Fundamental quantum noise in a fully quantized degenerate parametric amplification process

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.