Abstract

We theoretically investigate the optical memory in a nanofiber system via electromagnetically induced transparency (EIT) in a nonlinear region. Because of the tight transverse confinement, the light-atom interaction is significantly enhanced and thus, the EIT effect is enhanced. The inhomogeneous mode field distribution contributes spatially to the EIT dispersion. We develop a systematic analysis method to study the nonlinearity of the system and prove that the optical soliton is available in the system and can be stored and retrieved with high efficiency and stability. We also study a strategy to optimize the soliton optical memory. The results obtained in this study are promising for practical applications of all-optical information processing.

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

1. Introduction

In the past two decades, quantum memory, especially optical quantum memory, has attracted considerable attention, and tremendous efforts have been made to achieve quantum information processing and communication [13]. To realize quantum optical memory, many protocols have been developed, such as the gradient echo memory (GEM) scheme [4], the atomic frequency comb (AFC) scheme [5,6], the Duan-Lukin-Cirac-Zoller (DLCZ) protocol via the Raman scheme [7], or the electromagnetically induced transparency (EIT) scheme [8,9]. Among these, the EIT scheme is one of the most promising candidates because of its potential high efficiency, long lifetime, and high fidelity [3,8,9]. A number of novel nonlinear optical effects can be generated during the EIT process, which generalizes the EIT-based memory to a weak nonlinear regime.

To date, light storage and retrieval based on EIT have been widely studied. Most of these methods rely on ensembles of atoms in free space or in other optically dense coherent media [3], which can generate a collective enhancement to improve the storage efficiency. With the help of magneto-optical trapping (MOT) techniques, an efficient EIT memory can be obtained [10]. However, these kinds of systems suffer from several imperfections, including Doppler broadening (for warm atoms), limited optical depth, and leakage of signal light. Moreover, a practical optical memory system should be easy to couple with other devices; consequently, the collection of emitted retrieval light is an additional problem for free space EIT-based optical memory.

Fortunately, a number of techniques have been developed to optimize the effect of EIT-based memory, such as confining light in the waveguide [1113], introducing nonlinearity [14,15], and optimizing with microwaves [16]. A promising solution is to use a nanofiber waveguide. Compared with free space, the light field is tightly confined at the interface of the nanofiber and the light-matter interaction is strongly enhanced. The atomic ensemble can be trapped near the nanofiber interface using two-color optical dipole trapping techniques [17]; thus, the collective enhancement is more effective [18]. Furthermore, the emitted photons can be coupled to the guide mode of the nanofiber efficiently [19], which makes it easy to collect the retrieval light. The complex polarization of the guide mode and the orbital angular momentum of the photon also add extra degrees of freedom to the light field, which can be used to achieve a new optical memory scheme [20,21].

Nanofiber-based coherent optical memory has been demonstrated experimentally in several different mechanisms, such as EIT [22,23], GEM [24], and DLCZ [18]. Since 2008, the EIT phenomenon has been observed both in warm [25,26] and cold [27] atoms near a nanofiber interface, which has accelerated the research of EIT-based optical memory in nanofiber systems. Up to 2015, two independent teams–B. Gouraud and colleagues [22] and C. Sayrin and colleagues [23]–presented experimental realizations of optical memory within the EIT scheme in two different nanofiber-based interface setups. Both teams demonstrated that nanofiber-based EIT optical memory can be realized at the single-photon level, which is promising for building quantum information networks using this technique. However, most of the theoretical and experimental work in this field is concentrated in the linear response region, which causes the retrieved light pulse to undergo a strong dispersion, while studies of the nonlinear response, which may improve the memory quality [14], are relatively few.

In this study, we propose a scheme to store and retrieve optical solitons in a nanofiber system via EIT. We show that the transverse confinement of the nanofiber significantly enhances the EIT effect and provides a spatial modulation of the linear dispersion of the stored pulse. We also develop a set of formalism theories to describe the nonlinear pulse storage and retrieval. The storage and retrieval of optical solitons proved to be efficient and robust.

This article is organized as follows. In Sec. II, the theoretical model and method are described. In Sec. III, the EIT characteristics in the nanofiber system are reported. In Sec. IV, the storage and retrieval of optical pulses are studied under a nonlinear theory framework. Finally, in Sec. V, a summary of the main results obtained in this work is provided.

2. Model

The nanofiber used in this study is shown in Fig. 1(a). The radius of the core is $r_0=125~\textrm {nm}$ with a refractive index $n_1=1.456$, and the cladding is a vacuum with refractive index $n_2=1$. Both the control field $\mathbf {E}_c$ and signal field $\mathbf {E}_s$ are tightly guided by the nanofiber interface in the fundamental mode (i.e., the $\textrm {HE}_{11}$ mode. The normalized field distribution is shown in Fig. 1(b), and the detailed expression of the field mode is given in Appendix A). The core is surrounded by a cold atomic gas ensemble. The atomic ensemble is a three-level system with a ladder-type configuration, as illustrated in Fig. 1(c), where the $|1\rangle$, $|2\rangle$, and $|3\rangle$ are the ground state, intermediate state, and upper state, respectively. The upper state $|3\rangle$ could be a Rydberg state with a long lifetime. The intermediate state is coupled to the ground state (upper) by a weak signal (strong control) field with half Rabi frequency $\Omega _s$ ($\Omega _c$). There exists a spontaneous emission decay rate $\Gamma _{23}$ ($\Gamma _{12}$) from the upper excited state (intermediate state) to the intermediate state (ground state). $\Delta _2$ and $\Delta _3$ are the detunings of the signal and control fields, respectively.

For simplicity, we choose to work in cylindrical coordinates ($\rho , \theta , z$), and assuming all fields propagate along the $z$ direction, the electrical field can be expressed as

$$\mathbf{E}=\mathbf{E}_s+\mathbf{E}_c=\sum_{l=s,c}\mathcal{E}_l(z,t)\mathbf{u}_l(\rho,\theta)e^{i(\beta_lz-\omega t)}+c.c.,$$
where $\mathcal {E}_l(z,t)=[\hbar \omega _l/2\varepsilon _0V_l^{eff}]^{1/2}a(\omega _l,z,t)$ is the slowvary envelope, $\omega _l$ is the optical frequency of the driving field, $V_l^{eff}$ is the effective mode volume, $\mathbf {u}_l(\rho ,\theta )$ is the fundamental mode distribution function, $\beta _l=\beta (\omega _l)$ is the propagation constant, $a(\omega _l,z,t)$ is the annihilation operator (because the photon number is large enough that it can be treated as a function of $\omega _l,z,t$), and the subscript $l$ indicates either the signal field ($s$) or control field ($c$) Under electric-dipole and rotating-wave approximations (RWA), the Hamiltonian of the system in the interaction picture is
$$\hat{H}_{int}={-}\hbar\sum_{j=2}^{3}\Delta_j|j\rangle\langle j| -\hbar[\zeta_s(\rho,\theta)\Omega_s|2\rangle\langle 1|, +\zeta_c(\rho,\theta)\Omega_c|3\rangle\langle 2|+h.c.],$$
where $\zeta _s(\rho ,\theta )=\mathbf {e}_{21}\cdot \mathbf {u}_s(\rho ,\theta )$ ($\zeta _c(\rho ,\theta )=\mathbf {e}_{32}\cdot \mathbf {u}_c(\rho ,\theta )$) is the mode functions of the signal field (control field), $\Omega _s=|\mathbf {p}_{21}|\mathcal {E}_s/\hbar$ ($\Omega _c=|\mathbf {p}_{32}|\mathcal {E}_c/\hbar$) is the Rabi frequency of the signal field (control field), with $\mathbf {e}_{ij}$ being the unit vector of the electric-dipole moment associated with the state $|i\rangle$ and state $|j\rangle$, i.e., $\mathbf {p}_{ij}=|p_{ij}|\mathbf {e}_{ij}$.

 

Fig. 1. (a) A scheme for optical memory via EIT using a nanofiber. The control field (red) and signal field (blue) are tightly guided by the nanofiber. The cold atomic gas ensemble is trapped near the nanofiber and strongly interacts with evanescent light fields. (b) Fundamental guided field mode distribution of the nanofiber. (c) Ladder-type energy level configuration of cold atomic gas ensemble. $\Delta _j(j =2,3)$ is the light field detuning, $\Gamma _{ij}(i<j)$ is the spontaneous emission rate from level $|j\rangle$ to level $|i\rangle$, $\Gamma _{31}$ is the external incoherent pumping , and $\Omega _c$ and $\Omega _s$ are the half Rabi frequencies of the control and signal fields, respectively.

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In this study, we use the density matrix $\sigma$ to analyze the system. Under the RWA, the equation of motion for $\sigma$ in the interaction picture is given by [28]

$$i\hbar{\big(}\frac{\partial}{\partial t}+\Gamma{\big)}\sigma={\big[}\hat{H}_{int},\sigma{\big]},$$
where $\sigma$ is a $3\times 3$ density matrix with element $\sigma _{ij}=|i\rangle \langle j|$, and $\Gamma$ is a $3\times 3$ relaxation matrix that donates the effects of spontaneous emission and dephasing. The details of Eq. (3) are given in Appendix B. Note that the control field $\Omega _c$ in our system is strong enough; thus, the change in the decay rate resulting from the Purcell effect is neglected.

The evolution of the electric field is described by the Maxwell equation $\nabla ^2\mathbf {E}-\partial ^2\mathbf {E}/(c^2\partial ^2 t) =\partial ^2\mathbf {P}/(\varepsilon _0c^2\partial ^2 t),$, where $\mathbf {P}=\mathbf {P}_{host}+\mathcal {N}\{\mathbf {p}_{12}\sigma _{21}exp[i(\beta _sz-\omega _st)], +\mathbf {p}_{23}\sigma _{32}exp[i(\beta _cz-\omega _ct)]+c.c.\}$, is the electric polarization intensity, and $\mathbf {P}_{host}=\varepsilon _0\chi _{host}\mathbf {E}$, is the electric polarization intensity of the system in the absence of atomic gas, with $\chi _{host}$ the susceptibility (i.e., $n_2=\sqrt {1+\chi _{host}^2}$), and $\mathcal {N}$ is the atomic density. Under a slowly varying envelope and mean-field approximation, the Maxwell equation reduces to

$$i{\big(}\frac{\partial}{\partial z}+\frac{n_2^2}{cn_{eff}^s}\frac{\partial}{\partial t}{\big)}\Omega_s, +\kappa_{12}\langle\sigma_{21}\rangle=0,,$$
$$i{\big(}\frac{\partial}{\partial z}+\frac{n_2^2}{cn_{eff}^c}\frac{\partial}{\partial t}{\big)}\Omega_c, +\kappa_{23}\langle\sigma_{32}\rangle=0,$$
where $n_{eff}^l=c\beta _l/\omega _l$ ($l=s,c$) is the effective reflective index of the signal field ($l=s$) and the control field ($l=c$), $\kappa _{12}=\mathcal {N}\omega _s|\mathbf {p}_{12}|^2/(2\varepsilon _0cn_{eff}^s\hbar )$. $\kappa _{23}=\mathcal {N}\omega _c|\mathbf {p}_{23}|^2/(2\varepsilon _0cn_{eff}^c\hbar )$ are the coupling strength coefficients. The expectation operator $\langle \rangle$ is defined as follows:
$$\langle f(\rho,\theta)\rangle=\frac{\int\int |\zeta_c(\rho,\theta)|^2f(\rho,\theta)\rho d\rho d\theta} {\int\int |\zeta_c(\rho,\theta)|^2\rho d\rho d\theta},$$
where $f(\rho ,\theta )$ is a function of $\rho$ and $\theta$. Note that the light field propagates along the nanofiber without any other geometry constraint in the transverse direction; thus, the diffraction effect plays no significant role. Therefore, we have neglected the diffraction terms in Eqs. (4a) and (4b).

Equations (3) and (4) are known as Maxwell-Bloch (MB) equations and contain the interaction information of the system. Using the multi-scale method developed in singular perturbation theory [29], the MB equations can be solved order by order. To this end, we regard all quantities in the MB equations as functions of multi-scale variables $z_l=\epsilon ^lz(l=0,1,2)$ and $t_l=\epsilon ^lt(l=0,1)$, where $\epsilon$ is a small parameter characterizing the typical amplitude of the signal field. Then, we perform the asymptotic expansion $\sigma _{ij}=\sum _l\epsilon ^l\sigma _{ij}^{(l)}(l=0,1,2,3)$ and $\Omega _s=\sum _l\epsilon ^l\Omega _s^{(l)}(l=1,2,3)$. By substituting the expansion into the MB equations and collecting the coefficients of $\epsilon ^l$, we can obtain a series of linear but inhomogeneous equations for each order. The explicit form of the MB equations and the solution of each order are shown in Appendix B.

Here, we assume that the atomic gas are cold and dilute enough, thus the dephasing rates, which are caused by collisions between atoms and nanofiber interface, are much smaller than the spontaneous decay rates. Note that the upper excited state $|3\rangle$ is chosen as Rydberg state (the energy level is given latter), the dilute gas condition also ensure that there is only one Rydberg atom in each blockade area, thus the interaction of Rydberg atoms is neglected.

With above assumptions, the ladder-type system is chosen from cold $^{87}{Rb}$ atomic gas ensemble, and the energy levels are [16]$|1\rangle =|5S_ {1/2},F=2\rangle ,|2\rangle =|5P_{3/2},F=3\rangle ,|3\rangle =|60S_{1/2}\rangle$, with decay parameters [30] $\Gamma _{12}/2\pi =6~\textrm {MHz}, \Gamma _{23}/2\pi =3.2~\textrm {kHz}$. Thus, the wavelengths of the control and signal fields are $480~\textrm {nm}$ and $780~\textrm {nm}$, respectively. The corresponding effective reflective indices are $n_{eff}^s\approx 1.006$ and $n_{eff}^c\approx 1.104$. The density of atom gas is chosen as $\mathcal {N}=1.1\times 10^{12}~\textrm {cm}^{-3}$, and the transition dipole moments are [30] $|\mathbf {p}_{12}|=3.584\times 10^{-29}~\textrm {C}\cdot \textrm {m}$, $|\mathbf {p}_{23}|=5.087\times 10^{-32}~\textrm {C}\cdot \textrm {m}$, thus corresponding coupling strength coefficients are $\kappa _{12}=6.1\times 10^{10}~\textrm {cm}^{-1}\cdot \textrm {s}^{-1}$ and $\kappa _{23}=2.0\times 10^{5}~\textrm {cm}^{-1}\cdot \textrm {s}^{-1}$.

3. EIT characteristics in nanofiber waveguide system

3.1 Base state

When the signal field is absent (i.e., $\Omega _s=0$), we can obtain the base state of the system from Eqs. (3) and (4). The base state is

$$\begin{aligned}\sigma_{11}^{(0)}=\frac{\Gamma_{12}\Gamma_{23}|d_{32}|^2+2\gamma_{32} \Gamma_{12}|\zeta_c(\rho,\theta)\Omega_c|^2}{\Gamma|d_{32}|^2+2\gamma_{32}|\zeta_c(\rho,\theta)\Omega_c|^2 (\Gamma_{12}+2\Gamma_{31})}, \end{aligned}$$
$$\begin{aligned}\sigma_{22}^{(0)}=\frac{\Gamma_{31}}{\Gamma_{12}}\sigma_{11}^{(0)}, \end{aligned}$$
$$\begin{aligned}\sigma_{33}^{(0)}=1-(\sigma_{11}^{(0)}+\sigma_{22}^{(0)}), \end{aligned}$$
$$\begin{aligned}\sigma_{32}^{(0)}=\frac{\zeta_c(\rho,\theta)\Omega_c}{d_{32}}(\sigma_{33}^{(0)}-\sigma_{22}^{(0)}), \end{aligned}$$
$$\begin{aligned}\sigma_{31}^{(0)}=\sigma_{21}^{(0)}=0. \end{aligned}$$
Here, we have defined $\Gamma =\Gamma _{12}\Gamma _{23}+\Gamma _{23}\Gamma _{31}+\Gamma _{31}\Gamma _{12}$, and $d_{32}$ is given in Appendix B.

It is helpful to analyze the base state, which can provide useful information about the initial state of the system. We note that if there is no incoherent pumping in the system ($i.e., \Gamma _{31}=0$), the system only populates in state $|1\rangle$, and the coherence $\langle \sigma _{32}^{(0)}\rangle$ vanishes. However, when introducing incoherent pumping into the system, state $|2\rangle$ can be populated, i.e., $\langle \sigma _{22}^{(0)}\rangle \neq 0$, and coherence $\langle \sigma _{32}^{(0)}\rangle \neq 0$, which implies that a Raman-like gain will appear in the signal field. This gain can be used to overcome the absorption of the signal field in a lossy medium waveguide system (such as a nanowire) during the propagation and storage processes.

3.2 Linear dispersion and slow light effect

In the first-order approximation of the MB equations, we obtain the linear dispersion relation of the signal field:

$$K(\omega)=\frac{\omega}{c}\frac{n_2^2}{n_{eff}}+, \kappa_{12}\left\langle{\frac{(\omega+d_{31})(\sigma_{22}^{(0)}-\sigma_{11}^{(0)})-\zeta_c^{*}(\rho,\theta)\Omega_{c}^{*}\sigma_{32}^{(0)}}{(\omega+d_{21})(\omega+d_{31})-|\zeta_c(\rho,\theta)\Omega_c|^2}}\right\rangle.$$
where $\omega$ is the frequency shift to the center frequency of the signal field; thus, $\omega =0$ corresponds to the center frequency. Because the field near the nanofiber interface is an evanescent field, the EIT effect in the nanofiber system is different from that in free space. The interaction is significantly enhanced, and there is a spatial modulation of the EIT effect.

Atom parameters is given in Sec. 2, other system parameters used in this section are $\Omega _c^\textrm {free}/2\pi =1~\textrm {MHz}$, and $\Delta _2=\Delta _3=0$. $\Omega _c^\textrm {free}(\Omega _c^\textrm {nano})$ is the half Rabi frequency in free space (nanofiber).

Figure 2 shows the real part (dispersion) and imaginary part (absorption) of $K(\omega )$ as a function of $\omega$. The red and blue curves represent the free space and nanofiber interface, respectively. As shown in Fig. 2(a), a transparency window is opened near the center frequency (i.e., $\omega =0$) in the absorption spectrum, which is a result of quantum interference. However, the width of the transparency window is much wider for the nanofiber system (note that the comparison is based on the same energy of the control field, which gives $|\Omega _c^\textrm {nano}/\Omega _c^\textrm {free}|\approx 3.55$, and the ratio is calculated via set the power in the cross section equal in the two cases, the power can be obtained by integrating Poynting vector over the cross-section), which means the interaction between light and atoms and the EIT effect are enhanced. Figure 2(b) shows the EIT dispersion feature, where it is clear that the curve corresponding to the nanofiber (blue curve, compared with the red curve) is broadened. Then, the slope of the curve near the center frequency becomes smaller, which means that the group velocity ($\textrm {Re}[\partial K(\omega )/\partial \omega ]^{-1}$) is greater. This agrees with the fact that the interaction is enhanced, which is a result of the mode confinement effect of the nanofiber. We observe that both the absorption and dispersion spectra are broadened in the nanofiber system. The physical reason for this is that the atomic gas ensemble interacts with the evanescent field; thus, the EIT effect is modulated by the inhomogeneous field distribution. Note that the modulation of EIT effect is related the integration over the cross section, this means the enhancement of EIT in nanofiber system is an average effect. In addition, we have taken the incoherent pumping $\Gamma _{31}$, which introduced to overcome the loss of the system [31], to be $0$, since the nanofiber system is lossless in linear region.

 

Fig. 2. (a) Absorption spectrum $\textrm {Im}(K)$; (b) dispersion relation $\textrm {Re}(K)$ as a function of frequency shift $\omega$. The red and blue curves correspond to the free space and nanofiber system at the same energy power level of the control field, respectively.

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4. Ultraslow soliton at optical nanofiber interface

4.1 Nonlinear theory for EIT memory

The dark-state polariton theory provides a clear physical picture of the optical memory; however, it also uses many approximations, such as neglecting the nonlinearity. In this subsection, we introduce nonlinearity into the EIT memory model by combining the multi-scale method and the dark state polariton theory.

Using a multi-scale method, we obtain the second-order approximation based on the lower-order approximations. The solvable condition of the second-order approximation is $i[\partial F/\partial z_1+\partial F/(V_g\partial t_1)]=0,$, where $V_g=[\partial K(\omega )/\partial \omega ]^{-1}$ is the complex group velocity of the signal field. The explicit form of the second-order approximation is given in Appendix B. Similarly, the solvable condition of the third-order approximation is $i\partial F/\partial z_2-K_2\partial ^2 F/(2\partial t_1^2)-W|F|^2Fe^{-2\bar {\alpha }z_2}=0,$, where $K_2=\partial ^2 K(\omega )/\partial \omega ^2$ is the group velocity dispersion and $W$ is the self-phase modulation coefficient (the explicit form is given in Appendix B). Combining the solvability conditions for all order approximations and returning to the original variables, we obtain the nonlinear evolution equation of the signal field envelope

$$i(\frac{\partial}{\partial z}+\alpha)U-\frac{K_2}{2}\frac{\partial^2U}{\partial\tau^2}-W|U|^2U=0,$$
with $\alpha =\textrm {Im}[K(\omega =0)],U=\epsilon Fe^{-\alpha z},\tau =t-z/V_g$. Typically, the coefficient in Eq. (8) is a complex number; however, we can make the imaginary part of these coefficients significantly smaller than their corresponding real part with appropriate parameters in the EIT condition [12,14]. The physical reason for this is that quantum interference reduces the absorption in the system and maintains a strong resonance at the same time.

We introduce the symbols $\mathcal {U}_0, L_\textrm {D}$, and $\tau _0$, which are the typical half Rabi frequency, typical dispersion length, and typical time scale, respectively. Now, we define the dimensionless variables as $u=U/\mathcal {U}_0, s=-z/(2L_\textrm {D})$, and $\xi =\tau /\tau _0$. Then, Eq. (8) can be written in dimensionless form: $i\partial u/\partial s+\partial ^2u/\partial \xi ^2+2|u|u=0$, which is known as the nonlinear Schrödinger equation. Note that we neglect the small absorption term proportional to $\alpha$. The dispersion length is $L_\textrm {D}=\tau _0^2/\tilde {K}_2$, while the nonlinear length is $L_N=1/(\tilde {W}|\mathcal {U}_0|^2)$. The dimensionless nonlinear Schrödinger equation is valid when the dispersion and nonlinear effects are balanced, that is, $L_\textrm {D}=L_N$, which yields $\mathcal {U}_0= (1/\tau _0)\sqrt {\tilde {K}_2/\tilde {W}}$. Here, the tilde above the symbol denotes the real part: $\tilde {K}_2=\textrm {Re}(K_2), \tilde {W}=\textrm {Re}(W)$. A single soliton solution of the nonlinear Schrödinger equation reads $u(s,\xi )=2\beta \mathrm {sech}[2\beta (\xi -\xi _{0}+4\delta s)]\exp [-2i\delta \xi -4i(\delta ^{2}-\beta ^{2})s-i\phi _{0}]$, where $\beta$, $\delta$, $\xi _{0}$, and $\phi _{0}$ are free parameters that determine the amplitude (also width), propagating velocity, initial position, and initial phase of the soliton, respectively. For simplicity, we take $\beta =1/2$ and $\delta =\xi _{0}=\phi _{0}=0$; then, the solution reads $u=\mathrm {sech}\xi \exp (is)$, Returning to the form of the signal field

$$\begin{aligned} \mathbf{E}_s(\mathbf{r},t)&=\frac{\hbar}{|\mathbf{p}_{12}|\tau_{0}}\sqrt{\frac{\widetilde{K_{2}}}{\widetilde{W}}} \mathbf{u}_s(\rho,\theta)\cdot\mathrm{sech}\left[ \frac{1}{\tau_{0}}(\tau-\frac{z}{\widetilde{V}_{g}})\right]\\ &\times\exp\left[ i\left(k(\omega_s)+\widetilde{K_{0}}+\frac{1}{2L_{D}}\right)z-\omega_st\right] +c.c. \end{aligned}$$
with $\widetilde {K_{0}}=\mathrm {Re}[K(0)]$.

The above analysis can be achieved using a set of realistic parameters: $\Delta _2=-8\times 10^8~\textrm {s}^{-1}, \Delta _3=1.1\times 10^6~\textrm {s}^{-1}, \Omega _c=1\times 10^{8}~\textrm {s}^{-1}, \tau _0=1\times 10^{-7}~\textrm {s}$. The other system parameters are the same as those used in the Sec. 2. Using these values, we obtain $K_0=(33.834+2.591i)~\textrm {cm}^{-1}, K_1=(2.125+0.173i)\times 10^{-6}~\textrm {cm}^{-1}\textrm {s}, K_2=(-7.860+1.532i)\times 10^{-13}~\textrm {cm}^{-1}\textrm {s}^2$, and $W=(1.667+0.121i)\times 10^{-13}~\textrm {cm}^{-1}\textrm {s}^2$. All of these complex coefficients have a much larger real part than corresponding imaginary part. Then, we obtain $\mathcal {U}_0=2.17\times 10^{7}~\textrm {s}^{-1}$, $L_D=L_N=0.0127~\textrm {cm}$. Note that the absorption length $L_A=1/[2\textrm {Im}(K_0)]=0.19~\textrm {cm}$ is much larger than $L_D,L_N$; thus, when the propagation length is within the absorption length, the above analysis is valid.

To store and retrieve the optical soliton, we introduce an adiabatic switch control field, which in the form of the half Rabi frequency reads:

$$\Omega_c(0,t)=\Omega_{c0}\left\{1-\frac{1}{2}\cdot\tanh\left[\frac{t-T_\textrm{off}}{T_\textrm{s}}\right]+\frac{1}{2}\cdot\tanh\left[\frac{t-T_\textrm{on}}{T_\textrm{s}}\right]\right\},$$
where $T_\textrm {s}$ is the adiabatic parameter, and $T_\textrm {off}$ and $T_\textrm {on}$ are the switch-off and switch-on moments of the control field, respectively. According to the dark state polariton theory, before, during, and after the storage of the soliton, the soliton can be written as
$$\Omega_s(z,t)\approx \left\{ \begin{array}{c} \frac{A}{\tau_{0}}\sqrt{\frac{\widetilde{K_{2}}}{\widetilde{W}}} \mathrm{sech}[\frac{1}{\tau_{0}}(\tau-\frac{z}{\widetilde{V}_{g}})] \exp[i\widetilde{K_{0}}z+i\frac{z}{2L_{D}}],~~~~t<T_\textrm{off} \\ 0,~~~~T_\textrm{off}<t<T_\textrm{on} \\ \frac{B}{\tau_{0}}\sqrt{\frac{\widetilde{K_{2}}}{\widetilde{W}}} \mathrm{sech}[\frac{1}{\tau_{0}}(\tau-\frac{z}{\widetilde{V}_{g}}-T_\textrm{D})] \exp[i\widetilde{K_{0}}z+i\frac{z}{2L_{D}}+i\varphi_0] ,~~~~t>T_\textrm{on} \end{array} \right.$$
Here, $A,B$ are constants related to the initial signal pulse, $\varphi _0$ is a constant phase factor, and $T_\textrm {D}\approx T_\textrm {on}-T_\textrm {off}$ is the time delay of the storage process.

4.2 Storage and retrieval of ultraslow solitons

In this subsection, we use a numerical method to verify our former theory and study the storage of the optical solitons on the nanofiber interface. For simplicity but without distortion, we assume that the field mode will not change during the propagation of light in the nanofiber. With this assumption, we simplify the MB equations into the effective MB equation (the explicit form is given in Appendix C), and our numerical analysis is based on the effective MB equation. Before studying the storage and retrieval optical solitons, we first define several indices that measure the efect of optical storage:

$$\begin{aligned}\eta = \frac{\int_{-\infty}^{+\infty}|\Omega_s(L,t)|^2dt}{\int_{-\infty}^{+\infty}|\Omega_s(z=0,t)|^2dt} \end{aligned}$$
$$\begin{aligned}J^2 = \frac{|\int_{-\infty}^{+\infty}\Omega_s(z=0,t)\cdot\Omega_s(z=L,t+T_\textrm{D}')dt|^2}{\int_{-\infty}^{+\infty}|\Omega_s(z=0,t)|^2dt\cdot\int_{-\infty}^{+\infty}|\Omega_s(z=L,t+T_\textrm{D}')|^2dt}. \end{aligned}$$
$\eta$ is the storage efficiency, measuring the energy loss of the output pulse; $J^2$ is the overlap integral, measuring the waveform overlap between the output pulse and the input pulse, where the parameters $L,T_\textrm {D}'$ are the length of the nanofiber and the time delay between the output and input pulse, respectively. Thus, the quantity $\eta J^2$ characterizes the fidelity of the light memory.

The parameters are $L=1~\textrm {mm}$, $\Omega _{c0}=1\times 10^{8}~\textrm {s}^{-1}$, and $T_\textrm {off}=3\tau _0, T_\textrm {on}=13\tau _0$. $\Delta _2, \Delta _3$ are consistent with these in last subsection. The other system parameters are the same as those used in the Sec. 2. The control field takes the form of Eq. (10), and the input signal pulse takes the form $\Omega _{s0}\cdot \mathrm {sech}[1.763t/\tau _0]$, that is, a hyperbolic secant pulse with pulse duration $\tau _0$ and amplitude $\Omega _{s0}$.

Figure 3 shows the variation of signal pulse $\Omega _s\tau _0$ (color filled areas) and control pulse $\Omega _c\tau _0$ (color lines) in three optical memory processes, where the different colors indicate that the fields propagate to a corresponding position in the nanofiber. The signal pulse is launched into the nanofiber at $z=0$ and read out at $z=1~\textrm {mm}$ after a 1 $\mu s$ storage and retrieval process.

 

Fig. 3. Temporal and spatial evolution of $|\Omega _c\tau _0|$ and $|\Omega _s\tau _0|$ for the EIT memory process in the nanofiber system. Memory process for (a) a relatively weak signal pulse with an input pulse $\Omega _{s}(0,t)\tau _0=1.8\mathrm {sech}[1.763t/\tau _0]$; (b)a soliton signal pulse with input pulse $\Omega _{s}(0,t)=3.0\mathrm {sech}[1.763t/\tau _0]$; (c) a relatively strong signal pulse with input pulse $\Omega _{s}(0,t)=4.0\mathrm {sech}[1.763t/\tau _0]$. The initial amplitude of the switch control field is $\Omega _{c0}\tau _0=10$.

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Figure 3(a) illustrates the optical memory process for a weak signal field with input pulse $\Omega _{s}(0,t)\tau _0=1.8\mathrm {sech}[1.763t/\tau _0]$. It is clear that the signal pulse width broadened, which results from the linear dispersive feature of the system. In our numerical simulation, the detuning $\Delta _{2,3}$ does not vanish to maintain a realistic condition; thus, linear dispersion is unavoidable. The storage efficiency in this simulation is $\eta \approx 82.5\%$, and the dispersion-caused pulse deformation yields an overlap integral $J^2\approx 84.7\%$, which finally gives a low memory fidelity $\eta J^2\approx 69.9\%$.

Figure 3(b) illustrates the optical memory process for a soliton signal field with input pulse $\Omega _{s}(0,t)\tau _0=3.0\mathrm {sech}[1.763t/\tau _0]$. In this case, $\Omega _{s0}$ is similar to the soliton amplitude we calculated in the previous subsection; we observe that the signal pulse deforms to a sharp soliton before we turn off the control field, and the retrieval signal keeps almost the same profile, except for a slight decrease in amplitude. Although the pulse still deforms compared with the input pulse, the deformation is much less than that in the weak signal field case because of the weak nonlinearity counteracting the dispersion effect. The efficiency $\eta \approx 90.8\%$ and overlap integral $J^2\approx 96.5\%$, and the memory fidelity improves to $\eta J^2\approx 87.7\%$.

Figure 3(c) illustrates the optical memory process for a strong signal field with input pulse $\Omega _{s}(0,t)\tau _0=4.0\mathrm {sech}[1.763t/\tau _0]$. For such a signal field, a huge nonlinearity can be obtained, which can lead the signal pulse to propagate unstably. The signal pulse becomes significantly distorted after storage, and higher harmonics are generated. The strong nonlinearity causes not only significant distortion (overlap integral is $J^2\approx 76.0\%$), but also a low efficiency $\eta \approx 76.8\%$ because of the nonlinear absorption. Finally, the memory fidelity is $\eta J^2\approx 58.4\%$.

The above discussion indicates that the optical soliton memory has a relatively high stability and efficiency. This inspired us to optimize the nanofiber optical memory using the nonlinearity provided by the confinement waveguide system. To verify this, we simulated the nanofiber memory process with the same parameters as above, but varied the input signal amplitude $|\Omega _{s0}\tau _0|$ and input control field amplitude $|\Omega _{c0}\tau _0|$.

Figure 4(a) shows the dependence of the memory fidelity $\eta J^2$ on $|\Omega _{s0}\tau _0|$ and $|\Omega _{c0}\tau _0|$. There is a band region where the memory fidelity is higher than $80\%$. Near this band, the memory fidelity decays quickly. In the band region, the amplitude of the control field $|\Omega _{c0}\tau _0|$ is relatively large, and we can obtain a relatively high memory fidelity, even if we vary the signal amplitude $|\Omega _{s0}\tau _0|$ over a wide range. Notice that the nonlinearity of the system is associated with both the control field amplitude and the signal field amplitude, but the dispersion effect is mainly determined by the control field amplitude as discussed in the previews theoretical analysis. Thus the region of high fidelity is sensitive to the control field amplitude but not the signal field amplitude.

 

Fig. 4. Optimizing the nanofiber optical memory via nonlinearity. (a) Memory fidelity $\eta J^2$ as a function of the input signal amplitude $|\Omega _{s0}\tau _0|$ and input control field amplitude $|\Omega _{c0}\tau _0|$. (b) Readout signal pulse $\Omega _s(L,t)\tau _0$ as a function of $|\Omega _{c0}\tau _0|$ when fixing $|\Omega _{s0}\tau _0|=2.1$.

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Figure 4(b) shows the readout signal pulse $\Omega _s(L,t)\tau _0$ as a function of $|\Omega _{c0}\tau _0|$ when fixing $|\Omega _{s0}\tau _0|=3.1$. At the beginning, the readout pulse has a broadened width. With increasing $|\Omega _{c0}\tau _0|$, the broadening of the readout pulse is reduced; at the same time, the pulse delay is reduced. We can understand these results as follow: for small $|\Omega _{c0}\tau _0|$, the dispersion effect is strong and the pulse gains a large pulse broadening; as $|\Omega _{c0}\tau _0|$ become bigger, the dispersion effect is reduced and can be balanced by the weak nonlinearity; the overlap integral $J^2$ becomes bigger, which improves the memory fidelity $\eta J^2$; as $|\Omega _{c0}\tau _0|$ keeps increasing, the read out pulse is partly retrieved and a leakage signal appears, which decreases the storage efficiency $\eta$ as well as $\eta J^2$. As the $|\Omega _{c0}\tau _0|$ increases, the group velocity also increases. Thus, the time delay of the readout pulse decreases, and a leakage signal appears when $|\Omega _{c0}\tau _0|$ is too large.

5. Conclusions

In conclusion, we propose a scheme for storage and retrieval of optical solitons in nanofibers based on EIT. In the linear region, our analysis shows that the evanescent field at the interface of the nanofiber can significantly enhance the interaction of light and cold atoms in the system, thereby enhancing the EIT effect. Moreover, the line shape of the EIT spectrum is modulated by the spatial distribution of the evanescent field mode. In the nonlinear region, we demonstrate that the optical soliton is available in nanofiber systems. The stability of optical soliton storage, retrieval, and propagation in the nanofiber system is further analyzed using numerical methods. The results show that the storage and retrieval of optical solitons in the nanofiber demonstrate high storage quality and stability. We also study a strategy to optimize the soliton optical memory. Our results demonstrate that the nonlinearity in the nanofiber EIT system is a powerful method to improve the optical memory, and the nanofiber EIT system may has great potential for building future high performance optical quantum information networks.

Appendix

A. Electrical field mode of nanofiber

The mode function $\mathbf {u}_l(\rho ,\theta )$ is a vector function in cylindrical coordinates, $\mathbf {u}_l(\rho ,\theta )=(u_{l,\rho },u_{l,\theta },u_{l,z})$, and it satisfies the Helmholtz equation in cylindrical coordinates.

$$\frac{\partial^2\mathbf{u}_l(\rho,\theta)}{\partial \rho^2}+\frac{\partial\mathbf{u}_l(\rho,\theta)}{\partial \rho}+\frac{1}{\rho^2}\frac{\partial^2\mathbf{u}_l(\rho,\theta)}{\partial \theta^2}+(n^2k_0^2-\beta_l^2)\mathbf{u}_l(\rho,\theta)=0$$
where $n$ is the refractive index($n=n_1$ for $\rho <a$ and $n=n_2$ for $\rho >a$, $a$ is the radius of the nanofiber), $\beta _l$ is the propagation constant, and $k_0=\omega _l/c$ is the free-space wave number. It can be solved using the method of separation of variables, and the variation of variable $\rho$ is a modified Bessel function of the first or second kind. The variation of variable $\theta$ is a cosine or sine function. For the region $\rho <a$,
$$\begin{aligned}u_{l,\rho}=\frac{i\beta_l}{2\kappa J_m(U)}\left[(1-s)J_{m-1}-(1+s)J_{m+1}\right], \left\{ \begin{array}{c} \cos(m\theta)\\ i\sin(m\theta), \end{array} \right\}, \end{aligned}$$
$$\begin{aligned}u_{l,\theta}=-\frac{i\beta_l}{2\kappa J_m(U_0)}\left[(1-s)J_{m-1}+(1+s)J_{m+1}\right], \left\{ \begin{array}{c} \sin(m\theta)\\ -i\cos(m\theta), \end{array} \right\}, \end{aligned}$$
$$\begin{aligned}u_{l,z}=\frac{1}{J_m(U_0)}J_m, \left\{ \begin{array}{c} \cos(m\theta)\\ i\sin(m\theta), \end{array} \right\}. \end{aligned}$$
For the region $\rho >a$,
$$\begin{aligned}u_{l,\rho}=\frac{i\beta_l}{2\gamma K_m(W_0)}\left[(1-s)K_{m-1}+(1+s)K_{m+1}\right], \left\{ \begin{array}{c} \cos(m\theta)\\ i\sin(m\theta), \end{array} \right\}, \end{aligned}$$
$$\begin{aligned}u_{l,\theta}=-\frac{i\beta_l}{2\gamma K_m(W_0)}\left[(1-s)K_{m-1}-(1+s)K_{m+1}\right], \left\{ \begin{array}{c} \sin(m\theta)\\ -i\cos(m\theta), \end{array} \right\}, \end{aligned}$$
$$\begin{aligned}u_{l,z}=\frac{1}{K_m(W_0)}K_m, \left\{ \begin{array}{c} \cos(m\theta)\\ i\sin(m\theta), \end{array} \right\}. \end{aligned}$$
Here, $m$ is an integer; $J_q=J_q(\gamma \rho )$ and $K_q=K_q(\kappa \rho )$($q$ is an integer) are the modified Bessel functions of the first and second kind, respectively; $\gamma =\sqrt {n_1^2k_0^2-\beta _l^2}; \kappa =\sqrt {\beta _l^2-n_2^2k_0^2}$; $U_0=\gamma a; W_0=\kappa a$; and parameter $s$ is defined as
$$s=\frac{1/U_0^2+1/W_0^2}{J_m'(U_0)/J_m(U_0)U_0+K_m'(W_0)/K_m(W_0)W_0}$$
The propagation constant $\beta$ is determined by the eigen equation of the nanofiber
$$\left(\frac{n_1^2}{U_0}\frac{J'_m(U_0)}{J_m(U_0)}+\frac{n_2^2}{W_0}\frac{K'_m(W_0)}{K_m(W_0)}\right) \left(\frac{1}{U_0}\frac{J'_m(U_0)}{J_m(U_0)}+\frac{1}{W_0}\frac{K'_m(W_0)}{K_m(W_0)}\right) =m^2n_\textrm{eff}^2\left(\frac{V}{U_0W_0}\right)^4.$$
with $n_\textrm {eff}=\beta _l/k_0$, $V=\sqrt {U_0^2+W_0^2}$. For the fundamental mode of the nanofiber(i.e., the $HE_{11}$ mode), $m=1$.

B. Bloch equations and solutions for each order

B.1. Bloch equations

$$\begin{aligned} &i(\frac{\partial}{\partial t}+\Gamma_{31})\sigma_{11}-i\Gamma_{12}\sigma_{22}+\zeta_s^{*}(\rho,\theta)\Omega_{s}^{*}\sigma_{21}-\zeta_s(\rho,\theta)\Omega_{s}\sigma_{21}^{*}=0,. \end{aligned}$$
$$\begin{aligned} &i(\frac{\partial}{\partial t}+\Gamma_{12})\sigma_{22}-i\Gamma_{23}\sigma_{33}+\zeta_s(\rho,\theta)\Omega_{s}\sigma_{21}^{*}+\zeta_c^{*}(\rho,\theta)\Omega_{c}^{*}\sigma_{32}\\ & -\zeta_s^{*}(\rho,\theta)\Omega_{s}^{*}\sigma_{21}-\zeta_c(\rho,\theta)\Omega_{c}\sigma_{32}^{*}=0,. \end{aligned}$$
$$\begin{aligned} i(\frac{\partial}{\partial t}+\Gamma_{23})\sigma_{33}-i\Gamma_{31}\sigma_{11}+\zeta_c(\rho,\theta)\Omega_{c}\sigma_{32}^{*}-\zeta_c^{*}(\rho,\theta)\Omega_{c}^{*}\sigma_{32}=0,. \end{aligned}$$
$$\begin{aligned}(i\frac{\partial}{\partial t}+d_{21})\sigma_{21}+\zeta_c^{*}(\rho,\theta)\Omega_{c}^{*}\sigma_{31}+\zeta_s(\rho,\theta)\Omega_{s}(\sigma_{11}-\sigma_{22})=0. \end{aligned}$$
$$\begin{aligned}(i\frac{\partial}{\partial t}+d_{31})\sigma_{31}-\zeta_s(\rho,\theta)\Omega_{s}\sigma_{32}+\zeta_c(\rho,\theta)\Omega_{c}\sigma_{21}=0, \end{aligned}$$
$$\begin{aligned}(i\frac{\partial}{\partial t}+d_{32})\sigma_{32}-\zeta_s^{*}(\rho,\theta)\Omega_{s}^{*}\sigma_{31}+\zeta_c(\rho,\theta)\Omega_{c}(\sigma_{22}-\sigma_{33})=0. \end{aligned}$$
with $d_{21}=\Delta _2+i\gamma _{21},d_{31}=\Delta _3+i\gamma _{31},d_{32}=\Delta _3-\Delta _2+i\gamma _{32}$ and $\gamma _{ij}=(\Gamma _i+\Gamma _j)/2$.

B.2. First-order approximation of MB equations

$$\begin{aligned}\Omega_s^{(1)} = Fe^{i\Theta} \end{aligned}$$
$$\begin{aligned}\sigma_{21}^{(1)}=\frac{D_1}{D}\zeta_s(\rho,\theta)\Omega_s^{(1)} \end{aligned}$$
$$\begin{aligned}\sigma_{31}^{(1)}=\frac{D_2}{D}\zeta_s(\rho,\theta)\Omega_s^{(1)} \end{aligned}$$
where $F$ is the envelope function of the signal pulse, $\Theta =K(\omega )z_0-\omega t_0$, $D=(\omega +d_{21})(\omega +d_{31})-|\zeta _c(\rho ,\theta )\Omega _c|^2, D_1=(\omega +d_{31})(\sigma _{22}^{(0)}-\sigma _{11}^{(0)})-\zeta _c^{*}(\rho ,\theta )\Omega _{c}^{*}\sigma _{32}^{(0)}, D_2=(\omega +d_{21})\sigma _{32}^{(0)}-(\sigma _{22}^{(0)}-\sigma _{11}^{(0)})\zeta _c(\rho ,\theta )\Omega _{c}$. Moreover, $\sigma _{11}^{(1)}=\sigma _{22}^{(1)}=\sigma _{33}^{(1)}=\sigma _{32}^{(1)}=0$,

B.3. Second-order approximation of MB equations

$$\begin{aligned}\sigma_{11}^{(2)}&=\frac{[{-}i\Gamma_{23}+2|\zeta_c(\rho,\theta)\Omega_{c}|^2(\frac{1}{d_{32}}-\frac{1}{d_{32}^*})](\frac{D_1^*}{D^*}-\frac{D_1}{D})+i\Gamma_{12}(\frac{1}{d_{32}}\frac{D_2}{D}\zeta_c^{*}(\rho,\theta)\Omega_{c}^{*}-c.c.)}{\Gamma_{23}\Gamma_{31}+\Gamma_{12}\Gamma_{23}+i(2\Gamma_{31}+\Gamma_{12})|\zeta_c(\rho,\theta)\Omega_{c}|^2(\frac{1}{d_{32}}-\frac{1}{d_{32}^*})}\\ &\quad ~\times|\zeta_s(\rho,\theta)|^2|F|^2e^{{-}2\bar{\alpha}z_2}\\ &=a_{11}^{(2)}|\zeta_s(\rho,\theta)|^2|F|^2e^{{-}2\bar{\alpha}z_2} \end{aligned}$$
$$\begin{aligned} \sigma_{22}^{(2)} &=\frac{i}{\Gamma_{12}}[(\frac{D_1^*}{D^*}-\frac{D_1}{D})-i\Gamma_{31}a_{11}^{(2)}]|\zeta_s(\rho,\theta)|^2|F|^2e^{{-}2\bar{\alpha}z_2},\\ &=a_{22}^{(2)}|\zeta_s(\rho,\theta)|^2|F|^2e^{{-}2\bar{\alpha}z_2} \end{aligned}$$
$$\begin{aligned} \sigma_{32}^{(2)} &=\frac{1}{d_{32}}[\frac{D_2}{D}-\zeta(\rho,\theta)\Omega_{c}(2a_{22}^{(2)}+a_{11}^{(2)})]|\zeta_s(\rho,\theta)|^2|F|^2e^{{-}2\bar{\alpha}z_2}\\ &=a_{32}^{(2)}|\zeta_s(\rho,\theta)|^2|F|^2e^{{-}2\bar{\alpha}z_2} \end{aligned}$$
$$\begin{aligned} \sigma_{21}^{(2)} &=i\frac{D_2\zeta^{*}(\rho,\theta)\Omega_{c}^{*}-(\omega+d_{31})D_1}{D^2}\zeta_s(\rho,\theta)\frac{\partial F}{\partial t_1}e^{i\Theta}\\ &=a_{21}^{(2)}\zeta_s(\rho,\theta)\frac{\partial F}{\partial t_1}e^{i\Theta}, \end{aligned}$$
$$\begin{aligned} \sigma_{31}^{(2)} &=-\frac{1}{\zeta_c^{*}(\rho,\theta)\Omega_c^*}[i\frac{D_1}{D}+(\omega+d_{21})a_{21}^{(2)}]\zeta_s(\rho,\theta)\frac{\partial F}{\partial t_1}e^{i\Theta} \\ &=a_{31}^{(2)}\zeta_s(\rho,\theta)\frac{\partial F}{\partial t_1}e^{i\Theta}, \end{aligned}$$
and $\sigma _{33}^{(2)}=-(\sigma _{11}^{(2)}+\sigma _{22}^{(2)})$.

B.4. Third-order approximation of MB equations

The explicit forms of the coefficients $K_2$ and $W$ are

$$\begin{aligned}K_2&=2i\kappa_{12}\cdot\left\langle\frac{(\omega+d_{31})a_{21}^{(2)}-\zeta_c^{*}(\rho,\theta)\Omega_{c}^{*}a_{31}^{(2)}}{D}\right\rangle, \end{aligned}$$
$$\begin{aligned}W&=\kappa_{12}\cdot\left\langle\frac{(\omega+d_{31})(a_{11}^{(2)}-a_{22}^{(2)})+\zeta_c^{*}(\rho,\theta)\Omega_{c}^{*}a_{32}^{(2)}}{D}|\zeta_s(\rho,\theta)|^2\right\rangle, \end{aligned}$$

C. Effective MB equations

Taking the transformation $\tilde {\sigma }_{jj}(z,t)=\langle \sigma _{jj}(\rho ,\theta ,z,t)\rangle$, $\tilde {\sigma }_{31}(z,t)=\langle \sigma _{31}(\rho ,\theta ,z,t)\rangle$, $\tilde {\sigma }_{21}(z,t)=\sigma _{21}(\rho ,\theta ,z,t)/\zeta _{c}(\rho ,\theta )$, $\tilde {\sigma }_{32}(z,t)=\sigma _{32}(\rho ,\theta ,z,t)/\zeta _{c}(\rho ,\theta )$, the MB equations are reduced to the effective MB equations

$$\begin{aligned} i(\frac{\partial}{\partial t}+\Gamma_{31})\tilde{\sigma}_{11}-i\Gamma_{12}\tilde{\sigma}_{22}+\varrho_{s1}\Omega_{s}^{*}\tilde{\sigma}_{21}-\varrho_{s1}\Omega_{s}\tilde{\sigma}_{21}^{*}=0,, \end{aligned}$$
$$\begin{aligned} i(\frac{\partial}{\partial t}+\Gamma_{12})\tilde{\sigma}_{22}-i\Gamma_{23}\tilde{\sigma}_{33}+\varrho_{s1}\Omega_{s}\tilde{\sigma}_{21}^{*}+\varrho_{c}\Omega_{c}^{*}\tilde{\sigma}_{32}-\varrho_{s1}\Omega_{s}^{*}\tilde{\sigma}_{21}-\varrho_{c}\Omega_{c}\tilde{\sigma}_{32}^{*}=0,\end{aligned}$$
$$\begin{aligned} i(\frac{\partial}{\partial t}+\Gamma_{23})\tilde{\sigma}_{33}-i\Gamma_{31}\tilde{\sigma}_{11}+\varrho_{c}\Omega_{c}\tilde{\sigma}_{32}^{*}-\varrho_{c}\Omega_{c}^{*}\tilde{\sigma}_{32}=0,, \end{aligned}$$
$$\begin{aligned} (i\frac{\partial}{\partial t}+d_{21})\tilde{\sigma}_{21}+\Omega_{c}^{*}\tilde{\sigma}_{31}+\varrho_{s2}\Omega_{s}(\tilde{\sigma}_{11}-\tilde{\sigma}_{22})=0, \end{aligned}$$
$$\begin{aligned} (i\frac{\partial}{\partial t}+d_{31})\tilde{\sigma}_{31}-\varrho_{s1}\Omega_{s}\tilde{\sigma}_{32}+\varrho_{c}\Omega_{c}\tilde{\sigma}_{21}=0, \end{aligned}$$
$$\begin{aligned}(i\frac{\partial}{\partial t}+d_{32})\tilde{\sigma}_{32}-\varrho_{s2}\Omega_{s}^{*}\tilde{\sigma}_{31}+\Omega_{c}(\tilde{\sigma}_{22}-\tilde{\sigma}_{33})=0. \end{aligned}$$
with $\varrho _{s1}=\langle \zeta _c(\rho ,\theta )\cdot \zeta _s(\rho ,\theta )\rangle$, $\varrho _{s2}=\langle \zeta _s(\rho ,\theta )/\zeta _c(\rho ,\theta )\rangle$, $\varrho _{c}=\langle |\zeta _c(\rho ,\theta )|^2\rangle$.

Funding

National Natural Science Foundation of China (No.11604185, No.11804196).

Acknowledgments

The authors thank Datang Xu for useful discussions. Yong Zhou and Can Yi contribute equally to this work.

Disclosures

The authors declare no conflicts of interest.

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References

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  1. M. Afzelius, N. Gisin, and H. De Riedmatten, “Quantum memory for photons,” Phys. Today 68(12), 42–47 (2015).
    [Crossref]
  2. K. Heshami, D. G. England, P. C. Humphreys, P. J. Bustard, V. M. Acosta, J. Nunn, and B. J. Sussman, “Quantum memories: emerging applications and recent advances,” J. Mod. Opt. 63(20), 2005–2028 (2016).
    [Crossref]
  3. L. Ma, O. Slattery, and X. Tang, “Optical quantum memory based on electromagnetically induced transparency,” J. Opt. 19(4), 043001 (2017).
    [Crossref]
  4. M. P. Hedges, J. J. Longdell, Y. Li, and M. J. Sellars, “Efficient quantum memory for light,” Nature 465(7301), 1052–1056 (2010).
    [Crossref]
  5. M. Sabooni, Q. Li, S. Kröll, and L. Rippe, “Efficient quantum memory using a weakly absorbing sample,” Phys. Rev. Lett. 110(13), 133604 (2013).
    [Crossref]
  6. K. Kutluer, M. Mazzera, and H. de Riedmatten, “Solid-state source of nonclassical photon pairs with embedded multimode quantum memory,” Phys. Rev. Lett. 118(21), 210502 (2017).
    [Crossref]
  7. L.-M. Duan, J. I. Cirac, P. Zoller, and E. S. Polzik, “Quantum Teleportation with Atomic Ensembles and Coherent Light,” in Quantum Communication, Computing, and Measurement 3, (Kluwer Academic Publishers, Boston, 2006), pp. 351–357.
  8. G. Heinze, C. Hubrich, and T. Halfmann, “Stopped light and image storage by electromagnetically induced transparency up to the regime of one minute,” Phys. Rev. Lett. 111(3), 033601 (2013).
    [Crossref]
  9. D. Schraft, M. Hain, N. Lorenz, and T. Halfmann, “Stopped light at high storage efficiency in a pr3+:y2sio5 crystal,” Phys. Rev. Lett. 116(7), 073602 (2016).
    [Crossref]
  10. Y. Wang, J. Li, S. Zhang, K. Su, Y. Zhou, K. Liao, S. Du, H. Yan, and S. L. Zhu, “Efficient quantum memory for single-photon polarization qubits,” Nat. Photonics 13(5), 346–351 (2019).
    [Crossref]
  11. M. R. Sprague, P. S. Michelberger, T. F. M. Champion, D. G. England, J. Nunn, X.-M. Jin, W. S. Kolthammer, A. Abdolvand, P. S. J. Russell, and I. A. Walmsley, “Broadband single-photon-level memory in a hollow-core photonic crystal fibre,” Nat. Photonics 8(4), 287–291 (2014).
    [Crossref]
  12. D. Xu, Z. Chen, and G. Huang, “Ultraslow weak-light solitons and their storage and retrieval in a kagome-structured hollow-core photonic crystal fiber,” Opt. Express 25(16), 19094 (2017).
    [Crossref]
  13. D. Xu, Z. Bai, and G. Huang, “Storage and retrieval of light pulses in a fast-light medium via active raman gain,” Phys. Rev. A 94(6), 063857 (2016).
    [Crossref]
  14. Y. Chen, Z. Bai, and G. Huang, “Ultraslow optical solitons and their storage and retrieval in an ultracold ladder-type atomic system,” Phys. Rev. A 89(2), 023835 (2014).
    [Crossref]
  15. Z. Chen, Z. Bai, H.-j. Li, C. Hang, and G. Huang, “Storage and retrieval of (3+ 1)-dimensional weak-light bullets and vortices in a coherent atomic gas,” Sci. Rep. 5(1), 8211 (2015).
    [Crossref]
  16. D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and Control of Optical Photons Using Rydberg Polaritons,” Phys. Rev. Lett. 110(10), 103001 (2013).
    [Crossref]
  17. E. Vetsch, D. Reitz, G. Sagué, R. Schmidt, S. T. Dawkins, and A. Rauschenbeutel, “Optical Interface Created by Laser-Cooled Atoms Trapped in the Evanescent Field Surrounding an Optical Nanofiber,” Phys. Rev. Lett. 104(20), 203603 (2010).
    [Crossref]
  18. N. V. Corzo, J. Raskop, A. Chandra, A. S. Sheremet, B. Gouraud, and J. Laurat, “Waveguide-coupled single collective excitation of atomic arrays,” Nature 566(7744), 359–362 (2019).
    [Crossref]
  19. K. P. Nayak, P. N. Melentiev, M. Morinaga, F. L. Kien, V. I. Balykin, and K. Hakuta, “Optical nanofiber as an efficient tool for manipulating and probing atomic Fluorescence,” Opt. Express 15(9), 5431 (2007).
    [Crossref]
  20. Y.-W. Cho and Y.-H. Kim, “Atomic vapor quantum memory for a photonic polarization qubit,” Opt. Express 18(25), 25786 (2010).
    [Crossref]
  21. L. Veissier, A. Nicolas, L. Giner, D. Maxein, A. S. Sheremet, E. Giacobino, and J. Laurat, “Reversible optical memory for twisted photons,” Opt. Lett. 38(5), 712 (2013).
    [Crossref]
  22. B. Gouraud, D. Maxein, A. Nicolas, O. Morin, and J. Laurat, “Demonstration of a Memory for Tightly Guided Light in an Optical Nanofiber,” Phys. Rev. Lett. 114(18), 180503 (2015).
    [Crossref]
  23. C. Sayrin, C. Clausen, B. Albrecht, P. Schneeweiss, and A. Rauschenbeutel, “Storage of fiber-guided light in a nanofiber-trapped ensemble of cold atoms,” Optica 2(4), 353 (2015).
    [Crossref]
  24. Y.-W. Cho, G. T. Campbell, J. L. Everett, J. Bernu, D. B. Higginbottom, M. T. Cao, J. Geng, N. P. Robins, P. K. Lam, and B. C. Buchler, “Highly efficient optical quantum memory with long coherence time in cold atoms,” Optica 3(1), 100 (2016).
    [Crossref]
  25. S. M. Spillane, G. S. Pati, K. Salit, M. Hall, P. Kumar, R. G. Beausoleil, and M. S. Shahriar, “Observation of Nonlinear Optical Interactions of Ultralow Levels of Light in a Tapered Optical Nanofiber Embedded in a Hot Rubidium Vapor,” Phys. Rev. Lett. 100(23), 233602 (2008).
    [Crossref]
  26. D. E. Jones, J. D. Franson, and T. B. Pittman, “Ladder-type electromagnetically induced transparency using nanofiber-guided light in a warm atomic vapor,” Phys. Rev. A 92(4), 043806 (2015).
    [Crossref]
  27. R. Kumar, V. Gokhroo, and S. N. Chormaic, “Multi-level cascaded electromagnetically induced transparency in cold atoms using an optical nanofibre interface,” New J. Phys. 17(12), 123012 (2015).
    [Crossref]
  28. Q. Liu, N. Li, and C. Tan, “All-optical logic gate based on manipulation of surface polaritons solitons via external gradient magnetic fields,” Phys. Rev. A 101(2), 023818 (2020).
    [Crossref]
  29. G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E 72(1), 016617 (2005).
    [Crossref]
  30. D. A. Steck, “Rubidium 87 d line data, available online at http://steck.us/alkalidata ,” (2019).
  31. Q. Liu and C. Tan, “Coherent control of subluminal optical solitons by the incoherent pumping in a ladder-type atomic system,” Eur. Phys. J. D 72(6), 99 (2018).
    [Crossref]

2020 (1)

Q. Liu, N. Li, and C. Tan, “All-optical logic gate based on manipulation of surface polaritons solitons via external gradient magnetic fields,” Phys. Rev. A 101(2), 023818 (2020).
[Crossref]

2019 (2)

Y. Wang, J. Li, S. Zhang, K. Su, Y. Zhou, K. Liao, S. Du, H. Yan, and S. L. Zhu, “Efficient quantum memory for single-photon polarization qubits,” Nat. Photonics 13(5), 346–351 (2019).
[Crossref]

N. V. Corzo, J. Raskop, A. Chandra, A. S. Sheremet, B. Gouraud, and J. Laurat, “Waveguide-coupled single collective excitation of atomic arrays,” Nature 566(7744), 359–362 (2019).
[Crossref]

2018 (1)

Q. Liu and C. Tan, “Coherent control of subluminal optical solitons by the incoherent pumping in a ladder-type atomic system,” Eur. Phys. J. D 72(6), 99 (2018).
[Crossref]

2017 (3)

D. Xu, Z. Chen, and G. Huang, “Ultraslow weak-light solitons and their storage and retrieval in a kagome-structured hollow-core photonic crystal fiber,” Opt. Express 25(16), 19094 (2017).
[Crossref]

L. Ma, O. Slattery, and X. Tang, “Optical quantum memory based on electromagnetically induced transparency,” J. Opt. 19(4), 043001 (2017).
[Crossref]

K. Kutluer, M. Mazzera, and H. de Riedmatten, “Solid-state source of nonclassical photon pairs with embedded multimode quantum memory,” Phys. Rev. Lett. 118(21), 210502 (2017).
[Crossref]

2016 (4)

D. Schraft, M. Hain, N. Lorenz, and T. Halfmann, “Stopped light at high storage efficiency in a pr3+:y2sio5 crystal,” Phys. Rev. Lett. 116(7), 073602 (2016).
[Crossref]

K. Heshami, D. G. England, P. C. Humphreys, P. J. Bustard, V. M. Acosta, J. Nunn, and B. J. Sussman, “Quantum memories: emerging applications and recent advances,” J. Mod. Opt. 63(20), 2005–2028 (2016).
[Crossref]

D. Xu, Z. Bai, and G. Huang, “Storage and retrieval of light pulses in a fast-light medium via active raman gain,” Phys. Rev. A 94(6), 063857 (2016).
[Crossref]

Y.-W. Cho, G. T. Campbell, J. L. Everett, J. Bernu, D. B. Higginbottom, M. T. Cao, J. Geng, N. P. Robins, P. K. Lam, and B. C. Buchler, “Highly efficient optical quantum memory with long coherence time in cold atoms,” Optica 3(1), 100 (2016).
[Crossref]

2015 (6)

D. E. Jones, J. D. Franson, and T. B. Pittman, “Ladder-type electromagnetically induced transparency using nanofiber-guided light in a warm atomic vapor,” Phys. Rev. A 92(4), 043806 (2015).
[Crossref]

R. Kumar, V. Gokhroo, and S. N. Chormaic, “Multi-level cascaded electromagnetically induced transparency in cold atoms using an optical nanofibre interface,” New J. Phys. 17(12), 123012 (2015).
[Crossref]

B. Gouraud, D. Maxein, A. Nicolas, O. Morin, and J. Laurat, “Demonstration of a Memory for Tightly Guided Light in an Optical Nanofiber,” Phys. Rev. Lett. 114(18), 180503 (2015).
[Crossref]

C. Sayrin, C. Clausen, B. Albrecht, P. Schneeweiss, and A. Rauschenbeutel, “Storage of fiber-guided light in a nanofiber-trapped ensemble of cold atoms,” Optica 2(4), 353 (2015).
[Crossref]

Z. Chen, Z. Bai, H.-j. Li, C. Hang, and G. Huang, “Storage and retrieval of (3+ 1)-dimensional weak-light bullets and vortices in a coherent atomic gas,” Sci. Rep. 5(1), 8211 (2015).
[Crossref]

M. Afzelius, N. Gisin, and H. De Riedmatten, “Quantum memory for photons,” Phys. Today 68(12), 42–47 (2015).
[Crossref]

2014 (2)

Y. Chen, Z. Bai, and G. Huang, “Ultraslow optical solitons and their storage and retrieval in an ultracold ladder-type atomic system,” Phys. Rev. A 89(2), 023835 (2014).
[Crossref]

M. R. Sprague, P. S. Michelberger, T. F. M. Champion, D. G. England, J. Nunn, X.-M. Jin, W. S. Kolthammer, A. Abdolvand, P. S. J. Russell, and I. A. Walmsley, “Broadband single-photon-level memory in a hollow-core photonic crystal fibre,” Nat. Photonics 8(4), 287–291 (2014).
[Crossref]

2013 (4)

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and Control of Optical Photons Using Rydberg Polaritons,” Phys. Rev. Lett. 110(10), 103001 (2013).
[Crossref]

M. Sabooni, Q. Li, S. Kröll, and L. Rippe, “Efficient quantum memory using a weakly absorbing sample,” Phys. Rev. Lett. 110(13), 133604 (2013).
[Crossref]

G. Heinze, C. Hubrich, and T. Halfmann, “Stopped light and image storage by electromagnetically induced transparency up to the regime of one minute,” Phys. Rev. Lett. 111(3), 033601 (2013).
[Crossref]

L. Veissier, A. Nicolas, L. Giner, D. Maxein, A. S. Sheremet, E. Giacobino, and J. Laurat, “Reversible optical memory for twisted photons,” Opt. Lett. 38(5), 712 (2013).
[Crossref]

2010 (3)

Y.-W. Cho and Y.-H. Kim, “Atomic vapor quantum memory for a photonic polarization qubit,” Opt. Express 18(25), 25786 (2010).
[Crossref]

M. P. Hedges, J. J. Longdell, Y. Li, and M. J. Sellars, “Efficient quantum memory for light,” Nature 465(7301), 1052–1056 (2010).
[Crossref]

E. Vetsch, D. Reitz, G. Sagué, R. Schmidt, S. T. Dawkins, and A. Rauschenbeutel, “Optical Interface Created by Laser-Cooled Atoms Trapped in the Evanescent Field Surrounding an Optical Nanofiber,” Phys. Rev. Lett. 104(20), 203603 (2010).
[Crossref]

2008 (1)

S. M. Spillane, G. S. Pati, K. Salit, M. Hall, P. Kumar, R. G. Beausoleil, and M. S. Shahriar, “Observation of Nonlinear Optical Interactions of Ultralow Levels of Light in a Tapered Optical Nanofiber Embedded in a Hot Rubidium Vapor,” Phys. Rev. Lett. 100(23), 233602 (2008).
[Crossref]

2007 (1)

2005 (1)

G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E 72(1), 016617 (2005).
[Crossref]

Abdolvand, A.

M. R. Sprague, P. S. Michelberger, T. F. M. Champion, D. G. England, J. Nunn, X.-M. Jin, W. S. Kolthammer, A. Abdolvand, P. S. J. Russell, and I. A. Walmsley, “Broadband single-photon-level memory in a hollow-core photonic crystal fibre,” Nat. Photonics 8(4), 287–291 (2014).
[Crossref]

Acosta, V. M.

K. Heshami, D. G. England, P. C. Humphreys, P. J. Bustard, V. M. Acosta, J. Nunn, and B. J. Sussman, “Quantum memories: emerging applications and recent advances,” J. Mod. Opt. 63(20), 2005–2028 (2016).
[Crossref]

Adams, C. S.

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and Control of Optical Photons Using Rydberg Polaritons,” Phys. Rev. Lett. 110(10), 103001 (2013).
[Crossref]

Afzelius, M.

M. Afzelius, N. Gisin, and H. De Riedmatten, “Quantum memory for photons,” Phys. Today 68(12), 42–47 (2015).
[Crossref]

Albrecht, B.

Bai, Z.

D. Xu, Z. Bai, and G. Huang, “Storage and retrieval of light pulses in a fast-light medium via active raman gain,” Phys. Rev. A 94(6), 063857 (2016).
[Crossref]

Z. Chen, Z. Bai, H.-j. Li, C. Hang, and G. Huang, “Storage and retrieval of (3+ 1)-dimensional weak-light bullets and vortices in a coherent atomic gas,” Sci. Rep. 5(1), 8211 (2015).
[Crossref]

Y. Chen, Z. Bai, and G. Huang, “Ultraslow optical solitons and their storage and retrieval in an ultracold ladder-type atomic system,” Phys. Rev. A 89(2), 023835 (2014).
[Crossref]

Balykin, V. I.

Beausoleil, R. G.

S. M. Spillane, G. S. Pati, K. Salit, M. Hall, P. Kumar, R. G. Beausoleil, and M. S. Shahriar, “Observation of Nonlinear Optical Interactions of Ultralow Levels of Light in a Tapered Optical Nanofiber Embedded in a Hot Rubidium Vapor,” Phys. Rev. Lett. 100(23), 233602 (2008).
[Crossref]

Bernu, J.

Buchler, B. C.

Busche, H.

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and Control of Optical Photons Using Rydberg Polaritons,” Phys. Rev. Lett. 110(10), 103001 (2013).
[Crossref]

Bustard, P. J.

K. Heshami, D. G. England, P. C. Humphreys, P. J. Bustard, V. M. Acosta, J. Nunn, and B. J. Sussman, “Quantum memories: emerging applications and recent advances,” J. Mod. Opt. 63(20), 2005–2028 (2016).
[Crossref]

Campbell, G. T.

Cao, M. T.

Champion, T. F. M.

M. R. Sprague, P. S. Michelberger, T. F. M. Champion, D. G. England, J. Nunn, X.-M. Jin, W. S. Kolthammer, A. Abdolvand, P. S. J. Russell, and I. A. Walmsley, “Broadband single-photon-level memory in a hollow-core photonic crystal fibre,” Nat. Photonics 8(4), 287–291 (2014).
[Crossref]

Chandra, A.

N. V. Corzo, J. Raskop, A. Chandra, A. S. Sheremet, B. Gouraud, and J. Laurat, “Waveguide-coupled single collective excitation of atomic arrays,” Nature 566(7744), 359–362 (2019).
[Crossref]

Chen, Y.

Y. Chen, Z. Bai, and G. Huang, “Ultraslow optical solitons and their storage and retrieval in an ultracold ladder-type atomic system,” Phys. Rev. A 89(2), 023835 (2014).
[Crossref]

Chen, Z.

D. Xu, Z. Chen, and G. Huang, “Ultraslow weak-light solitons and their storage and retrieval in a kagome-structured hollow-core photonic crystal fiber,” Opt. Express 25(16), 19094 (2017).
[Crossref]

Z. Chen, Z. Bai, H.-j. Li, C. Hang, and G. Huang, “Storage and retrieval of (3+ 1)-dimensional weak-light bullets and vortices in a coherent atomic gas,” Sci. Rep. 5(1), 8211 (2015).
[Crossref]

Cho, Y.-W.

Chormaic, S. N.

R. Kumar, V. Gokhroo, and S. N. Chormaic, “Multi-level cascaded electromagnetically induced transparency in cold atoms using an optical nanofibre interface,” New J. Phys. 17(12), 123012 (2015).
[Crossref]

Cirac, J. I.

L.-M. Duan, J. I. Cirac, P. Zoller, and E. S. Polzik, “Quantum Teleportation with Atomic Ensembles and Coherent Light,” in Quantum Communication, Computing, and Measurement 3, (Kluwer Academic Publishers, Boston, 2006), pp. 351–357.

Clausen, C.

Corzo, N. V.

N. V. Corzo, J. Raskop, A. Chandra, A. S. Sheremet, B. Gouraud, and J. Laurat, “Waveguide-coupled single collective excitation of atomic arrays,” Nature 566(7744), 359–362 (2019).
[Crossref]

Dawkins, S. T.

E. Vetsch, D. Reitz, G. Sagué, R. Schmidt, S. T. Dawkins, and A. Rauschenbeutel, “Optical Interface Created by Laser-Cooled Atoms Trapped in the Evanescent Field Surrounding an Optical Nanofiber,” Phys. Rev. Lett. 104(20), 203603 (2010).
[Crossref]

de Riedmatten, H.

K. Kutluer, M. Mazzera, and H. de Riedmatten, “Solid-state source of nonclassical photon pairs with embedded multimode quantum memory,” Phys. Rev. Lett. 118(21), 210502 (2017).
[Crossref]

M. Afzelius, N. Gisin, and H. De Riedmatten, “Quantum memory for photons,” Phys. Today 68(12), 42–47 (2015).
[Crossref]

Deng, L.

G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E 72(1), 016617 (2005).
[Crossref]

Du, S.

Y. Wang, J. Li, S. Zhang, K. Su, Y. Zhou, K. Liao, S. Du, H. Yan, and S. L. Zhu, “Efficient quantum memory for single-photon polarization qubits,” Nat. Photonics 13(5), 346–351 (2019).
[Crossref]

Duan, L.-M.

L.-M. Duan, J. I. Cirac, P. Zoller, and E. S. Polzik, “Quantum Teleportation with Atomic Ensembles and Coherent Light,” in Quantum Communication, Computing, and Measurement 3, (Kluwer Academic Publishers, Boston, 2006), pp. 351–357.

England, D. G.

K. Heshami, D. G. England, P. C. Humphreys, P. J. Bustard, V. M. Acosta, J. Nunn, and B. J. Sussman, “Quantum memories: emerging applications and recent advances,” J. Mod. Opt. 63(20), 2005–2028 (2016).
[Crossref]

M. R. Sprague, P. S. Michelberger, T. F. M. Champion, D. G. England, J. Nunn, X.-M. Jin, W. S. Kolthammer, A. Abdolvand, P. S. J. Russell, and I. A. Walmsley, “Broadband single-photon-level memory in a hollow-core photonic crystal fibre,” Nat. Photonics 8(4), 287–291 (2014).
[Crossref]

Everett, J. L.

Franson, J. D.

D. E. Jones, J. D. Franson, and T. B. Pittman, “Ladder-type electromagnetically induced transparency using nanofiber-guided light in a warm atomic vapor,” Phys. Rev. A 92(4), 043806 (2015).
[Crossref]

Gauguet, A.

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and Control of Optical Photons Using Rydberg Polaritons,” Phys. Rev. Lett. 110(10), 103001 (2013).
[Crossref]

Geng, J.

Giacobino, E.

Giner, L.

Gisin, N.

M. Afzelius, N. Gisin, and H. De Riedmatten, “Quantum memory for photons,” Phys. Today 68(12), 42–47 (2015).
[Crossref]

Gokhroo, V.

R. Kumar, V. Gokhroo, and S. N. Chormaic, “Multi-level cascaded electromagnetically induced transparency in cold atoms using an optical nanofibre interface,” New J. Phys. 17(12), 123012 (2015).
[Crossref]

Gouraud, B.

N. V. Corzo, J. Raskop, A. Chandra, A. S. Sheremet, B. Gouraud, and J. Laurat, “Waveguide-coupled single collective excitation of atomic arrays,” Nature 566(7744), 359–362 (2019).
[Crossref]

B. Gouraud, D. Maxein, A. Nicolas, O. Morin, and J. Laurat, “Demonstration of a Memory for Tightly Guided Light in an Optical Nanofiber,” Phys. Rev. Lett. 114(18), 180503 (2015).
[Crossref]

Hain, M.

D. Schraft, M. Hain, N. Lorenz, and T. Halfmann, “Stopped light at high storage efficiency in a pr3+:y2sio5 crystal,” Phys. Rev. Lett. 116(7), 073602 (2016).
[Crossref]

Hakuta, K.

Halfmann, T.

D. Schraft, M. Hain, N. Lorenz, and T. Halfmann, “Stopped light at high storage efficiency in a pr3+:y2sio5 crystal,” Phys. Rev. Lett. 116(7), 073602 (2016).
[Crossref]

G. Heinze, C. Hubrich, and T. Halfmann, “Stopped light and image storage by electromagnetically induced transparency up to the regime of one minute,” Phys. Rev. Lett. 111(3), 033601 (2013).
[Crossref]

Hall, M.

S. M. Spillane, G. S. Pati, K. Salit, M. Hall, P. Kumar, R. G. Beausoleil, and M. S. Shahriar, “Observation of Nonlinear Optical Interactions of Ultralow Levels of Light in a Tapered Optical Nanofiber Embedded in a Hot Rubidium Vapor,” Phys. Rev. Lett. 100(23), 233602 (2008).
[Crossref]

Hang, C.

Z. Chen, Z. Bai, H.-j. Li, C. Hang, and G. Huang, “Storage and retrieval of (3+ 1)-dimensional weak-light bullets and vortices in a coherent atomic gas,” Sci. Rep. 5(1), 8211 (2015).
[Crossref]

Hedges, M. P.

M. P. Hedges, J. J. Longdell, Y. Li, and M. J. Sellars, “Efficient quantum memory for light,” Nature 465(7301), 1052–1056 (2010).
[Crossref]

Heinze, G.

G. Heinze, C. Hubrich, and T. Halfmann, “Stopped light and image storage by electromagnetically induced transparency up to the regime of one minute,” Phys. Rev. Lett. 111(3), 033601 (2013).
[Crossref]

Heshami, K.

K. Heshami, D. G. England, P. C. Humphreys, P. J. Bustard, V. M. Acosta, J. Nunn, and B. J. Sussman, “Quantum memories: emerging applications and recent advances,” J. Mod. Opt. 63(20), 2005–2028 (2016).
[Crossref]

Higginbottom, D. B.

Huang, G.

D. Xu, Z. Chen, and G. Huang, “Ultraslow weak-light solitons and their storage and retrieval in a kagome-structured hollow-core photonic crystal fiber,” Opt. Express 25(16), 19094 (2017).
[Crossref]

D. Xu, Z. Bai, and G. Huang, “Storage and retrieval of light pulses in a fast-light medium via active raman gain,” Phys. Rev. A 94(6), 063857 (2016).
[Crossref]

Z. Chen, Z. Bai, H.-j. Li, C. Hang, and G. Huang, “Storage and retrieval of (3+ 1)-dimensional weak-light bullets and vortices in a coherent atomic gas,” Sci. Rep. 5(1), 8211 (2015).
[Crossref]

Y. Chen, Z. Bai, and G. Huang, “Ultraslow optical solitons and their storage and retrieval in an ultracold ladder-type atomic system,” Phys. Rev. A 89(2), 023835 (2014).
[Crossref]

G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E 72(1), 016617 (2005).
[Crossref]

Hubrich, C.

G. Heinze, C. Hubrich, and T. Halfmann, “Stopped light and image storage by electromagnetically induced transparency up to the regime of one minute,” Phys. Rev. Lett. 111(3), 033601 (2013).
[Crossref]

Humphreys, P. C.

K. Heshami, D. G. England, P. C. Humphreys, P. J. Bustard, V. M. Acosta, J. Nunn, and B. J. Sussman, “Quantum memories: emerging applications and recent advances,” J. Mod. Opt. 63(20), 2005–2028 (2016).
[Crossref]

Jin, X.-M.

M. R. Sprague, P. S. Michelberger, T. F. M. Champion, D. G. England, J. Nunn, X.-M. Jin, W. S. Kolthammer, A. Abdolvand, P. S. J. Russell, and I. A. Walmsley, “Broadband single-photon-level memory in a hollow-core photonic crystal fibre,” Nat. Photonics 8(4), 287–291 (2014).
[Crossref]

Jones, D. E.

D. E. Jones, J. D. Franson, and T. B. Pittman, “Ladder-type electromagnetically induced transparency using nanofiber-guided light in a warm atomic vapor,” Phys. Rev. A 92(4), 043806 (2015).
[Crossref]

Jones, M. P. A.

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and Control of Optical Photons Using Rydberg Polaritons,” Phys. Rev. Lett. 110(10), 103001 (2013).
[Crossref]

Kien, F. L.

Kim, Y.-H.

Kolthammer, W. S.

M. R. Sprague, P. S. Michelberger, T. F. M. Champion, D. G. England, J. Nunn, X.-M. Jin, W. S. Kolthammer, A. Abdolvand, P. S. J. Russell, and I. A. Walmsley, “Broadband single-photon-level memory in a hollow-core photonic crystal fibre,” Nat. Photonics 8(4), 287–291 (2014).
[Crossref]

Kröll, S.

M. Sabooni, Q. Li, S. Kröll, and L. Rippe, “Efficient quantum memory using a weakly absorbing sample,” Phys. Rev. Lett. 110(13), 133604 (2013).
[Crossref]

Kumar, P.

S. M. Spillane, G. S. Pati, K. Salit, M. Hall, P. Kumar, R. G. Beausoleil, and M. S. Shahriar, “Observation of Nonlinear Optical Interactions of Ultralow Levels of Light in a Tapered Optical Nanofiber Embedded in a Hot Rubidium Vapor,” Phys. Rev. Lett. 100(23), 233602 (2008).
[Crossref]

Kumar, R.

R. Kumar, V. Gokhroo, and S. N. Chormaic, “Multi-level cascaded electromagnetically induced transparency in cold atoms using an optical nanofibre interface,” New J. Phys. 17(12), 123012 (2015).
[Crossref]

Kutluer, K.

K. Kutluer, M. Mazzera, and H. de Riedmatten, “Solid-state source of nonclassical photon pairs with embedded multimode quantum memory,” Phys. Rev. Lett. 118(21), 210502 (2017).
[Crossref]

Lam, P. K.

Laurat, J.

N. V. Corzo, J. Raskop, A. Chandra, A. S. Sheremet, B. Gouraud, and J. Laurat, “Waveguide-coupled single collective excitation of atomic arrays,” Nature 566(7744), 359–362 (2019).
[Crossref]

B. Gouraud, D. Maxein, A. Nicolas, O. Morin, and J. Laurat, “Demonstration of a Memory for Tightly Guided Light in an Optical Nanofiber,” Phys. Rev. Lett. 114(18), 180503 (2015).
[Crossref]

L. Veissier, A. Nicolas, L. Giner, D. Maxein, A. S. Sheremet, E. Giacobino, and J. Laurat, “Reversible optical memory for twisted photons,” Opt. Lett. 38(5), 712 (2013).
[Crossref]

Li, H.-j.

Z. Chen, Z. Bai, H.-j. Li, C. Hang, and G. Huang, “Storage and retrieval of (3+ 1)-dimensional weak-light bullets and vortices in a coherent atomic gas,” Sci. Rep. 5(1), 8211 (2015).
[Crossref]

Li, J.

Y. Wang, J. Li, S. Zhang, K. Su, Y. Zhou, K. Liao, S. Du, H. Yan, and S. L. Zhu, “Efficient quantum memory for single-photon polarization qubits,” Nat. Photonics 13(5), 346–351 (2019).
[Crossref]

Li, N.

Q. Liu, N. Li, and C. Tan, “All-optical logic gate based on manipulation of surface polaritons solitons via external gradient magnetic fields,” Phys. Rev. A 101(2), 023818 (2020).
[Crossref]

Li, Q.

M. Sabooni, Q. Li, S. Kröll, and L. Rippe, “Efficient quantum memory using a weakly absorbing sample,” Phys. Rev. Lett. 110(13), 133604 (2013).
[Crossref]

Li, Y.

M. P. Hedges, J. J. Longdell, Y. Li, and M. J. Sellars, “Efficient quantum memory for light,” Nature 465(7301), 1052–1056 (2010).
[Crossref]

Liao, K.

Y. Wang, J. Li, S. Zhang, K. Su, Y. Zhou, K. Liao, S. Du, H. Yan, and S. L. Zhu, “Efficient quantum memory for single-photon polarization qubits,” Nat. Photonics 13(5), 346–351 (2019).
[Crossref]

Liu, Q.

Q. Liu, N. Li, and C. Tan, “All-optical logic gate based on manipulation of surface polaritons solitons via external gradient magnetic fields,” Phys. Rev. A 101(2), 023818 (2020).
[Crossref]

Q. Liu and C. Tan, “Coherent control of subluminal optical solitons by the incoherent pumping in a ladder-type atomic system,” Eur. Phys. J. D 72(6), 99 (2018).
[Crossref]

Longdell, J. J.

M. P. Hedges, J. J. Longdell, Y. Li, and M. J. Sellars, “Efficient quantum memory for light,” Nature 465(7301), 1052–1056 (2010).
[Crossref]

Lorenz, N.

D. Schraft, M. Hain, N. Lorenz, and T. Halfmann, “Stopped light at high storage efficiency in a pr3+:y2sio5 crystal,” Phys. Rev. Lett. 116(7), 073602 (2016).
[Crossref]

Ma, L.

L. Ma, O. Slattery, and X. Tang, “Optical quantum memory based on electromagnetically induced transparency,” J. Opt. 19(4), 043001 (2017).
[Crossref]

Maxein, D.

B. Gouraud, D. Maxein, A. Nicolas, O. Morin, and J. Laurat, “Demonstration of a Memory for Tightly Guided Light in an Optical Nanofiber,” Phys. Rev. Lett. 114(18), 180503 (2015).
[Crossref]

L. Veissier, A. Nicolas, L. Giner, D. Maxein, A. S. Sheremet, E. Giacobino, and J. Laurat, “Reversible optical memory for twisted photons,” Opt. Lett. 38(5), 712 (2013).
[Crossref]

Maxwell, D.

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and Control of Optical Photons Using Rydberg Polaritons,” Phys. Rev. Lett. 110(10), 103001 (2013).
[Crossref]

Mazzera, M.

K. Kutluer, M. Mazzera, and H. de Riedmatten, “Solid-state source of nonclassical photon pairs with embedded multimode quantum memory,” Phys. Rev. Lett. 118(21), 210502 (2017).
[Crossref]

Melentiev, P. N.

Michelberger, P. S.

M. R. Sprague, P. S. Michelberger, T. F. M. Champion, D. G. England, J. Nunn, X.-M. Jin, W. S. Kolthammer, A. Abdolvand, P. S. J. Russell, and I. A. Walmsley, “Broadband single-photon-level memory in a hollow-core photonic crystal fibre,” Nat. Photonics 8(4), 287–291 (2014).
[Crossref]

Morin, O.

B. Gouraud, D. Maxein, A. Nicolas, O. Morin, and J. Laurat, “Demonstration of a Memory for Tightly Guided Light in an Optical Nanofiber,” Phys. Rev. Lett. 114(18), 180503 (2015).
[Crossref]

Morinaga, M.

Nayak, K. P.

Nicolas, A.

B. Gouraud, D. Maxein, A. Nicolas, O. Morin, and J. Laurat, “Demonstration of a Memory for Tightly Guided Light in an Optical Nanofiber,” Phys. Rev. Lett. 114(18), 180503 (2015).
[Crossref]

L. Veissier, A. Nicolas, L. Giner, D. Maxein, A. S. Sheremet, E. Giacobino, and J. Laurat, “Reversible optical memory for twisted photons,” Opt. Lett. 38(5), 712 (2013).
[Crossref]

Nunn, J.

K. Heshami, D. G. England, P. C. Humphreys, P. J. Bustard, V. M. Acosta, J. Nunn, and B. J. Sussman, “Quantum memories: emerging applications and recent advances,” J. Mod. Opt. 63(20), 2005–2028 (2016).
[Crossref]

M. R. Sprague, P. S. Michelberger, T. F. M. Champion, D. G. England, J. Nunn, X.-M. Jin, W. S. Kolthammer, A. Abdolvand, P. S. J. Russell, and I. A. Walmsley, “Broadband single-photon-level memory in a hollow-core photonic crystal fibre,” Nat. Photonics 8(4), 287–291 (2014).
[Crossref]

Paredes-Barato, D.

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and Control of Optical Photons Using Rydberg Polaritons,” Phys. Rev. Lett. 110(10), 103001 (2013).
[Crossref]

Pati, G. S.

S. M. Spillane, G. S. Pati, K. Salit, M. Hall, P. Kumar, R. G. Beausoleil, and M. S. Shahriar, “Observation of Nonlinear Optical Interactions of Ultralow Levels of Light in a Tapered Optical Nanofiber Embedded in a Hot Rubidium Vapor,” Phys. Rev. Lett. 100(23), 233602 (2008).
[Crossref]

Payne, M. G.

G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E 72(1), 016617 (2005).
[Crossref]

Pittman, T. B.

D. E. Jones, J. D. Franson, and T. B. Pittman, “Ladder-type electromagnetically induced transparency using nanofiber-guided light in a warm atomic vapor,” Phys. Rev. A 92(4), 043806 (2015).
[Crossref]

Polzik, E. S.

L.-M. Duan, J. I. Cirac, P. Zoller, and E. S. Polzik, “Quantum Teleportation with Atomic Ensembles and Coherent Light,” in Quantum Communication, Computing, and Measurement 3, (Kluwer Academic Publishers, Boston, 2006), pp. 351–357.

Pritchard, J. D.

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and Control of Optical Photons Using Rydberg Polaritons,” Phys. Rev. Lett. 110(10), 103001 (2013).
[Crossref]

Raskop, J.

N. V. Corzo, J. Raskop, A. Chandra, A. S. Sheremet, B. Gouraud, and J. Laurat, “Waveguide-coupled single collective excitation of atomic arrays,” Nature 566(7744), 359–362 (2019).
[Crossref]

Rauschenbeutel, A.

C. Sayrin, C. Clausen, B. Albrecht, P. Schneeweiss, and A. Rauschenbeutel, “Storage of fiber-guided light in a nanofiber-trapped ensemble of cold atoms,” Optica 2(4), 353 (2015).
[Crossref]

E. Vetsch, D. Reitz, G. Sagué, R. Schmidt, S. T. Dawkins, and A. Rauschenbeutel, “Optical Interface Created by Laser-Cooled Atoms Trapped in the Evanescent Field Surrounding an Optical Nanofiber,” Phys. Rev. Lett. 104(20), 203603 (2010).
[Crossref]

Reitz, D.

E. Vetsch, D. Reitz, G. Sagué, R. Schmidt, S. T. Dawkins, and A. Rauschenbeutel, “Optical Interface Created by Laser-Cooled Atoms Trapped in the Evanescent Field Surrounding an Optical Nanofiber,” Phys. Rev. Lett. 104(20), 203603 (2010).
[Crossref]

Rippe, L.

M. Sabooni, Q. Li, S. Kröll, and L. Rippe, “Efficient quantum memory using a weakly absorbing sample,” Phys. Rev. Lett. 110(13), 133604 (2013).
[Crossref]

Robins, N. P.

Russell, P. S. J.

M. R. Sprague, P. S. Michelberger, T. F. M. Champion, D. G. England, J. Nunn, X.-M. Jin, W. S. Kolthammer, A. Abdolvand, P. S. J. Russell, and I. A. Walmsley, “Broadband single-photon-level memory in a hollow-core photonic crystal fibre,” Nat. Photonics 8(4), 287–291 (2014).
[Crossref]

Sabooni, M.

M. Sabooni, Q. Li, S. Kröll, and L. Rippe, “Efficient quantum memory using a weakly absorbing sample,” Phys. Rev. Lett. 110(13), 133604 (2013).
[Crossref]

Sagué, G.

E. Vetsch, D. Reitz, G. Sagué, R. Schmidt, S. T. Dawkins, and A. Rauschenbeutel, “Optical Interface Created by Laser-Cooled Atoms Trapped in the Evanescent Field Surrounding an Optical Nanofiber,” Phys. Rev. Lett. 104(20), 203603 (2010).
[Crossref]

Salit, K.

S. M. Spillane, G. S. Pati, K. Salit, M. Hall, P. Kumar, R. G. Beausoleil, and M. S. Shahriar, “Observation of Nonlinear Optical Interactions of Ultralow Levels of Light in a Tapered Optical Nanofiber Embedded in a Hot Rubidium Vapor,” Phys. Rev. Lett. 100(23), 233602 (2008).
[Crossref]

Sayrin, C.

Schmidt, R.

E. Vetsch, D. Reitz, G. Sagué, R. Schmidt, S. T. Dawkins, and A. Rauschenbeutel, “Optical Interface Created by Laser-Cooled Atoms Trapped in the Evanescent Field Surrounding an Optical Nanofiber,” Phys. Rev. Lett. 104(20), 203603 (2010).
[Crossref]

Schneeweiss, P.

Schraft, D.

D. Schraft, M. Hain, N. Lorenz, and T. Halfmann, “Stopped light at high storage efficiency in a pr3+:y2sio5 crystal,” Phys. Rev. Lett. 116(7), 073602 (2016).
[Crossref]

Sellars, M. J.

M. P. Hedges, J. J. Longdell, Y. Li, and M. J. Sellars, “Efficient quantum memory for light,” Nature 465(7301), 1052–1056 (2010).
[Crossref]

Shahriar, M. S.

S. M. Spillane, G. S. Pati, K. Salit, M. Hall, P. Kumar, R. G. Beausoleil, and M. S. Shahriar, “Observation of Nonlinear Optical Interactions of Ultralow Levels of Light in a Tapered Optical Nanofiber Embedded in a Hot Rubidium Vapor,” Phys. Rev. Lett. 100(23), 233602 (2008).
[Crossref]

Sheremet, A. S.

N. V. Corzo, J. Raskop, A. Chandra, A. S. Sheremet, B. Gouraud, and J. Laurat, “Waveguide-coupled single collective excitation of atomic arrays,” Nature 566(7744), 359–362 (2019).
[Crossref]

L. Veissier, A. Nicolas, L. Giner, D. Maxein, A. S. Sheremet, E. Giacobino, and J. Laurat, “Reversible optical memory for twisted photons,” Opt. Lett. 38(5), 712 (2013).
[Crossref]

Slattery, O.

L. Ma, O. Slattery, and X. Tang, “Optical quantum memory based on electromagnetically induced transparency,” J. Opt. 19(4), 043001 (2017).
[Crossref]

Spillane, S. M.

S. M. Spillane, G. S. Pati, K. Salit, M. Hall, P. Kumar, R. G. Beausoleil, and M. S. Shahriar, “Observation of Nonlinear Optical Interactions of Ultralow Levels of Light in a Tapered Optical Nanofiber Embedded in a Hot Rubidium Vapor,” Phys. Rev. Lett. 100(23), 233602 (2008).
[Crossref]

Sprague, M. R.

M. R. Sprague, P. S. Michelberger, T. F. M. Champion, D. G. England, J. Nunn, X.-M. Jin, W. S. Kolthammer, A. Abdolvand, P. S. J. Russell, and I. A. Walmsley, “Broadband single-photon-level memory in a hollow-core photonic crystal fibre,” Nat. Photonics 8(4), 287–291 (2014).
[Crossref]

Steck, D. A.

D. A. Steck, “Rubidium 87 d line data, available online at http://steck.us/alkalidata ,” (2019).

Su, K.

Y. Wang, J. Li, S. Zhang, K. Su, Y. Zhou, K. Liao, S. Du, H. Yan, and S. L. Zhu, “Efficient quantum memory for single-photon polarization qubits,” Nat. Photonics 13(5), 346–351 (2019).
[Crossref]

Sussman, B. J.

K. Heshami, D. G. England, P. C. Humphreys, P. J. Bustard, V. M. Acosta, J. Nunn, and B. J. Sussman, “Quantum memories: emerging applications and recent advances,” J. Mod. Opt. 63(20), 2005–2028 (2016).
[Crossref]

Szwer, D. J.

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and Control of Optical Photons Using Rydberg Polaritons,” Phys. Rev. Lett. 110(10), 103001 (2013).
[Crossref]

Tan, C.

Q. Liu, N. Li, and C. Tan, “All-optical logic gate based on manipulation of surface polaritons solitons via external gradient magnetic fields,” Phys. Rev. A 101(2), 023818 (2020).
[Crossref]

Q. Liu and C. Tan, “Coherent control of subluminal optical solitons by the incoherent pumping in a ladder-type atomic system,” Eur. Phys. J. D 72(6), 99 (2018).
[Crossref]

Tang, X.

L. Ma, O. Slattery, and X. Tang, “Optical quantum memory based on electromagnetically induced transparency,” J. Opt. 19(4), 043001 (2017).
[Crossref]

Veissier, L.

Vetsch, E.

E. Vetsch, D. Reitz, G. Sagué, R. Schmidt, S. T. Dawkins, and A. Rauschenbeutel, “Optical Interface Created by Laser-Cooled Atoms Trapped in the Evanescent Field Surrounding an Optical Nanofiber,” Phys. Rev. Lett. 104(20), 203603 (2010).
[Crossref]

Walmsley, I. A.

M. R. Sprague, P. S. Michelberger, T. F. M. Champion, D. G. England, J. Nunn, X.-M. Jin, W. S. Kolthammer, A. Abdolvand, P. S. J. Russell, and I. A. Walmsley, “Broadband single-photon-level memory in a hollow-core photonic crystal fibre,” Nat. Photonics 8(4), 287–291 (2014).
[Crossref]

Wang, Y.

Y. Wang, J. Li, S. Zhang, K. Su, Y. Zhou, K. Liao, S. Du, H. Yan, and S. L. Zhu, “Efficient quantum memory for single-photon polarization qubits,” Nat. Photonics 13(5), 346–351 (2019).
[Crossref]

Weatherill, K. J.

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and Control of Optical Photons Using Rydberg Polaritons,” Phys. Rev. Lett. 110(10), 103001 (2013).
[Crossref]

Xu, D.

D. Xu, Z. Chen, and G. Huang, “Ultraslow weak-light solitons and their storage and retrieval in a kagome-structured hollow-core photonic crystal fiber,” Opt. Express 25(16), 19094 (2017).
[Crossref]

D. Xu, Z. Bai, and G. Huang, “Storage and retrieval of light pulses in a fast-light medium via active raman gain,” Phys. Rev. A 94(6), 063857 (2016).
[Crossref]

Yan, H.

Y. Wang, J. Li, S. Zhang, K. Su, Y. Zhou, K. Liao, S. Du, H. Yan, and S. L. Zhu, “Efficient quantum memory for single-photon polarization qubits,” Nat. Photonics 13(5), 346–351 (2019).
[Crossref]

Zhang, S.

Y. Wang, J. Li, S. Zhang, K. Su, Y. Zhou, K. Liao, S. Du, H. Yan, and S. L. Zhu, “Efficient quantum memory for single-photon polarization qubits,” Nat. Photonics 13(5), 346–351 (2019).
[Crossref]

Zhou, Y.

Y. Wang, J. Li, S. Zhang, K. Su, Y. Zhou, K. Liao, S. Du, H. Yan, and S. L. Zhu, “Efficient quantum memory for single-photon polarization qubits,” Nat. Photonics 13(5), 346–351 (2019).
[Crossref]

Zhu, S. L.

Y. Wang, J. Li, S. Zhang, K. Su, Y. Zhou, K. Liao, S. Du, H. Yan, and S. L. Zhu, “Efficient quantum memory for single-photon polarization qubits,” Nat. Photonics 13(5), 346–351 (2019).
[Crossref]

Zoller, P.

L.-M. Duan, J. I. Cirac, P. Zoller, and E. S. Polzik, “Quantum Teleportation with Atomic Ensembles and Coherent Light,” in Quantum Communication, Computing, and Measurement 3, (Kluwer Academic Publishers, Boston, 2006), pp. 351–357.

Eur. Phys. J. D (1)

Q. Liu and C. Tan, “Coherent control of subluminal optical solitons by the incoherent pumping in a ladder-type atomic system,” Eur. Phys. J. D 72(6), 99 (2018).
[Crossref]

J. Mod. Opt. (1)

K. Heshami, D. G. England, P. C. Humphreys, P. J. Bustard, V. M. Acosta, J. Nunn, and B. J. Sussman, “Quantum memories: emerging applications and recent advances,” J. Mod. Opt. 63(20), 2005–2028 (2016).
[Crossref]

J. Opt. (1)

L. Ma, O. Slattery, and X. Tang, “Optical quantum memory based on electromagnetically induced transparency,” J. Opt. 19(4), 043001 (2017).
[Crossref]

Nat. Photonics (2)

Y. Wang, J. Li, S. Zhang, K. Su, Y. Zhou, K. Liao, S. Du, H. Yan, and S. L. Zhu, “Efficient quantum memory for single-photon polarization qubits,” Nat. Photonics 13(5), 346–351 (2019).
[Crossref]

M. R. Sprague, P. S. Michelberger, T. F. M. Champion, D. G. England, J. Nunn, X.-M. Jin, W. S. Kolthammer, A. Abdolvand, P. S. J. Russell, and I. A. Walmsley, “Broadband single-photon-level memory in a hollow-core photonic crystal fibre,” Nat. Photonics 8(4), 287–291 (2014).
[Crossref]

Nature (2)

M. P. Hedges, J. J. Longdell, Y. Li, and M. J. Sellars, “Efficient quantum memory for light,” Nature 465(7301), 1052–1056 (2010).
[Crossref]

N. V. Corzo, J. Raskop, A. Chandra, A. S. Sheremet, B. Gouraud, and J. Laurat, “Waveguide-coupled single collective excitation of atomic arrays,” Nature 566(7744), 359–362 (2019).
[Crossref]

New J. Phys. (1)

R. Kumar, V. Gokhroo, and S. N. Chormaic, “Multi-level cascaded electromagnetically induced transparency in cold atoms using an optical nanofibre interface,” New J. Phys. 17(12), 123012 (2015).
[Crossref]

Opt. Express (3)

Opt. Lett. (1)

Optica (2)

Phys. Rev. A (4)

D. E. Jones, J. D. Franson, and T. B. Pittman, “Ladder-type electromagnetically induced transparency using nanofiber-guided light in a warm atomic vapor,” Phys. Rev. A 92(4), 043806 (2015).
[Crossref]

Q. Liu, N. Li, and C. Tan, “All-optical logic gate based on manipulation of surface polaritons solitons via external gradient magnetic fields,” Phys. Rev. A 101(2), 023818 (2020).
[Crossref]

D. Xu, Z. Bai, and G. Huang, “Storage and retrieval of light pulses in a fast-light medium via active raman gain,” Phys. Rev. A 94(6), 063857 (2016).
[Crossref]

Y. Chen, Z. Bai, and G. Huang, “Ultraslow optical solitons and their storage and retrieval in an ultracold ladder-type atomic system,” Phys. Rev. A 89(2), 023835 (2014).
[Crossref]

Phys. Rev. E (1)

G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E 72(1), 016617 (2005).
[Crossref]

Phys. Rev. Lett. (8)

S. M. Spillane, G. S. Pati, K. Salit, M. Hall, P. Kumar, R. G. Beausoleil, and M. S. Shahriar, “Observation of Nonlinear Optical Interactions of Ultralow Levels of Light in a Tapered Optical Nanofiber Embedded in a Hot Rubidium Vapor,” Phys. Rev. Lett. 100(23), 233602 (2008).
[Crossref]

M. Sabooni, Q. Li, S. Kröll, and L. Rippe, “Efficient quantum memory using a weakly absorbing sample,” Phys. Rev. Lett. 110(13), 133604 (2013).
[Crossref]

K. Kutluer, M. Mazzera, and H. de Riedmatten, “Solid-state source of nonclassical photon pairs with embedded multimode quantum memory,” Phys. Rev. Lett. 118(21), 210502 (2017).
[Crossref]

B. Gouraud, D. Maxein, A. Nicolas, O. Morin, and J. Laurat, “Demonstration of a Memory for Tightly Guided Light in an Optical Nanofiber,” Phys. Rev. Lett. 114(18), 180503 (2015).
[Crossref]

G. Heinze, C. Hubrich, and T. Halfmann, “Stopped light and image storage by electromagnetically induced transparency up to the regime of one minute,” Phys. Rev. Lett. 111(3), 033601 (2013).
[Crossref]

D. Schraft, M. Hain, N. Lorenz, and T. Halfmann, “Stopped light at high storage efficiency in a pr3+:y2sio5 crystal,” Phys. Rev. Lett. 116(7), 073602 (2016).
[Crossref]

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and Control of Optical Photons Using Rydberg Polaritons,” Phys. Rev. Lett. 110(10), 103001 (2013).
[Crossref]

E. Vetsch, D. Reitz, G. Sagué, R. Schmidt, S. T. Dawkins, and A. Rauschenbeutel, “Optical Interface Created by Laser-Cooled Atoms Trapped in the Evanescent Field Surrounding an Optical Nanofiber,” Phys. Rev. Lett. 104(20), 203603 (2010).
[Crossref]

Phys. Today (1)

M. Afzelius, N. Gisin, and H. De Riedmatten, “Quantum memory for photons,” Phys. Today 68(12), 42–47 (2015).
[Crossref]

Sci. Rep. (1)

Z. Chen, Z. Bai, H.-j. Li, C. Hang, and G. Huang, “Storage and retrieval of (3+ 1)-dimensional weak-light bullets and vortices in a coherent atomic gas,” Sci. Rep. 5(1), 8211 (2015).
[Crossref]

Other (2)

L.-M. Duan, J. I. Cirac, P. Zoller, and E. S. Polzik, “Quantum Teleportation with Atomic Ensembles and Coherent Light,” in Quantum Communication, Computing, and Measurement 3, (Kluwer Academic Publishers, Boston, 2006), pp. 351–357.

D. A. Steck, “Rubidium 87 d line data, available online at http://steck.us/alkalidata ,” (2019).

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Figures (4)

Fig. 1.
Fig. 1. (a) A scheme for optical memory via EIT using a nanofiber. The control field (red) and signal field (blue) are tightly guided by the nanofiber. The cold atomic gas ensemble is trapped near the nanofiber and strongly interacts with evanescent light fields. (b) Fundamental guided field mode distribution of the nanofiber. (c) Ladder-type energy level configuration of cold atomic gas ensemble. $\Delta _j(j =2,3)$ is the light field detuning, $\Gamma _{ij}(i<j)$ is the spontaneous emission rate from level $|j\rangle$ to level $|i\rangle$, $\Gamma _{31}$ is the external incoherent pumping , and $\Omega _c$ and $\Omega _s$ are the half Rabi frequencies of the control and signal fields, respectively.
Fig. 2.
Fig. 2. (a) Absorption spectrum $\textrm {Im}(K)$; (b) dispersion relation $\textrm {Re}(K)$ as a function of frequency shift $\omega$. The red and blue curves correspond to the free space and nanofiber system at the same energy power level of the control field, respectively.
Fig. 3.
Fig. 3. Temporal and spatial evolution of $|\Omega _c\tau _0|$ and $|\Omega _s\tau _0|$ for the EIT memory process in the nanofiber system. Memory process for (a) a relatively weak signal pulse with an input pulse $\Omega _{s}(0,t)\tau _0=1.8\mathrm {sech}[1.763t/\tau _0]$; (b)a soliton signal pulse with input pulse $\Omega _{s}(0,t)=3.0\mathrm {sech}[1.763t/\tau _0]$; (c) a relatively strong signal pulse with input pulse $\Omega _{s}(0,t)=4.0\mathrm {sech}[1.763t/\tau _0]$. The initial amplitude of the switch control field is $\Omega _{c0}\tau _0=10$.
Fig. 4.
Fig. 4. Optimizing the nanofiber optical memory via nonlinearity. (a) Memory fidelity $\eta J^2$ as a function of the input signal amplitude $|\Omega _{s0}\tau _0|$ and input control field amplitude $|\Omega _{c0}\tau _0|$. (b) Readout signal pulse $\Omega _s(L,t)\tau _0$ as a function of $|\Omega _{c0}\tau _0|$ when fixing $|\Omega _{s0}\tau _0|=2.1$.

Equations (49)

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E = E s + E c = l = s , c E l ( z , t ) u l ( ρ , θ ) e i ( β l z ω t ) + c . c . ,
H ^ i n t = j = 2 3 Δ j | j j | [ ζ s ( ρ , θ ) Ω s | 2 1 | , + ζ c ( ρ , θ ) Ω c | 3 2 | + h . c . ] ,
i ( t + Γ ) σ = [ H ^ i n t , σ ] ,
i ( z + n 2 2 c n e f f s t ) Ω s , + κ 12 σ 21 = 0 , ,
i ( z + n 2 2 c n e f f c t ) Ω c , + κ 23 σ 32 = 0 ,
f ( ρ , θ ) = | ζ c ( ρ , θ ) | 2 f ( ρ , θ ) ρ d ρ d θ | ζ c ( ρ , θ ) | 2 ρ d ρ d θ ,
σ 11 ( 0 ) = Γ 12 Γ 23 | d 32 | 2 + 2 γ 32 Γ 12 | ζ c ( ρ , θ ) Ω c | 2 Γ | d 32 | 2 + 2 γ 32 | ζ c ( ρ , θ ) Ω c | 2 ( Γ 12 + 2 Γ 31 ) ,
σ 22 ( 0 ) = Γ 31 Γ 12 σ 11 ( 0 ) ,
σ 33 ( 0 ) = 1 ( σ 11 ( 0 ) + σ 22 ( 0 ) ) ,
σ 32 ( 0 ) = ζ c ( ρ , θ ) Ω c d 32 ( σ 33 ( 0 ) σ 22 ( 0 ) ) ,
σ 31 ( 0 ) = σ 21 ( 0 ) = 0.
K ( ω ) = ω c n 2 2 n e f f + , κ 12 ( ω + d 31 ) ( σ 22 ( 0 ) σ 11 ( 0 ) ) ζ c ( ρ , θ ) Ω c σ 32 ( 0 ) ( ω + d 21 ) ( ω + d 31 ) | ζ c ( ρ , θ ) Ω c | 2 .
i ( z + α ) U K 2 2 2 U τ 2 W | U | 2 U = 0 ,
E s ( r , t ) = | p 12 | τ 0 K 2 ~ W ~ u s ( ρ , θ ) s e c h [ 1 τ 0 ( τ z V ~ g ) ] × exp [ i ( k ( ω s ) + K 0 ~ + 1 2 L D ) z ω s t ] + c . c .
Ω c ( 0 , t ) = Ω c 0 { 1 1 2 tanh [ t T off T s ] + 1 2 tanh [ t T on T s ] } ,
Ω s ( z , t ) { A τ 0 K 2 ~ W ~ s e c h [ 1 τ 0 ( τ z V ~ g ) ] exp [ i K 0 ~ z + i z 2 L D ] ,         t < T off 0 ,         T off < t < T on B τ 0 K 2 ~ W ~ s e c h [ 1 τ 0 ( τ z V ~ g T D ) ] exp [ i K 0 ~ z + i z 2 L D + i φ 0 ] ,         t > T on
η = + | Ω s ( L , t ) | 2 d t + | Ω s ( z = 0 , t ) | 2 d t
J 2 = | + Ω s ( z = 0 , t ) Ω s ( z = L , t + T D ) d t | 2 + | Ω s ( z = 0 , t ) | 2 d t + | Ω s ( z = L , t + T D ) | 2 d t .
2 u l ( ρ , θ ) ρ 2 + u l ( ρ , θ ) ρ + 1 ρ 2 2 u l ( ρ , θ ) θ 2 + ( n 2 k 0 2 β l 2 ) u l ( ρ , θ ) = 0
u l , ρ = i β l 2 κ J m ( U ) [ ( 1 s ) J m 1 ( 1 + s ) J m + 1 ] , { cos ( m θ ) i sin ( m θ ) , } ,
u l , θ = i β l 2 κ J m ( U 0 ) [ ( 1 s ) J m 1 + ( 1 + s ) J m + 1 ] , { sin ( m θ ) i cos ( m θ ) , } ,
u l , z = 1 J m ( U 0 ) J m , { cos ( m θ ) i sin ( m θ ) , } .
u l , ρ = i β l 2 γ K m ( W 0 ) [ ( 1 s ) K m 1 + ( 1 + s ) K m + 1 ] , { cos ( m θ ) i sin ( m θ ) , } ,
u l , θ = i β l 2 γ K m ( W 0 ) [ ( 1 s ) K m 1 ( 1 + s ) K m + 1 ] , { sin ( m θ ) i cos ( m θ ) , } ,
u l , z = 1 K m ( W 0 ) K m , { cos ( m θ ) i sin ( m θ ) , } .
s = 1 / U 0 2 + 1 / W 0 2 J m ( U 0 ) / J m ( U 0 ) U 0 + K m ( W 0 ) / K m ( W 0 ) W 0
( n 1 2 U 0 J m ( U 0 ) J m ( U 0 ) + n 2 2 W 0 K m ( W 0 ) K m ( W 0 ) ) ( 1 U 0 J m ( U 0 ) J m ( U 0 ) + 1 W 0 K m ( W 0 ) K m ( W 0 ) ) = m 2 n eff 2 ( V U 0 W 0 ) 4 .
i ( t + Γ 31 ) σ 11 i Γ 12 σ 22 + ζ s ( ρ , θ ) Ω s σ 21 ζ s ( ρ , θ ) Ω s σ 21 = 0 , .
i ( t + Γ 12 ) σ 22 i Γ 23 σ 33 + ζ s ( ρ , θ ) Ω s σ 21 + ζ c ( ρ , θ ) Ω c σ 32 ζ s ( ρ , θ ) Ω s σ 21 ζ c ( ρ , θ ) Ω c σ 32 = 0 , .
i ( t + Γ 23 ) σ 33 i Γ 31 σ 11 + ζ c ( ρ , θ ) Ω c σ 32 ζ c ( ρ , θ ) Ω c σ 32 = 0 , .
( i t + d 21 ) σ 21 + ζ c ( ρ , θ ) Ω c σ 31 + ζ s ( ρ , θ ) Ω s ( σ 11 σ 22 ) = 0.
( i t + d 31 ) σ 31 ζ s ( ρ , θ ) Ω s σ 32 + ζ c ( ρ , θ ) Ω c σ 21 = 0 ,
( i t + d 32 ) σ 32 ζ s ( ρ , θ ) Ω s σ 31 + ζ c ( ρ , θ ) Ω c ( σ 22 σ 33 ) = 0.
Ω s ( 1 ) = F e i Θ
σ 21 ( 1 ) = D 1 D ζ s ( ρ , θ ) Ω s ( 1 )
σ 31 ( 1 ) = D 2 D ζ s ( ρ , θ ) Ω s ( 1 )
σ 11 ( 2 ) = [ i Γ 23 + 2 | ζ c ( ρ , θ ) Ω c | 2 ( 1 d 32 1 d 32 ) ] ( D 1 D D 1 D ) + i Γ 12 ( 1 d 32 D 2 D ζ c ( ρ , θ ) Ω c c . c . ) Γ 23 Γ 31 + Γ 12 Γ 23 + i ( 2 Γ 31 + Γ 12 ) | ζ c ( ρ , θ ) Ω c | 2 ( 1 d 32 1 d 32 )   × | ζ s ( ρ , θ ) | 2 | F | 2 e 2 α ¯ z 2 = a 11 ( 2 ) | ζ s ( ρ , θ ) | 2 | F | 2 e 2 α ¯ z 2
σ 22 ( 2 ) = i Γ 12 [ ( D 1 D D 1 D ) i Γ 31 a 11 ( 2 ) ] | ζ s ( ρ , θ ) | 2 | F | 2 e 2 α ¯ z 2 , = a 22 ( 2 ) | ζ s ( ρ , θ ) | 2 | F | 2 e 2 α ¯ z 2
σ 32 ( 2 ) = 1 d 32 [ D 2 D ζ ( ρ , θ ) Ω c ( 2 a 22 ( 2 ) + a 11 ( 2 ) ) ] | ζ s ( ρ , θ ) | 2 | F | 2 e 2 α ¯ z 2 = a 32 ( 2 ) | ζ s ( ρ , θ ) | 2 | F | 2 e 2 α ¯ z 2
σ 21 ( 2 ) = i D 2 ζ ( ρ , θ ) Ω c ( ω + d 31 ) D 1 D 2 ζ s ( ρ , θ ) F t 1 e i Θ = a 21 ( 2 ) ζ s ( ρ , θ ) F t 1 e i Θ ,
σ 31 ( 2 ) = 1 ζ c ( ρ , θ ) Ω c [ i D 1 D + ( ω + d 21 ) a 21 ( 2 ) ] ζ s ( ρ , θ ) F t 1 e i Θ = a 31 ( 2 ) ζ s ( ρ , θ ) F t 1 e i Θ ,
K 2 = 2 i κ 12 ( ω + d 31 ) a 21 ( 2 ) ζ c ( ρ , θ ) Ω c a 31 ( 2 ) D ,
W = κ 12 ( ω + d 31 ) ( a 11 ( 2 ) a 22 ( 2 ) ) + ζ c ( ρ , θ ) Ω c a 32 ( 2 ) D | ζ s ( ρ , θ ) | 2 ,
i ( t + Γ 31 ) σ ~ 11 i Γ 12 σ ~ 22 + ϱ s 1 Ω s σ ~ 21 ϱ s 1 Ω s σ ~ 21 = 0 , ,
i ( t + Γ 12 ) σ ~ 22 i Γ 23 σ ~ 33 + ϱ s 1 Ω s σ ~ 21 + ϱ c Ω c σ ~ 32 ϱ s 1 Ω s σ ~ 21 ϱ c Ω c σ ~ 32 = 0 ,
i ( t + Γ 23 ) σ ~ 33 i Γ 31 σ ~ 11 + ϱ c Ω c σ ~ 32 ϱ c Ω c σ ~ 32 = 0 , ,
( i t + d 21 ) σ ~ 21 + Ω c σ ~ 31 + ϱ s 2 Ω s ( σ ~ 11 σ ~ 22 ) = 0 ,
( i t + d 31 ) σ ~ 31 ϱ s 1 Ω s σ ~ 32 + ϱ c Ω c σ ~ 21 = 0 ,
( i t + d 32 ) σ ~ 32 ϱ s 2 Ω s σ ~ 31 + Ω c ( σ ~ 22 σ ~ 33 ) = 0.

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