  Light drives offer a potential tool for the dynamical control of magnetic
interactions in matter. We theoretically investigate the indirect exchange
coupling between two parallel chains of magnetic impurities on the surface of a
topological insulator, driven by a time-periodic circularly polarized light
field in the high-frequency, off-resonant regime. We derive a closed-form
analytic expression for the spin susceptibility of the photon-dressed
topological insulator surface states and obtain the irradiation dependence of
the Ising, Heisenberg, and Dzyaloshinskii- Moriya exchange couplings between
the impurity chains. Our results show a two-pronged modification of these
exchange couplings by periodic drives. First, the Ruderman-Kittel-Kasuya-Yosida
(RKKY) oscillation period of the exchange couplings can be extended by
enhancing the driving strength. Second, increasing driving strength enhances
the envelope of RKKY oscillations of the Ising type while suppressing those of
the Heisenberg type and Dzyaloshinskii-Moriya type. Our work provides useful
insights for realizing Floquet engineering of collinear and noncollinear
indirect exchange interactions in topological insulating systems.
