Inversion-symmetry breaking in spin patterns by a weak magnetic field

Laser-driven cold atoms near a plane retroreflecting mirror exhibit self-organization above a pump threshold. We analyze the properties of self-organized spin patterns in the ground state of cold rubidium atoms. Antiferromagnetic patterns in zero magnetic field give way to ferrimagnetic patterns if a small longitudinal field is applied. We demonstrate how the experimental system can be modeled as spin-1 atoms diffractively coupled by the light reflected by the mirror. The roles of both dipolar and quadrupolar magnetization components in determining the threshold and symmetry variations with a weak longitudinal magnetic field are examined. Although the magnetic structures correspond dominantly to a lattice of magnetic dipoles, the symmetry breaking to ferrimagnetic structures in a finite field is mediated by the coupling to a homogenous quadrupole (alignment), which is not possible in a spin-1/2 system. Our study provides a basis for further exploration of instabilities in driven multilevel systems with feedback.

Figure 1

Simplified schematics of the experimental setup, adapted from Ref. [25]. A linearly polarized pump beam is reflected from a feedback mirror to drive the spin self-organization in the atomic cloud. A small transmitted part of the beam is used for polarization selective NF and FF (not shown) imaging. Inversion symmetry is broken by applying a small longitudinal magnetic field (B field), resulting in formation of hexagons and honeycombs in the $\sigma$ polarization channels (inset: NF data). Rb, cold cloud of ${}^{87}\mathrm{Rb}$; M, feedback mirror; $\lambda /4$, quarter-wave plate; PBS, polarizing beam-splitter cube; CCD, charge-coupled device camera.

Figure 2

Near-field images of the transmitted part of the reentrant beam intensities (see text) in the self-organized magnetization phases, for varying $B_z$. (a) Columns (left to right): $B_z=-0.09$ G, $B_z=0$, $B_z=0.12$ G. Rows (top to bottom): ${\sigma }^{+}$, ${\sigma }^{-}$ and ${\sigma }^{+}-{\sigma }^{-}$ (Strathclyde). The difference images are normalized to their respective maximum absolute values. The situations for $B_z\ne 0$ correspond to ferrimagnetic spin lattices, whereas the situation for $B_z=0$ corresponds to an antiferromagnetic spin lattice. (b) Top rows: ${\sigma }^{+}$, ${\sigma }^{-}$ images for $B_z=1.4$ G (Strathclyde). Bottom rows: ${\sigma }^{+}$, ${\sigma }^{-}$ images for $B_z=0.54$ G (Nice). Experimental parameters (Strathclyde): $b_0=27$, $\mathrm{\Delta }=-14{\mathrm{\Gamma }}_2$, $I=10\mathrm{mW}/{\mathrm{cm}}^2$, $d=-2.9$ mm, $R=0.95$, field of view $0.36\times 0.36{\mathrm{mm}}^2$. Experimental parameters (Nice): $b_0=110$, $\mathrm{\Delta }=-24{\mathrm{\Gamma }}_2$, $I=22\mathrm{mW}/{\mathrm{cm}}^2$, $d=-20$ mm, $R=0.99$, field of view $3.15\times 3.15{\mathrm{mm}}^2$.

Figure 3

Diffracted power at small longitudinal magnetic fields (Strathclyde). (a) $I=14\mathrm{mW}/{\mathrm{cm}}^2$. (b) $I=19\mathrm{mW}/{\mathrm{cm}}^2$. (c) $I=24\mathrm{mW}/{\mathrm{cm}}^2$. Dots, ${\sigma }^{+}$ light; circles, ${\sigma }^{-}$ light. Experimental parameters: $\mathrm{\Delta }=-14{\mathrm{\Gamma }}_2$, $b_0=27$, $d=-2.9$ mm.

Figure 4

Diffracted power and threshold intensity for positive $B_z$ (Strathclyde). (a) Threshold pump intensity. (b) Total diffracted power for varying input beam intensity. Triangles (black): $I=3.1\mathrm{mW}/{\mathrm{cm}}^2$. Diamonds (red): $I=5.5\mathrm{mW}/{\mathrm{cm}}^2$. Squares (blue): $I=7.1\mathrm{mW}/{\mathrm{cm}}^2$. Experimental parameters: $\mathrm{\Delta }=-14{\mathrm{\Gamma }}_2$, $b_0=27$, $d=1.3$ mm.

Figure 5

Model atomic system. (a) Zeeman sublevel structure of the $F=1\to F^{\prime }=2$ transition with the corresponding Clebsch-Gordan coefficients. (b) Illustration of symmetries of the tensors related to the magnetic moments $w$, $x$, $u$, $v$, from left to right.

Figure 6

Growth rate of the patterns for a one-dimensional (1D) numerical simulations (see text). (a) Growth rate $\lambda$ at $I_0=10\mathrm{mW}/{\mathrm{cm}}^2$ against the diffractive phase shift $\theta$ (see text) for $B_z=0$ (black, top circles), $B_z=20$ mG (blue, middle circles), and $B_z=1.5$ G (red, bottom circles). The values for $B_z=1.5$ G were multiplied by 10 for visibility. (b) Scan of the fastest growth rate ${\lambda }_m$ against $B_z$ for three input intensities $I_0$: $10I_s$ (red, solid), $15I_s$ (blue, dot-dashed), and $20I_s$ (black, dashed). Parameters: $b_0=30$, $\mathrm{\Delta }=-10{\mathrm{\Gamma }}_2$, $R=0.95$, $\gamma ={10}^{-4}{\mathrm{\Gamma }}_2$.

Figure 7

Near-field steady-state data (depicting intensity of the reentrant feedback field $B_{\pm }={\tilde{B}}_{\pm }(z=L)$ and atomic orientation $w$), taken from two-dimensional (2D) simulations. Columns (left to right): $B_z=-140$ mG, $B_z=0$, $B_z=140$ mG. Rows (top to bottom): $|B_{+}{|}^2$, $|B_{-}{|}^2$, $|B_{+}{|}^2-{\left|B_{-}\right|}^2$ and the $w$ variable. Simulation parameters: $b_0=60$, $I_0=15I_s$, $\mathrm{\Delta }=-10{\mathrm{\Gamma }}_2$, $R=0.95$, $\gamma ={10}^{-4}{\mathrm{\Gamma }}_2$, simulation time: $2\times {10}^4/{\mathrm{\Gamma }}_2$. The size of the numerical grid was adjusted to contain seven periods of the lattice. Periodic boundary conditions are used in the simulations.

Figure 8

Variation of threshold intensity with $B_z$. Red solid curve: full solution using (8). Green dot-dashed curve: solution for the model neglecting both linear and nonlinear Faraday effects [using (9)]. Blue dashed curve: solution of $w$-only model (see text). Parameters: $\mathrm{\Delta }=-14{\mathrm{\Gamma }}_2$, $R=0.95$, $b_0=30$, $\gamma =3\times {10}^{-4}{\mathrm{\Gamma }}_2$.

Figure 9

Homogeneous solutions of (1) at threshold, against $B_z$. Coherences (a) $u$ and (b) $v$, (c) alignment $x$ and (d) orientation $w$. The narrow features for small $B_z$ are caused by magnetic field coupling to the coherences $u$ and $v$ (i.e., coherent nonlinear Faraday rotation), whereas the features at large $B_z$ are caused by linear and nonlinear Faraday rotation due to detuning via Zeeman shifts (see text). This is confirmed by the solution of the $w$-only model, represented by the blue dash-dotted curve of panel (d). The scan is representative of the single-pass behavior for the $F=1\to F^{\prime }=2$ transition at low pump saturation. Parameter values are as in Fig. 8.

Figure 10

Dependence of the threshold $\eta$ coefficient on $B_z$. (a) Parameters as in Fig. 8. (b) Parameters: $\mathrm{\Delta }=-24{\mathrm{\Gamma }}_2$, $R=0.99$, $b_0=110$, $\gamma =1\times {10}^{-4}{\mathrm{\Gamma }}_2$. Red solid line: calculated from (14) for $p=1$. Blue dash-dotted line: calculated from the $w$-only model (see text).