Collective cavity quantum electrodynamics with multiple atomic levels

We study the transmission spectra of ultracold rubidium atoms coupled to a high-finesse optical cavity. Under weak probing with $\pi$-polarized light, the linear response of the system is that of a collective spin with multiple levels coupled to a single mode of the cavity. By varying the atom number, we change the collective coupling of the system. We observe the change in transmission spectra when going from a regime where the collective coupling is much smaller than the separation of the atomic levels to a regime where both are of comparable size. The observations are in good agreement with a reduced model we developed for our system.

Figure 1

Setup of the experiment. (a) We trap an unpolarized gas of ultracold ${}^{87}\mathrm{Rb}$ atoms in a two-dimensional lattice inside the mode of a high-finesse optical cavity and probe the system weakly with $\pi$-polarized light. (b) Under weak driving the system behaves as a collective spin with three transitions coupled to a single mode of the cavity. Of interest are the transitions to the excited states $|F^{\prime }=1,2,3\rangle$ of the $D2$ line. The corresponding splittings are separated by the hyperfine splitting of the excited states. Their size is given by the collective coupling, which in turn depends on the atom number $N$ and an effective single atom coupling constant ${\overline{g}}_{F^{\prime }}$. By varying $N$, we change the collective coupling and are able to explore different regimes of cavity QED with multiple atomic levels.

Figure 2

Theoretical transmission spectra for weak driving. Three avoided crossings between the energy of the bare cavity (solid diagonal line) and the energies of the bare atom (solid horizontal lines) contribute to the transmission spectrum. Transmission through the cavity is expected at the eigenenergies of the system. The amount of transmission is proportional to $\langle a^{†}a\rangle$ of the corresponding eigenstate, as indicated by color. For low atom numbers the spectrum consists of three separate splittings, while for higher atom numbers these merge into a single splitting. Calculations were performed for an equal distributions of $m_F$ states and a $\overline{g}$ of $6.5\mathrm{MHz}$. All detunings are with respect to the $|F=2\rangle$ to $|F^{\prime }=3\rangle$ transition.

Figure 3

Cavity transmission. For a given experiment we fix the cavity detuning ${\Delta }_C$ and the atom number $N$. The probe laser detuning is swept from $-700\mathrm{MHz}$ to $+700\mathrm{MHz}$ in $210\mathrm{ms}$, which is small compared to the measured atom number lifetime of 3 s. At the end of the sweep the atom number is measured by absorption imaging. The cavity output is directed onto a single photon counting module and the counts are binned in $100\mu \mathrm{s}$ intervals. All detunings are with respect to the $|F=2\rangle$ to $|F^{\prime }=3\rangle$ transition.

Figure 4

Experimental transmission spectra. By varying the cavity detuning we map out the full transmission spectra of the system for different average atom numbers $N$, with their standard deviation indicated in parentheses. The recorded transmission is indicated in color in terms of intracavity photon number $\overline{n}$, normalized to the average photon number inside the empty cavity on resonance ${\overline{n}}_{\text{empty}}$, where ${\overline{n}}_{\text{empty}}=0.25$ for the top left and bottom right plots and ${\overline{n}}_{\text{empty}}=0.35$ for the top right and bottom left plots. Points with counts below 2 are colored as 0. The gray solid lines indicate the positions of the theoretical eigenenergies for the measured average atom number $N$, an equal distributions of $m_F$ states and a $\overline{g}$ of $6.5\mathrm{MHz}$. All detunings are with respect to the $|F=2\rangle$ to $|F^{\prime }=3\rangle$ transition.