Emergent crystallinity and frustration with BECs in multimode cavities

We now justify the limit of approximately adiabatic switching (discussed in the Effective Action section of the main text), which underpins our functional-integral formalism. The extent to which the assumption of adiabatic switching fails is given by the flux of energy through the system, which is the sum of the irreversible contributions due to (i) spontaneous emission and (ii) cavity-photon loss, viz., Nγ(Ω/4∆A)+2κ  αnα [1]. Below the threshold for self-organization, nα ∝ NΩ; slightly above threshold, nα ∝ Ωρα, where ρα(∝ N) is the order parameter for self-organization. Our model, whose focus is to describe the threshold regime, is concerned only with the flux near threshold. We shall see that Ωth (i.e., the laser’s Rabi frequency at threshold) scales as N−1/2; therefore, below thresohld, the energy flux is independent of particle number, and contributes negligibly, per particle, as N → ∞. Above but near threshold, ρα/N  1, because the transition is weakly first-order; together with the assumption that κ/∆C  1 (see the section on Experimental Considerations in the main text), this implies that the dissipative dynamics due to cavity loss is dominated by the Hamiltonian dynamics of the atom-cavity coupling, and for sufficiently large particle number it is therefore valid to neglect the departure from equilibrium near the transition.