  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
(TASEP). 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.

Statistical Mechanics;Strongly Correlated Electrons;Quantum Physics