Nature Physics  Letter
Distribution of entropy production in a singleelectron box
 J. V. Koski^{1}^{, }
 T. Sagawa^{2}^{, }
 OP. Saira^{1, 3}^{, }
 Y. Yoon^{1}^{, }
 A. Kutvonen^{4}^{, }
 P. Solinas^{1, 4}^{, }
 M. Möttönen^{1, 5}^{, }
 T. AlaNissila^{4, 6}^{, }
 J. P. Pekola^{1}^{, }
 Journal name:
 Nature Physics
 Volume:
 9,
 Pages:
 644–648
 Year published:
 DOI:
 doi:10.1038/nphys2711
 Received
 Accepted
 Published online
Recently, the fundamental laws of thermodynamics have been reconsidered for small systems. The discovery of the fluctuation relations^{1, 2, 3, 4, 5} has spurred theoretical^{6, 7, 8, 9, 10, 11, 12, 13} and experimental^{14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25} studies. The concept of entropy production has been extended to the microscopic level by considering stochastic trajectories of a system coupled to a heat bath. However, this has not been studied experimentally if there are multiple thermal baths present. Here, we measure, with high precision, the distributions of microscopic entropy production in a singleelectron box consisting of two islands with a tunnel junction. The islands are coupled to separate heat baths at different temperatures, maintaining a steady thermal nonequilibrium. We demonstrate that stochastic entropy production^{8, 10, 11, 12, 17, 20, 25, 26} from trajectories of electronic transitions is related to thermodynamic entropy production from dissipated heat in the respective thermal baths. We verify experimentally that the fluctuation relations for both definitions are satisfied. Our results reveal the subtlety of irreversible entropy production in nonequilibrium.
Subject terms:
At a glance
Figures
Main
The second law of thermodynamics states that, on average, total entropy production is either zero or positive, the latter of which is a hallmark of irreversible processes. However, the socalled integral fluctuation theorem (IFT) reveals that negative entropy production is possible for an individual stochastic trajectory of the system. In systems that are not coupled to a thermal bath^{17, 20}, the stochastic entropy production and work seem to have no obvious connection to their thermodynamic counterparts. The situation is further complicated by the fact that stochastic trajectories actually depend on the scale of observation. If one only accesses mesoscopic degrees of freedom, one observes coarsegrained trajectories of mesoscopic states. The corresponding entropy production then differs from that without coarsegraining. Recent experiments^{25} with a driven colloidal particle in a bath of fluid have shown that coarsegraining of the slow background degrees of freedom may actually lead to a modification of the fluctuation relations. An important question that remains is what happens to a small system coupled to more than just one heat bath when the temperatures of the baths are not equal. Such a system goes naturally to a nonequilibrium state, but the dissipated heat of the baths can still be physically defined. To this end, we consider here the case of a twolevel system coupled to two such thermal baths. We measure the distributions of entropy production at different levels of description, and clarify the connection between the stochastic entropy production and that of the heat baths.
A singleelectron box (SEB) at low temperatures is an excellent test bench for thermodynamics in small systems^{24, 27, 28}. The SEB employed here is shown in Fig. 1a. The electrons in the normalmetal copper island (N) can tunnel to the superconducting Al island (S) through the aluminium oxide insulator (I). We denote by n the integer net number of electrons tunnelled from S to N relative to charge neutrality. As we can monitor the charge state n with a nearby singleelectron transistor (SET) shown in Fig. 1a, we take our classical system degree of freedom to be n.
The device in Fig. 1a can be represented with a classical electric circuit, in which the energy stored in the capacitors and the voltage sources is given by^{27, 29, 30}
where E_{C} is the characteristic charging energy, C_{g} is the gate capacitance, n_{g} = C_{g}V _{g}/e is the gate charge in units of the elementary charge e, and V _{g} is the gate voltage that drives the system externally. Equation (1) gives the internal energy of the system. In an instantaneous singleelectron tunnelling event from n = k to n = k+1, the drive parameters stay constant and hence the work done to the system vanishes. Thus the first law of thermodynamics states that the generated heat is given by
It has been recently demonstrated that when the SEB is in thermal equilibrium and the transition rates obey the detailed balance condition, the Jarzynski Equality e^{−βW} = e^{−βΔF}, relating the work done in the system W at inverse temperature β = 1/k_{B} T to its free energy change ΔF, can be verified both theoretically^{27, 28, 30} and experimentally^{24} to a high degree of accuracy. In the present work, however, the two environments consisting of the excitations in the normal metal and the superconductor are at different temperatures T_{N} = 1/(k_{B}β_{N}) and T_{S} = 1/(k_{B}β_{S}), respectively, and hence the Jarzynski Equality cannot be applied. Nevertheless, we expect that our system obeys the IFT^{8},
for the total entropy production Δs_{tot} = Δs+Δs_{m} given in terms of the increase of the system entropy Δs = ln{P[n(t_{f})]/P[n(0)]} and the medium entropy production Δs_{m}. Here, P[n(t)] is the directly measurable probability of the system to be in state n at time instant t given the initial condition and the drive n_{g}(t).
As each heat bath is described by a thermal distribution, the total thermodynamic entropy production in the medium is given by
where Q_{N} and Q_{S} are the heat dissipated along the trajectory in the normal metal and in the superconductor, respectively. We can measure the total dissipated heat Q = Q_{N}+Q_{S} directly by monitoring n(t) with the SET and using equation (2). The only essential assumption here is that the tunnelling is elastic because the parameters of the Hamiltonian equation (1) can be measured independently. Heat is dissipated on N and S whenever n changes. For instance, in a transition n:kk+1 an electron tunnels from S to N. As illustrated in Fig. 2a, −Q_{S} equals the energy carried by the electron, whereas Q_{N} = −Q_{S}+Q equals the sum of the electron energy and the energy it gains according to equation (2). We can further obtain the conditional probability of Q_{S} on Q by some additional assumptions (for technical details, see Supplementary Information) and hence the probability distribution of Δs_{m}^{th}.
We can take a different approach by eliminating the dependence on Q_{S} in entropy production by averaging. Consider a single transition n:kk±1. For this, we define a coarsegrained entropy production in the medium Δs_{m}^{cc} as
where ⋯_{Q} denotes an average over the heat dissipated in Q_{S} for a fixed Q. It is shown in the Supplementary Information that this definition of entropy coincides with stochastic entropy production in the medium^{8, 13}, which is defined as
where the system is taken to make transitions at time instants t_{j} from the state n_{−} to the state n_{+}, and Γ_{n−→n+}(t_{j}) and Γ_{n+→n−}(t_{j}) are the corresponding forward and backward transition rates, and Q_{j} is the total dissipated heat at that time instant. This derivation provides a physical interpretation for Δs_{m}^{st}, and by further noting that equation (4) implies Δs_{m}^{th}_{Q}≥Δs_{m}^{cc}(Q), we may conclude that
Note that, by introducing transition rates, we have implicitly assumed that the system is Markovian, a fact that can be experimentally verified in our setup.
By extracting the entropy production from each detector trace, we can experimentally obtain the probability distributions and of Δs_{tot}^{th} = Δs+Δs_{m}^{th} and Δs_{tot}^{st} = Δs+Δs_{m}^{st}, respectively, and hence access the IFT of equation (3), which should be satisfied by all the distributions. Here, P_{→} is the distribution for a forward driving protocol n_{g,→}(t) and corresponds to the backward protocol . In addition, we expect our system to satisfy the socalled detailed fluctuation relations^{5, 13} (DFR)
In our experiments, we drive the system with the gate charge n_{g}(t) = n_{0}−Acos(πft), where n_{0}≈A≈0.5. Figure 1b shows the applied drive and an example trace of the detector current. Clearly, two discrete current levels corresponding to the charge states n = 0 and n = 1 are observable. Owing to the low bath temperatures, 130–160 mK, the relatively high charging energy E_{C}≈162 μ eV = 1.88 K×k_{B}, and low driving frequencies f≤120 Hz, the system essentially always finds the minimumenergy state at the extrema of the drive. Thus we partition the continuous measurement into legs of forward and backward protocols, for which the charge state and gate charge change from 0 to 1 and 1 to 0, respectively. Conversion of the current trace from such a leg using threshold detection yields a realization for a system trajectory n(t), an ensemble of which is used to obtain the desired distributions. Moreover, the system entropy change Δs in equation (3) vanishes, and we thus need only to obtain Δs_{m} to assess the fluctuation relations.
Figure 2c shows the estimated Δs_{m}^{th} conditional probability distribution and Δs_{m}^{st} as functions of the drive n_{g}. To obtain Δs_{m}^{st}, the tunnelling rates Γ_{i→j}(n_{g}), shown in Fig. 2b, are measured and fitted by a standard sequential tunnelling model, see Supplementary Information for details. For n_{g}≈0.4–0.6, the tunnelling probability is primarily determined by the thermal excitations of the overheated superconductor and not by n_{g}. This leads to a nearly vanishing Δs_{m}^{st}, whereas the average of Δs_{m}^{th} remains positive as a sign of heat flow from hot S to cold N.
Table 1 presents a collection of measurement parameters as well as exponential averages for entropy production. Figures 3 and 4 show the experimentally obtained distributions for work and entropy production together with respective theoretical predictions for various T_{N} and f. As expected, the work distributions in Fig. 3a do not satisfy Jarzynski Equality. For comparison, the case T_{S} = T_{N}, where Jarzynski Equality is valid, is shown by the dashed lines. The difference between T_{N} and T_{S} decreases with increasing T_{N}, and hence the difference between data and the dashed lines decreases as well. Conversely, all the entropy distributions in Figs 3b,c and 4, obtained from the same trajectories as the work distributions, satisfy the IFT within the experimental error. As it is relatively common that the transition 01 or 10 occurs when n_{g}≈0.4–0.6, one observes peaks in the Δs_{tot}^{st} distributions (Figs 3b and 4a,b) in the vicinity of zero, whereas the Δs_{tot}^{th} distributions (Figs 3c and 4c) exhibit a tail for positive Δs_{tot}^{th}, as indicated by Fig. 2c.
Figure 5a,b show the DFR for Δs_{tot}^{st} and Δs_{tot}^{th}, respectively. The Δs_{tot}^{st} distributions for forward and backward protocls, shown in Fig. 4b, are overlapping, apart from the positions of the peaks near vanishing Δs_{tot}^{st}. The offset of the peaks is explained by different superconductor temperatures (Table 1) for different tunnelling directions, leading to Γ_{0→1}(n_{g} = 0.5)<Γ_{1→0}(n_{g} = 0.5). For the n:01 event, this corresponds to negative entropy production, whereas positive production is observed for n:10. Different temperatures for different tunnelling directions can be justified by the difference in the observed tunnelling rates in Fig. 2b. The SET current is higher for n = 1 than for n = 0 (Fig. 1c), inducing a higher excess heating power for the superconductor at n = 1. However, even with these offsets in the Δs_{tot}^{st} distributions, they do obey the DFR.
Our measured distributions satisfy the IFT and DFR, verifying the fluctuation relations in thermal nonequilibrium. The fluctuation relations can be used to determine thermodynamic quantities such as free energy. Moreover, these relations apply even beyond the linear response regime, whereas the conventional fluctuationdissipation theorem, widely used for instance in condensed matter physics, is a result of linear response theory with limited applicability to nonequilibrium processes.
Methods
The sample fabrication methods (see Supplementary Information for details) are similar to those in ref. 24, but the design is different such that the S side of the junction does not overlap with the normal conductor to intentionally weaken the relaxation of energy in S (ref. 31). Moreover, the main results in ref. 24 were extracted from measurements at a temperature of 220 mK, whereas the present measurements are conducted at 140 mK. Lower temperature further weakens the relaxation significantly^{31}, leading to a steady elevated temperature in S.
The tunnelling rates are solved by comparing the measured data to the outcome from the master equation, see Supplementary Information for details. The rates from the standard sequential tunnelling model are in agreement with the experimentally obtained data. Utilizing the model as a fit yields the charging energy, the tunnelling resistance of the junction R_{T}≈1.7 MΩ, and the excitation gap of the superconductor Δ≈224 μ eV. T_{N} is assumed to be the temperature of the cryostat, whereas T_{S} is obtained for each measurement from the fit as listed in Table 1.
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Acknowledgements
This work has been supported in part by the Academy of Finland though its LTQ (project no. 250280) and COMP (project no. 251748) CoE grants, the European Union Seventh Framework Programme INFERNOS (FP7/2007–2013) under grant agreement no. 308850, the Research Foundation of Helsinki University of Technology, and the Väisälä Foundation. We acknowledge Micronova Nanofabrication Centre of Aalto University for providing the processing facilities and technical support. We thank D. Averin, S. Gasparinetti, F. Hekking, K. Likharev, V. Maisi and M. Meschke for useful discussions.
Author information
Affiliations

Low Temperature Laboratory (OVLL), Aalto University, POB 13500, FI00076 AALTO, Finland
 J. V. Koski,
 OP. Saira,
 Y. Yoon,
 P. Solinas,
 M. Möttönen &
 J. P. Pekola

Department of Basic Science, The University of Tokyo, Komaba 381, Meguroku, Tokyo 1538902, Japan
 T. Sagawa

Kavli Institute of Nanoscience, Delft University of Technology, PO Box 5046, 2600 GA Delft, The Netherlands
 OP. Saira

COMP Centre of Excellence, Department of Applied Physics, Aalto University School of Science, PO Box 11000, FI00076 Aalto, Espoo, Finland
 A. Kutvonen,
 P. Solinas &
 T. AlaNissila

QCD Labs, COMP Centre of Excellence, Department of Applied Physics, Aalto University School of Science, PO Box 13500, FI00076 Aalto, Espoo, Finland
 M. Möttönen

Department of Physics, Brown University, Providence, Rhode Island 029121843, USA
 T. AlaNissila
Contributions
J.V.K., OP.S., Y.Y. and J.P.P. conceived and designed the experiments; J.V.K., OP.S. and Y.Y. performed the experiments; J.V.K. and M.M. analysed the data. All authors contributed with materials/analysis tools; J.V.K., T.S., M.M., T.AN. and J.P.P. wrote the paper.
Competing financial interests
The authors declare no competing financial interests.
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