Random quantum batteries

Quantum nanodevices are fundamental systems in quantum thermodynamics that have been the subject of profound interest in recent years. Among these, quantum batteries play a very important role. In this paper we lay down a theory of random quantum batteries and provide a systematic way of computing the average work and work fluctuations in such devices by investigating their typical behavior. We show that the performance of random quantum batteries exhibits typicality and depends only on the spectral properties of the time evolving operator, the initial state, and the measuring Hamiltonian. At given revival times a random quantum battery features a quantum advantage over classical random batteries. Our method is particularly apt to be used both for exactly solvable models like the Jaynes-Cummings model or in perturbation theory, e.g., systems subject to harmonic perturbations. We also study the setting of quantum adiabatic random batteries.

Figure 1

Average of $Q$ over 1000 samples for random matrices in the circulant unitary ensembles of dimensions $n=10,n=100,500,1000$. The peak of the distribution converges to zero for larger values of $n$.

Figure 2

Average work extraction for a random quantum battery made by an optical trap described by the Jaynes-Cummings model. (a) The function $Q\left(\alpha \right)$ as function of $\alpha$ for $n=2,10,20$. The maximum value of this function is 0.5. As the size increases, revivals become more peaked. (b) Work for the Jaynes-Cummings model as a function of time for $\rho =0.5$ for $n=2,10,20$ and $\text{Tr}\left(H_0\right)=90*n$ and $E_0=100$. The baseline represents the work extracted by a battery that brings the system in the completely mixed state.

Figure 3

Average work from Eq. (B74) for $n=3,10,20,100$ and $\omega =0.5$, against the baseline work $E_0-\frac{\text{Tr}\left(H_0\right)}{n}$, with $\text{Tr}\left(H_0\right)=90n$ and $E_0=100$.

Figure 4

Frequency distribution of the average $\langle e^{{\sum }_{j=1}^4\pm i{\theta }_j}\rangle$ for CUE over $M=1000$ samples. We see that the distribution is strongly peaked around the value of $\langle K^2\rangle =0$, for the three terms with three possible signatures in the exponent for (a)–(c), which is what is necessary for the proof of our concentration at least in the case of CUE. The last peak is just a binning artifact.