with the radiation intensity, see Fig. 2d. 

An extra modulation of the beam intensities h cos( ω$_{F}$ t/ 2+ φ$_{h}$ ) leads to factors exp( h$_{12}$ ) and exp( − h$_{12}$ ) in the switching rates W$_{12}$ and W$_{21}$ , with h$_{12}$ ∝ h/T [21]. In calculating these factors one can disregard the weak atom-atom interaction. As a result, P st 1 is multiplied by exp( N$_{tot}$h$_{12}$x ). This gives η ∝ h 1 / 3 at criticality ( θ = 0), in agreement with the experiment. If the frequency of the extra modulation is slightly detuned from ω$_{F}$ / 2, the linear response does not diverge at criticality. The result of the mean-field calculation in the Supplementary Ma- terial is in excellent agreement with the experiment, as seen from Fig. 3. 

An independent estimate of the interaction strength can be obtained from the dependence of the vibration amplitudes of the clouds on the cloud populations. For comparatively weak interaction, the interaction-induced change of the amplitude of an n th cloud should be pro- portional to the number of atoms in the other cloud $^{N$_{3}$}$− n ( n = 1 , 2). This is indeed seen in Fig. 1c. The main contribution comes from the long-range interaction, the shadow effect [4, 23, 24]. This interaction also gives the main contribution to the parameters α, β and thus deter- mines the critical total number of atoms N$_{c}$ . The value of N$_{c}$ obtained in a simple one-dimensional model [16, 21] that assumes sinusoidal vibrations is within a factor of 2 from the experimental data, as are also the slopes of the straight lines in Fig. 1c. 

In the experiment, the total number of trapped atoms was slowly fluctuating. The mean-field theory still re- mains applicable, but needs to be extended, see Sup- plementary Material. An important consequence is that in the low-symmetry phase, 1 ≫ θ ≫ $_{N}$− 1 / 2 c , the vari- 

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We thank M. C. Cross, G. S. Jeon, W. D. Phillips, D. L. Stein, and T. G. Walker for helpful comments. This work was supported by the Acceleration Research Pro- gram of Korean Ministry of Science and Technology. MD was supported in part by NSF grant PHY-0555346. WJ acknowledges support from Yeonam Foundation. 

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