Abstract We consider a quantum battery (QB) composed of a pair of qubits coupled with an Ising interaction in the usual NMR framework, where the longitudinal applied field is constant and the time-dependent variables controlling the system are the amplitude and phase of the transverse field, and use optimal control to derive fast charging protocols. We study both the cases where the Ising coupling is weaker and stronger than the longitudinal field. In the first case, where the lowest-energy state of the system is the spin-down state, the optimal charging protocol stipulates the transverse field amplitude to be constant and equal to its maximum allowed value, while the minimum time for full charging of the battery tends to zero as this maximum bound increases. In the second case, where the lowest-energy state is a maximally entangled Bell state, the optimal charging protocol includes a time interval where the transverse field amplitude is zero and its phase is immaterial, corresponding to singular control. In this case, the QB can be charged with higher levels of stored energy, while the minimum time for full charging tends to a nonzero limit proportional to the inverse Ising interaction, as the maximum bound of the control amplitude increases. We analyze intuitively and quantitatively the distinct behavior of the two cases and additionally use the dynamical Lie algebra of the system to elucidate the presence of a singular arc in the optimal pulse-sequence in the second case. The discovered interplay between the QB parameters, the stored energy and the minimum time for full charging, provides great flexibility for optimizing the performance of the device according to the operating constraints. The valuable insights gained regarding the design of QBs is expected to find immediate applications in modern quantum science and technology, while we aim to extend the proposed methods to larger spin chains.
Abstract We consider a quantum battery (QB) composed of a pair of qubits coupled with an Ising interaction in the usual NMR framework, where the longitudinal applied field is constant and the time-dependent variables controlling the system are the amplitude and phase of the transverse field, and use optimal control to derive fast charging protocols. We study both the cases where the Ising coupling is weaker and stronger than the longitudinal field. In the first case, where the lowest-energy state of the system is the spin-down state, the optimal charging protocol stipulates the transverse field amplitude to be constant and equal to its maximum allowed value, while the minimum time for full charging of the battery tends to zero as this maximum bound increases. In the second case, where the lowest-energy state is a maximally entangled Bell state, the optimal charging protocol includes a time interval where the transverse field amplitude is zero and its phase is immaterial, corresponding to singular control. In this case, the QB can be charged with higher levels of stored energy, while the minimum time for full charging tends to a nonzero limit proportional to the inverse Ising interaction, as the maximum bound of the control amplitude increases. We analyze intuitively and quantitatively the distinct behavior of the two cases and additionally use the dynamical Lie algebra of the system to elucidate the presence of a singular arc in the optimal pulse-sequence in the second case. The discovered interplay between the QB parameters, the stored energy and the minimum time for full charging, provides great flexibility for optimizing the performance of the device according to the operating constraints. The valuable insights gained regarding the design of QBs is expected to find immediate applications in modern quantum science and technology, while we aim to extend the proposed methods to larger spin chains.
We consider the problem of maximizing the stored energy for a given charging duration in a quantum battery composed of a pair of spins-1/2 with Ising coupling starting from the spin-down state, using bounded transverse field control. We map this problem to an optimal control problem on a single qubit and by using optimal control theory we show that, although a single bang pulse can quickly achieve considerable charging levels for relatively large upper control bounds, higher levels of stored energy including complete charging are accomplished by a bang-singular-bang pulse sequence, where the intermediate singular pulse is an off pulse. If the control is restricted between zero and a maximum value, the initial and final bang pulses attain the maximum bound but have different durations, while if it is restricted between symmetric negative and positive boundaries, the bang pulses have the same duration but opposite boundary values. For both cases we provide transcendental equations from which the durations of the individual pulses in the optimal pulse-sequence can be calculated. For the case of full charging we surprisingly find that the three ``switching'' functions for the equivalent qubit problem become zero while the adjoint ket does not, consistent with optimal control theory.
We derive fast protocols for charging a three-qubit Ising spin-chain quantum battery, using amplitude and phase control of the global transverse field. For weak coupling between the qubits in the chain, where the initial ground state of the system is the spin-down state, we find that the optimal strategy is to fix the control amplitude at its maximum value while properly varying the phase with time. For the strong coupling case, where the initial ground state is the one-excitation Dicke state [Formula: see text], we apply two related charging protocols. In the first approach we use a pulse-sequence derived numerically in the recent work (Stojanović and Nauth in Phys Rev A 108:012608, 2023. https://doi.org/10.1103/PhysRevA.108.012608 ), composed of delta Bang pulses and intervals where the control is switched off. We call this method fast charging via subspace decomposition, since it involves identifying and manipulating effective two-level subsystems. Here we find analytically the characteristics of the pulses composing the sequence, which were calculated numerically in the above reference. In the second approach, we apply a similar pulse-sequence with delta Bang pulses and Off intervals, but with a carrier frequency which brings on resonance the initial [Formula: see text] state and the target spin-up state of maximum energy. We call this method charging via resonant excitation. We calculate numerically the characteristics of the pulses in the sequence and find a shorter charging time compared to the non-resonant method inspired by the above reference. We also use numerical optimal control to solve the problem of full charging the quantum battery but for finite values of the control amplitude and find that, for some relatively large values of this bound, the optimal solution has similar structure with the sequence incorporating delta Bang pulses but longer duration, as expected. Although this study focuses on a three-spin chain, the proposed methodology can be also extended to larger spin-chains. The lowest and highest energy eigenstates need to be identified first, serving as the initial and target states. Then, optimal control can be exploited to find the amplitude and phase modulation of a field resonant to this transition, which maximize the stored energy in the battery for a given duration.
An open quantum battery (QB) model of a single qubit system charging in a coherent auxiliary bath (CAB) consisting of a series of independent coherent ancillae is considered. According to the collision charging protocol we derive a quantum master equation and obtain the analytical solution of QB in a steady state. We find that the full charging capacity (or the maximal extractable work (MEW)) of QB, in the weak QB-ancilla coupling limit, is positively correlated with the coherence magnitude of ancilla. Combining with the numerical simulations we compare with the charging properties of QB at finite coupling strength, such as the MEW, average charging power and the charging efficiency, when considering the bath to be a thermal auxiliary bath (TAB) and a CAB, respectively. We find that when the QB with CAB, in the weak coupling regime, is in fully charging, both its capacity and charging efficiency can go beyond its classical counterpart, and they increase with the increase of coherence magnitude of ancilla. In addition, the MEW of QB in the regime of relative strong coupling and strong coherent magnitude shows the oscillatory behavior with the charging time increasing, and the first peak value can even be larger than the full charging MEW of QB. This also leads to a much larger average charging power than that of QB with TAB in a short-time charging process. These features suggest that with the help of quantum coherence of CAB it becomes feasible to switch the charging schemes between the long-time slow charging protocol with large capacity and high efficiency and the short-time rapid charging protocol with highly charging power only by adjusting the coupling strength of QB-ancilla. This work clearly demonstrates that the quantum coherence of bath can not only serve as the role of “fuel” of QB to be utilized to improve the QB’s charging performance but also provide an alternative way to integrate the different charging protocols into a single QB.
Quantum batteries (QBs) exploit principles of quantum mechanics to accelerate the charging process and aim to achieve optimal energy storage. However, analytical results for investigating these problems remain lacking due to the challenges associated with nonequilibrium dynamics. In this work we analytically investigate a central-spin QB model in which Nb spin-21 battery cells interact with Nc spin-21 charger units, using m initially excited charger units as a resource. By employing the invariant subspace method and the shifted Holstein-Primakoff transformation, we identify four scenarios in which optimal energy storage can be achieved: (i) ${N}_{b}\ensuremath{\ll}m\ensuremath{\ll}{N}_{c}$, (ii) $m\ensuremath{\ll}{N}_{b}\ensuremath{\ll}{N}_{c}$, (iii) $m\ensuremath{\ll}{N}_{c}\ensuremath{\ll}{N}_{b}$, and (iv) ${N}_{b}\ensuremath{\ll}m=k{N}_{c} [k\ensuremath{\in}(0,1)]$. In these cases, optimal storage is ensured by the SU(2) symmetry emerging from the charging dynamics. The first three cases map the central-spin QB to different Tavis-Cummings (TC) QBs, while the fourth corresponds to the non-TC limit. We analytically determine the charging time and demonstrate that in the fully charging cases (i) and (iv), the collective charging exhibits an Nb-fold enhancement in speedup compared to the parallel charging scheme. Additionally, we numerically observe a unified charging behavior when m=Nc, showing that asymptotically optimal energy storage is possible when Nb=m=Nc. In this case, we find a collective charging enhancement scaling as Nb0.8264. The origin of the collective charging advantage in central-spin quantum batteries is also analyzed through the quantum speed limit and a multipartite entanglement witness. Our results highlight the crucial role of dynamically emergent SU(2) symmetry in providing an analytical understanding of nonequilibrium charging dynamics in QBs.
Reliable charging protocols are crucial for advancing quantum batteries toward practical use. We investigate a transverse-field Ising chain as a quantum battery, focusing on the combined role of qubit interactions in the battery model and finite charging time. This interplay yields smoother and more controllable charging compared to sudden protocols or non-interacting batteries. Introducing stochastic noise reveals a strong dependence on the charging trajectory. Protocols that weakly excite the system gain energy under noise but lose extractable work. In contrast, protocols that strongly excite many modes show the opposite trend: noise reduces stored energy yet improves efficiency, defined as the ratio of ergotropy to stored energy. These findings demonstrate that finite-time ramps stabilize charging and highlight that noise can either hinder or enhance quantum-battery performance depending on the protocol.
Quantum batteries represent a new and promising technological application of quantum mechanics, offering the potential for enhanced energy storage and fast charging. In this work, we study a quantum battery composed of three two-level systems with XXZ coupling operating under open boundary conditions. We investigate the role played by ferromagnetic and antiferromagnetic initial configurations on the charging dynamics of the battery. Two charging mechanisms are explored: static charging, where the battery interacts with a constant classical external field, and harmonic charging, where the field oscillates periodically over time. Our results demonstrate that static charging can be more efficient in the ferromagnetic case, achieving maximum energy due to complete population inversion between the ground and excited states. In contrast, harmonic charging excels in the antiferromagnetic case. By analyzing the stored energy and the average charging power in these two regimes, we highlight the impact of anisotropy on the performance of quantum batteries. Our findings provide valuable insights for optimizing quantum battery performance based on the system's initial state and coupling configuration, paving the way for the study of more efficient quantum devices for energy storage.