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The Jacobi elliptic functions have been instrumental for the discovery

Physicists at HSE University and NUST MISIS have formulated and solved equations for a quantum battery, a device capable of storing energy in the form of light. Their findings will facilitate precise calculations of the capacity, power, and duration required for optimal battery charging. Quantum batteries are expected to improve the performance of solar panels and electric vehicles, while also opening up new avenues for efficient energy transfer. The study has been published in *Physical Review A*.

Quantum batteries represent a novel class of devices that store energy through coherent quantum effects not present in conventional electrochemical batteries. To date, quantum batteries have solely existed in laboratory settings; however, physicists worldwide are engaged in research that holds the potential to revolutionise the energy domain.

Promising candidates for such devices are ensembles of two–level systems interacting with an electromagnetic wave in a resonant cavity. Quantum systems of this type exhibit two energy levels: a low state known as the ground state, and a higher state referred to as the excited state. Transitions between these levels occur through the absorption and emission of photons within the resonator linked to the ensemble.

When a photon is absorbed, the two-level system transitions from the lower to the upper level, and when a photon is emitted, the system moves from the upper to the lower level. Due to quantum effects, two-level systems within the ensemble behave coherently, enabling them to simultaneously emit and absorb light. Theoretically, these systems are described using the Dicke model.

In prior studies, the dynamics of absorption and emission of the electromagnetic field in quantum batteries were described solely through numerical modelling, without obtaining an analytical solution to the equations describing the battery's behaviour. For the first time, Russian scientists have described the dynamics of the electromagnetic field in a quantum battery using an analytical solution based on Jacobi elliptic functions. The authors have previously employed a similar solution in several of their other papers to describe the coherent state of an ensemble of two-level systems interacting with an electromagnetic wave in a resonator, a concept they termed 'bound luminosity.'

Descriptions which use mathematical equations are consistently preferred, as they are more rigorous and convenient for analysis. A numerical solution does not allow us to precisely determine the dependencies of the system parameters on each other and on time. Moreover, numerical modelling is inherently limited in its accuracy.

The researchers employed a quasiclassical approximation method wherein quantum dynamics mimic classical dynamics. This method of mathematical analysis is applied to problems in quantum mechanics, based on the premise that under specific conditions, the behaviour of quantum systems can be approximated to that of classical systems. Its application is feasible here due to the collective behaviour of two-level systems in a coherent state, alongside the formation of a macroscopic photonic condensate by the electromagnetic field within the resonator.

In the bound luminosity state, the electromagnetic field is coherently absorbed and emitted periodically by two-level systems. Our solution describes the form and periodicity of these oscillations, allowing us to calculate the battery's capacity, charging time and power.

Seidali Seidov

Co-author of the paper, Research Fellow at the Quantum Nanoelectronics Laboratory, MIEM HSE

Moreover, the new solution makes it possible to specify the battery charging protocol, particularly regarding the timing of interruptions coupling between the two-level systems and the electromagnetic resonator. This represents a specific moment when the accumulated energy reaches its peak. It can be determined from the solution provided by the authors of the paper.

The researchers also demonstrated that the battery's charging power grows in a superlinear fashion as a function of the number of two-level systems. For instance, doubling the number of two-level systems in the ensemble results in nearly triple charging power (2^{{3/2}} ≈ 2.83 times). This phenomenon is attributed precisely to the presence of coherent quantum effects in the battery and is referred to as supercharging. The scientists are confident that the proposed model will aid in designing quantum batteries and optimising their charging to achieve maximum energy capacity rapidly.

**IQ**

March 15

Seidali Seidov

Co-author of the paper, Research Fellow at the Quantum Nanoelectronics Laboratory, MIEM HSE