# Transient dynamics of the quantum light retrieved from Rydberg polaritons

Auxiliadora Padrón-Brito1, Roberto Tricarico1, Pau Farrera 1,2, Emanuele
Distante 1,2, Klara Theophilo1, Darrick Chang1,3 and Hugues de Riedmatten 1,3
1 ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and
Technology, 08860 Castelldefels, Barcelona, Spain  2 Max-Planck-Institut für
Quantenoptik, Garching, Germany  3 ICREA-Institució Catalana de Recerca i
Estudis Avaçats, 08015 Barcelona, Spain<EMAIL_ADDRESS>
###### Abstract

We study the photon statistics of weak coherent pulses propagating through a
cold Rydberg atomic ensemble in the regime of Rydberg electromagnetically
induced transparency. We show experimentally that the value of the second-
order autocorrelation function of the transmitted light strongly depends on
the position within the pulse and heavily varies during the transients of the
pulse. In particular, we show that the falling edge of the transmitted pulse
displays much lower values than the rest of the pulse. We derive a theoretical
model that quantitatively predicts our results and explains the physical
behavior involved. Finally, we use this effect to generate single photons
localized within a pulse from the atomic ensemble. We show that by selecting
only the last part of the transmitted pulse, the single photons show an
antibunching parameter as low as 0.12 and a generation efficiency per trial
larger than possible with probabilistic generation schemes with atomic
ensembles.

##  1 Introduction

Highly excited Rydberg atoms enable long-range controllable atom-atom
interactions [1]. This property has been used to demonstrate entanglement [2,
3] and quantum gates between individual atoms [4, 5] and more recently quantum
simulation in atomic arrays [6, 7]. Besides using individual atoms, another
approach currently investigated is to use ensembles of highly excited Rydberg
atoms, which on top of providing a medium with long-range controllable
interaction between atoms also allows an efficient quantum light-matter
interface. These properties lead to optical nonlinearities at the single-
photon level [8, 9, 10] that can be used for quantum nonlinear optics [11, 12]
and quantum information applications. The use of ensemble-based Rydberg
mediated interactions has been explored within a range of schemes, such as
single-photon level transistors and switches [13, 14, 15], photon-photon
interactions [16] and gates [17]. Ensembles of Rydberg atoms can also be used
to generate single photons in a quasi-deterministic fashion [18, 19, 20, 21],
contrary to ground state ensembles which usually rely on probabilistic or
heralded schemes [22].

The use of electromagnetically induced transparency (EIT) with Rydberg excited
levels [23] provides an efficient way of mapping light onto Rydberg
excitations, leading to the creation of strong optical nonlinearities in the
transmitted light thanks to the phenomenon of Rydberg blockade. Under EIT
conditions, probe photons ($\Omega_{p}$) resonant with a ground-to-excited
state transition $\mbox{$|g\rangle$}\to\mbox{$|e\rangle$}$ can coherently
propagate within a narrow transparency window with the use of an additional
resonant control field ($\Omega_{c}$) that couples $|e\rangle$ to a metastable
state $|r\rangle$ (see Fig. 1c). This allows probe photons to be mapped onto
dark-state polaritons, propagating with a group velocity $v_{gr}$ much smaller
than the velocity of light in vacuum $c$. However, if $|r\rangle$ is a Rydberg
level, the dipole-dipole interaction prevents the excitation of two atoms to
the Rydberg state if they are closer than a distance called the blockade
radius ($r_{b}$) [1], which destroys the EIT transparency for two propagating
probe photons within the distance $r_{b}$. As a consequence, strong scattering
and absorption associated with a resonant two-level medium occurs, whereby the
probability of detecting two photons at the same time in the transmitted light
mode is ideally reduced to zero. In practice, within $r_{b}$, the amount of
light transmitted in the two-level regime depends on the optical depth per
blockade radius $D_{b}$, as $\exp(-D_{b})$.

The transmission of continuous-wave weak coherent input states of light
through an ensemble of atoms under Rydberg EIT has been investigated
theoretically [8, 24, 25, 26, 27], and experimentally with the demonstration
of single-photon nonlinearity and strong antibunching of the output light [8,
10]. However, the use of CW light renders challenging the efficient creation
of single photons localized in time, as required for some applications in
quantum information science [28, 29, 30]. It has been shown that localized
single photons can be retrieved from collective Rydberg excitations, which can
be created by using an off-resonant two-photon excitation scheme [18, 20, 21,
31] or by storage of weak coherent light pulses under EIT conditions [19, 32].
However, this represents an additional experimental complexity and source of
inefficiency. The use of transmitted pulses (without storage) in Rydberg EIT
could therefore potentially lead to the generation of localized single photons
with higher efficiencies. Nevertheless, the quantum statistics of weak pulses
traveling under Rydberg EIT conditions has not been well studied yet.
Moreover, as shown in [18] reaching low values of $g^{(2)}(0)$ requires very
high Rydberg levels and high values of $D_{b}$, which are experimentally
challenging to achieve, and low values of $g^{(2)}(\tau)$ are only observed
for a small value of $\tau$ ($\sim$100 ns), such that reaching low values of
$g^{(2)}$ for the entire pulse is challenging.

In this paper, we study experimentally and theoretically the propagation of
weak coherent input pulses through a Rydberg EIT window. We measure the
second-order correlation function $g^{(2)}_{\Delta t}$ for short time windows
$\Delta t$ inside the pulse for various input pulse shapes. We show that the
values of $g^{(2)}_{\Delta t}$ strongly vary during the transient phases,
leading to very low values for the trailing edge, while the value for the full
output pulse is much higher. Recently, similar behavior of $g^{(2)}$ during
the EIT transients have been observed [33]. Here, we perform a systematic
experimental and theoretical investigation to better understand the underlying
physics. We develop a theoretical framework that can quantitatively predict
our experimental results and explain the physical behavior involved. In
addition, we investigate the use of the transient regime for the production of
higher quality narrowband single photons by measuring $g^{(2)}_{\Delta t}$ as
a function of the photon generation efficiency and comparing the results with
the single photons obtained via storage in the Rydberg states.

##  2 Experiment

Fig. 1 shows a scheme of the experimental setup. A cloud of cold 87Rb atoms is
loaded in a dipole trap, with an atomic peak density of $\sim 4\cdot 10^{11}$
cm-3 and a transversal size of 34 $\mu$m. The experimental sequence is the
following: we start by loading a magneto-optical trap (MOT) for 2 seconds,
followed by a compression of the MOT and a molasses period; the dipole trap is
switched on 500 ms after the beginning of the MOT and it is switched off
during the excitation of the atoms to the Rydberg level, to avoid losing atoms
and remove AC Stark shifts; the dipole trap is modulated with a period of 16
$\mu$s, which leaves a time of less than 8 $\mu$s to perform one experimental
trial. This is repeated 13000 times during the dipole trap lifetime, which
gives a total experimental rate of 5.73 kHz.

The atoms are initially prepared in the ground state
$|g\rangle=|5S_{1/2},F=2\rangle$ and resonantly coupled with the state
$|e\rangle=|5P_{3/2},F^{\prime}=3\rangle$ by a weak probe field of 780 nm. The
excited state is also resonantly coupled with the Rydberg state
$|r\rangle=|90S_{1/2}\rangle$ using a counter-propagating control beam at
479.4 nm (see Fig. 1.c). The probe beam is sent with an angle of $19^{o}$ with
respect to the dipole trap beam (see Fig. 1.a) and propagates through the
medium with an optical depth of $D\approx 10$. It is focused in the center of
the atomic medium with a beam waist of $w_{p}\approx 6.5\ \mu$m. As the
blockade radius is $r_{b}\approx 13\ \mu$m for our experimental parameters,
the parallel propagation of two polaritons in our atomic cloud is negligible
and the optical depth per blockade radius is $D_{b}\approx 1$. Fig. 1.d shows
an example of the transmission spectrum of Rydberg EIT in our cloud, for a
weak single-photon level probe. By fixing the spontaneous decay rate of the
excited level $\Gamma$ to be $\approx 2\pi\times$6 MHz [34], it results a
control Rabi frequency of $2\Omega_{c}\approx 2\pi\times$6.4 MHz, a dephasing
rate of the Rydberg level of $\gamma_{r}\approx 2\pi\times$0.8 MHz, and a full
width at half maximum of the transmission (EIT bandwidth) of
$\delta_{EIT}\approx 2\pi\times 2.3$ MHz.

Figure 1:  (a) Schematic representation of the experimental setup. An input
probe pulse (in red) is sent to the experiment with a spatial distribution
given by the probe beam (also in red), at a 19o angle to the dipole trap. The
control beam (in blue) is counter-propagated with the probe and both are
focused in the center of a dipole trap by two aspheric lenses. (b) The
transmitted photons are sent through a Hanbury Brown and Twiss setup
consisting of a beam-splitter and two single-photon detectors SPD1 and SPD2 to
measure the second-order autocorrelation function. $\Omega_{p}$: Probe Rabi
frequency, $\Omega_{c}$: Control Rabi frequency, DM: Dichroic mirror. (c)
Level scheme and transitions used in the experiment. (d) EIT transparency
window measurement (points) and theoretical simulation (dashed line) presented
in Sec. 4.

The statistics of the light is obtained from the second-order autocorrelation
function, defined as:

$g^{(2)}(t,\tau)=\frac{\left<E^{\dagger}(t)E^{\dagger}(t+\tau)E(t+\tau)E(t)\right>}{\left<E^{\dagger}(t)E(t)\right>\left<E^{\dagger}(t+\tau)E(t+\tau)\right>}\mathrm{,}$
(1)

where $E(t)$ is the electric field operator for the transmitted light mode at
time t. We measure the second-order autocorrelation function by sending the
output pulses through a Hanbury Brown and Twiss setup, which consists of a
beam splitter and two single-photon avalanche detectors (SPDs), as shown in
Fig. 1.b. The arrival times of the photons for each detector, together with
trigger times for each trial, are saved in a time-stamp file.

First, we select a detection time window $\Delta t$. Then, $g^{(2)}_{\Delta
t}(t,\tau)$ is obtained from the equation

$g^{(2)}_{\Delta t}(t,\tau)=\frac{P_{C}(\Delta t)}{P_{1}(\Delta t)P_{2}(\Delta
t)}\mathrm{,}$ (2)

where $t$ is the starting time of the detection window in the detector SPD1
and $t+\tau$ is the starting time in the detector SPD2, both measured with
respect to the trigger time. $P_{C}(\Delta t)$ is the coincidence probability
between the two detectors and $P_{1}(\Delta t)$ and $P_{2}(\Delta t)$ are the
detection probabilities in detector SPD1 and SPD2, respectively. The
normalization factor $P_{1}(\Delta t)P_{2}(\Delta t)$ is obtained by averaging
the number of coincidences between photons arriving to the first detector in
one trial and the photons arriving to the second detector from the 5th to the
20th following trials, where there is no correlation (see Fig. 2a). The
correlation between the first five trials can be explained by the creation of
long-lived pollutants [35, 36]. The transmission efficiency from the ensemble
to the first detector is $0.23\pm 0.02$, taking into account all the optical
elements, and the SPD1 detection efficiency is $0.43\pm 0.04$.

##  3 Time-resolved photon correlation

First, we study the photon statistics for the case of a square pulse with a
temporal length of $1\ \mu$s and a mean number of photons of $\approx 1.5$,
propagating through the Rydberg medium under EIT conditions. In the inset of
Fig. 2.a, we can see the temporal shape of the input and output pulses,
showing an EIT transmission efficiency of $\eta=N_{out}/N_{in}=0.285\pm
0.016$, where $N_{out(in)}$ is the total number of counts in the output(input)
pulse. Fig. 2.a, shows an example of the autocorrelation measurement for
$\Delta t=1.6\ \mu$s, sufficiently large to include the whole output pulse
(see the detection window delimited by the black dashed lines of the inset).
The normalized coincidences for the $n_{trial}=0$ leads to $g^{(2)}_{\Delta
t}(0,0)=0.908\pm 0.004$, which shows that while the full output pulse displays
quantum statistics, it is still far from being a single photon.

We then measure the time-resolved autocorrelation function $g^{(2)}(\tau)$,
where $\tau$ is the delay time between two-photon detections within the whole
output pulse. For that purpose, we measure the coincidences for different
$\tau$, inside a bin size of 10 ns, and normalize them by the coincidences
between the following 5th-20th trials. The results are shown in Fig. 2.b. For
$\tau=0$, we obtain $g^{(2)}(0)=0.31\pm 0.03$, demonstrating the single-photon
nature of the output light, as shown previously in [8, 19, 14, 13, 15].
However, we see that $g^{(2)}(\tau)$ quickly increases to 1 for $\tau\geq 350$
ns, much shorter than the pulse duration. This increase is attributed to the
fact that the compressed pulse is longer than the blockade radius.

Figure 2: Statistics for a square pulse. (a) Normalized coincidences between
the trial 0 and trial $n_{trial}$. Then, the first value, corresponding to
$n_{trial}=0$, shows the normalized coincidences in the same trial, which
gives a value of $g^{(2)}_{\Delta t}(0,0)=0.908\pm 0.004$. Inset in (a) shows
the temporal distribution of the input (in orange) and the output pulse (in
green). The solid lines indicate the experimental values and the dashed lines
the results of the theoretical simulations presented in Sec. 4. The black
lines delimit the temporal window with $\Delta t=1.6\ \mu s$, longer that the
output pulse duration. (b) Normalized coincidences as a function of the delay
time $\tau$ between the two-photon events, being the one at zero-delay time
$g^{(2)}(0)=0.31\pm 0.03$. These values are calculated with a bin size of 10
ns. Error bars in the plots correspond to one standard deviation.

To get a better understanding of the dynamics of Rydberg polaritons, we
perform a more detailed study of the second-order autocorrelation function
within a pulse. To this end, we select a detection window $\Delta t$ much
shorter than the pulse duration and we measure the zero-delay autocorrelation
function $g^{(2)}_{\Delta t}(t,0)$ as a function of the starting time of the
detection window $t$. For simplicity, $g^{(2)}_{\Delta t}(t,0)\equiv
g^{(2)}_{\Delta t}(t)$ henceforth. An example is shown in Fig. 3 where we show
the $g^{(2)}_{\Delta t}(t)$ (blue points), obtained by taking a window of
$\Delta t=200$ ns (green shadowed region), for different positions of this
window throughout the output pulse.

We find that the second-order autocorrelation function is not constant
throughout the pulse duration, but decreases towards the end of the pulse. We
observe three different regimes. For early times ($t<0.4\ \mu$s), a first
transient is observed, exhibited by a decrease in the $g^{(2)}_{\Delta t}(t)$
and an increase in the output intensity over time. We measure $g^{(2)}_{\Delta
t}(t)=0.8\pm 0.1$ for $t=0$ and $g^{(2)}_{\Delta t}(t)=0.53\pm 0.02$ for
$t=0.3\ \mu$s. In the steady-state region ($0.4\ \mu$s $<t<1\ \mu$s), both the
$g^{(2)}_{\Delta t}(t)$ and the output intensity are constant over time. Note
that in this region, the value of $g^{(2)}_{\Delta t}(t)$ decreases for lower
$\Delta t$, as shown in the inset of Fig. 3, since the probability of a two-
photon event decreases for lower delay times $\tau$ (see Fig. 2.b). When the
input pulse is switched off ($t>1\ \mu$s), a second transient is observed.
Although a decrease in the output intensity is expected, $g^{(2)}_{\Delta
t}(t)$ follows the same behavior in the same time range, decreasing until
reaching a value as low as $g^{(2)}_{\Delta t}(t)=0.12\pm 0.05$ for $t=1.4\
\mu$s. These changes of $g^{(2)}_{\Delta t}(t)$ along the pulse correspond to
three distinguishable processes, happening at different times during the
propagation of Rydberg polaritons, as we will describe in the sec. 4.

Figure 3: Transients for a square pulse. The counts per trial (right y-axis)
of the input (orange) and output (green) pulses are shown with respect to the
arrival time to the photodetector. An example of the 200 ns window that we
take to calculate the $g^{(2)}_{\Delta t}(t)$ is shadowed. The measured
second-order autocorrelation function (blue points, left y-axis) is shown for
different starting times $t$ of the 200 ns window along the output pulse. The
highlighted point corresponds to the shadowed region. The dashed blue line
shows the second-order correlation function obtained from the theoretical
simulation of $g^{(2)}(t)$ presented in Sec. 4, which corresponds to the case
with $\Delta t\to 0$. For comparison, in the inset plot, we show that the
average steady-state (ss) experimental value $g^{(2)}_{\Delta t}(t_{ss})$
tends to the theoretical $g^{(2)}_{ss}(t)\approx 0.3$ for smaller $\Delta t$
(the dashed orange line is a linear fit to guide the eye). Note that using a
$\Delta t>0$ would smooth out the discontinuities seen in the simulation of
the turn-on transient and could explain why they are not seen in the
experimental data. The error bars correspond to one standard deviation.

To show the dependence of these transients with the shape of the input pulse,
we study the propagation of pseudo-triangular pulses (see Fig. 4), following
the same method described above. In the case of a triangular shape with a
negative slope (see Fig. 4.a), the probe field is switched on abruptly but
slowly switched off. Here, we observe that $g^{(2)}_{\Delta t}(t)$ starts with
a value close to 1, but then it decreases rapidly towards smaller values,
remaining constant at the end of the pulse with $g^{(2)}_{\Delta t}(t)=0.52\pm
0.13$ for the last point ($t=0.8\ \mu$s). For a triangular shape with a
positive slope (see Fig. 4.b), i.e. slowly switched on and abruptly turned
off, we only observe a clear transient at the end of the pulse, since
$g^{(2)}_{\Delta t}(t)$ starts to decrease when the input pulse intensity goes
to zero ($t>1\ \mu$s). A value of $g^{(2)}_{\Delta t}(t)=0.05\pm 0.04$ is
obtained for the last point ($t=0.8\ \mu$s), which is much lower than the
observed value for the triangular shape with negative slope.

These results show that the appearance of the transients depends on how the
input pulse varies over time. Specifically, very low values of
$g^{(2)}_{\Delta t}(t)$ are observed at the end of the pulse only if the
decrease in intensity of the input pulse is fast enough.

Figure 4: Transients for triangular pulses. Second-order autocorrelation
function (in blue) for a window $\Delta t$ of 200 ns along the output pulse,
temporal distribution of the input (in orange) and output (in green) pulses
for an input pulse with a pseudo-triangular shape and a negative slope (a) and
for an input pulse with a positive slope (b). The error bars correspond to one
standard deviation.

##  4 Theoretical model

Here, we numerically analyze the transient behavior observed experimentally
and also provide an intuitive model that elucidates the underlying physics in
the simplest case of a fully blockaded medium. Afterward, we use this model to
understand the experimental results. We first briefly discuss our numerical
“spin model” technique. The features of the output field are expected to only
depend on the control field $\Omega_{c}$, on the optical depth $D$ of the
entire medium and on the optical depth $D_{b}$ per blockade radius, but not on
the total number of atoms or on the per-atom coupling efficiency of the probe
mode separately. Taking advantage of this observation, it is then possible to
investigate an artificial, quasi-one-dimensional system of a much smaller,
tractable number of atoms with increased coupling efficiency to the probe,
while maintaining the same $D$, $D_{b}$ of the experiment. In all subsequent
numerical simulations, we fix the optical depth per atom on the
$|g\rangle$-$|e\rangle$ transition (absent EIT) to be $D_{atom}\approx 0.36$
and vary the number of atoms to tune $D$. Furthermore, the dynamics of the
atomic internal degrees of freedom (“spins”) are encoded in an interacting
spin model, whose solution can be used to re-construct all field properties
via an input-output formalism. Details of this formalism can be found in Refs.
[37, 38] and its previous application toward modeling Rydberg EIT experiments
is described in Ref. [35]. Our numerical calculation truncates the Hilbert
space to two total atomic excitations (in the states $|e\rangle$ and/or
$|r\rangle$) and ignores quantum jumps, which is valid in the regime of weak
probe light [37]. Moreover, we will restrict ourselves to the case where the
probe field and the control field are resonant with the
$|g\rangle$-$|e\rangle$ and $|e\rangle$-$|r\rangle$ transitions, respectively.

Analytically, we consider the simplest model where similar transient behavior
can be observed, consisting of the limit where the probe input field
$E_{p}(t)$ approaches a square pulse with amplitude $E_{p_{0}}$ when turned on
and where the system is fully blockaded, so that two atoms cannot
simultaneously be in the Rydberg state $|r\rangle$. We consider a square pulse
long enough that all observables equilibrate to a steady-state value at some
point during the period where the pulse is turned on, and we assume that the
dephasing of the Rydberg state is negligible. In Fig. 5 we show a
representative plot versus time of the normalized output intensity
$\tilde{I}(t)=\mbox{$\langle\psi|$}E^{\dagger}(t)E(t)\mbox{$|\psi\rangle$}/E_{p_{0}}^{2}$,
the normalized “two-photon intensity”
$\tilde{G}^{(2)}=\mbox{$\langle\psi|$}E^{\dagger
2}(t)E^{2}(t)\mbox{$|\psi\rangle$}/E_{p_{0}}^{4}$ and the normalized second-
order correlation function at zero delay
$g^{(2)}(t,0)=\tilde{G}^{(2)}(t)/\tilde{I}^{2}(t)\equiv g^{(2)}(t)$. We can
clearly identify three separate regimes of behavior: the initial turn-on, the
steady state and the final turn-off. Below, we analyze each regime separately
in more detail.

Figure 5: Analysis of the dynamics of a weak square input pulse for a control
Rabi frequency of $\Omega_{c}=\Gamma/2$, $D\approx 3.6$ and a fully blockaded
system. (a) The normalized output intensity $\tilde{I}(t)$ as a function of
time (yellow) and the correlation function $\tilde{G}^{(2)}(t)$ (green). The
intensity of the input pulse $\tilde{E}_{p}$ (normalized to unity) is also
shown in black for reference. Three distinct regions in the dynamics can be
observed, turn-on (T-ON), steady state (SS) and turn-off (T-OFF). (b)
Normalized second-order correlation function $g^{(2)}(t)$ (red), output
intensity $\tilde{I}(t)$ (yellow) and $\tilde{G}^{(2)}(t)$ (green) in the
turn-on stage, with the input again shown for reference (black). (c)
$g^{(2)}(t)$ (red), $\tilde{I}(t)$ (yellow) and $\tilde{G}^{(2)}(t)$ (green)
in the turn-off stage. Figure 6: Schematic representation of the atomic level
scheme for a fully blockaded medium in the weak driving regime. Within the
single-excitation manifold, one atom can either be in the excited state
$|e\rangle$ or in the Rydberg state $|r\rangle$, as represented by the
subspaces $|\psi_{e,r}\rangle$, respectively. Similarly, in the two-excitation
manifold, either one atom can occupy the state $|e\rangle$ while another one
the state $|r\rangle$, or two atoms can simultaneously occupy the state
$|e\rangle$. The pathways by which the subspaces can become excited by either
the probe ($E_{p}$) or by the control field ($\Omega_{c}$) are indicated by
dashed arrows, while the free-space emission rates due to decay of the excited
state $|e\rangle$ by green wavy arrows.

###  4.1 Steady state

We begin with the steady-state properties, which have already been extensively
studied in Ref. [8]. First, we note that in the weak driving regime, the
output intensity is predominantly dictated by the single-excitation component
of the atomic system (since the one- and two-photon populations of the
incoming pulse scale like $E_{p}^{2}$ and $E_{p}^{4}$, respectively). As the
single-photon component does not experience Rydberg nonlinearities and sees
perfect transparency associated with EIT, it results that the normalized
intensity $\tilde{I}(t)\approx 1$ in the steady state (yellow curve in Fig.
5.a). On the other hand, the two-photon intensity $\tilde{G}^{(2)}$ is
efficiently attenuated due to the destruction of the EIT transparency
condition by the Rydberg blockade. For a fully blockaded medium, the steady-
state value of the output second-order correlation function was proved to be
decreasing with $D$ and increasing with $\Omega_{c}$ and to admit the
following approximate expression in the limit of high optical depth:
$g^{(2)}_{ss}\approx 4\frac{1+\left(\Omega_{c}/\Gamma\right)^{2}}{\pi
D}\exp\left[\frac{-D}{1+\left(\Omega_{c}/\Gamma\right)^{2}}\right]$ [8].

###  4.2 Turn-on

Once we switch the probe field suddenly on at $t=0$, the system evolves toward
the steady-state configuration. In Fig. 5.b, we show a representative plot of
$g^{(2)}(t)$ during this transient period. To further understand the behavior,
it is convenient to utilize an input-output relation, which formally allows
one to express the output field operator as a sum of the input field and the
field re-emitted by the atoms. In a quasi-one-dimensional propagation problem,
this takes the form [37, 38]:

$E=E_{p}(t)-i\sqrt{\frac{\Gamma_{1D}}{2}}\sum_{h=1}^{N}e^{ik_{p}z_{h}}\sigma_{ge}^{h}.$
(3)

Here, $\sigma_{ge}^{h}=\mbox{$|g\rangle$}\mbox{$\langle e_{h}|$}$ is the
$h$-th atomic lowering operator, where $|g\rangle$ is the ground state of the
system and $|e_{h}\rangle$ states for atom $h$ in the $|e\rangle$ state and
all the others in the ground state, $z_{h}$ is the position of atom $h$ and
$k_{p}\approx\omega_{eg}/c$ is the probe beam wavevector, where $c$ is the
speed of light in vacuum. We have normalized the field such that
$E^{\dagger}E$ has units of photon number per unit time and $\Gamma_{1D}$ is
the emission rate of atoms into the Gaussian mode defined by the input probe
beam. The spontaneous decay rate $\Gamma$ of the excited state is decomposed
into the sum of $\Gamma_{1D}$ and $\Gamma^{\prime}$, which is the decay into
noncollectable directions and represents the losses. The optical depth reads
$D=2N\mathrm{log}\left(\frac{\Gamma_{1D}+\Gamma^{\prime}}{\Gamma^{\prime}}\right)$,
and in our spin model simulations we fix $\Gamma_{1D}/\Gamma^{\prime}=0.2$, so
that the optical depth per atom results to be $D_{atom}\approx 0.36$. Since
the atoms are initially in the ground state and the atomic properties must
evolve continuously, one finds
$\sigma_{ge}^{h}\mbox{$|\psi(t=0^{+})\rangle$}=0$ at a time $t=0^{+}$
immediately after the turn-on of the pulse. From Eq. 3, this implies that the
output field is the same as the input field immediately after turn-on, and in
particular, $g^{(2)}(0^{+})=1$ reflects the coherent-state statistics of the
input field. This also causes the first intensity peak immediately after turn-
on, shown in Fig. 5.b, which can alternatively be thought as input light
components traveling through the ensemble too fast to interact with the atoms.

We now characterize the time scale $\tau_{0}$ over which $g^{(2)}(t)$ is
expected to approach its steady-state value, $g^{(2)}(\tau_{0})\sim
g^{(2)}_{ss}$. To do so, it is helpful to draw a schematic of the possible
atomic levels that can be excited in the weak-probe limit (up to two
excitations), as indicated in Fig. 6. Here, we denote $|\psi_{e,r}\rangle$ as
the manifold of states where only one atom is excited to states
$\mbox{$|e\rangle$},\mbox{$|r\rangle$}$, respectively. Similarly,
$|\psi_{er,ee}\rangle$ denote the manifold of states where one atom is in
$|e\rangle$ while another is in $|r\rangle$, or two atoms are in the state
$|e\rangle$ (recall that we consider a fully blockaded medium, so two atoms
cannot occupy state $|r\rangle$). We denote with dashed arrows the possible
paths by which these states can be excited by the probe and control fields. We
have also indicated by the wavy green arrows the rates of dissipation of these
states due to spontaneous emission into $4\pi$.

In the limit of a weak probe beam, the population of the two-excitation
manifold is sufficiently small that its back-action onto the evolution of the
single-excitation manifold can be neglected. Furthermore, since the coupling
of the probe and control fields between the ground state and the single-
excitation manifold simply behave under EIT in the linear optics regime, their
dynamics are straightforward to analyze. In particular, the square pulse
should propagate through the system at the reduced EIT group velocity $v_{gr}$
and its leading-edge should reach the end of the medium in a time
$\tau_{EIT}=L/v_{gr}=4D\Gamma^{\prime}/\Omega_{c}^{2}$. The single-excitation
manifold effectively acts as a source to populate the two-excitation one (see
Fig. 6), and the two-excitation manifold should reach the steady state within
a time $\sim\Gamma^{-1}$ of the single-excitation manifold doing so, due to
its natural dissipation. As this time is negligible, we then expect for
$\tau_{0}\approx\tau_{EIT}$.

Figure 7: Relationship between the turn-on transient time $\tau_{0}$ and the
EIT propagation time $\tau_{EIT}$ (black solid line). $\tau_{0}$ is defined as
the time after which
$\left|g^{(2)}(t)-g^{(2)}_{ss}\right|/g^{(2)}_{ss}<0.005$. Different values of
$\tau_{EIT}=4D\Gamma^{\prime}/\Omega_{c}^{2}$ are sampled by varying the
optical depth $D$ from $1.8$ to $9.1$ and the control field $\Omega_{c}$ from
$\Gamma/100$ to $\Gamma/2$. The values of $\Omega_{c}$ are indicated by the
different colors, as represented in the colorbar to the right.

In Fig. 7, we confirm this scaling numerically. In particular, we vary both
the control field amplitude $\Omega_{c}$ and optical depth $D$ over a large
range of values. We further define $\tau_{0}$ as the time in which the
numerically obtained fractional difference between the transient value of the
second-order correlation function and its steady-state value
$|g^{(2)}(\tau_{0})-g_{ss}^{(2)}|/g_{ss}^{(2)}$ drops below $0.005$. We see
that $\tau_{0}\approx\tau_{EIT}$ over the entire range of parameters studied,
and independent of the specific values of $D$ and $\Omega_{c}$ separately.
This is in agreement with the experimental data, where the duration of the
first transient is $\sim 400$ ns and we calculate $\tau_{EIT}\sim$ 450 ns,
similar to the time the output intensity takes to arrive at its steady-state
value (see Fig. 3).

###  4.3 Turn-off

In Fig. 5.c we show the output $\tilde{I}$, $\tilde{G}^{(2)}$ and $g^{(2)}$
when we switch the probe field suddenly off, starting from a steady-state
initial condition. The dynamics exhibits two notable features. First,
discontinuities can develop in the observables immediately after the shutoff.
Second, one sees that the second-order correlation function $g^{(2)}(t)$
approaches zero at sufficiently long times, indicating a stronger antibunching
than the one realizable in the steady state. We begin by analyzing the
dynamics in the vicinity of the shutoff of the probe, which we define to occur
at the time $\bar{t}$.

Using the input-output relation of Eq. (3), one sees that the output field
operator $E$ evolves discontinuously. In particular, the outgoing intensity at
a time $\bar{t}^{+}$ immediately following the shutoff, $\langle
E^{\dagger}(\bar{t}^{+})E(\bar{t}^{+})\rangle$, will only be due to purely
atomic emission and jump from its value at $\bar{t}^{-}$ immediately before
the shutoff. Furthermore, since this intensity is dominated by the single-
excitation component for weak driving, and as the $|e\rangle$-state component
of this manifold is unpopulated ($\mbox{$|\psi_{e}\rangle$}=0$) due to perfect
EIT, the atoms are not able to emit light instantaneously and one finds
$\tilde{I}(\bar{t}^{+})\approx 0$. Conversely, the normalized two-photon
intensity $\tilde{G}^{(2)}$ experiences a discontinuous increase. In steady-
state conditions and high optical depth, its value $\tilde{G}^{(2)}_{ss}\ll 1$
can be understood from Eq. 3 as nearly perfect destructive interference
between the incoming field and the field re-emitted by the atoms, since two
photons cannot be efficiently transmitted due to the Rydberg blockade.
Therefore, if the input field is instantly extinguished, the two-photon
outgoing intensity is due to a purely atomic emission which, for large $D$, is
almost equal in amplitude (but opposite in phase) to the incoming field. As a
consequence, one expects that $\tilde{G}^{(2)}(\bar{t}^{+})\rightarrow 1$ and
$g^{(2)}(\bar{t}^{+})\rightarrow\infty$ in the square pulse limit. For a
continuous switch off, a flash of bunched output light can still emerge if the
time scale of the shutoff is faster than the time needed by the atoms to
react, which is roughly $\sim\Gamma^{-1}$ [33].

Figure 8: Analysis of the single-excitation dynamics in the turn-off regime.
(a) Representative plot of the normalized output intensity $\tilde{I}(t)$ for
$D\approx 23$ and $\Omega_{c}=\Gamma/5$. The single-excitation characteristic
time $\tau_{I}$ is defined as the amount of time needed, following the turn-
off (occurring at time $\bar{t}\>$), for the normalized intensity to drop to
half of its steady-state value, $\tilde{I}(\bar{t}+\tau_{I})=0.5$. (b) Detail
of the short times dynamics. (c) Relationship between $\tau_{I}$ and the EIT
propagation time $\tau_{EIT}$ sampled by varying $D$ form $9.1$ to $27.3$ and
$\Omega_{c}$ from $\Gamma/100$ to $\Gamma/2$. The values of $\Omega_{c}$ are
indicated by the different colors, as represented in the colorbar to the
right.

The analysis of the transient dynamics following the instant shutoff of the
probe is made easier by noting that the single- and the two-excitation
components of the atomic system become decoupled when $E_{p}=0$ (see Fig. 6)
and we can thus investigate the dynamics in each manifold separately.
Immediately following the shutoff, any atomic excitation in the single-
excitation manifold takes the form of a spin wave in the Rydberg state
$\mbox{$|\psi_{1}(\bar{t}^{+})\rangle$}=\mbox{$|\psi_{1}^{ss}\rangle$}\sim\sum_{h}e^{ik_{p}z_{h}}\mbox{$|r_{h}\rangle$}$
due to the perfect EIT condition. This makes the subsequent dynamics
equivalent to the retrieval of a stored spin wave in an EIT-based quantum
memory. We will restrict ourselves to the regime of reasonably high $D$ and
small $\Omega_{c}$, such that the retrieved field takes a similar spatial form
as the spin wave itself (in contrast when $\Omega_{c}\geq\Gamma$, the outgoing
pulse can oscillate due to Rabi flopping dynamics between $|e\rangle$ and
$|r\rangle$). In Fig. 8.a, we show a representative plot of the output
intensity as a function of time following the shutoff, for $D\approx 23$ and
$\Omega_{c}/\Gamma=0.2$. Due to the finite bandwidth of the EIT transparency
window, the shape of the retrieved output intensity is smoother than the flat
rectangular shape of the steady-state spin wave itself. We define a
characteristic time $\tau_{I}$ as the amount of time following the shutoff for
the normalized outgoing intensity to reach half of its steady-state value,
$\tilde{I}(\bar{t}+\tau_{I})=0.5$. We expect that $\tau_{I}\approx\tau_{EIT}$,
i.e. $\tau_{I}$ should approximately coincide with the time needed to
propagate across the medium at the reduced EIT group velocity. In Fig. 8.c, we
plot the numerically extracted $\tau_{I}$ over a broad range of optical depths
and control field amplitudes and see a good agreement with the expected
result. For the experimental data, the calculated value of $\tau_{EIT}$ is
also in agreement with the decay time of the second transient (see Fig. 3).
Finally, we remark that at short times following the shutoff, the outgoing
intensity can exceed the steady-state intensity of the original square wave
input, $\tilde{I}(t)>1$, as illustrated in Fig. 8.b. Physically, at short
times, the control field can drive the original spin wave stored in the state
$|r\rangle$ into a “bright” spin wave
$\sim\sum_{h}e^{ik_{p}z_{h}}\mbox{$|e_{h}\rangle$}$. It is known that such a
spin wave experiences collectively enhanced or superradiant emission into the
forward direction [39, 40], at a rate $\sim\Gamma D/4$, which physically
arises from the constructive interference between the light emitted by
different atoms in this direction due to their relative phases $\sim
e^{ik_{p}z_{h}}$.

We now turn to the two-photon intensity. In Fig. 9.a we show a representative
plot of the output $\tilde{G}^{(2)}(t)$ at the turn-off. One sees that at
short times it rapidly decays to zero with a hump-like profile, while at long
times it exhibits slower oscillating features. We define a characteristic
temporal length of the short time behavior, $\tau_{II}$, as the amount of time
for the normalized two-photon intensity to reach half of its value immediately
after turn-off,
$\tilde{G}^{(2)}(\bar{t}+\tau_{II})=\tilde{G}^{(2)}(\bar{t})/2$. Numerically,
we find that $\tau_{II}$ scales like $1/D$, as shown in Fig. 9.b. This can be
understood as both the $er$ and the $ee$ components of the two-excitation
steady-state wave function $|\psi_{2}^{ss}\rangle$ are phase-matched and emit
in a collectively enhanced fashion immediately after turn-off. However, as
$|\psi_{2}^{ss}\rangle$ is not an eigenstate of the system, it can undergo
spatio-temporal evolution. At long times, the emitted two-photon intensity
will thus no longer be collectively enhanced but will decay at a rate
$\sim\Gamma$ comparable to the single-atom decay rate. This behavior can be
seen in Fig. 9.c, by comparing the envelope of $\tilde{G}^{(2)}$ at long times
with $e^{-\Gamma t}$.

Figure 9: Analysis of the double-excitation dynamics in the turn-off regime.
(a) Representative plot of the normalized output two-photon intensity
$\tilde{G}^{(2)}(t)$ for $D\approx 7.3$ and $\Omega_{c}=\Gamma/2$. The double-
excitation characteristic time $\tau_{II}$ is defined as the amount of time
needed, following the turn-off (occurring at time $\bar{t}\>$), for the
normalized two-photon intensity to drop to half value immediately after turn-
off, $\tilde{G}^{(2)}(\bar{t}_{II})=\tilde{G}^{(2)}(\bar{t})/2$. (b) The
double-excitation characteristic time $\tau_{II}$ sampled by varying $D$ from
$2.9$ to $9.1$ for $\Omega_{c}=\Gamma/20,\>\Gamma/4,$ and $\Gamma/2$. (c) Long
time profile of $\tilde{G}^{(2)}(t)$ and reference spontaneous decay
$e^{-\Gamma t}$ (dashed yellow line).

We can now finally understand why
$g^{(2)}(t)=\tilde{G}^{(2)}(t)/\tilde{I}(t)^{2}$ decreases below its steady-
state value at long times after the turn-off. In particular, with increasing
optical depth, the numerator $\tilde{G}^{(2)}(t)$ describing the two-
excitation component rapidly decays on a time scale $\tau_{II}\propto 1/D$ due
to collective enhancement (with a small residual component decaying at the
single-atom rate $\sim\Gamma$), while the denominator persists for a longer
time $\tau_{I}\propto D$ due to the slow retrieval dynamics of the single-
excitation component.

###  4.4 Simulation of the experiment

To reproduce the experimental results, we insert into our spin model the
optical depth, the spontaneous decay rate and the control field values given
in Sec. 2. We also add the motional dephasing $\gamma_{r}$ of the Rydberg
atoms. The numerical results thus obtained are in good agreement with the
experimental EIT transmission and with the output light pulse, shown in Fig.
1.d and 2.a, respectively. We remove the fully blockaded hypothesis and model
the finite blockade with an interaction potential of the form
$V_{r}=V_{0}\left(r_{b}/r\right)^{6}$, where $r_{b}$ is the blockade radius
and
$V_{0}=2\Omega_{c}^{2}\left[\Gamma_{1D}\left(2\Gamma^{\prime}+\Gamma_{1D}\right)\right]^{-1/2}$
is the single-atom bandwidth. By setting an optical depth per blockade radius
of $D_{b}\approx 0.9$, one finds $g^{(2)}(t)\approx 0.3$, which is in
agreement with the $\tau=0$ value of Fig. 2.b. In Fig. 3, the resultant pulse
analysis is shown and it is worth noticing how the measured $g_{\Delta
t}^{(2)}(t)$ tends towards the numerically predicted $g^{(2)}(t)$ as one
decreases the length of the detection window $\Delta t$.

##  5 Single-photon generation and storage

The turn-off transient, where the $g^{(2)}_{\Delta t}(t)$ value strongly
decreases, opens the way for possible applications related to narrowband
single-photon generation. For that purpose, we could cut the output pulses and
exploit the single photons arriving in the last part of the pulse. To analyze
this proposal, we select a temporal window at the end of the pulse where
looking for detection counts and coincidences. Then, we measure the
$g^{(2)}_{\Delta t}(t)$ values of the photons arriving inside this window and
their corresponding detection and generation probability. The generation
probability is inferred as $P_{g}=N_{1}/\eta_{det}$, where $N_{1}$ is the
number of counts per trial arriving to SPAD1 in the selected time window and
$\eta_{det}$ is the detection efficiency, including fiber coupling,
transmission through all the optical elements and single-photon detector
efficiency. Fig. 10 shows an example of the single-photon generation for a
time window $\Delta t$ = 500 ns and $t=1.2\ \mu$s (see inset plot). When
measuring the normalized coincidences for consecutive pulses (see Fig. 10.a),
we get a value of $g^{(2)}_{\Delta t}=0.218\pm 0.015$ significantly lower than
for $\Delta t=1.6\mu s$. Moreover, the time resolved measurement within one
pulse (see Fig. 10.b) shows that the values of $g^{(2)}(\tau)$ remain low for
the full window, showing the generation of a localized single photon.

Figure 10: Single-photon generation: example of single-photon generation with
$\Delta t$ = 500 ns. (a) Normalized coincidence counts between different
trials, leading to a value of $g^{(2)}_{\Delta t}=0.218\pm 0.015$ for
$n_{trial}=0$. (b) Time-resolved measurement of $g^{(2)}(\tau)$ leading to a
zero-delay value of $g^{(2)}(0)=0.15\pm 0.03$. Error bars correspond to one
standard deviation. Figure 11: Single-photon generation. (a) Temporal
distribution of a Gaussian pulse after propagation through EIT (in blue) and
after storage (in green) and that of a square pulse after propagation through
EIT (in orange). The temporal window is varied in plots (b) and (c) for the
slow light case, as indicated by the arrow. (b) $g^{(2)}_{\Delta t}(t)$ as a
function of the number of counts per trial in SPAD1 (bottom axis) and the
probability to have a photon at the output of the cloud (upper axis), for an
input Gaussian pulse (in blue), the previously shown square pulse (in orange)
and after storing for 500 ns (in green). The generation probability is
increased by taking the same final time of the temporal window but changing
the starting time to increase $\Delta t$, as we can see in plot (a). (c)
$g^{(2)}_{\Delta t}(t)$ as a function of the inverse of the temporal window,
for the same cases as before. Vertical error bars correspond to one standard
deviation.

To further study the effect of the temporal window, we then vary $\Delta t$,
while keeping the final time $t+\Delta t=1.7\ \mu$s fixed. Fig. 11 shows the
results for different duration of the temporal window, for Gaussian and square
input pulses. When we increase the time window, the probability to have a
detection count increases, but at the expense of reducing the quality of the
single photons. For the previously studied square pulse, $g^{(2)}_{\Delta
t}(t)=0.48\pm 0.01$ is obtained for $\Delta t=0.68\ \mu$s and a generation
probability of $0.145\pm 0.014$, while $g^{(2)}_{\Delta t}(t)=0.147\pm 0.017$
is obtained for $\Delta t=0.45\ \mu$s and a generation probability of
$0.046\pm 0.004$. We compare the results for the square pulse with the case of
a Gaussian input pulse. We see that the Gaussian pulse leads to a higher value
of $g^{(2)}_{\Delta t}(t)$, for all generation efficiencies, compared to the
square pulse. This confirms that a fast shutoff of the input pulse is
beneficial for the production of single photons.

We also compare the results with the strategy consisting of storing the input
Gaussian pulse in the Rydberg state [19, 41, 42, 43]. This can be made by
switching off the control beam ($\Omega_{c}\rightarrow 0$) when polaritons are
traveling inside the medium. After a storage time of $t_{S}$ = 500 ns, we can
retrieve the output pulse by switching on again the control beam. This method
allows for an increase in the quantum character of the output pulse without
changing the experimental conditions. Due to our limited $D$, the entire input
pulse cannot be compressed inside the medium and only a part of the pulse can
be stored as Rydberg excitations, importantly reducing the output efficiency.
In Fig. 11, we show the $g^{(2)}_{\Delta t}$ obtained after storage and
retrieval (in green), for $\Delta t$ taking into account the whole retrieved
pulse (see Fig. 11.a). As we can observe, the $g^{(2)}_{\Delta t}$ in the
storage case is similar to the obtained in the turn-off transient of the
transmitted square pulse, for the same generation probability (see green and
orange points of Fig. 11.b). However, for lower quality of the output photons,
i.e higher values of $g^{(2)}_{\Delta t}$, the generation probability in the
transient case importantly increases.

Finally, in Fig. 11.c we plot $g^{(2)}_{\Delta t}$ as function of the inverse
of $\Delta t$, which is proportional to the photon bandwidth. Values of
$g^{(2)}(\Delta t)$ $<$ 0.2 can be achieved for $1/\Delta t\sim 2$ MHz,
showing that high quality, narrowband single photons can be generated with
this technique.

As shown in Sec. 4, increasing the optical depth and reducing the Rabi
frequency of the control field results in a better separation between the two-
photon dynamics and the single-photon one and therefore leads to a smaller
value of $g^{(2)}_{\Delta t}$ in the turn-off transient. However, the single-
photon generation efficiency is currently affected by the low value of the
transmission in the EIT transparency window, which is likely limited by the
decoherence rate of the $|g\rangle$ to $|r\rangle$ transition, which also
enforces a lower bound in the choice of $\Omega_{c}$. We expect that reducing
the laser linewidths (currently around 300 kHz) by active stabilization on an
optical cavity would bring to a significant increase in the EIT transmission
and also make lower choices of the Rabi frequency of the control laser
possible. Moreover, a larger value of $D_{b}$ would allow a more efficient
compression of the pulse within a blockade radius. The current values of
generation efficiency achieved (around 10 $\%$) are comparable with the values
reported in single-photon generation experiments using off-resonant (Raman)
excitation to the Rydberg states with Rydberg cold atomic ensembles [18]. It
is informative to compare this efficiency with techniques of probabilistic
single-photon generations with atomic ensembles, such as e.g. the Duan-Lukin-
Cirac-Zoller scheme [22]. This scheme generates probabilistically photon pairs
in a two-mode squeezed state with a probability per trial $p$, where one of
the photons is stored as a collective atomic spin excitation in the ensemble.
Upon detection of the first photon which provides a heralding signal, this
collective spin excitation can then be efficiently transferred into a single
photon in a well-defined spatio-temporal mode with an efficiency $\eta_{R}$.
The probability to generate a single photon per trial is therefore given by
$P_{DLCZ}=p\eta_{D}\eta_{R}$ where $\eta_{D}$ is the probability to detect the
first photon. For a perfect two-mode squeezed state and for $p\ll 1$, the
second-order autocorrelation of the retrieved photon is $g^{(2)}(0)$=4$p$. In
the best-case scenario (i.e. with unity detection and read-out efficiency), a
DLCZ source could generate a photon with $g^{(2)}(0)$=0.1 with a probability
of $P_{DLCZ}=0.025$ per trial. In practice, with finite detection and read-out
efficiencies, this value will be even lower. Therefore, even though the
single-photon generation efficiency demonstrated in this paper is quite modest
and could still be largely improved, it compare favorably to probabilistic
schemes.

##  6 Conclusions

In this paper, we investigated the propagation of weak coherent pulses in a
cold atomic ensemble in the regime of Rydberg electromagnetically induced
transparency. We found experimentally that the second-order correlation
function of the output pulse depends on the time throughout the pulse and
strongly varying during the transients of the pulse. In particular, the value
of $g^{(2)}_{\Delta t}$ strongly decreases towards the end of the traveling
pulse. Through a spin model analysis, we were able to quantitatively predict
the measured pulse dynamics, both at the linear and nonlinear level, and to
provide a qualitative explanation, in the test scenario of a perfectly square
pulse propagating in a fully blockaded medium. Taking advantage of this
behavior, we explored the possibility of using this effect to generate
localized single photons and showed that it has better efficiency than a
probabilistic DLCZ-like source.

##  7 Acknowledgment

This project received funding from the Government of Spain
(PID2019-106850RB-I00; Severo Ochoa CEX2019-000910-S), Fundació Cellex,
Fundació Mir-Puig, Generalitat de Catalunya (CERCA, AGAUR), Gordon and Betty
Moore Foundation through Grant No. GBMF7446 to H. d. R. and from the European
Union’s Horizon 2020 research and innovation program under Grant Agreement No.
899275 (DAALI).

DEC acknowledges support from the European Union’s Horizon 2020 research and
innovation programme, under FET-Open grant agreement No 899275 (DAALI), AEI
Europa Excelencia program (EUR2020-112155, project ENHANCE), and Quantum
Flagship project 820445 (QIA), MINECO Severo Ochoa program CEX2019-000910-S,
Generalitat de Catalunya through the CERCA program, Fundació Privada Cellex,
Fundació Mir-Puig, Plan Nacional Grant ALIQS (funded by MCIU, AEI, and FEDER),
and Secretaria d’Universitats i Recerca del Departament d’Empresa i
Coneixement de la Generalitat de Catalunya, co-funded by the European Union
Regional Development Fund within the ERDF Operational Program of Catalunya
(project QuantumCat, ref. 001-P-001644)

##  8 References

## References

  * [1] Saffman M, Walker T G and Mølmer K 2010 Rev. Mod. Phys. 82 2313–2363 URL https://link.aps.org/doi/10.1103/RevModPhys.82.2313
  * [2] Wilk T, Gaetan A, Evellin C, Wolters J, Miroshnychenko Y, Grangier P and Browaeys A 2010 Phys. Rev. Lett. 104 010502 URL https://link.aps.org/doi/10.1103/PhysRevLett.104.010502
  * [3] Levine H, Keesling A, Omran A, Bernien H, Schwartz S, Zibrov A S, Endres M, Greiner M, Vuletić V and Lukin M D 2018 Phys. Rev. Lett. 121 123603 URL https://link.aps.org/doi/10.1103/PhysRevLett.121.123603
  * [4] Isenhower L, Urban E, Zhang X L, Gill A T, Henage T, Johnson T A, Walker T G and Saffman M 2010 Phys. Rev. Lett. 104 010503 URL https://link.aps.org/doi/10.1103/PhysRevLett.104.010503
  * [5] Maller K M, Lichtman M T, Xia T, Sun Y, Piotrowicz M J, Carr A W, Isenhower L and Saffman M 2015 Phys. Rev. A 92 022336 URL https://link.aps.org/doi/10.1103/PhysRevA.92.022336
  * [6] Bernien H, Schwartz S, Keesling A, Levine H, Omran A, Pichler H, Choi S, Zibrov A S, Endres M, Greiner M, Vuletić V and Lukin M D 2017 Nature 551 579–584 ISSN 1476-4687 URL https://doi.org/10.1038/nature24622
  * [7] Barredo D, Lienhard V, de Léséleuc S, Lahaye T and Browaeys A 2018 Nature 561 79–82 ISSN 1476-4687 URL https://doi.org/10.1038/s41586-018-0450-2
  * [8] Peyronel T, Firstenberg O, Liang Q Y, Hofferberth S, Gorshkov A V, Pohl T, Lukin M D and Vuletić V 2012 Nature 488 57–60 ISSN 0028-0836
  * [9] Firstenberg O, Peyronel T, Liang Q Y, Gorshkov A V, Lukin M D and Vuletić V 2013 Nature 502 71–5 ISSN 1476-4687 URL http://www.ncbi.nlm.nih.gov/pubmed/24067613
  * [10] Paris-Mandoki A, Braun C, Kumlin J, Tresp C, Mirgorodskiy I, Christaller F, Büchler H P and Hofferberth S 2017 Phys. Rev. X 7 041010 URL https://link.aps.org/doi/10.1103/PhysRevX.7.041010
  * [11] Chang D E, Vuletić V and Lukin M D 2014 Nature Photonics 8 685–694 ISSN 1749-4893 URL https://doi.org/10.1038/nphoton.2014.192
  * [12] Firstenberg O, Adams C S and Hofferberth S 2016 J. Phys. B: At. Mol. Opt. Phys. 49 152003 ISSN 1361-6455 URL http://dx.doi.org/10.1088/0953-4075/49/15/152003
  * [13] Tiarks D, Baur S, Schneider K, Dürr S and Rempe G 2014 Phys. Rev. Lett. 113 1–5 ISSN 10797114
  * [14] Gorniaczyk H, Tresp C, Schmidt J, Fedder H and Hofferberth S 2014 Phys. Rev. Lett. 113 1–5 ISSN 10797114
  * [15] Baur S, Tiarks D, Rempe G and Dürr S 2014 Phys. Rev. Lett. 112 1–5 ISSN 00319007
  * [16] Busche H, Huillery P, Ball S W, Ilieva T, Jones M P and Adams C S 2017 Nature Physics 13 655–658 ISSN 1745-2481 URL https://doi.org/10.1038/nphys4058
  * [17] Tiarks D, Schmidt-Eberle S, Stolz T, Rempe G and Dürr S 2019 Nature Physics 15 124–126 ISSN 1745-2481 URL https://doi.org/10.1038/s41567-018-0313-7
  * [18] Dudin Y O and Kuzmich A 2012 Science 336 887–889 ISSN 0036-8075
  * [19] Maxwell D, Szwer D J, Paredes-Barato D, Busche H, Pritchard J D, Gauguet a, Weatherill K J, Jones M P a and Adams C S 2013 Phys. Rev. Lett. 110 1–7 ISSN 00319007
  * [20] Li L and Kuzmich A 2016 Nature Communications 7 13618 ISSN 2041-1723 URL https://doi.org/10.1038/ncomms13618
  * [21] Li J, Zhou M T, Jing B, Wang X J, Yang S J, Jiang X, Molmer K, Bao X H and Pan J W 2016 Phys. Rev. Lett. 117 180501 URL https://link.aps.org/doi/10.1103/PhysRevLett.117.180501
  * [22] Duan L M, Lukin M D, Cirac J I and Zoller P 2001 Nature 414 413–418 ISSN 0028-0836
  * [23] Mohapatra A K, Jackson T R and Adams C S 2007 Phys. Rev. Lett. 98 113003 URL https://link.aps.org/doi/10.1103/PhysRevLett.98.113003
  * [24] Petrosyan D, Otterbach J and Fleischhauer M 2011 Phys. Rev. Lett. 107 213601 URL https://link.aps.org/doi/10.1103/PhysRevLett.107.213601
  * [25] Gorshkov A V, Nath R and Pohl T 2013 Phys. Rev. Lett. 110 153601 URL https://link.aps.org/doi/10.1103/PhysRevLett.110.153601
  * [26] Moos M, Höning M, Unanyan R and Fleischhauer M 2015 Phys. Rev. A 92(5) 053846 URL https://link.aps.org/doi/10.1103/PhysRevA.92.053846
  * [27] Zeuthen E, Gullans M J, Maghrebi M F and Gorshkov A V 2017 Phys. Rev. Lett. 119 043602 URL https://link.aps.org/doi/10.1103/PhysRevLett.119.043602
  * [28] Sangouard N and Zbinden H 2012 Journal of Modern Optics 59 1458–1464
  * [29] Sangouard N, Simon C, Minář J c v, Zbinden H, de Riedmatten H and Gisin N 2007 Phys. Rev. A 76(5) 050301 URL https://link.aps.org/doi/10.1103/PhysRevA.76.050301
  * [30] Knill E, Laflamme R and Milburn G J 2001 nature 409 46–52
  * [31] Craddock A N, Hannegan J, Ornelas-Huerta D P, Siverns J D, Hachtel A J, Goldschmidt E A, Porto J V, Quraishi Q and Rolston S L 2019 Phys. Rev. Lett. 123 213601 URL https://link.aps.org/doi/10.1103/PhysRevLett.123.213601
  * [32] Maxwell D, Szwer D J, Paredes-Barato D, Busche H, Pritchard J D, Gauguet A, Jones M P A and Adams C S 2014 Phys. Rev. A 89 043827 URL https://link.aps.org/doi/10.1103/PhysRevA.89.043827
  * [33] Moehl C, Spong N L R, Jiao Y, So C, Ilieva T, Weidemueller M and Adams C S 2020 J. Phys. B: At. Mol. Opt. Phys. 53 084005 ISSN 1361-6455 URL http://dx.doi.org/10.1088/1361-6455/ab728f
  * [34] Steck D A 2001 Rubidium 87 d line data
  * [35] Bienias P, Douglas J, Paris-Mandoki A, Titum P, Mirgorodskiy I, Tresp C, Zeuthen E, Gullans M J, Manzoni M, Hofferberth S, Chang D and Gorshkov A V 2020 PRRESEARCH 2 033049 URL https://link.aps.org/doi/10.1103/PhysRevResearch.2.033049
  * [36] Ornelas-Huerta D P, Craddock A N, Goldschmidt E A, Hachtel A J, Wang Y, Bienias P, Gorshkov A V, Rolston S L and Porto J V 2020 Optica 7 813–819
  * [37] Caneva T, Manzoni M T, Shi T, Douglas J S, Cirac J I and Chang D E 2015 New Journal of Physics 17 113001 ISSN 1367-2630 URL http://dx.doi.org/10.1088/1367-2630/17/11/113001
  * [38] Manzoni M T, Chang D E and Douglas J S 2017 Nature Communications 8 1743 ISSN 2041-1723 URL https://doi.org/10.1038/s41467-017-01416-4
  * [39] Bromley S L, Zhu B, Bishof M, Zhang X, Bothwell T, Schachenmayer J, Nicholson T L, Kaiser R, Yelin S F, Lukin M D, Rey A M and Ye J 2016 Nature Communications 7 11039 ISSN 2041-1723 URL https://doi.org/10.1038/ncomms11039
  * [40] Svidzinsky A A, Chang J T and Scully M O 2010 Phys. Rev. A 81 053821 URL https://link.aps.org/doi/10.1103/PhysRevA.81.053821
  * [41] Distante E, Padrón-Brito A, Cristiani M, Paredes-Barato D and de Riedmatten H 2016 Phys. Rev. Lett. 117 113001 URL https://link.aps.org/doi/10.1103/PhysRevLett.117.113001
  * [42] Distante E, Farrera P, Padrón-Brito A, Paredes-Barato D, Heinze G and de Riedmatten H 2017 Nature Communications 8 14072 ISSN 2041-1723 URL https://doi.org/10.1038/ncomms14072
  * [43] Schmidt-Eberle S, Stolz T, Rempe G and Duerr S 2020 Phys. Rev. A 101 013421 URL https://link.aps.org/doi/10.1103/PhysRevA.101.013421