  We study the dynamics of two ensembles of atoms (or equivalently, atomic
clocks) coupled to a bad cavity and pumped incoherently by a Raman laser. Our
main result is the nonequilibrium phase diagram for this experimental setup in
terms of two parameters - detuning between the clocks and the repump rate.
There are three main phases - trivial steady state (Phase I), where all atoms
are maximally pumped, nontrivial steady state corresponding to monochromatic
superradiance (Phase II), and amplitude-modulated superradiance (Phase III).
Phases I and II are fixed points of the mean-field dynamics, while in most of
Phase III stable attractors are limit cycles. Equations of motion possess an
axial symmetry and a $\mathbb{Z}_{2}$ symmetry with respect to the interchange
of the two clocks. Either one or both of these symmetries are spontaneously
broken in various phases. The trivial steady state loses stability via a
supercritical Hopf bifurcation bringing about a $\mathbb{Z}_{2}$-symmetric
limit cycle. The nontrivial steady state goes through a subcritical Hopf
bifurcation responsible for coexistence of monochromatic and
amplitude-modulated superradiance. Using Floquet analysis, we show that the
$\mathbb{Z}_{2}$-symmetric limit cycle eventually becomes unstable and gives
rise to two $\mathbb{Z}_{2}$-asymmetric limit cycles via a supercritical
pitchfork bifurcation. Each of the above attractors has its own unique
fingerprint in the power spectrum of the light radiated from the cavity. In
particular, limit cycles in Phase III emit frequency combs - series of
equidistant peaks, where the symmetry of the frequency comb reflects the
symmetry of the underlying limit cycle. For typical experimental parameters,
the spacing between the peaks is several orders of magnitude smaller than the
monochromatic superradiance frequency, making the lasing frequency highly
tunable.
