Stabilizing open quantum batteries by sequential measurements

A quantum battery is a work reservoir that stores energy in quantum degrees of freedom. When immersed in an environment, an open quantum battery needs to be stabilized against free-energy leakage due to decoherence, unavoidably entailing entropy production. For this purpose we here propose a stabilization protocol given by a nonunitary open-loop control action able to compensate for the entropy increase and to maintain the open quantum battery in its highest ergotropy state. The protocol relies on nonselective, frequent, projective measurements that are interspersed by optimized time intervals. In accordance with a second-law-like inequality derived for the entropy production rate of the controlled battery, the proposed procedure results in minimized control power. The effectiveness of the method is finally tested on a qubit subject to decoherence, achieving an average fidelity value around $95\%$.

Figure 1

Stabilization scheme—single-run illustration. After the initialization step (with initial state ${\rho }_0$), the battery stabilization protocol consists of intermittent free evolutions and fast unitary controlled dynamics [dotted points, corresponding to steps (i) and (iii) of the procedure] and projective measurements [solid brown line, step (ii)] in time intervals of duration $\tau$. In particular, the green dots denote the maximum energy state ${\rho }_e$, while the blue dots represent the state ${\rho }_{\alpha }$ of the battery immediately before a projective measurement in the Zeno regime. ${\rho }_i$ and ${\rho }_i^{\prime }$ are the nearest states to ${\rho }_e$ on the unitary orbit of ${\rho }_0$ and ${\rho }_g$, respectively.

Figure 2

Stabilization scheme—numerical results for a qubit with internal Hamiltonian $H_0=3{\sigma }_x+{\sigma }_z$ (in natural units). (a) Average behavior over time of the battery density matrix, obtained by repeating the stabilization procedure 1000 times. (b), (c) Behavior over time of the battery density matrix in single realizations of the scheme: Being probabilistic, the charging process could require the application of more than one projective measurement. In the subplots, $\left({\rho }_e^{\left(11\right)},{\rho }_e^{\left(12\right)}\right)$ and $\left({\rho }_t^{\left(11\right)},{\rho }_t^{\left(12\right)}\right)$ are the top diagonal elements and the coherence terms, respectively, of the maximum energy state ${\rho }_e$ of the qubit battery and of the corresponding time-evolved density matrix ${\rho }_t$. (d) Zoom of (c) in the time interval [0.75,2.15], showing the occurrence of a failure collapse and the resulting reinitialization procedure. Further details can be found in Appendix pp2. (e) Stabilization fidelity $\mathcal{F}$ over 1000 realizations of the stabilization procedure. At $t=0, \mathcal{F}$ starts from a value in the range [0.3,0.4] since also the initialization step has been taken into account.

Figure 3

Entropic cost of the Zeno protection procedure: ${\sigma }_{\mathrm{Zeno}}$ as a function of the time interval $\tau$ between Zeno measurements. The results have been numerically derived for the same quantum system used in Fig. 2. Each curve has been obtained by choosing a fixed duration $T_{\mathrm{Zeno}}\equiv t_{\mathrm{fin}}-t_{\max }$, among a set of values (see the legend of the figure), and letting $\tau$ vary, so that also $\overline{m}\approx T_{\mathrm{Zeno}}/\tau$ of Zeno measurements changes every time. The integral ${\int }_0^{\tau }\dot{H}\left(P\right)dt$ (black line), numerically solved with the initial condition $\rho ={\rho }_e$, has a monotonically increasing behavior for greater values of $\tau$, thus identifying $\overline{m}$ as the dominant factor. Inset: Amount of not stored energy $\overline{m}{P}_g\left(\tau \right)$ (normalized by $E_{\max }$) as a function of $\tau$ for $T_{\mathrm{Zeno}}=0.4,$ 1, 2, 3, 4, 5, and 6, in natural units. Here, the black line denotes $P_g\left(\tau \right)$, and an unavoidable worsening of the battery stabilization is observed when $\tau$ increases.

Figure 4

(a) Behavior over time of the battery density matrix elements ${\rho }_t^{\left(11\right)}$ and ${\rho }_t^{\left(22\right)}=1-{\rho }_t^{\left(11\right)}$. (b) Behavior over time of the real and imaginary part of the battery density matrix element ${\rho }_t^{\left(12\right)}$.

Figure 5

(a) Behavior over time of the trace distance $T({\rho }_t,{\rho }_e)$. (b) Behavior over time of the probability $P_e$.