Optimal state for a Tavis-Cummings quantum battery via the Bethe ansatz method

In this paper, we investigate the effect of different optical field initial states on the performance of the Tavis-Cummings (TC) quantum battery. In solving the dynamical evolution of the system, we found a fast way to solve the Bethe ansatz equation. We find that the stored energy and the average charging power of the TC quantum battery are closely related to the probability distribution of the optical field initial state in the number states. We define a quantity called the number-state stored energy. With this prescribed quantity, we only need to know the probability distribution of the optical field initial state in the number states to obtain the stored energy and the average charging power of the TC quantum battery at any time. We propose an equal probability and equal expected value splitting method by which we can obtain two inequalities, and the two inequalities can be reduced to Jensen's inequalities. By this method, we found the optimal initial state of the optical field. We found that the maximum stored energy and the maximum average charging power of the TC quantum battery are proportional to the initial average photon number, and the quantum battery can be fully charged when the initial average photon number is large enough. We found two phenomena, which can be described by two empirical inequalities. These two phenomena imply the hypersensitivity of the stored energy of the TC quantum battery to the number-state cavity field. Finally, we discuss the impact of decoherence on battery performance.

Figure 1

Schematic diagram of the TC quantum battery. (a) The state of a quantum battery when it has no energy to output. All the atoms are in the ground state $|g\rangle$, and the battery is empty. (b) The quantum battery is in a fully charged state. $N$ identical atoms are all in the excited state $|e\rangle$

Figure 2

Running time as a function of the number $M$. Semilogarithmic plot giving the time (in hours) required to solve the BAE by the trial solution method (blue stars joined by dashed lines) and by the bertini software (red circles joined by solid lines). Here, we take $N=10$, and we did not use the bertini software to solve the BAE when $M\ge 8$ because it was too slow.

Figure 3

(a) The stored energy of the TCQB changes with time $t$ under different number states. (b) The average charging power of the TCQB varies with time $t$ under different number states. In (a) and (b), the marked lines and unmarked lines with different colors represent numerical and analytical solutions, respectively. (c) The energy acquired by the atomic ensemble varies with the average photon number $M$ of the number state at different times. In this paper, the number of atoms is $N=10$.

Figure 4

(a) The probability distributions of coherent states $|\sqrt{6}\rangle$ and $|\sqrt{4}\rangle$ in the number states, respectively. Here, we truncate the Hilbert space at the 16-photon Fock state $|M=16\rangle$. (b) The stored energy of the TCQB changes with time $t$ in the two initial coherent states, $|\sqrt{6}\rangle$ and $|\sqrt{4}\rangle$. (c) The average charging power of the TCQB varies with time $t$ in the two initial coherent states, $|\sqrt{6}\rangle$ and $|\sqrt{4}\rangle$. In (b) and (c), the marked lines and unmarked lines with different colors represent numerical and analytical solutions, respectively.

Figure 5

(a) The blue and red histograms respectively represent the probability distribution of number state $|10\rangle$ and the state $|\varphi \rangle$ in the number states. (b) The probability distributions of the number state $|10\rangle$ and the state $|\varphi \rangle$ after decomposition by the equal probability and equal expected value splitting method. (c) The variations of the stored energy and the average charging power of the TCQB with time $t$ under two different initial states, the number state $|10\rangle$ and the state $|\varphi \rangle$.

Figure 6

The average slope $F(M,t)/M$ as a function of the time $t$ and the average photon number $M$ of the initial number state.

Figure 7

The variations of $F(M,t)/F(10,t)$ and $F(M,t)$ with time $t$ when $M$ take different values are shown in (a) and (b), respectively.

Figure 8

(a) The variations of the stored energy and the average charging power of the TCQB with time $t$ for the different collective dephasings of the atoms and the different decays of the optical field are shown respectively in (a) and (b). Here, the relevant parameters are chosen from Ref. [47], and all parameters are in unit of ${\omega }_c$.