Coherent mixed states and a generalised P representation

The pure Glauber (harmonic oscillator) coherent states provide a very useful basis for many purposes. They are complete in the sense that an arbitrary state in the Hilbert space may be expanded in terms of them. Furthermore, the well known P representation provides a diagonal expansion of an arbitrary operator in the Hilbert space in terms of the projection operators onto the coherent states. The authors study the extensions of these results to the analogous mixed states which describe comparable harmonic oscillator systems in thermodynamic equilibrium at non-zero temperatures. Their results are given for the general density operator which describes the mixed squeezed coherent states of the displaced and squeezed harmonic oscillator. They show how these squeezed coherent mixed states similarly provide a very convenient complete description of a Hilbert space. In particular they show how the usual P and Q representations of operators in terms of pure states may be extended to finite temperatures with the corresponding mixed states, and various relations between them are demonstrated. The question of the existence of the generalised P representation for an arbitrary operator is further examined and some pertinent theorems are proven. They also show how their results relate to the Glauber-Lachs formalism in quantum optics for mixtures of coherent and incoherent radiation. Particular attention is focused both on the interplay between the quantum mechanical and thermodynamical uncertainties and on the entropy associated with such mixed states.