Fast protocols for charging a three-spin-chain quantum battery

Abstract

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 $\vert {W} \rangle$, 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 $\vert {W} \rangle$ 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.

Introduction

Quantum batteries (QBs)^1,2 have emerged as a significant research frontier in quantum technology, offering the prospect of energy storage devices that surpass their classical counterparts at microscopic scales. This potential for superior performance in energy storage and retrieval is rooted in distinctly quantum mechanical principles, including superposition and entanglement², a concept supported by extensive theoretical work^3,4,5,6,7. A broad spectrum of QB models has been put forward, from fundamental few-level systems to complex many-body constructions like spin chains, quantum oscillators, and interacting frameworks such as the Su–Schrieffer–Heeger and Sachdev–Ye–Kitaev models^{8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60}. In parallel, experimental efforts are materializing QB prototypes across diverse platforms like superconductors^39,54, quantum dots⁴⁰, organic microcavities⁴¹, and nuclear spin systems³¹.

Maximizing the energy stored in a QB hinges on the critical task of designing effective charging protocols. To this end, researchers have investigated a wide array of quantum control techniques to address this challenge⁶¹. Simple pulsed protocols have been used for two-level QBs²⁰ and harmonic oscillator QBs²⁵. Methods such as stimulated Raman adiabatic passage (STIRAP)^62,63 have been applied to charge three-level QBs in both open¹² and closed⁴² environments. Adiabatic passage has likewise been adapted for systems of non-interacting qubits¹⁵. To accelerate charging, protocols derived from shortcuts to adiabaticity⁶⁴ have been broadly applied, from simple few-level systems^43,44 to more elaborate ones like the Ising chain⁴⁵ and the Lipkin–Meshkov–Glick model⁴⁶. The quantum adiabatic brachistochrone method, a related approach, has been realized experimentally in superconducting qutrit QBs³⁹. Additionally, strong-pulse protocols have demonstrated rapid charging in many-body QBs⁴⁷, while the quantum speed limit formalism has refined the theoretical bounds on charging times^65,66. Optimal control theory^{67,68,69,70,71} has recently proven to be a particularly effective tool for determining the ideal driving fields for qubit-based QBs^49,50 and coupled oscillator systems⁵¹. Other sophisticated approaches like reinforcement learning have been used to successfully charge Dicke⁵² and cavity-Heisenberg spin-chain QBs⁵³. These developments highlight how essential advanced control strategies^72,73 are for realizing the full potential of QB technologies.

In a preceding work⁷⁴, we analyzed a QB model consisting of two spin-1/2 particles governed by an Ising interaction, with control exerted through the amplitude and phase of a transverse field in the presence of a static longitudinal field. Our findings revealed that the charging dynamics depend crucially on the relative strengths of the Ising coupling, J, and the longitudinal field, $\Omega _z$. Two distinct operational regimes were identified. For weak coupling ($J<\Omega _z$), where the initial ground state of the system is the spin-down state, the fastest charging is achieved by applying the transverse field at its maximum available strength with a properly modulated phase, allowing the minimum charging time to approach zero as this control bound increases. For strong coupling ($J>\Omega _z$), where the initial ground state is a maximally entangled Bell state, the time-optimal protocol is structurally different, incorporating a “singular arc” where the transverse control is switched off and the phase is immaterial. This feature imposes a finite, non-zero lower limit on the charging time even for large values of the maximum control amplitude, which time is inversely proportional to the Ising coupling strength.

The present work builds upon this foundation, extending our analysis to a more complex and scalable architecture: a three-qubit (N = 3) spin chain. This progression enables us to examine the scalability of the control schemes and reveal new dynamical features specific to a three-body interacting system. For weak coupling, where the initial ground state of the system is the spin down state, we find similarly to the spin-pair case that the optimal strategy is to keep the control amplitude fixed at its maximum value while properly varying with time the phase. Again, the full charging duration tends to zero as the maximum control amplitude increases. For the strong coupling case, where the initial ground state is now the one-excitation Dicke state $\vert {W} \rangle$, the heightened complexity of the three-qubit Hilbert space demands a more sophisticated control methodology. We therefore investigate two distinct charging strategies. The first approach is based on a recent work for fast Dicke state preparation through global transverse control of Ising coupled qubits⁷⁵. We call this method fast charging via subspace decomposition, since it involves identifying and manipulating effective two-level subsystems. Via this methodology, a pulse sequence of the form Bang-Off-Bang-Off-Bang accomplishing the desired state transfer is obtained in Ref.⁷⁵. Here, the term “Bang pulse” refers to a pulse equal to the maximum control amplitude. The sequence in Ref.⁷⁵ is composed of delta Bang pulses, with very high amplitude relative to the interaction and negligible duration, also known as “delta kicks” in the literature, and intervals where the transverse control is switched off and the system evolves only under the Ising interaction, the “Off” pulses. The rotation angles and phases of the Bang pulses and the durations of the Off pulses are calculated numerically in Ref.⁷⁵. Here we use this pulse sequence for the fast charging of our QB, while we also explain how the pulse characteristics of the sequence can be derived analytically. Although this method accomplishes the transfer between the $\vert {W} \rangle$ and the highest energy spin-up state, the corresponding framework arises when the carrier frequency of the applied control field is tuned to the transition from the spin-down to the spin-up state. In the second methodology we combine a control field resonant to the transition from the initial $\vert {W} \rangle$ to the target spin-up state with a Bang-Off-Bang-Off-Bang pulse-sequence inspired from Ref.⁷⁵; we call this approach charging via resonant excitation. We calculate numerically the characteristics of the pulses in the sequence and find a shorter charging time for the QB, compared to the previous non-resonant method motivated by the work in Ref.⁷⁵. Then, we use numerical optimal control to solve the problem of full charging the QB for several finite values of the maximum control amplitude. We find that, for some relatively large values of this bound, the Bang-Off-Bang-Off-Bang sequence arises as the optimal solution, where now the Bang pulses have finite height. Of course, the times needed for full charging with finite control amplitude are longer than the limit obtained with delta Bang pulses. Although the present study centers around a three-spin chain, the proposed technique can be also applied to larger spin-chains in the following way: the lowest and highest energy eigenstates are identified first, serving as the initial and target states. Then, a field resonant to this transition is used to drive the QB, and optimal control is exploited to find its amplitude and phase modulation which maximize the stored energy in the QB for a given duration.

This article is organized as follows. In “Optimal charging of an Ising N-qubit quantum battery” section is introduced the three-qubit QB model and framed the charging process as a problem of optimal control. In “Case $J<\Omega _z/2$” and “Case $J>\Omega _z/2$” sections are derived optimal charging protocols for the weak and strong Ising coupling regimes, respectively. Specifically, “Case $J>\Omega _z/2$” section provides a detailed analysis of two charging strategies: via subspace decomposition and through resonant excitation. In “Discussion” section are further discussed the obtained results while the paper concludes in “Conclusions” section.

Optimal charging of an Ising N-qubit quantum battery

Consider an Ising spin-chain with N spins-1/2 in the standard NMR framework, consisting of a constant longitudinal magnetic field $\Omega _z$ and time-dependent transverse fields $\Omega _x(t), \Omega _y(t)$ acting as the control functions, described by the Hamiltonian ($\hbar =1$)⁷⁵

$$\begin{aligned} \hat{H}(t) = \hat{H}_0+\hat{H}_1(t), \end{aligned}$$

(1)

with

$$\begin{aligned} \hat{H}_0&= \Omega _z \sum _{n=1}^{N} {\sigma _{nz}} + J \sum _{1 \le n < n' \le N} {\sigma _{nz} \sigma _{n'z}}, \end{aligned}$$

(2a)

$$\begin{aligned} \hat{H}_1(t)&= \Omega _x(t) \sum _{n=1}^{N} {\sigma _{nx}} + \Omega _y(t) \sum _{n=1}^{N} {\sigma _{ny}}, \end{aligned}$$

(2b)

where J is the Ising interaction. The system is schematically illustared in Fig. 1 for $N = 3$.

Fig. 1

Schematic illustration of the QB and the charging protocols. (a) A three-spin Ising chain with coupling J, driven by a global transverse field with bounded amplitude $0\le \Omega (t) \le \Omega _0$ and phase $\phi (t)$. (b,c) Representative charging protocol for weak and strong coupling, where note the different initial state in each case.

Charging of this quantum battery corresponds to transferring population from the initial state $\vert {\psi (0)} \rangle$, which is usually taken to be the ground state, to the excited states such that at the final time T the stored energy is maximized

$$\begin{aligned} \Delta E&= E(T) - E(0) \nonumber \\&= \langle {\psi (T)}\vert \hat{H}_0\vert {\psi (T)} \rangle - \langle {\psi (0)}\vert \hat{H}_0\vert {\psi (0)} \rangle . \end{aligned}$$

(3)

We consider transverse control fields with time-varying amplitude $\Omega (t)$ and phase $\phi (t)$,

$$\begin{aligned} \Omega _x(t)&= \Omega (t) \cos \left[ \omega _c t + \phi (t)\right] ,\end{aligned}$$

(4a)

$$\begin{aligned} \Omega _y(t)&= -\Omega (t) \sin \left[ \omega _c t + \phi (t)\right] , \end{aligned}$$

(4b)

while $\omega _c$ denotes the carrier frequency. We impose an upper bound $\Omega _0$ on the control amplitude

$$\begin{aligned} 0\le \Omega (t)\le \Omega _0, \end{aligned}$$

(5)

and under this restriction we find $\Omega (t), \phi (t)$ maximizing the stored energy (3) for a given duration T. As we increase T, the achieved stored energy also increases, until its maximum possible value is reached. We refer to this situation as full charging, and the corresponding duration is the minimum time for full charging.

In this work we concentrate on a spin-chain with $N=3$ qubits. In this case, Hamiltonian $\hat{H}_1$ couples only the states

$$\begin{aligned} \vert {\psi _0} \rangle&= \vert {000} \rangle , \end{aligned}$$

(6a)

$$\begin{aligned} \vert {\psi _1} \rangle&= \frac{1}{\sqrt{3}} \left( \vert {001} \rangle + \vert {010} \rangle + \vert {100} \rangle \right) , \end{aligned}$$

(6b)

$$\begin{aligned} \vert {\psi _2} \rangle&= \frac{1}{\sqrt{3}} \left( \vert {011} \rangle + \vert {101} \rangle + \vert {110} \rangle \right) , \end{aligned}$$

(6c)

$$\begin{aligned} \vert {\psi _3} \rangle&= \vert {111} \rangle , \end{aligned}$$

(6d)

where note that $\vert {0} \rangle =(0 \; 1)^T, \vert {1} \rangle =(1 \; 0)^T$ are the individual spin-down and spin-up states, respectively. Let

$$\begin{aligned} \vert {\Psi (t)} \rangle = c_0(t)e^{i \frac{3\omega _c }{2} t} \vert {\psi _0} \rangle + c_1(t)e^{i \frac{\omega _c }{2} t} \vert {\psi _1} \rangle + c_2(t)e^{-i \frac{\omega _c }{2} t} \vert {\psi _2} \rangle + c_3(t)e^{-i \frac{3\omega _c }{2} t} \vert {\psi _3} \rangle \end{aligned}$$

(7)

be the representation of the system state on this restricted manifold, with $c_i, i=0, 1, 2, 3$, the corresponding probability amplitudes. The Schrödinger evolution under Hamiltonian (1) leads to the system

$$\begin{aligned} i \frac{d}{dt}\vert {\psi } \rangle = \begin{bmatrix} \frac{3}{2} \left( \omega _c+2 J-2 \Omega _z\right) & \quad \sqrt{3}\Omega (t) e^{i \phi (t) } & \quad 0 & \quad 0 \\ \sqrt{3}\Omega (t) e^{-i \phi (t) } & \quad \frac{1}{2} \left( \omega _c-2 J-2 \Omega _z\right) & \quad 2 \Omega (t) e^{i \phi (t)} & \quad 0 \\ 0 & \quad 2 \Omega (t) e^{-i \phi (t)} & \quad -\frac{1}{2} \left( \omega _c + 2 J - 2 \Omega _z\right) & \quad \sqrt{3}\Omega (t) e^{i \phi (t) } \\ 0 & \quad 0 & \quad \sqrt{3}\Omega (t) e^{-i \phi (t) } & \quad -\frac{3}{2} \left( \omega _c-2 J-2 \Omega _z\right) \end{bmatrix} \vert {\psi } \rangle \nonumber \\ \end{aligned}$$

(8)

for the state vector $\vert {\psi (t)} \rangle =[c_0(t), c_1(t), c_2(t), c_3(t)]^T$, where we have also used Eq. (4).

To formulate the optimal control problem of maximizing the stored energy (3), we use Schrödinger equation (8) as state equation and denote with $\vert {\lambda (t)} \rangle =[\lambda _0(t), \lambda _1(t), \lambda _2(t), \lambda _3(t)]^T$ the corresponding adjoint ket state. By applying the optimal control formalism for quantum systems⁷⁶, we get the control Hamiltonian

$$\begin{aligned} H_c&= \Re {( \langle {\lambda | {\dot{\psi }}}\rangle )}, \nonumber \\&= \Omega (A \sin \phi + B \cos \phi ) + C, \end{aligned}$$

(9)

where $A, B \in {\mathbb {R}}$ depend on $\vert {\psi } \rangle , \vert {\lambda } \rangle$ and the off-diagonal elements of the matrix in Eq. (8), $C\in {\mathbb {R}}$ depends on $\vert {\psi } \rangle , \vert {\lambda } \rangle$ and the diagonal elements of the same matrix, while $\Re , \Im$ denote real and imaginary parts, respectively. If we define the derivative with respect to $\vert { \psi } \rangle$ as

$$\begin{aligned} \frac{\partial }{\partial \vert {\psi } \rangle } = \frac{1}{2} \left( \frac{\partial }{\partial \Re \left( \vert {\psi } \rangle \right) } - i \frac{\partial }{\partial \Im \left( \vert {\psi } \rangle \right) } \right) , \end{aligned}$$

(10)

then $\vert {\psi } \rangle , \vert {\lambda } \rangle$ satisfy the pair of Hamilton’s equation

$$\begin{aligned} \vert {{\dot{\psi }}} \rangle&= 2 \frac{\partial H_c}{\partial \langle {\lambda }\vert } = - i H \vert {\psi } \rangle , \nonumber \\ \langle {{\dot{\lambda }}}\vert&= - 2 \frac{\partial H_c}{\partial \vert {\psi } \rangle } = i \langle {\lambda }\vert H, \end{aligned}$$

(11)

where H denotes the matrix in Eq. (8). Note that the first equation is actually Schrödinger Eq. (8) as expected, while the second one leads to a similar equation for $\langle {\lambda }\vert$. The adjoint variables should also satisfy the terminal condition

$$\begin{aligned} \langle {\lambda (T)}\vert = 2 \frac{\partial (\Delta E(T))}{\partial \vert {\psi (T)} \rangle }, \end{aligned}$$

(12)

which connects the final adjoint state with the final state $\vert {\psi (T)} \rangle$ because of Eq. (3), defining thus a two-point boundary value problem.

Pontryagin’s maximum principle^67,68 states that the optimal controls $\Omega ^*(t), \phi ^*(t)$ should be chosen to maximize the control Hamiltonian (9). Given that $\phi$ is unrestricted, for $\Omega (t)\ne 0$ it is

$$\begin{aligned} \frac{\partial H_c}{\partial \phi }\bigg |_{\phi =\phi ^*} = 0 \Rightarrow \tan {\phi ^*}=\frac{A}{B}, \end{aligned}$$

(13)

which leads to

$$\begin{aligned} H_c = \Omega \sqrt{A^2+B^2}+ C. \end{aligned}$$

(14)

Observe from Eq. (14) that if either $A\ne 0$ or $B\ne 0$ then the optimal control amplitude maximizing $H_c$ is $\Omega ^*(t)=\Omega _0$. When $A=B=0$ for a finite duration then maximum principle provides a priori no information about the optimal $\Omega$. In this situation $\Omega ^*$ can take any value in the interval $[0, \Omega _0]$ and the corresponding optimal control is called singular. Special consideration require the time intervals for which $\Omega ^*(t)=0$, where the phase is immaterial. As we will see below, for the problem at hand numerical optimization indicates that the optimal amplitude take the boundary values 0 and $\Omega _0$, at least for the examples considered here. We use the numerical optimal control solver of Ref.⁷⁷, which does not need to formulate the two-point boundary value problem from scratch, just code the dynamics correctly and define the cost function to be maximized, here the stored energy. The program terminates when an optimal solution is found, according to its termination criteria.

From Eq. (3) we see that the stored energy also depends on the initial state $\vert {\psi (0)} \rangle$, usually taken to be the ground state. From Eq. (2a) for the QB Hamiltonian $\hat{H}_0$ we easily find that the energies of the states $\vert {\psi _i} \rangle$, $i = 0, 1, 2, 3$, are $E_0 = 3(J - \Omega _z)$, $E_1 = -(J+\Omega _z)$, $E_2 = \Omega _z - J$, $E_3 = 3(J + \Omega _z)$. Thus which one is the ground state of the battery depends on the ratio

$$\begin{aligned} \chi = \frac{J}{\Omega _z}. \end{aligned}$$

(15)

Based on this observation we need to distinguish two cases. Representative charging protocols for both cases are schematically illustrated in Fig. 1.

Case $J<\Omega _z/2$

In this case, the ground state is the spin-down state $\vert {\psi _0} \rangle$. Using it as the initial state $\vert {\psi (0)} \rangle =\vert {\psi _0} \rangle$ in Eq. (3), we obtain the following expression for the stored energy in terms of the final populations of the states with at least one excitation

$$\begin{aligned} \frac{\Delta E}{\Omega _z}= 2 (1 - 2 \chi ) \left| c_1(T)\right| ^2 +4 (1-\chi ) \left| c_2(T)\right| ^2 + 6 \left| c_3(T)\right| ^2, \end{aligned}$$

(16)

where $\chi =J/\Omega _z<1/2$ and for the closed system under study we have used $\left| c_0(T)\right| ^2$ = $1 - \left| c_1(T)\right| ^2$ − $\left| c_2(T)\right| ^2 - \left| c_3(T)\right| ^2$. The maximum stored energy is $\Delta E_{max}=6\Omega _z$.

We would like starting from state $\vert {\psi _0} \rangle$ to maximize the stored energy and since the spin-up state $\vert {\psi _3} \rangle$ has the highest energy we choose the carrier frequency in Eq. (8) as $\omega _c= 2 \Omega _z$ to bring them on resonance, ending up with the equation

$$\begin{aligned} i \frac{d\vert {\psi } \rangle }{dt} = \begin{bmatrix} 3J & \quad \sqrt{3} \Omega (t) e^{i \phi } & \quad 0 & \quad 0 \\ \sqrt{3} \Omega (t) e^{-i \phi } & \quad -J & \quad 2 \Omega (t) e^{i \phi } & \quad 0 \\ 0 & \quad 2 \Omega (t) e^{-i \phi } & \quad -J & \quad \sqrt{3} \Omega (t) e^{i \phi } \\ 0 & \quad 0 & \quad \sqrt{3} \Omega (t) e^{-i \phi } & \quad 3J \\ \end{bmatrix} \vert {\psi } \rangle . \end{aligned}$$

(17)

We use numerical optimal control to solve the problem of fully charging a quantum battery with ratio $\chi =J/\Omega _z=1/4$, for three values of the maximum control amplitude $\Omega _0/\Omega _z = 1, 1.5, 5$. The results are displayed in Figs. 2, 3 and 4, respectively, where for each case we plot the optimal control amplitude and phase, as well as the time evolution of charging energy and state populations. Observe that for all cases the optimal control amplitude is constant throughout the whole time interval. Further numerical simulations support that this is a general characteristic of the optimal control for the case under study, independent of the maximum control amplitude $\Omega _0$. A similar behavior has also been observed in the time-optimal control of two- and three-qubit gates for Rydberg atoms⁷⁸. The optimal phase on the other hand is time-dependent, with a steeper variation for the smaller control amplitudes $\Omega _0 = \Omega _z$ and $\Omega _0 = 1.5\Omega _z$, as shown in Figs. 2b and 3b, and smoother evolution for the larger control amplitude $\Omega _0 = 5\Omega _z$, as displayed in Fig. 4b. As a consequence, the evolution of stored energy and populations is more wavy in the former cases than in the latter.

In Fig. 5a we plot the minimum necessary time for full charging (black curve) versus the maximum control amplitude $\Omega _0$. We observe that this duration approaches zero as $\Omega _0$ increases. This behavior can be understood by considering a delta pulse with amplitude $\Omega _0\rightarrow \infty$, duration $T\rightarrow 0$ and constant phase $\phi$. Then, the J-term in Eq. (17) can be neglected and we get the solution

$$\begin{aligned} c_0(T)&= \cos ^3\left( \Omega _0 T\right) , \end{aligned}$$

(18a)

$$\begin{aligned} c_1(T)&= -i \sqrt{3} e^{-i \phi } \sin \left( \Omega _0 T\right) \cos ^2\left( \Omega _0 T\right) , \end{aligned}$$

(18b)

$$\begin{aligned} c_2(T)&= -\sqrt{3} e^{-2 i \phi } \sin ^2\left( \Omega _0 T\right) \cos \left( \Omega _0 T\right) , \end{aligned}$$

(18c)

$$\begin{aligned} c_3(T)&= i e^{-3 i \phi } \sin ^3\left( \Omega _0 T\right) . \end{aligned}$$

(18d)

If the pulse area is selected as $\Omega _0T=\pi /2$, then $|c_3(T)|=1$ and the battery is fully charged within $T\rightarrow 0$. In Fig. 5b we also plot the average charging power versus $\Omega _0$ (black curve).

Fig. 2

Optimal control amplitude (a) and phase (b), along with the corresponding time evolution of stored energy (c) and populations ($|c_0|^2$ black, $|c_1|^2$ red, $|c_2|^2$ green, $|c_3|^2$ blue) (d) for full charging in minimum time, when $\Omega _0/\Omega _z=1$ and $\chi =J/\Omega _z=1/4$.

Fig. 3

Optimal control amplitude (a) and phase (b), along with the corresponding time evolution of stored energy (c) and populations ($|c_0|^2$ black, $|c_1|^2$ red, $|c_2|^2$ green, $|c_3|^2$ blue) (d) for full charging in minimum time, when $\Omega _0/\Omega _z=1.5$ and $\chi =J/\Omega _z=1/4$.

Fig. 4

Optimal control amplitude (a) and phase (b), along with the corresponding time evolution of stored energy (c) and populations ($|c_0|^2$ black, $|c_1|^2$ red, $|c_2|^2$ green, $|c_3|^2$ blue) (d) for full charging in minimum time, when $\Omega _0/\Omega _z=5$ and $\chi =J/\Omega _z=1/4$.

Fig. 5

Minimum duration for full charging (a) and achieved charging power $\Delta E_{max}/T$ (b) as functions of the maximum control amplitude $\Omega _0$, for the case $J=\Omega _z/4<\Omega _z/2$ (black) and the case $J=\Omega _z>\Omega _z/2$ (red).

Case $J>\Omega _z/2$

In this case, the ground state is $\vert {\psi _1} \rangle$. Using it as the initial state $\vert {\psi (0)} \rangle =\vert {\psi _1} \rangle$ in (3), we obtain the following expression for the stored energy in terms of the final populations of the states with at least one excitation

$$\begin{aligned} \frac{\Delta E}{\Omega _z}=4 \chi - 2 + 2(1 - 2 \chi ) \left| c_1(T)\right| ^2 + 4(1 - \chi ) \left| c_2(T)\right| ^2 + 6 \left| c_3(T)\right| ^2, \end{aligned}$$

(19)

where $\chi =J/\Omega _z>1/2$ and for the closed system under study we have used $\left| c_0(T)\right| ^2$ = $1 - \left| c_1(T)\right| ^2$ − $\left| c_2(T)\right| ^2 - \left| c_3(T)\right| ^2$. The maximum stored energy is $\Delta E_{max}=4(1+\chi ) \Omega _z > 6\Omega _z$, since $\chi > 1/2$. Thus, the maximum stored energy achieved in the strong coupling case, where the system starts from the entangled state $\vert {\psi _1} \rangle$, is higher than that obtained for weak coupling, where the starting state is the spin-down state $\vert {\psi _0} \rangle$. This is obviously a quantum advantage of this type of QB.

The maximization of stored energy is achieved by transferring the total population of the system to the highest energy eigenstate, which is the three-excitation state $\vert {\psi _3} \rangle$. In “Fast charging via subspace decomposition” section we adopt an analytical approach, inspired by the work of Stojanović et al.⁷⁵, which leverages the underlying symmetries of system (17) to construct a fast and efficient charging protocol. Note that Eq. (17) is obtained by applying fields with carrier frequency $\omega _c= 2 \Omega _z$, which brings on resonance states $\vert {\psi _0} \rangle , \vert {\psi _3} \rangle$. In “Charging via resonant excitation” section we obtain a faster pulse-sequence by using a carrier frequency which brings on resonance the current initial state $\vert {\psi _1} \rangle$ with the target state $\vert {\psi _3} \rangle$.

Fast charging via subspace decomposition

We present a method to accomplish the transfer $\vert {\psi _1} \rangle \rightarrow \vert {\psi _3} \rangle$ in system (17), which is based on the work of Stojanović et al.⁷⁵. The foundation of this method lies in the decomposition of the 4D subspace, spanned by the basis of excitation number states $\{\vert {\psi _0} \rangle , \vert {\psi _1} \rangle , \vert {\psi _2} \rangle , \vert {\psi _3} \rangle \}$, into smaller, independent subspaces where the dynamics of Eq. (17) are simpler to analyze. This decomposition is possible because the Ising interaction, expressed by the diagonal part of the matrix in Eq. (17), as well as the x-component of the applied field, expressed by the real part of the off-diagonal elements in the same equation, both commute with the x-parity operator, $\hat{X}_p = {\hat{\sigma }}_x \otimes {\hat{\sigma }}_x \otimes {\hat{\sigma }}_x$. This operator has two eigenvalues, $+1$ and $-1$, and we can define a parity basis, $\{\vert {v_1} \rangle , \vert {v_2} \rangle , \vert {v_3} \rangle , \vert {v_4} \rangle \}$, where vectors $\vert {v_1} \rangle , \vert {v_2} \rangle$ span the $+1$ eigensubspace and $\vert {v_3} \rangle , \vert {v_4} \rangle$ span the $-1$ eigensubspace, which is related to the excitation basis by:

$$\begin{aligned} \vert {v_1} \rangle&= \frac{1}{\sqrt{2}}(\vert {\psi _0} \rangle + \vert {\psi _3} \rangle ) \qquad (+1 \text { subspace}), \end{aligned}$$

(20a)

$$\begin{aligned} \vert {v_2} \rangle&= \frac{1}{\sqrt{2}}(\vert {\psi _1} \rangle + \vert {\psi _2} \rangle ) \qquad (+1 \text { subspace}), \end{aligned}$$

(20b)

$$\begin{aligned} \vert {v_3} \rangle&= \frac{1}{\sqrt{2}}(\vert {\psi _0} \rangle - \vert {\psi _3} \rangle ) \qquad (-1 \text { subspace}), \end{aligned}$$

(20c)

$$\begin{aligned} \vert {v_4} \rangle&= \frac{1}{\sqrt{2}}(\vert {\psi _1} \rangle - \vert {\psi _2} \rangle ) \qquad (-1 \text { subspace}). \end{aligned}$$

(20d)

In this parity basis, evolution (17) is described by the matrix

$$\begin{aligned} H = H_0+H_1, \end{aligned}$$

(21)

where

$$\begin{aligned} H_0 = \begin{bmatrix} 3 J & \quad 0 & \quad 0 & \quad 0 \\ 0 & \quad -J & \quad 0 & \quad 0 \\ 0 & \quad 0 & \quad 3 J & \quad 0 \\ 0 & \quad 0 & \quad 0 & \quad -J \\ \end{bmatrix} \end{aligned}$$

(22)

and

$$\begin{aligned} H_1 = \begin{bmatrix} 0 & \quad \sqrt{3} \Omega (t) \cos (\phi ) & \quad 0 & \quad -i \sqrt{3} \Omega (t) \sin (\phi ) \\ \sqrt{3} \Omega (t) \cos (\phi ) & \quad 2 \Omega (t) \cos (\phi ) & \quad i \sqrt{3} \Omega (t) \sin (\phi ) & \quad 2 i \Omega (t) \sin (\phi ) \\ 0 & \quad -i \sqrt{3} \Omega (t) \sin (\phi ) & \quad 0 & \quad \sqrt{3} \Omega (t) \cos (\phi ) \\ i \sqrt{3} \Omega (t) \sin (\phi ) & \quad -2 i \Omega (t) \sin (\phi ) & \quad \sqrt{3} \Omega (t) \cos (\phi ) & \quad -2 \Omega (t) \cos (\phi ) \\ \end{bmatrix}. \end{aligned}$$

(23)

Observe that the Ising part $H_0$ is diagonal, thus it does not couple the two subspaces, while $H_1$ becomes block-diagonal when $\sin {\phi }=0 \rightarrow \phi =0$ or $\pi$, i.e. the field is along the x-axis:

$$\begin{aligned} H_{\pm 1x} = \pm \begin{bmatrix} 0 & \quad \sqrt{3} \Omega (t) & \quad 0 & \quad 0 \\ \sqrt{3} \Omega (t) & \quad 2 \Omega (t) & \quad 0 & \quad 0 \\ 0 & \quad 0 & \quad 0 & \quad \sqrt{3} \Omega (t) \\ 0 & \quad 0 & \quad \sqrt{3} \Omega (t) & \quad -2 \Omega (t) \\ \end{bmatrix}, \end{aligned}$$

(24)

with the ± sign corresponding to the two opposite phases. A state prepared in one of the subspaces will remain within that subspace for all time under the evolution of operators in Eqs. (22) and (24). This decoupling is the key to simplifying the control problem and forms the core geometric justification for the approach: by confining the evolution to a lower-dimensional invariant subspace, the control path becomes significantly less complex.

The proposed control protocol is structured as a sequence of extremely short, high-amplitude Bang pulses, approximated as Dirac delta-functions, separated by periods of free evolution (Off pulses). The Bangs are global rotational pulses generated by a strong control field $\Omega (t) = \Omega _0 \gg J$. Their duration is negligible and their function is to rotate the state vector instantaneously. The corresponding propagator is

$$\begin{aligned} \hat{U}_{\text {bang}}&= e^{-i H_1 \tau } \nonumber \\&= \begin{bmatrix} \cos ^3\theta & \quad -i \sqrt{3} e^{i \phi } \sin \theta \cos ^2\theta & \quad -\sqrt{3} e^{2 i \phi } \sin ^2\theta \cos \theta & \quad i e^{3 i \phi } \sin ^3\theta \\ -i \sqrt{3} e^{-i \phi } \sin \theta \cos ^2\theta & \quad \frac{1}{4} (\cos \theta +3 \cos 3 \theta ) & \quad \frac{1}{4} i e^{i \phi } (\sin \theta -3 \sin 3 \theta ) & \quad -\sqrt{3} e^{2 i \phi } \sin ^2\theta \cos \theta \\ -\sqrt{3} e^{-2 i \phi } \sin ^2\theta \cos \theta & \quad \frac{1}{4} i e^{-i \phi } (\sin \theta -3 \sin 3 \theta ) & \quad \frac{1}{4} (\cos \theta +3 \cos 3 \theta ) & \quad -i \sqrt{3} e^{i \phi } \sin \theta \cos ^2\theta \\ i e^{-3 i \phi } \sin ^3\theta & \quad -\sqrt{3} e^{-2 i \phi } \sin ^2\theta \cos \theta & \quad -i \sqrt{3} e^{-i \phi } \sin \theta \cos ^2\theta & \quad \cos ^3\theta \\ \end{bmatrix}, \end{aligned}$$

(25)

where the rotation angle $\theta = \Omega _0 \tau$ is finite and the phase $\phi$ is constant over the Bang “duration” $\tau \rightarrow 0$.

In the parity basis, the initial state $\vert {\psi _1} \rangle$ and the target state $\vert {\psi _3} \rangle$ are superpositions of vectors from both subspaces, as derived from Eq. (20):

$$\begin{aligned} \vert {\psi _1} \rangle&= \frac{1}{\sqrt{2}}(\vert {v_2} \rangle + \vert {v_4} \rangle ), \end{aligned}$$

(26)

$$\begin{aligned} \vert {\psi _3} \rangle&= \frac{1}{\sqrt{2}}(\vert {v_1} \rangle - \vert {v_3} \rangle ). \end{aligned}$$

(27)

To accomplish the transfer $\vert {\psi _1} \rangle \rightarrow \vert {\psi _3} \rangle$, a five-step pulse-sequence of the form Bang-Off-Bang-Off-Bang is used in Ref.⁷⁵. The corresponding path traveled can be expressed as $\vert {\psi _1} \rangle$ $\xrightarrow {\text {bang}}$ $\vert {A} \rangle$ $\xrightarrow {\text {off}} \vert {B} \rangle$ $\xrightarrow {\text {bang}} \vert {C} \rangle \xrightarrow {\text {off}} \vert {D} \rangle \xrightarrow {\text {bang}} \vert {\psi _3} \rangle$ , where states $\vert {A} \rangle , \vert {B} \rangle , \vert {C} \rangle , \vert {D} \rangle$ belong to one of the parity subspaces.

The first Bang rotates the initial state $\vert {\psi _1} \rangle$ so that it lies entirely within one of the invariant subspaces; here we choose the $(-1)$ subspace. If $\theta _1, \phi _1$ denote the rotation angle and phase of the Bang, respectively, the requirement that the rotated state $\hat{U}_{\text {bang}}\vert {\psi _1} \rangle$ has zero projection in the $(+1)$ subspace gives the conditions

$$\begin{aligned} \left\langle {v_1\left|\hat{U}_{\text {bang}}\right|\psi _1}\right\rangle &= e^{-i \phi _1} \sin \theta _1 \cos \theta _1 \left( e^{3 i \phi _1} \cos \theta _1-i \sin \theta _1\right) = 0, \end{aligned}$$

(28a)

$$\begin{aligned} \left\langle {v_2\left|\hat{U}_{\text {bang}}\right|\psi _1}\right\rangle&= \left( \sin \theta _1-3 \sin 3 \theta _1\right) \left( \sin \phi _1+i \cos \phi _1\right) +\cos \theta _1+3 \cos 3 \theta _1 = 0. \end{aligned}$$

(28b)

By processing the real and imaginary parts of the above relations it is not hard to end up with the solution $\theta _1=3\pi /4$ and $\phi _1=\pi /2$. The resulting state expressed in the parity basis is

$$\begin{aligned} \vert {A} \rangle = \frac{1}{2}\left( 0, 0, \sqrt{3}, 1\right) ^T. \end{aligned}$$

(29)

In analogy with the first Bang, the last Bang rotates the ultimate state within the $(-1)$ subspace to the final target state $\vert {\psi _3} \rangle$. We derive the corresponding conditions by taking the inverse rotation from the target state back to the $(-1)$ subspace. Similarly to the previous case we get

$$\begin{aligned} \left\langle {v_1\left|U_{\text {bang}}^{\dagger }\right|\psi _3}\right\rangle&= \cos ^3\theta _3-i e^{3 i \phi _3} \sin ^3\theta _3 = 0, \end{aligned}$$

(30a)

$$\begin{aligned} \left\langle {v_2\left|U_{\text {bang}}^{\dagger }\right|\psi _3}\right\rangle&= e^{i \phi _3} \sin \theta _3 \cos \theta _3 \left( e^{i \phi _3} \sin \theta _3-i \cos \theta _3\right) = 0. \end{aligned}$$

(30b)

The corresponding solution is $\theta _3=\pi /4$ and $\phi _3=\pi /2$, while the state just before the Bang is

$$\begin{aligned} \vert {D} \rangle = \frac{1}{2}\left( 0,0,-1,\sqrt{3}\right) ^T. \end{aligned}$$

(31)

Fig. 6

Minimum time trajectory on the two-level $(-1)$ subspace visualized on the corresponding Bloch sphere.

During the intermediate evolution from state $\vert {A} \rangle$ after the first Bang to state $\vert {D} \rangle$ before the last Bang the system remains within the $(-1)$ subspace. In this two-dimensional subspace the above states can be expressed as

$$\begin{aligned} \vert {A_s} \rangle = \frac{1}{2} \begin{pmatrix} \sqrt{3} \\ 1 \end{pmatrix}, \qquad \vert {D_s} \rangle = \frac{1}{2} \begin{pmatrix} -1 \\ \sqrt{3} \end{pmatrix}, \end{aligned}$$

(32)

while matrices (22) and (24) as

$$\begin{aligned} \hat{H}_0^{(s)} = \begin{bmatrix} 3 J & \quad 0 \\ 0 & \quad -J \\ \end{bmatrix} = J(I + 2\sigma _z), \qquad \hat{H}_{\pm 1x}^{(s)} = \pm \begin{bmatrix} 0 & \quad \sqrt{3} \Omega \\ \sqrt{3} \Omega & \quad -2 \Omega \end{bmatrix} = \pm \Omega (-I + 2\hat{n}\cdot {\hat{\sigma }}), \end{aligned}$$

(33)

where

$$\begin{aligned} \hat{n} = n_x \hat{x}+n_z \hat{z} = \frac{1}{2}(\sqrt{3}\hat{x}+\hat{z}) \end{aligned}$$

(34)

and the ± sign corresponds to the field phase $\phi =0$ or $\pi$. State $\vert {A_s} \rangle$ is transferred to state $\vert {D_s} \rangle$ by the intermediate part Off-Bang-Off of the pulse-sequence, thus

$$\begin{aligned} \vert {D_s} \rangle = \hat{U}_{\text {2,off}} \hat{U}_{\text {2,bang}} \hat{U}_{\text {1,off}}\vert {A_s} \rangle . \end{aligned}$$

(35)

Let $\xi _{1, 2} = Jt_{1,2}$ denote the rotation angles corresponding to the first and second Off pulses with duration $t_{1,2}$, respectively, and $\theta _2 = \lim _{\tau \rightarrow 0}(\Omega _0\tau )$ the rotation angle corresponding to the middle Bang pulse. For the phase of the Bang pulse, among the possible choices $0, \pi$ which ensure that the state remains within the subspace we select $\phi _2=\pi$ so the matrix $\hat{H}_{-1x}^{(s)}$ is applied and as we will see leads to a positive angle $\theta _2$. Parameters $\xi _{1, 2}, \theta _2$ can be found by using Eq. (35). To simplify the calculations we exploit the observation that states $\vert {A_s} \rangle , \vert {D_s} \rangle$ correspond to antipodal points on the Bloch sphere, see Fig. 6. We can use the diameter connecting these points as the new $z'$ axis of the rotated coordinate system displayed in Fig. 6. In this new frame, the rotation axes corresponding to matrices $\hat{H}_0^{(s)}$ and $\hat{H}_{\pm 1x}^{(s)}$ become

$$\begin{aligned} \hat{z} = \hat{n}' = -n_x \hat{x}'+n_z \hat{z}'=\frac{1}{2}(-\sqrt{3}\hat{x}'+\hat{z}'), \quad \pm \hat{n} = \pm \hat{z}', \end{aligned}$$

(36)

while the initial and final states within the subspace

$$\begin{aligned} \vert {A_s} \rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \qquad \vert {D_s} \rangle = e^{i\gamma } \begin{pmatrix} 0 \\ 1 \end{pmatrix}, \end{aligned}$$

(37)

with $\gamma$ being an overall phase accounting for the effect of identity matrices in Eq. (33). Condition (35) can be expressed as

$$\begin{aligned} e^{i2\xi _2\hat{n}'\cdot {\hat{\sigma }}}\vert {D_s} \rangle = e^{i2\theta _2\hat{z}'\cdot {\hat{\sigma }}} e^{-i2\xi _1\hat{n}'\cdot {\hat{\sigma }}}\vert {A_s} \rangle , \end{aligned}$$

(38)

where note that the effect of the second Off pulse is applied to state $\vert {D_s} \rangle$ to get a simpler relation. From Eq. (38) we obtain

$$\begin{aligned} -in_x\sin 2\xi _2 \, e^{i\gamma }&= (\cos 2\xi _1-in_z\sin 2\xi _1)\,e^{i2\theta _2}, \end{aligned}$$

(39a)

$$\begin{aligned} (\cos 2\xi _2-in_z\sin 2\xi _2)\,e^{i\gamma }&= in_x\sin 2\xi _1 \,e^{-i2\theta _2}. \end{aligned}$$

(39b)

By taking the absolute square of both sides of any of the above equations we get

$$\begin{aligned} \sin ^22\xi _2 = \frac{n_z^2}{n_x^2}+\cos ^22\xi _1. \end{aligned}$$

(40)

The total duration of the pulse sequence is determined by the durations of the Off pulses, since the Bang pulses are instantaneous, thus $T=t_1+t_2= (\xi _1+\xi _2)/J$. To minimize T we take the derivative with respect to $\xi _1$ and find the condition

$$\begin{aligned} \frac{dT}{d\xi _1} = 0 \Rightarrow \frac{d\xi _2}{d\xi _1} = -1. \end{aligned}$$

(41)

By taking the derivative of Eq. (40) with respect to $\xi _1$ we easily find

$$\begin{aligned} \frac{d\xi _2}{d\xi _1} = - \frac{\sin 2\xi _1\cos 2\xi _1}{\sin 2\xi _2\cos 2\xi _2} = -\frac{\sin 4 \xi _1}{\sin 4 \xi _2}, \end{aligned}$$

(42)

and if we plug it in Eq. (41) we get the condition

$$\begin{aligned} \sin 4 \xi _1 = \sin 4 \xi _2. \end{aligned}$$

(43)

The acceptable solution of Eq. (43) is $\xi _2=\xi _1$, since the other solution $\xi _2 = \pi /4 - \xi _1$ leads to a false equality when plugged in Eq. (40). Using the equality of $\xi _i$ in Eq. (40) we find

$$\begin{aligned} \cos 4\xi _1 = \cos ^22\xi _1-\sin ^22\xi _1 = -\frac{n_z^2}{n_x^2} = -\frac{1}{3}, \end{aligned}$$

(44)

thus

$$\begin{aligned} \xi _1 = \xi _2 = \frac{1}{4}\left[ \pi -\arccos \left( \frac{1}{3}\right) \right] . \end{aligned}$$

(45)

To find angle $\theta _2$, we multiply Eq. (39a) with the complex conjugate of Eq. (39b) to get

$$\begin{aligned} e^{i4\theta _2}&= \frac{\cos 2\xi _2+in_z\sin 2\xi _2}{\cos 2\xi _1-in_z\sin 2\xi _1} \nonumber \\&= \frac{1}{3}(1+i2\sqrt{2}), \end{aligned}$$

(46)

which gives

$$\begin{aligned} \theta _2 = \frac{1}{4}\arccos {\left( \frac{1}{3}\right) }. \end{aligned}$$

(47)

We have calculated analytically the optimal pulse sequence which was derived numerically in Ref.⁷⁵, where note that $\theta _2$ is the rotation angle around the $-z'$ axis. Within the reduced subspace we found the minimum time trajectory displayed in Fig. 6, while the initial and final Bang pulses serve to instantaneously enter and exit this subspace. The corresponding total duration is

$$\begin{aligned} T = \frac{\xi _1+\xi _2}{J} = \frac{1}{2J}\left[ \pi -\arccos \left( \frac{1}{3}\right) \right] \approx \frac{0.9553}{J}. \end{aligned}$$

(48)

Note that this time limit has been derived using delta Bang pulses.

Charging via resonant excitation

To implement more efficiently the transfer $\vert {\psi _1} \rangle \rightarrow \vert {\psi _3} \rangle$, we choose in Eq. (8) the carrier frequency as $\omega _c = 2 (J +\Omega _z)$, which brings the initial and target states on resonance, ending up with the equation

$$\begin{aligned} i \frac{d}{dt}\vert {\psi } \rangle = \begin{bmatrix} 6 J & \quad \sqrt{3} \Omega (t) e^{i \phi } & \quad 0 & \quad 0 \\ \sqrt{3} \Omega (t) e^{-i \phi } & \quad 0 & \quad 2 \Omega (t) e^{i \phi } & \quad 0 \\ 0 & \quad 2 \Omega (t) e^{-i \phi } & \quad -2 J & \quad \sqrt{3} \Omega (t) e^{i \phi } \\ 0 & \quad 0 & \quad \sqrt{3} \Omega (t) e^{-i \phi } & \quad 0 \\ \end{bmatrix} \vert {\psi } \rangle . \end{aligned}$$

(49)

We first show that, contrary to the case studied in “Case $J<\Omega _z/2$” section, a single delta Bang pulse cannot achieve the desired transfer instantaneously. We consider such a pulse with amplitude $\Omega _0\rightarrow \infty$ and duration $T\rightarrow 0$ so the rotation angle $\theta = \Omega _0 T$ is finite, and constant phase $\phi$. Then, the J-term in Eq. (49) can be neglected and we get the solution

$$\begin{aligned} c_0(T)&= -i \sqrt{3} e^{i \phi } \sin \theta \cos ^2\theta , \end{aligned}$$

(50a)

$$\begin{aligned} c_1(T)&= \frac{1}{4} (\cos \theta +3 \cos 3 \theta ), \end{aligned}$$

(50b)

$$\begin{aligned} c_2(T)&= \frac{1}{4} i e^{-i \phi } (\sin \theta -3 \sin 3 \theta ), \end{aligned}$$

(50c)

$$\begin{aligned} c_3(T)&= -\sqrt{3} e^{-2 i \phi } \sin ^2\theta \cos \theta . \end{aligned}$$

(50d)

From these expressions it is clear that whenever $c_0(T) = 0$, we necessarily also have $c_3(T) = 0$, thus a complete population transfer to state $\vert {\psi _3} \rangle$ cannot be achieved with this method.

The next thought is to consider a Bang-Off-Bang sequence with delta Bangs; we show that this approach does not work either. Propagating the initial state forward by Bang-Off and the final state backward by the terminal Bang and demanding the equality of the resultant states,

$$\begin{aligned} \hat{U}^{\dagger }_{\text {2,bang}}\vert {\psi _3} \rangle = \hat{U}_{\text {off}} \hat{U}_{\text {1,bang}}\vert {\psi _1} \rangle , \end{aligned}$$

(51)

we get the system of equations

$$\begin{aligned} \sqrt{3} \sin \theta _1 \cos ^2\theta _1 e^{i \phi _1-6 i \xi }&= \sin ^3\theta _2 e^{3 i \phi _2}, \end{aligned}$$

(52a)

$$\begin{aligned} \frac{1}{4} (\cos \theta _1+3 \cos 3 \theta _1)&= -\sqrt{3} \sin ^2\theta _2 \cos \theta _2 e^{2 i \phi _2}, \end{aligned}$$

(52b)

$$\begin{aligned} \frac{1}{4} (\sin \theta _1-3 \sin 3 \theta _1) e^{2 i \xi -i \phi _1}&= \sqrt{3} \sin \theta _2 \cos ^2\theta _2 e^{i \phi _2}, \end{aligned}$$

(52c)

$$\begin{aligned} -\sqrt{3} \sin ^2\theta _1 \cos \theta _1 e^{-2 i \phi _1}&= \cos ^3\theta _2, \end{aligned}$$

(52d)

where $\theta _1, \theta _2$ and $\phi _1, \phi _2$ are the rotation angles and phases of the two Bangs and $\xi =JT$ is the normalized duration of the Off pulse. If we multiply Eqs. (52a) and (52d) and divide the result with the product of Eqs. (52b) and (52c) then we obtain, after some simplifications, the following equation

$$\begin{aligned} 1-\frac{16}{9 \cos 4 \theta _1+7}=e^{8 i \xi }. \end{aligned}$$

(53)

Now observe that, since one side of Eq. (52b) is real, it is necessarily $e^{2 i \phi _2} = \pm 1$ and similarly $e^{2 i \phi _1} = \pm 1$ from Eq. (52d) and $e^{i (2\xi -\phi _1-\phi _2)} = \pm 1$ from Eq. (52c). From the first two relations we get $e^{4 i \phi _1} = e^{4 i \phi _2} = 1$ and from the last $e^{i (8\xi -4\phi _1-4\phi _2)} = 1$, which give $e^{i8\xi }=1$ when combined. As is evident from Eq. (53), this would require the fractional term to be zero. Since the numerator is a non-zero constant, this yields a mathematical impossibility. Therefore, the Bang-Off-Bang sequence cannot accomplish the desired transfer.

Inspired by the previous work⁷⁵, we try next a Bang-Off-Bang-Off-Bang sequence. Since Eq. (49) does not possess the symmetry of Eq. (17), here we cannot apply the procedure of “Fast charging via subspace decomposition” section to derive the pulse parameters analytically. We thus calculate them using numerical optimization. Let $\theta _i$ and $\phi _i$, $i = 1, 2, 3$, denote the rotation angles and phases of the three delta Bang pulses and $\xi _{1, 2}$ the normalized durations of the Off pulses. We propagate the initial state using this sequence and ask to find the pulse parameters which minimize the total duration $T=(\xi _1+\xi _2)/J$ while achieving full charging of the quantum battery, $|c_3(T)|=1$. The corresponding optimal parameters obtained from the numerical solution are presented in Table 1.

Table 1 Numerically obtained parameters for the Bang-Off-Bang-Off-Bang sequence yielding the minimum duration for full charging $T \approx 0.7774$ J⁻¹.

The corresponding total duration is

$$\begin{aligned} T = \frac{\xi _1+\xi _2}{J} \approx \frac{0.7774}{J}. \end{aligned}$$

(54)

Note that this duration, obtained with resonant excitation between the initial and target states, is shorter than the one previously found in Eq. (48), as anticipated. The duration for full charging given in Eq. (54) is obtained using delta Bang pulses, thus imposes a lower limit on the necessary time when using the same sequence but with Bang pulses of finite amplitude.

We next use numerical optimal control to solve the problem of full charging a quantum battery with ratio $\chi =J/\Omega _z=1$, but for finite maximum control amplitude, using the same upper bound values as in the case $J < \Omega _z/2$, i.e. $\Omega _0/\Omega _z = 1, 1.5, 5$. The results are presented in Figs. 7, 8 and 9, respectively. Observe that for small maximum control amplitude, $\Omega _0=\Omega _z$, the optimal $\Omega (t)$ is constant and equal to the maximum value, see Fig. 7a, as in the case $J < \Omega _z/2$. But for larger $\Omega _0 = 1.5\Omega _z$, the optimal $\Omega (t)$ develops the structure Bang-Off-Bang, see Fig. 8a, while for even larger $\Omega _0 = 5\Omega _z$ the sequence Bang-Off-Bang-Off-Bang arises, as shown in Fig. 9a. In Fig. 10 we plot the normalized maximum achievable stored energy $\Delta E(T)/\Delta E_{max}$ obtained with numerical optimization, where $\Delta E_{max}=4(1+\chi ) \Omega _z = 8\Omega _z$ for $\chi = 1$, versus the available charging time T, for the same three values of the maximum control amplitude: $\Omega _0/\Omega _z=1$ (black line), $\Omega _0/\Omega _z=1.5$ (red line), and $\Omega _0/\Omega _z=5$ (green line). The vertical line indicates the time limit given in Eq. (54), obtained with the Bang-Off-Bang-Off-Bang sequence with delta Bang pulses. Since for the finite value $\Omega _0/\Omega _z=5$ the optimal pulse sequence has the Bang-Off-Bang-Off-Bang form displayed in Fig. 9a, the corresponding green curve intersects the full charging horizontal axis $\Delta E(T)/\Delta E_{max} = 1$ at a duration longer than this limit. In Fig. 5a we plot the minimum time for full charging versus the maximum control amplitude (red curve), up to the value $\Omega _0/\Omega _z = 10$. Note that it approaches the horizontal line corresponding to the limit (54), indicating that the optimal control maintains the Bang-Off-Bang-Off-Bang shape up to this value. We point out that for larger values of the maximum control amplitude $\Omega _0$ more complicated optimal sequences may arise, with shorter charging times than the limit (54) obtained for this particular form. Finally, in Fig. 5b we display the corresponding average charging power as a function of $\Omega _0$ (red curve).

Fig. 7

Optimal control amplitude (a) and phase (b), along with the corresponding time evolution of stored energy (c) and populations ($|c_0|^2$ black, $|c_1|^2$ red, $|c_2|^2$ green, $|c_3|^2$ blue) (d) for full charging in minimum time, when $\Omega _0/\Omega _z=1$ and $\chi =J/\Omega _z=1$.

Fig. 8

Optimal control amplitude (a) and phase (b), along with the corresponding time evolution of stored energy (c) and populations ($|c_0|^2$ black, $|c_1|^2$ red, $|c_2|^2$ green, $|c_3|^2$ blue) (d) for full charging in minimum time, when $\Omega _0/\Omega _z=1.5$ and $\chi =J/\Omega _z=1.$.

Fig. 9

Optimal control amplitude (a) and phase (b), along with the corresponding time evolution of stored energy (c) and populations ($|c_0|^2$ black, $|c_1|^2$ red, $|c_2|^2$ green, $|c_3|^2$ blue) (d) for full charging in minimum time, when $\Omega _0/\Omega _z=5$ and $\chi =J/\Omega _z=1$.

Fig. 10

Maximum achievable stored energy as a function of the available charging time, for three different values of the maximum control amplitude: $\Omega _0/\Omega _z=1$ (black line), $\Omega _0/\Omega _z=1.5$ (red line), and $\Omega _0/\Omega _z=5$ (green line).

Discussion

We begin the discussion by elucidating the relation between the charging protocols obtained in this work and the conventional coherent driving $\Omega _0 \cos (\omega _c t)(\sigma _++\sigma _-)$ with constant amplitude applied on resonance. This conventional protocol is actually optimal in the case of a QB composed of non-interacting spins. The protocols derived here are essentially generalizations of this protocol for QBs with interacting spins. This is obvious in the weak coupling case, where the constant control amplitude is reinforced with a time-dependent phase (a chirp), which is capable to drive all the spins simultaneously to the spin-up target state, despite their different frequencies due to the interactions. A constant amplitude protocol without the proper phase modulation would fail to handle these interactions. For the strong coupling case where the starting state is not the conventional spin-down state but an entangled state, the additional feature of Off intervals appears in the optimal amplitude.

Next, it is instructive to explain how the obtained results are reconciled with the quantum speed limit. To demonstrate the argument is more convenient to work with the quantum speed limit for large control bound $\Omega _0$. For a constant control with amplitude $\Omega (t)=\Omega _0$ and phase $\phi$, in the limit $\Omega _0\gg J$, the Hamiltonian matrices for both the weak and strong coupling cases, Eqs. (17) and (49), are essentially proportional to $\Omega _0$ and the same is true for the eigenenergies. Consequently, the quantum speed limit is proportional to $1/\Omega _0$, which tends to zero for $\Omega _0\rightarrow \infty$, This is consistent with our finding regarding the full charging time for weak coupling, but for strong coupling our analysis suggests a finite charging time even for very large $\Omega _0$. The resolution comes when we realize that the quantum speed limit provides the minimum time for a quantum system to evolve from some initial state to an orthogonal state. For weak coupling the orthogonal state is indeed the target spin-up state, as discussed below Eqs. (18). But for strong coupling where the system starts from the entangled state, the orthogonal state where it evolves within the vanishingly small quantum speed limit is not the target spin-up state but a superposition state. This can be seen from Eqs. (50), where the choice $\theta =\arccos (\sqrt{2/3})$ that nullifies the probability amplitude of the starting state, $c_1(T) = 0$, does not lead to $|c_3(T)| = 1$ but to a state with nonzero $c_0(T), c_2(T) \ne 0$ also. The target spin-up state can be reached by exploiting the coupling, and this leads to a nonzero lower bound on the minimum-time, no matter how large is $\Omega _0$. We close the discussion about speed limits observing that for the weak coupling case, where the system starts from the spin-down state, we expect an increase in the charging time with the number of spins. For the strong coupling case the situation is more complicated because the lowest energy starting state depends on the coupling and the charging time depends from where the system sets off.

Another interesting observation is regarding the symmetry of obtained protocols. For the case of weak coupling, the symmetry of the optimal amplitude and phase may be attributed to the symmetry of the Hamiltonian matrix in the corresponding Eq. (17) and the symmetry between the initial and target states. Even for the case with strong coupling, where the initial and target states are not symmetric, the symmetry of system (17) leads to equal durations of the Off pulses in the subspace decomposition method, which uses that system. On the other hand, when Eq. (49) is used for strong coupling in the resonant method, the symmetry is broken and this is reflected on the solutions obtained with either finite or delta bang pulses.

We also discuss the possible effect of dissipation on the optimal charging protocols. Dissipation processes limit the charging time window. If the minimum time for full charging is shorter than this window then the protocols will not be affected; otherwise, full charging is unattainable and an efficient protocol should do the best possible within the available time frame determined by dissipation. In the light of this observation, the Off intervals in the charging protocol for the strong coupling case may be reduced or even eliminated. The unavoidable presence of dissipation provides a serious motivation for deriving fast and efficient charging protocols.

Regarding the experimental implementation of the proposed protocols, note that from the amplitude $\Omega (t)$ and phase $\phi (t)$ the transverse component $\Omega _x(t), \Omega _y(t)$ of the applied field can be obtained using Eq. (4). The implementation of such control fields in the NMR framework considered here is a routine procedure, see for example Ref.⁷⁹.

Finally, note that although here we concentrate on the stored energy, a similar procedure can be followed for maximizing ergotropy, the maximum amount of work that can be extracted from the battery via a unitary process¹. Specifically, after having charged the battery with the highest stored energy, the lowest energy eigenstate should be set as the target state in the optimal control problem, to extract the maximum possible work.

Conclusions

In this work, we obtained some fast protocols for charging a three-qubit Ising spin-chain quantum battery, using amplitude and phase control of the global transverse field. Two distinct cases were identified. For weak coupling between the qubits in the chain, where the initial ground state of the system is the spin-down state, we found that the optimal strategy is to keep the control amplitude fixed at its maximum value while properly modulate the phase with time. For the strong coupling case, where the initial ground state is the one-excitation Dicke state $\vert {W} \rangle$, we employed two related charging protocols. In the first approach we used a pulse-sequence derived numerically in the recent work⁷⁵, consisting of delta Bang pulses and intervals where the control is switched off. We named this method fast charging via subspace decomposition, because it involves identifying and manipulating effective two-level subsystems. In this article we derived analytically the characteristics of the pulses composing the sequence, which were calculated numerically in Ref.⁷⁵. In the second approach, we used a similar pulse-sequence with delta Bang pulses and Off intervals, but with a carrier frequency which brings on resonance the initial $\vert {W} \rangle$ state and the target spin-up state having maximum energy; we named this method charging via resonant excitation. We calculated numerically the characteristics of the pulses in the sequence and found a shorter charging time compared to the non-resonant method motivated by Ref.⁷⁵. We also used numerical optimal control to solve the problem of full charging the quantum battery but with finite values of the control amplitude and found that, for some relatively large values of this bound, the optimal solution has similar structure with the sequence using delta Bang pulses but longer duration, as expected. The present work concentrates on a three-spin chain but the suggested approach can be also extended to larger systems. The lowest and highest energy eigenstates must be identified first, serving as the initial and target states. Then, optimal control can be used to calculate the amplitude and phase modulation of a control field resonant to this transition, to maximize the stored energy in the battery for a given duration.