Test of the Jarzynski and Crooks Fluctuation Relations in an Electronic System

Recent progress on micro- and nanometer-scale manipulation has opened the possibility to probe systems small enough that thermal fluctuations of energy and coordinate variables can be significant compared with their mean behavior. We present an experimental study of nonequilibrium thermodynamics in a classical two-state system, namely, a metallic single-electron box. We have measured with high statistical accuracy the distribution of dissipated energy as single electrons are transferred between the box electrodes. The obtained distributions obey Jarzynski and Crooks fluctuation relations. A comprehensive microscopic theory exists for the system, enabling the experimental distributions to be reproduced without fitting parameters.

Figure 1

(a) Scanning electron micrograph of the active area of the measured sample, which shows metallic films fabricated on an oxidized silicon wafer by $e$-beam lithography and shadow evaporation technique [18]. Two shifted copies of the original resist mask pattern lie on the surface: copper layer appears brighter compared to oxidized aluminum. Tunnel junctions are formed in the overlap regions between the two films. The single-electron box is located on the left, and the SET electrometer is at the top. The tips of two gate electrodes that are used to control the electrostatics of the box and the electrometer are visible at the left and right edges. (b) Simplified circuit diagram of the system. The galvanically isolated single-electron box is connected capacitively to its environment via $C_L$ and $C_R$ and to the electrometer as illustrated by the dashed gray line. (c) Full period of the sinusoidal drive signal (top) applied to the control gate, and one instance of electrometer response (bottom). The drive frequency is 1 Hz, and the amplitude is equal to one gate modulation period of the box. (d) Energy-level diagram of the system for the two lowest-energy charge states. Black parabolas represent the charging energy of the system in the states $n=0$ and $n=1$ as a function of the externally controlled gate charge $n_g$. The arrows indicate possible combinations of initial and final states (open and filled circles, respectively). The expression for dissipated work $W-\Delta F$ is given next to the arrow for each trajectory type, illustrating that work separates into dissipated heat $Q$ and change of internal energy.

Figure 2

(a) Measured distribution of the generated heat at drive frequencies 1 Hz (black squares), 2 Hz (red circles), and 4 Hz (blue diamonds). The solid lines are exact theoretical predictions for the independently determined sample parameter values. The dashed lines show the results of Monte Carlo simulations, where the finite bandwidth of the detector was included in the model. Inset: $P(Q)/P(-Q)$ ratio for the experimental distributions. Solid line shows the result from the Crooks fluctuation theorem. [(b) and (c)] First and second moments, respectively, of the $Q$ distribution at different drive frequencies and bath temperatures. Markers are experimental data, and solid lines are exact theoretical predictions as in part (a). For the lowest bath temperatures, the theoretical curves have been calculated using a slightly elevated electron temperature to account for nonideal thermalization as discussed in the text. The temperature used in the calculation is given in parentheses if it differs from the sample stage temperature. In panel (c), dashed lines represent the distribution width inferred from the FDT formula $⟨(Q-⟨Q⟩{)}^2⟩=2kT⟨Q⟩$ using the theoretical value of the first moment $⟨Q⟩$.