As an alternative to state-of-the-art laser frequency stabilisation usingultra-stable cavities, it has been proposed to exploit the non-linear effectsfrom coupling of atoms with a narrow transition to an optical cavity. Here wehave constructed such a system and observed non-linear phase shifts of a narrowoptical line by strong coupling of a sample of strontium-88 atoms to an opticalcavity. The sample temperature of a few mK provides a domain where the Dopplerenergy scale is several orders of magnitude larger than the narrow linewidth ofthe optical transition. This makes the system sensitive to velocity dependentmulti-photon scattering events (Dopplerons) that affect the cavity fieldtransmission and phase. By varying the number of atoms and the intra-cavitypower we systematically study this non-linear phase signature which displaysroughly the same features as for much lower temperature samples. Thisdemonstration in a relatively simple system opens new possibilities foralternative routes to laser stabilization at the sub 100 mHz level andsuperradiant laser sources involving narrow line atoms. The understanding ofrelevant motional effects obtained here has direct implications for otheratomic clocks when used in relation with ultranarrow clock transitions.
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