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)
Sanjib Dey, Andreas Fring
We find that real and complex Bohmian quantum trajectories resulting from well-localized Klauder coherent states in the quasi-Poissonian regime possess qualitatively the same type of trajectories as those obtained from a purely classical analysis of the corresponding Hamilton-Jacobi equation. In the complex cases treated the quantum potential results to a constant, such that the agreement is exact. For the real cases we provide conjectures for analytical solutions for the trajectories as well as the corresponding quantum potentials. The overall qualitative behaviour is governed by the Mandel parameter determining the regime in which the wavefunctions evolve as soliton like structures. We demonstrate these features explicitly for the harmonic oscillator and the Poeschl-Teller potential.
http://arxiv.org/abs/1305.4619
Quantum Physics (quant-ph); Mathematical Physics (math-ph)