  The absorption by neutral hydrogen in the intergalactic medium (IGM) produces
the Ly$\alpha$ forest in the spectra of quasars. The Ly$\alpha$ forest
absorbers have a broad distribution of neutral hydrogen column density $N_{\rm
HI}$ and Doppler $b$ parameter. The narrowest Ly$\alpha$ absorption lines (of
lowest $b$) with neutral hydrogen column density above $\sim 10^{13}{\rm
cm^{-2}}$ are dominated by thermal broadening, which can be used to constrain
the thermal state of the IGM. Here we constrain the temperature-density
relation $T=T_0(\rho/\bar{\rho})^{\gamma-1}$ of the IGM at $1.6<z<3.6$ by using
$N_{\rm HI}$ and $b$ parameters measured from 24 high-resolution and
high-signal-to-noise quasar spectra and by employing an analytic model to model
the $N_{\rm HI}$-dependent low-$b$ cutoff in the $b$ distribution. In each
$N_{\rm HI}$ bin, the $b$ cutoff is estimated using two methods, one
non-parametric method from computing the cumulative $b$ distribution and a
parametric method from fitting the full $b$ distribution. We find that the IGM
temperature $T_0$ at the mean gas density $\bar{\rho}$ shows a peak of $\sim
1.5\times 10^4$K at $z\sim $2.7-2.9. At redshift higher than this, the index
$\gamma$ approximately remains constant, and it starts to increase toward lower
redshifts. The evolution in both parameters is in good agreement with
constraints from completely different approaches, which signals that He II
reionization completes around $z\sim 3$.
