### Abstract:

An exact analytical solution of the time-dependent Schroedinger equation for a fully quantized
degenerate parametric amplification process generated by quantized multi-mode pump photons
provides a quantized multi-photon squeeze operator which generates dynamics characterized by
interaction frequency mixing specified by quantized anti-normal order Rabi frequency operator
Rˆ = g
√
N −ˆ k
2 and normal order Rabi frequency operator Rˆ = g
√
Nˆ − k
2, where g is the pump
photon coupling constant, Nˆ is the anti-normal order and Nˆ is the normal order pump photon
number operator, while k is the frequency de-tuning parameter. Due to the quantum frequency
mixing mechanism, the time evolving annihilation and creation operators of the initially degenerate signal-idler photon modes become effectively non-degenerate. The full quantum interaction
thus destroys the initial degeneracy of the signal-idler photon modes. The loss of degeneracy
means that all time evolving signal-idler photon operators become complex, thus developing
purely quantum Hermitian imaginary parts. Fluctuations of these purely quantum Hermitian
operators generate fundamental quantum noise effects. General fundamental quantum noise
arising from cross-correlations of the Hermitian real and imaginary parts of the complex time
evolving operators modify the minimum values of the corresponding Heisenberg uncertainty
products. The full quantum squeezing phenomenon signified by the smaller, purely quantum
fluctuations, of the imaginary parts, compared to the larger fluctuations of the corresponding
Hermitian parts, takes the form of a sum and difference of two time evolving interaction variables
and does not display extreme behavior such as exponential growth or exponential decay of one
or the other quadrature fluctuation usually obtained in the corresponding semiclassical model.