The quantum mechanical orbital angular momentum operator for light

begin{align}
hat L_j & = -ihbar int d^3k hat a_m(mathbf k)^dagger (mathbf k wedge tilde{mathbf nabla})_j hat a_m(mathbf k),

hat S_j & = -ihbar int d^3k hat a_m(mathbf k)^dagger epsilon_{jmn} , hat a_n(mathbf k),
tag{3.10}
end{align}

I am confused on how exactly the $hat L$ operator, particularly the $mathbf k wedge tilde{mathbf nabla}$ part acts on the photon state. Please help me figure this out, maybe with an example where it acts on a photon state (say single photon state). Any insight on both $hat L$ and $hat S$ is welcome.