Single particle microscopy and spectroscopy strategies reveal hidden relationships between the surface plasmon resonances (SPRs) and the sizes, shapes, and arrangements of gold nanoparticles (Au NPs). The SPR, the coherent oscillation of the conduction electrons, leads to intense absorption and scattering of light at frequencies satisfying the resonance condition determined by the size, shape, and spacings between NPs. Growing and assembling NPs through wet chemistry yields a diversity of geometries. Together, optical spectroscopy, scanning electron microscopy (SEM), and computational modeling of individual NPs and NP assemblies elucidate the resulting variety of SPRs. Strong coupling of the SPRs in linear assemblies provokes particular interest for tunable structures that will benefit surface enhanced spectroscopies and optical computing. The influence of the constituents and imperfections in such assemblies, which deviate from idealized model systems, must be established one assembly at a time. This thesis demonstrates previously unknown and sensitive relationships between the SPRs and these geometric parameters through systematic single particle experiments of self-assembled ring superstructures, nanorod dimers, individual nanorods populating different size regimes, and short linear chains of Au NPs through single particle spectroscopy. Dark-field scattering of self-assembled ring superstructures of 40 nm Au NPs reveals new plasmon modes that are redshifted from the single NP SPR by hundreds of nanometers, highly polarized along the axis of alignment, and indifferent to irregularities in the NP arrangement. SPRs of Au nanorod dimers, however, are dramatically altered by NP size heterogeneity, reduced symmetry, and metallic contact, consistent with previous studies of small assemblies. Broad band single particle extinction measurements of individual Au nanorods and short chains of 200-1000 nm long demonstrate the importance of the overall dimensions of an NP or an assembly of NPs. Finally, extinction measurements of these chains provide a compelling comparison to chemical polymers via the redshifting of the lowest energy SPR, tolerance to disorder, and the influence of the repeat unit. This result extends already well-defined analogies between plasmonic assemblies and chemical molecules to the ‘plasmonic polymer’. The findings presented in this thesis bring deeper and more detailed understanding to the tunable optical properties of real NP assemblies.

This dissertation introduces some new properties of both superfluid phases of fermions with spin-orbit coupling (SOC) and superradiant phases of photons in an optical cavity. The effects of SOC on the phase transition between normal and superfluid phase are revealed; an unconventional crossover driven by SOC from the Bardeen-Cooper-Schrieffer (BCS) state to the Bose-Einstein condensate (BEC) state is verified in three different systems; and two kinds of excitations, a Goldstone mode and a Higgs mode, are demonstrated to occur in a quantum optical system. We investigate the BCS superfluid state of two-component atomic Fermi gases in the presence of three kinds of SOCs. We find that SOC drives a class of BCS to BEC crossover that is different from the conventional one without SOC. Here, we extend the concepts of the coherence length and Cooper-pair size in the absence of SOC to Fermi systems with SOC. We study the dependence of chemical potential, coherence length, and Cooper-pair size on the SOC strength and the scattering length in three dimensions (3D) (or the twobody binding energy in two dimensions (2D)) for three attractively interacting Fermi gases with 3D Rashba, 3D Weyl, and 2D Rashba SOC respectively. By adding a population imbalance to a Fermi gas with Rashba-type SOC, we also map out the finite-temperature phase diagram. Due to a competition between SOC and population imbalance, the finite-temperature phase diagram reveals a large variety of new features, including the expanding of the superfluid state regime and the shrinking of both the phase separation and the normal regimes. We find that the tricritical point moves toward a regime of low temperature, high magnetic field, and high polarization as the SOC strength increases. Besides Fermi fluids, this dissertation also gives a new angle of view on the superradiant phase in the Dicke model. Here, we demonstrate that Goldstone and Higgs modes can be observed in an optical system with only a few atoms inside a cavity. The model we study is the U(1)/Z2 Dicke model with N qubits (two-level atoms) coupled to a single photon mode.