The guidance and tracking system as a whole is an incredibly complex system.
The system itself can be described as a fusion of inter-operable systems with
a common goal. This makes the system a system of systems (SoS).

System Breakdown

[scale=0.25]Pictures/missileInterceptDiagram.png

Figure : Ballistic Missile Acqusition and Engagement

The Aegis system utilizes both local and remote subsystems (Figure ). The
capabilities of these subsystems are realized during the acquisition,
planning, tracking, and engagement of a ballistic threat. The system utilizes
external systems such as satellites, sea-based radar, ground-based tracking
radar, and forward-based radar systems. Once a threat is detected, these
external subsystems transfer data to shipboard systems for threat tracking and
engagement. Local systems include radar, electro-optic/infrared sensors,
passive electronic warfare sensors, communications/link architecture, command
and control systems, decoys, electronic countermeasures [sysArch]. Once the
threat is tracked, it is intercepted utilizing shipboard weapons systems. For
Aegis ballistic missile defense systems, standard Missile or SM-2-Block IV
missiles are utilized to engage and destroy the threat [sysArch]. Furthermore,
the system communicates with various entities such as U.S. Strategic Command,
U.S. Pacific Command, and fire control suites across the nation. These systems
must work in unison to achieve a common goal (threat detection and
engagement).

##  SoS Entites

Four entities comprise a SoS.

  1. 1.

Services: Services of a SoS are the services they provide. Such services are
the direct applications of the system (Ballistic missile defense).
Furthermore, the system provides indirect benefits. These are the
technological advancements made in the development and application of the SoS.
For guidance and tracking subsystems, advancements would improve a variety of
civilian services. Some of these services include the optimization of driver
routes, reduction of theft and control of vehicle functions, mobility pattern
recognition, improved vehicle navigation, and fleet management.

  2. 2.

Service Owners: Entities that have final say over service tasking. Service
owners also uphold resource allocation. These entities are also the primary
motivation in the development of the SoS. This can be through policy
decisions, technical, and performance requirements as stated in the contract.

  3. 3.

Stakeholders: SoS stakeholders are those that benefit from the capabilities of
the system. For the vehicle guidance and tracking subsystems, stakeholders are
both military and civilian entities. For civilians, the technological
advancements of the system have the potential of influencing the development
of civilian technology. For both civilian and defense entities, the system
provides a technological advantage over adversaries and better defense
potential.

  4. 4.

Contracts: Contracts are essential to a SoS because they establish the
framework in which the SoS operates. Contracts dictate the allocations of
resources throughout the life cycle of the system. They also organize the
roles of the SoS entites.

##  Boardman and Sauser SoS Characteristics

In order to better understand the composition of the SoS, the relationships
and roles of subsystems will be investigated. Boardman and Sauser identify a
specific set of characteristics that define a SoS. These are autonomy,
belonging, connectivity, diversity, and emergence [bsSoS]. Autonomy is the
capability of subsystems to complete tasks without outside control. Some
systems that comprise the guidance and tracking systems are operationally
independent. An example of this is the forward-based radar system (AN/TPY-2).
The unit itself was designed as a multi-functional system. This system can be
self-cued or accept cues from outside systems (Aegis or other satellites)
[antpy2]. These independent systems are networked together to create the Aegis
system. The system’s contribution to national defense cements its belonging.
Guidance and tracking systems are essential when it comes to ballistic missile
defense. Connectivity measures the interoperability between the system and new
systems. The Aegis system fulfills this characteristic because of the network
of interconnected systems it uses. As mentioned previously, the system has a
network of satellites for threat detection. Upon threat detection, the data is
transferred to the appropriate system for further analysis. Boardman and
Sauser describe that it is necessity for a SoS to ”be incredibly diverse in
its capability as a system” [bsSoS]. Ballistic missile defense systems are
incredibly diverse in their capabilities. Not only is the Aegis implemented in
seagoing vessels, but land based systems as well [missileDefenseAgency].
Emergent capabilities are designed into a SoS by a virtue of factors. These
factors are defined as ”preservation of constituent systems autonomy, choosing
to belong, enriched connectivity, and commitment to diversity of SoS
manifestations and behavior” [bsSoS] The Aegis system is continuously evolving
and flaws in the system are being discovered and resolved. The developments
are essential to keep a technological advantage over adversaries.


# Control of the Purcell effect via unexcited atoms and exceptional points

G. S. Agarwal<EMAIL_ADDRESS>Institute for Quantum Science and
Engineering, Department of Physics and Astronomy, Department of Biological and
Agricultural Engineering, Texas A&M University, College Station, Texas 77843,
USA

###### Abstract

We examine the possible control of the celebrated Purcell effect in cavity
quantum electrodynamics. We demonstrate that the presence of an unexcited atom
can significantly alter the Purcell decay depending on the strength of
coupling of the unexcited atom with the cavity mode though the excited atom
has to be weakly coupled for it to be in the Purcell regime. This is distinct
from the nonradiative nature of the singlet state which is an entangled state
of the two atom system. We present physical interpretation for inhibition as
due to interference between two polariton channels of decay. We bring out
connection to exceptional points in the cavity QED system as the unexcited
atom and cavity mode can produce a second order exceptional point. We further
show how two unexcited atoms can create a third order exceptional point
leading to inhibition of Purcell effect. We also discuss the case when the
Purcell effect can be enhanced.

Introduction.– The Purcell effect has been the hallmark of cavity quantum
electrodynamics [1, 2, 3, 4, 5, 6, 7, 8]. It brought for the first time the
role of coupling to the cavity and how this coupling can lead to the decay of
the atom even if the decay of the atom otherwise is negligible. The enhanced
decay results due to enhancement of the density of states in the cavity.
Analogues of similar enhancement in the density of states and the resulting
Purcell effect are known in other contexts, for example in the decay of the
atom located in the close vicinity of the metal [5, 6, 8] and in other
photonic structures [4, 9]. Cavities thus provide an effective way to deal
with the manipulation of spontaneous emission and these can also lead to
inhibition of spontaneous emission [3, 10] if the density of states at the
location of the atom is negligible. Other methods to control spontaneous
emission are based on photonic crystals [11, 12, 13], whispering gallery modes
[14] etc; one dimensional photonic waveguides [4]. A new direction in recent
years, is to use exceptional points [EP] for studying various effects like
enhanced sensing capabilities of parameters when perturbed in the vicinity of
the EP [15, 16, 17, 18, 19, 20, 21]. Such enhanced capabilities have been
demonstrated especially in systems with PT symmetry i.e. with both gain and
loss [22, 23]. Enhanced sensing at both second order [16] and higher order
EP’s [17] have been studied. While these are semiclassical studies and there
exist plenty of such studies; the QED effects near EP’s are still scant [24].
Early studies focused on spontaneous generation of photons [25], where as more
recent ones examined the effects of quantum fluctuations on signal to noise
ratio in sensing [19, 20, 21]. Other applications of EP in the context of
quantum physics are starting to appear [26, 24, 27]. A recent one presents
speed up of the generation of quantum entanglement near EP [26].

In this work we consider how the presence of an unexcited atom can influence
significantly the Purcell effect of an excited atom. The two atoms are
differently coupled to the cavity. The excited atom is weakly coupled to the
cavity and this can be achieved by placing it close to the node of the cavity
mode. We consider several different scenarios for strong inhibition – The
first scenario is such that the atom’s coupling leads to large cooperativity
parameter. In other scenarios we can use unexcited atoms to create EP’s of
second and higher orders. It is easy to create second order EP whereas it is
more involved to create higher order EP’s. We create third order EP by
choosing two unexcited atoms with appropriate decay parameters and then
examine the decay of a weakly coupled third atom. In each case we present
approximate analytical results for the inhibition of the Purcell decay. We
also present physical interpretation of the inhibition in terms of the cavity
polaritons [28]. The approximate results are confirmed from the solution of
full quantum master equation describing the dynamics of atoms in a cavity. The
inhibition effects are especially pronounced for Rydberg atoms which were used
in the first study of the Purcell effect [2]. For detuned excited atom, the
Purcell decay can be enhanced by the unexcited atom. In this case, the
enhancement arises as the detuned excited atom can come in resonance with one
of the cavity polariton states. Thus in general, the unexcited atom can lead
to the significant inhibition or enhancement of the Purcell decay. The results
that we report are quite generic to any cavity QED system and can be studied
experimentally for example with superconducting qubits in cavities [29] or
with atoms trapped in cavities [30].

Control of the Purcell Decay.– In order to demonstrate how the Purcell decay
can be controlled, we start with the simplest model [Fig. 1(a)] with two atoms
in a cavity with frequency $\displaystyle\omega_{c}$. For simplicity we assume
that both atoms are on resonance with the cavity mode. Let $\displaystyle
2\kappa$ be the leakage rate of photons from the cavity and let $\displaystyle
g_{A}$ be the coupling constant of the excited atom A to the cavity mode. Then
the Purcell decay of the excited atom is given by $\displaystyle
2\Gamma=2g_{A}^{2}/\kappa$ if $\displaystyle g_{A}^{2}/\kappa^{2}\ll 1$. Let
us now introduce an unexcited atom B with a coupling constant $\displaystyle
g_{B}$ with the cavity mode. We will keep $\displaystyle g_{B}$ flexible – we
will examine both cases when $\displaystyle g_{B}$ is large and when
$\displaystyle g_{B}$ is small. This can be achieved by moving the position of
the atom B along the cavity mode function. The interaction Hamiltonian in a
frame rotating with the frequency $\displaystyle\omega_{c}$ is given by

Figure 1: Purcell decay of an excited atom A in presence of an unexcited atom
B and the relevant quantum states participating in dynamical evolution.

$H=\hbar g_{A}(a^{\dagger}S_{A}^{-}+aS_{A}^{+})+\hbar
g_{B}(a^{\dagger}S_{B}^{-}+aS_{B}^{+}),$ (1)

where $\displaystyle S_{A}^{+}$ and $\displaystyle S_{B}^{+}$ are the
excitation operators for atoms A and B respectively. The master equation of
the atoms and the cavity mode is

$\displaystyle\frac{\partial\rho}{\partial t}$
$\displaystyle=-\frac{i}{\hbar}[H,\rho]-\kappa(a^{\dagger}a\rho-2a\rho
a^{\dagger}+\rho a^{\dagger}a)$ (2)
$\displaystyle\qquad-\gamma_{A}(S_{A}^{+}S_{A}^{-}\rho-2S_{A}^{-}\rho
S_{A}^{+}+\rho S_{A}^{+}S_{A}^{-})$
$\displaystyle\qquad-\gamma_{B}(S_{B}^{+}S_{B}^{-}\rho-2S_{B}^{-}\rho
S_{B}^{+}+\rho S_{B}^{+}S_{B}^{-}),$

where atom A [B] decays from the excited state $\displaystyle|e\rangle$ to the
ground state $\displaystyle|g\rangle$ at the rate $\displaystyle 2\gamma_{A}$
[$\displaystyle 2\gamma_{B}$]. The initial state of the system consisting of
two atoms and cavity mode is $\displaystyle|e_{A},g_{B},0\rangle$, i.e., atom
A is excited and both atom B and the cavity mode are in ground state. Since
there is only one excitation in the system, during dynamical evolution we have
the following possibilities as illustrated in Fig. 1(b):
$\displaystyle|\psi_{1}\rangle=|e_{A},g_{B},0\rangle\stackrel{{\scriptstyle
g_{A}}}{{\rightleftarrows}}|g_{A},g_{B},1\rangle=|\psi_{2}\rangle$;
$\displaystyle|g_{A},g_{B},1\rangle\stackrel{{\scriptstyle
g_{B}}}{{\rightleftarrows}}|g_{A},e_{B},0\rangle=|\psi_{3}\rangle$;
$\displaystyle|e_{A},g_{B},0\rangle\stackrel{{\scriptstyle\gamma_{A}}}{{\rightarrow}}|g_{A},g_{B},0\rangle=|\psi_{4}\rangle$;
$\displaystyle|g_{A},g_{B},1\rangle\stackrel{{\scriptstyle\kappa}}{{\rightarrow}}|g_{A},g_{B},0\rangle$;
$\displaystyle|g_{A},e_{B},0\rangle\stackrel{{\scriptstyle\gamma_{B}}}{{\rightarrow}}|g_{A},g_{B},0\rangle$.
Thus the time evolution is restricted to only four states
$\displaystyle|\psi_{j}\rangle$, $\displaystyle j=1,\ldots,4$. It is to be
noted that no population is transferred back from the state
$\displaystyle|\psi_{4}\rangle$ to the states $\displaystyle|\psi_{i}\rangle$,
$\displaystyle i=1,2,3$. This observation allows us to express elements of the
density matrix as $\displaystyle(\rho(t))_{ij}=\psi_{i}(t)\psi_{j}^{*}(t)$,
$\displaystyle i=1,2,3$, $\displaystyle j=1,2,3$. This procedure is exact.
Using Eq. (2), the time evolution is then given by
$\displaystyle\dot{\psi}_{i}(t)=-iM\psi$, with

$M=\left(\begin{array}[]{ccc}-i\gamma_{A}&g_{A}&0\\\ g_{A}&-i\kappa&g_{B}\\\
0&g_{B}&-i\gamma_{B}\end{array}\right).$ (3)

The eigenvalues $\displaystyle\lambda$ of $\displaystyle M$ are given by

$(\lambda+i\kappa)(\lambda+i\gamma_{A})(\lambda+i\gamma_{B})-g_{B}^{2}(\lambda+i\gamma_{A})-g_{A}^{2}(\lambda+i\gamma_{B})=0.$
(4)

To see the emergence of the Purcell effect, we check the relevant eigenvalues
for $\displaystyle g_{B}=0$, $\displaystyle\gamma_{A}=0$. These are
$\displaystyle-\frac{i\kappa}{2}\pm\frac{i}{2}\sqrt{\kappa^{2}-4g_{A}^{2}}\rightarrow-i\kappa;-\frac{ig_{A}^{2}}{\kappa}$
in the limit of $\displaystyle\frac{4g^{2}}{\kappa^{2}}\ll 1$. The eigenvalue
$\displaystyle-i\kappa$ is the cavity decay whereas the eigenvalue
$\displaystyle-ig_{A}^{2}/\kappa$ gives the Purcell decay. In presence of the
unexcited atom B, $\displaystyle g_{B}\neq 0$, we can evaluate perturbativity
the Purcell decay as $\displaystyle 4g_{A}^{2}/\kappa^{2}$ is small. Note that
for $\displaystyle g_{A}=0$, the two eigenvalue are given by

$\lambda_{\pm}=-\frac{i(\kappa+\gamma_{B})}{2}\pm\sqrt{g_{B}^{2}-\left(\frac{\kappa-\gamma_{B}}{2}\right)^{2}},$
(5)

which give the energies and decays of the well known dressed states, also
known as cavity polariton states. The eigenvalues associated with Eq. (4) will
have a perturbative correction term of order of $\displaystyle g_{A}^{2}$. The
perturbative corrections to Eq. (5) can be ignored because of the large term
$\displaystyle-i(\kappa+\gamma_{B})/2$. The other eigenvalue which was
$\displaystyle g_{A}^{2}/\kappa$ if $\displaystyle g_{B}=0$, now becomes

$\tilde{\Gamma}=\Gamma/(1+\frac{g_{B}^{2}}{\gamma_{B}\kappa})=\Gamma/(1+C),\qquad\Gamma=\frac{g_{A}^{2}}{\kappa},$
(6)

where $\displaystyle C$ is the coorperativity parameter $\displaystyle
g_{B}^{2}/\gamma_{B}\kappa$. We thus find that the Purcell decay constant of
the excited state of the atom A can be inhibited by a factor $\displaystyle
1+C$, i.e., by a large amount if the cooperativity parameter $\displaystyle C$
is large. For strong coupling of the unexcited atom, the inhibition is quite
large. In such a situation the population remains trapped in excited state! We
remind the reader that this is different from subradiance which requires two
atoms to be prepared in a singlet state which is an entangled state, in
addition to the requirement that the two atoms have identical coupling to the
cavity. In contrast for our case $\displaystyle g_{A}\neq g_{B}$ and that the
initial state is an unentangled state $\displaystyle|e_{A},g_{B},0\rangle$.
While we have derived the result given by Eq. (6) from the consideration of
the eigenvalues of the matrix $\displaystyle M$ which gives the evolution of
populations in the excited states, we have also confirmed Eq. (6) from the
full time dependent solution of the master equation (2). We evaluated the
population in $\displaystyle|e_{A},g_{B},0\rangle$, i.e.,
$\displaystyle\rho_{11}(t)$ and fit it to an exponential function
$\displaystyle\exp(-2\tilde{\Gamma}t)$. This fitting yields the result Eq. (6)
– for instance, $\displaystyle\tilde{\Gamma}/\Gamma=1$, $\displaystyle 0.1$,
$\displaystyle 0.04$ for $\displaystyle g_{B}/\kappa=0$, $\displaystyle 3$ and
$\displaystyle 5$ respectively and for $\displaystyle\gamma_{B}/\kappa=1$. We
also note in passing that Eq. (4) admits other behavior as well. For example,
for identical atoms, if $\displaystyle\gamma_{A}=\gamma_{B}\sim 0$,
$\displaystyle g_{A}=g_{B}=g$, then eigenvalues are $\displaystyle\lambda=0$,
$\displaystyle\lambda(\lambda+i\kappa)-2g^{2}=0$, which lead to subradiant and
superradiant character [29].

Purcell Decay at the second ord EP.–

Figure 2: Decay channels for the excited state of atom A via the cavity
polaritons/dressed states $\displaystyle|\psi_{\pm}\rangle$ with separation
$\displaystyle 2\sqrt{g_{B}^{2}-(\frac{\kappa-\lambda}{2})^{2}}$ if
$\displaystyle g_{B}>\left|\frac{\kappa-\lambda}{2}\right|$.

Note that the cavity QED system can produce an EP if $\displaystyle
g_{B}=|\frac{\kappa-\gamma_{B}}{2}|$, then from Eq. (5) the two degenerate
eigenvalues are $\displaystyle-\frac{i(\kappa+\gamma_{B})}{2}$ and the two
eigenfunctions coalesce into $\displaystyle(1,i{\rm
sign}(\kappa-\gamma_{B}))^{T}/\sqrt{2}$, where $\displaystyle T$ stands for
transpose. This is the second order EP which is different from what happens in
PT symmetric systems where the two eigenvalues are strictly zero. At the
second order EP created by the unexcited atom B, the result Eq. (6) becomes

$\tilde{\Gamma}=\Gamma(1-(\frac{\kappa-\gamma_{B}}{\kappa+\gamma_{B}})^{2}),$
(7)

which can become much smaller than $\displaystyle\Gamma$ if
$\displaystyle\kappa/\gamma_{B}\ll 1$ or $\displaystyle\kappa/\gamma_{B}\gg
1$. Again the validity of the result (7) has been confirmed from the solution
of the master equation (2). For instance
$\displaystyle\tilde{\Gamma}/\Gamma=0.555$ for
$\displaystyle\gamma_{B}/\kappa=5$ or $\displaystyle\kappa/\gamma_{B}=5$ and
with $\displaystyle g_{B}$ given in terms of $\displaystyle\gamma_{B}$ and
$\displaystyle\kappa$, i.e., $\displaystyle
g_{B}=\frac{|\gamma_{B}-\kappa|}{2}$.

The inhibition can also be understood in terms of the dressed states
$\displaystyle|\psi_{\pm}\rangle$ or cavity polaritons created by the atom B
in the absence of atom A(see Fig. 2). These correspond to the eigenvalues
$\displaystyle\lambda_{\pm}$ (Eq. (5)). The states
$\displaystyle|\psi_{\pm}\rangle$ are linear combinations of the states
$\displaystyle|e_{B},0\rangle$ and $\displaystyle|g_{B},1\rangle$, i.e.,
$\displaystyle(|g_{B},1\rangle\pm|e_{B},0\rangle)/\sqrt{2}$, where
$\displaystyle|e_{B}\rangle$ and $\displaystyle|g_{B}\rangle$ are the excited
and ground states of atom B. The excited states $\displaystyle|e_{A}\rangle$
couples to both $\displaystyle|\psi_{\pm}\rangle$ via the component
$\displaystyle|g_{B},1\rangle$. Thus in the Purcell decay of atom A, there
will be interference effects between the two transition amplitudes
corresponding to channels $\displaystyle|e_{A}\rangle\stackrel{{\scriptstyle
g_{A}}}{{\rightarrow}}|\psi_{\pm}\rangle|g_{A}\rangle\stackrel{{\scriptstyle\gamma_{B},\kappa}}{{\rightarrow}}|g_{A},g_{B},0\rangle$
leading to the inhibition of decay of atom A.

We like to add that spontaneous emission at EP has been considered before for
two different models – Agarwal [31] used a single Lambda atomic scheme to
create double pole in S matrix via coupling to a laser field and then
considered spontaneous emission under weak coupling on the transition
uncoupled to the laser field [32, 33]. Khanbekyan and Wiersig [24] considered
suppression of spontaneous emission from an atom coupled strongly to a high
quality double mode cavity. The two modes of the cavity created a second order
pole in the S matrix. The model that we present is quite different we use an
unexcited atom or even an unexcited ensemble to inhibit Purcell effect. Our
work is based on coupling to a single mode of the cavity. This is quite
counterintuitive. In our model the two atoms are differently coupled. Thus
standard subradiant effects [34] are not the reason for the inhibition of the
Purcell effect as the conditions for subradiant effects are rigid and require
preparation of the entangled state of the two atoms coupled equally to the
cavity mode.

Purcell decay at the third order EP.–

Figure 3: Creation of the third order EP in cavity QED under conditions given
by Eq. (9) with two atoms coupled to a cavity mode via the single excitation
manifold.

We will first discuss how a third order EP can be created in cavity QED. We
will now use two unexcited atoms B and C coupled similarly to the cavity mode.
Let us first assume that the excited atom A is absent, but assume one photon
in the cavity. This is to bring out the dressed state structure and the third
order exception point in the cavity QED system. In terms of the states
$\displaystyle|e_{B},g_{C},0\rangle$, $\displaystyle|g_{B},g_{C},1\rangle$ and
$\displaystyle|g_{B},e_{C},0\rangle$, the matrix giving time evolution, of the
system consisting of atoms B, C and the cavity mode, would be similar to Eq.
(3)

$\displaystyle M_{0}$
$\displaystyle=\left(\begin{array}[]{ccc}-i(\gamma_{B}-\kappa)&g_{B}&0\\\
g_{B}&0&g_{B}\\\ 0&g_{B}&-i(\gamma_{C}-\kappa)\end{array}\right)-i\kappa$ (8)
$\displaystyle=\left(\begin{array}[]{ccc}i\tilde{\gamma}&g_{B}&0\\\
g_{B}&0&g_{B}\\\ 0&g_{B}&-i\tilde{\gamma}\end{array}\right)-i\kappa,$

where we choose decays $\displaystyle\gamma_{B}$ and $\displaystyle\gamma_{C}$
so that matrix in Eq. (8) has structure similar to PT symmetric systems [17],

$\tilde{\gamma}=\gamma_{C}-\kappa,\qquad\gamma_{B}=2\kappa-\gamma_{C},\qquad\frac{\gamma_{C}}{2}<\kappa<\gamma_{C}.$
(9)

The eigenvalues of $\displaystyle M_{0}$ are given by

$\lambda=-i\kappa,\qquad-i\kappa\pm\sqrt{2}g_{B}\sqrt{1-(\frac{\tilde{\gamma}}{\sqrt{2}g_{B}})^{2}},$
(10)

the corresponding eigenfunctions of $\displaystyle M_{0}$ will give three
polariton states. For $\displaystyle\tilde{\gamma}=\sqrt{2}g_{B}$, we have a
third order EP which is created by two atoms interacting with the cavity mode
as shown in Fig. 3. We need to choose two atoms with different decay constants
$\displaystyle\gamma_{B}\neq\gamma_{C}$, so that conditions in Eq. (9) are
satisfied. Note that all three polaritons have the same decay constant
$\displaystyle\kappa$ with energies given by $\displaystyle 0$,
$\displaystyle\pm\sqrt{2}g_{B}\sqrt{1-(\tilde{\gamma}/\sqrt{2}g_{B})^{2}}$.
These coalesce into degenerate states if
$\displaystyle\tilde{\gamma}=\sqrt{2}g_{B}$.

We can now consider Purcell decay of an excited atom A in presence of two
unexcited atoms B and C. We start with the generalization of the master
equation (2) and consider the relevant excited states
$\displaystyle|e_{A},g_{B},g_{C},0\rangle$,
$\displaystyle|g_{A},e_{B},g_{C},0\rangle$,
$\displaystyle|g_{A},g_{B},e_{C},0\rangle$,
$\displaystyle|g_{A},g_{B},g_{C},1\rangle$. Then instead of the dynamical
matrix given by Eq. (3), we get the $\displaystyle 4\times 4$ matrix

$M=\left(\begin{array}[]{cccc}0&0&g_{A}&0\\\ 0&-i\gamma_{B}&g_{B}&0\\\
g_{A}&g_{B}&-i\kappa&g_{B}\\\ 0&0&g_{B}&-i\gamma_{C}\end{array}\right).$ (11)

Following the same procedure as employed in connection with Eq. (3), we find
that

$\bar{\Gamma}=\Gamma/[1+\frac{2g_{B}^{2}}{\gamma_{C}(2\kappa-\gamma_{C})}],$
(12)

which at third order EP simplifies to

$\tilde{\Gamma}=\Gamma(\frac{\gamma_{C}}{\kappa})(2-\frac{\gamma_{C}}{\kappa})=\Gamma(\frac{\gamma_{B}}{\kappa})(2-\frac{\gamma_{B}}{\kappa}),\qquad\frac{\gamma_{C}}{2}<\kappa<\gamma_{C}.$
(13)

Thus at third order EP, we can again have significant inhibition of Purcell
emission. For instance for $\displaystyle\gamma_{C}/\kappa=1.95$,
$\displaystyle\tilde{\Gamma}/\Gamma=0.0975$. These results can be confirmed
from the full solution of the master equation for three atoms interacting with
the cavity mode. In terms of the cavity polaritons, we have a picture similar
to Fig. 2, i.e., the excited atom has two channels of decay:
$\displaystyle|e_{A}\rangle\rightarrow|\psi_{\pm}\rangle|g_{A}\rangle\rightarrow|g_{A},g_{B},g_{C},0\rangle$;
$\displaystyle|\psi_{\pm}\rangle=\frac{1}{2}(|e_{B},g_{C},0\rangle+|g_{B},e_{C},0\rangle\pm\sqrt{2}|g_{B},g_{C},1\rangle)$.
The excited atom does not couple to the other polariton state
$\displaystyle|\psi_{0}\rangle=\frac{1}{\sqrt{2}}(|e_{B},g_{C},0\rangle-|g_{B},e_{C},0\rangle)$
as it has no cavity photon component.

Enhanced Purcell decay by an unexcited atom.–

Figure 4: Enhancement of the Purcell decay in presence of an unexcited atom
coupled strongly to the cavity, $\displaystyle\Delta=5$, with all frequencies
in units of $\displaystyle\kappa$.

We now demonstrate how the Purcell effect can be enhanced due to the presence
of an unexcited atom B. We will assume that the atom B is on resonance with
the cavity whereas the atom A is detuned by an amount $\displaystyle\Delta$.
The well known Purcell decay rate when $\displaystyle\Delta\neq 0$, is

$\Gamma_{\Delta}=g_{A}^{2}\kappa/(\kappa^{2}+\Delta^{2}),$ (14)

which is negligible if $\displaystyle\Delta/\kappa$ is large.

We can now repeat the analysis that led to Eq. (6) to obtain modification of
Eq. (14) due to the presence of an unexcited atom. The eigenvalues of the
matrix describing the evolution of the states $\displaystyle|\psi_{i}\rangle$
given before Eq. (11) are given by ($\displaystyle\gamma_{A}=0$)

$(\lambda+i\kappa)(\lambda-\Delta)(\lambda+i\gamma_{B})-g_{B}^{2}(\lambda-\Delta)-g_{A}^{2}(\lambda+i\gamma_{B})=0.$
(15)

An analysis of Eq. (15) shows that

$\displaystyle\tilde{\Gamma}_{\Delta}$ $\displaystyle=g_{A}^{2}{\rm
Im}\\{\frac{g_{B}^{2}}{\Delta+i\gamma_{B}}-i\kappa-\Delta\\}^{-1}$ (16)
$\displaystyle=-g_{A}^{2}{\rm
Im}\\{\frac{\Delta+i\gamma_{B}}{(\Delta-\lambda_{+})(\Delta-\lambda_{-})}\\},$

where $\displaystyle\lambda_{\pm}$ are given by Eq. (5), i.e., these are the
complex energies of the two polariton states formed by the interaction of atom
B with the cavity. For large $\displaystyle\Delta$, comparable to the
polariton energy $\displaystyle{\rm Re}\lambda_{+}$,
$\displaystyle\tilde{\Gamma}_{\Delta}$ can be significantly enhanced over
$\displaystyle\Gamma_{\Delta}$. This is confirmed by the plot in Fig. 4. The
actual amount of enhancement depends on the decay parameter
$\displaystyle\gamma_{B}$ of the unexcited atom. We note that with
$\displaystyle\Delta\neq 0$, the density of states available for Purcell decay
is less than for $\displaystyle\Delta=0$. With the presence of the atom B, the
atom A can interact resonantly with the polariton states
$\displaystyle|\psi_{\pm}\rangle$ leading to much higher density of states and
hence to enhanced emission.

Conclusions.– We have demonstrated how the Purcell decay of an atom coupled
weakly to a cavity can be controlled by the presence of unexcited atoms. We
have presented various physical mechanisms for the inhibition or enhancement
of the Purcell decay. We bring out the role of the polariton states created by
unexcited atoms and the coupling of these polariton states to the excited
atom. These polariton states provide several decay channels to the excited
atom and depending on the strength of coupling of the unexcited atoms to the
cavity, there could be interferences among the decay channels. We also show
that the unexcited atoms can create exceptional points in the cavity QED set
up. The decay of the excited atom can then be related to decay at second and
higher order exceptional points. The effects that we report are different from
the more traditional superradiant and subradiant effects which require
entangled states.

Acknowledgments.– We thank the Air Force Office of Scientific Research (Award
No. FA-9550-20-1-0366) and the Robert A. Welch Foundation (Grant No.
A-1943-20210327) for supporting this work. We thank Q. Miao for help in
preparing this manuscript.

## References

  * Purcell [1946] E. M. Purcell, Spontaneous emission probabilities at radio frequencies, Phys. Rev. 69, 681 (1946).
  * Goy  _et al._ [1983] P. Goy, J. Raimond, M. Gross, and S. Haroche, Observation of cavity-enhanced single-atom spontaneous emission, Phys. Rev. Lett. 50, 1903 (1983).
  * Angelakis  _et al._ [2004] D. G. Angelakis, P. L. Knight, and E. Paspalakis, Photonic crystals and inhibition of spontaneous emission: an introduction, Contemp. Phys. 45, 303 (2004).
  * Kockum  _et al._ [2018] A. F. Kockum, G. Johansson, and F. Nori, Decoherence-free interaction between giant atoms in waveguide quantum electrodynamics, Phys. Rev. Lett. 120, 140404 (2018).
  * Collison  _et al._ [2021] R. Collison, J. B. Pérez-Sánchez, M. Du, J. Trevino, J. Yuen-Zhou, S. O’Brien, and V. M. Menon, Purcell effect of plasmonic surface lattice resonances and its influence on energy transfer, ACS Photonics 8, 2211 (2021).
  * Krishnamoorthy  _et al._ [2012] H. N. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, Topological transitions in metamaterials, Science 336, 205 (2012).
  * Krasnok  _et al._ [2015] A. E. Krasnok, A. P. Slobozhanyuk, C. R. Simovski, S. A. Tretyakov, A. N. Poddubny, A. E. Miroshnichenko, Y. S. Kivshar, and P. A. Belov, An antenna model for the purcell effect, Sci. Rep. 5, 12956 (2015).
  * Agarwal [1975] G. S. Agarwal, Quantum electrodynamics in the presence of dielectrics and conductors. iv. general theory for spontaneous emission in finite geometries, Phys. Rev. A 12, 1475 (1975).
  * Rao and Hughes [2007] V. M. Rao and S. Hughes, Single quantum-dot purcell factor and $\displaystyle\beta$ factor in a photonic crystal waveguide, Phys. Rev. B 75, 205437 (2007).
  * Kleppner [1981] D. Kleppner, Inhibited spontaneous emission, Phys. Rev. Lett. 47, 233 (1981).
  * Yablonovitch [1987] E. Yablonovitch, Inhibited spontaneous emission in solid-state physics and electronics, Phys. Rev. Lett. 58, 2059 (1987).
  * John and Wang [1990] S. John and J. Wang, Quantum electrodynamics near a photonic band gap: Photon bound states and dressed atoms, Phys. Rev. Lett. 64, 2418 (1990).
  * Leistikow  _et al._ [2011] M. Leistikow, A. Mosk, E. Yeganegi, S. Huisman, A. Lagendijk, and W. L. Vos, Inhibited spontaneous emission of quantum dots observed in a 3d photonic band gap, Phys. Rev. Lett. 107, 193903 (2011).
  * He  _et al._ [2011] L. He, Ş. K. Özdemir, J. Zhu, W. Kim, and L. Yang, Detecting single viruses and nanoparticles using whispering gallery microlasers, Nat. Nanotechnol. 6, 428 (2011).
  * Wiersig [2014] J. Wiersig, Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: application to microcavity sensors for single-particle detection, Phys. Rev. Lett. 112, 203901 (2014).
  * Chen  _et al._ [2017] W. Chen, Ş. Kaya Özdemir, G. Zhao, J. Wiersig, and L. Yang, Exceptional points enhance sensing in an optical microcavity, Nature 548, 192 (2017).
  * Hodaei  _et al._ [2017] H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, Enhanced sensitivity at higher-order exceptional points, Nature 548, 187 (2017).
  * Wiersig [2020] J. Wiersig, Review of exceptional point-based sensors, Photonics Res. 8, 1457 (2020).
  * Lau and Clerk [2018] H.-K. Lau and A. A. Clerk, Fundamental limits and non-reciprocal approaches in non-hermitian quantum sensing, Nat. Commun. 9, 4320 (2018).
  * Chen  _et al._ [2019] C. Chen, L. Jin, and R.-B. Liu, Sensitivity of parameter estimation near the exceptional point of a non-hermitian system, New J. Phys. 21, 083002 (2019).
  * Anderson  _et al._ [2023] D. Anderson, M. Shah, and L. Fan, Clarification of the exceptional-point contribution to photonic sensing, Phys. Rev. Appl. 19, 034059 (2023).
  * El-Ganainy  _et al._ [2018] R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, Non-hermitian physics and pt symmetry, Nat. Phys. 14, 11 (2018).
  * Özdemir  _et al._ [2019] Ş. K. Özdemir, S. Rotter, F. Nori, and L. Yang, Parity–time symmetry and exceptional points in photonics, Nat. Mater. 18, 783 (2019).
  * Khanbekyan and Wiersig [2020] M. Khanbekyan and J. Wiersig, Decay suppression of spontaneous emission of a single emitter in a high-q cavity at exceptional points, Phys. Rev. Research 2, 023375 (2020).
  * Agarwal and Qu [2012] G. S. Agarwal and K. Qu, Spontaneous generation of photons in transmission of quantum fields in pt-symmetric optical systems, Phys. Rev. A 85, 031802 (2012).
  * Li  _et al._ [2023] Z.-Z. Li, W. Chen, M. Abbasi, K. W. Murch, and K. B. Whaley, Speeding up entanglement generation by proximity to higher-order exceptional points, Phys. Rev. Lett. 131, 100202 (2023).
  * Mukamel  _et al._ [2023] S. Mukamel, A. Li, and M. Galperin, Exceptional points treatment of cavity spectroscopies, J. Chem. Phys. 158, 154106 (2023).
  * Hopfield [1958] J. Hopfield, Theory of the contribution of excitons to the complex dielectric constant of crystals, Phys. Rev. 112, 1555 (1958).
  * Mlynek  _et al._ [2014] J. A. Mlynek, A. A. Abdumalikov, C. Eichler, and A. Wallraff, Observation of dicke superradiance for two artificial atoms in a cavity with high decay rate, Nat. Commun. 5, 5186 (2014).
  * Liu  _et al._ [2023] Y. Liu, Z. Wang, P. Yang, Q. Wang, Q. Fan, S. Guan, G. Li, P. Zhang, and T. Zhang, Realization of strong coupling between deterministic single-atom arrays and a high-finesse miniature optical cavity, Phys. Rev. Lett. 130, 173601 (2023).
  * Agarwal [2023] G. S. Agarwal, Laser field initiation of higher order poles of s-matrix-optical realization of field theoretic models, arXiv preprint arXiv:2301.05179 (2023).
  * Harris  _et al._ [1990] S. E. Harris, J. Field, and A. Imamoğlu, Nonlinear optical processes using electromagnetically induced transparency, Phys. Rev. Lett. 64, 1107 (1990).
  * Liang  _et al._ [2023] C. Liang, Y. Tang, A.-N. Xu, and Y.-C. Liu, Observation of exceptional points in thermal atomic ensembles, Phys. Rev. Lett. 130, 263601 (2023).
  * Scully [2015] M. O. Scully, Single photon subradiance: quantum control of spontaneous emission and ultrafast readout, Phys. Rev. Lett. 115, 243602 (2015).