1. INTRODUCTION

High-dimensional spatiotemporal optical solitons, alias light bullets (LBs) [1], are solitary nonlinear waves localized in $m$-spatial dimensions and one time axis $[(m+1)\mathrm{D};m=1,2,3]$. In recent years, the study of LBs has attracted intensive theoretical and experimental interests [2] because of their rich nonlinear physics and technological applications [3–8]. Experimental signatures of LBs in different types of optical media, such as $(2+1)D$ LBs in ${\mathrm{LiIO}}_3$ crystals [9] and quasi-$(3+1)D$ LBs in waveguide arrays [10,11], have been reported. Recently, trains of $(3+1)D$ dark solitons have been observed in a photo-refractive material by Lahav et al. [12]. These LBs, however, generally suffer severe instability, which typically propagate just a few diffraction lengths. To alleviate the rapid distortion of the LBs, short pulses (on the order of femtosecond) with high laser powers were commonly used experimentally. It is difficult to apply these LBs in optical information processing, where a key requirement is the on-demand generation, storage, and retrieval of low power single LBs [13]. A remaining challenge is to identify optical media with tunable dispersion, diffraction, and optical nonlinearities, such that stable single LBs can be manipulated.

Theoretical efforts have attempted to study the formation of LBs with different mechanisms. A commonly used approach for creating stable $(3+1)D$ LBs is to exploit local and nonlocal optical nonlinearities with quite different response times. These nonlinearities have been examined, e.g., in nematic liquid crystals [14–16] and lead glass [17], in which the nonlocal nonlinearity (resulted from the reorientation [14–16] or thermal motion [17] of molecules) has a very long response time (typically on the order of second or even longer), while the local nonlinearity (resulted from the electronic Kerr effect) has a very short response time (on the order of femtosecond). Due to the mismatch between slow response and short propagation time, LBs were realized in the form of pulse-train solitons but not single LBs [12]. A different approach is based on electromagnetically induced transparency (EIT) in atomic gases [18], in which optical absorption can be largely suppressed due to quantum interference. Together with Kerr nonlinearities [19,20] induced by resonant laser fields, it has been shown that stable $(1+1)D$ temporal [21–23] and spatial [24,25] optical solitons can form. Though promising, the local nature of the Kerr nonlinearity does not support stable LBs, which suffer unavoidable catastrophic collapse during propagation.

On the other hand, recent theoretical [26] and experimental [27] studies revealed that strong and long-range optical nonlinearities can be built with Rydberg atoms [20,28,29], which are in electronically high-lying states with large principal quantum number $n$ [30]. Large dipole moments in Rydberg states render a strong, long-range Rydberg–Rydberg interaction between remote atoms. Rydberg interactions find applications in quantum information, precision measurement, quantum simulation, and many-body physics [20,28,29,31,32]. Importantly, the Rydberg–Rydberg interaction can be mapped to a nonlocal optical nonlinearity through EIT, which is strong even at the single photon level [20,28,29]. Long lifetimes (on the order of tens of microseconds) guarantee that the induced optical nonlinearities are largely coherent during light propagation [33–35]. This provides a new platform to study quantum nonlinear optics [28] and develop new photon devices, such as single-photon switches and transistors [36–38], quantum memories, and phase gates [39–44].

In this work, we present a scheme for the generation and storage of stable $(3+1)D$ LBs in highly tunable cold gases of Rydberg atoms. A key element is the co-existence of giant nonlocal and local optical Kerr nonlinearities. The former features a fast (sub-microsecond) response [45], which is complemented by the latter [46,47], whose response is relatively slow (on the order of microseconds). In conjunction with tunable dispersion and diffraction, this allows us to precisely control dynamics of LBs. To go beyond the commonly used mean-field theory [48–52], we derive a nonlinear $(3+1)D$ light propagation equation taking into account many-body correlations. We show that the synergetic nonlocal and local Kerr nonlinearities in the system permit us to obtain stable single $(3+1)D$ LBs as well as LBs that carry definite orbital angular momentum [i.e., light vortices (LVs)] [53–58]. We reveal that the stability of such single LBs and LVs is achieved via a two-step self-trapping mechanism (see below). As a result, stable single bright $(3+1)D$ LBs are generated with ultraslow propagation velocity, extremely low light power, and narrow bandwidth. More importantly, $(3+1)D$ LBs and LVs obtained can be stored and retrieved in the system with high efficiency and fidelity, which may have potential applications in optical information processing and transformation.

2. MODEL

We consider a three-level atomic system with a ladder-type level configuration, shown in Fig. 1(a). A weak, pulsed probe field of angular frequency ${\omega }_p$ (half Rabi frequency ${\mathrm{\Omega }}_p$) couples the ground state $|1⟩$ and intermediate state $|2⟩$, and a strong, continuous-wave control field of angular frequency ${\omega }_c$ (half Rabi frequency ${\mathrm{\Omega }}_c$) couples state $|2⟩$ and a Rydberg state $|3⟩$. The probe field has a pulse length ${\tau }_0$ at the entrance of the medium. The electric-field vector of the system is $\mathbf{E}(\mathbf{r},t)=\sum _{l=p,c}{\mathbf{e}}_l{\mathcal{E}}_l\exp [i({\mathbf{k}}_l·\mathbf{r}-{\omega }_{l}t)]+\text{c.c.}$, where ${\mathbf{e}}_l({\mathbf{k}}_l)$ is the unit polarization vector (wavevector) of the electric-field component with envelope ${\mathcal{E}}_l(l=p,c)$. A possible experimental arrangement of beam geometry is shown in Fig. 1(b).

 

Fig. 1. Rydberg atomic model. (a) EIT level scheme, where the ground state $|1⟩$, intermediate state $|2⟩$, and Rydberg state $|3⟩$ are, respectively, driven by a pulsed probe field (with pulse duration ${\tau }_0$) and a strong control field. State $|2⟩$ has a large spontaneous decay rate ${\mathrm{\Gamma }}_{12}\sim \mathrm{MHz}$. The weak decay ${\mathrm{\Gamma }}_{23}\sim \mathrm{KHz}$ from $|3⟩$ to $|2⟩$ is also taken into account. The van der Waals interaction $\hslash V(\mathbf{r}-{\mathbf{r}}^{\prime })$ between the two atoms in Rydberg states, respectively located at $\mathbf{r}$ and ${\mathbf{r}}^{\prime }$, shifts the Rydberg state energy. (b) Geometry of the system. The probe and control laser fields counter-propagate in the Rydberg gas. Depletion of the strong control field is neglected. (c) Storage and retrieval of a $(3+1)D$ light bullet, illustrated by an isosurface plot of the light intensity of the light bullet before storage ($z=0$), at the beginning of the storage ($z=5.4\mathrm{mm}$), and after the storage ($z=10.8\mathrm{mm}$); see text for details.

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The system works at an ultracold temperature, and the probe and control fields counter-propagate [i.e., ${\mathbf{k}}_p=(0,0,k_p)$, ${\mathbf{k}}_c=(0,0,-k_c)$], so that the center-of-mass motion of the atoms and dephasing due to the atomic collisions are negligible. Under the electric-dipole approximation, the system Hamiltonian is ${\widehat{H}}_H={\mathcal{N}}_a\int {\mathrm{d}}^3\mathbf{r}{\widehat{\mathcal{H}}}_H(\mathbf{r},t)$ with

To study propagation of the probe field, we assume the size of the atomic gas is much larger than the Rydberg blockade radius (given later). Depletion of the control field will be neglected (except in the discussion of LB memories), additionally assuming that the weak probe pulse is still a classical field (consisting of many photons), so that a semi-classical approach can be used. With slowly varying envelope approximation, the propagation of the probe field is governed by the Maxwell equation [18]:

3. $(3+1)D$ NONLINEAR ENVELOPE EQUATION

Due to the Rydberg–Rydberg interaction, the Bloch Eq. (2) for one-body correlators ${\rho }_{\alpha \beta }(\mathbf{r},t)=⟨{\widehat{S}}_{\alpha \beta }(\mathbf{r},t)⟩$ involves two-body correlators ${\rho }_{33,3\alpha }({\mathbf{r}}^{\prime },\mathbf{r},t)\equiv ⟨{\widehat{S}}_{33}({\mathbf{r}}^{\prime },t){\widehat{S}}_{3\alpha }(\mathbf{r},t)⟩$, for which three-body correlators ${\rho }_{\alpha \beta ,\mu \nu ,\zeta \eta }({\mathbf{r}}^{\prime \prime },{\mathbf{r}}^{\prime },\mathbf{r})\equiv ⟨{\widehat{S}}_{\alpha \beta }({\mathbf{r}}^{\prime \prime },t){\widehat{S}}_{\mu \nu }({\mathbf{r}}^{\prime },t){\widehat{S}}_{\zeta \eta }(\mathbf{r},t)⟩$ will be involved, etc. (see Section 1 of Supplement 1). Thus, we obtain an infinite hierarchy of equations for correlators of one-body, two-body, three-body, and so on. It is difficult to solve the hierarchy of equations for many-body correlators by employing conventional techniques. Fortunately, since in our consideration the probe field is relatively weak, we can solve the hierarchy of equations by using the method of multiple scales [22,23,60]. Our calculations for solving the Maxwell–Bloch (MB) Eqs. (1) and (2) are exact to third order (i.e., up to ${\mathrm{\Omega }}_p^3$), and hence $n$-body correlators ($n\ge 3$) can be neglected safely. By assuming ${\mathrm{\Omega }}_p\sim F\exp [i(Kz-\omega t)]$ [61], it turns out that the envelope $F$ of the probe pulse satisfies the $(3+1)D$ nonlinear equation (see Section 2 of Supplement 1):

Note that the second and third terms in Eq. (3) represent, respectively, the dispersion and diffraction, and the fourth and fifth terms are the local and nonlocal Kerr nonlinearities of the system. Different from Ref. [49], in our approach, a non-zero two-photon detuning ${\mathrm{\Delta }}_3$ is assumed, which is needed to obtain the local Kerr nonlinearity (i.e., $W_1\ne 0$). As will be illustrated explicitly below, the local Kerr nonlinearity is very crucial for the formation of stable, single $(3+1)D$ LBs and LVs in the system. Furthermore, equations for the diagonal elements of the density matrix $\widehat{\rho }$ and those for related two-body correlators are taken into account in our approach. Because the diagonal elements of $\widehat{\rho }$ and related two-body correlators contribute also to both the local and nonlocal Kerr nonlinearities, such consideration is not only for consistency in theory but also for giving reasonable results for both the local and nonlocal Kerr nonlinearities. In addition, our approach is semi-classical in nature, thus valid only for classical probe regime. It is different from that used in Ref. [51], where the Kerr nonlinearity in a quantum probe regime was studied for a $(1+1)D$ system, and no LBs or LVs or their storage and retrieval were considered.

4. ULTRASLOW WEAK LIGHT BULLETS AND VORTICES

We now discuss possible LBs in the system. To be concrete, we consider strontium atoms (${}^{88}\mathrm{Sr}$) in this work, although our approach is valid for other Rydberg atomic gases. The energy levels shown in Fig. 1(a) are selected as $|1⟩=|5s^2{}^1{S}_0⟩$, $|2⟩=|5s5p{}^1{P}_1⟩$, $|3⟩=|5sns{}^1{S}_0⟩$, with $n$ the principal quantum number [62]. The spontaneous emission rates of the atoms are given by ${\mathrm{\Gamma }}_{12}=2\pi \times 16\mathrm{MHz}$, ${\mathrm{\Gamma }}_{23}=2\pi \times 16.7\mathrm{kHz}$, so one has ${\gamma }_{21}={\mathrm{\Gamma }}_{12}/2$, ${\gamma }_{31}={\mathrm{\Gamma }}_{23}/2$, ${\gamma }_{32}=({\mathrm{\Gamma }}_{12}+{\mathrm{\Gamma }}_{23})/2$. For this choice, the vdW interaction in ${}^1{S}_0$ states is isotropically attractive ($C_6>0$), which is important to realize a self-focusing nonlinearity.

The form of the solution of the $(3+1)D$ envelope Eq. (3) depends heavily on the property of the nonlocal Kerr nonlinearity, which is characterized by the nonlocality degree of the system. The nonlocality degree can be described by using the parameter $R_b/R_0$. Here, $R_b$ is the spatial width of the effective interaction potential $G_2({\mathbf{r}}_{\perp }-{\mathbf{r}}_{\perp }^{\prime })$, i.e., the Rydberg blockade radius, defined by [49] $R_b={[|C_6{d}_{21}|/(2{|{\mathrm{\Omega }}_c|}^2)]}^{1/6}$; $R_0$ is the transverse spatial width of the probe-pulse envelope $U$, which can be measured by the transverse beam radius of the incident probe pulse. For example, a Gaussian-type incident pulse $U=U_0\exp [-{(r_{\perp }/R_0)}^2]$, with $r_{\perp }={[x^2+y^2]}^{1/2}$. Hence, the nonlocality degree can be tuned by adjusting $R_0$ or $R_b$.

In the following, we take the system parameters in the dispersive regime (i.e., $|{\mathrm{\Delta }}_2|\gg {\mathrm{\Gamma }}_{12}$). Exact values of parameters will be given later. This allows us to divide the nonlocality degree of the system into three typical regions [29,49]: (i) $R_b/R_0\ll 1$ (local response region); (ii) $R_b/R_0\sim 1$ (nonlocal response region); and (iii) $R_b/R_0\gg 1$ (strong nonlocal response region). We will discuss different types of LB solutions supported by the Rydberg medium.

A. Local Response Region

We first consider the case when the range of the Rydberg–Rydberg interaction (equivalently, the spatial width of the effective atomic interaction potential $G_2$) is much smaller than that of the beam radius of the probe pulse), i.e., $R_b/R_0\ll 1$. Figure 2(a) shows $\mathrm{Re}[G_2({\mathbf{r}}_{\perp }^{\prime }-{\mathbf{r}}_{\perp })]$ (real part of $G_2$), $\mathrm{Im}[G_2({\mathbf{r}}_{\perp }^{\prime }-{\mathbf{r}}_{\perp })]$(imaginary part of $G_2$), and ${|U/U_0|}^2$ (the intensity profile of the probe field) as functions of $r_{\perp }^{\prime }/R_b$ by taking $R_0={300}^{-}\mathrm{m}$ (and hence $R_b/R_0=0.019$). In the figure, we see that ${|U|}^2$ varies very slowly compared with $G_2$, and hence the integral $\int {\mathrm{d}}^2{\mathbf{r}}_{\perp }^{\prime }{G}_2({\mathbf{r}}_{\perp }^{\prime }-{\mathbf{r}}_{\perp }){|U({\mathbf{r}}_{\perp }^{\prime },z,\tau )|}^{2}U(\mathbf{r},\tau )$ can be approximated by $[\int {\mathrm{d}}^2{\mathbf{r}}_{\perp }^{\prime }{G}_2({\mathbf{r}}_{\perp }^{\prime }-{\mathbf{r}}_{\perp })]{|U(\mathbf{r},\tau )|}^{2}U(\mathbf{r},\tau )=W_2{|U(\mathbf{r},\tau )|}^{2}U(\mathbf{r},\tau )$. Here, $W_2=\int {\mathrm{d}}^2{\mathbf{r}}_{\perp }^{\prime }{G}_2({\mathbf{r}}_{\perp }^{\prime })=P/Q$, with $P\simeq -i4{\pi }^2{\kappa }_{12}{\mathcal{N}}_a\sqrt{C_6}{|{\mathrm{\Omega }}_c|}^4(2\omega +d_{21}+d_{31})$, $Q\simeq 3D(\omega ){|D(\omega )|}^2\sqrt{D_3(\omega )}$, and $D_3(\omega )=D_2[2(\omega +d_{21}{|{\mathrm{\Omega }}_c|}^2)-D_2(2\omega +d_{31})]$. This means that for a large probe-field radius, the nonlocal Kerr effect contributed by the Rydberg–Rydberg interaction reduces into a local Kerr nonlinearity. As a result, Eq. (3) is simplified into a local nonlinear Schrödinger (NLS) equation with the Kerr coefficient given by $W_1+W_2$. Since the probe-beam radius in this response region is large, the diffraction effect of the system is negligible; the problem is simplified into a $(1+1)D$ one, and hence $(1+1)D$ optical solitons with ultraslow propagating velocity and extremely low light power can be easily obtained. For a detailed discussion of the soliton solutions and their properties in this region, see Section 3 of Supplement 1.

 

Fig. 2. Effective atomic interaction potential $G_2$ as functions of $r_{\perp }^{\prime }/R_b$. We show the local response region (a) with $R_0=300\mathrm{\mu m}$, nonlocal response region; (b) with $R_0=10\mathrm{\mu m}$, and strong nonlocal response region; and (c) with $R_0=1\mathrm{\mu m}$. In all situations, real parts (blue solid line) dominate the imaginary part (orange dashed line). For a better visualization, $G_2$ has been amplified ${10}^8$ times. We also show the intensity profile of the probe field ${|U/U_0|}^2$ (black dotted-dashed line). The purple dashed line in (c) is for the function $G_2(0)+[{\partial }^2{G}_2(0)/\partial r_{\perp }^2]r_{\perp }^2/2$. Parameters are ${\mathrm{\Delta }}_2=-15{\mathrm{\Gamma }}_{12}$ and $R_b={[|C_6{\mathrm{\Delta }}_2|/(2{|{\mathrm{\Omega }}_c|}^2)]}^{1/6}=5.8\mathrm{\mu m}$. These parameters guarantee that the system is in the dispersive nonlinearity regime (i.e., $|{\mathrm{\Delta }}_2|\gg {\mathrm{\Gamma }}_{12}$).

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B. Nonlocal Response Region

This region corresponds to $R_b/R_0\sim 1$, which can be achieved by decreasing the probe-beam radius $R_0$ or increasing the principal quantum number $n$ (thus $R_b$). One example is shown in Fig. 2(b). In the figure, we see that ${|U|}^2$ varies the same way as $G_2$. In this case, the nonlinear optical response contributed by the Rydberg–Rydberg interaction depends not only on a particular point on the light intensity, but also on a certain neighborhood of the field.

To seek possible LBs in the system, we write Eq. (3) into the dimensionless form

However, as indicated in the first section, different from the systems considered in Refs. [9–12,14–17], stable, single LBs bounded in all spatial directions and in time can be realized in the present Rydberg-EIT system. The reasons are the following: (i) the nonlocal Kerr nonlinearity coming from the Rydberg–Rydberg interaction [described by the last term on the left hand side of Eq. (4)] has very fast response speed (with response time on the order of 0.1 μs) [45], and this is very different from the slow nonlocal optical nonlinearity in the systems studied in Refs. [12,14–17], where the response time of the nonlocal optical nonlinearity is in the range of 1 s or even longer; and (ii) the nonlocal optical nonlinearity (${\chi }_p^{(3)}\sim {10}^{-8}{\mathrm{m}}^2{\mathrm{V}}^{-2}$) by the Rydberg–Rydberg interaction is much stronger and possesses a faster response speed than the local optical nonlinearity (${\chi }_p^{(3)}\sim {10}^{-11}{\mathrm{m}}^2{\mathrm{V}}^{-2}$) by the photon–atom interaction [described by the fourth term on the left hand side of Eq. (4), which has a response time on the order of 1 μs]. Based on these important properties (absent in the systems considered in Refs. [9–12,14–17]), a single $(3+1)D$ LB can form in the system via the following two-step mechanism for self-trapping: when a single $(3+1)D$ probe pulse is incident into the system, it is first self-trapped in the two transverse (i.e., $x$, $y$) dimensions via the balance between the diffraction and the fast nonlocal Kerr nonlinearity; then it is further stabilized in the longitudinal (i.e., $z$) direction by the balance between the dispersion and the slow local Kerr nonlinearity. Through such two-step self-trapping processes, a stable, single LB very different from those obtained in Refs. [9–12,14–17] can be realized by the synthetic nonlocal and local Kerr nonlinearities in the Rydberg atomic gas. Such LB can form in a very short distance and extremely low light power due to the giant Kerr nonlinearities and ultraslow propagation velocity of the probe pulse resulted from the Rydberg-based EIT effect.

Before presenting the LB solution, we make an estimation on numerical values of the coefficients in Eq. (4). We choose $R_0=10\mathrm{\mu m}$, ${\tau }_0=9\times {10}^{-7}\mathrm{s}$, $U_0=0.3{\mathrm{\Gamma }}_{12}$, ${\mathrm{\Delta }}_2=-15{\mathrm{\Gamma }}_{12}$, ${\mathrm{\Delta }}_3=-0.02{\mathrm{\Gamma }}_{12}$, ${\mathrm{\Omega }}_c={\mathrm{\Gamma }}_{12}$, and $N_a=8.2\times {10}^{10}{\mathrm{cm}}^{-3}$. Then we obtain $R_b/R_0=0.6$, $g_d=0.134$, and $g_1=0.27$. We thus have the diffraction length $L_{\mathrm{diff}}=1.36\mathrm{mm}$ and the dispersion length $L_D(=L_{\mathrm{LN}})=10\mathrm{mm}$. As expected, $L_{\mathrm{diff}}\ll L_D$, which supports the two-step mechanism for self-trapping described above (i.e., the diffraction plays its role earlier than the dispersion).

According to the above analysis, we can assume the LB solution takes the form

We employ a variational method to solve Eq. (4) by taking the ansatz (5) to be a LB solution [63]. Through a Ritz optimization procedure, the LB energy $E$, defined by $E=\iiint {|u|}^2\mathrm{d}\xi \mathrm{d}\eta \mathrm{d}\sigma =2\pi A^2{w}_s^2{w}_t$, is calculated as a function of the transverse beam width $w_s$, with the result shown in Fig. 3(a). We observe that there are three branches for the LB solution, i.e., curves $C_1$, $C_2$, $C_3$. In the region where $\partial E/\partial w_s<0$ (curve $C_2$), the LB is stable. Yet, in regions $\partial E/\partial w_s>0$ (curves $C_1$ and $C_3$), the LB is unstable. This conclusion is verified by linearizing variational equations around the LB solution and examining their eigenvalues and eigenfunctions; it is also checked by using a numerical simulation on Eq. (4) directly. Figures 3(b)–3(d) show results for pulse energy $E$, beam width $w_s$, and pulse duration $w_t$ as functions of $z/(2L_{\mathrm{diff}})$, obtained, respectively, by choosing initial conditions from curves $C_1$, $C_2$, and $C_3$ in Fig. 3(a), i.e., $(w_s,w_t,E)=(0.08,0.06,1.15)$ [Fig. 3(b)], $(w_s,w_t,E)=(0.66,0.41,0.44)$ [Fig. 3(c)], $(w_s,w_t,E)=(1.5,1.1,0.55)$ [Fig. 3(d)]. We see that only in the case in Fig. 3(c) do the LB’s beam width $w_s$, pulse duration $w_t$, and energy $E$ remain almost unchanged, which means that the LB in the region of curve $C_2$ is indeed stable during propagation. Note that the stable single LB solution obtained here is localized in all three spatial dimensions and also in time.

 

Fig. 3. Stability of light bullets. (a) Light bullet energy $E$ as a function of the transverse beam width $w_s$. In the region where $\partial E/\partial w_s<0$ (i.e., curve $C_2$), the light bullet is stable; in regions $\partial E/\partial w_s>0$ (i.e., curves $C_1$ and $C_3$), the light bullet is unstable. Panels (b), (c), and (d) are numerical results of $E$, $w_s$ (transverse beam width), and $w_t$ (pulse duration) as a functions of $z/(2L_{\mathrm{diff}})$, obtained by choosing initial conditions from curves $C_1$, $C_2$, and $C_3$ in panel (a). Stability of parameter set $(w_s,w_t,E)$ with initial conditions (0.08, 0.06, 1.15) (b), (0.66, 0.41, 0.44) (c), and (1.5, 1.1, 0.55) (d).

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Moreover, Eq. (4) admits stable $(3+1)D$ LBs carrying orbital angular momenta (i.e., LVs). To demonstrate this, we take $u=u_{mp}(\xi ,\eta ,\sigma ,\varphi )$ as a test solution, with

In Fig. 4, we illustrate the evolution of ${|u|}^2$. Panel (a) is for the LB [i.e., the fundamental mode ${(\mathrm{LG})}_0^0$] with atomic density ${\mathcal{N}}_a=3\times {10}^{10}{\mathrm{cm}}^{-3}$; panel (b) is for the light vortex corresponding to mode ${(\mathrm{LG})}_0^1[{\mathcal{N}}_a=4.95\times {10}^{10}{\mathrm{cm}}^{-3}]$. In the simulation, the beam radius $R_0$ for both modes is taken to be 10 μm. In Figs. 4(a) and 4(b), we see that for the lower-order modes ${(\mathrm{LG})}_0^0$ and ${(\mathrm{LG})}_0^1$, the pulses have nearly no deformation after propagating to $z=8L_{\mathrm{diff}}(\approx 10.9\mathrm{mm})$.

 

Fig. 4. Evolution of light bullets and vortices in the nonlocal response region. (a) Evolution of ${|u|}^2$ with the fundamental mode ${(\mathrm{LG})}_0^0$ (i.e., light bullet), as a function of $x/R_b$ and $y/R_b$ when propagating to the distance, respectively, at $z/(2L_{\mathrm{diff}})=0$, 1, 2, 3, and 4 for atomic density ${\mathcal{N}}_a=3\times {10}^{10}{\mathrm{cm}}^{-3}$. (b) Evolution of ${|u|}^2$ with the higher-order mode ${(\mathrm{LG})}_0^1$ (i.e., light vortex) for ${\mathcal{N}}_a=4.95\times {10}^{10}{\mathrm{cm}}^{-3}$.

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However, pulses corresponding to the higher-order LG modes may not keep their shape in the nonlocal response region. To show this, we carried out the simulation on the evolution of the LV corresponding to ${(\mathrm{LG})}_1^2$ by choosing $R_0=1.67R_b$, $0.83R_b$, and $0.42R_b$, with the result shown in panels (a), (b), and (c) in Fig. 5, respectively. We see that for a small beam radius [$R_0=0.42R_b$; Fig. 5(c)], the vortex is relatively stable compared with that having a large beam radius [$R_0=1.67R_b$; Fig. 5(a)]. This result means that if the degree of nonlocality in the system increases (i.e., $R_0$ is reduced for fixed blockade radius $R_b$), the lifetime of the vortex pulse may be increased significantly.

 

Fig. 5. Evolution of light vortices corresponding to the mode ${(\mathrm{LG})}_1^2$ in the nonlocal response region. Evolution of ${|u|}^2$ as a function of $x/R_b$ and $y/R_b$ when propagating to the distance at $z/(2L_{\mathrm{diff}})=0,1,2,3$, and 4, respectively. Parameters are $R_0=1.67R_b$, ${\mathcal{N}}_a=9.9\times {10}^{10}{\mathrm{cm}}^{-3}$ (a), $R_0=0.83R_b$, ${\mathcal{N}}_a=1.46\times {10}^{11}{\mathrm{cm}}^{-3}$ (b), and $R_0=0.42R_b$, ${\mathcal{N}}_a=5.3\times {10}^{11}{\mathrm{cm}}^{-3}$ (c) with $R_b=6\mathrm{\mu m}$. Other system parameters are the same as those used in Fig. 4.

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C. Strongly Nonlocal Response Region

We now turn to consider the situation when the range of the Rydberg–Rydberg interaction is much larger than that of the beam radius of the probe pulse, i.e., $R_b/R_0\gg 1$, which corresponds to the case shown in Fig. 2(c), where $\mathrm{Re}(G_2)$, $\mathrm{Im}(G_2)$, and ${|U/U_0|}^2$ as functions of $r_{\perp }^{\prime }/R_b$ are shown by taking $R_0=1\mathrm{\mu m}$ (and hence $R_b/R_0=5.8$). We see that, compared with ${|U/U_0|}^2$, the response function $G_2$ is very flat near $r_{\perp }^{\prime }=r_{\perp }$, which means that $G_2({\mathbf{r}}_{\perp }^{\prime }-{\mathbf{r}}_{\perp })\approx G_2(r_{\perp })\approx G_2(0)+G_2^{\prime \prime }(0)r_{\perp }^2/2$[64,65], plotted by the purple dashed line in Fig. 2(c), agreeing well with the exact one near $r_{\perp }=0$ [i.e., $\mathrm{Re}(G_2)$ (blue solid line); $\mathrm{Im}(G_2)$ is very small]. In this case, Eq. (3) can be reduced into the dimensionless form

Before presenting LB and LV solutions, we first give an example of numerical values of the coefficients in Eq. (7). We choose ${\mathrm{\Delta }}_3=-0.003{\mathrm{\Gamma }}_{12}$, ${\mathrm{\Delta }}_2=-10{\mathrm{\Gamma }}_{12}$, $R_0=1.86\mathrm{\mu m}$, ${\tau }_0=2.6\times {10}^{-7}\mathrm{s}$, ${\mathcal{N}}_a=5.8\times {10}^{12}{\mathrm{cm}}^{-3}$, $U_0=0.15{\mathrm{\Gamma }}_{12}$, and $C_6\simeq 2\pi \times 167\mathrm{THz}{\mathrm{\mu m}}^6$ (for the principal quantum number $n=120$). With these parameters, we obtain diffraction length $L_{\mathrm{diff}}=0.047\mathrm{mm}$, and hence the blockade radius $R_b=19\mathrm{\mu m}(R_b/R_0=10.2)$, $g_d=1.0-0.07i$, $g_1=2.0-0.014i$, $d_0=-0.03$, $g_4=2+0.1i$. Thus, the system is within the strongly nonlocal region with a very small dissipation.

Similarly, a variational method is employed to solve Eq. (7) by using the ansatz (6) as test LB and LV solutions; their stability is investigated through a linear stability analysis. Then a numerical simulation starting directly from Eq. (4) working in this strong nonlocal response region is carried out by taking the solutions obtained via the variational method as an initial condition, together with a random disturbance to it. Specifically, we take $U(z=0,x,y,t)=U_0(1+\epsilon f_R)u_{mp}(z=0,x,y,t)\exp (i{g}_3)$, with $\epsilon$ being a typical amplitude of the perturbation, and $f_R$ being a random variable uniformly distributed in the interval [0, 1]. We find that the system allows indeed stable LBs and LVs. Shown in Figs. 6(a) and 6(b) are time evolution of the $(3+1)D$ LB and LV [corresponding to the fundamental mode ${(\mathrm{LG})}_0^0$ and the higher-order mode ${(\mathrm{LG})}_1^2$] by taking $\epsilon =0.1$, where isosurface plots are illustrated of these pulses when propagating to distances $s=z/(2L_{\mathrm{diff}})=0,1,2,3,4$, respectively. We observe that these $(3+1)D$ nonlinear optical pulses relax to self-cleaned forms quite close to the unperturbed ones, and their shapes undergo no apparent change during propagation.

 

Fig. 6. Evolution of light bullets and light vortices in the strongly nonlocal response region. Evolution of ${|u|}^2$ with ${(\mathrm{LG})}_0^0$ mode [panel (a); light bullet] and ${(\mathrm{LG})}_1^2$ mode [panel (b); light vortex], obtained based on solving Eq. (3). Isosurface plots of the pulses are shown when propagating to distances $s=z/(2L_{\mathrm{diff}})=0,1,2,3,4$, respectively.

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Based on the solution (6) and using the parameters given above, we get the propagating velocity of the LBs and LVs:

5. STORAGE AND RETRIEVAL OF $(3\mathbf{+}1)D$ WEAK LIGHT BULLETS AND VORTICES

One of main advantages of EIT is the possibility for realizing an active manipulation of optical pulses by tuning system parameters. Especially, optical pulses can be stored and retrieved through switching off and on of the control field. In recent years, a number of studies have been carried out to build light memories using EIT [18,23,66], including ones performed in Rydberg atomic systems [39,41]. However, it is generally difficult to realize the memory of high-dimensional nonlinear optical pulses via conventional EIT because of the catastrophic collapse during propagation. Nevertheless, we will show that the storage and retrieval of the LBs and LVs with high efficiency and high fidelity are possible in the present Rydberg-EIT system.

To this end, we investigate the evolution of the LBs and LVs described above through solving the MB Eqs. (1) and (2) and by using a control field that is switched on and off adiabatically, which can be described by the switching function with the form ${\mathrm{\Omega }}_c(t)={\mathrm{\Omega }}_{c0}\{1-1/2\tanh [(t-T_{\mathrm{off}})/T_s]+1/2\tanh [t-T_{\mathrm{on}}/T_s]\}$, where $T_{\mathrm{off}}$ and $T_{\mathrm{on}}$ are times at which the control field is switched off and on. The duration of the switching time is $T_s$, and the storage time of the probe pulse is approximately given by $T_{\mathrm{on}}-T_{\mathrm{off}}$.

A. Memory in the Nonlocal Response Region

We first study the storage and retrieval of the $(3+1)D$ LB in the nonlocal response regime. The spatial waveshape of the input probe pulse is taken as a fundamental LG mode [i.e., ${(\mathrm{LG})}_0^0$ mode], and the temporal profile is assumed as a hyperbolic secant one. Figure 1(c) shows the numerical result of the probe-pulse intensity ${|U/U_0|}^2$ during the process of storage and retrieval. The pulse waveshapes for $z=0$ (before storage), $z=4L_{\mathrm{diff}}$ (at the beginning of storage), and $z=8L_{\mathrm{diff}}$ (after storage), with $L_{\mathrm{diff}}=1.36\mathrm{mm}$, are plotted. We see that when the control field ${\mathrm{\Omega }}_c$ is switched on, the $(3+1)D$ LB is created; by switching off the control field, the LB is stored in the atomic medium; then the LB is retrieved when ${\mathrm{\Omega }}_c$ is switched on again. Note that the retrieved LB has nearly the same waveshape as that before storage. During storage, the information of the LB is converted into that of atomic spin wave (i.e., the coherence ${\rho }_{13}$). The slight deformation of the LB after storage is due to dissipation (dephasing and spontaneous emission) as well as the small imbalance between the diffraction, dispersion, and nonlinearities in the system.

It is possible to store LVs in the nonlocal response regime in the same way. As an example, we show the storage and retrieval of a $(3+1)D$ LV of ${(\mathrm{LG})}_0^1$ mode in Fig. 7. The black dashed line in the figure shows the process of the switching on, switching off, and re-switching on of the control field $|{\mathrm{\Omega }}_c{\tau }_0|$. Curves 1, 2, and 3 are temporal profiles of the probe pulse $|{\mathrm{\Omega }}_p{\tau }_0|$ when the LV propagates at $z=0$ (before storage), $z=4L_{\mathrm{diff}}$ (at the beginning of storage), and $8L_{\mathrm{diff}}$ (after storage), respectively. Corresponding isosurface plots with $|{\mathrm{\Omega }}_p{\tau }_0|=0.1$ illustrate the storage of retrieval of the LV.

 

Fig. 7. Storage and retrieval of $(3+1)D$ light vortices with the ${(\mathrm{LG})}_0^1$ mode in the nonlocal response region. The black dashed line shows the switching on, switching off, and re-switching on of the control field $|{\mathrm{\Omega }}_c{\tau }_0|$. The curves 1, 2, and 3 are temporal profiles of the probe pulse $|{\mathrm{\Omega }}_p{\tau }_0|$, respectively, at $z=0$ (before the storage), $z=4L_{\mathrm{diff}}$ (at the beginning of the storage), and $8L_{\mathrm{diff}}$ (after the storage), with $L_{\mathrm{diff}}=1.36\mathrm{mm}$; the corresponding isosurface plots for $|{\mathrm{\Omega }}_p{\tau }_0|=0.1$are also shown. In the calculation, we have set $R_0=10\mathrm{\mu m}$, $C_6\simeq 2\pi \times 81.6\mathrm{GHz}{\mathrm{\mu m}}^6$, ${\mathrm{\Omega }}_p{\tau }_0=27$, ${\mathrm{\Delta }}_2{\tau }_0=-1340$, ${\mathrm{\Delta }}_3{\tau }_0=-1.8$, and ${\mathrm{\Omega }}_{c0}{\tau }_0=90$ with ${\tau }_0=9\times {10}^{-7}\mathrm{s}$. The control parameters are $T_s=0.2{\tau }_0$, $T_{\mathrm{off}}=5{\tau }_0$, $T_{\mathrm{on}}=15{\tau }_0$.

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The quality of the storage and retrieval of optical pulses can be characterized by two parameters, i.e., memory efficiency $\eta$ and memory fidelity $\zeta$, which are defined in Section 4 of Supplement 1. Based on the results obtained above, for LB [LV with ${(\mathrm{LG})}_0^1$ mode], we have $\eta =93.12\%$ and $\zeta =90.57\%$ ($\eta =92.38\%$ and $\zeta =89.01\%$) for $L_z=8L_{\mathrm{diff}}=10.9\mathrm{mm}$, which means that the memory of the LBs and LVs in the Rydberg atomic system is quite high.

For comparison, Fig. 8 shows the result of the memory for the LB when the Rydberg–Rydberg interaction is absent. We see that the LB suffers a significant deformation after storage, with the fidelity of memory only 6.3% (for the memory of LVs without the Rydberg–Rydberg interaction, the fidelity is even lower), which cannot be applied for light information processing because the information is lost during storage.

 

Fig. 8. Memory of light bullets for the case without the Rydberg–Rydberg interaction. Storage and retrieval of $(3+1)D$ LB at $z=0$(before storage), $z=L_{\mathrm{diff}}$ (at the beginning of storage), and $2L_{\mathrm{diff}}$ (after storage), with $L_{\mathrm{diff}}=1.36\mathrm{mm}$; the corresponding isosurface plots for $|{\mathrm{\Omega }}_p{\tau }_0|=0.1$ are also shown.

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B. Memory in the Strong Nonlocal Response Region

We now turn to the strong nonlocal response region and explore the storage and retrieval of the $(3+1)D$ LBs and LVs. Shown in Fig. 9(a) [Fig. 9(b)] is the numerical result of the storage and retrieval of a $(3+1)D$ LB [LV with the ${(\mathrm{LG})}_1^2$ mode]. In the figure, the black dashed line is the time sequence of the switching on and switching off of the control field $|{\mathrm{\Omega }}_c{\tau }_0|$. Curves 1, 2, and 3 are temporal profiles of the probe pulse $|{\mathrm{\Omega }}_p{\tau }_0|$, respectively, at $z=0$ (before storage), $z=6L_{\mathrm{diff}}$ (at the beginning of storage), and $12L_{\mathrm{diff}}$ (after storage), with $L_{\mathrm{diff}}=0.047\mathrm{mm}$; the corresponding isosurface plots for $|{\mathrm{\Omega }}_p{\tau }_0|=0.1$ are also illustrated. Here, system parameters used in the calculation are the same as those given in the nonlocal regime discussed above but with $R_0=1.86\mathrm{\mu m}$, $R_b=19\mathrm{\mu m}$, and ${\tau }_0=2.7\times {10}^{-7}\mathrm{s}$. In the figure, we see that the LB (LV) after storage retains nearly the same waveshape as that before storage. The efficiency and fidelity of the LB memory (LV memory) in this strong nonlocal region can reach to $\eta =93.97\%$ and $\zeta =91.86\%$ ($\eta =93.14\%$ and $\zeta =89.11\%$). Even higher values can be reached by tuning systemic parameters (i.e., transverse beam radius $R_0$ or principal quantum number $n$) to enhance the nonlocality.

 

Fig. 9. Storage and retrieval of $(3+1)D$ light bullets and light vortices in the strong nonlocal response region. (a) Memory of the LB [${(\mathrm{LG})}_0^0$ mode]. The black dashed line shows the switching on and off of the control field $|{\mathrm{\Omega }}_c{\tau }_0|$. Curves 1, 2, and 3 are temporal profiles of the probe pulse $|{\mathrm{\Omega }}_p{\tau }_0|$, respectively, at $z=0$ (before storage), $z=6L_{\mathrm{diff}}$ (at the beginning of storage), and $12L_{\mathrm{diff}}$ (after storage), with $L_{\mathrm{diff}}=0.047\mathrm{mm}$; the corresponding isosurface plots for $|{\mathrm{\Omega }}_p{\tau }_0|=0.1$ are also shown. (b) The same as (a) but for the memory of the LV with the ${(\mathrm{LG})}_1^2$ mode.

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The calculation on the storage and retrieval of other (higher-order) LVs in the strong nonlocal response region is also carried out (not illustrated here). The result shows that optical memories of higher-order LVs can also be realized with high efficiency and fidelity. All these results indicate that the Rydberg medium supports high-quality memory for various LG modes. Since LG modes carry definite orbital angular momentum and hence more information, the use of the strong, nonlocal Rydberg–Rydberg interaction provides the possibility to realize a high-quality multi-dimensional light memory.

6. CONCLUSION

In summary, we have carried out a detailed investigation on the formation, propagation, and storage of ultraslow weak-light bullets and vortices via Rydberg-EIT in a cold atomic gas. By using an approach beyond mean-field theory, we have shown that the system may acquire two types of Kerr nonlinearities, i.e., the fast-responding giant nonlocal Kerr nonlinearity (contributed by the Rydberg–Rydberg interaction) and the relatively weak, slow-responding local Kerr nonlinearity (contributed by the photon–atom interaction); both of them are enabled by the Rydberg-EIT. We have derived a $(3+1)D$ nonlinear envelope equation governing the spatiotemporal evolution of the high-dimensional probe pulse and present various $(3+1)D$ LB and LV solutions. The resulting LBs and LVs have very slow propagating velocity and extremely low generation power, and can be stabilized by the interplay between the synergetic local and nonlocal Kerr nonlinearities. These $(3+1)D$ nonlinear optical pulses can be dynamically manipulated based on the active character of the system; especially, they can be stored and retrieved with high efficiency and fidelity.

Our study expands the breadth in the study of higher-dimensional, nonlocal nonlinear optics with Rydberg atomic gases. Recent experiments have reported the proof-of-principle demonstration of storing light modes with orbital angular momentum [54–58]. The highly efficient method proposed in this work has potential applications in topological quantum information processing. We expect that our predictions can be verified by combining experimental techniques on Rydberg-EIT and light storage developed in recent years with the study of cold Rydberg gases.