Thermalization of Squeezed States

We first of all propose a simple definition of a thermal squeezed state. Previous definitions of thermal coherent states and squeezed states[1, 2] have used the thermofield approach of Takahashi and Umezawa[3]. An early paper by Vourdas[4] discusses the superposition of squeezed states with thermal photons; a more recent paper[5] considers only the case of thermal coherent states. An early study of canonically transformed thermal states similar to the approach of the present work was made in [6], and also in [7, 8]. Two further papers adopt essentially the same definition as the present note s[9, 10] using a slightly different approach. The method that we take here is the straightforward one of defining the hamiltonian whose eigenstates are squeezed states (sometimes known as squeezed-coherent states to distinguish them from squeezed vacuum states). The complete set of eigenstates are in fact squeezed displaced number states. These are the states SD|n〉, where D is the displacement operator, and S the squeezing operator. The thermal squeezed state we define is simply a mixed state given as a thermal sum of these states. The simplicity of this approach in, for example, calculating thermal averages is due to the group-theoretical structure of these states, which essentially enables such calculations to be reduced to free hamiltonian averages. This structure is a consequence of the definition of the squeezed-coherent states as transformed vacua under the automorphism group of the canonical commutation relations, which we shall describe in the next section. To exemplify how the physical properties are degraded by the thermalization process, we calculate the important signal-to-quantum noise ratio for these thermal squeezed states. Finally, we use the Kraus formalism to propose a model for the thermalization.