\(E_2\)-quasi-exact solvability for non-Hermitian models

Andreas Fring

We propose the notion of \(E_2\)-quasi-exact solvability and apply this idea to find explicit solutions to the eigenvalue problem for a non-Hermitian Hamiltonian system depending on two parameters. The model considered reduces to the complex Mathieu Hamiltonian in a double scaling limit, which enables us to compute the exceptional points in the energy spectrum of the latter as a limiting process of the zeros for some algebraic equations. The coefficient functions in the quasi-exact eigenfunctions are univariate polynomials in the energy obeying a three-term recurrence relation. The latter property guarantees the existence of a linear functional such that the polynomials become orthogonal. The polynomials are shown to factorize for all levels above the quantization condition leading to vanishing norms rendering them to be weakly orthogonal. In two concrete examples we compute the explicit expressions for the Stieltjes measure.

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

Quantum star-graph analogues of PT-symmetric square wells. II: Spectra

Miloslav Znojil

For non-Hermitian equilateral q-pointed star-shaped quantum graphs of paper I [Can. J. Phys. 90, 1287 (2012), arXiv 1205.5211] we show that due to certain dynamical aspects of the model as controlled by the external, rotation-symmetric complex Robin boundary conditions, the spectrum is obtainable in a closed asymptotic-expansion form, in principle at least. Explicit formulae up to the second order are derived for illustration, and a few comments on their consequences are added.

Quantum Physics (quant-ph); Spectral Theory (math.SP)

Pseudo PT-symmetric lattice

C. Yuce

We study pseudo PT symmetry for a tight binding lattice with a general form of the modulation. Using high-frequency Floquet method, we show that the critical non-Hermitian degree for the reality of the spectrum can be manipulated by varying the parameters of the modulation. We study the effect of periodical and quasi-periodical nature of the modulation on the pseudo PT symmetry.

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

Real discrete spectrum in complex non-PT-symmetric Scarf II potential

Zafar Ahmed, Amal Nathan Joseph

Hitherto, it is well known that complex PT-symmetric Scarf II has real discrete spectrum in the parametric domain of unbroken PT-symmetry. We reveal new interesting complex, non-PT-symmetric parametric domains of this versatile potential, \(V(x)\), where the spectrum is again discrete and real. Showing that the Hamiltonian, \(p^2/2m+V(x)\), is pseudo-Hermitian could be challenging, if possible.

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

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)

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)