An Ideal Mean-Field Transition in a Modulated Cold Atom System Myoung-Sun Heo, 1 Yonghee Kim, 1 Kihwan Kim, 1 Geol Moon, 1 Junhyun Lee, 1 Heung-Ryoul Noh, 2 M. I. Dykman, 3, ∗ and Wonho Jhe 1, † $^{1}$Department of Physics and Astronomy, Seoul National University, Seoul 151-747, Korea $^{2}$Department of Physics, Chonnam National University, Gwangju 500-757, Korea $^{3}$Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA (Dated: November 17, 2018) We show that an atomic system in a periodically modulated optical trap displays an ideal mean- field symmetry-breaking transition. The symmetry is broken with respect to time translation by the modulation period. The transition is due to the interplay of the long-range interatomic interaction and nonequilibrium fluctuations. The observed critical dynamics, including anomalous fluctuations in the symmetry broken phase, are fully described by the proposed microscopic theory. PACS numbers: 05.70.Fh, 05.70.Ln, 32.10.Ee, 05.40.-a The mean-field approach has been instrumental for de- veloping an insight into symmetry breaking transitions in thermal equilibrium systems [1]. It has been broadly ap- plied also to nonequilibrium systems, the studies of pat- tern formation being an example [2]. However, close to the phase transition point the mean field approximation usually breaks down. This happens even in finite-size systems provided they are sufficiently large. In this paper we study a nonequilibrium system of ∼ 10 7 particles which, as we show, displays an ideal mean-field symmetry breaking transition. It is accompa- nied by anomalous fluctuations, which are also described by a mean-field theory. The system is formed by moder- ately cold atoms in a magneto-optical trap (MOT) [3, 4]. The atoms are periodically modulated in time [5]. Peri- odically modulated systems form one of the most impor- tant classes of nonequilibrium systems, both conceptually and in terms of applications. They have discrete time- translation symmetry: they are invariant with respect to time translation by modulation period τ$_{F}$ . Nevertheless, they may have stable vibrational states with periods that are multiples of τ$_{F}$ , in particular 2 τ$_{F}$ . Period doubling is well-known from parametric resonance, where a system has two identical vibrational states shifted in phase by π . It is broadly used in classical and quantum optics and has attracted significant attention recently in the context of nano- and micro-mechanical resonators [6–9]. In a many-body system, dynamical period doubling in itself does not break the time translation symmetry. This is a consequence of fluctuations. Even though each vi- brational state has a lower symmetry, fluctuations make the states equally populated and the system as a whole remains symmetric. However, if as a result of the inter- action the state populations become different, the sym- metry is broken. It is reminiscent of the Ising transition where the interaction leads to preferred occupation of one of the two equivalent spin orientations, except that The beam intensities were modulated at frequency ω$_{F}$ = 2 ω$_{0}$ by acousto-optic modulators. When the modu- lation amplitude exceeded a threshold value, after a tran- sient, the atomic cloud split into two clouds [5], which were vibrating in counter-phase with frequency ω$_{F}$ / 2, see Fig. 1a. We took snapshots at the maximum separation of the clouds at the frame rate of 1-50 Hz to obtain the In the experiment, $^{85}$Rb atoms in a MOT were cooled down to ≈ 0.4 mK. Full three-dimensional confinement was achieved with three pairs of counter-propagating laser beams. The intensity of the beams along the MOT axis z was 0.19 mW/cm $^{2}$, the transverse beam intensities were 5 to 10 times larger. The magnetic field gradient at the trap center was 10 G/cm. The transverse beams were detuned from the atomic transition by δ ≈ − 2 . 3Γ$_{p}$, the longitudinal beams were further detuned by 5 MHz (the atomic decay rate is Γ$_{p}$ / 2 π ≈ 5 . 9 MHz). The atomic cloud motion was essentially one-dimensional, along z - axis, with frequency ω$_{0}$/ 2 π ≈ 45 Hz and the damping rate Γ ≈ 36 s − $^{1}$. The total number of trapped atoms N$_{tot}$ was varied by changing the hyperfine-repumping laser in- tensity at a decrease rate of 0.5% per second. For the or- der parameter and the variance measurements very close to criticality, the decrease rate was reduced to 0.03%. Here we show that the time-translation symmetry breaking in a modulated MOT results from the coop- eration and competition between the inter-particle inter- action and the fluctuations that lead to atom switching between the vibrational states. The interaction is weak. On its own, it cannot change the state populations. How- ever, it may become strong enough to change the rates of fluctuation induced switching, which in turn will cause the population change. We provide a quantitative theory of the phenomenon. We measure the critical exponents and the frequency dispersion of the susceptibility. The observations are consistent with the mean-field behavior and are in good agreement with the theory. the symmetry is broken in time. For atoms in a para- metrically modulated MOT spontaneous breaking of the discrete time-translation symmetry was observed exper- imentally by Kim et al. [10]. arXiv:1001.0749v1 [cond-mat.stat-mech] 5 Jan 2010 ∗ Electronic address: dykman@pa.msu.edu † Electronic address: whjhe@snu.ac.kr