Hybrid systems made of quantum emitters and plasmonic nanoresonators offer a unique platform to implement artificial atoms with completely novel optical responses that are not available otherwise. However, their theoretical analysis is difficult, and since many degrees of freedom have to be explored, engineering their optical properties remains challenging. Here, we propose a new formalism that removes most limitations encountered in previous analytical treatments and allows a flexible and efficient study of complex nanoresonators with arbitrary shapes in an almost fully analytically way. The formalism brings accurate closed-form expressions for the hybrid-system optical response and provides an intuitive description based on the coupling between the quantum emitters and the resonance modes of the nanoresonator. The ability to quickly predict light-scattering properties of hybrid systems paves the way to a deep exploration of their fascinating properties and may enable rapid optimization of quantum plasmonic metamaterials or quantum information devices.
We consider here a model previously introduced to describe the collective behavior of an ensemble of cold atoms interacting with a coherent electromagnetic field. The atomic motion along the self-generated spatially periodic force field can be interpreted as the rotation of a phase oscillator. This suggests a relationship with synchronization transitions occurring in globally coupled rotators. In fact, we show that whenever the field dynamics can be adiabatically eliminated, the model reduces to a self-consistent equation for the probability distribution of the atomic "phases." In this limit, there exists a formal equivalence with the Kuramoto model, though with important differences in the self-consistency conditions. Depending on the field-cavity detuning, we show that the onset of synchronized behavior may occur through either a first- or second-order phase transition. Furthermore, we find a secondary threshold, above which a periodic self-pulsing regime sets in, that is immediately followed by the unlocking of the forward-field frequency. At yet higher, but still experimentally meaningful, input intensities, irregular, chaotic oscillations may eventually appear. Finally, we derive a simpler model, involving only five scalar variables, which is able to reproduce the entire phenomenology exhibited by the original model.
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