Spatial propagation of squeezed light in a degenerate parametric amplifier

Differential equations which describe the steady state spatial evolution of nonclassical light are established using standard quantum field theoretic techniques. A Schrodinger equation for the state vector of the optical field is derived using the quantum analog of the lowly varying envelope approximation (SVEA). The steady sate solutions are those that satisfy the time independent Schrodinger equation. The resulting eigenvalue problem then leads to the spatial propagation equations. For the degenerate parametric amplifier this method shows that the squeezed state is the ground state of the squeezing Hamiltonian. The magnitude and phase of the squeezing parameter obey nonlinear differential equations coupled by the amplifier gain constant and phase mismatch. The solution to these differential equations is equivalent to one obtained from the classical three wave mixing steady solution to the parametric amplifier with a nondepleted pump.