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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 

<img src='content_image/422580.jpg'>

where 

<img src='content_image/422579.jpg'>

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 

<img src='content_image/422577.jpg'>

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, 

<img src='content_image/422576.jpg'>

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 

<img src='content_image/422578.jpg'>

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 ). 

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 . 

## A. The variance of the order parameter 

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 

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 | − $^{1}$. 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 

<img src='content_image/422575.jpg'>

