Quantum supercapacitors

Recently, there has been a great deal of interest in the possibility to exploit quantum-mechanical effects to increase the performance of energy storage systems. Here, we introduce and solve a model of a quantum supercapacitor. This consists of two chains, one containing electrons and the other one holes, hosted by arrays of double quantum dots, the latter being a building block of experimental architectures for realizing charge and spin qubits. The two chains are in close proximity and embedded in the same photonic cavity, which is responsible for long-range coupling between all the qubits, in the spirit of the Dicke model. By employing a variational approach, we find the phase diagram of the model, which displays ferromagnetic and antiferromagnetic phases for suitable pseudospin degrees of freedom, together with phases characterized by collective superradiant behavior. Importantly, we show that when transitioning from the ferromagnetic/antiferromagnetic to the superradiant phase, the quantum capacitance of the model is enhanced. Our work offers opportunities for the experimental realization of a novel class of quantum supercapacitors with an enhanced capacitance stemming from quantum-mechanical effects.

Figure 1

(a) Voltage profile of a double quantum dot occupied by a single electron. The ground (${|g\rangle }_e$) and excited (${|e\rangle }_e$) states are geometrically separated, and the energy gap between them is $\varepsilon$. The ${|g\rangle }_e\to {|e\rangle }_e$ transition can be induced by the absorption of photons (and vice versa the ${|e\rangle }_e\to {|g\rangle }_e$ transition occurs through photon emission). (b) Schematic top view of the two-chain system. Here, one has a top chain (red) and a bottom chain (light blue) made up of double quantum dots. Each of them is singly occupied with electrons (dark red) and holes (dark blue), respectively. Two dominant contributions to the electrostatic interaction have been analyzed in this work: (i) an interchain attractive interaction of strength $\mathcal{U}$ (green arrow) between an electron and a hole in their respective ground states and (ii) an intrachain repulsive interaction of strength $\mathcal{V}$ (orange arrow) between electrons (holes) in either the ${|g\rangle }_e-{|g\rangle }_e$ (${|g\rangle }_h-{|g\rangle }_h$) or ${|e\rangle }_e-{|e\rangle }_e$ (${|e\rangle }_h-{|e\rangle }_h$) configuration.

Figure 2

(a) At $\tilde{\mathcal{V}}=0$ the phase diagram shows a continuous phase transition between the ferromagnetic-normal (blue) and the ferromagnetic-superradiant (red) ordering, occurring at ${\mathrm{\Lambda }}^2=(1+\tilde{\mathcal{U}})/8$. The situation remains qualitatively analogous up to $\tilde{\mathcal{V}}\lesssim 0.5$. By further increasing $\tilde{\mathcal{V}}$, $\tilde{\mathcal{V}}=0.7$ in (b) and $\tilde{\mathcal{V}}=1.0$ in (c), one observes the emergence of both an antiferromagnetic-normal (green) and a narrow antiferromagnetic-superradiant (yellow) phase at the expense of the previously discussed ones. The ferromagnetic-normal/antiferromagnetic-normal transition is first order and occurs at $\tilde{\mathcal{U}}=4\tilde{\mathcal{V}}-2$. Notice that this phase diagram has been deduced from the global analysis of the order parameters $\mathcal{A}$, $s$, and $m$, introduced in Eqs. (12), which are reported in Fig. 3. The present phase diagram has been calculated for the resonant condition $\varepsilon =\hslash {\omega }_{\mathrm{c}}$, where the coupling between DQDs and radiation is optimal [60].

Figure 3

Density plots of the order parameters $\mathcal{A}=\langle {\widehat{a}}^{†}\widehat{a}\rangle /N$, $s=\langle \left({\widehat{\sigma }}_1^z+{\widehat{\sigma }}_2^z\right)\rangle /2$, and $m=\langle \left({\widehat{\sigma }}_1^z-{\widehat{\sigma }}_2^z\right)\rangle /2$ as functions of $\tilde{\mathcal{U}}$ and ${\mathrm{\Lambda }}^2$ for (a)–(c) $\tilde{\mathcal{V}}=0.7$ and (d)–(f) $\tilde{\mathcal{V}}=1.0$. Notice that from the top row one can reconstruct the phase diagram of Fig. 2, while the bottom row leads to Fig. 2. All data in this figure refer to the resonant condition $\varepsilon =\hslash {\omega }_{\mathrm{c}}$.

Figure 4

Density plots of the ratio $\kappa =C/\overline{C}-1$ (with $C$ being the capacitance of the system and $\overline{C}$ being its value at ${\mathrm{\Lambda }}^2=0$) as a function of $\tilde{\mathcal{U}}$ and ${\mathrm{\Lambda }}^2$. (a) $\tilde{\mathcal{V}}=0$, corresponding to the phase diagram in Fig. 2, where, in the ferromagnetic-superradiant phase, one has $\kappa =(1-s^2)/(1+s^2)$. This ratio asymptotically approaches $\kappa =1$ (doubling of the capacitance) at high values of ${\mathrm{\Lambda }}^2$. (b) $\tilde{\mathcal{V}}=1.0$, corresponding to the phase diagram in Fig. 2, where $\kappa =(2+\tilde{\mathcal{U}})/\left(1+s^2\right)\tilde{\mathcal{V}}-1$ crucially depends on the geometry of the device. Here, for the considered values of the parameters, the ratio can exceed $\kappa >2.5$ (red region), leading to a remarkable enhancement of the capacitance with respect to the reference case in the absence of radiation.