We investigate how particle pair creation and annihilation, within the quantum transverse XY model, affects the non-equilibrium steady state (NESS) and Liouvillian gap of the stochastic totally asymmetric exclusion process. By utilising operator quantization we formulate a perturbative description of the NESS. Furthermore, we estimate the Liouvillian gap by exploiting a Majorana canonical basis as the basis of super-operators. In this manner we show that the Liouvillian gap can remain finite in the thermodynamic limit provided the XY model anisotropy parameter remains non-zero. Additionally, we show that the character of the gap with respect to the anisotropy parameter differs depending on the phase of the XY model. The change of character corresponds to the quantum phase transition of the XY model.