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 so-called 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 bath17,20, 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 coarse-grained trajectories of mesoscopic states. The corresponding entropy production then differs from that without coarse-graining. Recent experiments25 with a driven colloidal particle in a bath of fluid have shown that coarse-graining 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 non-equilibrium state, but the dissipated heat of the baths can still be physically defined. To this end, we consider here the case of a two-level 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 single-electron box (SEB) at low temperatures is an excellent test bench for thermodynamics in small systems24,27,28. The SEB employed here is shown in Fig. 1a. The electrons in the normal-metal 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 single-electron transistor (SET) shown in Fig. 1a, we take our classical system degree of freedom to be n.

Figure 1: Measurement set-up.
figure 1

a, Sketch of the measured system together with a scanning electron micrograph of a typical sample. The colours on the micrograph indicate the correspondingly coloured circuit elements in the sketch. b, Example trace of the measured detector signal under a sinusoidal protocol for the drive V g, plotted in green. This trace covers three realizations of the forward protocol (V g from −0.1 to 1 mV), and three realizations of the backward protocol (V g from 1 to −0.1 mV). The SET current Idet, plotted in black, indicates the charge state of the box. The output of the threshold detection is shown in solid blue, with the threshold level as indicated by the dashed red line.

Full size image

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 by27,29,30

where EC is the characteristic charging energy, Cg is the gate capacitance, ng = CgV 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 single-electron 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/kB T to its free energy change ΔF, can be verified both theoretically27,28,30 and experimentally24 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 TN = 1/(kBβN) and TS = 1/(kBβS), respectively, and hence the Jarzynski Equality cannot be applied. Nevertheless, we expect that our system obeys the IFT8,

for the total entropy production Δstot = Δs+Δsm given in terms of the increase of the system entropy Δs = ln{P[n(tf)]/P[n(0)]} and the medium entropy production Δsm. 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 ng(t).

As each heat bath is described by a thermal distribution, the total thermodynamic entropy production in the medium is given by

where QN and QS are the heat dissipated along the trajectory in the normal metal and in the superconductor, respectively. We can measure the total dissipated heat Q = QN+QS 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:k→k+1 an electron tunnels from S to N. As illustrated in Fig. 2a, −QS equals the energy carried by the electron, whereas QN = −QS+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 QS on Q by some additional assumptions (for technical details, see Supplementary Information) and hence the probability distribution of Δsmth.

Figure 2: Evaluation of entropy production.
figure 2

a, Energy diagram of a transition 0→1, where an electron tunnels from S to N. The event removes an energy −QS from S with respect to its Fermi level and similarly adds QN to N. Δsmth depends on both QN and QS, whereas the total dissipated heat Q is always determined by the control parameter ng. The energy gap in the superconductor density of states is not shown for simplicity. b, The tunnelling rates Γ0→1 (triangles pointing up) and Γ1→0 (triangles pointing down) for measurements 1–3 (dark blue, light blue and green symbols) and 8–10 (dark red, light red and pink symbols) with their corresponding fits (solid lines; for parameters see Table 1). c, Entropy produced by the transition 0→1 at 153 mK as a function of ng at the time instant of the transition. Probability density of Δsmth along the y axis is shown in red, 〈Δsmth〉Q is shown in brown, the Δsmst extracted from b is shown in blue, and βNQ is shown in green.

Full size image

We can take a different approach by eliminating the dependence on QS in entropy production by averaging. Consider a single transition n:k→k±1. For this, we define a coarse-grained entropy production in the medium Δsmcc as

where 〈⋯〉Q denotes an average over the heat dissipated in QS for a fixed Q. It is shown in the Supplementary Information that this definition of entropy coincides with stochastic entropy production in the medium8,13, which is defined as

where the system is taken to make transitions at time instants tj 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 Qj is the total dissipated heat at that time instant. This derivation provides a physical interpretation for Δsmst, and by further noting that equation (4) implies 〈Δsmth〉Q≥Δsmcc(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 set-up.

By extracting the entropy production from each detector trace, we can experimentally obtain the probability distributions and of Δstotth = Δs+Δsmth and Δstotst = Δs+Δsmst, 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 ng,→(t) and corresponds to the backward protocol . In addition, we expect our system to satisfy the so-called detailed fluctuation relations5,13 (DFR)

In our experiments, we drive the system with the gate charge ng(t) = n0−Acos(πf t), where n0≈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 EC≈162 μ eV = 1.88 K×kB, and low driving frequencies f≤120 Hz, the system essentially always finds the minimum-energy 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 Δsm to assess the fluctuation relations.

Figure 2c shows the estimated Δsmth conditional probability distribution and Δsmst as functions of the drive ng. To obtain Δsmst, the tunnelling rates Γi→j(ng), shown in Fig. 2b, are measured and fitted by a standard sequential tunnelling model, see Supplementary Information for details. For ng≈0.4–0.6, the tunnelling probability is primarily determined by the thermal excitations of the overheated superconductor and not by ng. This leads to a nearly vanishing Δsmst, whereas the average of Δsmth 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 TN and f. As expected, the work distributions in Fig. 3a do not satisfy Jarzynski Equality. For comparison, the case TS = TN, where Jarzynski Equality is valid, is shown by the dashed lines. The difference between TN and TS decreases with increasing TN, 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 0→1 or 1→0 occurs when ng≈0.4–0.6, one observes peaks in the Δstotst distributions (Figs 3b and 4a,b) in the vicinity of zero, whereas the Δstotth distributions (Figs 3c and 4c) exhibit a tail for positive Δstotth, as indicated by Fig. 2c.

Table 1 Measurement parameters and obtained averages for work and entropy production.
Full size table
Figure 3: Distributions of entropy production at different temperatures.
figure 3

a, βN(W−ΔF) distributions for a 40 Hz forward protocol at different bath temperatures. The symbols show measured values (key in b applies to all panels), solid lines are numerical expectations (all panels), and dashed lines demonstrate what the distribution would be for TS = TN, such that Jarzynski Equality would be satisfied. b, Corresponding Δstotst distributions. c, Δstotth distributions for single jump trajectories.

Full size image
Figure 4: Distributions of entropy production at different frequencies.
figure 4

a, Δstotst distributions obtained at TN = 142 mK with different drive frequencies. b, Distributions of an individual measurement for forward and backward protocols. c, Corresponding Δstotth distributions for single jump trajectories.

Full size image

Figure 5a,b show the DFR for Δstotst and Δstotth, respectively. The Δstotst distributions for forward and backward protocls, shown in Fig. 4b, are overlapping, apart from the positions of the peaks near vanishing Δstotst. The offset of the peaks is explained by different superconductor temperatures (Table 1) for different tunnelling directions, leading to Γ0→1(ng = 0.5)<Γ1→0(ng = 0.5). For the n:0→1 event, this corresponds to negative entropy production, whereas positive production is observed for n:1→0. 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 Δstotst distributions, they do obey the DFR.

Figure 5: Test of the DFR.
figure 5

a, DFR for Δstotst. Despite the asymmetry of forward and backward protocols due to detector back-action, the relation is satisfied. b, DFR for Δstotth of the forward protocol. In both a and b, the expected dependence given by equation (5) is shown as a solid black line.

Full size image

Our measured distributions satisfy the IFT and DFR, verifying the fluctuation relations in thermal non-equilibrium. 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 fluctuation-dissipation theorem, widely used for instance in condensed matter physics, is a result of linear response theory with limited applicability to non-equilibrium 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 significantly31, 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 RT≈1.7 MΩ, and the excitation gap of the superconductor Δ≈224 μ eV. TN is assumed to be the temperature of the cryostat, whereas TS is obtained for each measurement from the fit as listed in Table 1.