Pure and mixed states in degenerate parametric oscillation

The degree of purity of the quantum state of electromagnetic field at the output of a degenerate optical parametric oscillator is theoretically analyzed. It is shown that, in the subthreshold and significantly suprathreshold modes, the degree of purity of the state of pairs of waves with frequencies ω _s ± Ω (ω _s is the signal mode frequency) is unity. Based on this fact, it is concluded that these pairs of waves parametrically develop independent of any other such pairs. At the same time, in the suprathreshold mode near the oscillation threshold, the degree of purity of wave pairs can be much smaller than unity. However, consideration of a tetrad of waves with frequencies ω _s ± Ω and ω _p ± Ω shows that this system is in a pure state, which indicates its independent development.