Mohammed F. Saleh, Andrea Marini, Fabio Biancalana
We have investigated the interaction between a strong soliton and a weak probe with certain configurations that allow optical trapping in gas-filled hollow-core photonic crystal fibers in the presence of the shock effect. We have shown theoretically and numerically that the shock term can lead to an unbroken parity-time (PT) symmetry potential in these kinds of fibers. Reciprocity breaking, a remarkable feature of the PT symmetry, is also demonstrated numerically. Our results will open different configurations and avenues for observing PT-symmetry breaking in optical fibers, without the need to resort to cumbersome dissipative structures.
http://arxiv.org/abs/1310.7497
Optics (physics.optics)
Ananya Ghatak, Raka Dona Ray Mandal, Bhabani Prasad Mandal (BHU)
We complexify a 1-d potential which exhibits bound, reflecting and free states to study various properties of a non-Hermitian system. This potential turns out a PT-symmetric non-Hermitian potential when one of the parameters becomes imaginary. For one PT-symmetric case we have entire real bound state spectrum. Explicit scattering states are constructed to show reciprocity at certain discrete values of energy even though the potential is not parity symmetric. Coexistence of deep energy minima of transmissivity with the multiple spectral singularities (MSS) is observed. We further show that this potential becomes invisible from left (or right) at certain discrete energies. The penetrating states in the other PT-symmetric configuration are always reciprocal even though it is PT-invariant and no spectral singularity (SS) is present in this case. Presence of MSS and reflectionlessness are also discussed for the free states in the later case.
http://arxiv.org/abs/1310.7752
Quantum Physics (quant-ph)
H.Xu, P.G.Kevrekidis, Q.Zhou, D.J.Frantzeskakis, V.Achilleos, R.Carretero-Gonzalez
We study the nonlinear Schrodinger equation with a PT-symmetric potential. Using a hydrodynamic formulation and connecting the phase gradient to the field amplitude, allows for a reduction of the model to a Duffing or a generalized Duffing equation. This way, we can obtain exact soliton solutions existing in the presence of suitable PT-symmetric potentials, and study their stability and dynamics. We report interesting new features, including oscillatory instabilities of solitons and (nonlinear) PT-symmetry breaking transitions, for focusing and defocusing nonlinearities.
http://arxiv.org/abs/1310.7635
Pattern Formation and Solitons (nlin.PS)
Paul Fendley
The spectrum of the quantum Ising chain can be found by expressing the spins in terms of free fermions. An analogous transformation exists for clock chains with Zn symmetry, but is of less use because the resulting parafermionic operators remain interacting. Nonetheless, Baxter showed that a certain non-hermitian (but PT-symmetric) clock Hamiltonian is “free”, in the sense that the entire spectrum is found in terms of independent energy levels, with the striking feature that there are n possibilities for occupying each level. Here I show this directly explicitly finding shift operators obeying a Zn generalization of the Clifford algebra. I also find higher Hamiltonians that commute with Baxter’s and prove their spectrum comes from the same set of energy levels. This thus provides an explicit notion of a “free parafermion”. A byproduct is an elegant method for the solution of the Ising/Kitaev chain with spatially varying couplings.
http://arxiv.org/abs/1310.6049
Statistical Mechanics (cond-mat.stat-mech); Strongly Correlated Electrons (cond-mat.str-el); High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph)
Dmitry E. Pelinovsky, Dmitry A. Zezyulin, Vladimir V. Konotop
We generalize a finite parity-time (PT-)symmetric network of the discrete nonlinear Schrodinger type and obtain general results on linear stability of the zero equilibrium, on the nonlinear dynamics of the dimer model, as well as on the existence and stability of large-amplitude stationary nonlinear modes. A result of particular importance and novelty is the classification of all possible stationary modes in the limit of large amplitudes. We also discover a new integrable configuration of a PT-symmetric dimer.
http://arxiv.org/abs/1310.5651
Pattern Formation and Solitons (nlin.PS); Dynamical Systems (math.DS)
S. Wimberger, C. A. Parra-Murillo, G. Kordas
A paradigm model of modern atom optics is studied, strongly interacting ultracold bosons in an optical lattice. This many-body system can be artificially opened in a controlled manner by modern experimental techniques. We present results based on a non-hermitian effective Hamiltonian whose quantum spectrum is analyzed. The direct access to the spectrum of the metastable many-body system allows us to easily identify relatively stable quantum states, corresponding to previously predicted solitonic many-body structures.
http://arxiv.org/abs/1310.5937
Pattern Formation and Solitons (nlin.PS); Quantum Gases (cond-mat.quant-gas); Quantum Physics (quant-ph)
A. V. Yulin, V. V. Konotop
Stable discrete compactons in arrays of inter-connected three-line waveguide arrays are found in linear and nonlinear limits in conservative and in parity-time PT symmetric models. The compactons result from the interference of the fields in the two lines of waveguides ensuring that the third (middle) line caries no energy. PT-symmetric compactons require not only the presence of gain and losses in the two lines of the waveguides but also complex coupling, i.e. gain and losses in the coupling between the lines carrying the energy and the third line with zero field. The obtained compactons can be stable and their branches can cross the branches of the dissipative solitons. Unusual bifurcations of branches of solitons from linear compactons are described.
http://arxiv.org/abs/1310.5328
Optics (physics.optics); Pattern Formation and Solitons (nlin.PS)
Stefano Longhi
We investigate the onset of parity-time (PT) symmetry breaking in non-Hermitian tight-binding lattices with spatially-extended loss/gain regions in presence of an advective term. Similarly to the instability properties of hydrodynamic open flows, it is shown that PT-symmetry breaking can be either absolute or convective. In the former case, an initially-localized wave packet shows a secular growth with time at any given spatial position, whereas in the latter case the growth is observed in a reference frame moving at some drift velocity while decay occurs at any fixed spatial position. In the convective unstable regime, PT-symmetry is restored when the spatial region of gain/loss in the lattice is limited (rather than extended). We consider specifically a non-Hermitian extension of the Rice-Mele tight binding lattice model, and show the existence of a transition from absolute to convective symmetry breaking when the advective term is large enough. An extension of the analysis to ac-dc-driven lattices is also presented, and an optical implementation of the non-Hermitian Rice-Mele model is suggested, which is based on light transport in an array of evanescently-coupled optical waveguides with a periodically-bent axis and alternating regions of optical gain and loss.
http://arxiv.org/abs/1310.5004
Quantum Physics (quant-ph); Mesoscale and Nanoscale Physics (cond-mat.mes-hall)
Fabio Bagarello, Andreas Fring
We demonstrate that a non self-adjoint Hamiltonian of harmonic oscillator type defined on a two-dimensional noncommutative space can be diagonalized exactly by making use of pseudo-bosonic operators. The model admits an antilinear symmetry and is of the type studied in the context of PT-symmetric quantum mechanics. Its eigenvalues are computed to be real for the entire range of the coupling constants and the biorthogonal sets of eigenstates for the Hamiltonian and its adjoint are explicitly constructed. We show that despite the fact that these sets are complete and biorthogonal, they involve an unbounded metric operator and therefore do not constitute (Riesz) bases for the Hilbert space \(\Lc^2(\Bbb R^2)\), but instead only D-quasi bases. As recently proved by one of us (FB), this is sufficient to deduce several interesting consequences.
http://arxiv.org/abs/1310.4775
Quantum Physics (quant-ph); Mathematical Physics (math-ph)
Jianke Yang
For the one-dimensional nonlinear Schroedinger equation with a complex potential, it is shown that if this potential is not parity-time (PT) symmetric, then no continuous families of solitons can bifurcate out from linear guided modes, even if the linear spectrum of this potential is all real. Both localized and periodic non-PT-symmetric potentials are considered, and the analytical conclusion is corroborated by explicit examples. Based on this result, it is argued that PT-symmetry of a one-dimensional complex potential is a necessary condition for the existence of soliton families.
http://arxiv.org/abs/1310.4490
Pattern Formation and Solitons (nlin.PS); Optics (physics.optics)