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population of each cloud. For a small total number of trapped atoms N$_{tot}$ the populations were equal and the cloud vibrations had equal amplitudes. 

Once N$_{tot}$ exceeded a critical value N$_{c}$ ( N$_{c}$ ∼ 10 7 in our experiment), the average populations of the clouds became different, as observed earlier [10]. The vibra- tion amplitudes became different, too, see Fig. 1b. This is spontaneous breaking of the discrete time-translation symmetry, the system becomes invariant with respect to time translation by 2 τ$_{F}$ , not the modulation period τ$_{F}$ . 

As phase transitions in equilibrium systems, the ob- served transition is a many-body effect. This is evidenced by the required critical number of atoms, which is finite but large, typical of phase transitions in cold atom sys- tems [11–15]. However, in contrast to systems in ther- mal equilibrium, our system is not characterized by the free energy and has a discrete time-translation symme- try. Most importantly, there are no long-range corre- lations of spatial density fluctuations. The intra-cloud density fluctuations are irrelevant, they are uncorrelated with the inter-cloud fluctuations, which, as we show, are responsible for the transition and display critical slowing 

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down. As a result, the symmetry breaking is quantita- tively described by the mean-field theory. 

The irrelevance of intra-cloud density fluctuations is a consequence of the time scale separation and the weak- ness of the interaction. The decay of correlations of intr- acloud density fluctuations is characterized by the damp- ing rate Γ of atomic motion, which is determined by the Doppler effect in the MOT [4] and does not change at the phase transition. On the other hand, switching be- tween periodic states in modulated systems requires a large fluctuation and involves overcoming an effective ac- tivation barrier [8, 16–20]. For low temperature, switch- ing is a rare event. We measured the rates W$_{nm}$ of inter- cloud n → m switching ( n, m = 1 , 2) directly for low N$_{tot}$ from the decay time of the difference of the cloud pop- ulations, which gave W$_{12}$ = W$_{21}$ ∼ 1 s − $^{1}$. Therefore, in the experiment there is a strong inequality between the characteristic reciprocal times, ω$_{F}$ ≫ Γ ≫ W$_{nm}$ . During switching an atom is far from the clouds. Its 

interaction energy with individual atoms in the clouds is much smaller than k$_{B}$T . However, the total energy of interaction with all atoms in the clouds may exceed k$_{B}$T , and then it changes the inter-cloud switching rate. The interaction depends on the density distribution within the clouds and ultimately on the cloud populations N$_{1}$$_{,}$$_{2}$ . 

The switching rates modification by the atom-atom in- teraction is the underlying mechanism of the symmetry breaking. Since these are the cloud populations N$_{1}$$_{,}$$_{2}$ that control the switching rates and that are changed as a re- sult of switching, we choose their mean relative difference η = 〈 N$_{2}$ − N$_{1}$ 〉 /N$_{tot}$ as the order parameter. A natural control parameter is the reduced total number of atoms 

θ = ( N$_{tot}$ − N$_{c}$ ) /N$_{c}$ . Our main observations are presented in Figs. 2 and 3. They show that all measured quantities display mean- field critical behavior. The order parameter scales as | η | ∝ θ β for θ > 0, with β ≈ 1 / 2, see Fig. 2a, and η = 0 for θ < 0 (this scaling was also seen earlier [10], but now we have come much closer to the critical point, and the precision is much higher). Fluctuations of the population difference are characterized by the variance σ 2 = $_{N}$− 2 tot 〈 ( N$_{2}$ − N$_{1}$ ) $^{2}$〉 − η $^{2}$. It scales as σ ∝ | θ | − ˜ γ with ˜ γ ≈ 1 for positive and negative θ , see Fig. 2b. Equi- librium systems at criticality show a strongly nonlinear response to the symmetry-breaking static field, like a magnetic field at the Ising transition. For our system an analogue of such field is an additional periodic driv- ing at half the strong-field frequency. With this field, the overall time-translation period is 2 τ$_{F}$ , and the vibra- tional states of period 2 τ$_{F}$ become non-equivalent [21]. We studied the effect of the dynamic symmetry breaking field at criticality, θ = 0, by asymmetrically modulat- ing the counter propagating beams. Fig. 2c shows that the cloud population difference scales with the additional field amplitude h as | η | ∝ h 1 /δ with δ ≈ 3, in agreement with the mean field theory. 

The critical slowing down should lead to a strong re- sponse of the population difference to a field detuned by 

