10 − 10 10 − 8 10 − 6 10 − 4 10 − 2 10 0 20000 24000 28000 32000 36000 population (a) β 2 0 β 2 1 10 − 10 10 − 8 10 − 6 10 − 4 10 − 2 10 0 20000 24000 28000 32000 36000 populatio n (b) 10 − 10 10 − 8 10 − 6 10 − 4 10 − 2 10 0 20000 24000 28000 32000 36000 population Δ$_{C}$ in units of ω$_{R}$ (c) 9 part of the fluctuation spectrum changes its sign. Fluc- tuations grow exponentially on this side of the resonance, that is, the slightly retarded cavity field dynamics heats the atomic motion instead of cooling it. Therefore the solution above the resonance ( δ > 0) is dynamically un- stable. It is only due to the adiabatic elimination of the photon field that the numerical method finds this solu- tion as well. Note, that the parameters of our calculations mimics those of Ref. [4]. The agreement in the photon number with Fig. 3 of Ref. [4] is well within the system- atic uncertainty of the photon number estimation of the experiments (25%). With the use of the recoil frequency for Rubidium atoms, ω$_{R}$ = 2 π × 3 . 57 kHz, we can com- pare the location of the resonance points to the experi- mental findings, too. In panel (a), the resonance point is at around 28800 ω$_{R}$ ≈ 2 π × 103 MHz, in panel (b) the two instability points are at around 26700 ω$_{R}$ ≈ 2 π × 95 MHz and 27700 ω$_{R}$ ≈ 2 π × 99 MHz. All these are within 5% of the respective experimental value. In panel (c) the resonance points are at around 21500 ω$_{R}$ ≈ 2 π × 77 MHz and 27000 ω$_{R}$ ≈ 2 π × 96 MHz. This latter is only 2% higher than in the experiment, but the lower point is contrasted to the 2 π × 84 MHz value (about 10 % de- viation). In our case the bistable region is a bit wider than in the experiments. This discrepancy might be at- tributed mainly to the uncertainty in the effective de- tuning δ which we obtain by an estimation of the actual atom number and overlap between the cavity potential and the atomic cloud. FIG. 3: (color online) The squares of the condensate com- ponents β 2 0 and β 2 1 as a functions of the detuning Δ$_{C}$ in a semi-log scale. The parameter settings are the same as for Fig. 2. For η > η$_{c}$ (Δ$_{C}$ , κ, U$_{0}$, N ), the threshold value of the pumping strength depending on the detuning, the cavity loss, the light shift and also the atom number, there is a region where three solutions exist for the photon num- ber and also for the condensate wavefunction. Two of them are stable solutions for the numerical method we use. These solutions are plotted in panels b) and c) for η = 283 . 8 ω$_{R}$ and η = 549 . 5 ω$_{R}$ , respectively. The un- stable solution is not plotted in the figure. We note, however, that the upper one of the two plotted solutions corresponds to the δ > 0, cavity heating, situation. So this solution is also unstable, dynamically. But since this instability is related to photon dynamics (neglected at this level) one can still find this solution by integrating the GPE for the atoms in imaginary time. a lot even if the variation of 〈 U 〉 is small, because of the large number N multiplying it. The linear stability analysis of the mean-field solution is done with the help of the linear equations (7) for the case of the full model, and with the help of Eqs. (32) for the optomechanical approximation. Once the con- densate wavefunction and the steady-state value of the mean cavity field is known, one can construct the ma- trix M and calculate its eigenvalues and eigenvectors. In the full model, discretized on a grid with 200 sites, M is a 402 × 402 matrix, since it acts on a row vector hav- ing 2 elements for the photon fluctuation operator and its hermitian adjoint, and also having 400 elements for the discretized field fluctuation operator and on its her- mitian adjoint. For the optomechanical model one has a much smaller, 4 × 4 matrix, since the zero mode is al- ready separated in this case, so one has 2 components for the photon fluctuations and 2 components for the atomic motion orthogonal to the condensate wavefunction. The numerical diagonalization was obtained with the help of the LAPACK package. The fraction of condensate atoms occupying the ho- mogeneous and the cos k$_{0}$x states are plotted on Fig. 3. The parameter settings are the same as for Fig. 2. Both the resonance feature and the bistabily is exhib- ited in the condensate component of the cos k$_{0}$x mode. The condensate is almost homogeneous for the whole range, β 2 1 ≪ β 2 0 ≈ 1, and β 2 1 shows a very similar de- pendence as the photon number. It is remarkable that a very small change in the shape of the condensate wave- function causes a drastic increase or fall of the photon number. This can be elucidated by the collective cou- pling of the atoms to the resonator field. The detuning δ = Δ$_{C}$ − N 〈 U 〉 , appearing in the denominator of the steady-state value of the resonator field (5), can change B. Fluctuation spectrum