1. Introduction

Entanglement, one of the most intriguing features of quantum mechanics, has attracted widespread attention since its discovery. For realizing the applications of entanglement in the quantum communication [1] and quantum computation [2], the reversible storage and manipulation of entanglement must be implemented with high fidelity. Two kinds of popular schemes for this purpose have been proposed, and enormous successes have been achieved. One of them is the probabilistic scheme following the measurement induced approach developed by Duan et al. [3], the other is the deterministic one of Choi et al. [4] based on dynamic electromagnetically induced transparency (EIT) [5–7]. Based on both the schemes, the investigation has been extended from the stored entanglement shared by two atomic ensembles to that by four or more ensembles [8]. Entanglement encoded in different degrees of freedom (DOFs), such as polarization [9,10] and energy-time [11], has been stored in different media, such as the atomic ensemble [3,4] and solid-state spin qubits [12].

For realizing large-scale quantum networks in future applications, the capability of reversibly storing and manipulating the higher-dimensional states of light has received high attentions. The orbital angular momentum (OAM) of a photon with inherent infinite dimension has proven to be an outstanding DOF for achieving this purpose [13,14]. The reversible storage of entanglement encoded in the DOF of OAM has been implemented experimentally [15,16]. Acting as information carriers, photons can be entangled not only in a single DOF, but also in multiple DOFs [17–19]. The storage and retrieval of vector beams of light [20] and entanglement [21] encoded in multiple DOFs has also been achieved experimentally for making full use of the advantages of different DOFs, for example, photons entangled in the polarization can be efficiently transmitted through an optical fibre, whereas photons encoded in OAM can enhance the channel capacity and improve the efficiency of a network [22]. In addition, entanglement in multiple DOFs can also find advantages in Bell measurements [23,24], quantum logic gate operations [25–27], superdense coding [28], and asymmetrical optical quantum networks [29], etc.

All the above achievements are realized by mapping the entanglement into and out the quantum memory by switching off and on a traveling-wave controlling field, i.e., no optical component exists during entanglement storage. In 2002, André and Lukin first proposed the concept of stationary light pulse (SLP) [30], whose effective group velocity is zero, by replacing the traveling-wave controlling field with a standing-wave controlling field in the EIT medium. SLP, the duration of which is no longer restricted by the atomic spin coherence, can remarkably increase the interaction time between light and atoms, and can find important potential applications in the fields of low-light-level nonlinear optics and photonic quantum information processing without a cavity [31]. Because of the significant advantages of the SLP, a lot of meaningful results have been achieved in the candidate media [32–36]. For attaining robust SLP, the schemes implementing in a hot atomic vapor with large Doppler broadening [37], or employing the counter-propagating controlling fields with different frequencies [34–36] have been proposed. Very recently, quantum SLP has also been demonstrated experimentally in an atomic ensemble [38].

In the present paper, we propose a quantum memory, each subsystem of which is comprised of two double M-type system of cold atoms, for the first generation of entangled stationary photons (ESPs) encoded in DOFs of polarization and OAM. We send the entangled two photons into two spatially separated atomic ensembles. Through the active operation of two pairs of counter-propagating controlling fields in time, both the single photon pulse and the entangled photon pairs can be mapped into and out of the quantum memory, and ESPs can be generated deterministically. By introducing the reduced density matrix in the photon-polarization basis, we also discuss the dynamics evolution of entanglement in terms of the concurrence. We think such a multiple DOFs dependent scheme could pave the way toward quantum nonlinear optics without a cavity, and could greatly enhance the tunability and capacity for the quantum information processing.

2. Theoretical model and equations of motion

The scheme under consideration is composed of two identical but spatially separated photon storage systems (see Fig. 1). Each subsystem has a double M-type atom-field interaction configuration as shown in Fig. 2(a) and can be realized in the $D$1 or $D$2 transitions of $^{87}$Rb atoms. According to the polarization of the input signal photons, the interaction between the signal photons and the atomic ensemble can separate into two classes of double $\Lambda$-type systems, which are illustrated in Figs. 2(b) and 2(c). In the Fig. 2(b), a forward (FW) and a backward (BW) propagating quantum fields with horizontal polarization $\widehat {E}_{H\pm }(z,\;t)=\sqrt {\frac { \hbar \omega _{H\pm }}{2\varepsilon _{0}V_{\pm }}}\widehat {\mathcal {E}} _{H\pm }(z,\;t)e^{-i\omega _{H\pm }t\pm ik_{H\pm }z}+H.c.$ first pass through a $\lambda /4$ plate and become left circular polarization, then they are applied to the transitions $\left \vert 1\right \rangle \rightarrow \left \vert 3\right \rangle$ and $\left \vert 1\right \rangle \rightarrow \left \vert 4\right \rangle$, respectively, where $\omega _{H\pm }$ ($k_{H\pm }$) stands the carrier frequency (wave number), $V_{\pm }$ is the quantization volume, and $\widehat {\mathcal {E}}_{H\pm }$ is the slowly varying dimensionless operator. The other two dipole-allowed transitions $\left \vert 2\right \rangle \rightarrow \left \vert 3\right \rangle$ and $\left \vert 2\right \rangle \rightarrow \left \vert 4\right \rangle$ are coupled, respectively, by a FW and a BW propagating classical controlling fields with $\pi$ polarization and Rabi frequency (wave number) $\Omega _{1\pm }$ ($k_{1\pm }$). The configuration illustrated in Fig. 2(c) is similar to the Fig. 2(b) except that it is driven by two classical controlling fields $\Omega _{2\pm }$ with $\pi$ polarization and two quantum fields with vertical polarization $\widehat {E}_{V\pm }(z,\;t)=\sqrt {\frac {\hbar \omega _{V\pm }}{2\varepsilon _{0}V_{\pm }}}\widehat {\mathcal {E}}_{V\pm }(z,\;t)e^{-i\omega _{V\pm }t\pm ik_{V\pm }z}+H.c.$, where the two quantum fields will also first pass through a $\lambda /4$ plate and become right circular polarization. The two double $\Lambda$-type systems share the same ground state $\left \vert 1\right \rangle$, and can be employed for coherent generation and manipulation of ESPs encoded in DOF of polarization. Without misunderstanding, we will still use horizontal and vertical polarizations to represent the quantum fields in the following discussion.

 

Fig. 1. Illustration of entangled photon pairs $W_{in}$ stored into and $W_{out}$ retrieved from two spatially separated atomic ensembles A and B.

Download Full Size | PDF

 

Fig. 2. (a) Schematic of a double-M linkage structure, which can be realized in the $^{87}$Rb $D1$ or $D2$ transitions. (b) and (c) are the corresponding decompositions of atom-field interaction in (a) according to the polarization of the signal photons ((b) for the horizontal polarization, (c) for the vertical polarization).

Download Full Size | PDF

We assume that, without loss of generality, all the signal photons and the controlling fields are coupled to the corresponding atomic transitions resonantly. Then, under the electric-dipole and rotating-wave approximations, the interaction Hamiltonian for each subsystem can be written as

For simplicity, the ensemble of $^{87}$Rb atoms can be described in the conventional definition of the collective atomic operator [39]

Under the assumption of weak signal photon, we can solve the complete Heisenberg-Langevin equations of each subsystem to the first order in the signal photon. The reduced dynamic equations are shown as follows

The propagation dynamics of the signal photons in the atomic ensemble are governed by the Maxwell wave equations. We derive the equations of motion for signal photons $\widehat {\mathcal {E}}_{H(V)\pm }$ with horizontal (vertical) polarization in the medium and shown as follows

As the four controlling fields $\Omega _{1(2)\pm }$ change slowly enough, i.e., the adiabatic condition is fulfilled, similar systems can be described by the shape-preserving propagation of the dark-state polariton (DSP) [5], which does not involve excited states and is thus immune to spontaneous emission, can describe clearly the reversible mapping of quantum states between the signal photon and the atomic ensemble. In the present subsystem, the horizontal (vertical) polarization dependent DSP can be written as [40]

3. Generation and manipulation of SP

First, we consider the simplest case, i.e., only a signal photon with specific polarization, horizontal polarization for example, is input into one subsystem, where all the atoms have been pumped into the level $\left \vert 1\right \rangle$. According to the above analysis, the effective schematic of atom-field interaction is shown in Fig. 2(b). For generating the SP on demand, we assume that the counter-propagating controlling fields $\Omega _{1+}$ and $\Omega _{1-}$ vary regularly in terms of the formulas $\Omega _{1+}=\Omega _{1}\left [ 1-0.5\tanh \frac {t-t_{1}}{t_{s}}+0.5\tanh \frac {t-t_{2}}{t_{s}}-0.5\tanh \frac {t-t_{3}}{t_{s}}+0.5\tanh \frac {t-t_{4}}{ t_{s}}\right ]$ and $\Omega _{1-}=\Omega _{1}\left [ 0.5\tanh \frac {t-t_{2}}{ t_{s}}-0.5\tanh \frac {t-t_{3}}{t_{s}}\right ]$. The varying curves of both the controlling fields are shown in Fig. 3(a).

 

Fig. 3. Time evolution of the controlling fields $\Omega _{1+}$ (black solid curves) and $\Omega _{1-}$ (red dashed curves) in an adiabatic way (a) and (d). Dynamic propagation and evolution of the signal photon pulse with horizontal polarization propagating in the $+\vec {z}$ (b), (e) and $-\vec { z}$ (c), (f) directions. The other parameters are $\lambda _{H\pm }=\lambda _{1\pm }=795nm$, $\gamma _{31}=\gamma _{41}=\gamma _{61}=\gamma _{71}=1.5\Gamma$, $\gamma _{21}=\gamma _{51}=0.001\Gamma$, $g_{H}\sqrt {N}=3\Gamma$, $\Omega _{1}=10\Gamma$, where $\Gamma$ is the population decay rate from the excited level to the ground level, and $\Gamma =5.75MHz$.

Download Full Size | PDF

In a real situation, a spectral line of a light expands more or less for various reasons. In this sense, saying a photon as a wavepacket, instead of as an ideal particle, is more accurate. Based on this consideration, we assume that the temporal envelope of the incident signal photon has the form $f(t)=f_0e^{-(t/\sigma )^2}$ , where $\sigma$ is the half-width of the signal photon, and $f_0$ is the constant normalized by $\int _{-\infty }^{+\infty }\left \vert f(t)\right \vert ^{2}dt=1$. As the FW propagating signal photon pulse $\widehat {\mathcal {E}}_{H+}(t)$ and the controlling field $\Omega _{1+}$ enter the atomic medium, according to the DSP theory, the information carried by the signal photon pulse $\widehat {\mathcal {E}}_{H+}$ can be mapped into the atomic spin coherence $\rho _{12}$ when the controlling field is switched off at $t=t_{1}$ ($\Gamma t_{1}=40$) adiabatically. After a decent interval, at $t=t_{2}$ ($\Gamma t_{2}=80$), a pair of counter-propagating controlling fields $\Omega _{1+}$ and $\Omega _{1-}$ with the same intensity are switched on simultaneously, as a result of four-wave mixing, two signal photon pulses $\widehat {\mathcal {E}}_{H+}$ and $\widehat { \mathcal {E}}_{H-}$ with the same intensity propagating in the opposite directions are retrieved from the atomic spin coherence $\rho _{12}$ determinately. As the result of the tight coupling and balanced competition between the two retrieved signal photon pulses, SP is generated (see Figs. 3(b) and 3(c)). The SP can also be released by the reverse process, i.e., firstly switching off both the controlling fields $\Omega _{1+}$ and $\Omega _{1-}$ at $t=t_{3}$ ($\Gamma t_{3}=120$) to transfer the information carried by the SP into the atomic spin coherence $\rho _{12}$ again, then switching on the controlling field $\Omega _{1+}$ at $t=t_{4}$ ($\Gamma t_{4}=160$) to retrieve the signal photon pulse $\widehat {\mathcal {E}}_{H+}$ propagating in the $+\vec {z}$ direction (see Figs. 3(b) and 3(c)).

There are two questions that need to be emphasized. First, compared to the above case, if only the controlling field $\Omega _{1-}$ is switched on for releasing the SP at $t=t_{4}$ (the varying curves of both the controlling fields are shown in Fig. 3(d)), then only the signal photon pulse $\widehat { \mathcal {E}}_{H-}$ propagating in the $-\vec {z}$ direction can be retrieved (see Figs. 3(e) and 3(f)). Second, as the signal photon pulse $\widehat {\mathcal {E}}_{V+}$ with vertical polarization is input into the atomic ensemble, it undergoes a process similar to that of the signal photon pulse $\widehat {\mathcal {E}}_{H+}$ on the basis of the varying of the controlling fields $\Omega _{2+}$ and $\Omega _{2-}$. In a word, the proposed scheme not only can be employed for the reversible storage of the signal photon with horizontal or vertical polarization, but also can be used for coherent generation and manipulation of polarization dependent SP.

4. Generation of ESPs and discussion

Further, we consider the generation and manipulation of ESPs. In the present scheme, entanglement between two SPs is created by mapping two entangled photons into and out of the two subsystems of the quantum memory, respectively. As the two optical modes of a pair of photons with polarization entanglement, $\left \vert \psi \right \rangle =\frac {1}{\sqrt {2}}\left ( \left \vert HV\right \rangle +\left \vert VH\right \rangle \right )$ for example, are respectively input into the two subsystems, according to the above analysis, SPs with horizontal and vertical polarizations can be respectively generated in the two subsystems through the active operation of the counter-propagating controlling fields $\Omega _{1\pm }$ and $\Omega _{2\pm }$ in time. When the four controlling fields $\Omega _{1\pm }$ and $\Omega _{2\pm }$ are switched off and on synchronously, ESPs can be created, i.e., trapping two entangled photons with zero group velocity in an atomic ensemble without a cavity. Of course, the ESPs also can be released by turning off all the controlling fields and then switching on the FW or BW controlling fields only on demand. The whole process is obtained based on dynamic EIT, and the approach has the characteristic of inherent determinacy.

For discussing the entanglement dynamics in this process, we follow the protocol introduced in Ref. [41] and reconstruct a reduced density matrix $\rho _{c}$ in the photon-polarization basis, which provides a lower bound for any purported entanglement. The reduced density matrix $\rho _{c}(t)$ at time $t$ is written as

The two-body entanglement can be quantified admittedly in terms of concurrence, $C(t)$, given by $C(t)=\max \left \{ 0,\sqrt {\lambda _{1}(t)}- \sqrt {\lambda _{2}(t)}-\sqrt {\lambda _{3}(t)}-\sqrt {\lambda _{4}(t)}\right \}$ with $\lambda _{i}(t)$ ($i=1,2,3,4$) being the eigenvalues in decreasing order of magnitude of the density matrix operator $R(t)=\rho _{c}(t)(\sigma _{y}\otimes \sigma _{y})\rho _{c}^{\ast }(t)(\sigma _{y}\otimes \sigma _{y})$ at time $t$, where $\sigma _{y}$ is the Pauli $Y$ matrix [42]. The value of the concurrence $C(t)$ ranges from $0$ for a separable state to $1$ for a maximally entangled state. For the above reduced density matrix $\rho _{c}(t)$, the expression of the concurrence can be calculated, i.e., $C(t)= \max \left ( 0,2\left \vert d(t)\right \vert -2\sqrt { p_{HH}(t)p_{VV}(t)}\right )$ [41].

Figure 4 shows the time-dependent photon entanglement when the Rabi frequencies of the controlling fields $\Omega _{1(2)+}$ and $\Omega _{1(2)-}$ vary respectively in accordance with the solid and the dashed curves in Fig. 3(a). There is a common feature of the curves similar to that seen in the generation and manipulation of SPs (see Figs. 3(b) and 3(c)). As the controlling fields are switched off at the time $t=t_{1}$ ($\Gamma t_{1}=40$), the entanglement is transferred from the two polarization-entangled photons into the two distant atomic ensembles. Then, as the four controlling fields are switched on at the same time $t=t_{2}$ ($\Gamma t_{2}=80$), SPs with horizontal and vertical polarizations are respectively generated in the two subsystems. The entanglement is also retrieved from the atomic ensembles, and ESPs are generated coherently. Then, by firstly switching off the four controlling fields at $t=t_{3}$ ($\Gamma t_{3}=120$) to transfer the entanglement information carried by the SPs into the atomic spin coherence again, and switching on the controlling fields propagating in the $+\vec {z}$ direction at $t=t_{4}$ ($\Gamma t_{4}=160$), the two entangled photons can be released from the exit of the atomic ensembles. Of course, if the controlling fields propagating in the $-\vec { z}$ direction at $t=t_{4}$ ($\Gamma t_{4}=160$) are turned on, the entangled photon pairs can also be retrieved from the entrance of the atomic ensembles. Thus, we confirm that the present scheme can be used for the reversible storage of the entanglement and the generation of the ESPs. The ESPs may find important potential applications in the fields of few photon level nonlinear optics and photonic quantum information processing. Due to the presence of first order Fourier components of spin coherence and optical coherence in the present scheme, the SPs experience decay and diffusion, which lead to the decrease of the concurrence of the entangled photon pairs.

 

Fig. 4. Time evolution of the concurrence $C$ for the signal photons propagating in the $+\vec {z}$ (a) and $-\vec {z}$ (b) directions. The solid (black), dashed (red) and dash-dotted (green) curves are related to $g_{H}\sqrt {N}=g_{V}\sqrt {N}=\Gamma , \sqrt {3/2}\Gamma , 3\Gamma$. $V_i=0.91$ [4]. The other parameters are the same as in Fig. 3.

Download Full Size | PDF

5. ESPs encoded in multiple DOFs

The present scheme can also be employed for the reversibly storing and manipulating the photons encoded in multiple DOFs, for example, photons encoded in DOFs of polarization and OAM. The capability are important for realizing large-scale quantum networks in future applications.

Under the slowly varying envelope approximation, the equations of motion for the signal photons encoded in DOFs of polarization and OAM in the medium are governed by the Maxwell equations, and can be reduced as

Based on the basis of the Laguerre-Gaussian (LG) modes, the signal photon operator $\widehat {\mathcal {E}}_{H(V)\pm }(r,\;t)$ and the density matrix element $\widehat {\rho }_{\mu \nu }$ can be expanded into the form as follows

Substituting Eq. (8) into Eqs. (3) and (7), we can obtain the equations for $\widehat {\mathcal {E}}_{H(V)\pm }^{mn}(z,\;t)$ and $\widehat {\rho }_{\mu \nu }^{mn}$ as follows:

6. Conclusions

In conclusion, we have proposed a multiple modes and multiple DOFs quantum memory, each subsystem of which is comprised of two double M-type systems of cold atoms. We have demonstrated that both the single photon pulse and the entangled photon pairs, which can be encoded in the DOFs of polarization and OAM, can be mapped into and out of the quantum memory deterministically, and the SPs and ESPs can be generated and manipulated coherently and all-optically through the active operation of the two pairs of counter-propagating controlling fields in time. By introducing the reduced density matrix in the photon-polarization basis, the dynamics evolution of entanglement is discussed by numerical simulation. Our protocol with the characteristic of multiple modes and multiple DOFs will make possible the generation, storage and distribution of entanglement among remote quantum memories for scalable quantum networks with high-capacity, and the created ESPs also may find important applications in quantum nonlinear optics without a cavity.