  Non-equilibrium ionization effects are important in cosmological
hydrodynamical simulations but are computationally expensive. We study the
effect of non-equilibrium ionization evolution and UV ionizing background (UVB)
generated with different quasar spectral energy distribution (SED) on the
derived physical conditions of the intergalactic medium (IGM) at $2\leq z \leq
6$ using our post-processing tool 'Code for Ionization and Temperature
Evolution' (CITE). CITE produces results matching well with self-consistent
simulations more efficiently. The HeII reionization progresses more rapidly in
non-equilibrium model as compared to equilibrium models. The redshift of HeII
reionization strongly depends on the quasar SED and occurs earlier for UVB
models with flatter quasar SEDs. During this epoch the normalization of
temperature-density relation, $T_0(z)$, has a maximum while the slope,
$\gamma(z)$, has a minimum, but occurring at different redshifts. The $T_0$ is
higher in non-equilibrium models using UVB obtained with flatter quasar SEDs.
While our models produce the observed median HeII effective optical depth
evolution and its scatter for equilibrium and non-equilibrium considerations,
to explain the observed cumulative distributions we may need to consider
fluctuating UVB. For a given UVB model, the redshift dependence of the HI
photo-ionization rate derived from the observed HI effective optical depth
($\tau_{\rm eff,HI}$) for the equilibrium model is different from that for the
non-equilibrium model. This may lead to different requirements on the evolution
of ionizing emissivities of sources. We show that, in the absence of strong
differential pressure smoothing effects, it is possible to recover the $T_0$
and $\gamma$ realised in non-equilibrium model from the equilibrium models
generated by rescaling photo-heating rates while producing the same $\tau_{\rm
eff,HI}$.
