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## PT-symmetry broken by point-group symmetry

Francisco M Fernández, Javier Garcia

We discuss a PT-symmetric Hamiltonian with complex eigenvalues. It is based on the dimensionless Schrodinger equation for a particle in a square box with the PT-symmetric potential $$V(x,y)=iaxy$$. Perturbation theory clearly shows that some of the eigenvalues are complex for sufficiently small values of $$|a|$$. Point-group symmetry proves useful to guess if some of the eigenvalues may already be complex for all values of the coupling constant. We confirm those conclusions by means of an accurate numerical calculation based on the diagonalization method. On the other hand, the Schrodinger equation with the potential $$V(x,y)=iaxy^{2}$$ exhibits real eigenvalues for sufficiently small values of $$|a|$$. Point group symmetry suggests that PT-symmetry may be broken in the former case and unbroken in the latter one.

http://arxiv.org/abs/1308.6179
Quantum Physics (quant-ph)

## Modulation theory in PT-Symmetric Magnetic Metamaterial Arrays in the continuum limit

Danhua Wang, Alejandro B. Aceves

We present results on the dynamics of split-ring dimers having both gain and loss in one dimensional nonlinear parity-time- (PT-)symmetric magnetic metamaterials. For the longwave (continuum) limit approximation and in the weakly nonlinear limit, we show analytic results on the existence of gap soliton solutions and on symmetry breaking phenomenon at a critical value of the gain/loss term.

http://arxiv.org/abs/1308.5362
Optics (physics.optics)

## Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities

Bo Peng, Sahin Kaya Ozdemir, Fuchuan Lei, Faraz Monifi, Mariagiovanna Gianfreda, Gui Lu Long, Shanhui Fan, Franco Nori, Carl M. Bender, Lan Yang

Optical systems combining balanced loss and gain profiles provide a unique platform to implement classical analogues of quantum systems described by non-Hermitian parity-time- (PT-) symmetric Hamiltonians and to originate new synthetic materials with novel properties. To date, experimental works on PT-symmetric optical systems have been limited to waveguides in which resonances do not play a role. Here we report the first demonstration of PT-symmetry breaking in optical resonator systems by using two directly coupled on-chip optical whispering-gallery-mode (WGM) microtoroid silica resonators. Gain in one of the resonators is provided by optically pumping Erbium (Er3+) ions embedded in the silica matrix; the other resonator exhibits passive loss. The coupling strength between the resonators is adjusted by using nanopositioning stages to tune their distance. We have observed reciprocal behavior of the PT-symmetric system in the linear regime, as well as a transition to nonreciprocity in the PT symmetry-breaking phase transition due to the significant enhancement of nonlinearity in the broken-symmetry phase. Our results represent a significant advance towards a new generation of synthetic optical systems enabling on-chip manipulation and control of light propagation.

http://arxiv.org/abs/1308.4564
Optics (physics.optics); Materials Science (cond-mat.mtrl-sci); Mathematical Physics (math-ph); Classical Physics (physics.class-ph); Quantum Physics (quant-ph)

## PT-symmetry Management in Oligomer Systems

R.L. Horne, J. Cuevas, P.G. Kevrekidis, N. Whitaker, F.Kh. Abdullaev, D.J. Frantzeskakis

We study the effects of management of the PT-symmetric part of the potential within the setting of Schrodinger dimer and trimer oligomer systems. This is done by rapidly modulating in time the gain/loss profile. This gives rise to a number of interesting properties of the system, which are explored at the level of an averaged equation approach. Remarkably, this rapid modulation provides for a controllable expansion of the region of exact PT-symmetry, depending on the strength and frequency of the imposed modulation. The resulting averaged models are analyzed theoretically and their exact stationary solutions are translated into time-periodic solutions through the averaging reduction. These are, in turn, compared with the exact periodic solutions of the full non-autonomous PT-symmetry managed problem and very good agreement is found between the two.

http://arxiv.org/abs/1308.3738
Pattern Formation and Solitons (nlin.PS)

## Geometric phase and phase diagram for non-Hermitian quantum XY model

X. Z. Zhang, Z. Song

We study the geometric phase for the ground state of a generalized one-dimensional non-Hermitian quantum XY model, which has transverse-field-dependent intrinsic rotation-time reversal symmetry. Based on the exact solution, this model is shown to have full real spectrum in multiple regions for the finite size system. The result indicates that the phase diagram or exceptional boundary, which separates the unbroken and broken symmetry regions corresponds to the divergence of the Berry curvature. The scaling behaviors of the groundstate energy and Berry curvature are obtained in an analytical manner for a concrete system.

http://arxiv.org/abs/1308.4057
Quantum Physics (quant-ph)

## Biorthogonal Quantum Mechanics

Dorje C. Brody

The Hermiticity condition in quantum mechanics required for the characterisation of (a) physical observables and (b) generators of unitary motions can be relaxed into a wider class of operators whose eigenvalues are real and whose eigenstates are complete. In this case, the orthogonality of eigenstates is replaced by the notion of biorthogonality that defines the relation between the Hilbert space of states and its dual space. The resulting quantum theory, which might appropriately be called ‘biorthogonal quantum mechanics’, is developed here in some detail in the case for which the Hilbert space dimensionality is finite. Specifically, characterisations of probability assignment rules, observable properties, pure and mixed states, spin particles, measurements, combined systems and entanglements, perturbations, and dynamical aspects of the theory are developed. The paper concludes with a brief discussion on infinite-dimensional systems.

http://arxiv.org/abs/1308.2609
Quantum Physics (quant-ph); Mathematical Physics (math-ph)

## PT-symmetry and Transparency

Zafar Ahmed

It is known that the perfect absorption of two identical waves incident on a complex potential from left and right can occur at a fixed real energy and that the time-reversed setting of this system would act as a laser at threshold at the same energy. Here, we argue and show that PT-symmetric potentials are exceptional in this regard which do not allow Coherent Perfect Absorption without lasing as the modulus of the determinant of the $$S$$-matrix, $$|\det S|$$, becomes 1, for all positive energies. Next we show that in the parametric regimes where the PT-symmetry is unbroken, the eigenvalues, $$s_\pm$$ of $$S$$, can become unitary (uni-modular) for all energies. Then the potential becomes coherent perfect emitter on both sides for any energy of coherent injection. We call this property Transparency.

http://arxiv.org/abs/1308.1270
Quantum Physics (quant-ph); Mathematical Physics (math-ph)

## A solvable model for solitons pinned to a PT-symmetric dipole

Thawatchai Mayteevarunyoo, Boris A. Malomed, Athikom Reoksabutr

We introduce the simplest one-dimensional nonlinear model with the parity-time (PT) symmetry, which makes it possible to find exact analytical solutions for localized modes (“solitons”). The PT-symmetric element is represented by a point-like (delta-functional) gain-loss dipole $$\delta^\prime(x)$$, combined with the usual attractive potential $$\delta$$(x). The nonlinearity is represented by self-focusing (SF) or self-defocusing (SDF) Kerr terms, both spatially uniform and localized ones. The system can be implemented in planar optical waveguides. For the sake of comparison, also introduced is a model with separated $$\delta$$-functional gain and loss, embedded into the linear medium and combined with the $$\delta$$-localized Kerr nonlinearity and attractive potential. Full analytical solutions for pinned modes are found in both models. The exact solutions are compared with numerical counterparts, which are obtained in the gain-loss-dipole model with the $$\delta^\prime$$- and $$\delta$$- functions replaced by their Lorentzian regularization. With the increase of the dipole’s strength, $$\gamma$$, the single-peak shape of the numerically found mode, supported by the uniform SF nonlinearity, transforms into a double-peak one. This transition coincides with the onset of the escape instability of the pinned soliton. In the case of the SDF uniform nonlinearity, the pinned modes are stable, keeping the single-peak shape.

http://arxiv.org/abs/1308.0426
Pattern Formation and Solitons (nlin.PS)

## Type II Perfect Absorption and Amplification Modes with Controllable Bandwidth in PT-Symmetric/Traditional Bragg Grating Combined Structures

C. Y. Huang, R. Zhang, J. L. Han, J. Zheng, J. Q. Xu

We reveal previously unnoticed Type II perfect absorption and amplification modes of optical scattering system. These modes, in contrast to the counterparts in recent works [Phys. Rev. A 82, 031801 (2010); Phys. Rev. Lett. 106, 093902 (2011).] which could be referred as Type I modes, appear at a continuous region along the real frequency axis with any frequency. The Type II modes can be demonstrated in the PT-symmetric/traditional Bragg grating combined structures. A series of exotic nonreciprocal absorption and amplification behaviours are observed in the combined structures, making them have potential for versatile devices acting simultaneously as a perfect absorber, an amplifier, and an ultra-narrowband filter. Based on the properties of Type II modes, we also propose structures with controllable perfect absorption and amplification bandwidth at any single or multiple wavelengths.

http://arxiv.org/abs/1307.7583
Optics (physics.optics)

## Localized exact solutions of $$\mathcal{PT}$$ symmetric nonlinear Schrödinger equation with space and time modulated nonlinearities

L. E. Arroyo Meza, M. B. Hott, A. de Souza Dutra, P. Roy

Using canonical transformations we obtain localized (in space) exact solutions of the nonlinear Schrodinger equation (NLSE) with space and time modulated nonlinearity and in the presence of an external potential depending on space and time. In particular we obtain exact solutions of NLSE in the presence of a number of non Hermitian $$\mathcal{PT}$$ symmetric external potentials.

http://arxiv.org/abs/1307.7591

Pattern Formation and Solitons (nlin.PS); Mathematical Physics (math-ph)