Observation of Motion-Dependent Nonlinear Dispersion with Narrow-Linewidth Atoms in an Optical Cavity

As an alternative to state-of-the-art laser frequency stabilization using ultrastable cavities, it has been proposed to exploit the nonlinear effects from coupling of atoms with a narrow transition to an optical cavity. Here, we have constructed such a system and observed nonlinear phase shifts of a narrow optical line by a strong coupling of a sample of strontium-88 atoms to an optical cavity. The sample temperature of a few mK provides a domain where the Doppler energy scale is several orders of magnitude larger than the narrow linewidth of the optical transition. This makes the system sensitive to velocity dependent multiphoton scattering events (Dopplerons) that affect the cavity field transmission and phase. By varying the number of atoms and the intracavity power, we systematically study this nonlinear phase signature which displays roughly the same features as for much lower temperature samples. This demonstration in a relatively simple system opens new possibilities for alternative routes to laser stabilization at the sub–100 mHz level and superradiant laser sources involving narrow-line atoms. The understanding of relevant motional effects obtained here has direct implications for other atomic clocks when used in relation to ultranarrow clock transitions.

Figure 1

(a) Experimental setup. A sample of cold atoms (MOT) is prepared inside a low finesse cavity ($F=85$) which is held at resonance with the probe laser. We probe the atoms on the intercombination line $|5s^2{}^1{S}_0⟩-|5s5p{}^3{P}_1⟩$ at 689 nm ($\mathrm{\Gamma }/2\pi =7.6\mathrm{kHz}$). Both intensity and phase shift of the transmitted probe light are recorded. The phase is measured relative to the input field by employing cavity-enhanced heterodyne spectroscopy (NICE-OHMS). (b) Energy levels of the ${}^{88}\mathrm{Sr}$ atom and transitions relevant to this work. (c) Relation between the spectral components in the experiment. The probe laser frequency ${\omega }_l$ (and consequently the cavity resonance ${\omega }_c$) is detuned a variable amount $\mathrm{\Delta }$ with respect to the atomic resonance ${\omega }_0$.

Figure 2

(a) Illustration of the Doppleron multiphoton processes that take place in our system. We consider a given atom with velocity component $v$ in the direction of the cavity axis. The first resonance condition (the top equation) involves only one photon and corresponds to the usual Doppler effect. The next resonance involves two photons absorbed and one photon emitted, and so forth. (b),(c) Typical frequency scan without any averaging across the atomic resonance for an input power of 975 nW and a total number of atoms in the MOT of $N=4.4\times {10}^8$. The data in (b) display the transmission of the probe light through the cavity normalized to a signal with no atoms in the cavity. The data in (c) is the phase shift of the cavity-transmitted field obtained using the NICE-OHMS method. The solid lines are theoretical predictions based on our theoretical model, which includes the Doppler effect and the spatial overlap of the thermal cloud (here with temperature $T=2.3\mathrm{mK}$) with the cavity field. At maximum phase shift (around detunings of $\mathrm{\Delta }\simeq \pm 1\mathrm{MHz}$), our detection system starts to saturate, giving a slightly flatter appearance of the phase data. (Inset) Zoom on central phase feature with similar experimental parameters [data are identical to Fig. 4]. Here, we have included a theoretical plot that does not take the Dopplerons into account (black, dashed curve). The effect of the Dopplerons is readily apparent. Units on axes are the same as in (c).

Figure 3

Measured phase shift of the cavity-transmitted field when scanned across the atomic resonance. The input probe laser power $P_{\mathrm{in}}$ is progressively decreased from 1950 nW (a), 900 nW (b), 700 nW (c) to 650 nW (d). The number of atoms is about $N_{\text{cavity}}=2.5\times {10}^7$. Each point is an average of three data points. The solid lines are theoretical predictions based on our theoretical model.

Figure 4

Measured phase shift of the probe light when scanned across the atomic resonance. The number of atoms in the cavity is progressively decreased from $2.5\times {10}^7$ (a), $2.0\times {10}^7$ (b), $1.7\times {10}^7$ (c) to $1.2\times {10}^7$ (d). The input power used for all plots was 650 nW. Each point is an average of three data points. The solid lines are theoretical predictions based on our theoretical model. The central slope scales linearly with atom number.