  We theoretically investigate coherent optical excitations of collective modes
in two-band BCS superconductors, which accommodate two Higgs modes and one
Leggett mode corresponding, respectively, to the amplitude and relative-phase
oscillations of the superconducting order parameters associated with the two
bands. We find, based on a mean-field analysis, that each collective mode can
be resonantly excited through a nonlinear light-matter coupling when the
doubled frequency of the driving field coincides with the frequency of the
corresponding mode. Among the two Higgs modes, the higher-energy one exhibits a
sharp resonance with light, while the lower-energy mode has a broadened
resonance width. The Leggett mode is found to be resonantly induced by a
homogeneous ac electric field because the leading nonlinear effect generates a
potential offset between the two bands that couples to the relative phase of
the order parameters. The resonance for the Leggett mode becomes sharper with
increasing temperature. All of these light-induced collective modes along with
density fluctuations contribute to the third-harmonic generation. We also
predict an experimental possibility of optical detection of the Leggett mode.
