  We analyze the spectrum of spatially uniform, single-particle fluctuation
modes in the linear electromagnetic response of a semiconductor laser. We show
that if the decay rate of the interband polarization, $\gamma_p$, and the
relaxation rate of the occupation distribution, $\gamma_f$, are different, a
gapless regime exists in which the order parameter $\Delta^{(0)} (k)$ (linear
in the coherent photon field amplitude and the interband polarization) is
finite but there is no gap in the real part of the single-particle fluctuation
spectrum. The laser being a pumped-dissipative system, this regime may be
considered a non-equilibrium analog of gapless superconductivity. We analyze
the fluctuation spectrum in both the photon laser limit, where the interactions
among the charged particles are ignored, and the more general model with
interacting particles. In the photon laser model, the order parameter is
reduced to a momentum-independent quantity, which we denote by $\Delta$. We
find that, immediately above the lasing threshold, the real part of the
fluctuation spectrum remains gapless when $0 < | \Delta | < \sqrt{2 / 27} \, |
\gamma_f - \gamma_p |$ and becomes gapped when $| \Delta |$ exceeds the upper
bound of this range. Viewed as a complex function of $|\Delta|$ and the
electron-hole energy, the eigenvalue set displays some interesting exceptional
point (EP) structure around the gapless-gapped transition. The transition point
is a third-order EP, where three eigenvalues (and eigenvectors) coincide.
Switching on the particle interactions in the full model modifies the spectrum
of the photon laser model and, in particular, leads to a more elaborate EP
structure. However, the overall spectral behavior of the continuous
(non-collective) modes of the full model can be understood on the basis of the
relevant results of the photon laser model.
