  By coupling the asymmetric three-terminal mesoscopic dielectric system with a
temperature probe, at low temperature, the ballistic heat flux flow through the
other two asymmetric terminals in the nonlinear response regime is studied
based on the Landauer formulation of transport theory. The thermal
rectification is attained at the quantum regime. It is a purely quantum effect
and is determined by the dependence of the ratio
$\tau_{RC}(\omega)/\tau_{RL}(\omega)$ on $\omega$, the phonon's frequency.
Where $\tau_{RC}(\omega)$ and $\tau_{RL}(\omega)$ are respectively the
transmission coefficients from two asymmetric terminals to the temperature
probe, which are determined by the inelastic scattering of ballistic phonons in
the temperature probe. Our results are confirmed by extensive numerical
simulations.
