Quench dynamics in the Jaynes-Cummings-Hubbard and Dicke models Andrew R. Hoganand Andy M. Martin School of Physics University of Melbourne Parkville 3010 Australia

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Quench dynamics in the Jaynes-Cummings-Hubbard and Dicke models

Andrew R. Hogan∗and Andy M. Martin

School of Physics, University of Melbourne, Parkville, 3010, Australia

(Dated: Feb 2022)

Both the Jaynes-Cummings-Hubbard (JCH) and Dicke models can be thought of as idealised

models of a quantum battery. In this paper we numerically investigate the charging properties of

both of these models. The two models diﬀer in how the two-level systems are contained in cavities.

In the Dicke model, the Ntwo-level systems are contained in a single cavity, while in the JCH model

the two-level systems each have their own cavity and are able to pass photons between them. In

each of these models we consider a scenario where the two-level systems start in the ground state

and the coupling parameter between the photon and the two-level systems is quenched. Each of

these models display a maximum charging power that scales with the size of the battery Nand no

super charging was found. Charging power also scales with the square root of the average number of

photons per two-level system mfor both models. Finally, in the JCH model, the power was found

to charge inversely with the photon-cavity coupling κ.

I. INTRODUCTION

Energy storage capabilities and eﬃciency by electro-

chemical batteries have rapidly improved in recent times,

pushed by the need to robustly deal with the ever in-

creasing energy demands of daily life. As we advance

technologically in the search for faster charging batter-

ies, recently the idea of a quantum battery has become a

more heavily researched topic [1–24]. The goal underpin-

ning the exploration of a battery made of single quantum

bits each with a single excited state is to use quantum

phenomena to engineer a greatly improved energy stor-

age device. Some limiting factors for classical electro-

chemical batteries are their thermodynamic energy loss

due to heat and their increasing charging times for scaled

up batteries [25–28]. Investigating ways that a quantum

battery can deal with these issues has lead to the desire

to understand how quantum states might be utilised to

produce a battery with minimal energy loss and how the

system can be built to minimise its charging time [29–40].

Previous theoretical work [29] found that quantum bat-

teries can display a super-charging characteristic. They

found that as the number of two-level systems (N) in

the battery increased, the speed with which the battery

charged increased at a rate of N√N. This result has

ignited signiﬁcant interest in quantum batteries and in-

spired us to explore quantum batteries in the context of

the Dicke model [41] and the Jaynes-Cummings-Hubbard

(JCH) model [42].

Functionally, a quantum battery can be thought of as

idealised two-level system inside a cavity whose mode

is able to excite the two-level system. For such a sys-

tem the battery can be thought of as being charged

(uncharged) when the two-level system is in the excited

(ground) state. Figure 1 schematically describes the two

systems we will consider in this work. Speciﬁcally the

JCH model, Fig. 1(a) and the Dicke model, Fig. 1(b),

under the charging protocol shown in Fig 1(c). In each

∗arhogan@student.unimelb.edu.au

case we sonsider a scenario where we have Nelements

in the quantum battery. The system is initialised such

that the two-level systems are in the ground state. At

t= 0 the coupling between the two-level systems and

the photons is quenched from 0to β. We will ﬁrst con-

sider the charging in the JCH model in sections II &III.

For the JCH system we ﬁnd that the maximum charg-

ing power, Pmax, is proportional to the number of the

cavities in the JCH system. Additionally, we ﬁnd that

the maximum charging power is (inversely) proportional

to square root of the number photons initially in each

cavity (the photon coupling between individual cavities).

The result that the maximum charging power is propor-

tional to the number of two-level systems in the JCH

model then prompts us to revisit, in sections IV &V, re-

sults for the Dicke model, where we construct the Dicke

Hamiltonian to ensure that the thermodynamic limit is

bounded. For such a regime we regain a scaling for Pmax

proportional to the number of two-level systems in the

Dicke cavity.

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(a)

(b)

(c)

FIG. 1. (a) Schematic for the JCH model. Nidentical two-

level systems each occupying their own cavity, with photons

coupling between cavities with strength κ. (b) Schematic for

the Dicke model. Two-level systems the same as above except

that they are all in the one cavity. (c) Representation of

the charging sequence of the quantum battery. Initially the

photon coupling to the two-level system βis zero, then it is

quenched to a value β > 0, where charging begins.

arXiv:2210.01355v2 [quant-ph] 10 May 2023

II. JCH QUANTUM BATTERIES

The JCH model can be thought of as representing an

atom with a single excited state in the presence of n

photons inside a cavity. The two-level atomic system

is coupled to the photons in the cavity via β, and the

photons with frequency ωcare coupled between the N

identical cavities via κ. Speciﬁcally the JCH Hamiltonian

[43] is (~= 1)

HJCH =

n=1

ωca†

nan+

n=1

ωaσ+

nσ−

n+β

n=1

(anσ†

n+a†

nσ−

−κ

n=1

(a†

n+1an+a†

nan+1)(1)

where ωais the energy of separation between the energy

levels of the TLS, a†and aare the photonic raising and

lowering operators, and σ+and σare the spin raising

and lowering operators.

Diagonalising the JCH Hamiltonian allows the Time

Dependent Schrodinger Equation (TISE) to be solved

and the dynamics analysed. Starting with the system

in the lowest energy eigenstate, the atom-photon cou-

pling is quenched from β= 0 to β > 0at time t= 0. In

doing so, the two-level systems are taken from a parame-

ter space where they cannot charge, and instantaneously

quenched to one where they are able to begin charging.

In order to quantify the charging rate we deﬁne that the

energy of the system is the diﬀerence between the energy

of the time varying energy and that of the initial state,

Eβ(t) = ωc{hψN

β(t)|ˆ

Jz|ψN

β(t)i−hψN(0)|ˆ

Jz|ψN(0)i},(2)

where the energy operator for the atomic spin is

Jz=ωa

n=1

σ+

nσ−

n.(3)

With the time varying energy we ﬁnd the maximum

charging power of the battery by taking the maximum

rate of change of the energy with respect to time,

Pmax = maxEβ(t)

t,(4)

which has a charging time to reach Pmax of τ. This

deﬁnition of power has been used to make a direct com-

parison with with existing literature [29]. Alternatively,

the time to charge the battery to its maximum energy

was explored, with both methods returning results with

the same scaling factors. The two ways to analyse the

power of the quantum battery are to consider how long

it takes to fully charge the battery, which has a strong

analogous relationship between classical batteries, or to

consider the best possible charging power and consider

how that scales. In the rest of this paper we will use the

later deﬁnition, as in equation (4).

The limit κ= 0 represents the case where individual

elements of the cavities are not coupled to each other,

and there is no photon transfer between them. We will

use this as the baseline by which we analyse how diﬀerent

parameters may change the charging rate of the battery,

with express interest in whether increasing the size of the

battery improves the charging power. With a system of

isolated (κ= 0) JCH two-level systems, the behaviour

reduces to that of individual Rabi two-level systems with

Hamiltonian,

HJCH

m=m+ ∆ √mβ

√mβ mω (5)

where ∆ = ωa−ωcand the average number of photons

per two-level system is m. In this regime the JCH model

can be solved analytically and has its ﬁrst maximum en-

ergy at time

τ=π

2Ω,(6)

where the Rabi frequency is

Ω = p∆2+ 4mβ2

2.(7)

It can be seen from equation (6) that when the energy

separation between the two energy levels and the photon

mode energy is zero (∆=0), the charging time will scale

with the number of photons according to τ∝1/√m, and

Escales proportional to N. It follows that Pmax ∝N

and Pmax ∝√m. It is therefore of interest to explore how

this relationship changes when the two-level systems are

able to interact. Allowing the cavities in the quantum

battery to interact via photon coupling (κ > 0) makes it

possible to analyse how κ,Nand meﬀect it’s charging

power.

III. JCH RESULTS

In this paper we present results in natural units where

~= 1, and for a resonant regime where the dimension-

less photon mode energy and the dimensionless atomic

energy separation are both 1, and hence ∆=0. In Fig.2,

the eﬀect of increasing battery size is shown for diﬀerent

values of the photon mode coupling parameter κ. When

κ= 0, the JCH model has an analytical solution. The

present simulation results overlap exactly with the ana-

lytical results obtained from the Rabi matrix of equation

(6). This serves as the starting point for the compari-

son of the power for larger JCH systems. It can be seen

in this ﬁgure, that the charging power of the quantum

battery for any value of κnever exceeds that of the com-

pletely uncoupled κ= 0 case. The quantum battery has

the largest maximal charging power when it acts as if

it was Nindependent single atom batteries. With the

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摘要：

QuenchdynamicsintheJaynes-Cummings-HubbardandDickemodelsAndrewR.HoganandAndyM.MartinSchoolofPhysics,UniversityofMelbourne,Parkville,3010,Australia(Dated:Feb2022)BoththeJaynes-Cummings-Hubbard(JCH)andDickemodelscanbethoughtofasidealisedmodelsofaquantumbattery.Inthispaperwenumericallyinvestigatethech...

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Quench dynamics in the Jaynes-Cummings-Hubbard and Dicke models Andrew R. Hoganand Andy M. Martin School of Physics University of Melbourne Parkville 3010 Australia

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