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## PT Metamaterials via Complex-Coordinate Transformation Optics

Giuseppe Castaldi, Silvio Savoia, Vincenzo Galdi, Andrea Alu’, Nader Engheta

We extend the transformation-optics paradigm to a complex spatial coordinate domain, in order to deal with electromagnetic metamaterials characterized by balanced loss and gain, giving special emphasis to parity-time (PT) symmetry metamaterials. We apply this general theory to complex-source-point radiation and unidirectional invisibility, illustrating the capability and potentials of our approach in terms of systematic design, analytical modeling and physical insights into complex-coordinate wave-objects and resonant states.

http://arxiv.org/abs/1210.7629

Optics (physics.optics)

## The physics of exceptional points

W.D. Heiss

A short resume is given about the nature of exceptional points (EPs) followed by discussions about their ubiquitous occurrence in a great variety of physical problems. EPs feature in classical as well as in quantum mechanical problems. They are associated with symmetry breaking for ${\cal PT}$-symmetric Hamiltonians, where a great number of experiments have been performed in particular in optics, and to an increasing extent in atomic and molecular physics. EPs are involved in quantum phase transition and quantum chaos, they produce dramatic effects in multichannel scattering, specific time dependence and more. In nuclear physics they are associated with instabilities and continuum problems. Being spectral singularities they also affect approximation schemes.

http://arxiv.org/abs/1210.7536
Quantum Physics (quant-ph); Mathematical Physics (math-ph)

## Non-Hermitian anisotropic XY model with intrinsic rotation-time reversal symmetry

X. Z. Zhang, Z. Song

We systematically study the non-Hermitian version of the one-dimensional anisotropic XY model, which in its original form, is a unique exactly solvable quantum spin model for understanding the quantum phase transition. The distinguishing features of this model are that it has full real spectrum if all the eigenvectors are intrinsic rotation-time reversal (RT) symmetric rather than parity-time reversal (PT) symmetric, and that its Hermitian counterpart is shown approximately to be an experimentally accessible system, an isotropic XY spin chain with nearest neighbor coupling. Based on the exact solution, exceptional points which separated the unbroken and broken symmetry regions are obtained and lie on a hyperbola in the thermodynamic limit. It provides a nice paradigm to elucidate the complex quantum mechanics theory for a quantum spin system.

http://arxiv.org/abs/1210.5613
Quantum Physics (quant-ph)

## Time Dependent PT-Symmetric Quantum Mechanics

Jiangbin Gong, Qing-hai Wang

The so-called parity-time-reversal- (PT-) symmetric quantum mechanics (PTQM) has developed into a noteworthy area of research. However, to date most known studies of PTQM focused on the spectral properties of non-Hermitian Hamiltonian operators. In this work, we propose an axiom in PTQM in order to study general time-dependent problems in PTQM, e.g., those with a time-dependent PT-symmetric Hamiltonian and with a time-dependent metric. We illuminate our proposal by examining a proper mapping from a time-dependent Schroedinger-like equation of motion for PTQM to the familiar time-dependent Schroedinger equation in conventional quantum mechanics. The rich structure of the proper mapping hints that time-dependent PTQM can be a fruitful extension of conventional quantum mechanics. Under our proposed framework, we further study in detail the Berry phase generation in a class of PT-symmetric two-level systems. It is found that a closed adiabatic path in PTQM is often associated with an open adiabatic path in a properly mapped problem in conventional quantum mechanics. In one interesting case we further interpret the Berry phase as the flux of a continuously tunable fictitious magnetic monopole, thus highlighting the difference between PTQM and conventional quantum mechanics despite the existence of a proper mapping between them.

http://arxiv.org/abs/1210.5344
Quantum Physics (quant-ph)

## Linear and Nonlinear PT-symmetric Oligomers: A Dynamical Systems Analysis

M. Duanmu, K. Li, R. L. Horne, P. G. Kevrekidis, N. Whitaker

In the present work we focus on the cases of two-site (dimer) and three-site (trimer) configurations, i.e. oligomers, respecting the parity-time (PT) symmetry, i.e., with a spatially odd gain-loss profile. We examine different types of solutions of such configurations with linear and nonlinear gain/loss profiles. Solutions beyond the linear PT-symmetry critical point as well as solutions with asymmetric linearization eigenvalues are found in both the nonlinear dimer and trimer. The latter feature is absent in linear PT-symmetric trimers, while both of them are absent in linear PT symmetric dimers. Furthermore, nonlinear gain/loss terms enable the existence of both symmetric and asymmetric solution profiles (and of bifurcations between them), while only symmetric solutions are present in the linear PT-symmetric dimers and trimers. The linear stability analysis around the obtained solutions is discussed and their dynamical evolution is explored by means of direct numerical simulations. Finally, a brief discussion is also given of recent progress in the context of PT-symmetric quadrimers.

http://arxiv.org/abs/1210.3871
Quantum Physics (quant-ph)

## Gain-Driven Discrete Breathers in PT-Symmetric Nonlinear Metamaterials

N. Lazarides, G. P. Tsironis

We introduce a one dimensional parity-time (PT)-symmetric nonlinear magnetic metamaterial consisted of split ring dimers having both gain and loss. When nonlinearity is absent we find a transition between an exact to a broken PT-phase; in the former the system features a two band gapped spectrum with shape determined by the gain and loss coefficients as well as the inter-unit coupling. In the presence of nonlinearity we show numerically that as a result of the gain/dissipation matching a novel type of long-lived stable discrete breathers can form below the lower branch of the band with no attenuation. In these localized modes the energy is almost equally partitioned between two adjacent split rings on the one with gain and the other one with loss.

http://arxiv.org/abs/1210.2410
Materials Science (cond-mat.mtrl-sci); Pattern Formation and Solitons (nlin.PS); Optics (physics.optics)

## Effective spontaneous PT-symmetry breaking in hybridized metamaterials

Ming Kang, Fu Liu, Jensen Li

We show that metamaterials can be used as a testing ground to investigate spontaneous symmetry breaking associated with non-Hermitian quantum systems. By exploring the interplay between near-field dipolar coupling and material absorption or gain, we demonstrate various spontaneous breaking processes of the $$\mathcal{PT}$$-symmetry for a series of effective Hamiltonians associated to the scattering matrix. By tuning the coupling parameter, coherent perfect absorption, laser action and gain-induced complete reflection ($$\pi$$ reflector) by using an ultra-thin metamaterial can be obtained. Moreover, an ideal $$\mathcal{PT}$$-symmetry can be established effectively in a passive system by using metamaterials.

http://arxiv.org/abs/1210.2027
Optics (physics.optics); Mesoscale and Nanoscale Physics (cond-mat.mes-hall); Classical Physics (physics.class-ph)

## Infinite families of (quasi)-Hermitian Hamiltonians associated with exceptional $$X_m$$ Jacobi polynomials

Bikashkali Midya, Barnana Roy

Using an appropriate change of variable, the Schroedinger equation is transformed into a second-order differential equation satisfied by recently discovered Jacobi type $$X_m$$ exceptional orthogonal polynomials. This facilitates the derivation of infinite families of exactly solvable Hermitian as well as non-Hermitian trigonometric and hyperbolic Scarf potentials whose bound state solutions are associated with the aforesaid exceptional orthogonal polynomials. These infinite families of potentials are shown to be extensions of the conventional trigonometric and hyperbolic Scarf potentials by the addition of some rational terms characterized by the presence of classical Jacobi polynomials. All the members of a particular family of these `rationally extended polynomial-dependent’ potentials have the same energy spectrum and possess translational shape invariant symmetry. The obtained non-Hermitian potentials are shown to be quasi-Hermitian in nature ensuring the reality of the associated energy spectra.

http://arxiv.org/abs/1210.0119
Mathematical Physics (math-ph); Quantum Physics (quant-ph)

## Comment on “Numerical estimates of the spectrum for anharmonic PT symmetric potentials” by Bowen et al

Carl M. Bender, Stefan Boettcher

The paper by Bowen, Mancini, Fessatidis, and Murawski (2012 Phys. Scr. {\bf 85}, 065005) demonstrates in a dramatic fashion the serious difficulties that can arise when one rushes to perform numerical studies before understanding the physics and mathematics of the problem at hand and without understanding the limitations of the numerical methods used. Based on their flawed numerical work, the authors conclude that the work of Bender and Boettcher is wrong even though it has been verified at a completely rigorous level. Unfortunately, the numerical procedures performed and described in the paper by Bowen et al are incorrectly applied and wrongly interpreted.

http://arxiv.org/abs/1210.0426
Quantum Physics (quant-ph); Mathematical Physics (math-ph)

## Generic examples of PT-symmetric qubit (spin 1/2) Liouvillians

Tomaz Prosen

We outline two general classes of examples of PT-symmetric quantum Liouvillian dynamics of open many qubit systems, namely interacting hard-core bosons (or more general XYZ-type spin 1/2 systems) with, either (i) pure dephasing noise, or (ii) having solely single particle/spin injection/absorption incoherent processes.

http://arxiv.org/abs/1209.6588
Quantum Physics (quant-ph); Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph)