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a small frequency Ω from ω$_{F}$ / 2. This is an analog of ap- plying a weak slowly varying field to thermal equilibrium systems. The amplitude and phase of the forced oscilla- tions of η at frequency Ω as functions of θ are shown in Fig. 3. They display characteristic asymmetric resonant features, again in excellent agreement with the mean-field theory. 

In general, one should not expect that our oscillating system can be described by the Landau-type theory with standard critical exponents, and in fact the theory has to be extended. We explain the observations as due to kinetic many-body effects. They turn out to be strong as a consequence of the exponential sensitivity of the inter- cloud switching rates W$_{nm}$ . 

If the atom-atom coupling can be disregarded, the switching rates are equal by symmetry, W (0) 12 = W (0) 21 ≡ W (0) = C exp ( − R $^{(0)}$/k$_{B}$ T ) with C ∼ ω$_{cl}$ , where ω$_{cl}$ ≈ 3Γ is the frequency of small-amplitude damped atom vibrations about the cloud center. Of primary inter- est is the activation energy R $^{(0)}$. For weekly non- sinusoidal period-2 vibrations it was calculated and mea- sured for single-oscillator systems with cubic nonlinear- 

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ity [8, 16, 17]. If we use the same model, for the present experimental parameters the theory [16] gives R $^{(0)}$/k$_{B}$ ≈ 3.4 mK. This is within a factor of 2 of the value extracted from the measured W (0) and estimated T , which is satisfactory given that in the present case the parametric modulation was comparatively strong (the squared MOT eigenfrequency ω 2 0 was modulated with rel- ative amplitude 0.9), and therefore the vibrations were noticeably non-sinusoidal. 

The atom-atom interaction changes the switching ac- tivation energy. The change becomes significant only because of the cumulative effect of the interaction of a switching atom with the atoms in the clouds. For not too strong interaction the rate of 1 → 2 switching be- comes 

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with α, β ∝ 1 /T (to obtain W$_{12}$ one should replace N$_{1}$ ↔ N$_{2}$ ). The parameters α, β are determined by the generalized work of the interatomic force done on an atom during its intercloud switching, see Supplemen- tary Material. The probability P$_{1}$ to have N$_{1}$ atoms in cloud 1 for given N$_{tot}$ can be found from the bal- ance equation. For N$_{tot}$ ≫ 1, the stationary probabil- ity P st 1 ( N$_{1}$ ) is a function of the reduced population dif- ference x = ( N$_{2}$ − N$_{1}$ ) /N$_{tot}$ and the control parameter θ = αN$_{tot}$ − 1. It has a sharp maximum whose position gives the order parameter η = 〈 x 〉 . For fixed N$_{tot}$ , near criticality ( x , θ 1) the station- 

| | | | ≪ ary distribution has a characteristic Landau mean-field form P st 1 ∝ exp[ − N$_{tot}$ ( x 4 − 6 θx $^{2}$) / 12] [22]. For θ < 0 it has a peak at x = η = 0, which corresponds to equal mean cloud populations. For θ > 0 and | θ | ≫ $_{N}$− 1 / 2 tot the symmetry is broken and 〈 x 〉 = η = ± (3 θ ) 1 / $^{2}$. This describes the dependence of η on θ in Fig. 2a. The form of P st 1 explains also the scaling σ 2 ∝ | θ | − 1 of the variance σ 2 = 〈 x $^{2}$〉 − 〈 x 〉 2 in Fig. 2b. The critical total number of atoms (where θ = 0) is 

N$_{c}$ = α − $^{1}$. Thus, N$_{c}$ ∝ T . This dependence was tested by heating up the atoms with additional resonant radi- ation. Indeed, N$_{c}$ was found to increase almost linearly 

