Figure 1

A two-level atom in an optical cavity interacting with different cavity modes depending on the spatially dependent couplings. Shown here are two higher-order transversal modes (modes 1 and 2) and one two-level atom exchange excitation with rates $g_1$ and $g_2$, respectively. The coherent process is damped by radiative coupling to the environment either via cavity decay of each mode through the mirrors (with rates ${\kappa }_1$ and ${\kappa }_2$) or atomic polarization decay (with rate $\gamma$).Reuse & Permissions

Figure 2

Typical images of the cavity transmission and their fits. (a) Transmission through the empty cavity dominated by the TEM${}_{00}$ mode and (b) corresponding fit. (c) Transmission through the atom-filled cavity with a strong TEM${}_{20}$ component and (d) corresponding fit.Reuse & Permissions

Figure 3

Left: Mode resolved intracavity photon number with atoms (red filled circles) and without atoms (black filled squares) in the cavity for different detunings ${\Delta }_C/\left(2\pi \right)$. The number of atoms in the fundamental mode TEM${}_{00}$ of the cavity as derived from fluorescence images is $N_{00}=(100\pm 8)\times {10}^3$. The data points correspond to the absolute squared amplitude $|{\alpha }_{nm}{|}^2$ of the corresponding two-dimensional Hermite Gaussian mode function as derived from image fits with the function Eq. (1). The data were then fitted with a Lorentzian transmission function of an empty cavity and the theoretical transmission function of the coupled cavity-atom system, as from Eq. (2). All the displayed higher-order transversal modes exhibit the characteristic normal-mode splitting. We stress that for one of the data sets (the third figure from top, on the left) the deviation of the data from the fitting function is due to experimental limitations and does not constitute evidence of a further splitting. Right: The effective coupling for the $nm$ mode plotted as a function of the square root of the number of atoms contained in that mode.Reuse & Permissions

Figure 4

Numerical results for the intracavity photon number as a function of ${\Delta }_C$ for the the case of 1000 (left) and 16 000 (right) atoms Gaussian distributed in a realistic three-dimensional (3D) multimode cavity field with the available TEM modes 00, 20, 02, and 11. The (black) dots are the actual numerical results from the simulation, while the solid (red) lines are the best fits with Eq. (2). Dashed (black) lines are results for an empty cavity, reported for comparison. The parameters of the calculation are as follows. The maximum value for the coupling constant for the 00 mode is $g=\kappa$, with the individual couplings for the different atoms calculated depending on their position (see text). $\gamma =3.25\kappa$. The mode pumped is the TEM${}_{00}$ mode with ${\eta }_{00}=0.1\kappa$, and the displayed steady-state photon number is multiplied by ${10}^6$. The atomic distribution is similar to the experiment following a cloud with Gaussian width ${\sigma }_z=1.25w_0$ and ${\sigma }_r=1.6{\sigma }_z$, where $w_0$ is the waist of the cavity field and $z$ is perpendicular to the cavity axis.Reuse & Permissions

Figure 5

Value for the parameter ${\kappa }_{\mathrm{eff}}$ as derived from fitting the numerically simulated transmission data of the TEM${}_{00}$ mode data with Eq. (2) as a function of the number of atoms $N$. The atoms are randomly distributed in a 3D Gaussian centered with the center of the cavity and with width ${\sigma }_x=1.3w_0$ and ${\sigma }_y={\sigma }_z=1.6{\sigma }_x$. The cavity has four transverse modes: 00, 11, 20, and 02. The mode pumped is the 00 mode with ${\eta }_{00}=0.1\kappa$. Since the eigenvalues of the system are quasidegenerate the resulting effective $\kappa$ scales linearly with $\sqrt{N}$, where $N$ is the total atom number in the distribution.Reuse & Permissions

Figure 6

Numerical simulations for the cavity transmission spectroscopy for different spatial extensions of the atomic cloud. All the data sets are for a sample of 16 000 atoms and a cavity with four degenerate modes. The atoms are 3D Gaussian distributed with width $(s_x,2s_x,2s_x)$, expressed in the figure in terms of the fundamental Gaussian cavity mode waist $w_{00}$. The different data plots refer to different widths $s_x$ of the atomic distribution. The mode pumped is the 00 mode with ${\eta }_{00}=0.1\kappa$.Reuse & Permissions