  Despite a Curie-Weiss temperature $\theta_{\rm CW} \sim -14$ K, the Tb2Ti2O7
pyrochlore magnetic material lacks long range magnetic order down to at least
$T^*\approx 50$ mK. It has recently been proposed that the low temperature
collective paramagnetic or spin liquid regime of this material may be akin to a
spin ice state subject to both thermal and quantum fluctuations $-$ a {\it
quantum spin ice} (QSI) of sorts. Here we explore the effect of a magnetic
field ${\bm B}$ along the $[111]$ direction on the QSI state. To do so, we
investigate the magnetic properties of a microscopic model of Tb2Ti2O7 in an
independent tetrahedron approximation in a finite ${\bm B}$ along $[111]$. Such
a model describes semi-quantitatively the collective paramagnetic regime where
nontrivial spin correlations start to develop at the shortest lengthscale, that
is over a single tetrahedron, but where no long range order is yet present. Our
results show that a magnetization plateau develops at low temperatures as the
system develops ${\bm B}=0$ ferromagnetic spin-ice-like "two-in/two-out"
correlations at the shortest lengthscale. From these results, we are led to
propose that the observation of such a [111] magnetization plateau in Tb2Ti2O7
would provide compelling evidence for a QSI at ${\bm B}=0$ in this material and
help guide the development of a theory for the origin of its spin liquid state.
