  The retrieval approach is currently a standard method for deriving
atmospheric properties from observed spectra of exoplanets. However, the
approach ignores disequilibrium chemistry in most current retrieval codes,
which can lead to misinterpretation of the metallicity or elemental abundance
ratios of the atmosphere. We have implemented the disequilibrium effect of
vertical mixing or quenching for the major species in hydrogen/helium-dominated
atmospheres, namely $\mathrm{CH_4}$, $\mathrm{CO}$, $\mathrm{H_2O}$,
$\mathrm{NH_3}$, $\mathrm{N_2}$, and $\mathrm{CO_2}$, for the spectral
retrieval code ARCiS with a physical basis using the chemical relaxation
method. Then, using ARCiS updated with this module, we have performed
retrievals of the observed transmission spectra of 16 exoplanets with sizes
ranging from Jupiter to mini-Neptune. As a result, we find indications of
disequilibrium chemistry for HD 209458b ($\geq 4.1\sigma$) and WASP-39b ($\geq
2.7\sigma$). The retrieved spectrum of HD 209458b exhibits a strong
$\mathrm{NH_3}$ absorption feature at 10.5 $\mu$m accessible by JWST owing to
an enhanced abundance of $\mathrm{NH_3}$ due to the quenching effect. This
feature is absent in the spectrum retrieved assuming equilibrium chemistry,
which makes HD 209458b an ideal target for studying disequilibrium chemistry in
exoplanet atmospheres. Moreover, for HAT-P-11b and GJ 436b, we obtain
relatively different results than for the retrieval with the equilibrium
assumption, such as a $2.9\sigma$ difference for the C/O ratio. We have also
examined the retrieved eddy diffusion coefficient, but could not identify a
trend over the equilibrium temperature, possibly due to the limits of the
current observational precision.
