^{1,a)}, C. Rössler

^{1}, M. Beck

^{2}, M. Marthaler

^{3}, D. S. Golubev

^{4}, Y. Utsumi

^{5}, T. Ihn

^{1}and K. Ensslin

^{1}

### Abstract

Using time-resolved charge detection in a double quantum dot, we present an experimental test of the fluctuation theorem. The fluctuation theorem, a result from nonequilibrium statistical mechanics, quantifies the ratio of occurrence of fluctuations that drive a small system against the direction favored by the second law of thermodynamics. Here, these fluctuations take the form of single electrons flowing against the source–drain bias voltage across the double quantum dot. Our results, covering configurations close to as well as far from equilibrium, agree with the theoretical predictions, when the finite bandwidth of the charge detection is taken into account. In further measurements, we study a fluctuation relation that is a generalization of the Johnson–Nyquist formula and relates the second-order conductance to the voltage dependence of the noise. Current and noise can be determined with the time-resolved charge detection method. Our measurements confirm the fluctuation relation in the nonlinear transport regime of the double quantum dot.

The authors thank M. Büttiker, K. Kobayashi, A. Yacoby, C. Flindt, and G. Schön for discussions. Sample growth and processing was mainly carried out at FIRST laboratory, ETH Zurich. Financial support from the Swiss National Science Foundation (Schweizerischer Nationalfonds) is gratefully acknowledged.

I. INTRODUCTION

II. EXPERIMENTAL SETUP

III. TEMPERATURE DEPENDENCE

IV. FINITE-BANDWIDTH CORRECTION

V. HIGH-BIAS REGIME

VI. FLUCTUATION RELATIONS

VII. CONCLUSION

### Key Topics

- Electric measurements
- 11.0
- Entropy
- 9.0
- Quantum dots
- 9.0
- Quantum fluctuations
- 8.0
- Thermal noise
- 5.0

## Figures

(a) Configuration described by the FT. The essential part is a dissipative system (here, an electrical resistor) which is brought out of equilibrium by an external force (here a voltage) performing work. The dissipated heat flows into a thermal bath. (b) Atomic-force micrograph of the sample. Electrons can travel between the source and drain via the two quantum dots marked by disks. The conductance of the quantum point contact serves to read out the charge state of the quantum dots. Fig. 1(b) reprinted with permission from Küng *et al.*, Phys. Rev. X **2**, 011001 (2012). Copyright (2012) by the American Physical Society.

(a) Configuration described by the FT. The essential part is a dissipative system (here, an electrical resistor) which is brought out of equilibrium by an external force (here a voltage) performing work. The dissipated heat flows into a thermal bath. (b) Atomic-force micrograph of the sample. Electrons can travel between the source and drain via the two quantum dots marked by disks. The conductance of the quantum point contact serves to read out the charge state of the quantum dots. Fig. 1(b) reprinted with permission from Küng *et al.*, Phys. Rev. X **2**, 011001 (2012). Copyright (2012) by the American Physical Society.

(a) time trace recorded at a positive DQD source–drain voltage. The three discrete levels are assigned to the DQD charge states L, R, and 0. (b) Diagram of the DQD states and transitions. To count the number *n* of electrons that pass the center DQD barrier, we count the number of transitions minus . (c) Histogram of the electron number *n* obtained from the analysis of 3000 time segments of length . The histogram was measured at finite DQD source–drain voltage and has a nonzero mean value which converts into a nonzero DQD current . The variance of the histogram, , determines the zero-frequency current noise spectral density of the DQD. Fig. 2(a) reprinted with permission from Küng *et al.*, Phys. Rev. X **2**, 011001 (2012). Copyright (2012) by the American Physical Society.

(a) time trace recorded at a positive DQD source–drain voltage. The three discrete levels are assigned to the DQD charge states L, R, and 0. (b) Diagram of the DQD states and transitions. To count the number *n* of electrons that pass the center DQD barrier, we count the number of transitions minus . (c) Histogram of the electron number *n* obtained from the analysis of 3000 time segments of length . The histogram was measured at finite DQD source–drain voltage and has a nonzero mean value which converts into a nonzero DQD current . The variance of the histogram, , determines the zero-frequency current noise spectral density of the DQD. Fig. 2(a) reprinted with permission from Küng *et al.*, Phys. Rev. X **2**, 011001 (2012). Copyright (2012) by the American Physical Society.

(a)–(c) Comparison of experimental data with theory for three different bath temperatures. The data points correspond to the left-hand side of Eq. (2) and describe the probability ratio of forward ( , entropy-producing) and backward ( , entropy-consuming) processes for a given *n*. The solid lines mark the expected exponential behavior for the two source–drain voltages (dark blue) and (red). If the finite bandwidth of the detector is taken into account ^{ 30,35 } (dashed lines), experiment and theory agree within the statistical uncertainty of the data (error bars: estimated standard deviation). Reprinted with permission from Küng *et al.*, Phys. Rev. X **2**, 011001 (2012). Copyright (2012) by the American Physical Society.

(a)–(c) Comparison of experimental data with theory for three different bath temperatures. The data points correspond to the left-hand side of Eq. (2) and describe the probability ratio of forward ( , entropy-producing) and backward ( , entropy-consuming) processes for a given *n*. The solid lines mark the expected exponential behavior for the two source–drain voltages (dark blue) and (red). If the finite bandwidth of the detector is taken into account ^{ 30,35 } (dashed lines), experiment and theory agree within the statistical uncertainty of the data (error bars: estimated standard deviation). Reprinted with permission from Küng *et al.*, Phys. Rev. X **2**, 011001 (2012). Copyright (2012) by the American Physical Society.

The red data points show the -depencence of the left-hand side of Eq. (4) , the ratio of entropy-consuming vs. entropy-producing cycles, with error bars indicating its estimated standard deviation. The blue circles show the right-hand side, the average of the Boltzmann factor among the entropy-consuming cycles. The FT (4) is satisfied if the finite detector bandwidth is taken into account in the form of a correction factor to the exponent in Eq. (4) (crosses). The uncertainty in the finite-bandwidth correction is comparable with the error on for all bias voltages. Reprinted with permission from Küng *et al.*, Phys. Rev. X **2**, 011001 (2012). Copyright (2012) by the American Physical Society.

The red data points show the -depencence of the left-hand side of Eq. (4) , the ratio of entropy-consuming vs. entropy-producing cycles, with error bars indicating its estimated standard deviation. The blue circles show the right-hand side, the average of the Boltzmann factor among the entropy-consuming cycles. The FT (4) is satisfied if the finite detector bandwidth is taken into account in the form of a correction factor to the exponent in Eq. (4) (crosses). The uncertainty in the finite-bandwidth correction is comparable with the error on for all bias voltages. Reprinted with permission from Küng *et al.*, Phys. Rev. X **2**, 011001 (2012). Copyright (2012) by the American Physical Society.

Measurement of fluctuation relation between DQD current and noise. The upper plot shows the bandwidth-corrected DQD current ( , 260 time bins per point) along with a fit to a second-order polynomial in . Using the linear and quadratic coefficients of the fit, the fluctuation relations (6) and (7) are used to calculate the expected (linear) dependence of on . This is plotted as a black line along with the data in the lower plot.

Measurement of fluctuation relation between DQD current and noise. The upper plot shows the bandwidth-corrected DQD current ( , 260 time bins per point) along with a fit to a second-order polynomial in . Using the linear and quadratic coefficients of the fit, the fluctuation relations (6) and (7) are used to calculate the expected (linear) dependence of on . This is plotted as a black line along with the data in the lower plot.

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