  Despite decades of effort, the timing and duration of He II reionization, as
well as its morphology and the properties of the quasars believed to drive it,
are still not well constrained. In this paper we present a new method to study
both He II reionization and quasars via the thermal proximity effect -- the
photoelectric heating of the intergalactic medium around quasars when their
hard radiation doubly ionizes helium. We post-process a SPH simulation with 1D
radiative calculations, and study how the thermal proximity effect depends on
the amount of singly ionized helium, $x_{\rm HeII,0}$, which prevailed in the
IGM before the quasar turned on, and the characteristic lifetime $t_{\rm Q}$
for which quasars shine. We find that the amplitude of the temperature boost in
the quasar environment depends on $x_{\rm HeII,0}$, with a characteristic value
of $\Delta T \simeq 10^4\,{\rm K}$ for an initially singly ionized IGM ($x_{\rm
HeII,0} = 1.0$), whereas the size of the thermal proximity zone is sensitive to
quasar lifetime $t_{\rm Q}$, with typical sizes of ~100 cMpc for luminous
quasars shining for $t_{\rm Q}=10^8$ yr. This temperature boost is manifest as
a change in the thermal broadening of H I absorption lines near the quasar. We
introduce a new method based on measuring the Ly$\alpha$ forest power spectrum
as a function of distance from the quasar, and conduct Bayesian MCMC analysis
to demonstrate that the thermal proximity effect should be easily detectable.
For a mock dataset of 50 quasars at z~4, we predict that one can measure
$x_{\rm HeII,0}$ to a precision $\approx 0.04$, and $t_{\rm Q}$ to a precision
of $\approx 0.1$ dex. By applying our formalism to existing high-resolution
Ly$\alpha$ forest spectra of quasars at $3.1 \lesssim z \lesssim 5.0$, one
should be able to detect the thermal proximity effect, and reconstruct the full
reionization history of He II.
