Multiple Self-Organized Phases and Spatial Solitons in Cold Atoms Mediated by Optical Feedback

We study the transverse self-structuring of cold atomic clouds with effective atomic interactions mediated by a coherent driving beam retroreflected by means of a single mirror. The resulting self-structuring due to optomechanical forces is much richer than that of an effective-Kerr medium, displaying hexagonal, stripe and honeycomb phases depending on the interaction strength parametrized by the linear susceptibility. Phase domains are described by Ginzburg-Landau amplitude equations with real coefficients. In the stripe phase the system recovers inversion symmetry. Moreover, the subcritical character of the honeycomb phase allows for light-density feedback solitons functioning as self-sustained dark atomic traps with motion controlled by phase gradients in the driving beam.

Figure 1

Optomechanical SFM scheme. A far detuned input beam of amplitude ${\mathcal{E}}_{+,0}$ and wave number $k_0$ illuminates a cloud of rubidium atoms of thickness $L$, optical density $b_0$, and temperature $T$. The reflected ${\mathcal{E}}_{-}$ provides feedback by means of the dipole potential, leading to self-structuring with critical wavelength $\mathrm{\Lambda }$ [3].

Figure 2

Optomechanical self-structured phases obtained at fixed $b_0=110$ and $T=300\mu \mathrm{K}$. (a),(d) ${\mathbf{H}}^{-}$ phase at $\mathrm{\Delta }=25$. (b),(e) $\mathbf{S}$ phase at $\mathrm{\Delta }=55$. (b),(e) ${\mathbf{H}}^{+}$ phase at $\mathrm{\Delta }=90$.

Figure 3

Numerically observed stability domains of the $\mathbf{S}$, ${\mathbf{H}}^{\pm }$ phases at fixed $I/I_0$. The observed boundaries match the values of the susceptibility $\chi$ from Eqs. (15)–(16). The $\mathbf{S}$ phase (gray) is absolutely stable on a domain sandwiched between the lines corresponding to the ${\chi }_{1,2}^{\mathbf{S}}$ points (red), around $\chi =1$ (black). Moreover, the $\mathbf{S}$ phase coexists with ${\mathbf{H}}^{\pm }$ and minimizes the free energy in the region between the ${\chi }_{1,2}^{\mathbf{H}}$ (dashed-black) and ${\chi }_{1,2}^{*}$ points (dashed-green). ${\mathbf{H}}^{\pm }$ phases are stable within the yellow and cyan domains and absolutely stable outside ${\chi }_{1,2}^{\mathbf{H}}$.

Figure 4

(a) Lyapunov functionals for the three phases at $p=1.2$: ${\mathcal{F}}_{{\mathbf{H}}^{+}}(\chi )$ (blue), ${\mathcal{F}}_{{\mathbf{H}}^{-}}(\chi )$ (orange), ${\mathcal{F}}_{\mathbf{S}}(\chi )$ (green). The resulting minimum determines the observed self-structured phase while ${\chi }_1^{*}$ and ${\chi }_2^{*}$ identify the boundaries in Fig. 3. Note that ${\mathcal{F}}_{{\mathbf{H}}^{+}}={\mathcal{F}}_{{\mathbf{H}}^{-}}$ for $\chi =1$ (dotted line). (b) Critical ${\mu }_{\mathbf{S}}^{>}$ and ${\mu }_{{\mathbf{H}}^{\pm }}^{>}$ (dashed-black and red lines) and phase boundaries (dashed-green) as functions of $\chi$. Intersections with $\mu$ (blue) determine the size of the $\mathbf{S}/\mathbf{H}$ competition region.

Figure 5

Optomechanical solitons for blue detuning. (a) Amplitude of the ${\mathbf{H}}^{\pm }$, $\mathbf{S}$ branches as functions of $p$ for $\chi \approx 0.31$ ($b_0=50$, $\mathrm{\Delta }=80$, $\sigma \approx 78.3$), plotted together with the numerical amplitude (black and red triangles). (b)–(c) Dark backwards intensity and bright density profiles at $p=0.98$.

Figure 6

1D angular dynamics simulation of $4\times {10}^4$ atoms with a driving beam possessing OAM (index $l=1$) [40]. (a) Density evolution $n(\overline{x},\overline{t})$, numerically reconstructed from particle trajectories. (b) Phase space distribution at $\overline{t}=120$.