# Remote Charging and Degradation Suppression for the Quantum Battery

Wan-Lu Song* — Department of Physics, Hubei University, Wuhan 430062, China
Hai-Bin Liu  Department of Physics, Hubei University, Wuhan 430062, China  Bin
Zhou* —<EMAIL_ADDRESS>Department of Physics, Hubei University, Wuhan
430062, China  Wan-Li Yang<EMAIL_ADDRESS>State Key Laboratory of Magnetic
Resonance and Atomic and Molecular Physics, Innovation Academy for Precision
Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430071,
China  Jun-Hong An* —<EMAIL_ADDRESS>Key Laboratory of Quantum Theory and
Applications of MoE, Lanzhou Center for Theoretical Physics, and Key
Laboratory of Theoretical Physics of Gansu Province, Lanzhou University,
Lanzhou 730000, China

###### Abstract

The quantum battery (QB) makes use of quantum effects to store and supply
energy, which may outperform its classical counterpart. However, there are two
challenges in this field. One is that the environment-induced decoherence
causes the energy loss and aging of the QB, the other is that the decreasing
of the charger-QB coupling strength with increasing their distance makes the
charging of the QB become inefficient. Here, we propose a QB scheme to realize
a remote charging via coupling the QB and the charger to a rectangular hollow
metal waveguide. It is found that an ideal charging is realized as long as two
bound states are formed in the energy spectrum of the total system consisting
of the QB, the charger, and the electromagnetic environment in the waveguide.
Using the constructive role of the decoherence, our QB is immune to the aging.
Additionally, without resorting to the direct charger-QB interaction, our
scheme works in a way of long-range and wireless-like charging. Effectively
overcoming the two challenges, our result supplies an insightful guideline to
the practical realization of the QB by reservoir engineering.

##  I Introduction

The miniaturization of electronic devices has inspired a rapid development of
quantum technologies, which are expected to bring a lot of technological
innovations [1, 2, 3]. Among them, quantum thermodynamics [4, 5, 6] has
emerged as a field aiming to reconstruct thermodynamics via basic laws of
quantum mechanics and to break through the constraint of classical physics to
device performance by quantum effects [7, 8, 9, 10, 11, 12]. As an energy-
storing and -converting device in atomic size, the quantum battery (QB)
promotes the application of quantum effects in thermodynamics [13, 14]. With
the aid of quantum resources, such as quantum coherence and entanglement, the
QB possesses a potential of stronger charging power, higher charging capacity,
and larger work extraction than its classical counterpart [15, 16, 17, 18, 19,
20, 21].

An efficient QB scheme is a prerequisite for its realization and further
applications. The widely studied QB model is based on two-level systems
(TLSs), which are charged by the coherent coupling to other TLSs [22, 23, 24,
25, 26] or external fields [27, 28, 29, 30, 31, 32, 33]. Experiments on the
single-body QB were carried out in nuclear magnetic resonance [34],
superconducting circuit [35, 36], semiconducting quantum dot [37, 38], and
photonic [39] systems. It was also found that the collection of more TLSs
exhibits more energy storage capacity and the collective charging benefiting
from quantum correlation induces stronger charging power of the QB [40, 41,
42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
61]. However, the practical performance of the QB is challenged by two facts.
One is that its energy during the long-time storage process is depleted by the
environment-induced decoherence, which is called the aging of the QB [62]. The
other is that its charging becomes inefficient when the charger-QB coupling is
weakened due to certain reasons. Several schemes using feedback control [63],
dark states [64], Floquet engineering [65], and environment engineering [66,
67, 68, 69, 69, 70, 71, 72, 73, 74, 75, 76, 77], have been proposed to beat
the decoherence effects. However, how to suppress the decoherence effects on
the QB and improve the charging efficiency simultaneously is still an open
question.

Figure 1: Scheme of a remote-charging QB. The noninteracting charger and QB
formed by TLSs are separated in a distance $\Delta_{z}$ in a rectangular
hollow metal waveguide. An efficient remote charging to the QB is realized by
the mediation role of the EMF in the waveguide.

Inspired by the development of quantum interconnect to establish correlation
among distant quantum entities [78], we propose a remote-charging QB scheme
via coupling the QB and the charger to a rectangular hollow metal waveguide
[79, 80, 81]. Utilizing the non-Markovian decoherence induced by the
electromagnetic field (EMF) in the waveguide, we find a mechanism that allows
for a persistent energy exchange between the noninteracting charger and QB in
the steady state. It is due to the formation of two bound states in the energy
spectrum of the total system formed by the charger, the QB, and the EMF. Not
causing the aging of the QB, the decoherence in our scheme plays a
constructive role in the charging. Without resorting to the direct charger-QB
coupling, our charging works in a long-range and wireless way. Overcoming the
aging and low charging efficiency in the weak charger-QB coupling condition
simultaneously, our scheme supplies an insightful guideline to the practical
realization of the QB.

##  II Remote-charging QB model

We consider a long-range charging QB scheme, which consists of two TLSs acting
as the charger and the QB residing in a rectangular hollow metal waveguide
(see Fig. 1). Without direct interaction, the charger and QB couple to a
common EMF in the waveguide. The Hamiltonian reads [82, 81, 80]

$\displaystyle\hat{H}$ $\displaystyle=$
$\displaystyle\omega_{0}\sum_{j=1,2}\hat{\sigma}^{\dagger}_{j}\hat{\sigma}_{j}+\sum^{+\infty}_{k=-\infty}\big{[}\omega_{k}\hat{a}^{\dagger}_{k}\hat{a}_{k}$
(1)
$\displaystyle+\sum_{j=1,2}\big{(}g_{k,j}\hat{\sigma}^{\dagger}_{j}\hat{a}_{k}+\text{H.c.}\big{)}\big{]},$

where $\hat{\sigma}_{j}=|g_{j}\rangle\langle e_{j}|$ is the transition
operator from the excited state $|e_{j}\rangle$ to the ground state
$|g_{j}\rangle$ with frequency $\omega_{0}$ of the charger when $j=1$ and of
the QB when $j=2$ and $\hat{a}_{k}$ is the annihilation operator of the $k$th
mode with frequency $\omega_{k}$ of the EMF. The coupling strength between the
$j$th TLS and the $k$th mode of the EMF is
$g_{k,j}=\sqrt{\omega_{k}/(2\epsilon_{0})}\mathbf{d}_{j}\cdot\mathbf{u}_{k}(\mathbf{r}_{j})$,
where $\epsilon_{0}$ is the vacuum permittivity, $\mathbf{d}_{j}$ is the
dipole moment of the $j$th TLS, and $\mathbf{u}_{k}(\mathbf{r}_{j})$ is the
spatial function of the EMF. The dispersion relation of the EMF is
$\omega_{k}=[(ck)^{2}+\omega^{2}_{mn}]^{1/2}$, where $c$ is the speed of
light, $k$ is the longitudinal wave number, and
$\omega_{mn}=c[(m\pi/a)^{2}+(n\pi/b)^{2}]^{1/2}$, with $a$ and $b$ being the
transverse lengths of the waveguide, is the cutoff frequency of the transverse
mode of the EMF. Assuming that the two TLSs are polarized in the $z$
direction, i.e., $\mathbf{d}_{1}=\mathbf{d}_{2}=d_{z}\mathbf{z}$, and
considering only the dominated EMF mode with $m=n=1$, the coupling strength is
further characterized by the spectral density

$J_{|j-j^{\prime}|}(\omega)=\frac{\Gamma_{11}}{2\pi}\frac{\cos[\frac{z_{j}-z_{j^{\prime}}}{c}\sqrt{\omega^{2}-\omega^{2}_{11}}]}{\sqrt{(\omega/\omega_{11})^{2}-1}}\Theta(\omega-\omega_{11}).$
(2)

Here,
$\Gamma_{11}=\frac{4\omega_{11}d_{z}^{2}}{\epsilon_{0}abc}\sin^{2}\left(\pi
x_{0}/a\right)\sin^{2}\left(\pi y_{0}/b\right)$ is the radiation rate of the
TLS with transverse position $\left(x_{0},y_{0}\right)$ into the waveguide and
$\Theta(\omega-\omega_{11})$ is the Heaviside step function. The spectral
density $J_{0}(\omega)$ induces an individual spontaneous emission and Lamb
frequency shift to the $j$th TLS, while $J_{1}(\omega)$ induces a correlated
spontaneous emission and dipole-dipole interaction between the two TLSs.
Different from Refs. [72, 59, 24, 47, 54], we use such an incoherent
correlation to charge the QB.

Our scheme begins from a pump laser to initialize the charger to its excited
state. When a charging to the QB is needed, the QB is taken to the waveguide
not necessarily too near to the charger. The EMF in the waveguide is expected
to mediate a charger-QB energy transfer. This process under the condition
$|\Psi(0)\rangle=|e_{1},g_{2};\\{0_{k}\\}\rangle$ is described by
$|\Psi(t)\rangle=[\sum_{j}c_{j}(t)\hat{\sigma}_{j}^{\dagger}+\sum_{k}d_{k}(t)\hat{a}_{k}^{\dagger}]|g_{1},g_{2},\\{0_{k}\\}\rangle$.
Here, $c_{j}(t)$ and $d_{k}(t)$ are the excited-state probability amplitudes
of the TLSs and the EMF with only one photon in the $k$th mode, respectively.
From the Schrödinger equation, we obtain [83]

$\dot{c}_{j}(t)+i\omega_{0}c_{j}(t)+\sum_{j^{\prime}=1,2}\int^{t}_{0}f_{|j-j^{\prime}|}(t-\tau)c_{j^{\prime}}(\tau)d\tau=0,$
(3)

where
$f_{|j-j^{\prime}|}(x)=\int^{\infty}_{0}J_{|j-j^{\prime}|}(\omega)e^{-i\omega
x}d\omega$, $c_{1}(0)=1$, and $c_{2}(0)=0$. Equation (3) reveals that,
although the direct charger-QB interaction is absent, an effective charger-QB
coupling is induced by the mediation role of the EMF in the waveguide. The
convolution in Eq. (3) renders the dynamics non-Markovian. The energy of the
QB is

$\mathcal{E}(t)=\text{Tr}[\rho_{B}(t)\hat{H}_{B}],$ (4)

where $\hat{H}_{B}=\omega_{0}\hat{\sigma}_{2}^{\dagger}\hat{\sigma}_{2}$ and
$\rho_{B}(t)=\text{Tr}_{1}[|\Psi(t)\rangle\langle\Psi(t)|]$. According to the
second law of thermodynamics, not all of $\mathcal{E}(t)$ can be converted
into work [84]. The maximum of the extractable energy called ergotropy is
defined as [18, 19]

$\displaystyle\mathcal{W}(t)=\text{Tr}[\rho_{B}(t)\hat{H}_{B}]-\text{Tr}[\tilde{\rho}_{B}(t)\hat{H}_{B}],$
(5)

where
$\tilde{\rho}_{B}(t)=\sum_{k}{r_{k}(t)|\varepsilon_{k}\rangle\langle\varepsilon_{k}|}$
is the passive state, $r_{k}(t)$ are the eigenvalues of $\rho_{B}(t)$ ordered
in a descending sort, and $|\varepsilon_{k}\rangle$ are the eigenstates of
$\hat{H}_{B}$ with the corresponding eigenvalues $\varepsilon_{k}$ ordered in
an ascending sort.

In the special case of the weak TLS-EMF coupling, we apply the Markovian
approximation to neglect the memory effect and extend the upper limit of the
$\tau$ integral in Eq. (3) to infinity. The approximate solution is

$\displaystyle
c_{j}^{\text{MA}}(t)=e^{-(i\omega_{0}+\Upsilon_{0})t}[e^{-\Upsilon_{1}t}-(-1)^{j}e^{\Upsilon_{1}t}]/2,$
(6)

where $\Upsilon_{j}=\pi J_{j}(\omega_{0})+i\delta_{j}$, with
$\delta_{j}=\mathcal{P}\int_{0}^{\infty}d\omega{J_{j}(\omega)\over\omega_{0}-\omega}$
and $\mathcal{P}$ denoting the Cauchy principal value. Equation (6) reveals
that the EMF in the waveguide indeed can mediate an energy exchange between
the charger and the QB. However, it is transient and the energies in both of
them exponentially damp to zero in the long-time condition due to the
destructive role of the Markovian decoherence.

In contrast to the above approximate result, the mediated charger-QB coupling
can induce a persistently reversible energy exchange between them even in the
steady state in the non-Markovian dynamics. Equation (3) is only numerically
solvable. However, its long-time form is analytically obtainable by the
Laplace transform method. It converts Eq. (3) to
$\tilde{c}_{j}(s)=[{1\over\Xi(s)+\tilde{f}_{1}(s)}-{(-1)^{j}\over\Xi(s)-\tilde{f}_{1}(s)}]/2$,
where $\Xi(s)=s+i\omega_{0}+\tilde{f}_{0}(s)$ and
$\tilde{f}_{|j-j^{\prime}|}(s)=\int_{0}^{\infty}\frac{J_{|j-j^{\prime}|}(\omega)}{i\omega+s}d\omega$.
$c_{j}(t)$ is obtained by making the inverse Laplace transform to
$\tilde{c}_{j}(s)$, which needs finding the poles of $\tilde{c}_{j}(s)$ from

$Y_{\pm}(E)\equiv\omega_{0}-\int^{\infty}_{0}\frac{J_{0}(\omega)\pm
J_{1}(\omega)}{\omega-E}d\omega=E,(E=is).$ (7)

Note that the roots $E$ of Eq. (7) are just the eigenenergies of Eq. (1). To
prove this, we expand the eigenstate as
$|\Phi\rangle=[\sum_{j}\alpha_{j}\hat{\sigma}_{j}^{\dagger}+\sum_{k}\beta_{k}\hat{a}_{k}^{\dagger}]|g_{1},g_{2},\\{0_{k}\\}\rangle$.
From $\hat{H}|\Phi\rangle=E|\Phi\rangle$, we obtain
$(E-\omega_{0})\alpha_{j}=\sum_{j^{\prime}}\int_{0}^{\infty}{J_{|j-j^{\prime}|}(\omega)\alpha_{j^{\prime}}\over
E-\omega}d\omega$, which readily results in Eq. (7) after eliminating
$\alpha_{j}$ [83]. Thus, on one hand, Eq. (7) governs the dynamical evolution
of $c_{j}(t)$, and on the other hand, it determines the eigenenergies of the
total system. This means that the dynamics of the TLSs is essentially
determined by the energy-spectrum feature of the total system. $Y_{\pm}(E)$ is
ill-defined and jumps rapidly between $\pm\infty$ in the regime
$E>\omega_{11}$ because of the divergence of the integrand. Thus, Eq. (7) has
infinite roots in $E>\omega_{11}$, which form a continuous energy band. In the
regime $E<\omega_{11}$, $Y_{\pm}(E)$ are monotonic decreasing functions of $E$
and an isolated root $E_{+}^{\text{b}}$ or $E_{-}^{\text{b}}$ is formed
provided $Y_{+}(\omega_{11})<\omega_{11}$ or $Y_{-}(\omega_{11})<\omega_{11}$.
We call the eigenstates corresponding to $E_{\pm}^{\text{b}}$ bound states.
Substituting these poles from Eq. (7) into the inverse Laplace transform, the
bound-state energies $E^{\text{b}}_{\pm}$ contribute two nontrivial residues;
while the band energies in the continuum contribute a branch cut, which tends
to zero in the long-time limit due to the out-of-phase interference [85, 86].
Thus, we have

$c_{j}(\infty)=\begin{cases}0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{0
BS},\\\
Z_{+}e^{-iE^{\text{b}}_{+}t},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{1
BS},\\\
Z_{+}e^{-iE^{\text{b}}_{+}t}-(-1)^{j}Z_{-}e^{-iE^{\text{b}}_{-}t},\text{2
BSs},\end{cases}$ (8)

where $Z_{\pm}={1\over 2}\big{[}1+\int^{\infty}_{0}\frac{J_{0}(\omega)\pm
J_{1}(\omega)}{(E_{\pm}^{\text{b}}-\omega)^{2}}d\omega\big{]}^{-1}$ is the
residue contributed by $E^{\text{b}}_{\pm}$ to $c_{j}(t)$. Equation (8)
indicates the decisive role played by the energy-spectrum feature of the total
TLS-EMF system in the non-Markovian dynamics of the TLSs. It is remarkable to
see that the formation of the bound states prevents $|c_{j}(t)|^{2}$ from
damping to zero, which is not captured by the Markovian result in Eq. (6) and
can be used to remotely charge the QB. When the QB is not needed for supplying
energy to other devices, it is kept in the waveguide. Reversibly exchanging
energy with the charger, the QB does not experience the aging caused by the
energy loss in this storage process [83].

Equation (8) reveals that the QB is dynamically synchronous with the charger
in the steady state. In the absence of the bound state, no energy is left in
both the QB and the charger and the energy completely relaxes into the EMF of
the waveguide. If one bound state with eigenenergy $E^{\text{b}}_{+}$ is
formed, then a stable energy $\mathcal{E}(\infty)=\omega_{0}Z_{+}^{2}$ is
preserved in the QB, which is equal to the one in the charger. As long as two
bound states with eigenenergies $E^{\text{b}}_{\pm}$ are formed, a periodic
energy exchange of the QB with the charger in a frequency
$E^{\text{b}}_{+}-E^{\text{b}}_{-}$, i.e.,
$\mathcal{E}(\infty)=\omega_{0}\\{Z^{2}_{+}+Z^{2}_{-}-2Z_{+}Z_{-}\cos[(E_{+}^{\text{b}}-E_{-}^{\text{b}})t]\\}$,
is kept in the steady state. Such a Rabi-like oscillation is naturally
expected in the case of the direct charger-QB coupling. However, the charging
realized by the direct coupling is generally fragile to the environment-
induced decoherence, which results in the aging of the QB. Our result
indicates that, different from the destructive role in the direct-coupling
case, the non-Markovian decoherence caused by the common EMF can be used to
realize an ideal charging to the QB in the open system framework. It is just
the constructive role played by the decoherence that generates such an
efficient charging in our scheme.

The persistent Rabi-like charger-QB energy exchange endows our scheme with a
substantial difference from previous environment-assisted charging schemes
either in the Markovian approximation [66, 67, 68, 69, 69, 70, 71, 72, 73] or
in the non-Markovian dynamics [74, 75, 76, 77]. Our charging performance,
which is as perfect as the ideal charging case by direct charge-QB
interaction, is guaranteed by the feature of the energy spectrum of the total
system consisting of the charger, the QB, and the EMF in the waveguide. It
makes our scheme immune to the environment-induced aging. Our scheme works in
a way of long-range and wireless-like charging, while the spatial charger-QB
distance has never been investigated. As an attractive feature in future
applications, it overcomes the low-charging-efficiency problem experienced by
the conventional QB schemes when the charger-QB coupling is weak.

Figure 2: Energy spectrum $E$ of the total system via solving Eq. (7) and
evolution of the QB energy $\mathcal{E}(t)$ via solving Eq. (3) in different
$\omega_{0}$ in the presence of zero (red dashed lines), one (blue solid
lines), and two (green dot-dashed lines) bound states. The black dots mark the
results evaluated from the analytical equation (8). We use
$\Delta_{z}=0.1\lambda_{11}$ and $\Gamma_{11}=0.5\omega_{11}$. Figure 3: (a)
Bound-state eigenenergies $E_{\pm}^{\text{b}}$ (red dashed lines) and long-
time evolution of $\mathcal{E}(t)$, and (b) extremal values of
$\mathcal{E}(\infty)$ as a function of $\Delta_{z}$ when
$\omega_{0}=1.4\omega_{11}$. The cases of $\omega_{0}=1.2\omega_{11}$ (blue
blocks) and $1.0\omega_{11}$ (green triangles) are also given in (b). The
other parameters are the same as Fig. 2.

##  III Numerical results

Choosing the charger-QB distance
$\Delta_{z}\equiv|z_{1}-z_{2}|=0.1\lambda_{11}$, with $\lambda_{11}=2\pi
c/\omega_{11}$, which is much larger than the typical range of the dipole-
dipole interactions, we plot in Fig. 2 the energy spectrum of the total system
formed by the QB, the charger, and the EMF and the evolution of the QB energy
$\mathcal{E}(t)$ in different frequency $\omega_{0}$ of the TLSs. It is
interesting to find that, in contrast to the damping to zero in the Markovian
case, $\mathcal{E}(t)$ approaches a finite value for a moderate $\omega_{0}$,
while it exhibits a lossless Rabi-like oscillation for a small $\omega_{0}$.
These diverse behaviors can be explained by the energy-spectrum feature. We
see that two branches of bound states in the band gap region separate the
energy spectrum into three regimes. When $\omega_{0}\geq 2.8\omega_{11}$, no
bound state is formed and $\mathcal{E}(t)$ damps to zero exclusively. When
$1.1\omega_{11}\leq\omega_{0}<2.8\omega_{11}$, one bound state is formed and
$\mathcal{E}(t)$ tends to a finite value. As long as two bound states are
present when $\omega_{0}<1.1\omega_{11}$, $\mathcal{E}(t)$ exhibits a lossless
Rabi-like oscillation. The matching of the long-time behaviors in the three
regimes with the analytical result in Eq. (8) verifies the distinguished role
played by the bound states and non-Markovian effect in the charging
performance of our scheme. The result reveals that we can properly choose the
working frequency $\omega_{0}$ of the QB and charger such that a remote
charging to the QB is dynamically realized without resorting to the direct
charger-QB interactions.

Figure 4: (a) Number $M$ of the bound states, (b) maxima of the ergotropy
$\mathcal{W}(\infty)$ in different $\omega_{0}$, and (c) maxima of the
ergotropy $\text{max}_{\omega_{0}}\mathcal{W}(\infty)$ via optimizing
$\omega_{0}$ as a function of $\Delta_{z}$. 2D means two degenerate bound
states. The other parameters are the same as Fig. 2.

To verify the tolerance of the scheme to the increase of the charger-QB
distance, we plot in Fig. 3 the eigenenergies of bound states, the long-time
evolution of $\mathcal{E}(t)$, and the extremal values of
$\mathcal{E}(\infty)$ as a function of $\Delta_{z}$. The result for
$\omega_{0}=1.4\omega_{11}$ shows that when $\Delta_{z}<0.3\lambda_{11}$, one
bound state is present and $\mathcal{E}(t)$ evolves to a finite
$\Delta_{z}$-dependent value. When
$0.3\lambda_{11}\leq\Delta_{z}<1.7\lambda_{11}$, two bound states are formed
and $\mathcal{E}(t)$ tends to a persistent Rabi-like oscillation in a
frequency $E_{+}^{\text{b}}-E_{-}^{\text{b}}$ characterizing a coherent energy
exchange between the QB and the charger. When $\Delta_{z}\geq
1.7\lambda_{11}$, the eigenenergies of the two bound states become degenerate
and $\mathcal{E}(t)$ tends to zero according to Eq. (8). Therefore, our
charging scheme can work well when $\Delta_{z}$ is freely changed within the
range from $0.3\lambda_{11}$ to $1.7\lambda_{11}$. The result in Fig. 3(b)
also reveals that the maximal stored energy $\max\mathcal{E}(\infty)$ can be
enhanced by properly choosing the working frequency $\omega_{0}$ of the QB and
the charger. This gives us sufficient room to optimize the performance of our
remote-charging scheme. Note that the presence of two bound states in Fig.
4(a) and nonzero $\mathcal{E}(\infty)$ are not always accompanied by an
extractable energy of the QB described by the nonzero ergotropy
$\mathcal{W}(\infty)$, see Fig. 4(b). Via optimizing $\omega_{0}$, we plot
$\max_{\omega_{0}}\mathcal{W}(\infty)$ in Fig. 4(c) in different $\Delta_{z}$,
which exhibits a good performance of our QB in quite a large charger-QB
distance regime.

##  IV Discussions and conclusion

Although only the rectangular hollow metal waveguide is considered, our result
is applicable to other systems. Applying it in the surface plasmon polariton
waveguide [87, 88], we are even not bothered to put the charger and the QB
inside the waveguide, which reduces the experimental difficulty. Current
experimental advances of the waveguide QED provide support for our QB scheme
[89, 90, 91, 92]. Especially, based on the solid-state emitter, i.e.,
nitrogen-vacancy centers (NVC), a series of integrated platforms have ensured
the feasibility of our proposal, such as microwave coplanar [93], silicon
nitride [94], laser-written [95], and nanowire [96] waveguides. The long-range
interaction of NVCs in magnon waveguide QED has been proposed [97]. In our
setup, proper values of the working frequency $\omega_{0}$ and the charger-QB
distance $\Delta_{z}$ are required. NVC is of the capability for coherent
manipulation by either light or microwave. In the optical frequency regime, we
can focus on the working frequency $\lambda_{0}=637$ nm. To match
$\omega_{0}=1.0\omega_{11}$ for the maximal stored energy
$\mathcal{E}=0.6\omega_{11}$, the transverse lengths of waveguide are set as
$a=b=0.45$ $\mu$m. To match $\omega_{0}=0.6\omega_{11}$ for the maximal
extractable energy $\mathcal{W}=0.3\omega_{11}$, they are set as $a=b=0.27$
$\mu$m. In the microwave frequency regime, the working frequency of NVC with
zero-field splitting $2.87$ GHz has a tremendous tunability by an external
field. Generally speaking, the effective charging distance could range from
$1$ $\mu$m in the optical waveguide to $0.1$ m in the microwave one, which are
far beyond the typical length of the dipole-dipole coupling. The bound state
and its distinguished role in the non-Markovian dynamics have been observed in
both photonic crystal [98] and ultracold-atom [99, 100] systems, which sets a
solid foundation to the realizability of our scheme. Also, presenting a
scalable and tunable framework, our scheme is easily extended to the multi-TLS
scenarios.

In conclusion, we have proposed a remote-charging scheme via coupling the QB
and the charger to a rectangular hollow metal waveguide. It is found that,
contrary to one’s belief that decoherence generally leads to the aging of a
QB, the non-Markovian decoherence induced by the EMF in the waveguide can
realize a persistent Rabi-like energy exchange between the QB and the charger.
Our analysis reveals that such an ideal charging performance is caused by the
formation of two bound states in the energy spectrum of the total system
consisting of the QB, the charger, and the EMF in the waveguide. Without
resorting to direct charger-QB coupling, our scheme avoids the insufficient-
charging difficulty encountered in the case when the charger-QB coupling is
weak. It provides a useful path to the practical realization of antiaging QB.

This work is supported by the National Natural Science Foundation of China
(Grants No. 12105089, No. 12074107, No. 12275109, No. 12274422, and No.
12247101), the innovation group project (Grant No. 2022CFA012) of Hubei
Province, the program of outstanding young and middle-aged scientific and
technological innovation team of colleges and universities in Hubei Province
(Grant No. T2020001), and the Hubei Province Science Fund for Distinguished
Young Scholars under Grant No. 2020CFA078.

## References

  * Bharti  _et al._ [2022] K. Bharti, A. Cervera-Lierta, T. H. Kyaw, T. Haug, S. Alperin-Lea, A. Anand, M. Degroote, H. Heimonen, J. S. Kottmann, T. Menke, W.-K. Mok, S. Sim, L.-C. Kwek, and A. Aspuru-Guzik, Noisy intermediate-scale quantum algorithms, Rev. Mod. Phys. 94, 015004 (2022).
  * Xu  _et al._ [2020] F. Xu, X. Ma, Q. Zhang, H.-K. Lo, and J.-W. Pan, Secure quantum key distribution with realistic devices, Rev. Mod. Phys. 92, 025002 (2020).
  * Degen  _et al._ [2017] C. L. Degen, F. Reinhard, and P. Cappellaro, Quantum sensing, Rev. Mod. Phys. 89, 035002 (2017).
  * Mayer  _et al._ [2023] D. Mayer, E. Lutz, and A. Widera, Generalized Clausius inequalities in a nonequilibrium cold-atom system, Commun. Phys. 6, 61 (2023).
  * Maslennikov  _et al._ [2019] G. Maslennikov, S. Ding, R. Hablützel, J. Gan, A. Roulet, S. Nimmrichter, J. Dai, V. Scarani, and D. Matsukevich, Quantum absorption refrigerator with trapped ions, Nat. Commun. 10, 202 (2019).
  * Alicki and Kosloff [2018] R. Alicki and R. Kosloff, Introduction to quantum thermodynamics: History and prospects, in  _Thermodynamics in the Quantum Regime: Fundamental Aspects and New Directions_, edited by F. Binder, L. A. Correa, C. Gogolin, J. Anders, and G. Adesso (Springer International Publishing, Cham, 2018) pp. 1–33.
  * Klatzow  _et al._ [2019] J. Klatzow, J. N. Becker, P. M. Ledingham, C. Weinzetl, K. T. Kaczmarek, D. J. Saunders, J. Nunn, I. A. Walmsley, R. Uzdin, and E. Poem, Experimental demonstration of quantum effects in the operation of microscopic heat engines, Phys. Rev. Lett. 122, 110601 (2019).
  * Campaioli  _et al._ [2018] F. Campaioli, F. A. Pollock, and S. Vinjanampathy, Quantum batteries, in  _Thermodynamics in the Quantum Regime: Fundamental Aspects and New Directions_, edited by F. Binder, L. A. Correa, C. Gogolin, J. Anders, and G. Adesso (Springer International Publishing, Cham, 2018) pp. 207–225.
  * Roßnagel  _et al._ [2014] J. Roßnagel, O. Abah, F. Schmidt-Kaler, K. Singer, and E. Lutz, Nanoscale heat engine beyond the Carnot limit, Phys. Rev. Lett. 112, 030602 (2014).
  * Alicki and Fannes [2013] R. Alicki and M. Fannes, Entanglement boost for extractable work from ensembles of quantum batteries, Phys. Rev. E 87, 042123 (2013).
  * Linden  _et al._ [2010] N. Linden, S. Popescu, and P. Skrzypczyk, How small can thermal machines be? the smallest possible refrigerator, Phys. Rev. Lett. 105, 130401 (2010).
  * Scully  _et al._ [2003] M. O. Scully, M. S. Zubairy, G. S. Agarwal, and H. Walther, Extracting work from a single heat bath via vanishing quantum coherence, Science 299, 862 (2003).
  * Campaioli  _et al._ [2023] F. Campaioli, S. Gherardini, J. Q. Quach, M. Polini, and G. M. Andolina, Colloquium: Quantum batteries (2023), arXiv:2308.02277 [quant-ph] .
  * Bhattacharjee and Dutta [2021] S. Bhattacharjee and A. Dutta, Quantum thermal machines and batteries, Eur. Phys. J. B 94, 239 (2021).
  * Yang  _et al._ [2023] X. Yang, Y.-H. Yang, M. Alimuddin, R. Salvia, S.-M. Fei, L.-M. Zhao, S. Nimmrichter, and M.-X. Luo, Battery capacity of energy-storing quantum systems, Phys. Rev. Lett. 131, 030402 (2023).
  * Šafránek  _et al._ [2023] D. Šafránek, D. Rosa, and F. C. Binder, Work extraction from unknown quantum sources, Phys. Rev. Lett. 130, 210401 (2023).
  * Shi  _et al._ [2022] H.-L. Shi, S. Ding, Q.-K. Wan, X.-H. Wang, and W.-L. Yang, Entanglement, coherence, and extractable work in quantum batteries, Phys. Rev. Lett. 129, 130602 (2022).
  * Francica  _et al._ [2020] G. Francica, F. C. Binder, G. Guarnieri, M. T. Mitchison, J. Goold, and F. Plastina, Quantum coherence and ergotropy, Phys. Rev. Lett. 125, 180603 (2020).
  * Çakmak [2020] B. Çakmak, Ergotropy from coherences in an open quantum system, Phys. Rev. E 102, 042111 (2020).
  * Chitambar and Gour [2019] E. Chitambar and G. Gour, Quantum resource theories, Rev. Mod. Phys. 91, 025001 (2019).
  * Hovhannisyan  _et al._ [2013] K. V. Hovhannisyan, M. Perarnau-Llobet, M. Huber, and A. Acín, Entanglement generation is not necessary for optimal work extraction, Phys. Rev. Lett. 111, 240401 (2013).
  * Salvia  _et al._ [2023] R. Salvia, M. Perarnau-Llobet, G. Haack, N. Brunner, and S. Nimmrichter, Quantum advantage in charging cavity and spin batteries by repeated interactions, Phys. Rev. Res. 5, 013155 (2023).
  * Chen  _et al._ [2022] P. Chen, T. S. Yin, Z. Q. Jiang, and G. R. Jin, Quantum enhancement of a single quantum battery by repeated interactions with large spins, Phys. Rev. E 106, 054119 (2022).
  * Seah  _et al._ [2021] S. Seah, M. Perarnau-Llobet, G. Haack, N. Brunner, and S. Nimmrichter, Quantum speed-up in collisional battery charging, Phys. Rev. Lett. 127, 100601 (2021).
  * Liu  _et al._ [2021] J.-X. Liu, H.-L. Shi, Y.-H. Shi, X.-H. Wang, and W.-L. Yang, Entanglement and work extraction in the central-spin quantum battery, Phys. Rev. B 104, 245418 (2021).
  * Andolina  _et al._ [2018] G. M. Andolina, D. Farina, A. Mari, V. Pellegrini, V. Giovannetti, and M. Polini, Charger-mediated energy transfer in exactly solvable models for quantum batteries, Phys. Rev. B 98, 205423 (2018).
  * Santos  _et al._ [2023] T. F. F. Santos, Y. V. de Almeida, and M. F. Santos, Vacuum-enhanced charging of a quantum battery, Phys. Rev. A 107, 032203 (2023).
  * Hao  _et al._ [2023] X. Hao, K. Yan, J. Tan, and Q.-Y. Wu, Quantum work extraction of an accelerated Unruh-Dewitt battery in relativistic motion, Phys. Rev. A 107, 012207 (2023).
  * Abah  _et al._ [2022] O. Abah, G. De Chiara, M. Paternostro, and R. Puebla, Harnessing nonadiabatic excitations promoted by a quantum critical point: Quantum battery and spin squeezing, Phys. Rev. Res. 4, L022017 (2022).
  * Arjmandi  _et al._ [2022a] M. B. Arjmandi, A. Shokri, E. Faizi, and H. Mohammadi, Performance of quantum batteries with correlated and uncorrelated chargers, Phys. Rev. A 106, 062609 (2022a).
  * Maffei  _et al._ [2021] M. Maffei, P. A. Camati, and A. Auffèves, Probing nonclassical light fields with energetic witnesses in waveguide quantum electrodynamics, Phys. Rev. Res. 3, L032073 (2021).
  * Caravelli  _et al._ [2020] F. Caravelli, G. Coulter-De Wit, L. P. García-Pintos, and A. Hamma, Random quantum batteries, Phys. Rev. Res. 2, 023095 (2020).
  * Santos  _et al._ [2019] A. C. Santos, B. i. e. i. f. m. c. Çakmak, S. Campbell, and N. T. Zinner, Stable adiabatic quantum batteries, Phys. Rev. E 100, 032107 (2019).
  * Joshi and Mahesh [2022] J. Joshi and T. S. Mahesh, Experimental investigation of a quantum battery using star-topology NMR spin systems, Phys. Rev. A 106, 042601 (2022).
  * Zheng  _et al._ [2022] R.-H. Zheng, W. Ning, Z.-B. Yang, Y. Xia, and S.-B. Zheng, Demonstration of dynamical control of three-level open systems with a superconducting qutrit, New J. Phy. 24, 063031 (2022).
  * Hu  _et al._ [2022] C.-K. Hu, J. Qiu, P. J. P. Souza, J. Yuan, Y. Zhou, L. Zhang, J. Chu, X. Pan, L. Hu, J. Li, Y. Xu, Y. Zhong, S. Liu, F. Yan, D. Tan, R. Bachelard, C. J. Villas-Boas, A. C. Santos, and D. Yu, Optimal charging of a superconducting quantum battery, Quantum Sci. Technol. 7, 045018 (2022).
  * Maillette de Buy Wenniger  _et al._ [2023] I. Maillette de Buy Wenniger, S. E. Thomas, M. Maffei, S. C. Wein, M. Pont, N. Belabas, S. Prasad, A. Harouri, A. Lemaître, I. Sagnes, N. Somaschi, A. Auffèves, and P. Senellart, Experimental analysis of energy transfers between a quantum emitter and light fields, Phys. Rev. Lett. 131, 260401 (2023).
  * Gemme  _et al._ [2022] G. Gemme, M. Grossi, D. Ferraro, S. Vallecorsa, and M. Sassetti, IBM quantum platforms: A quantum battery perspective, Batteries 8, 43 (2022).
  * Zhu  _et al._ [2023] G. Zhu, Y. Chen, Y. Hasegawa, and P. Xue, Charging quantum batteries via indefinite causal order: Theory and experiment, Phys. Rev. Lett. 131, 240401 (2023).
  * Huang  _et al._ [2023] X. Huang, K. Wang, L. Xiao, L. Gao, H. Lin, and P. Xue, Demonstration of the charging progress of quantum batteries, Phys. Rev. A 107, L030201 (2023).
  * Quach  _et al._ [2022] J. Q. Quach, K. E. McGhee, L. Ganzer, D. M. Rouse, B. W. Lovett, E. M. Gauger, J. Keeling, G. Cerullo, D. G. Lidzey, and T. Virgili, Superabsorption in an organic microcavity: Toward a quantum battery, Sci. Adv. 8, eabk3160 (2022).
  * Zhao  _et al._ [2022] F. Zhao, F.-Q. Dou, and Q. Zhao, Charging performance of the Su-Schrieffer-Heeger quantum battery, Phys. Rev. Res. 4, 013172 (2022).
  * Dou  _et al._ [2022a] F.-Q. Dou, Y.-Q. Lu, Y.-J. Wang, and J.-A. Sun, Extended Dicke quantum battery with interatomic interactions and driving field, Phys. Rev. B 105, 115405 (2022a).
  * Konar  _et al._ [2022] T. K. Konar, L. G. C. Lakkaraju, S. Ghosh, and A. Sen De, Quantum battery with ultracold atoms: Bosons versus fermions, Phys. Rev. A 106, 022618 (2022).
  * Ghosh and Sen [De] S. Ghosh and A. Sen(De), Dimensional enhancements in a quantum battery with imperfections, Phys. Rev. A 105, 022628 (2022).
  * Dou  _et al._ [2022b] F.-Q. Dou, H. Zhou, and J.-A. Sun, Cavity Heisenberg-spin-chain quantum battery, Phys. Rev. A 106, 032212 (2022b).
  * Gyhm  _et al._ [2022] J.-Y. Gyhm, D. Šafránek, and D. Rosa, Quantum charging advantage cannot be extensive without global operations, Phys. Rev. Lett. 128, 140501 (2022).
  * Arjmandi  _et al._ [2022b] M. B. Arjmandi, H. Mohammadi, and A. C. Santos, Enhancing self-discharging process with disordered quantum batteries, Phys. Rev. E 105, 054115 (2022b).
  * Barra  _et al._ [2022] F. Barra, K. V. Hovhannisyan, and A. Imparato, Quantum batteries at the verge of a phase transition, New J. Phys. 24, 015003 (2022).
  * Zhao  _et al._ [2021] F. Zhao, F.-Q. Dou, and Q. Zhao, Quantum battery of interacting spins with environmental noise, Phys. Rev. A 103, 033715 (2021).
  * Huangfu and Jing [2021] Y. Huangfu and J. Jing, High-capacity and high-power collective charging with spin chargers, Phys. Rev. E 104, 024129 (2021).
  * Ghosh  _et al._ [2020] S. Ghosh, T. Chanda, and A. Sen(De), Enhancement in the performance of a quantum battery by ordered and disordered interactions, Phys. Rev. A 101, 032115 (2020).
  * Kamin  _et al._ [2020a] F. H. Kamin, F. T. Tabesh, S. Salimi, and A. C. Santos, Entanglement, coherence, and charging process of quantum batteries, Phys. Rev. E 102, 052109 (2020a).
  * Rossini  _et al._ [2020] D. Rossini, G. M. Andolina, D. Rosa, M. Carrega, and M. Polini, Quantum advantage in the charging process of Sachdev-Ye-Kitaev batteries, Phys. Rev. Lett. 125, 236402 (2020).
  * Rosa  _et al._ [2020] D. Rosa, D. Rossini, G. M. Andolina, M. Polini, and M. Carrega, Ultra-stable charging of fast-scrambling SYK quantum batteries, J. High Energy Phys. 2020 (11), 67.
  * Rossini  _et al._ [2019] D. Rossini, G. M. Andolina, and M. Polini, Many-body localized quantum batteries, Phys. Rev. B 100, 115142 (2019).
  * Andolina  _et al._ [2019] G. M. Andolina, M. Keck, A. Mari, V. Giovannetti, and M. Polini, Quantum versus classical many-body batteries, Phys. Rev. B 99, 205437 (2019).
  * Le  _et al._ [2018] T. P. Le, J. Levinsen, K. Modi, M. M. Parish, and F. A. Pollock, Spin-chain model of a many-body quantum battery, Phys. Rev. A 97, 022106 (2018).
  * Ferraro  _et al._ [2018] D. Ferraro, M. Campisi, G. M. Andolina, V. Pellegrini, and M. Polini, High-power collective charging of a solid-state quantum battery, Phys. Rev. Lett. 120, 117702 (2018).
  * Campaioli  _et al._ [2017] F. Campaioli, F. A. Pollock, F. C. Binder, L. Céleri, J. Goold, S. Vinjanampathy, and K. Modi, Enhancing the charging power of quantum batteries, Phys. Rev. Lett. 118, 150601 (2017).
  * Binder  _et al._ [2015] F. C. Binder, S. Vinjanampathy, K. Modi, and J. Goold, Quantacell: powerful charging of quantum batteries, New J. Phys. 17, 075015 (2015).
  * Pirmoradian and Mølmer [2019] F. Pirmoradian and K. Mølmer, Aging of a quantum battery, Phys. Rev. A 100, 043833 (2019).
  * Yao and Shao [2022] Y. Yao and X. Q. Shao, Optimal charging of open spin-chain quantum batteries via homodyne-based feedback control, Phys. Rev. E 106, 014138 (2022).
  * Quach and Munro [2020] J. Q. Quach and W. J. Munro, Using dark states to charge and stabilize open quantum batteries, Phys. Rev. Appl. 14, 024092 (2020).
  * Bai and An [2020] S.-Y. Bai and J.-H. An, Floquet engineering to reactivate a dissipative quantum battery, Phys. Rev. A 102, 060201(R) (2020).
  * Xu  _et al._ [2022] K. Xu, H.-G. Li, Z.-G. Li, H.-J. Zhu, G.-F. Zhang, and W.-M. Liu, Charging performance of quantum batteries in a double-layer environment, Phys. Rev. A 106, 012425 (2022).
  * Carrasco  _et al._ [2022] J. Carrasco, J. R. Maze, C. Hermann-Avigliano, and F. Barra, Collective enhancement in dissipative quantum batteries, Phys. Rev. E 105, 064119 (2022).
  * Mayo and Roncaglia [2022] F. Mayo and A. J. Roncaglia, Collective effects and quantum coherence in dissipative charging of quantum batteries, Phys. Rev. A 105, 062203 (2022).
  * Xu  _et al._ [2021] K. Xu, H.-J. Zhu, G.-F. Zhang, and W.-M. Liu, Enhancing the performance of an open quantum battery via environment engineering, Phys. Rev. E 104, 064143 (2021).
  * Ghosh  _et al._ [2021] S. Ghosh, T. Chanda, S. Mal, and A. Sen(De), Fast charging of a quantum battery assisted by noise, Phys. Rev. A 104, 032207 (2021).
  * Yao and Shao [2021] Y. Yao and X. Q. Shao, Stable charging of a Rydberg quantum battery in an open system, Phys. Rev. E 104, 044116 (2021).
  * Barra [2019] F. Barra, Dissipative charging of a quantum battery, Phys. Rev. Lett. 122, 210601 (2019).
  * Farina  _et al._ [2019] D. Farina, G. M. Andolina, A. Mari, M. Polini, and V. Giovannetti, Charger-mediated energy transfer for quantum batteries: An open-system approach, Phys. Rev. B 99, 035421 (2019).
  * Song  _et al._ [2022] M.-L. Song, L.-J. Li, X.-K. Song, L. Ye, and D. Wang, Environment-mediated entropic uncertainty in charging quantum batteries, Phys. Rev. E 106, 054107 (2022).
  * Qi and Jing [2021] S.-f. Qi and J. Jing, Magnon-mediated quantum battery under systematic errors, Phys. Rev. A 104, 032606 (2021).
  * Tabesh  _et al._ [2020] F. T. Tabesh, F. H. Kamin, and S. Salimi, Environment-mediated charging process of quantum batteries, Phys. Rev. A 102, 052223 (2020).
  * Kamin  _et al._ [2020b] F. H. Kamin, F. T. Tabesh, S. Salimi, F. Kheirandish, and A. C. Santos, Non-Markovian effects on charging and self-discharging process of quantum batteries, New J. Phys. 22, 083007 (2020b).
  * Awschalom  _et al._ [2021] D. Awschalom, K. K. Berggren, H. Bernien, S. Bhave, L. D. Carr, P. Davids, S. E. Economou, D. Englund, A. Faraon, M. Fejer, S. Guha, M. V. Gustafsson, E. Hu, L. Jiang, J. Kim, B. Korzh, P. Kumar, P. G. Kwiat, M. Lončar, M. D. Lukin, D. A. Miller, C. Monroe, S. W. Nam, P. Narang, J. S. Orcutt, M. G. Raymer, A. H. Safavi-Naeini, M. Spiropulu, K. Srinivasan, S. Sun, J. Vučković, E. Waks, R. Walsworth, A. M. Weiner, and Z. Zhang, Development of quantum interconnects (quics) for next-generation information technologies, PRX Quantum 2, 017002 (2021).
  * Tabares  _et al._ [2023] C. Tabares, A. Muñoz de las Heras, L. Tagliacozzo, D. Porras, and A. González-Tudela, Variational quantum simulators based on waveguide QED, Phys. Rev. Lett. 131, 073602 (2023).
  * Bai and An [2021] S.-Y. Bai and J.-H. An, Generating stable spin squeezing by squeezed-reservoir engineering, Phys. Rev. Lett. 127, 083602 (2021).
  * Song  _et al._ [2017] W. Song, W. Yang, J. An, and M. Feng, Dissipation-assisted spin squeezing of nitrogen-vacancy centers coupled to a rectangular hollow metallic waveguide, Opt. Express 25, 19226 (2017).
  * Shahmoon and Kurizki [2013] E. Shahmoon and G. Kurizki, Nonradiative interaction and entanglement between distant atoms, Phys. Rev. A 87, 033831 (2013).
  * [83] See the Supplemental Material that contains Ref. [PhysRevLett.125.040601] for a detailed derivation of the energy spectrum, the dynamics and the steady-state solution, and the detailed analysis of the full cycle of the QB scheme.
  * Monsel  _et al._ [2020] J. Monsel, M. Fellous-Asiani, B. Huard, and A. Auffèves, The energetic cost of work extraction, Phys. Rev. Lett. 124, 130601 (2020).
  * Liu  _et al._ [2016] H.-B. Liu, W. L. Yang, J.-H. An, and Z.-Y. Xu, Mechanism for quantum speedup in open quantum systems, Phys. Rev. A 93, 020105(R) (2016).
  * Ji  _et al._ [2022] F.-Z. Ji, S.-Y. Bai, and J.-H. An, Strong coupling of quantum emitters and the exciton polariton in MoS2 nanodisks, Phys. Rev. B 106, 115427 (2022).
  * Falk  _et al._ [2009] A. L. Falk, F. H. L. Koppens, C. L. Yu, K. Kang, N. de Leon Snapp, A. V. Akimov, M.-H. Jo, M. D. Lukin, and H. Park, Near-field electrical detection of optical plasmons and single-plasmon sources, Nat. Phys. 5, 475 (2009).
  * Gonzalez-Tudela  _et al._ [2011] A. Gonzalez-Tudela, D. Martin-Cano, E. Moreno, L. Martin-Moreno, C. Tejedor, and F. J. Garcia-Vidal, Entanglement of two qubits mediated by one-dimensional plasmonic waveguides, Phys. Rev. Lett. 106, 020501 (2011).
  * Sheremet  _et al._ [2023] A. S. Sheremet, M. I. Petrov, I. V. Iorsh, A. V. Poshakinskiy, and A. N. Poddubny, Waveguide quantum electrodynamics: Collective radiance and photon-photon correlations, Rev. Mod. Phys. 95, 015002 (2023).
  * Foster  _et al._ [2019] A. P. Foster, D. Hallett, I. V. Iorsh, S. J. Sheldon, M. R. Godsland, B. Royall, E. Clarke, I. A. Shelykh, A. M. Fox, M. S. Skolnick, I. E. Itskevich, and L. R. Wilson, Tunable photon statistics exploiting the fano effect in a waveguide, Phys. Rev. Lett. 122, 173603 (2019).
  * Mirhosseini  _et al._ [2019] M. Mirhosseini, E. Kim, X. Zhang, A. Sipahigil, P. B. Dieterle, A. J. Keller, A. Asenjo-Garcia, D. E. Chang, and O. Painter, Cavity quantum electrodynamics with atom-like mirrors, Nature (London) 569, 692 (2019).
  * Sipahigil  _et al._ [2016] A. Sipahigil, R. E. Evans, D. D. Sukachev, M. J. Burek, J. Borregaard, M. K. Bhaskar, C. T. Nguyen, J. L. Pacheco, H. A. Atikian, C. Meuwly, R. M. Camacho, F. Jelezko, E. Bielejec, H. Park, M. Lončar, and M. D. Lukin, An integrated diamond nanophotonics platform for quantum-optical networks, Science 354, 847 (2016).
  * Amsüss  _et al._ [2011] R. Amsüss, C. Koller, T. Nöbauer, S. Putz, S. Rotter, K. Sandner, S. Schneider, M. Schramböck, G. Steinhauser, H. Ritsch, J. Schmiedmayer, and J. Majer, Cavity qed with magnetically coupled collective spin states, Phys. Rev. Lett. 107, 060502 (2011).
  * Mouradian  _et al._ [2015] S. L. Mouradian, T. Schröder, C. B. Poitras, L. Li, J. Goldstein, E. H. Chen, M. Walsh, J. Cardenas, M. L. Markham, D. J. Twitchen, M. Lipson, and D. Englund, Scalable integration of long-lived quantum memories into a photonic circuit, Phys. Rev. X 5, 031009 (2015).
  * Hoese  _et al._ [2021] M. Hoese, M. K. Koch, V. Bharadwaj, J. Lang, J. P. Hadden, R. Yoshizaki, A. N. Giakoumaki, R. Ramponi, F. Jelezko, S. M. Eaton, and A. Kubanek, Integrated magnetometry platform with stackable waveguide-assisted detection channels for sensing arrays, Phys. Rev. Appl. 15, 054059 (2021).
  * Humbert  _et al._ [2022] M. Humbert, P. R. Wiecha, G. Colas des Francs, X. Yu, N. Mallet, A. Lecestre, G. Larrieu, V. Larrey, F. Fournel, L. Ressier, C. Girard, V. Paillard, and A. Cuche, Tailoring wavelength- and emitter-orientation-dependent propagation of single photons in silicon nanowires, Phys. Rev. Appl. 17, 014008 (2022).
  * Fukami  _et al._ [2021] M. Fukami, D. R. Candido, D. D. Awschalom, and M. E. Flatté, Opportunities for long-range magnon-mediated entanglement of spin qubits via on- and off-resonant coupling, PRX Quantum 2, 040314 (2021).
  * Liu and Houck [2017] Y. Liu and A. A. Houck, Quantum electrodynamics near a photonic bandgap, Nat. Phys. 13, 48 (2017).
  * Krinner  _et al._ [2018] L. Krinner, M. Stewart, A. Pazmiño, J. Kwon, and D. Schneble, Spontaneous emission of matter waves from a tunable open quantum system, Nature (London) 559, 589 (2018).
  * Kwon  _et al._ [2022] J. Kwon, Y. Kim, A. Lanuza, and D. Schneble, Formation of matter-wave polaritons in an optical lattice, Nat. Phys. 18, 657 (2022).