Supplementary Material for “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 4 $^{1}$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 $^{4}$Department of Physics and Astronomy, Seoul National University, Seoul 151-747, Korea (Dated: November 17, 2018) arXiv:1001.0749v1 [cond-mat.stat-mech] 5 Jan 2010 PACS numbers: I. THEORY OF INTER-CLOUD SWITCHING A. Single-atom switching B. Switching rate modification by the atom-atom interaction Here we provide a detailed closed-form description of the critical phenomena in a parametrically modulated MOT. We start with the single-atom dynamics for reso- nant MOT modulation, where the modulation frequency ω$_{F}$ is close to twice the eigenfrequency ω$_{0}$ of atomic vi- brations in the MOT. The modulation can lead to the onset of period-two vibrations at frequency ω$_{F}$ / 2. These vibrations are close to sinusoidal for moderately strong modulation, there is no dynamical chaos in the range of phase space of interest. In the single-atom approximation the equation of motion can be written as where r is the position vector of an atom, m$_{a}$ is the atomic mass, and G is the well-understood force from the laser beams [1–3]; for modulated beams it periodically depends on time, G ( t + τ$_{F}$ ) = G ( t ), with τ$_{F}$ = 2 π/ω$_{F}$ being the modulation period. The force f ( t ) in Eq. (1) is the zero-mean noise from spontaneous light emis- sion, which is often modeled by white Gaussian noise, 〈 f$_{κ}$ ( t ) f$_{κ}$$_{′}$ ( t ′ ) 〉 = 4Γ m$_{a}$k$_{B}$T δ$_{κκ}$$_{′}$δ ( t − t ′ ), where Γ is the effective friction coefficient, T is the characteristic gas temperature, and κ, κ ′ = 1 , 2 , 3 enumerate the atomic coordinates. We are interested in the parameter range where, in the absence of noise, Eq. (1) has two stable periodic so- lutions r$_{1}$$_{,}$$_{2}$ ( t ), with r$_{n}$ ( t + 2 τ$_{F}$ ) = r$_{n}$ ( t ) ( n = 1 , 2) and r$_{2}$ ( t ) = r$_{1}$ ( t + τ$_{F}$ ). These solutions give the positions of the centers of the vibrating atomic clouds in Fig. 1a of the main paper. By linearizing Eq. (1) about r$_{n}$ ( t ) one finds the typical frequencies ω$_{cl}$ and relaxation rates ( ∼ Γ) of the intracloud motion. Noise leads to fluctua- tions about r$_{n}$ ( t ) with variance ∝ T that determines the size of the clouds in Fig. 1. Switching between the clouds results from large rare fluctuations. It is central for the analysis that the switch- ing rates are much smaller than ω$_{F}$ and than ω$_{cl}$, Γ. Therefore of interest are period-averaged rates W$_{nm}$ of The interaction between atoms leads to an extra force G$_{int}$ ( r ) in Eq. (1). This force is weak in the sense that it weakly affects the intracloud atomic dynamics. In the experiment, the radius of the clouds changed by ∼ 10% where the number of atoms changed by a factor of 2. The change of the vibration amplitude was also small, see Fig. 1 of the main paper. The force G$_{int}$ depends on the positions of all atoms. In the analysis of the effect of the interaction on the in- tracloud dynamics it can be separated into the mean- field and fluctuating parts. The mean-field part is due Since in the single-atom case the period-two vibra- tional states differ only by phase, the activation ener- gies R$_{1}$$_{,}$$_{2}$ are equal, R$_{1}$ = R$_{2}$ = R $^{(0)}$. The variational problem is significantly simplified if the vibrations r$_{n}$ ( t ) are almost sinusoidal, which corresponds to small damp- ing and comparatively weak nonlinearity, see below. For a one-dimensional weakly nonlinear oscillator R (0) was found and measured earlier [7–9]. Here, the term quadratic in f determines the probability density of a realization of the random force that leads to switching [6]; λ ( t ) is the Lagrange multiplier that relates the dynamics of the atom and the force to each other. For switching from n th periodic state, the trajectory r ( t ) should go from r$_{n}$ ( t ) for t → −∞ to the unstable periodic saddle-type state r$_{b}$ ( t ) on the boundary of the basin of attraction to r$_{n}$ ( t ) for t → ∞ ; λ ( t ) , f ( t ) → 0 for t → ±∞ . where the prefactor C$_{W}$ ∼ max( ω$_{cl}$, Γ). The most im- portant in W$_{nm}$ is the exponential factor, which has the activation form. The activation energy R$_{n}$ of switching from the periodic state n is given by the solution of a variational problem R$_{n}$ = min R , where the functional R is [5], n → m switching, n, m = 1 , 2. Quite generally [4, 5] W$_{nm}$ = C$_{W}$ exp( − R$_{n}$/k$_{B}$T ) , (2) R = (8Γ m$_{a}$ ) − 1 ∫ dt f ( t ) 2 + ∫ dt λ ( t ) [ m ¨ r − G − f ( t )] (3) m$_{a}$ ¨ r = G ( r , ˙ r ; t ) + f ( t ) (1)