5 the density of outside atoms and choose parameter N$_{0}$ in such a way that it provides the typical scale of N$_{tot}$ , as will be seen below. The master equation for P ( N ) has the form and δ$_{1}$ = (1 , 0) and δ$_{2}$ = (0 , 1) [the inter-cloud switching rate ˜ W is defined in Eqs. (6), (16)]. The terms ˆ L$_{cl}$ and ˆ L$_{out}$ describe inter-cloud transitions and exchange with atoms outside the trap, respectively. We assume that the typical rate of this exchange W$_{out}$ ≪ ˜ W . We now consider the critical region where N$_{tot}$ is close to N$_{c}$ . We will be interested in the range of N$_{1}$, N$_{2}$ where P ( N$_{1}$, N$_{2}$ ) is close to maximum, | N$_{1}$ − N$_{2}$ | ≪ N$_{tot}$ , and introduce quasi-continuous variables As we will see, variables x and θ coincide with the vari- ables used earlier if fluctuations of the total number of trapped atoms can be disregarded; variable u gives the deviation of N$_{tot}$ from N$_{0}$ . For | x | ≪ 1 , | θ | ≪ 1, and | u | ≪ 1 to leading order in x, θ, u , and $_{N}$− 1 0 the operators ˆ L$_{cl}$ and ˆ L$_{out}$ become Equations (20) - (22) allow one to study the stationary distribution of the modulated system in the presence of fluctuations of the total population. We note first that exchange of atoms with the surrounding leads to effec- tive exchange of atoms between the clouds. In turn, this leads to renormalization of inter-cloud transition rates and, effectively, of the control parameter, The change of θ is negative, which should be expected, since exchange with the surrounding should stabilize the symmetric phase, and therefore a larger number of atoms is required for the symmetry breaking transition than in It is seen from Eq. (24) that, for θ ≫ W$_{out}$/ 4 W , the variance of the order parameter fluctuations in the broken-symmetry state scales with the control parame- ter in the same way as without fluctuations of the total population, N$_{c}$σ 2 ≈ 5 / 4 θ . However, the factor in front of 1 /θ is now larger than in the symmetric phase. This corresponds to what was seen in the experiment. As θ Fluctuations of u and x (i.e., of the total population of the clouds and the population difference) are corre- lated in the broken-symmetry phase, with 〈 u ( x − x$_{0}$ ) 〉 ≈ (2 x$_{0}$/N$_{c}$ )[4 θ + ( W$_{out}$/ ˜ W )] − $^{1}$, where x$_{0}$ = ± (3 θ ) 1 / $^{2}$. In contrast, in the symmetric phase for 1 ≫ − θ ≫ $_{N}$− 1 / 2 c fluctuations of u and x are uncorrelated, with N$_{c}$σ 2 ≈ | θ | − 1 [ 1 + ( W$_{out}$/ 2 ˜ W ) ] ≈ | θ | − $^{1}$. ˜ The major effect of the coupling between fluctuations of x and u is the change of the variance σ 2 = 〈 x $^{2}$〉 − 〈 x 〉 $^{2}$. The latter occurs in the broken-symmetry state where θ ≫ $_{N}$− 1 / 2 c . It can be calculated in the stationary regime by decoupling the chain of equations for the moments of the stationary distribution 〈 x $^{k}$u$^{l}$〉 with integer k, l . For $_{N}$− 1 / 2 c ≪ θ ≪ 1 In the stationary regime fluctuations of the total num- ber of atoms in the trap are small, 〈 u $^{2}$〉 ∼ 1 /N$_{0}$ . How- ever, the distribution over the scaled difference of the cloud populations x differs from the standard Landau- type distribution (9). This is in spite the fact that the system is described by the mean-field theory, no spatial correlations are involved. The dynamics of x and u are very different. There is no critical slowing down for u , and in the critical region | θ | ≲ $_{N}$− 1 / 2 c the width of the distribution over x is ∼ $_{N}$− 1 / 4 c ≫ $_{N}$− 1 / 2 c . the case where there is no such exchange. N$_{c}$ is now the critical value of the parameter N$_{0}$ , the mean number of trapped atoms is 〈 N$_{tot}$ 〉 = N$_{0}$ (1+ 〈 u 〉 ). The coefficient of diffusion over x is also renormalized, 2 $_{N}$− 1 0 → 2 $_{N}$− 1 0 (1 + W$_{out}$/ 2 ˜ W ). N$_{c}$σ 2 ≈ 1 2 θ + 3 4 θ + ( W$_{out}$/ ˜ W ) . (24) θ → θ − W$_{out}$ 2 ˜ W , N$_{c}$ → α − 1 ( 1 + W$_{out}$ 2 ˜ W ) . (23) ˆ L$_{cl}$P = − 2 ∂$_{x}$ {[ x ( θ + u ) − 1 3 x 3 ] P } +2 $_{N}$− 1 0 ∂ 2 $_{x}$P, (22) ˆ L$_{out}$P = ∂$_{x}$ ( xP ) + ∂$_{u}$ ( uP ) + $_{N}$− 1 0 [ ( ∂ 2 x + ∂ 2 $_{u}$) P ] . x = N$_{2}$ − N$_{1}$ N$_{0}$ , u = N$_{1}$ + N$_{2}$ N$_{0}$ − 1 , (21) θ = αN$_{0}$ − 1 . µ ′ ( N$_{n}$ ; N$_{tot}$ ) = N$_{n}$ exp ( αN$_{tot}$ − 2 αN$_{n}$ ) ≡ ˜ W − $^{1}$µ ( N$_{n}$ ) ˙ P ( N ) = ˜ W ˆ L$_{cl}$P + W$_{out}$ ˆ L$_{out}$P, ˆ L$_{cl}$P = − ∑ $_{n}$µ ′ ( N$_{n}$ ; N$_{tot}$ ) P ( N ) + µ ′ ( N$_{1}$ + 1; N$_{tot}$ ) P ( N$_{1}$ + 1 , N$_{2}$ − 1) + µ ′ ( N$_{2}$ + 1; N$_{tot}$ ) P ( N$_{1}$ − 1 , N$_{2}$ + 1) , (20) ˆ L$_{out}$P = − ( N$_{1}$ + N$_{2}$ + N$_{0}$ ) P ( N ) + ∑ $_{n}$( N$_{n}$ + 1) P ( N + δ$_{n}$ ) + ( N$_{0}$/ 2) ∑ $_{n}$P ( N − δ$_{n}$ ) , where A. The variance of the order parameter