Engineering Giant Nonlinearities in Quantum Nanosystems

We describe a method to engineer giant nonlinearities in, and probes to measure nonlinear observables of, mesoscopic quantum resonators. This involves tailoring the Hamiltonian of a simple auxiliary system perturbatively coupled to the resonator, and has the potential to engineer a wide range of nonlinearities to high accuracy. We give a number of explicit examples, including a readily realizable two-qubit auxiliary system that creates an $x^4$ potential and a ${\chi }^{(3)}$ (Kerr) nonlinearity, valid to fifth order in the perturbative coupling.