Fluctuations in Extractable Work Bound the Charging Power of Quantum Batteries

We study the connection between the charging power of quantum batteries and the fluctuations of the extractable work. We prove that in order to have a nonzero rate of change of the extractable work, the state ${\rho }_{\mathcal{W}}$ of the battery cannot be an eigenstate of a “free energy operator,” defined by $\mathcal{F}\equiv H_{\mathcal{W}}+{\beta }^{-1}\log ({\rho }_{\mathcal{W}})$, where $H_{\mathcal{W}}$ is the Hamiltonian of the battery and $\beta$ is the inverse temperature of a reference thermal bath with respect to which the extractable work is calculated. We do so by proving that fluctuations in the free energy operator upper bound the charging power of a quantum battery. Our findings also suggest that quantum coherence in the battery enhances the charging process, which we illustrate on a toy model of a heat engine.

Figure 1

Fluctuations in extractable work limit the charging power of batteries. A quantum battery is a system that follows the laws of quantum physics, and that can store energy to be extracted later in order to perform work. The most important quantity in a battery is the maximum amount of work $W$ that can be extracted from it, i.e., the extractable work, which in certain cases is directly characterized by the free energy. However, due to the quantum nature of the battery, this free energy is a random quantity with uncertainties. We prove that the rate at which any battery can be charged is linked to such fluctuations, via fundamental bounds that hold for any charging protocol. For a battery with a well-defined value of the extractable work $W$, depicted on the left, the maximum rate at which the extractable work changes is zero. Fluctuations allow for faster charging rate, as illustrated on the right.

Figure 2

Charging power and extractable work. Charging power $P(t)$ (left) and change in extractable work $\mathrm{\Delta }W\equiv W_{\max }(t)-W_{\max }(0)$ (right) as a function of time for various initial states (for the values of the parameters used see Ref. [70]). The incoherent state ${\rho }_{\mathrm{diag}}$ (red asterisk) has null charging power initially, as expected from bound (12). This charging power is slowly increased as the state supports higher fluctuations in extractable work. A significant increase in the charging power is seen for a coherent superposition of eigenstates of $\mathcal{F}$. For intermediate times this initial increase is penalized with a regime in which the battery is being discharged. Nevertheless, there is a net increase in the extractable work with respect to an incoherent initial state. Taking coherent superpositions between more than two states gives a further increase in charging power and extractable work.