Extreme reductions of entropy in an electronic double dot

We experimentally study negative fluctuations of stochastic entropy production in an electronic double dot operating in nonequilibrium steady-state conditions. We record millions of random electron tunneling events at different bias points, thus collecting extensive statistics. We show that for all bias voltages, the experimental average values of the minima of stochastic entropy production lie above $-k_{\mathrm{B}}$, where $k_{\mathrm{B}}$ is the Boltzmann constant, in agreement with recent theoretical predictions for nonequilibrium steady states. Furthermore, we also demonstrate that the experimental cumulative distribution of the entropy production minima is bounded, at all times and for all bias voltages, by a universal expression predicted by the theory. We also extend our theory by deriving a general bound for the average value of the maximum heat absorbed by a mesoscopic system from the environment and compare this result with experimental data. Finally, we show by numerical simulations that these results are not necessarily valid under nonstationary conditions.

Figure 1

Experimental realization of a double dot. (a) Scanning electron micrograph of the sample with false color identifying its different components. The device consists of two leads, left (purple) and right (turquoise), two islands, left (green) and right (orange), and two single-electron transistor (SET) detectors, left (blue) and right (red). (b) Sketch of the circuit elements of the sample (N, normal metal; S, superconductor; I, insulator) with colors corresponding to the ones in (a). An external dc voltage $V_b$ controls the net current through the double dot. (c) Zoomed view of the yellow rectangular region in (a). Electrons (yellow circles) can tunnel between the leads and the islands in the directions indicated by the arrows. Details on fabrication techniques and measurement setup can be found in Appendices pp1 and pp2, respectively.

Figure 2

Experimental measurements on a double dot. (a) Top: time trace of left $(I_{\text{det},L}$, blue) and right $(I_{\text{det},R}$, red) detector currents for $V_b=90\mu \mathrm{V}$. Bottom: corresponding time trace for the charge state $\left(n_L,n_R\right)$ of the double dot, where $n_L\left(n_R\right)=0$ or 1 implies the absence or presence of one extra electron in the left (right) island, respectively (see Appendix pp4 for details). (b) Probability density of the normalized detector currents $\left(i_L,i_R\right)$ [22] obtained from a $15\text{−}\mathrm{s}$ time trace for $V_b=90\mu \mathrm{V}$, showing four densely populated charge states $\left(n_L,n_R\right)$. The black arrows represent the possible transitions between the states with the numbers indicating the number of jumps per second averaged over $7.5\mathrm{h}$.

Figure 3

Fluctuations in transition rates over time. The transition rates ${\mathrm{\Gamma }}_{(1,0),(0,1)}$ (solid lines) and ${\mathrm{\Gamma }}_{(0,1),(1,0)}$ (dotted lines) as a function of file numbers for different bias voltages [see legend in (a)]. Each file is a $15\text{−}\mathrm{s}$ time evolution of system charge state. The transition rates are evaluated using Eq. (D1) (as described in Sec. 2 and Appendix pp4) and the variables $N_{n\to n^{\prime }},P^{\mathrm{st}}\left(n\right)$ computed for the individual files with time duration $\tau =15\mathrm{s}$. The solid line in (b) is the average transition rate (shown in Table 1) used for computing the entropy production using Eqs. (1, 2, 3, 4).

Figure 4

Experimental measurement of stochastic entropy production. (a) Sample traces of stochastic entropy production as a function of time, for different values of the bias voltage (see legend). The straight red line is the average Joule heating for $90\mu \mathrm{V}$ normalized by $T_{\mathrm{eff}}$. Inset: steady-state average Joule dissipation rate $\langle I\rangle V_b$ as a function of the steady-state entropy-production rate $\langle \dot{S}\rangle$ for different bias voltages. The dashed line is a linear fit with slope $T_{\mathrm{eff}}=(1.01\pm 0.16)\mathrm{K},y$ intercept $y_0=(0.14\pm 0.40)\text{K}/s$, and $R^2=0.990$ [see Eq. (6)]. (b) Zoomed view of the shaded region in (a) with finite-time minima $S_{\min }\left(t\right)$ of each trace represented with symbols.

Figure 5

Experimental cumulative distribution of the finite-time entropy production minima in the double dot with bias voltage $V_b=-50\mu \mathrm{V}$ (a), $-25\mu \mathrm{V}$ (b), $25\mu \mathrm{V}$ (c), $50\mu \mathrm{V}$ (d), and $90\mu \mathrm{V}$ (e) for different values of the observation time (see legend). A horizontal line is set to one to guide the eye and corresponds to the cumulative distribution at $t=0$. The black curve is the theoretical bound given by the right-hand side in Eq. (7).

Figure 6

Finite-time average minimum of stochastic entropy production $\langle S_{\min }\left(t\right)\rangle$ (a) and entropy outflow $\langle S_{\min }^{\mathrm{e}}\left(t\right)\rangle$ (b) in the double dot as a function of time, for different values of the bias voltage (shown in different colors). The gray area indicates the events where entropy extrema are below $-1$ and the dashed lines in (b) are given by the right-hand side of Eq. (9). The inset in (a) shows the value of $\langle S_{\min }\left(\tau \right)\rangle$ as a function of $\tau =t\langle \dot{S}\rangle$, where the black curve is the average entropy-production minimum for a driven colloidal particle in a ring with equal drift and diffusion coefficients $v=D=1$, given by $-\text{erf}(\sqrt{\tau }/2)+(\tau /2)\text{erfc}(\sqrt{\tau }/2)-\sqrt{\tau /\pi }exp(-\tau /4)$.

Figure 7

Finite-time average minimum of stochastic entropy production $\langle S_{\min }\left(t\right)\rangle$ as a function of time (symbols) obtained from numerical simulations of a double dot driven out of equilibrium and with time-dependent rate ${\mathrm{\Gamma }}_{(0,1),(1,0)}\left(t\right)={\mathrm{\Gamma }}_0[1+\gamma \theta (N\left(t\right))]$. Here, ${\mathrm{\Gamma }}_0=27.1\mathrm{Hz},N\left(t\right)$ is the total number of jumps between any two states up to time $t$ and $\theta \left(N\right)$ denotes the parity of $N$, i.e., $\theta \left(N\right)=1\left[\theta \right(N)=-1]$ for $N$ even (odd). The parameter $\gamma >0$ is fixed in the simulations and controls the amplitude of the variation of the time-dependent rate (see legend). The values of the time-independent rates (in Hz) are ${\mathrm{\Gamma }}_{(0,0),(0,1)}=101.8,{\mathrm{\Gamma }}_{(0,1),(0,0)}=2,{\mathrm{\Gamma }}_{(0,0),(1,1)}=1.6,{\mathrm{\Gamma }}_{(1,1),(0,0)}=74.1,{\mathrm{\Gamma }}_{(0,0),(1,0)}=29.3,{\mathrm{\Gamma }}_{(1,0),(0,0)}=31,{\mathrm{\Gamma }}_{(0,1),(1,1)}=22.5,{\mathrm{\Gamma }}_{(1,1),(0,1)}=44,{\mathrm{\Gamma }}_{(1,0),(1,1)}=73.9,{\mathrm{\Gamma }}_{(1,1),(1,0)}=101.4,{\mathrm{\Gamma }}_{(0,1),(1,0)}=28$ and the total number of simulations is $5\times {10}^4$ for all cases.

Figure 8

Sample and measurement scheme. Left: sample micrograph with false color identifying the different stages of fabrication process. Right: schematic of the electronic circuit corresponding to the sample together with the measurement setup. The orange colored structures are ground plane, mainly $30\mathrm{nm}$ of Au, evaporated before the $50\mathrm{nm}$ of ${\mathrm{Al}}_2{\mathrm{O}}_3$ dielectric layer. The orange rectangles are capacitive couplers between the double-dot islands and the detectors' islands, with capacitances $C_{c,L}$ and $C_{c,R}$. The remaining orange structures are gates to the double-dot and detectors' islands, each connected to a dc voltage source $V_{g,L},V_{g,R},V_{g,\text{det},L}$, and $V_{g,\text{det},R}$ via the capacitances $C_{g,L},C_{g,R},C_{g,\text{det},L}$, and $C_{g,\text{det},R}$, respectively. The blue colored structures are Al evaporated as a first layer on the top of ${\mathrm{Al}}_2{\mathrm{O}}_3$. This Al is oxidized and covered by Cu (red) to make detectors' junctions, with capacitances $C_{\text{det},L,1},C_{\text{det},L,2},C_{\text{det},R,1}$, and $C_{\text{det},R,2}$. The Cu (green) is evaporated last to make high-resistance double-dot junctions, with capacitances $C_L,C_m,C_R$ and resistances $R_L,R_m,R_R$, corresponding to left, middle, and right junctions, respectively. During the measurement, dc voltages $V_b,V_{b,\text{det},L}$, and $V_{b,\text{det},R}$ are applied across the double dot and detectors, through their left lead. The currents $I_{\text{det},L}^{\mathrm{meas}}$ and $I_{\text{det},R}^{\mathrm{meas}}$ through detectors are recorded with sampling frequency $f_{\mathrm{s}}=25\mathrm{kHz}$, by connecting an amplifier (triangles) and analog to digital converter (ADC) to the right lead of each of the detectors. The circles represent the charge states of the dots $(n_L$ and $n_R$, yellow), the gates $(n_{g,L}$ and $n_{g,R}$, orange), the detector dots $(n_{\text{det},L}$ and $n_{\text{det},R}$, red), and the gates to the detectors $(n_{g,\text{det},L}$ and $n_{g,\text{det},R}$, brown).

Figure 9

Stability diagram of the double dot. Electric current across the double-dot structure as a function of the two gate voltages $V_{g,L}$ and $V_{g,R}$ measured at high magnetic field, $H=0.144\mathrm{T}$, in which superconductivity of aluminum is suppressed. (a) Bias voltage $V_b$ is close to zero. A honeycomb pattern typical for double-dot devices [56] is visible. The high current spots correspond to the triple points at which the energies of three charge states are degenerate and the electric current can flow through the device. (b) $V_b=120\mu \mathrm{V}$. The triple points grow into triangles because finite bias allows electrons to pass through the double dot away from degeneracy. The shape of the honeycomb structure and of the triangles allows us to determine the charging energies of the islands [56].

Figure 10

Filtering effect on current and use of dwell time as a corrective measure. (a) Smearing of detector output currents introduced by filtering high-frequency noise. Top: measured detector currents $I^{\mathrm{meas}}$ for left detector in blue and right detector in red. Middle: the corresponding filtered current. Bottom: the corresponding charge state of the system island $n$. Close to 2 ms, both the detector currents jump instantaneously from higher state to the lower state in the measured current $I^{\mathrm{meas}}$ but, after filtering, there is a delay introduced in the left detector current (blue curve) because of smearing of the transition point. (b) Jump counts for each transition (see legend) as a function of the dwell-time threshold used to correct for the stochastic jitter in the traces $\left\{\widehat{n}\left(t\right)\right\}$ of the charge state of the double dot. The dwell-time threshold ${\tau }_{\mathrm{th}}$ is determined empirically as the minimal time threshold beyond which the jump counts for all transitions are barely affected upon small increments of the threshold. For bias voltage $V_b=50\mu \mathrm{V}$ shown in the figure, the dwell-time threshold is estimated to be ${\tau }_{\mathrm{th}}=0.28\mathrm{ms}$, illustrated by the black vertical dashed line. Duration of the time trace analyzed here is 15 s.

Figure 11

Experimental waiting-time distributions ${\psi }_{(n_L,n_R)}\left(t\right)$ for the different states $(n_L,n_R)$ in the double dot. Empirical waiting-time distributions (symbols) calculated from the traces $\left\{n_{\mathrm{tot}}\left(t\right)\right\}$ for $V_b=25\mu \mathrm{V}$ (top) and $V_b=90\mu \mathrm{V}$ (bottom). The solid lines are exponential fits.

Figure 12

Maximum entropy jump for different bias. The maximum entropy change in a jump, $|S_{n,n^{\prime }}{|}_{\max }={\max }_{n,n^{\prime }}\left|S_{n,n^{\prime }}\right|$, as a function of bias voltage. Table 2 lists $|S_{n,n^{\prime }}|$ for all possible jumps for different bias voltages.