Bilinear and quadratic Hamiltonians in two-mode cavity quantum electrodynamics

In this work we show how to engineer bilinear and quadratic Hamiltonians in cavity quantum electrodynamics through the interaction of a single driven two-level atom with cavity modes. The validity of the engineered Hamiltonians is numerically analyzed even when considering the effects of both dissipative mechanisms, the cavity field and the atom. The present scheme can be used, in both optical and microwave regimes, for quantum state preparation, the implementation of quantum logical operations, and fundamental tests of quantum theory.