Full distribution of work done on a quantum system for arbitrary initial states
FFull distribution of work done on a quantum system for arbitrary initial states
P. Solinas and S. Gasparinetti SPIN-CNR, Via Dodecaneso 33, 16146 Genova, Italy Department of Physics, ETH Z¨urich, CH-8093 Z¨urich, Switzerland (Dated: September 12, 2018)We propose a novel approach to define and measure the statistics of work, internal energy and dissipated heatin a driven quantum system. In our framework the presence of a physical detector arises naturally and work andits statistics can be investigated in the most general case. In particular, we show that the quantum coherence ofthe initial state can lead to measurable effects on the moments of the work done on the system. At the sametime, we recover the known results if the initial state is a statistical mixture of energy eigenstates. Our methodcan also be applied to measure the dissipated heat in an open quantum system. By sequentially coupling thesystem to a detector, we can track the energy dissipated in the environment while accessing only the systemdegrees of freedom.
I. INTRODUCTION
One of the cornerstones of the Copenhagen interpretationof quantum mechanics is the measurement postulate: after aprojective measurement, the wave function collapses into aneigenstate of the measured observable. In this framework, twosubsequent measurements of an observable are not indepen-dent, as the first measurement perturbs the state of the systemand thereby affects the result of the second [1]. Still, thereare quantities in classical physics which are not ”local” intime and need two (or more) observations to be determined.Among them are the charge flowing through and the workdone on a system. In such cases, the extension of classicaldefinitions and protocols to the quantum realm is not straight-forward. Recently, the statistics of the work done on a quan-tum system and, more generally, its energy exchanges haveattracted much attention [2–9]. Besides a fundamental inter-est, the thermodynamics of quantum systems has importantimplications to the energetic performance of quantum devices[10] and quantum heat engines [11].An established protocol to measure work involves a dou-ble projective measurement of the energy of the system atthe beginning and at the end of the evolution. Such a two-measurement protocol (TMP) can be described in terms ofclassical conditional probabilities [2, 3, 12, 13]. It has provensuccessful in formulating quantum fluctuation relations in asetting where the system is initially in a statistical mixture ofenergy eigenstates. However, TMP has limitations that haveso far failed to receive the due attention. These limitations be-come apparent when one tries to apply TMP to a more generalclass of processes, namely, those in which the system is ini-tially in a quantum-coherent superposition of different energyeigenstates. Most quantum gates developed in the contextof quantum information and computation [10] belong to thisclass. The problem with TMP is that the initial measurementforces the system into an eigenstate of the initial Hamiltonian.The ultimate result of this operation is to reduce the dynamicsto a classical statistical one [7] and to destroy the interferenceeffects that generate, in Feynman’s words, the ”interfering al-ternatives” in the dynamics [14]. In this respect, TMP failsto answer in a general way the most straightforward and im-portant question [15, 16]: how much energy is it needed in order to perform a given quantum operation on an arbitrarilyprepared quantum system?In this Article, we address the key question we have justposed by proposing a measurement protocol that is meantto preserve the quantum-mechanical nature of the work per-formed on a quantum system. In this protocol, a quantumdetector is coupled to the system at the beginning and at theend of the evolution. The information on the energy is storedin a phase shift that can be measured, for example, by inter-ferometric means. Our protocol can be formally derived froma path-integral description of the dynamics by adding a con-straint on the admissible paths [17–19]. Its predictions coin-cide with that of TMP for a mixed initial state. However, assoon as we introduce quantum coherence in the initial state,we find a stark disagreement between the two protocols, evenat the level of the first moment of the distribution, that is, theaverage work. We discuss the reasons for this disagreementand set the stage for further investigation. We also discusshow to extend our protocol to measure the dissipated work indriven, open quantum systems, by accessing the degrees offreedom of the system only. As compared to previous propos-als relying on measurements of the environment [6, 20, 21],our protocol may provide an experimentally more accessibleway to measure the statistics of heat and work in this case.
II. GENERAL FORMALISM
It is known that the work done on a quantum system cannotbe associated to a hermitian operator and, therefore, it is notan observable [3, 22]. In general, the work performed on anopen system depends on the full evolution of the system andnot only on its initial and final state. As quantum trajectories(or paths) play a key role in determining energy exchangesand dissipation, we find it natural to tackle the work measure-ment problem by using a path integral approach [14]. Theformalism we describe in this section is an adaptation of thatdeveloped by Sokolovski in a series of papers [17–19]. Werefer to the Appendix A and to the original papers for moretechnical details.We consider a closed quantum system whose dynamicsis generated by a time-dependent Hamiltonian ˆ H S ( t ) . The a r X i v : . [ qu a n t - ph ] A ug drive starts at t = t = T . The correspondingunitary evolution operator can be approximated as U ( T ) = −→ T exp ( − i (cid:82) T dt ˆ H S ( t )) ≈ Π Nk = e − i ∆ t ˆ H kS where −→ T denotes thetime-ordering product and in the second writing we have dis-cretized the time in N + ∆ t and used the no-tation ˆ H S ( k ∆ t ) = ˆ H kS .The probability amplitude to go from the initial state | ψ (cid:105) to a given final state | ψ T (cid:105) can be decomposed into Feynmanpaths [14]. Differently from the usual approach, in which thedynamics is described in the position-momentum basis, weexploit the freedom to choose any complete basis at each timeto decompose the paths. If we are interested in the behavior ofa time dependent operator ˆ A ( t ) , the preferred basis is the onecomposed of its eigenstates, i.e., ˆ A ( t ) | a i ( t ) (cid:105) = a i ( t ) | a i ( t ) (cid:105) .The idea behind this choice is that it allows us to associateto ˆ A ( t ) a value a ( t ) depending on the path traversed duringthe evolution. In a more formal way, we write t k = k ∆ t and {| a i ( t k ) (cid:105)} ≡ {| a ki k (cid:105)} for a complete basis set. By insertingthe completeness relation for | a ki k (cid:105) into the expression for theprobability amplitude to go from | ψ (cid:105) to | ψ T (cid:105) , we obtain (seeA) (cid:104) ψ T | U ( T ) | ψ (cid:105) ≈ (cid:104) ψ T | Π Nk = e − i ∆ t ˆ H kS | ψ (cid:105) = ∑ all P A P (1)where P is the path defined by the sequence of states {| a i (cid:105) , | a i (cid:105) , ..., | a Ni N (cid:105)} (see Fig. 1) and A P is the probabilityamplitude to go from | ψ (cid:105) to | ψ T (cid:105) , associated to that path.Along the path P , the operator ˆ A ( t ) takes the set of values (cid:16) a i , a i , ..., a Ni N (cid:17) ≡ a ( t ) . Thus, we can also associate to P anyfunctional F [ P ] of a ( t ) .At this point, we add a constraint to the evolution by re-quiring F [ P ] to take the value f . The constrained probabil-ity amplitude reads A [ f ] = ∑ P δ ( F [ P ] − f ) A P . As in [19],we consider functionals of the form F [ P ] = (cid:82) T dt β ( t ) a ( t ) = ∆ t ∑ Nk = β k a ki k , where β ( t ) is an arbitrary function. The Diracdelta in the expression for A [ f ] can be written as a Fouriertransform in a conjugate space described by the variable λ ,as follows: δ ( F [ P ] − f ) = (cid:82) d λ exp [ i λ ( F [ P ] − f )] . Notice[17] that λ and f can be thought of as eigenvalues of conju-gate operators ˆ λ and ˆ f acting on an additional Hilbert space,their corresponding eigenstates | λ (cid:105) and | f (cid:105) satisfying the re-lation (cid:104) λ | f (cid:105) = e − i λ f . Denoting with ˆ A k ≡ ˆ A ( k ∆ t ) and recall-ing that ˆ A N | a Ni N (cid:105) = a Ni N | a Ni N (cid:105) , we can write (see Appendix A) A [ f ] = (cid:82) d λ ∑ P A λ f P , where A λ f P = (cid:104) ψ T , λ | e − i ∆ t ( ˆ H NS − ˆ λβ N ˆ A N ) | a Ni N (cid:105) · . . . ·(cid:104) a i | e − i ∆ t ( ˆ H S − ˆ λβ ˆ A ) | a i (cid:105) · (cid:104) a i | ψ , f (cid:105) . (2)is the probability amplitude to go from the state | ψ , f (cid:105) to thestate | ψ T , λ (cid:105) [17–19]. The evolution described by A λ f P is gen-erated by the effective Hamiltonianˆ H ( t ) = ˆ H S ( t ) − ˆ λ β ( t ) ˆ A ( t ) . (3)Equation (3) plays a central role in our work and it is worth afew comments. (i) The additional Hilbert space we introduced E ne r g y (cid:72) a . u . (cid:76) Ε (cid:72) t (cid:76) Ε (cid:72) t (cid:76) Ε (cid:72) t (cid:76) Ε (cid:72) t N (cid:45) (cid:76) t t t N (cid:45) t time (cid:72) a.u. (cid:76) t N (cid:45) Ε (cid:72) t N (cid:45) (cid:76) FIG. 1.
Quantum work and path integral.
Pictorial representationof the unitary evolution of a quantum system from the initial state ψ (in this case an eigenstate fo the initial Hamiltonian) to the genericfinal state ψ T , described in terms of paths in energy space. The time-dependent energy spectrum ε i ( t ) of the system Hamiltonian is plottedin black. Quantum trajectories (blue, red) consists of a sequence ofjumps between different eigenstates. The red trajectory satisfies theconstraint (in this case ∆ U =
0) while the blue ones do not. can be related to a detector in the von Neumann measurementscheme [23]. Therefore, requiring that the functional F as-sumes the value f along the evolution is equivalent to intro-ducing a detector and coupling it to the observable we wishto measure. Here, ˆ λ and ˆ f act as the momentum and posi-tion operator of the detector, respectively. (ii) The interactiondescribed by (3) does not induce any transition between theeigenstates | λ (cid:105) of the detector momentum. (iii) The informa-tion about the system-detector interaction – and hence aboutthe value taken by the functional F – is encoded in the phaseaccumulated between the eigenstates | λ (cid:105) and | λ (cid:48) (cid:105) .Observation (iii) suggests that the statistics of the integratedobservable ˆ A ( t ) can be determined by measuring the phase ofthe detector, as done in the full-counting-statistics approach(FCS) [24–26]. Let the composite system be initially de-scribed by the factorized density operator ρ = ρ S ⊗ ρ D , where ρ S and ρ D are the density operators of the system and the de-tector, respectively. Then the phase difference acquired be-tween the eigenstates | λ / (cid:105) and |− λ / (cid:105) of the detector reads G λ = (cid:104) λ / | ρ D ( t ) |− λ / (cid:105)(cid:104) λ / | ρ D |− λ / (cid:105) = Tr S (cid:104) U λ / ( t ) ρ S U † − λ / ( t ) (cid:105) (4)where U λ ( t ) = −→ T exp [ − i (cid:82) t dt (cid:48) ( ˆ H S − λ β ˆ A )] is the evolutionoperator generated by (3). The function G λ plays the role of amoment generating function, as the n -th moment of A is givenby (cid:104) A n (cid:105) = ( − i ) n d n G λ / d λ n | λ = [27, 28]. III. INTERNAL ENERGY OF A CLOSED SYSTEM
We now have the instruments to determine the variation ofthe internal energy of a driven closed system. Starting fromEq. (3), we take ˆ A ( t ) = ˆ H S ( t ) and β ( t ) = δ ( t − T ) − δ ( t ) [19].This corresponds to coupling the detector and the system onlyat the beginning and at the end of the drive. (More precisely,we couple the system and the detector at time t = − and attime t = T + , i.e., immediately before and after the starting andending drive.) The resulting G λ is given by Eq. (4), with theevolution operator (see Appendix A) U λ / ( T ) = e i λ H S ( T ) U ( T ) e − i λ H S ( ) . (5)The so-obtained G λ is a measurable quantity and can beused to determine all moments of the internal-energy varia-tion ∆ U in the system. However, the interpretation of thisresult presents some subtleties, which we are now going todiscuss.It is known from previous work [26–29] that, in general, theFourier transform of the G λ in Eq. (4) cannot be associated toa probability distribution. A similar problem is encounteredwhen defining the FCS of electron transfer across a supercon-ducting device [30]. If a probability distribution cannot bedefined for the variation of the internal energy, the questionthen arises what is the meaning of the moments generated by G λ . To clarify this point, let us first analyze the first moment,which for a closed system corresponds to the average workperformed on the system. A physical expectation for the re-sult can be developed by considering the following gedanken-experiment . We repeatedly prepare the system in the same ini-tial state ρ S ( ) . Half of times we just measure ˆ H S ( ) and de-termine its average (cid:104) ˆ H S ( ) (cid:105) . The remaining times we first ap-ply the desired evolution to arrive at ρ S ( t ) = U ( t ) ρ S ( ) U † ( t ) and then measure ˆ H S ( T ) to determine (cid:104) ˆ H S ( T ) (cid:105) . According tothis procedure, we estimate variation of the internal energy as ∆ U = (cid:104) ˆ H S ( T ) (cid:105) − (cid:104) ˆ H S ( ) (cid:105) . This result is the same as obtainedfrom Eq. (4); by contrast, it cannot be reproduced by TMP.To pinpoint the differences between the two methods, let usexplicitly write ∆ U as obtained from Eq. (4) : ∆ U = ∑ i ρ S , i , i ∑ k W k , i ( ε T k − ε i ) + ∑ k , i (cid:54) = j ρ S , i , j ε T k U k , i U † j , k , (6)where ρ S , i , j = (cid:104) ε i | ρ S | ε j (cid:105) , U k , i = (cid:104) ε T k | U ( T ) | ε i (cid:105) , U † j , k = (cid:104) ε j | U † ( T ) | ε T k (cid:105) , W k , i = |(cid:104) ε T k | U ( T ) | ε i (cid:105)| , and | ε k (cid:105) and | ε T k (cid:105) arethe eigenstates of the Hamiltonian at the beginning and at theend of the evolution, respectively. The first term in (6) is thesame as in TMP [2, 3, 31] and can be straightforwardly inter-preted in terms of classical conditional probabilities. On thecontrary, the remaining terms, which depend on the initial co-herences ρ S , i , j , are of a purely quantum nature. These termsare destroyed by the initial measurement of ˆ H S ( ) performedin TMP. The fact that the interfering terms can have importanteffect in the statistics of the work was first pointed out in Ref.[15, 32].The situation is well exemplified by the cyclic evolution ofa coherent superposition of energy eigenstates into itself. Asboth the initial and final state and the initial and final Hamil-tonians are the same, we would naturally expect ∆ U = H S ( T ) = ˆ H S ( ) . We initializethe system in a state | ψ (cid:105) that, a part from a phase factor,in left unchanged by the evolution generated by ˆ H S ( T ) , i.e., U ( T ) | ψ (cid:105) = e i ξ | ψ (cid:105) . The existence of such a state is guar-anteed, for instance, by Floquet theorem [33]. Clearly, theinternal energy of the system does not change and, therefore, ∆ U =
0. This is correctly predicted by the first moment cal-culated from G λ .But, in general, | Ψ (cid:105) needs not be an eigenstate of H S ( ) .We consider the case in which | Ψ (cid:105) = cos α | ε (cid:105) + sin α | ε (cid:105) where | ε i (cid:105) ( i = ,
2) are eigenstate of the initial (and final)Hamiltonian. If we take α to be a free parameter, then therequirement of cyclic evolution for | Ψ (cid:105) forces the evolutionoperator to take the form in the {| ε (cid:105) , | ε (cid:105)} U ( T ) = (cid:18) cos ξ + i cos 2 α sin ξ i sin 2 α sin ξ i sin 2 α sin ξ cos ξ − i cos 2 α sin ξ (cid:19) . With the TMP, after the first measurement, the system is foundin | ε (cid:105) with probability cos α and in | ε (cid:105) with probabilitysin α . These two states now evolve independently as the “in-terfering alternatives” have been destroyed by the projectivemeasurement. The final result for the work distribution canbe computed in terms of classical conditional probabilities P i j for the system to make a transition between states i and j . Inparticular, for the average change in the internal energy, onefinds ∆ U = ∆ E ( P − P ) = ∆ E cos 2 α sin α sin ξ , where ∆ E = (cid:104) ε | H S ( ) | ε (cid:105) − (cid:104) ε | H S ( ) | ε (cid:105) . We have thusfound that ∆ U is generally nonzero, except in the cases ξ = α = , π / α = π / G λ is a quasi-probability which can assume neg-ative values. A probability distribution can be retrieved insome cases after partial integration of the relevant Wignerfunction [26, 27]. Ultimately, these complications are rootedin the full quantum treatment of the detector [27]. Indeed,different types of measurements performed at the end of theevolution yield different distributions for the same quantity,each of which must be interpreted accordingly. The mea-surement of the phase of the detector has the advantage that,since ˆ λ is a constant of motion, preserves the “quantumness”of the evolution and leads to Eq. (4). It is our belief that thequantum correlations stemming from Eq. (4) should not be ig-nored; instead, they deserve further exploration. For instance,the negativity in the quasi-probability distribution of work canbe thought of as due to nonclassical temporal correlations ofthe energy, leading to the violation of a Leggett-Garg-type in-equality [27, 28, 34]. Further progress in this direction willhopefully appear in future work. IV. OPEN SYSTEM AND HEAT STATISTICS
We now turn our attention to the more general case in whichthe system is coupled to an environment during the drive. Inorder to determine the work performed on the system, we needto complement the measurement of the internal energy dis-cussed above with one of the dissipated heat. To this end,different approaches have been proposed, including the mea-surement of an engineered environment [3, 20, 21, 35, 36].Yet measuring the environment is a challenging task, restrict-ing the applicability of these proposal to specific physical re-alizations. In the following, we describe an extension of ourmeasurement protocol that allows one to obtain the statisticsof the work and dissipated heat by accessing only the systemdegrees of freedom.We describe the open system by the Hamiltonian ˆ H = ˆ H S + ˆ H SE + ˆ H E where ˆ H E and ˆ H SE are the environment andsystem-environment coupling Hamiltonians, respectively, andassume weak coupling between the system and environment.We first take both ˆ H and ˆ H S to be time independent and con-sider a measurement of ˆ H S . Then (5) simplifies into U − λ / = e − i λ ˆ H S e − i T ˆ H e i λ ˆ H S . As ˆ H and ˆ H S are constant, no externalwork is done on the system and the variation of internal energymust correspond to the dissipated heat. We can also show (seeAppendix B) that the statistics obtained from the above equa-tion is the same as the one obtained by measuring the environ-ment degrees of freedom [2, 35, 37, 38]. We conclude that foran open system with constant Hamiltonian, the scheme givesthe statistics of the dissipated heat Q (Appendix B).For a time-dependent ˆ H S ( t ) , we discretize the evolutionin N time intervals ∆ t , denote ˆ H k = ˆ H kS + ˆ H SE + ˆ H E with U k = e − i ∆ t ˆ H k . Within each time interval ∆ t , the Hamiltonianis constant. At the beginning and at the end of each interval,we instantaneously couple our detector to ˆ H kS . In analogy with(5), the evolution operator for each interval reads U k λ / = e − i λ ˆ H kS e − i ∆ t ˆ H k e i λ ˆ H kS . (7)Each U k λ is defined so that we keep track of the heat Q k dis-sipated in the time interval ( k − ) ∆ t ≤ t ≤ k ∆ t . As a result,the information on the dissipated heat along the evolution isstored in the phase of the detector. Notice the opposite signin the exponents with respect to Eq. (3) takes into account thefact that an emission (absorption) by the environment, i.e., de-creasing (increasing) of the environment energy, correspondsto an absorption (emission) process of the system, i.e., in-creasing (decreasing) of the system energy.In order to account for the variation of the internal energy aswell, we must add a measurement of H S at the beginning andend of the evolution (Appendix C). Putting things together,the total evolution operator reads U λ / = e i λ ˆ H NS Π Nk = U k λ / e − i λ ˆ H S . (8)A pictorial representation of the scheme described by (8) ispresented in Fig. 2. In the case of unitary evolution, ˆ H k = ˆ H kS and we immediately recover the closed-system result for H H (cid:68)Ρ (cid:68) H H Τ (cid:68) t FIG. 2.
Measuring work and dissipation in open quantum sys-tems.
Schematic representation of the sequence of driven evolutionsand interactions with the detector with the open system protocol inEq. (8). The evolution steps exp ( − i ∆ t ˆ H k ) are represented by the flatline and characterized by the Hamiltonian H k . Each coupling withthe detector is represented by a dot. The coupling is either of theform exp ( − i λ ˆ H kS ) (red dots), or exp ( i λ ˆ H kS ) (blue dots). In the blue-shadowed region, the evolution is frozen ( ∆ ρ =
0) and the Hamilto-nian changes by ∆ H . In the red-shadowed region, the Hamiltonian isconstant ( ∆ H =
0) while the density operator changes by ∆ ρ . the variation of the internal energy. The moment generatingfunction is the same as in (4) with U λ given by (8). Let uscalculate its first moment, which gives the average work W = − idG λ / d λ | λ = . We find (Appendix C) W = Tr S (cid:104) ˆ H NS ρ S , N − ˆ H S ρ S , − Σ k ˆ H kS ∆ ρ S , k (cid:105) , (9)where ∆ ρ S , k = ρ S , k − ρ S , k − . In the first two terms werecognize the variation of the internal energy of the sys-tem: ∆ U = Tr S (cid:104) ˆ H NS ρ S , N − ˆ H S ρ S , (cid:105) . Accordingly, we iden-tify the remaining term with the dissipated heat: Q = Σ k Q k = Σ k Tr S (cid:104) ˆ H kS ∆ ρ S , k (cid:105) . Notice that while ∆ U depends only on theinitial and final state of the system, Q is determined by the fulldissipative evolution as in the classical counterpart (AppendixC).In the fast-decoherence limit, the dissipated heat takes anilluminating form. When energy-relaxation processes aremuch faster than the dynamics induced by the drive, we al-ways find the system in its instantaneous thermal equilibriumstate ρ S ( t ) = exp [ − ˆ H S ( t ) / k B T ] / Z S ( t ) where Z S ( t ) is the par-tition function of the system and T is the temperature of theenvironment. In other words, the system evolves throughstates of quasi-equilibrium. Defining the Von Neumann en-tropy as S = − Tr [ ρ S log ρ S ] , we can show (see Appendix D)that the variation of the density operator is related to it byTr S (cid:104) ˆ H kS ∆ ρ k (cid:105) = k B T ∆ S k where ∆ S k is the variation of entropyat time t k . Then we can link the variation of entropy to thedissipated heat by the relation Q k = k B T ∆ S k , confirming theabove interpretation of Q as the dissipated heat.There is an alternative way to interpret Eq. (9). Takingthe time derivative of the average internal energy, we have d (cid:104) ˆ H S ( t ) (cid:105) / dt = (cid:104) ˙ˆ H S ( t ) ρ ( t ) (cid:105) + (cid:104) ˆ H S ( t ) ˙ ρ ( t ) (cid:105) [15]. If the evolu-tion is unitary, the second contribution vanishes and we canrelate the variation of the system Hamiltonian to the instan-taneous work done on the system. By expanding the productin (8), we identify pairs of sequential system-detector interac-tions of the form exp ( i λ ˆ H k + S / ) exp ( − i λ ˆ H kS / ) . Each suchpair effectively keeps track of a variation in the Hamiltonian.As the variation is instantaneous, the system has no dynamics.We can interpret the action of the pairs as a “measurement”of the work done on the system by an external force. Thisinterpretation is strengthened by the analysis of Eq. (9). Byregrouping the terms, we can write it as W = Tr S (cid:104) Σ k ∆ ˆ H kS ρ S , k (cid:105) where ∆ ˆ H kS = ˆ H k + S − ˆ H kS .One may wonder whether the repeated coupling to the de-tector can ’freeze’ the dynamics of the system (dynamic Zenoeffect). This turns out not to be the case: a dynamic Zeno ef-fect would require λ → ∞ , while we derive our physical quan-tities, i.e., the moments of the work done, in the opposite limit λ →
0. Our protocol can instead be regarded as a noninvasivemeasurement [28, 39] of the work distribution. In fact, themoments generated by G λ depend on evolution operators thatdescribe the dynamics of the open system without a detector. V. CONCLUSIONS AND OUTLOOK
In summary, we have shown that the statistics of work per-formed on a quantum system exhibits nonclassical correla-tions in a deeper and more fundamental way that it had so farbeen appreciated. Such correlations become apparent onceone replaces the customary double projective measurementwith a less-invasive coupling to a quantum detector. The re-sulting protocol is immediately applicable to the case of uni-tary evolution and can be suitably extended to treat open quan-tum systems. Our approach puts the problem of work undera new perspective and leads the way toward further investiga-tions. In particular, the links between quantum-mechanicalwork, Leggett-Garg-type inequalities [27, 34], weak mea-surements [39], and stochastic quantum trajectories [40–42],await to be fully elucidated. An experimental test of our pre-dictions is in reach of state-of-the art quantum technology.Among different architectures, superconducting quantum cir-cuits in combination with nearly-quantum-limited parametricamplifiers are a first choice, given the high degree on controlachieved in recent experiments [41, 43–45].
ACKNOWLEDGMENTS
We gratefully acknowledge A. Braggio and M. Carrega forfruitful discussions. P.S. has received funding from the Eu-ropean Union FP7/2007-2013 under REA grant agreementno 630925 – COHEAT and from MIUR-FIRB2013 – ProjectCoca (Grant No. RBFR1379UX). S.G. acknowledges finan-cial support from the Swiss National Science Foundation(SNF) Project 150046.
Appendix A: Probability amplitude in the path integralrepresentation
We consider a closed quantum system whose dynamics isgenerated by a time-dependent Hamiltonian ˆ H S ( t ) . The drivestarts at t = t = T . The corresponding unitaryevolution operator is U ( T ) = −→ T e − i (cid:82) T dt ˆ H S ( t ) ≈ Π Nk = e − i ∆ t ˆ H kS (A1)where −→ T denotes the time-ordering product and in the secondwriting we have discretized the time in N + ∆ t and used the notation ˆ H S ( k ∆ t ) = ˆ H kS . Our goal is to writethe probability amplitude to go from | ψ (cid:105) to | ψ T (cid:105) in terms ofFeynman paths [14]. Equation (A1) is approximated to theorder ∆ t and it is a convenient way to describe the evolutionin terms of path integral. As a final step, we will take thelimit ∆ t → At any time t k we can find a basis {| a i ( t k ) (cid:105)} ≡ {| a ki k (cid:105)} suchthat the completeness relation ∑ i k | a ki k (cid:105)(cid:104) a ki k | = holds. In thisnotation, k is a time index and i k denotes the eigenstate ba-sis index at time t k . By inserting the completeness relationfor | a ki k (cid:105) into the expression for the probability amplitude weobtain (cid:104) ψ T | U ( T ) | ψ (cid:105) = N ∑ k = ∑ i k (cid:104) ψ T | e − i ∆ t ˆ H NS | a Ni N (cid:105)(cid:104) a Ni N | e − i ∆ t ˆ H N − S | a N − i N − (cid:105) ... (cid:104) a i | e − i ∆ t ˆ H S | a i (cid:105)(cid:104) a i | e − i ∆ t ˆ H S | a i (cid:105) (cid:104) a i | ψ (cid:105) . (A2)The term (cid:104) a ki k | e − i ∆ t ˆ H NS | a k − i k − (cid:105) is the probability amplitude to gofrom | a k − i k − (cid:105) to | a ki k (cid:105) . Then, A P = (cid:104) ψ T | e − i ∆ t ˆ H NS | a Ni N (cid:105)(cid:104) a Ni N | e − i ∆ t ˆ H N − S | a N − i N − (cid:105) ... (cid:104) a i | e − i ∆ t ˆ H S | a i (cid:105)(cid:104) a i | e − i ∆ t ˆ H S | a i (cid:105) (cid:104) a i | ψ (cid:105) (A3)is the probability amplitude to go from | ψ (cid:105) to | ψ T (cid:105) passingthrough the sequence of states: | a i (cid:105) , | a i (cid:105) , ..., | a Ni N (cid:105) . This se-quence define a path P in the basis space {| a ki k (cid:105)} . We interpret(A2) as the sum over all the possible paths of the probabilityamplitudes: (cid:104) ψ T | U ( T ) | ψ (cid:105) = ∑ all P A P . (A4)In the limit ∆ t →
0, we obtain a continuous path a ( t ) . In thisway, we can associate to the path P a physical quantity F [ P ] depending on it. Mathematically, F is then a functional of P .We now add a constraint and select only the paths that sat-isfy some properties. We are asking which is the probabilityamplitude A [ f ] to go from | ψ (cid:105) to | ψ T (cid:105) though a path P de-termined by a ( t ) for which the functional F [ P ] assumes thevalues f . The constrained probability amplitude reads A [ f ] = ∑ P δ ( F [ P ] − f ) A P (A5)where the delta function restricts the admissible paths to thosefor which F [ P ] = f . F [ P ] could a generic functional of P .However, as in Ref. [19], we assume that the functional de-pends on the integral of the path a ( t ) (or | a i (cid:105) , | a i (cid:105) , ..., | a Ni N (cid:105) in the discretized expression) F [ P ] = (cid:90) T dt β ( t ) a ( t ) = ∆ t N ∑ k = β k a ki k (A6)where β ( t ) is an arbitrary function.By writing the Dirac delta in terms of Fourier Transform δ ( F [ P ] − f ) = (cid:82) d λ exp [ − i λ ( f − F [ P ])] inserting it in thepath integral representation, and splitting the term F [ P ] withrespect to the corresponding time interval, we obtain A [ f ] = (cid:90) d λ e − i λ f ∑ P (cid:104) ψ T | e − i ∆ t ˆ H NS e i ∆ t λβ N a NiN | a Ni N (cid:105) ... (cid:104) a i | e − i ∆ t ˆ H S e i ∆ t λβ a i | a i (cid:105) (cid:104) a i | ψ (cid:105) . (A7)We must choose the time-dependent basis set {| a ki k (cid:105)} consid-ering the observable ˆ A ( t ) we are interested in. In particular,we must take them in order that ˆ A k | a ki k (cid:105) = a ki k | a ki k (cid:105) . For small ∆ t [46], we can write A [ f ] ≈ (cid:90) d λ e − i λ f ∑ P (cid:104) ψ T | e − i ∆ t ( ˆ H NS − λβ N ˆ A N ) | a Ni N (cid:105) ... (cid:104) a i | e − i ∆ t ( ˆ H S − λβ ˆ A ) | a i (cid:105) (cid:104) a i | ψ (cid:105) = (cid:90) d λ e − i λ f ∑ P A λ P . (A8)Therefore, the constrained amplitude probability A [ f ] can bewritten as the sum of the path amplitudes A λ P , which are gen-erated by the effective Hamiltonian ˆ H S ( t ) − λ β ( t ) ˆ A ( t ) . Bydefining the corresponding unitary operator as U λ ( t ) = −→ T e − i (cid:82) t dt (cid:48) [ ˆ H S ( t ) − λβ ( t ) ˆ A ( t )] , (A9)we have ∑ P e i λ F [ P ] A P = ∑ P A λ P = (cid:104) ψ T | U λ ( t ) | ψ (cid:105) . (A10)We can go further with the interpretation of the con-straint. The parameter λ and f can be thought of as eigen-values of conjugate operators ˆ λ and ˆ f satisfying the rela-tion (cid:104) λ | f (cid:105) = e − i λ f . Using the relation e − i ∆ t ( ˆ H k − λβ k ˆ A k ) (cid:104) λ | = (cid:104) λ | e − i ∆ t ( ˆ H k − ˆ λβ k ˆ A k ) , we can write A [ f ] = (cid:90) d λ ∑ P (cid:104) ψ T , λ | e − i ∆ t ( ˆ H N − ˆ λβ N ˆ A N ) | a Ni N (cid:105) ... (cid:104) a i | e − i ∆ t ( ˆ H − ˆ λβ ˆ A ) | a i (cid:105) (cid:104) a i | ψ , f (cid:105) (A11)Based on Eq. (A11), we can interpret A [ f ] is the probabilityamplitude to go from the state | ψ , f (cid:105) to the state | ψ T , λ (cid:105) ,where ˆ λ and ˆ f are conjugate operator acting on an additionalHilbert space. The latter is interpreted as the Hilbert spaceof the detector needed to measure the special observable [17–19]. The effective Hamiltonian describing the system and aquantum detector dynamics isˆ H ( t ) = ˆ H S ( t ) − ˆ λ β ( t ) ˆ A ( t ) . (A12) The approach outlined above applies to any time-dependentobservable ˆ A ( t ) . In this work we take ˆ A ( t ) to be the time-dependent Hamiltonian. The power operator considered inRef. [15] would be another meaningful choice.To determine the variation of the internal energy, we takeˆ A ( t ) = ˆ H S ( t ) and β ( t ) = δ ( T − t ) − δ ( t ) . When using adiscretized evolution, we assume that δ ( t k − t ) = / ∆ t for t k ≤ t ≤ t k + = t k + ∆ t and 0 elsewhere. From Eqs. (B1) wehave, in the limit of ∆ t → U λ ( T ) ≈ e − i ∆ t ( ˆ H N − λβ N ˆ A N ) e − i ∆ t ( ˆ H N − − λβ N − ˆ A N − ) ... e − i ∆ t ( ˆ H − λβ ˆ A ) → e i λ ˆ H ( T ) U ( T ) e − i λ ˆ H ( ) . (A13)Therefore, the total unitary evolution corresponds to two fastcouplings with the detector with an central driven evolution ofthe system. From this we immediately arrive to the momentgenerating function G λ discussed in the main text. Appendix B: System versus environment measurement.
In the main text we have discussed how to measure the dis-sipated heat statistics though the system degrees of freedom.Here we show that this statistics is the same as we would ob-tain by measuring directly the environment [3, 35].We make the standard assumption that the system andthe environment are weakly coupled. This allows us toneglect the energy related to system-environment couplingHamiltonian. We consider ˆ H S to be time-independent asin the fundamental interaction block discussed in the maintext. The total Hamiltonian reads ˆ H = ˆ H S + ˆ H E + ˆ H SE .If we measure the degrees of freedom of the environment,we obtain a ¯ G λ that has the form of G λ with ¯ U λ ( T ) = exp ( i λ ˆ H E / ) exp ( − i ˆ HT ) exp ( − i λ ˆ H E / ) [35].In the weak coupling limit, exp ( i λ ˆ H E / ) = exp [ i λ ( ˆ H − ˆ H S − ˆ H SE ) / ] ≈ exp [ i λ ( ˆ H − ˆ H S ) / ] and [ ˆ H , ˆ H S ] ≈
0. Therefore,¯ U λ ( T ) ≈ e − i λ ˆ H S / e i λ ˆ H / e − i ˆ HT e − i λ ˆ H / e i λ ˆ H S / = e − i λ ˆ H S / e − i ˆ HT e i λ ˆ H S / = U − λ ( T ) . (B1)From Eq. (B1) it follows that ¯ G λ = G − λ and the statistics gen-erated by measuring ˆ H E is equal to the one obtained by mea-suring ˆ H S with opposite sign. The opposite sign in the expo-nents with respect to Eq. (B1) takes into account the fact thatan emission (absorption) by the environment, i.e., decreasing(increasing) of the environment energy, corresponds to an ab-sorption (emission) process of the system, i.e., increasing (de-creasing) of the system energy. Appendix C: First moment of the work done on a quantumsystem
We first restrict our attention to the dissipated heat and cal-culate the first moment of the moment generating function G λ − Q = − i dG λ d λ (cid:12)(cid:12)(cid:12) λ = = Tr S + E (cid:104) dU λ d λ ρ U † − λ + U λ ρ dU † − λ d λ (cid:105)(cid:12)(cid:12)(cid:12) λ = (C1)where U λ = Π Nk = U k λ , U k λ = e − i λ ˆ H kS / e − i ∆ t ˆ H k e i λ ˆ H kS / , and wefollow the convention that the heat flowing into the system isgiven a positive sign [3].The first term in Eq. (C1) reads − i dU λ d λ (cid:12)(cid:12)(cid:12) λ = = − (cid:104) ˆ H NS U − U N ( ˆ H NS − ˆ H N − S ) U N − ... U + ... − U N ... U ( ˆ H S − ˆ H S ) U − U ˆ H S (cid:105) . (C2)where we used the compact notation U k = e − i ∆ t ˆ H k . In an anal-ogous way, the second term reads − i dU † − λ d λ (cid:12)(cid:12)(cid:12) λ = = (cid:104) ˆ H S U † + U †0 ( ˆ H NS − ˆ H N − S ) U †1 ... U † N − + ... + U †0 ... U † N − ( ˆ H S − ˆ H S ) U † N − U † ˆ H NS (cid:105) . (C3)Putting everything together in Eq. (C1) and using the cyclicproperty of the trace, it is possible to simplify some of theevolution operators U k . After defining ρ k = U k ... U ρ U †0 ... U † k and ˜ ρ = U ρ U †0 , we have − Q = Tr S + E (cid:104) − ˆ H NS ( ρ N − − ρ N − ) ... − ˆ H S ( ˜ ρ − ρ ) (cid:105) . (C4)The trace over the system and environment can beseparated by observing that Tr S + E (cid:104) ˆ H NS ( ρ N − ρ N − ) (cid:105) = Tr S (cid:104) ˆ H NS Tr E (cid:16) ρ N − ρ N − (cid:17)(cid:105) = Tr S (cid:104) ˆ H NS ( ρ S , N − ρ S , N − ) (cid:105) . There-fore, the dissipated heat written in terms of the system degreesof freedom reads Q = Tr S (cid:104) Σ k H kS ( ρ S , k − ρ S , k − ) (cid:105) = Σ k Tr S (cid:104) H kS ∆ ρ S , k (cid:105) = Σ k Q k (C5)where Q k is the dissipated heat in the time interval t k − ≤ t ≤ t k . Let us now introduce a coupling between the system andthe detector at the beginning and at the end of the evolution.The unitary operator then reads U λ = e i ˆ λ ˆ H NS / Π Nk = U k λ e − i ˆ λ ˆ H S / . (C6)Accordingly, the calculation in Eq. (C2) is modified as − i dU λ ( t ) d λ (cid:12)(cid:12)(cid:12) λ = = ˆ H NS U − U ˆ H S − i dd λ (cid:16) Π Nk = U k λ (cid:17)(cid:12)(cid:12)(cid:12) λ = , (C7)which differs from Eq. (C2) by the addition of the termˆ H NS U − U ˆ H S .We find that the average work W is W = Tr S (cid:104) H NS ρ S , N − H S ρ S , − Σ k H kS ( ρ S , k − ρ S , k − ) (cid:105) (C8)which differs from Eq. (C5) by the variation of the internalenergy ∆ H = Tr S (cid:104) H NS ρ S , N − H S ρ S , (cid:105) . Therefore, we have ob-tained the usual result W = ∆ H − Q [3].The heat contributions Q k in (C5) are related to the variationof the system density operator ∆ ρ k during infinitesimal evolu-tions generated by constant Hamiltonians. We can check that if the evolution is unitary, i.e., H k = H kS , then the Q k vanishand no heat is dissipated. In fact, we have ρ S , k = U k ρ S , k − U † k , [ H S , k , U k ] =
0, and Q k = Tr S (cid:104) H kS ρ S , k − H kS ρ S , k − (cid:105) = Tr S (cid:104) U k H kS ρ S , k − U † k − H kS ρ S , k − (cid:105) = . (C9)The interpretation of the Q k as the dissipative contributionto the dynamics is strengthened by the following observation.The dissipated heat depends on the variation of the densityoperator ∆ ρ S , k , which, in turn, can be due to both unitary andnon-unitary dynamics. However, the unitary contribution tothe change of ρ , i.e., the one given by [ H S , ρ S ] , vanishes iden-tically when we calculate H kS ( ρ S , k − ρ S , k − ) . Thus, the Q k arerelated solely to the dissipative dynamics. This result is anal-ogous to the one obtained in Ref. [15].The expressions for Q and W can be written in anothermeaningful way as follows. Instead of grouping ∆ ρ k , we cankeep the terms ∆ H kS = ˆ H kS − ˆ H k − S as written in Eq. (C2). Thenthe dissipated heat in Eq. (C4) reads Q = Tr S (cid:104) − H NS ρ S , N + H S ρ S , + Σ k ∆ H kS ρ S , k (cid:105) . (C10)As the contributions of the initial and final measurements arethe same, we have that W = Tr S (cid:104) Σ k ∆ H kS ρ S , k (cid:105) . (C11)This confirms the interpretation discussed in the main text thatthe work can be seen as the instantaneous energy injected inthe system due to the variation of the Hamiltonian in time. Appendix D: Work, heat and entropy in quantum system anddynamics
The Von Neumann entropy in a quantum system is definedas S = − Tr [ ρ S log ρ S ] . Writing it in the basis {| i (cid:105)} in which ρ S is diagonal, we obtain S = − ∑ i ρ S , ii log ρ S , ii . If we take thetime derivative of the entropy, we have dSdt = − ∑ i ( ˙ ρ S , ii log ρ S , ii + ˙ ρ S , ii ) = − ∑ i ˙ ρ S , ii log ρ ii (D1)where the last equation comes from the fact that ∑ i ˙ ρ S , ii = ρ S ( t ) = exp [ − β ˆ H S ( t )] / Z S ( t ) where β = / ( k B T ) is the inverse tem-perature of the environment and Z S ( t ) is the partition func-tion. In addition, we have log [ ρ S ( t )] = − ˆ H S ( t ) − log [ Z S ( t )] .Again, this must be intended in terms of component in the ba-sis in which ρ S and ˆ H S are diagonal. If ε i is the energy ofthe state | i (cid:105) , we have ρ S = ∑ i ρ S , ii | i (cid:105)(cid:104) i | = ∑ i e − βε i / Z S | i (cid:105)(cid:104) i | andlog ρ S , ii = − β ε i − log [ Z S ( t )] . With Eq. (D1), we can write dSdt = ∑ i [ β ˙ ρ ii ε i + log Z S ( t ) ˙ ρ S , ii ] = k B T ∑ i ˙ ρ S , ii ε i . (D2)Keeping in mind that H S = ∑ k ε k | k (cid:105)(cid:104) k | , we have that ∑ i [ ˙ ρ S , ii ε i ] = Tr [ d ρ S dt H S ] and we can rewrite the above equation as Tr (cid:104) d ρ S dt H S (cid:105) = T dSdt . (D3)The variation of the entropy can be related, as in the classicalcase, to the dissipated heat. With this identification we find Q k = Tr S (cid:104) ˆ H kS ∆ ρ k (cid:105) = k B T ∆ S k . [1] P. A. M. Dirac, The principles of quantum mechanics 4th ed (theinternational series of monographs on physics no 27) (1967).[2] M. Esposito, U. Harbola, and S. Mukamel, Rev. Mod. Phys. ,1665 (2009).[3] M. Campisi, P. H¨anggi, and P. Talkner, Rev. Mod. Phys. , 771(2011).[4] R. Dorner, S. R. Clark, L. Heaney, R. Fazio, J. Goold, and V. Ve-dral, Phys. Rev. Lett. , 230601 (2013).[5] L. Mazzola, G. De Chiara, and M. Paternostro, Phys. Rev. Lett. , 230602 (2013).[6] M. Campisi, R. Blattmann, S. Kohler, D. Zueco, and P. H¨anggi,New J. Phys. , 105028 (2013).[7] T. B. Batalh˜ao, A. M. Souza, L. Mazzola, R. Auccaise, R. S.Sarthour, I. S. Oliveira, J. Goold, G. De Chiara, M. Paternostro,and R. M. Serra, Phys. Rev. Lett. , 140601 (2014).[8] A. J. Roncaglia, F. Cerisola, and J. P. Paz, Phys. Rev. Lett. ,250601 (2014).[9] S. An, J.-N. Zhang, M. Um, D. Lv, Y. Lu, J. Zhang, Z.-q. Yin,H. Quan, and K. Kim, Nat. Phys. (2014).[10] M. A. Nielsen and I. L. Chuang, Quantum computation andquantum information (Cambridge university press, 2010).[11] R. Kosloff and A. Levy, Annu. Rev. Phys. Chem. , 365(2014), pMID: 24689798.[12] J. Kurchan, arXiv preprint cond-mat/0007360 (2000).[13] H. Tasaki, arXiv preprint cond-mat/0009244 (2000).[14] R. P. Feynman and A. R. Hibbs, Quantum mechanics and pathintegrals (McGraw-Hill, 1965).[15] P. Solinas, D. V. Averin, and J. P. Pekola, Phys. Rev. B ,060508 (2013).[16] J. Salmilehto, P. Solinas, and M. M¨ott¨onen, Phys. Rev. E ,052128 (2014).[17] D. Sokolovski and R. S. Mayato, Phys. Rev. A , 042101(2005).[18] D. Sokolovski and R. S. Mayato, Phys. Rev. A , 052115(2006).[19] D. Sokolovski, Phys. Rev. D , 076001 (2013).[20] S. Gasparinetti, K. L. Viisanen, O. P. Saira, T. Faivre, M. Arzeo,M. Meschke, and J. P. Pekola, Phys. Rev. Appl. , 014007(2015).[21] J. Pekola, P. Solinas, A. Shnirman, and D. Averin, New J. Phys. , 115006 (2013).[22] P. Talkner, E. Lutz, and P. H¨anggi, Phys. Rev. E , 050102(2007).[23] J. V. Neumann, Mathematical foundations of quantum mechan- ics , 2 (Princeton university press, 1955).[24] L. Levitov and G. Lesovik, JETP Lett. , 230 (1993).[25] L. S. Levitov, H. Lee, and G. B. Lesovik, J. Math. Phys. ,4845 (1996).[26] Y. Nazarov and M. Kindermann, EPJ B , 413 (2003).[27] A. A. Clerk, Phys. Rev. A , 043824 (2011).[28] A. Bednorz, W. Belzig, and A. Nitzan, New J. Phys. , 013009(2012).[29] A. Bednorz and W. Belzig, Phys. Rev. Lett. , 106803 (2010).[30] W. Belzig and Y. V. Nazarov, Phys. Rev. Lett. , 197006(2001).[31] A. Engel and R. Nolte, Europhys. Lett. , 10003 (2007).[32] A. E. Allahverdyan, Phys. Rev. E , 032137 (2014).[33] M. Grifoni and P. H¨anggi, Phys. Rep. , 229 (1998).[34] A. J. Leggett and A. Garg, Phys. Rev. Lett. , 857 (1985).[35] S. Gasparinetti, P. Solinas, A. Braggio, and M. Sassetti, New J.Phys. , 115001 (2014).[36] M. Carrega, P. Solinas, A. Braggio, M. Sassetti, and U. Weiss,New J. Phys. , 045030 (2015).[37] M. Campisi, P. Talkner, and P. H¨anggi, Phys. Rev. Lett. ,210401 (2009).[38] M. Silaev, T. T. Heikkil¨a, and P. Virtanen, Phys. Rev. E ,022103 (2014).[39] C. Gardiner and P. Zoller, Quantum Noise (Springer, 2004), 3rded.[40] H. Carmichael,
An Open Systems Approach to Quantum Optics (Springer, 1993).[41] K. W. Murch, S. J. Weber, C. Macklin, and I. Siddiqi, Nature , 211 (2013).[42] S. J. Weber, a. Chantasri, J. Dressel, a. N. Jordan, K. W. Murch,and I. Siddiqi, Nature , 570 (2014).[43] R. Vijay, C. Macklin, D. H. Slichter, S. J. Weber, K. W. Murch,R. Naik, a. N. Korotkov, and I. Siddiqi, Nature , 77 (2012).[44] M. Hatridge, S. Shankar, M. Mirrahimi, F. Schackert, K. Geer-lings, T. Brecht, K. M. Sliwa, B. Abdo, L. Frunzio, S. M.Girvin, et al., Science , 178 (2013).[45] D. Rist`e, M. Dukalski, C. a. Watson, G. de Lange, M. J. Tiggel-man, Y. M. Blanter, K. W. Lehnert, R. N. Schouten, and L. Di-Carlo, Nature , 350 (2013), 1306.4002.[46] H. Kleinert,