Optimal Control Methods for Quantum Batteries Francesco Mazzoncini1 2Vasco Cavina2 3Gian Marcello Andolina2 4Paolo Andrea Erdman5and Vittorio Giovannetti2

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Optimal Control Methods for Quantum Batteries

Francesco Mazzoncini,1, 2, ∗Vasco Cavina,2, 3 Gian Marcello

Andolina,2, 4 Paolo Andrea Erdman,5and Vittorio Giovannetti2

1T´el´ecom Paris-LTCI, Institut Polytechnique de Paris,

19 Place Marguerite Perey, 91120 Palaiseau, France

2NEST, Scuola Normale Superiore, I-56126 Pisa, Italy

3Department of Physics and Materials Science, University of Luxembourg, L-1511 Luxembourg, Luxembourg

4ICFO-Institut de Ci`encies Fot`oniques, The Barcelona Institute of Science and Technology,

Av. Carl Friedrich Gauss 3, 08860 Castelldefels (Barcelona), Spain

5Freie Universit¨at Berlin, Department of Mathematics and Computer Science, Arnimallee 6, 14195 Berlin, Germany

(Dated: October 11, 2022)

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 the time-dependent external ﬁelds. The technique presented here is general,

and we apply it to speciﬁc cases in which the energy is both pumped into the battery by external

forces (direct charging) or transferred into it from an external charger (mediated charging). In this

article 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.

I. INTRODUCTION

In recent years, with the rapid development of new

quantum technologies [1,2], there has been a worldwide

interest in exploiting quantum phenomena that arise at a

microscopic level. Here, we will focus on studying the so-

called ”quantum batteries”[3–10], i.e. quantum mechani-

cal systems employed for energy storage, where quantum

eﬀects can be used to obtain more eﬃcient and faster

charging processes than classical systems.

This blossoming research ﬁeld has to address many dif-

ferent questions, such as the stabilization of stored energy

[11,12], the practical implementation of quantum batter-

ies [13,14], and the study of the optimal charging pro-

cesses [15–17], oﬀering a vast research panorama on both

theoretical [11–13,18–27] and experimental ends [14,28].

Within this framework, we will derive optimal charging

strategies for quantum batteries using techniques from

Quantum Control Theory [29–31], a powerful mathemat-

ical tool that has many applications in diﬀerent ﬁelds of

physics such as quantum optics [32] and physical chem-

istry [33–35]. Quantum control theory has contributed to

understanding interesting aspects of quantum mechanics

such as the quantum speed limit [36–39] and to generate

eﬃcient quantum gates in open quantum systems [40,41].

In this work, we study how a qubit or a quantum har-

monic oscillator can be optimally charged with a modu-

lation of an external Hamiltonian. In order to ﬁnd the

best charging protocol we will use the Pontryagin’s Min-

imum Principle (PMP) [42,43], a very useful theorem

∗mazzoncini@telecom-paris.fr

of Classical Optimal Control Theory, which is frequently

used also in Quantum Control Theory [44,45]. We show

that, in most cases that we consider, quantum batteries

can be optimally charged through diﬀerent variants of a

so called Bang-Bang modulation of the intensity of an

external Hamiltonian.

Our paper is organized as follows. In section II we in-

troduce two general charging protocols to inject energy in

a quantum battery. In section III we present a brief intro-

duction to Pontryagin’s Minimum Principle, highlighting

the main tools that we shall use throughout the paper.

In section IV we focus on the ﬁrst charging protocol, con-

sisting of a closed system charged by the modulation of

an external Hamiltonian. Section VI is devoted to ana-

lyzing a second charging process, where we make use of

the coupling between a quantum battery and an auxil-

iary quantum system. Finally, a brief summary of our

main conclusions is reported in section VII, while useful

technical details can be found in the appendix.

II. CHARGING OF A QUANTUM BATTERY

We start deﬁning two general protocols for the charg-

ing process of a quantum battery, see Fig. 1for a pictorial

representation.

a. Direct Charging Process (DCP): The ﬁrst charg-

ing model consists of a single closed quantum system ini-

tialized in a state ρ(0) that evolves in time under the

action of a time-dependent Hamiltonian of the form

H(t) = H0+λ(t)·H:= H0+

i=1

λi(t)Hi.(1)

arXiv:2210.04028v1 [quant-ph] 8 Oct 2022

(a)

(b)

FIG. 1. a) Direct Charging Process: charging model for a

closed system through the modulation of an external control

for a ﬁnite amount of time τ. b) Mediated Charging Process:

it consists in letting two systems A and B interact through

an Hamiltonian H1.

In this expression H0is the intrinsic Hamiltonian con-

tribution which deﬁnes the energy content of the sys-

tem before and after the charging process and H:=

(H1,··· , Hm) is a collection of charging Hamiltonian

terms which are modulated by control functions λ(t) :=

(λ1(t),··· , λm(t)) that we assume to be active (i.e. dif-

ferent from zero) only over a limited time interval [0, τ].

They can take values that are determined by some as-

signed constraint, i.e. λ(t)∈D[0, τ], where τ > 0 is

the total duration of the charging process and D[0, τ]

is a proper subset of the real functions F[0, τ] mapping

[0, τ] into Rm. Our goal is hence to ﬁnd an optimal

λ?(t)∈D[0, τ] that, given an assigned τ, maximizes the

mean energy of the system at the end of the process.

Introducing

Uτ:= Texp[−iZτ

dt H(t)] ,(2)

the time-ordered unitary evolution operator associated

with the time-dependent Hamiltonian (1), and

ρ(τ) = Uτρ(0)U†

τ,(3)

the evolved state of the system at time τ, we aim to

determine the quantity

Emax(τ) := E(τ)λ?(t)= max

λ(t)∈D[0,τ]E(τ),(4)

where using h·i as a short-hand notation to indicate the

trace operator, we set

E(τ) := hρ(τ)H0i,(5)

(notice that hereafter we have set ~= 1). It is worth

pointing out that since the DCP models considered here

rely on closed dynamical evolutions (no interactions with

external degrees of freedom being allowed), the DCP op-

timization we are targeting corresponds also to maximiz-

ing the amount of extractable work we can store in the

system as measured by the ergotropy, the total ergotropy,

or the thermal free-energy [46]. To see this explicitly we

recall that given a quantum system with Hamiltonian

H(t) and state ρ(t), all these quantities can be computed

W[ρ(t), H(t)] := hρ(t)H(t)i−F(sρ(t), sH(t)),(6)

where F(sρ(t), sH(t)) is a functional that only depends

upon the collections sρ(t)={η1(t), η2(t),···} and

sH(t)={1(t), 2(t),···} of the eigenvalues of ρand H

respectively (see App. Afor details). Since the unitary

evolution (3) preserves sρ(t), and sH(τ)=sH0in the DCP,

F(sρ(τ), sH(τ)) = F(sρ(0), sH0) so that this quantity plays

no role in the optimization procedure.

b. Mediated Charging Process (MCP): Although

the DCP is of undoubted theoretical interest, a closed

evolution of a unique system is not genuinely realistic

from the physical implementation’s point of view. Such

unitary evolution regime occurs only when the dynamics

of the energy source are very slow compared to the Quan-

tum Battery dynamics (i.e. in the Born-Oppenheimer

limit). Therefore, we also consider a second charging

model, called charger-mediated process [4,5], that in-

volves instead two separate elements: an auxiliary quan-

tum system A, called charger, and a quantum battery B.

In the MCP we aim at maximizing the energy stored in

B by suitably modulating its interaction with A in ﬁnite

time τ. For this sake we replace the DCP hamiltonian

(1) with

H(t) := HA+HB+λ(t)·H,(7)

where HA,HBare local operators of Aand Brespec-

tively and His now free to act on both the battery and

the auxiliary system. The quantity to optimize is now

given by

EB(τ) := hρB(τ)HBi,(8)

where ρB(τ) is the reduced density matrix of the battery

at time τ. Since ρB(τ) does not follow a unitary tra-

jectory in the MCP scenario, sρB(τ)is typically diﬀerent

from sρB(0), this implies that W[ρB(τ), HB] is consider-

ably more challenging to optimize. We shall see however

that by choosing wisely the global initial state ρAB (0),

we can reduce our analysis to simpler DCPs, as shown in

Sec. VI.

III. PONTRYIAGIN’S MINIMUM PRINCIPLE

The Pontryiagin’s Minimum Principle (PMP) [43] is

the main tool we will use in the optimization of DCPs

and MCPs and will allow us to formally identify necessary

conditions for the optimality of λ?(t). Here we introduce

the approach to optimal control problems provided by

PMP using a general formalism, that will be adapted to

both DCP and MCP problems afterwards. Consider a

set of state variables at a given time t, represented by

the elements of a vector v(t) := (v1(t),··· , vn(t)) which

evolves via a dynamical equation represented by nﬁrst-

order diﬀerential equations of the form

v(t) = f(v(t),λ(t), t),(9)

with fa vectorial function. The quantity to optimize,

also called perfomance criterion is evaluated in terms of

a cost function written as

J=Zτ

g(v(t),λ(t), t)dt, (10)

with ga scalar function. Deﬁning the pseudo-

Hamiltonian Has

H:= g(v(t),λ(t), t) + p(t)·f(v(t),λ(t), t),(11)

with p(t) the n-dimensional row vector of Lagrange mul-

tipliers, called costates, the PMP states that necessary

conditions for an optimal control λ?(t)∈D[0, τ] to min-

imize Jare that for all t∈[0, τ]:











v(t) = ∂H

∂p(v(t),λ?(t),p(t), t),

p(t) = −∂H

∂v(v(t),λ?(t),p(t), t),

H(v(t),λ?(t),p(t), t)≤ H(v(t),λ(t),p(t), t),

∀λ(t)∈D[0, τ].

(12)

Moreover, the PMP gives additional constraints based

on the boundary conditions of our problem, i.e. whether

the ﬁnal state and the ﬁnal time are ﬁxed or free. In

particular:

•if the ﬁnal time τis ﬁxed and no constraint is posed

on the ﬁnal state v(τ), then

p(τ) = (0,0,··· ,0) ; (13)

•if the ﬁnal time τis free while the ﬁnal state v(τ)

is ﬁxed, then

H(v(τ),λ?(τ),p(τ), τ) = 0 .(14)

We ﬁnally highlight that the PMP is not the only pos-

sible optimization method to analyze charging processes

for quantum batteries. For instance, Ref. [17] deploys an

iterative approach to minimize the distance between the

target state and the ﬁnal state, considering a variant of

our charger-mediated process where a ﬁeld is modulated

only acting on the charger, considered in this case as an

open dissipative system. However, since PMP gives nec-

essary conditions for optimality, any other optimization

method must eventually satisfy those conditions.

IV. DCP OPTIMAL SOLUTIONS

In this section we will derive the optimal solutions for

DCPs considering two diﬀerent settings: ﬁrst in Sec. IV A

we ﬁx the total duration of the charging event τand try

to identify the optimal pulse λ?(t) which, starting from a

given initial conﬁguration ρ(0), produces the maximum

value of the output energy Emax(τ); then in Sec. IV B we

analyze the opposite problem: that is we ﬁx a target out-

put state that ensures a certain value of the ﬁnal energy,

and try ﬁnd the optimal control λ?(t) that enable us to

reach it in the minimum time τ.

A. Maximum output energy at ﬁxed time τ

To begin with, we observe that if i) the charging Hamil-

tonian terms Hi’s are generators of the group Uof the

unitary operators on the system, and ii) no restrictions

are imposed on the choice of the control vector λ(t), al-

lowing D[0, τ] to include all possible elements F[0, τ], then

the dynamical evolutions (2) can span the entire set U

of unitary transformation on the system. Accordingly

under conditions i) and ii) we can write

Emax(τ) = max

λ(t)∈F[0,τ]hUτρ(0)U†

τH0i

= max

U∈U hUρ(0)U†H0i=: Emax ,(15)

where Emax is a τindependent constant that represents

the maximum amount of energy we can force into the sys-

tem via arbitrary unitary manipulations. The constant

Emax can be explicitly evaluated as

Emax =X

i=1

η↑

i(0) ↑

i(0) ,(16)

with s↑

ρ(0)={η↑

1(0), η↑

2(0), . . . }and s↑

={↑

(0), ↑

(0), . . . }

being the spectra of ρ(0) and H0, rearranged in increasing

order. Note that it is possible to establish a direct con-

nection between Emax and the anti-ergotropy [47] of the

system (see App. A1 for details). Apart from this spe-

cial case, the explicit evaluation of Emax(τ) is typically

rather demanding and does not admit a closed analytical

solution. One possible approach to tackle it is to make

use of optimal control techniques. In particular, in what

follows we shall rely on the PMP we have reviewed in

Sec. III. For this purpose we rewrite the ﬁnal energy (5)

E(τ) = Zτ

0H0˙ρ(t)dt +E(0) ,(17)

where ˙ρ(t) = N[ρ(t)] = −i[H(t), ρ(t)]. Accordingly, we

can study the optimization of the charging process as a

minimization problem of the cost function

J:= −Zτ

0H0N[ρ(t)]dt . (18)

The optimization task can then be translated into a PMP

problem by introducing the following arrangements

v(t)→ρ(t),λ(t)→λ(t),p(t)→π(t),

f(v(t),λ(t), t)→ N[ρ(t)] ,

g(v(t),λ(t), t)→ −hH0N[ρ(t)],

p(t)·f(x(t),λ(t), t)→ hπ(t)N[ρ(t)]i,(19)

with π(t) being a self-adjoint operator of the same di-

mension of ρ(t), and deﬁning the pseudo-Hamiltonian

H(ρ(t),λ(t), π(t), t) := h(π(t)−H0)N[ρ(t)]i(20)

=λ(t)·G(t)−ihπ0(t)[H0, ρ(t)]i,

with π0(t) := π(t)−H0and G(t) := (G1(t),··· , Gm(t))

being a column-vector of elements

Gj(t) := −ihπ0(t)[Hj, ρ(t)]i.(21)

This allows us to express the necessary conditions (12)

for the optimal control vector λ?(t) as











˙ρ(t) = −iH?(t), ρ(t),

˙π0(t) = −i[H?(t), π0(t)] ,

λ?(t)·G(t)≤λ(t)·G(t),∀λ(t)∈D[0, τ],

(22)

where H?(t) represents the Hamiltonian (1) evaluated on

the optimal control pulse, i.e.

H?(t) := H0+λ?(t)·H.(23)

In the third line of Eq. (22) we exploited Eq. (20) and

the fact that the term −ihπ0(t)[H0, ρ(t)]idoes not depend

explicitly on λ(t). In the case of a charging process with

ﬁxed time τand unknown optimal ﬁnal state ρ(τ), the

list (22) has to be completed with the extra condition (13)

which in the present case becomes

π(τ)=0 ⇐⇒ π0(τ) = −H0.(24)

The ﬁrst two equations in (22) simply tell us that ρ(t)

and π0(t) represent the state and the costate operator

of the system evolved under the action of the Hamil-

tonian (23). What ultimately decides whether a given

λ?(t) has a chance of being an optimal solution is the

inequality in Eq. (22) which, unfortunately, due to the

implicit dependence upon λ?(t) of G(t), is typically not

analytically treatable. Nonetheless, in the special spe-

cial case where we have a unique control function (i.e.

m= 1) and the allowed domain D[0, τ] is chosen to simply

force the intensity of λ1(t) to belong to a given interval

I1= [λmin

1, λmax

2], the inequality in Eq. (22) translates

into a series of (simpliﬁed) conditions which provide us

with a nice guidance on how to construct the optimal

control pulse, i.e.

a) λ?

1(t) can take the minimum allowed value λmin

1iﬀ

the associated G1(t) function is strictly positive, i.e.

λ?

1(t) = λmin

1⇐⇒ G1(t)>0 ;

FIG. 2. Example of the relationship between a time-optimal

control λ?

1(t) and the G1(t) function for the case in which the

system is characterized by a single control function (m= 1).

The region with a question mark is a singular interval, where

the value of the optimal control is not determined by the

conditions in Eq. (22).

b) λ?

1(t) can take the maximum allowed value λmax

1iﬀ

the associated G1(t) function is strictly negative,

i.e.

λ?

1(t) = λmax

1⇐⇒ G1(t)<0 ;

c) λ?

1(t) can take arbitrary values in the allowed do-

main I1:= [λmin

1, λmax

1] iﬀ the associated G1(t) is

equal to zero.

From the above analysis it emerges that natural candi-

dates for λ?

1(t) are Bang-Bang-like step functions similar

to the one shown in Fig. 2which alternate their values

among the allowed extreme λmin

1and λmax

1with switch-

ing points corresponding to the zeros of the associated

G1(t) function. The only allowed exceptions to this rule

is when G1(t) is zero over an extended interval (singular

interval scenario): in this case the necessary conditions

in Eq. (22) give no information about how to select λ?

1(t)

without specifying the nature of the system.

B. Minimum charging time at ﬁxed ﬁnal state

Another problem that we can tackle using the PMP

method is to determine the minimum value of the charg-

ing time τthat allows us to move our initial state ρ(0)

into a ﬁnal target conﬁguration ρ— for instance a state

in which the eigenvalues are sorted in increasing order

which according to Eq. (16) grants us the maximum value

of the stored ﬁnal energy Emax allowed by the most gen-

eral DCP process. The new cost function of the problem

can be written as

J:= −Zτ

1dt , (25)

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OptimalControlMethodsforQuantumBatteriesFrancescoMazzoncini,1,2,VascoCavina,2,3GianMarcelloAndolina,2,4PaoloAndreaErdman,5andVittorioGiovannetti21TelecomParis-LTCI,InstitutPolytechniquedeParis,19PlaceMargueritePerey,91120Palaiseau,France2NEST,ScuolaNormaleSuperiore,I-56126Pisa,Italy3DepartmentofP...

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Optimal Control Methods for Quantum Batteries Francesco Mazzoncini1 2Vasco Cavina2 3Gian Marcello Andolina2 4Paolo Andrea Erdman5and Vittorio Giovannetti2

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