# Frequency-resolved Purcell effect for the dissipative generation of steady-
state entanglement

Alejandro Vivas-Viaña  Departamento de Física Teórica de la Materia Condensada
and Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid,
28049 Madrid, Spain  Diego Martín-Cano  Departamento de Física Teórica de la
Materia Condensada and Condensed Matter Physics Center (IFIMAC), Universidad
Autónoma de Madrid, 28049 Madrid, Spain  Carlos Sánchez Muñoz
<EMAIL_ADDRESS>Departamento de Física Teórica de la Materia
Condensada and Condensed Matter Physics Center (IFIMAC), Universidad Autónoma
de Madrid, 28049 Madrid, Spain

###### Abstract

We report a driven-dissipative mechanism to generate stationary entangled $W$
states among strongly-interacting quantum emitters placed within a cavity.
Driving the ensemble into the highest energy state—whether coherently or
incoherently—enables a subsequent cavity-enhanced decay into an entangled
steady state consisting of a single de-excitation shared coherently among all
emitters, i.e., a $W$ state, well known for its robustness against qubit loss.
The non-harmonic energy structure of the interacting ensemble allows this
transition to be resonantly selected by the cavity, while quenching subsequent
off-resonant decays. Evidence of this purely dissipative mechanism should be
observable in state-of-the-art cavity QED systems in the solid-state, enabling
new prospects for the scalable stabilization of quantum states in dissipative
quantum platforms.

Light-matter interfaces consisting of quantum emitters in solid-state cavity
QED setups stand as promising platforms for the implementation of quantum
technologies [1, 2, 3, 4]. Examples of quantum emitters with optical
transitions in the solid state include semiconductor quantum dots (QDs) [3],
single molecules [4], or color centres [5, 6]. A key advantage of these
systems is that the emitters can be integrated at fixed positions into
photonic chips [2], which enhances their prospects for scalability. It has
been demonstrated that, by integrating these emitters in tailored photonic
environments, such as photonic or plasmonic cavities and waveguides [7, 8, 9,
10], the resulting collective dissipation can be used to stabilize entangled
states of ensembles of emitters [11, 12, 13, 14, 15, 16, 17, 10, 18, 19],
which represents a key hallmark for the development of quantum technologies,
with key applications in quantum information [20, 1, 21] and metrology [22,
23]. However, the generation of entangled states in these platforms faces
important challenges, such as the uncontrolled location of the emitters and
the inhomogeneous broadening of their transition frequencies [24], which
prevents the emergence of collective phenomena either by direct dipole-dipole
coupling or mediated by the photonic structures.

Despite these challenges, recent years have witnessed a considerable progress
in the fabrication of solid-state systems featuring collective quantum
phenomena. Advances in the nanofabrication of photonic environments have
enabled the observation of collective super- and subradiant dynamics between
distant QDs [25] and among molecules [26], as well as photon-mediated coherent
interactions between color centres in a nanocavity [27]. The eigenstates of
strongly interacting quantum emitters offer a direct way to realize
entanglement. In this direction, the small size of molecules and the
possibility to synthesize bridged dimers with controllable intermolecular
distances [28] allows to tackle the challenge of inherently weak dipole-dipole
coupling by realizing strongly interacting emitters separated by sub-
wavelength distances and featuring sub- and superradiant eigenstates [29, 30,
31]. These advances are complemented by the development of scalable methods to
address inohomogenous broadening by optically tuning different emitters into
resonance [32]

The progress in synthesizing strongly interacting quantum emitters concurs
with advancements in the design of light-matter interfaces with unprecedented
radiative rate enhancements, which includes technologies such as plasmonic
nanoantennas [33, 34, 35] or hybrid cavity-antenna systems [36, 37]. This
aligned development has called for further exploration into the phenomenology
of strongly interacting quantum emitters when they are coupled to photonic
structures [38, 39], with prospects such as enabling novel quantum-optical
phases [40, 41, 18] or the fast and robust preparation of the emitter ensemble
into its entangled eigenstates [42, 12].

In this Letter, we delve into this question and report a driven-dissipative
protocol to prepare steady entangled states in an ensemble of $N$ interacting
quantum emitters coupled to a single mode cavity. This effect occurs when the
coupling between emitters (in the absence of the cavity) yields hybridized
eigenstates with a non-harmonic energy spectrum, such as the sub- and
superradiant eigenstates for $N=2$. When this ensemble is pumped into its
highest excited state ($|\frac{N}{2},\frac{N}{2}\rangle$ in the Dicke basis),
the non-harmonic energy structure allows for a cavity to resonantly enhance a
subsequent transition into an entangled $W$ state [43] in the form of a
symmetric superposition of all the one-de-excitation states,
$|\frac{N}{2},\frac{N}{2}-1\rangle$. Additional decay transitions, which fall
out of resonance with the cavity, become suppressed, aiding in stabilizing the
system into the desired $W$ state. This particular amplification of specific
transitions within the ensemble is the reason behind terming it a frequency-
resolved Purcell effect. The $W$ states prepared here have garnered particular
interest and efforts for their generation [44, 45], since they represent a
fundamental class of entangled states [43] which are persistent and robust
against particle loss [46] and may display stronger non-classicality than GHZ
states [47].

The mechanism that we propose works provided that the cavity linewidth is
sufficiently narrow to resolve the energy structure of the interacting
ensemble, while remaining in the bad-cavity limit with high cooperativity. The
process operates under continuous driving, which can be both coherent (for
$N=2$) and incoherent (for any $N$). The case of coherent driving of two
emitters is extremely rich in phenomenology, giving rise to a variety of
mechanisms of generation of entanglement which we discuss in more detail in
the accompanying paper [48].

Figure 1: Entanglement generation for $N=2$ emitters. (a) Sketch of the
system. (b) Energy diagram and incoherent decay processes in the diagonal
basis. (c) Entanglement (measured by the concurrence) versus cavity detuning.
(d) Time evolution of the fidelity $F(|S\rangle)$ when $\Delta_{a}=-J$. Dashed
lines correspond to analytical predictions. Parameters: $\kappa=10^{4}\gamma$,
$P=40\gamma$, $\Omega=10^{4}\gamma$.

Model.— Each of the $N$ emitters is described as two-level system (TLS), i.e.,
a qubit, with annihilation operator $\hat{\sigma}_{i}$. We start the paper by
focusing on the case $N=2$, see Fig. 1(a). For non-degenerate emitters, we
express their natural frequencies as $\omega_{1}=\omega_{0}-\delta$ and
$\omega_{2}=\omega_{0}+\delta$. The single mode cavity is defined by its
frequency $\omega_{a}$, annihilation operator $\hat{a}$, and coupling rate to
the emitters $g$. The coherent drive has a Rabi frequency $\Omega$ and it is
resonant with the average qubit frequency, $\omega_{L}=\omega_{0}$. In the
frame rotating at $\omega_{0}$ and under the rotating wave approximation, the
system Hamiltonian reads (we set $\hbar=1$ henceforth)
$\hat{H}=\hat{H}_{q}+\hat{H}_{a}+\hat{H}_{d}$, where $\hat{H}_{q}$ is the
qubit-qubit Hamiltonian,
$H_{q}=-\delta\hat{\sigma}_{1}^{\dagger}\hat{\sigma}_{1}+\delta\hat{\sigma}_{2}^{\dagger}\hat{\sigma}_{2}+J(\hat{\sigma}_{1}^{\dagger}\hat{\sigma}_{2}+\text{H.c.})$,
$\hat{H}_{a}$ is the cavity Hamiltonian,
$\hat{H}_{a}=\Delta_{a}\hat{a}^{\dagger}\hat{a}+g[\hat{a}^{\dagger}(\hat{\sigma}_{1}+\hat{\sigma}_{2})+\text{H.c.}]$,
and $\hat{H}_{d}$ is the coherent drive Hamiltonian,
$\hat{H}_{d}=\Omega(\hat{\sigma}_{1}+\hat{\sigma}_{2}+\text{H.c.})$, where
$\Delta_{a}\equiv\omega_{a}-\omega_{0}$ is the average qubit-cavity detuning,
$J$ is the coherent coupling rate between the TLSs, and H.c. denotes Hermitian
conjugate.

The dynamics of the density matrix is described, in the Markovian regime, by
the master equation [49, 50, 51],
$d\hat{\rho}/dt=-i[\hat{H},\hat{\rho}]+\frac{\kappa}{2}\mathcal{D}[\hat{a}]\hat{\rho}+\sum_{i,j=1}^{N=2}\frac{\gamma_{ij}}{2}\mathcal{D}[\hat{\sigma}_{i},\hat{\sigma}_{j}]\hat{\rho}+\frac{P_{i}}{2}\mathcal{D}[\hat{\sigma}_{i}^{\dagger}]\hat{\rho}$,
where we have defined the Lindblad superoperators,
$\mathcal{D}[\hat{A},\hat{B}]\hat{\rho}\equiv
2\hat{A}\hat{\rho}\hat{B}^{\dagger}-\\{\hat{B}^{\dagger}\hat{A},\hat{\rho}\\}$
and $\mathcal{D}[\hat{A}]\equiv\mathcal{D}[\hat{A},\hat{A}].$ Here,
$\gamma_{ii}=\gamma$ and $P_{i}=P$ are the local rates of spontaneous emission
and incoherent pumping, respectively, which we take equal for all the
emitters; $\gamma_{12}=\gamma_{21}$ is the collective dissipative coupling
rate between them; and $\kappa$ is the photon leakage rate. The two pumping
schemes considered in this work are set by fixing: (i) $\Omega=0$, $P\neq 0$
(incoherent), and (ii) $\Omega\neq 0$, $P=0$ (coherent), see Fig. 1(b). We
note that the coherent drive excites directly the state $|ee\rangle$ via a
two-photon resonance with an effective two-photon Rabi frequency,
$\Omega_{\text{2p}}\equiv 2\Omega^{2}/J$ [52, 48]. We assume the bad-cavity
limit [53, 54, 55] ($\kappa\gg g\gg\gamma$), meaning that the qubit-cavity
dynamics is irreversible and that the qubit decay is enhanced by the cavity,
as quantified by the cooperativity
$C\equiv\Gamma_{P}/\gamma=4g^{2}/\kappa\gamma\gg 1$, where $\Gamma_{P}$ is the
Purcell rate [56, 57].

The mechanism that we report manifests in the regime of strongly interacting
emitters $J\gg\delta$. For $N=2$, the basis of hybridized excitonic
eigenstates is then the natural choice to describe the system,
$\\{|gg\rangle,|A\rangle,|S\rangle,|ee\rangle\\}$, where
$|S/A\rangle\equiv(|eg\rangle\pm|ge\rangle)/\sqrt{2}$ are entangled super- and
subradiant eigenstates of the single-excitation subspace [58] [see Fig. 1(a)].
The corresponding eigenenergies and decay rates are then
$E_{\pm}\approx\omega_{0}\pm J$ and $\gamma_{S/A}\approx\gamma\pm\gamma_{12}$,
respectively. Unless stated otherwise, in all the results we set the following
parameters: $\ J=9.18\times 10^{4}\gamma$, $\gamma_{12}=0.999\gamma,$
$\delta=10^{-2}J$ , $g=10^{-1}\kappa$. These values of $J$ and $\gamma_{12}$
correspond to what would be obtained from the dipole-dipole interaction
between two emitters in a H-aggregate configuration in free space, separated
by $r_{12}=2.5\ \text{nm}$ and with emission wavelength
$\lambda_{0}=780\,\text{nm}$ [29, 30, 50, 59, 48]. These high coupling values
may only be applicable to a limited number of situations involving interacting
molecules at nanometer distances [29, 28]. While we intentionally chose these
values to clearly demonstrate the effect, it is important to note that
entanglement can also be produced with less stringent parameters. We explore
this aspect in more detail later in our work and in the Supplemental Material
111See Supplemental Material for further information..

Frequency-Resolved Purcell enhancement.— Once the stationary density matrix of
the system ($\hat{\rho}_{\text{ss}}$) is obtained by solving the master
equation, we quantify the degree of steady-state entanglement between the two
qubits by means of the concurrence $\mathcal{C}$ [61, 62, 63, 64]. The
mechanism to generate entanglement requires that the cavity enhances
resonantly, via a frequency-resolved Purcell effect, the decay from the
highest excited state to an entangled state where a single de-excitation is
symmetrically distributed among all the emitters, which for $N=2$ is the
superradiant state $|S\rangle$ [see Fig. 1(a-b)]. The process is enabled when
the cavity frequency is set to $\Delta_{a}\approx-J$, matching the energy of
the transition $|ee\rangle\rightarrow|S\rangle$. Figure 1(c) displays
$\mathcal{C}$ versus the cavity detuning $\Delta_{a}$, confirming that, for
both driving schemes, the system exhibits high values of concurrence at this
resonance.

This mechanism requires a cavity with high cooperativity $C\gg 1$ and a
linewidth small enough to resolve the different energy transitions taking
place within the dimer, $J\gg\kappa$. Indeed, under these conditions, and
provided that the fine splitting due to two-photon dressing is not resolved by
the cavity in the case of coherent driving, $\kappa\gg\Omega_{\mathrm{2p}}$
[48], the cavity can be adiabatically eliminated in both pumping schemes,
yielding a simple effective Lindblad term that neatly describes the Purcell
effect brought in by the cavity,
$\mathcal{L}_{\mathrm{cav}}^{\mathrm{eff}}\approx\Gamma_{P}\mathcal{D}[\hat{\xi}_{S}]\hat{\rho}$,
with the effective jump operator $\hat{\xi}_{S}\equiv|S\rangle\langle ee|$
[60, 48]. Crucially, activating the desired decay
$|ee\rangle\rightarrow|S\rangle$ also requires an efficient excitation towards
the state $|ee\rangle$, which we can achieve with both driving mechanisms,
thanks to the two-photon resonant excitation in the coherent case, and to a
sufficiently large $P\gg\gamma$ in the incoherent case. The combination of
this excitation and the subsequent cavity-enhanced decay gives rise to an
effective incoherent pump of $|S\rangle$.

Figure 2:  Stationary concurrence versus cooperativity $C$ and driving
intensity: (a) incoherent excitation, (b) coherent excitation, for $N=2$ and
$\Delta_{a}\approx-J$. White straight lines mark the conditions outlined in
the text and bound the regions of formation of entanglement. Blue dot-dashed
lines indicate the optimal driving strength that maximizes the concurrence.
Black straight contours correspond to numerical calculations with a full model
that perfectly match thee grey-dashed lines, obtained from an effective model
where the cavity is adiabatically eliminated, see Ref. [60]. Top panels
correspond to a cut of the concurrence (black) for a fixed cooperativity:
$C=100$. Exact (blue) and analytical (dot-dashed blue) predictions of the
population of $|S\rangle$ are depicted.

Such an incoherent mechanism may enable fast state preparations [65], in
contrast to the resonant driving of the transition
$|gg\rangle\rightarrow|S\rangle$ [30], that can never reach a stationary
population inversion of the entangled state $|S\rangle$ and, furthermore, has
a speed that is ultimately limited by the breakdown of the two-level system
approximation through the excitation of higher electronic states. The
saturation of the stationary occupation probability to values close to unity
will occur whenever the dissipative timescale of preparation, $\tau_{S}$, is
much faster than the timescale of decay from $|S\rangle$ to other states, set
by $\gamma^{-1}$. The resulting dynamics is shown in Fig. 1(d) in terms of the
population of $|S\rangle$, which corresponds to the fidelity
$\mathcal{F}(|S\rangle)\equiv\langle S|\hat{\rho}|S\rangle$ [20].

We obtained analytical estimations of $\tau_{S}$ by assuming the effective
cavity-enabled decay given by $\hat{\xi}_{S}$. These are given by
$1/\tau_{S}^{\text{Incoh.}}\approx
P+\Gamma_{P}-\sqrt{\Gamma_{P}^{2}+P\gamma_{S}},$ and
$1/\tau_{S}^{\text{Coh.}}\approx(\Gamma_{P}-\text{Re}\sqrt{\Gamma_{P}^{2}-4\Omega_{\text{2p}}^{2}})/2$,
for incoherent and coherent excitation respectively [60, 48]. Figure 1(d)
confirms the good agreement with exact numerical simulations. The mechanism is
most efficient when $\Gamma_{P}$ is the dominant dissipative rate; in that
case, both timescales reduce to ${1/\tau_{S}^{\text{Incoh.}}\approx P}$ and
${1/\tau_{S}^{\text{Coh.}}\approx P_{\mathrm{S}}}$, where
$P_{S}\equiv\Omega_{\mathrm{2p}}^{2}/\Gamma_{P}$ can be considered an
effective incoherent pumping rate of the state $|S\rangle$ in the case of
coherent excitation [48]. The ability to make $\tau_{S}$ small by designing
systems with large Purcell factor and fixing high pumping rates has important
implications for the generation of entanglement in the presence of extra
decoherence channels, as we consider in detail in Ref. [60]. We find that
entanglement can be consistently generated provided
$\tau_{S}\ll\gamma_{\text{d}}^{-1}$, where $\gamma_{\text{d}}$ is the relevant
decoherence rate. This condition also ensures that entanglement survives as a
steady state, with the exception of the particular case of local pure
dephasing.

Our understanding of the mechanism suggests a hierarchy of rates that should
be respected for it to occur. A general condition is
$J\gg\kappa\gg\Gamma_{P}\gg\gamma$, which ensures a frequency-dependent
Purcell effect ($J\gg\kappa\gg\Gamma_{P}$) with high cooperativity
($\Gamma_{P}\gg\gamma$). Furthermore, there are conditions that depend on the
driving mechanism and that ensure that the target entangled state is
efficiently populated for (i) incoherent excitation: $\Gamma_{P}\gg
P\gg\gamma$, and (ii) coherent excitation: $P_{S}\gg\gamma$ and
$\kappa\gg\Omega_{\mathrm{2p}}$.

Figure 1(c) also shows that, in the coherent excitation scheme, the system
features an antisymmetric entanglement resonance $\Delta_{a}\approx J$ due to
the population of the subradiant eigenstate $|A\rangle$ when $\delta$ is small
but finite. We discuss this process in more detail in Ref. [48]. This
mechanism is highly inneficient for incoherent excitation due the depletion of
the antisymmetric state by the incoherent repumping.

Dependence with pumping strength.— The mechanism we present is optically
tunable via the intensity of the pump. To explore this dependence on a range
of platforms, we compute the stationary concurrence in terms of the driving
intensity ($P$ or $\Omega$) and the cooperativity ($C$) see Fig. 2. The
regions of high entanglement are enclosed within the white lines that
delineate the set of conditions outlined previously, confirming their
validity. Notably, we find high values of the concurrence in the range of
cooperativities $C\sim 10-10^{2}$, accessible by state-of-the-art open-cavity
configurations used, e.g., in reports of strong coupling with single molecules
[66]. A key parameter is the optimal pumping rate that maximizes entanglement
for a given cavity setup. For the incoherent pumping case, we find that this
is given by $P_{\text{opt}}=(\Gamma_{P}/2)\sqrt{(\kappa/J)^{2}+16/C}$ (see
Ref. [60] for details), whose accuracy is confirmed in Fig. 2(a). The
numerically determined optimum value for the coherent drive,
$\Omega_{\text{opt}}$, is shown in Fig. 2(b).

Figure 3: (a) Sketch of the setup of entanglement generation for $N>2$. (b)
Enhanced dissipative transition in the Dicke ladder. (c) Fidelity with the $W$
state versus cavity detuning for $N=5$ for two values of $J$. The
corresponding cavity population is shown in the inset. Dashed lines show the
fidelities for a post-selected state, heralded by the detection of a photon
from the cavity. (d) Fidelity with the $W$ state for $N=50$ versus cavity
cooperativity and incoherent pumping rate.

Generalization to $N$ emitters.— Figure 2 suggests that, for a fixed set of
cavity parameters, coherent driving yields higher values of entanglement than
the incoherent case. However, incoherent driving allows to straightforwardly
extend the scheme to the case of $N>2$ [Fig. 3(a)]. As an illustrative
example, we consider $N$ degenerate emitters with an all-to-all interaction
Hamiltonian, as could be provided by the coupling to a common photonic
environment [67, 27, 10]. In the frame rotating at their bare frequency, the
Hamiltonian reads $\hat{H}_{q}=J\hat{S}^{+}\hat{S}^{-}$, with
$\hat{S}^{-}\equiv\sum_{i}^{N}\hat{\sigma}_{i}$. In addition, we assume a
coupling to the cavity $g\hat{a}^{\dagger}\hat{S}^{-}+\text{H.c.}$, and the
same type of dissipators considered before. The eigenstates of $\hat{H}_{q}$
are the Dicke states $|j,m\rangle$ [68, 69], with eigenvalues
$\lambda_{j,m}=J[j(j+1)-m(m-1)]$.

As for $N=2$, qubit-qubit interactions provide a non-equidistant set of
eigenvalues, enabling the tuning of the cavity to enhance specific transitions
within the ladder, energetically separated from each other by values $\propto
J$. Following our previous discussion, a strong pump $P$ will tend to populate
the highest excited state, $|N/2,N/2\rangle=|ee\ldots e\rangle$, allowing the
cavity to successively enhance the transition from this state to a $W$
entangled state of the form $|W_{N}\rangle\equiv|N/2,N/2-1\rangle=(|ge\ldots
e\rangle+|eg\ldots e\rangle)+\ldots+|ee\ldots g\rangle)/\sqrt{N}$ [see Fig.
3(b)]. This specific transition is selected by setting the cavity frequency at
$\Delta_{a}=J(2-N)$. Figure 3(c) shows the fidelity between the steady-state
and the $W$ state, $\mathcal{F}(|W_{N}\rangle)$, for $N=5$ versus cavity
detuning, confirming that high fidelity values are achieved at resonance.
Notably, the activation of this mechanism translates into measurable
properties of the emission by the cavity. This is shown in the inset of Fig.
3, which depicts how the resonant profile of the fidelity correlates with the
same profile in the cavity population, proportional to emission intensity.
This change is accompanied by a change of the photon counting statistics, that
becomes antibunched at resonance even when many emitters are involved (shown
in Ref. [60]), suggesting that these features could work as evidence of the
formation of entanglement.

The plot also illustrates how a decrease of two orders of magnitude in $J$
results in a moderate decrease in fidelity, from approximately $0.9$ to about
$0.6$ (a more systematic analysis of how the efficiency of the mechanism
depends on $J$ is provided in Ref. [60]).

Furthermore, we show evidence that this mechanism can be used in a projective
preparation scheme based on post-selection [45]. In particular, we show
fidelities for the conditional state heralded by the detection of a photon in
the steady-state,
$\hat{\rho}_{c}=\hat{a}\hat{\rho}_{\text{ss}}\hat{a}^{\dagger}/\text{Tr}[\hat{a}\rho_{\text{ss}}\hat{a}^{\dagger}]$,
which always shows a significant increase in fidelity with respect to the non-
post-selected case. Figure 3(d) demonstrates how high values of the
fidelity—that follow the patterns previously discussed in Fig. 2(a)—can be
achieved for a number of emitters as high as $N=50$ (simulations were done
using the PIQS library in QuTiP [69, 70, 71]).

Our results also imply that strongly interacting molecules located at
subwavelength distances, which have been shown to support entangled
eigenstates [72, 30, 31], could serve as suitable systems to observe these
effects when coupled to state-of-the-art Fabry-Perot open cavities [66]. We
anticipate that one of the primary challenges in observing the mechanism
described here is the necessity for a substantial disparity between the
coherent coupling ($J$) and spontaneous emission ($\gamma$) rates, which
occupy opposite ends of the required rate hierarchy. Our results prompt
further investigation into the exotic non-equilibrium states that can arise in
novel light-matter interfaces where the coupling between quantum emitters
and/or the supression of undesired emission plays a crucial role, such as
emitter interactions within a photonic band gap [73, 10], implementations of
giant atoms [74, 75] or subwavelength atomic arrays [40].

Acknowledgments.— We acknowledge financial support from the Proyecto Sinérgico
CAM 2020 Y2020/TCS- 6545 (NanoQuCo-CM), and MCINN projects
PID2021-126964OB-I00 (QENIGMA) and TED2021-130552B-C21 (ADIQUNANO). C. S. M.
and D. M. C. acknowledge the support of a fellowship from la Caixa Foundation
(ID 100010434), from the European Union’s Horizon 2020 Research and Innovation
Programme under the Marie Sklodowska-Curie Grant Agreement No. 847648, with
fellowship codes LCF/BQ/PI20/11760026 and LCF/BQ/PI20/11760018. D. M. C.
acknowledges support from the Ramon y Cajal program (RYC2020-029730-I).

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Supplemental Material:
Frequency-resolved Purcell effect for the dissipative generation of steady-
state entanglement

Alejandro Vivas-Viaña, Diego Martín-Cano, and Carlos Sánchez Muñoz.

###### Contents

  1. I Adiabatic elimination of the cavity
  2. II Analytical solutions of the master equation under incoherent excitation
  3. III Dependence of the entanglement generation on the coupling rate between emitters
  4. IV Effect of additional decoherence channels
    1. General conditions
    2. Local pure dephasing with coherent excitation
    3. Collective pure dephasing and incoherent excitation
  5. V Entanglement in large emitter ensembles
  6. VI Entanglement based on post-selection
  7. VII Optical signatures of entanglement

##  I Adiabatic elimination of the cavity

Here, we prove that the cavity contribution to the dynamics can be described
by the effective Lindblad term,
$\mathcal{L}_{\text{cav}}^{\text{eff}}\approx\Gamma_{P}\mathcal{D}[\hat{\xi}_{S}]\hat{\rho}$,
in the case of two quantum emitters, $N=2$. We focus on the case of a
incoherent excitation, where the drive does not dress the energy level
structure of the emitter ensemble. The case of coherent driving is discussed
in more detail in Ref. [48].

Under a Born-Markov approximation, the effective contribution of the cavity is
accurately described by the following Bloch-Redfield term [48],

$\mathcal{L}_{\text{cav}}^{\text{eff}}\equiv\sum_{i,j,m,n=1}^{4}\left(\frac{g_{ij}g_{mn}^{*}}{\kappa/2+i(\Delta_{a}-\omega_{ij})}\left[\hat{\sigma}_{ij}\hat{\rho},\hat{\sigma}_{mn}^{\dagger}\right]+\text{H.c.}\right),$
(S1)

where we defined $\hat{\sigma}_{ij}=|j\rangle\langle i|$, $g_{ij}=g\langle
j|\hat{\sigma}_{1}+\hat{\sigma}_{2}|i\rangle$, and
$\omega_{ij}\equiv\lambda_{i}-\lambda_{j}$, where $|i\rangle$ and
$\lambda_{i}$, $i=1,\ldots,4$, are the eigenvectors and eigenvalues of the
qubit-qubit system. In order to keep the discussion as general as possible, we
consider in this Section the possibility of a finite qubit-qubit detuning
$\delta$, so that the single-excitation eigenstates are given by
$|\pm\rangle\equiv
1/\sqrt{2}(\sqrt{1\mp\sin\beta}|eg\rangle\pm\sqrt{1\pm\sin\beta}|ge\rangle)$,
where $\beta\equiv\arctan(\delta/J)$ denotes a mixing angle. The corresponding
eigenvalues are $\omega_{\pm}=\Delta\pm R$, where $R=\sqrt{J^{2}+\delta^{2}}$
is the Rabi frequency of the emitter-emitter coupling. We assume in the
follwing that $\delta$ is finite but much smaller than $J$, so that $R\approx
J$. The eigenstates of the qubit-qubit system are labeled as

$|1\rangle=|gg\rangle;\quad|2\rangle=|-\rangle;\quad|3\rangle=|+\rangle;\quad|4\rangle=|ee\rangle.$
(S2)

By setting the cavity frequency at the symmetric resonance, the terms where
$\omega_{ij}\approx\Delta_{a}\approx-J$ will be dominant in the sum over
$(i,j)$ in Eq.(S1). These terms are given by the following set of enhanced
transitions,

$\mu_{S}\equiv\\{(2,1),(4,3)\\},$ (S3)

corresponding to $|-\rangle\rightarrow|gg\rangle$, and
$|ee\rangle\rightarrow|+\rangle$, respectively. The rest of the terms in the
sum can be neglected since they are proportional to $g^{2}/J\rightarrow 0$,
provided the conditions for a frequency-resolved Purcell effect hold,

$J\gg\kappa\gg g.$ (S4)

By performing a rotating-wave approximation, terms not belonging to $\mu_{S}$
in the sum over $(m,n)$ can also be neglected, since these will rotate at
frequencies $J\gg g^{2}/\kappa$ in the Heisenberg picture [the inequality
being a consequence of Eq. (S4)].

Under these conditions, the cavity contribution in Eq. (S1) can be reduced to
the sum

$\mathcal{L}_{\text{cav}}^{\text{eff}}\approx\frac{g_{21}}{\kappa/2}[\hat{\sigma}_{21}\hat{\rho},\hat{\Lambda}]+\frac{g_{43}}{\kappa/2}[\hat{\sigma}_{43}\hat{\rho},\hat{\Lambda}]+\text{H.c.},$
(S5)

where

$\hat{\Lambda}\equiv\sum_{(m,n)\in\mu_{S}}g_{mn}^{*}\hat{\sigma}_{mn}^{\dagger}=g_{21}^{*}\hat{\sigma}_{21}^{\dagger}+g_{43}^{*}\hat{\sigma}_{43}^{\dagger}\approx\beta\frac{g}{\sqrt{2}}|-\rangle\langle
gg|+\sqrt{2}g|ee\rangle\langle+|,$ (S6)

where $g_{21}\approx\beta g/\sqrt{2}$ (assuming $\delta\ll J$),
$g_{43}\equiv\sqrt{2}g$, $\hat{\sigma}_{21}\equiv|gg\rangle\langle-|$, and
$\hat{\sigma}_{43}\equiv|+\rangle\langle ee|$. Rearranging these terms, we
obtain an effective Lindblad-like term of the form,

$\mathcal{L}_{\text{cav}}^{\text{eff}}\approx\Gamma_{P}\mathcal{D}[\hat{\xi}_{S}]\hat{\rho},$
(S7)

with $\Gamma_{P}\equiv 4g^{2}/\kappa$ and a single jump operator given by

$\hat{\xi}_{S}\equiv|+\rangle\langle ee|+\beta|gg\rangle\langle-|.$ (S8)

Notice that, in the limit $\beta\rightarrow 0$ considered in the main text, we
obtain $\hat{\xi}_{S}\approx|S\rangle\langle ee|$. The effective dynamics
$\mathcal{L}^{\text{eff}}_{\text{cav}}$ thus induces an enhanced superradiant
transition from the doubly-excited state to the symmetric state,
$|ee\rangle\rightarrow|S\rangle$, with a rate $2\Gamma_{P}$. For non-
neglibible $\delta$, the second term in Eq. (S8) corresponds to a weakly
enhanced transition from the subradiant state to the ground state,
$|-\rangle\rightarrow|gg\rangle$, with a rate $\beta^{2}\Gamma_{P}$. Given its
quadratic dependence with the small parameter $\beta$, this second type of
transition will clearly be negligible in the dimer configuration $\delta\ll
J$, where $\beta=\tan^{-1}(\delta/J)\ll 1$.

##  II Analytical solutions of the master equation under incoherent excitation

In the case of an incoherent drive and $N=2$, we can obtain good analytical
approximations of the dynamics based on the adiabatic elimination of the
cavity discussed above. A first approach is the direct diagonalization of the
Bloch-Redfield equation in Eq. (S1), which allows to obtain analytical
solutions of the steady state of the emitters, provided $\kappa\gg g$. In the
limit $\beta\approx 0$ and setting $\Delta_{a}\approx-J$, the steady-state
populations read (assuming $P\gtrsim\gamma$)

$\displaystyle\rho_{S,\text{ss}}$
$\displaystyle\approx\frac{P^{2}J^{2}(8g^{2}+\gamma_{S}\kappa)}{8g^{2}PJ^{2}(P+\gamma_{S})+4g^{4}P\kappa+J^{2}\kappa
P^{3}},$ (S9) $\displaystyle\rho_{A,\text{ss}}$
$\displaystyle\approx\frac{P\left[4g^{2}J^{2}\gamma_{S}+2g^{4}\kappa+J^{2}\kappa\gamma_{S}\gamma\right]}{8g^{2}PJ^{2}(P+\gamma_{S})+4g^{4}P\kappa+J^{2}\kappa
P^{3}},$ (S10) $\displaystyle\rho_{ee,\text{ss}}$
$\displaystyle\approx\frac{P^{2}J^{2}\kappa(P+\gamma)}{8g^{2}PJ^{2}(P+\gamma_{S})+4g^{4}P\kappa+J^{2}\kappa
P^{3}}.$ (S11)

We can rewrite Eq. (S9), which corresponds to the relevant fidelity
$\mathcal{F}(|S\rangle)$ used as a figure of merit in the text, in terms of
the Purcell rate $\Gamma_{P}$. In the limit $C\gg 1$, it takes the following
transparent form

$\rho_{S,\text{ss}}\approx\left[1+\frac{P}{2\Gamma_{P}}+\frac{\gamma_{S}}{P}+\frac{1}{8}\frac{\Gamma_{P}}{P}\left(\frac{\kappa}{J}\right)^{2}\right]^{-1}.$
(S12)

This expression clearly shows how, under the conditions established in the
main text for the efficient generation of entanglement,
$J\gg\kappa\gg\Gamma_{P}\gg P\gg\gamma$, the population saturates to its
maximum possible value $\rho_{S,\text{ss}}\approx 1$.

The expressions above are general and apply beyond the conditions of the
frequency-resolved Purcell effect in Eq. (S4). However, using the simple form
of the cavity-induced Lindblad term in Eq. (S7), which applies in the
frequency-resolved Purcell regime, allows us to obtain simpler expressions for
the steady state and the dynamics. In particular, in the dimer configuration
$\beta\approx 0$, the system can be reduced to a cascaded three-level system
composed of $\\{|gg\rangle,|S\rangle,|ee\rangle\\}$, since the subradiant
state, $|A\rangle$, can be neglected as it is essentially decoupled from the
system dynamics due to its dark nature [50]. Consequently, the system is
effectively described by a density matrix $\hat{\rho}_{\text{eff}}$ whose
dynamics is governed by the following master equation

$\frac{d\hat{\rho}_{\text{eff}}}{dt}=\left(\frac{\gamma_{S}}{2}\mathcal{D}[\hat{\sigma}_{gg,S}]+\Gamma_{P}\mathcal{D}[\hat{\sigma}_{S,ee}]+\frac{P}{2}\mathcal{D}[\hat{\sigma}_{gg,S}^{\dagger}]+\frac{P}{2}\mathcal{D}[\hat{\sigma}_{S,ee}^{\dagger}]\right)\hat{\rho}_{\text{eff}}.$
(S13)

where we have defined $\hat{\sigma}_{gg,S}\equiv|gg\rangle\langle S|$ and
$\hat{\sigma}_{S,ee}\equiv|S\rangle\langle ee|$. From this equation, we can
straightforwardly compute analytical expressions for the qubit-qubit dynamics,
such as the symmetric state population $\rho_{S}(t)\equiv\langle
S|\hat{\rho}_{\text{eff}}(t)|S\rangle$,

$\rho_{S}(t)\approx\rho_{S,\text{ss}}\left(1-e^{-t/\tau_{S}}\right),$ (S14)

where $\rho_{S,\text{ss}}$ is its stationary population,

$\rho_{S,\text{ss}}\approx\left[1+\frac{P}{2\Gamma_{P}}+\frac{\gamma_{S}}{P}\right]^{-1},$
(S15)

and $\tau_{S}$ denotes the stabilization timescale,

$1/\tau_{S}^{\text{Incoh.}}\approx
P+\Gamma_{P}+\frac{\gamma_{S}}{2}-\sqrt{P\gamma_{S}+\frac{1}{4}(\gamma_{S}-2\Gamma_{P})^{2}}\approx
P+\Gamma_{P}-\sqrt{\Gamma_{P}^{2}+P\gamma_{S}}.$ (S16)

The last approximation corresponds to the expression presented in the text
under the assumption $\Gamma_{P}\gg\gamma_{S}$. We note that Eq. (S12)
corresponds to the limit $J\gg\kappa$ of Eq. (S15). For analytical expressions
of the stationary density matrix under coherent excitation, we refer the
reader to Ref. [48].

These results allow us to obtain an analytical expression for the optimal
incoherent pumping rate $P_{\text{opt}}$, that maximizes the population of the
symmetric state $|S\rangle$. To do so, we find the $P$ that maximizes the
expression for $\rho_{S,\text{ss}}$ in Eq. (S9), which has a wider range of
validity than Eq. (S15). The optimal pumping rate thus reads, assuming
$\gamma_{S}=2\gamma$ and the cooperativity $C\equiv\Gamma_{P}/\gamma$,

$P_{\text{opt}}=\frac{\Gamma_{P}}{2}\sqrt{\left(\frac{\kappa}{J}\right)^{2}+\frac{16}{C}}.$
(S17)

Under the conditions that we established for the efficient generation of
entanglement, $J\gg\kappa$ and $C\gg 1$ (equivalent to $\Gamma_{P}\gg\gamma$),
this equation shows that the optimum pumping consistently fulfills the
condition $P\ll\Gamma_{P}$. By substituting $P_{\text{opt}}$ back in Eq.
(S12), we can obtain $\rho_{S,\text{ss}}^{\text{max}}$, the maximum achievable
population of $|S\rangle$:

$\rho_{S,\text{ss}}^{\text{max}}=\left[1+\frac{1}{2}\sqrt{\left(\frac{\kappa}{J}\right)^{2}+\frac{16}{C}}\right]^{-1}.$
(S18)

There are two different limits to this maximum value that can be considered
when approaching the optimum conditions of entanglement, namely (i)
$C\gg\frac{J}{\kappa}\gg 1$ or (ii) $\frac{J}{\kappa}\gg C\gg 1$. In the first
case, we find that the maximum population for the entangled state is given by

$\rho_{S,\text{ss}}^{\text{max}}(C\gg
J/\kappa)=\left(1+\frac{\kappa}{2J}\right)^{-1},$ (S19)

while in the second case,

$\rho_{S,\text{ss}}^{\text{max}}(J/\kappa\gg
C)=\left(1+2/\sqrt{C}\right)^{-1}.$ (S20)

Equations (S19) and (S20) serve as guidelines that inform one about the
maximum fidelity to $|S\rangle$ attainable (and thus the maximum values of
stationary entanglement) when the limiting factor are given by the inter-
emitter coupling rate, $J$, or the cavity cooperativity $C$, respectively.

##  III Dependence of the entanglement generation on the coupling rate between
emitters

Figure S1:  (a) Maximum achievable stationary concurrence and (b) fidelity
with $|S\rangle$ versus cooperativity $C$ and dipole-dipole coupling $J$. In
each panel the driving intensity optimizes the generation of entanglement,
that is, $P=P_{\text{opt}}$ and $\Omega=\Omega_{\text{opt}}$, respectively.
The black straight lines correspond to numerical computations, and grey-dashed
lines are numerical predictions from the Bloch-Redfield master equation.
Parameters: $\delta=10^{-3}J$, $\Omega=\Omega_{\text{opt}}$,
$P=P_{\text{opt}}$.

The finite value of $J$ can be a limiting factor for the generation of
entanglement, as we just saw in Eq. (S19). Assuming optimal driving
intensities, $P_{\text{opt}}$ and $\Omega_{\text{opt}}$, we extend here the
numerical analysis presented in the main text to different values of the
coupling rate between emitters, $J$.

In the simplest scenario, $J$ is set by dipole-dipole interactions, and
therefore different value of $J$ correspond to different inter-emitter
distances. With this situation in mind, we also vary $\gamma_{12}$ accordingly
[18, 50, 52]. Nevertheless, we note that $\gamma_{12}$ is not a critical
parameter (since we work in a regime in which it is overcome by the Purcell-
enhanced decay rate $\Gamma_{P}\gg\gamma$) and that, in general, $J$ could be
provided by mechanisms other than dipole-dipole interaction, e.g., mediated by
photonic structures [67, 27, 10].

In Fig. S1, we show the stationary concurrence,
$\text{max}_{P/\Omega}(\mathcal{C})$ and the fidelity to $|S\rangle$,
$\text{max}_{P/\Omega}[\mathcal{F}(|S\rangle)]$, both maximized over the
driving strength, plotted against $J$ and the cavity cooperativity $C$. In
this plot, we see wide regions of values of $J$ and $C$ where high values of
stationary entanglement are obtained, consistent with the regions where the
symmetric state $|S\rangle$ is highly populated. We observe that these regions
are well bounded by the condition $\kappa\ll J$ that allows for the frequency-
resolved Purcell effect to take place. In both schemes, we observe that
interaction rates of at least $J\sim 10^{3}\gamma$ are necessary to achieve
sizable values of the concurrence, with values as high as $\mathcal{C}\sim
0.7$ within reach for cooperativities in the range $C\sim 10-100$. These
results underscore the generality of the reported mechanism and its potential
application for obtaining maximum entanglement in different solid-state cavity
QED platforms.

##  IV Effect of additional decoherence channels

In the main text, the main sources of decoherence taken into account are local
and collective spontaneous decay and cavity leakage, stemming from the
interaction with a vacuum electromagnetic bath [50, 59, 48]. However, in order
to assess whether our findings can be observed in realistic setups, additional
decoherence channels must be considered [69]. For instance, coupling to phonon
baths or to lossy nanophotonic structures with specific geometries can yield
additional spontaneous channels beyond the zero-phonon line [17, 76], pure
dephasing (ubiquitous in semiconductor quantum dots or in molecular systems)
[77, 78, 79]; or collective pure dephasing [80, 81]. To assess the robustness
of our mechanism for generating high stationary entanglement, we consider
individually (for the case of two emitters $N=2$) the impact of additional
decoherence in these three distinct scenarios: (i) extra spontaneous decay,
(ii) local pure dephasing, and (iii) collective pure dephasing.

These additional decoherence channels translate into three extra Lindblad
terms in the master equation:

$\frac{d\hat{\rho}}{dt}=\mathcal{L}[\hat{\rho}]+\underbrace{\frac{\Gamma}{2}(\mathcal{D}[\hat{\sigma}_{1}]\hat{\rho}+\mathcal{D}[\hat{\sigma}_{2}]\hat{\rho})}_{\begin{subarray}{c}\text{Extra
spontaneous}\\\ \text{decay},\
\mathcal{L}_{\Gamma}[\hat{\rho}]\end{subarray}}+\underbrace{\frac{\gamma_{\phi}}{2}(\mathcal{D}[\hat{\sigma}_{z,1}]\hat{\rho}+\mathcal{D}[\hat{\sigma}_{z,2}]\hat{\rho})}_{\text{Local
pure dephasing},\
\mathcal{L}_{\gamma_{\phi}}[\hat{\rho}]}+\underbrace{\frac{\Gamma_{\phi}}{2}\mathcal{D}[\hat{\sigma}_{z,1}+\hat{\sigma}_{z,2}]\hat{\rho}}_{\begin{subarray}{c}\text{Collective
pure}\\\ \text{dephasing},\
\mathcal{L}_{\Gamma_{\phi}}[\hat{\rho}]\end{subarray}},$ (S21)

where $\hat{\sigma}_{z,i}\equiv
2\hat{\sigma}_{i}^{\dagger}\hat{\sigma}_{i}-\mathbb{1}$ is the z-component of
the Pauli matrices, and $\mathcal{L}[\hat{\rho}]$ denotes the original
Liouvillian,

$\mathcal{L}[\hat{\rho}]\equiv-i[\hat{H},\hat{\rho}]+\frac{\kappa}{2}\mathcal{D}[\hat{a}]\hat{\rho}+\sum_{i,j=1}^{N=2}\frac{\gamma_{ij}}{2}\mathcal{D}[\hat{\sigma}_{i},\hat{\sigma}_{j}]\hat{\rho}+\frac{P_{i}}{2}\mathcal{D}[\hat{\sigma}_{i}^{\dagger}]\hat{\rho}.$
(S22)

We will exhibit the effects of the additional decoherence channels
independently, resulting in a master equation
$d\hat{\rho}/dt=\mathcal{L}[\hat{\rho}]+\mathcal{L}_{\gamma_{\text{d}}}[\hat{\rho}]$,
where $\gamma_{\text{d}}=\\{\Gamma,\gamma_{\phi},\Gamma_{\phi}\\}$. Figure
S2(a-c) depicts the concurrence versus the cooperativity and the additional
decoherence rate for these three cases and both coherent and incoherent
excitation schemes.

### General conditions

Figure S2:  Robustness test of the mechanism of entanglement generation for
both excitation schemes: incoherent (upper row) and coherent (lower row). The
considered additional channels are: (a) extra spontaneous decay; (b) local
pure dephasing; and (c) collective pure dephasing. Each panel depicts the
concurrence as a function of the cavity cooperativity $C$ and the
corresponding extra decoherence rate. Parameters: $J=9.18\times 10^{4}\gamma,\
\gamma_{12}=0.999\gamma,\ \delta=10^{-2}J,\ \ g=10^{-1}\kappa,\
\Delta_{a}\approx-J,\ P=40\gamma,\ \Omega=10^{4}\gamma$

The frequency-resolved Purcell effect is efficient when the Purcell rate
$\Gamma_{P}$ is the dominant dissipative rate. Consequently, for the mechanism
to be robust against additional decoherence, the Purcell rate should exceed
the corresponding decoherence rates, i.e., $\Gamma_{P}\gg\gamma_{\text{d}}$.
This condition is illustrated in Fig. S2(a-c) by a solid-white line. This line
bounds the region of entanglement in most cases, with some exceptions that we
discuss below. Additionally, for entanglement to survive in the long-time
limit, the stabilization timescale $\tau_{S}$ must also be faster than the
additional decoherence rates. This translates into the condition
$\tau_{S}\ll\gamma_{\text{d}}^{-1}$, depicted in Fig. S2(a-c) by dashed white
lines. In summary, we require:

  1. (i)

$\Gamma_{P}\gg\gamma_{\text{d}}$: The Purcell rate must dominate over all
decoherence rates.

  2. (ii)

$\tau_{S}\ll\gamma_{\text{d}}^{-1}$: The stabilization timescale must be
faster than the decoherence.

There are two exceptions to these general observations: (i) local pure
dephasing with coherent drive; (ii) collective pure dephasing with incoherent
drive.

### Local pure dephasing with coherent excitation

This first exception arises from the nature of the subradiant state,
$|-\rangle$. As proven in Ref. [48], the effective Lindblad term in the
presence of coherent drive is given by a jump operator
$\hat{\xi}_{S}=-|+\rangle\langle ee|+\beta/2|gg\rangle\langle-|$, resulting in

$\mathcal{L}_{\text{cav}}^{\text{eff}}=\Gamma_{P}\mathcal{D}[\hat{\xi}_{S}]\hat{\rho}\approx\Gamma_{P}\mathcal{D}[|+\rangle\langle
ee|]\hat{\rho}+\frac{(\beta^{2}/4)\Gamma_{P}}{2}\mathcal{D}[|gg\rangle\langle-|]\hat{\rho}.$
(S23)

Local pure dephasing tends to mix the states $|+\rangle$ and $|-\rangle$ by
transferring population from one to the other. In order for entanglement due
to the population of $|+\rangle\approx|S\rangle$ to survive, the subradiant
state $|-\rangle$ must be depleted faster than it is populated, which sets the
condition $\beta^{2}\Gamma_{P}\gg\gamma_{\phi}$. When this condition does not
hold, the mechanism of entanglement generation breaks down since the
population transfer between the super- and subradiant states occur with a rate
$\propto\gamma_{\phi}$, destroying any possible entanglement. Notably, the
generation of entanglement exhibits greater robustness against local pure
dephasing under incoherent excitation. In contrast to the coherent excitation
scenario [48], once the subradiant state acquires population due to the
population transfer induced by this decoherence term, the incoherent
excitation is able to deplete this state by repumping towards the doubly-
excited state, leading it back into the symmetric state by means of the
frequency-resolved Purcell effect.

If this condition does not hold, but the formation of the entangled state is
faster than the decoherence timescale, $\tau_{S}\ll\gamma_{\phi}^{-1}$, we can
still observe the formation of entanglement as a metastable state that
survives for a time $\gamma_{\phi}^{-1}$ (not shown).

### Collective pure dephasing and incoherent excitation

The second exception is the specific case of collective pure dephasing and
incoherent excitation, where the stationary entanglement is maintained even
when the conditions (i-ii) do not hold. This type of decoherence essentially
degrades the coherence between the ground state $|gg\rangle$ and the doubly-
excited state $|ee\rangle$. Contrary to the coherent excitation scheme, in the
case of incoherent excitation, population transfer among states occurs via
incoherent one-photon processes, and does not rely on any type of coherence
between $|gg\rangle$ and $|ee\rangle$ Therefore, the mechanism of entanglement
generation via incoherent excitation proves to be highly robust against
collective pure dephasing, maintaining entanglement even at very high values
of this decoherence term ($\Gamma_{\phi}\sim 10^{5}\gamma$). This type of
dephasing only becomes detrimental when its rate becomes comparable to the
coherent coupling, $J\approx\Gamma_{\phi}$, since then the relevant
transitions are no longer resolved due to the dephasing-induced broadening.

##  V Entanglement in large emitter ensembles

In this Section, we provide further numerical evidence of the prospects of the
proposed mechanism to generate entangled $W$ states in large emitter ensembles
$N\gg 1$. To do this, we make use of the master equation generalized to $N$
emitters,

$\frac{d\hat{\rho}}{dt}=-i[\hat{H},\hat{\rho}]+\frac{\kappa}{2}\mathcal{D}[\hat{a}]\hat{\rho}+\sum_{i,j}^{N}\frac{\gamma_{ij}}{2}\mathcal{D}[\hat{\sigma}_{i},\hat{\sigma}_{j}]\hat{\rho}+\sum_{i}^{N}\frac{P_{i}}{2}\mathcal{D}[\hat{\sigma}_{i}^{\dagger}]\hat{\rho},$
(S24)

where the Hamiltonian is now given by

$\hat{H}=J\hat{S}^{+}\hat{S}^{-}+\Delta_{a}\hat{a}^{\dagger}\hat{a}+g(\hat{a}^{\dagger}\hat{S}^{-}+\hat{a}\hat{S}^{+}),$
(S25)

having defined $S^{-}\equiv\sum_{i}^{N}\hat{\sigma}_{i}$. As it is considered
in the case of $N=2$ emitters, $\gamma_{ii}=\gamma$, $P_{i}=P$, and
$\gamma_{ij}=\gamma_{ji}$ (with $i\neq j$) for all $(i,j)=1,\ldots,N$, then we
denote the collective dissipative coupling simply as
$\gamma_{ij}\rightarrow\gamma_{\text{col}}$.

In Fig. S3, the fidelity $\mathcal{F}(|W_{N}\rangle)$ for $N=3,25,50$ emitters
is plotted versus cooperativity and the incoherent pumping rate $P$ when the
cavity frequency is placed in resonance at $\Delta_{a}=J(N-2)$. These
calculations make use of the PIQS library in QuTiP [69, 70, 71]). Our
numerical simulations confirm the stabilization of $W$ states with fidelities
exceeding $0.8$ for numbers of emitters as high as $N=50$, which is the case
shown in the main text. The addition of extra emitters increases the minimum
pumping rate required to generate entanglement. For instance, in the case of
$N=3$, we observe that $P>\gamma$, while for $N=50$, a much higher pumping
intensity is needed, $P\gtrsim 10^{2}\gamma$. This observation is consistent
with the intuition that introducing more emitters increases the pumping
intensity required to bring the system into a fully excited state $|ee\ldots
e\rangle$, from where the relevant cavity-enhanced decay takes place.

Figure S3:  Fidelity to the $W$ state for $N=3,25,50$ emitters versus cavity
cooperativity and incoherent pumping rate. Parameters: $J=10^{5}\gamma,\
\gamma_{\text{col}}=0.999\gamma,\ g=10^{-1}\kappa,\ \Delta_{a}=J(2-N)$.

##  VI Entanglement based on post-selection

Figure S4:  Comparison of fidelities between non- (upper row,
$\mathcal{F}(|W_{25}\rangle)$) and post-selection (lower row,
$\mathcal{F}^{\text{H}}(|W_{25}\rangle)$) measurement schemes for detecting
$W$ states. In both rows, the fidelity to the $W$ state for $N=25$ emitters is
depicted versus cavity cooperativity and incoherent pumping rate for different
coherent coupling rates are considered, from left to right,
$J/\gamma=10^{2},10^{3},10^{4},10^{5}$. Parameters:
$\gamma_{\text{col}}=0.999\gamma,\ g=10^{-1}\kappa,\ \Delta_{a}=J(2-N)$.

Here we explore further the prospects of enhancing entanglement via post-
selection—where the entangled state is heralded by the detection of a photon
emitted by the cavity—and how this allows to relax the condition of high
coherent coupling between emitters. For this, we perform calculations with a
fixed number of emitters, $N=25$, and vary the coherent coupling rate from a
relatively small value of $J=10^{2}\gamma$ to the case presented in the text
of $J=10^{5}\gamma$. We then compare the fidelity to $|W\rangle$ obtained in
the steady-state and heralded by the detection of a photon.

Figure S4 illustrates the fidelity to the $W$ state versus cooperativity and
incoherent pumping rate. The upper and lower rows represent the non-post-
selected ($\mathcal{F}(|W_{25}\rangle)$) and post-selected
($\mathcal{F}^{H}(|W_{25}\rangle)$) detection schemes, respectively. These
results highlight the significant advantage of using post-selection
measurement techniques to achieve higher fidelities to the $W$ state. For
instance, in the case of $J=10^{4}$, the non-post-selected fidelity reaches a
maximum achievable value of approximately $0.5$, while in the post-selected
measurement, this quantity increases to around $0.8$

##  VII Optical signatures of entanglement

Figure S5:  Correspondence between entanglement and the properties of the
emitted light by the cavity. In both columns, corresponding to the case of
$N=2$ (a-c) and $N=5$ (d-f) emitters, from top to bottom, the fidelity to the
entangled state $|S\rangle$ (left) and $|W_{5}\rangle$, emission intensity
$\langle\hat{a}^{\dagger}\hat{a}\rangle$, and second order correlations at
zeroy delay $g^{2}(0)$, are computed versus cavity frequency $\Delta_{a}$. We
note that in the case of $N=2$ emitters, both excitation schemes are
illustrated. We use a solid filling for incoherent excitation and patterned
filling for the coherent case. Parameters (Case $N=2$ qubits): (a-c) $\
J=9.18\times 10^{4}\gamma$, $\gamma_{12}=0.999\gamma,$ $\delta=10^{-2}J$,
$g=10^{-1}\kappa$, $C=500$, $P_{\text{opt}}=66.39\gamma$,
$\Omega_{\text{opt}}=8504.99\gamma$; (d-f) $J=10^{5}\gamma$,
$\gamma_{\text{col}}=0.999\gamma$, $g=10^{-1}\kappa$,
$P_{\text{opt}}=132.3\gamma$, $C=496.84$.

Here we show that the resonant activation of the frequency-resolved Purcell
effect leading to entanglement correlates with features in the and classical
and quantum observables of the light emitted by the system. From top to
bottom, Fig. S5 depicts, for $N=2$ (left) and $N=5$ (right), the fidelity
$\mathcal{F}(|W_{N}\rangle)$, the emission intensity
$\langle\hat{a}^{\dagger}\hat{a}\rangle$, and the second-order correlations at
zero delay $g^{(2)}(0)$. We observe that a region of high fidelity—i.e., high
degree of entanglement—correlates with a peak in the emission intensity, as
well as an antibunching resonance in the photon statistics at frequencies
$\Delta_{a}\approx-J,-3J$ for the case of $N=2,5$, respectively. In the case
of incoherent pumping, the antibunching resonance remains even as the number
of emitters is increased, in contrast with the general tendency of light
emitted from an ensemble to thermalize. Optical signatures in the case
coherent driving are discussed in more depth in Ref. [48].