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Month December 2011

Matrix algorithm for solving Schroedinger equations with position-dependent mass or complex optical potentials

Johann Foerster, Alejandro Saenz, Ulli Wolff

We represent low dimensional quantum mechanical Hamiltonians by moderately sized finite matrices that reproduce the lowest O(10) boundstate energies and wave functions to machine precision. The method extends also to Hamiltonians that are neither Hermitian nor PT symmetric and thus allows to investigate whether or not the spectra in such cases are still real. Furthermore, the approach is especially useful for problems in which a position-dependent mass is adopted, for example in effective-mass models in solid-state physics or in the approximate treatment of coupled nuclear motion in molecular physics or quantum chemistry. The performance of the algorithm is demonstrated by considering the inversion motion of different isotopes of ammonia molecules within a position-dependent-mass model and some other examples of one- and two-dimensional Hamiltonians that allow for the comparison to analytical or numerical results in the literature.

http://arxiv.org/abs/1112.5294
Quantum Physics (quant-ph); High Energy Physics – Lattice (hep-lat); Mathematical Physics (math-ph)

Conservation relations and anisotropic transmission resonances in one-dimensional PT-symmetric photonic heterostructures

Li Ge, Y. D. Chong, A. D. Stone

We analyze the optical properties of one-dimensional (1D) PT-symmetric structures of arbitrary complexity. These structures violate normal unitarity (photon flux conservation) but are shown to satisfy generalized unitarity relations, which relate the elements of the scattering matrix and lead to a conservation relation in terms of the transmittance and (left and right) reflectances. One implication of this relation is that there exist anisotropic transmission resonances in PT-symmetric systems, frequencies at which there is unit transmission and zero reflection, but only for waves incident from a single side. The spatial profile of these transmission resonances is symmetric, and they can occur even at PT-symmetry breaking points. The general conservation relations can be utilized as an experimental signature of the presence of PT-symmetry and of PT-symmetry breaking transitions. The uniqueness of PT-symmetry breaking transitions of the scattering matrix is briefly discussed by comparing to the corresponding non-Hermitian Hamiltonians.

http://arxiv.org/abs/1112.5167
Optics (physics.optics)

Disorder effects in tunable waveguide arrays with parity-symmetric tunneling

Clinton Thompson, Yogesh N. Joglekar, Gautam Vemuri

We investigate the effects of disorder on single particle time-evolution and two-particle correlations in an array of evanescently coupled waveguides with position-dependent tunneling rates. In the clean limit, the energy spectrum of such an array is widely tunable. In the presence of a Hermitian on-site or tunneling disorder, we find that the localization of a wave packet is highly sensitive to this energy spectrum. In particular, for an input confined to a single waveguide, we show that the fraction of light localized to the original waveguide depends on the tunneling profile. We compare the two-particle intensity correlations in the presence of Hermitian, tunneling disorder and non-Hermitian, parity-and-time-reversal \{\mathcal{PT}\} symmetric, on-site potential disorder. We show the two-particle correlation function in both cases is qualitatively similar, since both disorders preserve the particle-hole symmetric nature of the energy spectrum.

http://arxiv.org/abs/1112.4720
Quantum Physics (quant-ph); Optics (physics.optics)

Exceptional Point Dynamics in Photonic Honeycomb Lattices with PT Symmetry

Hamidreza Ramezani, Tsampikos Kottos, Vassilios Kovanis, Demetrios N. Christodoulides

We theoretically investigate the flow of electromagnetic waves in complex honeycomb photonic lattices with local PT symmetries. Such PT structure is introduced via a judicious arrangement of gain or loss across the honeycomb lattice, characterized by a gain/loss parameter \{\gamma\}. We found a new class of conical diffraction phenomena where the formed cone is brighter and travels along the lattice with a transverse speed proportional to \{\sqrt{\gamma}\}.

http://arxiv.org/abs/1112.4734
Optics (physics.optics); Quantum Physics

Pseudo Hermitian formulation of Black-Scholes equation

T. K. Jana, P. Roy

We show that the non Hermitian Black-Scholes Hamiltonian and its various generalizations are eta-pseudo Hermitian. The metric operator eta is explicitly constructed for this class of Hamitonians. It is also shown that the e?ective Black-Scholes Hamiltonian and its partner form a pseudo supersymmetric system.

http://arxiv.org/abs/1112.3217
General Finance (q-fin.GN)

Scattering along a complex loop in a solvable PT-symmetric model

Miloslav Znojil

A non-unitary version of quantum scattering is studied via an exactly solvable toy model. The model is merely asymptotically local since the smooth path of the coordinate is admitted complex in the non-asymptotic domain. At any real angular-momentum-like parameter the reflection R and transmission T are shown to change with the winding number (i.e., topology) of the path. The points of unitarity appear related to the points of existence of quantum-knot bound states.

http://arxiv.org/abs/1112.2644
Mathematical Physics (math-ph); Quantum Physics (quant-ph)