Phase Space Representations of the Density Operator

In Chapter 4, we introduced the concept of the photon number

representation of the electromagnetic (EM) field that is based on

the number states, the eigenstates of the Hamiltonian of the EM

field. However, this is not the only possible representation, in fact

not always the most convenient. In this chapter, we introduce the

concept of phase space representations of a quantum system which

is in a state described by the density operator ρˆ. The density

operator of a given system encodes classical as well as non-classical

(quantum) properties of the system. How to distinguish these

two kinds of properties is of basic importance in quantum optics.

Therefore, we shall address this issue by using representations of

the density operator that are based on the parametric space of

complex eigenvalues of the annihilation operator aˆ in a coherent state |α〉. We first determine what we mean by the density operator of a quantum system and discuss its basic properties. Next, we

introduce different representations of the density operator in terms

of coherent state projectors |α〉〈α|. The representations show that

a state of the EM field or of an arbitrary quantum system may

be regarded as a mixture of coherent states. Along the way, we

will learn how to find the P , Q and Wigner representations of the density operator, what are their properties, how to interpret the

properties and how to calculate relations between the different

representations. Most importantly, certain field states exhibit non-

classical features and these non-classical features can be manifested

in the phase space representations. Accordingly, we will learn how

the representations are very convenient tools to describe quantum

states of simple systems.