K. Li, D. A. Zezyulin, V. V. Konotop, P. G. Kevrekidis
In this work, we propose a PT-symmetric coupler whose arms are birefringent waveguides as a realistic physical model which leads to a so-called quadrimer i.e., a four complex field setting. We seek stationary solutions of the resulting linear and nonlinear model, identifying its linear point of PT symmetry breaking and examining the corresponding nonlinear solutions that persist up to this point, as well as, so-called, ghost states that bifurcate from them. We obtain the relevant symmetry breaking bifurcations and numerically follow the associated dynamics which give rise to growth/decay even within the PT-symmetric phase. Our obtained stationary nonlinear solutions are found to terminate in saddle-center bifurcations which are analogous to the linear PT-phase transition.
http://arxiv.org/abs/1212.1676
Quantum Physics (quant-ph)
D. A. Zezyulin, V. V. Konotop
By rearrangements of waveguide arrays with gain and losses one can simulate transformations among parity-time (PT-) symmetric systems not affecting their pure real linear spectra. Subject to such transformations, however, the nonlinear properties of the systems undergo significant changes. On an example of an array of four waveguides described by the discrete nonlinear Schr\”odinger equation with dissipation and gain, we show that the equivalence of the underlying linear spectra implies similarity of neither structure nor stability of the nonlinear modes in the arrays. Even the existence of one-parametric families of nonlinear modes is not guaranteed by the PT symmetry of a newly obtained system. Neither the stability is directly related to the PT symmetry: stable nonlinear modes exist even when the spectrum of the linear array is not purely real. We use graph representation of PT-symmetric networks allowing for simple illustration of linearly equivalent networks and indicating on their possible experimental design.
http://arxiv.org/abs/1202.3652
Pattern Formation and Solitons (nlin.PS); Optics (physics.optics)
D. A. Zezyulin, V. V. Konotop
By similarity transformations a parity-time (PT-) symmetric Hamiltonian can be reduced to a Hermitian or to another PT-symmetric Hamiltonian having the same linear spectrum. On an example of a PT-symmetric quadrimer we show that the spectral equivalence of different PT-symmetric and Hermitian systems implies neither similarity of the nonlinear modes nor their stability properties. Even the existence of one-parametric families of nonlinear modes is not guaranteed by the PT symmetry of the underlying linear system. Neither the stability is directly related to the PT symmetry: stable nonlinear modes exist beyond the PT symmetry breaking threshold. A “phase diagram” of a general PT-symmetric quadrimer allows for existence of “triple” points, where three different phases meet. We use graph representation of PT-symmetric networks giving simple illustration of linearly equivalent networks and indicating on their possible experimental design.
http://arxiv.org/abs/1202.3652
Pattern Formation and Solitons (nlin.PS); Optics (physics.optics)
D. A. Zezyulin, Y. V. Kartashov, V. V. Konotop
We report on detailed investigation of the stability of localized modes in the nonlinear Schrodinger equations with a nonlinear parity-time (alias PT) symmetric potential. We are particularly focusing on the case where the spatially-dependent nonlinearity is purely imaginary. We compute the Evans function of the linear operator determining the linear stability of localized modes. Results of the Evans function analysis predict that for sufficiently small dissipation localized modes become stable when the propagation constant exceeds certain threshold value. This is the case for periodic and \(\tanh\)-shaped complex potentials where the modes having widths comparable with or smaller than the characteristic width of the complex potential are stable, while broad modes are unstable. In contrast, in complex potentials that change linearly with transverse coordinate all modes are stable, what suggests that the relation between width of the modes and spatial size of the complex potential define the stability in the general case. These results were confirmed using the direct propagation of the solutions for the mentioned examples.
http://arxiv.org/abs/1111.0898
Pattern Formation and Solitons (nlin.PS); Mathematical Physics (math-ph); Optics (physics.optics)