February 2012
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Month February 2012

Nonlinear modes in finite-dimensional PT-symmetric systems

D. A. Zezyulin, V. V. Konotop

By similarity transformations a parity-time (PT-) symmetric Hamiltonian can be reduced to a Hermitian or to another PT-symmetric Hamiltonian having the same linear spectrum. On an example of a PT-symmetric quadrimer we show that the spectral equivalence of different PT-symmetric and Hermitian systems implies neither similarity of the nonlinear modes nor their stability properties. Even the existence of one-parametric families of nonlinear modes is not guaranteed by the PT symmetry of the underlying linear system. Neither the stability is directly related to the PT symmetry: stable nonlinear modes exist beyond the PT symmetry breaking threshold. A “phase diagram” of a general PT-symmetric quadrimer allows for existence of “triple” points, where three different phases meet. We use graph representation of PT-symmetric networks giving simple illustration of linearly equivalent networks and indicating on their possible experimental design.

Pattern Formation and Solitons (nlin.PS); Optics (physics.optics)

Path Integrals for (Complex) Classical and Quantum Mechanics

Ray J. Rivers

An analysis of classical mechanics in a complex extension of phase space shows that a particle in such a space can behave in a way redolant of quantum mechanics; additional degrees of freedom permit ‘tunnelling’ without recourse to instantons and lead to time/energy uncertainty. In practice, ‘classical’ particle trajectories with additional degrees of freedom have arisen in several different formulations of quantum mechanics. In this talk we compare the extended phase space of the closed time-path formalism with that of complex classical mechanics, to suggest that $\hbar$ has a role in our understanding of the latter. However, differences in the way that trajectories are used make a deeper comparison problematical. We conclude with some thoughts on quantisation as dimensional reduction.


Quantum Physics (quant-ph); Classical Physics (physics.class-ph)

Optical Modes in PT-Symmetric Double-Channel Waveguides

Li Chen, Rujiang Li, Na Yang, Da Chen, Lu Li

We investigate the unique properties of various analytical optical modes, including the fundamental modes and the excited modes, in a double-channel waveguide with parity-time (PT) symmetry. Based on these optical modes, the dependence of the threshold values for the gain/loss parameter, i.e., PT symmetry breaking points, on the structure parameters is discussed. We find that the threshold value for the excited modes is larger than that of the fundamental mode. In addition, the beam dynamics in the double-channel waveguide with PT symmetry is also investigated.

Optics (physics.optics)

Entangled Quantum State Discrimination using Pseudo-Hermitian System

Ananya Ghatak, Bhabani Prasad Mandal

We demonstrate how to discriminate two non-orthogonal, entangled quantum state which are slightly different from each other by using pseudo-Hermitian system. The positive definite metric operator which makes the pseudo-Hermitian systems fully consistent quantum theory is used for such a state discrimination. We further show that non-orthogonal states can evolve through a suitably constructed pseudo-Hermitian Hamiltonian to orthogonal states. Such evolution ceases at exceptional points of the pseudo-Hermitian system.

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

On elements of the Lax-Phillips scattering scheme for PT-symmetric operators

Sergio Albeverio, Sergii Kuzhel

Generalized PT-symmetric operators acting an a Hilbert space $\mathfrak{H}$ are defined and investigated. The case of PT-symmetric extensions of a symmetric operator $S$ is investigated in detail. The possible application of the Lax-Phillips scattering methods to the investigation of PT-symmetric operators is illustrated by considering the case of 0-perturbed operators.

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

Dark solitons and vortices in PT-symmetric nonlinear media: from spontaneous symmetry breaking to nonlinear PT phase transitions

V. Achilleos, P.G. Kevrekidis, D.J. Frantzeskakis, R. Carretero-González

We consider the nonlinear analogues of Parity-Time (\(\mathcal{PT}\)) symmetric linear systems exhibiting defocusing nonlinearities. We study the ground state and excited states (dark solitons and vortices) of the system and report the following remarkable features. For relatively weak values of the parameter \(\varepsilon\) controlling the strength of the \(\mathcal{PT}\)-symmetric potential, excited states undergo (analytically tractable) spontaneous symmetry breaking; as \(\varepsilon\) is further increased, the ground state and first excited state, as well as branches of higher multi-soliton (multi-vortex) states, collide in pairs and disappear in blue-sky bifurcations, in a way which is strongly reminiscent of the linear \(\mathcal{PT}\)-phase transition —thus termed the nonlinear \(\mathcal{PT}\)-phase transition. Past this critical point, initialization of, e.g., the former ground state leads to spontaneously emerging “soliton (vortex) sprinklers”.

Pattern Formation and Solitons (nlin.PS); Soft Condensed Matter (cond-mat.soft)

PT Symmetry on 60 Minutes

An article on a young mathematical prodigy on the 60 minutes news programme on US television shows him working on PT Symmetry with Prof YN Joglekar at IUPUI.

Their research article is available on arXiv at http://arxiv.org/abs/1108.6083, was published in Physical Review A (http://pra.aps.org/abstract/PRA/v84/i2/e024103), was reported on the PT-Symmeter (http://ptsymmetry.net/?p=552), and can now be seen as part of a documentary on the CBS 60 minutes website here:


Dirac Equation with Imaginary Mass and Helicity-Dependence

U. D. Jentschura

In the matter wave equations describing spin one-half particles, one can either enforce superluminal propagation by an explicit substitution of the real mass term for an imaginary mass, or one can use a matrix representation of the imaginary unit that multiplies the mass term. The latter leads to thetachyonic Dirac equation, while the equation obtained by the substitution m -> i*m in the Dirac equation is naturally referred to as the imaginary-mass Dirac equation. Both the tachyonic as well as the imaginary-mass Dirac Hamiltonians commute with the helicity operator. Both Hamiltonians are pseudo-Hermitian and also possess additional modified pseudo-Hermitian properties, leading to constraints on the resonance eigenvalues. The spectrum is found to consist of well-defined real energy eigenvalues and complex resonance and anti-resonance energies. The quantization of the tachyonic Dirac field has recently been discussed, and we here supplement a discussion of the quantized imaginary-mass Dirac field. Just as for the tachyonic Dirac Hamiltonian, we find that one-particle states of right-handed helicity acquire a negative norm and can be excluded from the physical spectrum by a Gupta–Bleuler type condition. This observation may indicate a deeper, general connection of superluminal propagation and helicity-dependent interactions.

High Energy Physics – Phenomenology (hep-ph); High Energy Physics – Theory (hep-th)