6 decreases below W$_{out}$/ 4 ˜ W , there occurs a crossover to σ 2 ≈ 1 / 2 N$_{c}$θ , i.e., σ 2 scales with θ with the same scaling exponent, but with smaller amplitude than in the sym- metric phase. We note that there are other fluctuations that can con- tribute to the variance σ $^{2}$. For example, slow fluctuations of the trap parameters may lead to small and slow fluc- tuations of the total population, which can be thought of as fluctuations of the control parameter θ . They give an extra contribution 3 〈 ( δθ ) $^{2}$〉 / 4 〈 θ 〉 to σ 2 in Eq. (12) for the symmetry-broken phase, where 〈 δθ $^{2}$〉 is the variance of θ . Interestingly, this contribution also scales as θ − 1 away from the critical point. It further increases the dif- ference between the variance of the order parameter in the broken-symmetry and symmetric phase. approximation (RWA), the equations for slow variables q ≡ ( q$_{1}$, q$_{2}$ ) in slow time τ are Here, f ( τ ) is white Gaussian noise with two asymp- totically independent components, 〈 f$_{κ}$ ( τ ) f$_{κ}$$_{′}$ ( τ ′ ) 〉 = 2 D$_{τ}$ k$_{B}$T δ$_{κκ}$$_{′}$δ ( τ − τ ′ ), with κ, κ ′ = 1 , 2 and D$_{τ}$ = 6 γ Γ /m$_{a}$ω 5 $_{0}$ϵ 2 $_{F}$. We use vector notations here formally , q is not a vector in real space, its components are com- binations of atomic displacement and momentum along the MOT axis. Functions K ( q ) are cubic polynomials and are given explicitly in Ref. 14 [they were denoted by K $^{(0)}$( q )]. The zeros of K occur at the stationary states in the rotating frame and, in turn, at the period-two vibrational states in the laboratory frame. We will be interested in the param- eter range where Eq. (27) has two attractors q A 1 = − q A 2 . Functions K are also equal to zero for q = 0 , which corre- sponds to the unstable state of zero-amplitude period-two vibrations in the laboratory frame. Calculation of the switching rates and response coef- ficients is simplified if the atomic motion in MOT is as- sumed one-dimensional, along the MOT axis z . This cor- responds to averaging over the motion transverse to the axis, which is largely small-amplitude fluctuations about the vibrating centers of the clouds. In the experiment the radiation that confined the atoms transverse to the axis had large intensity, so that the correlation time of these fluctuations was short. In the single-atom approxi- mation the atomic motion can be modeled by the Duffing oscillator, Here, ω$_{0}$ and Γ are the MOT eigenfrequency and viscous friction coefficient, respectively, γ is the nonlinearity pa- rameter, and ϵ$_{F}$ is determined by the amplitude of the laser beam modulation [16]. These parameters are given in the main paper. The terms nonlinear in the veloc- ity, which are disregarded in Eq. (25), are comparatively small for the experimental conditions. The function f ( t ) is white thermal Gaussian noise, cf. Eq. (1). For the reduced modulation strength ϵ$_{F}$ ω$_{0}$/ 4Γ > 1, resonant modulation ( | ω$_{F}$ − 2 ω$_{0}$ | ≪ ω$_{0}$ ) can excite period- two atomic vibrations [17]. The vibrations are nearly si- nusoidal in the absence of noise, if the modulation is not too strong. The atomic dynamics can be conveniently de- scribed by switching to the rotating frame using a stan- dard transformation To estimate f$_{sh}$ for a switching atom we have to take into account that the atomic clouds are 3-dimensional. We consider a simple model in which a laser beam with The major inter-atomic interaction that affects the switching rate is the long-range interaction, since in switching atoms move far away from the clouds. It comes from the shadow effect, which is due to “shielding” of atoms from the laser light by other atoms [1, 2, 18–20]. In the one-dimensional picture, the force on i th atom with coordinate z i from j th atom with coordinate z j can be approximately modeled as F i = − f$_{sh}$ ∑ j sgn( z $^{i}$− z $^{j}$). This force is weak, much smaller than the Doppler force that confines atoms to the trap. Multiple scattering of light is disregarded in the above expression. The major distinction from the general formulation (3) is that function K does not explicitly depend on time. This made it possible to solve the variational problem (28) and to find both the optimal trajectories followed in switching from n th stable state q (opt) n ( τ ) [7] and functions λ$_{n}$ ( τ ), which are equal to the logarithmic susceptibilities of Ref. 14 multiplied by − 1 /D$_{τ}$ . For low temperatures, the motion described by Eq. (27) is primarily small-amplitude fluctuations about states q A 1 , $_{2}$. Switching between the states requires a large rare fluctuation. The single-atom switching rate has ac- tivation form W (0) = C exp( − R $^{(0)}$/k$_{B}$ T ), Eq. (2), where as in Eq. (3), R (0) = min R (0) with B. Interatomic interaction IV. ONE-DIMENSIONAL MODEL A. The rotating wave approximation ¨ z + 2Γ ˙ z + ω 2 $_{0}$(1 + ϵ$_{F}$ cos ω$_{F}$ t ) z + γz 3 = f ( t ) /m$_{a}$, (25) R (0) = (4 D$_{τ}$ ) − 1 ∫ dτ f $^{2}$( τ ) + ∫ dτ λ ( τ ) [ d q dτ − K − f ( τ ) ] . (28) d q dτ = K ( q ) + f ( τ ) , τ = ω$_{0}$ϵ$_{F}$ t/ 4 . (27) z = C$_{RWA}$ [ q$_{2}$ cos( ω$_{F}$ t/ 2) − q$_{1}$ sin( ω$_{F}$ t/ 2)] , (26) ˙ z = − C$_{RWA}$ ( ω$_{F}$ / 2) [ q$_{2}$ sin( ω$_{F}$ t/ 2) + q$_{1}$ cos( ω$_{F}$ t/ 2)] with C$_{RWA}$ = (2 ω 2 $_{0}$ϵ$_{F}$ / 3 γ ) 1 / 2 (the modulation sign is cho- sen in such a way that ϵ$_{F}$ /γ > 0). In the rotating wave