Quantum states talk via the environment

Abstract: The states of an open quantum system interact ("talk") with one another via the extended environment into which the localized system is embedded. This interaction is mediated by the source term of the Schr\"odinger equation which describes the coupling between system and environment. The source term is nonlinear and causes width bifurcation and, respectively, level repulsion. It is strong only in the neighborhood of singular (exceptional) points. We provide typical results for the phase rigidity and the mixing of the biorthogonal eigenfunctions of the Hamiltonian. A completely unexpected result is that the phase rigidity approaches a value near to one (characteristic of orthogonal eigenfunctions) when width bifurcation (or level repulsion) becomes maximum. This behavior of the phase rigidity is caused exclusively by the nonlinearity of the source term of the Schr\"odinger equation. The eigenfunctions remain mixed in the set of original wavefunctions also under these critical conditions. Eventually, a dynamical phase transition occurs. This process is irreversible. It allows, among others, a physical interpretation of the well-known resonance trapping phenomenon. Our results for the eigenvalues and eigenfunctions of a non-Hermitian Hamiltonian are supported by experimental results obtained in different systems. The relation of our results for open quantum systems under critical conditions to those in optics and photonics with PT-symmetry breaking is considered. As a result, the balance between gain and loss is a very interesting general phenomenon that may occur in many different systems.