  We investigate the nonlinear optical response of a four-level double-V-type
quantum system interacting with a pair of weak probe fields while located near
a two-dimensional array of metal-coated dielectric nanospheres. Such a quantum
system contains a V-type subsystem interacting with surface plasmons, and
another V-type subsystem interacting with the free-space vacuum. A distinctive
feature of the proposed setup is its sensitivity to the relative phase of the
applied fields when placed near the plasmonic nanostructure. We demonstrate
that due to the presence of the plasmonic nanostructure, the third-order
(Kerr-type) susceptibility for one of the laser fields can be significantly
modified while another probe field is acting. Moreover, the Kerr nonlinearity
of the system can be controlled and even enhanced by varying the distance of
the quantum system from the plasmonic nanostructure.We also show that the Kerr
nonlinearity of such a system can be controlled by adjusting the relative phase
of the applied fields. The results obtained may find potential applications in
on-chip nanoscale photonic devices. We also study the light-matter interaction
in the case where one probe field carries an optical vortex, and another probe
field has no vortex. We demonstrate that due to the phase sensitivity of the
closed-loop double V-type quantum system, the linear and nonlinear
susceptibility of the nonvortex probe beam depends on the azimuthal angle and
orbital angular momentum (OAM) of the vortex probe beam. This feature is
missing in open four-level double V-type quantum system interacting with
free-space vacuum, as no quantum interference occurs in this case. We use the
azimuthal dependence of optical susceptibility of the quantum system to
determine the regions of spatially-structured transmittance.
