1 Introduction

Entanglement has been regarded as a key source in the tasks of quantum computation and quantum communication, for it can hold the power for the quantum nonlocality [1] and provide wide applications [2–25]. In all the applications, the ideal entangled state is the maximally entangled state. For example, in the quantum teleportation [12–14], quantum dense coding [15], quantum secret sharing and quantum state sharing [16–19], quantum communication [20–22], and entanglement-based quantum key distribution [23–25], the parties need to use the maximally entangled state to set up the quantum entanglement channel. Unfortunately, during the transmission and storage process, the maximally entangled state may inevitably interact with the channel noise from the environment, which can make the maximally entangled state degrade to the mixed state or pure less-entangled state. In the application process, the mixed state and pure less-entangled state may decrease after the entanglement swapping and cannot ultimately set up the high quality quantum entanglement channel [26], so that we need to recover the mixed state or pure less-entangled state into the maximally entangled state.

Entanglement concentration, which will be detailed here, is a powerful way to recover the pure less-entangled state into the maximally entangled state probabilistically [27–48]. In 1996, Bennett et al. proposed the first entanglement concentration protocol (ECP), which is called the Schmidt projection method [27]. Since then, various ECPs have been put forward successively. For example, in 1999, Bose et al. proposed the ECP based on entanglement swapping [28], which was improved by the group of Shi later [29]. In 2001, Zhao et al. and Yamamoto et al. proposed two similar ECPs independently with linear optical elements [30, 31], both of which were realized in experiment later. During the past few years, Sheng et al. have improved the two ECPs with the help of the cross-Kerr nonlinearity [32–34], and further proposed an ECP for the less-entangled single-photon state [35]. Recently, Deng proposed an efficient ECP for the multiphoton systems in a partially entangled pure state with the help of the projection measurement on an additional photon. This protocol has the maximal success probability, which is just equivalent to the entanglement of the partially entangled state [36]. In 2013, Ren et al. put forward a series of high efficient linear optical ECPs for two-photon four-qubit systems in partially hyperentangled states in both the spatial mode and the polarization mode [37]. So far, most ECPs have focused on photon state, for photon is the best candidate for optimal transmission. Actually, the solid atom is also a good candidate for quantum communication and computation. During the past decade, the cavity quantum electrodynamics (QED) have become a powerful platform for the QIP of the photon-atom states, due to the controllable interaction between atoms and photons [49–57]. Especially, many researchers have showed that with the atoms strongly interacting with local high-quality (Q) cavities, the spatially separated cavities could serve as quantum nodes, and construct a quantum network assisted by the photons acting as a quantum bus [57–60]. However, the requirements for high-Q cavities and strong coupling to the confined atoms are stringent for current techniques. Moreover, the high-Q cavity, which needs to be well isolated from the environment, seems unsuitable for efficiently accomplishing the input-output process of the photons.

In 2008, Dayan et al. successfully realized the microtoroidal resonator (MTR), for which some theoretical QIP works can be carried out with moderate cavity-atom coupling [61]. It provides us a possible way to construct the quantum information processing with optical low-Q cavities. If we can combine the input-output process with low-Q cavities, we can accomplish high-quality QIP tasks with currently available techniques. In 2009, An et al. put forward an innovative scheme to implement the QIP tasks by moderate cavity-atom coupling with low-Q cavities [62]. They have shown that when a photon interacts with an atom trapped in a low-Q cavity, different polarization of the input photon can cause different phase rotation on the output photon, which is called the photonic Faraday rotation. The photonic Faraday rotation has attracted great attention, for it only works in low-Q cavities and is insensitive to both cavity decay and atomic spontaneous emission [62]. Based on the Faraday rotation, the protocols for entanglement generation [63], quantum teleportation [64], controlled teleportation [65], quantum logic gates [66], and entanglement swapping [67] were proposed successively. In 2012, Peng et al. proposed an ECP for the less-entangled N-atom Greenberger-Horne-Zeilinger (GHZ) state with the help of the photonic Faraday rotation [48]. In the ECP, they require two pairs of the less-entangled GHZ states, and after the ECP, at least one pair of maximally entangled GHZ state can be successfully distilled. However, this ECP is not optimal for two reasons. First, two pairs of less-entangled atom states are not necessary. Second, the success probability of this ECP is relatively low. In this paper, we will put forward an improved ECP for less-entangled N-atom GHZ state. In our protocol, we only require a pair of less-entangled N-atom GHZ state and an auxiliary atom, all of which are trapped in the low-Q cavities. With the help of the photonic Faraday rotation, we can successfully distill the maximally entangled atom state with the same success probability as Ref. [48]. Moreover, our ECP can be used repeatedly to further concentrate the discarded items in Ref. [48] and obtain a higher success probability. Especially, if we consider the practical operation and imperfect detection, our ECP is more powerful.

This paper is organized as follows: In Sect. 2, we first explain the basic principle of the photonic Faraday rotation. In Sect. 3, we explain our ECP for the less-entangled two-atom state. In Sect. 4, we extend this ECP to concentrate the N-atom GHZ state, which shows that this ECP is more convenient in practical experiment. In Sect. 5, we make a discussion and summary.

2 Photonic Faraday Rotation

As the photonic Faraday rotation is the key operation in our ECP, before explaining the details of our ECP, we introduce the basic principle of the photonic Faraday rotation firstly. As shown in Fig. 1(a), we suppose that a three-level atom is trapped in the low-Q cavity (one side), where the states |g L 〉 and |g R 〉 represent two Zeeman sublevels of its degenerate ground state, and |e〉 represents its excited state. A single photon pulse with the frequency ω p enters the cavity and reacts with the three-level atom. The single photon state can be written as

$$\begin{aligned} |\varphi_{in}\rangle=\frac{1}{\sqrt{2}}(|L\rangle+|R \rangle), \end{aligned}$$
(1)

where |L〉 and |R〉 represent the left-circularly polarization and right-circularly polarization, respectively. The interaction between the single photon and the three-level atom can lead the transitions |g L 〉↔|e〉 ( |g R 〉↔|e〉) for the atom absorbing or emitting a |L〉 ( |R〉) circularly polarized photon. The Hamiltonian of the whole system can be described as [13, 14, 62, 64, 68],

$$\begin{aligned} H=H_{0}+\hbar\lambda\sum_{j=L,R} \bigl(a^{\dagger}_{j}\sigma_{j-}+a_{j}\sigma _{j+}\bigr)+H_{R}, \end{aligned}$$
(2)

with

$$\begin{aligned} H_{0}=\sum_{j=L,R}\biggl[\frac{\hbar\omega_{0}}{2} \sigma_{jz}+\hbar\omega _{c}a^{\dagger}_{j}a_{j} \biggr], \end{aligned}$$
(3)

and

$$\begin{aligned} H_{R} =&H_{R0}+i\hbar\biggl[\int_{-\infty}^{\infty}d \omega\sum_{j=L,R}\alpha (\omega) \bigl(b^{\dagger}_{j}( \omega)a_{j}+b_{j}(\omega)a^{\dagger }_{j} \bigr) \\ &{}+\int_{-\infty}^{\infty}d\omega\sum _{j=L,R}\bar{\alpha}(\omega ) \bigl(c^{\dagger}_{j}( \omega)\sigma_{j-}+c_{j}(\omega)\sigma_{j+}\bigr) \biggr]. \end{aligned}$$
(4)

Here, λ is the atom-field coupling constant, while \(a^{\dagger}_{j}\) and a j (j=L,R) are the creation and annihilation operators of the filed-mode in the cavity, respectively. σ L− and σ L+ (σ R− and σ R+) are the lowering and raising operators of the transition L (R), respectively, and ω c (ω 0) is the atomic (field) frequency. In Eq. (4), H R0 represents the Hamiltonian of the free reservoirs, while b j and c j (\(b^{\dagger}_{j}\) and \(c^{\dagger}_{j}\)) are the annihilation (creation) operators of the reservoirs.

Fig. 1
figure 1

A schematic drawing of the interaction between the photon pulse and the three-level atom in the low-Q cavity. (a) the three-level atom trapped in the low-Q cavity. |g L 〉 and |g R 〉 represent two Zeeman sublevels of its degenerate ground state, and |e〉 represents its excited state. (b) The interaction between the photon pulse and the three-level atom. The state |g L 〉 and |g R 〉 couple with a left (L) polarized and a right (R) polarized photon, respectively

Full size image

Considering the low-Q cavity limit and the weak excitation limit, we can solve the Langevin equations of motion for cavity and atomic lowering operators analytically, and obtain a single relation between the input and output single-photon state in the form of [62]

$$\begin{aligned} r(\omega_{p})\equiv\frac{a_{out,j(t)}}{a_{in,j(t)}}=\frac{[i(\omega _{c}-\omega_{p})-\frac{\kappa}{2}][i(\omega_{0}-\omega_{p})+\frac{\gamma }{2}]+g^{2}}{ [i(\omega_{c}-\omega_{p})+\frac{\kappa}{2}][i(\omega_{0}-\omega _{p})+\frac{\gamma}{2}]+g^{2}}, \end{aligned}$$
(5)

where κ and γ are the cavity damping rate and atomic decay rate, respectively, and g is the atom-cavity coupling strength. Equation (5) is a general expression. In the case of the atom uncoupled to the cavity, which makes g=0, Eq. (5) can be simplified as

$$\begin{aligned} r_{0}(\omega_{p})=\frac{i(\omega_{c}-\omega_{p})-\frac{\kappa }{2}}{i(\omega_{c}-\omega_{p})+\frac{\kappa}{2}}. \end{aligned}$$
(6)

It is obvious that Eq. (6) can be written as a pure phase shift as \(r_{0}(\omega_{p})=e^{i\phi_{0}}\). On the other hand, during the interaction process in the cavity, the photon experiences an extremely weak absorption, so that we can consider that the output photon only experiences a pure phase shift without any absorption for a good approximation. In this case, with strong κ, weak γ and g, Eq. (5) can be rewritten as r(ω p )≃e iϕ. In this way, if the photon pulse takes action, the output photon state will convert to |φ out 〉=r(ω p )|L(R)〉≃e iϕ|L(R)〉, otherwise, the single-photon pulse would only sense the empty cavity, and the output photon state will convert to \(|\varphi _{out}\rangle=r_{0}(\omega_{p})|L (R)\rangle= e^{i\phi_{0}}|L (R)\rangle \). Therefore, for an input single-photon state as Eq. (1), if the initial atom state is |g L 〉, the output photon state can be described as

$$\begin{aligned} |\varphi_{out}\rangle_{-}=\frac{1}{\sqrt{2}} \bigl(e^{i\phi}|L\rangle+e^{i\phi _{0}}|R\rangle\bigr), \end{aligned}$$
(7)

while if the initial atom state is |g R 〉, the output photon state is

$$\begin{aligned} |\varphi_{out}\rangle_{+}=\frac{1}{\sqrt{2}} \bigl(e^{i \phi_{0}}|L\rangle +e^{i \phi}|R\rangle\bigr). \end{aligned}$$
(8)

Finally, it can be found the polarization direction of the output photon rotates an angle as \(\varTheta^{-}_{F}=\frac{\phi_{0}-\phi}{2}\) or \(\varTheta^{+}_{F}=\frac{\phi-\phi_{0}}{2}\), which is called as the photonic Faraday rotation.

Based on the photonic Faraday rotation, it can be seen that in a certain case, i.e., ω 0=ω c , \(\omega_{p}=\omega _{c}-\frac{\kappa}{2}\), and \(g=\frac{\kappa}{2}\), we can get ϕ=π and \(\phi_{0}=\frac{\pi}{2}\), so that the relation between the input and output photon state can be simplified as [48]

$$\begin{aligned} \begin{aligned} &|L\rangle|g_{L}\rangle\rightarrow-|L\rangle|g_{L} \rangle,\qquad |R\rangle|g_{L}\rangle\rightarrow i|R \rangle|g_{L}\rangle, \\ &|L\rangle|g_{R}\rangle\rightarrow i|L\rangle|g_{R} \rangle,\qquad |R\rangle|g_{R}\rangle\rightarrow-|R \rangle|g_{R}\rangle. \end{aligned} \end{aligned}$$
(9)

Under this special case, the photonic Faraday rotation can be used to perform the entanglement concentration.

3 The ECP for Less-Entangled Two-Atom State

We firstly explain our ECP for the simplest case, that is, the arbitrary less-entangled two-atom state. The schematic drawing of our ECP is shown in Fig. 2. We suppose the two parties Alice and Bob share a pair of less-entangled two-atom state with the form

$$\begin{aligned} |\psi\rangle_{12}=\alpha|g_{L}g_{R} \rangle_{12}+\beta|g_{R}g_{L}\rangle _{12}, \end{aligned}$$
(10)

where the two atoms are marked as 1 and 2, respectively. α and β are the initial entanglement coefficients of the atom state, |α|2+|β|2=1.

Fig. 2
figure 2

A schematic drawing of our ECP for the pure less-entangled two-atom state. The target less-entangled atoms 1 and 2 are trapped in the low-Q cavity b and c, while the auxiliary atom 3 is trapped in the low-Q cavity a. A single-photon pulse passes through the low-Q cavities a and b, and interacts with the atom 3 and 1, successively. By measuring the states of the auxiliary three-level atom and the output photon, we can distill the maximally entangled atom state with some success probability. Moreover, our ECP can be used repeatedly to further concentrate the less-entangled atom state

Full size image

In our ECP, Alice and Bob make the two atoms trapped in the low-Q cavities b and c, respectively. Alice firstly introduces an auxiliary three-level atom 3 with the form

$$\begin{aligned} |\psi\rangle_{3}=\beta|g_{L}\rangle_{3}+ \alpha|g_{R}\rangle_{3}, \end{aligned}$$
(11)

and makes it trapped in the low-Q cavity a. Here, for preparing the auxiliary single-atom state, we need to know the exact value of α and β in advance. According to the previous research results, we can obtain the exact value of α and β by measuring an enough amount of the target less-entangled states [29, 32–34, 36, 43, 44].

Then, Alice makes a single-photon pulse with the form of \(|\phi\rangle =\frac{1}{\sqrt{2}}(|L\rangle+|R\rangle)\) pass through the cavities a and b, sequentially. Based on Eq. (9), we can get the relation between the input and the output photon state as

$$\begin{aligned} \begin{aligned} &|L\rangle|g_{L}g_{L}\rangle\rightarrow|L \rangle|g_{L}g_{L}\rangle ,\qquad|R\rangle|g_{L}g_{L} \rangle\rightarrow-|R\rangle |g_{L}g_{L}\rangle, \\ &|L\rangle|g_{L}g_{R}\rangle\rightarrow-i|L \rangle|g_{L}g_{R}\rangle ,\qquad|R\rangle|g_{L}g_{R} \rangle\rightarrow-i|R\rangle |g_{L}g_{R}\rangle, \\ &|L\rangle|g_{R}g_{L}\rangle\rightarrow-i|L \rangle|g_{R}g_{L}\rangle ,\qquad|R\rangle|g_{R}g_{L} \rangle\rightarrow-i|R\rangle |g_{R}g_{L}\rangle, \\ &|L\rangle|g_{R}g_{R}\rangle\rightarrow-|L \rangle|g_{R}g_{R}\rangle ,\qquad|R\rangle|g_{R}g_{R} \rangle\rightarrow|R\rangle |g_{R}g_{R}\rangle. \end{aligned} \end{aligned}$$
(12)

In this way, after the photon pulse emitting from the low-Q cavity b, the photo state combined with the three-atom states can evolve to

$$\begin{aligned} |\varPsi\rangle_{out} =&\alpha\beta|L\rangle|g_{L}g_{R}g_{L} \rangle _{123}+(-\alpha\beta)|R\rangle|g_{L}g_{R}g_{L} \rangle_{123} +\bigl(-i\beta^{2}\bigr)|L\rangle|g_{R}g_{L}g_{L} \rangle_{123} \\ &{}+\bigl(-i\beta^{2}\bigr)|R\rangle|g_{R}g_{L}g_{L} \rangle_{123}+\bigl(-i\alpha ^{2}\bigr)|L\rangle|g_{L}g_{R}g_{R} \rangle_{123} +\bigl(-i\alpha^{2}\bigr)|R\rangle|g_{L}g_{R}g_{R} \rangle_{123} \\ &{}+(-\alpha\beta)|L\rangle|g_{R}g_{L}g_{R} \rangle_{123}+\alpha\beta |R\rangle|g_{R}g_{L}g_{R} \rangle_{123}. \end{aligned}$$
(13)

Then Alice performs the Hadamard operation on the atom 3 by driving atom 3 with an external classical field (polarized lasers), which makes

$$\begin{aligned} |g_{L}\rangle\rightarrow\frac{1}{\sqrt{2}}(|g_{L} \rangle+|g_{R}\rangle ), \qquad|g_{R}\rangle\rightarrow \frac{1}{\sqrt{2}}(|g_{L}\rangle -|g_{R}\rangle). \end{aligned}$$
(14)

After the Hadamard operation, Eq. (13) can evolve to

$$\begin{aligned} |\varPsi\rangle_{out} \rightarrow&\bigl(\alpha\beta|L-R\rangle |g_{L}g_{R}\rangle_{12}-i\beta^{2}|L+R \rangle|g_{R}g_{L}\rangle_{12} -i \alpha^{2}|L+R\rangle|g_{L}g_{R} \rangle_{12} \\ &{}-\alpha\beta|L-R\rangle|g_{R}g_{L}\rangle_{12} \bigr)|g_{L}\rangle _{3}+\bigl(\alpha\beta|L-R \rangle|g_{L}g_{R}\rangle_{12} -i \beta^{2}|L+R\rangle|g_{R}g_{L} \rangle_{12} \\ &{}+i\alpha^{2}|L+R\rangle|g_{L}g_{R} \rangle_{12}+\alpha\beta|L-R\rangle |g_{R}g_{L} \rangle_{12}\bigr)|g_{R}\rangle_{3}. \end{aligned}$$
(15)

After that, Alice makes the output photon pass through a quarter-wave plate (QWP), which can make

$$\begin{aligned} |L\rangle\rightarrow\frac{1}{\sqrt{2}}(|H\rangle+|V\rangle), \qquad |R\rangle \rightarrow\frac{1}{\sqrt{2}}(|H\rangle-|V\rangle). \end{aligned}$$
(16)

After the QWP, Eq. (15) can ultimately evolve to

$$\begin{aligned} |\varPsi_{1}\rangle_{out} =&\alpha\beta(|g_{L}g_{R} \rangle _{12}-|g_{R}g_{L}\rangle_{12})|V \rangle|g_{L\rangle_{3}} -\bigl(i\alpha^{2}|g_{L}g_{R} \rangle_{12}+i\beta^{2}|g_{R}g_{L} \rangle _{12}\bigr)|H\rangle|g_{L}\rangle_{3} \\ &{}+\alpha\beta(|g_{L}g_{R}\rangle_{12}+|g_{R}g_{L} \rangle_{12})|V\rangle |g_{R\rangle_{3}} +\bigl(i\alpha^{2}|g_{L}g_{R} \rangle_{12}-i\beta^{2}|g_{R}g_{L} \rangle _{12}\bigr)|H\rangle|g_{R}\rangle_{3}. \end{aligned}$$
(17)

Finally, Alice makes the photon pass through the PBS, which can transmit the |H〉 photon and reflect the |V〉 photon. After the PBS, both the output photon state and the auxiliary atom state are detected by the detectors. If the measurement result is |V〉|g R 〉3, Eq. (17) will collapse to

$$\begin{aligned} |\psi_{1}\rangle_{12}=\frac{1}{\sqrt{2}}(|g_{L}g_{R} \rangle _{12}+|g_{R}g_{L}\rangle_{12}), \end{aligned}$$
(18)

while if the measurement result is |V〉|g L 〉3, Eq. (17) will collapse to

$$\begin{aligned} |\psi'_{1}\rangle_{12}=\frac{1}{\sqrt{2}}(|g_{L}g_{R} \rangle _{12}-|g_{R}g_{L}\rangle_{12}). \end{aligned}$$
(19)

It can be found that both the Eq. (18) and Eq. (19) are the maximally-entangled atom states, and there is only a phase difference between them. Equation (19) can be easily converted to Eq. (18) by the phase flip operation. So far, our ECP is completed and we have successfully distilled the maximally-entangled atom state, with the success probability of P=2|αβ|2, which is the same as that in Ref. [48]. On the other hand, if the measurement result is |H〉|g L 〉3, Eq. (17) will collapse to

$$\begin{aligned} |\psi_{2}\rangle_{12}=\alpha^{2}|g_{L}g_{R} \rangle_{12}+\beta ^{2}|g_{R}g_{L} \rangle_{12}, \end{aligned}$$
(20)

while if the result is |H〉|g R 〉3, Eq. (17) will collapse to

$$\begin{aligned} |\psi'_{2}\rangle_{12}=\alpha^{2}|g_{L}g_{R} \rangle_{12}-\beta ^{2}|g_{R}g_{L} \rangle_{12}. \end{aligned}$$
(21)

Equation (21) can also be converted to Eq. (20) by the phase flip operation. Under these two cases, our ECP fails. However, it is interesting that Eq. (20) has the similar form as Eq. (10), that is to say, Eq. (20) is a new less-entangled atom state and can be reconcentrated for the next round. According to the concentration step described above, in the second concentration round, Alice introduces another auxiliary atom 3′ with the form

$$\begin{aligned} |\psi\rangle_{3'}=\beta^{2}|g_{L} \rangle_{3'}+\alpha^{2}|g_{R}\rangle _{3'}, \end{aligned}$$
(22)

and also makes it trapped in the low-Q cavity a. By making a single-photon pulse with the form of \(|\phi\rangle=\frac{1}{\sqrt {2}}(|L\rangle+|R\rangle)\) pass through the cavities a and b successively, the new less-entangled atom state combined with the single-photon state can evolve to

$$\begin{aligned} |\varPsi'\rangle_{out} =&\alpha^{2} \beta^{2}|L\rangle |g_{L}g_{R}g_{L} \rangle_{123'}+\bigl(-\alpha^{2}\beta^{2}\bigr)|R \rangle |g_{L}g_{R}g_{L}\rangle_{123'} + \bigl(-i\beta^{4}\bigr)|L\rangle|g_{R}g_{L}g_{L} \rangle_{123'} \\ &{}+\bigl(-i\beta^{4}\bigr)|R\rangle|g_{R}g_{L}g_{L} \rangle_{123'} +\bigl(-i\alpha^{4}\bigr)|L\rangle|g_{L}g_{R}g_{R} \rangle_{123'} +\bigl(-i\alpha^{4}\bigr)|R\rangle|g_{L}g_{R}g_{R} \rangle_{123'} \\ &{}+\bigl(-\alpha^{2}\beta^{2}\bigr)|L\rangle|g_{R}g_{L}g_{R} \rangle_{123'}+\alpha ^{2}\beta^{2}|R \rangle|g_{R}g_{L}g_{R}\rangle_{123'}. \end{aligned}$$
(23)

In this way, Alice performs the Hadamard operation on the atom 3′ and the single photon, and measures the quantum states of the auxiliary atom and the output photon. If the measurement result is |V〉|g R 〉3′, Eq. (23) will collapse to Eq. (18), while if the measurement result is |V〉|g L 〉3′, Eq. (23) will collapse to Eq. (19). Therefore, in the second concentration round, we can successfully distill the maximally entangled atom state with the probability \(P_{2}=\frac{2|\alpha\beta |^{4}}{|\alpha|^{4}+|\beta|^{4}}\), where the subscript 2′ means in the second concentration round. On the other hand, if the measurement result is |H〉|g L 〉3′ or |H〉|g R 〉3′, the second concentration round fails. However, by preparing another auxiliary atom, its discarded items can also be further concentrated in the third concentration round. Therefore, our ECP can be used repeatedly to further concentrate the less-entangled two-atom state.

4 The ECP for Less-Entangled N-Atom GHZ State

Our ECP can be extended to concentrate the less-entangled N-atom GHZ state. As shown in Fig. 3, we suppose N entangled three-level atoms marked as 1,2,…,N are possessed by N parties, say Alice, Bob, Charlie and so on, respectively. Each of the N parties makes an atom trapped in a low-Q cavity in his or her position. Here, the low-Q cavity in Alice’s position is marked as a, while other N−1 low-Q cavities in other parties’ position are marked as b 1, … , b N−1, respectively. In this way, we suppose the less-entangled N-atom GHZ state can be written as

$$\begin{aligned} |\psi\rangle_{N}=\alpha|g_{L}g_{R}\cdots g_{R}\rangle_{ab_{1}\cdots b_{N-1}}+\beta|g_{R}g_{L} \cdots g_{L}\rangle_{ab_{1}\cdots b_{N-1}}. \end{aligned}$$
(24)
Fig. 3
figure 3

A schematic drawing of our ECP for the pure less-entangled N-atom GHZ state. The less-entangled N atoms are trapped in N low-Q cavities with the name of a, b 1, b 2,… , b N−1, which are hold by N parties, respectively. Alice introduces an auxiliary single atom and makes it trapped in the low-Q cavity a 1 in her hand. Then, she makes a single-photon pulse pass through the low-Q cavities a 1 and a, successively. After the interaction between the photon and the atoms, by measuring the states of the auxiliary atom and the output photon, we can successfully distill the maximally entangled N-atom state. Our ECP can also be used repeatedly to further concentrate the less-entangled N-atom GHZ state

Full size image

Similarly, Alice prepares an auxiliary single atom with the form of Eq. (11) and makes it trapped in another low-Q cavity a 1 in her position. Then, Alice makes a single-photon pulse with the form \(|\phi\rangle=\frac{1}{\sqrt{2}}(|L\rangle+|R\rangle)\) pass through the cavity a 1 and a, successively. After the photon passing through the two cavities, the output photon state combined with the whole N+1 atom state can evolve to

$$\begin{aligned} |\varPsi\rangle_{outN} =&\alpha\beta|L\rangle|g_{L}g_{L} \rangle _{a_{1}a}|g_{R}\cdots g_{R} \rangle_{b_{1}\cdots b_{N-1}}+(-\alpha\beta )|R\rangle|g_{L}g_{L} \rangle_{a_{1}a}|g_{R}\cdots g_{R}\rangle _{b_{1}\cdots b_{N-1}} \\ &{}+(-\alpha\beta)|L\rangle|g_{R}g_{R}\rangle_{a_{1}a}|g_{L} \cdots g_{L}\rangle_{b_{1}\cdots b_{N-1}}+\alpha\beta|R\rangle |g_{R}g_{R}\rangle_{a_{1}a}|g_{L}\cdots g_{L}\rangle_{b_{1}\cdots b_{N-1}} \\ &{}+\bigl(-i\alpha^{2}\bigr)|L\rangle|g_{R}g_{L} \rangle_{a_{1}a}|g_{R}\cdots g_{R} \rangle_{b_{1}\cdots b_{N-1}} \\ &{}+\bigl(-i\alpha^{2}\bigr)|R\rangle |g_{R}g_{L}\rangle_{a_{1}a}|g_{R}\cdots g_{R}\rangle_{b_{1}\cdots b_{N-1}} \\ &{}+\bigl(-i\beta^{2}\bigr)|L\rangle|g_{L}g_{R} \rangle_{a_{1}a}|g_{L}\cdots g_{L} \rangle_{b_{1}\cdots b_{N-1}} \\ &{}+\bigl(-i\beta^{2}\bigr)|R\rangle |g_{L}g_{R}\rangle_{a_{1}a}|g_{L}\cdots g_{L}\rangle_{b_{1}\cdots b_{N-1}}. \end{aligned}$$
(25)

Then, Alice performs the Hadamard operation on the auxiliary single atom and Eq. (25) will convert to

$$\begin{aligned} |\varPsi\rangle_{outN} \rightarrow&\bigl(\alpha\beta|L-R\rangle |g_{L}g_{R}\cdots g_{R}\rangle_{ab_{1}\cdots b_{N-1}}- \alpha\beta |L-R\rangle|g_{R}g_{L}\cdots g_{L} \rangle_{ab_{1}\cdots b_{N-1}} \\ &{}-i\alpha^{2}|L+R\rangle|g_{L}g_{R}\cdots g_{R}\rangle_{ab_{1}\cdots b_{N-1}}-i\beta^{2}|L+R \rangle|g_{R}g_{L}\cdots g_{L}\rangle _{ab_{1}\cdots b_{N-1}}\bigr)|g_{L}\rangle \\ &{}+\bigl(\alpha\beta|L-R\rangle|g_{L}g_{R}\cdots g_{R}\rangle_{ab_{1}\cdots b_{N-1}}+\alpha\beta|L-R\rangle|g_{R}g_{L} \cdots g_{L}\rangle _{ab_{1}\cdots b_{N-1}} \\ &{}+i\alpha^{2}|L+R\rangle|g_{L}g_{R}\cdots g_{R}\rangle_{ab_{1}\cdots b_{N-1}}-i\beta^{2}|L+R \rangle|g_{R}g_{L}\cdots g_{L}\rangle _{ab_{1}\cdots b_{N-1}}\bigr)|g_{R}\rangle. \end{aligned}$$
(26)

Alice makes the output photon pass through the HWP and the PBS, successively. After the PBS, Eq. (25) can finally evolve to

$$\begin{aligned} |\varPsi_{1}\rangle_{outN} =&\alpha\beta(|g_{L}g_{R} \cdots g_{R}\rangle _{ab_{1}\cdots b_{N-1}}-|g_{R}g_{L} \cdots g_{L}\rangle_{ab_{1}\cdots b_{N-1}})|V\rangle|g_{L}\rangle \\ &{}-\bigl(i\alpha^{2}|g_{L}g_{R}\cdots g_{R}\rangle_{ab_{1}\cdots b_{N-1}}+i\beta^{2}|g_{L}g_{R} \cdots g_{R}\rangle_{ab_{1}\cdots b_{N-1}}\bigr)|H\rangle|g_{L} \rangle \\ &{}+\alpha\beta(|g_{L}g_{R}\cdots g_{R} \rangle_{ab_{1}\cdots b_{N-1}}+|g_{R}g_{L}\cdots g_{L} \rangle_{ab_{1}\cdots b_{N-1}})|V\rangle |g_{R}\rangle \\ &{}+\bigl(i\alpha^{2}|g_{L}g_{R}\cdots g_{R}\rangle_{ab_{1}\cdots b_{N-1}}-i\beta^{2}|g_{L}g_{R} \cdots g_{R}\rangle_{ab_{1}\cdots b_{N-1}}\bigr)|H\rangle|g_{H} \rangle. \end{aligned}$$
(27)

Finally, Alice measures the state of the auxiliary atom and the output photon. It can be found that there are still four possible cases based on different measurement results. If the result is |V〉|g R 〉 , Eq. (27) will collapse to

$$\begin{aligned} |\psi_{1}\rangle_{N}=\frac{1}{\sqrt{2}}(|g_{L}g_{R} \cdots g_{R}\rangle _{ab_{1}\cdots b_{N-1}}+|g_{R}g_{L} \cdots g_{L}\rangle_{ab_{1}\cdots b_{N-1}}), \end{aligned}$$
(28)

while if the result is |V〉|g L 〉, Eq. (27) will collapse to

$$\begin{aligned} |\psi'_{1}\rangle_{N}=\frac{1}{\sqrt{2}}(|g_{L}g_{R} \cdots g_{R}\rangle _{ab_{1}\cdots b_{N-1}}-|g_{R}g_{L} \cdots g_{L}\rangle_{ab_{1}\cdots b_{N-1}}). \end{aligned}$$
(29)

Equation (29) can also be converted to Eq. (28) easily by the phase flip operation. So far, our ECP is completed, and we successfully distill the maximally entangled N-atom GHZ state from the less-entangled GHZ state, with the success probability of P=2|αβ|2.

On the other hand, if the measurement result is |H〉|g L 〉, Eq. (27) will collapse to

$$\begin{aligned} |\psi_{2}\rangle_{N}=\alpha^{2}|g_{L}g_{R} \cdots g_{R}\rangle _{ab_{1}\cdots b_{N-1}}+\beta^{2}|g_{L}g_{R} \cdots g_{R}\rangle _{ab_{1}\cdots b_{N-1}}, \end{aligned}$$
(30)

while if the result is |H〉|g R 〉, Eq. (27) will collapse to

$$\begin{aligned} |\psi'_{2}\rangle_{N}=\alpha^{2}|g_{L}g_{R} \cdots g_{R}\rangle _{ab_{1}\cdots b_{N-1}}-\beta^{2}|g_{L}g_{R} \cdots g_{R}\rangle _{ab_{1}\cdots b_{N-1}}. \end{aligned}$$
(31)

When they get these two cases, the ECP fails. However, similar to the Sect. 3, Eq. (31) can be converted to Eq. (30) by the phase flip operation, and Eq. (30) is a new less-entangled N-atom GHZ state. Based on the concentration step described above, Alice only needs to prepare a new auxiliary single three-level atom with the form

$$\begin{aligned} |\psi_{1}\rangle_{a_{1}}=\beta^{2}|g_{L} \rangle_{a_{1}}+\alpha ^{2}|g_{R}\rangle_{a_{1}}, \end{aligned}$$
(32)

and Eq. (31) can be reconcentrated for the next round. Therefore, our ECP can also be used repeatedly to further concentrate the less-entangled N-atom GHZ state and the success probability in each concentration round is the same as that in Sect. 3.

5 Discussion and Summary

In the paper, we proposed an efficient ECP for arbitrary less-entangled N-atom GHZ state using the photonic Faraday rotation. In the ECP, we only require a pair of less-entangled N-atom GHZ state and one auxiliary atom, all of which are trapped in low-Q cavities. In the concentration process, Alice makes a single-photon pulse pass through two low-Q cavities, sequentially. With the photonic Faraday rotation, she can get the input-output relations of the single photon pulse. By measuring the auxiliary atom state and the output photon state, she can finally distill the maximally entangled N-atom GHZ state with some success probability. Comparing with the ECP in Ref. [48], our ECP has some advantages. First, in Ref. [48], two pairs of less-entangled GHZ states are required, while our ECP only requires one. As the entanglement source is precious, our ECP is more economical. Second, in Ref. [48], each of the N parties needs to measure the atom state in his or her position to decide whether to remain or discard the obtained GHZ state. In this way, when the atom number N is large, the operations will be extremely complicated. While in our ECP, all the operations are performed by Alice. After all the operations, according to her measurement results, Alice only needs to tell the other parties to remain or discard their atom states. Therefore, the operations of our ECP are much more simple. Third, the ECP in Ref. [48] can not be repeated, while our ECP can be repeated to further concentrate the discarded items in Ref. [48]. In this way, our ECP can obtain a higher success probability.

Here, it is interesting to calculate the total success probability of our ECP. According to the concentration step in Sects. 3 and 4, we can calculate the success probability in each concentration round as

$$\begin{aligned} P_{1} =&2|\alpha\beta|^{2}, \\ P_{2} =&\frac{2|\alpha\beta|^{4}}{|\alpha|^{4}+|\beta|^{4}}, \\ P_{3} =&\frac{2|\alpha\beta|^{8}}{(|\alpha|^{4}+|\beta|^{4})(|\alpha |^{8}+|\beta|^{8})}, \\ P_{4} =&\frac{2|\alpha\beta|^{16}}{(|\alpha|^{4}+|\beta|^{4})(|\alpha |^{8}+|\beta|^{8})(|\alpha|^{16}+|\beta|^{16})}, \\ \vdots& \\ P_{K} =&\frac{2|\alpha\beta|^{2^{K}}}{(|\alpha|^{4}+|\beta|^{4})(|\alpha |^{8}+|\beta|^{8})\cdots(|\alpha|^{2^{K}}+|\beta|^{2^{K}})}, \end{aligned}$$
(33)

where the success probability of the ECP in Ref. [48] only equals the P 1 of our ECP.

As our ECP can be used indefinitely in theory, the total success probability P total equals the sum of the probability in each concentration round, which can be written as

$$\begin{aligned} P_{total}=P_{1}+P_{2}+\cdots+ P_{K}+ \cdots=\sum\limits _{K=1}^{\infty} P_{K}. \end{aligned}$$
(34)

It is obvious that if the original state is the maximally entangled N-atom state, where \(\alpha=\beta=\frac{1}{\sqrt{2}}\), the probability \(P_{total}=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots+\frac {1}{2^{K}}+\cdots=1\), while if the original state is the less-entangled N-atom state, where α≠β, the P total <1. Figure 4 shows the P total of our ECP and the ECP in Ref. [48] as a function of the initial entanglement coefficient α. It can be found that P total largely depends on the original entanglement state. The higher initial entanglement lead to the higher P total . Meanwhile, by repeating the our ECP for 5 times, the P total of our ECP is much higher than that of Ref. [48].

Fig. 4
figure 4

The success probability (P total ) of both ECPs for obtaining a maximally entangled N-atom GHZ state under ideal conditions. (a) The P total after our ECP being operated for 5 times. (b) The P total of the ECP in Ref. [48]. It can be seen that the value of P total largely depends on the initial coefficient α. Moreover, by repeating our ECP for 5 times, the P total of our ECP is much higher than that in Ref. [48]

Full size image

In the concentration process, the atom state detection and photon state detection play prominent roles. In Fig. 4, we have assumed that both the two detections are perfect with the detection efficiency η=100 %. Actually, in current experimental conditions, the detection efficiency η<100 %. Therefore, it is worthy to compare the success probability of the two ECPs under practical experimental conditions. In the ECP of Ref. [48], the output single photon state and a pair of N-atom state need to be detected. We suppose that the single photon detection efficiency and the single atom detection efficiency are η p and η a , respectively. Therefore, the success probability of the ECP in Ref. [48] can be revised as

$$\begin{aligned} P'_{total} =&\eta_{p}\eta_{a}^N2| \alpha\beta|^{2}, \end{aligned}$$
(35)

which indicates that the success probability of the ECP in Ref. [48] shows an exponential decay with the atom number N.

On the other hand, in our ECP, we only need to detect the output single photon state and the auxiliary single atom state in each concentration round, so that the success probability in each concentration round can be described as

$$\begin{aligned} P'_{K}=\eta_{p}\eta_{a}P_{K}= \eta_{p}\eta_{a}\frac{2|\alpha\beta |^{2^{K}}}{(|\alpha|^{4}+|\beta|^{4})(|\alpha|^{8}+|\beta|^{8})\cdots (|\alpha|^{2^{K}}+|\beta|^{2^{K}})}. \end{aligned}$$
(36)

Therefore, the total success probability can be revised as

$$\begin{aligned} P'_{1total}=\eta_{p}\eta_{a}(P_{1}+P_{2}+ \cdots + P_{K}+\cdots)=\eta _{p}\eta_{a}P_{total}, \end{aligned}$$
(37)

so that the success probability of our ECP is independent of the atom number N.

Figure 5 shows the values of \(P'_{total}\) and \(P'_{1total}\) as a function of the entanglement coefficient α. For numerical simulation, we assume η p =90 % and η a =90 % for approximation. In our ECP, we choose the repeating number K=5 (Fig. 5(a)), while in the ECP in Ref. [48], we choose the atom number N=5 (Fig. 5(b)) and N=10 (Fig. 5(c)). It is obvious that \(P'_{total}\) of the ECP in Ref. [48] reduces largely with the increasing of the atom number N. Especially, according to (35), when the atom number N is large, the \(P'_{total}\rightarrow0\). However, our ECP can still get high success probability. Therefore, under practical experiment conditions, especially when the atom number N is large, our ECP shows greater advantage.

Fig. 5
figure 5

The success probability (\(P'_{total}\)) for both two ECPs under the imperfect detection, where the efficiencies of both the photon state detection and atom state detection are set as 90 % for approximation. (a) The \(P'_{total}\) after our ECP being operated for 5 times. (b) The \(P'_{total}\) of the ECP in Ref. [48], when the atom number N is 5. (c) The \(P'_{total}\) of the ECP in Ref. [48], when the atom number N is 10. It can be seen under imperfect detection, the \(P'_{total}\) of the ECP in Ref. [48] reduces largely with the atom number N, while the \(P'_{total}\) of our ECP can remains a relatively high level

Full size image

In conclusion, in the paper, we put forward an efficient ECP for arbitrary less-entangled N-atom GHZ state with the help of the photonic Faraday rotation. In the ECP, we only require a pair of the less-entangled GHZ state and one auxiliary atom, and can successfully distill the maximally entangled N-atom GHZ state with high success probability. Comparing with the previous ECP in Ref. [48], our ECP has some obvious advantages. Firstly, our ECP reduces one pair of less-entangled N-atom GHZ state but can obtain the same success probability, which indicates our ECP is much more economical. Second, in the ECP of Ref. [48], each of the N parties need to operate his or her atom, simultaneously, to ultimately obtain the maximally entangled N-atom GHZ state, which will increase the operating complexity greatly, especially when the atom number N is large. However, our ECP can be operated by Alice alone. After the concentration, she only needs to tell others the final results, which can greatly reduce the practical operations. Third, by repeating our ECP, the discarded items in Ref. [48] can be reused, so that our ECP can obtain a higher success probability. Moreover, if we consider the imperfect detection, the success probability of the ECP in Ref. [48] will show an exponential decay with the atom number N, while our ECP can still obtain a relatively high success probability independent of N. Therefore, our ECP is efficient and may be useful and convenient in the current quantum information processing.