Fluctuation Theorems for Continuously Monitored Quantum Fluxes

It is shown that quantum fluctuation theorems remain unaffected if measurements of any kind and number of observables are performed during the action of a force protocol. That is, although the backward and forward probabilities entering the fluctuation theorems are both altered by these measurements, their ratio remains unchanged. This observation allows us to describe the measurement of fluxes through interfaces and, in this way, to bridge the gap between the current theory, based on only two measurements performed at the beginning and end of the protocol, and experiments that are based on continuous monitoring.

Figure 1

Schematic representation of a bidirectional counting statistics experiment. Two leads (large semicircles) with different electronic chemical potentials (${\mu }_1\ne {\mu }_2$) and same inverse temperature (${\beta }_1={\beta }_2$) are connected through a double quantum dot (small circles), whose quantum state is continuously monitored. The state (1,0), i.e., “one electron in the left dot, no electrons in the right dot” is depicted. The transition from this state to the state (0,1) signals the exchange of one electron from subsystem 1 to subsystem 2. $H_{1,2}$ and ${\mathcal{N}}_{1,2}$ denote the Hamiltonian and number of electrons operators of each subsystem, respectively.