Christopher Birchall, Henning Schomerus

We formulate gaussian and circular random-matrix models representing a coupled system consisting of an absorbing and an amplifying resonator, which are mutually related by a generalized time-reversal symmetry. Motivated by optical realizations of such systems we consider a PT or a PTT’ time-reversal symmetry, which impose different constraints on magneto-optical effects, and then focus on five common settings. For each of these, we determine the eigenvalue distribution in the complex plane in the short-wavelength limit, which reveals that the fraction of real eigenvalues among all eigenvalues in the spectrum vanishes if all classical scales are kept fixed. Numerically, we find that the transition from real to complex eigenvalues in the various ensembles display a different dependence on the coupling strength between the two resonators. These differences can be linked to the level spacing statistics in the hermitian limit of the considered models.

http://arxiv.org/abs/1208.2575

Quantum Physics (quant-ph); Disordered Systems and Neural Networks (cond-mat.dis-nn); Optics (physics.optics)

Christopher Birchall, Henning Schomerus

We find that in nonhermitian PT-symmetric systems (as realized in resonators with balanced absorption and amplification), a mechanism based on quantum-to-classical correspondence reduces the occurrence of strongly amplified states. The reduction arises from semiclassically emerging hierarchical phase-space structures that are associated with the coupling of the amplifying and absorbing regions (forward and backward-trapped sets and their complements), and amounts to a generalization of the fractal Weyl law, earlier proposed for ballistically open systems. In the context of the recently introduced class of PT-symmetric laser-absorbers, this phenomenon reduces the number of states participating in the mode competition.

http://arxiv.org/abs/1208.2259

Quantum Physics (quant-ph); Optics (physics.optics)

Petr Siegl, David Krejcirik

We show that the eigenvectors of the PT -symmetric imaginary cubic oscillator are complete, but do not form a Riesz basis. This results in the existence of a bounded metric operator having intrinsic singularity reflected in the inevitable unboundedness of the inverse. Moreover, the existence of nontrivial pseudospectrum is observed. In other words, there is no quantum-mechanical Hamiltonian associated with it via bounded and boundedly invertible similarity transformations. These results open new directions in physical interpretation of PT -symmetric models with intrinsically singular metric, since their properties are essentially different with respect to self-adjoint Hamiltonians, for instance, due to spectral instabilities.

http://arxiv.org/abs/1208.1866

Quantum Physics (quant-ph); Mathematical Physics (math-ph)

Physical Review D **86**, 121702 (2012)

Mohammad-Ali Miri, Alois Regensburger, Ulf Peschel, Demetrios N. Christodoulides

We investigate a new class of optical mesh periodic structures that are discretized in both the transverse and longitudinal directions. These networks are composed of waveguide arrays that are discretely coupled while phase elements are also inserted to discretely control their effective potentials and can be realized both in the temporal and the spatial domain. Their band structure and impulse response is studied in both the passive and parity-time (PT) symmetric regime. The possibility of band merging and the emergence of exceptional points along with the associated optical dynamics are considered in detail both above and below the PT-symmetry breaking point. Finally unidirectional invisibility in PT-synthetic mesh lattices is also examined along with possible superluminal light transport dynamics.

http://arxiv.org/abs/1208.1722

Quantum Physics (quant-ph)