Ground-state coherence versus orientation: Competing mechanisms for light-induced magnetic self-organization in cold atoms

We investigate the interplay between two mechanisms for magnetic self-organization in a cloud of cold rubidium atoms subjected to a retroreflected laser beam. The transition between two different phases, one linked to a spontaneous spatial modulation of the $\mathrm{\Delta }m=2$ ground-state coherence and the other to that of the magnetic orientation (spin), can be induced by tuning either a weak transverse magnetic field or the laser intensity. We observe both first- and second-order transitions depending on the presence of the magnetic field. The experimental observations are successfully compared to extended numerical simulations based on a spin-1 model.

Figure 1

Sketch of the experiment. A red-detuned laser beam passes through a large cold ${}^{87}\mathrm{Rb}$ cloud. The beam is then retroreflected, and its intensity distributions in near field and far field are recorded. Two orthogonal polarization channels are simultaneously analyzed. The temporal dynamics is recorded with a photodiode. See text for details.

Figure 2

Analysis of temporal dynamics (experimental). We show an example of the time dependence of the diffracted power. The intensity profile of the laser is shown as a dashed line. Using a linear fit of the initial growth, we extract the growth rate (GR) and the delay ${\tau }_d$.

Figure 3

Near-field distribution of the GSC phase in the lin $\perp$ channel (experimental). The four panels illustrate shot-to-shot fluctuations of the observed patterns. The field of view is $5\times 5$ mm. Parameters: $B_x=1$ G, $I=2$ mW/${\mathrm{cm}}^2$.

Figure 4

Polarization distribution of the GSC phase (experimental). We show FF images (with the zeroth order blocked) recorded in each of the four polarization channels: (a) ${\sigma }^{+}$, (b) ${\sigma }^{-}$, (c) lin $//$, and (d) lin $\perp$. The intensity scale is the same within each pair of orthogonal channels, but different for circular and linear channels. The field of view is 16 mrad $\times$ 16 mrad. Parameters: $B_x=1$ G, $I=2$ mW/${\mathrm{cm}}^2$.

Figure 5

Observation of the GSC phase (numerical). (a) NF intensity (mW/${\mathrm{cm}}^2$) in lin $\perp$ channel, (b) $w$ distribution, (c) $u$ distribution, and (d) $v$ distribution. The field of view is 1.8 mm $\times$ 1.8 mm. Parameters: plane-wave input, $B_x=0.76$ G, $I=6.68\mathrm{mW}/{\mathrm{cm}}^2$.

Figure 6

Domain of observation of the GSC phase in the magnetic-field space (experimental). (a) GSC intensity threshold versus $B_z$ and $B_x$ ($B_y=0$). (b) GSC intensity threshold versus $B_z$ and $B_y$ ($B_x=0.71$ G).

Figure 7

Transition from AFM to GSC phase via $B_x$ tuning (experimental). We plot in panel (a) the diffracted energy and (b) the delay as a function of $B_x$ (in logarithmic scale). The squares and stars correspond to the AFM and GSC phases, respectively. The dashed line indicates the boundary between phases. The laser intensity is $I=16.3$ mW/${\mathrm{cm}}^2$.

Figure 8

Role of OD on AFM and GSC phases (experimental). We plot the scaled diffracted energy (see text) of the AFM (squares) and GSC (stars) phases versus $B_x$, for different values of OD: (1) OD = 110, (2) OD = 86, (3) OD = 57, (4) OD = 52, (5) OD = 41, (6) OD = 31, and (7) OD = 26. Note the double logarithmic scale. The laser intensity is $I=3.6$ mW/${\mathrm{cm}}^2$.

Figure 9

Transition from GSC to AFM phase via laser intensity tuning (experimental). We plot the diffracted energy $E_d$ at beam center as a function of increasing laser intensity, and we observe the transition from the GSC phase (stars, $\times 5$) to the AFM phase (squares). The circles correspond to $E_d=0$. The dashed line indicates the approximate position of the intensity threshold for the AFM phase. The insets illustrate the “blinking” behavior of the AFM phase in the bistability region (see text). The magnetic field is $B_x=0.71$ G.

Figure 10

Behavior of GSC and AFM phases versus laser intensity for different $B_x$ values (experimental). We report (a) the growth rate and (b) the delay for different values of $B_x$: (1) $B_x=0$, (2) $B_x=0.14$ G, (3) $B_x=0.28$ G, (4) $B_x=0.5$ G, (5) $B_x=0.71$ G, (6) $B_x=1.06$ G, (7) $B_x=1.56$ G, and (8) $B_x=2.27$ G. The stars (squares) correspond to the GSC (AFM) phase. The vertical dashed lines indicate the positions of AFM thresholds.

Figure 11

Transition from GSC to AFM phase via laser intensity tuning (numerical). (a) Steady-state diffracted power $P_d$ as a function of laser intensity for different $B_x$ values: (1) $B_x=0.2$ G, (2) $B_x=0.4$ G, (3) $B_x=0.6$ G, (4) $B_x=0.76$ G, and (5) $B_x=1$ G. (b) NF image just above the AFM threshold (simulation with Gaussian beam), showing the coexistence of AFM (center) and GSC (wings).

Figure 12

Critical slowing down and scaling exponents. (a) GSC delay for $B_x=0.71$ G plotted versus distance from threshold $I-I_{\mathrm{th}}$ (experimental). Note the double logarithmic scale. The stars and circles respectively correspond to the measured and rescaled threshold intensity (see text). (b) Scaling exponent for the GSC phase (stars) and the AFM phase (squares) versus $B_x$. The solid and open symbols correspond to experimental and numerical data, respectively. The dashed and dotted lines show theoretically expected values for the two types of bifurcations (see text).