Dynamical invariants for pseudo-Hermitian Hamiltonians

Abstract

We derive the dynamical invariants for a general $N$-state quantum system described by a pseudo-Hermitian Hamiltonian. Explicit expressions are presented for two- and three-state systems, which are exemplified by explicit analytic solutions for constant couplings. In the two-state case, we derive non-Hermitian analogs of the Bloch vector and the Bloch equation, customary for Hermitian quantum systems. We suggest possible physical implementations of the dynamical invariants in waveguide optics and frequency conversion.