We investigate a cavity quantum electrodynamic effect, where the alignment of two-dimensional freely rotating optical dipoles is driven by their collective coupling to the cavity field. By exploiting the formal equivalence of a set of rotating dipoles with a polymer we calculate the partition function of the coupled light-matter system and demonstrate it exhibits a second order phase transition between a bunched state of isotropic orientations and a stretched one with all the dipoles aligned. Such a transition manifests itself as an intensity-dependent shift of the polariton mode resonance. Our work, lying at the crossroad between cavity quantum electrodynamics and quantum optomechanics, is a step forward in the on-going quest to understand how strong coupling can be exploited to influence matter internal degrees of freedom.
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