Kazuki Kanki

Department of Physical Science
Osaka Prefecture University
Japan

Non-Hermitian Liouvillian dynamics in an unstable bosonic system with quadratic Hermitian Hamiltonian

Non-Hermitian nature of an unstable quantum system is analyzed in terms of microscopic laws of physics by using a specific model. A quadratic bosonic Hamiltonian that does not conserve particle number gives rise to non-Hermitian dynamics, i.e. the Liouvillian in the Heisenberg equation is not Hermitian but pseudo-Hermitian, in spite of the fact that the Hamiltonian is Hermitian. For a parametric amplifier the Liouvillian has a complex conjugate pair of eigenfrequencies and the Hamiltonian has a pair of complete sets of eigenstates with complex eigenvalues. Although all the eigenvalues in the complex spectral decomposition are complex, time evolution with the Hamiltonian is unitary. The complex eigenvalues are interpreted in terms of squeezing. They appear also as poles of the S matrix for a particle under an inverted harmonic potential.