Fig. 1. (a) Excitation scheme of the Rydberg EIT. |1〉, |2〉, and |3〉 are, respectively, the ground, intermediate, and Rydberg states; Ω_p (Ω_c) is the half-Rabi frequency of the probe (control) laser field; Γ₁₂ (∼MHz) and Γ₂₃ (∼kHz) are, respectively, decay rates from |2〉 to |1〉 and |3〉 to |2〉; Δ₂ = ω_p − (ω₂ − ω₁) and Δ₃ = ω_p + ω_c − (ω_c − ω₁) are, respectively, the one- and two-photon detunings. ℏV(r − r^′) is the vdW interaction between the two atoms in Rydberg states, respectively, located at r and r^′. (b) Geometry of the system. The probe and control fields counter-propagate in the Rydberg atomic gas. (c) Normalized χ_p,2⁽³⁾ (i.e., the coefficient of nonlocal Kerr nonlinearity) as a function of coordinate x, with the solid black (dashed red) line representing its real part Re(χ_p,2⁽³⁾) [imaginary part Im(χ_p,2⁽³⁾)] for coordinate y = 0 (see text for more details). Evolution of a (d1) nonlocal LB and (d2) vortex in the system.

Fig. 2. Stern–Gerlach deflections of nonlocal LBs. (a) 3D motion trajectory of an LB as a function of x/R₀, y/R₀, and z/(2L_diff) in the presence of the gradient magnetic field (B₁,B₂) = (3.2, 0) mG cm⁻¹; (c) 3D motion trajectory of the LB for (B₁, B₂) = (6.4, 0) mG cm⁻¹. (b) and (d) are trajectories of the LB in the x–z plane, corresponding, respectively, to panels (a) and (c).

Fig. 3. Motion trajectory of the LB in the presence of a time-varying gradient magnetic field. (a) Trajectory of the LB as a function of x/R₀, y/R₀, and z/(2L_diff) when the time-varying gradient magnetic field of Eq. (9) is present. (b) The corresponding sinusoidal trajectory of the LB in the x–z plane.