PT-symmetric Complex Potentials Research Articles

PT-symmetry rules applied to a class of real potentials

Extending the functional space to complex eigenfunctions R J Lombard et al (2022, Rom. J. Phys. 67, 104), we have shown that infinitely negative potentials at large distances admit finite energy states. The used techniques are similar to the ones applied in the case of PT symmetric complex potentials with real eigenvalues. We present the lowest part of the spectra for −∣x∣ n potentials with 4 ≤ n ≤ 8. We also discuss the norm and orthogonality of the wave functions.

Majorana tunneling in a one-dimensional wire with non-Hermitian double quantum dots

The combination of non-Hermitian physics and Majorana fermions can give rise to new effects in quantum transport systems. In this work, we investigate the interplay of PT-symmetric complex potentials, Majorana tunneling and interdot tunneling in a non-Hermitian double quantum dots system. It is found that in the weak-coupling regime the Majorana tunneling has pronounced effects on the transport properties of such a system, manifested as splitting of the single peak into three and a reduced 1/4 peak in the transmission function. In the presence of the PT-symmetric complex potentials and interdot tunneling, the 1/4 central peak is robust against them, while the two side peaks are tuned by them. The interdot tunneling only induces asymmetry, instead of moving the conductance peak, due to the robustness of the Majorana modes. There is an exceptional point induced by the union of Majorana tunneling and interdot tunneling. With increased PT-symmetric complex potentials, the two side peaks will move towards each other. When the exceptional point is passed through, these two side peaks will disappear. In the strong-coupling regime, the Majorana fermion induces a 1/4 conductance dip instead of the three-peak structure. PT-symmetric complex potentials induce two conductance dips pinned at the exceptional point. These effects should be accessible in experiments.

Majorana tunneling in a non-Hermitian double-quantum-dot structure

In this paper, we investigate a non-Hermitian quantum transport system, which is formed by two quantum dots possessing PT-symmetric complex potentials and side coupled to Majorana fermions. We study the transmission coefficient first. It is found that the conductance is reduced by a factor of 1/4, i.e., G=e2/4h, which can be seen as a signature of Majorana fermion in this double quantum dots structure. It is also found that the joint effects of Majorana tunneling, PT-symmetric complex potentials and dot energy level are mainly as follows: merging process, asymmetry and exceptional point. Then we study the current and conductance driven by a voltage or temperature gradient. Due to the merging process and asymmetry, the thermal current and conductance change sign drastically near the exceptional point. With large fixed temperature difference, the relationship between thermal current and dot level becomes linear, which is a manifestation of Majorana fermion’s robustness. These results can be useful in understanding the quantum transport behaviors modified by the interplay between PT symmetry and Majorana fermions.

Scattering of spin-1/2 particles from a PT-symmetric complex potential

In this letter, we study the scattering of particles from a spin-independent parity time complex potential, and for the first time, theoretically demonstrate the coexistence of and -broken phases for broadband energy spectra in this system. We also show the existence of anisotropic transmission resonances, accessible through the tuning of energy. Our results are promising for applications in spintronics, semiconductor-based devices, and a better understanding of the topological surface states.

Influence of [formula omitted]-symmetric complex potentials on the decoupling mechanism in quantum transport processes

We consider one system in which the terminal dots of a one-dimensional quantum-dot chain couple equally to the left and right leads and study the influence of PT-symmetric complex potentials on the quantum transport process. It is found that in the absence of the complex potentials, remarkable decoupling and antiresonance phenomena are able to co-occur in the transport process. For the chains with odd(even) dots, their even(odd)-numbered molecular states decouple from the leads. Meanwhile, antiresonance occurs at the positions of the even(odd)-numbered eigenenergies of the sub-chains without terminal dots. When the PT-symmetric complex potentials are applied on the terminal dots, the decoupling phenomenon is found to transform into the Fano antiresonance. These results can helpful in understanding the quantum transport modified by the PT symmetry in non-Hermitian discrete systems.

A class of exactly solvable rationally extended non-central potentials in two and three dimensions

We start from a seven parameter (six continuous and one discrete) family of non-central exactly solvable potentials in three dimensions and construct a wide class of ten parameters (six continuous and four discrete) family of rationally extended exactly solvable non-central real as well as PT symmetric complex potentials. The energy eigenvalues and the eigenfunctions of these extended non-central potentials are obtained explicitly and it is shown that the wave eigenfunctions of these potentials are either associated with the exceptional orthogonal polynomials or some type of new polynomials which can be further re-expressed in terms of the corresponding classical orthogonal polynomials. Similarly, we also construct a wide class of rationally extended exactly solvable non-central real as well as complex PT-invariant potentials in two dimensions.

Transport through a non-Hermitian parallel double-quantum-dot structure in the presence of interdot Coulomb interaction

In this work, we investigate the effect of PT-symmetric complex potentials on the transport properties of one non-Hermitian system, which is formed by a parallel coupled double-quantum-dot geometry. It is found that the PT-symmetric complex potentials take pronounced effects to the transport properties of such a system, manifested as the occurrence of antiresonance. In the presence of the interdot Coulomb interaction, such potentials lead to the shift and increase of the antiresonance points, dependent on the Coulomb interaction strength. This study exhibits the special impact of the interplay between the interdot Coulomb interaction and PT-symmetric potentials on the quantum transport behaviors.

The analytical transfer matrix method for PT-symmetric complex potential

We have extended the analytical transfer matrix (ATM) method to solve quantum mechanical bound state problems with complex PT-symmetric potentials. Our work focuses on a class of models studied by Bender and Jones, we calculate the energy eigenvalues, discuss the critical values of g and compare the results with those obtained from other methods such as exact numerical computation and WKB approximation method.

Comparison of Finite Differences and WKB approximation Methods for PT symmetric complex potentials

We consider the one dimensional schrödinger eigenvalue problem on a finite domain (Strum-Liouville problem) for several PT-symmetric complex potentials, studied by Bender and Jones using the WKB approximation method. We make a comparison between the solutions of theses PT-symmetric complex potentials using both the finite difference method (FDM) and the WKB approximation method and show quantitative and qualitative agreement between the two methods.

Scattering amplitudes for the rationally extended [formula omitted] symmetric complex potentials

In this paper, we consider the rational extensions of two different classes of PT symmetric complex potentials namely the asymptotically vanishing Scarf II and asymptotically non-vanishing Rosen–Morse II [ RM-II] and obtain the accompanying bound state eigenfunctions in terms of the exceptional Xm Jacobi polynomials and a certain class of orthogonal polynomials. By considering the asymptotic behavior of the exceptional polynomials, we also derive the reflection and transmission amplitudes for them and discuss the various novel properties of the corresponding amplitudes.

Modulational instability in a [formula omitted]-symmetric vector nonlinear Schrödinger system

A class of exact multi-component constant intensity solutions to a vector nonlinear Schrödinger (NLS) system in the presence of an external PT-symmetric complex potential is constructed. This type of uniform wave pattern displays a non-trivial phase whose spatial dependence is induced by the lattice structure. In this regard, light can propagate without scattering while retaining its original form despite the presence of inhomogeneous gain and loss. These constant-intensity continuous waves are then used to perform a modulational instability analysis in the presence of both non-hermitian media and cubic nonlinearity. A linear stability eigenvalue problem is formulated that governs the dynamical evolution of the periodic perturbation and its spectrum is numerically determined using Fourier–Floquet–Bloch theory. In the self-focusing case, we identify an intensity threshold above which the constant-intensity modes are modulationally unstable for any Floquet–Bloch momentum belonging to the first Brillouin zone. The picture in the self-defocusing case is different. Contrary to the bulk vector case, where instability develops only when the waves are strongly coupled, here an instability occurs in the strong and weak coupling regimes. The linear stability results are supplemented with direct (nonlinear) numerical simulations.

Parametric symmetries in exactly solvable real and PT symmetric complex potentials

In this paper, we discuss the parametric symmetries in different exactly solvable systems characterized by real or complex PT symmetric potentials. We focus our attention on the conventional potentials such as the generalized Pöschl Teller (GPT), Scarf-I, and PT symmetric Scarf-II which are invariant under certain parametric transformations. The resulting set of potentials is shown to yield a completely different behavior of the bound state solutions. Further, the supersymmetric partner potentials acquire different forms under such parametric transformations leading to new sets of exactly solvable real and PT symmetric complex potentials. These potentials are also observed to be shape invariant (SI) in nature. We subsequently take up a study of the newly discovered rationally extended SI potentials, corresponding to the above mentioned conventional potentials, whose bound state solutions are associated with the exceptional orthogonal polynomials (EOPs). We discuss the transformations of the corresponding Casimir operator employing the properties of the so(2, 1) algebra.

The asymmetric solitons in two-dimensional parity–time-symmetric potentials

We investigate the properties in two-dimensional (2D) special parity–time (PT) symmetric complex potentials. The linear case of this special 2D PT-symmetric complex potential and self-focusing nonlinear cases are discussed. For linear case, the eigenvalues and eigenfunction for different loss or gain level of the PT-symmetric complex potentials are obtained numerically. For nonlinear cases, the existence of asymmetric solitons and PT-symmetric solitons is studied in this PT symmetric system. The eigenvalue for linear case is equal to the critical propagation constant bc of existing PT-symmetric solitons. When the PT-symmetric soliton's propagation constant reaches a certain threshold bc1, a branch of asymmetric solitons can bifurcate out from the branch of PT-symmetric solitons.

Interplay between Fano resonance andPTsymmetry in non-Hermitian discrete systems

We study the effect of PT-symmetric complex potentials on the transport properties of non-Hermitian systems, which consist of an infinite linear chain and two side-coupled defect points with PT-symmetric complex on-site potentials. By analytically solving the scattering problem of two typical models, which display standard Fano resonances in the absence of non-Hermitian terms, we find that the PT-symmetric imaginary potentials can lead to some pronounced effects on transport properties of our systems, including changes from the perfect reflection to perfect transmission, and rich behaviors for the absence or existence of the prefect reflection at one and two resonant frequencies. Our study can help us to understand the interplay between the Fano resonance and PT symmetry in non-Hermitian discrete systems, which may be realizable in optical waveguide experiments.

Group theoretic approach to rationally extended shape invariant potentials

The exact bound state spectrum of rationally extended shape invariant real as well as PT symmetric complex potentials is obtained by using potential group approach. The generators of the potential groups are modified by introducing a new operator U(x,J3±12)to express the Hamiltonian corresponding to these extended potentials in terms of Casimir operators. Connection between the potential algebra and the shape invariance is elucidated.

Reciprocity in Parity Violating Non-Hermitian Systems

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.

Stable 2D localized modes in anisotropic media with harmonic and [formula omitted]-symmetric potentials

The analytical spatial localized mode solutions of a two-dimensional nonlinear Schrödinger equation with constant diffraction and nonlinearity in PT-symmetric potential are obtained, and the linear stability of these solutions is discussed. Then, the analytical spatial localized mode solutions in anisotropic media with harmonic and PT-symmetric potentials are also derived based on the relation between two nonlinear Schrödinger equations with variable and constant diffraction and nonlinearity. At last, the diffraction management of spatial localized modes in PT-symmetric complex potentials is investigated.

Supersymmetry, PT-symmetry and spectral bifurcation

We demonstrate that large class of PT-symmetric complex potentials, which can have isospectral real partner potentials, possess two different superpotentials. In the parameter domain, where the superpotential is unique, the spectrum is real and shape-invariant, leading to translational shift in a suitable parameter by real units. The case of two different superpotentials, leading to same potential, yields broken PT-symmetry, the energy spectra in the two phases being separated by a bifurcation. Interestingly, these two superpotentials generate the two disjoint sectors of the Hilbert space. In the broken case, shape invariance produces complex parametric shifts.

Level Spacing Distributions and Quantum Chaos in Hermitian and non-HermitianSystems

The correspondence between quantum level spacing distributions and classicalmotion of 1-D PT symmetric non-Hermitian systems is investigated using two PT symmetric complex potentials: complex rational power potentialV1(x) = (ix)(2n + 1)/m and general polynomial potential V2(x) =x2M + ib1 x2M − 1 + b2 x2M − 2 + ... + ib2M − 1x. The levelspacing distribution of V1 has two forms. When 2n + 1 − 2m ispositive, the level spacing distribution of real eigen values assumes adecreasing power function, while it behaves as an increasing power functionwhen 2n + 1 − 2m is negative. The PT symmetry of this system isspontaneously broken as 2n + 1 − 2m becomes negative. This changemanifests itself in classical mechanics as it is found by Bender et al. However, it was found that the change in the form of level spacingdistribution mentioned above is not due to the spontaneous breaking down of PT symmetry. Level spacing distribution of V2 assumes an increasing powerfunction when order of the polynomial is greater than two.

COMPLEXIFIED PSUSY AND SSUSY INTERPRETATIONS OF SOME PT-SYMMETRIC HAMILTONIANS POSSESSING TWO SERIES OF REAL ENERGY EIGENVALUES

We analyze a set of three PT-symmetric complex potentials, namely harmonic oscillator, generalized Pöschl–Teller and Scarf II, all of which reveal a double series of energy levels along with the corresponding superpotential. Inspired by the fact that two superpotentials reside naturally in order-two parasupersymmetry (PSUSY) and second-derivative supersymmetry (SSUSY) schemes, we complexify their frameworks to successfully account for the three potentials.