Daniel Leykam, Vladimir V. Konotop, Anton S. Desyatnikov
We study the effect of lifting the degeneracy of vortex modes with a PT symmetric defect, using discrete vortices in a circular array of nonlinear waveguides as an example. When the defect is introduced, the degenerate linear vortex modes spontaneously break PT symmetry and acquire complex eigenvalues, but nonlinear propagating modes with real propagation constants can still exist. The stability of nonlinear modes depends on both the magnitude and the sign of the vortex charge, thus PT symmetric systems offer new mechanisms to control discrete vortices.
http://arxiv.org/abs/1301.1052
Optics (physics.optics); Pattern Formation and Solitons (nlin.PS)
Chao Hang, Guoxiang Huang, Vladimir V. Konotop
We show that a vapor of multilevel atoms driven by far-off resonant laser beams, with possibility of interference of two Raman resonances, is highly efficient for creating parity-time (PT) symmetric profiles of the probe-field refractive index, whose real part is symmetric and imaginary part is anti-symmetric in space. The spatial modulation of the susceptibility is achieved by proper combination of standing-wave strong control fields and of Stark shifts induced by a far-off-resonance laser field. As particular examples we explore a mixture of isotopes of Rubidium atoms and design a PT-symmetric lattice and a parabolic refractive index with a linear imaginary part.
http://arxiv.org/abs/1212.5486
Optics (physics.optics)
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)
Yuliy V. Bludov, Vladimir V. Konotop, Boris A. Malomed
We construct dark solitons in the recently introduced model of the nonlinear dual-core coupler with the mutually balanced gain and loss applied to the two cores, which is a realization of parity-time symmetry in nonlinear optics. The main issue is stability of the dark solitons. The modulational stability of the CW (continuous-wave) background, which supports the dark solitons, is studied analytically, and the full stability is investigated in a numerical form, via computation of eigenvalues for modes of small perturbations. Stability regions are thus identified in the parameter space of the system, and verified in direct simulations. Collisions between stable dark solitons are briefly considered too.
http://arxiv.org/abs/1211.3746
Optics (physics.optics); Pattern Formation and Solitons (nlin.PS)
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)
Vladimir V. Konotop, Valery S. Shchesnovich, Dmitry A. Zezyulin
The combination of the interference with the amplification of modes in a waveguide with gain and losses can result in a giant amplification of the propagating beam, which propagates without distortion of its average amplitude. An increase of the gain-loss gradient by only a few times results in a magnification of the beam by a several orders of magnitude.
http://arxiv.org/abs/1207.1792
Optics (physics.optics)
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. Kochan, D. Krejcirik, R. Novak, P. Siegl
We consider one-dimensional Pauli Hamiltonians in a bounded interval with possibly non-self-adjoint Robin-type boundary conditions. We study the influence of the spin-magnetic interaction on the interplay between the type of boundary conditions and the spectrum. A special attention is paid to PT-symmetric boundary conditions with the physical choice of the time-reversal operator T.
http://arxiv.org/abs/1203.5011
Quantum Physics (quant-ph); High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph)
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)