1. Introduction

Quantum entanglement is a key resource in quantum information processing (QIP), especially in quantum communication, such as quantum teleportation [1], quantum dense coding [2], quantum secret sharing [3–5], quantum key distribution (QKD) [6–8], quantum secure direct communication (QSDC) [9–12], and so on. In order to perform quantum communication safely and perfectly, remote parties usually require the maximally entangled states to set up quantum channel. However, the maximally entangled quantum systems will inevitably suffer from environment noise, which will decrease the quality of entanglement and make the quantum communication insecure. Therefore, in long-distance quantum communication and quantum communication network, we need the quantum repeater to link two remote quantum nodes. The concept of quantum repeater was proposed by Briegel et al.[13] in 1998, whose basic ideal is to divide the total transmission line into segments with a shorter length at the order of the attenuation length, then entanglement purification and entanglement swapping can be used to depress the effect of noise and extend the entanglement to longer distance. In 2001, Duan et al.[14] proposed the famous DLCZ protocol and entanglement purification is an essential requirement in almost all the proposed quantum repeater protocols.

Entanglement purification is used to distill some high-fidelity maximally entangled quantum systems from a mixed state ensemble. In 1996, Bennett et al.[15] proposed the first entanglement purification protocol (EPP) to purify a Werner state, resorting to quantum controlled-NOT (CNOT) gates and bilateral rotations. Subsequently, Deutsch et al.[16] modified this protocol by adding two specific unitary transformations. As photons are often considered as the perfect flying qubits in entanglement distribution, EPP for entangled photonic systems have been widely discussed. In 2001, Pan et al.[17] proposed an EPP with only linear optical elements and single-photon detectors. They also accomplished the experimental demonstration later [18]. In 2002, Simon and Pan [19] introduced an EPP for currently available optical parametric down-conversion (PDC) source. In 2008, Sheng et al.[20] proposed an efficient EPP for a PDC source using cross-Kerr nonlinearity. The concept of deterministic entanglement purification protocol (DEPP) was proposed in 2010 based on the hyerentanglement [21–23]. In 2011, Wang et al.[24] proposed an interesting EPP using cross-Kerr nonlinearity by identifying the intensity of probe coherent beams. There are also some important EPPs for multipartite systems in Ref. [25, 26].

Entanglement concentration, on the other hand, is used to distill a subset system in a maximally entangled state from less-entangled pure state systems. In 1996, Bennett et al.[27] proposed the first entanglement concentration protocol (ECP) based on the Schmidt projection method and some collective measurements. In 1999, Bose et al.[28] presented an ECP via entanglement swapping. In 2000, Shi et al.[29] developed this method with collective unitary evaluation. Focusing on entangled photonic systems, in 2001, Zhao et al.[30] and Yamamoto et al.[31] proposed two similar ECPs with only linear optics, respectively. In 2008, Sheng et al.[32] proposed an ECP using cross-Kerr nonlinearity. Recently, two efficient single-photon-assisted ECPs for less-entangled photon pairs were proposed in Ref. [33, 34], which are far different from the previous ECPs mostly based on the Schmidt projection method. These two protocols can obtain an optimal yield by iterating the concentration process several rounds with the assistance of ancillary single photons and cross-Kerr nonlinearities.

Recent years, entanglement purification and concentration for solid state systems have been widely discussed as the solid state qubits are easy to be stored and promising candidate for QIP which exhibits longer coherence time and more scalable. For example, in 2005, Yang proposed an EPP [35] for arbitrary unknown ionic states via linear optics and an ECP [36] for unknown atomic states via entanglement swapping. Feng et al.[37] proposed an EPP for conduction electrons with charge detection. In 2006, an ECP for unknown atomic states via cavity decay was proposed by Cao et al.[38]. Reichle et al.[39] also reported their experiment for two-atom entanglement purification. In 2007, Ogden et al.[40] introduced the entanglement concentration and purification for qubit systems encoded in flying atomic pairs. In 2011, Wang et al.[41, 42] proposed an EPP and an ECP for entangled electrons based on quantum-dot spins in optical micro-cavities. In 2012, Peng et al.[43] proposed a concentration protocol for entangled atomic and photonic systems via photonic Faraday rotations.

Cavity quantum electrodynamics (QED) systems are fundamental systems in quantum optics as well as excellent candidates for solid-based QIP [44]. By exploiting the atoms interacting with local cavities as quantum nodes and the photons transmitting between remote nodes as quantum-bus, we can set up a quantum network to realize large-scale QIP. In the past decades, many theoretical and experimental works have been reported in this field [45–53]. However, most of them rely on the efficient single-photon input-output process in a high-Q cavity and the strongly coupling between the confined atoms and the high-Q cavity field. In 2009, An et al.[55] reported their work to implement QIP using single-photon input-output process with respect to low-Q cavity based on the photonic Faraday rotation, which is considerably different from the high-Q cavity and strong coupling cases. Following this scheme, various works have been presented, such as quantum logic gate [56], QIP in decoherence-free subspace [57], quantum teleportation [58], entanglement concentration [43], and so on. We noticed that the coherent optical pulses are good candidate to replace the single photons to perform cavity input-output process in cavity QED [59, 60]. Especially in 2010, Mei et al.[61] proposed an efficient entanglement distribution protocol among different single atoms confined in separated low-Q cavities by means of bringing cavity QED systems and coherent states together. Compared with other protocols also using cavity input-output process, this protocol does not need the confined atom strong coupling to a high-Q cavity, and extends the earlier protocols with single-photon to continuous variable regime, which could greatly relax the experimental requirement.

Motivated by recent progress on solid-based QIP with low-Q cavity, we investigate an atomic EPP based on the coherent state input-output process by working in low-Q cavity in the atom-cavity intermediate coupling region. The information of the entangled states are encoded in three-level configured single atoms confined in separated one-side optical micro-cavities. Through the coherent state input-output process, remote parties can construct the two-qubit parity check module (PCM) for the local atomic states, respectively, and then utilize the PCMs to distill a high-fidelity atomic entangled ensemble from an initial mixed state ensemble nonlocally. The scheme can further be used for atomic entanglement concentration, in which the two remote parties can concentrate the atom pairs in less-entangled pure states efficiently. Also by exploiting the PCM, we describe a modified ECP by introducing the ancillary single atoms in cavity QED. Compared with previous works, we use coherent optical pulse rather than single-photon to perform quantum channel, meanwhile, the coherent state input-output process is robust and scalable by working in low-Q cavity in the atom-cavity intermediate coupling region. Furthermore, the detection is based on the intensity of outgoing coherent states of the PCM, which is more efficient than single photon detection and can be easily achieved with current technologies. This paper is organized as follows: in Sec.II, we briefly introduced the coherent state input-output relation in low-Q cavity. In Sec.III and Sec.IV, we show the entanglement purification and concentration protocol between remote parties in detail. In Sec.V, we will discuss the experimental feasibility in realistic applications with current technologies. And the last section is the summary.

2. Coherent state input-output process in low-Q cavity QED regime

The idea to use the light-matter interaction in cavity QED based QIP was once presented in Ref. [54, 55] based on the state-dependent phase shift of an optical pulse reflected from a cavity coupled to a matter qubit. The theoretical analysis of the input-output relation in [54, 55] is based on the Jaynes-Cummings model. Here we replace the single-photon with coherent optical pulse to perform cavity input-output process. Suppose a three-level configured atom interacting with a one-side low-Q cavity driven by an input coherent optical pulse. The atom has two degenerate ground states |0〉 and |1〉, and an excited state |e〉. The qubit is encoded by different hyperfine levels |0〉 and |1〉. The transition |1〉 ↔ |e〉 for the atom is coupled to the cavity mode a and driven by the input field a_in, while the state |0〉 is decoupled from the cavity mode due to the large hyperfine splitting. The principle is shown in Fig. 1.

 

Fig. 1 Schematic diagram showing the principle of a coherent state |α〉 input-output process in low-Q cavity. |e〉, |0〉 and |1〉 are level structures of the atom. g is the coupling between the atom and the micro-cavity.

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According to the discussion in Ref. [55], under the assumption that the cavity decay rate κ is large enough to guarantee there is only a weak excitation by the input optical pulse on the atom initially prepared in the ground state. When ${\omega }_0-{\omega }_p\gg \frac{\gamma }{2}$, the Heisenberg-Langevin equations for the internal cavity field and the atomic operation can be adiabatically solved, and the reflection coefficient of the input optical pulse are

Suppose the initial input coherent optical pulse is prepared in the state |α〉 and the atom is prepared in a superposition state $\frac{1}{\sqrt{2}}\left(|0〉+|1〉\right)$. The evolution of the whole state with the cavity-assisted transformation can be described as:

3. Atomic entanglement purification using coherent input-output process in low-Q cavity

Now we start to explain our atomic EPP for bit-flip errors using the coherent state input-output process by working in low-Q cavity in the atom-cavity intermediate coupling region. The principle is shown in Fig. 2(a).

 

Fig. 2 (a) Schematic diagram showing the principle of the atomic EPP based on the coherent state input-output process in low-Q cavity QED system. BS denotes the 50:50 beam splitter, which transforms |α〉 |β〉 to $|\frac{\alpha -\beta }{\sqrt{2}}〉|\frac{\alpha +\beta }{\sqrt{2}}〉$. D_i (i=1,2,3 and 4) is the detector on the ith output port. Delay is time delay setup. The two devices surrounded by the dashed line are two two-qubit parity check modules (PCMs) constructed by Alice and Bob, respectively. (b) The fidelity of the present EPP in each iteration round by iterating the entanglement purification process N (N = 1,2,3 and 4) rounds. F is the initial fidelity of the state |ϕ⁺〉. It is obvious to see that the new fidelity is large than F when $F>\frac{1}{2}$, altered with the increment of N, and approximately approaches to 1 when N ≥ 4.

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Suppose the initial mixed state ensemble shared by two remote parties Alice and Bob can be described as

For example, if the four atoms are in the state |ϕ⁺〉_a1b1|ϕ⁺〉_a2b2, the evolution of the whole system is

Similarly, the evolution of the other three cases can be described by:

Alice and Bob discard all the items only if both their detected results on outgoing coherent states are $|-\sqrt{2}\alpha 〉|0〉$ (or $|0〉|\sqrt{2}\alpha 〉$). Finally, four atoms in the state |ψ⁺〉_a1b1|ψ⁺〉_a2b2 cannot be discarded as Alice and Bob can not distinguish the two cases that a₁b₁ and a₂b₂ both contain bit-flip errors. So the two cases |ϕ⁺〉_a1b1|ϕ⁺〉_a2b2 and |ψ⁺〉_a1b1|ψ⁺〉_a2b2 are preserved with probabilities of F² and (1 − F)², respectively. That is, based on the post-selection principle according to the detected results on outgoing coherent states, Alice and Bob can eventually preserved a new mixed state ensemble with a fidelity F′ = F²/[F² + (1 − F)²], which is larger than F when $F>\frac{1}{2}$.

Up to now, we have briefly introduced our atomic EPP for bit-flip errors. In this protocol, the PCM is designed to replace the CNOT gate in original EPP [15], it can be used to distinguish |0〉|0〉 and |1〉|1〉 states from |0〉|1〉 and |1〉|0〉 states of the two atoms. The advantage of our PCM is the use of coherent state pulse from standard stabilized laser source as quantum channel, which do not need the atoms to interact with each other directly. Thus, this protocol is quantum nondestructive like the protocols with cross-Kerr nonlinearity [20]. At the output ports, the detection is based on the intensity of outgoing coherent states, after two coherent states in two arms of the PCM interfered with each other on a 50:50 BS. Ideally, any response of the detector indicates that the coherent state is $|\pm \sqrt{2}\alpha 〉$, which is different from |0〉, and the error probability of the coherent states comparison is P_error = exp(−2|α|²). Such detector could be simple photodiode.

Based on our PCM, the remaining entangled atoms can be expected to obtain a higher fidelity. That is, by iterating the purification process for the preserved atom pairs, the entanglement fidelity can be improved further. We calculate the fidelity in each iteration round by iterating the entanglement purification process N rounds and numerically simulated them in Fig. 2(b). We can conclude that it is altered with the increment of N and approximately approaches to 1 when N ≥ 4. Phase-flip errors may also occur during the interaction with environment noise. As discussed in Ref. [17, 18], phase-flip errors can be transformed into bit-flip errors with Hadamard operations. So we only discuss bit-flip errors entanglement purification here and the phase-flip errors can be purified in a next round. That is, this atomic EPP can purify an arbitrary mixed state ensemble.

4. Atomic entanglement concentration using coherent input-output process in low-Q cavity

The scheme in Fig. 2(a) can further be employed for atomic entanglement concentration, based on the discussion in Ref. [31]. Suppose the two pairs of unknown less-entangled pure state atoms shared by Alice and Bob can be described as:

Moreover, if Alice know the coefficients m and n beforehand, the ECP can be further simplified with an ancillary particle, by exploiting a previously known protocol in Ref. [33]. The principle of the modified ECP is shown in Fig. 3(a). We can introduce an ancillary atom a2 confined in a low-Q cavity A2 on Alice’s side, whose parameters are same as the atom a1 confined in cavity A1 and the initial state is prepared in

 

Fig. 3 (a) Schematic diagram showing the principle of the modified ECP based on coherent state input-output process in low-Q cavity. a2 is an ancillary atom confined in a low-Q cavity A2, whose parameters are same as the atom a1 confined in cavity A1. BS denotes a 50:50 beam splitter, which transforms |α〉 |β〉 to $|\frac{\alpha -\beta }{\sqrt{2}}〉|\frac{\alpha +\beta }{\sqrt{2}}〉$D_i (i = 1, 2) is detector on the ith output port. Delay is time delay setup. (b) The success probability of the modified ECP in each iteration round by iterating the entanglement concentration process N (N=1,2,3 and 4) rounds. m is the coefficient of the initial state. It is obvious to see that the success probability is relatively high when N ≥ 4.

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Fig. 4 The success probability of the present EPP and ECP with respect to $\frac{\gamma }{g}=0.05$, 0.1 0.15 and 0.2 when η_D = 0.7, $\eta =\frac{1}{3}$, α = 3, the transmission rate through other optical components is 0.9. Here we only perform the entanglement purification and concentration process one round. The success probability of entanglement purification in an ideal condition is F² +(1 − F)². After considering the error probabilities P_e1 and P_e2, the success probability will decrease as shown in Fig. 4(a). The success probability of entanglement concentration is 2|αβ|², which will decrease such as in Fig. 4(b). As discuss above, the error probabilities P_e1 and P_e2 do not affect the fidelity of the mixed state based on the post-selection principle in the EPP, and the success probabilities in the ECP can be improved further by iterating the modified protocol several rounds.

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5. Experimental feasibility

The key element of the EPP and ECP is the PCM constructed by low-Q cavities with coherent optical pulse. According to the discussions above, the detuning of the input coherent state with respect to the atomic resonance frequency and the cavity mode are required to be the same as ${\omega }_0-{\omega }_p={\omega }_c-{\omega }_p=\frac{\kappa }{2}$. Thus, the frequency of the input coherent state ω_p should be set to satisfy this condition when the atom and cavity mode have been adjusted in resonant interaction. Suppose the cavities in our protocols are Fabry-Perot (F-P) cavities, then the atom-cavity coupling strength depends on the atomic position, which can be described as

Furthermore, though our protocols can be realized in the context of low-Q cavities (κ ≫ γ), the value of the Q should not be too low for avoiding reducing the strength of the atom-field coupling. In order to guarantee a higher efficiency, the ultralow-Q cavity may not be suitable for our protocols, though the condition $g=\frac{\kappa }{\sqrt{2}}$ may also be satisfied in this case. In the experiment, the Q value and the decay rate κ are closely related to the cavity finesse f. For the F-P cavity such as in Ref. [65], we can set the coupling strength to peak coupling strength g₀, then decrease the cavity finesse to 4×10³ magnitude to satisfy $g=\frac{\kappa }{\sqrt{2}}$, and the relevant Q factor $Q=\frac{{\omega }_c}{2\kappa }$ will decreases to 4 × 10⁵ magnitude. In Ref. [66], Dayan et al. realized intermediate atom-cavity coupling in the micro-toroidal resonator (MTR) “bad cavity”. The Q value in this work is more than 10⁴ in the low-Q cavity limit ( $\kappa \gg \frac{g^2}{\kappa }\gg \gamma$), which is also very suitable in our protocols. The requirement of the Q value in our protocols should be larger than 10². Moreover, we would like to point out that although the Q can be low, a high Q/V is usually required in cavity QED systems, in which V is the mode volume. So semiconductor systems may be better than atomic systems in some cavity QED based QIP for the mode volume is usually much smaller in semiconductor systems [59, 60].

There are still some imperfections in realistic applications. For example, small perturbations of the parameters of the cavity may lead to detuning of the cavity mode with respect to the atomic resonance frequency (ω_c ∼ ω₀). The variation of the atomic position in the cavity may lead to the mismatch of the coupling strength ( $g~\frac{\kappa }{\sqrt{2}}$). These will slightly change the phase shifts θ₀ and θ₁ in the coherent state input-output process. We detect the intensity of outgoing coherent state instead of the direct homodyne measurement to overcome this imperfection. Experimental schemes, such as optical lattice [67] or electromagnetic field [68] can be used to fix the atoms stably in the cavities. The photon loss occurs due to the absorption and scattering of the cavity mirror and the fiber. After considering the photon loss rate η and the efficiency of the detectors η_D, the error probability coming from the detections can be defined to P_e1 = exp[−2η_D(1 − η)|α|²]. Moreover, the atomic decay rate γ may also effect the success probability of the protocols. The error probability caused by atomic decay rate in the intermediate coupling region in low-Q cavity is $P_{e2}=\frac{\gamma }{g}$[55,56,62]. Suppose η_D = 0.7, $\eta =\frac{1}{3}$, α = 3, the transmission rate through other optical components is 0.9, then we can numerically simulated the success probability in the present EPP and ECP with respect to $\frac{\gamma }{g}$ in Fig. 4. Here we only perform the entanglement purification and concentration process one round. In our EPP, the success probability in an ideal condition is F² + (1 − F)². After considering the error probabilities P_e1 and P_e2, the success probability will decrease as shown in Fig. 4(a). However, is does not affect the fidelity of the mixed state based on the post-selection principle. In our ECP, the success probability 2|mn|² will decrease such as shown in Fig. 4(b), while we can utilize our modified protocol by introducing ancillary atoms in cavity QED to improve the success probability further.

Our protocols are based on the PCM by working in low-Q cavity in the atom-cavity intermediate coupling region, which is considerably different from the high-Q cavity and strong coupling cases [36, 38, 40]. In the previous works [36, 40], the atom is used as the flying qubit, where in our protocol, the atom is used as stationary qubit. Meanwhile, the sophisticated single-photon source and single-photon detector have been successfully avoided. The present protocols can further be extended to multi-atom Greenberger-Horne-Zeilinger (GHZ) state or W state purification or concentration, following some ideas in multipartite EPP and ECP for entangled photonic systems.

6. Summary

In summary, we have proposed an atomic entanglement purification protocol based on the coherent state input-output process in low-Q cavity QED system. The information of the entangled states are encoded in three-level configured single atoms confined inside separate one-side optical micro-cavities. Through the coherent state input-output process, remote parties can construct the two-qubit PCM for the local atomic states, respectively. And then they can utilize the PCMs to distill a high-fidelity atomic entangled ensemble from a initial mixed state ensemble nonlocally. The scheme can also be used for atomic entanglement concentration, in which the two remote parties can concentrate the atom pairs in less-entangled pure states efficiently. Also by exploiting the PCM, we describe a modified ECP by introducing the ancillary single atoms in cavity QED. The proposed protocols only require the coherent states as quantum channel, and the detection is based on the intensity of outgoing coherent states, which can be easily achieved with current technologies. Meanwhile, the coherent state input-output process is robust and scalable by working in low-Q cavity in the atom-cavity intermediate coupling region. Thus, the atomic EPP and ECP may be widely used in large-scaled and solid-based quantum repeater and QIP protocols. Moreover, the PCM is exploited instead of the CNOT gates, which could be widely used in entanglement generation, Bell-state analysis, quantum cloning, and so on.