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## Periodic and Hyperbolic Soliton Solutions of a Number of Nonlocal PT-Symmetric Nonlinear Equations

Avinash Khare, Avadh Saxena

For a number of nonlocal nonlinear equations such as nonlocal, nonlinear Schrodinger equation (NLSE), nonlocal Ablowitz-Ladik (AL), nonlocal, saturable discrete NLSE (DNLSE), coupled nonlocal NLSE, coupled nonlocal AL and coupled nonlocal, saturable DNLSE, we obtain periodic solutions in terms of Jacobi elliptic functions as well as the corresponding hyperbolic soliton solutions. Remarkably, in all the six cases, we find that unlike the corresponding local cases, all the nonlocal models simultaneously admit both the bright and the dark soliton solutions. Further, in all the six cases, not only $$\rm{Dn}(x,m)$$ and $$\rm{Cn}(x,m)$$ but even their linear superposition is shown to be an exact solution. Finally, we show that the coupled nonlocal NLSE not only admits solutions in terms of Lame polynomials of order 1, but it also admits solutions in terms of Lame polynomials of order 2, even though they are not the solutions of the uncoupled nonlocal problem. We also remark on the possible integrability in certain cases.

http://arxiv.org/abs/1405.5267
Pattern Formation and Solitons (nlin.PS)

## Unique optical characteristics of a Fabry-Perot resonator with embedded PT-symmetrical grating

Mykola Kulishov, Bernand Kress, H. F. Jones

We explore the optical properties of a Fabry-Perot resonator with an embedded Parity-Time (PT) symmetrical grating. This PT-symmetrical grating is non diffractive (transparent) when illuminated from one side and diffracting (Bragg reflection) when illuminated from the other side, thus providing a unidirectional reflective functionality. The incorporated PT-symmetrical grating forms a resonator with two embedded cavities. We analyze the transmission and reflection properties of these new structures through a transfer matrix approach. Depending on the resonator geometry these cavities can interact with different degrees of coherency: fully constructive interaction, partially constructive interaction, partially destructive interaction, and finally their interaction can be completely destructive. A number of very unusual (exotic) nonsymmetrical absorption and amplification behaviors are observed. The proposed structure also exhibits unusual lasing performance. Due to the PT-symmetrical grating, there is no chance of mode hopping; it can lase with only a single longitudinal mode for any distance between the distributed reflectors.

http://arxiv.org/abs/1405.6024
Optics (physics.optics)

## PT-Symmetry in Non-Hermitian Su-Schrieffer-Heeger model with complex boundary potentials

Baogang Zhu, Rong Lu, Shu Chen

We study the parity- and time-reversal PT symmetric non-Hermitian Su-Schrieffer-Heeger (SSH) model with two conjugated imaginary potentials $$\pm i\gamma$$ at two end sites. The SSH model is known as one of the simplest two-band topological models which has topologically trivial and nontrivial phases. We find that the non-Hermitian terms can lead to different effects on the properties of the eigenvalues spectrum in topologically trivial and nontrivial phases. In the topologically trivial phase, the system undergos an abrupt transition from unbroken PT-symmetry region to spontaneously broken $$\mathcal{PT}$$-symmetry region at a certain $$\gamma_{c}$$, and a second transition occurs at another transition point $$\gamma_{c^{‘}}$$ when further increasing the strength of the imaginary potential $$\gamma$$. But in the topologically nontrivial phase, the zero-mode edge states become unstable for arbitrary nonzero $$\gamma$$ and the $$\mathcal{PT}$$-symmetry of the system is spontaneously broken, which is characterized by the emergence of a pair of conjugated imaginary modes.

http://arxiv.org/abs/1405.5591
Other Condensed Matter (cond-mat.other); Quantum Physics (quant-ph)

## Dynamics of mode entanglement in a system of cavities coupled with a chiral mirror

Ali Ü. C. Hardal

We investigate the Hermitian and the non-Hermitian dynamics of the mode entanglement in two identical optical cavities coupled by a chiral mirror. By employing the non-Hermitian quantum evolution, we calculate the logarithmic negativity measure of entanglement for initially Fock, coherent and squeezed states, separately. We verify the non-conservation of mean spin for the initially coherent and squeezed states when the coupling is non-reciprocal and report the associated spin noise for each case. We examine the effects of non-conserved symmetries on the mode correlations and determine the degree of non-reciprocal coupling to establish robust quantum entanglement.

http://arxiv.org/abs/1405.5079
Quantum Physics (quant-ph)

## Parity-Time Synthetic Laser

Liang Feng, Zi Jing Wong, Renmin Ma, Yuan Wang, Xiang Zhang

Parity-time (PT) symmetry is a fundamental notion in quantum field theories1,2. It has opened a new paradigm for non-Hermitian Hamiltonians ranging from quantum mechanics, electronics, to optics. In the realm of optics, optical loss is responsible for power dissipation, therefore typically degrading device performance such as attenuation of a laser beam. By carefully exploiting optical loss in the complex dielectric permittivity, however, recent exploration of PT symmetry revolutionizes our understandings in fundamental physics and intriguing optical phenomena such as exceptional points and phase transition that are critical for high-speed optical modulators3-9. The interplay between optical gain and loss in photonic PT synthetic matters offers a new criterion of positively utilizing loss to efficiently manipulate gain and its associated optical properties10-19. Instead of simply compensating optical loss in conventional lasers, for example, it is theoretically proposed that judiciously designed delicate modulation of optical loss and gain can lead to PT synthetic lasing20,21 that fundamentally broadens laser physics. Here, we report the first experimental demonstration of PT synthetic lasers. By carefully exploiting the interplay between gain and loss, we achieve degenerate eigen modes at the same frequency but with complex conjugate gain and loss coefficients. In contrast to conventional ring cavity lasers with multiple modes, the PT synthetic micro-ring laser exhibits an intrinsic single mode lasing: the non-threshold PT broken phase inherently associated in such a photonic system squeezes broadband optical gain into a single lasing mode regardless of the gain spectral bandwidth. This chip-scale semiconductor platform provides a unique route towards fundamental explorations of PT physics and next generation of optoelectronic devices for optical communications and computing.

http://arxiv.org/abs/1405.2863
Optics (physics.optics)

## Is space-time symmetry a suitable generalization of parity-time symmetry?

Paolo Amore, Francisco M. Fernández, Javier Garcia

We discuss space-time symmetric Hamiltonian operators of the form $$H=H_{0}+igH^{\prime}$$, where $$H_{0}$$ is Hermitian and $$g$$ real. $$H_0$$ is invariant under the unitary operations of a point group $$G$$ while $$H^\prime$$ is invariant under transformation by elements of a subgroup $$G^\prime$$ of $$G$$. If $$G$$ exhibits irreducible representations of dimension greater than unity, then it is possible that $$H$$ has complex eigenvalues for sufficiently small nonzero values of $$g$$. In the particular case that $$H$$ is parity-time symmetric then it appears to exhibit real eigenvalues for all $$0<g<g_c$$, where $$g_{c}$$ is the exceptional point closest to the origin. Point-group symmetry and perturbation theory enable one to predict whether $$H$$ may exhibit real or complex eigenvalues for $$g>0$$. We illustrate the main theoretical results and conclusions of this paper by means of two- and three-dimensional Hamiltonians exhibiting a variety of different point-group symmetries.

http://arxiv.org/abs/1405.5234
Quantum Physics (quant-ph)

## Invisibility and PT Symmetry: A Simple Geometrical Viewpoint

Luis L. Sanchez-Soto, Juan J. Monzon

We give a simplified account of the properties of the transfer matrix for a complex one-dimensional potential, paying special attention to the particular instance of unidirectional invisibility. In appropriate variables, invisible potentials appear as performing null rotations, which lead to the helicity-gauge symmetry of massless particles. In hyperbolic geometry, this can be interpreted, via Mobius transformations, as parallel displacements, a geometric action that has no Euclidean analogy.

http://arxiv.org/abs/1405.4791

Quantum Physics (quant-ph)

## Generalized Unitarity and Reciprocity Relations for PT-symmetric Scattering Potentials

We derive certain identities satisfied by the left/right-reflection and transmission amplitudes, $$R^{l/r}(k)$$ and $$T(k)$$, of general $${\cal PT}$$-symmetric scattering potentials. We use these identities to give a general proof of the relations, $$|T(-k)|=|T(k)|$$ and $$|R^r(-k)|=|R^l(k)|$$, conjectured in [Z. Ahmed, J. Phys. A 45 (2012) 032004], establish the generalized unitarity relation: $$R^{l/r}(k)R^{l/r}(-k)+|T(k)|^2=1$$, and show that it is a common property of both real and complex $${\cal PT}$$-symmetric potentials. The same holds for $$T(-k)=T(k)^*$$ and $$|R^r(-k)|=|R^l(k)|$$.

http://arxiv.org/abs/1405.4212
Quantum Physics (quant-ph); Mathematical Physics (math-ph); Optics (physics.optics)

## Exponential asymptotics for solitons in PT-symmetric periodic potentials

Sean Nixon, Jianke Yang

Solitons in one-dimensional parity-time (PT)-symmetric periodic potentials are studied using exponential asymptotics. The new feature of this exponential asymptotics is that, unlike conservative periodic potentials, the inner and outer integral equations arising in this analysis are both coupled systems due to complex-valued solitons. Solving these coupled systems, we show that two soliton families bifurcate out from each Bloch-band edge for either self-focusing or self-defocusing nonlinearity. An asymptotic expression for the eigenvalues associated with the linear stability of these soliton families is also derived. This formula shows that one of these two soliton families near band edges is always unstable, while the other can be stable. In addition, infinite families of PT-symmetric multi-soliton bound states are constructed by matching the exponentially small tails from two neighboring solitons. These analytical predictions are compared with numerics. Overall agreements are observed, and minor differences explained.

http://arxiv.org/abs/1405.2827
Pattern Formation and Solitons (nlin.PS); Optics (physics.optics)

## Modulation instability in nonlinear complex parity-time (PT) symmetric periodic structures

Amarendra K. Sarma

We carry out a modulation instability (MI) analysis in nonlinear complex parity-time (PT) symmetric periodic structures. All the three regimes defined by the PT-symmetry breaking point or threshold, namely, below threshold, at threshold and above threshold are discussed. It is found that MI exists even beyond the PT-symmetry threshold indicating the possible existence of solitons or solitary waves, in conformity with some recent reports. We find that MI does not exist at the PT-symmetry breaking point in the case of normal dispersion below a certain nonlinear threshold. However, in the case of anomalous dispersion regime, MI does exist even at the PT-symmetry breaking point.

http://arxiv.org/abs/1405.2706
Optics (physics.optics)