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.
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