| Strong interactions can dramatically change the essence of a physical system. The behavior of strongly interacting systems can be fundamentally different than those where the interaction is absent or treated perturbatively. Many examples are known in solid-state physics including the superconductivity and the Fractional Quantum Hall Effect. At the same time, tremendous developments have been made in manipulating the interaction between light and matter. These advances have paved the way to explore the strongly interacting many-body physics in new regimes. This thesis explores two novel avenues to study strongly interacting systems. First, we investigate the effect of strong interaction between bosons subjected to an effective magnetic field. We show that how a Fractional Quantum Hall state of bosons in an optical lattice can be created, characterized and detected in a realistic experiment. Moreover, we demonstrate that Chern numbers can unambiguously characterize the topological order of such systems. Second, we investigate the effect of strong interaction between photons on their transport properties. We theoretically study the transmission of few-photon quantum fields through a strongly nonlinear optical medium. We develop a general approach to investigate non-equilibrium quantum transport of bosonic fields through a finite-size nonlinear medium and apply it to a recently demonstrated experimental system where cold atoms are loaded in a hollow-core optical fiber. We show that the photonic field can exhibit either anti-bunching or bunching, associated with the resonant excitation of bound states of photons by the input field. These effects can be observed by probing statistics of photons transmitted through the nonlinear fiber. As an application, we propose a scheme to realize a single-photon gate, where the presence or absence of a single "control" photon regulates the propagation of a "target" photon. Finally, we study optical nonlinearities due to the interaction of weak optical fields with the collective motion of a strongly dispersive ultracold gas. We present a theoretical model that is in good agreement with our experimental observations. |
