Semi-Deterministic Entanglement between a Single Photon and an Atomic Ensemble

TL;DR

This paper demonstrates a high-efficiency scheme for entangling a single photon with an atomic ensemble using Rydberg blockade, achieving 50% efficiency and high fidelity, advancing quantum network capabilities.

Contribution

The authors propose and experimentally realize a novel scheme for atom-photon entanglement with 50% efficiency using Rydberg blockade and momentum encoding.

Findings

Achieved 50% entanglement creation efficiency.

Measured fidelity of 0.901 for atom-photon entanglement.

Verified the single-photon nature of retrieved optical fields.

Abstract

Entanglement between a single photon and a matter qubit is an indispensable resource for quantum repeater and quantum networks. With atomic ensembles, the entanglement creation probability is typically very low to inhibit high-order events. In this paper, we propose and experimentally realize a scheme which creates atom-photon entanglement with an intrinsic efficiency of 50%. We make use of Rydberg blockade to generate two collective excitations, lying in separate internal states. By introducing the momentum degree of freedom for the excitations, and interfering them via Raman coupling, we entangle the two excitations. Via retrieving one excitation, we create the entanglement between the polarization of a single photon and the momentum of the remaining atomic excitation, with a measured fidelity of 0.901(8). The retrieved optical field is verified to be genuine single photons. The realized entanglement may be employed to create entanglement between two distant nodes in a fully heralded way and with a much higher efficiency.

Figures

Tables

Table 1. Table 1: Single-photon quality measurement, g 2 ​ ( 0 ) superscript 𝑔 2 0 g^{2}(0) , of each read-out field.

Basis	H/V	$+/-$	${\sigma }^{+}/{\sigma }^{-}$
1st	0.090(25)	0.124(24)	0.127(28)
2nd	0.129(33)	0.124(28)	0.106(29)

Equations

$$|\psi {}_{jk}(t)⟩$$

$$-i\sin (\frac{\sqrt{2}\Omega t}{2})|s_j{s}_k⟩].$$

$$|\Psi (t)⟩$$

$$=\frac{1}{\sqrt{2}}[|{\Psi }^{-}⟩+\cos (\frac{\sqrt{2}\Omega t}{2})|{\Psi }^{+}⟩$$

$$-i\sin (\frac{\sqrt{2}\Omega t}{2})|S_1,S_2⟩],$$

Full text

Quantum communication mostly makes use of photons as information carrier Gisin et al. (2002); Yuan et al. (2010). Scalable extension to ultra long distance Briegel et al. (1998) and to numerous nodes Wallnöfer et al. (2016) is prohibited due to losses and the probabilistic character of photon sources and detections Lvovsky et al. (2009). It is very promising of incorporating matter qubits into quantum communication to solve the scalability issue Duan et al. (2001). In this joint approach, entanglement between a single photon and a matter qubit is an indispensable resource Kimble (2008); Afzelius et al. (2015). Multiple pairs of such entanglement may be connected to build a quantum repeater Briegel et al. (1998) which scales much better than direct transmission of photons through a lossy channel, or to build a quantum network of matter qubits Wehner et al. (2018); Jing et al. (2019) which may enable multiparty quantum communication, synchronization of atomic clocks Kómár et al. (2014), construction of telescopes with an ultra long baseline Gottesman et al. (2012), etc.

Many physical systems Simon et al. (2010) have been studied as matter qubits for quantum communication, such as laser cooled ions Duan and Monroe (2010) and neutral atoms Sangouard et al. (2011); Reiserer and Rempe (2015), impurities Gao et al. (2015); Atatüre et al. (2018) and rare earth ions Tittel et al. (2009) in a crystal at cryogenic temperature. To prepare entanglement between a single photon and a matter qubit, it is ubiquitous of using spontaneous emission, which inevitably makes the preparation process probabilistic. In the case of single particles, the emission of photons is typically isotropic and only a very small proportion can be collected into a single-mode fiber Hensen et al. (2015). While in the case of an ensemble of particles, the creation probability of a single photon in one specific mode has to be kept very low to limit the contribution of high-order events Sangouard et al. (2011). To solve this probabilistic issue, a cavity which couples strongly with a single photon may be employed for the single-particle approaches Reiserer and Rempe (2015). While for an ensemble-based system, a mechanism of nonlinear interaction is required to inhibit high-order excitations. Rydberg blockade is such a mechanism Saffman et al. (2010). Harnessing Rydberg blockade, single collective excitations can be prepared in a deterministic fashion and later retrieved as single photons into a well-defined mode via collective enhancement Saffman and Walker (2002). In previous experiments, preparation and manipulation of single Dudin and Kuzmich (2012); Li et al. (2013) and double Ebert et al. (2014); Li et al. (2016) excitations have been demonstrated. Nevertheless, entanglement between a single photon and a Rydberg atomic ensemble is still yet to be realized.

In this paper, we propose and experimentally realize an atom-photon entanglement scheme which has an intrinsic efficiency of 50%. Our scheme is illustrated in Fig. 1. We consider a mesoscopic atomic ensemble composed of $N$ neutral atoms in the regime of ensemble blockade. All atoms are initially prepared in a ground state of $|g\rangle$. We make use of Rydberg blockade to prepare two collective excitations, with one in a ground state $|s\rangle$ and another in a Rydberg state $|r\rangle$. The momentums of the two excitations are set to be different, i.e. $\hbar\textbf{k}_{\rm{1}}$ for $|s\rangle$ and $\hbar\textbf{k}_{\rm{2}}$ for $|r\rangle$ respectively (hereafter $\hbar$ is assumed to be 1 for simplicity). Thus, the joint state for the two excitations can be expressed as $|\Psi(0)\rangle=M\Sigma_{jk}^{N}{\rm{e}}^{i(\textbf{k}_{\rm{1}}\cdotp\textbf{x}_{j}+\textbf{k}_{\rm{2}}\cdotp\textbf{x}_{k})}|s_{j}r_{k}g^{N-2}\rangle\equiv|R_{2},S_{1}\rangle$, where $\textbf{x}_{j(k)}$ is the coordinate of the $j(k)$th atom, $M=1/\sqrt{N(N-1)}$ is a normalizing coefficient. Since all the atoms locate inside the Rydberg blockade radius, the momentum-distinguishable collective excitations can be regarded as two super atoms, and the momentum symbols are just their sequence numbers. Then we apply a driving laser field which resonantly couples the transition $|s\rangle\leftrightarrow|r\rangle$ with a Rabi frequency of $\Omega$, and for simplicity we assume the driving laser induces no momentum kick. Different from noninteracting linear systems Pan et al. (2012); Lester et al. (2018), Rydberg interaction between the super atoms prohibits double Rydberg excitations and it results in a nonlinear process with a $\sqrt{2}$ enhancement Gaëtan et al. (2009); Ebert et al. (2014). At ${\rm t}=\pi/(\sqrt{2}\Omega)$, the collective state evolves into $|\Psi({\rm t})\rangle=(1\sqrt{2})(|\Psi^{-}\rangle-i|S_{1},S_{2}\rangle)$, where $|\Psi^{-}\rangle=(1/\sqrt{2})(|R_{2},S_{1}\rangle-|R_{1},S_{2}\rangle)$ (See Supplemental Material for details). Thus there is a probability of 50% to obtain $|\Psi^{-}\rangle$ with two momentum-entangled collective excitations remaining in $|r\rangle$ and $|s\rangle$ respectively. By mapping the Rydberg excitations onto phase-matched single photons, as shown in Fig. 1(b), we can prepare the atom-photon entanglement with an efficiency of 50%, which we refer to semi-deterministic.

The atomic energy levels and the experimental configuration are depicted in Fig. 2. Here $|g\rangle=|5S_{1/2},F=2,m_{F}=+2\rangle$ and $|s\rangle=|5S_{1/2},F=1,m_{F}=0\rangle$ are the hyperfine Zeeman sublevels of the ground states of ${}^{87}\!\rm{Rb}$ atoms, and $|r\rangle$ represents the Rydberg state $\left|81S_{1/2},m_{J}=+1/2\right>$. The atoms are manipulated within a small volume $\sim$10 $\mu$m in all three dimensions (see Ref. Li et al. (2016) and the caption of Fig. 2 for details). Using Rydberg blockade we can deterministically create $|\Psi(0)\rangle$, with the $\pi$ pulse sequences ($A$+$B$)$\to$($C$+$D$)$\to$($A$+$B$) as shown in Fig. 2(a). As a result, $|S_{1}\rangle$ has a momentum of $\textbf{k}_{\rm{1}}=\textbf{k}_{\rm{A}}+\textbf{k}_{\rm{B}}-\textbf{k}_{\rm{C}}-\textbf{k}_{\rm{D}}$, and $|R_{2}\rangle$ has a momentum of $\textbf{k}_{\rm{2}}=\textbf{k}_{\rm{A}}+\textbf{k}_{\rm{B}}$. Next we apply the Raman beams ($C$+$E$) to couple the two excitations as shown in Fig. 2(b). Considering the momentum kick of $\Delta\textbf{k}=\textbf{k}_{\rm{C}}+\textbf{k}_{\rm{E}}$ given by the Raman lights, the atomic state after Raman coupling is $|\Psi({\rm t})\rangle^{\prime}=(1\sqrt{2})(|\Psi^{-}\rangle^{\prime}-i|S_{1},S_{4}\rangle)$, with $|\Psi^{-}\rangle^{\prime}=(1/\sqrt{2})(|R_{2},S_{1}\rangle-|R_{3},S_{4}\rangle)$, $\textbf{k}_{\rm{3}}=\textbf{k}_{\rm{1}}+\Delta\textbf{k}$, $\textbf{k}_{\rm{4}}=\textbf{k}_{\rm{2}}-\Delta\textbf{k}$. For the validity of our scheme, one has to ensure $\textbf{k}_{\rm{2}}\neq\textbf{k}_{\rm{3}}$ and $\textbf{k}_{\rm{1}}\neq\textbf{k}_{\rm{4}}$, which is achieved by injecting beam $D$ and $E$ through slightly different directions, $\textbf{k}_{\rm{D}}\neq\textbf{k}_{\rm{E}}$. To generate atom-photon entanglement, we retrieve the Rydberg excitation by applying the read beam as shown in Fig. 2(c), and the momentum entangled atomic state $|\Psi^{-}\rangle^{\prime}$ is converted to $|\Psi\rangle=(1/\sqrt{2})(|\textbf{k}_{\uparrow}\rangle|S_{1}\rangle-|\textbf{k}_{\downarrow}\rangle|S_{4}\rangle)$, where $\textbf{k}_{\uparrow}=\textbf{k}_{\rm{2}}-\textbf{k}_{\rm{Re}}$ and $\textbf{k}_{\downarrow}=\textbf{k}_{\rm{3}}-\textbf{k}_{\rm{Re}}$ denote the momentum states of the read-out photons with $\textbf{k}_{\rm{Re}}$ being the momentum of the read beam. In our experiment, beam $A$ and $D$ are in the same direction, $\textbf{k}_{\rm A}\approx\textbf{k}_{\rm D}$, and the read beam and beam $B$ are in the same direction, $\textbf{k}_{\rm Re}\approx\textbf{k}_{\rm B}$. Thus we have $\textbf{k}_{\uparrow}\approx\textbf{k}_{\rm A}$ and $\textbf{k}_{\downarrow}\approx\textbf{k}_{\rm E}$. After a configurable delay of 300 ns, we retrieve the ground state excitation as well to verify the atom-photon entanglement by applying ($C$+$E$) and the read beam as shown in Fig. 2(d). Thus the atom-photon entanglement is converted into a photon pair entanglement $|\Psi\rangle_{\rm{mo}}=(1/\sqrt{2})(|\textbf{k}_{\uparrow}\rangle|\textbf{k}_{\downarrow}\rangle-|\textbf{k}_{\downarrow}\rangle|\textbf{k}_{\uparrow}\rangle)$. Note the Rydberg excitation is always the first to be read out. By the use of waveplates and a polarization beam splitter (PBS), as shown in Fig. 2(e), the momentums of photons can be conveniently transformed as polarizations. For the detection and analysis of the entanglement, we collect the photons with a single mode fiber and use an acousto-optic modulator (AOM) to switch the detection channels between the two photons for independent measurement. A phase shifter between horizontal polarization (H) and vetical polarization (V) is inserted in the path of the first photon, to introduce a variable phase term in the detected photonic state $|\Psi\rangle_{\rm pol}=(1/\sqrt{2})(|{\rm{H}}\rangle|{\rm{V}}\rangle-{\rm{e}}^{i{\rm{\phi}}}|{\rm{V}}\rangle|{\rm{H}}\rangle)$.

We first verify the success of Rydberg blockade by measuring the second-order autocorrelation $g^{2}$ of the photons converted from the Rydberg excitation $|R_{2}\rangle$. The result $g^{2}(0)$=0.062(7), which is primarily limited by the noise of the single photon detectors (SPDs), proves the effectiveness of Rydberg blockade and lays the foundation for the following experimental operations. And also we measure the Rabi oscillations during the preparation and manipulation of single collective excitations, with details given in the Supplemental Material. In particular we shows the Rabi oscillation between $|R_{2}\rangle$ and $|S_{4}\rangle$ as an inset of Fig. 2. In this measurement, a $|R_{2}\rangle$ excitation is first prepared with a 200 ns long exciting pulse, later manipulated by the Raman beams ($C$+$E$) with a tunable duration, and finally retrieved as single photons. Our numerical simulation shows that the damping is mainly due to Rydberg spin-wave dephasing and photon scattering from intermediate states (more details are given in the Supplemental Material).

Next, we measure the dynamics of two excitations under Raman manipulations via polarization correlation measurement of the read-out photons. At $t=0$, the corresponding read-out photonic state should be $|{\rm{H}}\rangle|{\rm{V}}\rangle$, while at $t=\pi/(\sqrt{2}\Omega)$, the corresponding photonic state is $(1/\sqrt{2})(|{\rm{H}}\rangle|{\rm{V}}\rangle+|{\rm{V}}\rangle|{\rm{H}}\rangle)$ (for $\phi=\pi$ in $|\Psi\rangle_{\rm pol}$), and the polarization correlations measured in the $|\pm\rangle=(1/\sqrt{2})(|H\rangle\pm|V\rangle)$ basis will differ radically between the product state and the entangled state. To demonstrate this process, we apply the following pulse sequence: $A$+$B$ for 200 ns, $C$+$D$ for 230 ns, and another $A$+$B$ for 200 ns, to prepare $|\Psi(0)\rangle$. And then variable C+E pulse follows, executing the Raman coupling. In the end we read out each excitation sequentially with a 300 ns interval between them, and record the coincidence counts of all possible polarization pair sets $C_{\rm{++}}$, $C_{\rm{--}}$, $C_{\rm{+-}}$ and $C_{\rm{-+}}$ in the $+/-$ basis. With $\phi\simeq\pi$, the results of coincidence count summations $C_{\rVert(\rm{+/-})}=C_{\rm++}+C_{\rm--}$ and $C_{\bot(\rm{+/-})}=C_{\rm+-}+C_{\rm-+}$ are shown in Fig. 3, which have been normalized by setting the summations of all coincidence counts as 1 unit. The sinusoidal oscillations reveal the atomic state evolution, and the fitting oscillation period is T′ = 339(5) ns, in good agreement with the theoretical description with a $\sqrt{2}$ factor compared with the single excitation result T= 492(5) ns. The two-excitation measurement verifies the successful Rydberg blockade betweeen the collective atomic excitaitons.

According to the above result, we set the Raman pulse duration fixed at 170 ns, corresponding to the $\pi/\sqrt{2}$ pulse, and rotate the half-wave plate in the phase shifter, to vary the relative phase $\phi$ in $|\Psi\rangle_{\rm pol}$. it is not difficult to see in H/V basis there should only be $C_{\rm{HV}}$ and $C_{\rm{VH}}$ components, and $C_{\rm{HH}}$ and $C_{\rm{VV}}$ should always be zero, while in $+/-$ basis the results of $C_{\rm{++}}$, $C_{\rm{--}}$, $C_{\rm{+-}}$ and $C_{\rm{-+}}$ depend on the value of $\phi$. The coincidence patterns are shown in Fig. 4. We define the polarization visibility as $V=|(C_{\bot}-C_{\rVert})/(C_{\bot}+C_{\rVert})|$ at the maximum oscillation points, and therefore we measure them at $\phi=\pi$ in H/V, $+/-$ and circular $\sigma^{+}$/$\sigma^{-}$ bases to comprehensively characterize the quality of the entanglement. The results are $V_{\rm(H/V)}=0.897(17)$, $V_{\rm(+/-)}=0.828(19)$ and $V_{\rm(\sigma^{+}/\sigma^{-})}=0.879(19)$ respectively, corresponding to an entanglement fidelity of $F=(1/4)[1+V_{\rm(H/V)}+V_{\rm(+/-)}+V_{\rm(\sigma^{+}/\sigma^{-})}]=0.901(8)$, which is primarily limited by the background noise of the detectors arising from stray lights and dark counts.

Finally, we characterize the single-photon quality of each read-out photon, by analyze their $g^{2}(0)$ in H/V, $+/-$ and $\sigma^{+}$/$\sigma^{-}$ bases respectively. The results are listed in Table 1. Though degraded by lower signal-to-noise ratio compared with single Rydberg excitation measurement, the values here are still well below 1. Averaging the results in the three bases, $g^{2}(0)$ of each photon are 0.114(15) and 0.120(17) respectively. It proves both read-out fields are genuine single photons, which is very advantageous when implementing quantum repeater and quantum networks using multiple pairs of entanglement. In comparison, the photonic fields of traditional probabilistic source of atom-photon entanglement with atomic ensembles Sangouard et al. (2011) are generally not single photons, and typically have $g^{2}(0)$ value around 2. Thus creating entanglement of remote Yuan et al. (2010) or multiple Jing et al. (2019) atomic ensembles with probabilistic schemes is usually spoiled with unwanted contributions of high-order events, which makes heralded entanglement creation difficult, while the genuine single-photon feature of our Rydberg-based scheme makes heralded creation of remote entanglement straightforward and enables further extension of node numbers more efficiently (See Supplemental Material for discussions).

Albeit the new scheme has many benefits, successful application of it in quantum repeater and quantum networks requires further improvements of overall efficiency and memory lifetime. At present the entanglement detection rate is limited by the overall detection efficiencies in each channel of $\sim$0.2%, which includes the phase-matching read-out efficiency less than 3% due to the small optical depth, the transmission and collection efficiency of 50%, the SPD efficiency of 60%, the manipulation efficiencies of laser pulses of 65% $\sim$ 85%, the effect of spin-wave dephasing, and so on. Much effort is needed to improve the experimental techniques in the future. For instance, by combining the small atomic ensemble with an optical cavity, we can expect the establishment of an effecient photon-atom inetrface Bao et al. (2012), enhancing the read-out effeciency by an order of magnitude. Replacing the SPDs with state-of-the-art ones, the single photon detection efficiency can exceed 90%. And with smaller atomic ensembles, better laser systems, as well as new energy configurations Levine et al. (2018), the losses during state manipulations will be greatly inhibited, due to more ideal experimental settings. In addition, the entanglement lifetime is limited by the coherence of the ground-state excitation $|S_{4}\rangle$, which is subject to motion-induced dephasing with a lifetime of $\sim$30 $\mu$s at present Li et al. (2016). It will be prolonged markedly to subsecond level Yang et al. (2016) with the use of an optical lattice in the future.

In conclusion, we have proposed and realized a Rydberg-based scheme which creates entanglement between a single photon and a collective atomic excitation by harnessing the momentum degree of freedom. Rydberg blockade is employed to create single collective excitations deterministically and let the excitations interact. The scheme succeeds with an intrinsic efficiency of 50%, which enables more efficient creation of remote entanglement. The photons created are genuine single photons, which makes heralded creation of remote entanglement straightforward. Limitations in current experimental demonstration can be overcome by incorporating the existing techniques, such as using a ring cavity with moderate finesse to improve the photonic retrieval efficiency Yang et al. (2015) and using optically lattice to improve the memory lifetime Yang et al. (2016). With these developments, our new source of atom-photon entanglement may become a fundamental building block for future quantum repeater and quantum networks Wehner et al. (2018).

This work was supported by National Key R&D Program of China (No. 2017YFA0303902), Anhui Initiative in Quantum Information Technologies, National Natural Science Foundation of China, and the Chinese Academy of Sciences. J. L. acknowledges support from China Postdoctoral Science Foundation.

I SUPPLEMENTAL MATERIAL

I.1 Detailed Derivation of the State Evolution

For simplicity, we assume the driving laser field will not introduce any additional momentum kick. As it couples $|s\rangle$ and $|r\rangle$ resonantly and large-detuning coupling between $|g\rangle$ and $|r\rangle$ is negligible, we only need to consider the response of the $j$th and the $k$th atoms residing in $|s\rangle$ and $|r\rangle$ respectively. Starting from $|\psi\!{{}_{jk}}(0)\rangle=|s_{j}r_{k}\rangle$, we can write it as $|\psi\!{{}_{jk}}(0)\rangle=(1/\sqrt{2})(|\psi^{-}_{jk}\rangle+|\psi^{+}_{jk}\rangle)$ where $|\psi^{\pm}_{jk}\rangle=(1/\sqrt{2})(|s_{j}r_{k}\rangle\pm|r_{j}s_{k}\rangle)$. The following derivation apears straightforward as $|\psi^{+}_{jk}\rangle$ is definitely coupled with $|s_{j}s_{k}\rangle$ under Rydberg blockade with an enhanced Rabi frequency of $\sqrt{2}\Omega$, while $|\psi^{-}_{jk}\rangle$ is decoupled from any other states smiroshnychenko2009observation :

[TABLE]

With the phase information of individual atoms preserved in the frozen atomic gas, now we can turn to consider the collective state with the two excitations. Recalling $|\Psi(0)\rangle=M\Sigma_{jk}^{N}{\rm{e}}^{i(\textbf{k}_{\rm{1}}\cdotp\textbf{x}_{j}+\textbf{k}_{\rm{2}}\cdotp\textbf{x}_{k})}|s_{j}r_{k}g^{N-2}\rangle=|R_{2},S_{1}\rangle$, we have

[TABLE]

with $|\Psi^{\pm}\rangle=(1/\sqrt{2})(|R_{2},S_{1}\rangle\pm|R_{1},S_{2}\rangle)$. This is the collective state evolution of the complete system of N atoms, and at ${\rm t^{\prime}}=\pi/(\sqrt{2}\Omega)$, we obtain $|\Psi({\rm t^{\prime}})\rangle=(1\sqrt{2})(|\Psi^{-}\rangle-i|S_{1},S_{2}\rangle)$.

I.2 Rabi Oscillation between Collective States

The collective Rabi oscillation between the initial state $|G\rangle=|g\rangle^{N}$ and the Rydberg excitation $|R_{2}\rangle$ is shown in Fig. S1(a), by recording the phase-matched single photons converted from the Rydberg excitations. The atom-field coupling is collectively enhanced by a factor of $\sqrt{N_{e}}$ due to Rydberg blockade, with the effective atom number $N_{e}\simeq 150$. A collective $\pi$ pulse will generate the single excitation in the Rydberg state deterministically. However, the pulse fidelity is experimentally limited by fluctuations of the atom number and the intensities of the laser fields, and the decay of read-out efficiency comes from spin-wave dephasing due to atomic motion and double Rydberg excitations dudin2012observation . Through the envelope of the oscillation fitting, we infer the generation efficiency of the single excitations is better than 80% with the collective $\pi$ pulse in our experiment.

Fig. S1(b) shows the Rabi oscillation between $|R_{2}\rangle$ and $|S_{1}\rangle$ with $C$+$D$ pulses, which we exploit to prepare the ground-state excitations from the Rydberg ones. Note there is only one atom among $N$ resonantly coupled with the light fields and no collective enhancement in this process. Compared with the collective Rabi oscillation between $|G\rangle$ and $|R_{2}\rangle$, the read-out signal of single-excitation Rabi oscillation between $|R_{2}\rangle$ and $|S_{1}\rangle$ declines much more rapidly. The manipulation losses between the single excitations is a significant factor affecting our experimental efficiency, and thus we clarify the main mechanisms of this damping by numerical simulation, and demonstrate them in the next section below.

I.3 Sources of Read-Out Losses during Single-Excitation Manipulation

The Rydberg state $|r\rangle$ and the ground state $|s\rangle$ are coupled by $C$+$D$ (or $C$+$E$) light fields (Fig. 2 in the main text). With the specified intermediate-state detuning $\Delta_{2}$, the coupling between $|r\rangle$ and $|s\rangle$ is constructively enhanced through $|e_{1}\rangle$ and $|e_{2}\rangle$, and the spatially inhomogeneous light shifts of $C$ are automatically cancelled. However, there are still some sources of Rydberg spin-wave dephasing de2018analysis leading to read-out losses. We use QuTiP johansson2012qutip ; johansson2013qutip2 , an open-source Python framework for solving the dynamics of open quantum systems, to simulate the single-excitation evolution with various experimental conditions, and try to reveal the causes of the losses.

In the simulations, the propagting directions of the laser beams are set generally the same as in the experiment, while the angles between them are simply ignored. The central Rabi frequencies are kept the same as in the experiment when the beam waists are adjusted for comparison. Other parameters not mentioned all follows the experiment settings, and the initial state is the collective Rydberg excitation, or spin wave, $|R_{2}\rangle$, with a Gaussian spatial mode determined by the 7 $\mu$m focused excitation beams. Fig. S2 demonstrates the simulation results with various experimental conditions, where the x axes are the light pulse duration and the y axes are the normalized Rydberg component. The yellow solid lines simply denote the population in $|r\rangle$, while the gray dashed lines denote the state projection onto the collective state $|R_{2}\rangle$. In Fig. S2(a), we consider only the effect of atomic motion, by setting the temperature as 150 $\mu$K and the driving laser beams nealy homogeneous with waists of 500 $\mu$m. We see the Rydberg spin wave dephases gradually, though the probability of finding the atom in the Rydberg state oscilates much better. The spin-wave oscillation has a $1/e^{2}$ decay time of 3.2 $\mu$s, exactly two times the free evolution lifetime of the Rydberg spin wave without any manipulation light fields (green dotted line). It is consistent with our intuition that the excitation stays in the ground state half of the time during the oscillation with a much longer spin-wave lifetime, and the overall decay time is an average of both. It should be noted that 150 $\mu$K is an effective value, deduced from the measured free Rydberg spin-wave lifetime of 1.6 $\mu$s. There may be some other dephasing machanisms associated with the Rydberg states rather than atomic motion, which results in this lifetime value, such as stray electric fields dudin2012strongly . However, it does not change the conclusion, that Rydberg spin-wave dephasing is one of the sources of read-out losses, and it sets the upper limit as twice the free Rydberg spin-wave lifetime.

In Fig. S2(b), we simulate the system with much smaller beam waist of $C$, 10 $\mu$m, approximately as used in the experiment, and the temperature is set to be zero. Though the inhomogeneous light shifts of $C$ have been eliminated, the spatial inhomogeneity of Rabi frequency is always an experimental defect, and it smears the oscillation significantly as shown in the figure.

Fig. S2(c) shows the effect of the scattering from the intermediate states. It is an additional source of the manipulation losses, which is not included in the previous two simulations.

Fig. S2(d) brings all the effects together and shows the final simulation result, which is also shown in the inset of Fig. 2 in the main text. From the comparisons above, we conclude that Rydberg-sensitive dephasing, light inhomogeneity, and intermediate-state scaterring are the main sources leading to the manipulation losses, and the light inhomogeneity is the dominant one. And our further experimental measurements with lower Rydberg levels and larger beam waists support this judgement. However, in the high-level cases, there has to be a trade-off between the homogeneity and the Rabi frequency limited by the laser power. Due to the post selection of the phase-matched read-out photons, the dephasing and losses do not degrade the entanglement fidelity markedly in our experiment. However, high-efficiency light-atom interface is crucial in the application of realistic quantum repeaters. Improvements in laser systems as well as new energy configurations slevine2018high will be pursued in our future studies.

I.4 Advantage of the Semi-Deterministic Atom-Photon Entanglement

Using the semi-deterministic atom-photon entanglement, we can herald the establishment of entanglement between remote quantum memory nodes without any spurious contributions. The simple fact is illustrated in Fig. S3. For simplicity, we do not consider photon losses in the fibers here. There are four possible cases when connecting the two quantum nodes 1 and 2 by entanglement swapping. Only case $a$, with both of the atom-photon entanglement pairs generated successfully, can obtain the expected two-photon coincidence counts in the middle. And then the remote entanglement between nodes 1 and 2 is prepared successfully for further extensions. In practice, single photons in long-distance fibers may well be absorbed before arriving at the detectors. But the photon losses or nonunit detection efficiency will only bring down the coincidence count rate but can never lead to spurious heraldings. In contrast, due to the existence of sencond-order excitations, the DLCZ-based protocols can not avoid generation of complex states that are not maximally entangled zhao2007robust ; chen2007fault ; jiang2007fast ; sangouard2008robust . And thus the measurement results at the intermediary connection nodes can not ensure the generation of remote entanglement after the first-level entanglement swapping. And following attempts on higher-level entanglement swapping with more classical communication and feedbacks may prove to be futile. As a result, the DLCZ-based quantum repeaters work in a low efficiency.