Figure 1

Schematic diagram of associated stages in Eq. (1): (a) A system is in equilibrium with a thermal reservoir, where the system Hamiltonian is given by $H\left(0\right)$ and the corresponding free energy is $F_i=-k_{B}T\ln Z$ with $Z=\text{Tr}{e}^{-\beta H\left(0\right)}$. For the time span $t\in (0,\mathcal{T})$, the system is disconnected from the thermal reservoir, and the work is done on the system by changing the Hamiltonian from $H\left(0\right)$ to $H\left(\mathcal{T}\right)$ along a prescribed path. If the system with $H\left(\mathcal{T}\right)$ is then brought in contact with the thermal reservoir, heat is dissipated into the reservoir; after a sufficiently long time $(t_f\gg \mathcal{T})$, the system reaches the final equilibrium state with $H\left(\mathcal{T}\right)$, and the free energy $F_f$ for this state gives the free energy difference, $\Delta F=F_f-F_i$ in Eq. (1). (b) The only difference from setup in (a) is that the work input is realized not by changing the Hamiltonian but by performing projective quantum measurements for $t\in (0,\mathcal{T})$. Here the system Hamiltonian remains constant, but projective measurements interrupting the unitary time evolution of the system at the moments marked by the crosses bring about an energy change of the system.Reuse & Permissions