Maximum precision charging of multi-qubit quantum batteries

TL;DR

The paper addresses precise energy charging of a multi-qubit quantum battery modeled by Jaynes-Cummings and Tavis-Cummings interactions, using Full Counting Statistics to quantify energy-exchange fluctuations. It demonstrates that charging the qubits sequentially with a genuine quantum non-Gaussian (N-photon Fock) field state can achieve maximal precision, even under sub-optimal conditions, while parallel charging cannot reach perfect charging. The work provides analytic TC-model solutions, FCS-based reports of mean energy transfer and variance, and robustness analyses against attenuation and detuning, establishing a robust quantum-non-Gaussian precision advantage. These results have practical implications for circuit QED, trapped ions, and quantum thermodynamics, offering a concrete route to high-precision quantum charging and energy-monitoring technologies.

Abstract

Precision, robustness, and efficiency are crucial aspects in the design of quantum technologies. Here, we show how genuine quantum features, together with non-Gaussianity, can be the key elements to achieve the best of these three aspects during a quantum battery-charging process. Taking inspiration from a light-matter interaction paradigm, i.e., the Jaynes-Cummings model, we employ the Full Counting Statistics to study the stochastic exchanges of energy between an entire stack of qubits and a single-mode electromagnetic field (or mechanical oscillator). Our study allows to conclude that charging the battery through a sequential protocol involving a quantum non-Gaussian field state guarantees extremely high-performances in the charging process, whose precision is maximized even under sub-optimal operating conditions. These results highlight the potential of non-Gaussian quantum state charging to achieve a robust quantum precision advantage over Gaussian states of the field by suppressing detrimental quantum fluctuations, thus making it suitable to ultimate tasks for which a significant degree of accuracy is required.

Figures (10)

• Figure 1: Pictorial scheme of a multi-qubit JC quantum battery, where a precision advantage in the qubits charging can be obtained by exploiting a Fock state cavity. Under optimal model conditions (e.g., zero noise and perfect resonance), by coupling the cavity with a single qubit at a time, the entire stack of $M$ qubits can be perfectly charged with a $N$-photon cavity Fock state, where $N = M$. Even in presence of noise or sub-optimal model parameters, a quantum non-Gaussian cavity state is still crucial to achieve an appreciable result.
• Figure 2: Sequential charging of $M=5$ qubits, where $\text{SNR}(\Delta U_{\tau_i})$ is calculated for each qubit charge. The fidelity $F = F(t)$ between $\hat{\rho}^{(j)}_{\text{qub}}(t)$ and $\hat{\rho}^{(j)}_{\text{target}} = \ket{e}_j\bra{e}_j$ is depicted in the inset. While the peaks associated with the Fock state SNR can be arbitrarily high, the ones related to the other states decrease exponentially in time. The fidelity can reach its maximum value $F=1$ only for the Fock state protocol; for the others, we always have $F<1$ and the peaks decrease in time as $F\propto-(gt)^\gamma$, with $\gamma>0$. For that simulation, we used $\langle n \rangle = N=5$, $\alpha = \sqrt{5}$ for the coherent state, $\zeta = 0.6$ and $\tilde{\alpha} \simeq 3.905$ for the squeezed coherent state, and $g/\omega_{\text{qub}} = 10^{-2}$. Here, we considered the resonant condition, i.e., $\Delta\omega = 0$.
• Figure 3: SNR of the energy injected in a single qubit during the parallel charging of $M=5$ qubits, based on a numerical solution of the TC model. The upper inset shows the maximum SNR for $M\in\{2,3,4,5\}$ qubits, while the lower one depicts the maximum fidelity $F$ between the state of the first qubit state $\hat{\rho}^{(1)}_{\text{qub}}(1)$ and $\hat{\rho}^{(1)}_{\text{target}} = \ket{e}_1\bra{e}_1$, varying the number of qubits. From the insets, we observe that the SNR and the fidelity provided by the three different cavity states seem to approach each other in the high-$M$ limit. For that simulation, we used $\langle n \rangle = N=5$, $\alpha = \sqrt{3}$ for the coherent state, $\zeta = 0.6$ and $\tilde{\alpha} \simeq 2.935$ for the squeezed coherent state, $g/\omega_{\text{qub}} = 10^{-2}$, and $\Delta\omega = 0$.
• Figure 4: Effect of the presence of $\bar{n}_{\text{th}}$ thermal photons on average in a Fock state cavity. The sequential charging of 5 qubits has been simulated for each value of the initial number of photons in the cavity state, $\langle n \rangle \in\{1,...,5\}$. The figure of merit is the difference $D_{\bar{n}_{\text{th}},\langle n \rangle}$ as defined in (\ref{['eq: Figure of merit for noise analysis']}). Notice that $D_{\bar{n}_{\text{th}},\langle n \rangle}$ is always positive, meaning that the Fock state precision advantage is still persistent, even if affected by a low number of thermal photons. To compare with phase-insensitive Fock states, we used a Gaussian coherent squeezed state $\ket{\zeta, \tilde{\alpha}}\bra{\zeta, \tilde{\alpha}}$ with randomized phases pradana2019quantum (for details, we refer to the Supplemental Material). For that simulation, as before, we used $N = \langle n \rangle$ for the Fock state; $\zeta$ and $\tilde{\alpha}$ have been optimized to give the best performances for the squeezed coherent state. Besides, $g/\omega_{\text{qub}} = 10^{-2}$, and $\Delta\omega = 0$.
• Figure S.1: Fidelity $F$ (discrete points) calculated for the single-qubit state, compared to the SNR of the two-qubits energy exchange $\Delta U^{\text{Q1,Q2}}_\tau$, normalized to its maximum value (black lines). As it can be observed, the maximum of the SNR does not occur in corrispondence of the maximum of $F$: this means that the variance on the qubits collective energy is increased when the qubits are nearly charged. For that simulation, we used $\langle n \rangle = N=2$, $\alpha = \sqrt{2}$ for the coherent state, $\zeta = 0.6$ and $\tilde{\alpha} \simeq 2.3$ for the squeezed coherent state, and $g/\omega_{\text{qub}} = 10^{-2}$. Here, we considered the perfect-resonance condition, i.e., $\Delta\omega = 0$.