D.E. Pelinovsky, P.G. Kevrekidis, D.J. Frantzeskakis

We illustrate, through a series of prototypical examples, that linear parity-time (PT) symmetric lattices with extended gain/loss profiles are generically unstable, for any non-zero value of the gain/loss coefficient. Our examples include a parabolic real potential with a linear imaginary part and the cases of no real and constant or linear imaginary potentials. On the other hand, this instability can be avoided and the spectrum can be real for localized or compact PT-symmetric potentials. The linear lattices are analyzed through discrete Fourier transform techniques complemented by numerical computations.

http://arxiv.org/abs/1211.5815

Exactly Solvable and Integrable Systems (nlin.SI); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

Rujiang Li, Pengfei Li, Lu Li

We design and optimize a waveguide structure consisted of three segments of PT-symmetric coupler. It can amplify the difference mode with the attenuation of the sum mode, so performs an asymmetric optical amplifier. Also, the dependence of the amplification factor for the difference mode and the attenuation factor for the sum mode on the gain/loss is illustrated.

http://arxiv.org/abs/1211.4296

Optics (physics.optics)

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)

Alexander I. Nesterov, Juan Carlos Beas Zepeda, Gennady P. Berman

We create a non-Hermitian quantum optimization algorithm to find the ground state of an Ising model with up to 1024 spins (qubits). Our approach leads to significant reduction of the annealing time. Analytic and numerical results demonstrate that the total annealing time is proportional to ln N, where N is the number of spins. This encouraging result is important for the rapid solution of NP-complete problems. Additional research is proposed for extending our dissipative algorithm to more complicated problems.

http://arxiv.org/abs/1211.3178

Quantum Physics (quant-ph); Mathematical Physics (math-ph); Computational Physics (physics.comp-ph)

P. E. G. Assis

We study the spectral correspondence between a particular class of Schrodinger equations and supersymmetric quantum integrable model (QIM). The latter, a quantized version of the Ablowitz-Kaupp-Newell-Segur (AKNS) hierarchy of nonlinear equations, corresponds to the thermodynamic limit of the Perk-Schultz lattice model. By analyzing the symmetries of the ordinary differential equation (ODE) in the complex plane, it is possible to obtain important objects in the quantum integrable model in exact form, under an exact spectral correspondence. In this manuscript our main interest lies on the set of nonlocal conserved inte- grals of motion associated to the integrable system and we provide a systematic method to compute their values evaluated on the vacuum state of the quantum field theory.

http://arxiv.org/abs/1211.2397

High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

I. V. Barashenkov, Sergey V. Suchkov, Andrey A. Sukhorukov, Sergey V. Dmitriev, Yuri S. Kivshar

We show that the parity-time (PT) symmetric coupled optical waveguides with gain and loss support localised oscillatory structures similar to the breathers of the classical \(\phi^4\) model. The power carried by the PT-breather oscillates periodically, switching back and forth between the waveguides, so that the gain and loss are compensated on the average. The breathers are found to coexist with solitons and be prevalent in the products of the soliton collisions. We demonstrate that the evolution of the small-amplitude breather’s envelope is governed by a system of two coupled nonlinear Schrodinger equations, and employ this Hamiltonian system to show that the small-amplitude PT-breathers are stable.

http://arxiv.org/abs/1211.1835

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

H. F. Jones

By the general theory of PT-symmetric quantum systems, their energy levels are either real or occur in complex-conjugate pairs, which implies that the secular equation must be real. However, for periodic potentials it is by no means clear that the secular equation arising in the Floquet method is indeed real, since it involves two linearly independent solutions of the Schrodinger equation. In this brief note we elucidate how that reality can be established.

http://arxiv.org/abs/1211.1560

Mathematical Physics (math-ph); Quantum Physics (quant-ph)

Abouzeid Shalaby

Based on the realization that, in \(\mathcal{PT}\)-symmetric quantum mechanics, the analytic continuation of the eigen value problem into the complex plane is equivalent to the known canonical point transformation, we raise the question why then a theory selects some specific canonical transformations represented by contours within the Stokes wedges of the theory and rejects others represented by contours outside the Stokes wedges? To answer this question, we show that the transition amplitudes are the same either calculated within or out of the Stokes wedges but with related metric operators. To illustrate our idea, we reinvestigated the \(\mathcal{PT}\)-symmetric \(-x^{4}\) theory by selecting a complex contour outside the Stokes wedges. Following orthogonal polynomials studies, we were able to reproduce exactly the same equivalent Hermitian Hamiltonian obtained before in the literature. Since the metric is implicit in algorithms employing the Heisenberg picture, we assert the importance of this trend for the research in \(\mathcal{PT}\)-symmetric field theories. Regarding this, we select a simple \(Z_{2}\) symmetry breaking contour, regardless of being inside or outside the Stokes wedges, to investigate the \(\mathcal{PT}\)-symmetric \(-\phi^{4}\) field theory. We follow the famous effective action approach, up to two loops, to obtain very accurate results for the vacuum energy and vacuum condensate compared to previous calculations carried out for the equivalent Hermitian theory.

http://arxiv.org/abs/1211.0272

Mathematical Physics (math-ph)