Experimental Verification of a Jarzynski-Related Information-Theoretic Equality by a Single Trapped Ion

Most nonequilibrium processes in thermodynamics are quantified only by inequalities; however, the Jarzynski relation presents a remarkably simple and general equality relating nonequilibrium quantities with the equilibrium free energy, and this equality holds in both the classical and quantum regimes. We report a single-spin test and confirmation of the Jarzynski relation in the quantum regime using a single ultracold ${{}^{40}\mathrm{Ca}}^{+}$ ion trapped in a harmonic potential, based on a general information-theoretic equality for a temporal evolution of the system sandwiched between two projective measurements. By considering both initially pure and mixed states, respectively, we verify, in an exact and fundamental fashion, the nonequilibrium quantum thermodynamics relevant to the mutual information and Jarzynski equality.

Figure 1

(I) The first measurement: Measurement $P_n$ on the state $\rho$ to obtain $p_n$. (II) The second measurement: Measurement pulse under the operator $U_C$ yielding $\tilde{\rho }=U_C\rho U_C^{†}$, along with a measurement $Q_m$ to obtain $q_m$. (III) Process for conditional probability: Measurement $Q_m$ on the state ${\rho }_n$ [produced from (I)], conditional on a previous measurement $P_n$, to obtain the conditional probability $p_{m|n}$.

Figure 2

Experimental results of the associated probabilities for pure states. In each panel, $n_1=1$, $n_2=1$, 2 and $n_1=2$, $n_2=1$, 2 denote the probabilities of $P_{-}$, $P_{+}$ and $Q_{-}$, $Q_{+}$, respectively, and $n_1=3$, $n_2=1$, 2, 3, 4 correspond to the conditional probabilities $p_{-|-}$, $p_{-|+}$, $p_{+|-}$, $p_{+|+}$, respectively. The evolution time is set to (a) $t=\tau$, (b) $t=2\tau$, (c) $t=3\tau$, and (d) $t=4\tau$, with $\tau =\pi /5\mathrm{\Omega }$. The initial state is $|\psi ⟩=(\sqrt{6}|\downarrow ⟩-i\sqrt{3}|\uparrow ⟩)/3$, and we obtain the data with the rms error $\le 0.02$ for individual points, under measurement repetition of 40 000 times.