Nonequilibrium Fluctuation Relations in a Quantum Coherent Conductor

We experimentally demonstrate the validity of nonequilibrium fluctuation relations by using a quantum coherent conductor. In equilibrium the fluctuation-dissipation relation leads to the correlation between current and current noise at the conductor, namely, the Johnson-Nyquist relation. When the conductor is voltage biased so that the nonlinear regime is entered, the fluctuation theorem has predicted similar nonequilibrium fluctuation relations, which hold true even when the Onsager-Casmir relations are broken in magnetic fields. Our experiments qualitatively validate the predictions as the first evidence of this theorem in the nonequilibrium quantum regime.

Figure 1

(a) AFM image of the AB ring with the dc and noise measurement setup in the dilution refrigerator whose base temperature is 45 mK. The in-plane gates defined by the oxide lines are grounded in this experiment. (b) Conductance of the AB ring as a function of $V_g$ and $B$. (c) $G_1$ (left axis) and $S_0$ (right axis) as a function of $V_g$ at $B=0\mathrm{T}$. (d) $S_0$ is plotted as a function of $G_1$. The solid line indicates the JN relation of $S_0=4k_{B}T{G}_1$ with $T=125\mathrm{mK}$. (e) $G_2$ (left axis) and $S_1$ (right axis) as a function of $V_g$. (f) $S_1$ is plotted as a function of $G_2$. The solid line is the result of the fitting ($S_1=3.64\times 4k_{B}T{G}_2$ with $T=125\mathrm{mK}$).

Figure 2

(a),(b) (Lower panels) Two examples of $I\mathrm{\text{−}}V$ characteristics obtained at different $V_g$’s, where $\frac{1}{2}{G}_2{V}^2$ components deduced from the numerical fitting are superposed as dashed curves. (Upper panels) The corresponding $S$, where the results of the polynomial fitting and $S_{1}V$ components are superposed as solid and dashed curves, respectively. The nonequilibrium noises are smaller than the equilibrium ones at certain bias voltages indicated by the arrows. The signs of $G_2$ and $S_1$ are correlated [negative in (a) and positive in (b)].

Figure 3

(a) $G_1$ (left axis) and $S_0$ (right axis) as a function of $B$ at $V_g=0.02\mathrm{V}$. (b) $S_0$ as a function of $G_1$. The solid line $S_0=4k_{B}T{G}_1$ is the JN relation at $T=125\mathrm{mK}$. (c) Symmetric components in the second-order coefficients $G_2^S$ (left axis) and $S_1^S$ (right axis) as a function of $B$. (d) $S_1^S$ as a function of $G_2^S$. The solid line is the result of the fitting corresponding to $S_1^S=2.69\times 4k_{B}T{G}_2^S$ for $T=125\mathrm{mK}$. (e) Antisymmetric components in the second-order coefficients $G_2^A$ (left axis) and $S_1^A$ (right axis) as a function of $B$. (f) $S_1^A$ as a function of $G_2^A$. The solid line is the result of the fitting corresponding to $S_1^A=2.18\times 4k_{B}T{G}_2^A$ for $T=125\mathrm{mK}$.

Figure 4

(a) Values of $S_0/G_1$ obtained at 125, 300, and 450 mK. The solid line shows the JN relation. (b) Values of $S_1^S/G_2^S$ at 125, 300, and 450 mK. The solid line is the theoretical expectation ($S_1^S/G_2^S=2k_{B}T$) [15, 16, 17, 18, 19]. The dashed line is the result of the linear fitting that gives $S_1^S/G_2^S=4.6\times 2k_{B}T$. (c) Values of $S_1^A/G_2^A$ at 125, 300, and 450 mK. The solid line is the theoretical expectation ($S_1^A/G_2^A=6k_{B}T$) [16, 19].