Theory of inelastic light scattering in spin-1 systems: resonant regimes and detection of quadrupolar order

Motivated by the lack of an obvious spectroscopic probe to investigate non-conventional order such as quadrupolar orders in spin S>1/2 systems, we present a theoretical approach to inelastic light scattering for spin-1 quantum magnets in the context of a two-band Hubbard model. In contrast to the S=1/2 case, where the only type of local excited state is a doubly occupied state of energy $U$, several local excited states with occupation up to 4 electrons are present. As a consequence, we show that two distinct resonating scattering regimes can be accessed depending on the incident photon energy. For $\mathrm{\hslash }{\omega }_{in}\lesssim U$, the standard Loudon-Fleury operator remains the leading term of the expansion as in the spin-1/2 case. For $\mathrm{\hslash }{\omega }_{in}\lesssim 4U$, a second resonant regime is found with a leading term that takes the form of a biquadratic coupling $\sim({\bf S}{i}\cdot{\bf S}{j)^{2}$. Consequences for the Raman spectra of S=1 magnets with magnetic or quadrupolar order are discussed. Raman scattering appears to be a powerful probe of quadrupolar order.