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## Nonlinear localized modes in PT-symmetric optical media with competing gain and loss

Bikashkali Midya, Rajkumar Roychoudhury

The existence and stability of the nonlinear spatial localized modes are investigated in parity-time symmetric optical media characterized by a generic complex hyperbolic refractive index distribution with competing gain and loss profile. The exact analytical expressions of the localized modes are found for all values of the competing parameter and in the presence of both the self-focusing and self-defocusing Kerr nonlinearity. The effect of competing gain/loss profile on the stability structure of these localized modes are discussed with the help of linear stability analysis followed by the direct numerical simulation of the governing equation. The spatial localized modes in two-dimensional geometry as well as the transverse power-flow density associated with these localized modes are also examined.

http://arxiv.org/abs/1306.5983

Optics (physics.optics)

## Nonlinear modes and integrals of motion in finite PT-symmetric systems

Dmitry A. Zezyulin, Vladimir V. Konotop

We investigate bifurcations of nonlinear modes in parity-time (PT-) symmetric discrete systems. We consider a general class of nonlinearities allowing for existence of the nonlinear modes and address situations when the underlying linear problem is characterized by the presence of degenerate eigenvalues or exceptional-point singularity. In each of the cases we construct formal expansions for small-amplitude nonlinear modes. We also report a class of nonlinearities allowing for a system to admit one or several integrals of motion, which turn out to be determined by the pseudo-Hermiticity of the nonlinearity.

http://arxiv.org/abs/1306.5286
Pattern Formation and Solitons (nlin.PS)

## Eigenvalue structure of a Bose-Einstein condensate in a PT-symmetric double well

Dennis Dast, Daniel Haag, Holger Cartarius, Jörg Main, Günter Wunner

We study a Bose-Einstein condensate in a PT-symmetric double-well potential where particles are coherently injected in one well and removed from the other well. In mean-field approximation the condensate is described by the Gross-Pitaevskii equation thus falling into the category of nonlinear non-Hermitian quantum systems. After extending the concept of PT symmetry to such systems, we apply an analytic continuation to the Gross-Pitaevskii equation from complex to bicomplex numbers and show a thorough numerical investigation of the four-dimensional bicomplex eigenvalue spectrum. The continuation introduces additional symmetries to the system which are confirmed by the numerical calculations and furthermore allows us to analyze the bifurcation scenarios and exceptional points of the system. We present a linear matrix model and show the excellent agreement with our numerical results. The matrix model includes both exceptional points found in the double-well potential, namely an EP2 at the tangent bifurcation and an EP3 at the pitchfork bifurcation. When the two bifurcation points coincide the matrix model possesses four degenerate eigenvectors. Close to that point we observe the characteristic features of four interacting modes in both the matrix model and the numerical calculations, which provides clear evidence for the existence of an EP4.

http://arxiv.org/abs/1306.3871

Quantum Physics (quant-ph)

## Revisiting the PT-symmetric Trimer: Bifurcations, Ghost States and Associated Dynamics

K. Li, P. G. Kevrekidis, D. J. Frantzeskakis, C. E. Ruter, D. Kip

In this paper, we revisit one of the prototypical PT-symmetric oligomers, namely the trimer. We find all the relevant branches of “regular” solutions and analyze the bifurcations and instabilities thereof. Our work generalizes the formulation that was proposed recently in the case of dimers for the so-called “ghost states” of trimers, which we also identify and connect to symmetry-breaking bifurcations from the regular states. We also examine the dynamics of unstable trimers, as well as those of the ghost states in the parametric regime where the latter are found to exist. Finally, we present the current state of the art for optical experiments in PT-symmetric trimers, as well as experimental results in a gain-loss-gain three channel waveguide structure.

http://arxiv.org/abs/1306.2255
Quantum Physics (quant-ph)

## Scattering in a varying mass PT symmetric double heterojunction

Anjana Sinha, R. Roychoudhury

We observe that the reflection and transmission coefficients of a particle within a double, PT-symmetric heterojunction with spatially varying mass, show interesting features, depending on the degree of non Hermiticity, although there is no spontaneous breakdown of PT symmetry. The potential profile in the intermediate layer is considered such that it has a non vanishing imaginary part near the heterojunctions. Exact analytical solutions for the wave function are obtained, and the reflection and transmission coefficients are plotted as a function of energy, for both left as well as right incidence. As expected, the spatial dependence on mass changes the nature of the scattering solutions within the heterojunctions, and the space-time (PT) symmetry is responsible for the left-right asymmetry in the reflection and transmission coefficients. However, the non vanishing imaginary component of the potential near the heterojunctions gives new and interesting results.

http://arxiv.org/abs/1306.2226

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

## Momentum-independent reflectionless transmission in the non-Hermitian time-reversal symmetric system

X. Z. Zhang, Z. Song

We theoretically study the non-Hermitian systems, the non-Hermiticity of which arises from the unequal hopping amplitude (UHA) dimers. The distinguishing features of these models are that they have full real spectra if all of the eigenvectors are time-reversal (T) symmetric rather than parity-time-reversal (PT) symmetric, and that their Hermitian counterparts are shown to be an experimentally accessible system, which have the same topological structures as that of the original ones but modulated hopping amplitudes within the unbroken region. Under the reflectionless transmission condition, the scattering behavior of momentum-independent reflectionless transmission (RT) can be achieved in the concerned non-Hermitian system. This peculiar feature indicates that, for a certain class of non-Hermitian systems with a balanced combination of the RT dimers, the defects can appear fully invisible to an outside observer.

http://arxiv.org/abs/1306.1969

Quantum Physics (quant-ph)

## The Sign Problem, PT Symmetry and Abelian Lattice Duality

Peter N. Meisinger, Michael C. Ogilvie

Lattice field theories with complex actions are not easily studied using conventional analytic or simulation methods. However, a large class of these models are invariant under CT, where C is charge conjugation and T is time reversal, including models with non-zero chemical potential. For Abelian models in this class, lattice duality maps models with complex actions into dual models with real actions. For extended regions of parameter space, calculable for each model, duality resolves the sign problem for both analytic methods and computer simulations. Explicit duality relations are given for models for spin and gauge models based on Z(N) and U(1) symmetry groups. The dual forms are generalizations of the Z(N) chiral clock model and the lattice Frenkel-Kontorova model, respectively. From these equivalences, rich sets of spatially-modulated phases are found in the strong-coupling region of the original models.

http://arxiv.org/abs/1306.1495
High Energy Physics – Lattice (hep-lat)

## Spectral and transport properties of time-periodic PT-symmetric tight-binding lattices

Giuseppe Della Valle, Stefano Longhi

We investigate the spectral properties and dynamical features of a time-periodic PT-symmetric Hamiltonian on a one-dimensional tight-binding lattice. It is shown that a high-frequency modulation can drive the system under a transition between the broken-PT and the unbroken-PT phases. The time-periodic modulation in the unbroken-PT regime results in a significant broadening of the quasi-energy spectrum, leading to a hyper-ballistic transport regime. Also, near the PT-symmetry breaking the dispersion curve of the lattice band becomes linear, with a strong reduction of quantum wave packet spreading.

http://arxiv.org/abs/1306.1048
Quantum Physics (quant-ph)

## $$\mathcal{PT}$$-symmetric strings

Paolo Amore, Francisco M. Fernández, Javier Garcia, German Gutierrez

We study both analytically and numerically the spectrum of inhomogeneous strings with $$\mathcal{PT}$$-symmetric density. We discuss an exactly solvable model of $$\mathcal{PT}$$-symmetric string which is isospectral to the uniform string; for more general strings, we calculate exactly the sum rules $$Z(p) \equiv \sum_{n=1}^\infty 1/E_n^p$$, with $$p=1,2,\dots$$ and find explicit expressions which can be used to obtain bounds on the lowest eigenvalue. A detailed numerical calculation is carried out for two non-solvable models depending on a parameter, obtaining precise estimates of the critical values where pair of real eigenvalues become complex.

http://arxiv.org/abs/1306.1419
Mathematical Physics (math-ph)