Thermodynamics of quantum-jump-conditioned feedback control

We consider open quantum systems weakly coupled to thermal reservoirs and subjected to quantum feedback operations triggered with or without delay by monitored quantum jumps. We establish a thermodynamic description of such systems and analyze how the first and second law of thermodynamics are modified by the feedback. We apply our formalism to study the efficiency of a qubit subjected to a quantum feedback control and operating as a heat pump between two reservoirs. We also demonstrate that quantum feedbacks can be used to stabilize coherences in nonequilibrium stationary states which in some cases may even become pure quantum states.

Figure 1

Illustration of a possible transition between two energy levels $E_i$ and $E_j$ of the system by absorption of an energy quantum (in black) or an emission of an energy quantum (in gray) from or into the thermal reservoir $\nu$.

Figure 2

Summary of the modified first and second law for a system at steady state subjected to feedback control in contact with two reservoirs. Both the first and second law are modified due to the rate of energy injection ${\dot{\mathcal{F}}}_E$ by the feedback and the information flow ${\dot{\mathcal{F}}}_S$.

Figure 3

(a) Qubit model: ${\kappa }_C$, Eq. (46), (solid line) and $\kappa$, Eq. (44) (dashed line), as a function of the delay time $\tau$ for $T_L=1$ and $T_R=1/2$ (thus, ${\eta }_C^{-1}=2$). The dotted line represents ${\dot{Q}}_L$. As the feedback parameters we choose ${\alpha }_{+}^{\nu }=0$ and ${\alpha }_{-}^{\nu }=\pi /2$. Inset: Energy and entropy injected by the feedback ${\dot{\mathcal{F}}}_E$ (solid line) and ${\dot{\mathcal{F}}}_S$ (dashed line) as a function of the delay time $\tau$. We also choose $\Omega ={\Gamma }_L={\Gamma }_R=1$. (b) Qutrit model (see Appendix ): No time delay $\tau =0$. Plot of ${\dot{Q}}^L$ as a function of $T_L$, for $T_R=1$ and $\alpha =\pi /3$. Insets: Plot of ${\kappa }_C$ (solid), $\kappa$ (dashed), and ${\eta }_C^{-1}$ (dotted) in the regions where ${\dot{Q}}^L<0$. We choose ${\Gamma }_L^1={\Gamma }_R^2=0.1,{\Gamma }_L^2={\Gamma }_R^1=2,{\Gamma }_L^{\Delta }={\Gamma }_R^{\Delta }=0.01,{\Omega }_2=1.1,{\Omega }_1=1.0$.

Figure 4

Contour plot of the distance $\mathfrak{D}[{\varrho }_{\mathrm{target}},\varrho ]$ for varying $\beta$ and $\alpha$ (left side). The horizontal (green) lines corresponds to a plot of the distance for $\alpha =\pi /20$ (dashed), $\pi /4$ (solid), and $3/2$ (dotted). The vertical (red) lines corresponds to plots of the energy injection rate due to the feedback for $\beta =0.5$ (solid), $1.5$ (dotted), and 5 (dashed). Further values were chosen as ${\Gamma }_L={\Gamma }_R=1$ and $\Omega =1$.

Figure 5

Plot of $1-\mathfrak{D}[{\varrho }_{\mathrm{target}},\varrho ]$ for $\alpha =\pi /4$ over the common reservoir temperature $T$. All the other parameters are chosen as in Fig. 4.

Figure 6

Contour plot of the distance $\mathfrak{D}[\varrho ,{\varrho }_{\mathrm{target}}(\pi /4)]$ over three different bath temperatures ${\beta }_L={\beta }_R=\beta$ where the steady state $\varrho$ is obtained for varying ${\alpha }_{+}$ and ${\alpha }_{-}$. The dashed diagonal line corresponds to the previous setting ${\alpha }_{+}=\alpha$ and ${\alpha }_{-}=\alpha +\pi /2$ and the dot marks the case $\alpha =\pi /4$.