Bosonic System and Quantum Optics

Abstract

When more than one particles can simultaneously share the same state of the quantum system, the particle is called a boson. A photon is a typical example of a boson. In the bosonic system, a unitary representation of Heisenberg group is given. In the photonic case, the coherent state of the representation is called coherent light and is physically implementable. The operation corresponding to the representation of $\mathop {\mathrm{SU}}\nolimits (1,1)$ is called squeezing. When we apply the squeezing to the vacuum state, which is the state with no photon, the resultant state is called a squeezed state and plays an important role in the quantum optical system. In fact, the Heisenberg group corresponds to the transformation of the position and the momentum in analytical mechanics. The canonical transformation is a typical transformation preserving the product of the Heisenberg group. This chapter discusses the representation that describes the transformation for the position and the momentum by Heisenberg group as well as the canonical transformation.