Solvable dilation model of time-dependent PT -symmetric systems

The dilation method is a practical way to experimentally simulate non-Hermitian, especially $\mathcal{PT}$-symmetric, quantum systems. However, the time-dependent dilation problem cannot be explicitly solved in general. In this paper, we present a simple yet nontrivial exactly solvable dilation problem for two-dimensional time-dependent $\mathcal{PT}$-symmetric Hamiltonians. Our system is initially set in the unbroken $\mathcal{PT}$-symmetric phase, then goes across the so-called exceptional point, and ends in the broken $\mathcal{PT}$-symmetric phase. For this system, the dilated Hamiltonian and the evolution of $\mathcal{PT}$-symmetric system are analytically worked out. By investigating the large-time behaviors, we give an effective method to choose and adjust the dilation parameters. Our result also shows that the exceptional points do not have much physical relevance in a time-dependent system.