Information retrieval and eigenstate coalescence in a non-Hermitian quantum system with anti- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">PT</mml:mi></mml:math> symmetry

Non-Hermitian systems with parity-time reversal ($\mathcal{PT}$) or anti-$\mathcal{PT}$ symmetry have attracted a wide range of interest owing to their unique characteristics and counterintuitive phenomena. One the most extraordinary features is presence an exceptional point, across which phase transition spontaneously broken $\mathcal{PT}$ takes place. We implement Floquet Hamiltonian single qubit by periodically driving dissipative quantum system trapped ion. With stroboscopic emission state tomography, we obtain time evolution density matrix for arbitrary initial state, directly demonstrate information retrieval, eigenstate coalescence, topological energy spectra as non-Hermitian systems.