Abstract

We propose an efficient scheme for realizing all-optical router or beam splitter (BS) by employing a double tripod-type atomic system, where the ground levels are coupled by two additional intensity-dependent weak microwave fields. We show that the high-dimensional probe field encoded in a degree of freedom of orbital angular momentum can be stored, retrieved, and manipulated. Due to the constructive or destructive interference between the introduced microwave fields and the atomic spin coherence, the generated stationary light pulses and the retrieved probe fields can be increased or decreased with high efficiency and fidelity in a controllable manner. On the basis of the results and a general extension, a tunable all-optical router or BS, which can split a high-dimensional probe field into two or more ones, can be achieved by actively operating the controlling fields and the microwave fields. The current scheme, integrating multiple functions and showing excellent performance, could greatly enhance the tunability and capacity for the all-optical information processing.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Electromagnetically induced transparency (EIT) [13] is a powerful quantum interference effect that allows light to pass through an opaque medium without absorption. Based on the EIT, a variety of fascinating physics phenomena have been found. The storage and retrieval of light is a prominent example, and has attracted widespread attention since it is proposed due to its inherent determinacy. The concept of dark-state polarization (DSP) [4] is introduced first by Lukin for describing the dynamic EIT process. By the adiabatic operation of the traveling-wave controlling field in time, the reversible mapping of information between the light and the atomic spin coherence can be realized. By now, researchers have demonstrated the storage of classical light pulses, single-photon states, squeezed vacuum states, and images in a series of impressive experiments [511]. The entangled state of light has also been mapped into and retrieved from two spatially separated interaction regions [1216]. By substituting the standing-wave controlling field for the traveling-wave controlling field in the EIT medium, stationary light pulse (SLP) has been observed by Bajcsy in 2003 [17]. Different from the above principle, the effective group velocity of the light can be zero in the scheme, then the interaction time between light and atoms can be remarkably increased. Because of the significant advantages and its potential applications in the fields of low-light-level nonlinear optics and photonic quantum information processing without a cavity [18], a lot of meaningful results have been achieved since its discovery [1925].

In quantum information processing and all-optical network [26], there are growing needs of all-optical devices, such as quantum repeater, all-optical router, phase shifter, beam splitter (BS), optical isolator and circulator. All-optical router and BS [2739], where the optical information can be transferred and distributed in a controlled fashion between different light channels, are a kind of extremely essential element in modern physics. At present, the implementation of the all-optical router and BS in actual physical system is mainly based on the interaction between light and matter or between light and light. A variety of platforms, such as linear optical system [27,28], quantum dot system [29,30], cavity QED system [31], and optomechanical system [3234], have been investigated for realizing the all-optical router and BS theoretically and experimentally. The all-optical routers with single output port or multi ones have been proposed successively, the work efficiency of which has been significantly improved. By employing EIT-based light storage, multi-channel all-optical router and BS have been studied in atomic gases or solid medium [3538]. Recently, controllable spatial-frequency router of image via atomic spin coherence has also been reported experimentally [39].

Orbital angular momentum (OAM) with inherent infinite dimension has proven to be an outstanding degree of freedom (DOF) for carrying high-dimensional states and dramatically improving channel capacity [40,41]. The coherent and nonlinear interaction of lights carrying OAMs with atomic systems has been reported previously using different experimental schemes [4244], and tremendous developments have also been realized in the reversible storage of OAM states [45,46]. In this paper, we propose an efficient double tripod-type atomic system, the ground levels of which are coupled by two additional intensity-dependent weak microwave fields, for realizing all-optical router or BS of high-dimensional probe field with OAM. Through theoretical analysis and numerical simulation, we will show that the probe fields can be stored, retrieved, and manipulated with high efficiency and fidelity. In addition, due to the constructive or destructive interference between the introduced microwave fields and the atomic spin coherence, the generated SLPs and the retrieved probe fields can be increased or decreased in a controllable fashion. On the basis of the efficient scheme of light storage, a tunable all-optical router or BS, which can split one high-dimensional probe field into two or more ones, can be realized by actively operating the controlling fields and the microwave fields. The current scheme, integrating of optimal storage, tunable all-optical router or BS, and efficient generation of SLP, can function as several optical elements, and may find important applications in scalable quantum networks.

2. Theoretical model and equations of motion

The scheme under consideration is a five-level double tripod-type system as shown in Fig. 1, which can be achieved in $^{87}$Rb atoms, for example, the levels $\left \vert 0\right \rangle$, $\left \vert 1\right \rangle$, $\left \vert 2\right \rangle$, $\left \vert e\right \rangle$, and $\left \vert f\right \rangle$ can refer to $\left \vert 5^{2}S_{1/2},F=1,m_{F}=-1\right \rangle$, $\left \vert 5^{2}S_{1/2},F=2,m_{F}=-1\right \rangle$, $\left \vert 5^{2}S_{1/2},F=2,m_{F}=+1\right \rangle$, $\left \vert 5^{2}P_{1/2},F=1,m_{F}=0\right \rangle$, and $\left \vert 5^{2}P_{1/2},F=2,m_{F}=0\right \rangle$ of $^{87}$Rb atoms, respectively. A weak probe field $\Omega _{p+}$ (wave number $k_{p+}$) and two strong controlling fields $\Omega _{1(2)+}$ (wave number $k_{1(2)+}$) propagating in the $+ \vec {z}$ direction are resonantly coupling to one tripod-type system ($\left \vert e\right \rangle ,\left \vert 0\right \rangle ,\left \vert 1\right \rangle ,\left \vert 2\right \rangle$), the other tripod-type system ($\left \vert f\right \rangle ,\left \vert 0\right \rangle ,\left \vert 1\right \rangle ,\left \vert 2\right \rangle$) interacts resonantly with another three fields (a weak probe field $\Omega _{p-}$ with wave number $k_{p-}$ and two strong controlling fields $\Omega _{1(2)-}$ with wave number $k_{1(2)-}$) propagating in the $-\vec {z}$ direction. Two additional intensity-modulated weak microwave fields with Rabi frequencies $\Omega _{m1}$ and $\Omega _{m2}$, which are employed during the time of the controlling fields are switched off, are employed for driving the atomic transitions $\left \vert 0\right \rangle \rightarrow \left \vert 1\right \rangle$ and $\left \vert 0\right \rangle \rightarrow \left \vert 2\right \rangle$, respectively. The microwave fields are chosen to be so weak that it alters the corresponding atomic spin coherence without changing the populations of the levels [47].

 figure: Fig. 1.

Fig. 1. Schematic diagram of a five-level double tripod-type system of cold $^{87}$Rb atoms.

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Under the electric-dipole and rotating-wave approximations, the Hamiltonian for the present system in the interaction picture can be written as

$$\begin{aligned} H_{int}=&-\hbar \Big[ \Omega _{p+}\left\vert e\right\rangle \left\langle 0\right\vert +\Omega _{p-}\left\vert f\right\rangle \left\langle 0\right\vert +\Omega _{1+}\left\vert e\right\rangle \left\langle 1\right\vert +\Omega _{1-}\left\vert f\right\rangle \left\langle 1\right\vert +\Omega _{2+}\left\vert e\right\rangle \left\langle 2\right\vert\\ & +\Omega _{2-}\left\vert f\right\rangle \left\langle 2\right\vert +\Omega _{m1}e^{i\Phi _{1}}\left\vert 1\right\rangle \left\langle 0\right\vert +\Omega _{m2}e^{i\Phi _{2}}\left\vert 2\right\rangle \left\langle 0\right\vert +h.c.\Big] , \end{aligned}$$
here $\Phi _{1(2)}=\Phi _{1(2)+}-\Phi _{p+}+\Phi _{p-}-\Phi _{1(2)-}+\Phi _{m1(m2)}$, where $\Phi _{1(2)\pm }$, $\Phi _{p\pm }$, and $\Phi _{m1(m2)}$ are the phases of the corresponding light fields $\Omega _{1(2)\pm }$, $\Omega _{p\pm }$, and $\Omega _{m1(m2)}$, respectively. Due to the existence of the two closed-loop interaction formed by the probe fields and the controlling fields, the scheme is sensitive to the relative phase $\Phi _{1(2)}$, which can be set $\Phi _{1(2)}=\Phi _{m1(m2)}$ under the condition of phase matching $\Phi _{1(2)+}+\Phi _{p-}=\Phi _{p+}+\Phi _{1(2)-}$ for the generation of SLP in the following discussion. Obviously, no matter what kind of probe fields, the condition of phase matching can be easily met.

The atomic dynamics can be described by a $5\times 5$ density matrix $\rho$, obeying the optical Bloch equation $\frac {\partial \rho }{\partial t}=- \frac {i}{\hbar }\left [ H_{int},\rho \right ] -\Gamma \left [ \rho \right ]$, where $\Gamma$ is a relaxation matrix characterizing the spontaneous emission and dephasing in the system. Under the assumption of weak probe fields and weak microwave fields [47], we can solve the complete Bloch equations of the system to the first order in the probe fields. The reduced dynamic equations are shown as follows

$$\begin{aligned} \frac{\partial \rho _{10}}{\partial t} &=i\left[ \Omega _{1+}^{\ast }\rho _{e0}+\Omega _{1-}^{\ast }\rho _{f0}+\Omega _{m1}e^{i\Phi _{1}}\right] -\gamma _{01}\rho _{10},\\ \frac{\partial \rho _{20}}{\partial t} &=i\left[ \Omega _{2+}^{\ast }\rho _{e0}+\Omega _{2-}^{\ast }\rho _{f0}+\Omega _{m2}e^{i\Phi _{2}}\right] -\gamma _{02}\rho _{20}, \\ \frac{\partial \rho _{e0}}{\partial t} &=i\left[ \Omega _{1+}\rho _{10}+\Omega _{2+}\rho _{20}+\Omega _{p+}\right] -\gamma _{0e}\rho _{e0},\\ \frac{\partial \rho _{f0}}{\partial t} &=i\left[ \Omega _{1-}\rho _{10}+\Omega _{2-}\rho _{20}+\Omega _{p-}\right] -\gamma _{0f}\rho _{f0}, \end{aligned}$$
where $\gamma _{0\nu }$ is the decay rate of atomic coherence on the transition $\left \vert 0\right \rangle \rightarrow \left \vert \nu \right \rangle$ ($\nu =1,2,e,f$).

The equations of motion for the probe fields $\Omega _{p\pm }$ encoded in the DOF of OAM in the medium are governed by the Maxwell equations. Under the slowly varying envelope approximation, the equations can be reduced to

$$\begin{aligned} \frac{\partial \Omega _{p+}}{\partial z}+\frac{1}{c}\frac{\partial \Omega _{p+}}{\partial t} &=\frac{i}{2k_{p+}}\nabla _{\perp }^{2}\Omega _{p+}+i \frac{Nd_{0e}^{2}k_{p+}}{2\varepsilon _{0}\hbar }\rho _{e0},\\ \frac{\partial \Omega _{p-}}{\partial z}-\frac{1}{c}\frac{\partial \Omega _{p-}}{\partial t} &=\frac{-i}{2k_{p-}}\nabla _{\perp }^{2}\Omega _{p-}-i \frac{Nd_{0f}^{2}k_{p-}}{2\varepsilon _{0}\hbar }\rho _{f0}, \end{aligned}$$
where $\nabla _{\perp }^{2}=\partial ^{2}/\partial x^{2}+\partial ^{2}/\partial y^{2}$, $d_{0\nu }$ is the electric-dipole moment on the transition $\left \vert 0\right \rangle \rightarrow \left \vert \nu \right \rangle$.

For intuitively revealing the dynamic behavior of the probe fields with OAMs, we expand their Rabi frequencies $\Omega _{p\pm }$ into the following form

$$\Omega _{p\pm }(r,t)=\sum_{m,n}\mathcal{L}^{mn}_{\pm}(r,\psi ,z)\Omega _{p\pm }^{mn}(z,t),$$
here $r=(x^{2}+y^{2})^{1/2}$ ($\psi$) is radial coordinate (azimuthal angle) in a frame of cylindrical coordinate system, $\Omega _{p\pm }^{mn}(z,t)$ is expansion coefficient, and $\mathcal {L}^{mn}_{\pm }(r,\psi ,z)$ satisfies the equation $2ik_{p\pm }\partial \mathcal {L}^{mn}_{\pm }(r,\psi ,z)/\partial z\pm \nabla _{\perp }^{2}\mathcal {L}^{mn}_{\pm }(r,\psi ,z)=0$, the eigen solutions of which are Laguerre-Gaussian (LG)$_{n}^{m}$ modes with OAM $m\hbar$ along the ${\pm }\vec {z}$ direction, and have the approximate expression under the condition of Rayleigh length being large enough [48,49]
$$\mathcal{L}^{mn}(r,\psi )=\frac{C_{mn}}{\sqrt{w_{0}}}\left[ \frac{\sqrt{2}r}{ w_{0}}\right] ^{\left\vert m\right\vert }\exp \left[ -\frac{r^{2}}{w_{0}^{2}} \right] L_{n}^{\left\vert m\right\vert }\left[ \frac{2r^{2}}{w_{0}^{2}} \right] \exp (im\psi ),$$
where $C_{mn}=\sqrt {2^{\left \vert m\right \vert +1}n!/[\pi (\left \vert m\right \vert +n)!]}$ is the normalization constant, $w_{0}$ ($L_{n}^{\left \vert m\right \vert }$) is the beam waist (generalized LG polynomials), and $m$ ($n$) is azimuthal (radial) indice. The profile of the (LG)$_{n}^{m}$ modes shows concentric rings, the number of which is determined by the mode index $n$. The mode index $m$ is contained in the azimuthal phase term $\exp (im\psi )$, which gives rise to $\left \vert m\right \vert$ intertwined helical wave-fronts. The handedness of these helixes is determined by the sign of $m$.

Following the above idea, the density matrix element $\rho _{kj}$ and the microwave field $\Omega _{mx}$ ($x=1,2$) can also be expanded based on the basis $\mathcal {L}^{mn}(r,\psi )$. Then, we can obtain the equations for $\Omega _{p\pm }^{mn}(z,t)$ as follows

$$\begin{aligned} \frac{\partial \Omega _{p+}^{mn}}{\partial z}+\frac{1}{c}\frac{\partial \Omega _{p+}^{mn}}{\partial t} &=i\frac{Nd_{0e}^{2}k_{p+}}{2\varepsilon _{0}\hbar }\rho _{e0}^{mn},\\ \frac{\partial \Omega _{p-}^{mn}}{\partial z}-\frac{1}{c}\frac{\partial \Omega _{p-}^{mn}}{\partial t} &=-i\frac{Nd_{0f}^{2}k_{p-}}{2\varepsilon _{0}\hbar }\rho _{f0}^{mn}, \end{aligned}$$
here $\rho _{\nu 0}^{mn}$ is the expansion coefficient of the corresponding density matrix element $\rho _{\nu 0}$. Obviously, the form of the above Eq. (6) is the same as the Maxwell equation describing the propagation dynamics of the general light pulse in the atomic ensemble. Thus, we can firmly believe that the present scheme can be suitable for the probe fields with OAMs.

3. Numerical simulation and discussion

Now, let’s take a look at how the all-optical router or BS works. We assume that all the atoms have been initially pumped into the state $\left \vert 0\right \rangle$, and the Rabi frequencies of the two microwave fields take the same form of Gaussian type for simplicity, i.e., $\Omega _{m1}=\Omega _{m2}=\Omega _{m}T_{m}(t)\exp \left [ -(z-z_{0})^{2}/z_{t}^{2}\right ]$, and the controlling field $\Omega _{1(2)\pm }=\Omega _{1(2)}T_{1(2)\pm }(t)$, where $z_{0}$ ($z_{t}$) is the peak position (half-width), $\Omega _{m,1,2}$ is constant, and $T_{m}(t)$ ($T_{1(2)\pm }(t)$) describes the time evolution of the microwave fields (the controlling fields $\Omega _{1(2)\pm }$). In the following discussion, we give two different ways to control the dynamic propagation and evolution of the probe fields as presented in Fig. 2(b-d) and 2(f-h), the corresponding varying curves of the controlling fields ($T_{1\pm }(t)$ and $T_{2\pm }(t)$) and the microwave fields ($T_{m}(t)$) are shown in Fig. 2(a1, a2) and 2(e1, e2), respectively. We can clearly see that, through adiabatically switching off and on the controlling fields, the probe field incident into the atomic medium along $+ \vec {z}$ direction experiences four processes: slowly moving, storage, stationary, and releasing. The processes can be understood based on the well-known DSP

$$\Psi _{\pm }(z,t)=\cos \theta _{\pm }(t)\Omega _{p\pm }-g\sqrt{N}\sin \theta _{\pm }(t)(\cos \phi _{\pm }\rho _{01}+\sin \phi _{\pm }\rho _{02}),$$
where $\tan \theta _{\pm }=g_{\pm }\sqrt {N}/\sqrt {\Omega _{1\pm }^{2}+\Omega _{2\pm }^{2}}$, $\tan \phi _{\pm }=\Omega _{2\pm }/\Omega _{1\pm }$, and $g_{+(-)}$ is the coupling constant between the atoms and the probe field propagating in the $+\vec {z}$ ($-\vec {z}$) direction.

 figure: Fig. 2.

Fig. 2. Time evolution of the controlling fields $T_{1\pm }$ (a1, e1), $T_{2\pm }$ (a2, e2), and microwave fields $T_{m}$ ((a1, a2) corresponding to (b, c, d), (e1, e2) corresponding to (f, g, h)) in an adiabatic way. The dynamic propagation and evolution of the normalized probe field with $\Omega _{m}=0$ (b, f), $2\times 10^{-5}\Gamma$ (c, g), and $4\times 10^{-5}\Gamma$ (d, h) propagating in the $+\vec {z}$ direction. The half-width of the microwave field is equal to that of the initial signal field. The other parameters are $\Omega _{1}=\Omega _{2}=10\Gamma$, $\gamma _{0e}=\gamma _{0f}=1.5\Gamma$, $\gamma _{01}=\gamma _{02}=0.0001\Gamma$, where $\Gamma$ is the population decay rate from the excited level to the ground level, and $\Gamma =5.75MHz$.

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At the beginning time, the two controlling fields $\Omega _{1+}$ and $\Omega _{2+}$ (propagating in the $+\vec {z}$ direction) with the same Rabi frequencies are employed for reducing the velocity of the probe field in the medium to $v=c\cos ^{2}\theta _{+}$, then the information carried by the probe field is transferred into the two atomic spin coherence $\rho _{01}$ and $\rho _{02}$ by simultaneously switching off $\Omega _{1+}$ and $\Omega _{2+}$. Of course, due to $\rho _{01}/\rho _{02}=\Omega _{1}/\Omega _{2}$, we can conveniently map the information carried by the probe field into $\rho _{01}$ and $\rho _{02}$ in a controllable proportion by adjusting the two controlling fields $\Omega _{1+}$ and $\Omega _{2+}$. After a decent interval, through simultaneously switching on the controlling fields $\Omega _{1\pm }$ ($\Omega _{1+}=\Omega _{1-}$), two probe fields, propagating in the opposite directions and being the same in strength, are retrieved from the atomic spin coherence $\rho _{01}$. As the results of the tight coupling and balanced competition between the two probe fields, a SLP is generated. If the controlling fields $\Omega _{2\pm }$ are adjusted in the same way of $\Omega _{1\pm }$, two probe fields can be retrieved from $\rho _{02}$, and another SLP can be generated. When the two pairs of controlling fields are switched on simultaneously (see Fig. 2(a1) and 2(a2)) or separately (see Fig. 2(e1) and 2(e2)), the two SLPs can be generated at the same (see Fig. 2(b-d)) or different (see Fig. 2(f-h)) time. Finally, the both SLPs not only can be released respectively from the entrance and the exit of the medium at the same time (see Fig. 2(b-d)) by switched off the controlling fields $\Omega _{1-}$ and $\Omega _{2+}$ (or $\Omega _{1+}$ and $\Omega _{2-}$) simultaneously, but also can be released from the entrance (exit) at different time (see Fig. 2(f-h)) by switched off the controlling fields $\Omega _{1+}$ and $\Omega _{2+}$ ($\Omega _{1-}$ and $\Omega _{2-}$) successively. From the figures and above analysis, we easily see that the input probe field can indeed be stored, stationary, and partially retrieved by active operation the four controlling fields, and the scheme can behavior as a BS or all-optical router.

Compared to Fig. 2(b) and 2(f), the generated SLPs and the retrieved probe fields in Fig. 2(c-d) and Fig. 2(g-h) are enhanced due to the application of the additional intensity-dependent weak microwave fields. For further revealing the effect of the microwave fields on the atomic spin coherence, we solve the Eqs. (2) during the storage time. The expression of $\rho _{1(2)0}$ is

$$\rho _{1(2)0}(t_{1}+t_{m})=\rho _{1(2)0}(t_{1})e^{-\gamma _{01(2)}t_{m}}+ie^{i\Phi _{1(2)}}\frac{1-e^{-\gamma _{01(2)}t_{m}}}{\gamma _{01(2)}}\Omega _{m}.$$
For obtaining the above result, we have assumed that the duration of the microwave fields is $t_{m}$, the controlling fields are switched off at the time $t_{1}$, the Rabi frequencies of the microwave fields are constant, and the independent phase factors are ignored. Obviously, due to the additional microwave fields, the atomic spin coherence is indeed modulated. The modulation can be attributed to the constructive or destructive interference between the microwave fields and the atomic spin coherence, so $\rho _{1(2)0}$ can be controllably increased or decreased by changing the relative phase $\Phi _{1(2)}$. For example, with the increasing of $\Omega _{m}$, as $\Phi _{1(2)}=0,\pi /2$, $\left \vert \rho _{1(2)0}(t_{1}+t_{m})\right \vert$ will be magnified, and the magnification in the case with $\Phi _{1(2)}=\pi /2$ is greater than that with $\Phi _{1(2)}=0$, while it will be weakened first and then enhanced as $\Phi _{1(2)}=3\pi /2$. As a result, the retrieved probe fields, which are proportional to $\rho _{1(2)0}(t_{1}+t_{m})$ as we all know, can be modulated flexibly and expediently.

Next, by numerical simulation, we intuitively take a look at the modulation of the microwave field on the atomic spin coherence according to the retrieval efficiency defined as $\eta _{x}=\int _{-\infty }^{+\infty }\left \vert \Omega _{x}(t)\right \vert dt/\int _{-\infty }^{+\infty }\left \vert \Omega _{p}(0)\right \vert dt$, where $\Omega _{x}(t)$ and $\Omega _{p}(0)$ are, respectively, the Rabi frequencies of the retrieved probe fields at the time $t$ and its initial counterpart. In the following discussion, we assume that the $T_{1\pm }(t)$ and $T_{2\pm }(t)$ vary in accordance with Fig. 2(a1) and 2(a2). Figure 3 shows the retrieval efficiency $\eta _{p+}$ via $\Omega _{m}$ (a) and $\Phi _{1}$ (b) when the retrieval process has completely finished ($\Gamma t=160$). Apparently, the features of the Fig. 3 are agree well with our above analytical solution.

 figure: Fig. 3.

Fig. 3. The retrieval efficiency $\eta _{p+}$ of the signal field via $\Omega _{m}$ (a) with $\Phi _{1}=0$ (black squares), $\pi /2$ (red circles), $3\pi /2$ (green triangles), and via $\Phi _{1}$ (b) with $\Omega _{m}=0$ (black squares), $2\times 10^{-5}\Gamma$ (red circles) and $4\times 10^{-5}\Gamma$ (green triangles) at the time $t=160/\Gamma$. The other parameters are the same as in Fig. 2.

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On the basis of the above discussion, the two retrieved probe fields can be regulated independently by adjusting the controlling fields and the microwave fields. In view of the significant advantages of the current scheme, the tunable BS or all-optical router can be realized. Figure 4 presents the retrieval efficiencies $\eta _{p+}$ (black squares) and $\eta _{p-}$ (red circles) via $\Omega _{1}/(\Omega _{1}+\Omega _{2})$ (a) and $\Omega _{m}$ (b) by numerical simulation. It is clear that the ratio of the retrieved two probe fields can be continuously tunable and can cover a wide range. The retrieved visibility (defined as $(\eta _{p+}-\eta _{p-})/(\eta _{p+}+\eta _{p-})$) in Fig. 4(a) can reach $100\%$, and it can also reach $83\%$ in Fig. 4(b). Of course, the retrieved visibility of the two retrieved probe fields can be adjusted more flexibly and expediently through the combination management of the controlling fields and the microwave fields.

 figure: Fig. 4.

Fig. 4. The retrieval efficiency $\eta _{p+}$ (black squares) and $\eta _{p-}$ (red circles) of the signal field as a function of $\Omega _{1}/(\Omega _{1}+\Omega _{2})$ (a) with $\Omega _{m}=0$, and $\Omega _{m}$ (b) with $\Phi _{1}=\pi /2$, $\Phi _{2}=3\pi /2$. The other parameters are the same as in Fig. 2.

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Last, the probe field with OAMs is investigated for the purpose of the paper. The intensity distributions of the probe field with the superposition of a set of LG modes [(LG)$_{0}^{2}$+(LG)$_{0}^{-2}$] in the $x-y$ plane before and after the storage are shown in Fig. 5. The first column ($\Gamma t=0$) is the patterns before the storage, and the second one ($\Gamma t=60$) is obtained as both the controlling fields ($\Omega _{1+}$ and $\Omega _{2+}$) are switched off completely, then we can find that the optical components disappear because the information of the patterns has been mapped into the atomic spin coherence $\rho _{01}$ and $\rho _{02}$. The third column ($\Gamma t=100$) gives the patterns of the generated SLPs, and the fourth (propagating in $+ \vec {z}$ direction) and fifth (propagating in $-\vec {z}$ direction) ones ($\Gamma t=160$) are the released patterns. We can easily see that the probe field with OAMs can be stored, retrieved, and manipulated with high fidelity by actively adjusting the controlling fields and the microwave fields. In the first line, corresponding to the condition of $\Omega _{1}=\Omega _{2}$ and $\Omega _{m}=0$, the intensity of the generated SLPs and the both retrieved patterns is reduced, and the two patterns are exactly the same in the intensity distribution. In the second one, the intensity of the retrieved pattern propagating in the $+\vec {z}$ ($-\vec {z}$) direction is increased (decreased) under the condition of $\Omega _{1}=\Omega _{2}$, $\Omega _{m}=2\times 10^{-5}\Gamma$ and $\Phi _{1}=\pi /2$ ($\Phi _{2}=3\pi /2$) due to the constructive (destructive) interference between the microwave fields and the atomic spin coherence. Except of adjusting the microwave fields, the intensity of the retrieved two patterns can also be reconfigured by operating the controlling fields $\Omega _{1}$ and $\Omega _{2}$ as shown in the last line, which is obtained as $\Omega _{1}=2\Omega _{2}$, $\Omega _{m}=2\times 10^{-5}\Gamma$ and $\Phi _{1}=\Phi _{2}=\pi /2$. Thus, a tunable BS or all-optical router for probe field with OAM can be realized based on our current scheme.

 figure: Fig. 5.

Fig. 5. The normalized intensity patterns for the superposition of LG modes [(LG)$_{0}^{2}$+(LG)$_{0}^{-2}$] at time $\Gamma t=0$ (the first column), $60$ (the second column), $100$ (the third column), $160$ (the fourth and fifth columns) under the conditions of $\Omega _{1}=\Omega _{2}$, $\Omega _{m}=0$ (the first line), $\Omega _{1}=\Omega _{2}$, $\Omega _{m}=2\times 10^{-5}\Gamma$, $\Phi _{1}=\pi /2$, $\Phi _{2}=3\pi /2$ (the second line), and $\Omega _{1}=2\Omega _{2}$, $\Omega _{m}=2\times 10^{-5}\Gamma$, $\Phi _{1}=\Phi _{2}=\pi /2$ (the third line). The fourth (fifth) column is corresponding to the retrieved patterns propagating in $+\vec {z}$ ($-\vec {z}$) direction. The other parameters are the same as in Fig. 2.

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It should be noted that the current scheme proposed above has two optical excitation channels, the incident probe fields can only be divided into two light fields. By introducing additional ground levels, the scheme can be generalized to systems with multiple optical excitation channels. For example, a scheme with $N$ ground states and two excited states, which is driven by two probe fields and $2(N-1)$ controlling fields, has $N-1$ optical excitation channels, and can divide one probe field into $N-1$ light fields. For retrieving the $N-1$ light fields separately, we can make full use of the time and the space modulation on the probe fields by flexible operation of the controlling fields in direction and time, as shown in Fig. 1. Considering the advantages mentioned above, the generalized scheme can be employed for designing more powerful BS or all-optical router.

4. Conclusions

In conclusion, we have proposed a double tripod-type atomic system for realizing all-optical router or BS. Two additional intensity-dependent weak microwave fields are introduced to drive the ground levels during the time of storage, then the atomic spin coherence can be modulated efficiently. Theoretical analysis and numerical simulation show that, due to the constructive or destructive interference between the microwave fields and the atomic spin coherence, the generated SLP and the retrieved probe fields can be amplified or decreased with high fidelity in a controllable manner. On the basis of the flexible and convenient manipulation of the controlling fields and the microwave fields on the probe field, a tunable all-optical router or BS splitting one high-dimensional probe field into two or more ones can be achieved. All the advantages make the current scheme open a route for the applications in all-optical information processing and testing some fundamental quantum physics.

Funding

National Natural Science Foundation of China (11604174, 11704214, 11975132, 61772295); Natural Science Foundation of Shandong Province (ZR2019YQ001).

Disclosures

The authors declare no conflicts of interest.

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8. K. Honda, D. Akamatsu, M. Arikawa, Y. Yokoi, K. Akiba, S. Nagatsuka, T. Tanimura, A. Furusawa, and M. Kozuma, “Storage and retrieval of a squeezed vacuum,” Phys. Rev. Lett. 100(9), 093601 (2008). [CrossRef]  

9. J. Appel, E. Figueroa, D. Korystov, M. Lobino, and A. I. Lvovsky, “Quantum memory for squeezed light,” Phys. Rev. Lett. 100(9), 093602 (2008). [CrossRef]  

10. G. Heinze, A. Rudolf, F. Beil, and T. Halfmann, “Storage of images in atomic coherences in a rare-earth-ion-doped solid,” Phys. Rev. A 81(1), 011401 (2010). [CrossRef]  

11. D. S. Ding, J. H. Wu, Z. Y. Zhou, Y. Liu, B. S. Shi, X. B. Zou, and G. C. Guo, “Multimode image memory based on a cold atomic ensemble,” Phys. Rev. A 87(1), 013835 (2013). [CrossRef]  

12. K. S. Choi, H. Deng, J. Laurat, and H. J. Kimble, “Mapping photonic entanglement into and out of a quantum memory,” Nature (London) 452(7183), 67–71 (2008). [CrossRef]  

13. T. H. Qiu, G. J. Yang, and M. Xie, “Reversible storage of photon entanglement in thermal atomic ensembles,” J. Opt. Soc. Am. B 30(3), 687–690 (2013). [CrossRef]  

14. S. B. Papp, K. S. Choi, H. Deng, P. Lougovski, S. J. van Enk, and H. J. Kimble, “Characterization of multipartite entanglement for one photon shared among four optical modes,” Science 324(5928), 764–768 (2009). [CrossRef]  

15. C. H. Yuan, L. Q. Chen, and W. P. Zhang, “Storage of polarizationencoded cluster states in an atomic system,” Phys. Rev. A 79(5), 052342 (2009). [CrossRef]  

16. E. Togan, Y. Chu, A. S. Trifonov, L. Jiang, J. Maze, L. Childress, M. V. G. Dutt, A. S. Sørensen, P. R. Hemmer, A. S. Zibrov, and M. D. Lukin, “Quantum entanglement between an optical photon and a solid-state spin qubit,” Nature (London) 466(7307), 730–734 (2010). [CrossRef]  

17. M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature (London) 426(6967), 638–641 (2003). [CrossRef]  

18. A. Andre, M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Nonlinear optics with stationary pulses of light,” Phys. Rev. Lett. 94(6), 063902 (2005). [CrossRef]  

19. S. A. Moiseev, A. I. Sidorova, and B. S. Ham, “Stationary and quasistationary light pulses in three-level cold atomic systems,” Phys. Rev. A 89(4), 043802 (2014). [CrossRef]  

20. Y. Xue and B. S. Ham, “Investigation of temporal pulse splitting in a three-level cold-atom ensemble,” Phys. Rev. A 78(5), 053830 (2008). [CrossRef]  

21. Y. W. Lin, W. T. Liao, T. Peters, H. C. Chou, J. S. Wang, H. W. Cho, P. C. Kuan, and I. A. Yu, “Stationary light pulses in cold atomic media and without bragg gratings,” Phys. Rev. Lett. 102(21), 213601 (2009). [CrossRef]  

22. Y. Zhang, Y. Zhang, X. H. Zhang, M. Yu, C. L. Cui, and J. H. Wu, “Efficient generation and control of robust stationary light signals in a double-Λ system of cold atoms,” Phys. Lett. A 376(4), 656–661 (2012). [CrossRef]  

23. S. A. Moiseev and B. S. Ham, “Quantum manipulation of two-color stationary light: quantum wavelength conversion,” Phys. Rev. A 73(3), 033812 (2006). [CrossRef]  

24. M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature (London) 426(6967), 638–641 (2003). [CrossRef]  

25. K. K. Park, Y. W. Cho, Y. T. Chough, and Y. H. Kim, “Experimental demonstration of quantum stationary light pulses in an atomic ensemble,” Phys. Rev. X 8(2), 021016 (2018). [CrossRef]  

26. H. J. Kimble, “The quantum internet,” Nature (London) 453(7198), 1023–1030 (2008). [CrossRef]  

27. K. Lemr and A. Cernoch, “Linear-optical programmable quantum router,” Opt. Commun. 300, 282–285 (2013). [CrossRef]  

28. M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73(1), 58–61 (1994). [CrossRef]  

29. D. Loss and D. P. Divincenzo, “Quantum computation with quantum dots,” Phys. Rev. A 57(1), 120–126 (1998). [CrossRef]  

30. R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha, and L. M. K. Vandersypen, “Spins in few-electron quantum dots,” Rev. Mod. Phys. 79(4), 1217–1265 (2007). [CrossRef]  

31. K. Hammerer, A. S. Sorensen, and E. S. Polzik, “Quantum interface between light and atomic ensembles,” Rev. Mod. Phys. 82(2), 1041–1093 (2010). [CrossRef]  

32. P. Rabl, “Photon blockade effect in optomechanical systems,” Phys. Rev. Lett. 107(6), 063601 (2011). [CrossRef]  

33. T. Botter, D. W. C. Brooks, N. Brahms, S. Schreppler, and D. M. Stamper-Kurn, “Linear amplifier model for optomechanical systems,” Phys. Rev. A 85(1), 013812 (2012). [CrossRef]  

34. M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, “Enhanced quantum nonlinearities in a two-mode optomechanical system,” Phys. Rev. Lett. 109(6), 063601 (2012). [CrossRef]  

35. K. K. Park, T. M. Zhao, J. C. Lee, Y. T. Chough, and Y. H. Kim, “Coherent and dynamic beam splitting based on light storage in cold atoms,” Sci. Rep. 6(1), 34279 (2016). [CrossRef]  

36. H. H. Wang, Y. F. Fan, R. Wang, D. M. Du, X. J. Zhang, Z. H. Kang, Y. Jiang, J. H. Wu, and J. Y. Gao, “Three-channel all-optical routing in a Pr3+:Y2SiO5 crystal,” Opt. Express 17(14), 12197–12202 (2009). [CrossRef]  

37. H. H. Wang, X. G. Wei, L. Wang, Y. J. Li, D. M. Du, J. H. Wu, Z. H. Kang, Y. Jiang, and J. Y. Gao, “Optical information transfer between two light channels in a Pr3+:Y2SiO5 crystal,” Opt. Express 15(24), 16044–16050 (2007). [CrossRef]  

38. H. H. Wang, A. J. Li, D. M. Du, Y. F. Fan, L. Wang, Z. H. Kang, Y. Jiang, J. H. Wu, and J. Y. Gao, “All-optical routing by light storage in a Pr3+:Y2SiO5 crystal,” Appl. Phys. Lett. 93(22), 221112 (2008). [CrossRef]  

39. L. Wang, J. X. Sun, M. X. Luo, Y. H. Sun, X. X. Wang, Y. Chen, Z. H. Kang, H. H. Wang, J. H. Wu, and J. Y. Gao, “Image routing via atomic spin coherence,” Sci. Rep. 5(1), 18179 (2015). [CrossRef]  

40. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature (London) 412(6844), 313–316 (2001). [CrossRef]  

41. J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329(5992), 662–665 (2010). [CrossRef]  

42. J. W. R. Tabosa and D. V. Petrov, “Optical pumping of orbital angular momentum of light in cold cesium atoms,” Phys. Rev. Lett. 83(24), 4967–4970 (1999). [CrossRef]  

43. S. Barreiro and J. W. R. Tabosa, “Generation of light carrying orbital angular momentum via induced coherence grating in cold atoms,” Phys. Rev. Lett. 90(13), 133001 (2003). [CrossRef]  

44. D. Akamatsu and M. Kozuma, “Coherent transfer of orbital angular momentum from an atomic system to a light field,” Phys. Rev. A 67(2), 023803 (2003). [CrossRef]  

45. A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photonics 8(3), 234–238 (2014). [CrossRef]  

46. Z. Q. Zhou, Y. L. Hua, X. Liu, G. Chen, J. S. Xu, Y. J. Han, C. F. Li, and G. C. Guo, “Quantum storage of three-dimensional orbital-angular-momentum entanglement in a crystal,” Phys. Rev. Lett. 115(7), 070502 (2015). [CrossRef]  

47. H. B. Li, V. A. Sautenkov, Y. V. Rostovtsev, G. R. Welch, P. R. Hemmer, and M. O. Scully, “Electromagnetically induced transparency controlled by a microwave field,” Phys. Rev. A 80(2), 023820 (2009). [CrossRef]  

48. D. L. Andrews and M. Babiker, The angular momentumof light (Cambridge University Press, 2012).

49. M. Moos, M. Honing, R. Unanyan, and M. Fleischhauer, “Many-body physics of Rydberg dark-state polaritons in the strongly interacting regime,” Phys. Rev. A 92(5), 053846 (2015). [CrossRef]  

References

  • View by:

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  2. M. D. Lukin, “Colloquium: Trapping and manipulating photon states in atomic ensembles,” Rev. Mod. Phys. 75(2), 457–472 (2003).
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  3. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77(2), 633–673 (2005).
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  7. H. H. Wang, Y. F. Fan, R. Wang, L. Wang, D. M. Du, Z. H. Kang, Y. Jiang, J. H. Wu, and J. Y. Gao, “Slowing and storage of double light pulses in a Pr3+:Y2SiO5 crystal,” Opt. Lett. 34(17), 2596–2598 (2009).
    [Crossref]
  8. K. Honda, D. Akamatsu, M. Arikawa, Y. Yokoi, K. Akiba, S. Nagatsuka, T. Tanimura, A. Furusawa, and M. Kozuma, “Storage and retrieval of a squeezed vacuum,” Phys. Rev. Lett. 100(9), 093601 (2008).
    [Crossref]
  9. J. Appel, E. Figueroa, D. Korystov, M. Lobino, and A. I. Lvovsky, “Quantum memory for squeezed light,” Phys. Rev. Lett. 100(9), 093602 (2008).
    [Crossref]
  10. G. Heinze, A. Rudolf, F. Beil, and T. Halfmann, “Storage of images in atomic coherences in a rare-earth-ion-doped solid,” Phys. Rev. A 81(1), 011401 (2010).
    [Crossref]
  11. D. S. Ding, J. H. Wu, Z. Y. Zhou, Y. Liu, B. S. Shi, X. B. Zou, and G. C. Guo, “Multimode image memory based on a cold atomic ensemble,” Phys. Rev. A 87(1), 013835 (2013).
    [Crossref]
  12. K. S. Choi, H. Deng, J. Laurat, and H. J. Kimble, “Mapping photonic entanglement into and out of a quantum memory,” Nature (London) 452(7183), 67–71 (2008).
    [Crossref]
  13. T. H. Qiu, G. J. Yang, and M. Xie, “Reversible storage of photon entanglement in thermal atomic ensembles,” J. Opt. Soc. Am. B 30(3), 687–690 (2013).
    [Crossref]
  14. S. B. Papp, K. S. Choi, H. Deng, P. Lougovski, S. J. van Enk, and H. J. Kimble, “Characterization of multipartite entanglement for one photon shared among four optical modes,” Science 324(5928), 764–768 (2009).
    [Crossref]
  15. C. H. Yuan, L. Q. Chen, and W. P. Zhang, “Storage of polarizationencoded cluster states in an atomic system,” Phys. Rev. A 79(5), 052342 (2009).
    [Crossref]
  16. E. Togan, Y. Chu, A. S. Trifonov, L. Jiang, J. Maze, L. Childress, M. V. G. Dutt, A. S. Sørensen, P. R. Hemmer, A. S. Zibrov, and M. D. Lukin, “Quantum entanglement between an optical photon and a solid-state spin qubit,” Nature (London) 466(7307), 730–734 (2010).
    [Crossref]
  17. M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature (London) 426(6967), 638–641 (2003).
    [Crossref]
  18. A. Andre, M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Nonlinear optics with stationary pulses of light,” Phys. Rev. Lett. 94(6), 063902 (2005).
    [Crossref]
  19. S. A. Moiseev, A. I. Sidorova, and B. S. Ham, “Stationary and quasistationary light pulses in three-level cold atomic systems,” Phys. Rev. A 89(4), 043802 (2014).
    [Crossref]
  20. Y. Xue and B. S. Ham, “Investigation of temporal pulse splitting in a three-level cold-atom ensemble,” Phys. Rev. A 78(5), 053830 (2008).
    [Crossref]
  21. Y. W. Lin, W. T. Liao, T. Peters, H. C. Chou, J. S. Wang, H. W. Cho, P. C. Kuan, and I. A. Yu, “Stationary light pulses in cold atomic media and without bragg gratings,” Phys. Rev. Lett. 102(21), 213601 (2009).
    [Crossref]
  22. Y. Zhang, Y. Zhang, X. H. Zhang, M. Yu, C. L. Cui, and J. H. Wu, “Efficient generation and control of robust stationary light signals in a double-Λ system of cold atoms,” Phys. Lett. A 376(4), 656–661 (2012).
    [Crossref]
  23. S. A. Moiseev and B. S. Ham, “Quantum manipulation of two-color stationary light: quantum wavelength conversion,” Phys. Rev. A 73(3), 033812 (2006).
    [Crossref]
  24. M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature (London) 426(6967), 638–641 (2003).
    [Crossref]
  25. K. K. Park, Y. W. Cho, Y. T. Chough, and Y. H. Kim, “Experimental demonstration of quantum stationary light pulses in an atomic ensemble,” Phys. Rev. X 8(2), 021016 (2018).
    [Crossref]
  26. H. J. Kimble, “The quantum internet,” Nature (London) 453(7198), 1023–1030 (2008).
    [Crossref]
  27. K. Lemr and A. Cernoch, “Linear-optical programmable quantum router,” Opt. Commun. 300, 282–285 (2013).
    [Crossref]
  28. M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73(1), 58–61 (1994).
    [Crossref]
  29. D. Loss and D. P. Divincenzo, “Quantum computation with quantum dots,” Phys. Rev. A 57(1), 120–126 (1998).
    [Crossref]
  30. R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha, and L. M. K. Vandersypen, “Spins in few-electron quantum dots,” Rev. Mod. Phys. 79(4), 1217–1265 (2007).
    [Crossref]
  31. K. Hammerer, A. S. Sorensen, and E. S. Polzik, “Quantum interface between light and atomic ensembles,” Rev. Mod. Phys. 82(2), 1041–1093 (2010).
    [Crossref]
  32. P. Rabl, “Photon blockade effect in optomechanical systems,” Phys. Rev. Lett. 107(6), 063601 (2011).
    [Crossref]
  33. T. Botter, D. W. C. Brooks, N. Brahms, S. Schreppler, and D. M. Stamper-Kurn, “Linear amplifier model for optomechanical systems,” Phys. Rev. A 85(1), 013812 (2012).
    [Crossref]
  34. M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, “Enhanced quantum nonlinearities in a two-mode optomechanical system,” Phys. Rev. Lett. 109(6), 063601 (2012).
    [Crossref]
  35. K. K. Park, T. M. Zhao, J. C. Lee, Y. T. Chough, and Y. H. Kim, “Coherent and dynamic beam splitting based on light storage in cold atoms,” Sci. Rep. 6(1), 34279 (2016).
    [Crossref]
  36. H. H. Wang, Y. F. Fan, R. Wang, D. M. Du, X. J. Zhang, Z. H. Kang, Y. Jiang, J. H. Wu, and J. Y. Gao, “Three-channel all-optical routing in a Pr3+:Y2SiO5 crystal,” Opt. Express 17(14), 12197–12202 (2009).
    [Crossref]
  37. H. H. Wang, X. G. Wei, L. Wang, Y. J. Li, D. M. Du, J. H. Wu, Z. H. Kang, Y. Jiang, and J. Y. Gao, “Optical information transfer between two light channels in a Pr3+:Y2SiO5 crystal,” Opt. Express 15(24), 16044–16050 (2007).
    [Crossref]
  38. H. H. Wang, A. J. Li, D. M. Du, Y. F. Fan, L. Wang, Z. H. Kang, Y. Jiang, J. H. Wu, and J. Y. Gao, “All-optical routing by light storage in a Pr3+:Y2SiO5 crystal,” Appl. Phys. Lett. 93(22), 221112 (2008).
    [Crossref]
  39. L. Wang, J. X. Sun, M. X. Luo, Y. H. Sun, X. X. Wang, Y. Chen, Z. H. Kang, H. H. Wang, J. H. Wu, and J. Y. Gao, “Image routing via atomic spin coherence,” Sci. Rep. 5(1), 18179 (2015).
    [Crossref]
  40. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature (London) 412(6844), 313–316 (2001).
    [Crossref]
  41. J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329(5992), 662–665 (2010).
    [Crossref]
  42. J. W. R. Tabosa and D. V. Petrov, “Optical pumping of orbital angular momentum of light in cold cesium atoms,” Phys. Rev. Lett. 83(24), 4967–4970 (1999).
    [Crossref]
  43. S. Barreiro and J. W. R. Tabosa, “Generation of light carrying orbital angular momentum via induced coherence grating in cold atoms,” Phys. Rev. Lett. 90(13), 133001 (2003).
    [Crossref]
  44. D. Akamatsu and M. Kozuma, “Coherent transfer of orbital angular momentum from an atomic system to a light field,” Phys. Rev. A 67(2), 023803 (2003).
    [Crossref]
  45. A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photonics 8(3), 234–238 (2014).
    [Crossref]
  46. Z. Q. Zhou, Y. L. Hua, X. Liu, G. Chen, J. S. Xu, Y. J. Han, C. F. Li, and G. C. Guo, “Quantum storage of three-dimensional orbital-angular-momentum entanglement in a crystal,” Phys. Rev. Lett. 115(7), 070502 (2015).
    [Crossref]
  47. H. B. Li, V. A. Sautenkov, Y. V. Rostovtsev, G. R. Welch, P. R. Hemmer, and M. O. Scully, “Electromagnetically induced transparency controlled by a microwave field,” Phys. Rev. A 80(2), 023820 (2009).
    [Crossref]
  48. D. L. Andrews and M. Babiker, The angular momentumof light (Cambridge University Press, 2012).
  49. M. Moos, M. Honing, R. Unanyan, and M. Fleischhauer, “Many-body physics of Rydberg dark-state polaritons in the strongly interacting regime,” Phys. Rev. A 92(5), 053846 (2015).
    [Crossref]

2018 (1)

K. K. Park, Y. W. Cho, Y. T. Chough, and Y. H. Kim, “Experimental demonstration of quantum stationary light pulses in an atomic ensemble,” Phys. Rev. X 8(2), 021016 (2018).
[Crossref]

2016 (1)

K. K. Park, T. M. Zhao, J. C. Lee, Y. T. Chough, and Y. H. Kim, “Coherent and dynamic beam splitting based on light storage in cold atoms,” Sci. Rep. 6(1), 34279 (2016).
[Crossref]

2015 (3)

L. Wang, J. X. Sun, M. X. Luo, Y. H. Sun, X. X. Wang, Y. Chen, Z. H. Kang, H. H. Wang, J. H. Wu, and J. Y. Gao, “Image routing via atomic spin coherence,” Sci. Rep. 5(1), 18179 (2015).
[Crossref]

Z. Q. Zhou, Y. L. Hua, X. Liu, G. Chen, J. S. Xu, Y. J. Han, C. F. Li, and G. C. Guo, “Quantum storage of three-dimensional orbital-angular-momentum entanglement in a crystal,” Phys. Rev. Lett. 115(7), 070502 (2015).
[Crossref]

M. Moos, M. Honing, R. Unanyan, and M. Fleischhauer, “Many-body physics of Rydberg dark-state polaritons in the strongly interacting regime,” Phys. Rev. A 92(5), 053846 (2015).
[Crossref]

2014 (2)

A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photonics 8(3), 234–238 (2014).
[Crossref]

S. A. Moiseev, A. I. Sidorova, and B. S. Ham, “Stationary and quasistationary light pulses in three-level cold atomic systems,” Phys. Rev. A 89(4), 043802 (2014).
[Crossref]

2013 (3)

K. Lemr and A. Cernoch, “Linear-optical programmable quantum router,” Opt. Commun. 300, 282–285 (2013).
[Crossref]

D. S. Ding, J. H. Wu, Z. Y. Zhou, Y. Liu, B. S. Shi, X. B. Zou, and G. C. Guo, “Multimode image memory based on a cold atomic ensemble,” Phys. Rev. A 87(1), 013835 (2013).
[Crossref]

T. H. Qiu, G. J. Yang, and M. Xie, “Reversible storage of photon entanglement in thermal atomic ensembles,” J. Opt. Soc. Am. B 30(3), 687–690 (2013).
[Crossref]

2012 (3)

Y. Zhang, Y. Zhang, X. H. Zhang, M. Yu, C. L. Cui, and J. H. Wu, “Efficient generation and control of robust stationary light signals in a double-Λ system of cold atoms,” Phys. Lett. A 376(4), 656–661 (2012).
[Crossref]

T. Botter, D. W. C. Brooks, N. Brahms, S. Schreppler, and D. M. Stamper-Kurn, “Linear amplifier model for optomechanical systems,” Phys. Rev. A 85(1), 013812 (2012).
[Crossref]

M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, “Enhanced quantum nonlinearities in a two-mode optomechanical system,” Phys. Rev. Lett. 109(6), 063601 (2012).
[Crossref]

2011 (1)

P. Rabl, “Photon blockade effect in optomechanical systems,” Phys. Rev. Lett. 107(6), 063601 (2011).
[Crossref]

2010 (4)

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329(5992), 662–665 (2010).
[Crossref]

K. Hammerer, A. S. Sorensen, and E. S. Polzik, “Quantum interface between light and atomic ensembles,” Rev. Mod. Phys. 82(2), 1041–1093 (2010).
[Crossref]

E. Togan, Y. Chu, A. S. Trifonov, L. Jiang, J. Maze, L. Childress, M. V. G. Dutt, A. S. Sørensen, P. R. Hemmer, A. S. Zibrov, and M. D. Lukin, “Quantum entanglement between an optical photon and a solid-state spin qubit,” Nature (London) 466(7307), 730–734 (2010).
[Crossref]

G. Heinze, A. Rudolf, F. Beil, and T. Halfmann, “Storage of images in atomic coherences in a rare-earth-ion-doped solid,” Phys. Rev. A 81(1), 011401 (2010).
[Crossref]

2009 (6)

S. B. Papp, K. S. Choi, H. Deng, P. Lougovski, S. J. van Enk, and H. J. Kimble, “Characterization of multipartite entanglement for one photon shared among four optical modes,” Science 324(5928), 764–768 (2009).
[Crossref]

C. H. Yuan, L. Q. Chen, and W. P. Zhang, “Storage of polarizationencoded cluster states in an atomic system,” Phys. Rev. A 79(5), 052342 (2009).
[Crossref]

H. H. Wang, Y. F. Fan, R. Wang, L. Wang, D. M. Du, Z. H. Kang, Y. Jiang, J. H. Wu, and J. Y. Gao, “Slowing and storage of double light pulses in a Pr3+:Y2SiO5 crystal,” Opt. Lett. 34(17), 2596–2598 (2009).
[Crossref]

Y. W. Lin, W. T. Liao, T. Peters, H. C. Chou, J. S. Wang, H. W. Cho, P. C. Kuan, and I. A. Yu, “Stationary light pulses in cold atomic media and without bragg gratings,” Phys. Rev. Lett. 102(21), 213601 (2009).
[Crossref]

H. H. Wang, Y. F. Fan, R. Wang, D. M. Du, X. J. Zhang, Z. H. Kang, Y. Jiang, J. H. Wu, and J. Y. Gao, “Three-channel all-optical routing in a Pr3+:Y2SiO5 crystal,” Opt. Express 17(14), 12197–12202 (2009).
[Crossref]

H. B. Li, V. A. Sautenkov, Y. V. Rostovtsev, G. R. Welch, P. R. Hemmer, and M. O. Scully, “Electromagnetically induced transparency controlled by a microwave field,” Phys. Rev. A 80(2), 023820 (2009).
[Crossref]

2008 (6)

H. H. Wang, A. J. Li, D. M. Du, Y. F. Fan, L. Wang, Z. H. Kang, Y. Jiang, J. H. Wu, and J. Y. Gao, “All-optical routing by light storage in a Pr3+:Y2SiO5 crystal,” Appl. Phys. Lett. 93(22), 221112 (2008).
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J. Appel, E. Figueroa, D. Korystov, M. Lobino, and A. I. Lvovsky, “Quantum memory for squeezed light,” Phys. Rev. Lett. 100(9), 093602 (2008).
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K. S. Choi, H. Deng, J. Laurat, and H. J. Kimble, “Mapping photonic entanglement into and out of a quantum memory,” Nature (London) 452(7183), 67–71 (2008).
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2007 (2)

H. H. Wang, X. G. Wei, L. Wang, Y. J. Li, D. M. Du, J. H. Wu, Z. H. Kang, Y. Jiang, and J. Y. Gao, “Optical information transfer between two light channels in a Pr3+:Y2SiO5 crystal,” Opt. Express 15(24), 16044–16050 (2007).
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R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha, and L. M. K. Vandersypen, “Spins in few-electron quantum dots,” Rev. Mod. Phys. 79(4), 1217–1265 (2007).
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2006 (1)

S. A. Moiseev and B. S. Ham, “Quantum manipulation of two-color stationary light: quantum wavelength conversion,” Phys. Rev. A 73(3), 033812 (2006).
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2005 (2)

A. Andre, M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Nonlinear optics with stationary pulses of light,” Phys. Rev. Lett. 94(6), 063902 (2005).
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M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77(2), 633–673 (2005).
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2003 (5)

M. D. Lukin, “Colloquium: Trapping and manipulating photon states in atomic ensembles,” Rev. Mod. Phys. 75(2), 457–472 (2003).
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M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature (London) 426(6967), 638–641 (2003).
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M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature (London) 426(6967), 638–641 (2003).
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S. Barreiro and J. W. R. Tabosa, “Generation of light carrying orbital angular momentum via induced coherence grating in cold atoms,” Phys. Rev. Lett. 90(13), 133001 (2003).
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D. Akamatsu and M. Kozuma, “Coherent transfer of orbital angular momentum from an atomic system to a light field,” Phys. Rev. A 67(2), 023803 (2003).
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2001 (3)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature (London) 412(6844), 313–316 (2001).
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D. F. Phillips, A. Fleischhauer, A. Mair, R. L. Walsworth, and M. D. Lukin, “Storage of light in atomic vapor,” Phys. Rev. Lett. 86(5), 783–786 (2001).
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2000 (1)

M. Fleischhauer and M. D. Lukin, “Dark-state polaritons in electromagnetically induced transparency,” Phys. Rev. Lett. 84(22), 5094–5097 (2000).
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1999 (1)

J. W. R. Tabosa and D. V. Petrov, “Optical pumping of orbital angular momentum of light in cold cesium atoms,” Phys. Rev. Lett. 83(24), 4967–4970 (1999).
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1998 (1)

D. Loss and D. P. Divincenzo, “Quantum computation with quantum dots,” Phys. Rev. A 57(1), 120–126 (1998).
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1997 (1)

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997).
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1994 (1)

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M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature (London) 426(6967), 638–641 (2003).
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M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature (London) 426(6967), 638–641 (2003).
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K. Lemr and A. Cernoch, “Linear-optical programmable quantum router,” Opt. Commun. 300, 282–285 (2013).
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C. H. Yuan, L. Q. Chen, and W. P. Zhang, “Storage of polarizationencoded cluster states in an atomic system,” Phys. Rev. A 79(5), 052342 (2009).
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L. Wang, J. X. Sun, M. X. Luo, Y. H. Sun, X. X. Wang, Y. Chen, Z. H. Kang, H. H. Wang, J. H. Wu, and J. Y. Gao, “Image routing via atomic spin coherence,” Sci. Rep. 5(1), 18179 (2015).
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Y. W. Lin, W. T. Liao, T. Peters, H. C. Chou, J. S. Wang, H. W. Cho, P. C. Kuan, and I. A. Yu, “Stationary light pulses in cold atomic media and without bragg gratings,” Phys. Rev. Lett. 102(21), 213601 (2009).
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K. K. Park, Y. W. Cho, Y. T. Chough, and Y. H. Kim, “Experimental demonstration of quantum stationary light pulses in an atomic ensemble,” Phys. Rev. X 8(2), 021016 (2018).
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K. K. Park, T. M. Zhao, J. C. Lee, Y. T. Chough, and Y. H. Kim, “Coherent and dynamic beam splitting based on light storage in cold atoms,” Sci. Rep. 6(1), 34279 (2016).
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K. S. Choi, H. Deng, J. Laurat, and H. J. Kimble, “Mapping photonic entanglement into and out of a quantum memory,” Nature (London) 452(7183), 67–71 (2008).
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D. S. Ding, J. H. Wu, Z. Y. Zhou, Y. Liu, B. S. Shi, X. B. Zou, and G. C. Guo, “Multimode image memory based on a cold atomic ensemble,” Phys. Rev. A 87(1), 013835 (2013).
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Fan, Y. F.

Figueroa, E.

J. Appel, E. Figueroa, D. Korystov, M. Lobino, and A. I. Lvovsky, “Quantum memory for squeezed light,” Phys. Rev. Lett. 100(9), 093602 (2008).
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D. F. Phillips, A. Fleischhauer, A. Mair, R. L. Walsworth, and M. D. Lukin, “Storage of light in atomic vapor,” Phys. Rev. Lett. 86(5), 783–786 (2001).
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M. Moos, M. Honing, R. Unanyan, and M. Fleischhauer, “Many-body physics of Rydberg dark-state polaritons in the strongly interacting regime,” Phys. Rev. A 92(5), 053846 (2015).
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M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77(2), 633–673 (2005).
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M. Fleischhauer and M. D. Lukin, “Dark-state polaritons in electromagnetically induced transparency,” Phys. Rev. Lett. 84(22), 5094–5097 (2000).
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Z. Q. Zhou, Y. L. Hua, X. Liu, G. Chen, J. S. Xu, Y. J. Han, C. F. Li, and G. C. Guo, “Quantum storage of three-dimensional orbital-angular-momentum entanglement in a crystal,” Phys. Rev. Lett. 115(7), 070502 (2015).
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D. S. Ding, J. H. Wu, Z. Y. Zhou, Y. Liu, B. S. Shi, X. B. Zou, and G. C. Guo, “Multimode image memory based on a cold atomic ensemble,” Phys. Rev. A 87(1), 013835 (2013).
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S. A. Moiseev, A. I. Sidorova, and B. S. Ham, “Stationary and quasistationary light pulses in three-level cold atomic systems,” Phys. Rev. A 89(4), 043802 (2014).
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Y. Xue and B. S. Ham, “Investigation of temporal pulse splitting in a three-level cold-atom ensemble,” Phys. Rev. A 78(5), 053830 (2008).
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S. A. Moiseev and B. S. Ham, “Quantum manipulation of two-color stationary light: quantum wavelength conversion,” Phys. Rev. A 73(3), 033812 (2006).
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R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha, and L. M. K. Vandersypen, “Spins in few-electron quantum dots,” Rev. Mod. Phys. 79(4), 1217–1265 (2007).
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Harris, S. E.

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997).
[Crossref]

Hau, L. V.

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature (London) 409(6819), 490–493 (2001).
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G. Heinze, A. Rudolf, F. Beil, and T. Halfmann, “Storage of images in atomic coherences in a rare-earth-ion-doped solid,” Phys. Rev. A 81(1), 011401 (2010).
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E. Togan, Y. Chu, A. S. Trifonov, L. Jiang, J. Maze, L. Childress, M. V. G. Dutt, A. S. Sørensen, P. R. Hemmer, A. S. Zibrov, and M. D. Lukin, “Quantum entanglement between an optical photon and a solid-state spin qubit,” Nature (London) 466(7307), 730–734 (2010).
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K. Honda, D. Akamatsu, M. Arikawa, Y. Yokoi, K. Akiba, S. Nagatsuka, T. Tanimura, A. Furusawa, and M. Kozuma, “Storage and retrieval of a squeezed vacuum,” Phys. Rev. Lett. 100(9), 093601 (2008).
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Honing, M.

M. Moos, M. Honing, R. Unanyan, and M. Fleischhauer, “Many-body physics of Rydberg dark-state polaritons in the strongly interacting regime,” Phys. Rev. A 92(5), 053846 (2015).
[Crossref]

Hua, Y. L.

Z. Q. Zhou, Y. L. Hua, X. Liu, G. Chen, J. S. Xu, Y. J. Han, C. F. Li, and G. C. Guo, “Quantum storage of three-dimensional orbital-angular-momentum entanglement in a crystal,” Phys. Rev. Lett. 115(7), 070502 (2015).
[Crossref]

Imamoglu, A.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77(2), 633–673 (2005).
[Crossref]

Ireland, D. G.

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329(5992), 662–665 (2010).
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J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329(5992), 662–665 (2010).
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J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329(5992), 662–665 (2010).
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E. Togan, Y. Chu, A. S. Trifonov, L. Jiang, J. Maze, L. Childress, M. V. G. Dutt, A. S. Sørensen, P. R. Hemmer, A. S. Zibrov, and M. D. Lukin, “Quantum entanglement between an optical photon and a solid-state spin qubit,” Nature (London) 466(7307), 730–734 (2010).
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Jiang, Y.

Kang, Z. H.

Kim, Y. H.

K. K. Park, Y. W. Cho, Y. T. Chough, and Y. H. Kim, “Experimental demonstration of quantum stationary light pulses in an atomic ensemble,” Phys. Rev. X 8(2), 021016 (2018).
[Crossref]

K. K. Park, T. M. Zhao, J. C. Lee, Y. T. Chough, and Y. H. Kim, “Coherent and dynamic beam splitting based on light storage in cold atoms,” Sci. Rep. 6(1), 34279 (2016).
[Crossref]

Kimble, H. J.

S. B. Papp, K. S. Choi, H. Deng, P. Lougovski, S. J. van Enk, and H. J. Kimble, “Characterization of multipartite entanglement for one photon shared among four optical modes,” Science 324(5928), 764–768 (2009).
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K. S. Choi, H. Deng, J. Laurat, and H. J. Kimble, “Mapping photonic entanglement into and out of a quantum memory,” Nature (London) 452(7183), 67–71 (2008).
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H. J. Kimble, “The quantum internet,” Nature (London) 453(7198), 1023–1030 (2008).
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Korystov, D.

J. Appel, E. Figueroa, D. Korystov, M. Lobino, and A. I. Lvovsky, “Quantum memory for squeezed light,” Phys. Rev. Lett. 100(9), 093602 (2008).
[Crossref]

Kouwenhoven, L. P.

R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha, and L. M. K. Vandersypen, “Spins in few-electron quantum dots,” Rev. Mod. Phys. 79(4), 1217–1265 (2007).
[Crossref]

Kozuma, M.

K. Honda, D. Akamatsu, M. Arikawa, Y. Yokoi, K. Akiba, S. Nagatsuka, T. Tanimura, A. Furusawa, and M. Kozuma, “Storage and retrieval of a squeezed vacuum,” Phys. Rev. Lett. 100(9), 093601 (2008).
[Crossref]

D. Akamatsu and M. Kozuma, “Coherent transfer of orbital angular momentum from an atomic system to a light field,” Phys. Rev. A 67(2), 023803 (2003).
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Kuan, P. C.

Y. W. Lin, W. T. Liao, T. Peters, H. C. Chou, J. S. Wang, H. W. Cho, P. C. Kuan, and I. A. Yu, “Stationary light pulses in cold atomic media and without bragg gratings,” Phys. Rev. Lett. 102(21), 213601 (2009).
[Crossref]

Laurat, J.

A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photonics 8(3), 234–238 (2014).
[Crossref]

K. S. Choi, H. Deng, J. Laurat, and H. J. Kimble, “Mapping photonic entanglement into and out of a quantum memory,” Nature (London) 452(7183), 67–71 (2008).
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Figures (5)

Fig. 1.
Fig. 1. Schematic diagram of a five-level double tripod-type system of cold $^{87}$ Rb atoms.
Fig. 2.
Fig. 2. Time evolution of the controlling fields $T_{1\pm }$ (a1, e1), $T_{2\pm }$ (a2, e2), and microwave fields $T_{m}$ ((a1, a2) corresponding to (b, c, d), (e1, e2) corresponding to (f, g, h)) in an adiabatic way. The dynamic propagation and evolution of the normalized probe field with $\Omega _{m}=0$ (b, f), $2\times 10^{-5}\Gamma$ (c, g), and $4\times 10^{-5}\Gamma$ (d, h) propagating in the $+\vec {z}$ direction. The half-width of the microwave field is equal to that of the initial signal field. The other parameters are $\Omega _{1}=\Omega _{2}=10\Gamma$ , $\gamma _{0e}=\gamma _{0f}=1.5\Gamma$ , $\gamma _{01}=\gamma _{02}=0.0001\Gamma$ , where $\Gamma$ is the population decay rate from the excited level to the ground level, and $\Gamma =5.75MHz$ .
Fig. 3.
Fig. 3. The retrieval efficiency $\eta _{p+}$ of the signal field via $\Omega _{m}$ (a) with $\Phi _{1}=0$ (black squares), $\pi /2$ (red circles), $3\pi /2$ (green triangles), and via $\Phi _{1}$ (b) with $\Omega _{m}=0$ (black squares), $2\times 10^{-5}\Gamma$ (red circles) and $4\times 10^{-5}\Gamma$ (green triangles) at the time $t=160/\Gamma$ . The other parameters are the same as in Fig. 2.
Fig. 4.
Fig. 4. The retrieval efficiency $\eta _{p+}$ (black squares) and $\eta _{p-}$ (red circles) of the signal field as a function of $\Omega _{1}/(\Omega _{1}+\Omega _{2})$ (a) with $\Omega _{m}=0$ , and $\Omega _{m}$ (b) with $\Phi _{1}=\pi /2$ , $\Phi _{2}=3\pi /2$ . The other parameters are the same as in Fig. 2.
Fig. 5.
Fig. 5. The normalized intensity patterns for the superposition of LG modes [(LG) $_{0}^{2}$ +(LG) $_{0}^{-2}$ ] at time $\Gamma t=0$ (the first column), $60$ (the second column), $100$ (the third column), $160$ (the fourth and fifth columns) under the conditions of $\Omega _{1}=\Omega _{2}$ , $\Omega _{m}=0$ (the first line), $\Omega _{1}=\Omega _{2}$ , $\Omega _{m}=2\times 10^{-5}\Gamma$ , $\Phi _{1}=\pi /2$ , $\Phi _{2}=3\pi /2$ (the second line), and $\Omega _{1}=2\Omega _{2}$ , $\Omega _{m}=2\times 10^{-5}\Gamma$ , $\Phi _{1}=\Phi _{2}=\pi /2$ (the third line). The fourth (fifth) column is corresponding to the retrieved patterns propagating in $+\vec {z}$ ( $-\vec {z}$ ) direction. The other parameters are the same as in Fig. 2.

Equations (8)

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H i n t = [ Ω p + | e 0 | + Ω p | f 0 | + Ω 1 + | e 1 | + Ω 1 | f 1 | + Ω 2 + | e 2 | + Ω 2 | f 2 | + Ω m 1 e i Φ 1 | 1 0 | + Ω m 2 e i Φ 2 | 2 0 | + h . c . ] ,
ρ 10 t = i [ Ω 1 + ρ e 0 + Ω 1 ρ f 0 + Ω m 1 e i Φ 1 ] γ 01 ρ 10 , ρ 20 t = i [ Ω 2 + ρ e 0 + Ω 2 ρ f 0 + Ω m 2 e i Φ 2 ] γ 02 ρ 20 , ρ e 0 t = i [ Ω 1 + ρ 10 + Ω 2 + ρ 20 + Ω p + ] γ 0 e ρ e 0 , ρ f 0 t = i [ Ω 1 ρ 10 + Ω 2 ρ 20 + Ω p ] γ 0 f ρ f 0 ,
Ω p + z + 1 c Ω p + t = i 2 k p + 2 Ω p + + i N d 0 e 2 k p + 2 ε 0 ρ e 0 , Ω p z 1 c Ω p t = i 2 k p 2 Ω p i N d 0 f 2 k p 2 ε 0 ρ f 0 ,
Ω p ± ( r , t ) = m , n L ± m n ( r , ψ , z ) Ω p ± m n ( z , t ) ,
L m n ( r , ψ ) = C m n w 0 [ 2 r w 0 ] | m | exp [ r 2 w 0 2 ] L n | m | [ 2 r 2 w 0 2 ] exp ( i m ψ ) ,
Ω p + m n z + 1 c Ω p + m n t = i N d 0 e 2 k p + 2 ε 0 ρ e 0 m n , Ω p m n z 1 c Ω p m n t = i N d 0 f 2 k p 2 ε 0 ρ f 0 m n ,
Ψ ± ( z , t ) = cos θ ± ( t ) Ω p ± g N sin θ ± ( t ) ( cos ϕ ± ρ 01 + sin ϕ ± ρ 02 ) ,
ρ 1 ( 2 ) 0 ( t 1 + t m ) = ρ 1 ( 2 ) 0 ( t 1 ) e γ 01 ( 2 ) t m + i e i Φ 1 ( 2 ) 1 e γ 01 ( 2 ) t m γ 01 ( 2 ) Ω m .

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