Defect modes in defective parity-time symmetric periodic complex potentials

We investigate the effect of parity-time ()-symmetric complex potentials on the transmission in a non-Hermitian system, which consists of an infinite dimerized chain and two side-coupled defects with -symmetric complex on-site potentials. It is found that the -symmetric imaginary potentials have pronounced effects on the transmission behavior of such a system, including the shift of the normal antiresonance and the occurrence or disappearance of the Fano resonance. These phenomena depend significantly on the coupling manners and strengths between the defects and the dimerized chain. Our study can be helpful in understanding the influences of the symmetry of the dimerized systems on transmission properties.

Fundamental, dipole, and vortex solitons in fractional nonlinear Schrödinger equation with a parity-time-symmetric periodic potential

Defect modes of defective parity-time symmetry in one-dimensional fractional Schrödinger equation are theoretically investigated. Both positive and negative defect modes are extremely sensitive to the defect strength and amplitude of periodic potential. Their eigenvalues increase almost linearly with the defect strength and continuously jump among neighboring gaps in the entire defect strength domain. There exists a phase transition point where two adjacent defect mode curves merge into a single one. Below which the eigenvalues are real, and their profiles have the same symmetry as the parity-time symmetric potential. Above the transition point, the defect modes give rise to complex eigenvalues, and the above symmetric property is destroyed. The transition point also grows linearly with the defect strength. As defect strength increases, much more positive defect mode branches merge together and more phase transition points appear in each gap. However, only one phase transition point exists for negative defect modes, and moves into the higher gaps.

Near-surface Brownian motor with synchronously fluctuating symmetric potential and applied force

We investigate an overdamped Brownian motion in symmetric periodic potential switching by Markovian dichotomous noise between two configurations. Second configuration differs from the first one by the half of spatial period displacement when the maxima of a potential profile become the minima and vice versa. We establish a validity of the formula for effective diffusion constant previously obtained in the case of fixed periodic potentials and, thus, reduce the problem to calculation of mean first-passage times (MFPTs). At the same time the MFPT in this formula should be substituted for the semi-sum of MFPTs regarding two initial configurations of potential profile. A set of equations for MFPTs in flipping potentials can be solved for the sawtooth periodic potential. As a result, after cumbersome calculations we obtain the exact complex expression for effective diffusion coefficient of Brownian particles in such a medium which is valid for arbitrary mean frequency of potential switchings. We detect the acceleration of diffusion in comparison with the case of fixed sawtooth potential profile as it was demonstrated for a symmetric periodic potential modulating by external Gaussian white noise.

Under investigation in this work is a newly proposed nonlocal long-wave–short-wave resonance interaction (LSRI) equation with the self-induced parity-time (PT) symmetric potential. This equation offers PT symmetry analogues of the classical integrable LSRI equation and may be important for the occurence of such equations in nonlinear optics as the nonlocal NLS equation. General soliton solutions to the nonlocal LSRI equation with nonzero boundary condition are derived by using the Hirota’s bilinear method combined with the Kadomtsev–Petviashvili (KP) hierarchy reduction method. These solutions are expressed in terms of Gramian determinants and include dark–dark solitons, dark–antidark solitons and antidark–antidark solitons. Three typical cases of the two solitons, namely two dark–dark solitons, two dark–antidark solitons and two antidark–antidark solitons, are demonstrated.

The existence and stability of fundamental, dipole, and tripole solitons in Kerr nonlinear media with parity-time symmetric Gaussian complex potentials are reported. Fundamental solitons are stable not only in deep potentials but also in shallow potentials. Dipole and tripole solitons are stable only in deep potentials, and tripole solitons are stable in deeper potentials than for dipole solitons. The stable regions of solitons increase with increasing potential depth. The power of solitons increases with increasing propagation constant or decreasing modulation depth of the potentials.

Collective directional transport of particles in symmetric periodic potentials is studied. An example is given to reveal the directed motion of a single particle in a symmetric periodic potential and subject to an asymmetric ac force with zero mean. It is then shown that the asymmetric coupling can give rise to a directed transport. The transport failure (pinning of the lattice) is related to the phase delocalization (crisis) in the circle map. For symmetric couplings, it is found that a train of plane wave can also lead to a directed transport. The mode-locking steps of the transport velocity is found and analyzed. The collective transport can be well optimized by adjusting parameters in the system.

Ratchet effect in an underdamped periodic potential and its characterisation

In this work we present a method of investigation of nonlinear optical beams generated from non-Hermitian optical systems1 . This method can be applied in the development of optical filters and optical sensors to process, analyze and choose the passband of the propagation modes of an optical pulse from an non-Hermitian optical system. Non-Hermitian optical systems can be used to develop optical fiber sensors that suppress certain propagation modes of optical pulses that eventually behave as quantum noise. Such systems are described by the Nonlinear Schrodinger-like Equation with Parity-Time (PT) Symmetric Optical Potentials. There are optical fiber sensors that due to high laser intensity and frequency can produce quantum noise, such as Raman and Brillouin scattering. However, the optical fiber, for example, can be designed so that its geometry suppress certain propagation modes of the beam. We apply some results of non- Hermitian optical systems with PT symmetry to simulate optical lattice by a appropriate potential function, which among other applications, can naturally suppress certain propagation modes of an optical beam propagating through a waveguide. In other words, the optical system is modeled by a potential function in the Nonlinear Schrodinger-like Equation that one relates with the geometric aspects of the wave guides and with the optical beam interacting with the waveguide material. The paper is organized as follows: sections 1 and 2 present a brief description about nonlinear optical systems and non-Hermitian optical systems with PT symmetry. Section 3 presents a description of the dynamics of nonlinear optical pulses propagating through optical networks described by a optical potential non-Hermitian. Sections 4 and 5 present a general description of this non-Hermitian optical systems and how to get them from a more general model. Section 6 presents some conclusions and comment and the final section presents the references. Begin the abstract two lines below author names and addresses. © (2015) COPYRIGHT Society of Photo-Optical Instrumentation Engineers (SPIE). Downloading of the abstract is permitted for personal use only.

Optical solitons in the parity-time (PT)-symmetric Bessel complex potential are studied, including the linear case, and self-focusing and self-defocusing nonlinear cases. For the linear case, the PT-symmetric breaking points, eigenvalues and the eigenfunction for different modulated depths of the PT-symmetric Bessel complex potential are obtained numerically. The PT-symmetric breaking points increase linearly with increasing the real part of the modulated depths of the PT potential. Below the PT-symmetric breaking points, the eigenfunctions of linear modes are symmetrical; however, the symmetries of the eigenfunction break above the PT-symmetric breaking points. For nonlinear cases, the existence and stability of fundamental and multipole solitons are studied in self-focusing and self-defocusing media. The eigenvalue for the linear case is equal to the critical propagation constant bc of the existing soliton. Fundamental solitons are stable in the whole region and multipole solitons are stable with the propagation constants being close to bc both for self-focusing and self-defocusing nonlinearities. The range of solitons’ stability decreases with an increase of the number of the intensity peaks of the solitons.

Chaos-like properties such as noise or fluctuations arising from quantum effects have recently investigated on a single barrier potential and on a one-dimensional periodic potential barrier. They used either the planewave of light or a wavepacket (pulse) passing through the potential barrier, demonstrating the chaos created by the bounded one-dimensional multibarrier potential (Bar and Horwitz, 2002; Hondou and Sawada, 1995), and proved it is an ordered and periodic phenomenon. Also another group demonstrated unexpected behavior of a dissipative particle in simple multiscale systems subject to chaotic noise and clarified the reason for the particular behavior because it is under a periodic potential (Chialvo et al., 1997). They concluded the occurrence of drift in a symmetric periodic potential would be expected to be a common feature of noise-driven systems. Essentially this effect is quite similar to the problem of a particle placed in a potential well of finite barrier (Cohen-Tannoudji et al., 1977) and the present issue of the photon distribution on a crystallite surface. When the crystallite size decreases to a few nanometers, the light absorption or photoluminescence in air moves to the direction of smaller wavelength, which is known as the blueshift (Kale and Lokhande, 2000; Nanda et al., 1999; Tsunekawa et al., 2003; Von Behren et al., 2000). The basic theory for this phenomenon is the quantum confinement effect (Andersen et al., 2002; Sharma et al., 2005). According to this theory the number of photons confined in each unit cell increases with the decreasing crystallite size. It is also said to enlarge the band gap between the valence band and the conduction band. Another approach is to take crystalline unit cells as numerical elements and employ different potential energies over a unit cell (Choi, 2007; Choi and Kim, 2007), realizing light intensity fluctuations. The effect of the film thickness on the blueshift was investigated by one of the authors (Choi and Pyun, 2008) about cellular crystalline surfaces with periodic potential, getting the blueshift numerically demonstrated in good agreement with experiments. In this paper we deal with surface nanostructures and try to prove the blueshift numerically by predicting the location of absorption wavelength. One of the best ways to uphold the present numerical method would be to demonstrate the blueshift phenomenon observed widely experimentally (Lu et al., 2008; Miyake et al., 1999; Tan et al., 2005; Tsunekawa et al., 2000). They demonstrated experimentally the dependence of the blueshift on the crystallite size for CeO and CdS. One of the authors has been studying this problem recently, publishing a few papers (Choi, 2007; Choi and Kim, 2007; Choi and Pyun, 2008), and this paper may be taken as an extension of these consecutive efforts. Another publication (Choi and Choi, 2009) predicting the blueshift with respect to the shell layer thickness has been already accomplished and led the present investigation to be made. 17

In this paper, we have studied the single-hump and double-hump solitons having parity - time () symmetric complex potential in a medium with cubic nonlinearity. The system has been analyzed for the stationary states and the dynamical states in the linear and nonlinear regime. The system undergoes spontaneous symmetric phase transition from the symmetric region to the broken symmetric region depending on the strength of the imaginary part of the potential. The coefficient of the imaginary part of the potential at which the phase transition occurs is known as the threshold value. The symmetric phase transition point has been studied by varying the strength of real part of the potential and the coefficient of the nonlinearity. The stable solitons are formed for low values of the coefficients of imaginary part of the potential and nonlinearity. We have analyzed the stability of the solitons using Bogoliubov–De–Genes equations.

We show that defect modes (DMs) in optical lattices are possible in biased centrosymmetric photorefractive crystals. These DMs exist in different bandgaps due to the change of the defect strength. For a positive defect, DMs exist in each bandgap. The strongest confinement of DMs appears in the semi-infinite bandgap when the positive defect strength is fixed. For a negative defect, DMs exist in each bandgap except the semi-infinite bandgap. When the negative defect strength is fixed, the strongest confinement of DMs appears in the first bandgap. For a given bandgap, the strongest confinement of DMs appears when the negative defect strength is equal to a certain value. The strong negative defects do not favor the creation of DMs in the first bandgap.

A -symmetric optically active medium that lases at the threshold gain also acts as a complete perfect absorber at the laser wavelength. This is because spectral singularities of -symmetric complex potentials are always accompanied by their time-reversal dual. We investigate the significance of -symmetry for the appearance of these self-dual spectral singularities. In particular, using a realistic optical system we show that self-dual spectral singularities can emerge also for non--symmetric configurations. This signifies the existence of non--symmetric coherent perfect absorbing lasers.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Quantum physics with non-Hermitian operators’.