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.
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