Energy storage in a continuous-variable quantum battery with nonlinear coupling

Abstract

In the quantum world, the process of energy storage can be enhanced thanks to various nonclassical phenomena. This inspiring fact suggests quantum batteries as plausible sources of power for future quantum devices, at least in principle. However, thermodynamically not all of the energy stored in a quantum battery is useful for doing work. By considering a class of models based upon quantum continuous variables, here we show how the maximum extractable energy from a bosonic quantum battery can be intimately related to Heisenberg's uncertainty principle. We found that realizing minimum uncertainty essentially guarantees that all of the energy stored in a Gaussian quantum battery can be withdrawn and used to do work. For a standard system where the charger and battery are coupled linearly, this criterion is satisfied rather trivially. However, our theoretical results demonstrate that - for a quantum battery with nonlinear coupling - a state of minimum uncertainty can also be achieved nontrivially via the generation of quantum squeezing. We characterize the charging performance of our proposed continuous variable quantum batteries in detail, and we hope that our theory may be useful in the design of a new generation of efficient quantum batteries harnessing bosonic excitations, such as those built with photonic architectures.

Figures (4)

• Figure 1: The bosonic quantum battery as a driven-dissipative system. Panel (a): the general setup of the two-component quantum battery, complete with its charger and battery pieces (modes $a$ and $b$). The charger is driven by laser light with an amplitude $\Omega$ (green region), and it suffers from dissipation at the rate $\gamma$ into its surrounding heat bath (blue region). Two flavours of interaction (orange region) are considered: either (i) linear coupling (of strength $g$) or (ii) nonlinear coupling (of strength $J$). These couplings enable excitation transfer from the time $t = 0$ from the charger mode $a$ into the battery mode $b$, where the energy is eventually stored in once the interaction is turned off at $t = T$. Panel (b): The nonlinear quantum battery is modelled as two quantum harmonic oscillators interacting via a nonlinear coupling of strength $J$, which is mediated by (for example) a $\chi^{(2)}$ nonlinear crystal. The emission of one photon of frequency $2\omega_b$ from the charger (mode $a$) leads to two photons of frequency $\omega_b$ impinging on the battery (mode $b$). Panel (c): the quadrature variances $\sigma_x^2$ and $\sigma_p^2$ in the steady state ($t \to \infty$) as a function of the dimensionless ratio $\Omega/J$ for the nonlinear quantum battery [cf. Eq. \ref{['eq:xfgxsdfdfg']} and Eq. \ref{['eq:xfgxsdfsdddfg']}]. Dashed green line: guide for the eye at the minimum uncertainty.
• Figure 2: Performance of the quantum battery with linear coupling. Panel (a): the energy $E$ (cyan line) stored in the quantum battery, in units of the asymptotic result $\lim_{t \to \infty} (E)$, as a function of time $t$ (in units of $1/g$) [cf. Eq. \ref{['eq:sdfssfddfgdfgfsdfsdf']}]. Dotted line: the steady state energy [cf. Eq. \ref{['eq:dfgdfgdg2']}]. The instant in time (pink line) and specific energy (orange line) of the energetic maximum is marked. In this column we take the charger--battery coupling strength $g = \gamma/2$, where $\gamma$ is the decay rate of the charger. Panel (b): the optimal charging time $t_E$ (pink line) in units of $1/g$, as a function of the dimensionless ratio $g/\gamma$ [cf. Eq. \ref{['eq:sdfsdfsdfsdf']} and Eq. \ref{['eq:sdfsdfsdfsdfcccc']}]. Dotted line: the asymptote $t_E = \pi/g$ of the dissipationless limit. Panel (c): the maximum energy $E(t_E)$ (orange line) as a function of $g/\gamma$ [cf. Eq. \ref{['eq:jhfghfgfgh']} and Eq. \ref{['eq:jhfghfgfgh2']}]. Dashed line: the steady state result [cf. Eq. \ref{['eq:dfgdfgdg2']}]. Dotted line: in the dissipationless limit, the result is four times the steady state result [cf. Eq. \ref{['eq:dfgdfgdg']} when $t = \pi/g$]. Panel (d): the charging power $P$ (green line) of the quantum battery, in units of $\omega_b \Omega^2/g$, as a function of time $t$ [cf. Eq. \ref{['eq:sfdsdfsf33']} with Eq. \ref{['eq:sdfssfddfgdfgfsdfsdf']}]. The instant in time (pink line) and specific power (orange line) of the power maximum is marked. Panel (e): the optimal time $t_P$ (pink line) as a function of $g/\gamma$. Dashed line: the weak coupling limit [cf. Eq. \ref{['eq:sfeessefesfsdfsdf']}]. Dotted line: the strong coupling limit [cf. Eq. \ref{['eq:sfeessefesfsdfsdf111']}]. Panel (f): the maximum power $P(t_P)$ (orange line) as a function of $g/\gamma$. Dashed line: the weak coupling limit [cf. Eq. \ref{['eq:sdfsdfssefdfsdf']}]. Dotted line: the strong coupling limit [cf. Eq. \ref{['eq:fdsvsfffffb']}]. In this figure the vertical grey lines in panels (b, c, e, f) mark the location of the exceptional point $g_{\mathrm{EP}}$ [cf. Eq. \ref{['eq:sdfsf']}], and the energies $E$ referred to in the top row of panels are exactly equivalent to ergotropies $\mathcal{E}$ since the quantum battery is in a minimum uncertainty state [cf. Eq. \ref{['eq:sdfdffhghfssfdfsdfsdf']}].
• Figure 3: Dynamics of the quantum battery with nonlinear coupling. Panel (a): the energy $E$ stored in the quantum battery in the dissipationless limit (solid cyan line) and in units of the energy level spacing $\omega_b$, as a function of time $t$ (in units of $1/J$, the inverse of the coupling rate $J$). Dashed, dotted and dash-dotted lines: three approximate results, derived with perturbation theory of increasing order [cf. Eq. \ref{['eq:sdfsdfs']}, along with Eq. \ref{['eq:dfgsdfdsdfgdg']}, Eq. \ref{['eq:wilf']} and Eq. \ref{['eq:sfvgvwwwq']}]. Panel (b): the stored energy $E$ (solid cyan line) in units of the steady state energy, within the weak driving regime $\Omega \ll J$. Dashed blue line: the approximate behaviour [cf. Eq. \ref{['eq:ssssdfsdfsdfdfsfd']}]. Vertical pink line: the optimal charging time $t_E$ [cf. Eq. \ref{['eq:sfdsfdfdf']}]. Horizontal orange line: the approximate maximum stored energy $E(t_E)$ [cf. Eq. \ref{['eq:sfdsfdfdffkuuuf']}]. Thin grey line: the approximate steady state energy $\lim_{t \to \infty} (E)$ [cf. Eq. \ref{['eq:sfdssdsdffsdf']}]. Panel (c): a semi-logarithmic plot of the dynamic energy $E$ in units of $\omega_b$. Panels (d, e, f): the charging power $P$ of the quantum battery, in units of $\omega_b J$, corresponding to the stored energies of the above panels (a, b, c) respectively [cf. Eq. \ref{['eq:sfdsdfsf33']}]. In panel (e), the vertical pink line corresponds to the optimal charging time $t_P$ [cf. Eq. \ref{['eq:zvxxczxczcx']}] and the horizontal orange line represents the maximum power $P(t_p)$ [cf. Eq. \ref{['eq:gfgdbdg']}]. Within this figure, in the first two columns $\Omega = J/4$ and in the final two columns $\gamma = J/2$.
• Figure 4: Performance of the quantum battery with nonlinear coupling. Panel (a): the energy $E$ (thick cyan line) and ergotropy $\mathcal{E}$ (thin pink line) stored in the quantum battery in the steady state, shown as a function of the drive amplitude $\Omega$ (in units of the coupling rate $J$). Dashed orange line: the cumulant approximation [cf. Eq. \ref{['eq:sssdfsfd']}]. Dotted green line: the weak driving approximation [cf. Eq. \ref{['eq:sfdssdsdffsdf']}]. Panel (b): the optimal charging times for the the quantum battery in order to maximize the energy $t_E$ (thick cyan line) and power $t_P$ (thin orange line) [cf. Eq. \ref{['eq:sfdsdfsf22']} and Eq. \ref{['eq:sfdsdfsf44']}]. Dashed blue line: the weak driving approximation for energy [cf. Eq. \ref{['eq:sfdsfdfdf']}]. Dotted red line: the weak driving approximation for power [cf. Eq. \ref{['eq:zvxxczxczcx']}]. Panel (c): the maximum energy $E(t_E)$ (thick cyan line) and the peak power $P(t_P)$ (thin orange line) that can be achieved in the nonlinear quantum battery [cf. Eq. \ref{['eq:sfdsdfsf22']} and Eq. \ref{['eq:sfdsdfsf44']}]. Dashed blue line: the weak driving approximation for energy [cf. Eq. \ref{['eq:sfdsfdfdffkuuuf']}]. Dotted red line: the weak driving approximation for power [cf. Eq. \ref{['eq:gfgdbdg']}]. Upper row: we consider the damping rate $\gamma = J/2$. Panels (d, e, f): as for panels (a, b, c), but with the loss rate increased fourfold up to $\gamma = 2J$. In this figure we utilize log--log plots and the calculations are performed such the results are convergent up to the energy $10\omega_b$.