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arXiv:2412.17087 · Physical Review A

Fast charging of an Ising-spin-pair quantum battery using optimal control

Vasileios Evangelakos, Emmanuel Paspalakis, Dionisis Stefanatos

Published Nov 4, 2024• 22 citations
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Abstract

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.

DOI
10.1103/physreva.110.052601

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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.

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Minimum-time generation of a uniform superposition in a qubit with only transverse field control

Vasileios Evangelakos, Emmanuel Paspalakis, Dionisis Stefanatos

We consider a two-level system with a fixed energy spacing (detuning) between the two levels and a single transverse control field which can take values between zero and a maximum amplitude. Using Pontryagin's maximum principle, we completely solve the problem of generating in minimum time a uniform superposition of the two quantum states when starting from one of them, for all the values of the ratio between the maximum control amplitude and the detuning. For each value of this ratio we find the optimal pulse sequence to have the bang-bang form and calculate the durations of the pulses composing it. The suggested framework is not only restricted to the problem at hand, but it can be also exploited in the problem of fast charging a quantum battery based on a two-level system, as well as for the optimization of pulse sequences used for the controlled preparation of the excited state in a quantum emitter, which is a prerequisite for its usage as a single-photon source.

arXiv:2312.16744•Physical Review A•Dec 26, 2023•11 citations
We consider the problem of maximizing the stored energy for a given charging duration in a quantum battery composed of a pair of spins-1/21/21/2 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.
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 study a minimal model for charging a quantum battery consisting of a two-level system (TLS) acting as a charger, coupled to a harmonic oscillator that serves as the quantum battery. A single-photon quantum pulse of light excites the TLS, which subsequently transfers its excitation to the isolated battery. The TLS may also decay into the electromagnetic environment. We obtain analytical solutions for the dynamics of the battery and determine the optimal pulse shape that maximizes the stored energy. The optimal pulse saturates a universal bound for the stored energy, determined by the TLS decay rates into the pulse and the environment. Furthermore, we derive the minimum charging time and establish a quantum speed limit at the exceptional point, where a critical transition occurs in the system's dynamics. We also present analytical expressions for the charging power and investigate the pulse duration that maximizes it.
We investigate the optimal charging processes for several models of quantum batteries, finding how to maximize the energy stored in a given battery with a finite-time modulation of a set of external fields. We approach the problem using advanced tools of optimal control theory, highlighting the universality of some features of the optimal solutions, for instance the emergence of the well-known bang-bang behavior of time-dependent external fields. The technique presented here is general, and we apply it to specific cases in which the energy is either pumped into the battery by external forces (direct charging) or transferred into it from an external charger (mediated charging). In this paper we focus on particular systems that consist of coupled qubits and harmonic oscillators, for which the optimal charging problem can be explicitly solved using a combined analytical-numerical approach based on our optimal control techniques. However, our approach can be applied to more complex setups, thus fostering the study of many-body effects in the charging process.
Abstract We study a minimal model for charging a quantum battery consisting of a two-level system (TLS) acting as a charger, coupled to a harmonic oscillator that serves as the quantum battery. A single-photon quantum pulse of light excites the TLS, which subsequently transfers its excitation to the isolated battery. The TLS may also decay into the electromagnetic environment. We obtain analytical solutions for the battery dynamics and determine the optimal pulse shape that maximizes the stored energy. The optimal pulse saturates a universal bound for the stored energy, determined by the TLS decay rates into the pulse and the environment. Furthermore, we derive the minimum charging time and establish a quantum speed limit at the exceptional point, where a critical transition occurs in the system’s dynamics. We also present analytical expressions for the charging power and investigate the pulse duration that maximizes it.
We consider a two-level system with a fixed energy spacing (detuning) between the two levels and a single transverse control field which can take values between zero and a maximum amplitude. Using Pontryagin's maximum principle, we completely solve the problem of generating in minimum time a uniform superposition of the two quantum states when starting from one of them, for all the values of the ratio between the maximum control amplitude and the detuning. For each value of this ratio we find the optimal pulse sequence to have the bang-bang form and calculate the durations of the pulses composing it. The suggested framework is not only restricted to the problem at hand, but it can be also exploited in the problem of fast charging a quantum battery based on a two-level system, as well as for the optimization of pulse sequences used for the controlled preparation of the excited state in a quantum emitter, which is a prerequisite for its usage as a single-photon source.