L. Boivin et al., ANALYTICAL SOLUTION TO THE QUANTUM-FIELD THEORY OF SELF-PHASE MODULATION WITH A FINITE RESPONSE-TIME, Physical review letters, 73(2), 1994, pp. 240-243
The quantum theory of a field in 1+1 dimensions coupled to localized o
scillators is developed. The solution to the Heisenberg equation for t
he field is given in closed form. It is shown that the nonlinearity of
the medium is inevitably accompanied by phase noise of the field. Thi
s noise explains the preferential growth of the Stokes wave for short
propagation distances.