# Optimal energy storage in the Tavis-Cummings quantum battery

Hui-Yu Yang  School of Physics, Northwest University, Xi’an 710127, China
Hai-Long Shi<EMAIL_ADDRESS>QSTAR and INO-CNR, Largo Enrico Fermi 2, 50125
Firenze, Italy  Hefei National Laboratory, Hefei 230088 China  Qing-Kun Wan
Innovation Academy for Precision Measurement Science and Technology, Chinese
Academy of Sciences, Wuhan 430071, China  University of Chinese Academy of
Sciences, Beijing 100049, China  Kun Zhang<EMAIL_ADDRESS>School of
Physics, Northwest University, Xi’an 710127, China  Shaanxi Key Laboratory for
Theoretical Physics Frontiers, Xi’an 710127, China  Peng Huanwu Center for
Fundamental Theory, Xi’an 710127, China  Xiao-Hui Wang<EMAIL_ADDRESS>School of Physics, Northwest University, Xi’an 710127, China  Shaanxi Key
Laboratory for Theoretical Physics Frontiers, Xi’an 710127, China  Peng Huanwu
Center for Fundamental Theory, Xi’an 710127, China  Wen-Li Yang  Institute of
Modern Physics, Northwest University, Xi’an 710127, China  Shaanxi Key
Laboratory for Theoretical Physics Frontiers, Xi’an 710127, China  Peng Huanwu
Center for Fundamental Theory, Xi’an 710127, China

(August 28, 2024)

###### Abstract

The Tavis-Cummings (TC) model, which serves as a natural physical realization
of a quantum battery, comprises $N_{b}$ atoms as battery cells that
collectively interact with a shared photon field, functioning as the charger,
initially containing $n_{0}$ photons. In this study, we introduce the
invariant subspace method to effectively represent the quantum dynamics of the
TC battery. Our findings indicate that in the limiting case of
$n_{0}\\!\gg\\!N_{b}$ or $N_{b}\\!\gg\\!n_{0}$, a distinct SU(2) symmetry
emerges in the dynamics, thereby ensuring the realization of optimal energy
storage. We also establish a negative relationship between the battery-charger
entanglement and the energy storage capacity. As a result, we demonstrate that
the asymptotically optimal energy storage can be achieved in the scenario
where $N_{b}\\!=\\!n_{0}\\!\gg\\!1$. Our approach not only enhances our
comprehension of the algebraic structure inherent in the TC model but also
contributes to the broader theoretical framework of quantum batteries.
Furthermore, it provides crucial insights into the relation between energy
transfer and quantum correlations.

##  I Introduction

Embracing the promising trend of device miniaturization, quantum batteries
(QBs) have been proposed to exploit quantum features, thereby accelerating
charging rates compared to their classical counterparts. From a theoretical
perspective, QBs offer fundamental insights into the influence of quantum
correlations on extractable work, a pivotal concept in quantum thermodynamics
Perarnau-Llobet  _et al._ (2015); Andolina  _et al._ (2019); Shi  _et al._
(2022); Liu  _et al._ (2021); Alicki and Fannes (2013); Allahverdyan  _et al._
(2004); Kamin  _et al._ (2020); Korzekwa  _et al._ (2016); Caravelli  _et al._
(2021); Francica  _et al._ (2020); Çakmak (2020); Manzano  _et al._ (2018);
Francica  _et al._ (2017); Francica (2022); Yang  _et al._ (2023); Seah  _et
al._ (2021). QBs, in conjunction with quantum heat engines, offer a tangible
approach to incorporate quantum correlations, such as quantum entanglement and
quantum coherence, into the field of quantum thermodynamics Binder  _et al._
(2018); Oppenheim  _et al._ (2002); Lostaglio  _et al._ (2015); Uzdin  _et
al._ (2015); Brandão  _et al._ (2013); Shi  _et al._ (2020); Barrios  _et al._
(2017); Ji  _et al._ (2022); Altintas  _et al._ (2014); Park  _et al._ (2013).

One notable characteristic of QBs is the phenomenon known as “charging
speedup” Hovhannisyan  _et al._ (2013); Binder  _et al._ (2015); Campaioli
_et al._ (2017). When connecting a group of $N_{b}$ QB cells to a shared
charger, the charging rate of the QB can potentially scale up to $N_{b}^{2}$
Gyhm  _et al._ (2022), indicating a significant quantum advantage over
classical batteries, whose charging speed scales linearly with $N_{b}$. This
quantum advantage has been verified in various models, including Dicke QBs
Ferraro  _et al._ (2018); Dou  _et al._ (2022a); Zhang and Blaauboer (2023),
Sachdev-Ye-Kitaev QBs Rossini  _et al._ (2020), spin-chain QBs Le  _et al._
(2018), cavity spin-chain QBs Dou  _et al._ (2022b) and central-spin QBs Peng
_et al._ (2021). Nevertheless, it is imperative to emphasize that a charging
speedup does not inherently ensure the optimal energy storage. Here, the
optimal energy storage denotes the complete transfer of input energy from the
charger to the battery. To enrich the theory of QBs, we will shift our focus
towards an important cavity quantum electrodynamics (cavity-QED) system, the
Tavis-Cummings (TC) model, to address the challenge of achieving optimal
energy storage.

Figure 1:  Schematic illustration of two charging schemes for the Tavis-
Cummings battery. (a) Collective charging scheme: $N_{b}$ battery cells
(atoms) are commonly interacted with a charger (a cavity photon field). (b)
Parallel charging scheme: $N$ single-atom TC models, known as the Jaynes-
Cummings model, are placed in parallel. At $t<0$, the battery is in the ground
state with the lowest energy. From $0<t<\tau$, the interaction Hamiltonian
$H_{I}$ is activated to charge the battery. Finally, at the charging time
$\tau$, the interaction is turned off and the energy is stored in the battery.

The TC model serves as a foundational framework for investigating the
collective behavior of atoms or qubits in the presence of electromagnetic
fields Tavis and Cummings (1968); TAVIS and CUMMINGS (1969); Hepp and Lieb
(1973a, b). It constitutes a theoretical underpinning for the fields of
quantum optics and cavity-QED Blais  _et al._ (2021); Zare Rameshti  _et al._
(2022); Garraway (2011). Furthermore, it enhances our comprehension of the
qubit manipulation and the generation of entangled states, both of which are
pivotal components in the advancement of quantum computing and quantum
information processing technologies Tessier  _et al._ (2003); Retzker  _et
al._ (2007); Fink  _et al._ (2009); López  _et al._ (2007); Steffen  _et al._
(2013); Wang  _et al._ (2020); Leek  _et al._ (2010); Higgins  _et al._
(2014). Moreover, the TC model naturally lends itself to describing a QB,
wherein each of the $N_{b}$ atoms serves as an individual battery cell and the
optical field serves as the charger, as illustrated in Fig. 1(a). The
eigenproblem associated with the TC model can be exactly solved using the
algebraic Bethe ansatz Bogolyubov (2000). However, this approach does not
alleviate the inherent challenges,exponential wall Kohn (1999) , associated
with calculating quantum dynamics, which is at the core of investigating the
energy storage problem. Transitioning from the eigenproblem to quantum
dynamics is inherently challenging, not to mention the complexity of solving
the Bethe ansatz equations. For instance, in Ref. Lu  _et al._ (2021)
employing the Bethe ansatz approach, numerical calculations were conducted for
a scenario involving 10 QB cells. The study concluded that Fock state is the
optimal initial charger state for the maximum stored energy capacity.
Reference Andolina  _et al._ (2019) utilized the PYTHON toolbox QuTiP2 to
numerically investigate cases in the TC battery where the initial photon
number equals the number of battery cells. Their findings revealed that a
polynomial increase in the support of the battery state guarantees that the
stored work can be completely extracted. For a fixed number of QB cells, Ref.
Shi  _et al._ (2022) established a general inverse relationship between the
battery-charger entanglement and the extracted work in incoherent QBs. Despite
these productive findings, there remains a need for a deeper understanding of
the optimal energy storage problem in the TC battery: 1) What conditions must
be met for a Fock state to achieve optimal energy storage? 2) How should we
address scenarios where the number of battery cells is significantly greater
than the initial photon number? 3)Does the negative relation between the
entanglement and the energy storage capacity hold as the number of QB cells
varies?

In this paper, we address the aforementioned questions for the TC battery. In
Sec. II, we begin by introducing the TC model and proceed to formulate the
energy storage problem mathematically. Subsequently, we introduce the
invariant subspace method to overcome the exponential wall obstruction,
allowing us to deal with the dynamics of the TC battery when the charger is
chosen in a Fock state. In Sec. III, we first apply the invariant subspace
method to analyze the parallel charging scheme. Subsequently, we delve into
three collective charging scenarios: $n_{0}\\!>\\!N_{b}$, $N_{b}\\!>\\!n_{0}$,
and $N_{b}\\!=\\!n_{0}$. Our analysis rigorously establishes that the
emergence of SU(2) symmetry in the limit of $n_{0}\\!\gg\\!N_{b}$ or
$N_{b}\\!\gg\\!n_{0}$ guarantees the realization of optimal energy storage.
Additionally, we provide a verification of the negative correlation between
the entanglement and the energy storage capacity. Furthermore, we present
numerical evidence that, in the limit of $N_{b}\\!=\\!n_{0}\\!\gg\\!1$, the
asymptotically optimal energy storage can be attained. In Sect. IV, we provide
a discussion regarding the experimental implementation of optimal energy
storage within the TC battery and make a conclusion.

##  II Tavis-Cummings battery

The TC battery consists of $N_{b}$ battery cells, represented by the total
spin operators $S^{\alpha}\\!=\\!\sum_{j=1}^{N_{b}}\sigma_{j}^{\alpha}/2$
$(\alpha\\!=\\!x,y,z)$, and a charger described by bosonic operators $a$ and
$\hat{a}$. The TC Hamiltonian is given by:

$\displaystyle H=H_{b}+H_{c}+\lambda(t)H_{I},$ (1) $\displaystyle
H_{b}=\omega_{b}S^{z},\quad H_{c}=\omega_{c}a^{\dagger}a,$ $\displaystyle
H_{I}=g(S^{+}a+S^{-}a^{\dagger}),$

where $H_{b}$, $H_{c}$, and $H_{I}$ correspond to the Hamiltonian of the
battery, charger and their interaction, respectively. Here, the parameter $g$
characterizes the flip-flop interaction and $S^{\pm}\\!=\\!S^{x}\pm iS^{y}$
represents the spin ladder operators. We always consider the resonance
condition, specifically $\omega_{b}\\!=\\!\omega_{c}\\!\equiv\\!\omega$,
thereby guaranteeing the conservation of the total energy, i.e.,
$[H,H_{b}+H_{c}]\\!=\\!0$.

The QB is initially prepared in a product state
$|\psi(0)\rangle\\!=\\!|0\rangle_{b}|n_{0}\rangle_{c}$, where the battery is
in the ground state of $H_{b}$, i.e.,
$|0\rangle_{b}\equiv|\downarrow,\ldots,\downarrow\rangle$ and the charger is
in a Fock state $|n_{0}\rangle_{c}$. At time $t>0$, the interaction term
$H_{I}$ is activated to facilitate the transfer of energy from the charger to
the battery. Over a time interval $[0,\tau]$, our objective is to maximize the
amount of energy stored in the battery. This optimal time duration $\tau$ is
referred to as the “charging time”.

A pivotal step in our analysis involves the calculation of quantum dynamics
for the battery state. Due to the U(1) symmetry
$[H,S^{z}\\!+\\!a^{\dagger}a]\\!=\\!0$ and the local SU(2) symmetries
$[H,S^{\alpha}]\\!=\\!0$ $(\alpha\\!=\\!x,y,z)$, we observe that the TC
battery dynamics can be effectively reformulated within the following
subspace:

$\displaystyle\mathcal{B}\\!=\\!{\rm
span}\\{|0\rangle_{b}|n_{0}\rangle_{c},|1\rangle_{b}|n_{0}\\!-\\!1\rangle_{c},\cdots,|d\rangle_{b}|n_{0}\\!-\\!d\rangle_{c}\\},$
(2)

where $|m\rangle_{b}$ denotes the Dicke state
$|N_{b}/2,-N_{b}/2+m\rangle_{b}$, with $m=0,\ldots,N_{b}$ for the battery and
$|n\rangle_{c}$ signifies the Fock state for the charger. The parameter $d$ is
defined as $d\\!=\\!\min\\{N_{b},n_{0}\\}$.

Within the basis (2), we can express the Hamiltonian (1) as a
$(d+1)\times(d+1)$ matrix details are available in the Appendix A):

$\displaystyle\bm{H}=\begin{pmatrix}0&u_{1}&\\\ u_{1}&0&u_{2}&\\\
&\ddots&\ddots&\ddots&\\\ &&u_{d-1}&0&u_{d}\\\ &&&u_{d}&0\end{pmatrix},$ (3)

where $u_{j}\\!=\\!g\sqrt{j(N_{b}-j+1)(n_{0}-j+1)}$. Assuming that $\bm{H}$ is
diagonalized by a unitary matrix $\bm{U}$, i.e.,
$\bm{H}=\bm{U}\bm{E}\bm{U}^{\dagger}$ where $\bm{E}$ is a diagonal matrix, the
matrix representation of the wavefunction for the entire system at time $t$ is
given by:

$\displaystyle\bm{\psi}(t)=\bm{U}e^{-i\bm{E}t}\bm{U}^{\dagger}(1\ 0\ \ldots\
0)^{T}.$ (4)

Consequently, the corresponding quantum state is

$\displaystyle|\psi(t)\rangle=\bm{\psi}_{1}(t)|0\rangle_{b}|n_{0}\rangle_{c}+\cdots+\bm{\psi}_{d+1}(t)|d\rangle_{b}|n_{0}-d\rangle_{c}.$

This leads to the reduced density matrix of the battery state:

$\displaystyle\rho_{b}(t)$ $\displaystyle\equiv$
$\displaystyle\mathrm{Tr}_{c}(|\psi(t)\rangle\langle\psi(t)|)$ (6)
$\displaystyle=$ $\displaystyle|\bm{\psi}_{1}(t)|^{2}|0\rangle\langle
0|+\cdots+|\bm{\psi}_{d+1}|^{2}|d\rangle\langle d|.$

At time $t$, the energy stored in battery is given by

$\displaystyle\Delta E(t)$ $\displaystyle\equiv$
$\displaystyle\mathrm{Tr}(H_{b}\rho_{b}(t))-\mathrm{Tr}(H_{b}\rho_{b}(0))$ (7)
$\displaystyle=$
$\displaystyle\frac{N_{b}}{2}\omega+\sum_{j=1}^{d+1}\left(j-\frac{N_{b}}{2}-1\right)|\bm{\psi}_{j}(t)|^{2}\omega.$

The definition of _optimal energy storage_ encompasses the following two
scenarios:
I) The battery absorbs all the energy initially stored in the charger,
resulting in zero charger energy after the charging process.
II) The battery absorbs some energy from the charger and reaches a fully
charged state $|\uparrow,\ldots,\uparrow\rangle_{b}$.
Case I corresponds to a situation where the initial energy of the charger is
less than the energy storage capacity of the battery, i.e.,
$n_{0}\\!<\\!N_{b}$, while Case II corresponds to the opposite case where
$n_{0}\\!>\\!N_{b}$. Referring to Eq. (II), optimal energy storage implies
that the evolved quantum state is
$|\psi(\tau)\rangle=|d\rangle_{b}|n_{0}-d\rangle_{c}$. In other words, the
optimal condition can also be expressed as

$\displaystyle\exists\tau\ {\rm such\ that}\ \Delta E(\tau)=\omega d,$ (8)

where $d\\!=\\!\min\\{N_{b},n_{0}\\}$ and $\Delta E(\tau)\equiv\max_{t}\Delta
E(t)$ is called the energy storage capacity. From the perspective of quantum
correlations, thus, optimal energy transport occurs when there is no
entanglement between the battery and the charger. We will further elaborate on
this point in the ensuing discussion.

It is worth noting that the dimension of the matrix (3) no longer exhibits
exponential growth with the number of battery cells $N_{b}$ but rather depends
linearly on the minimum of $N_{b}$ and $n_{0}$. As a result, compared to the
Bethe ansatz approach Lu  _et al._ (2021), our method offers greater numerical
tractability. In the subsequent sections, we will demonstrate that our
approach also provides profound insights into the algebraic structure of the
TC model, enabling us to analytically address the optimal energy storage
problem.

##  III energy transport and Entanglement

###  III.1 Parallel charging: $N_{b}=1$

The parallel charging scheme involves the assembly of $N$ Jaynes-Cummings (JC)
models, each of which contains only a single battery cell, as depicted in Fig.
1(b). The JC model is a special case of the TC model by taking
$N_{b}\\!=\\!1$. In this scenario, a straightforward calculation yields the
following evolved quantum state (II):

$\displaystyle|\psi(t)\rangle\\!=\\!\cos(g\sqrt{n_{0}}t)|0\rangle_{b}|n_{0}\rangle_{c}\\!-\\!{\rm
i}\sin(g\sqrt{n_{0}}t)|1\rangle_{b}|n_{0}\\!-\\!1\rangle_{c}.$

It thus follows from Eq. (7) that the energy stored in battery is given by

$\displaystyle\Delta E(t)=\omega\sin^{2}(g\sqrt{n_{0}}t),$ (9)

which indicates that the battery state can be excited to its highest energy
state, $|\uparrow\rangle$, at time $\tau=\pi/(2g\sqrt{n_{0}})$. Hence, we can
conclude that the optimal energy storage can be achieved in the parallel
charging scheme. However, we know that the collective charging scheme offers a
charging speedup Hovhannisyan  _et al._ (2013); Binder  _et al._ (2015);
Campaioli  _et al._ (2017). Therefore, our primary focus will be on seeking
optimal energy storage within the collective charging scheme, i.e., when
$N_{b}\\!>\\!1$.

###  III.2 Collective charging: $n_{0}\\!>\\!N_{b}$

Figure 2:  Negative relation between the battery-charger entanglement
$S(\tau)$ and the energy storage capacity $\Delta E(\tau)$ in the case of
$n_{0}\\!\geq\\!N_{b}\\!=\\!2$. Other parameters are set to $\omega\\!=\\!1$
and $g\\!=\\!1$.

In this section, we consider the case where $n_{0}>N_{b}$, with the
expectation that the battery can be fully charged. For this purpose, we first
examine the simplest scenario of $N_{b}=2$ and $n_{0}>N_{b}$ to illustrate our
analytical approach. Under this circumstance, the Hamiltonian (3) can be
represented as a three-dimensional matrix:

$\displaystyle\bm{H}=\begin{pmatrix}0&g\sqrt{2n_{0}}&0\\\
g\sqrt{2n_{0}}&0&g\sqrt{2(n_{0}-1)}\\\ 0&g\sqrt{2(n_{0}-1)}&0\\\
\end{pmatrix}.$ (10)

Its eigenproblem (4) can be solved straightforward as $\bm{E}={\rm
diag}(0,\delta,-\delta)$ and

$\displaystyle\bm{U}\\!=\\!\frac{1}{2\sqrt{2n_{0}-1}}\begin{pmatrix}2\sqrt{n_{0}-1}&\sqrt{2n_{0}}&\sqrt{2n_{0}}\\\
0&\sqrt{4n_{0}-2}&-\sqrt{4n_{0}-2}\\\
-2\sqrt{n_{0}}&\sqrt{2(n_{0}-1)}&\sqrt{2(n_{0}-1)}\end{pmatrix},$

where $\delta=g\sqrt{4n_{0}-2}$. Substituting the above solution into Eq. (4),
we obtain

$\displaystyle|\bm{\psi}_{1}(t)|^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{(2n_{0}-1)^{2}}[n_{0}-1+n_{0}\cos(\delta t)]^{2},$
$\displaystyle|\bm{\psi}_{2}(t)|^{2}$ $\displaystyle=$
$\displaystyle\frac{n_{0}}{2n_{0}-1}[1-\cos^{2}(\delta t)],$
$\displaystyle|\bm{\psi}_{3}(t)|^{2}$ $\displaystyle=$
$\displaystyle\frac{n_{0}(n_{0}-1)}{(2n_{0}-1)^{2}}[1-\cos(\delta t)]^{2}.$
(12)

Then, according to Eq. (7), we have

$\displaystyle\Delta E(t)$ $\displaystyle=$
$\displaystyle\frac{\omega}{(2n_{0}-1)^{2}}[-n_{0}\cos^{2}(\delta t)$ (13)
$\displaystyle+4n_{0}(1-n_{0})\cos(\delta t)+n_{0}(4n_{0}-3)].$

The maximum of $\Delta E(t)$ is thus given by

$\displaystyle\Delta E(\tau)=\omega\left[2-\frac{2}{(2n_{0}-1)^{2}}\right],$
(14)

occurring at $\tau=\pi/[g\sqrt{4n_{0}-2}]$. From Eq. (14), it is evident that
the energy storage capacity $\Delta E(\tau)$ becomes larger as the number of
initial photons, $n_{0}$, increases. If $n_{0}\to\infty$, optimal energy
storage can be realized.

As demonstrated in Ref. Shi  _et al._ (2022), the energy storage is
constrained by the battery-charger entanglement. Hence, it is anticipated that
as $n_{0}$ increases, the entanglement will decrease and eventually vanish as
$n_{0}\to\infty$. Utilizing the von Neumann entropy and the expression for the
reduced density matrix of the battery in Eq. (6), we find that at time $\tau$,
the entanglement entropy is given by

$\displaystyle S(\tau)$ $\displaystyle\equiv$
$\displaystyle-\mathrm{Tr}(\rho_{b}(\tau)\log_{2}\rho_{b}(\tau))$ (15)
$\displaystyle=$ $\displaystyle h\left(\frac{1}{(2n_{0}-1)^{2}}\right)$

where $h(x)=-x\log_{2}x-(1-x)\log_{2}(1-x)$ represents the binary Shannon
entropy function. Since $h(x)$ is an increasing function when $0\leq x\leq
1/2$ and $n_{0}>N_{b}=2$, we can conclude that entanglement decreases as
$n_{0}$ increases. Therefore, we validate the negative relation between the
energy storage capacity and the entanglement, see Fig. 2.

Figure 3:  Comparison of analytical results of Eqs. (20) and (24) with the
numerical results. This comparison is valid for both the cases of
$n_{0}\\!=\\!N_{b}/2\\!=\\!5$ and $N_{b}\\!=\\!n_{0}/2\\!=\\!5$. Other
parameters are set to $\omega\\!=\\!1$ and $g\\!=\\!1$.

A natural question that arises is whether optimal energy storage can be
achieved under the condition $n_{0}\\!\gg\\!N_{b}$, not limited to the
specific case of $N_{b}\\!=\\!2$ as discussed earlier. It appears challenging
to address this question by directly solving the eigenproblem of Eq. (3) for
arbitrary values of $N_{b}$. Fortunately, we observe that under the condition
$n_{0}\\!\gg\\!N_{b}$ the matrix element
$u_{j}\\!=\\!g\sqrt{j(N_{b}-j+1)(n_{0}-j+1)}$ in Eq. (3) can be approximated
as $u_{j}\\!\simeq\\!g\sqrt{n_{0}-N_{b}/2+1/2}\sqrt{j(N_{b}-j+1)}$. With this
approximation, the Hamiltonian (3) can be mapped to the spin-$N_{b}/2$
operator, see the Appendix B:

$\displaystyle\bm{H}\simeq\Omega\bm{J}^{x},$ (16)

where $\Omega$ is the generalized Rabi frequency:

$\displaystyle\Omega=2g\sqrt{n_{0}-\frac{N_{b}-1}{2}}.$ (17)

Now, the Hamiltonian Eq. (16) is just one of the generators of the su(2) Lie
algebra. Therefore, we can assert that SU(2) symmetry emerges in this limiting
case. By using Eq. (16), we can reformulate Eq. (7) as

$\displaystyle\Delta E$ $\displaystyle=$
$\displaystyle\frac{N_{b}\omega}{2}+\bm{\psi}^{\dagger}(0)\bm{J}^{z}(t)\bm{\psi}(0)\omega,$
(18)

where $\bm{J}^{z}(t)=\exp({\rm i}t\Omega\bm{J}^{x})\bm{J}^{z}\exp(-{\rm
i}t\Omega\bm{J}^{x})$.

Now, it is evident that our question has been reduced to a Lie algebra
problem, thanks to the emergence of SU(2) symmetry in the limit
$n_{0}\\!\gg\\!N_{b}$. Using the Baker-Campbell-Hausdorff formula, we have

$\displaystyle\bm{J}^{z}(t)=\sin(\Omega t)\bm{J}^{y}+\cos(\Omega
t)\bm{J}^{z}.$ (19)

By substituting Eq. (19) into Eq. (18), we obtain

$\displaystyle\Delta E(t)=\frac{N_{b}\omega}{2}[1-\cos(\Omega t)],$ (20)

which is extremely accurate even in the finite case
$N_{b}\\!=\\!n_{0}/2\\!=\\!5$, see Fig 3. Equation (20) indicates that optimal
energy storage $\Delta E=N_{b}\omega$ can be achieved at $\tau=\pi/\Omega$
when $n_{0}\\!\gg\\!N_{b}$.

###  III.3 Collective charging: $N_{b}>n_{0}$

In Sec. III.2, we employed the invariant subspace method, as given in Eq. (3),
to reveal the fundamental algebraic structure that characterizes the quantum
dynamical behavior of the TC model. This approach was entirely applicable to
the case where $N_{b}>n_{0}$. To delve deeper into the core of our
investigation, we will now proceed directly to explore the scenario where
$N_{b}\gg n_{0}$ and claim that the optimal energy storage can be realized.

For $N_{b}\\!\gg\\!n_{0}$, we can approximate $u_{j}$ as
$u_{j}\\!\simeq\\!g\sqrt{N_{b}-n_{0}/2+1/2}\sqrt{j(n_{0}-j+1)}$ and then the
Hamiltonian (3) can be approximated as a spin-$n_{0}/2$ operator
$\bm{\tilde{J}}^{\alpha}$, i.e.,

$\displaystyle\bm{H}\simeq\tilde{\Omega}\bm{\tilde{J}}^{x},$ (21)

where

$\displaystyle\tilde{\Omega}=2g\sqrt{N_{b}-\frac{n_{0}-1}{2}}.$ (22)

Thus, the energy transfer Eq. (7) can be repressed as

$\displaystyle\Delta E$ $\displaystyle=$
$\displaystyle\frac{n_{0}\omega}{2}+\bm{\psi}^{\dagger}(0)\bm{\tilde{J}}^{z}(t)\bm{\psi}(0)\omega,$
(23)

where $\bm{\tilde{J}}^{z}(t)=\exp({\rm
i}t\tilde{\Omega}\bm{\tilde{J}}^{x})\bm{\tilde{J}}^{z}\exp(-{\rm
i}t\tilde{\Omega}\bm{\tilde{J}}^{x})$. We see that these equations are highly
similar to the $n_{0}\gg N_{b}$ case if we exchange $n_{0}$ and $N_{b}$.
Therefore, we can deduce that

$\displaystyle\Delta E(t)=\frac{n_{0}\omega}{2}[1-\cos(\tilde{\Omega}t)],$
(24)

which suggests that the optimal energy storage $\Delta E=n_{0}\omega$ can be
realized at charing time $\tau=\pi/\tilde{\Omega}$ in the case of $N_{b}\gg
n_{0}$. Figure 3 shows the consistency of the limit result with the numerical
result obtained also in the finite case $n_{0}\\!=\\!N_{b}/2\\!=\\!5$.

###  III.4 Collective charging: $n_{0}=N_{b}$

The most economical charging scheme is preparing the initial charger with the
same energy as the energy storage capacity of the battery, i.e.,
$n_{0}\\!=\\!N_{b}$. As we can see in the case of
$n_{0}\\!\geq\\!N_{b}\\!=\\!2$ (Fig. 2), however, the energy storage capacity
is weakest for the $n_{0}\\!=\\!N_{b}\\!=\\!2$ case. Hence, we inquire
whether, in the limiting case of $n_{0}=N_{b}\gg 1$, we can approach the
optimal energy storage. Within this parameter range, no evident approximations
for Eq. (3) can be employed to obtain analytical results. Nevertheless, we can
also perform an effective numerical calculation based on Eq. (3) to analyze
the charging behavior in the case of $n_{0}\\!=\\!N_{b}$.

Figure 4:  Negative relation between average entanglement $s(\tau)\equiv
S(\tau)/N_{b}$ and average energy storage capacity
$\Delta\varepsilon(\tau)\equiv\Delta E(\tau)/N_{b}$ in the case of
$N_{b}\\!=\\!n_{0}$. Other parameters are set to $\omega\\!=\\!1$ and
$g\\!=\\!1$.

Figure 4 illustrates the average entanglement
$s(\tau)\\!\equiv\\!S(\tau)/N_{b}$ and the average stored energy
$\Delta\varepsilon(\tau)\\!\equiv\\!\Delta E(\tau)/N_{b}$ at the charging time
$\tau$ for the scenario where $N_{b}\\!=\\!n_{0}$. The motivation for
averaging these quantities stems from the fact that the support of the reduced
density matrix, Eq. (6), is determined by $\min\\{n_{0},N_{b}\\}+1$, which
equals $(N_{b}+1)$ in this scenario. Therefore, average quantities enable us
to compare cases with varying values of $N_{b}$. Figure 4 shows that the
average energy storage capacity $\Delta\varepsilon(\tau)$ initially decreases
and then increases with an increase in the size of the QB. Conversely, the
average entanglement between the battery and the charger displays the opposite
behavior compared to $\Delta\varepsilon(\tau)$. We have previously established
a negative relation between the energy storage and the entanglement for a
fixed QB size in Ref. Shi  _et al._ (2022). Figure 4 demonstrates that this
negative relation persists even when the QB size is not fixed. It is important
to emphasize that as $N_{b}$ becomes sufficiently large $(>100)$, the
asymptotically optimal energy transfer $(\Delta\varepsilon(\tau)>0.95)$ can be
achieved, as depicted in Fig. 4. It is worth noting that a specific
perturbation theory, based on polynomially deformed $su$(2) algebras Vadeiko
_et al._ (2003), can be applied to address the eigenproblem of the TC model in
the case of $N_{b}=n_{0}$. Exploring the potential applicability of this
method to address QBs problems is an interesting topic for further discussion

##  IV Discussion

In this work, we have investigated the realization of optimal energy storage
in the TC battery. By selecting the Fock state as the initial state of the
charger, we have introduced the invariant subspace method to investigate the
quantum dynamics of the TC model. This approach has enabled us to demonstrate
that SU(2) symmetry emerges in case where either $n_{0}\gg N_{b}$ or $N_{b}\gg
n_{0}$. Consequently, under these two conditions, the optimal energy storage
in the TC battery can be achieved. In the scenario where $N_{b}\\!=\\!n_{0}$,
we have conducted numerical investigations into the energy storage problem,
confirming the existence of a negative relation between the battery-charger
entanglement and the energy storage capacity. Our numerical results have
indicated the possibility of asymptotically achieving optimal energy storage
in the limit $N_{b}\\!=\\!n_{0}\\!\gg\\!1$.

It is worth noting that the TC battery is possibly realized using the state-
of-the-art solid-state technology through the circuit/cavity QED Fink  _et
al._ (2009); Yang  _et al._ (2020); Leek  _et al._ (2009); Lolli  _et al._
(2015); Liu  _et al._ (2023). Reference Fink  _et al._ (2009) reported an
ideal realization of the TC model in the absence of atom number and coupling
fluctuations by embedding a discrete number of fully controllable
superconducting qubits at fixed positions into a transmission line resonator.
Reference Liu  _et al._ (2023) realized strong coupling between deterministic
single-atom arrays and a high-finesse miniature optical cavity. While our
theoretical starting point has been the Fock state as the initial charge, in
experimental settings, coherent light can be used as an approximation to the
Fock state. This simplifies the difficulty of experimentally preparing Fock
states. Consequently, our theoretical approaches to the optimal energy storage
can be implemented in a well-designed circuit/cavity QED system. Our work not
only contributes to a profound understanding of the algebraic structure of the
TC model but also offers significant insights into the problem of the optimal
energy storage. We anticipate that our approach can be extended to address
other quantum battery-related challenges in the future.

###### Acknowledgements.

This work was supported by the NSFC (Grants No. 12275215, No. 12305028, No.
12247103, and No. 12175178), the Major Basic Research Program of Natural
Science of Shaanxi Province (Grant No. 2021JCW-19), Shaanxi Fundamental
Science Research Project for Mathematics and Physics (Grant No. 22JSZ005) and
the Youth Innovation Team of Shaanxi Universities. Q.-K. Wan and H.-L. Shi
received partial support from the NSFC key grants No. 12134015 and No.
92365202. H.-L. Shi was supported by the European Commission through the H2020
QuantERA ERA-NET Cofund in Quantum Technologies project “MENTA”. The authors
H.-L. Shi and H.-Y. Yang contributed equally to this work.

## Appendix A: Matrix expression of Hamiltonian (1) in the invariant subspace

According to the algebraic relations:

$\displaystyle S^{+}|m\rangle_{b}\\!=\\!\sqrt{(N_{b}-m)(m+1)}|m+1\rangle_{b},$
$\displaystyle S^{-}|m\rangle_{b}\\!=\\!\sqrt{m(N_{b}-m+1)}|m-1\rangle_{b},$
$\displaystyle a^{\dagger}|n\rangle_{c}=\sqrt{n+1}|n+1\rangle_{c},$
$\displaystyle a|n\rangle_{c}=\sqrt{n}|n-1\rangle_{c},$ (25)

the matrix elements of the Hamiltonian (1) expressed in the invariant subspace
(2) are given by

$\displaystyle\bm{H}_{jk}$ $\displaystyle\\!=\\!$ $\displaystyle\langle
j\\!-\\!1|_{b}\langle
n_{0}\\!-\\!j\\!+\\!1|_{c}H|k\\!-\\!1\rangle_{b}|n_{0}\\!-\\!k\\!+\\!1\rangle_{c}$
(26) $\displaystyle\\!=\\!$ $\displaystyle g\langle j\\!-\\!1|_{b}\langle
n_{0}\\!-\\!j\\!+\\!1|_{c}(S^{+}a+{\rm
H.C.})|k\\!-\\!1\rangle_{b}|n_{0}\\!-\\!k\\!+\\!1\rangle_{c}$
$\displaystyle\\!=\\!$ $\displaystyle g\langle
j\\!-\\!1|_{b}S^{+}|k\\!-\\!1\rangle_{b}\langle
n_{0}\\!-\\!j\\!+\\!1|_{c}a|n_{0}\\!-\\!k\\!+\\!1\rangle_{c}$
$\displaystyle+g\langle j\\!-\\!1|_{b}S^{-}|k\\!-\\!1\rangle_{b}\langle
n_{0}\\!-\\!j\\!+\\!1|_{c}a^{\dagger}|n_{0}\\!-\\!k\\!+\\!1\rangle_{c}$
$\displaystyle\\!=\\!$ $\displaystyle
u_{j-1}\delta_{j-1,k}+u_{j}\delta_{j+1,k},$

where $u_{j}\\!=\\!g\sqrt{j(N_{b}-k+1)(n_{0}-j+1)}$ and
$j,k\\!=\\!1,2,\ldots,\min\\{N_{b},n_{0}\\}$.

## Appendix B: Generators of su(2) algebra

The spin-$d$ irreducible representation of the generators of su(2) algebra is
given by

$\displaystyle\bm{S}^{x}=\frac{1}{2}\begin{pmatrix}0&f_{1}&\\\
f_{1}&0&f_{2}&\\\ &\ddots&\ddots&\ddots&\\\ &&f_{d-1}&0&f_{d}\\\
&&&f_{d}&0\end{pmatrix},$ $\displaystyle\bm{S}^{y}=\frac{1}{2\rm
i}\begin{pmatrix}0&-f_{1}&\\\ f_{1}&0&-f_{2}&\\\ &\ddots&\ddots&\ddots&\\\
&&f_{d-1}&0&-f_{d}\\\ &&&f_{d}&0\end{pmatrix},$
$\displaystyle\bm{S}^{z}=\begin{pmatrix}-d&0&\\\ 0&-d\\!+\\!1&0&\\\
&\ddots&\ddots&\ddots&\\\ &&0&d\\!-\\!1&0\\\ &&&0&d\end{pmatrix}.$ (27)

where $f_{j}\equiv\sqrt{j(2d-j+1)}$. If we take $d\\!=\\!N_{b}/2$, then
$\bm{S}^{\alpha}$ ($\alpha\\!=\\!x,y,z$) correspond to the operators
$\bm{J}^{\alpha}$, see Eqs. (16), (18), and (19). If we take
$d\\!=\\!n_{0}/2$, then $\bm{S}^{\alpha}$ correspond to the operators
$\bm{\tilde{J}}^{\alpha}$, see Eqs. (21) and (23).

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# Absolute absorption on the potassium D lines: theory and experiment

Ryan K Hanley, Philip D Gregory, Ifan G Hughes and Simon L Cornish  Joint
Quantum Centre (JQC) Durham-Newcastle, Department of Physics, Durham
University, South Road, Durham, DH1 3LE, United Kingdom
<EMAIL_ADDRESS>
###### Abstract

We present a detailed study of the absolute Doppler-broadened absorption of a
probe beam scanned across the potassium D lines in a thermal vapour. Spectra
using a weak probe were measured on the 4S $\rightarrow$ 4P transition and
compared to the theoretical model of the electric susceptibility detailed by
Zentile et al(2015) in the code named ElecSus. Comparisons were also made on
the 4S $\rightarrow$ 5P transition with an adapted version of ElecSus. This is
the first experimental test of ElecSus on an atom with a ground state
hyperfine splitting smaller than that of the Doppler width. An excellent
agreement was found between ElecSus and experimental measurements at a variety
of temperatures with rms errors $\sim 10^{-3}$. We have also demonstrated the
use of ElecSus as an atomic vapour thermometry tool, and present a possible
new measurement technique of transition decay rates which we predict to have a
precision of $\sim$ $3\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$.

##  1 Introduction

The study of thermal vapours is a rapidly expanding field of physics with many
applications. These range from all optical delay lines [1], quantum atomic
memories [2, 3, 4, 5], clocks [6, 7] and compact magnetometry [8, 9, 10] to
name but a few. The knowledge of the absorptive and dispersive properties of
an atomic vapour has many benefits for existing and potential applications.
This has been shown to facilitate the design of optical components, such as
optical isolators [11] and atomic Farday filters [12, 13, 14], without the
need for exhaustive trial and error. Furthermore, spectroscopy of alkali-metal
elements, such as rubidium and caesium, is used for many laser-locking schemes
[15, 16, 17]. Knowledge of the response of a thermal vapour to a weak resonant
beam can aid the effective design of such systems.

The absolute Doppler-broadened absorption of a probe beam resonant on the D
lines of rubidium has been extensively studied by Siddons et al[18]. They
presented a theoretical framework which was utilised to correctly predict the
absorption profile of a weak probe beam resonant on the D lines. Adapted from
this theoretical framework, Zentile et aldeveloped ‘ElecSus’ [19]. ElecSus is
a program which calculates the electric susceptibility of a thermal vapour in
the vicinity of the alkali-metal D lines which in turn allows one to calculate
many optical measurables, such as the transmission of a weak probe beam, the
dispersive properties of the medium [20] and the magnetic shifts of spectra
[21]. The absolute Doppler-broadened absorption on the potassium D lines is
yet to be studied extensively. The D lines of potassium are currently
extensively used for gray molasses cooling [22, 23, 24, 25] to create
ultracold ensembles of atoms, which are used to simulate many-body problems in
physics [26, 27]. It has also been shown that using the 4S $\rightarrow$ 5P D
line transitions allows for greater cooling of the potassium atoms due to the
reduced linewidth [28, 29]. Potassium is of particular interest due to the
existence of both bosonic and fermionic species [30, 31, 32, 33] which allows
one to study interactions in Bose-Fermi mixtures [34].

Spectroscopy of potassium is different from that of rubidium due to the small
spacing of its hyperfine energy levels. In particular, the ground state
hyperfine splitting in potassium is less than the Doppler width and hence only
one absorption profile is seen on each of the D lines [35, 36]. The aim of
this work is to establish the weak-probe regime for the 4S $\rightarrow$ 4P
and 4S $\rightarrow$ 5P D lines in potassium in order to test the quality of
ElecSus on an atom where the ground state hyperfine splitting is less than the
Doppler width.

The paper is structured as follows. In section 2, we give a brief description
of the weak-probe regime and the physics behind ElecSus. Section 3 provides
details of the experimental apparatus and procedure. Section 4 details the
atomic structure of the potassium D line transitions along with our
experimental results. Section 5 outlines a possible new experiment technique
to measure transition decay rates. We conclude our findings in section 6.

##  2 Theory

###  2.1 Weak-probe regime

The weak-probe regime of a resonant probe beam is defined as the incident
intensity $I_{0}$ at which a the probe beam intensity is sufficiently weak
that the absorption coefficient is independent of incident intensity. This
regime in rubidium is typically several orders of magnitude smaller than the
saturation intensity $I_{\rm{sat}}$ of the transition [18]. The saturation
intensity of a two-level system is defined by

$I_{\rm{sat}}=\frac{\pi hc\Gamma}{3\lambda^{3}}~{},$ (1)

where $h$ is Planck’s constant, $c$ is the speed of light, $\Gamma$ is the
linewidth and $\lambda$ is the wavelength of the transition [37].

Despite their simple atomic structure, alkali-metal atoms cannot be described
by a simple two-level model. In particular, the presence of multiple hyperfine
levels in the ground state can significantly alter the transmission of
resonant probe light. Consider a system with two ground states
$\left|g_{\rm{1}}\right\rangle$ and $\left|g_{\rm{2}}\right\rangle$ and one
excited state $\left|e\right\rangle$ where the transitions
$\left|g_{\rm{1}}\right\rangle\rightarrow\left|e\right\rangle$ and
$\left|g_{\rm{2}}\right\rangle\rightarrow\left|e\right\rangle$ are allowed and
$\left|g_{\rm{1}}\right\rangle\rightarrow\left|g_{\rm{2}}\right\rangle$ is
forbidden. If the probe beam is resonant with the
$\left|g_{\rm{1}}\right\rangle\rightarrow\left|e\right\rangle$ transition,
atoms are excited to $\left|e\right\rangle$ but they may spontaneously decay
from $\left|e\right\rangle\rightarrow\left|g_{\rm{1}}\right\rangle$ or
${\left|e\right\rangle\rightarrow\left|g_{\rm{2}}\right\rangle}$. As the light
is off-resonance with the
${\left|g_{\rm{2}}\right\rangle\rightarrow\left|e\right\rangle}$ transition,
the atoms which have decayed into $\left|g_{\rm{2}}\right\rangle$ remain
there. This leads to a depletion in the ground state population of
${\left|g_{\rm{1}}\right\rangle}$ and hence an increase in transmission of the
probe beam. Redistributing the ground state populations using light is known
as optical pumping [38]. This system therefore has non-trivial dependencies on
the determination of the weak-probe regime due to optical pumping.

###  2.2 ElecSus

The frequency dependent complex electric susceptibility of a dielectric
$\chi\left(\Delta\right)$, where $\Delta$ is the detuning from resonance,
connects the macroscopic polarisation to the applied electric field [39]. The
real part $\chi_{\rm{R}}\left(\Delta\right)$ describes the dispersive
properties and the imaginary part $\chi_{\rm{I}}\left(\Delta\right)$ describes
the absorptive properties.

In order to calculate $\chi\left(\Delta\right)$ in the vicinity of the D line
transitions, ElecSus first calculates the transition strengths of allowed
electric-dipole transitions between all hyperfine sub-levels. To account for
Doppler broadening due to the line-of-sight thermal motion of the atoms, a
lineshape of the resonance is applied to each transition. This lineshape is a
convolution of the Lorentzian atomic lineshape, dependent on $\Gamma$, and the
Gaussian velocity distribution which is a function of temperature. Finally,
$\chi\left(\Delta\right)$ is calculated by summing over all transitions. From
$\chi\left(\Delta\right)$, many optical observables may be predicted.

As $\chi_{\rm{I}}\left(\Delta\right)$ describes the absorptive properties of
the medium, the absorption coefficient $\alpha$ is given by
$\alpha\left(\Delta\right)=k\chi_{\rm{I}}\left(\Delta\right)$ [18] where $k$
is the wavevector of the incident radiation. The absorption coefficient is
dependent on the linewidth of the transition and the atomic number density
${\cal{N}}$ in the vapour which is exponential in the temperature, $T$. For a
particular hyperfine transition, the absorption coefficient is given by [18]

$\alpha\left(\Delta\right)=kC_{\rm{F}}^{2}d^{2}{\cal{N}}\frac{1}{2\left(2{\cal{I}}+1\right)}\frac{1}{\hbar\epsilon_{0}}s^{\rm{I}}\left(\Delta\right)~{},$
(2)

where $C_{\rm{F}}$ is the total transition strength, ${\cal{I}}$ is the
nuclear spin, $\hbar$ is the reduced Planck’s constant, $\epsilon_{0}$ is the
permittivity, $s^{\rm{I}}\left(\Delta\right)$ is the transition lineshape and
$d$ is given by

$d=\sqrt{3}\sqrt{\frac{3\epsilon_{0}\hbar\Gamma\lambda^{3}}{8\pi^{2}}}~{},$
(3)

where $\lambda$ is the wavelength of the transition. Therefore, using the
Beer-Lambert law, one is able to predict the transmission of a weak probe beam
resonant on the alkali-metal D lines. For rubidium and caesium a vapour cell
of a few cm length at room temperature has sufficient number density to
observe significant absorption features. However, due to the higher melting
point of potassium [40], the vapour cell must be heated to ensure a large
enough atomic vapour number density such that absorption features are
observed.

##  3 Experimental details

The experimental configuration is shown in Figure 1. A home-built external
cavity diode laser (ECDL) in the Littrow configuration [41] was the light
source. An Eagleyard EYP-RWE-0790-04000-0750-SOT01-0000 laser diode was used
for the 4S $\rightarrow$ 4P transition and a Sanyo DL5146-101S laser diode was
used for the 4S $\rightarrow$ 5P transition. The output beam from the laser
passed through an optical isolator (OI) before impinging on a 70:30
beamsplitter (BS). The stronger beam was directed into a scanning Fabry-Perot
etalon, which was used for frequency calibration. A Toptica FPI-100-0750-y
scanning Fabry-Perot etalon with a free spectral range of $1.00\pm
0.01\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ and a Thorlabs SA200-3B scanning
Fabry-Perot etalon with a free spectral range of $1.50\pm
0.01\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ were used for the 4S
$\rightarrow$ 4P and 4S $\rightarrow$ 5P transitions respectively. The weaker
beam was sent through a variable neutral density filter (ND), to ensure
control of the probe beam intensity, before passing through a resistively
heated, $10\text{\,}\mathrm{c}\mathrm{m}$, natural abundance potassium vapour
cell (K Cell). The magnetic field over the cell was measured to be of the
order of ${10}^{-1}\text{\,}\mathrm{G}$. Note that no attempt was made to null
the field. The probe beam was then focused onto a calibrated photodiode (PD)
which was integrated into a low-pass filter circuit in order to remove high-
frequency noise and allow for greater sensitivity. The minimum detection power
was $\sim$1\text{\,}\mathrm{n}\mathrm{W}$$.

Figure 1: Experimental configuration for the potassium spectroscopy. ECDL -
extended cavity diode laser; OI - optical isolator; BS - 70:30 beam splitter;
ND - neutral density filter; PD - photodiode; K Cell - heated potassium vapour
cell (see main text). The $100\text{\,}\mathrm{m}\mathrm{m}$ and
$50\text{\,}\mathrm{m}\mathrm{m}$ refer to the focal lengths of the lenses.

The wavelength of light from the ECDL was tuned to the D lines by adjusting
the angle of the diffraction grating. The wavelength was measured using a
HighFinesse WS-5 wavelength meter to ensure the correct transitions were being
probed. The laser was then scanned by applying a triangular ramp voltage at
$10\text{\,}\mathrm{H}\mathrm{z}$ to a piezo stack actuator controlling the
grating position. This allowed the laser to be scanned approximately
$8\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ before a mode hop. This was
sufficiently large to scan the whole Doppler-broadened transition. Note that
current feed-forward was employed on the blue 4S $\rightarrow$ 5P transition
in order to increase the scan range of the laser.

The voltage output from the calibrated photodiode was recorded on a digital
oscilloscope in order to record the transmission spectrum. The ramp voltage
and etalon transmission peaks were also recorded. The etalon transmission
peaks were used to convert from the measurement time into frequency, as well
as linearising the scan. The spacing between the etalon peaks was plotted as a
function of observed time. A polynomial was then fitted to a plot of the
difference between observed and expected transmission peak times and
subsequently used to linearise the scan.

The weak-probe regime was established by varying the intensity of the probe
beam incident on the cell from ${1\times 10^{-3}~{}I_{\text{sat}}\rightarrow
1\times 10^{2}~{}I_{\text{sat}}}$ by inserting a variety of neutral density
filters into the path of the probe beam and measuring the minimum
transmission. This was taken to be the mean over several points at the minimum
of the transmission spectrum. Using a CCD camera, the $1/e^{2}$ widths of the
probe beam were measured by taking cross sectional images of the probe beam
before the cell and fitting Gaussian profiles to the images. The incident
power was measured using the photodiode in the wings of the transmission
spectrum. Spectra in the weak-probe regime were recorded at a variety of
temperatures. Theoretical spectra generated by ElecSus were fit to the
experimental spectra. A linear fit was made to the wings of the spectrum in
order normalise the data to account for a weak variation in the laser power
during the laser scan.

##  4 Results and Discussion

###  4.1 4S $\rightarrow$ 4P D lines

In order to have measurable absorption of a probe beam with a good signal-to-
noise ratio, the potassium vapour cell was heated to a typical temperature of
$\sim$40\text{\,}\mathrm{\SIUnitSymbolCelsius}$$, as at this temperature there
is sufficient atomic vapour number density to absorb a significant proportion
of the incident beam. At a temperature of
$40\text{\,}\mathrm{\SIUnitSymbolCelsius}$, the Doppler width is
$\sim$0.8\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$. This is much larger than
the ground state hyperfine splitting of the potassium isotopes shown in Figure
2. Consequently, one would expect to observe only a single Doppler-broadened
transmission spectrum for each of the D lines with contributions from both
ground states. This can be seen in plots (i) and (iii) of Figure 2. The
saturation intensity and decay rates for the D1 and D2 transitions are
$1.71\text{\,}\mathrm{m}\mathrm{W}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$,
$1.75\text{\,}\mathrm{m}\mathrm{W}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$,
${\Gamma/2\pi=$5.956\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$}$ and
${\Gamma/2\pi=$6.035\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$}$ respectively
[42].

Figure 2: An energy level diagram of the 4S $\rightarrow$ 4P D lines of 39K
and 41K along with their natural abundance. Note that we have neglected 40K
due to its natural abundance of $0.0117\text{\,}\%$. Data taken from [30, 43,
44, 45, 46, 47]. Plots (i) and (iii) show the calculated contributions to the
total transmission spectrum of a probe beam resonant with the D1 and D2
transition passing through a $10\text{\,}\mathrm{c}\mathrm{m}$ natural
abundance potassium vapour cell at $40\text{\,}\mathrm{\SIUnitSymbolCelsius}$
respectively. The blue (dotted) and green (solid) spectra correspond to
transitions from $F=1\rightarrow F^{\prime}$ and $F=2\rightarrow F^{\prime}$
for 39K respectively. The red (dashed) and cyan (dot-dashed) spectra
correspond to transitions from $F=1\rightarrow F^{\prime}$ and $F=2\rightarrow
F^{\prime}$ for 41K respectively. The thick black profile is the total
transmission spectrum. Plots (ii) and (iv) show the transition strengths,
weighted by the isotopic abundance, where the red and green lines correspond
to 39K and 41K respectively. The linear detuning ($\Delta/2\pi$) is referenced
to the weighted line-centre of the transition.

In order to test ElecSus, the weak-probe regime was experimentally determined.
In Figure 3, the main figure shows the minimum transmission of a probe beam as
a function of incident intensity, normalised to the saturation intensity, of
the D2 transition. The red data points show the minimum transmission of a
probe beam, with $1/e^{2}$ widths of ${w_{x}=$0.341\pm
0.001\text{\,}\mathrm{m}\mathrm{m}$}$ and ${w_{y}=$0.351\pm
0.002\text{\,}\mathrm{m}\mathrm{m}$}$, through the vapour cell at a
temperature of ${T=$45\pm 1\text{\,}\mathrm{\SIUnitSymbolCelsius}$}$. The
temperature was measured by inserting a thermocouple between the vapour cell
and the surrounding metal heater assembly. The results demonstrate the
significance of optical pumping. At low intensities, the probe beam is weak
enough such that optical pumping effects are negligible and hence a change in
probe beam intensity does not change the transmission. However, even at
intensities $\sim I_{\rm{sat}}$ the scattering rate is sufficient to lead to a
noticeable modification of the transmission. It is striking that at high
intensities the amount of absorption is negligible compared to that of the
weak-probe regime. From the transmission data, we experimentally established
the weak-probe regime to be at an incident intensity of $\sim
10^{-1}~{}I/I_{\text{sat}}$. The larger error bars at the low incident
intensities are a consequence of low light levels and hence a poor signal-to-
noise ratio.

The red data points in the inset of Figure 3 show a transmission spectrum in
the weak-probe regime. The theory (blacked dashed line) is fitted to the data
along with the transmission spectra for probe intensities higher than that of
the weak probe to demonstrate the effect of optical pumping. The four
transmission spectra colours represent the different probe intensities where
$I_{\text{red}},~{}I_{\text{blue}},~{}I_{\text{orange}}~{}\text{and}~{}I_{\text{pink}}$
correspond to incident intensities of
$\sim\left($8\text{\times}{10}^{-3}\text{\,}$,~{}$5\text{\times}{10}^{-1}\text{\,}$,~{}$4\text{\,}$~{}\text{and}~{}$100\text{\,}$\right)$
$I_{\text{sat}}$ respectively. The residuals for the theoretical fit to the
red data show limited structure and hence the quality of the theoretical
model.

Figure 3: The main figure shows a plot of the minimum transmission as a
function of probe intensity, normalised to the saturation intensity, for a
natural abundance potassium cell with light resonant on the D2 transition. The
data corresponds to ${T=$43.3\pm 0.2\text{\,}\mathrm{\SIUnitSymbolCelsius}$}$
and probe beam widths ${w_{x}=$0.341\pm 0.001\text{\,}\mathrm{m}\mathrm{m}$}$
and ${w_{y}=$0.351\pm 0.002\text{\,}\mathrm{m}\mathrm{m}$}$. The red dashed
line is the theoretical value predicted by ElecSus. The inset is a plot of the
transmission profile as a function of detuning from line centre, along with a
theoretical fit (black dashed line). The four colours represent different
probe intensities where
$I_{\text{red}},~{}I_{\text{blue}},~{}I_{\text{orange}}~{}\text{and}~{}I_{\text{pink}}$
correspond to incident intensities of
$\sim\left($8\text{\times}{10}^{-3}\text{\,}$,~{}$5\text{\times}{10}^{-1}\text{\,}$,~{}$4\text{\,}$~{}\text{and}~{}$100\text{\,}$\right)$
$I_{\text{sat}}$ respectively. The residuals for the weak-probe regime are
shown below.

A weak probe on the D1 and D2 transitions was passed through the cell at a
variety of temperatures as shown in Figure 4, in order to further test the
predictions of ElecSus. Plot (i) shows transmission spectra on the D1
transition at temperatures of $34.0\pm
0.7\text{\,}\mathrm{\SIUnitSymbolCelsius}$, $43.8\pm
0.4\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and $56.0\pm
0.2\text{\,}\mathrm{\SIUnitSymbolCelsius}$, as shown by the blue, green and
red data points, with rms errors between theory and experiment of
$5\text{\times}{10}^{-3}\text{\,}$, $5\text{\times}{10}^{-3}\text{\,}$ and
$8\text{\times}{10}^{-3}\text{\,}$ respectively. Here the temperature of the
vapour was measured by fitting ElecSus to a weak-probe transmission spectrum,
using a Marquardt-Levenberg algorithm to minimise the weighted squares of the
residuals [48]. The errors on the temperatures were determined by calculating
the $\chi^{2}_{\nu}+1$ statistic from the fits [48]. The fitted temperatures
compare favourably with the thermocouple measurements, being consistently
$\sim$3\text{\,}\mathrm{\SIUnitSymbolCelsius}$$ higher due to the fact that
the thermocouple was poorly contacted to the cell. Plot (ii) shows
transmission spectra on the D2 transition at temperatures of $26.8\pm
0.7\text{\,}\mathrm{\SIUnitSymbolCelsius}$, $43.3\pm
0.5\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and $51.7\pm
0.4\text{\,}\mathrm{\SIUnitSymbolCelsius}$ with rms errors between theory and
experiment of $5\text{\times}{10}^{-3}\text{\,}$,
$2\text{\times}{10}^{-3}\text{\,}$ and $1\text{\times}{10}^{-2}\text{\,}$
respectively. The residuals for each of the three temperatures are shown below
the main figures. All of the fits have a low rms errors and the residuals show
limited structure which highlights the quality of the fit. The residuals for
the red data points in (ii) show some undulating structure. This is believed
to be due to an inadequate linearisation of the frequency scan as a
consequence of a small scan range producing a limited number of etalon peaks.
This therefore shows that the fitting procedure is sensitive to the scan
linearisation. In rubidium for example, sub-Doppler spectroscopy is used to
create many frequency markers to linearise the scan [18]. However, sub-Doppler
spectroscopy of potassium does not lead to clear frequency markers due to the
small hyperfine spacing [49, 50] and hence this method was not employed.
However, our method of linearisation appears to be sufficiently accurate, as
shown by the quality of the other fits.

Figure 4: The main figure shows a plot of the transmission as a function of
temperature for a natural abundance potassium cell with a probe intensity in
the weak-probe regime with light resonant on the D1 (i) and D2 (ii)
transitions. The black dashed lines are the theoretical spectra predicted by
ElecSus. The blue, green and red spectra correspond to temperatures of
$34.0\pm 0.7\text{\,}\mathrm{\SIUnitSymbolCelsius}$, $43.8\pm
0.4\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and $56.0\pm
0.2\text{\,}\mathrm{\SIUnitSymbolCelsius}$ with rms errors between theory and
experiment of $5\text{\times}{10}^{-3}\text{\,}$,
$5\text{\times}{10}^{-3}\text{\,}$ and $8\text{\times}{10}^{-3}\text{\,}$
respectively for (i) and $26.8\pm 0.7\text{\,}\mathrm{\SIUnitSymbolCelsius}$,
$43.3\pm 0.5\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and $51.7\pm
0.4\text{\,}\mathrm{\SIUnitSymbolCelsius}$ with rms errors between theory and
experiment of $5\text{\times}{10}^{-3}\text{\,}$,
$2\text{\times}{10}^{-2}\text{\,}$ and $1\text{\times}{10}^{-2}\text{\,}$
respectively for (ii). The residuals for each temperature are shown below.

###  4.2 4S $\rightarrow$ 5P D lines

The atomic structure of 39K and 41K on the 4S $\rightarrow$ 5P D lines is
shown in Figure 5. One should note that to the best of our knowledge, the
isotope shift for the D2 transition has never been experimentally measured.
Hence we have adopted the notation of $\delta$ to label the unknown isotope
shift. The 4S $\rightarrow$ 5P transition is similar in structure to that of
the 4S $\rightarrow$ 4P as an increase in principal quantum number does not
change the angular components of the system. However, we note a different
isotope shift and a smaller hyperfine splitting. The saturation intensity and
decay rates for the D1 and D2 transitions are
$58.8\text{\,}\mathrm{m}\mathrm{W}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$,
$57.6\text{\,}\mathrm{m}\mathrm{W}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$,
${\Gamma/2\pi=$170.3\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$}$ and
${\Gamma/2\pi=$184.6\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$}$ respectively
[47]. One should note that equation 1 is not applicable here in calculating
the saturation intensity as the 5P states may decay to alternative states.
Therefore, all transitions must be taken into consideration [51]. This leads
to a saturation intensity of the 4S $\rightarrow$ 5P transition of

$I_{\rm{sat}}=2\pi\frac{hc}{\lambda^{3}}\frac{\Gamma_{\rm{T}}^{2}}{\Gamma}\frac{1}{1.58}~{},$
(4)

where $\Gamma_{\rm{T}}$ is the total linewidth of the transition. The factor
of $1/1.58$ is a consequence of considering all decay paths from the 5P state.

ElecSus was adapted to model the ${4\text{S}\rightarrow 5\text{P}}$
transitions. This was easily implemented by changing the hyperfine constants,
isotope shifts and decay rates. However, two decay rates were included in the
code. $\Gamma$ was used to calculate the strength of the D line transitions,
as shown in equation 3, but the width of the excited state was controlled via
$\Gamma_{\rm{T}}$. The total linewidth for the D1 and D2 transitions are
$\Gamma_{\rm{T}}/2\pi=$1.16\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$$ [29] and
$\Gamma_{\rm{T}}/2\pi=$1.19\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$$ [28]
respectively. As shown in equation 2, the absorption coefficient is directly
proportional to $\Gamma$ and ${\cal{N}}$. As $\Gamma$ for the 4S $\rightarrow$
5P transition is much smaller than that of the 4S $\rightarrow$ 4P transition,
in order to see a significant absorption feature, the vapour cell was heated
to a typical temperature of $100\text{\,}\mathrm{\SIUnitSymbolCelsius}$ to
increase ${\cal{N}}$. The transmission profile for a probe beam resonant with
the D lines at a vapour temperature of
$100\text{\,}\mathrm{\SIUnitSymbolCelsius}$ is shown in Figure 5, which is
similar to that of Figure 2 and follows the same labelling conventions. We see
a single Doppler-broadened spectrum with contributions from each hyperfine
transition. However, the Doppler width is much larger,
$\sim$1.7\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$ compared to
$\sim$0.8\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$, as the Doppler width is
directly proportional to the angular frequency of the transition.

Figure 5: An energy level diagram of the 4S $\rightarrow$ 5P D lines of 39K
and 41K along with their natural abundance. Data taken from [30, 46, 28, 47,
52, 53, 54]. Plots (i) and (iii) show the calculated contributions to the
total transmission spectrum of a probe beam resonant with the D1 and D2
transition passing through a $10\text{\,}\mathrm{c}\mathrm{m}$ natural
abundance potassium vapour cell at $100\text{\,}\mathrm{\SIUnitSymbolCelsius}$
respectively. The colours correspond to those in Figure 2. Plots (ii) and (iv)
show the transition strengths, weighted by the isotopic abundance, where the
red and green lines correspond to 39K and 41K respectively. Detuning is
reference to the weighted line-centre of the transition.

Following the same method as before, the weak probe regime was experimentally
determined to be $\sim$1\text{\times}{10}^{-2}\text{\,}$~{}I_{\text{sat}}$ for
both the D1 and D2 transitions using probe beams of widths ${w_{x}=$0.309\pm
0.001\text{\,}\mathrm{m}\mathrm{m}$}$, ${w_{y}=$0.907\pm
0.008\text{\,}\mathrm{m}\mathrm{m}$}$ and $w_{x}=$0.269\pm
0.001\text{\,}\mathrm{m}\mathrm{m}$$, $w_{y}=$0.623\pm
0.006\text{\,}\mathrm{m}\mathrm{m}$$ at temperatures of $T=$104\pm
1\text{\,}\mathrm{\SIUnitSymbolCelsius}$$ and $107\pm
1\text{\,}\mathrm{\SIUnitSymbolCelsius}$ respectively where the temperatures
were measured using the thermocouple.

Figure 6: The main figure shows a plot of the transmission as a function of
temperature for a natural abundance potassium vapour cell with a probe
intensity in the weak-probe regime with light resonant on the D1 (i) and D2
(ii) transitions. The black dashed lines are the theoretical spectra predicted
by ElecSus. The blue, green and red spectra correspond to temperatures of
$101.0\pm 0.8\text{\,}\mathrm{\SIUnitSymbolCelsius}$, $110.7\pm
0.4\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and $125.4\pm
0.3\text{\,}\mathrm{\SIUnitSymbolCelsius}$ with rms errors between theory and
experiment of $5\text{\times}{10}^{-3}\text{\,}$ for all three data sets for
(i) and temperatures of $83.1\pm 0.4\text{\,}\mathrm{\SIUnitSymbolCelsius}$,
$103.0\pm 0.2\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and $117.0\pm
0.7\text{\,}\mathrm{\SIUnitSymbolCelsius}$ with rms errors between theory and
experiment of $2\text{\times}{10}^{-3}\text{\,}$,
$4\text{\times}{10}^{-3}\text{\,}$ and $1\text{\times}{10}^{-2}\text{\,}$
respectively for (ii). The residuals for each temperature are shown below. The
inset shows the rms error between the red data and ElecSus as a function of
the D2 isotope shift, $\delta$.

Figure 6 shows the transmission of a weak probe on the D1 and D2 transitions
at a variety of temperatures. Plot (i) shows transmission spectra on the D1
transition at temperatures of $101.0\pm
0.8\text{\,}\mathrm{\SIUnitSymbolCelsius}$, $110.7\pm
0.4\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and $125.4\pm
0.3\text{\,}\mathrm{\SIUnitSymbolCelsius}$, as shown by the blue, green and
red data points, with rms errors between theory and experiment of
$5\text{\times}{10}^{-3}\text{\,}$ for all three data sets. Here the
temperatures and their respective errors were measured using the fitting
procedure detailed earlier. Plot (ii) shows transmission spectra on the D2
transition at temperatures of $83.1\pm
0.4\text{\,}\mathrm{\SIUnitSymbolCelsius}$, $103.0\pm
0.2\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and $117.0\pm
0.7\text{\,}\mathrm{\SIUnitSymbolCelsius}$ with rms errors between theory and
experiment of $2\text{\times}{10}^{-3}\text{\,}$,
$4\text{\times}{10}^{-3}\text{\,}$ and $1\text{\times}{10}^{-2}\text{\,}$
respectively. One should note that the isotope shift used for the D2
transition was $\delta=$454.2\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$$ which
is the isotope shift of the D1 transition. By considering Figure 2, we made
the approximation that the difference between the isotope shifts of the D1 and
the D2 transitions in Figure 5 would be of the order of
$undef\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$. The small rms errors and the
lack of structure in the residuals shows the quality of the agreement between
theory and experimental data. Similar to that displayed in Figure 4, there is
a small amount of undulation in some of the residuals. This is once again
believed to be attributed to the quality of the linearisation of the laser
scan. However, these data indicate that the theory and experiment are in
agreement, considering the scale of the residuals and the rms errors. The
level of noise appears to vary across different spectra. This is believed to
be attributed to the scale on the oscilloscope at which the data was recorded.
This is a significant result as it is indicative that the adapted version of
ElecSus is a very accurate model of the 4S $\rightarrow$ 5P transition which
could prove fruitful in future experiments.

The inset in Figure 6 plot (ii) shows the rms error between the red data and
ElecSus as a function of the D2 isotope shift, $\delta$. This procedure was
carried out in an attempt to measure $\delta$ as one would expect that the rms
error would be minimised for the correct value of $\delta$. This procedure was
also carried out on the other two fits in plot (ii), however the values for
$\delta$ determined were not in agreement with each other. We believe this is
a manifestation of poor scan linearisation. Although we are not able to
identify $\delta$, this is significant as it does highlight that the quality
of the agreement between the transmission spectra and ElecSus is sensitive to
isotope shifts and hence, may be used to measure them. We believe that if one
were to improve the signal-to-noise ratio and the scan linearity, it should be
possible to measure $\delta$. One possibility for improving the scan
linearisation would be to scan the probe light across the transition using a
broadly tunable waveguide EOM driver by an rf synthesiser, avoiding the non-
linearities associated with scanning the ECDL.

##  5 Future Experiment

The plots in Figure 6 show that ElecSus can be adapted to accurately model the
4S $\rightarrow$ 5P transition. This leads one to consider an experiment where
two parallel beams resonant with the 4S $\rightarrow$ 4P and 4S $\rightarrow$
5P transitions propagate though the same vapour cell. The 4S $\rightarrow$ 4P
beam may be used to accurately determine the temperature of the atoms to a
precision of $\sim$0.2\text{\,}\mathrm{\SIUnitSymbolCelsius}$$. The other beam
may be used for spectroscopy of the atoms. For example, the line centre
transmission is dependent on the strength of the transition which is directly
proportional to $\Gamma$. Therefore, having characterised the atoms with one
probe beam, we would be able to fit to experimental data of the transmission
of the other probe beam in order to measure the decay rate of the transition.
This two beam method could also prove useful in experiments in which the
temperature of an atomic vapour must be known accurately, such as experiments
attempting to measure the Boltzmann constant from the Doppler-width of an
absorption feature [55, 56].

By varying $\Gamma$ for the 4S $\rightarrow$ 5P transition in simulations and
assuming the best signal-to-noise we experimentally obtained in our
experiment, we believe one should be able to determine $\Gamma$ to a precision
of $\sim$3\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$ at a temperature of
$100\text{\,}\mathrm{\SIUnitSymbolCelsius}$ which is comparable to that given
in the literature which is of the order of
$\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$ [47]. If the improvements to
experimental noise were implemented, then it should be possible to surpass the
experimental error on $\Gamma$ given in the literature.

##  6 Concluding Remarks

Transmission spectra using a weak probe were measured on the 4S $\rightarrow$
4P and 4S $\rightarrow$ 5P transitions. Theoretical spectra generated by
ElecSus, and its adapted version, were fit to the experimental spectra. This
is the first experimental test of ElecSus on an atom with a ground state
hyperfine splitting smaller than that of the Doppler width. An excellent
agreement was found between ElecSus and experimental measurements at a variety
of temperatures on both transitions with rms errors
$\sim${10}^{-3}\text{\,}$$. This also shows that ElecSus may be adapted with
relative ease to model higher principal quantum number transitions with
similar accuracy. This validation of ElecSus on potassium will enable accurate
design and implementation of future experiments where the absorptive and
dispersive properties of a potassium vapour are required, such as optical
isolators, Faraday filters and laser locking schemes. Furthermore, we have
shown the sensitivity of the agreement between spectra and ElecSus to isotope
shifts and proposed improvements to the current experiment which should
facilitate more accurate measurements. We have also discussed the possibility
of an experiment where two probe beams could be used to simultaneously
interrogate an atomic vapour on different transitions. Using one beam for
thermometry enhances the accuracy of the information which can be determined
with the second beam, allowing accurate decay rate measurements for example.

We are grateful for the loan of equipment and discussion from Steve Hopkins
and Stefan Kemp, as well as the detailed help with the core of ElecSus from
Mark Zentile. We would also like to thank Michael Köppinger for the design of
the potassium vapour cell heater. This work was financed by Durham University
and EPSRC (grants EP/L023024/1 and EP/I012044/1).

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