  Quantum Hall (QH) systems consist of many-body electron and nuclear spins.
They are coupled so weakly through the hyperfine interaction that normally
electron spin dynamics are scarcely affected by the nuclear spins. The dynamics
of the QH systems, however, may drastically change when the nuclear spins
interact with low-energy collective excitation modes of the electron spins. We
theoretically investigate the nuclear-electron spin interaction in the QH
systems as hybrid quantum systems driven by the hyperfine interaction. In
particular, we study the interaction between the nuclear spins and the
Nambu-Goldstone (NG) mode with the linear dispersion relation associated with
the U(1) spin rotational symmetry breaking. We show that such an interaction is
described as nuclear spins collectively coupled to the NG mode, and can be
effectively described by the Dicke model. Based on the model we suggest that
various collective spin phenomena realized in quantum optical systems also
emerge in the QH systems.
