Self-Organization Threshold Scaling for Thermal Atoms Coupled to a Cavity

We make a detailed experimental study of the threshold for the self-organization of thermal ${}^{87}\mathrm{Rb}$ atoms coupled to a high-finesse cavity over a range of atom numbers and cavity detunings. We investigate the difference between probing with a traveling wave and a retroreflected lattice. These two scenarios lead to qualitatively different behavior in terms of the response of the system as a function of cavity detuning with respect to the probe. In both cases, we confirm a $N^{-1}$ scaling of the threshold with atom number.

Figure 1

Schematic representation of self-organization in the lattice (top) and traveling wave (bottom) geometry. (a),(c) Below threshold, the atoms (black) are confined by the intracavity 1560-nm FORT, indicated by light gray (yellow online). (b),(d) Above threshold, the atoms organize into a $\lambda$-spaced lattice trapped by the probe and scattered fields, indicated by dark gray (red online). (b) For a lattice probe, the atoms can form one of two possible $\lambda$-spaced lattices (filled or open circles). (d) For a traveling wave probe, interference between the probe and scattered fields results in a $\lambda$-spaced transverse lattice out of phase with the atoms by $\theta$.

Figure 2

Threshold measurement results for the traveling wave and lattice probe geometries, in the left and right columns, respectively. Top: The self-organization threshold is determined from the sudden increase in cavity transmission as the probe intensity is ramped up over 10 ms; example traces are shown for a traveling wave (a) and lattice (b) probe. Middle: One set of threshold measurements at fixed ${\Delta }_c$, for traveling wave (c) and lattice (d) probe. The black lines are an ab initio calculation of the threshold from Eqs. (1, 2) for a lattice (traveling wave) probe. Bottom: All measurements from the data sets at various ${\Delta }_c$ near the minimum threshold, which for the traveling wave is where $|{\tilde{\Delta }}_c|\le \kappa /2$ (e), and for the lattice is where $|{\tilde{\Delta }}_c+\kappa |\le \kappa /2$ (f). The black lines are the calculated minimum threshold for the respective cases. In [(c)–(f)] the black (red online) circles are data taken at the atomic detuning $-265\mathrm{GHz}$, and gray (cyan online) circles at $-110\mathrm{GHz}$.

Figure 3

Indication of excess heating due to nonadiabatic dynamics after self-organization. (a) Measured temperature after self-organization by probing above threshold at constant power for 0.8 ms plotted against the depth of the scattered field in the cavity as determined by the cavity output at the time of measurement. (b) Two example traces of the cavity output scaled the peak output corresponding to data points in (a). For the black line and circles (red online) the cavity output was above 50% of its peak at the measurement time, whereas for gray line and circles (cyan online) the transmission was already below 50% and falling. For reference, an intra-cavity photon number of $4\times {10}^6$ corresponds to a $530\mu \mathrm{K}$ deep optical lattice potential ($\Delta =-265\mathrm{GHz}$).