  The phase conjugated response from nonmagnetic multi-level metallic quantum
wells is analyzed and an essentially complete analytical solution is presented
and discussed. The description is based on a semi-classical local-field theory
for degenerate four-wave mixing in mesoscopic interaction volumes of condensed
media developed by the present authors [T. Andersen and O. Keller, Phys.
Scripta 58, 132 (1998)]. The analytical solution is supplemented by a numerical
analysis of the phase conjugated response from a two-level quantum well in the
case where one level is below the Fermi level and the other level is above.
This is the simplest configuration of a quantum well phase conjugator in which
the light-matter interaction can be tuned to resonance. The phase conjugated
response is examined in the case where all the scattering takes place in one
plane, and linearly polarized light is used in the mixing. In the numerical
work we study a two-monolayer thick copper quantum well using the infinite
barrier model potential. Our results show that the phase conjugated response
from such a quantum-well system is highly dependent on the spatial dispersion
of the matter response. The resonances showing up in the numerical results are
analytically identified from the expressions for the linear and nonlinear
response tensors. In addition to the general discussion of the phase conjugated
response with varying frequency and parallel component of the wavevector, we
present the phase conjugated response in the special case where the light is in
resonance with the interband transition.
