Equilibrium free energy measurement of a confined electron driven out of equilibrium

The Jarzynski equality relates nonequilibrium thermodynamics and equilibrium states. We realize a coupled quantum dot-electron reservoir system in which the time resolved observation of the tunneling dynamics is used to explicitly measure the exerted work and dissipated heat per single charge. We determine accurate values of the equilibrium free energy change over a large range of final state energies by driving the system far from equilibrium.

Figure 1

The device used in the experiments described in the text together with an energy diagram showing a single tunneling event. (a) Scanning electron micrograph of the device formed on a GaAs/AlGaAs heterostructure with a 2DEG 90 nm below the surface (dark gray area). Ti/Au gates (bright fingers) are patterned with electron-beam lithography to capacitively deplete the 2DEG. The QD is formed in the place indicated by the blue circle and is tunnel coupled to a reservoir, as shown by the red arrow, while the other barrier is fully closed. An arbitrary waveform generator is connected to the plunger gate to drive the QD. (b) An energy diagram indicates a tunneling process between the QD and the lead reservoir at energy $E$ measured from the Fermi energy $\mu$.

Figure 2

Time trace of the charge detector signal showing single electron tunneling events as well as the extracted tunnel frequency and occupation probability. (a) A typical time trace of the charge detector current $\left({\mathrm{I}}_{C}D\right)$. The current oscillates around 4 nA, and the difference in current between the two states is $\approx 100$ pA. (b) Tunnel rates (filled blue for tunneling in and open blue dots for tunneling out) and occupation probabilities (green squares, filled for occupied and empty for nonoccupied) extracted from CD time traces of 60 s. The abscissa is the negative plunger gate voltage measured from the Fermi energy and scaled by the temperature, $T=223\mu \mathrm{V}$ in units of $V_{PG}$. The error bars indicate statistical errors assuming Gaussian fluctuations. The solid blue lines are weighted least-mean-square fits to Eq. (2), from which ${\mathrm{\Gamma }}_{i}n=41.8\pm 1.6$ Hz and ${\mathrm{\Gamma }}_{o}ut=21.2\pm 0.6$ Hz and the temperature are determined. The dashed green lines are fits to Eq. (4). Inset: The ratio of the two tunnel rates (black circles) is plotted as a function of energy from $\mu$, together with the exponential behavior as expected from the detailed balance condition (red line).

Figure 3

Statistics of the heat dissipated in the reservoir and the work done on the quantum dot. (a) Examples of drive realizations in the upward (blue) and downward (green) direction, plotted as solid lines. The applied drive signals are shown as dashed lines. The driving is done by applying a voltage $V_{PG}\propto -E$ to the plunger gate through an arbitrary waveform generator. From the position of the tunneling events (gray dashed lines), the heat and work of each realization are extracted. Bottom: The distribution for $\mathrm{\Delta }Q$ (left) and $\mathrm{\Delta }W$ (right) measured from approximately $20000$ realizations. Blue circles originate from upward drive and green squares from downward drive. The solid lines are rate equation calculations without free fitting parameters.

Figure 4

The free energy of the QD during the drive, obtained with nonequilibrium measurement, measured with respect to the Fermi energy. Experimental results are plotted in green for the downward and in blue for the upward drive direction. Different markers denote independent experiments with different drive parameters. Each experiment consists of approximately 20 000 repetitions. The theory curve, Eq. (8), is plotted as a solid red line, for $d=2$. Dashed gray lines indicate the expected linear increase at low $E_1$ as well as the saturation value for $E_1>E_F$. Inset: One full period $-V_{PG}\left(t\right)\propto E$ is plotted in blue. A waiting time of 0.5 s ensures thermal equilibration of the initial state. Green dashed lines indicate the values of $E_1$ for which we extract $\mathrm{\Delta }F$ from the experiment.