Formation and propagation of stable high-dimensional soliton molecules and breather molecules in a cold Rydberg atomic gas

Abstract

We investigate the mechanisms of formation of stable (2+1)-dimensional optical soliton molecules (SMs) and breather molecules (BMs) in a Rydberg atomic gas, highlighting the distinct roles of nonlocality. The underlying giant, nonlocal nonlinearity induced via Rydberg electromagnetically induced transparency (EIT), supports diverse, large-size lattice SMs (rhombic, square, checkerboard, hexagonal lattice SMs). Crucially, we identify two distinct formation regimes: In the nonlocal regime, long-range interactions alone stabilize the SMs without requiring initial motion. In contrast, within the strongly nonlocal regime, an initial velocity is essential to generate a centrifugal force that counteracts the strong attraction, resulting in rotating SMs. Furthermore, specific initial velocities can induce a periodic breathing instability, leading to the formation of BMs. Our study offers a new scheme for engineering SMs with diverse configurations and opens new avenues for data processing and transmission in optical systems.

Figures (7)

• Figure 1: (a) Energy-level diagram and excitation scheme for ladder-type three-level atoms. A weak probe laser field (half-Rabi frequency $\Omega_{p}$) couples transition $|1\rangle\leftrightarrow|2\rangle$. A strong control laser field (half-Rabi frequency ${\Omega}_c$) couples transition $|2\rangle\leftrightarrow|3\rangle$. In Rydberg state $|3\rangle$, atoms interact strongly through the van der Waals interaction $V_{\mathrm{vdW}}(\mathbf{r'} - \mathbf{r}) = -\hbar C_6 / |\mathbf{r'} - \mathbf{r}|^6$. Here $\Delta_{\alpha}$ are detuning and $\Gamma_{\alpha\beta}$ ($\alpha<\beta$) are spontaneous emission decay rates. (b) The long-range interaction between Rydberg atoms blocks the excitation of the atoms within blockade spheres [the blockade spheres boundary indicated by the red dashed lines]. In each blocked sphere only one Rydberg atom (small dark blue sphere) is excited and other atoms (small dark yellow spheres) are prevented to be excited. The red and purple arrow indicates the propagating direction of the probe and control fields. (c) The contactless interaction between four optical solitons, which form a stable $2\times2$ rhombic-lattice SM and undergoes no apparent distortion during propagation.
• Figure 2: Binding energy. (a) Under local nonlinearity ($\sigma = 0.1$), the binding energy lacks a minimum, indicating no stable SMs. In contrast, under strong nonlocal nonlinearity ($\sigma = 8$), a minimum occurs at $d = 0.61$, suggesting the formation of a small-sized SM. Parameters: $A = 4$, $a = 1$, $b = 0$, and $v = 0$. (b) In the nonlocal nonlinear regime ($\sigma = 1.75$), a stable equilibrium position for SMs is observed at $d = 3.87$. Parameters: $A = 4$, $a = 0.95$, and $v = 0$. The inset shows a rhombic-lattice type four-soliton molecule. (c) Under strong nonlocal nonlinearity ($\sigma = 8$), varying the initial velocity leads to different equilibrium positions: $d = 3.6$ for $v = 0.8$; $d = 3.23$ for $v = 1$; and $d = 2.94$ for $v = 1.2$. Other parameters: $A = 4$, $a = 1$, and $b=0$.
• Figure 3: The propagation of rhombic-lattice SMs. The initial conditions of the solitons are set as $A = 4$, $a = 0.95$, $d = 3.87$, $b = 0$, and a 5% random perturbation is introduced. (a) Under the condition of local nonlinearity, with $\sigma = 0.1$, the solitons are unstable because of diffraction. (b) In the case of nonlocal nonlinearity, with $\sigma=1.75$, the SM propagate stably and remain unaffected by the perturbation. (c) When a angular speed $v = 0.1$ is added, the SM exhibit centrifugal motion. The red dashed circle marks the initial separation $d$; the white solid curve traces the instantaneous radius of the center-of-mass trajectory.
• Figure 4: The propagation dynamics of $2\times2$ square-lattice SM, $3\times3$ checkerboard-lattice SM, hexagonal-lattice SM in nonlocal nonlinearity regime, i.e., $\sigma = 1.75$. (a) Stable propagation of the $2\times2$ square-lattice SM with $d = 4$. (b) For the $3\times3$ checkerboard-lattice SM, stable propagation was achieved when the equilibrium position was set at $d = 5.68$. (c) Solitons arranged in a hexagonal-lattice SM also demonstrated stable propagation at $d = 5.57$. The parameters of the initial input are taken as $A = 4$, $a = 0.95$, and $b=0$.
• Figure 5: (Color online) The propagation of rhombic-lattice SMs and BMs. (a) During propagation, the SM first executes a centripetal motion until it reaches an extremum, then switches to a centrifugal motion until it returns to the initial radius, with initial velocity $v = 0.8$. This periodic breathing repeats indefinitely with a period of $\pi/2$. Simultaneously, the BM rotates, completing one full turn every $\pi$. As the tangent velocity is directed counter-clockwise, the BM rotates counter-clockwise. The rotation period is $\pi$. (b) When initial velocity $v = 1$, the SM exhibit a counterclockwise rotational motion. The rotation circle's radius is exactly that of the initial ring. (c) Same as (a), but for $v=1.2$. The SM first executes a centrifugal motion until it reaches an maximum, then switches to a centripetal motion until it returns to the initial radius. The rotation and breathing periods are the same as those in panel (a). In the $\xi$-$\eta$ cross-sections, $d$ and $d_1$ represent the radius at the corresponding propagation distance $\zeta$. In the last cross-section, the solid circle marks the initial radial position of the solitons ($d$); the dashed circle indicates the extremal radial position reached during the breathing cycle ($d_1$). The four circles mark the spatial positions of the first soliton at $\zeta = 0,\, \pi/4,\,\pi/2,$ and $3\pi/4$, respectively. At $\zeta = \pi$, the first soliton returns to its initial position. The other parameters same as Fig. \ref{['bl']}(c). Panels (a1), (b1), and (c1) display the trajectory equations corresponding to Figs. (a), (b), and (c), respectively; (a1) and (c1) are ellipses with eccentricities 0.60 and 0.55, while (b1) is a circle ($e=0$).