  We investigate steady-state thermal transport and photon statistics in a
nonequilibrium hybrid quantum system, in which a qubit is longitudinally and
quadratically coupled to an optical resonator. Our calculations are conducted
with the method of the quantum dressed master equation combined with full
counting statistics. The effect of negative differential thermal conductance is
unravelled at finite temperature bias, which stems from the suppression of
cyclic heat transitions and large mismatch of two squeezed photon modes at weak
and strong qubit-resonator hybridizations, respectively. The giant thermal
rectification is also exhibited at large temperature bias. It is found that the
intrinsically asymmetric structure of the hybrid system and negative
differential thermal conductance show the cooperative contribution. Noise power
and skewness, as typical current fluctuations, exhibit global maximum with
strong hybridization at small and large temperature bias limits, respectively.
Moreover, the effect of photon quadrature squeezing is discovered in the strong
hybridization and low-temperature regime, which shows asymmetric response to
two bath temperatures. These results would provide some insight to thermal
functional design and photon manipulation in qubit-resonator hybrid quantum
systems.
