1. Introduction

Realization of strong nonlinear interactions between dilute atomic ensembles and weak optical fields have attracted growing attentions in recent years due to diverse applications in quantum computation and quantum information, high-precision sensing, simulation of many-body physics, and so on [1,2]. Particularly, light propagation through near-resonance atomic gases with the use of coherent multi-photon process (such as Raman process and electromagnetically induced transparency) has been proved to be a very promising approach for realizing the strong nonlinear interactions between photons [3–11]. In addition, under the excitation of a laser field, collective atomic recoil motion, also known as matter-wave superradiance, may occur [12–14]. The collective center-of-mass recoil motion of ultracold atoms is an intriguing process, where a group of atoms in the same electronic state can recoil coherently; the recoiled atoms can mediate the conversion of atomic kinetic energy into radiation, resulting in the generation of scattered laser fields.

In early experiments on the matter-wave superradiance [15–17], an elongated Bose-Einstein condensate (BEC) was used, and a long-duration pump laser field was sent along the short transverse direction of the BEC. In such a scheme, the scattered laser field is dominated by the axial or end-fire modes, while the recoiled atoms from the BEC form special fan-like or X-shaped scattering patterns in the momentum space (alias as the side-mode distribution) depending on the duration and strength of the incident laser pulse. Since then, comprehensive studies for understanding the underlying physics of these experimental observations were carried out, providing further insights into the features of the atomic momentum distribution [18–24] and the coupled dynamics of matter waves and laser fields [25,26]. Meanwhile, superradiant scattering and polariton formation from an elongated BEC with the incident laser along the long axis was also investigated, where the asymmetric backward scattering pattern and multi-matter-optical wave-mixing process were found [27–31]. Recent experimental and theoretical works also revealed that matter-wave superradiance can be achieved in degenerate fermion gases [30]. In addition to its fundamental role for atom optics [32] and related research fields, the study of matter-wave superradiance opens possibilities for many practical applications such as matter-wave amplifiers, highly sensitive matter-wave interferometers, quantum information processors based on ultracold atoms, and so on. Up to now, however, the collective atomic recoil motion has been demonstrated for ultracold atoms with only short-range contact interactions.

On the other hand, much attention has been paid to the study of ultracold Rydberg atomic gases [33,34] in recent years, which exhibit strong and long-range interactions. Due to the large electric-dipole moment of Rydberg states, the interaction between two Rydberg atoms can reach several megahertz at their separation of several micrometers [35,36], which may generate giant nonlocal optical Kerr nonlinearity when such atoms are coupled with laser fields [37–44]. However, lifetimes of Rydberg states are typically 10-100 $\mu$s, which are much shorter than BEC lifetimes and are not long enough to explore a wave propagation phenomenon. In order to overcome this drawback, Rydberg dressing, in which a far-detuned laser field couples electronic ground states to Rydberg states, is proposed [45–49]. The Rydberg dressing renders a long-range, soft-core type interaction between Rydberg-dressed atoms, whose coherence time and interaction strength can be controlled by the coupling laser. Particularly, the coherence time of Rydberg-dressed atoms can be prolonged to 10-100 ms, which is comparable with BEC lifetimes. Recent experiments have successfully demonstrated Rydberg-dressing in optical tweezers [50], optical lattices [51–53], and traps [54–56]; many interesting physics associated with Rydberg-dressing such as magnets [52], transport [57], wave mixing [58], supersolids [46,59–61], and experimental observation [62] have also been studied. Different from BECs with dipolar interactions, the Rydberg-dressing in the BECs without dipolar interaction gives rise to isotropic interaction potentials, which is crucial for the stabilization of BECs [45,46].

In this article, we present a scheme to create giant nonlocal optical Kerr nonlinearity and ultraslow polaritonic solitons via matter-wave superradiance in a Rydberg-dressed BEC. The system under consideration is a cigar-shaped BEC, which interacts with a pump and a control laser fields, propagating along the long-axis direction of the BEC. The pump field couples the ground state and an excited state, while the control field couples the excited state and a high-lying Rydberg state. Due to such a coupling configuration, the BEC acquires a collective atomic recoil motion from the pump and control lasers and a scattered laser field is spontaneously generated and propagates along the same direction with the pump field (i.e. along the long-axis direction of the BEC).

We show that, due to the existence of the long-range Rydberg-Rydberg interaction between the atoms, the polariton spectrum (linear dispersion relation) of the scattered laser field generated by the superradiance is changed significantly comparing with the case without the Rydberg-Rydberg interaction. Particularly, when the strength of the Rydberg-Rydberg interaction increases, the spectrum can develop a maxon maximum followed by a roton minimum, which ultimately results in a roton instability. Furthermore, we find that the BEC structure factor has a strong dependence on the property of the Rydberg-dressing, which is much easier to be manipulated than that in conventional BECs with only short-range interactions.

We demonstrate that by using such a system a giant nonlocal optical Kerr nonlinearity can be achieved, which is also contributed by the strong and long-range Rydberg-Rydberg interaction. Based on such result, the formation and robust propagation of polaritonic solitons (i.e. collective nonlinear excitations of the Rydberg-dressed BEC coupled with the scattered laser field) through a backward scattering process can be realized. Moreover, such solitons have ultraslow propagation velocity and can be generated with ultralow light power. The results reported here are useful for the understanding of the unique properties of Rydberg-dressing in ultracold atomic gases (BECs) and have potential applications in optical information processing and transmission.

The remainder of the paper is organized as follows. In Sec. 2, we describe the physical model under study. In Sec. 3, we make the asymptotic expansion on the model equations and calculate the polariton spectrum and the structure factor of Rydberg-dressed BEC. In Sec. 4, we derive the nonlinear envelope equation controlling the evolution of nonlinear polaritons and investigate the nonlocal Kerr nonlinearity and polaritonic solitons. The last section contains the summary of the main results of the present work.

2. Physical model

We consider a ultracold three-state atomic gas with a ladder-shaped level configuration where the atoms are initially populated in the ground state $|1\rangle$. We assume that the atoms are trapped in a highly anisotropic trapping potential $V(\mathbf {r})$, and they form a cigar-shaped BEC, with its long axis along $z$ direction; see Fig. 1(a). We also assume that a pump laser field $\mathbf {E}_P$ (with the angular frequency $\omega _{P}$ and the wavevector $\mathbf {k}_{P}$) and a control laser field $\mathbf {E}_C$ (with the angular frequency $\omega _{C}$ and the wavevector $\mathbf {k}_{C}$) are applied to the BEC and they are all linearly polarized along the $x$ axis and propagate along the $z$ direction. The pump laser field couples the transition from the ground state to an excited state $|2\rangle$, while the control laser field couples the transition from the excited state to a high-lying Rydberg state $|3\rangle$; see Fig. 1(c).

 

Fig. 1. Excitation scheme, level diagram, and nonlocal response function of the cigar-shaped Rydberg-dressed BEC, which interacts with a pump laser field $\mathbf {E}_P$ (wavevector $\mathbf {k}_{P}$) and a control laser field $\mathbf {E}_C$ (wavevector $\mathbf {k}_{C}$), propagating along the long-axis ($z$) direction of the BEC. A scattered laser field $\mathbf {E}_S$ (wavevector $\mathbf {k}_{S}$) can spontaneously emerge in the system, propagating co-linearly or anti-co-linearly with the pump field. (a) Excitation scheme for the forward scattering. Inset: relation between $\mathbf {k}_{P}$ and $\mathbf{k}_S$, with $\Delta \mathbf{k}=\mathbf{k}_P-\mathbf{k}_S$. (b) The same as (a) but for the excitation scheme of backward scattering, with $\Delta \mathbf{k}=\mathbf{k}_P+\mathbf{k}_S$. (c) Ladder-type energy-level diagram of the atoms, in which the pump field $\mathbf {E}_P$ couples the ground state $|1\rangle$ to the excited state $|2\rangle$ (with detuning $\Delta _1$) and the control field $\mathbf {E}_C$ far-off-resonantly couples $|2\rangle$ to the Rydberg state $|3\rangle$ (with detuning $\Delta _R$). A weak scattered field $\mathbf {E}_S$ radiates from the excited state $|2\rangle$ to the ground state $|1\rangle$ (with detuning $\Delta _2$). The interaction between Rydberg atoms is described by the van der Waals potential $U_{\rm VdW}=\hbar C_6/|\mathbf{r}-\mathbf{r}'|^6$ ($C_6$ is the dispersion parameter). (d) Long-range soft-core interaction potential $U_1(r)$, scaled by $\tilde {C}_6/R_c^6$ ($\tilde {C}_6$ is the modified dispersion parameter; $R_c$ is the Rydberg blockade radius), as a function of the separation $r=|\mathbf{r}-\mathbf{r}'|$ between two Rydberg atoms (locate at $\mathbf{r}$ and $\mathbf{r}'$, respectively) for the principle quantum number $n=60$.

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To make the system work in a Rydberg-dressed regime, we further assume that the control field is sufficiently detuned from the related one-photon resonance, so that the excitation of the atoms in the Rydberg state $|3\rangle$ can be significantly suppressed. In such a regime, the excited state is dressed by the Rydberg state through the far-off-resonant coupling of the control field; the atomic gas can be described by a reduced two-state system (see the Supplement 1).

Due to the gain provided by the pump laser field, a weak scattered laser field (with the center angular frequency $\omega _S$ and the center wavenumber $\mathbf {k}_S$) will emerge, which radiates from the excited state $|2\rangle$ to the ground state $|1\rangle$. Thereby, the total electric-field vector in the system can be written as $\mathbf {E}=\sum _{\alpha =P,C,S}\mathbf {E}_{\alpha }=\sum _{\alpha =P,C,S}{\mathbf {e}_{\alpha } \mathcal {E}}_{\alpha } e^{i(\mathbf {k}_\alpha \cdot \mathbf {r}-\omega _\alpha t)}+{\rm c.c.}$, where $\mathbf {e}_P$, $\mathbf {e}_C$ and $\mathbf {e}_S$ ($\mathcal {E}_P$, $\mathcal {E}_C$ and $\mathcal {E}_S$) are respectively polarization unit vectors and amplitudes of the pump, control and scattered fields; c.c. denotes the corresponding complex conjugate terms. Because of the cigar-shaped geometry of the BEC, the scattered field has nearly the same polarization direction with the pump and control fields, i.e. $\mathbf {e}_S=\mathbf {e}_P=\mathbf {e}_C=\mathbf {e}_x$, and propagates along the long-axis of the BEC, i.e. $\mathbf {k}_S \cdot \mathbf {r}\approx k_S z$. Generally, the spontaneously generated scattered field propagates co-linearly and anti-co-linearly with the pump field, as illustrated in Fig. 1(a) and Fig. 1(b), respectively. The wavevector of the recoiled atoms is then determined by $\Delta \mathbf{k}=\mathbf{k}_P-\mathbf{k}_S$ for the forward scattering and $\Delta \mathbf{k}=\mathbf{k}_P+\mathbf{k}_S$ for the backward scattering.

The dynamics of the BEC and the laser fields can be described by the Hamiltonian $\hat {H}=\hat {H}_{\rm atom}+\hat {H}_{\rm laser}+\hat {H}_{\rm int}$, where

From the Hamiltonian, one can obtain the Heisenberg equations for $\hat {\Psi }_{\alpha }$, i.e. $i\hbar \partial \hat {\Psi }_{\alpha }/\partial t=[\hat {\Psi }_{\alpha }, \hat {H}]$ ($\alpha =1,2,3$). We emphasize that the present system is studied under the following conditions: (i) The BEC is initially prepared in the ground state $|1\rangle$; (ii) The two-photon detuning $\Delta _R$ is enough large (order of GHz) so that the atomic excitation of the Rydberg state $|3\rangle$ is not significant and the system works in a Rydberg-dressed regime; (iii) The one-photon detuning $\Delta _1$ is also large (order of tens of MHz), and hence the spontaneous emission from the excited state $|2\rangle$ can be largely suppressed and the system works in a dispersive nonlinearity regime; (iv) The interatomic interactions are described by the two-body potential with the form $U_{\alpha \beta }(\mathbf {r}-\mathbf {r}')=\delta _{\alpha \, 1}\delta _{\beta \, 1}U_{0}\delta (\mathbf {r}-\mathbf {r}')+\delta _{\alpha \, 3}\delta _{\beta \, 3}U_{\textrm {vdW}}(\mathbf {r}-\mathbf {r}')$. Here, the first (second) term on the right hand side describes the contact [van der Waals (vdW)] interaction between the atoms in the ground state $|1\rangle$ (Rydberg state $|3\rangle$) located at $\mathbf {r}$ and $\mathbf {r}'$, respectively [45,46]. The parameter $U_{0}= 4\pi \hbar ^2a_s/M$, with $a_s$ the s-wave scattering length; the vdW interaction potential has the form of $U_{\textrm {vdW}}(\mathbf{r},\mathbf{r}')=\hbar C_6/|\mathbf{r}-\mathbf{r}'|^6$, with $C_6$ the dispersion parameter ($C_6\propto n^{11}$; $n$ is the principle quantum number of the Rydberg state) [34–36].

Under the above assumptions, we adiabatically eliminate the equations for the motion of $\hat {\Psi }_2$ and $\hat {\Psi }_3$. Then, by employing mean-field approximation to $\hat {\Psi }_1$, i.e. $\hat {\Psi }_1 \rightarrow \langle \hat {\Psi }_1 \rangle \equiv \Psi$, we can obtain the Gross-Pitaevskii (GP) equation for the condensate wave function $\Psi$, reading

The equation of motion for the electric field can be obtained from the Maxwell equation $\partial ^2 \mathbf {E}^{(+)}/\partial t^2=c^2\triangledown ^2\mathbf {E}^{(+)}-(1/\epsilon _0)\partial ^2\mathbf {P}^{(+)}/\partial t^2$, where $\mathbf {P}^{(+)}=\mathbf {p}_{12}(\mathbf {p}_{21}\cdot \mathbf {E}^{(+)})|\Psi |^2/(\hbar \Delta _1)$ is the electric polarization vector related to $\mathbf {E}^{(+)}$ [25–27]. Under the slowly varying envelope approximation, the Maxwell equation is reduced to

The GP Eq. (2) and Maxwell Eq. (3) are both (3+1)D. To simplify the problem, we assume that the Rydberg-dressed BEC is trapped in a highly anisotropic potential of the form $V(\mathbf {r})=(M/2)[\omega _{\bot }^2(x^2+y^2)+\omega _z^2z^2]$, with $\omega _{\bot }\gg \omega _z$. Thus, the BEC is elongated along the $z$ axis but remains symmetric in the transverse ($x$ and $y$) directions. By introducing the dimensionless variables $(\xi,\eta )=a_{\bot }^{-1}(x,y)$ (with $a_{\bot }=\sqrt {\hbar /(M\omega _{\bot })}$ the harmonic oscillator length in the transverse directions), $\zeta =a_{z}^{-1}z$ (with $a_z=\sqrt {\hbar /(M\omega _z)}$ the harmonic oscillator length in the longitudinal direction), $\tau =\omega _{\bot }t$, $\psi =\Psi /\sqrt {N_0}$ (with $N_0=N/a_{\bot }^3$ the atomic density), and $\varepsilon _S=\mathcal {E}_{S}/\mathcal {E}_{P}$, Eq. (2) can be written into a dimensionless form

Here $\tilde {\nabla }=(\partial _\xi,\partial _\eta,\partial _\zeta )$, $\Omega _P=|\mathbf {p}_{12}\cdot \mathbf {e}_x|\mathcal {E}_{P}/\hbar$ is the half Rabi frequency of the pump field, and $\Phi =\Delta \tilde {k} \zeta -\Delta \tilde {\omega } \tau$ is the phase, with $\Delta \tilde {k}=(k_P \mp k_S) a_{\bot }$ (“$-$” for the forward scattering; “$+$” for the backward scattering) and $\Delta \tilde {\omega }=\Delta _2/\omega _{\bot }$. Note that since $\omega _{\bot }\gg \omega _z$, one has $a_{\bot }\ll a_{z}$.

Due to the strong transverse confinement provided by the trapping potential, it is possible to factor the solution of Eq. (4) as $\psi =\varphi (\xi,\eta )F(\zeta,\tau )e^{-i\mu \tau }$ [63], with $\mu$ the chemical potential and $\varphi (\xi,\eta )$ the transverse mode, governed by the equation

Note that Eq. (5) has the same form with the eigenvalue equation of a two-dimensional harmonic oscillator in quantum mechanics. When the trapping potential is deep enough, no collective excitations of the Rydberg-dressed BEC can occur in the $x$ and $y$ (transverse) directions, but the occurrence of low-energy excitations induced by the atomic interactions is allowed in the $z$ direction. Furthermore, one can set $\varphi (\xi,\eta )=\varphi _0(\xi,\eta )=e^{-(\xi ^2+\eta ^2)/2}$, which is the ground-state solution of Eq. (5) with the eigenvalue $\nu =1$. In addition, we assume that the scattered field has the same Gaussian transverse distribution with that of $\varphi$, with the form ${\cal E}_{S}=g(\zeta,\tau ){\cal E}_{P}e^{-(\xi ^2+\eta ^2)/2}$, where $g(\zeta,\tau )$ is a dimensionless function characterizing the amplitude of the scattered field. Since the scattered field is much weaker than the pump one, the magnitude of $g(\zeta,\tau )$ should be much smaller than one.

With the above analysis, Eq. (4) can be simplified into the following (1+1)D form

By using the same approach, Eq. (3) can be converted into the following (1+1)D form

The model we considered above is rather general. It can be realized by using a Bose-condensed Strontium atomic gas [64] (e.g. $^{88}$Sr), with the atomic internal states assigned to be $|1\rangle =|5s^2\,^{1}S_0\rangle$, $|2\rangle =|5s5p\,^{1}P_1\rangle$ and $|3\rangle =|5sns\,^{1}S_0\rangle$. The s-wave scattering length of the BEC is $a_s = 94.8a_0$, with the Bohr radius $a_0\approx 53$ pm. The spontaneous emission rates of atoms are given by $\Gamma _{12}\approx 2\pi \times 16$ MHz, and $\Gamma _{23}\approx 2\pi \times 16.7$ kHz. The dispersion parameter of the Rydberg state for $n=60$ is $C_6\approx 2\pi \times 10.9$ GHz $\mu$m$^6$. Since $C_6$ is positive, the vdW interaction between atoms in the Rydberg state is attractive, useful to realize a self-focusing Kerr nonlinearity and realize polaritonic solitons. In the following calculations, we assume $\omega _{\bot }/(2\pi )=100$ Hz, which results the harmonic oscillator length in the transverse directions $a_{\bot }\approx 1.1$ $\mu$m, $\omega _z/(2\pi )=0.1$ Hz, which corresponds to the harmonic oscillator length in the longitudinal direction $a_z\approx 32a_{\bot }\approx 35$ $\mu$m, the atomic density $n_0\approx 2.4\times 10^{13}$ cm$^{-3}$, the one-photon (two-photon) detuning $\Delta _1 \approx 50$ MHz ($\Delta _R \approx 18$ GHz), and $\Omega _P\approx \Omega _C \approx 10$ MHz, which gives the Rydberg blockade radius $R_c\approx 4.5$ $\mu$m. All calculations given below will be based on these parameters.

3. Linear excitation spectrum and structure factor of the Rydberg-dressed BEC

Now we turn to solve the reduced GP-Maxwell Eqs. (6) and (7). To understand the physical property of the excitations in the system, it is necessary to know the stationary background of the condensate when the scattered laser is absent, which constitutes the base state of the system. Such a based state can be obtained from Eqs. (6) and (7) by neglecting the derivative terms and setting $F=f_0$, $g=0$, yielding the equations

Now we apply the method of multiple scales [31] to investigate the linear and nonlinear excitations and the propagation behavior of the scattered field in the system. To this end, we make the asymptotic expansions $F=f_0+\sum _{j=1}^\infty \epsilon ^j f^{(j)}$ and $g=\sum _{j=1}^\infty \epsilon ^j g^{(j)}$. Here $\epsilon$ is a small parameter characterizing the typical amplitude of the scattered field relative to the pump field; all quantities on the right hand side of the expansions are functions of the multi-scale variables $\tau _{\alpha }=\epsilon ^{\alpha } \tau$ ($\alpha =0,\,1$) and $\zeta _{\beta }=\epsilon ^{\beta } \zeta$ ($\beta =0,\,1,\,2$). The damping parameter $\gamma$ can be assumed to be the order of $\epsilon$ (i.e. $\gamma \sim {\cal O}(\epsilon )$), which is reasonable because the spontaneous emission is greatly suppressed by the use of the large one-photon detuning $\Delta _1$. With such expansions, Eqs. (6) and (7) are transferred into the form

For seeking the density fluctuations of the Rydberg-dressed BEC and the scattered field induced by the pump field, we further assume $f^{(j)}=f_+^{(j)}e^{i\varphi }+f_-^{(j)\ast }e^{-i\varphi }$, with $\varphi =q\zeta _0-\omega _q\tau _0$. Here $q$ and $\omega _q$ are respectively the dimensionless atomic momentum and the corresponding atomic energy of density fluctuations induced by the light scattering process. Then, Eqs. (9a) and (9b) can be recast into the following equations for $f_+^{(j)}$, $f_-^{(j)}$ and $g^{(j)}$:

At the first-order ($j=1$) approximation, under the phase-matching condition ($\phi \approx \varphi$) and assuming that $f_+^{(1)}$, $f_-^{(1)}$ and $g^{(1)\ast }$ are all proportional to $e^{i(k\zeta _0-\omega \tau _0)}$, we obtain the linear dispersion relation

Particularly, when $|q|R_c\ll 1$, one has $f(q)\approx 1-q^2R_c^2/4$. Since $k$ is a small deviation from $q$, we can approximate $k+q\approx q$. As a result, Eq. (11) can be simplified as

Shown in Fig. 2(a) is the Bogoliubov excitation spectrum of the Rydberg-dressed BEC by taking the oscillating frequency $\omega _B$ as a function of $q R_c$ ($q$ is the atomic recoil momentum and $R_c$ is the Rydberg blockade radius). The dashed black line, solid red line, and dotted blue line in the figure are for the interaction parameter $\alpha \equiv 2n_0c_1f_0^2Z_0=10$, $80$, and $180$, respectively. Particularly, we have $f_0=0.32$, $Z_0=1.18\times 10^{-8}$ GHz $\mu$m, with the parameters given in the last paragraph of Section 2. Different form conventional BECs with contact interactions, in the Rydberg-dressed BEC the presence of long-range Rydberg-Rydberg interaction can result a significant effect on the the polariton spectrum. Particularly, a maxon maximum followed by a roton minimum can be observed for a moderate value of $\alpha$, which ultimately results in a roton instability when further increasing $\alpha$.

 

Fig. 2. Linear excitations of the Rydberg-dressed BEC. (a) Linear (Bogoliubov) excitation spectrum $\omega _B$ as a function of $q R_c$ ($q$ is the atomic recoil momentum and $R_c$ is the Rydberg blockade radius). The dashed black line, solid red line, and dotted blue line are for the interaction parameter $\alpha =10$, $80$, and $180$, respectively; $\delta$ denotes the frequency gap for the occurrence of polaritonic roton. (b) Roton gap $\delta$ as a function of $\alpha$. The vertical dashed lines correspond to $\alpha _{\textrm {min}}\approx 10$, $\alpha _{\textrm {max}}\approx 38$ and $\alpha _{\textrm {inst}}\approx 118$, respectively.

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The parameter $\delta$ in Fig. 2(a) indicates a frequency gap for the local roton minumim, for which $\omega _B$ displays the roton minimum at a non-zero value of $q$. Figure 2(b) shows $\delta$ as a function of $\alpha$. The vertical dashed lines are for $\alpha _{\textrm {min}}\approx 10$, $\alpha _{\textrm {max}}\approx 38$ and $\alpha _{\textrm {inst}}\approx 118$, respectively. We find that $\delta$ firstly increases with growth of $\alpha$ until arriving at a critical value $\alpha =\alpha _{\textrm {max}}\approx 38$; then it decreases as $\alpha$ increases further, and ultimately vanishes at $\alpha =\alpha _{\textrm {inst}}\approx 118$ which marks the onset of a roton instability.

The structure factor $S(q)$ can significantly affect the dispersive property of the system. Specifically, the system is weakly dispersive when $S(q)$ is small, while it is strongly dispersive when $S(q)$ is large. From the definition of $S(q)$, we have $S(q)\approx q/(2\sqrt {c_0+c_1Z_0}f_0)\ll 1$ when $q^2/4 \ll c_0f_0^2+c_1f_0^2\tilde {U}(q)$ and $S(q)\approx 1$ when $q^2/4 \gg c_0f_0^2+c_1f_0^2\tilde {U}(q)$. Consequently, if the scattered field propagates co-linearly with the pump field (i.e. the forward scattering), one has $q\approx \Delta \tilde {k}\approx (k_P-k_S)a_{\bot }$, resulting in a small $q$ and hence $S(q)\ll 1$. In such a case, the system is weakly dispersive and has the linear dispersion relation $k\approx \omega /v$. However, if the scattered field propagates anti-co-linearly with the pump field (i.e. the backward scattering), one has $q\approx \Delta \tilde {k}\approx (k_P+k_S)a_{\bot }$, resulting in a large $q$ and hence $S(q)\approx 1$. In such a scenario, the system is strongly dispersive and has the linear dispersion relation $k\approx \bar {c}_2\bar {d}_2f_0^2/(\omega +\omega _q+i\gamma -\omega _B)$. Since the structure factor of the Rydberg-dressed BEC depends on more system parameters (such as the control-field intensity $|\Omega _c|^2$ and the principal quantum number of Rydberg states $n$) than that of conventional BECs, the dispersive property of the Rydberg-dressed BEC is more controllable than that of conventional BECs.

Shown in Fig. 3(a) is the structure factor $S(q)$ as a function of the atomic recoil momentum $q$, for the principle quantum number $n=30$, 60, 90, and 120, respectively. Note that with the parameters given in the last paragraph of Section 2, we get $\mu \approx -3.64$, $f_0\approx 0.32$, and $\gamma \approx 0.06$. We see that $S(q)\approx 0$ when $q\rightarrow 0$ and $S(q)$ increases rapidly when $q$ has a small value. However, $S(q)\approx 1$ when $q\rightarrow \infty$ and $S(q)$ increases very slowly when $q$ has a large value. Thereby, in the case of forward scattering $q\approx 0$ one has $S(q)\approx 0$ whereas in the case of backward scattering $q\approx 17$ one has $S(q)\approx 1$. Figure 3(b) is an illustration of $S(q)$ as a function of $n$ for different values of $q$. It is seen that although $S(q)$ always decreases as $n$ increases, it decreases fast (slowly) when $q$ has a small (large) value.

 

Fig. 3. Structure factor $S(q)$ of the Rydberg-dressed BEC when the roton instability does not occur. (a) $S(q)$ as a function of the atomic recoil momentum $q$, for the principle quantum number $n=30$, 60, 90, and 120, respectively. The regions of the forward and backward scattering are indicated. (b) $S(q)$ as a function of $n$, for $q=1$ (solid line), 1.5 (dash-dotted line), 2 (dotted line) and 2.5 (dashed line), respectively.

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At the first-order approximation, we get solutions $f_+^{(1)}=Ae^{i(k\zeta _0-\omega \tau _0)}$, $f_-^{(1)}=a Ae^{i(k\zeta _0-\omega \tau _0)}$, and $g^{(1)\ast }=b Ae^{i(k\zeta _0-\omega \tau _0)}$, with

Here, $X=\omega +\omega _q+i\gamma -q^2/2-c_0f_0^2-c_1f_0^2\tilde {U}(q) -\bar {c}_2\bar {d}_2f_0^2/(k-\omega /v)$ and $A$ is a yet to be determined envelope function. The propagation property of the polariton in the linear regime is determined by the linear dispersion relation $k(\omega )$. Particularly, the polariton is stable during the propagation if the imaginary part of $k(\omega )$ [Im$(k)$] is zero; otherwise, it undergoes an attenuation (amplification) if Im$(k)$ is positive (negative).

Shown in Fig. 4 are real and imaginary parts of $k(\omega )$ [Re($k$) and Im($k$)] as functions of $\omega$ for $\omega _q\approx \omega _B$. It is seen that Re$(k)\approx 0$ and Im$(k)$ arrives at a negative maximum at $\omega \approx 0$, corresponding to a negligible phase shift and a large gain. Thus, the scattered field and hence the polariton can be spontaneously generated and amplified at this point. However, when $|\omega |\gtrsim 1$, one has Re$(k)\gg$Im$(k)\approx 0$, corresponding to a large phase shift and a negligible gain. In these regions, the stable propagation of the scattered field and hence the polariton can be achieved, which will be crucial for the robust propagation of polaritonic solitons.

 

Fig. 4. The linear dispersion relation $k(\omega )$ without occurrence of roton instability as a function of $\omega$. Solid and dashed lines denote the real [Re($k$)] and imaginary [Im($k$)] parts of $k(\omega )$, respectively.

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4. Giant nonlocal Kerr nonlinearity and ultraslow polaritonic solitons

The study of Kerr effect in ultracold atoms is of great interest in nonlinear optics because the Kerr effect is essential for the realization of most nonlinear optical processes and related to many applications [65]. By means of the matter-wave superradiance, it is possible to achieve a significant enhancement of the optical Kerr nonlinearity with controllable nonlocality in the Rydberg-dressed BEC. In order to demonstrate this, we need to go to the high-order solutions of Eq. (10).

At the second order ($j=2$) approximation, the solvability condition of Eq. (10) requires the condition

Note that the imaginary part of $V_g$ is much smaller than the corresponding real part under the condition $\gamma \ll \omega _q-\omega _B$ and hence it plays a negligible role.

The evolution of the envelope function $A$ in the nonlinear regime can be obtained by the solvability condition of Eq. (10) at the third-order ($j=3$) approximation, which yields the nonlinear envelope equation

Here, $\alpha _1=[c_0+c_1+\bar {c}_2\bar {d}_2/(k-\omega /v)]f_0^2/[Y-\bar {c}_2\bar {d}_2f_0^2/(k-\omega /v)]$, $\beta _1=-\bar {d}_2(\alpha _1+1)/(k-\omega /v)$, $Y=\omega +\omega _q+i\gamma +q^2/2+c_1f_0^2$, $Z=k\alpha _1-\bar {c}_2f_0/(k-\omega /v)$, and $\tilde {\alpha }=\epsilon ^{-2}\alpha =\epsilon ^{-2}{\textrm {Im}}(k)$. The second term on the left hand side of Eq. (17) related to the coefficient $D$ describes the group-velocity dispersion; the first two terms in the parenthesis (related to coefficients $W_1$ and $W_2$) characterize the nonlocal Kerr nonlinearity, originated from the long-range Rydberg-Rydberg interaction; the last term in the parenthesis (related to the coefficient $W_3$) characterizes the local Kerr nonlinearity, originated from the short-range contact interaction between the atoms. The coefficients $W_1 \approx 0.7$, $W_2 \approx 0.3$, and $W_3 \approx -0.005$, with the given parameters.

The nonlinear optical effect of the system can be described by the refractive index $n$, defined by

From the above result, it is obvious that $n_{2}$ is two orders of magnitude larger than $n_3$, i.e. the nonlocal Kerr nonlinearity contributed by the Rydberg-Rydberg interaction is much stronger than that of the local one contributed by short-range contact interaction. Moreover, $n_{2}$ is about 15 orders of magnitude larger than that obtained in usual nonlinear optical materials, such as optical fibers [66], characterizing a giant nonlocal Kerr nonlinearity.

Now we turn to seek polaritonic soliton solutions based on the nonlinear envelope Eq. (17). To avoid a large gain, one can make $\omega _q$ and $\omega _B$ slightly different so that the gain peak can be moved away from $\omega =0$ (see Fig. 4). Compared with the forward scattering, the backward scattering can provide a significant dispersion, which is necessary for balancing the Kerr nonlinearity and hence is crucial for the formation and stable propagation of polaritonic solitons. Combining the Eqs. (15) and (17), we have

Since Eq. (20) is not integrable, we employ a variational method to search possible polaritonic soliton solutions. The Lagrangian of Eq. (20) of the system described by Eq. (20) for $\alpha \approx 0$ is given by $L=\int _{-\infty }^{\infty } \mathcal {L} d\zeta$, where $\mathcal {L}$ is the Lagrangian density, given by $\mathcal {L}=(SS_\zeta ^\ast -S^\ast S_\zeta )+(D/2)|S_T|^2-[(W_1+W_2)/2]|S|^2\int U(\zeta -\zeta ')|S(\zeta ',T)|^2d\zeta '-(W_3/2)|S|^4$. Then, the soliton solutions are sought by the ansatz with a Gaussian wavepacket

We are interested in the stable propagation of the polaritonioc solitons in the system. Returning to the original (i.e., dimensional) variables, the soliton solution for the condensate wave function and the scattered field are respectively given by

From the solution (23) we see that the collective wave of recoiled atoms consists of two parts, i.e. the left- and right-moving waves along the $z$ axis. With the given system parameters, we have Re$(a)\approx -1.69\times 10^{-4}$ and Re$(b)\approx 0.98$, with their imaginary parts being two orders of magnitude smaller than their corresponding real parts. Since $|a|\ll 1$ and $|b|\sim 1$, almost all recoiled atoms are moving to the right due to the presence of the pump field, while the scattered field is backward propagating. The propagation velocity of the soliton is approximately given by the linear group velocity, estimated to be

The input light power of the pump laser for generating the polaritonic soliton can be estimated by computing the Poynting’s vector integrated over the cross-sectional area of the condensate, which is given by

Shown in Fig. 5 is the numerical result of polaritonic soliton, where the soliton amplitude $|E_s/E_0|$ as a function of $T=[\omega _\bot t-z/(a_\bot V_g)]$ and $\zeta =z/a_\bot$ is illustrated. The initial condition used in the simulation is the ansatz (21), with $a_s$ being obtained through solving Eq. (22) numerically. Figure 5(a) shows the case of $(|E_s(\zeta =0)/E_0|,q)=(1,4)$. We see that the soliton is robust during propagation. This is because of the exact balance between the dispersion and nonlinearity in the system. Figure 5(b) is for the case $(|E_s(\zeta =0)/E_0|,q)=(1,22)$. One sees that the soliton displays significant dispersion, and hence its amplitude (width) decreases (broadens) rapidly during propagation. The reason is that the dispersion (originated by the large atomic recoil momentum $q$) is much larger than the nonlinearity. The panel (c) of the figure is for $(|E_s(\zeta =0)/E_0|,q)=(8.5, 1)$. In this situation, the soliton undergoes a collapse due to the fact that the nonlinearity is much larger than the dispersion.

 

Fig. 5. The formation and propagation of a polaritonic soliton, by taking soliton amplitude $|E_s/E_0|$ as a function of $T=\omega _\bot t-z/(a_\bot V_g)$ and $\zeta =z/a_\bot$. (a) The stable polaritonic soliton solution for the parameters $(|E_s(\zeta =0)/E_0|,q)=(1,4)$. Insert: top view of the soliton. (b) The same as (a) but for $(|E_s(\zeta =0)/E_0|,q)=(1,22)$. The soliton undergoes significant dispersion during propagation. (c) Also the same as (a) but for $(|E_s(\zeta =0)/E_0|,q)=(8.5,1)$. The soliton collapses during propagation. (d) Stability diagram of polaritonic solitons by measuring the propagation fidelity in the parameter plane of the input soliton amplitude $|E_s(\zeta =0)/E_0|$ and the atomic recoil momentum $q$. Dots A, B and C represent the cases of panels (a), (b) and (c), respectively. The fidelity shown by the white dashed line is ${\mathcal {J}}=0.9$.

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The domains of stable (or unstable) propagation of polaritonic solitons can be obtained by calculating the propagation fidelity, defined by

It is obvious that ${\mathcal {J}}=1$ at $\zeta =0$ and ${\mathcal {J}}\in (0,1)$ for $0<\zeta <\ell$. If ${\mathcal {J}}$ is very close to one in the distance of $\ell$, the shape of the soliton can keep nearly invariant during the propagation in this distance. However, if ${\mathcal {J}}$ is close to zero, the shape of the soliton suffers a large distortion during the propagation, i.e. no stable polaritonic solitons are achievable in the system. In Fig. 5(d), we show the stability diagram of polaritonic solitons by measuring the fidelity as a function of the input soliton amplitude $|E_s/E_0|$ and the atomic recoil momentum $q$. When plotting this figure, polaritonic solitons are assumed to be stable if ${\mathcal {J}}\gtrsim 0.9$ at $\zeta =\ell =10$ (corresponding to the length of the BEC $\approx 700$ $\mu$m), whereas they are unstable if ${\mathcal {J}}<0.9$. Dots A, B and C in the figure represent the cases of stable soliton [${\mathcal {J}}\approx 1$; corresponding to Fig. 5(a)], dispersive wave [${\mathcal {J}}\approx 0$; corresponding to Fig. 5(b)], and collapsed wave [${\mathcal {J}}\approx 0$; corresponding to Fig. 5(c)], respectively. We see that polaritonic solitons are stable (unstable) when both $|E_s/E_0|$ and $q$ are small (either $|E_s/E_0|$ or $q$ is large). The boundary between the stable and unstable regions is highlighted by the white dashed line in the figure.

5. Discussion and summary

In this work the Rydberg-dressed BEC is assumed to be described by a single wave function, which is valid only if the recoil energy of the atoms in the BEC is much smaller than that of the atom-photon interaction and that of the atom-atom interaction. In this small recoil-energy regime, excitations of the atoms to high excited states are highly suppressed so that the recoiled atoms mainly remain in the same internal state and the condensate. If the recoil energy is large, the atoms can be scattered from the condensate and generate new side modes. In this case, the description by a single wave function is not applicable. Moreover, the theoretical approach above is built upon a semiclassical description, which is valid only if the numbers of the scattered photons and recoiled atoms are large enough. In addition, to investigate the initial stage of the superradiance process where scattered photons and recoiled atoms are not large, a fully quantum approach should be used [25–30], which is a topic deserving to be explored in future.

In conclusion, we have presented a scheme for generating nonlocal optical Kerr nonlinearity and polaritonic solitons via matter-wave superradiance in a Rydberg-dressed BEC. We have shown that, due to the strong and long-range interactions between Rydberg atoms, the polariton spectrum of the scattered field is qualitatively different; moreover, the BEC structure factor depends significantly on the Rydberg-dressing and hence is more controllable than that in conventional BECs. We have also shown that such a Rydberg-dressed BEC system supports a giant nonlocal Kerr nonlinearity, and hence allows the formation and stable propagation of polaritonic solitons having ultraslow propagation velocity and ultralow generation power. The results reported here are useful for understanding the unique properties of Rydberg-dressed BECs and have potential applications in optical information processing and transmission.