October 2014
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Month October 2014

Localization of the states of a PT-symmetric double well

Riccardo Giachetti, Vincenzo Grecchi

We make a nodal analysis of the processes of level crossings in a model of quantum mechanics with a PT-symmetric double well. We prove the existence of infinite crossings with their selection rules. At the crossing, before the PT-symmetry breaking and the localization, we have a total P-symmetry breaking of the states.

Mathematical Physics (math-ph)

Reciprocity in parity violating non-Hermitian systems

Ananya Ghatak, Brijesh Kumar Mourya, Raka Dona Ray Mandal, Bhabani Prasad Mandal (BHU)

Reciprocity is shown so far only when the scattering potential is either real or parity symmetric complex. We extend this result for parity violating complex potential by considering several explicit examples: (i) we show reciprocity for a PT symmetric (hence parity violating) complex potential which admits penetrating state solutions analytically for all possible values of incidence energy and (ii) reciprocity is shown to hold at certain discrete energies for two other parity violating complex potentials.

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

Loss-induced suppression and revival of lasing

B. Peng, S. K. Ozdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, L. Yang

Controlling and reversing the effects of loss are major challenges in optical systems. For lasers losses need to be overcome by a sufficient amount of gain to reach the lasing threshold. We show how to turn losses into gain by steering the parameters of a system to the vicinity of an exceptional point (EP), which occurs when the eigenvalues and the corresponding eigenstates of a system coalesce. In our system of coupled microresonators, EPs are manifested as the loss-induced suppression and revival of lasing. Below a critical value, adding loss annihilates an existing Raman laser. Beyond this critical threshold, lasing recovers despite the increasing loss, in stark contrast to what would be expected from conventional laser theory. Our results exemplify the counterintuitive features of EPs and present an innovative method for reversing the effect of loss.

Optics (physics.optics); Quantum Physics (quant-ph)

Parity-time (PT) symmetric Double-lambda optical quadrimer waveguides in linear and nonlinear regimes

Samit Kumar Gupta, Jyoti Prasad Deka, Amarendra K. Sarma

We report a study on a closed-form optical quadrimer waveguides system. We have studied the beam dynamics of the system below, at and above the PT-threshold in both the linear and nonlinear regimes. We have also explored the effects of gain/loss parameter and the strength as well as the nature of the nonlinearity (i.e. focusing or defocusing) upon the evolution of optical intensity in each of the four sites. We observe saturation behaviors in the spatial power evolution, when nonlinearity is incorporated into the system, in the gain-guides, a feature that could be exploited for various practical applications.

Optics (physics.optics)

Transparency of PT-symmetric complex potentials for coherent injection

Zafar Ahmed, Joseph Amal Nathan

It is known that when two identical waves are injected from left and right on a complex PT-symmetric scattering potential the two-port s-matrix can have uni-modular eigenvalues. If this happens for all energies, there occurs a perfect emission of waves at both ends. We call this phenomenon transparency. Using the versatile PT-Symmetric complex Scarf II potential, we demonstrate analytically that the transparency occurs when the potential has real discrete spectrum i.e., when PT-symmetry is exact(unbroken). Next, we find that exactness of PT-symmetry is only sufficient but not necessary for the transparency. Two other PT-symmetric domains of Scarf II reveal transparency without the PT-symmetry being exact. In these two cases there exist only scattering states. In one case the real part of the potential is a well devoid of real discrete spectrum and in the other real part is a barrier. Other numerically solved models also support our findings.

Quantum Physics (quant-ph)

A unidirectional invisible PT-symmetric complex crystal with arbitrary thickness

Stefano Longhi

We introduce a new class of \(\cal{PT}\)-symmetric complex crystals which are almost transparent and one-way reflectionless over a broad frequency range around the Bragg frequency, i.e. unidirectionally invisible, regardless of the thickness \(L\) of the crystal. The \(\cal{PT}\)-symmetric complex crystal is synthesized by a supersymmetric transformation of an Hermitian square well potential, and exact analytical expressions of transmission and reflection coefficients are given. As \(L\) is increased, the transmittance and reflectance from one side remain close to one and zero, respectively, whereas the reflectance from the other side secularly grows like ~\(L^2\) owing to unidirectional Bragg scattering. This is a distinctive feature as compared to the previously studied case of the complex sinusoidal \(\cal{PT}\)-symmetric potential \(V(x)=V_0\exp(−2ik_ox)\) at the symmetry breaking point, where transparency breaks down as \(L\to\infty\).

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

PT symmetry and a dynamical realization of the SU(1,1) algebra

Rabin Banerjee, Pradip Mukherjee

We show that the elementary modes of the planar harmonic oscillator can be quantised in the framework of quantum mechanics based on pseudo-hermitian hamiltonians. These quantised modes are demonstrated to act as dynamical structures behind a new Jordan – Schwinger realization of the SU(1,1) algebra. This analysis complements the conventional Jordan – Schwinger construction of the SU(2) algebra based on hermitian hamiltonians of a doublet of oscillators.

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

Action-at-a-distance in a solvable quantum model

Miloslav Znojil

Among quantum systems with finite Hilbert space a specific role is played by systems controlled by non-Hermitian Hamiltonian matrices \(H\neq H^\dagger\) for which one has to upgrade the Hilbert-space metric by replacing the conventional form \(\Theta^{(Dirac)}=I\) of this metric by a suitable upgrade \(\Theta^{(non−Dirac)}\neq I\) such that the same Hamiltonian becomes self-adjoint in the new, upgraded Hilbert space, \(H=H\ddagger=\Theta^{−1}H^\dagger\Theta\). The problems only emerge in the context of scattering where the requirement of the unitarity was found to imply the necessity of a non-locality in the interaction, compensated by important technical benefits in the short-range-nonlocality cases. In the present paper we show that an why these technical benefits (i.e., basically, the recurrent-construction availability of closed-form Hermitizing metrics \(\Theta^{(non−Dirac)}\) can survive also in certain specific long-range-interaction models.

Quantum Physics (quant-ph)

Resonant mode conversion in the waveguides with an unbroken and broken PT-symmetry

Victor A. Vysloukh, Yaroslav V. Kartashov

We study resonant mode conversion in the PT-symmetric multimode waveguides, where symmetry breaking manifests itself in sequential destabilization (appearance of the complex eigenvalues) of the pairs of adjacent guided modes. We show that the efficient mode conversion is possible even in the presence of the resonant longitudinal modulation of the complex refractive index. The distinguishing feature of the resonant mode conversion in the PT-symmetric structure is a drastic growth of the width of the resonance curve when the gain/losses coefficient approaches a critical value, at which symmetry breaking occurs. We found that in the system with broken symmetry the resonant coupling between exponentially growing mode with stable higher-order one effectively stabilizes dynamically coupled pair of modes and remarkably diminishes the average rate of the total power growth.

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

Integrable Spatiotemporally Varying NLS, PT-Symmetric NLS, and DNLS Equations: Generalized Lax Pairs and Lie Algebras

Matthew Russo, S. Roy Choudhury

This paper develops two approaches to Lax-integrbale systems with spatiotemporally varying coefficients. A technique based on extended Lax Pairs is first considered to derive variable-coefficient generalizations of various Lax-integrable NLPDE hierarchies recently introduced in the literature. As illustrative examples, we consider generalizations of the NLS and DNLS equations, as well as a PT-symmetric version of the NLS equation. It is demonstrated that the techniques yield Lax- or S-integrable NLPDEs with both time- AND space-dependent coefficients which are thus more general than almost all cases considered earlier via other methods such as the Painleve Test, Bell Polynomials, and various similarity methods. However, this technique, although operationally effective, has the significant disadvantage that, for any integrable system with spatiotemporally varying coefficients, one must ‘guess’ a generalization of the structure of the known Lax Pair for the corresponding system with constant coefficients. Motivated by the somewhat arbitrary nature of the above procedure, we therefore next attempt to systematize the derivation of Lax-integrable sytems with variable coefficients. We attempt to apply the Estabrook- Wahlquist (EW) prolongation technique, a relatively self-consistent procedure requiring little prior infomation. However, this immediately requires that the technique be significantly generalized or broadened in several different ways. The new and extended EW technique which results is illustrated by algorithmically deriving generalized Lax-integrable versions of NLS, PT-symmetric NLS, and DNLS equations.

Analysis of PDEs (math.AP); Exactly Solvable and Integrable Systems (nlin.SI)