  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. 
Implementation of disequilibrium chemistry to spectral retrieval code   ARCiS and application to sixteen exoplanet transmission spectra: Indication   of disequilibrium chemistry for HD 209458b and WASP-39b