Figure 8: Threshold as a function of transverse pump power ramp rate.

(a) A comparison between the thresholds for the l+m=0 (blue) and l+m=1 (red) modes for a BEC placed at the centre of the cavity. In both cases the BEC is pumped at a large detuning of Δ_c=−30 MHz. The l+m=0 mode has an antinode at the centre of the cavity, whereas the l+m=1 mode has a node. The overall higher threshold of the odd mode is indicative of the worse overlap between the atomic wavefunction and the node of the optical mode. The greater rate of threshold increase with pump power ramp rate indicates that the atoms need more time to adapt to the sign-flip of the odd mode than can be provided during the timescale of the faster ramp. Higher threshold pump power is needed to compensate. A consequence of this can be seen in the slow turn-on of superradiance in the data of Fig. 7b. This dynamical effect hints that motion of the atoms following the turn-on of the cavity light may in some cases lead to effects beyond the linear stability analysis presented in the Methods section, which produced the green curve in Fig. 2m. (b) An analogous behaviour is seen in the l+m=2 family, for the TEM₀₂ (blue) and TEM₁₁ (red) modes. The former has an antinode at the cavity centre, whereas the latter, although still an even-parity mode, has a node at the cavity centre. The thresholds were measured by pumping at a detuning Δ_c=−2 MHz from the respective cavity mode. Lines are guides to the eye. Error bars represent 1 s.e.