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 μm. 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 μm–100 μm 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_c, both are guided inside a kagome-structured HC-PCF with (hollow) core radius r₀ and pitch Λ₀ [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_α are the polarization unit vector, eigenfunction, and propagation constant for the mode with index 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 r₀ = Λ₀ = 13 μm 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, Δ₃ and Δ₂ are respectively one- and two-photon detunings, Γ_jl (j = 1, 2; l = 3) denotes the spontaneous emission rate from |l〉 to |j〉; ω_s and Ω_s [ω_c 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 z direction, with the form $\mathbf{E}={\mathbf{E}}_s+{\mathbf{E}}_c={\sum }_{l=s,c}{\mathbf{e}}_l\sqrt{\frac{\hslash {\omega }_l}{2{∊}_0{V}_l^{\text{eff}}}}{ℰ}_l(z,t)q_l(x,y)e^{i\left[{\beta }_{l}z-{\omega }_{l}t\right]}+\text{c}.\text{c}.$, where β_l ≡ β(ω_l), $V_l^{\text{eff}}\equiv L_z\iint {\left|q_l(x,y)\right|}^{2}dxdy$ is the effective mode volume for the mode ω = ω_l, L_z is the fiber length, the subscript s (c) denotes the signal (control) field, and c.c. represents complex conjugate.

 

Fig. 1 (a) Kagome-structured HC-PCF with core radius r₀ and pitch Λ₀. (b) Atoms of a lambda-type three-level configuration are filled within the hollow core of the fiber, and initially prepared in the metastable |1〉 for suppressing four-wave mixing effect. (c) Numerical result of the electric-field distribution of the normalized fundamental-mode amplitude as a function of radius coordinate r in the HC-PCF made of silica when the atoms are absent. (d) Absorption spectrum Im(K) of the signal field as a function of Δω for different core radius r₀. The red, blue and green lines are for r₀ = 13 μm, 17 μm, and 100 μm, respectively. The other notations in the figure are explained in the text.

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We choose a cesium atomic gas with the energy levels selected as |1〉 = |6²S_1/2, F = 4〉, |2〉 = |6²S_1/2, F = 3〉, and |3〉 = |6²P_3/2, F = 4〉. For such level configuration the wavelengths of the signal and control fields are approximately equal (i.e. λ_s ≈ λ_c = 852 nm) [28], which means 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) = J₀(kr) when r ≤ r₀ and q(x, y) = 0 when r > r₀ with k ≈ 2.405/r₀ [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 μm [31], which is chosen in the following discussion). Then the electric field can be expressed as

The equation of motion for Ω_s can be obtained by the Maxwell equation ∇²E − (1/c²)∂²E/∂t² = [1/(∊₀c²)]∂²P_p/∂t², with the electric polarization intensity given by P_p = P_host + 𝒩p₁₃σ₃₁ exp[i(β_sz − ω_st)] + c.c., where 𝒩 is the atomic density, P_host = ∊₀χ_hostE 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

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 σ₁₁ = 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 σ₁₁ = 1, σ_j1 ∼ exp(iθ) (j = 2, 3), Ω_s ∼ exp(iθ), and σ₂₂ = σ₃₃ = σ₃₂ = 0, with θ = K(Δω)z − Δωt [34]. The linear dispersion relation is given by

When plotting Fig. 1(d), the system parameters are chosen as Ω_c = 1.0 × 10⁷ s⁻¹, Δ₂ = Δ₃ = 0, |p₁₃| ≃ |p₂₃| = 3.8 × 10⁻²⁷ C · cm [28], 𝒩 = 2.0 × 10¹⁰ cm⁻³, R_⊥ = 100 μm, Γ₃ ≃ 2γ₃₁ = π × 5.23 MHz, γ₃₁ = γ₃₂. 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 ${\sigma }_{jk}={\sum }_{l=0}^{\infty }{∊}^l{\sigma }_{jk}^{(l)}$ (j, k =1, 2, 3; ${\sigma }_{jk}^{(0)}={\delta }_{j1}{\delta }_{k1}$), and ${\mathrm{\Omega }}_s={\sum }_{l=1}^{\infty }{∊}^l{\mathrm{\Omega }}_s^{(l)}$. 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 = ∊^lz (l = 0, 1, 2, · · ·) and t_l = ∊^lt (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

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 K₂ ≡ ∂²K/∂Δω² 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

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

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 + ∂²u/∂σ² + 2|u|²u = 0, where u = U/U₀, s = −z/(2L_D), and σ = τ/τ₀, with $L_D={\tau }_0^2/{\tilde{K}}_2$ (τ₀ is the pulse duration of the signal field) the typical dispersion length of the system. Note that we have taken L_N = L_D 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 K₂ and W means taking the real part, K̃₂ = Re(K₂) and W̃ = Re(W). A single-soliton solution reads u(s, σ) = 2ϑsech[2ϑ(σ − σ₀ + 4δs)] exp [−2iδσ − 4i(δ² − ϑ²)s − iϕ₀], where ϑ, σ₀, δ, and ϕ₀ 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, σ₀ = δ = ϕ₀ = 0. Then, when returning to original variables, we obtain the expression of the signal field E_s = e_sE_s, with

We now give a set of system parameter for the formation of the optical soliton given above. By taking Ω_c = 3.6 × 10⁶ s⁻¹, τ₀ = 1.0 × 10⁻⁷ s, Δ₂ = 2.5 × 10⁷ s⁻¹, Δ₃ = 3.75 × 10⁸ s⁻¹, 𝒩 = 2.0 × 10¹⁰ cm⁻³ and other parameters are the same as those given in the above text, we obtain (evaluated at Δω = 0) K₀ = (−3.23 + 0.09i) cm⁻¹, K₁ = (4.05 − 0.33i) × 10⁻⁸ cm⁻¹ s, K₂ = (−3.45 + 0.39i) × 10⁻¹⁵ cm⁻¹ s², W = (−5.38 + 0.16i) × 10⁻¹³ cm⁻¹ s², U₀ = 0.8×10⁶ s⁻¹, and L_A = 1/Im(K₀) = 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_z, where e_z is the unit vector in the propagation direction [40]. At leading order the corresponding magnetic field of the soliton, H_s, is transverse and in the e_s × e_z direction. Thus if e_s = e_y then H_s = e_xH_s with H_s ≈ n_eff∊₀cE_s. After averaging over the carrier-wave period, we obtain P = P_max sech² [(t − z/Ṽ_g)/τ₀], where P_max = 2∊₀cn_eff(ħU₀/|p₁₃|)² ∬ |ζ(x, y)|²dxdy, 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)τ₀| as a function of z/L_D and t/τ₀. The comparison of waveshapes at z = 0 and at z = 2L_D 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_s(0, τ)τ₀ = 0.08 sech(t/τ₀). 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 = 2L_D, the soliton can keep its shape well for a propagation distance up to 2L_D = 5.8 cm.

 

Fig. 2 (a) The propagation of the ultraslow optical soliton, with |Ω_s(z, τ)τ₀| as a function of z/L_D and t/τ₀ (L_D is dispersion length and τ₀ is pulse duration). (b) The comparison of waveshapes between z = 0 (blue curve) and z = 2L_D (red curve).

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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

We first consider the kagome-structured HC-PCF with a relatively small core radius r₀ (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 r₀ = Λ₀ = 13 μm, where |Ω_sτ₀| 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_cτ₀| 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 $t=T_c^{\text{on}}=20\hspace{0.17em}{\tau }_0$.

 

Fig. 3 Numerical results on the storage and retrieval of an optical soliton pulse and a linear optical pulse for r₀ = Λ₀ = 13 μm. (a) [(b)] Evolution of the dimensionless half Rabi frequency |Ω_sτ₀| of the signal field (atomic coherence σ̃₂₁) as a function of time t for different propagation distance z in the soliton regime. (c) [(d)] Evolution of the dimensionless half Rabi frequency |Ω_sτ₀| of the signal field (atomic coherence σ̃₂₁) as a function of t for different z in the linear regime. In each panel, the black, red, green, yellow, sky blue, and blue solid lines are for z=0, 1, 2, 3, 4 and 5 cm, respectively; the purple solid line represents the dimensionless half Rabi frequency |Ω_cτ₀| of the control field.

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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 |σ̃₂₁| as a function of z and t. One sees that σ̃₂₁ is nonzero during the switch-off of the control field. Since the signal pulse is stored in the form of atomic coherence σ̃₂₁ when the control field is switched off and is retained until the control field is switched on again, the atomic coherence σ̃₂₁ 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.,

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_sτ₀| 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 × 10⁴ and its width is more than 2.0τ₀. 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 ηJ² = 62.9% for L_z = 5 cm, lower than the optical soliton memory shown in Fig. 3(a). The atomic coherence |σ̃₂₁| 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 r₀ when the other system parameters are fixed. Fig. 4 shows the result of η as a function of the fiber length L_z 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 r₀ taking 13, 15, and 17 μm (the pitch Λ₀ is fixed to be 13 μm). From the figure we can obtain the following conclusions: (i) For any core radius r₀, the memory efficiency η grows rapidly in the small L_z region (i.e. the region on the left side of its peak value), then as L_z increases it reaches to the peak value, and finally drops down slowly in the large L_z region (i.e. the region on the right side of the peak value); (ii) In the large (small) L_z region, the smaller the core radius r₀ 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_z 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 L_z 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.

 

Fig. 4 Memory efficiency η of the optical soliton pulse as a function of the fiber length L_z. Solid lines are fitted curves based on numerical calculation. The blue, red and green solid lines are for the fiber core radius r₀=13, 15 and 17 μm, respectively. Dashed-dotted lines for each core radius are the memory efficiencies when a microwave field is coupled to the two lower atomic levels [i.e. |1〉 and |2〉 in Fig. 1(b)].

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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 σ₂₁ [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 r₀.

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 r₀ of kagome-structured HC-PCF has been extended to 50 μm [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 r₀ = 50 μm and the pitch Λ₀ = 25 μm. 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, 3.5τ₀, 15τ₀, 25τ₀, and 28.5τ₀ 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 ηJ² = 71.2% for L_z = 5 cm, respectively.

 

Fig. 5 (a) Storage and retrieval of the optical soliton pulse in the HC-PCF with core radius r₀ = 50 μm and pitch Λ₀ = 25 μm. The subplots from (i) to (v) give the intensity distribution |E_s|² of the soliton pulse in the (x, y) plane when propagating to the distance z = 0, 1, 2.5, 4, and 5 cm, respectively. t = 0, 3.5τ₀, 15τ₀, 25τ₀, and 28.5τ₀ shown respectively in (i) to (v) are the times when soliton’s peak arrives at the positions z = 0, 1, 2.5, 4, and 5 cm, respectively. (b) Storage and retrieval of a linear optical pulse in the same fiber.

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Fig. 6 Numerical result for the effective refractive index n_eff of the kagome-structured HC-PCF as a function of frequency ω with pitch Λ₀ = 13 μm and three different core radius r₀. The solid (dashed) line is for the fundamental (first higher-order) mode; the blue, red, and green colors are for r₀ = 13 μm, r₀ = 15 μm, r₀ = 17 μm, respectively.

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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 (ηJ² = 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) = J₀(kr), the characteristic diffraction length of the system reads $L_{\mathit{Dif}}=\sqrt{3}\pi r_0^2/{\lambda }_s$. If r₀ = 13 μm (r₀ = 50 μm), one has L_Dif = 0.1 cm (L_Dif = 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 r₀ = 13 μm (r₀ = 50 μm), the light memory efficiency and fidelity are respectively given by η = 0.997% and ηJ² = 0.786% (η = 16.65% and ηJ² = 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 r₀ = 5 μm and the atomic density 𝒩 = 3.0 × 10⁹ 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 + 0.31i) × 10⁻¹⁶ cm⁻¹ s² and hence the nonlinearity length (describing the formation distance of the optical solitons) L_N = 8.62 cm. Nevertheless, using the core radius r₀ = 13 μm and 𝒩 = 1.1 × 10¹¹ cm⁻³ in the kagome-structured HC-PCF given in Ref. [33], one obtains W = (2.17 + 0.44i) × 10⁻¹⁴ cm⁻¹ s² and hence L_N = 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 L_N 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.

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.