# 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. 

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 

PACS numbers: 05.70.Fh, 05.70.Ln, 32.10.Ee, 05.40.-a 

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]. 

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. 

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%. 

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 

