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)
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)
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)
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)
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)
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)
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)
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)