On-demand indistinguishable single photons from an efficient and pure source based on a Rydberg ensemble

Single photons coupled to atomic systems have shown to be a promising platform for developing quantum technologies. Yet a bright on-demand, highly pure and highly indistinguishable single-photon source compatible with atomic platforms is lacking. In this work, we demonstrate such a source based on a strongly interacting Rydberg system. The large optical nonlinearities in a blockaded Rydberg ensemble convert coherent light into a single-collective excitation that can be coherently retrieved as a quantum field. We observe a single-transverse-mode efficiency up to 0.18(2), $g^{(2)}=2.0(1.5)\times10^{-4}$, and indistinguishability of 0.982(7), making this system promising for scalable quantum information applications. Accounting for losses, we infer a generation probability up to 0.40(4). Furthermore, we investigate the effects of contaminant Rydberg excitations on the source efficiency. Finally, we introduce metrics to benchmark the performance of on-demand single-photon sources.


INTRODUCTION
Engineering single-photon sources with high efficiency, purity, and indistinguishability is a longstanding goal for applications such as linear optical quantum computation [1], boson sampling [2], quantum networks [3] and quantum metrology [4]. Atomic systems have shown significant progress towards quantum light-matter interfaces, including efficient quantum memories [5], quantum networks [6], high-fidelity light-matter entanglement [7], atomic gates [8], and quantum simulators [9]. Atomic platforms require spectrally matched single photons that can coherently couple with atomic processors, provided with high-efficiency generation, purity, and indistinguishability.
We describe here an efficient single-photon source based on collective excitation and de-excitation of a cold, trapped ensemble of atoms through a highly excited Rydberg state [14,15,30]. During two-photon excitation from the ground to the Rydberg state via an intermediate state [see Fig. 1(a)], long-range van der Waals interactions suppress multiple Rydberg excitations within a blockade radius, r b [31]. The resulting single, col-lective atomic excitation is coherently shared among N atoms as a spin wave [30]. Due to the collective nature of the excitation, if the initial phase coherence of the spin wave is maintained, the subsequent coupling of the Rydberg state to the intermediate state can efficiently map the excitation onto a single photon in a well-defined mode [32]. Our system produces single photons with repetition rates up to 400 kHz, a generation probability up to 0.40 (4), g (2) = 2.0(1.5) × 10 −4 , and indistinguishability of 0.982 (7). We model the write and retrieval process, including the measured spin-wave dephasing rate. We identify long-lived-contaminant Rydberg states [33] as a limiting factor on the source efficiency for increasing production rates.
Given the requirements for most quantum information applications, the single-mode efficiency, rate, and quality of single-photon sources are of key importance since successful scaling of these systems involves detection of multiple identical photons. Thus, we introduce metrics to describe the probability, rate, and fidelity of producing a single photon in a single-mode, which includes the contributions from the commonly used metrics: overall collection efficiency, purity, indistinguishability, and repetition rate [34].

EXPERIMENTAL APPARATUS AND PROCEDURE
We start the experiment with a magneto-optical trap of 87 Rb atoms and further laser cool the atoms with a Λgray molasses down to ≈ 10 µK. We load the atoms into a 1003-nm wavelength optical dipole trap. To write the   Figure 1(a). The probe beam coupling |g to |e is focused into the atom cloud with a waist of ≈ 3.3 µm, with a Rabi frequency Ω p ≈ 2π × 1 MHz. The counter-propagating control beam coupling |e to |r has a larger, ≈ 19 µm waist and peak Rabi frequency Ω c ≈ 2π × 7 MHz.
The van der Waals coefficient of the Rydberg state 139S 1/2 is C 6 ≈ −2π × 2.5 × 10 6 GHz µm 6 [35], which results in a blockade radius r b ≈ 60 µm during the spinwave writing. Since r b is larger than the probe beam waist and the atomic cloud extension in the propaga-  (2) (τ ) with 5 µs cycle. (b) Normalized coincidences for g (2) (τ ) around τ = 0, grey line represents the background coincidences with 20-ns bins. The shape of this profile arises from the convolution of the photon pulse shape with a constant background within the gate window, and the pedestal asymmetry is because the background rate is not the same for each channel. All data shown were taken with 60% duty cycle. tion direction, σ z ≈ 27 µm, the excitation volume is blockaded. The effective two-photon Rabi frequency, Ω 2ph = ΩpΩc 2∆p is enhanced by a factor √ N ≈ 20 from the N atoms participating in the collective excitation [30,36].
After a spin-wave storage time t s > 350 ns [see Fig. 1(d)], we turn back on the control field with a detuning ∆ c ≈ 2π × 7 MHz that maximizes the retrieval efficiency of the spin wave into a single photon. We can vary the repetition rate of the write-retrieval sequence up to 400 kHz, with interrogation times up to 600 ms (0.6 duty cycle) before we need to reload the optical dipole trap.

SINGLE-PHOTON SOURCE PURITY AND INDISTINGUISHABILITY
We use Hanbury Brown-Twiss and Hong-Ou-Mandel interferometers to characterize the purity and indistinguishability of our single photons [see Fig. 1(b)]. We define the purity of our single-photon source as 1 − g (2) (0), where g (2) (τ ) is the second-order autocorrelation function. We apply a 1.4 µs long software gate window, containing more than 99.9% of the pulse [see Fig. 1(c)]. Coincidences at zero time delay are substantially suppressed, as shown in Figure 2(a), with strong antibunching g (2) raw (0) = 0.0145(2), integrating the area around τ = 0 and without background subtraction. The background coincidence rate is dominated by coincidences involving photon events with background counts unrelated to the single-photon generation, coming from detector dark counts and room light leakage. The independently measured background rate, photon shape, and photon rate are constant throughout each experimental run, from which we determine that the accidental coincidences contribute to g (2) back (0) = 0.0143. The gray curve in Figure 2(b) shows the background coincidence profile within the gate window (see [37] for details). After background subtraction, our single-photon source has g (2) We use a Hong-Ou-Mandel interferometer (HOM) to measure the photon indistinguishability. We implement a fiber-based 4.92 µs delay in one arm to temporally overlap adjacently produced photons. Additionally, there is a polarizing beam splitter (PBS) at the output of each fiber to account for any polarization rotation due to the fibers. At the exit of the short arm, there is a halfwave plate (HWP) to rotate the polarization and control the degree of distinguishability of the photons. Figure 3(a) shows the normalized coincidences for orthogonal and parallel polarizations. Integrating the number of coincidences in a window around τ = 0 for the two cases, we measure a raw HOM interference visibility V raw = 1 − C /C ⊥ = 0.894 (6). Accounting for the accidental coincidences with background events and the slight differences in the transmission and reflection coefficients of our combining beamsplitter gives a mode overlap of 0.982(7) (see [37]).

SOURCE EFFICIENCY
We measure a peak probability of 0.18 (2) to generate a single photon into a single-mode fiber after polarization filtering and averaged for a 20% duty cycle. Accounting for optical losses and assuming that the single-photon has the same spatial mode as the 780-nm-write beam, we estimate a generation probability of 0.40(4) immediately after the atomic ensemble. The average probabilities go down to 0.14(1) and 0.31(1), respectively for a 60% duty cycle.
We calculate P th = η w η s η r as a product of the writing, η w , storage, η s , and retrieval, η r , efficiencies to estimate the theoretical probability of generating a photon. Referring the reader to the Supplement [37] for the details of the theoretical analysis, we summarize it here only briefly. We simulate the writing of the spin wave using a Lindblad master equation to estimate the writing efficiency and the storage efficiency. We calculate the retrieval efficiency using the optical Maxwell-Bloch equations with the formalism in Ref. [38]. Using independently measured experimental values as input parameters, we obtain a theoretical prediction of P th ≈ 0.42(3) (see Supplement [37]). This value is consistent with the measured generation probability for the longest pulsing periods, t p . We observed that the average photon production efficiency decreased at higher repetition rates, as shown in Figure 4(a). (Here the photon probability is determined immediately after the atom cloud by accounting for independently measured optical losses.) The initial pulse in a pulse series had higher efficiency, however, the efficiency of subsequent pulses decreased exponentially to the steady-state value on a ≈ 60 µs time scale [see These observations are consistent with the creation of contaminant atoms in other long-lived Rydberg states that are not removed by the retrieval field. These states interact strongly with the target Rydberg state, affecting subsequent writing events. Similar contaminant states have been observed in previous experiments [33,39,40], and have been analyzed extensively [41][42][43][44]. Once a contaminant is in the medium, it disables the writing of a spin wave for the later pulses. However, contaminants have a finite lifetime in the medium, therefore, the pho-ton generation probability decreases for shorter pulse periods.
We use a simple model to capture the effect of contaminants on photon production (see [37] for details). We assume that for any given pulse, there is a probability P c of creating a contaminant. If the contaminant state has a lifetime τ c , then the probability P n of having a contaminant in the n-th pulse of a pulse series with period t p is For τ c t p , the average contaminant probability as n → ∞ can be significant, even if P c is small. The probability P g (n) of successfully generating a single-photon on the n-th pulse in the presence of a contaminant is decreased according to P g (n) = P max (1 − P n ), where P max is the probability of photon generation in the absence of contaminants. The steady state efficiency is given by P g (n → ∞). Fitting this equation to pulse sequence data as shown in Fig. 4(b), we determine P c = 1.9(3) × 10 −2 , and τ c = 65(8) µs, which is in good agreement with the data in Fig. 4(a).
We find that P c increases linearly with atomic density ρ [see Fig. 4(c)], which suggests that the source of contaminants is ground-Rydberg interactions. For high principal quantum number, n, collisionally produced contaminants were identified in Ref [42] to be Rydberg states with principal quantum number n − 4 and quantum angular momentum l > 2. Furthermore, we find that P c increases with storage time t s at a rate ≈ 3 × 10 −2 µs −1 , which gives a contaminant generation time-scale of ≈ 33 µs for a density ≈ 4 × 10 11 cm −3 . Contaminants are not a fundamental limitation since strong electric field pulses between writing pulses could be used to remove them.
We also note that for interrogation times longer than 100 ms, other effects such as heating and atom depolarization from rescattering become more significant, further reducing the photon generation for shorter t p . However, these effects can be mitigated by detuning farther from the intermediate state.

SINGLE-MODE EFFICIENCY, RATE AND FIDELITY
There are many metrics used to quantify the various properties of single-photon sources. Optical quantum information schemes are susceptible to errors if they are not implemented with highly pure and indistinguishable single photons. In addition, scaling up quantum information protocols needs high generation efficiency, since any inefficiency will lead to an exponential decrease of the success probability with system size. Finally, the rate of single-photon production provides a limitation on the practicality of any protocol. To that end, we define three metrics that quantify these properties: F, the single-photon fidelity, which is the fraction of emission that consists of a single photon in a single spectral, temporal, polarization, and spatial mode; η, the probability of generating a single photon in the desired mode; and R, the brightness, which the rate of photon production in the desired mode.
Assuming that the probability of multi-photon events greater than two is negligible, the only outcomes from a source are: single photons in the desired mode with probability η, single photons in an undesirable mode with probability P 1 , two photons with probability P 2 , and null events with probability P 0 . Experimentally, we measure the following quantities: the overall emission efficiency, P = 1 − P 0 ; the HOM visibility, V; and the measure of the single-photon purity, g (2) . These are given by:  [45], multiplexed-heraldedsingle-photon source (MUX-HSPS) [46,47] and quantum dots (QD) [48][49][50][51][52]. Atomic systems considered are single atoms in free-space [53,54], atoms in cavities [55][56][57][58], and the Rydberg ensemble studied in this work (indicated in the purple line) accounting for the effect of different repetition rates for a duty cycle of 0.6. (For details on these sources, see tables in [37]). (a) Fidelity vs. single-mode efficiency. (b) Brightness vs. single-mode efficiency.
where we have assume that the visibility V is compensated for multi-photon events [37], and that these measurements are taken with standard non-number resolving photon counting detectors.
Solving the system of equation for η to second order in g (2) , we get the single-mode efficiency η: (2) .
We report the source brightness as R = R eff η, where R eff , is the clock rate weighted by the experimental duty cycle. Apart from source brightness, the rate at which undesirable emission is produced also matters for applications. We characterize this rate by the fidelity, which is the fraction of collected emission that is made up of single photons in the correct mode. In Fig. 5 we show η, F, and R for a sample of different single-photon sources. Narrow bandwidth sources naturally compatible with coherent atomic systems are indicated with filled symbols.

CONCLUSION
By using the quantum nonlinearities of strongly interacting Rydberg states in a cold atomic ensemble, we demonstrated a single-photon source, operating with a 60% duty cycle, single-mode efficiency η = 0.139(5), a single-mode brightness of R = 840(70) s −1 , and singlemode fidelity F = 0.982 (7), this fidelity is the highest reported to our knowledge for an atomic-based source. Furthermore, we investigated the limitations of our current setup arising from nearby long-lived contaminant states.
Implementing feasible improvements to the current experiment we estimate that we can achieve up to η ≈ 0.4 and moreover, ionizing pulses after each write-retrieval pulse to remove atoms in pollutant states may increase the brightness up to R ≈ 1.2 × 10 5 s −1 without decreasing the duty cycle or the fidelity (see [37] for details). The efficiency could be further improved if the ensemble were coupled to a cavity [59]. Given their high efficiency, brightness, and fidelity, we have shown that single-photon sources based on Rydberg-atomic ensembles provide a promising platform for scalable quantum photonics. Furthermore, they are inherently compatible with narrow-bandwidth atomic platforms that have shown significant progress towards quantum information applications. We are grateful to Mary Lyon for her significant contributions to the design and construction of the apparatus and Patrick Banner for his contributions to data collection. We also want to thank Luis A. Orozco for fruitful discussions.

Detailed experimental configuration
All the experiments are carried out with ≈ 10 4 87 Rb atoms trapped in a three-beam-crossed optical dipole trap with 1003-nm wavelength. Two of the beams form a ≈ ±11 • with respect to the x-axis (along the probe direction), while a third elliptical shaped beam travels in the y-axis, with all beams in the same (x-y) plane. The relative powers of the dipole beams are adjusted so that the RMS dimensions of the trapped atomic cloud are σ r = 20 µm in the radial direction and σ x = 27 µm.
The initial trapping and cooling take place in a magneto-optical trap (MOT). For most experiments, we load for 250 ms; if we need to adjust the atomic medium optical density (OD), we change the loading time, ranging from 50 ms to 1500 ms (with OD up to ≈16). Afterward, we perform a compressed-MOT stage by ramping-up the magnetic field gradient, while at the same time slowly ramping-up the dipole trap power.
We further cool the atoms to ≈ 10 µK using a gray molasses [S1]. Next, we optically pump the atoms into the 5S 1/2 , F = 2, m F = 2 state, using σ + polarized light blue-detuned from the F = 2 to F = 2, D1 transition. We then couple the ground and Rydberg state with a two-photon transition. A 780-nm weak-probe field addresses the transition from the ground state, 5S 1/2 , F = 2, m F = 2 to the intermediate state, 5P 3/2 , F = 3, m F = 3 ; a strong-control field addresses the transition from the intermediate state to the Rydberg state, 139S 1/2 , J = 1/2, m J = 1/2 with a wavelength of 479 nm.
Both the probe and control lasers are frequency stabilized via an ultra-low expansion (ULE) cavity with a linewidth < 10 kHz. We use probe light that has been transmitted and filtered by the ULE cavity to reduce phase noise [S2], .
There are eight electrodes in vacuum that allow for control of local electric fields. With this configuration, we cancel DC-Stark shifts to tens of kHz level in all three directions, shifts that would otherwise tune the Rydberg state out of resonance due to the large polarizability of the 139S state, α 139S ≈ 61 GHz/(V/cm) 2 [S3].
The axial RMS of the atomic cloud, σ x ≈ 27 µm is smaller than the blockade radius, r b ≈ 60 µm to suppress the creation of multiple Rydberg atoms. Additionally, we focus the probe beam down to a 1/e 2 waist of w p ≈ 3.3 µm to ensure the system is effectively uni-dimensional (w p ≤ r b ). The control beam is counter-propagating to the probe and focused to a beam waist of w c ≈ 19 µm. The larger beam waist provides an approximately uniform control field across the probe area. After exiting the chamber, the probe light passes through a polarization beam splitter (PBS), and a set of bandpass filters centered at 780-nm, a narrow 1-nm bandwidth filter (Alluxa 780-1 OD6[? ]), and a broader 12.5-nm bandwidth filter (Semrock LL01-780-12.5), before being coupled into a single-mode polarization-maintaining fiber (PMF). Then, the light is sent to a Hong-Ou-Mandel (HOM) interferometer, which has another set of broad filters in front of the single-mode fibers (SMF) that send the light to the single-photon avalanche detectors (SPAD) (Excelitas SPCM-780-13).
We write a spin wave by pulsing the probe and the control field for ≈ 370 ns. The peak Rabi frequencies are Ω p ≈ 2π × 1 MHz and Ω c ≈ 2π × 7 MHz, respectively. Both fields are detuned from the intermediate state by ∆ p ≈ 2π ×50 MHz, with the two-photon transition close to resonance. Due to the collective nature from the blockaded excitation [S4], there is a √ N ≈ 20 enhancement to the two-photon Rabi frequency, , inferred from the π-time. This enhancement corresponds to an OD≈ 13 given the blockaded volume.
After writing, we turn off the addressing lasers and hold (store) the spin wave in the medium for ≈ 350 ns; this is the minimum time required to switch the control acousto-optic modulator (AOM) frequency. We turn on the control field blue-detuned from the intermediate state by ∆ c ≈ 2π × 7 MHz to map the spin wave into a single photon. We use an AOM before the PMF as a hardware gate to avoid saturating the SPADs from the initial write pulse.
We measure the optical losses along the path of the probe light to characterize the generation efficiency in Table S1. The propagation efficiency includes all the optical elements, such as filters, dichroics, mirrors, polarizing beam splitters, mirrors, and lenses. With realistic improvements on higher transmission coatings and using an electrooptical modulator instead of an AOM, we could get an efficiency up to 0.65 after the PMF, from the current 0.44.

Background subtraction
For all our single-photon measurements, we use two SPADs, with average background rates of ≈ 80 s −1 , and, ≈ 100 s −1 . This count rate is due to detector dark counts and leakage of ambient light.
Since the photons arrive at the detectors at a known time, we apply a gate corresponding to a 1.4 µs time window, which contains more than 99.9% of the pulse. We implement this in software to extract the background-photon and background-background coincidence rates from counts outside this window. With this information, we can determine the temporal profile of the accidental coincidences, which we subtract from the data. The probability of a background coincidence, c back , is the sum of the products of single event rates: where t 1 and t 2 are absolute times relative to some clock, for SPAD 1 and 2 respectively. P i (t i ), is the probability per unit time of a photon detection event at detector i, and B i (t i ) is the probability per unit time of a background. Changing to the relative time coordinate, τ = t 2 − t 1 , the background coincidence probability is, We integrate t 1 over a time window t end − t start to obtain the total background coincidence rate as a function of the relative time, τ : where, t start , is synchronized to the photon arrival. With the gate, the background and pulse probability have a time dependence With the independently measured single event rates P i (t) and B i , we calculate C back (τ ). This process is shown graphically in Figure S1, where C back are the total coincidences rate from photon-background and background-background around τ = 0. Finally Figure S1(h) shows the background subtracted coincidences rate, C s (τ ), within the gate window.

HOM visibility discussion
If two single photons are incident simultaneously on separate ports a 1 and a 2 of a perfect 50:50 beamsplitter (BS) the initial state |1 1 , 1 2 , becomes: where a 3 , a 4 are the output ports and we assumed that the input photons are in pure states and indistinguishable from each other. In this case, the probability of a coincidence detection is zero and the HOM visibility is one. In practice, the following factors reduce the visibility from its maximum value [S5]: • one or both photons are not in a pure state, • there is more than one photon at either BS input port, • an imperfect 50:50 BS.
We will focus on the effect of the last two conditions: multi-photon events and imperfect BS. Following the discussion from [S6], we define the scattering matrix, S for a general BS as, where r 1 (r 2 ), t 1 (t 2 ), are the reflection and transmission amplitudes with a relative phase φ 1 (φ 2 ) for port 1 (2). Then the input-output relations of the BS, ignoring any frequency dependence: whereâ i are the photon ladder operator for the input and output ports. Generally, the scattering matrix, S, is not unitary.
For a lossy BS, where the output fields total energy is lower than the input fields energy, the following inequality holds: where α = φ 1 + φ 2 , affects the maximum value that the visibility can attain. The phase, α, is constrained by energy conservation, and we assume α = π. The number operator for the input ports 1 and 2 (output 3 and 4) isn i =â † iâ i . Assuming that the probability of states with more than two photons is negligible, the coincidence probability, P (1 3 , 1 4 ), Here P 1 is the probability of a single photon, P 2 is the probability of two photons at one input port, and c is the mode overlap of the two incident photons. Following the assumption that the probability of more than two-photon states is negligible, we can rewrite P 2 as a function of the correlation function g (2) (0) and P 1 , as P 2 ≈ g (2) (0)P 2 1 /2. The coincidence probability: For the more general case, where the BS coefficients are not the same for orthogonal polarizations, H, and V where we assume that in the case of P (1 3 , 1 4 ) HV , the photon at port 1 has H-polarization and the photon at port 2 has V -polarization, similarly for P (1 3 , 1 4 ) HH , both incoming photons have H−polarization.
In the following We measured a background-subtracted visibility to be V = 0.966 (6), and using equation (13) to take into account the imperfect BS, we find a mode overlap of 0.982 (7).

Contaminants
We use a simple model to characterize the effects of the contaminants on the photon generation, where there is a probability that a stored spin wave is converted to a contaminant. Once a contaminant is present in the medium, it disables the writing and storing of a spin wave until the contaminant decays, with a time constant τ c . If P c is the probability of creating a contaminant on a given pulse, then the probability, P n , of a contaminant being present at pulse n depends on whether one was created in one of the previous pulses and remained to the n-th pulse P n = P n−1 e −tp/τc + (1 − P n−1 )P c , (S12) where t p is the pulse spacing. If we set the initial condition to be P 1 = P c , and use the identity, (1−x) n−1 j=0 x j = 1−x n , we get the expression: Then, the probability of successfully generating a photon, P g (n) is where P max is the maximum probability of generating a photon. For n → ∞, the steady state probability P s , We also model how the correlation function, g (2) (m t p ) for integer m = 0, is modified due to contaminants: this manifests as a bunching feature around τ = 0.

Write and storage efficiency
We model the spin-wave as a super-atom with N -atoms being collectively driven into a single excitation to the Rydberg state, for the writing and storage time. The energy levels and decay rates of the super-atom are shown in Figure S2. We simulated the writing stage as driving the super-atom from the ground to the Rydberg state, with √ N -enhanced Rabi frequency. During the writing time, t w , the Rabi frequencies, Ω p ≈ 2π × 1.0(2) MHz and Ω c ≈ 2π × 6.8(3) MHz are kept constant. For the storage time, t s , these driving frequencies are set to zero.
The Hamiltonian describing the the system depicted in Fig S2 in the rotating wave approximation is given by: in the basis of |g , |e , |r , |c , for the ground, intermediate, Rydberg and contaminant state, respectively. Using the Python package QuTip [S7], we calculated the non-unitary dynamics of this first stage using the master equation for the four level density matrix ρ: where C 1 = √ γ ge |g e|, C 2 = √ γ gr |g r|, C 3 = √ γ cr |c r|, and C 4 = √ γ gc |g c| are the jump operators.

Retrieval efficiency
We follow the derivations in Ref. [S8] to compute the retrieval efficiency. In the rescaled unit-less coordinates,z = 0 andz = 1 represent the front and the end of the atomic cloud, respectively. Suppose all atoms are in the |r state in the beginning of the retrieval stage at timet = 0, the shape of the spin wave is given by S(z,t = 0) = 1 forz ∈ [0, 1] and S(z,t = 0) = 0 forz elsewhere. The retrieval efficiency can be expressed in terms of the photon field E(z,t) emitted by the stored spin wave at the end of the atomic cloud: (S19) E(1,t) can be calculated as: where we define dimensionless parameters d =OD/2,γ s = (γ gr + γ cr )/γ ge ,∆ = 2∆ p /γ ge ,Ω(t) = Ω c (t)/γ ge . h(t,t ) = t t |Ω(t )| 2 dt and I 0 is the 0th-order modified Bessel function of the first kind. When the control field Ω c is constant in time, we define the dimensionless parameter x s = 2γ s /|Ω c | 2 which characterizes the strength of the decay rate compared to the control field. (S19) can be evaluated as where K r is given by and f (x s ) = 2 2+xs(1+∆ 2 ) . Evaluating the integral in Eq. (S21) numerically, we obtain the retrieval efficiency η r = 0.63 (2). With these results, we estimate that the photon generation probability at the end of the cloud is P th = 0.42 (3).

Possible improvements
With conservative feasible experimental improvements, such as implementing a ground-state blue-detuned optical dipole trap, as well as increasing the following parameters: Ω c = 2π × 10 MHz, ∆ p = 2π × 100 MHz and OD=20, while decreasing the spin wave dephasing by a factor of two, we estimate that we could increase our probabilities up to η w η s = 0.86 and η r = 0.72, while maintaining a relatively low contaminant probability, P c ≈ 3 × 10 −2 .
From the theoretical model, the main limiting factor is the retrieval process; in principle, the retrieval efficiency increases with higher OD; however, the contaminant production also grows with OD. A Rydberg ensemble with low OD coupled to a cavity could further increase light-matter interactions and therefore increase the overall photon production probability, making it a promising platform for scalable quantum information applications.

Single-photon sources
In Tables S3 and S4, there is detailed information about the properties of a representative sample of single-photon sources plotted in Fig. 5. in the main text. The notation, R, repetition rate, P is the probability of coupling a singlephoton into a single-mode fiber, V , is the indistinguishability, η is the single-mode probability, R is the brightness, and F is the fidelity. [S1] S. Rosi, A. Burchianti, S. Conclave, D. S. Naik, G. Roati, C. Fort, and F. Minardi, "λ-enhanced grey molasses on the d2 transition of rubidium-87 atoms," Scientific reports 8, 1301 (2018).