10 0.01 0.1 1 10 100 1000 10000 20000 22000 24000 26000 28000 30000 32000 34000 36000 ϵ k in units of ω R cavity mode, full GPE first excited mode, full GPE cavity mode, 2 mode model first excited mode, 2 mode model 0.1 1 10 100 1000 10000 20000 22000 24000 26000 28000 30000 32000 34000 36000 ϵ k in units of ω R 0.1 1 10 100 1000 10000 20000 22000 24000 26000 28000 30 0 00 32000 34000 36000 ϵ k in units of ω R Δ$_{C}$ in units of ω$_{R}$ 20000 22000 24000 26000 28000 30000 32000 34000 36000 10 − 10 10 − 8 10 − 6 10 − 4 10 − 2 10 0 10 2 10 4 | γ k | in units of ω R (a) γ$_{k}$ < 0 γ$_{k}$ > 0 20000 22000 24000 26000 28000 30000 32000 34000 36000 10 − 8 10 7 10 − 6 10 5 10 − 4 10 3 10 − 2 10 1 10 0 10 1 10 2 10 3 | γ k | in units of ω R (b) γ$_{k}$ < 0 γ$_{k}$ > 0 20000 22000 24000 26000 28000 30000 32000 34000 36000 10 − 7 10 6 10 − 5 10 4 10 − 3 10 2 10 − 1 10 0 10 1 10 2 10 3 | γ k | in units of ω R Δ$_{C}$ in units of ω$_{R}$ (c) γ$_{k}$ < 0 γ$_{k}$ > 0 FIG. 4: (color online) The fluctuation sp ectrum ω$_{k}$ = ϵ$_{k}$ + iγ$_{k}$ of the atom-cavity dynamics. The real and imaginary parts of the eigenvalues are plotted as functions of Δ$_{C}$ . All quantities are measured in units of the recoil frequency. The parameter settings are the same as for Fig. 2. Note, that the fluctuation spectrum of the full model, discretized on a 200 grid, contains 201 pairs of eigenvalues. We only plot those 2, which correspond to those of the optomechanical system. Note that M is a general complex matrix. Its eigenval- ues are complex. However, due to the sym metry relation (9) the eigenvalues of M come in pairs. Each element of the pair have the same imaginary part, while the real parts are just the opposite of each other. Figure 4 shows the eigenvalues of M for the same parameter settings as that of Figs. 2 and 3. The left panels sho w the real part ϵ$_{k}$ of the k th eigenvalue, while the right panels show the modulus of the imaginary parts γ$_{k}$ for ω$_{k}$ = ϵ$_{k}$ + iγ$_{k}$ . Only those two eigenvalues are plotted, which are present in the optomechanical model and have p ositiv e real parts. The other two eigenvalues of the optomec hanical model can be obtained simply by changing the sign of the real parts to negative: ω$_{k}$ → − ϵ$_{k}$ + iγ$_{k}$ . In the bistability regime, the excitation frequencies are presen ted for both mean field solutions. the resonance. The other eigenvalue, having a real part around 4 ω$_{R}$ corresponds mainly to atomic fluctuations. The imaginary part of this eigenvalue changes sign at res- onance. Below resonance the imaginary part is negative (fluctuations in the atomic motion are damped due to the interaction with the resonator field [39, 40]), while above resonance the imaginary part becomes positive (fluctua- tions in the atomic motion are exponentially growing in time). The situation is clear on panel a), where the reso- nance point can be exactly defined. On panels b) and c) the coexistence of the two solutions makes the definition of the resonance point ambiguous. Nevertheless, on pan- els b) and c) the curves which continue to the left hand side of the figure correspond to the solutions with cavity cooling and those which continue to the right hand side correspond to the heating solution. The dynamical cool- ing and heating effects are closely related to the same effects at the single atom level [41–45]. Again, all figures show that the optome chanical ap- proximation very well reproduces the re sults of the full GPE simulation even for dynamical quan tities. The two modes are easy to be physically in terpreted. The first eigenvalue, having a bigger real part around δ = Δ$_{C}$ − N 〈 U 〉 and an imaginary part almost exactly at − κ , corresponds to a mainly photonic mode. Note that the imaginary part of this mode remains also negative above In Refs. [3, 4] the coherent atomic dynamics was also studied. The harmonic oscillator behavior of the low-energy atomic dynamics caused periodic oscillations in the output photon signal with a frequency close to 35 kHz, as reported and explained in Ref. [3], or close to around 42 kHz in Ref. [4]. In our model the dynam-