## Abstract

We investigate the formation and propagation of ultraslow weak-light solitons and their memory in the atomic gas filled in a kagome-structured hollow-core photonic crystal fiber (HC-PCF) via electromagnetically induced transparency (EIT). We show that, due to the strong light-atom coupling contributed by the transverse confinement of the HC-PCF, the EIT and hence the optical Kerr nonlinearity of the system can be largely enhanced, and hence optical solitons with very short formation distance, ultraslow propagation velocity, and extremely low generation power can be realized. We also show that the optical solitons obtained can not only be robust during propagation, but also be stored and retrieved with high efficiency through the switching off and on of a control laser field. The results reported herein are promising for practical applications of all-optical information processing and transmission via the ultraslow weak-light solitons and the kagome-structured HC-PCF.

© 2017 Optical Society of America

## 1. Introduction

In recent years, tremendous efforts have been paid to the investigation on optical pulse memory, which is important for the realization of fast optical information processing. One of the main techniques to realize optical pulse memory is the utilization of electromagnetically induced transparency (EIT) [1,2], an interesting quantum interference effect typically occurring in three-level atomic systems. EIT can be used not only to cancel optical absorption in resonant quantum systems, but also to bring many novel nonlinear optical effects at weak-light level, including the production of weak-light solitons [3–5]. Based on the dark-state polariton [6] inherent in EIT systems, a signal optical mode can be mapped into an atomic mode, stored temporarily, and then retrieved from atoms by switching off and on of a control laser field [7–10].

However, nearly all researches reported up to now on the storage and retrieval of optical pulses via EIT are limited within linear optical regime. Due to the significant dispersion inherent in resonant systems, linear optical pulses in EIT-based atomic ensembles are not stable, resulting in a serious deformation for retrieved pulses and hence the loss of optical information. For practical applications of light memory, it is desirable to obtain a signal pulse that is robust not only during propagation, but also during storage and retrieval, and hence to acquire high efficiency and fidelity.

Recently, the EIT-based memory has been generalized to weak nonlinear optical regime, where the storage and retrieval of optical soliton pulses have been analyzed [11]. Nevertheless, because the optical pulses and atomic gases considered in Refs. [11] work in free space, the diffraction effect in those systems can not be neglected, which makes the optical soliton pulses unstable during propagation as well as during storage and retrieval. In addition, a larger power is needed for generating the optical soliton pulses in those systems since the coupling between light and atoms is weak in free space.

In this article, we propose a scheme for the formation, propagation, storage and retrieval of the optical soliton pulses in an atomic gas filled in a kagome-structured hollow-core photonic crystal fiber (HC-PCF) [12–16] via EIT. Because of the transverse confinement provided by the fiber, the diffraction effect of the optical pulses is completely eliminated. We show that the EIT and hence the Kerr nonlinearity may be largely enhanced, which can be used to balance the dispersion and hence supports the production of the optical soliton pulses in the system. We prove that the optical soliton pulses obtained with such a scheme have very short formation distance (< 2 cm), ultraslow propagation velocity (∼ 10^{−5} *c*), and extremely low generation power (∼ nW); in addition, they are robust during propagation and can be stored and retrieved with high efficiency through the switching on and off of a control laser field. The results reported here are promising for practical applications of all-optical information processing and transmission.

Before preceding, we note that bandgap-structured HC-PCFs [15–17] can also be used to perform EIT (see Refs. [18–25]), and to form an optical soliton pulse as shown recently by Facão *et al.* [26] where the fiber-core diameter is less than 4 *μ*m. However, the use of the kagome-structured HC-PCF possesses many advantages, including: (i) The kagome-structured HC-PCF has a large core diameter (26 *μ*m–100 *μ*m in our scheme), which is easier not only for fabrication and but also for loading more atoms in the fiber core; (ii) Because of the larger atomic number in the kagome-structured HC-PCF, the coupling between the light field and the atomic gas is stronger, which is desirable for forming optical solitons within a shorter distance and with a lower generation power. (iii) Comparing with bandgap-structured HC-PCFs, the kagome-structured HC-PCF has a larger transverse transit time for the atoms in the fiber core, which admits a smaller dephasing and hence a higher efficiency for the storage and retrieval of the optical soliton pulses (see a further discussion in Sec. 5).

The rest of the article is arranged as follows. In Sec. II, the theoretical model is described. In Sec. III, the formation and propagation of ultraslow weak-light solitons is investigated. In Sec. 4, the storage and retrieval of optical soliton pulses are studied. Finally, in Sec. IV a discussion and a summary on the main results obtained in this work is given.

## 2. Model

We consider a pulsed signal laser field **E*** _{s}* and a continuous-wave (CW) control laser field

**E**

*, both are guided inside a kagome-structured HC-PCF with (hollow) core radius*

_{c}*r*

_{0}and pitch Λ

_{0}[Fig. 1(a)]. The core of the HC-PCF is filled with an atomic gas with a lambda-type level configuration [Fig. 1(b)]. In the absence of the atomic gas, the HC-PCF allows many eigenmodes of electric field

**E**with the form

**e**

_{α}*q*(

_{α}*x*,

*y*) exp{

*i*[

*β*(

_{α}*ω*)

*z*−

*ωt*]}, here

**e**

*,*

_{α}*q*(

_{α}*x*,

*y*), and

*β*are the polarization unit vector, eigenfunction, and propagation constant for the mode with index

_{α}*α*(for a simple introduction of the eigenmodes in the kagome-structured HC-PCF, see Appendix A). In the present study, we are interested in the fundamental mode of the HC-PCF made of silica. The blue solid line in Fig. 1(c) is the numerical result of the electric-field distribution in the radius direction (with $r\equiv \sqrt{{x}^{2}+{y}^{2}}$ the radius coordinate of the system) of the fundamental mode for

*r*

_{0}= Λ

_{0}= 13

*μ*m when the atomic gas is absent. The shape of the fundamental mode can be fitted well by the zeroth-order Bessel function [27] [the red dashed curve in Fig. 1(c)]. One sees that the fundamental mode is well confined in the fiber core, as shown in the inset of Fig. 1(c). In Fig 1(b), |1〉, |2〉, and |3〉 are respectively the metastable, ground, and excited states, Δ

_{3}and Δ

_{2}are respectively one- and two-photon detunings, Γ

*(*

_{jl}*j*= 1, 2;

*l*= 3) denotes the spontaneous emission rate from |

*l*〉 to |

*j*〉;

*ω*and Ω

_{s}*[*

_{s}*ω*and Ω

_{c}*] are respectively the angular and half Rabi frequencies of the signal (control) field, which is coupled to the transition |1〉 ↔ |3〉 (|2〉 ↔ |3〉). Both of the signal and control fields belong to the fundamental mode of the HC-PCF and propagate along*

_{c}*z*direction, with the form $\mathbf{E}={\mathbf{E}}_{s}+{\mathbf{E}}_{c}={\sum}_{l=s,c}{\mathbf{e}}_{l}\sqrt{\frac{\hslash {\omega}_{l}}{2{\u220a}_{0}{V}_{l}^{\text{eff}}}}{\mathcal{E}}_{l}(z,t){q}_{l}(x,y){e}^{i\left[{\beta}_{l}z-{\omega}_{l}t\right]}+\text{c}.\text{c}.$, where

*β*≡

_{l}*β*(

*ω*), ${V}_{l}^{\text{eff}}\equiv {L}_{z}\iint {\left|{q}_{l}(x,y)\right|}^{2}dxdy$ is the effective mode volume for the mode

_{l}*ω*=

*ω*,

_{l}*L*is the fiber length, the subscript

_{z}*s*(

*c*) denotes the signal (control) field, and c.c. represents complex conjugate.

We choose a cesium atomic gas with the energy levels selected as |1〉 = |6^{2}*S*_{1/2}, *F* = 4〉, |2〉 = |6^{2}*S*_{1/2}, *F* = 3〉, and |3〉 = |6^{2}*P*_{3/2}, *F* = 4〉. For such level configuration the wavelengths of the signal and control fields are approximately equal (i.e. *λ _{s}* ≈

*λ*= 852 nm) [28], which means

_{c}*q*(

_{c}*x*,

*y*) ≈

*q*(

_{s}*x*,

*y*) ≡

*q*(

*x*,

*y*). For an analytical description, in the following the fundamental mode is approximated by the zero-order Bessel function

*q*(

*x*,

*y*) =

*J*

_{0}(

*kr*) when

*r*≤

*r*

_{0}and

*q*(

*x*,

*y*) = 0 when

*r*>

*r*

_{0}with

*k*≈ 2.405/

*r*

_{0}[27], which gives ${V}_{c}^{\text{eff}}\approx {V}_{s}^{\text{eff}}=\pi {r}_{0}^{2}{J}_{1}^{2}(k{r}_{0}){L}_{z}\equiv {V}^{\text{eff}}$. In order to have a quantitative discussion on light-beam confinement effect in the HC-PCF, following the approach in Refs. [29, 30] we define a reference mode volume of free space, i.e. ${V}^{\text{ref}}=\pi {R}_{\perp}^{2}{L}_{z}$. The quantity

*R*

_{⊥}(determined by input condition) is the transverse radius of laser beams in free space, introduced here for illustrating the confinement effect due to the HC-PCF by comparing the EIT in the HC-PCF and the EIT in free space. Note that experimentally

*R*

_{⊥}is usually large (e.g.,

*R*

_{⊥}= 100

*μ*m [31], which is chosen in the following discussion). Then the electric field can be expressed as

*= |*

_{s}**p**

_{13}|

*ℰ*/

_{s}*ħ*(Ω

*= |*

_{c}**p**

_{23}|

*ℰ*/

_{c}*ħ*) is the half Rabi frequency of the signal (control) field, with

**p**

*the electric-dipole moment associated with the state |*

_{jl}*j*〉 and the state |

*l*〉. The equation of motion for the density matrix

*σ*in the interaction picture reads [32]

*σ*is a 3 × 3 density matrix with matrix elements

*σ*≡ |

_{jl}*j*〉 〈

*l*|, Γ is a 3 × 3 relaxation matrix denoting the spontaneous emission and dephasing. The explicit expression of Eq. (2) is presented in Appendix B.

The equation of motion for Ω* _{s}* can be obtained by the Maxwell equation ∇

^{2}

**E**− (1/

*c*

^{2})

*∂*

^{2}

**E**/

*∂t*

^{2}= [1/(

*∊*

_{0}

*c*

^{2})]

*∂*

^{2}

**P**

_{p}/∂t^{2}, with the electric polarization intensity given by

**P**

*=*

_{p}**P**

_{host}+

*𝒩*

**p**

_{13}

*σ*

_{31}exp[

*i*(

*β*−

_{s}z*ω*)] + c.c., where

_{s}t*𝒩*is the atomic density,

**P**

_{host}=

*∊*

_{0}

*χ*

_{host}

**E**is the electric polarization intensity, with

*χ*

_{host}=

*χ*

_{host}(

*x*,

*y*) is the susceptibility of HC-PCF in the absence of the atomic gas. According to the expression of the electric field [i.e. the Eq. (1)] and using Eq. (18), the Maxwell equation under a slowly varying envelope approximation is reduced to

*n*≡ [1 +

*χ*

_{host}(

*x*,

*y*)]

^{1/2}is the refractive index [

*χ*

_{host}(

*x*,

*y*) is the electric susceptibility] of the HC-PCF in the absence of the atomic gas,

*κ*

_{13}≡

*𝒩ω*|

_{s}**p**

_{13}|

^{2}/(2

*ħ∊*

_{0}

*cn*

_{eff}) is light-atom coupling coefficient with

*n*

_{eff}the effective refractive index. In Eq. (3) we have defined

*σ*

_{31}=

*σ̃*

_{31}

*ζ*(

*x*,

*y*) and 〈

*ψ*〉 = ∬ |

*ζ*(

*x*,

*y*)|

^{2}

*ψ*(

*x*,

*y*)

*dxdy*/ ∬ |

*ζ*(

*x*,

*y*)|

^{2}

*dxdy*, with

*ψ*an arbitrary function of

*x*and

*y*(for the detailed derivation of Eq. (3), see Appendix A). We see that, the diffraction effect of the signal field is effectively canceled due to the transverse (mode) confinement of the HC-PCF.

For simplicity, the following assumptions are made for our theoretical calculations presented below: (i) To suppress Doppler effect, the atoms are assumed to be cooled to low temperature and the signal and the control fields are injected with opposite directions. (ii) To reduce the dephasing rate caused by the adsorption of atoms to the inner walls of the fiber core, the inner walls are coated with some materials, or a light-induced atomic desorption process is used to release the atoms into the center of fiber core, or a dipole trap along the fiber axis is employed to attract the atoms away from the inner walls of the fiber. (iii) The atoms are initially populated on the metastable state |1〉 by an optical pumping to suppress four-wave mixing effect. These assumptions are realistic because related experiments have already been done in recent years [19,21,33].

## 3. Ultraslow weak-light solitons in the HC-PCF

#### 3.1. Linear dispersion relation and EIT enhancement

We first solve the Maxwell-Bloch (MB) equations (2) and (3) in the linear regime and discuss the linear property of the system. The steady state before the signal field opens is *σ*_{11} = 1 and all other *σ _{jl}* are zero. When the signal field is switched on but very weak, the system will evolve into a time-dependent state and the solution of the MB equations (2) and (3) have the solution

*σ*

_{11}= 1,

*σ*

_{j1}∼ exp(

*iθ*) (

*j*= 2, 3), Ω

*∼ exp(*

_{s}*iθ*), and

*σ*

_{22}=

*σ*

_{33}=

*σ*

_{32}= 0, with

*θ*=

*K*(Δ

*ω*)

*z*− Δ

*ωt*[34]. The linear dispersion relation is given by

*d*

_{21}and

*d*

_{31}have been given in Appendix B. Because the signal field is resonant with the atoms in the fiber, in usual cases one expects that a significant absorption will happen. However, the absorption can be made very low in the present system. Fig. 1(d) shows the calculation result on the imaginary part of

*K*, i.e. Im(

*K*), as a function of Δ

*ω*, where the blue, red and green lines are respectively for the core radius

*r*

_{0}= 13

*μ*m, 17

*μ*m, and 100

*μ*m. From Fig. 1(d) we see that: (i) A transparency window (i.e., the dip near Δ

*ω*= 0) is opened, which is due to the quantum interference (i.e. EIT) effect induced by the control field; (ii) For different core radius

*r*

_{0}, the width of the transparency window (called EIT transparency window) is different. The smaller the value of

*r*

_{0}, the wider the EIT transparency window, which means that the EIT effect is enhanced when the core radius

*r*

_{0}is reduced. The physical reason of the enhancement of EIT effect is due to the existence of the (geometrical) confinement effect in the kagome-structured HC-PCF. By the confinement, the control field Ω

*has an enhanced factor (*

_{c}*R*

_{⊥}/

*r*

_{0})[

*J*

_{1}(

*kr*

_{0})]

^{−1}, which leads to a larger EIT transparency window for smaller

*r*

_{0}. For instance, the transparency window is very wide for

*r*

_{0}= 13

*μ*m, which will be used in our following calculations.

When plotting Fig. 1(d), the system parameters are chosen as Ω* _{c}* = 1.0 × 10

^{7}s

^{−1}, Δ

_{2}= Δ

_{3}= 0, |

**p**

_{13}| ≃ |

**p**

_{23}| = 3.8 × 10

^{−27}C · cm [28],

*𝒩*= 2.0 × 10

^{10}cm

^{−3},

*R*

_{⊥}= 100

*μ*m, Γ

_{3}≃ 2

*γ*

_{31}=

*π*× 5.23 MHz,

*γ*

_{31}=

*γ*

_{32}. Note that the ground-state dephasing rate ${\gamma}_{21}^{\text{dep}}$ for the atoms filled in the HC-PCF may be caused by many physical factors, widely discussed in literature (see, e.g., [18, 20, 25, 35–37]). Here we take ${\gamma}_{21}^{\text{dep}}={\gamma}_{21}\simeq 2\pi \times 0.05\hspace{0.17em}\text{MHz}$ in our numerical calculations carried out here and below, which is slightly larger than that given in the recent experiment [25] in order to account for the factors causing the dephasing not included in our model.

#### 3.2. Nonlinear envelope equation

The approach above is valid not only for CW but also for pulsed signal fields. However, vanishing one- and two-photon detunings assumed there usually cannot be satisfied in reality, which will result in a dispersion effect in the system and hence the spreading of the signal pulse. In particular, in light memory what we need is to store and retrieve a signal pulse within a finite time duration. The sideband components of the pulse are detuned from the energy-level difference of the related transition (|1〉 to |3〉 in the present system), which brings a non-negligible dispersion effect to the system. Such effect brings not only a deformation of the signal pulse but also a reduction of the quality of light memory in the system.

In order to suppress the dispersion effect and obtain a shape-preserving signal pulse useful for light memory, one can exploit the Kerr effect of the system to balance the dispersion [3, 4]. In fact, the Kerr nonlinearity of the system can be largely enhanced in the present system by both the EIT effect and the confinement effect of the HC-PCF, as shown below.

To this end, based on the MB equations (2) and (3) we derive a nonlinear envelope equation describing the nonlinear evolution of the signal field by using the method developed in Ref. [4]. Taking the asymptotic expansion ${\sigma}_{jk}={\sum}_{l=0}^{\infty}{\u220a}^{l}{\sigma}_{jk}^{(l)}$ (*j*, *k* =1, 2, 3; ${\sigma}_{jk}^{(0)}={\delta}_{j1}{\delta}_{k1}$), and ${\mathrm{\Omega}}_{s}={\sum}_{l=1}^{\infty}{\u220a}^{l}{\mathrm{\Omega}}_{s}^{(l)}$. Here *∊* is a dimensionless small parameter characterizing the typical amplitude of the signal pulse. All the quantities on the right-hand side of the expansion are considered as functions of the multi-scale variables *z _{l}* =

*∊*(

^{l}z*l*= 0, 1, 2, · · ·) and

*t*=

_{l}*∊*(

^{l}t*l*= 0, 1, 2, · · ·). Substituting the expansions into the Eqs. (2) and (3) and comparing the coefficients of

*∊*(

^{m}*m*=1, 2, 3), we obtain a set of linear but inhomogeneous equations which can be solved order by order.

The first-order (*m* = 1) solution is given by

*θ*≡

*K*(Δ

*ω*)

*z*

_{0}− Δ

*ωt*

_{0}, with

*F*a yet to be determined envelope function depending on the indicated slow variables

*t*

_{1},

*z*

_{1}, and

*z*

_{2}and

*K*(Δ

*ω*) the same as (4). At the second order (

*m*= 2), we obtain the solvability condition i[

*∂F/∂z*

_{1}+ (

*∂K/∂*Δ

*ω*)

*∂F/∂t*

_{1}] = 0, which shows that the signal-pulse envelope

*F*travels with the group velocity ${V}_{g}\equiv {K}_{1}^{-1}={\left(\partial K/\partial \mathrm{\Delta}\omega \right)}^{-1}$, given by

With the above results we go to the third order (*m* = 3). A solvability condition in this order yields the equation for the signal-pulse envelope $i\partial F/\partial {z}_{2}-(1/2)\left[{\partial}^{2}K/\partial \mathrm{\Delta}{\omega}^{2}\right]{\partial}^{2}F/\partial {t}_{1}^{2}-W{\left|F\right|}^{2}F{e}^{-2\overline{a}{z}_{2}}=0$, here *K*_{2} ≡ *∂*^{2}*K/∂*Δ*ω*^{2} describes the second-order dispersion, and the nonlinear coefficient

Combining the solvability conditions in all orders and returning to the original variables, we obtain the equation

*τ*=

*t*−

*z/V*,

_{g}*U*=

*∊Fe*

^{−âz2}, and

*a*=

*∊*

^{2}

*ā*, with the definition of

*ā*presented in Appendix C. Note that Eq. (8) is valid for Δ

*ω*≪

*ω*. Since we are interested in the evolution of the signal pulse at the center frequency

_{s}*ω*, in the calculations given below all the coefficients in Eq. (8) will be evaluated at Δ

_{s}*ω*= 0.

#### 3.3. Ultraslow weak-light solitons

The third-order nonlinear optical susceptibility *χ*^{(3)} of the signal field is proportional to the self-phase modulation coefficient *W* in Eq. (8) via the relation

*= 3.6 × 10*

_{c}^{6}s

^{−1}, Δ

_{2}= 2.5 × 10

^{7}s

^{−1}, Δ

_{3}= 3.75 × 10

^{8}s

^{−1},

*𝒩*= 2.0 × 10

^{10}cm

^{−3}and

*r*

_{0}= 13

*μ*m, we obtain

*W*= (−5.38+0.16

*i*)×10

^{−13}cm

^{−1}s

^{2}and Re(

*χ*

^{(3)}) = −1.56×10

^{−2}cm

^{2}V

^{−2}, which is many orders of magnitude larger than that in conventional fibers [38]. Hence the HC-PCF with the filled atomic gas via EIT possesses greatly enhanced Kerr nonlinearity [39], contributed from the quantum interference effect in the (resonant) atomic gas, which is very useful for practical applications of many nonlinear optical processes, including effective four-wave mixing and formation of weak-light solitons.

Since the system under study is a resonant one, the coefficients in Eq. (8) are generally complex. However, when the system works under the EIT condition, the absorption of the signal field can be largely suppressed, and therefore the imaginary part of these coefficients can be made to be much smaller. Thus Eq. (8), when writing into the dimensionless form, can be approximated by the nonlinear Schrödinger equation *i∂u/∂s* + *∂*^{2}*u/∂σ*^{2} + 2|*u*|^{2}*u* = 0, where *u* = *U/U*_{0}, *s* = −*z*/(2*L _{D}*), and

*σ*=

*τ/τ*

_{0}, with ${L}_{D}={\tau}_{0}^{2}/{\tilde{K}}_{2}$ (

*τ*

_{0}is the pulse duration of the signal field) the typical dispersion length of the system. Note that we have taken

*L*=

_{N}*L*with ${L}_{N}=1/\left(\tilde{W}{U}_{0}^{2}\right)$ the typical nonlinear length, and thus ${U}_{0}=(1/{\tau}_{0})\sqrt{{\tilde{K}}_{2}/\tilde{W}}$ (typical Rabi frequency of the signal field in free space). Here the tilde above

_{D}*K*

_{2}and

*W*means taking the real part,

*K̃*

_{2}= Re(

*K*

_{2}) and

*W̃*= Re(

*W*). A single-soliton solution reads

*u*(

*s*,

*σ*) = 2

*ϑ*sech[2

*ϑ*(

*σ*−

*σ*

_{0}+ 4

*δs*)] exp [−2

*iδσ*− 4

*i*(

*δ*

^{2}−

*ϑ*

^{2})

*s*−

*iϕ*

_{0}], where

*ϑ*,

*σ*

_{0},

*δ*, and

*ϕ*

_{0}are real free parameters that determine the amplitude (also width), propagating velocity, initial position, and initial phase of the soliton, respectively. For simplicity, we take

*ϑ*= 1/2,

*σ*

_{0}=

*δ*=

*ϕ*

_{0}= 0. Then, when returning to original variables, we obtain the expression of the signal field

**E**

*=*

_{s}**e**

*, with*

_{s}E_{s}*K̃*

_{0}≡ Re(

*K*

_{0}), with

*K*

_{0}=

*K*(Δ

*ω*)|

_{Δω=0}.

We now give a set of system parameter for the formation of the optical soliton given above. By taking Ω* _{c}* = 3.6 × 10

^{6}s

^{−1},

*τ*

_{0}= 1.0 × 10

^{−7}s, Δ

_{2}= 2.5 × 10

^{7}s

^{−1}, Δ

_{3}= 3.75 × 10

^{8}s

^{−1},

*𝒩*= 2.0 × 10

^{10}cm

^{−3}and other parameters are the same as those given in the above text, we obtain (evaluated at Δ

*ω*= 0)

*K*

_{0}= (−3.23 + 0.09

*i*) cm

^{−1},

*K*

_{1}= (4.05 − 0.33

*i*) × 10

^{−8}cm

^{−1}s,

*K*

_{2}= (−3.45 + 0.39

*i*) × 10

^{−15}cm

^{−1}s

^{2},

*W*= (−5.38 + 0.16

*i*) × 10

^{−13}cm

^{−1}s

^{2},

*U*

_{0}= 0.8×10

^{6}s

^{−1}, and

*L*= 1/Im(

_{A}*K*

_{0}) = 10.39 cm (typical absorption length). In particular, the nonlinearity length of the system is given by

The energy flux of the ultraslow optical soliton in the HC-PCF predicted by Eq. (10) can be calculated by using the Poynting vector integrated over the hollow core cross-section, i.e., *P* = ∬ *dxdy*(**E*** _{s}* ×

**H**

*) ·*

_{s}**e**

*, where*

_{z}**e**

*is the unit vector in the propagation direction [40]. At leading order the corresponding magnetic field of the soliton,*

_{z}**H**

*, is transverse and in the*

_{s}**e**

*×*

_{s}**e**

*direction. Thus if*

_{z}**e**

*=*

_{s}**e**

*then*

_{y}**H**

*=*

_{s}**e**

*with*

_{x}H_{s}*H*≈

_{s}*n*

_{eff}

*∊*

_{0}

*cE*. After averaging over the carrier-wave period, we obtain

_{s}*P*=

*P*

_{max}sech

^{2}[(

*t*−

*z/Ṽ*)/

_{g}*τ*

_{0}], where

*P*

_{max}= 2

*∊*

_{0}

*cn*

_{eff}(

*ħU*

_{0}/|

**p**

_{13}|)

^{2}∬ |

*ζ*(

*x*,

*y*)|

^{2}

*dxdy*, with

*n*

_{eff}the effective refractive index of the signal field, given in the Appendix A [i.e. Fig. (6)]. Using the parameters given above, we obtain

Shown in Fig. 2(a) is the propagation of the ultraslow optical soliton with |Ω* _{s}* (

*z*,

*τ*)

*τ*

_{0}| as a function of

*z/L*and

_{D}*t/τ*

_{0}. The comparison of waveshapes at

*z*= 0 and at

*z*= 2

*L*is given by Fig. 2(b). The solution is obtained by numerically solving Eq. (8) with all the imaginary parts of the complex coefficients taken into account. In the simulation, the initial condition is chosen as Ω

_{D}*(0,*

_{s}*τ*)

*τ*

_{0}= 0.08 sech(

*t*/

*τ*

_{0}). One sees that, although due to the dissipative effect (mainly contributed by the dephsing ${\gamma}_{21}^{\text{dep}}$) there is a weak amplitude decay when propagating to

*z*= 2

*L*, the soliton can keep its shape well for a propagation distance up to 2

_{D}*L*= 5.8 cm.

_{D}## 4. Storage and retrieval of the ultraslow weak-light solitons in kagome structured HC-PCF

#### 4.1. Storage and retrieval of the optical soliton pulse in a small-core kagome-structured HC-PCF

Although many schemes have been proposed and realized in the atomic gases in free space (see Refs. [41–51] and references therein), the light memory using an atomic gas filled in HC-PCFs is more desirable. The reasons are the following. First, guided-wave optics can confine optical modes within a small area over a distance longer than that is possible with diffractive optics in free space, thus the light power required to obtain strong light-atom coupling can be largely reduced, which can increase light memory efficiency. Second, an integrated platform is easier to interface with existing photonic architectures and also easier to scale up. Thus it is nature to seek the possibility of the storage and retrieval of the ultraslow weak-light solitons obtained in the last section.

The principle of EIT-based light memory is as follows [6]. When switching on the control field, the signal pulse propagates in the atomic gas with nearly vanishing absorption; by slowly switching off the control field the signal pulse disappears and gets stored in the atomic gas in the form of atomic coherence; when the control field is switched on again the signal pulse appears again. However, this principle was usually applied for linear optical pulses, which are not stable during propagation and suffer serious deformation due to the dispersion and/or diffraction. In the following, we show that it is available to realize the memory of stable optical soliton pulses in the kagome-structured HC-PCF obtained in the last section.

To this end, we numerically solve the MB equations (2) and (3) by taking the control field to be adiabatically changed with time to realize the function of its switching on and off. The switching-on and the switching-off of the control field are modeled by the combination of two hyperbolic tangent functions with the form

*T*and the storage time of the signal pulse is approximately given by ${T}_{c}^{\text{on}}-{T}_{c}^{\text{off}}$. We choose

_{s}*T*= 0.5

_{s}*τ*

_{0}, ${T}_{c}^{\text{off}}=10.0{\tau}_{0}$, ${T}_{c}^{\text{on}}=20.0{\tau}_{0}$, with

*τ*

_{0}= 1.0 × 10

^{−7}s. Note that due to the light confinement in the transverse directions, the MB equations [i.e. (2) and (3)] controlling the motion of the atoms filled in the HC-PCF are different from those in free space. In particular, the mode function

*ζ*(

*x*,

*y*) appears in the coefficients of these equations, making the calculation of light memory more complicated than that in free space.

We first consider the kagome-structured HC-PCF with a relatively small core radius *r*_{0} (but much larger than that of the bandgap-structured HC-PCF in Ref. [26]). In this case, the system can be well approximated by a quasi-one-dimensional waveguide since the variation of the electric field on *x* and *y* is much faster than that on *z*, and hence the light memory can be studied by using the Eq. (3) and effective Bloch equation (23) (see Appendixes B and D). Shown in Fig. 3(a) is the result of storage and retrieval of an optical soliton pulse for *r*_{0} = Λ_{0} = 13 *μ*m, where |Ω_{s}τ_{0}| is taken as a function of the propagation distance *z* and the evolution time *t*. The wave shape of the input signal pulse is taken as a hyperbolic secant one with a larger amplitude for forming soliton, i.e., ${\mathrm{\Omega}}_{s}^{\text{in}}(t){\tau}_{0}=0.1\text{sech}(1.5t/{\tau}_{0})$. The black, red, green, yellow, sky blue, and blue solid lines in each panel are for the soliton pulse propagating to the distance *z*=0, 1, 2, 3, 4 and 5 cm, respectively; the purple solid line is for the dimensionless half Rabi frequency |Ω_{c}τ_{0}| of the control field. From the figure, we see that in this soliton regime the pulse is narrowed (i.e. the soliton is indeed formed) before its storage because of the balance between the dispersion and nonlinearity in the system. When the control field is switched off at $t={T}_{c}^{\text{off}}=10\hspace{0.17em}{\tau}_{0}$, the soliton pulse disappears, and then appears again when the control field is switched on at $t={T}_{c}^{\text{on}}=20\hspace{0.17em}{\tau}_{0}$.

During the storage stage, the energy (information) of the optical soliton pulse is transferred into the atomic gas. Fig. 3(b) shows the result of the atomic coherence |*σ̃*_{21}| as a function of *z* and *t*. One sees that *σ̃*_{21} is nonzero during the switch-off of the control field. Since the signal pulse is stored in the form of atomic coherence *σ̃*_{21} when the control field is switched off and is retained until the control field is switched on again, the atomic coherence *σ̃*_{21} can be taken as the intermediary for the memory of the signal pulse.

The efficiency *η* of the light memory can be described by the energy ratio between the retrieved pulse and the input pulse [52], i.e.,

*L*the fiber length. Our calculation gives that the optical pulse memory efficiency in this soliton regime is

_{z}*η*= 86.3% for

*L*= 5 cm. The fidelity of the light memory can be characterized by the quantity

_{z}*ηJ*

^{2}, here

*J*

^{2}is the overlap integral

*T*is the time interval between the peak of the input signal pulse ${E}_{s}^{\text{in}}$ and the peak of the retrieved signal pulse ${E}_{s}^{\text{out}}$. From Fig. 3(a) we see that the peak of the input signal pulse (the black solid curve) is at

*t*= 0 and the peak of the retrieved signal pulse (the blue solid curve) is at

*t*= 28

*τ*

_{0}, and hence Δ

*T*= 28

*τ*

_{0}. Using the formula (16) we obtain

*J*

^{2}= 91.9% for

*L*= 5 cm. Thus, the fidelity of the optical soliton memory is

_{z}*ηJ*

^{2}= 79.4%.

For comparison, the optical pulse memory in a linear regime is also calculated. Fig. 3(c) shows the result of numerical simulation on the evolution of |Ω_{s}τ_{0}| as function of *z* and *t*. The input signal pulse is also taken as a hyperbolic secant function but with a much smaller amplitude, i.e., ${\mathrm{\Omega}}_{s}^{\text{in}}(t){\tau}_{0}=0.01\text{sech}(1.5t/{\tau}_{0})$. The colored solid lines are for *z*=0, 1, 2, 3, 4 and 5 cm, respectively. At beginning, the pulse profile for red line (*z* = 1 cm) in Fig. 3(b) is similar to that in Fig. 3(a) with almost same peak amplitude and pulse width. However, after 5 cm propagation, the peak value of blue line (*z* = 5 cm) decreases to 6.8 × 10^{4} and its width is more than 2.0*τ*_{0}. The blue line in Fig. 3(a) still keeps its amplitude and pulse width. From the figure, we see that the retrieved signal pulse is significantly broadened and its amplitude decreases rapidly. Based on Eqs. (15) and (16), we obtain the memory efficiency and fidelity of the linear optical pulse, respectively given by *η* = 79.8% and *ηJ*^{2} = 62.9% for *L _{z}* = 5 cm, lower than the optical soliton memory shown in Fig. 3(a). The atomic coherence |

*σ̃*

_{21}| of the linear optical pulse during the storage and retrieval is presented in Fig. 3(d).

For a given storage time, the memory efficiency of the optical soliton pulse depends on the fiber length *L _{z}* and the fiber core radius

*r*

_{0}when the other system parameters are fixed. Fig. 4 shows the result of

*η*as a function of the fiber length

*L*for the storage time ${T}_{c}^{\text{on}}-{T}_{c}^{\text{off}}=10{\tau}_{0}$. The blue, red and green solid curves in the figure are respectively for

_{z}*r*

_{0}taking 13, 15, and 17

*μ*m (the pitch Λ

_{0}is fixed to be 13

*μ*m). From the figure we can obtain the following conclusions: (i) For any core radius

*r*

_{0}, the memory efficiency

*η*grows rapidly in the small

*L*region (i.e. the region on the left side of its peak value), then as

_{z}*L*increases it reaches to the peak value, and finally drops down slowly in the large

_{z}*L*region (i.e. the region on the right side of the peak value); (ii) In the large (small)

_{z}*L*region, the smaller the core radius

_{z}*r*

_{0}the higher (lower) the memory efficiency

*η*. The physical reasons are the following. Since the signal pulse in the fiber with a larger core radius has a smaller group velocity, in the small

*L*region (where damping due to the spontaneous emission and dephasing is negligible) the compression of the signal pulse is more effective and hence the memory efficiency for the large-core fiber is larger than that of the small-core fiber. However, in the large

_{z}*L*region (where damping plays a significant role), the attenuation of the signal pulse in the small-core fiber is relatively smaller than in the large-core fiber and thus the memory efficiency is higher for the small-core fiber. Consequently, one can acquire a large memory efficiency of the optical soliton pulse by suitably selecting the core radius and the fiber length.

_{z}The main factor affecting the light memory efficiency and fidelity is the value of the dephasing rate ${\gamma}_{21}^{\text{dep}}$. To improve the efficiency and fidelity, one can employ a microwave field to couple the two lower atomic levels [i.e. |1〉 and |2〉 in Fig. 1(b)]. The microwave field is applied within the time interval in which the control field is switched off, which can provide a gain to the atomic coherence *σ*_{21} [53]. We have carried out a numerical simulation on the memory of an optical soliton pulse by adding such a microwave field into the system, with the result shown by the dashed-dotted lines in Fig. 4. We see that by using the microwave field the soliton memory efficiency can be increased by 20%, which is valid for the all values of the core-radius *r*_{0}.

#### 4.2. Storage and retrieval of the optical soliton pulse for a large-core kagome-structured HC-PCF

As pointed out previously, a large-core fiber is desirable to fill more atoms in its core to obtain a strong light-atom coupling in the system. Recently, the core radius *r*_{0} of kagome-structured HC-PCF has been extended to 50 *μ*m [54, 55], which encourages us to consider the memory of optical soliton pulse with such large-core fiber.

To this end, we make a numerical simulation to investigate the memory of optical soliton pulse in a kagome-structured HC-PCF with the core radius *r*_{0} = 50 *μ*m and the pitch Λ_{0} = 25 *μ*m. Note that in this situation the distribution range of the mode function *ζ*(*x*, *y*) is relatively large, the effective Bloch equation [i.e. Eq. (23)] is not a good approximation. Nevertheless, the (reduced) Maxwell equation (3) is still valid since the distribution range of *ζ*(*x*, *y*) is still much smaller than the spatial length of the signal pulse in the propagation (i.e. *z*) direction [56]. Thus the simulation is carried out based on the Eqs. (3) and (19).

Shown in Fig. 5(a) is the result of the storage and retrieval of an optical soliton pulse in the large-core fiber. The switching on and off of the control field is still modeled by the function (14). Panels (i) to (v) in the figure give the electric-field intensity distribution |*E _{s}*|

^{2}of the soliton pulse in the (

*x*,

*y*) plane for the soliton propagating to the distance

*z*= 0, 1, 2.5, 4, and 5 cm, respectively.

*t*= 0, 3.5

*τ*

_{0}, 15

*τ*

_{0}, 25

*τ*

_{0}, and 28.5

*τ*

_{0}shown respectively in (i) to (v) are the times when the peak of the soliton arrives at the positions

*z*= 0, 1, 2.5, 4, and 5 cm, respectively. We see that the optical soliton pulse can still be stored and retrieved in the system and can keep well its shape during the storage process. The efficiency and fidelity of the light memory in the present case are

*η*= 77.6% and

*ηJ*

^{2}= 71.2% for

*L*= 5 cm, respectively.

_{z}For comparison, Fig. 5(b) shows the storage and retrieval of a linear optical pulse in the same large-core fiber. One sees that during the storage process the linear optical pulse undergoes a significant decrease of the amplitude. By the formulas (15) and (16) one obtains small memory efficiency (*η* = 63.8%) and small fidelity (*ηJ*^{2} = 50.3%), lower than the optical soliton memory shown in Fig. 5(a).

## 5. Discussion and summary

From the results described above, we see that the optical soliton pulses in the kagome-structured HC-PCF can not only be robust during propagation, but also be stored and retrieved with relatively higher efficiency and fidelity.

We note that there are two types of HC-PCFs, i.e., bandgap-structured HC-PCFs and kagome-structured HC-PCFs [15–17,57–59]. The bandgap-structured HC-PCFs have a true bandgap for eigenfrequency, which can be taken as a guidance mechanism preventing light from propagating in fibre cladding and confining the light within the fiber core. The EIT and related phenomena based on such HC-PCFs were demonstrated in many previous studies (see, e.g., Refs. [18–25]). The kagome-structured HC-PCFs do not possess a bandgap, so the light is guided by different mechanisms (e.g., anti-resonant reflection) [15]. The EIT and related Raman memory based on such HC-PCFs were also reported [33,60,61].

Our work presented above is different from that considered in Ref. [26], where the authors employed a bandgap-structured HC-PCF with a small core radius and did not discuss the storage and retrieval of optical solitons. In contrast, in our work not only the formation and propagation but also the storage and retrieval of the optical solitons are investigated, both are based on the kagome HC-PCF. In addition, our work is also at variance with those studied in Refs. [18–25,33, 60,61], where no optical solitons and their storage and retrieval were studied. We stress that the light memory using the atomic gas filling into the kagome HC-PCF has higher efficiency and fidelity than those using an atomic gas in free space. For comparison, consider the same cesium atomic gas (used above) works in a free space with the same control field Ω* _{c}*. If the input signal pulse has the transverse mode profile

*q*(

*x*,

*y*) =

*J*

_{0}(

*kr*), the characteristic diffraction length of the system reads ${L}_{\mathit{Dif}}=\sqrt{3}\pi {r}_{0}^{2}/{\lambda}_{s}$. If

*r*

_{0}= 13

*μ*m (

*r*

_{0}= 50

*μ*m), one has

*L*= 0.1 cm (

_{Dif}*L*= 1.6 cm), which is much shorter than the medium length (5 cm) and thus the diffraction effect in the system is very significant. Due to the diffraction, the signal pulse spreads fast in the transverse directions and its amplitude decreases rapidly. As a result, one obtains, for

_{Dif}*r*

_{0}= 13

*μ*m (

*r*

_{0}= 50

*μ*m), the light memory efficiency and fidelity are respectively given by

*η*= 0.997% and

*ηJ*

^{2}= 0.786% (

*η*= 16.65% and

*ηJ*

^{2}= 14.39%), which are much lower than those obtained by using the kagome-structured HC-PCF considered above.

Note that in principle one can take bandgap-structured HC-PCFs [15–17] for generating the ultraslow weak-light soliton pulses and for realizing their storage and retrieval. However, the kagome-structured HC-PCFs, as indicated in Sec. 1, are better for implementing such tasks. For instance, using the core radius *r*_{0} = 5 *μ*m and the atomic density *𝒩* = 3.0 × 10^{9} cm^{−3} in the bandgap-structured HC-PCF given in Ref. [21], under the same control field one obtains the nonlinear coefficient describing Kerr effect [defined by (9)] *W* = (5.05 + 0.31*i*) × 10^{−16} cm^{−1} s^{2} and hence the nonlinearity length (describing the formation distance of the optical solitons) *L _{N}* = 8.62 cm. Nevertheless, using the core radius

*r*

_{0}= 13

*μ*m and

*𝒩*= 1.1 × 10

^{11}cm

^{−3}in the kagome-structured HC-PCF given in Ref. [33], one obtains

*W*= (2.17 + 0.44

*i*) × 10

^{−14}cm

^{−1}s

^{2}and hence

*L*= 0.3 cm. Thus, comparing with the bandgap-structured HC-PCF, the Kerr effect in the kagome-structured HC-PCF can be made much larger, and hence the formation distance of the optical soliton pulses can be made much smaller, which is desirable for generating the solitons with a lower power and a smaller system size. In addition, due to the longer

_{N}*L*and larger dephasing caused by larger transverse transit time, the memory efficiency of the soliton storage in the bandgap-structured HC-PCF is less than 22%, much lower than that in the kagome-structured HC-PCF (> 77%). Due to these reasons, the kagome-structured HC-PCF is chosen in our model. We expect that the result and method reported herein will be useful not only for the understanding of nonlinear optical property of gas-filled HC PCFs, but also for practical applications for manipulating optical information at weak-light level.

_{N}In summary, in this work we have investigated the formation and propagation of ultraslow weak-light solitons and their memory in cold atomic gas filled in a kagome-structured HC-PCF via EIT. We have shown that, due to the strong light-atom coupling provided by the transverse confinement of the HC-PCF, the EIT and thus the Kerr nonlinearity of such system can be largely enhanced. As a result, stable optical solitons with ultralow generation power down to nW and the propagation velocity slow to 10^{−5} *c* can be realized. We have demonstrated that the optical solitons obtained can not only be robust during propagation, but also be stored and retrieved with high memory efficiency and fidelity through the switching off and on of a control laser field. The energy to generate soliton signals in HC-PCF is very low thus it is possible to realize such light memory in weak nonlinear regime under single photon level. The results reported herein are promising for practical applications of optical information processing and transmission based on the ultraslow weak-light solitons and kagome-structured HC-PCFs.

## Appendix

## A. Electric-field mode functions of the kagome-structured HC-PCF

When in the absence of the atomic gas, the electric field **E** of the kagome-structured HC-PCF satisfies the Maxwell equation ∇^{2}**E** − (1/*c*^{2})*∂*^{2}**E**/*∂t*^{2} = [1/(*∊*_{0}*c*^{2})]*∂*^{2}**P**/*∂t*^{2}, here **P** is electric polarization intensity. Assuming that the frequency of **E** is far from the resonant frequency of the HC-PCF material, the electric polarization intensity has the from **P** = *∊*_{0}*χ*_{host}(*x*, *y*)**E**, where *χ*_{host}(*x*, *y*) is the electric susceptibility of the HC-PCF. Then the Maxwell equation is simplified into ∇^{2}**E** − (*n*^{2}/*c*^{2})*∂*^{2}**E**/*∂t*^{2} = 0, with *n* = *n*(*x*, *y*) = [1 + *χ*_{host}(*x*, *y*)]^{1/2} the refractive index of the HC-PCF without the atomic gas.

The general solution of the Maxwell equation can be written as

**e**

*,*

_{α}*ℰ*,

_{α}*q*(

_{α}*x*,

*y*), and

*β*are the polarization unit vector, mode amplitude, eigenfunction, and propagation constant for the mode with index

_{α}*α*. The eigenfunction

*q*satisfies the equation

_{α}*n*

_{eff}(

*ω*) of the kagome-structured HC-PCF as a function of frequency

*ω*. The solid (dashed) line is for the fundamental (first higher-order) mode; the blue, red, and green colors are for

*r*

_{0}= 13

*μ*m,

*r*

_{0}= 15

*μ*m,

*r*

_{0}= 17

*μ*m, respectively. In the calculation, we assume the fiber is made of silica and the pitch Λ

_{0}(defined by the distance between two adjacent holes) of the HC-PCF [see Fig. 1(a)] is fixed to be Λ

_{0}= 13

*μ*m. Form the figure we see that the first higher-order mode has a lower effective refractive index than the fundamental mode. Based on the effective refractive indexes obtained we can find the corresponding eigenfunctions.

We are interested in the fundamental mode, whose electric-field distribution in the radial direction (i.e. as a function of $r=\sqrt{{x}^{2}+{y}^{2}}$) is shown by the blue solid line in Fig. 1(c). When obtaining the figure, *r*_{0} = Λ_{0} = 13 *μ*m is chosen. The shape of the fundamental mode can be fitted well by the zeroth-order Bessel function *J*_{0}(2.405*r*/*r*_{0}) [the red dashed curve in Fig. 1(c)] [27]. One sees that the fundamental mode is well confined in the fiber core, as shown in the inset of Fig. 1(c).

## B. Equations of motion for the density-matrix elements

The equation of motion for the density-matrix elements *σ _{jl}* reads [32]

*d*= Δ

_{jl}*− Δ*

_{j}*+*

_{l}*iγ*, Δ

_{jl}_{3}=

*ω*−

_{s}*ω*

_{31}and Δ

_{2}=

*ω*

_{12}− (

*ω*−

_{c}*ω*) are respectively one- and two-photon detuings, with

_{s}*ω*= (

_{jl}*E*−

_{j}*E*)/

_{l}*ħ*with

*E*the eigenenergy of the state |

_{j}*j*〉. The dephasing rates are defined as ${\gamma}_{jl}=({\mathrm{\Gamma}}_{j}+{\mathrm{\Gamma}}_{l})/2+{\gamma}_{jl}^{\text{dep}}$, with ${\mathrm{\Gamma}}_{j}={\sum}_{{E}_{l}<{E}_{j}}{\Gamma}_{lj}$ representing the rate of spontaneous emission of the state |

*j*〉 to all lower energy states |

*l*〉, and ${\gamma}_{jl}^{\text{dep}}$ being the dephasing rate reflecting the loss of phase coherence between |

*j*〉 and |

*l*〉.

Under the slowly varying envelope approximation, the equation of motion for Ω* _{s}* reads

*n*

^{2}= 1 +

*χ*

_{host}(

*x*,

*y*) the refractive index of the HC-PCF when the atomic gas is absent. When the spatial distribution range of

*ζ*(

*x*,

*y*) is much smaller than in the spatial length of the signal pulse in the propagation (i.e.

*z*) direction, by using the definition

*σ*

_{31}=

*σ̃*

_{31}

*ζ*(

*x*,

*y*) and 〈

*ψ*〉 = ∬ |

*ζ*(

*x*,

*y*)|

^{2}

*ψ*(

*x*,

*y*)

*dxdy*/ ∬ |

*ζ*(

*x*,

*y*)|

^{2}

*dxdy*, Eq. (20) is reduced into Eq. (3) given in the main text.

## C. Second-order approximation solution

The second-order approximation solution reads ${\sigma}_{21}^{(2)}={a}_{21}^{(2)}i\frac{\partial}{\partial {t}_{1}}F{e}^{i\theta}$, ${a}_{31}^{(2)}={a}_{31}^{(2)}i\frac{\partial}{\partial {t}_{1}}F{e}^{i\theta}$, ${\sigma}_{jj}^{(2)}={a}_{jj}^{(2)}{\left|F\right|}^{2}{e}^{-2\overline{a}{z}_{2}}$ (*j* = 1, 2, 3), ${\sigma}_{32}^{(2)}={a}_{32}^{(2)}{\left|F\right|}^{2}{e}^{-2\overline{a}{z}_{2}}$, with *ā* = *∊*^{−2}Im[*K*(Δ*ω*)|_{Δω=0}] and

*D*= |

*ζ*(

*x*,

*y*)Ω

*|*

_{c}^{2}− (Δ

*ω*+

*d*

_{21})(Δ

*ω*+

*d*

_{31}).

## D. Effective Bloch equation

Making the transformation [40]

## Funding

National Natural Science Foundation of China (No. 11474099) and Science and Technology Project of Department of Education of Jiangxi Province (No. GJJ160576).

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