2 to the long-range interaction and is described in terms of the continuous atomic density [1]. The fluctuating part is dominated by short-range elastic collisions. For com- paratively low atomic densities and high temperatures used in the experiment the random part of G$_{int}$ is small compared to the thermal noise and will be disregarded. The inter-cloud interaction is determined by long-range forces and can be described in the mean-field approxima- tion: the force G$_{int}$ ( r ) from another cloud changes only slightly when atoms in this cloud shift by an interatomic distance. From the above arguments, the interatomic force on an atom in cloud n , G$_{int}$ ( r ; n ), can be written as where N$_{m}$ is the number of atoms in cloud m . The force G$_{int}$ ( r ; n | m ) is determined by the density distribution in the cloud m , and the terms with n = m and n = m describe the force from the atoms in the same and in the different cloud, respectively. Even though G$_{int}$ is small compared to the single- atom force G , it may strongly affect the switching rate, leading to the spontaneous symmetry breaking. This is because G$_{int}$ changes the switching activation energy R$_{n}$ . Extending to the many-atom system the approach of Ref. 5, to first order in the interaction we obtain R$_{n}$ = R (0) + R (1) n with Here, r (opt) n ( t ) and λ (opt) n ( t ) describe the trajectory that minimizes functional R , Eq. (3), for switching from state n . Function r (opt) n ( t ) gives the most probable path fol- lowed in switching; for a parametrically excited oscillator this path was recently observed in experiment [10]. Since the states 1 and 2 differ only in phase, we have α$_{11}$ = α$_{22}$ and α$_{12}$ = α$_{21}$ . The following arguments were used in deriving Eq. (5). First, because the interatomic interaction is weak, atoms switch one by one, not in groups; switchings of individual atoms are uncorrelated. Second, and most importantly, in the optimal fluctuation leading to switching of an atom the atomic clouds practically do not change their shape. This is also because the interatomic interaction is weak, Because atoms move fast within the clouds compared to inter-cloud switching, on time scale ∼ W − 1 nm the state of the system is fully characterized by the numbers of atoms in each cloud N$_{1}$,N$_{2}$ . For a given N$_{tot}$ , the system is described by probability P$_{1}$ ( N$_{1}$ ) to have N$_{1}$ atoms in cloud 1 at time t . This probability can be found from the master equation where W (0) = C$_{W}$ exp( − R $^{(0)}$/T ) is the switching rate in the neglect of the interatomic interaction; α = ( α$_{11}$ − α$_{12}$ ) / 2 k$_{B}$T and β = − ( α$_{11}$ + α$_{12}$ ) / 2 k$_{B}$T . Equation (6) coincides with Eq. (1) of the paper and provides a general form of the coefficients α,β . Equation (5) shows that the effective switching activa- tion energy linearly depends on the numbers of atoms in the clouds, for weak interatomic coupling. Even though | R (1) n | is small compared to the single-atom activation en- ergy R $^{(0)}$, it can largely exceed k$_{B}$T , in which case the change of the switching rate will be large. From Eqs. (2) and (5) we obtain for the many-atom switching rate and the force from a single atom in the cloud very weakly affects the motion of the switching atom, | α$_{nm}$ | ≪ k$_{B}$T . Only the effect from all atoms in the cloud is apprecia- ble, the product α$_{nm}$N$_{m}$ ( N$_{1}$$_{,}$$_{2}$ ≫ 1) can become larger than k$_{B}$T . On the other hand, a significant change of the shapes of the clouds, even though it would change the switching activation energy, has an exceedingly small probability, which can not compensate the decrease of R (1) 1 , $_{2}$. Indeed, if the cloud radius is a$_{0}$ ∼ ( k$_{B}$T/m$_{a}$ω 2 $_{cl}$) 1 / $^{2}$, the probability density to have the atoms in n th cloud displaced by Δ r$_{n}$ ≫ a$_{0}$ is ∝ exp( − N$_{n}$ Δ r 2 $_{n}$/ 2 a 2 $_{0}$). There- fore in calculating G$_{int}$ ( r (opt) n ( t ); n | m ) one should assume that the atoms in clouds 1 and 2 are at their stable equi- librium positions r$_{1}$ ( t ) and r$_{2}$ ( t ). In other words, there is no fluctuational barrier preparation [11, 12] for intercloud switching, even though the switching barrier is strongly affected by the many-body interaction. This significantly simplifies the explicit calculation. G$_{int}$ ( r ; n ) = ∑ m =1 , $_{2}$N$_{m}$ G$_{int}$ ( r ; n | m ) , (4) R (1) n = ∑ m α$_{nm}$N$_{m}$, α$_{nm}$ = − ∫ ∞ −∞$^{dt}$$^{λ}$$^{(opt)}$ n ( t ) G$_{int}$ ( r (opt) n ( t ); n | m ) . (5) W$_{nm}$ ≡ W$_{nm}$ ( N$_{n}$ ; N$_{tot}$ ) = W $^{(0)}$e( α + β ) N$_{tot}$ − 2 $^{αN$_{n}$}$, N$_{tot}$ = N$_{1}$ + N$_{2}$, (6) II. THE SYMMETRY BREAKING TRANSITION A. The master equation ˙ P$_{1}$ ( N$_{1}$ ) = − [ µ ( N$_{1}$ ) + ν ( N$_{1}$ )] P$_{1}$ ( N$_{1}$ ) + ν ( N$_{1}$ − 1) P$_{1}$ ( N$_{1}$ − 1) + µ ( N$_{1}$ + 1) P$_{1}$ ( N$_{1}$ + 1) , µ ( N$_{1}$ ) = N$_{1}$W$_{12}$ ( N$_{1}$ ; N$_{tot}$ ) , ν ( N$_{1}$ ) = ( N$_{tot}$ N$_{1}$ ) W$_{12}$ ( N$_{tot}$ − N$_{1}$ ; N$_{tot}$ ) . (7) Here we have taken into account that, as a result of a n → m transition, the number of atoms in cloud n de- creases by 1 and in cloud m increases by 1, and that the