Exact coherent state transition elements for the squeezed harmonic oscillator
Richard B. Sohn, Mark S. Swanson · Journal of Physics A Mathematical and General · 2005
The squeezed harmonic oscillator Hamiltonian H = ωa†a + αa2 + α*a†2 is analysed, where a† and a are harmonic oscillator creation and annihilation operators, ω is real, and α is a time independent constant. For the case that ω2 − 4|α|2 > 0 it is known that the Hamiltonian possesses real and positive eigenvalues. The exact eigenstates are constructed by using a Bogoliubov transformation to express them in terms of the algebra and states of the harmonic oscillator. Known properties of these states are reviewed and new results are derived. Modified coherent states are constructed from the exact eigenstates and shown to yield a unit projection operator for the original harmonic oscillator Hilbert space via generalized Gaussian integrals. Transition elements between harmonic oscillator coherent states are then evaluated exactly by using the properties of the modified coherent state projection operator.