Experimental Reconstruction of Work Distribution and Study of Fluctuation Relations in a Closed Quantum System

We report the experimental reconstruction of the nonequilibrium work probability distribution in a closed quantum system, and the study of the corresponding quantum fluctuation relations. The experiment uses a liquid-state nuclear magnetic resonance platform that offers full control on the preparation and dynamics of the system. Our endeavors enable the characterization of the out-of-equilibrium dynamics of a quantum spin from a finite-time thermodynamics viewpoint.

Figure 1

Scheme for the interferometric reconstruction of ${\chi }_{\alpha }(u)$ ($\alpha =F,B$) [14, 15]. We show the conditional joint gates given in Eq. (2) and the single-spin operations needed to complete the protocol. Here, ${\rho }_S$ is a generic initial state of a driven system (the ${}^{13}\mathrm{C}$ nuclear spin in our experiment), while ${|0⟩}_A$ is an initial preparation for the ancilla (the ${}^1\mathrm{H}$ nuclear spin). We have ${\widehat{\text{R}}}_0=\widehat{\text{K}}$ (${\widehat{\text{R}}}_0=\widehat{\text{L}}$) and ${\widehat{\text{R}}}_f=\widehat{\text{L}}$ (${\widehat{\text{R}}}_f=\widehat{\text{K}}$) for the forward and backward protocols, respectively, with $\widehat{\text{K}}=({\widehat{\sigma }}_y+{\widehat{\sigma }}_z)/\sqrt{2}$ and $\widehat{\text{L}}=({\widehat{\sigma }}_x+{\widehat{\sigma }}_z)/\sqrt{2}$. Finally, ${\widehat{U}}^{\alpha }(t)$ is the time evolution operator determined by the time-dependent Hamiltonian ${\widehat{\mathcal{H}}}^{\alpha }(t)$ of the $\alpha =F,B$ protocol.

Figure 2

(a) and (b): Experimental data for the $x$ and $y$ components of the ${}^1\mathrm{H}$ transverse magnetization (blue circles and red squares, respectively) at two different values of the spin pseudotemperature, plotted against the time length $s=u\pi {\nu }_2/J$ of the controlled operations ${\widehat{G}}_1$ and ${\widehat{G}}_2$. The solid lines show Fourier fittings, which are in agreement with the theoretical simulation of the protocol. The error bars are smaller than the size of the symbols and are not shown (cf. Ref. [25] for the definition of $\epsilon ,\gamma$ and $B_0$). (c) and (d): The experimental points for the distribution corresponding to the forward (backward) protocol are shown as red squares (blue circles).

Figure 3

(a): The ratio $P^F(W)/P^B(-W)$ is plotted in logarithmic scale for four values of the spin pseudotemperature. The data are determined using the values of $P^F(W)$ and $P^B(-W)$ at the peaks shown in Figs. 2 and 2 and in Ref [25]. (b): Mean values and uncertainties for $\mathrm{\Delta }F$ and $\beta$ obtained using a linear fit of the data corresponding to $T>0$ in panel (a). The full red line represents the theoretical expectation, $\mathrm{\Delta }F=(1/\beta )\ln (\cosh (\beta {\nu }_1)/\cosh (\beta {\nu }_2))$. (c): We report the experimental values of the left- and right-hand sides of the Jarzynski identity, measured for three choices of pseudotemperature, together with the respective uncertainties. The experimental results are compared to the theoretical predictions for $\ln (Z_{\tau }/Z_0)$.