Vladimir V. Konotop, Dmitry A. Zezyulin

We introduce a stochastic PT-symmetric coupler, which is based on dual-core waveguides with fluctuating parameters, such that the gain and the losses are exactly balanced in average. We consider different parametric regimes which correspond to the broken and unbroken PT symmetry, as well as to the exceptional point of the underlying deterministic system. We demonstrate that in all the cases the statistically averaged intensity of the field grows. This result holds for either linear or nonlinear coupler and is independent on the type of fluctuations.

http://arxiv.org/abs/1401.6352

Optics (physics.optics)

D. Borisov

We consider a model of planar PT-symmetric waveguide and study the phenomenon of the eigenvalues collision under the perturbation of boundary conditions. This phenomenon was discovered numerically in previous works. The main result of this work is an analytic explanation of this phenomenon.

http://arxiv.org/abs/1401.6316

Spectral Theory (math.SP); Mathematical Physics (math-ph); Analysis of PDEs (math.AP)

E. M. Ferreira, J. Sesma

The spectrum of a one-dimensional Hamiltonian with potential V(x)=ix2 for negative x and \(V(x)=−ix^2\) for positive x is analyzed. The Schrodinger equation is algebraically solvable and the eigenvalues are obtained as the zeros of an expression explicitly given in terms of Gamma functions. The spectrum consists of one real eigenvalue and an infinite set of pairs of complex conjugate eigenvalues.

http://arxiv.org/abs/1401.5937

Quantum Physics (quant-ph)

Soumendu Sundar Mukherjee, Pinaki Roy

We investigate some questions on the construction of \(\eta\) operators for pseudo-Hermitian Hamiltonians. We give a sufficient condition which can be exploited to systematically generate a sequence of \(\eta\) operators starting from a known one, thereby proving the non-uniqueness of \(\eta\) for a particular pseudo-Hermitian Hamiltonian. We also study perturbed Hamiltonians for which \(\eta\)’s corresponding to the original Hamiltonian still work.

http://arxiv.org/abs/1401.5255

Quantum Physics (quant-ph)

Sanjib Dey, Andreas Fring, Thilagarajah Mathanaranjan

We study several classes of non-Hermitian Hamiltonian systems, which can be expressed in terms of bilinear combinations of Euclidean Lie algebraic generators. The classes are distinguished by different versions of antilinear (PT)-symmetries exhibiting various types of qualitative behaviour. On the basis of explicitly computed non-perturbative Dyson maps we construct metric operators, isospectral Hermitian counterparts for which we solve the corresponding time-independent Schroedinger equation for specific choices of the coupling constants. In these cases general analytical expressions for the solutions are obtained in the form of Mathieu functions, which we analyze numerically to obtain the corresponding energy eigenspectra. We identify regions in the parameter space for which the corresponding spectra are entirely real and also domains where the PT symmetry is spontaneously broken and sometimes also regained at exceptional points. In some cases it is shown explicitly how the threshold region from real to complex spectra is characterized by the breakdown of the Dyson maps or the metric operator. We establish the explicit relationship to models currently under investigation in the context of beam dynamics in optical lattices.

http://arxiv.org/abs/1401.4426

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

Ali Mostafazadeh

The transfer matrix of a possibly complex and energy-dependent scattering potential can be identified with the S-matrix of a two-level time-dependent non-Hermitian Hamiltonian H(t). We show that the application of the adiabatic approximation to H(t) corresponds to the semiclassical description of the original scattering problem. In particular, the geometric part of the phase of the evolving eigenvectors of H(t) gives the pre-exponential factor of the WKB wave functions. We use these observations to give an explicit semiclassical expression for the transfer matrix. This allows for a detailed study of the semiclassical unidirectional reflectionlessness and invisibility. We examine concrete realizations of the latter in the realm of optics.

http://arxiv.org/abs/1401.4315

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

A. A. Zyablovsky, A. P. Vinogradov, A. V. Dorofeenko, A. A. Pukhov, A. A. Lisyansky

We discuss phase transitions in PT-symmetric optical systems. We show that due to frequency dispersion of the dielectric permittivity, an optical system can have PT-symmetry at isolated frequency points only. An assumption of the existence of a PT-symmetric system in a continuous frequency interval violates the causality principle. Therefore, the ideal symmetry-breaking transition cannot be observed by simply varying the frequency.

http://arxiv.org/abs/1401.4043

Optics (physics.optics); Mesoscale and Nanoscale Physics (cond-mat.mes-hall); Quantum Physics (quant-ph)

Daniel Haag, Holger Cartarius, Günter Wunner

We consider the linear and nonlinear Schrodinger equation for a Bose-Einstein condensate in a harmonic trap with PT-symmetric double-delta function loss and gain terms. We verify that the conditions for the applicability of a recent proposition by Mityagin and Siegl on singular perturbations of harmonic oscillator type self-adjoint operators are fulfilled. In both the linear and nonlinear case we calculate numerically the shifts of the unperturbed levels with quantum numbers n of up to 89 in dependence on the strength of the non-Hermiticity and compare with rigorous estimates derived by those authors. We confirm that the predicted 1/n1/2 estimate provides a valid upper bound on the the shrink rate of the numerical eigenvalues. Moreover, we find that a more recent estimate of log(n)/n3/2 is in excellent agreement with the numerical results. With nonlinearity the shrink rates are found to be smaller than without nonlinearity, and the rigorous estimates, derived only for the linear case, are no longer applicable.

http://arxiv.org/abs/1401.2896

Quantum Physics (quant-ph); Mathematical Physics (math-ph); Spectral Theory (math.SP)

Andreas Löhle, Holger Cartarius, Daniel Haag, Dennis Dast, Jörg Main, Günter Wunner

A Bose-Einstein condensate trapped in a double-well potential, where atoms are incoupled to one side and extracted from the other, can in the mean-field limit be described by the nonlinear Gross-Pitaevskii equation (GPE) with a \({\mathcal PT}\)-symmetric external potential. If the strength of the in- and outcoupling is increased two \({\mathcal PT}\) broken states bifurcate from the \({\mathcal PT}\)-symmetric ground state. At this bifurcation point a stability change of the ground state is expected. However, it is observed that this stability change does not occur exactly at the bifurcation but at a slightly different strength of the in-/outcoupling effect. We investigate a Bose-Einstein condensate in a \({\mathcal PT}\)-symmetric double-\(\delta\) potential and calculate the stationary states. The ground state’s stability is analysed by means of the Bogoliubov-de Gennes equations and it is shown that the difference in the strength of the in-/outcoupling between the bifurcation and the stability change can be completely explained by the norm-dependency of the nonlinear term in the Gross-Pitaevskii equation.

http://arxiv.org/abs/1401.2354

Quantum Physics (quant-ph); Quantum Gases (cond-mat.quant-gas); Chaotic Dynamics (nlin.CD)

Marco Ornigotti, Alexander Szameit

The concept of quasi-PT symmetry in optical wave guiding system is elaborated by comparing the evolution dynamics of a PT-symmetric directional coupler and a passive directional coupler. In particular we show that in the low loss regime, apart for an overall exponentially damping factor that can be compensated via a dynamical renormalization of the power flow in the system along the propagation direction, the dynamics of the passive coupler fully reproduce the one of the PT-symmetric system.

http://arxiv.org/abs/1401.2299

Optics (physics.optics)