Debdeep Sinha, Pijush K. Ghosh
A class of non-local non-linear Schrodinger equations(NLSE) is considered in an external potential with space-time modulated coefficient of the nonlinear interaction term as well as confining and/or loss-gain terms. This is a generalization of a recently introduced integrable non-local NLSE with self induced potential that is PT symmetric in the corresponding stationary problem. Exact soliton solutions are obtained for the inhomogeneous and/or non autonomous non-local NLSE by using similarity transformation and the method is illustrated with a few examples. It is found that only those transformations are allowed for which the transformed spatial coordinate is odd under the parity transformation of the original one. It is shown that the non-local NLSE without the external potential and a \(d+1\) dimensional generalization of it, admits all the symmetries of the \(d+1\) dimensional Schrodinger group. The conserved Noether charges associated with the time-translation, dilatation and special conformal transformation are shown to be real-valued in spite of being non-hermitian. Finally, dynamics of different moments are studied with an exact description of the time-evolution of the “pseudo-width” of the wave-packet for the special case when the system admits a \(O(2,1)\) conformal symmetry.
http://arxiv.org/abs/1408.0954
Exactly Solvable and Integrable Systems (nlin.SI); High Energy Physics – Theory (hep-th)
Jianke Yang
Symmetry breaking of solitons in a class of one-dimensional parity-time (PT) symmetric complex potentials with cubic nonlinearity is reported. In generic PT symmetric potentials, such symmetry breaking is forbidden. However, in a special class of PT-symmetric potentials \(V(x)=g^2(x)+αg(x)+ig′(x)\), where \(g(x)\) is a real and even function and α a real constant, symmetry breaking of solitons can occur. That is, a branch of non-PT-symmetric solitons can bifurcate out from the base branch of PT-symmetric solitons when the base branch’s power reaches a certain threshold. At the bifurcation point, the base branch changes stability, and the bifurcated branch can be stable.
http://arxiv.org/abs/1408.0687
Optics (physics.optics); Pattern Formation and Solitons (nlin.PS)
Bing He, Shu-Bin Yan, Jing Wang, Min Xiao
It is generally difficult to study the dynamical properties of a quantum system with both inherent quantum noises and non-perturbative nonlinearity. Due to the possibly drastic intensity increase of an input coherent light in the gain-loss waveguide couplers with parity-time (PT) symmetry, the Kerr effect from a nonlinearity added into the systems can be greatly enhanced, and is expected to create the macroscopic entangled states of the output light fields with huge photon numbers. Meanwhile, the quantum noises also coexist with the amplification and dissipation of the light fields. Under the interplay between the quantum noises and nonlinearity, the quantum dynamical behaviors of the systems become rather complicated. However, the important quantum noise effects have been mostly neglected in the previous studies about nonlinear PT symmetric systems. Here we present a solution to this non-perturbative quantum nonlinear problem, showing the real-time evolution of the system observables. The enhanced Kerr nonlinearity is found to give rise to a previously unknown decoherence effect that is irrelevant to the quantum noises, and imposes a limit on the emergence of macroscopic nonclassicality. In contrast to what happen in the linear systems, the quantum noises exert significant impact on the system dynamics, and can create the nonclassical light field states in conjunction with the enhanced Kerr nonlinearity. This first study on the noise involved quantum nonlinear dynamics of the coupled gain-loss waveguides can help to better understand the quantum noise effects in the broad nonlinear systems.
http://arxiv.org/abs/1408.0565
Quantum Physics (quant-ph)
Sanjib Dey, Andreas Fring, Thilagarajah Mathanaranjan
We propose a noncommutative version of the Euclidean Lie algebra \(E_2\). Several types of non-Hermitian Hamiltonian systems expressed in terms of generic combinations of the generators of this algebra are investigated. Using the breakdown of the explicitly constructed Dyson maps as a criterium, we identify the domains in the parameter space in which the Hamiltonians have real energy spectra and determine the exceptional points signifying the crossover into the different types of spontaneously broken PT-symmetric regions with pairs of complex conjugate eigenvalues. We find exceptional points which remain invariant under the deformation as well as exceptional points becoming dependent on the deformation parameter of the algebra.
http://arxiv.org/abs/1407.8097
Quantum Physics (quant-ph); Mathematical Physics (math-ph)
Richard J. Potton
In crystal optics the special status of the rest frame of the crystal means that space-time symmetry is less restrictive of electrodynamic phenomena than it is of static electromagnetic effects. A relativistic justification for this claim is provided and its consequences for the analysis of optical activity are explored. The discrete space-time symmetries P and T that lead to classification of static property tensors as polar or axial, time-invariant (-i) or time-change (-c) are shown to be connected by orientation considerations. The connection finds expression in the dynamic phenomenon of gyrotropy in certain, symmetry determined, crystal classes. In particular, the degeneracies of forward and backward waves in optically active crystals arise from the covariance of the wave equation under space-time (PT) reversal.
http://arxiv.org/abs/1407.6797
Optics (physics.optics)
Asiri Nanayakkara, Thilagarajah Mathanaranjan
In this paper we show that the non-Hermitian Hamiltonians \(H=p^{2}-gx^{4}+a/x^2\) and the conventional Hermitian Hamiltonians \(h=p^2+4gx^{4}+bx\) (\(a,b\in \mathbb{R}\)) are isospectral if \(a=(b^2-4g\hbar^2)/16g\) and \(a\geq -\hbar^2/4\). This new class includes the equivalent non-Hermitian – Hermitian Hamiltonian pair, \(p^{2}-gx^{4}\) and \(p^{2}+4gx^{4}-2\hbar \sqrt{g}x\), found by Jones and Mateo six years ago as a special case. When \(a=\left(b^{2}-4g\hbar ^{2}\right) /16g\) and \(a<-\hbar^2/4\), although \(h\) and \(H\) are still isospectral, \(b\) is complex and \(h\) is no longer the Hermitian counterpart of \(H\).
http://arxiv.org/abs/1407.4633
Mathematical Physics (math-ph)
Yogesh N. Joglekar, Rahul Marathe, P. Durganandini, Rajeev K. Pathak
We investigate the effects of a time-periodic, non-hermitian, PT-symmetric perturbation on a system with two (or few) levels, and obtain its phase diagram as a function of the perturbation strength and frequency. We demonstrate that when the perturbation frequency is close to one of the system resonances, even a vanishingly small perturbation leads to PT symmetry breaking. We also find a restored PT-symmetric phase at high frequencies, and at moderate perturbation strengths, we find multiple frequency windows where PT-symmetry is broken and restored. Our results imply that the PT-symmetric Rabi problem shows surprisingly rich phenomena absent in its hermitian or static counterparts.
http://arxiv.org/abs/1407.4535
Optics (physics.optics); Other Condensed Matter (cond-mat.other); Quantum Physics (quant-ph)
C. Poli, M. Bellec, U.Kuhl, F. Mortessagne, H. Schomerus
An attractive mechanism to induce robust spatially confined states utilizes interfaces between regions with topologically distinct gapped band structures. For electromagnetic waves, this mechanism can be realized in two dimensions by breaking symmetries in analogy to the quantum Hall effect or by employing analogies to the quantum spin Hall effect, while in one dimension it can be obtained by geometric lattice modulation. Induced by the presence of the interface, a topologically protected, exponentially confined state appears in the middle of the band gap. The intrinsic robustness of such states raises the question whether their properties can be controlled and modified independently of the other states in the system. Here, we draw on concepts from passive non-hermitian parity-time (PT)-symmetry to demonstrate the selective control and enhancement of a topologically induced state in a one-dimensional microwave set-up. In particular, we show that the state can be isolated from losses that affect all other modes in the system, which enhances its visibility in the temporal evolution of a pulse. The intrinsic robustness of the state to structural disorder persists in the presence of the losses. The combination of concepts from topology and non-hermitian symmetry is a promising addition to the set of design tools for optical structures with novel functionality.
http://arxiv.org/abs/1407.3703
Optics (physics.optics); Mesoscale and Nanoscale Physics (cond-mat.mes-hall)
P.A. Kalozoumis, G. Pappas, F.K. Diakonos, P. Schmelcher
Recently [Phys. Rev. Lett. 106, 093902 (2011)] it has been shown that PT-symmetric scattering systems with balanced gain and loss, undergo a transition from PT-symmetric scattering eigenstates, which are norm preserving, to symmetry broken pairs of eigenstates exhibiting net amplification and loss. In the present work we derive the existence of an invariant non-local current which can be directly associated with the observed transition playing the role of an “order parameter”. The use of this current for the description of the PT-symmetry breaking allows the extension of the known phase diagram to higher dimensions incorporating scattering states which are not eigenstates of the scattering matrix.
http://arxiv.org/abs/1407.2655
Optics (physics.optics); Quantum Physics (quant-ph)
João-Paulo Dias, Mário Figueira, Vladimir V. Konotop, Dmitry A. Zezyulin
We prove finite time supercritical blowup in a parity-time-symmetric system of the two coupled nonlinear Schrodinger (NLS) equations. One of the equations contains gain and the other one contains dissipation such that strengths of the gain and dissipation are equal. We address two cases: in the first model all nonlinear coefficients (i.e. the ones describing self-action and non-linear coupling) correspond to attractive (focusing) nonlinearities, and in the second case the NLS equation with gain has attractive nonlinearity while the NLS equation with dissipation has repulsive (defocusing) nonlinearity and the nonlinear coupling is repulsive, as well. The proofs are based on the virial technique arguments. Several particular cases are also illustrated numerically.
http://arxiv.org/abs/1407.2438
Analysis of PDEs (math.AP); Optics (physics.optics)