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 The critical slowing down should lead to a strong re- sponse of the population difference to a field detuned by 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 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 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 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$_{nm}$ . down. As a result, the symmetry breaking is quantita- tively described by the mean-field theory. 2 FIG. 1: (Color) (a) The two-dimensional density profiles of the vibrating clouds for the maximal displacement along the MOT axis ( ≈ 4.5 mm). The lower, middle, and upper panels refer to the symmetric, close to critical, and broken-symmetry states, with N$_{tot}$ ≈ 1 . 5 × 10 $^{6}$, 5 . 2 × 10 $^{6}$, and 6 . 7 × 10 6 , respec- tively; N$_{c}$ ≈ 5 . 6 × 10 $^{6}$. The cloud centers are indicated by the ticks at the bottom of the panels and are sketched in the right panel. (b) The amplitude changes of the cloud vibra- tions as functions of the population of the other cloud. The red dashed lines show the expected linear dependence. The black dashed line shows N$_{1}$ = N$_{2}$ = N$_{c}$/ 2. The amplitude of the cloud with smaller population beyond the transition, cloud 1, monotonically increases with N$_{tot}$ , whereas that of the larger cloud, cloud 2, is maximal at criticality. The error bars show the standard deviation for 20 measurements.