#  $N$ Scaling of Large-Sample Collective Decay in Inhomogeneous Ensembles

Sergiy Stryzhenko  Institut für Angewandte Physik, Technische Universität
Darmstadt, Hochschulstraße 6, 64289 Darmstadt, Germany  Institute of Physics,
National Academy of Science of Ukraine, Nauky Avenue 46, Kyiv 03028, Ukraine
Alexander Bruns  Institut für Angewandte Physik, Technische Universität
Darmstadt, Hochschulstraße 6, 64289 Darmstadt, Germany  Thorsten Peters
Institut für Angewandte Physik, Technische Universität Darmstadt,
Hochschulstraße 6, 64289 Darmstadt, Germany<EMAIL_ADDRESS>darmstadt.de

###### Abstract

We experimentally study collective decay of an extended disordered ensemble of
$N$ atoms inside a hollow-core fiber. We observe up to $300$-fold enhanced
decay rates, strong optical bursts and a coherent ringing. Due to
inhomogeneities limiting the synchronization of atoms, the data does not show
the typical scaling with $N$. We show that an effective number of collective
emitters can be determined to recover the $N$ scaling known to homogeneous
ensembles over a large parameter range. This provides physical insight into
the limits of collective decay and allows for its optimization in extended
ensembles as used, e.g., in quantum optics, precision time-keeping or
waveguide QED.

##  I Introduction

The rate $\Gamma$ at which an individual two-level system (TLS) emits
radiation depends on its coupling to the available vacuum modes and can thus
be altered when, e.g., placed inside a cavity [1]. For an ensemble of $N$
identical TLSs that are dipole-dipole coupled to each other via the emitted
radiation, the decay rate can change significantly even without a cavity, as
first noted by Dicke [2]. Here, coherence can potentially build up between the
emitters, leading to their spontaneous synchronization [3], and an enhanced
collective decay rate $\Gamma_{N}>\Gamma$. This is termed superradiance (SR)
[4, 5], if there is some initial coherence present, and superfluorescence (SF)
[6] if the ensemble is initially completely inverted and coherence is
initiated by vacuum fluctuations. Since its first observation [7], studies of
enhanced collective decay were reported for various experimental platforms
such as free-space [8, 9, 10, 11] and waveguide-coupled atomic ensembles [12,
13, 14, 15], solid-state materials [16], superconducting qubits [17] and
biological systems [18]. Besides being interesting due to its fundamental
nature, applications of collective decay can be found, e.g., in quantum optics
[19], waveguide QED [20], astrophysics [21], atomic clocks [22, 23], and
chemistry [18, 24].

SR/SF can be observed in small, dense samples [25] as well as in large, dilute
samples, where propagation effects have to be considered. Both cases exhibit
several $N$ scaling signatures [4]. In the small-sample case, when $N$
emitters are confined to a volume of dimensions smaller than defined by the
emitted wavelength $\lambda$, the initial decay rate is enhanced and given by
$\Gamma_{N}=N\Gamma$. In the large-sample case, where the $N$ emitters are
distributed in a volume $>\lambda^{3}$, excitation of a homogeneously
delocalized coherence is crucial, taking into account a position-dependent
phase factor. In the regime of linear-optics SR this is reflected in the
timed-Dicke state [26]. Due to the extended ensemble volume, a geometric
factor $\mu$ has to be introduced, describing the fraction of spontaneous
radiation that can coherently couple to the ensemble of $N$ emitters in the
detected mode [4]. The initial decay rate is then $\Gamma_{N}=\mu N\Gamma$,
which is a factor $\mu$ smaller than in the small-sample case. $\mu$ and $\mu
N$ ($\equiv N_{\mu}$) are the single-atom and collective cooperativities [27],
respectively, and $N_{\mu}$ becomes the new scaling parameter (see, e.g., [28,
29]). Besides enabling collectively-enhanced decay rates, also bursts can be
generated in SR/SF if the ensemble is initially inverted, with a delay $t_{D}$
of the burst after the inversion. The photon rate of the burst evolves
nonlinearly, reaching its peak at $t_{D}$ with a value $\propto N^{2}$. This
burst occurs when $t_{D},\Gamma_{N}^{-1}\ll\gamma^{-1}$, where $\gamma$ is the
decoherence rate of the TLSs, which defines a threshold condition. We note
that recently a general condition for the occurrence of bursts in waveguide
QED systems was derived [30].

As homogeneous excitation conditions in large samples can be challenging to
realize (see e.g. [16, 31]), the effect of inhomogeneities on collective
scattering are still investigated (see, e.g., [32, 33, 34])–specifically in
optically-dense ensembles [35, 28, 36, 37, 11, 38]. In such inhomogeneous
conditions, collective decay occurs only in sub-systems of the ensemble for
which the threshold condition is surpassed, while each of them radiates
independently [33]. This leads to a breakdown of the textbook $N$ scaling [39,
28, 40, 11, 41], potentially hiding the signature of collective decay. As
noted by Arecchi and Courtens [42], however, a maximum cooperation number
(MCN) $N_{\text{mc}}\leq N_{\mu}$ can be determined when propagation losses,
delays and inhomogeneous broadening become relevant, which replaces $N_{\mu}$
in the equations characterizing collective emission. This concept of a MCN
allows for understanding the physical limits to collective decay and has been
applied, e.g., to cases where propagation time through the ensemble is longer
than the timescale of cooperation build-up [39, 40]. As we will show in the
following, a MCN can also be determined in cases where spectral and spatial
inhomogeneities are relevant, thereby recovering the scaling expected for
homogeneous ensembles.

In this paper we experimentally study large-sample SF for a broad range of
inhomogeneous excitation conditions. For $N\leq 2.2\times 10^{5}$ atoms
confined within a hollow-core photonic bandgap fiber (HCPBGF), we determine
collective cooperativities up to $\sim 300$. Due to intrinsic spatial and
spectral inhomogeneities, the experimental data does not exhibit a textbook
$N$ scaling. However, by accounting for inhomogeneities in a simple model we
determine a MCN which recovers the textbook scaling over a large range of
experimental parameters when dispersion is negligible. Our model allows for
physical insight into the limits of collective decay in large ensembles, which
is relevant to understand and optimize collective effects in, e.g., free-space
[8, 9, 10, 11, 43] or waveguide-coupled ensembles [13, 44, 45, 14, 46, 20], or
solids [16].

The paper is organized as follows: In Sec. II we describe the model system
used to study SF. Section III presents an overview of the experimental setup
followed by the experimental results in Sec. IV. In Sec. V we then describe
the model used to determine a MCN and compare its results to the experimental
data. Following a discussion of the MCN model in Sec. VI, we conclude with a
summary and outlook in Sec. VII.

##  II Model System

We study SF for a disordered, cold ensemble of 87Rb atoms loaded into a HCPBGF
(NKT Photonics, HC-800-02) [47, 48]. The ensemble can be considered dilute as
even at the peak atomic density $\mathcal{N}\sim 10^{12}~{}$cm-3 we have
$\lambda^{3}\mathcal{N}<1$. The single-mode waveguide provides long-range
coupling between distant atoms, thereby facilitating the observation of
collective effects [13, 49] while avoiding the use of a multi-mode theory to
analyze the data [14]. The atomic ensemble possesses an inhomogeneous radial
density profile due to the guiding potential, which is of similar width as the
excitation field [see Fig. 1(c)] [48]. Moreover, as the ensemble exhibits a
large optical depth (OD) [47], attenuation of the excitation field has to be
considered. Thus, the system shows large radial and longitudinal
inhomogeneities. Radiation trapping is expected to be negligible due to an
ensemble aspect ratio of order $10^{4}:1$. In contrast to typical realizations
of SR/SF, where the excited state has a finite lifetime of $\Gamma^{-1}$, we
here study the decay of an effective TLS comprised of two long-lived ground
states [see Fig. 1(a,b)] [50, 51].

Figure 1:  (a) Simplified three level system with
$|1\rangle\,\hat{=}\,{}^{2}S_{1/2}\,F{=}1$,
$|2\rangle\,\hat{=}\,{}^{2}S_{1/2}\,F{=}2$, and
$|3\rangle\,\hat{=}\,{}^{2}P_{1/2}\,F^{\prime}{=}2$. The excited state decay
rate is $\Gamma=2\pi\cdot\qty{5.75}{}$ [52]. (b) Corresponding effective TLS
comprised of the ground states $|1\rangle$, $|2\rangle$. (c) Simplified
experimental setup.

An off-resonant excitation field (termed pump) scatters off the ensemble
prepared in $|1\rangle$. Once the pump is on, spontaneous Raman scattering
induced by vacuum fluctuations creates a delocalized coherence between states
$|1\rangle$ and $|2\rangle$. This initiates the decay, followed by a buildup
of radiation bursts in the Stokes field. We note that this quasi-continuous,
off-resonant driving of the system is different from the strong, resonant
driving in [29]. The single-atom scattering rate of such an effective TLS is
for large detunings given by

$\Gamma_{R}=R_{B}\Gamma\frac{\Omega_{p}^{2}}{4\Delta^{2}},$ (1)

where $R_{B}(=0.5)$ is the branching ratio into the ground state $|2\rangle$,
$\Gamma$ is the excited state decay rate, and $\Omega_{p}$ is the Rabi
frequency of the pump field effectively detuned from the transition
$|1\rangle\leftrightarrow|3\rangle$ by $\Delta=\Delta_{p}+2S$. Here,
$\Delta_{p}$ is the externally applied detuning and
$2S=\Omega_{p}^{2}/(2\Delta_{p})$ is the AC Stark shift of the
$|1\rangle\leftrightarrow|3\rangle$ transition frequency due to the strong and
detuned pump field.

Studying SF in an effective TLS has several advantages over real TLSs. First,
$\Gamma_{R}$ can be tuned by the intensity and detuning $\Delta_{p}$ of the
pump field. Second, $\Gamma_{R}$ can be made much smaller than the excited
state decay rate $\Gamma$, which sets the width of the dispersive resonance.
Thus, even for large decay rates $\Gamma_{N}\gg\Gamma_{R}$, we still can have
$\Gamma_{N}<\Gamma$, and dispersive effects as discussed in [38] are more
likely to be negligible. This leads to a significant simplification when
modeling the MCN. Third, the modulation bandwidth of the excitation field has
to be larger than $\Gamma_{N}$. This can be technically challenging if the
single-atom decay rate $\Gamma$ is already quite large [28].

##  III Experimental Setup

Figure 1(c) shows our simplified experimental setup. A detailed description
including the procedure to load atoms into the HCPBGF can be found in [48].
Therefore, we here only give a brief summary. First, we load a magneto-optical
trap with around $10^{7}$ 87Rb atoms. Then, we shift the atom cloud towards
the HCPBGF tip while compressing it radially using magnetic fields. Up to
$2.5\%$ of the atoms are then loaded into a red-detuned Gaussian-shaped far-
off-resonant trap (FORT) emerging from the HCPBGF with a numerical aperture
$\textrm{NA}=0.092(6)$. The FORT guides the atoms radially inside the fiber,
while they are basically free-falling in the longitudinal (vertical)
direction, resulting in a radial $1/e$ half-width $\sigma_{a}\sim\qty{1.7}{}$
and a length $L\sim\qty{3}{}$ of the ensemble. We prepare the atoms in state
$|1\rangle$ by optical pumping without specific population of Zeeman levels.
The linear-polarized pump beam is launched into the fiber in such a way that
its polarization is maintained at the exit to allow for efficient polarization
filtering [53]. Its temporal intensity is controlled by an acousto-optic
modulator with a rise time of $\tau_{p}=\qty{130(5)}{}$. The radial pump
intensity has a near-Gaussian profile with a $1/e^{2}$ half-width of
$\sigma_{p}\sim\qty{2.75(20)}{}$. The shot-to-shot intensity fluctuations are
$\lesssim 5\%$. We detect the transmitted light using an avalanche photodiode
(APD) and a digital oscilloscope.

##  IV Experimental Studies

In the following we study SF in the effective TLS shown in Fig. 1. Before each
measurement, we switch off the trapping potential to avoid inhomogeneous
broadening by the FORT. When the pump is resonant with
$|1\rangle\leftrightarrow|3\rangle$ we can measure $N$ by time-resolved
optical pumping [48]. The measured shot-to-shot fluctuations of $N$ are
$\lesssim$3.5\text{\,}\mathrm{\char 37\relax}\text{/}$$. We then detune the
pump field from $|1\rangle\leftrightarrow|3\rangle$ by a variable amount
$\Delta_{p}$. As confirmed by measurements and a numerical simulation, the
polarization of the Stokes field is predominantly linear and orthogonal to the
pump field. We thus observe the temporally-resolved Stokes radiation by
detecting light orthogonally polarized to the pump using a polarizing beam
splitter (PBS). We show in Fig. 2(a) exemplary data for the transmitted pump
power through the HCPBGF without atoms (blue) and for the transmitted Stokes
power with $N=$83\text{\times}{10}^{3}$$ atoms loaded into the HCPBGF (orange)
versus time.

Figure 2:  (a) Exemplary traces of the transmitted pump power without atoms
(blue, 16 averages, scaled down by factor $0.5$) and the Stokes power with
atoms loaded into the HCPBGF (single-shot data). The peak pump Rabi frequency
is $\Omega_{p}^{(0)}=6.4\Gamma$, $\Delta_{p}=18.4\Gamma$ with
$N=$83\text{\times}{10}^{3}$$ (orange) and $N=$220\text{\times}{10}^{3}$$
(green). $t=0$ corresponds to the moment the pump power reaches $50\%$ of its
mean maximum value. The extracted delay $t_{D}$ of the orange signal is shown
by the grey shaded area. (b) Measured mean delays of the first emitted SF
burst vs. $N_{\mu}$ for $\Delta_{p}=18.4\Gamma$, with error bars representing
the standard deviation. The expected theoretical dependence [Eq. (2)] is shown
considering $N_{\mu}$ (dashed green line) as well as the MCN $N_{\text{mc}}$
(solid orange line) for $\beta=0.07$.

No Stokes light is detected for several hundred nanoseconds after the pump is
switched on. Then, a short burst of peak power $P_{s}$ is emitted, followed by
another burst of smaller peak power due to a coherent ringing [4, 54]. This
trace, showing a large first burst followed sometimes by a smaller one,
represents the dominant temporal shape of the emitted Stokes light for
$N<$220\text{\times}{10}^{3}$$ and agrees with earlier observations of SF [4,
54, 28]. We note that the transmitted pump power shows complementary dynamics
to the Stokes field, i.e., fast absorption, which was termed ‘superabsorption’
in [55]. For our maximum attainable $N=$220\text{\times}{10}^{3}$$, however,
the dominant shape changes [green trace in Fig. 2(a)]. Here, the ringing is
more pronounced and the second burst is largest which is usually not the case
[56]. As to obtain good statistics for the Stokes signal, we repeat the
measurement for each set of experimental parameters $(N,\Delta_{p})$ $100$
times (see video [57]). The Stokes signal exhibits huge shot-to-shot
fluctuations in amplitude, delay, intensity and shape. The burst dynamics
occur on a timescale much shorter than expected for the calculated single-atom
scattering rate of $\langle\Gamma_{R}\rangle_{r}=2\pi\times\qty{45}{}$, where
$\langle...\rangle_{r}$ denotes a radial average over both the pump intensity
and atomic density distribution (see Appendix C.2). From each single dataset
we extract the delay $t_{D}$ and peak power $P_{s}$ of the first emitted
Stokes burst by an automated algorithm to obtain their mean values $\langle
t_{D}\rangle$ and $\langle P_{s}\rangle$ and the corresponding standard
deviations.

Figure 2(b) shows exemplary data for the measured mean delays vs. reduced atom
number $N_{\mu}$ (symbols). Here, $\Omega_{p}$ and $\Delta_{p}$ are the same
as in Fig. 2(a) and we expect the system to be well-described by an effective
TLS. The standard deviations from the mean delay are a significant $\sim
38\%$, which is much larger than both the $\sim 4\%$ uncertainty of
determining $t_{D}$ in a single shot and the expected uncertainty of $\sim
4\%$ due to fluctuating atom number and pump intensity (see Appendix A). As
the relative delay fluctuations from the mean $\langle t_{D}\rangle$ due to
vacuum fluctuations can be estimated as $2.6/\ln{N}$ [4] ($\sim 0.2..0.3$), we
attribute the dominant contribution to the observed fluctuations to vacuum
fluctuations initiating the decay process [58, 59]. We note that the standard
error of the mean is a factor 10 smaller than the standard deviation shown by
the error bars. In the same plot we show the expected dependence (dashed green
line) for a homogeneous ensemble of Fresnel number $F\sim 1$ (see Appendix F)
[60]

$\langle t_{D}(N)\rangle=\frac{1}{4\mu N\Gamma_{R}}\left[\ln{\sqrt{2\pi
N}}\right]^{2}.$ (2)

The measured mean delays are larger than Eq. (2) predicts, as expected for
inhomogeneously-broadened systems [61].

To extract the initial collective decay rate $\Gamma_{N}$ from our SF data, we
cannot resort to determining the decay rate as, e.g., in linear-optics SR due
to the non-exponential dynamics. An order-of-magnitude estimation can be done
by measuring the temporal widths $\tau_{b}$ of the first SF bursts, as
$\Gamma_{N}\propto\tau_{b}^{-1}$ is a reasonable assumption [4]. However, as
there is, to the best of our knowledge, no precise theoretical prediction for
$\Gamma_{N}(\tau_{b})$ in our considered case of SF bursts with ringing, we
proceed as follows. From Eq. (2) we determine the collective decay rate as
$\Gamma_{N}=[\ln{\sqrt{2\pi N}}]^{2}/(4\langle t_{D}\rangle)$ using the
measured $\langle t_{D}\rangle$ and $N$. Fig. 3(a) depicts the measured
relative collective decay rate $\Gamma_{N}/\langle\Gamma_{R}\rangle_{r}$
(symbols) vs. $N_{\mu}$ for a large range of detunings, along with the
expected theoretical dependence
$\Gamma_{N}/\langle\Gamma_{R}\rangle_{r}=N_{\mu}$ (blue line), where
$\mu\approx\textrm{NA}^{2}/4=2.1\times 10^{-3}$.

Figure 3:  (a,b) Experimentally determined initial relative collective decay
rate $\Gamma_{N}/\langle\Gamma_{R}\rangle_{r}$ vs. $N_{\mu}=\mu N$ (a) and MCN
$N_{\text{mc}}=\eta\mu N$ (b) for $2\Gamma\leq\Delta_{p}\leq 26.4\Gamma$ at
$\Omega_{p}^{(0)}=6.4\Gamma$. The symbol size is proportional to $N$. The
solid blue lines show the dependence
$\Gamma_{N}/\langle\Gamma_{R}\rangle_{r}=N_{\mu}$ (a) and
$\Gamma_{N}/\langle\Gamma_{R}\rangle_{r}=N_{\text{mc}}$ (b) for homogeneous
conditions. (c,d) Measured mean peak power of the first SF burst vs.
$N_{\mu}^{2}$ (c) and $N_{\text{mc}}^{2}$ (d) for $\Delta_{p}=18.4\Gamma$. The
green (orange) dashed line in (c) represents a linear (quadratic) scaling with
$N_{\mu}$ for reference. The orange solid line in (d) represents a least-
squares linear fit to the data, excluding the last data point (see text).

The collective decay rate is significantly enhanced by more than two orders of
magnitude compared to the single-atom case. Only at the largest detunings it
scales approximately linearly with $N_{\mu}$, but with a detuning-dependent
slope. Moreover, the experimental data does not collapse onto
$\Gamma_{N}/\langle\Gamma_{R}\rangle_{r}=N_{\mu}$ expected for homogeneous
conditions, except for the lowest $N$ and largest $\Delta_{p}$. Close to
resonance, $\Gamma_{N}/\langle\Gamma_{R}\rangle_{r}$ is approximately
independent of $N$. Thus, $N_{\mu}$ is not a good scaling parameter. We note
that we obtain quite similar values for
$\Gamma_{N}/\langle\Gamma_{R}\rangle_{r}$ (differing by a factor of $\sim
1.7$) when determining $\Gamma_{N}$ from the burst widths $\tau_{b}$ instead
of the mean delays $\langle t_{D}\rangle$ (see Appendix B), indicating that a
determination of $\Gamma_{N}$ from $\langle t_{D}\rangle$ is indeed correct.
Next, we plot in Fig. 3(c) the measured mean peak Stokes power (symbols) as a
function of $N_{\mu}^{2}$. We observe a $N_{\mu}^{2}$ scaling (dashed orange
line) only for small $N$. As $N$ becomes larger, the scaling becomes
approximately linear (dashed green line). This change in scaling indicates a
significant reduction in cooperativity for increasing $N$. A similar behavior
is seen for other detunings with a breakdown of $N_{\mu}^{2}$ scaling
occurring for smaller $N$ as $\Delta_{p}$ becomes smaller. Our results clearly
demonstrate strong collective decay for our dilute, disordered atomic ensemble
coupled to an HCPBGF. However, the in part significant deviations from theory
for homogeneous conditions evoke a deeper investigation as to understand the
limits to collective emission in our system and to optimize it.

##  V Determining the MCN

Based on the concept of MCN [42], we determine in the following the fraction
of atoms $N_{\text{mc}}=\eta\mu N$ which can radiate collectively, i.e., can
be viewed as homogeneously driven. We identify two major mechanisms affecting
the MCN. (i) Inhomogeneous spectral broadening, and (ii) pump attenuation. Due
to the radial as well as longitudinal dependence of the pump intensity
determining decay rate, AC Stark shift, and Stokes gain, we factorize the
problem: we first determine radially-averaged quantities and then apply them
to average longitudinally (see the Appendix D for further details).

First, we consider inhomogeneous spectral broadening. As noted in [42], the
number of cooperative scatterers is reduced by the ratio
$\sigma_{\text{hom}}/\sigma_{\text{inh}}$ of (homogeneous) excitation
bandwidth $\sigma_{\text{hom}}\sim\tau_{p}^{-1}$ and inhomogeneous linewidth
$\sigma_{\text{inh}}$ if $\sigma_{\text{hom}}<\sigma_{\text{inh}}$. We
estimate $\sigma_{\text{inh}}=\delta S+\delta\Gamma_{R}$, where $\delta S$ and
$\delta\Gamma_{R}$ are standard deviations of the AC Stark shift $S$ and the
effective scattering rate $\Gamma_{R}$ from their average values $\langle
S\rangle_{r}$, $\langle\Gamma_{R}\rangle_{r}$. This yields a first correction
factor

$\eta_{\text{inh}}=\frac{1}{\tau_{p}\sigma_{\text{inh}}}$ (3)

to the MCN. Next, we consider pump attenuation along the propagation direction
$z$ through the optically-dense ensemble. Such attenuation is detrimental to
excitation of a timed-Dicke state [37, 35], and in addition directly affects
the decay rate of the effective TLS. We first take (off-)resonant absorption
according to the Beer-Lambert law into account by approximating the absorption
coefficient by its initial value

$\alpha(\Delta_{p})\approx\langle\alpha(\Delta_{p},0)\rangle_{r}=\left\langle\alpha_{0}\frac{\Gamma^{2}}{4\left[\Delta_{p}+2S(r,0)\right]^{2}}\right\rangle_{r},$
(4)

where $\alpha_{0}$ is the resonant OD, and we performed a radial average.
Noting further that the Stokes intensity eventually grows as
$I_{s}(\Delta_{p},z^{\prime})\sim\exp[G_{s}(\Delta_{p})z^{\prime}]$ [50, 60],
where $z^{\prime}=z/L$ and the Stokes gain
$G_{s}(\Delta_{p},0)\approx\left\langle
2\alpha_{0}\Gamma_{R}(\Delta_{p},0)/\gamma\right\rangle_{r}$, with $\gamma$
being the ground state decoherence rate, we next consider pump attenuation due
to conversion into the Stokes field. We thus make the ansatz
$I_{p}(\Delta_{p},z^{\prime})\approx
I_{p}^{(0)}\exp[-\tilde{\alpha}(\Delta_{p})z^{\prime}]$ for the pump
intensity, where the total attenuation factor is

$\tilde{\alpha}(\Delta_{p})=\alpha(\Delta_{p},0)+\beta(\Delta_{p})G_{s}(\Delta_{p},0).$
(5)

In a first approximation we assumed steady-state conditions for both
absorption and conversion and that $\tilde{\alpha}(\Delta_{p})$ is given by
its initial radially-averaged value at $z^{\prime}=0$. This assumption of
steady-state conditions is a good approximation for pump absorption as the
duration of transients is roughly given by $\Gamma^{-1}\ll t_{D}$. However, as
the transient regime during buildup of the Stokes field is roughly determined
by $\gamma^{-1}\sim t_{D}$, this ansatz is probably less accurate for pump
conversion. We thus introduced a scaling factor $\beta(\Delta_{p})$ for the
Stokes gain as the only free parameter in our model (see Appendix E for a more
detailed discussion). This parameter accounts for neglecting transient
conditions during Stokes field buildup exhibiting a time-dependent Stokes
intensity evolution [50]. We next calculate the effective collective decay
rate as

$\left\langle\Gamma_{N}^{\text{eff}}\right\rangle_{r,z}=\mu
N\left\langle\Gamma_{R}\left[\Omega_{p}(r,z^{\prime})\right]\right\rangle_{r,z}$
(6)

by averaging along the $r-$ and $z-$coordinates and using
$\Gamma_{R}\left[\Omega_{p}(r,z^{\prime})\right]$ from Eq. (1) with

$\Omega_{p}(r,z^{\prime})=\Omega_{p}^{(0)}e^{-\frac{r^{2}}{\sigma_{p}^{2}}}e^{-\frac{1}{2}\tilde{\alpha}(\Delta_{p})z^{\prime}}.$
(7)

Following [62, 63] we can now determine a shadow factor

$\eta_{S}=\frac{\left\langle\Gamma_{N}^{\text{eff}}\right\rangle_{r,z}}{\mu
N\langle\Gamma_{R}\rangle_{r}}$ (8)

which incorporates pump attenuation. Here, $\langle\Gamma_{R}\rangle_{r}$ is
the radially-averaged single-atom scattering rate at $z^{\prime}=0$ used as a
reference without attenuation. Combining inhomogeneous broadening and pump
attenuation, we finally obtain the MCN as

$N_{\text{mc}}=\eta_{\text{inh}}\cdot\eta_{s}\cdot\mu\cdot N.$ (9)

To test our simple model, we determine $\langle t_{D}(N_{\text{mc}})\rangle$
using $N_{\text{mc}}$ instead of $N_{\mu}$ in Eq. (2) by manually adjusting
$\beta$ as the only free parameter to yield the best agreement with the
experimental data [see orange line in Fig. 2(b)]. In the range of detunings
$\Delta_{p}\gtrsim\Omega_{p}$ where our effective TLS model should be valid,
we determine $\beta(\Delta_{p})=0.182-6.1\times 10^{-3}\Delta_{p}/\Gamma$ from
a linear fit. Although we currently have no exact explanation for this
dependence on $\Delta_{p}$, we suspect that it is due to a larger temporal
ratio of transient to steady-state conditions at larger detunings (see
Appendix E for more details). For smaller $\Delta_{p}$, $\beta$ increases
dramatically, indicating that our model is inapplicable in this regime. We use
$\beta(\Delta_{p})$ to calculate $N_{\text{mc}}(\Delta_{p},N)$ for all our
data and plot the results for the relative MCN
$N_{\text{mc}}(\Delta_{p},N)/N_{\mu}$ in Fig. 4.

Figure 4:  Relative MCN $N_{\text{mc}}(\Delta_{p},N)/N_{\mu}$ vs. $N_{\mu}$
and detuning $\Delta_{p}$. The black line shows where
$\tilde{\alpha}(\Delta_{p},N)=1$ and the white line where $\eta_{\text{inh}}$
is as large as $\eta_{s}(\tilde{\alpha}{=}1)$.

Only for the smallest $N$ and largest $\Delta_{p}$ the MCN is close to
$N_{\mu}$. As $N$ increases and $\Delta_{p}$ decreases, however, the MCN
becomes smaller and only $\sim 10\%$ of the usual collective cooperativity
$N_{\mu}$ contributes to collective scattering. We can now plot
$\Gamma_{N}(N_{\text{mc}})/\langle\Gamma_{R}\rangle_{r}$ and $\left\langle
P_{s}(N_{\text{mc}}^{2})\right\rangle$, respectively, to test their scaling
[see Fig. 3(b,d)]. In contrast to Fig. 3(a) where $N_{\mu}$ was used,
$\Gamma_{N}(N_{\text{mc}})/\langle\Gamma_{R}\rangle_{r}$ collapses onto the
linear dependence $N_{\text{mc}}=\eta\mu N$ over a large range of detunings
and atom numbers. Moreover, we also observe a quadratic scaling of $\langle
P_{s}\rangle$ with the MCN over almost the full range of atom numbers. Only at
the largest $N_{\text{mc}}$ the scaling of $\langle P_{s}\rangle$ changes. In
this regime [green trace in Fig. 2(a)], however, the second instead of the
first burst exhibits the largest power in the dominant fraction of
measurements, which is not the case in standard SR theory [4]. Therefore, the
breakdown of $N^{2}$ scaling is not surprising, and we excluded this data
point from the fit.

##  VI Discussion

Our results demonstrate that the concept of MCN can recover the scaling of
collective decay in homogeneous ensembles for inhomogeneous conditions. This
provides insight into the limits of collective decay in our setup. To this end
we plot in Fig. 4 the boundaries for which pump attenuation (black line) [35,
11] and inhomogeneous broadening (white line) are becoming detrimental to
collective decay. Our system is obviously limited by pump attenuation for
large detunings. Only for $\Delta_{p}\lesssim 9\Gamma$ inhomogeneous spectral
broadening is becoming relevant. Besides supporting the picture of linear-
optics SR presented in [38, 64], our results demonstrate that one can find a
regime that renders the radial inhomogeneities insignificant for atomic
ensembles in a hollow-core fiber. For appropriately large detunings, almost
all atoms contribute to collective scattering, leading to a pronounced
enhancement of the collective decay rate, at least for negligible dispersion.

Despite its simplicity, our model seems to capture the relevant physics. Only
for detunings $\Delta_{p}\lesssim 7\Gamma$, where the approximation of an
effective TLS is not valid anymore for $\Omega_{p}^{(0)}=6.4\Gamma$,
deviations from the experimental data become more significant. Furthermore, an
estimation shows that dispersion is likely to become relevant here as
frequency components of the SF bursts are lying within the dispersive
bandwidth of the excited state [38]. This is currently not captured by our
model.

##  VII Conclusion and Outlook

We presented an experimental study of superfluorescence in a disordered
ensemble of cold atoms coupled to a hollow-core fiber. We demonstrated a decay
rate collectively-enhanced by more than two orders of magnitude for
$N\lesssim$2.2\text{\times}{10}^{5}$$ atoms. By implementing an effective TLS
we were able to study the decay with negligible dispersion for a large range
of detunings and optical depths. Thereby we could tune the amount of
inhomogeneity, studying its effect on the SF decay and developing a simple
theoretical model considering inhomogeneous broadening and attenuation of the
excitation field. Using this model, we determined a maximum cooperation number
$N_{\text{mc}}$ of atoms that can be treated homogeneously and thus decay
collectively. We demonstrated that using $N_{\text{mc}}$ instead of $\mu N$,
the $N$ scaling known to collective decay in homogeneous ensembles could be
recovered. This allowed for a physical understanding of the limits to enhanced
collective decay in our waveguide-coupled ensemble. Our results provide a
simple tool to understand and optimize collective scattering in inhomogeneous
extended ensembles, as can be found, e.g., in astrophysics [21], solids [16,
31], waveguide QED [20], novel atomic clocks [22, 23], and quantum optics
[19].

Future studies could try determining a further scaling factor for the MCN
incorporating dispersion which could help finding an analytical model of
collective decay for dispersive ensembles [65]. Also, applying the here
demonstrated approach to other types of experimental systems exhibiting
inhomogeneities would be interesting as to, e.g., determine their limits to
support collective decay. Furthermore, it might be interesting to study the
regime where the SF bursts exhibit an unusual dominant temporal shape [see
Fig. 2(a)]. Finally, our demonstration of collectively-enhanced Stokes
scattering rates could be used during the pair-generation process in quantum
light sources based on spontaneous four-wave mixing [66] thereby reducing
their required pump power.

###### Acknowledgements.

The authors would like to thank W. Guerin and L.P. Yatsenko for discussions,
E. Giese, R. Walser, and V. Stojanovic for discussions and comments on the
manuscript, and T. Halfmann for his continuous support and comments on the
manuscript. We acknowledge the group of T. Walther for providing us with their
home-built ultra-low noise laser diode driver with high modulation bandwidth.
The project received funding from the European Union’s Horizon 2020 research
and innovation programme under the Marie Skłodowska-Curie grant agreement No.
765075 and by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) project number 410249930.

##  Appendix A Determination of Experimental Parameters for the Theoretical
Model

Comparing the experimental data to the theoretical model requires knowledge of
several experimental parameters. The geometric factor describing the fraction
of spontaneous light emitted into the fiber core mode is approximately given
by $\mu\approx\textrm{NA}^{2}/4$, where NA is the numerical aperture of the
HCPBGF. Taking the measured $1/e^{2}$ mode field radius
$\sigma_{p}=\qty{2.75(20)}{}$ of the near-Gaussian-shaped intensity
distribution, we determine $\textrm{NA}=\lambda/(\pi\sigma_{p})=0.092(6)$ with
$\lambda=\qty{795}{}$. This results in $\mu=2.1(3)\times 10^{-3}$. The decay
rate of the effective TLS depends on the branching ratio $R_{B}=$0.5$$ [52]
into the ground state $|2\rangle$, the Rabi frequency $\Omega_{p}$ of the
pump, and its detuning $\Delta_{p}$, as does the AC Stark shift $S$. We
measure the pump power ($\sim$$5\text{\,}\mathrm{\char 37\relax}\text{/}$
shot-to-shot fluctuations) at a certain location inside the experimental setup
and correct for transmission losses from the HCPBGF to this location. Then,
using the measured mode field diameter, we calculate the peak Rabi frequency
$\Omega_{p}^{(0)}$ according to the parameters given in [52]. The detuning
$\Delta_{p}$ is controlled by the frequency of the acousto-optic modulator
(AOM) modulating the pump with a rise time $\tau_{p}=\qty{130(5)}{}$. The
frequency uncertainty of the AOM is negligible leaving only the uncertainty of
the pump laser lock-point. We estimate this uncertainty as
$\lesssim\Gamma/10$, which can therefore be neglected as well in our analysis.
The total atom number $N$ is determined as described in Sec. IV with a shot-
to-shot uncertainty of $3.5\text{\,}\mathrm{\char 37\relax}\text{/}$. The
radial atomic density profile was investigated in [48] from which we assume a
Gaussian-shaped radial density with a $1/e$ half width of
$\sigma_{a}=\qty{1.7}{}$. Using an atomic density
$\mathcal{N}(r,z)=\mathcal{N}(r)$ and $\Omega_{p}(r)$ we determine the peak
optical depth as

$\alpha_{0}=\frac{4}{\sigma_{p}^{2}}L\int_{0}^{r_{c}}dr\,r\,\sigma_{13}\,\mathcal{N}(r)\,e^{-\frac{2r^{2}}{\sigma_{p}^{2}}},$
(10)

where $\sigma_{13}$ is the resonant absorption cross-section of the transition
$|1\rangle\leftrightarrow|3\rangle$ according to [52], $r_{c}$ is the core
radius of the HCPBGF and we determined $\mathcal{N}(r)$ such that $2\pi\,L\int
dr\,r\,\mathcal{N}(r)=N$. As the final parameter to determine, there is a
residual decoherence rate between the two ground states due to residual
magnetic field gradients and time-of-flight broadening once the guiding
potential is off for the measurements. This value was determined from light
storage and retrieval experiments as $\gamma_{0}\sim 0.057(6)\Gamma$ [66].

##  Appendix B Alternative Method of Determining the Collective Decay Rate

Determination of the collective decay rate for our experimental data is not as
straightforward as, e.g., in the linear-optics SR regime, where a fit to the
exponentially decreasing detected power yields the decay rate [11]. In our
case of SF with strong bursts and ringing we thus have to apply another
method. As discussed in Sec. IV, we determined $\Gamma_{N}$ from the mean
delays $\langle t_{D}\rangle$ of the first SF bursts for which a theoretical
prediction exists [see Eq. (2)]. As this method is somewhat indirect, we apply
in the following an alternative method. As was done, e.g., by Okaba et al. in
the case of SR with bursts and ringing [14], the width of the first burst can
be used as a measure for the initial collective decay rate. In the case of
small-sample SR with a single burst the relation between the temporal burst
width $\tau_{b}$ (FWHM) and the initial collective decay rate is given by
$\Gamma_{N}=3.5/\tau_{b}$ [4]. For large-sample SR/SF, corresponding to our
experiment, we are, however, not aware of an analogous derivation for
$\Gamma_{N}(\tau_{b})$. Despite the lack of of a precise theoretical
prediction, we determined $\Gamma_{N}$ from our data using the first burst
widths and the small-sample relation $\Gamma_{N}=3.5/\tau_{b}$. The results
for $\Gamma_{N}/\langle\Gamma_{R}\rangle_{r}$ vs. $N_{\text{mc}}$ are plotted
in Fig. 5.

Figure 5: Same as Fig. 3(b) but using the widths of the first bursts instead
of the mean delays to determine $\Gamma_{N}$.

Comparing this plot to Fig. 3(b), we note the following: (a) The order of
magnitude of $\Gamma_{N}$ is the same. (b) Plotting $\Gamma_{N}$ vs. the MCN
recovers a near-linear scaling in both cases. (c) Scaling $\Gamma_{N}$ in Fig.
5 by a factor of $\sim 1.7$ results in a scattering of the datapoints around
$\Gamma_{N}/\langle\Gamma_{R}\rangle_{r}=N_{\text{mc}}$ (blue line). (d) The
scattering of the data is larger when using the burst widths compared to the
mean delays.

In view of these results we conclude that the initial collective decay rate of
SF can be determined either from the mean burst delays or from their temporal
widths. The former method produces more reliable results and provides a direct
comparison to theory. Using rather the burst delays instead of their widths is
also suggested by the uncertainties in the fitting process. Our data shows
that the delays can be determined with a smaller relative uncertainty than the
widths. This is intuitive as the width is also dependent on the fitted
amplitude. Finally we note that the widths determined from the fits to the
single-shot data are in general larger than the raw data would suggest. This
might explain why the datapoints in Fig. 5 are below the theoretical line.
Another reason for this discrepancy could, however, be that the relation
$\Gamma_{N}=3.5/\tau_{b}$, valid for small samples, cannot be directly applied
to the large-sample case.

##  Appendix C Determination of the Maximum Cooperation Number

In the following we present the model used to calculate the maximum
cooperation number $N_{\textrm{mc}}$. It is mainly based on the ideas
presented by Arecchi and Courtens [42], Bienaimé et al. [35], and Bachelard et
al. [62]. First, inhomogeneous broadening reduces the number of cooperative
atoms [42]. Second, attenuation of the pump field as it propagates through the
optically-dense ensemble reduces the effective scattering/decay rate
$\Gamma_{N}$, which can be viewed as a reduction of the number of synchronized
scatterers. We account for this by calculating a factor $\eta_{s}$ from an
effective scattering rate in the spirit of the shadow effect introduced by
Bachelard et al. [62]. We note that in our case the decay rate $\Gamma_{R}$
itself depends on the (attenuated) excitation field, which is in contrast to
decay in a true TLS where the decay rate is constant and only the excitation
is inhomogeneous.

To find a simple analytical model, we make several assumptions, as the actual
system exhibits complicated nonlinear spatial and temporal dynamics. First,
there is a radial variation of the atomic density distribution
$\mathcal{N}(r)=\mathcal{N}_{0}\exp\left(-r^{2}/\sigma_{a}^{2}\right)$, and we
assume a constant density along the propagation direction $z$. Second, the
pump intensity $I_{p}(r,z)\propto\Omega_{p}^{2}(r,z)$, determining both Raman
scattering rate and AC Stark shift of the ground state, is potentially
attenuated along the propagation direction $z$ and also exhibits a radial
variation
$\Omega_{p}^{2}(r,0)=\Omega_{p}^{2}(0,0)\exp\left(-2r^{2}/\sigma_{p}^{2}\right)$
. Its $1/e^{2}$ half width is only slightly larger than $\sigma_{a}$. We use
the fact that $\Omega_{p}^{2}(r,z)$ can be factorized as

$\Omega_{p}^{2}(r,z)=\Omega_{p}^{2}(0,0)\cdot g_{p}(r)\cdot h_{p}(z),$ (11)

Furthermore, inhomogeneous broadening due to the radial variations is expected
to dominate over longitudinal variations as it is also present for large
detunings. These assumptions allow for considering the radial and longitudinal
variations independently. In the following we start by calculating the radial
averages weighed by the atomic density distribution at the start of the medium
at $z=0$ for obtaining the inhomogeneous linewidth. Then, we take the radial
and longitudinal average to determine the effective scattering rate. Using
these results we then determine a MCN.

###  C.1 Inhomogeneous Broadening

As discussed by Arecchi and Courtens, inhomogeneous broadening reduces the
number of cooperative atoms by the ratio of (homogeneous) excitation bandwidth
$\sigma_{\text{hom}}$ to inhomogeneous linewidth $\sigma_{\text{inh}}$ [42].
The excitation bandwidth can be estimated by the rise time $\tau_{p}$ of the
pump field, $\sigma_{\text{hom}}=1/\tau_{p}$. The inhomogeneous linewidth of
the effective TLS can be estimated as follows. The pump intensity inside the
HCPBGF determines the AC Stark shift $S$ of the ground state $F{=}1$

$S(r,z)=\frac{\Omega_{p}^{2}(r,z)}{4\Delta_{p}}$ (12)

and the decay of the effective TLS

$\Gamma_{R}(r,z)=\Gamma\frac{\Omega_{p}^{2}(r,z)}{8\left[\Delta_{p}+2S(r,z)\right]^{2}}.$
(13)

We therefore calculate the radial averages of $\Omega_{p}^{2}(r,0)$, $S(r,0)$,
and $\Gamma_{R}(r,0)$ at the beginning $z=0$ of the medium, weighed by the
atomic density distribution as

$\langle
f(r)\rangle_{r}=\frac{2}{\sigma_{a}^{2}}\int_{0}^{\infty}dr\,r\,e^{-\frac{r^{2}}{\sigma_{a}^{2}}}f(r),$
(14)

where $f(r)=\left[\Omega_{p}^{2}(r,0),\,S(r,0),\,\Gamma_{R}(r,0)\right]$.

Using the radially-averaged values, we now estimate the inhomogeneous
linewidth $\sigma_{\text{inh}}$ as follows. Due to the radial Gaussian
dependence of the strong pump field, atoms in the ensemble of radial width
$\sigma_{a}$ will experience an inhomogeneous AC Stark shift $S(r)$ of the
ground state $F{=}1$ as well as an inhomogeneous decay rate $\Gamma_{R}(r)$.
As a measure for the corresponding linewidth we calculate the standard
deviation

$\delta f=\sqrt{\left\langle\left[f(r)-\langle
f(r)\rangle_{r}\right]^{2}\right\rangle_{r}}$ (15)

for $f(r)=\left[S(r,0),\,\Gamma_{R}(r,0)\right]$ and set

$\sigma_{\text{inh}}=\delta S+\delta\Gamma_{R}.$ (16)

Using the measured rise time $\tau_{p}$ of the pump pulse, we obtain the first
of two scaling factors,

$\eta_{\text{inh}}=\frac{1}{\tau_{p}\sigma_{\text{inh}}},$ (17)

to calculate the MCN. Note that we only apply this scaling factor if the
inhomogeneous bandwidth is larger than the homogeneous bandwidth. Considering
only inhomogeneous broadening in the MCN, we obtain the results shown in Fig.
6(a) which have to be compared to those shown in Fig. 3(a).

Figure 6: Initial relative collective decay rate
$\Gamma_{N}/\langle\Gamma_{R}\rangle_{r}$ vs. corresponding MCN when
considering only inhomogeneous broadening (a), inhomogeneous broadening and
pump absorption (without conversion) (b), pump attenuation (absorption and
conversion) without inhomogeneous broadening (c). Compare to Fig. 3(a,b).

Obviously, taking inhomogeneous broadening into account slightly improves the
scaling but is only relevant in the range of $\Delta_{p}\lesssim 9\Gamma$.

###  C.2 Effective Scattering Rate

Having determined a scaling factor which accounts for inhomogeneous broadening
(in the radial direction), we next turn to the variations in the longitudinal
direction $z$. This is relevant for our considered system, as propagation
through the optically-dense atomic ensemble can attenuate the pump field. To
this end, we calculate an effective collective scattering rate weighed by the
radial atomic density distribution $\mathcal{N}(r)$

$\displaystyle\left\langle\Gamma_{N}^{\text{eff}}\right\rangle_{r,z}$
$\displaystyle=\mu
N\Gamma\cdot\frac{2}{\sigma_{a}^{2}}\int_{0}^{1}dz^{\prime}\int_{0}^{\infty}dr\,r\,e^{-\frac{r^{2}}{\sigma_{a}^{2}}}\times$
(18)
$\displaystyle\quad\times\frac{\Omega_{p}^{2}(r,z^{\prime})}{8\left[\Delta_{p}+2S(r,z^{\prime})\right]^{2}},$

where

$\Omega_{p}(r,z^{\prime})=\Omega_{p}^{(0)}e^{-\frac{r^{2}}{\sigma_{p}^{2}}}e^{-\frac{1}{2}\tilde{\alpha}(\Delta_{p})z^{\prime}},$
(19)

and $\Omega_{p}^{(0)}$ $[=\Omega_{p}(0,0)]$ is the peak Rabi frequency of the
pump. We assume attenuation of the pump intensity according to Beer-Lambert’s
law along the dimensionless distance $z^{\prime}=z/L$ with an attenuation
factor $\tilde{\alpha}(\Delta_{p})$ affecting also $S(r,z^{\prime})$ in Eq.
(12). We obtain the corresponding collective scattering rate
$\langle\Gamma_{N}\rangle_{r}$ without pump attenuation by setting
$\tilde{\alpha}(\Delta_{p})\equiv 0$ in Eq. (18). Similar to Bachelard et al.,
where a shadow factor was defined to determine an effective reduction in the
total scattering cross-section of an optically-dense ensemble [62], we now
define a shadow factor

$\eta_{s}=\frac{\left\langle\Gamma_{N}^{\text{eff}}\right\rangle_{r,z}}{\mu
N\langle\Gamma_{R}\rangle_{r}},$ (20)

where $\langle\Gamma_{R}\rangle_{r}$ is the radially-averaged single-atom
scattering rate at $z^{\prime}=0$ used as reference without attenuation.
$\eta_{s}$ is thus a measure for the reduction of the collective scattering
rate due to pump attenuation. As
$\left\langle\Gamma_{N}^{\text{eff}}\right\rangle_{r,z}\propto N$ we identify
$\eta_{s}\mu N$ as an effective number of atoms that will scatter collectively
when considering attenuation of the pump field.

###  C.3 Maximum Cooperation Number

Finally, combining inhomogeneous broadening, longitudinal pump attenuation,
and the geometric factor $\mu$ we determine the MCN as

$N_{\textrm{mc}}=\eta_{\text{inh}}\cdot\eta_{s}\cdot\mu\cdot N.$ (21)

##  Appendix D Modeling Pump Attenuation

In order to apply the general model described above to our experimental data,
we need to identify which factors cause attenuation of the pump, i.e., we need
to model the (dimensionless) attenuation factor $\tilde{\alpha}(\Delta_{p})$.
The first obvious consideration is (off-)resonant absorption. Our experiment
allows for loading $N\lesssim 2.2\times 10^{5}$ atoms into the HCF. With an OD
per atom of $\alpha^{*}=2.75\times 10^{-3}$ on the transition
${}^{2}S_{1/2}\,F{=}1\leftrightarrow\,^{2}P_{1/2}\,F^{\prime}{=}2$ we
therefore achieve resonant optical depths of $\alpha_{0}\lesssim 600$. The
detuning-dependent optical depth at $z=0$ for large detunings is approximately
given by

$\alpha(\Delta_{p})\approx\alpha_{0}\left\langle\frac{\Gamma^{2}}{4\left[\Delta_{p}+2S(r,0)\right]^{2}}\right\rangle_{r},$
(22)

where we have taken a radial average (see Appendix C.1). For our experimental
parameters we reach $\alpha(\Delta_{p})\gtrsim 1$ for $\Delta_{p}\lesssim
12\Gamma$. Setting $\tilde{\alpha}(\Delta_{p})\equiv\alpha(\Delta_{p})$ in Eq.
(18), where we assumed in a first approximation that the attenuation factor
does not depend on $z$ and is given by its initial value at $z=0$, yields a
MCN
$N_{\textrm{mc}}^{\text{abs}}=\eta_{\text{inh}}\cdot\eta_{s}^{\text{abs}}\cdot\mu\cdot
N$ for each data point in Fig. 3(a). Plotting the initial relative collective
decay rate $\Gamma_{N}/\langle\Gamma_{R}\rangle_{r}$ vs.
$N_{\textrm{mc}}^{\text{abs}}$ we obtain the plot shown in Fig. 6(b). Note
that so far there is no free fitting parameter. Compared to Fig. 6(a), where
only inhomogeneous broadening was considered, $N_{\textrm{mc}}^{\text{abs}}$
is a further improved scaling parameter compared to the collective
cooperativity $N_{\mu}$. Yet the data points do not collapse onto the blue
line showing
$\Gamma_{N}/\langle\Gamma_{R}\rangle_{r}=N_{\textrm{mc}}^{\text{abs}}$.

We therefore consider additional attenuation of the pump. To this end we note
that the Stokes buildup happens on a timescale determined by the Stokes gain
[50] and that the derivation of Eq. (2) was done in a regime of exponential
growth of the SF burst [60]. We thus make the ansatz

$I_{p}(z^{\prime})\approx
I_{p}^{(0)}e^{-{\tilde{\alpha}(\Delta_{p})}z^{\prime}},$ (23)

where the total attenuation factor is given by

$\tilde{\alpha}(\Delta_{p})=\alpha(\Delta_{p})+\beta(\Delta_{p})\cdot
G_{s}(\Delta_{p},0).$ (24)

We thus assume that the pump intensity $I_{p}$ decreases exponentially due to
absorption and conversion into the exponentially-growing Stokes field
$I_{s}(z^{\prime})\propto\exp{[\beta(\Delta_{p})G_{s}(\Delta_{p},0)z^{\prime}]}$
with a scaling factor $\beta(\Delta_{p})$.

The initial radially-averaged Stokes gain can be approximately written as

$G_{s}(\Delta_{p},z^{\prime}{=}0)\approx\alpha_{0}\left\langle\frac{2\Gamma_{R}(\Delta_{p},0)}{\gamma}\right\rangle_{r},$
(25)

where $\gamma$ is the ground state decoherence rate. For our all our
experimental parameters we determine $G_{s}(\Delta_{p},z^{\prime}{=}0)>1$.
This ansatz corresponds to steady-state conditions in [50] while neglecting
transient conditions during Stokes field buildup, as during the latter one the
Stokes intensity evolution is additionally time-dependent which complicates
the theoretical description [50]. To account for this simplification we
introduced the scaling factor $\beta(\Delta_{p})$ for the Stokes gain, which
is allowed to vary with $\Delta_{p}$ (see Appendix E). We note that this
scaling factor accounting for partially transient conditions during the
buildup of coherence is only applied to the Stokes gain but not for
absorption. This is motivated as follows: Although absorption also exhibits a
transient after switch-on of the pump, the timescale is determined by the
excited state decay rate $\Gamma$, corresponding to $\sim 30~{}$ns which is
much smaller than the timescale of SF emission. On the other hand, the
transient timescale for generation of the Stokes field is determined by the
ground state coherence time $\gamma^{-1}$ [50], which is comparable to the
timescale of SF emission. Thus, transient conditions are likely more relevant
in the gain than in the absorption term.

In order to calculate the initial Stokes gain, we have to determine the
decoherence rate $\gamma$. We identify three major contributions to $\gamma$
for our effective TLS. First, there is a residual decoherence rate between the
two ground states determined by residual magnetic field gradients and time-of-
flight broadening. This value was determined from light storage and retrieval
experiments as $\gamma_{0}\sim 0.057(6)\Gamma$ [66]. Second, the inhomogeneous
AC Stark shift of the the ground state by the pump field causes decoherence.
We approximate this rate by the previously calculated standard deviation
$\delta S$ (see Appendix C.1). Third, the Raman scattering rate $\Gamma_{R}$
leads to decoherence due to decay between the two ground states. Therefore, we
set the total decoherence rate as

$\gamma=\gamma_{0}+\delta S+\langle\Gamma_{R}\rangle_{r}.$ (26)

Combining these results, we determine the MCN $N_{\textrm{mc}}$ for all
measurement parameters $(\Delta_{p},N)$. The results are shown in Fig. 4.
Complementary to Fig. 6(a), where we only considered inhomogeneous broadening,
we plot in Fig. 6(c) the results when considering only pump attenuation while
neglecting inhomogeneous broadening. As we can see, attenuation of the pump
field is relevant across the whole range of detunings, however, it fails to
provide the correct scaling for $\Delta_{p}\lesssim 9\Gamma$. Only a
combination of pump attenuation and inhomogeneous broadening yields the
expected scaling.

##  Appendix E Scaling Factor $\beta$

The scaling factor $\beta$ is the only free parameter in our model introduced
to account for the transient regime during buildup of the Stokes field. In
this regime, the Stokes intensity does not simply grow exponentially as
$\exp{(G_{s}z^{\prime})}$ as in the steady-state regime, but exhibits a time
dependence [50] which is difficult to handle in our simple model. Using a
constant $\beta\sim 0.07$ for all detunings and atom numbers works already
satisfactory for detunings $\Delta_{p}>7\Gamma$. However, not all measured
delays are as well reproduced as shown in Fig. 2(b) for
$\Delta_{p}=18.4\Gamma$. As $\beta$ accounts for temporal dynamics during
Stokes field buildup, it seems reasonable to assume that $\beta$ varies with
the detuning (and atom number). Therefore, we plotted $\langle
t_{D}(N)\rangle$ and determined $\beta(\Delta_{p})$ by manually fitting the
calculated mean delays [Eq. (2)] to the data. The resultant
$\beta(\Delta_{p})$ (see Fig. 7) varies approximately linearly for
$\Delta_{p}>6\Gamma$.

Figure 7: Scaling factor $\beta(\Delta_{p})$ as determined from manual fits to
the measured $\langle t_{D}(N)\rangle$. The solid line shows a linear fit for
$\Delta_{p}>6\Gamma$ with $\beta(\Delta_{p})=0.182-6.1\times
10^{-3}\Delta_{p}/\Gamma$.

The exact reason for such a linear dependence is currently unknown and we plan
to further investigate it in the future trying to find a model for $\beta$. We
currently speculate that the variation is due to a change in the relative time
spent in the transient regime: Our experimental data shows that the mean delay
$\langle t_{D}\rangle$ increases continuously from $\sim 0.7\gamma_{0}^{-1}$
at $\Delta_{p}=6\Gamma$ to $\sim 0.95\gamma_{0}^{-1}$ at
$\Delta_{p}=26.4\Gamma$. Thus, it seems plausible that at larger detunings the
temporal ratio of transient to steady-state conditions during coherence
buildup gets larger, resulting in a smaller $\beta$. For $\Delta_{p}\lesssim
6\Gamma$, $\beta$ increases strongly up to 1, indicating that our model is not
valid anymore in this range. This comes as no surprise as this is
approximately the regime where the effective TLS model breaks down, i.e.,
$\Delta_{p}\lesssim\Omega_{p}$ and dispersion should become relevant. For
calculating the MCN (see Sec. V), we fit $\beta(\Delta_{p})$ for
$\Delta_{p}>6\Gamma$ with a linear function yielding
$\beta(\Delta_{p})=0.182-6.1\times 10^{-3}\Delta_{p}/\Gamma$. We use this
$\beta(\Delta_{p})$ dependence for all measured detunings.

##  Appendix F Validity of the Pencil-Shaped Model for an Ensemble Coupled to
a HCPBGF

The pencil-shaped model can be applied when the field propagates along a
single direction and transverse field variations can be neglected [4]. In free
space, this is the case when the Fresnel number $F=\pi\sigma_{a}^{2}/(\lambda
L)\sim 1$. In our case, the Fresnel number is $F\sim 10^{-4}\ll 1$ so a 3D
description would be appropriate. However, the waveguide collects radiation
emitted into a cone of opening angle $\textrm{NA}=\lambda/(\pi\sigma_{p})$.
Thus, our effective Fresnel number is $F^{\prime}\sim\sigma_{a}/\sigma_{p}\sim
1$ and the pencil-shaped model can be applied.

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