Thermodynamics of Weakly Measured Quantum Systems

We consider continuously monitored quantum systems and introduce definitions of work and heat along individual quantum trajectories that are valid for coherent superposition of energy eigenstates. We use these quantities to extend the first and second laws of stochastic thermodynamics to the quantum domain. We illustrate our results with the case of a weakly measured driven two-level system and show how to distinguish between quantum work and heat contributions. We finally employ quantum feedback control to suppress detector backaction and determine the work statistics.

Figure 1

First law for a weakly measured qubit. (a) Infinitesimal change of work, heat, and energy along a single quantum trajectory ${\tilde{\rho }}_t$; for each realization $d{\tilde{U}}_t=\delta {\tilde{W}}_t+\delta {\tilde{Q}}_t$, Eq. (4). (b) Corresponding signal $I(t)$ in the detector. Parameters are $S_0/\mathrm{\Delta }{I}^2=2.5\times {10}^{5}dt$, $\hslash /g=1.6\times {10}^{2}dt$, $\hslash /\epsilon ={10}^{3}dt$, $\nu =8$, and $\tau =3\times {10}^{3}dt$ (see the main text).

Figure 2

(a) Averaged final transition probabilities $P_{m,n}^{\tau }$ (yellow) for a continuously monitored qubit with their work and heat contributions, $\delta P_{m,n}^W$ (blue), and $\delta P_{m,n}^Q$ (red), and the initial transition probability, $P_{m,n}^0={\delta }_{mn}$ (purple). The first law–like equation $d{P}_{m,n}=\delta P_{m,n}^W+\delta P_{m,n}^Q$ is verified. (b) Work and heat contributions $\delta {\tilde{P}}_{m,n}^W$ and $\delta {\tilde{P}}_{m,n}^Q$ at the single trajectory level. Same parameters as in Fig. 1.

Figure 3

Transition probabilities $P_{mn}^{\tau }$ for the weakly measured qubit with (orange) and without (yellow) feedback control for (a) ${\tau }_1=1.4\times {10}^{3}dt$ and (b) ${\tau }_2=2.5\times {10}^{3}dt$. The feedback strength is $f=3$. The isolated, unitary case (red) is shown as a reference. The feedback loop effectively suppresses the detector backaction and the associated heat exchange, achieving thermal isolation.