April 2013
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Month April 2013

Instabilities, solitons, and rogue waves in PT-coupled nonlinear waveguides

Yu.V. Bludov, R. Driben, V.V. Konotop, B.A. Malomed

We considered the modulational instability of continuous-wave backgrounds, and the related generation and evolution of deterministic rogue waves in the recently introduced parity-time (PT)-symmetric system of linearly-coupled nonlinear Schr\”odinger equations, which describes a Kerr-nonlinear optical coupler with mutually balanced gain and loss in its cores. Besides the linear coupling, the overlapping cores are coupled through cross-phase-modulation term too. While the rogue waves, built according to the pattern of the Peregrine soliton, are (quite naturally) unstable, we demonstrate that the focusing cross-phase-modulation interaction results in their partial stabilization. For PT-symmetric and antisymmetric bright solitons, the stability region is found too, in an exact analytical form, and verified by means of direct simulations.

Optics (physics.optics); Pattern Formation and Solitons (nlin.PS)

Surface solitons in PT-symmetric potentials

Huagang Li, Zhiwei Shi, Xiujuan jiang, Xing Zhu, Tianshu Lai

We investigate light beam propagation along the interface between linear and nonlinear media with parity-time PT symmetry, and derive an equation governing the beam propagation. A novel class of two-dimensional PT surface solitons are found analytically and numerically, and checked to be stable over a wide range of PT potential structures and parameters. These surface solitons do not require a power threshold. The transverse power flow across beam is examined within the solitons and found to be caused by the nontrivial phase structure of the solitons. PT surface solitons are possibly observed experimentally in photorefractive crystals.

Optics (physics.optics); Pattern Formation and Solitons (nlin.PS)

BCS model with asymmetric pair scattering: a non-Hermitian, exactly solvable Hamiltonian exhibiting generalised exclusion statistics

Jon Links, Amir Moghaddam, Yao-Zhong Zhang

We demonstrate the occurrence of free quasi-particle excitations obeying generalised exclusion statistics in a BCS model with asymmetric pair scattering. The results are derived from an exact solution of the Hamiltonian, which was obtained via the algebraic Bethe ansatz utilising the representation theory of an underlying Yangian algebra. The free quasi-particle excitations are associated to highest-weight states of the Yangian algebra, corresponding to a class of analytic solutions of the Bethe ansatz equations.

Exactly Solvable and Integrable Systems (nlin.SI); Superconductivity (cond-mat.supr-con); Mathematical Physics (math-ph)

A ‘Dysonization’ Scheme for Identifying Particles and Quasi-Particles Using Non-Hermitian Quantum Mechanics

Katherine Jones-Smith

In 1956 Dyson analyzed the low-energy excitations of a ferromagnet using a Hamiltonian that was non-Hermitian with respect to the standard inner product. This allowed for a facile rendering of these excitations (known as spin waves) as weakly interacting bosonic quasi-particles. More than 50 years later, we have the full denouement of non-Hermitian quantum mechanics formalism at our disposal when considering Dyson’s work, both technically and contextually. Here we recast Dyson’s work on ferromagnets explicitly in terms of two inner products, with respect to which the Hamiltonian is always self-adjoint, if not manifestly “Hermitian”. Then we extend his scheme to doped antiferromagnets described by the t-J model, in hopes of shedding light on the physics of high-temperature superconductivity.

Quantum Physics (quant-ph)

Vector Models in PT Quantum Mechanics

Katherine Jones-Smith, Rudolph Kalveks

We present two examples of non-Hermitian Hamiltonians which consist of an unperturbed part plus a perturbation that behaves like a vector, in the framework of PT quantum mechanics. The first example is a generalization of the recent work by Bender and Kalveks, wherein the E2 algebra was examined; here we consider the E3 algebra representing a particle on a sphere, and identify the critical value of coupling constant which marks the transition from real to imaginary eigenvalues. Next we analyze a model with SO(3) symmetry, and in the process extend the application of the Wigner-Eckart theorem to a non-Hermitian setting.

Quantum Physics (quant-ph)

PT asymmetry in viscous fluids with balanced inflow and outflow

Huidan (Whitney)Yu, Xi Chen, Nan Chen, Yousheng Xu, Yogesh N. Joglekar

In recent years, open systems with equal loss and gain have been investigated via their symmetry properties under combined parity and time-reversal (\(\mathcal{PT}\)) operations. We numerically investigate \(\mathcal{PT}\)-symmetry properties of an incompressible, viscous fluid with “balanced” inflow-outflow configurations. We define configuration-dependent asymmetries in velocity, kinetic energy density, and vorticity fields, and find that all asymmetries scale quadratically with the Reynolds number. Our proposed configurations have asymmetries that are orders of magnitude smaller than the asymmetries that occur in traditional configurations at low Reynolds numbers. Our results show that \(\mathcal{PT}\)-symmetric fluid flow configurations, which are defined here for the first time, offer a hitherto unexplored avenue to tune fluid flow properties.


Fluid Dynamics (physics.flu-dyn); Quantum Physics (quant-ph)

Non-Hermitian Hamiltonians in decoherence and equilibrium theory

Mario Castagnino, Sebastian Fortin

There are many formalisms to describe quantum decoherence. However, many of them give a non general and ad hoc definition of “pointer basis” or “moving preferred basis”, and this fact is a problem for the decoherence program. In this paper we will consider quantum systems under a general theoretical framework for decoherence and we will present a tentative definition of the moving preferred basis. These ideas are implemented in a well-known open system model. The obtained decoherence and the relaxation times are defined and compared with those of the literature for the Lee- Friedrichs model.

Quantum Physics (quant-ph)

Nonlinear localized modes in PT-symmetric Rosen-Morse potential well

Bikashkali Midya, Rajkumar Roychoudhury

We report the existence and properties of localized modes described by nonlinear Schroedinger equation with complex PT-symmetric Rosen-Morse potential well. Exact analytical expressions of the localized modes are found in both one dimensional and two-dimensional geometry with self-focusing and self-defocusing Kerr nonlinearity. Linear stability analysis reveals that these localized modes are unstable for all real values of the potential parameters although corresponding linear Schroedinger eigenvalue problem possesses unbroken PT-symmetry. This result has been verified by the direct numerical simulation of the governing equation. The transverse power flow density associated with these localized modes has also been examined.

Quantum Physics (quant-ph); Pattern Formation and Solitons (nlin.PS); Optics (physics.optics)

PT-Symmetric Nonlinear Metamaterials and Zero-Dimensional Systems

G. P. Tsironis, N. Lazarides

A one dimensional, parity-time \({\cal PT}\)-symmetric magnetic metamaterial comprising split-ring resonators having both gain and loss is investigated. In the linear regime, the transition from the exact to the broken \({\cal PT}\)-phase is determined through the calculation of the eigenfrequency spectrum for two different configurations; the one with equidistant split-rings and the other with the split-rings forming a binary pattern (\({\cal PT}\) dimer chain). The latter system features a two-band, gapped spectrum with its shape determined by the gain/loss coefficient as well as the inter-element coupling. In the presense of nonlinearity, the \({\cal PT}\) dimer chain with balanced gain and loss supports nonlinear localized modes in the form of novel discrete breathers below the lower branch of the linear spectrum. These breathers, that can be excited from a weak applied magnetic field by frequency chirping, can be subsequently driven solely by the gain for very long times. The effect of a small imbalance between gain and loss is also considered. Fundamendal gain-driven breathers occupy both sites of a dimer, while their energy is almost equally partitioned between the two split-rings, the one with gain and the other with loss. We also introduce a model equation for the investigation of classical \({\cal PT}\) symmetry in zero dimensions, realized by a simple harmonic oscillator with matched time-dependent gain and loss that exhibits a transition from oscillatory to diverging motion. This behaviour is similar to a transition from the exact to the broken \({\cal PT}\) phase in higher-dimensional \({\cal PT}-\)symmetric systems. A stability condition relating the parameters of the problem is obtained in the case of piecewise constant gain/loss function that allows for the construction of a phase diagram with alternating stable and unstable regions.

Pattern Formation and Solitons (nlin.PS); Materials Science (cond-mat.mtrl-sci); Optics (physics.optics)