Transmission spectrum of an optical cavity containing $N$ atoms

The transmission spectrum of a high-finesse optical cavity containing an arbitrary number of trapped atoms is presented in the zero-temperature, low saturation limit. We take spatial and motional effects into account and show that in the limit of strong coupling, the important spectral features can be determined for an arbitrary number of atoms, $N$. We also show that these results have important ramifications in limiting our ability to determine the number of atoms in the cavity.

Figure 1

(a) Intrinsic transmission spectrum of atoms-cavity system neglecting spatial dependence of potential and atomic motion. (b) Transmission spectrum of spatially independent case including cavity decay.

Figure 2

(a) Intrinsic transmission spectrum of atoms-cavity system including spatial dependence of potential and atomic motion. (b) Corresponding transmission spectrum including cavity decay.

Figure 3

Plot of $⟨{\omega }_{-}⟩$ as a function of the trap tightness $ϵ=\exp \left(-k^2{\sigma }^2\right)$ for $N=8$ and $N=9$ and small ratio of atomic recoil energy to vacuum Rabi splitting, $\hslash k^2∕2mg=0.01$. The shaded regions indicate the intrinsic width of the red sideband, $\pm \sqrt{⟨{\left(\Delta {\omega }_{-}\right)}^2⟩}∕2$. In the tight-trap limit, $N=8$ and $N=9$ can be distinguished. In the loose-trap limit, the intrinsic width of the spectra render determination of atom number difficult.

Figure 4

Maximum limit $N_{max}$ on atom counting as a function of trap tightness $ϵ=\exp \left(-k^2{\sigma }^2\right)$ for several values of the decay parameter ${\kappa }^{\prime }$. $ϵ\to 0$ corresponds to the loose-trap limit while $ϵ\to 1$ corresponds to the tight-trap limit. Notice that for the infinitely tight trap, atom counting is limited only by ${\kappa }^{\prime }$.