Extracting Quantum Work Statistics and Fluctuation Theorems by Single-Qubit Interferometry

We propose an experimental scheme to verify the quantum nonequilibrium fluctuation relations using current technology. Specifically, we show that the characteristic function of the work distribution for a nonequilibrium quench of a general quantum system can be extracted by Ramsey interferometry of a single probe qubit. Our scheme paves the way for the full characterization of nonequilibrium processes in a variety of quantum systems, ranging from single particles to many-body atomic systems and spin chains. We demonstrate our idea using a time-dependent quench of the motional state of a trapped ion, where the internal pseudospin provides a convenient probe qubit.

Figure 1

(a) The Ramsey protocol represented as a quantum circuit. The probe qubit in the upper branch is prepared in the $|\downarrow ⟩$ state and the system of interest is prepared in the state ${\varrho }_{\beta }({\lambda }_{\mathrm{i}})$ defined in the text. A black (white) circle indicates that the operation is controlled on the probe qubit being in the $|\downarrow ⟩$ ($|\uparrow ⟩$) state. Similar schemes are used in protocols for quantum parameter estimation [28]. In (b) and (c) examples are shown of the time variation of the spin-dependent couplings $g_{\downarrow }(t)$ and $g_{\uparrow }(t)$ over the Ramsey scheme time $t_{\mathrm{R}}=t_{\mathrm{Q}}+u$ required to obtain the characteristic function ${\chi }_{\mathrm{F}}(u)$. In (b) a forward quench described by $\lambda (t)={\lambda }_{\mathrm{i}}+({\lambda }_{\mathrm{f}}-{\lambda }_{\mathrm{i}})[1+\tanh (t/T)]/2$, where $T$ is the switching time and the total quench time is $t_{\mathrm{Q}}=8T$, is shown. In (c) the forward quench is composed of repeated fast and slow $\tanh $ switches between ${\lambda }_{\mathrm{i}}$ and ${\lambda }_{\mathrm{f}}$, as in (b), before ending at ${\lambda }_{\mathrm{f}}$.

Figure 2

A ${}^{40}\mathrm{Ca}^{+}$ ion is assumed to be confined with an axial trapping frequency ${\omega }_0=300\mathrm{kHz}$ and prepared in a thermal state with $\overline{n}=1$. The standing wave optical dipole potential for both polarizations are taken as being generated by a 397 nm laser $\approx 30\mathrm{GHz}$ detuned from the $S_{1/2}-P_{1/2}$ transition, giving $\eta =0.33$ and a maximum Rabi frequency of $\Omega =150\mathrm{kHz}$ [21, 23]. A sampling rate of 2 MHz and a decoherence time scale of $\tau =15\times (2\pi /{\omega }_0)=50\mu \mathrm{s}$ has been used with measurements being performed up to a time $\approx 500\mu \mathrm{s}$, where the signal was completely damped. (a) The quantum work distributions $P_{\mathrm{F}}(W)$ (solid) and $P_{\mathrm{B}}(-W)$ (dashed) are plotted on a logarithmic scale for a single $\tanh $ switching of $\lambda (t)$, as shown in Fig. 1b, with $T=0.3\times (2\pi /{\omega }_0)=1\mu \mathrm{s}$. The horizontal lines denote the peak amplitudes identified from both $P_{\mathrm{F}}(W)$ and $P_{\mathrm{B}}(-W)$. The vertical dashed lies at 67 kHz corresponding to the exact $\Delta F$. (b) The plot as (a) for a repeated $\tanh $ switching quench, as shown in Fig. 1c, with $T_{\mathrm{fast}}=0.2\times (2\pi /{\omega }_0)=0.03\mu \mathrm{s}$ and $T_{\mathrm{slow}}=3\times (2\pi /{\omega }_0)=20\mu \mathrm{s}$. (c) The plot and quench as (b) with 0.5% Gaussian noise added to the signals for ${\chi }_{\mathrm{F}}(u)$ and ${\chi }_{\mathrm{B}}(u)$. (d) The ratio $P_{\mathrm{F}}(W)/P_{\mathrm{B}}(-W)$ was evaluated from the peaks identified in (a) ($+$), (b) (filled circle) and (c) ($\times$) is plotted against $W$. The solid line follows the Crooks relation for the exact $\beta$ and $\Delta F$. A best fit of the function $\exp (AW-B)$ for the noisy spectrum in (c) gives $A=0.72/{\omega }_0$ and $B/A=0.20{\omega }_0$, compared to the exact values $\beta =0.69/{\omega }_0$ and $\Delta F=0.22{\omega }_0$, respectively.