  The dilation method is a practical way to experimentally simulate
non-Hermitian, especially $\cal PT$-symmetric quantum systems. However, the
time-dependent dilation problem cannot be explicitly solved in general. In this
paper, we present a simple yet non-trivial exactly solvable dilation problem
with two dimensional time-dependent $\cal PT$-symmetric Hamiltonian. Our system
is initially set in the unbroken $\cal PT$-symmetric phase and later goes
across the so-called exceptional point and enters the broken $\cal
PT$-symmetric phase. For this system, the dilated Hamiltonian and the evolution
of $\cal PT$-symmetric system are analytically worked out. Our result clearly
showed that the exceptional points do not have much physical relevance in a
\textit{time-dependent} system.
