Introduction

The realization of long-distance entanglement plays an important role in quantum communication, such as quantum key distribution1,2,3, quantum dense coding4,5, quantum teleportation6, quantum secret sharing7, quantum secure direct communication8,9,10,11 and so on. However, due to the thermal fluctuation, vibrations and other imperfections, inevitable exponential scaling errors occur on the quantum state of photons with the transmission distance in the noisy channel. In order to construct a long-distance entangled channel, the concept of quantum repeater was originally proposed by Briegel et al.12 in 1998. Its basic idea is to divide the total transmission distance into several segments and then use entanglement purification and entanglement swap** to suppress the influence of environment noises. In 2001, Duan et al.13 presented a proposal for quantum repeaters with atomic ensembles as quantum memories, known as the DLCZ protocol. In 2006, using only two qubits at each station, Childress et al.14 constructed a fault-tolerant quantum repeater, which provides the possibility to realize repeaters in simple physical systems such as solid-state single-photon emitters. In 2007, with two-photon Hong-Ou-Mandel-type interference, Zhao et al.15 proposed a robust and feasible quantum repeater. Meanwhile, Jiang, Taylor and Lukin16 also put forward a robust scheme to construct a quantum repeater with atomic ensembles. In 2012, assisted by the spatial entanglement of photons and quantum-dot spins in optical microcavities, Wang, Song and Long17 presented an efficient scheme for robust quantum repeaters. In 2014, Wang et al.18 proposed a scheme for a quantum repeater based on a quantum dot in an optical microcavity system. In 2015, Li and Deng19 presented a heralded high-efficiency quantum repeater with atomic ensembles and faithful single-photon transmission. Recently, Li, Yang and Deng20 introduced another heralded quantum repeater for quantum communication network based on quantum dots embedded in optical microcavities, resorting to effective time-bin encoding. Furthermore, many experiments have been reported for building quantum repeaters and remarkable progress has been made21,22,23,24,25,26,27.

In recent years, the scattering of photons off single emitters in one-dimensional (1D) waveguides has attracted much attention28,29,30,31,32,33,34,35,36,37,38,39. Single emitters can strongly interact with electromagnetic modes and the scattering of photons off single emitters has been extensively explored. By employing various schemes with two- or three-level atoms, one can well control the propagation of single photon in 1D waveguides and the quantum gates for quantum information processing have been realized40,41,42. In 2005, Shen and Fan28 discussed the interesting transport properties of a single photon interfering with the two-level emitters coupled to the modes in 1D waveguides. In 2007, Chang et al.40 implemented a single-photon transistor using nanoscale surface plasmons, in which strong nonlinear interactions between nanowires and waveguides are realized. In 2010, Witthaut and Sørensen32 solved the scattering problem for a single photon in a 1D waveguide coupled to a three-level emitter and observed electromagnetically induced transparency for a driven Λ-system and V-system if both transitions couple to the waveguides. In 2012, based on the scattering of photons off single emitters in 1D waveguides, Li et al.43 presented an interesting scheme for realizing the robust-fidelity atom-photon entangling gate, in which the faulty events between photons and atoms can be turned into heralded losses.

In this paper, we exploit the scattering of photons off single emitters in 1D waveguides to construct a heralded quantum repeater, including robust nonlocal entanglement creation, entanglement swap** and entanglement purification modules. Although great progress has been made, it is still a big challenge to obtain a long storage time in realistic quantum systems. In our scheme, we use a parametric down-conversion (PDC) source to create entangled photon pairs under the consideration that PDC sources are easily available with compact setups. Since atoms can provide coherence times as long as seconds, we choose a four-level atom as the emitter. It’s worth pointing out that, in our protocols, the faulty events can be turned into the detection of photon polarization, which can be immediately discarded. That is, the quantum repeater either succeeds with perfect fidelity or fails in a heralded way, which is very important for realistic quantum communication. With the remarkable progress on manipulating waveguide QED systems, there is no major difficulty to realize our scheme and maybe it will have good applications in realistic long-distance quantum communication in future.

Results

The scattering of photons off single emitters in a 1D waveguide

As illustrated in Fig. 1(a), the quantum system we consider is composed of a single emitter coupled to a 1D waveguide via electromagnetic interactions. The emitter is actually a simple two-level atom with the frequency difference ωa between the ground state |g〉 and the excited state |e〉 and coupled to a set of traveling electromagnetic modes of the 1D waveguide. Under the Jaynes-Cummings model, the Hamiltonian for the system is28,40

Figure 1
figure 1

(a) The basic structure of a two-level atom (the black dot) embedded in a 1D waveguide (the cylinder). The atom acts as a photon mirror28, with its two levels |g〉 and |e〉 coupled via the waveguide. Under ideal resonance condition, an incident photon (black wave packet) is fully reflected (blue wave packet), or goes freely through (green wave packet) on the condition of detuning. (b) The heralded protocol for a robust-fidelity Z gate on an atom in a 1D waveguide. In fact, the emitter is a four-level atom, with degenerate ground states |g±〉 and degenerate excited states |e±〉. BS is a 50:50 beam splitter, M is a fully reflected mirror and the black lines denote the paths of the travelling photon.

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where xa is the position of the atom, ak () is the annihilation (creation) operator of the mode with the frequency ωk (ωk = c|k|, k is the wave vector), σ+ (σ−) is the atomic raising (lowering) operator and σee = |e〉〈e|. γ′ is the decay rate of the atom out of the waveguide and g is the coupling strength between the atom and the electromagnetic modes of the 1D waveguide, assumed to be same for all modes.

Here, we focus on the scattering of a single photon, as shown in Fig. 1(a). By solving the scattering eigenvalue equation of the system (see the Methods section), one can obtain the reflection coefficient for the incident photon40

where γ1D = 4πg2/c is the decay rate of the atom into the waveguide and Δ = ωk − ωa is the frequency detuning between the photon and the atom. The transmission coefficient is given by t = 1 + r.

From Eq. (2), one can conclude that when the input photon resonates with the emitter (i.e., Δ = 0), the reflection coefficient changes into r = −1/(1 + 1/P), where P = γ1D/γ′ is the Purcell factor. As the spontaneous emission rate γ1D into the 1D waveguide can be much larger than the emission rate γ′ into all other possible channels in a realistic atom-waveguide system28,40, one can get the reflection coefficient r ≈ −1. Therefore, for a large Purcell factor, the atom in state |g〉 acts as a nearly perfect mirror, which puts a π-phase shift on the reflected photon. Whereas, when the photon is decoupled from the two-level atom, nothing happens to the photon after the scattering process.

Now, we consider a four-level atom as the emitter in the 1D waveguide, as shown in Fig. 1(b). The atom has two degenerate ground states |g±〉 and two degenerate excited states |e±〉. The transition |g−〉 ↔ |e−〉 (|g+〉 ↔ |e+〉) is coupled to the L-polarized (R-polarized) photon, where L (R) denotes the left-circular (right-circular) polarization along the waveguide. Provided that the spatial wave function of the input photon from left is in the state |ψ〉, after the photon scatters with the atom, one gets the transformations as follows43:

where |ϕ〉 = |ϕt〉 + |ϕr〉 represents the spatial state of the photon component left in the waveguide after the scattering process. Here, the states |ϕt〉 = t|ψ〉 and |ϕr〉 = r|ψ〉 denote the transmitted and reflected parts of the photon, respectively. If the incident photon is in the horizontal linear-polarization state , the corresponding transformations change into43

where is the vertical linear-polarization state. It is meaningful that the scattering process generates a vertical-polarized component.

With the transformations discussed above, Li et al.43 presented a simple scheme for implementing a high-fidelity Z gate on an atom, as shown in Fig. 1(b). In detail, the incident photon in state |H〉 or |V〉 (from port 1) is first split by a 50 : 50 beam splitter (BS). The transmitted and reflected components scatter with the atom and exit the beam splitter simultaneously. Note that, due to quantum destructive interference, the two parts exit the beam splitter in port 1, without any photon component coming out from port 2. Finally, one obtains a high-fidelity atomic Z gate (marked by Za) as follows:

Here |ϕr〉 = (|ϕ〉 − |ψ〉)/2 refers to the reflected part of the incident photon and |μ〉a is an arbitrary atomic superposition state in the basis {|0〉a = |g−〉, |1〉a = |g+〉}. The perfect scattering process occurs with the condition P → ∞ and we can get |ϕr〉 = −|ψ〉. While for the imperfect situation with a finite P, |ϕr〉 ≠ −|ψ〉 and the detection of an incorrectly polarized output heralds the failure of the corresponding gate. That is, the protocol for atomic Z gate works in a heralded manner.

Robust entanglement creation for nonlocal atomic systems using a PDC source against collective noise

Now, let us describe the principle of our scheme for entanglement creation between two nonlocal atoms, as shown in Fig. 2. Here, two remote atom-photon subsystems are connected by a noisy quantum channel with a PDC source positioned at the middle point. Initially, in each subsystem, a stationary atom in the 1D waveguide, which is named as a (b) on the left (right) part of the setup, is prepared in the superposition state and a pair of photons A and B produced by the PDC source is in a common entangled state , where . The state of the atom-photon system is

Figure 2
figure 2

Schematic diagram for compactly implementing entanglement creation.

a and b stand for two nonlocal atoms in 1D waveguides owned by Alice and Bob, respectively. PBSi (i = 1, 2, 3, 1′, 2′, 3′) is a polarizing beam splitter which transmits the horizontal polarized photon |H〉 and reflects the vertical polarized photon |V〉. QWPi (i = 1, 1′) is a quarter-wave plate to implement the conversion of the photon polarization. PBS± transmits photons with polarization |+〉 and reflects photons with polarization |−〉, where . TRi (i = 1, 1′) is an optical device which can be controlled exactly as needed to transmit or reflect a photon, BSi (i = 1, 1′) is a 50:50 beam splitter and Di (i = +, −) is a single-photon detector.

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Our scheme for nonlocal entanglement creation works with the following steps.

First, the two entangled photons travel along the noisy channels in opposite directions. Each goes through a polarizing beam splitter (PBS) which transmits the photon component in state |H〉 and reflects the photon component in state |V〉. In detail, the photon A (B) in state |H〉 transmits through PBS1 (PBS1′), TR1 (TR1′) and goes directly into the noisy channel through the short path (S), while the photon A (B) in state |V〉 is reflected by the PBS1 (PBS1′) and passes through the quarter-wave plate QWP1 (QWP1′) to rotate its polarization. After the operation, the photon A (B) in the long path (L) is reflected by TR1 (TR1′) into the same noisy channel, but a little later than its early counterpart. TRi (i = 1, 1′) is an optical device which can be controlled exactly as needed to transmit or reflect a photon. The state of the whole system at the entrance of the noisy channels changes into

where or represents the state in which a photon travels along the short (S) or long (L) path. Henceforth, a state with superscript S or L follows the same regulation.

Second, the photons A and B, including their early component and late component , are transmitted to Alice and Bob via different noisy channels, respectively. Since the polarization states of the two components in photon A (B) are both |H〉, the influences of the collective noise in the noisy channel on them are the same ones44,45,46,49,50,51,52,53,54. The schematic diagram for our entanglement swap** protocol is shown in Fig. 3. Here, we consider two pairs of nonlocal atoms ac and db, which are both initially prepared in the maximally entangled states and , respectively. By performing a Bell-state measurement on local atoms c and d, we make the two nonlocal atoms ab collapse into the maximally entangled state , which indicates that a longer quantum channel is constructed. The principle of our entanglement swap** can be described as follows.

Figure 3
figure 3

Schematic diagram for implementing entanglement swap**.

HWPi (i = 1, 1′) is a half-wave plate to complete a Hadamard operation (Hp) on the polarization photon.

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Suppose that an entangled photon pair AB produced by a PDC source is in the state , where . The initial state of the whole system is |Ψ0〉, where

First, the photon A (B) passes through PBS1 (PBS1′) which transmits the photon component in state |H〉 and reflects the photon component in state |V〉. Owing to the fact that the interaction between photon A and atom c is identical to that between photon B and atom d, for simplicity, we just discuss the former part and actually the latter part accomplishes the same process simultaneously. For photon A, the part in state |H〉 transmits through PBS1, QWP1 and TR1 via the short path (S), while the part in state |V〉 is reflected by PBS1 and TR1 via the long path (L). Since the two parts have the same processes, we only describe the interaction of the photon in the short path (S) in the following section. Then, the part in the short path (S) travels through a 50:50 beam splitter (BS1). The reflected component of this part is reflected by PBS2 into the scattering setup containing atom c and travels through PBS2 and PBS3, while the transmitted component goes into PBS3 directly. The two parts of photon A are rejoined in PBS3. The same processes occur to the part in the long path (L) in a late time. After the nonlinear interaction, the state of the whole system is changed from |Ψ0〉 to |Ψ1〉. Here

Second, a Hadamard operation Ha (e.g., using a π/2 microwave pulse or optical pulse55,56) is performed on local atoms c and d in the waveguides, respectively. Subsequently, by passing through HWP1 (HWP1′), the photon A (B) also gains a Hadamard operation Hp. After that, the state of the whole system becomes

The photon A (B) travels through PBS4 (PBS4′) and is detected by single-photon detectors. Meanwhile, the state of atom c (d) is measured by external classical field.

Third, according to the outcomes of the photon detectors and the measurement of atom c (d), one can perform corresponding operations (see Table 2) on atom a to complete the entanglement swap**. Finally, after the processes mentioned above, the state of atoms a and b collapses into the maximally entangled state

Table 2 The operations on atom a corresponding to the results of the photon detectors and the states of atoms c and d.
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As the same as our entanglement creation scheme, in the quantum entanglement swap** protocol, the faulty scattering process between photons and atoms can also be heralded by single-photon detectors DH (DH′) and DV (DV′). Owing to the heralded mechanism, the overall success probability of our protocol may not be high, but the fidelity is 100%. Moreover, we make use of a usual PDC source to implement quantum swap**, which is easily available with compact setups in laboratory.

Entanglement purification of atomic systems with PDC sources

As mentioned above, we just care about the influence of noise on auxiliary photons in long quantum channels. However, the atomic qubits confined in 1D waveguides also inevitably suffer from noises, such as thermal fluctuation and the imperfection of the waveguides. In fact, utilizing entanglement concentration57,Fig. 6, we find that for a given value of P (P = 20), with our scheme one achieves p1 = 82.3% and p2 = 67.7% as Δ/γ1D = 0. While in the case P = 100 and Δ/γ1D = 0, the success probabilities of our protocols are p1 = 96.1% and p2 = 92.3%, respectively. If the Purcell factor is P = 100, with Δ/γ1D = 0.1, the corresponding success probabilities become p1 = 89.0% and p2 = 79.2%, respectively. The above observation is agreed with the prediction that the success probabilities of our protocols for the heralded quantum repeater will approach to 100% when P → ∞ and Δ/γ1D = 0.

Figure 5
figure 5

The success probability ps of the scattering process vs the Purcell factor P and the detuning parameter Δ/γ1D.

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Figure 6
figure 6

The success probabilities of our protocols vs the Purcell factor P and the detuning parameter Δ/γ1D.

(a) The success probability p1 of our protocol for entanglement creation (entanglement swap**). (b) The success probability p2 of our protocol for entanglement purification.

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Note that, in practical situation, the polarization of output photon is swapped but in Eq. (5), which causes a problem that the spatial wave functions in two arms of the interferometer no longer coincide. To solve the problem, we adopt a waveform corrector (WFC) in one arm of the interferometer. Actually, for successful events of imperfect processes, the waveform is with |k| < 1. If the photon-atom detuning Δ is zero, and the WFC can be realized by a beam splitter with the transmissivity k. When the photon-atom detuning Δ ≠ 0, the WFC may also consist of a delay to make the wave packets in two arms arrive at one place simultaneously.

Our scheme for the heralded quantum repeater based on atom-waveguide systems is particularly interesting because of its following characters. First, in our protocols, we make use of PDC sources to implement quantum communication. Nowdays, PDC sources are available with the current experimental technology and have been widely used in various situations where entangled photon pairs are needed. Utilizing PDC sources, we make it possible to double the distance between two repeater nodes without influence of the noise coming from the increased quantum channels. Second, our scheme can turn the judgment of faulty events into the detection of the output photon polarization, which makes the fidelity of quantum repeater 100% in principle. In other words, the error-heralding mechanism ensures that our protocols either succeed with perfect fidelity or fail in a heralded way. As we know, if the entangled pairs are faulty, the fidelity of a realistic quantum repeater will decrease exponentially with the distance. Third, our scheme is also feasible in artificial solid-state systems, such as quantum dots embedded in a nanowire, superconducting quantum circuit coupled to transmission lines and nitrogen-vacancy centers coupled to photonic-crystal waveguides. As mentioned above, our scheme is suitable for implementing realistic long-distance quantum communication. It is worth noting that the core of our scheme is the atom-waveguide system, in which a high Purcell factor has been obtained in experiment. With the great progress in the emitter-waveguide system75,76, there is no major technical obstacle to realize our scheme.

In summary, we have proposed a scheme for a heralded quantum repeater with the Z gate based on the scattering of photons off a four-level atom in 1D waveguides. In our protocols, we choose PDC sources to double the distance between two repeater nodes without increasing the negative influence of the collective noise in the channel. Moreover, the faulty scattering events can be abandoned by detecting the polarization of output photons, which ensures the fidelity to be 100% in principle. Benefiting from the great progress in controlling atom-waveguide systems, the atomic Z gate, i.e., the main component of our protocols, has been demonstrated. Therefore, our scheme for heralded quantum repeaters is feasible with current experimental technology. One may draw inspiration from our scheme in develo** a new quantum repeater with a solid-state quantum system, such as quantum dots77,78,79,80 or nitrogen-vacancy centers81,82. Our repeater scheme will be useful in long-distance quantum communication in the future.

Methods

Single-photon dynamics

Owing to the fact that we only care about the interactions of near-resonant photons with the emitter, the quantum fields containing right- and left-going photons are completely separable28,40. Under this approximation, we can replace ak in Eq. (1) with (aR,k + aL,k). To obtain the transport property of the photon scattering with the emitter in a 1D waveguide, we assume that the photon initially comes from left with energy Ek. The general wave function for the atom-photon system can be described by

where x is the spatial coordinate along the waveguide, taking the origin x = 0 at the position of the atom, with positive to the right and negative to the left. () is a bosonic operator creating a left-propagating (right-propagating) photon and is the ground state of the system, meaning that there is no photon in the field and the atom is unexcited. The amplitudes of the photon wave-packets and could be written as28

where θ(x) is Heaviside step function, r and t are the reflection and transmission coefficients, respectively. By solving the time-independent Schrödinger equation , one can obtain the transmission coefficient r in Eq. (2).

Realization of strong coupling between emitters and 1D waveguides

The atomic Z gate is an indispensable element in our scheme, which is based on the coupling between the emitters and 1D waveguides. In the past decade, great progress has been made to realize this strong coherent coupling in both theory and experiment. In 2005, Vlasov et al.83 reported that a Purcell factor P = 60 can be experimentally observed in low-loss silicon photonic crystal waveguides. In 2006, Chang et al.84 proposed a technique that realizes a dipole emitter coupled to a nanowire or a metallic nanotip, in which the Purcell factor reaches for a silver nanowire in principle. Subsequently, some similar schemes78,85 were demonstrated experimentally that a single optical plasmon in metallic nanowires is coupled to quantum dots. In 2008, Hansen et al.79 experimentally demonstrated that spontaneous emission from single quantum dots can be coupled very efficiently to a photonic crystal waveguide, where the emitter acts as a highly reflective mirror. In 2010, by coupling single InAs/GaAs semiconductor quantum dots to a photonic crystal waveguide mode, Thyrrestrup et al.80 measured a Purcell factor of P = 5.2 in experiment. In 2013, a Purcell factor of up to 8.3 was experimentally obtained by Kumar et al.82 with a propagating plasmonic gap mode residing in between two parallel silver nanowires. Meanwhile, Hung et al.36 put forward a scheme based on strong atom-photon interactions in 1D photonic crystal waveguides. In their proposal, one atom trapped in single nanobeam structure could provide a resonant probe with transmission in theory. In 2014, Goban et al.86 realized this scheme in experiment. Recently, Kolchin et al.76 presented a scheme in which a single emitter is coupled to a dielectric slot waveguide and a high Purcell factor P = 31 is experimentally obtained.

Additional Information

How to cite this article: Song, G.-Z. et al. Heralded quantum repeater based on the scattering of photons off single emitters using parametric down-conversion source. Sci. Rep. 6, 28744; doi: 10.1038/srep28744 (2016).