  Despite the enormous technological interest in micro and nanolasers,
surprisingly, no class-B quantum density-matrix model is available to date,
capable of accurately describing coherence and photon correlations within a
unified theory. In class-B lasers $-$applicable for most solid-state lasers at
room temperature$-$, the macroscopic polarization decay rate is larger than the
cavity damping rate which, in turn, exceeds the upper level population decay
rate. Here we carry out a density-matrix theoretical approach for generic
class-B lasers, and provide closed equations for the photonic and atomic
reduced density matrix in the Fock basis of photons. Such a relatively simple
model can be numerically integrated in a straightforward way, and exhibits all
the expected phenomena, from one-atom photon antibunching, to the well-known
S-shaped input-output laser emission and super-Poissonian autocorrelation for
many atoms ($1\leq g^{(2)}(0)\leq 2$), and from few photons (large spontaneous
emission factors, $\beta\sim1$) to the thermodynamic limit ($N\gg1$ and
$\beta\sim 0$). Based on the analysis of $g^{(2)}(\tau)$, we conclude that
super-Poissonian fluctuations are clearly related to relaxation oscillations in
the photon number. We predict a strong damping of relaxation oscillations with
an atom number as small as $N\sim 10$. This model enables the study of
few-photon bifurcations and non-classical photon correlations in class-B laser
devices, also leveraging quantum descriptions of coherently coupled nanolaser
arrays.
