Measuring the Characteristic Function of the Work Distribution

We propose an interferometric setting for the ancilla-assisted measurement of the characteristic function of the work distribution following a time-dependent process experienced by a quantum system. We identify how the configuration of the effective interferometer is linked to the symmetries enjoyed by the Hamiltonian ruling the process and provide the explicit form of the operations to implement in order to accomplish our task. We finally discuss two physical settings, based on hybrid optomechanical-electromechanical devices, where the theoretical proposals discussed in our work could find an experimental demonstration.

Figure 1

(a) Quantum circuit illustrating the protocol for the measurement of ${\chi }_s(u)$. The ancilla $A$ is a qubit initialized in $|0{⟩}_A$ and undergoing a Hadamard gate $\widehat{H}$. System $S$ is prepared in a thermal state ${\rho }_S^{\mathrm{th}}$ and is subjected to the local transformation $\widehat{V}$. See the body of the manuscript for the explicit form of the gates (whose dependence on $u$ has been omitted here). (b) Quantum circuit illustrating the scheme for the most general process undergone by $S$. In both panels we show the symbol for conditional $A\mathrm{\text{−}}S$ gates controlled by the state of the ancilla. In panel (b) we also picture the symbol for a full inversion gate as given by ${\widehat{\sigma }}_x$, $A$.

Figure 2

(a) Hybrid micro-optomechanical setting for the measurement of $\chi (u,\tau )$. The process is undergone by a system embodied by the movable cavity mirror. The ancilla is encoded in the ground-state doublet of a three-level atom. (b) Nanomechanical version of the setup. System $S$ is an electrically driven nanobeam (bias voltage $V_x$). The ancilla is a CPB coupled to $S$ via the capacitance $C_x$. The state of the CPB is controlled by the gate voltage $V_g$ (coupled to the box through the capacitance $C_g$) and the Josephson energy $E_J$. (c) Plot of $\chi (u,\tau )$ against ${\omega }_{S}u$ for $\overline{n}=1$, ${\lambda }_t=0.1{\omega }_S\tanh ({\omega }_{S}t)$, and $\tau =10{\omega }_S^{-1}$. The solid (dashed) lines show real and imaginary part of the ideal (damped with $\Gamma =5{\tau }^{-1}$) characteristic function.