Two models of PT Quantum Mechanics and their behavior in the vicinity of spectral singularities

The last ten years witnessed a strong research activity into so called PT Quantum Mechanics (PTQM) --- a Quantum Mechanics whose Hamiltonians are allowed to be non-Hermitian but PT-symmetric. In general, PTQM has sectors of exact PT-symmetry with purely real energy spectrum as well as sectors of spontaneously broken PT-symmetry with pairwise complex conjugate energy branches. Sectors of exact PT-symmetry can be isomorphically mapped into models of conventional (von Neumann) Quantum Mechanics (possibly of highly nonlocal type). Sectors of spontaneously broken PT-symmetry might have realizations as certain effective quantum systems. Varying the coupling parameters of PTQM models the corresponding quantum system can undergo phase transitions from exact PT symmetry to spontaneously broken PT-symmetry. Such PT phase transitions are associated with branch points (exceptional points) of the energy spectrum --- so called spectral singularities.

In the talk two PTQM models and their behavior in the vicinity of spectral singularities will be discussed.

In the first part of the talk, I will report on the quantum brachistochrone problem for PTQM, i.e. the problem of finding a PT-symmetric Hamiltonian which minimizes the time needed for evolving a given initial state into a predefined final state (e.g. for a spin-flip). This problem was solved by Bender, Brody, Jones and Meister in 2007. It turned out that in PTQM the evolution time for a spin flip can be tended toward zero so that the Aharonov-Anandan lower bound on the evolution time for Hermitian systems would be violated --- a kind of "quantum wormhole" effect. In 2007 we showed that this effect occurs in the vicinity of a PT phase transition (a spectral singularity). A still open problem at that time was a possible scheme for an experimental realization, because one somehow would have to switch between PTQM and experimental setups based on conventional QM. In [Phys. Rev. Lett. 101, 230404, 2008] we solved this switching problem by embedding the PTQM brachistochrone into a higher dimensional Hilbert space. In this way we found a realization of the PTQM brachistochrone evolution as a special kind of tuned unitary evolution in a highly asymmetrically entangled two-qubit setup --- which in principle might be realized experimentally in the nearest future.

The second part of the talk will be devoted to a non-Hermitian PT-symmetric two-mode Bose Hubbard model which might describe e.g. a BEC in a double-well potential with additional well-balanced gain-loss couplings. The main focus is laid on the unfolding of the higher-order spectral singularities typical for this kind of models. A perturbative Newton polygon technique allows us to qualitatively explain the numerically obtained branching behavior of the energy spectrum. It turns out that the Hessenberg type of the effective coupling matrix is responsible for the special Galois structure of the occurring eigenvalue rings in the complex energy plane.