A solvable dilation model of PT -symmetric systems

The dilation method is an important and useful way in experimentally simulating non-Hermitian, especially $cal PT$-symmetric systems. However, the time dependent dilation problem cannot be explicitly solved in general. In this paper, we consider a special two dimensional time dependent $cal PT$-symmetric system, which is initially set in the unbroken $cal PT$-symmetric phase and later goes across the exceptional point and enters the broken $cal PT$-symmetric phase. For this system, the dilation Hamiltonian and the evolution of $cal PT$-symmetric system are analytically worked out.