ a † α , ˆ ρ (cid:9)(cid:21) . (23)The quantity ˆ ρ ( χ ) is directly related to the cumulantgenerating function S ( χ ) = ln [ Tr { ˆ ρ ( χ ) } ] , (24)which in turn defines the heat distribution P ( q ) = (cid:90) π dχ c π (cid:90) π dχ h π e S ( χ )+ i χ · q . (25)Note that the π periodicity of ˆ ρ ( χ ) reflects the fact thatits Fourier transform, P ( q ) , is a discrete distribution,taking only finite values for integer values of q α . Fromthe cumulant generating function, the cumulants of thedistribution P ( q ) may be derived, with the first, second,and third cumulant providing the mean, variance, andskewness of the distribution. The k -th cumulant is ob-tained by (cid:104)(cid:104) q kα (cid:105)(cid:105) = i k ∂ kχ α S ( χ ) | χ =0 . (26)In the long-time limit, S ( χ ) , and thus all cumulants,become linear in time, see App. A. In this limit, anyentropy change in the system becomes negligible and thesecond law given in Eq. (20) puts a restriction on theaverage values (cid:104) ˙ Q h (cid:105) T h + (cid:104) ˙ Q c (cid:105) T c ≤ ⇔ (cid:104)(cid:104) q h (cid:105)(cid:105) Ω h T h + (cid:104)(cid:104) q c (cid:105)(cid:105) Ω c T c ≤ . (27)It is well known that this restriction only holds for themean values and does not carry over to fluctuating quan-tities, i.e., P ( q ) (cid:54) = P ( q ) θ ( − q h β h Ω h − q c β c Ω c ) , (28)where θ ( x ) denotes the Heaviside theta function thatis equal to one for x ≥ and zero otherwise. Indeed,for fluctuating systems, the second law is generalized bythe fluctuation theorem [6], which in the long-time limitreads (the validity of this equality is shown below andcan be anticipated from [44]) P ( − q ) P ( q ) = e q h β h Ω h + q c β c Ω c . (29)Using Jensen’s inequality, the second law, as stated inEq. (27), may be recovered from the fluctuation theorem. Finally, we note that in the long-time limit, we find(see App. A) P ( q ) ∝ δ q h , − q c . (30)Because there is no photon accumulation (or generation)within the system, the full counting statistics of heat isfully determined by the photons which traverse the sys-tem and it is sufficient to consider a single counting fieldfor heat. B. Work fluctuations - counting electrons
To define work fluctuations, we connect them to thefull counting statistics of electrons in phase-coherent sys-tems, where a quasi-probabilistic description has beenemployed successfully [16, 45, 46]. This approach, whichhas been used for work fluctuations before [20, 47, 48],will be further motivated by its connection to a weakmeasurement of the electronic current in Sec. VII. In oursystem work is exclusively performed by the supercurrentcarried by Cooper pairs tunneling across the Josephsonjunction. We denote the net number of Cooper pairs thattunneled against the voltage bias in the time-interval [0 , t ] by w . The work provided in this time-interval may thenbe written as W = 2 eV w implying P ( W ) = (cid:88) w δ ( W − eV w ) P ( w ) . (31)For phase-coherent systems, the full counting statistics isobtained by introducing a counting field λ in the masterequation as [20, 45, 49] ∂ t ˆ ρ ( λ ) = − i [ ˆ H, ˆ ρ ( λ )] − iλ { ˆ I, ˆ ρ ( λ ) } + (cid:88) α = c,h L α ˆ ρ ( λ ) , (32)where the current operator ˆ I is defined in Eq. (16). Inanalogy to Eqs. (24) and (25), we introduce a cumulantgenerating function S ( λ ) = ln [ Tr { ˆ ρ ( λ ) } ] , (33)which defines the work distribution P ( w ) = (cid:90) ∞−∞ dλ π e S ( λ )+ iλw . (34)The cumulants of P ( w ) can then be obtained in analogyto Eq. (26) (cid:104)(cid:104) w k (cid:105)(cid:105) = i k ∂ kλ S ( λ ) | λ =0 . (35)Note that in contrast to the full counting statistics ofphotons, P ( w ) is a continuous function. This is relatedto the fact that the tunneled Cooper pairs cannot becounted one by one in the present system, as this wouldresult in strong back-action effects (see Sec. VII). Fur-thermore, the distribution P ( w ) may take on negativevalues. Such negativities have been shown to arise fromquantum interference effects [50] and have been found be-fore in the charge transfer between superconductors [46].We note that a different approach for counting electronshas been employed in a similar system [51]. In Sec. VII,we motivate the approach taken here by connecting it toan explicit measurement of the electric current. C. Joint heat and work fluctuations
We now turn to the distribution P ( q , w ) , providing ajoint description of heat and work fluctuations. Becauseheat is carried only by photons and work only by elec-trons, this distribution may be calculated by includingcounting fields for both photons and electrons ∂ t ˆ ρ = − i [ ˆ H, ˆ ρ ] − iλ { ˆ I, ˆ ρ } + L χ h h ˆ ρ + L χ c c ˆ ρ, (36)where we dropped the counting field dependence of ˆ ρ forease of notation. The cumulant generating function isthen introduced in the usual way S ( χ , λ ) = ln [ Tr { ˆ ρ ( χ , λ ) } ] . (37)Setting λ = 0 ( χ = 0 ), we recover the cumulant gener-ating function for photons (electrons) respectively. Thejoint distribution can then be written as P ( q , w ) = (cid:90) π dχ c π dχ h π (cid:90) ∞−∞ dλ π e S ( χ ,λ )+ iλw + i χ · q , (38)and the cumulants may be obtained by (cid:104)(cid:104) q kh q lc w m (cid:105)(cid:105) = i k + l + m ∂ kχ h ∂ lχ c ∂ mλ S ( χ , λ ) | χ =0 ,λ =0 . (39)Throughout, we will be interested in the long-timelimit, where the cumulants grow linearly in time andwhere q h = − q c . The fluctuations in the internal energymay then safely be neglected (cf. App. A) and P ( q, w ) provides a complete description of the fluctuating energyflows through the system, where q = q h denotes the num-ber of photons that went from the hot to the cold reser-voir. In this case, the first law constrains the mean valuesto satisfy (cid:104) ˙ W (cid:105) = (cid:104) ˙ Q (cid:105) ⇔ (cid:104)(cid:104) w (cid:105)(cid:105) = (cid:104)(cid:104) q (cid:105)(cid:105) , (40)where Q = Q h + Q c = (Ω h − Ω c ) q . It is tempting to takeover this statement of energy conservation to fluctuatingquantities, assuming that the work fluctuations can becompletely described by the heat fluctuations. However, this is not generally possible in quantum systems and wefind probabilistic violations of the first law, i.e., P ( w, q ) (cid:54) = P ( q ) δ ( w − q ) . (41)This condition may also be expressed in terms of thecumulants ∃ k s . t . (cid:104)(cid:104) ( w − q ) k (cid:105)(cid:105) (cid:54) = 0 . (42)We stress that the internal energy fluctuations cannotrestore the first law because the corresponding cumulantsscale differently with time, see App. A.Investigating a particular heat engine allows for a bet-ter understanding of the origin of these violations. In thepresent system, work is probed via the power (or electri-cal current) operator, as is usually the case in thermo-electric devices. Because the power operator does notcommute with the Hamiltonian [ ˆ H, ˆ P ] (cid:54) = 0 , the Heisen-berg uncertainty principle implies that any informationacquired on the power restricts our knowledge on energy.The probabilistic violations of the first law reflects thisfundamental limitation in the joint knowledge of workand energy and should not be seen as a violation of energyconservation. As no such limitation applies for classi-cal systems, probabilistic first law violations are a purelyquantum phenomenon. In contrast to fluctuations, meanvalues can quite generally be obtained without disturbingthe dynamics, ensuring that the first law is still respectedon average. We note that the influence of measurementson conservation laws in quantum mechanics is the subjectof a recent work [52]. VI. RESULTS
In order to explicitly calculate the heat and work dis-tributions, we follow the work of Clerk and Utami [53]and cast the master equation including counting fields[cf. Eq. (36)] into an equation of motion for the Wignerfunction. A Gaussian ansatz then allows for reducing aninfinite amount of differential equations (one for each en-try of the density matrix) to four coupled, non-linear dif-ferential equations. This procedure is outlined in App. B.
A. Distributions of heat and work
From Eq. (36), we can derive a cumulant generatingfunction for the joint fluctuations of heat and work inthe long-time limit S ( χ, λ ) t = κ h + κ c − (cid:113) κ h + κ c − g (4 + λ ) + 2 (cid:112) [ g (4 + λ ) + κ h κ c ] − g (4 + λ ) κ h κ c Ψ( χ, λ ) , (43)
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FIG. 2. Joint distribution of heat and work together with its marginals for g/κ = 1 [(a) and (b)] and g/κ = 0 . [(c) and (d)].Here κ ≡ κ c = κ h . The joint distribution for heat ( q ) and work ( w ) [(a) and (c)] features negative values, as well as probabilisticfirst law violations (non-negative values away from the diagonal dashed line). As discussed in the main text, these featuresarise from quantum coherence, preventing a back-action-free and precise measurement of work. The marginal distributions ofheat and work [(b) and (d)] are described by a discrete and continuous distribution respectively. This is a consequence of thefact that photons can be counted one by one while Cooper pairs cannot. The dashed line and dark blue bars in (d) show theanalytic expressions obtained for g (cid:28) κ . For large g ( ∼ κ ), the distributions look qualitatively the same as for g ∼ κ (notshown). Parameters: n hB = 1 , n cB = 0 . , gt = κt = 150 [(a) and (b)], gt = 600 , κt = 12 (cid:48) [(c) and (d)], ensuring the samemean values in panels (a) and (c). where we introduced the function Ψ( χ, λ ) = n hB ( n cB + 1) (cid:18) e − iχ − iλ/
21 + iλ/ − (cid:19) + n cB ( n hB + 1) (cid:18) e iχ iλ/ − iλ/ − (cid:19) . (44)The quasi-probability distribution for heat and work isobtained from the cumulant generating function throughEq. (38) and illustrated in Fig. 2. This distribution ex-hibits two striking features:I The first law of thermodynamics can be violatedprobabilistically.II The distribution takes on negative values.The first law violations are visualized by finite values of P ( q, w ) for q (cid:54) = w and are a direct consequence of thefact that S ( χ, λ ) (cid:54) = S ( χ + λ ) . It is instructive to inspectthe function Ψ( χ, λ ) . The first (second) term in Eq. (44)corresponds to photons going from hot to cold (cold tohot). The terms in the brackets show how these photonscontribute to heat and work. In general, their contribu-tions to heat and work are different. Only for small λ do these counting terms reduce to exp [ ∓ i ( χ + λ )] − .Furthermore, Eq. (43) shows an additional influence onwork fluctuations arising from a rescaling of the interac-tion strength g → g (4 + λ ) . We note that while q = w is not enforced on the level of (quasi-)probabilities, thefirst law still holds on average as we find (cid:104)(cid:104) q (cid:105)(cid:105) = (cid:104)(cid:104) w (cid:105)(cid:105) .The negative values of the quasi-probability distribu-tion reflect the fact that it is impossible to measure heatand work simultaneously without influencing the dynam-ics of the heat engine. This will be further illustrated inSec. VII D, where we illustrate the outcome of an explicitmeasurement of work.Both features I and II become particularly apparentin the distribution describing the difference of work andheat defined as P (∆) = ∞ (cid:88) q = −∞ P ( q, q + ∆) = (cid:90) ∞−∞ dλ π e S ( − λ,λ )+ iλ ∆ . (45)This distribution is illustrated in Fig. 3. It exhibits os-cillations and negative values. The fact that it takes onnon-zero values for ∆ = w − q (cid:54) = 0 implies the presenceof probabilistic first law violations. Note that the con-dition for probabilistic violations of the first law can be cast into [cf. Eq. (42)] ∃ k s . t . (cid:104)(cid:104) ∆ k (cid:105)(cid:105) (cid:54) = 0 . (46)We note that the probabilistic violations of the first lawoccur almost exclusively for w < q . As shown in Sec. VII,this implies that measuring a work value less than theexpended heat is vastly more probable than measuringwork that exceeds the heat input. B. Cumulants
For understanding both features I and II, it is instruc-tive to consider the cumulants that follow from Eq. (43).The averages read (see also, [15, 54]) (cid:104)(cid:104) q (cid:105)(cid:105) t = (cid:104)(cid:104) w (cid:105)(cid:105) t = 4 g κ h κ c ( n hB − n cB )(4 g + κ h κ c ) ( κ h + κ c ) , (47)for the (co-)variances, we find (cid:104)(cid:104) q (cid:105)(cid:105) = (cid:104)(cid:104) w (cid:105)(cid:105) = (cid:104)(cid:104) qw (cid:105)(cid:105) = (cid:104)(cid:104) q (cid:105)(cid:105) coth (cid:18) β c Ω c − β h Ω h (cid:19) + 2 (cid:104)(cid:104) q (cid:105)(cid:105) t ( κ c + κ h ) + (4 g + κ c κ h )( κ c + κ h )(4 g + κ c κ h ) , (48)and the third order cumulants read (cid:104)(cid:104) q (cid:105)(cid:105) = (cid:104)(cid:104) q (cid:105)(cid:105) (cid:20) (cid:104)(cid:104) q (cid:105)(cid:105) t ( κ c + κ h ) + (4 g + κ c κ h )( κ c + κ h )(4 g + κ c κ h ) − (cid:104)(cid:104) q (cid:105)(cid:105) /t g + κ c κ h (cid:21) , (cid:104)(cid:104) q w (cid:105)(cid:105) = (cid:104)(cid:104) q (cid:105)(cid:105) , (cid:104)(cid:104) qw (cid:105)(cid:105) = (cid:104)(cid:104) q (cid:105)(cid:105) − (cid:104)(cid:104) q (cid:105)(cid:105) κ c κ h g + κ c κ h , (cid:104)(cid:104) w (cid:105)(cid:105) = (cid:104)(cid:104) q (cid:105)(cid:105) + (cid:104)(cid:104) q (cid:105)(cid:105) g − κ c κ h g + κ c κ h . (49)We thus find that for the cumulants up to second order,the distribution behaves as if q = w (in agreement withRef. [18]). However, starting from the third order cu-mulants, this is no longer the case. In particular, fromEq. (45), we find that the first non-vanishing cumulantsfor ∆ = w − q are (cid:104)(cid:104) ∆ (cid:105)(cid:105) = 12 (cid:104)(cid:104) q (cid:105)(cid:105) , (cid:104)(cid:104) ∆ (cid:105)(cid:105) = (cid:104)(cid:104) q (cid:105)(cid:105) g − κ c κ h g κ c κ h . (50)We note that for higher cumulants, also even cumulantsare non-vanishing. The distribution P (∆) thus has botha vanishing mean and a vanishing variance. For any non-negative distribution, vanishing mean and variance im-ply that all cumulants vanish (i.e., the distribution isthe Dirac delta distribution). The finite higher cumu-lants in Eq. (50) thus imply that P (∆) takes on nega-tive values and exhibits probabilistic first law violations[cf. Eq. (46)]. When the first law holds up to the secondorder cumulants, probabilistic first law violations are thusintimately linked to negative quasi-probabilities.Since higher cumulants capture the behavior of the tails of a distribution, one can usually reproduce the char-acteristic features by only keeping the lowest cumulantsin the generating function. We thus approximate P (∆) ≈ (cid:90) ∞−∞ dλ π e i ( (cid:104)(cid:104) q (cid:105)(cid:105) λ + λ ∆ )= (cid:18) |(cid:104)(cid:104) q (cid:105)(cid:105)| (cid:19) Ai (cid:20) ∆( (cid:104)(cid:104) q (cid:105)(cid:105) / (cid:21) , (51)where the Airy function of the first kind is defined as Ai( x ) = (cid:90) ∞−∞ dte i ( t + xt ) . (52)Equation (51) captures the behavior of P (∆) well forsmall values of ∆ , cf. Fig. 3. C. Limiting cases
To get a better analytical understanding of the heat-to-work conversion process, it is instructive to consider
30 25 20 15 10 5 0 5 100.20.10.00.10.20.3
FIG. 3. Distribution for the difference of work and heat ∆ = w − q . Having a vanishing mean and variance, this distributionis bound to have negative values if probabilistic violations ofthe first law are present (i.e., if it is not equal to a Diracdelta). The dotted line shows the Airy function in Eq. (52),exhibiting good agreement with P (∆) for small values of ∆ .Parameters: n hB = 1 , n cB = 0 . , gt = κt = 150 . the two limiting cases κ c , κ h (cid:28) g and g (cid:28) κ c , κ h . For κ c , κ h (cid:28) g , we find S ( χ, λ ) t = κ h + κ c (cid:34) − (cid:115) − κ h κ c ( κ h + κ c ) Ψ( χ, λ ) (cid:35) . (53)For λ = 0 , this equation reduces to the cumulant generat-ing function for a single resonator coupled to two baths,cf. Eq. (F14) in Ref. [55]. Due to the strong coupling be-tween the resonators, they act similar to one single res-onator. Note however that the frequencies associated tothe two reservoirs are different ( Ω c (cid:54) = Ω h ), reflecting thefact that photons change their energy when going fromone resonator to the other. The analogy with a singleresonator should thus be treated with care.In the opposite limit, g (cid:28) κ c , κ h , we find S ( χ, λ ) t = g (4 + λ ) κ h + κ c Ψ( χ, λ ) . (54)In this limit, we obtain the marginals analytically. Forthe heat distribution, we find a bi-directional Poissonianwith the distribution P ( q ) = e − (Γ ch +Γ hc ) t (cid:18) Γ ch Γ hc (cid:19) q I q (cid:104) t (cid:112) Γ ch Γ hc (cid:105) , (55)where I q denotes the modified Bessel function of the firstkind. The rate for photons to go from reservoir α toreservoir β is given by Γ βα = 4 g κ c + κ h n αB ( n βB + 1) . (56)The moments of this distribution are particularly simple and read (cid:104)(cid:104) q k (cid:105)(cid:105) /t = (cid:40) Γ ch − Γ hc for k odd , Γ ch + Γ hc for k even . (57)Such a bi-directional Poissonian describes particles beingpartitioned at a single junction. In the present case, theJosephson junction provides a bottleneck for the photons,making it the only junction that is relevant for transportstatistics.For the work distribution, we find a Gaussian distribu-tion with the same mean and average as for the heat P ( w ) = 1 (cid:112) π (cid:104)(cid:104) q (cid:105)(cid:105) e − ( w −(cid:104)(cid:104) q (cid:105)(cid:105) )22 (cid:104)(cid:104) q (cid:105)(cid:105) . (58) D. Out-of-equilibrium relations
Having access to the analytic expression of the cumu-lant generating function [cf. Eq. (43)], we may verify out-of-equilibrium relations that are expected to hold for thepresent scenario. In particular, we consider the fluctu-ation theorem [3] as well as the thermodynamic uncer-tainty relation (TUR) [56].From the symmetry S ( χ, λ ) = S ( χ + iβ h Ω h − iβ c Ω c , − λ ) , (59)a fluctuation theorem follows P ( − q, − w ) P ( q, w ) = e − q ( β c Ω c − β h Ω h ) . (60)While this implies a fluctuation theorem for heat, seeEq. (29), no simple fluctuation theorem for work can bederived from Eq. (60). Indeed, derivations for work fluc-tuation theorems usually rely on the first law in order torelate work to entropy. Not surprisingly, this approachbreaks down in the presence of probabilistic first law vi-olations.From the second cumulant given in Eq. (48), we findthat the TUR is obeyed in our system (cid:104)(cid:104) q (cid:105)(cid:105)(cid:104)(cid:104) q (cid:105)(cid:105) ≥ k B (cid:104)(cid:104) Σ (cid:105)(cid:105) , (61)where the average entropy production reads (cid:104)(cid:104) Σ (cid:105)(cid:105) = (cid:104)(cid:104) q (cid:105)(cid:105) ( β c Ω c − β h Ω h ) . Equation (61) can be proven by not-ing that the second term in the variance [cf. Eq. (48)] isstrictly positive, and by using the inequality x coth( x ) ≥ . Since heat fluctuations are equal to work fluctuationsin our system, the TUR implies a direct trade-off be-tween power, efficiency, and power fluctuations [57] forthe present quantum heat engine. The validity of theTUR is in agreement with Ref. [58], where it was shownthat the TUR is valid for harmonic oscillator junctions.0 VII. MEASURING HEAT AND WORK
So far, we considered fluctuations of heat and workwithout considering an explicit measurement of thesequantities. In quantum systems, measurements mayhave unavoidable consequences, altering the statistics ofthe observables being measured and should thus be ac-counted for [59]. In this section, we show how the frame-work introduced above may be connected to explicit mea-surements of heat and work.
A. Measuring heat
Measuring heat and its fluctuations is experimentallyvery challenging. A promising route towards measur-ing heat fluctuations is provided by detecting the singlephotons that are emitted from the system, which canpotentially be achieved by monitoring temperature [60–62]. Very recently, temperature fluctuations, which canbe connected to the second cumulant of heat fluctuations,have been observed experimentally [63]. To circumventthe limitations and challenges that come with any specificexperimental setup, we consider here a more abstract ap-proach provided by the two-point measurement scheme,focusing on weakly coupled reservoirs [43, 64]. To de-termine the heat exchanged with the thermal reservoirs,their energy is determined with a projective measurementat the beginning and at the end of the time interval [0 , t ] ,with outcomes ( E c , E h ) and ( E tc , E th ) respectively. Theheat exchanged with reservoir α is then simply given by Q α = E α − E tα . (62)The statistics of this two-point measurement scheme canbe shown to coincide with the statistics obtained fromthe master equation including counting fields given byEq. (22) [12–14]. Since the reservoirs are well describedby a thermal state at the initial time, the initial projec-tive measurement does not influence the dynamics. Al-beit idealized, the two-point measurement scheme thusprovides a way of accessing the heat fluctuations com-puted in our framework, without influencing the opera-tion of the engine. We expect any experimental approachfor determining heat fluctuations to produce the sameresults, as long as heat is measured accurately and theinfluence on the system dynamics may be neglected. B. Measuring work
While heat can be measured without disturbing the dy-namics of the heat engine, this is not possible for work inour system. To appreciate this, one may consider a two-point measurement scheme for counting Cooper pairs, inanalogy to the above discussion on measuring heat. Inthis approach, the total number of electrons on one sideof the Josephson junction is measured at the initial and final times. However, here we consider a Josephson junc-tion that has a well defined phase [65, 66]. As the phaseis conjugate to the particle number, a measurement ofthe number of electrons disturbs the phase and wouldstrongly affect the dynamics of the heat engine. It isthus not possible to count the tunneling Cooper pairsone by one.As a non-invasive measurement is not available, thechoice of the measurement will affect the observed workfluctuations. To decide on a measurement scheme, weturn to experiments, where the electrical current is usu-ally measured. Following Refs. [19, 49, 67], we modelthe measurement by a detector, described by the conju-gate variables ˆ r and ˆ π , which is coupled linearly to theelectrical current during the time interval [0 , t ] by theinteraction Hamiltonian ˆ H m = s ˆ I ˆ π, (63)where s denotes the coupling strength. At the end of thetime-interval [0 , t ] , the observable ˆ r is measured projec-tively, yielding information about the integrated currentand thus the performed work. In practice, the detectorcould be provided by an LC element [68]. For simplicity,we consider a detector that has no internal dynamics.This measurement scheme results in the measured dis-tribution [20, 49] P m ( w ) = (cid:90) dw (cid:48) dγ W ([ w − w (cid:48) ] s, γ/s ) P ( w (cid:48) ; γ ) , (64)where W denotes the initial Wigner function of the detec-tor, and P ( w ; γ ) is a quasi-probability distribution that isclosely related to the work distribution introduced above.It is obtained from Eq. (32) upon including the term γ ˆ I ,which captures measurement back-action, in the Hamil-tonian. The expression for the measured distribution inEq. (64) can be interpreted as the intrinsic fluctuations ofthe system, encoded in P ( w ) = P ( w ; 0) , being modifiedby the detector [69]. This modification presents a trade-off between measurement imprecision and back-action[19]. For a weak measurement (small s ), γ is restrictedto small values, implying small back-action. At the sametime, the integral over w (cid:48) describes a convolution with abroad distribution, implying large imprecision. A strongmeasurement (large s ) yields small imprecision, as w − w (cid:48) is restricted to small values. At the same time, large val-ues of γ contribute to the integral implying large back-action. It is interesting to note that while P ( w ; γ ) maybecome negative, back-action and imprecision conspirein a way that always ensures the measured distribution P m ( w ) to remain non-negative [20, 49].For simplicity, we consider a detector with a GaussianWigner function W ( x, p ) = 12 πσ x σ p e − x σ x e − p σ p . (65)To ensure an optimal trade-off between measurement im-precision and back-action, we choose σ x σ p = 1 / , satu-rating the uncertainty principle, and define σ ≡ σ x /s .1The measured distribution function then reduces to P m ( w ) = 1 π (cid:90) dw (cid:48) dγe − ( w − w (cid:48) )22 σ e − γ σ P ( w (cid:48) ; γ ) . (66)The measurement strength is now fully captured by σ .In the weak measurement limit ( σ (cid:29) ), the integrand inthe last equation is only non-zero for vanishingly small γ and we may replace P ( w (cid:48) ; γ ) by the work distribution P ( w (cid:48) ) . In this case, the cumulants of the measured dis-tribution are related to the cumulants of the work distri-bution by the simple relation (cid:104)(cid:104) w k (cid:105)(cid:105) m = δ k, σ + (cid:104)(cid:104) w k (cid:105)(cid:105) . (67)The only effect the measurement has is an increase ofthe variance by σ , which is a large number for the weakmeasurement considered here. We thus conclude that theinternal statistics of the system, described by P ( w ) , canbe recovered from the cumulants obtained in a weak mea-surement upon correcting for measurement imprecision.While we consider a particular measurement scheme here,we believe this conclusion to remain valid for any weakmeasurement.We note that for an optimal trade-off between measure-ment imprecision and back-action, the detector generallyhas to be in a pure state. The energetic cost of preparingsuch a state (which diverges due to the third law of ther-modynamics [70, 71]) is not taken into account here. Westress however that this does not hamper our conclusionsfor weak measurements, which can solely be derived fromEq. (65). C. The first law in the presence of the detector
The coupling to the detector represents a time-dependent contribution to the Hamiltonian, cf. Eq. (63).This term modifies the energy balance and indeed, usingEqs. (12), (17), (18), we find a modified first law (cid:104) ˙ U (cid:105) = −(cid:104) ˙ W (cid:105) − (cid:104) ˙ W d (cid:105) + (cid:104) ˙ Q h (cid:105) + (cid:104) ˙ Q c (cid:105) , (68)where we used the subscript d to label the power providedby the detector. The contribution from the detector tothe first law reads (cid:104) ˙ W d (cid:105) = − Tr { ˆ ρ∂ t ˆ H m } = − i Tr { [ ˆ H m , ˆ H TD ]ˆ ρ } . (69)Since the detector couples to the power operator in the absence of the detector itself, it only measures W , whichis no longer the total work. This is a general issue: when-ever a time-dependent power operator is being measured,the measurement Hamiltonian will necessarily also betime-dependent, providing an additional contribution tothe total power. With a linear detector, it is impossi-ble to include this contribution in the measurement. Ifwe however consider weak measurements, then W d cansafely be neglected. This can be seen by noting that theHamiltonian describing the coupling to the detector isproportional to s ˆ π . As the Hamiltonian commutes with FIG. 4. Distribution describing a measurement of heat andwork. While heat is assumed to be measured perfectly (e.g.,by monitoring the photons that enter and leave the system),the measurement of work is described by Eq. (66). The mea-surement strength is quantified by σ = 0 . . The rest of theparameters are the same as in Fig. 2 (a). ˆ π , the momentum values of the detector much larger than σ p are negligible, cf. Eq. (65), and the power provided bythe detector will be proportional to a factor smaller than sσ p = 1 / (2 σ ) , which vanishes for weak measurements.Just as in the absence of the detector, Eq. (69) holdsfor average values and does not take over to fluctuations.Indeed, when adding a second detector to measure W d ,the energy balance gets modified again. Including theenergetics of the detector does therefore not salvage thefirst law of thermodynamics beyond average values.We note that the (average) energy provided by a mea-surement may act as a resource, fueling engines and re-frigerators [25, 72–74]. D. The joint probability distribution for measuringheat and work
The probability distribution for the outcomes of ajoint measurement of work and heat is shown in Fig. 4.While measurement imprecision and back-action al-ter the quasi-probability distribution rendering it non-negative, similarities are clearly visible [cf. Fig. 2]. Inparticular, the oscillatory behavior which results fromthe coherent nature of the heat-to-work conversion pro-cess remains visible. Furthermore, the measured distri-bution exhibits probabilistic violations of the first law,just as the underlying quasi-probability distribution. Asa consequence, a work value that is considerably smallerthan the heat input may be measured.First law violations in the measurement are to be ex-pected because neither back-action nor imprecision aregenerally able to restore the first law on the level of prob-abilities. While there are some instances where back-action may ensure the validity of the first law (this hap-2pens for a two-point measurement of work), this is notgenerally the case. As discussed above, the back-actionof a linear detector may even alter the energy balanceon average. In addition, measurement imprecision intro-duces randomness to the measurement outcomes in a waythat is unrelated to energy conservation.
VIII. CONCLUSIONS & OUTLOOK
We provided a detailed investigation into the joint fluc-tuations of heat and work in a quantum heat engine.In this device, heat fluctuations behave classically, whilework fluctuations are non-classical in nature, in the sensethat a measurement of work generally influences the workvalue itself. This qualitative difference between heat andwork has the striking consequence that the first law ofthermodynamics does not hold on the level of the dis-tribution. Our results thus imply that care has to betaken when employing the first law for individual exper-imental runs. Here we focused on weak system-reservoircoupling, where heat and internal energy enjoy an unam-biguous definition. At strong coupling, a thermodynam-ically consistent definition of internal energy and heat isa subject of debate [27, 75, 76], further challenging thevalidity of the first law on the level of probabilities.In agreement with previous works [52, 77], our resultsimply that conservation laws, such as energy conserva-tion, do not enjoy the same impact in quantum systemsas they do in classical systems. The reason for this is thatwhen dividing a conserved quantity into parts, a Heisen-berg uncertainty relation may restrict our simultaneousknowledge of these parts. The quasi-probability distribu-tion used to describe work fluctuations here is a memberof a larger family, which have been called Keldysh quasi-probability distributions [20]. Another prominent mem-ber is the Wigner function and generalizations thereof[78]. Different Keldysh quasi-probability distributionsmay shed light onto the impact of different conservationlaws on measurable but non-commuting quantities.Our results open up several interesting avenues: Equa-tion (60) implies that a usual Crooks fluctuation theo-rem for work cannot be derived when the first law ofthermodynamics does not hold on the level of the dis-tribution. Nevertheless, fluctuation theorems includingquantum corrections have been derived [79, 80]. Thesequantum corrections may have a connection to the prob-abilistic violations of the first law investigated here.Another open question concerns the relation betweenfirst law violations and negative values in the Keldyshquasi-probability distribution. Such negative values haverecently been shown to be a necessary and sufficient con-dition for ruling out a classical description of the mea-surement outcomes [81]. Here we showed that if the firstlaw holds not only for the first but also for the secondcumulants, then first law violations are accompanied bynegative quasi-probabilities. While the first law is only astatement about average values, we may speculate that it also generally holds for the second cumulants. If thiswere true, it would strengthen the link between prob-abilistic violations of the first law and negative quasi-probabilities.Probabilistic violations of the second law are by nowwell understood, in particular because of fluctuation the-orems which are recognized as the generalization of thesecond law to fluctuating systems. By demonstratingprobabilistic first law violations, our results may opena quest for finding a generalization of the first law thatincludes quantum fluctuations. At this point, we mayonly speculate how such a generalization may look likeand what its potential impact may be.
ACKNOWLEDGMENTS
We acknowledge fruitful discussions with the audiencefrom the Quarantine Thermo seminar series as well asthe QTD2020 conference, in particular with P. Stras-berg and C. Elouard. We further thank P. Hänggi, G.Haack, M. Huber, A. Jordan, G. Manzano, M. Perarnau-Llobet, and R. Sánchez for valuable feedback on themanuscript. This work was supported by the SwedishResearch Council. T.K. acknowledges the Knut and Al-ice Walenberg Foundation (KAW) (project 2016.0089).P.P.P. acknowledges funding from the European Union’sHorizon 2020 research and innovation programme un-der the Marie Skłodowska-Curie Grant Agreement No.796700.
Appendix A: The long-time limit
In the main text, we argued that the internal energyof the system may be neglected in the long time limit.We further stated that the fluctuations of heat are com-pletely determined by a single variable q , describing thenet number of photons that went from the hot to the coldreservoir. Here we prove these two statements. We startwith the moment generating function for heat, work, andinternal energy changes e S ( χ ,λ,u ) = Tr (cid:110) e − iu ˆ H e L ( χ ,λ ) t (cid:16) e i u ˆ H ˆ ρe i u ˆ H (cid:17)(cid:111) , (A1)where the Liouvillian L ( χ , λ ) is determined by the right-hand side of Eq. (36). The counting field u correspondsto the internal energy fluctuations. Here we used a quasi-probability approach as in Ref. [47]. Note however thatif we suppress any off-diagonal elements (in the energyeigenbasis) of the initial density matrix ˆ ρ , then the abovemoment generating function corresponds to a two-pointmeasurement of the internal energy [82]. We may re-write Eq. (A1) using the spectral decomposition of theLiouvillian e S = (cid:88) i e λ i t Tr (cid:110) e − iu ˆ H P i (cid:16) e i u ˆ H ˆ ρe i u ˆ H (cid:17)(cid:111) , (A2)3The eigenvalues of the Liouvillian are denoted by λ i andthe projectors P i depend on the eigenstates of the Liou-villian. In the long-time limit, the sum is dominated bythe eigenvalue with the largest real part, denoted λ max ,and we find S ( χ , λ, u ) = λ max ( χ , λ ) t + S ( χ , λ, u ) , (A3)where the time-independent term is given by S = ln Tr (cid:110) e − iu ˆ H P max ( χ , λ ) (cid:16) e i u ˆ H ˆ ρe i u ˆ H (cid:17)(cid:111) . (A4)In the long-time limit, we may safely drop the term S inthe cumulant generating function. In particular, for thecumulants we find (cid:104)(cid:104) ( Q − W − ∆ U ) k (cid:105)(cid:105) /t = (cid:104)(cid:104) ( Q − W ) k (cid:105)(cid:105) /t + O (1 /t ) . (A5)In particular, this implies that the first law may not berecovered by taking into account the internal energy fluc-tuations, as these scale differently with time.We now show that the long-time limit of the full count-ing statistics of heat is fully determined by a single count-ing field, i.e., P ( q ) ∝ δ q h , − q c . To this end, we introducethe unitary superoperator U ˆ A = e i χc (ˆ a † c ˆ a c +ˆ a † h ˆ a h ) ˆ Ae − i χc (ˆ a † c ˆ a c +ˆ a † h ˆ a h ) . (A6)To show that this superoperator is unitary, we note thatits adjoint, U † , is defined by (cid:104)U † ˆ B, ˆ A (cid:105) = (cid:104) ˆ B, U ˆ A (cid:105) , (A7)where we introduced the inner product (cid:104) ˆ B, ˆ A (cid:105) =Tr { ˆ B † ˆ A } . We find U † ˆ A = e − i χc (ˆ a † c ˆ a c +ˆ a † h ˆ a h ) ˆ Ae i χc (ˆ a † c ˆ a c +ˆ a † h ˆ a h ) , (A8) and it follows that U † U = 1 .We now consider the Liouvillian, setting λ = 0 forsimplicity (the proof also holds for λ (cid:54) = 0 ) ˜ L ( χ h − χ c )ˆ ρ = U † L ( χ ) U ˆ ρ = − i [ ˆ H, ˆ ρ ] + L χ h − χ c h ˆ ρ + L c ˆ ρ. (A9)As ˜ L only depends on the difference of the counting fields,so do its eigenvalues. Since a unitary transformationleaves the eigenvalues invariant, the same holds for theeigenvalues of the original Liouvillian L ( χ ) . In the long-time limit, the cumulant generating function thus onlydepends on the difference in counting fields, cf. Eq. (A3).The relation P ( q ) ∝ δ q h , − q c then follows directly fromEq. (25). We note that for short times, the cumulantgenerating function also depends on the eigenvectors ofthe Liouvillian and therefore explicitly depends on both χ h and χ c . Appendix B: The Wigner function approach
In this appendix, we show how the distribution P ( q , w ; γ ) can be calculated. All results shown in themain text follow from this distribution. As shown inSec. V, the cumulant generating function of this distribu-tion can be written as S = ln[Tr { ˆ ρ } ] , where ˆ ρ is governedby the master equation ∂ t ˆ ρ = − i [ ˆ H + γ ˆ I, ˆ ρ ] − i λ { ˆ I, ˆ ρ } + L χ h h ˆ ρ + L χ c c ˆ ρ. (B1)Here the Hamiltonian is given in Eq. (6), the currentoperator in Eq. (16), and the superoperators responsi-ble for dissipation in Eq. (23). To get rid of the time-dependence, we perform a unitary transformation with ˆ U = e i Ω h t ˆ a † h ˆ a h + i Ω c t ˆ a † c ˆ a c . (B2)We then cast Eq. (B1) into an equation of motion for theWigner function W ( r c , r h ) = (cid:90) dy c π dy h π e i ( p c y c + p h y h ) (cid:68) x h + y h , x c + y c (cid:12)(cid:12)(cid:12) ˆ ρ (cid:12)(cid:12)(cid:12) x c + y c , x h + y h (cid:69) , (B3)where r α = ( x α , p α ) and ˆ x α | x h , x c (cid:105) = x α | x h , x c (cid:105) , with the quadrature operators defined by ˆ a α = (ˆ x α + i ˆ p α ) / √ .Introducing the nabla operators ∇ α = ( ∂ x α , ∂ p α ) , we may write the equation of motion for the Wigner function as ∂ t W ( r c , r h ) = (cid:26) ig [ r h ( σ y − iγ ) ∇ c + r c ( σ y + iγ ) ∇ h ] + (cid:88) α = c,h κ α (cid:2) ∇ α · r α + ( n αB + 1 / ∇ α (cid:3) + (cid:88) α = c,h κ α (cid:2) ξ α + ( r α + ∇ α /
4) + ξ α − ( ∇ α · r α − (cid:3) − λg ( r h σ y r c − ∇ h σ y ∇ c ) (cid:27) W ( r c , r h ) , (B4)where σ y denotes the Pauli y -matrix and we abbreviated ξ α ± = ( n αB + 1) (cid:0) e iχ α − (cid:1) ± n αB (cid:0) e − iχ α − (cid:1) . (B5)4Equation (B4) can be solved by the Gaussian ansatz W ( r c , r h ) = e S (2 π ) √ det Σ e − ( r c , r h ) T Σ − ( r c , r h ) , Σ = σ c σ ch s ch σ c − s ch σ ch σ ch − s ch σ h s ch σ ch σ h . (B6)The cumulant generating function is then determined by the coupled, non-linear differential equations ∂ t S = κ c ξ c + σ c − κ c ξ c − + κ h ξ h + σ h − κ h ξ h − − iλgs ch , (B7a) ∂ t σ c = 2 g ( s ch − γσ ch ) + κ c ( n cB + 1 / − σ c ) − iλgσ c s ch + κ c ξ c + ( σ c + 1 / − κ c ξ c − σ c + κ h ξ h + ( σ ch + s ch ) , (B7b) ∂ t σ h = − g ( s ch − γσ ch ) + κ h ( n hB + 1 / − σ h ) − iλgσ h s ch + κ h ξ h + ( σ h + 1 / − κ h ξ h − σ h + κ c ξ c + ( σ ch + s ch ) , (B7c) ∂ t s ch = g ( σ h − σ c ) − κ c + κ h s ch − iλg ( σ c σ h + s ch − σ ch − /
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