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Maxwell's demon in the quantum-Zeno regime and beyond

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Published 5 February 2018 © 2018 The Author(s). Published by IOP Publishing Ltd on behalf of Deutsche Physikalische Gesellschaft
, , Citation G Engelhardt and G Schaller 2018 New J. Phys. 20 023011DOI 10.1088/1367-2630/aaa38d

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Abstract

The long-standing paradigm of Maxwell's demon is till nowadays a frequently investigated issue, which still provides interesting insights into basic physical questions. Considering a single-electron transistor, where we implement a Maxwell demon by a piecewise-constant feedback protocol, we investigate quantum implications of the Maxwell demon. To this end, we harness a dynamical coarse-graining method, which provides a convenient and accurate description of the system dynamics even for high measurement rates. In doing so, we are able to investigate the Maxwell demon in a quantum-Zeno regime leading to transport blockade. We argue that there is a measurement rate providing an optimal performance. Moreover, we find that besides building up a chemical gradient, there can be also a regime where the feedback loop additionally extracts energy, which results from the energy non-conserving character of the projective measurement.

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1. Introduction

Maxwell's demon is the central character in a long-standing gedankenexperiment suggested by Maxwell in 1871, which challenges the validity of the second law of thermodynamics [1, 2]: a box containing an ensemble of particles is divided in two compartments. The Maxwell demon observes the particles and has the ability to open and close a door such that only fast particles can enter the 'left' compartment, while slow particles leave the 'left' compartment. In doing so, the demon can build up a thermal gradient, which can later be used to run a thermal engine. However, as the opening and closing does not consume energy in the ideal case, this procedure would violate the second law of thermodynamics saying that such a perpetuum mobile of the second kind is not possible. This paradox was resolved by Landauer by recognizing that the Maxwell demon has to delete information in order to perform its task. This is directly related to heat dissipation [36].

This article provides a quantum mechanical treatment of Maxwell's demon. In quantum mechanics the action of the Maxwell demon is more involved due to the special meaning of the observation or measurement of the particles, namely the wave-function-collapse postulate. A quantum measurement thus does not leave the system state unaffected so that the observation by the Maxwell demon necessarily affects the dynamics. Here we investigate these quantum implications at the example of a single-electron transistor (SET). We implement the Maxwell demon using a piecewise-constant feedback scheme, where the system state is observed after time periods τ and the system parameters are adjusted accordingly. This feedback scheme has been already successfully implemented in experiment [7]. For other experimental and theoretical feedback-related approaches in mesoscopic devices to extract work or to achieve similar objectives we refer to [827].

In this article, we harness a dynamical coarse-graining (DCG) method [28]. This method provides interesting properties which are perfectly suitable for the issues which we are interested in. The DCG is designed in a way so that it becomes exact for short evolution times in contrast to a Born–Markov master equation or other coarse-graining approaches [2931]. For this reason, it is favorable to use it to describe the piecewise-constant feedback protocol for high measurement rates, as in this case the time-evolution is repeatedly restarted after each measurement.

In contrast to other methods as, e.g., the so-called Redfield equation, the DCG ensures complete positivity for all times [28, 32, 33], so that thermodynamic quantities, e.g., the system entropy, are guaranteed to be well defined for all time instants. Moreover, [34] shows that this technique even accounts for highly non-Markovian effects observable in the coherence or the entanglement dynamics. Furthermore, this approach can be amended for a full-counting statistics treatment in systems under non-equilibrium conditions [35].

The DCG thus provides a reliable accuracy for the parameter range which we are interested in. This article goes beyond the treatment in [13], where the time dynamics has been approximated by a Born–Markov master equation.

While investigating short feedback times, we unavoidably run into another long-standing paradox of physics, namely the quantum-Zeno effect. Strictly following the principles of quantum mechanics one finds that the dynamics of a quantum system freezes when continuously measuring it with projective measurements [3638]. This paradigm is particularly interesting in the context of the classical Maxwell demon, who continuously observes the system of its interest. Indeed, we find with the system and methods at hand that the action of the Maxwell demon results in a blocking of the particle and heat currents between the reservoirs.

Moreover, due to the action of the demon in the quantum regime, we observe another side effect. Besides building up a chemical potential gradient between the two reservoirs which could be used to charge a battery, we argue that there can be also a net energy decrease of the system due to the feedback action. We explain that it is most convenient to run the Maxwell demon in such a regime, as we do not have to invest external power in order to make the demon work.

This article is organized as follows: in section 2, we explain the SET, which we describe by a Fano–Anderson model. We give a compact introduction to the DCG method applied throughout the article and prove its validity. In section 3, we explain the implementation of the Maxwell demon by a piecewise-constant feedback scheme and show how to model this on the level of the equation of motion for the reduced density matrix. We show how the DCG approach reveals the quantum-Zeno effect for a continuous measurement. In section 4, we discuss the thermodynamic properties of the system like electric power, gain, heat flow and entropy production in the quantum-Zeno regime and beyond. In section 5, we provide a concluding discussion of our results. Supplemental information is given in the appendix.

2. Model and methods

We implement the Maxwell demon in a SET, which is a mesoscopic transport setup consisting of two electron reservoirs coupled by a quantum dot. A sketch of the system is depicted in figure 1(a). We model the SET by a two-terminal Fano–Anderson model, whose Hamiltonian reads [3941]

where and are fermionic operators representing the central dot with on-site energy epsilon and the reservoir states with energies , respectively. Thereby, (right, left) labels the reservoirs and k are their internal states. The hopping amplitudes between dot and reservoir states are given by .

Figure 1. Refer to the following caption and surrounding text.

Figure 1. (a) Sketch of the SET under the action of the Maxwell demon. Two reservoirs, which are locally in thermal equilibrium, are connected via a quantum dot with on-site energy epsilon. The Maxwell demon monitors the occupation of the dot and adjusts the tunnel barriers according to its observation. In doing so, one can generate a current against the chemical potential bias even at equal temperatures. (b) As an implementation of the Maxwell demon, we consider a piecewise-constant feedback protocol. At times , we projectively measure the dot occupation (empty, filled) using, e.g., a quantum-point contact. Subsequently, we adjust the tunnel rates accordingly for the next feedback period . On the level of equations we model this with the propagators as explained in equation (15). (c) Temporal sketch of the feedback action in detail. The times denote the times (infinitesimally) before the measurement, after the measurement and after switching the tunnel rates, respectively.

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We have introduced the index ν to implement a feedback protocol: the Hamiltonian (or more precisely the hopping amplitudes) will be conditioned on the dot occupation (empty, filled). We provide more details in section 3.

The initial condition which we consider throughout the article is given by

where is the initial density matrix of the dot and are the initial density matrices of the reservoirs. Thus, the reservoirs are considered to be locally in a thermal equilibrium state with inverse temperatures and chemical potentials . Here, Zα denotes the partition function which ensures that and is the particle number operator of the reservoir α.

In the following, we apply a DCG method [32] to calculate the dynamics of the reduced density matrix of the quantum dot , where denotes the trace over the reservoir degrees of freedom. We represent the reduced density matrix of the quantum dot in the local basis and and introduce the notation . In doing so, the diagonal elements of the reduced density matrix of the system decouple from the coherences and approximately read as a function of time

where the coarse-grained Liouvillian reads

In equation (4) we additionally introduced the counting fields and , which allow for a determination of the number of particles and the amount of energy entering the reservoir α during the time interval . For brevity we thereby combine the counting fields in the (transposed) vector . The matrix entries read

where is the sinc function and we have used the abbreviations

with the Fermi function and the spectral coupling density .

In the numerical calculations throughout the article, we use

This is a Lorentz function which is centered around and has a width . The function is the Heavyside function which ensures a compact support of the spectral coupling density between and needed for numerical calculations.

In figure 2(a), the accuracy of the DCG method is benchmarked against the exact solution of the Fano–Anderson model in the absence of feedback action [40]. There, we depict the occupation of the dot, which is given by . The DCG approach is optimized to resemble the exact dynamics for short times t [28, 32], see figure 2(a). By construction, in the long-time limit the DCG dynamics converges to the dynamics of the Born–Markov-secular (BMS) master equation, which resembles the exact solution for the parameters under consideration. Consequently, the DCG method guarantees a good performance for short times in all parameter regimes and for long times in the weak-coupling limit. Importantly, due to its construction, the DCG method maintains a Lindblad form in equation (4) for all times t, which ensures positivity of and consequently guarantees well-defined thermodynamic calculations.

Figure 2. Refer to the following caption and surrounding text.

Figure 2. (a) Dot occupation as a function of time. The results of the DCG approach, the exact solution and the BMS master equation are depicted with a solid (green), dashed and dotted line, respectively. The parameters are , , , , , , , and . (b) Time-averaged currents under the action of the Maxwell demon in the stationary state as a function of the feedback time τ investigated in section 3. Overall parameters are as in panel (a) with , except that we have changed the cut-off frequencies to , . The curves with feedback parameter are depicted in blue, orange and black, respectively. The solid, dashed and dotted lines depict the solution with the DCG method, the linear expansion for short times τ and the BMS master equation result, respectively.

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Importantly, in the derivation of the DCG method it is assumed that the initial state is a product state of the central system and the reservoirs, which are assumed to be locally in a thermal equilibrium state [32]. As we explain below, this requirement is met for the feedback scheme which we apply here.

3. Feedback control

In order to implement the Maxwell demon we apply a projective measurement in combination with a piecewise-constant feedback scheme. This is sketched in figure 1(b). At times we conduct projective measurements of the dot occupation. According to the outcome, we adjust the system parameters which then remain constant for the following time interval . In particular, here we vary the tunnel barriers which are parameterized by with if the dot occupation has been empty or filled at time tn, respectively.

3.1. Action on the density matrix of the total system

The feedback interventions occur at times . Within the time intervals the total system (including the reservoirs) evolves under the Hamiltonian equation (1) conditioned on and is therefore conservative. The total energy can thus only change during the feedback interventions at times .

The intervention can be divided in two steps. To this end, we introduce the (virtual) times and as depicted in figure 1(c) for illustration. First, one has to measure the dot occupation. This measurement shall take place during the time interval . Is it know that a projective measurement can change the mean energy of the system [4244], which can even give rise to measurement induced extracted work in a Maxwell demon setup [45].

Second, according to the measurement outcome, we adjust the tunnel barriers . This step shall take place in the time interval . However, as we explain in section 4.2, for projective measurements the switching work can be here neglected, such that only the first step changes the energy of the total system.

As the total Hamiltonian equation (1) is quadratic, the most important observables as the total system energy can be determined by the single-particle density matrix

with .

The projectors which project the system state to the empty and filled quantum dot are given by

Their action on the total system density matrix is accordingly [46]

In consequence, it is easy to see that the single-particle density matrix elements become and . The two distinct measurement results can be found with probabilities , so that the density matrix of the total system after the measurement irrespective of the measurement outcome reads

In the subsequent time interval , the dynamics is determined by the Hamiltonian depending on the measurement result . Importantly, means that there are no system-reservoir coherences after the measurement. This is essential for the application of the DCG method in the subsequent time evolution.

3.2. Time evolution of the reduced density matrix

In the following, we describe the dynamics on the level of the reduced density matrix of the quantum dot . For its diagonal elements contained in the vector , the projective measurement in equation (11) translates into

where

are the measurement operators corresponding to the measurement results .

The projective measurement is subsequently followed by a time evolution which is conditioned on the measurement result. The conditioned time evolution can be described by the DCG approach in equation (3), so that the feedback time-evolution propagator reads [47]

where . The propagator evolves the reduced density matrix by one feedback period τ. We emphasize that the repeated application of the propagator is in agreement with the derivation of the DCG method, which assumed that the initial condition is given by a product state of system and reservoir density matrices. This requirement is fulfilled, as the projective measurements result in a destruction of the system-reservoir correlations as explained in section 3.1.

From the generalized propagator we obtain the moment generation function (MGF)

where denotes the stationary density matrix, as we are interested in the long-term dynamics. The stroboscopic stationary state at times is the eigenvector of with eigenvalue ,

which always exists and depends on the measurement rate τ. Furthermore, it can be shown that the second eigenvalue fulfills . It thus describes the relaxation dynamics towards the stationary state. In terms of the MGF, the number of particles entering reservoir α within the time interval τ is given by [29]

In the same way we obtain the change of energy in reservoir α by deriving with respect to instead.

In figure 2(b), we depict the time-averaged matter current as a function of τ. In doing so, we have chosen the parametrization of the spectral coupling density as in equation (8), but with the proportional parameter adjusted to

The parameter δ controls the feedback action. For , there is no feedback control. For , the feedback control supports a matter current from the left to the right reservoir, while for the feedback supports the opposite direction.

We depict for three different feedback strengths δ with solid lines. Here and in the following, we choose equal temperatures in order to exclude a thermoelectric effect [40]. For the chemical potentials we consider . Consequently, the time-averaged current is negative in the absence of feedback, as can be found for . For a positive feedback parameter , we can find a current against the bias and we generate a time-averaged electric power, which we define as

Using this definition, we generate electric power for and waste power for . For , our numerical calculations verify that the feedback protocol supports the current along the chemical potential bias.

For long feedback times the time-averaged current is always directed along the chemical potential bias irrespective of the feedback strength δ. This can be explained as follows. In the limit of long feedback times τ, the dynamics of the propagators conditioned on in equation (15) converges to the ones of the BMS master equation, respectively [28]. Regardless of the feedback time τ, the propagator in equation (15) describes an average of two distinct time evolutions with no feedback. For the BMS master equation (in the absence of feedback and at equal temperatures) it is known that the current always flows along the chemical potential bias in the long-time limit. This is a consequence of the second law of thermodynamics which is respected by the BMS master equation. Consequently, the measurement-averaged current becomes directed along the chemical potential gradient.

3.3. Maxwell demon in the quantum-Zeno regime

In the limit of continuous feedback , the current vanishes independent of the feedback parameter δ as can be observed in figure 2(b). This can be explained with the quantum-Zeno effect. For , the propagator calculated using the DCG approach in equation (15) becomes

This means that now both eigenvalues are . As a consequence, the system is now bistable: Either the dot is occupied or empty for all times. In addition, coherences are continuously projected to zero as discussed in section 3.1. Due to the infinite measurement rate, the dot dynamics gets thus frozen, so that no particle can enter or leave the reservoirs. Thus, the DCG method under consideration resembles the quantum-Zeno effect [3638].

Here, the Zeno suppression is solely induced by frequent projective measurements. By contrast, there is also the possibility to generate a Zeno effect by increasing the decoherence rate by increasing the coupling to the reservoirs as discussed in [4850]. In the DCG approach applied here, this effect is absent, as we treat the coupling to the reservoirs in lowest order (e.g., in the strong-coupling regime the treatment is only valid for very short coarse-graining times).

In order to find a compact approximation and an intuitive explanation for the behavior in the quantum-Zeno regime, we expand the MGF for short times up to the lowest non-vanishing order in τ which still contains a dependence on the counting field. In doing so, we find

where we have defined

and

We note that the spectral coupling density must ensure an appropriate frequency cut-off in order to avoid that higher derivatives with respect to diverge. The MGF in the short feedback time limit thus reads

where

denotes the corresponding occupation of the dot. We emphasize that the first non-vanishing order of the MGF is . In consequence, the time-averaged current and all higher cumulants vanish for . This behavior is a typical feature in the Zeno regime [37].

Generally, the BMS master equation results are not valid in the short-time regime. A short-time expansion as before reveals why the BMS treatment provides an inaccurate result as can be seen in figure 2(b). Formally, the time evolution of the BMS approach reads as in equation (3), but with the time-dependent matrices replaced by time-independent ones, thus . Performing the same expansion of the MGF, we find that . Consequently, the Born–Markov treatment leads to a finite time-averaged current even for vanishing feedback times.

Let us choose a positive feedback strength δ which implies a current against the bias for rather short τ, thus . As we explained in section 3.2 , for long feedback periods τ the current flows always along the chemical potential gradient, thus . Then there must be consequently a feedback time at which the time-averaged current vanishes, thus . As due to the Zeno effect, it is also clear that there must be a feedback time τ in the interval at which the current against the chemical potential reaches a maximum value, as can be seen in figure 2(b).

4. Power, gain and heat flow

4.1. Power

In figure 3(a), we depict the power as a function of the bias and the feedback time τ. The dashed lines depict levels of equal power. The solid lines show the set of . There are two ways to cross this boundary. At the line V = 0 the bias changes sign, while in the upper left region of the diagram there is a sign change of as the time-averaged current changes its direction. Overall, the power is close to zero in wide parts of the diagram, but shows more structure for small τ. Here, we see that the feedback scheme generates most power for large negative bias and intermediate feedback times . On the other hand, most power is wasted for a large positive power and an intermediate feedback time.

Figure 3. Refer to the following caption and surrounding text.

Figure 3. (a) Time-averaged power as a function of the chemical potential bias V and the feedback time τ. Dashed lines depict sets of equal power. The solid lines mark the set with . Overall parameters are as in figure 2(b) with . (b) Depicts the corresponding gain defined in equation (28). In the gray region we find and the gain is not interesting as we waste power. In the white region we find . The most efficient way to run the Maxwell demon is in the blue region, where so that one does not have to invest power to run the feedback protocol. (c) Total amount of heat defined in equation (36) which enters () or leaves () the reservoirs. The solid lines mark the set with . The regions of are strongly correlated to the regions of .

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4.2. Gain

In order to estimate the performance of Maxwell's demon, power is not the only decisive quantity. As the total process is not conservative, the energy of the total system changes. In particular, we find that the total energy can increase or even decrease on average. This amount of energy change is denoted with feedback energy in the following. For , the action of the Maxwell demon leads to an increase of the total system energy, while for , the total system energy decreases. We define the corresponding gain parameter

where we restrict the definition of the gain to positive power and feedback energy . The feedback energy can be calculated by considering in detail the feedback process as sketched in figures 1(b), (c) and explained in section 3.1.

We first determine the change of the mean energy of the total system due to the measurement in the virtual time interval . To this end, we have to determine the difference of the energies of the total system in equation (1) before and after the measurement,

Thus, we compare the energy of the state shortly before the measurement with the state after the measurement with regard to the total Hamiltonian before the measurement. As theses density matrices differ only in the dot-reservoir coherences , which vanish due to the measurement , we find

where we have introduced the notation

This can be evaluated as follows

From line one to line two we have used that the total system evolves under a conservative time evolution in the interval . Line two is equal to line three as there are no dot-reservoir coherences at time . Finally in line four, we have defined the energy differences corresponding to the dot and reservoir subsystem, respectively, thus , and accordingly for .

Equation (32) is an interesting result, as it relates the change of energy induced by the measurement at time with the energy-conserving time evolution in the preceding time interval . We emphasize that this result is exact and holds even for more complicated Hamiltonians under the assumption that all system-reservoir coherences vanish due to the projective measurement.

If we consider a stationary state which is characterized by , we find for the averaged energy change

where is the density matrix if the measurement outcome at time has been ν. The corresponding probability is denoted by . In the stationary state, the averaged dot energy is constant at times tn, so that we find

This is the energy entering the reservoir during the interval averaged over the measurement results ν in the stationary state. Using the DCG method, we can thus obtain the feedback energy by deriving the MGF

where we have finally partitioned the total energy change into energies entering the left and right reservoirs within the feedback period τ. We note, that this result is consistent with the first law of thermodynamics.

In figure 3(b) we depict the gain G in the same regime as in (a) where we generate power, . For regions where we use a color code. For a clear representation, we restrict the range to . In the blue regions, we find a negative feedback energy , thus, the total system energy decreases due to the measurement and the demon does not perform work on the system but extracts work. It is thus most profitable to operate the system in this region.

Close to the transition at the gain diverges. This line represents the original idea of the Maxwell demon that due to a energy conserving action of the demon (measurement, opening and closing the door) one can generate a thermal gradient or increase a chemical potential bias. However, we emphasize that even though in the quantum regime the measurement does not change the energy balance, it changes the state of the system. This is in contrast to the classical Maxwell demon, where the measurement leaves the system state unaffected.

In principle, one could argue that a negative feedback energy could be stored or used by a smart demon for another application. However, a detailed discussion of this issue could become possible when specifying the measurement apparatus [5153].

4.3. Heat, entropy and information efficiency

Next we discuss the heat flow and the thermodynamic consistency. In the stationary state, the change of heat entering the reservoirs within a feedback period τ reads

The heat is depicted in figure 3(c). The solid curves represent sets where the total heat change in the reservoirs vanishes . These curves resemble roughly the zero power curves . A power generation is thus correlated to an overall loss of heat in the reservoirs. The correlation of and becomes clear when considering the relation

As is small compared to for a wide parameter range (compare with figure 3(b)), we find , which explains the correspondence of the curves and .

The second law of thermodynamics says that on average the total entropy increases in time in the absence of feedback processes. In a stationary state, this relation reads

where denotes the entropy production and is the entropy change in the reservoirs within a time interval τ. The change of entropy in the system (i.e., quantum dot) is given by

In [54], Esposito and coworkers have derived a general relation for the entropy change which is valid for arbitrary system-reservoir setups. Under the assumption of a product initial state as in equation (2) and a unitary time evolution, the entropy change in the reservoirs reads

Similar considerations can be found in [12, 55, 56]. It is straightforward to generalize this to the feedback protocol considered here. To this end, we consider the change of entropy conditioned on the measurement outcome at time within the subsequent feedback period . For both measurement outcomes the second law in equation (38) together with equation (40) is fulfilled separately, so that we find for the measurement-averaged entropy change

The action of the measurement is to delete the entropy of the system by exactly the amount during the virtual time interval [7]. For this reason, one can interpret equation (41) in the following way: the amount of entropy reduced by the measurement is not completely transferred to the reservoirs [57]. Moreover, it is not hard to prove that , so that we recover the Landauer principle [6].

Equation (41) allows to define a coefficient which measures how efficient the information is used to decrease the entropy in the reservoir

which we denote as information efficiency in the following [16]. While , the entropy change in the reservoir can be both, positive or negative, so that the information efficiency is not bounded from below. For similar inequalities in other feedback systems we refer to [58, 59].

We depict η in figure 4. We observe that the information efficiency is bounded by , which is a sanity check for the applied DCG method. The information efficiency is rather similar to the heat entering the reservoirs. This is a consequence of the dot occupation, which is approximately for the considered parameters.

Figure 4. Refer to the following caption and surrounding text.

Figure 4. (a) Information efficiency η as defined in equation (42). The parameters are as in figure 3.

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4.4. Feedback energy and gain in the Zeno regime

As the general expression for the matter and energy current is rather involved, it is hard to understand under which circumstances the feedback energy is small or even negative. For this reason, we focus on the Zeno regime in the following, where the expressions are simpler. In this regime, the feedback energy reads

where . In order to keep the analysis simple, we focus on an extremal feedback case where . In most cases we numerically find that the stationary dot occupation is close to , so that we find a small or negative current if . This relation implies

This condition can be met if the temperatures in the reservoirs are rather high , so that the Fermi functions are rather close to around a broad range around . If additionally is large for large ω and is large for small ω, the quantity and consequently also the feedback energy can become rather small or even negative.

This is exactly the parameter range which we use in figure 3, although we do not work in an extremal feedback limit. There we have chosen a rather high temperature . In the parametrization of the spectral densities in equation (8), we use and .

Furthermore, we can infer from equation (20) that a large bias V results in a large time-averaged power . Although this has a detrimental effect on the averaged matter current , the overall effect is indeed a large power as we can see in figure 3(a).

5. Discussion and conclusions

We have harnessed a DCG approach in order to conveniently describe the dynamics of the SET under the action of the Maxwell demon. We have implemented the demon by a piecewise-constant feedback scheme, where the occupation of the quantum dot is projectively measured with frequency . The accuracy of the DCG has been tested by benchmarking it with the exact solution in the absence of feedback. For vanishing feedback times τ, which corresponds to a continuous observation of the system by the demon, we resembled the quantum-Zeno effect by which the current between the reservoirs is blocked. Moreover, we found that the power and efficiency are optimized for an intermediate feedback time τ outside of the quantum-Zeno regime. The performance of the system is also better for a large bias and higher temperatures. With the DCG method we could thus show that there is an intermediate regime between a genuine quantum effect and a classical rate equation dynamics to optimize the performance of a quantum device under dissipative conditions. This seems thus reminiscent to the interplay of quantum and dissipative effects in other transport scenarios [6063].

Furthermore, we have discovered a novel aspect appearing in the quantum treatment of Maxwell's demon. Due to the projective measurement of the system, there is a parameter regime where the total system energy decreases. It is the regime where it is most profitable to run the setup. However, whether or not this work can be stored or harnessed to run a third task lies outside the scope of our methods. To approach this question a microscopic implementation of the measurement apparatus would be necessary in contrast to the bare effective description of the projective measurement applied here. A possible and experimentally realistic way would be to describe the measurement process by an adjacent quantum-point contact [6466] or an autonomous feedback setup as investigated in [8, 57, 67].

Acknowledgments

Financial support by the DFG (SFB 910, GRK 1558, SCHA 1646/3-1, BR 1928/9-1), the WE-Heraeus foundation (WEH 640) and the Natural Science Foundation of China (under Grant No.:U1530401) is gratefully acknowledged. We thank Philipp Strasberg and Javier Cerrillo for constructive discussions.

: Appendix. Heat entering the reservoirs

In figure A1, we depict the amounts of heat transported to the single reservoirs . Overall, the respective heat amounts differ strongly from the total heat Q depicted in figure 3(c). In the region, where the transported amount of heat is overall negative , we find that the heat amounts into the reservoirs Qα is rather small or even negative. If the overall amount of heat is positive , then only either reservoir experiences a strong increase of heat. For , the heat in the right reservoir strongly increases and for , the heat in the left reservoir strongly increases.

Figure A1. Refer to the following caption and surrounding text.

Figure A1. (a) Amount of heat entering the left reservoir within one period. (b) Amount of heat entering the right reservoir within one feedback period. The parameters are as in figure 3.

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