Wigner function and transition amplitude of three mutually coupled oscillators

A full quantum mechanical treatment of three electromagnetic fields is considered. The proposed model consists of three different coupling parameters for which the rotating and counter-rotating terms are retained. An exact solution of the wave function in the Schrödinger picture is obtained and the connection with the coherent states wave function is given. The symmetrical ordered quasi-probability distribution function (W  -Wigner function) is calculated via the wave function in the coherent states representation. The squeezing phenomenon is also examined for both single mode and squared-amplitude, where the collapse and revival phenomena are observed. For the case in which λ3=0λ3=0 and ω1=ω2=ω3ω1=ω2=ω3 (exact resonances) we find that the late phenomenon is apparent but only after long period of the time considered. The transition amplitude between two different coherent states (a state in which all the coupling parameters are involved and a state when the coupling parameter λ3=0λ3=0) is calculated. It is shown that the probability amplitude is sensitive to the variation of the mean photon numbers, and the coupling parameters as well as to the field frequencies.