1. Introduction

The techniques for cooling and trapping atoms were initially achieved by utilizing the state-of-the-art laser technologies in the 1980s [1]. In particular, the rapid development of laser technologies has not only helped generate ultracold atoms including Bose-Einstein condensates (BECs) [1] but also contributed to the control and manipulation of their dynamics in optical lattices formed by counterpropagating laser beams [2]. In past decades, one of the great achievements of BECs loaded onto optical lattices was the realization of matter-wave gap solitons under self-defocusing nonlinearity induced by atom-atom interactions [3]. In recent years, much attention has been paid to revealing the formation, property and stability of gap solitons in various physical systems including BEC and optics [4–11]. For the latter, linear periodic potentials such as photonic crystals and lattices [4,7], prefabricated or wrote optically, inside which the optical gap solitons are localized, had been implemented experimentally [12–15]. Nonlinear lattices [6], the nonlinear counterparts of periodic potentials, can be induced, respectively in BECs and optics, via Feshbach resonance mediated by spatially periodic fields [16] and properly made nonlinear photonic crystals [4]. The nonlinear lattices [6,17–19] and the combined linear-nonlinear lattices [20,21] have recently been used to stabilize different kinds of solitons.

Self-induced transparency solitons [22,23] and gap ones [24–26] had been well studied respectively in uniform and periodic resonantly atomic systems decades ago. Coherent atomic systems in the form of BECs whose atoms are prepared under electromagnetically induced transparency (EIT) condition with all-optical ways have recently been a hot topic in soliton studies [27–41]. EIT is a typical quantum destructive interference effect emerging in the resonant multilevel systems with some suitable laser fields, which possesses many striking scientific properties, i.e., the large elimination of optical absorption [42], the obvious reduction of group velocity [43,44], a remarkable enhancement of Kerr nonlinearity with very low power light fields [45,46]. The study of EIT has led to a surge of research on various frontier domains due to its rich physical properties and important practical applications [47–51].

It is relevant to point out that the self-induced transparency gap solitons in a resonantly absorbing Bragg reflector with EIT turning on had also been investigated by Kurizki and his colleagues years ago [52]. In addition, optical solitons have been predicted in a resonant three-level atomic system with a standing-wave control field which forms an optical lattice and excites EIT [53–55], and in optical waveguides controlled by EIT [56], while the literal report on gap solitons in the coherent atomic system via EIT have yet to be explored, i.e., the stability and instability regions of localized gap modes in the band-gap spectrum. Theoretical and experimental works have demonstrated the ways for manipulating light pulses via dynamically controlled EIT-induced photonic band gap in coherently prepared atomic gases [57–60]. Various effects including solitons have been widely studied in multilevel atomic systems with electromagnetically induced lattices formed by EIT in recent years [31,34,61–63], while gap solitons are still missing. It is thus the main objective of this paper to survey localized gap modes in such coherent atomic ensembles with the EIT on.

In this paper, an one-dimensional (1D) coherent atomic system consisting of a $\Lambda$-type three-level atomic gases that are excited under EIT condition and trapped by an optical lattice formed by a pair of counterpropagating far-detuned Stark laser fields is proposed as a new platform to generate localized gap modes. The physical equation describing such system is in the framework of nonlinear Schrödinger equation, formulated and derived from the Maxwell-Bloch equations and the method of multiple scales [29,31] under suitable conditions. The model supports two types of localized gap modes, fundamental gap solitons and dipole ones, in the first two finite band gaps of the underlying linear spectrum. Both localized gap modes can be constructed as on-site and off-site modes, with their central profiles placing respectively into the maximum and minimum values of the optical lattice. The property and (in)stability of the predicted gap modes are evaluated by linear-stability analysis and direct perturbed simulations. The proposed physical scheme and the predicted gap modes therein can enlarge the nonlinear spectrum of coherent atomic gases and open up a new avenue for implications including optical communication and information processing.

The rest of the article is arranged as follows. The physical model under study is presented in Sec. 2. The nonlinear Schrödinger equation of the probe field is derived from Maxwell-Bloch equations, and its numerical schemes, including the linear-stability analysis and direct perturbed simulations, are introduced in Sec. 3. The existence and stability properties of localized gap modes in forms of fundamental gap solitons and dipole ones are presented numerically and discussed with an analysis in Sec. 4. A summary of this article is finally made in Sec. 5.

2. Physical model

We consider a lifetime-broadened three-level atomic gas in a $\Lambda$-type configuration interacting resonantly with two laser fields, i.e., the pulsed probe field $\textbf {E}_p$ couples the transitions $|1\rangle \leftrightarrow |3\rangle$ and the strong continuous-wave control field $\textbf {E}_c$ drives the transition $|2\rangle \leftrightarrow |3\rangle$, as depicted in Fig. 1(a). $\Gamma _{13}$ and $\Gamma _{23}$ are the spontaneous emission decay rate from $|3\rangle$ to $|1\rangle$ and $|3\rangle$ to $|2\rangle$, respectively. $\Delta _2$ and $\Delta _3$ are the two- and one-photon detunings, respectively. The atoms are firstly prepared in the ground state $|1\rangle$, and cooled to an extreme low temperature to restrain the relevant center-of-mass motion. Under these conditions, the physical system of such an atomic ensemble interacting with two laser fields constitutes the above-mentioned EIT where the absorption of the probe field can be greatly suppressed due to the quantum interference effect induced by the control field [42].

 

Fig. 1. (a) Theoretical scheme of a $\Lambda$-type three-level atomic system for setting on EIT, whose excitation mechanism is described in text. $\Gamma _{13}$ ($\Gamma _{23}$) is the spontaneous emission decay rate from $|3\rangle$ to $|1\rangle$ ($|3\rangle$ to $|2\rangle$). $\Delta _j$ ($j=2,\,3$) is the detuning. (b) Possible experimental arrangement of laser beams geometry. The weak probe and strong control fields (with angular frequency $\omega _p$ and $\omega _c$, respectively) propagate along the $z$ direction. The (purple) thick arrows denote a pair of counter-propagating Stark fields (both with angular frequency $\omega _s$), which form standing waves called optical lattice.

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Both the probe and control laser fields are assumed to propagate along the $z$ direction [see Fig. 1(b)], thus the electric-field vector can be written as $\textbf {E}=\textbf {E}_p+\textbf {E}_c=\hat {\mathbf {e}}_{p}{\cal E}_{p} e^{i(k_{p} z-\omega _{p} t)}+\hat {\mathbf {e}}_{c}{\cal E}_{c} e^{i(k_{c} z-\omega _{c} t)}+\textrm {c.c.}$. Here $\hat {\mathbf {e}}_{p}$ ($\hat {\mathbf {e}}_{c}$) is the unit vector of the probe (control) field with the envelope ${\cal E}_p$ (${\cal E}_c$), $\omega _p$ ($\omega _c$) is the angular frequency of the probe (control) field, and $k_{p}=\omega _{p}/c$ ($k_c=\omega _c/c$) is the wavenumber of the probe (control) field before entering the atomic gas. Furthermore, we apply two counter-propagating far-detuned laser fields (Stark fields), which have the same angular frequency $\omega _s$, to the EIT medium, to generate an optical lattice, onto which the optical localized gap modes can be loaded; See the possible experimental arrangement in Fig. 1(b). The Stark fields follow the form

Under the electric-dipole and rotating-wave approximations, the Hamiltonian of the system in the interaction picture is $\hat {\cal H}=-\sum _{j=1}^{3}\hbar \Delta _j^\prime |j\rangle \langle j|-\hbar \,\left [\Omega _c|3\rangle \langle 2|+\Omega _{p}|3\rangle \langle 1|+\textrm {H.c.}\right ],$ with $\Delta _j^\prime =\Delta _j+\frac {1}{2}\alpha _{j1}|E_s(x)|^2$ and $\alpha _{jl}=(\alpha _{j}-\alpha _{l})/\hbar$. Here $\Delta _1=0$, $\Delta _2=\omega _{p}-\omega _{c}-\omega _{21}$, and $\Delta _3=\omega _{p}-\omega _{31}$ are the detunings, where $\omega _{jl}=(E_j-E_l)/\hbar$ with $E_j$ being the eigen energy of the state $|j\rangle$. $\Omega _c=(\mathbf {p}_{23}\cdot {\hat {\mathbf {e}}}_c){\cal E}_c/\hbar$ and $\Omega _{p}=(\mathbf {p}_{13}\cdot {\hat {\mathbf {e}}}_{p}){\cal E}_{p}/\hbar$ are half Rabi frequencies of the control and probe fields, where $\textbf {p}_{jl}$ is the electric dipole matrix element related to the transition from $|j\rangle$ to $|l\rangle$. Thus the equation of motion for density matrix $\sigma$ in the interaction picture is given by

Utilizing the slowly varying envelope approximation, the Maxwell equation for the probe-field Rabi frequency $\Omega _{p}$ is described as

3. Nonlinear Schrödinger equation and numerical schemes

3.1 Derivation of a nonlinear Schrödinger equation

To investigate the EIT-based control of the nonlinear evolution properties of the probe field in the system, we have derived the relevant nonlinear envelope equation in the framework of Maxwell-Bloch equations by adopting the standard multiple scales method [29,31]. The asymptotic expansion has the following form: $\sigma _{jl}=\sum _{m=0}^{\infty }\epsilon ^m\sigma _{jl}^{(m)}$, with $\sigma _{jl}^{(0)}=\delta _{j1}\delta _{l1}$, $\Omega _{p}=\sum _{m=1}^{\infty }\epsilon ^m \Omega _{p}^{(m)}$, Here $\epsilon$ is the dimensionless small parameter characterizing the typical amplitude of the probe field. Note that all the quantities on the right-hand side of the expansion depend on the multi-scale variables $x_1=\epsilon x$, $z_{m}=\epsilon ^{m}z$, and $t_{m}=\epsilon ^{m}t$ ($m=0,\,2$). In addition, the space-dependent amplitude of Stark field [given by Eq. (1)] is assumed as $E_s=\epsilon E_s^{(1)}$. Thus $d_{jl}=d_{jl}^{(0)}+\epsilon ^{2}d_{jl}^{(2)}$, with $d_{jl}^{(0)}=\Delta _j-\Delta _l+i\gamma _{jl}$ and $d_{jl}^{(2)}= \frac {\alpha _{jl}}{2}|E_s^{(1)}|^2$. Substituting the expansion into Maxwell-Bloch equations and comparing the coefficients of $\epsilon ^m$ ($m=1,\,2,\,3,\,\ldots$), we obtain a set of linear but inhomogeneous equations for $\sigma _{jl}^{(m)}$ and $\Omega _{p}^{(m)}$, which can be solved order by order.

When carrying out the solution up to the third order, the Kerr nonlinearity plays an important role, and hence the propagation of the envelope $\mathcal {F}$ of the probe field is described by the nonlinear equation

Substituting the original variables to Eq. (4), one can obtain a dimensionless form

Through choosing other parameters $\mathcal {N}_a\approx 3.69\times 10^{10}$ cm$^{-3}$, $\Omega _c=7.5\times 10^{7}$ Hz, $\Delta _2=1.1\times 10^7$ Hz, and $\Delta _3=2.5\times 10^8$ Hz, the complex coefficients in Eq. (4) are $\mathcal {K}=(3.83+0.043i)$ cm$^{-1}$, $W_{1}=-(7.34+0.096i)\times 10^{-15}$ cm$^{-1}$ s$^{2}$, and $W_{2}=(5.62+0.13i)\times 10^{-9}$ cm V$^{-2}$, where the imaginary parts of these quantities are indeed much smaller than their corresponding real parts. Furthermore, taking $u_0=8.68\times 10^{6}$ Hz, $\tau _0=1.26\,\mu$s, $R=48$ $\mu$m, and $E_0=9.92\times 10^{3}$ V cm$^{-1}$, we obtain ${g}=1-0.02i$, ${g}_{1}=-(1+0.01i)$, ${g}_{2}=1+0.02i$, and $\mathcal {A}=0.077$. In order to obtain nonlinear localized gap modes of the system, we assume the Stark fields have the form of sinusoidal function, i.e., $E_s(\xi )=E_{s0}\sin (\xi )$ ($E_{s0}$ is a real constant). Therefore, Eq. (6) can be reduced into

3.2 Numerical schemes for stability analysis

We search stationary solution of field amplitude $v=U(\xi ) \exp (ib \tau )$ with $b$ being propagation constant, then Eq. (7) becomes

 

Fig. 2. (a) Band-gap structure of linear spectrum $b(K)$ characterized by the momentum $K$, which is induced by the EIT-based optical lattice $V_{\textrm {OL}}=-c_0 \sin ^2 (\xi )$ with strength $c_0=6$. The 1st BG and 2nd BG hereafter represent the first and second band gaps, respectively. (b) Power $P$ versus propagation constant $b$ for on-site gap solitons (red solid line) and off-site ones (blue dashed line). Panels (c) and (d) represent respectively the real parts of maximum eigenvalues, expressed by Re($\lambda$) as a function of $b$, for the corresponding off-site and on-site gap solitons. Profiles of gap solitons marked by dotted points (A1, A2, A3) and (B1, B2, B3) will be displayed below.

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The linear stability is a key to the question of whether the stationary solutions can be stable localized gap modes. And so, we take the perturbed amplitude as $v=[U+p(\xi )\exp {(\lambda \tau )}+q^{\ast }(\xi )\exp {(\lambda ^{\ast } \tau )}]\exp {(ib \tau )}$, here $U$ is the undisturbed field amplitude found from Eq. (8), and $p(\xi )$ and $q^{\ast }(\xi )$ are the small background perturbations at eigenvalue $\lambda$. Substituting such expression into Eq. (7) leads to the following linear eigenvalue problem:

4. Numerical results and discussions

Having obtained the band-gap structure of the considered physical model above, we now turn to study the existence of optical localized gap modes within the given finite band gaps, with an emphasis on their property and stability in the first two band gaps. Here the localized gap modes of two types, the fundamental gap solitons and dipole ones, are discussed. As usual, the optical gap solitons can be classified into off-site and on-site solitons, with their central parts populating, respectively, in the minimum and maximum values of the optical lattice.

4.1 Off-site and on-site optical gap solitons

As shown in Fig. 2(b), where the dependence between the power $P$ and propagation constant $b$ for the on-site solitons (red solid line) and off-site (blue dashed line) fundamental optical gap solitons pinned in the relevant first and the second finite band gaps has been accumulated. One can obtain that the power $P$ increases with the decrease of propagation constant $b$, showing the obeying of the well-known inverted Vakhitov-Kolokolov stability criterion $dP/db<0$, which is a necessary condition for the stability of gap solitons supported by self-repulsive (self-defocusing) nonlinearity [20,67]. Insight scrutinization into the stability property of both gap solitons employing the linear stability analysis based on eigenvalue Eq. (9), we have obtained the relation of the maximal real part of eigenvalues Re$(\lambda )$ as a function of $b$ in Figs. 2(c) and 2(d) for the off-site and on-site localized gap modes, suggesting that both modes are robustly stable physical objects in the first finite band gap [recall that Re$(\lambda )=0$ as mentioned above], and in the second gap, however, the stability region is very limit for the former and the latter only has instability region because of Re$(\lambda )\neq 0$.

Typical examples of the 1D off-site and on-site optical gap solitons, with different values of propagation constant $b$ in the $U(\xi )$ profile, are respectively depicted in Figs. 3(a, b, c) and Figs. 3(d, e, f). For all the gap solitons lying in the first finite band gap [Figs. 3(a, d)], they exhibit a cusplike shape, i.e., they are accompanied by a weak modulation and such a modulation (of the gap solitons) increases near the gap edge [Figs. 3(b, e)] and in the second gap [Figs. 3(c, f)], resembling the same feature of gap solitons in other periodic physical systems [9–11,21]. In addition to the linear stability analysis, the stability properties of the fundamental gap solitons have also been examined in direct perturbed simulation using Eq. (7), and an agreement has been obtained between both methods, exemplified by the perturbed evolutions in Fig. 4 of these six gap solitons, with their eigenvalues being marked by points (A1, A2, A3) and (B1, B2, B3) in Figs. 2(c) and 2(d). The stable evolutions of perturbed gap solitons of both off-site and on-site types, which keep their profile over long time, are displayed in Figs. 4(a, c, d); their unstable counterparts, however, subject to decay and finally immerse into the background noise as radiating waves during evolution.

 

Fig. 3. Characteristic examples of off-site gap solitons (left column) and on-site ones (right column) with different propagation constant $b$. (a, d): $b=3.0$; (b, e): $b=1.78$; (c, f): $b=0.8$. The gap solitons in panels (c, f), marked by (A3, B3), lie at the second band gap, while the other four points (A1, A2, B1, B2) belong to the first band gap. The direct perturbed dynamics of these six gap modes will be shown in the following figure. The black dashed lines (in each panel), here and below, denote the shape of the optical lattice.

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Fig. 4. Left column: Direct perturbed evolutions of stable off-site gap solitons (a, c) and unstable counterpart (b). Right column: Direct perturbed evolutions of unstable on-site gap solitons (e, f) and stable one (d). (a, d): $b=3.0$; (b, e): $b=1.78$; (c, f): $b=0.8$. $\xi \in [-20,\,20]$ for all panels.

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The cases we are talking about are so far limited to the strength of optical lattice $c_0=6$, revealing that there are no any stable on-site gap solitons in the second band gap. This may be explained by the fact that, a stronger Bragg scattering (in such gap), which impacts deeply on the double humps of on-site gap solitons in the evolution process, turn to destabilize rather them stabilize them. This can also be understood by the phenomenon that in the second band gap more optical waves are participating with the multiple constructive interference (Bragg scattering) so as to form gap solitons in the nonlinear regime, recalling that the power $P$ increases almost linearly through the first band gap to the second gap. We thus conjecture that an enhancement of the optical lattice strength $c_0$ may help to create stable on-site gap solitons in second gap. Given that $c_0=12$, whose band-gap structure of the underlying linear spectrum is shown in Fig. 5(a), suggesting wider gaps in both the first and second band gap as compared with the case of $c_0=6$ in Fig. 2(a). Our systematic numerical study, as those done in Fig. 2, has produced the dependence $P(b)$ for both the off-site and on-site gap solitons in Fig. 5(b), and their relevant eigenvalues via linear-stability analysis in Fig. 5(c) and in Fig. 5(d) respectively, demonstrating clearly that, in the second band gap, the on-site gap solitons are indeed stable and the off-site ones have a larger stability region. Direct simulations of the perturbed evolution of both gap solitons, stable and unstable cases, show that they share the similarities of those in Fig. 4, and it is unnecessary to show them here accordingly.

 

Fig. 5. (a) Band-gap structure of the EIT-based optical lattice $V_{\textrm {OL}}=-c_0 \sin ^2(\xi )$ with a larger strength $c_0=12$. (b) Power $P$ as a function of propagation constant $b$ for on-site gap solitons (red solid line) and off-site ones (blue dashed line). Panels (c) and (d) represent respectively the real parts of maximum eigenvalues, expressed by Re($\lambda$) versus $b$, for the corresponding off-site and on-site gap solitons.

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4.2 Dipole gap modes

The physical model may also upholds dipole gap modes, i.e., localized modes of two fundamental gap solitons arranged in a spatial distance $\bigtriangleup d$ and with opposite signs. Provided that such a distance $\bigtriangleup d$ doubling the period of the optical lattice, that is $\bigtriangleup d=2\pi$, the dipole gap modes can be stable objects, according to our systematic numerical calculations. Illustrated in Figs. 6(a) and 6(b) are, respectively, the power $P$ of off-site dipole gap localized modes versus propagation constant $b$ for the above-mentioned two different depths of the optical lattice depth $c_0=12$ and $c_0=6$. It is to observe once again that the dependence $P(b)$ follows the inverted Vakhitov-Kolokolov stability criterion $dP/db<0$ ($P$ increases when decreasing $b$) from the first band gap to the second gap, recall that which is a necessary but not sufficient condition for stable gap modes [20,67]. The corresponding relation between maximum real eigenvalues Re$(\lambda )$ and propagation constant $b$, obtained with linear-stability analysis in eigenvalue problem Eq. (9), are displayed in Figs. 6(c) and 6(d). The latter shows that the dipole gap solitons can not be stabilized in the second band gap at $c_0=6$, whereas their stable fundamental gap solitons are confined to a narrow region [see Fig. 2(c)]. Examples of such dipole gap solitons populating in the first and second band gaps for both cases [$c_0=12$ and $6$] are displayed in Figs. 6(e) and 6(f).

 

Fig. 6. Top: Power $P$ versus propagation constant $b$ of off-site dipole gap solitons under different optical lattice depth $c_0$: (a) $c_0=12$ and (b) $c_0=6$. Middle: The corresponding real part of maximum eigenvalue, Re($\lambda$), as a function of $b$ in (c) and (d). Below: Profiles of such dipole gap modes corresponding to the marked points (C1, D1) in the first band gap (e) and (C2, D2) in the second band gap (f). The relevant propagation constants are $b=8.0$ and $b=2.3$ for the marked points (C1, C2), and $b=3.5$ and $b=0.9$ for the marked points (D1, D2), respectively.

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We have found that the stable dipole gap modes keep their coherence during long time evolution, as depicted by Figs. 7(a) and 7(c). On the contrary, the unstable dipole gap solitons undergo weak oscillation initially and quickly transform themselves into radiating waves, see examples in Figs. 7(b) and 7(d). These systematic simulation results match up well with the relevant predictions from linear stability analysis in Figs. 6(c) and 6(d), demonstrating the efficiency of the numerical methods we adopted.

 

Fig. 7. Direct perturbed evolutions of stable off-site dipole gap solitons (a, c) and unstable ones (b, d). Depth of the optical lattices $c_0=12$ for (a, b) and $c_0=6$ for (c, d); propagation constant $b=8.0$, $b=2.3$, $b=3.5$ and $b=0.9$ in a sequential order for panels (a$\sim$d). $\xi \in [-20,\,20]$ for all.

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5. Conclusion

To summarize, we have presented a physical system consisting of a resonant atomic ensemble, which is in a $\Lambda$-type three-level configuration, interacting with two laser fields represented as control and probe fields to turn EIT on, and controlled by an optical lattice induced by a pair of counter-propagating far-detuned Stark laser fields. Using the typical Maxwell-Bloch equations and the multiple scales method, the governing model (equation) for the probe field envelope has been derived in the framework of nonlinear Schrödinger equation which, is found to support localized gap modes of two types, i.e., the fundamental gap solitons and dipole gap ones, in the first and second finite band gaps. Both gap modes can be constructed as on-site gap solitons and off-site ones, with their central humps residing on the maximum and minimum positions (values) of the optical lattice. Our systematic simulations basing on linear-stability analysis and direct perturbed simulations demonstrate the (in)stability regions of both localized gap modes in the respective linear band-gap spectrum. The theoretical results predicted here may enrich insights into the nonlinear properties of coherent atomic gases and open up a new door for implications ranging from optical communication to optical information processing.

An extension of this work is to consider soliton composites (alias multiple-peaked solitons) consisted of several or many gap solitons with equal spacing between each. It is also an interesting issue for the investigation of gap solitons and vortices in two-dimensional space.