arXiv subjects

D. A. Zezyulin

Publications and source records attributed to D. A. Zezyulin.

At least 19 recordsLinked to original sources

arXiv2026

Towards the complete description of stationary states of a Bose-Einstein condensate in a one-dimensional quasiperiodic lattice: A coding approach

We consider stationary states of an effectively one-dimensional Bose-Einstein condensate in a quasiperiodic lattice. We formulate sufficient conditions for a one-to-one correspondence between the stationary states with a fixed chemical potential and the set of bi-infinite sequences over a finite alphabet. These conditions can be checked numerically. A bi-infinite sequence can be interpreted as a code of the corresponding solution. A numerical example demonstrates the coding approach using an alphabet of three symbols.

nlin.PS↗

arXiv2023

Eigenvalues bifurcating from the continuum in two-dimensional potentials generating non-Hermitian gauge fields

It has been recently shown that complex two-dimensional (2D) potentials $V_\varepsilon(x,y)=V(y+\mathrm{i}\varepsilonη(x))$ can be used to emulate non-Hermitian matrix gauge fields in optical waveguides. Here $x$ and $y$ are the transverse coordinates, $V(y)$ and $η(x)$ are real functions, $\varepsilon>0$ is a small parameter, and $\mathrm{i}$ is the imaginary unit. The real potential $V(y)$ is required to have at least two discrete eigenvalues in the corresponding 1D Schrödinger operator. When both transverse directions are taken into account, these eigenvalues become thresholds embedded in the continuous spectrum of the 2D operator. Small nonzero $\varepsilon$ corresponds to a non-Hermitian perturbation which can result in a bifurcation of each threshold into an eigenvalue. Accurate analysis of these eigenvalues is important for understanding the behavior and stability of optical waves propagating in the artificial non-Hermitian gauge potential. Bifurcations of complex eigenvalues out of the continuum is the main object of the present study. We obtain simple asymptotic expansions in $\varepsilon$ that describe the behavior of bifurcating eigenvalues. The lowest threshold can bifurcate into a single eigenvalue, while every other threshold can bifurcate into a pair of complex eigenvalues. These bifurcations can be controlled by the Fourier transform of function $η(x)$ evaluated at certain isolated points of the reciprocal space. When the bifurcation does not occur, the continuous spectrum of 2D operator contains a quasi-bound-state which is characterized by a strongly localized central peak coupled to small-amplitude but nondecaying tails. The analysis is applied to the case examples of parabolic and double-well potentials $V(y)$. In the latter case, the bifurcation of complex eigenvalues can be dampened if the two wells are widely separated.

physics.optics↗

arXiv2022

Dynamical stabilization by vacuum fluctuations in a cavity: Resonant electron scattering in the ultrastrong light-matter coupling regime

We developed a theory of electron scattering by a short-range repulsive potential in a cavity. In the regime of ultrastrong electron coupling to the cavity electromagnetic field, the vacuum fluctuations of the field result in the dynamical stabilization of a quasistationary polariton state confined in the core of the repulsive potential. When the energy of a free electron coincides with the energy of the confined state, the extremely efficient resonant nonelastic scattering of the electron accompanied by emission of a cavity photon appears. This effect is discussed as a basis for possible free-electron sources of nonclassical light.

cond-mat.mes-hall↗

arXiv2022

Bright and dark solitons in the systems with strong light-matter coupling: exact solutions and numerical simulations

We theoretically study bright and dark solitons in an experimentally relevant hybrid system characterized by strong light-matter coupling. We find that the corresponding two-component model supports a variety of coexisting moving solitons including bright solitons on zero and nonzero background, dark-gray and gray-gray dark solitons. The solutions are found in the analytical form by reducing the two-component problem to a single stationary equation with cubic-quintic nonlinearity. All found solutions coexist under the same set of the model parameters, but, in a properly defined linear limit, approach different branches of the polariton dispersion relation for linear waves. Bright solitons with zero background feature an oscillatory-instability threshold which can be associated with a resonance between the edges of the continuous spectrum branches. `Half-topological' dark-gray and nontopological gray-gray solitons are stable in wide parametric ranges below the modulational instability threshold, while bright solitons on the constant-amplitude pedestal are unstable.

nlin.PS↗

arXiv2022

Floquet engineering of excitons in semiconductor quantum dots

Within the Floquet theory of periodically driven quantum systems, we demonstrate that a high-frequency electromagnetic field can be used as an effective tool to control excitonic properties of semiconductor quantum dots (QDs). It is shown, particularly, that the field both decreases the exciton binding energy and dynamically stabilizes the exciton, increasing its radiative lifetime. The developed theory can serve as a basis for the ultrafast method to tune spectral characteristics of the QD-based photon emitters by a high-frequency field.

cond-mat.mes-hall↗

arXiv2020

Bifurcations of thresholds in essential spectra of elliptic operators under localized non-Hermitian perturbations

We consider the operator $${\cal H} = {\cal H}' -\frac{\partial^2\ }{\partial x_d^2} \quad\text{on}\quadω\times\mathbb{R}$$ subject to the Dirichlet or Robin condition, where a domain $ω\subseteq\mathbb{R}^{d-1}$ is bounded or unbounded. The symbol ${\cal H}'$ stands for a second order self-adjoint differential operator on $ω$ such that the spectrum of the operator ${\cal H}'$ contains several discrete eigenvalues $Λ_{j}$, $j=1,\ldots, m$. These eigenvalues are thresholds in the essential spectrum of the operator ${\cal H}$. We study how these thresholds bifurcate once we add a small localized perturbation $ε{\cal L}(ε)$ to the operator ${\cal H}$, where $ε$ is a small positive parameter and ${\cal L}(ε)$ is an abstract, not necessarily symmetric operator. We show that these thresholds bifurcate into eigenvalues and resonances of the operator ${\cal H}$ in the vicinity of $Λ_j$ for sufficiently small $ε$. We prove effective simple conditions determining the existence of these resonances and eigenvalues and find the leading terms of their asymptotic expansions. Our analysis applies to generic non-self-adjoint perturbations and, in particular, to perturbations characterized by the parity-time ($PT$) symmetry. Potential applications of our result embrace a broad class of physical systems governed by dispersive or diffractive effects. We use our findings to develop a scheme for a controllable generation of non-Hermitian optical states with normalizable power and real part of the complex-valued propagation constant lying in the continuum. The corresponding eigenfunctions can be interpreted as an optical generalization of bound states embedded in the continuum. For a particular example, the persistence of asymptotic expansions is confirmed with direct numerical evaluation of the perturbed spectrum.

math-ph↗

arXiv2019

Sequences of closely spaced resonances and eigenvalues for bipartite complex potentials

We consider a Schroedinger operator on the axis with a bipartite potential consisting of two compactly supported complex-valued functions, whose supports are separated by a large distance. We show that this operator possesses a sequence of approximately equidistant complex-valued wavenumbers situated near the real axis. Depending on its imaginary part, each wavenumber corresponds to either a resonance or an eigenvalue. The obtained sequence of wavenumbers resembles transmission resonances in electromagnetic Fabry-Pérot interferometers formed by parallel mirrors. Our result has potential applications in standard and non-hermitian quantum mechanics, physics of waveguides, photonics, and in other areas where the Schroedinger operator emerges as an effective Hamiltonian.

math-ph↗

arXiv2019

Spacing gain and absorption in a simple $\mathcal{PT}$-symmetric model: spectral singularities and ladders of eigenvalues and resonances

We consider a parity-time ($\mathcal{PT}$-) symmetric waveguide consisting of a localized gain and loss elements separated by a variable distance. The situation is modelled by a Schrödiner operator with localized complex $\mathcal{PT}$-symmetric potential. Properties of the latter Hamiltonian are considered subject to the change of the gain-to-loss distance. Resonances, spectral singularities and eigenvalues are analyzed in detail and discussed in the context of the associated laser-absorber modes and $\mathcal{PT}$-symmetry breaking phase transition. Increasing gain-to-loss distance creates new resonances and spectral singularities which do not exist in the waveguide with adjacent gain and loss. In the limit of large gain-to-loss distance, the waveguide features a ladder of resonances which can be transformed to a ladder of complex eigenvalues by means of the change of the gain-and-loss amplitude.

math-ph↗

arXiv2019

Global search for localised modes in scalar and vector nonlinear Schrödinger-type equations

We present a new approach for search of coexisting classes of localised modes admitted by the repulsive (defocusing) scalar or vector nonlinear Schrödinger-type equations. The approach is based on the observation that generic solutions of the corresponding stationary system have singularities at finite points on the real axis. We start with establishing conditions on the initial data of the associated Cauchy problem that guarantee the formation of a singularity. Making use of these sufficient conditions, we identify the bounded, nonsingular, solutions --- and then classify them according to their asymptotic behaviour. To determine the bounded solutions, a properly chosen space of initial data is scanned numerically. Due to asymptotic or symmetry considerations, we can limit ourselves to a one- or two-dimensional space. For each set of initial conditions we compute the distances $X^{\pm}$ to the nearest forward and backward singularities; large $X^+$ or $X^-$ indicate the proximity to a bounded solution. We illustrate our method with the Gross-Pitaevskii equation with a $\PT$-symmetric complex potential, a system of coupled Gross-Pitaevskii equations with real potentials, and the Lugiato-Lefever equation with normal dispersion.

nlin.PS↗

arXiv2018

Localized modes in the Gross-Pitaevskii equation with a parabolic trapping potential and a nonlinear lattice pseudopotential

We study localized modes (LMs) of the one-dimensional Gross-Pitaevskii/nonlinear Schrödinger equation with a harmonic-oscillator (parabolic) confining potential, and a periodically modulated coefficient in front of the cubic term (nonlinear lattice pseudopotential). The equation applies to a cigar-shaped Bose-Einstein condensate loaded in the combination of a magnetic trap and an optical lattice which induces the periodic pseudopotential via the Feshbach resonance. Families of stable LMs in the model feature specific properties which result from the interplay between spatial scales introduced by the parabolic trap and the period of the nonlinear pseudopotential. Asymptotic results on the shapes and stability of LMs are obtained for small-amplitude solutions and in the limit of a rapidly oscillating nonlinear pseudopotential. We show that the presence of the lattice pseudopotential may result in: (i) creation of new LM families which have no counterparts in the case of the uniform nonlinearity; (ii) stabilization of some previously unstable LM species; (iii) evolution of unstable LMs into a pulsating mode trapped in one well of the lattice pseudopotential.

nlin.PS↗

arXiv2018

Chiral solitons in spinor polariton rings

We consider theoretically one-dimensional polariton ring accounting for both longitudinal-transverse (TE-TM) and Zeeman splitting of spinor polariton states and spin dependent polariton-polariton interactions. We present the novel class of solutions in the form of the localized defects rotating with constant angular velocity and analyze their properties for realistic values of the parameters of the system. We show that the effects of the geometric phase arising from the interplay between external magnetic field and TE-TM splitting introduce chirality in the system and make solitons propagating in clockwise and anticlockwise directions non equivalent. This can be interpreted as solitonic analog of Aharonov-Bohm effect.

cond-mat.mes-hall↗

arXiv2017

Dynamical design of spatial patterns of colloidal suspensions

We study the collective dynamics of colloidal suspensions in the presence of a time-dependent potential, by means of dynamical density functional theory. We consider a non-linear diffusion equation for the density and show that spatial patterns emerge from a sinusoidal external potential with a time-dependent wavelength. These patterns are characterized by a sinusoidal density with the average wavelength and a Bessel-function envelope with an induced wavelength that depends only on the amplitude of the temporal oscillations. As a generalization of this result, we propose a design strategy to obtain a family of spatial patterns using time-dependent potentials of practically arbitrary shape.

cond-mat.soft↗

arXiv2015

Localised nonlinear modes in the PT-symmetric double-delta well Gross-Pitaevskii equation

We construct exact localised solutions of the PT-symmetric Gross-Pitaevskii equation with an attractive cubic nonlinearity. The trapping potential has the form of two $δ$-function wells, where one well loses particles while the other one is fed with atoms at an equal rate. The parameters of the constructed solutions are expressible in terms of the roots of a system of two transcendental algebraic equations. We also furnish a simple analytical treatment of the linear Schrödinger equation with the PT-symmetric double-$δ$ potential.

nlin.PS↗

arXiv2015

Exactly solvable Wadati potentials in the PT-symmetric Gross-Pitaevskii equation

This note examines Gross-Pitaevskii equations with PT-symmetric potentials of the Wadati type: $V=-W^2+iW_x$. We formulate a recipe for the construction of Wadati potentials supporting exact localised solutions. The general procedure is exemplified by equations with attractive and repulsive cubic nonlinearity bearing a variety of bright and dark solitons.

nlin.PS↗

arXiv2013

${\cal PT}$-symmetric coupler with $χ^{(2)}$ nonlinearity

We introduce the notion of a ${\cal PT}$-symmetric dimer with a $χ^{(2)}$ nonlinearity. Similarly to the Kerr case, we argue that such a nonlinearity should be accessible in a pair of optical waveguides with quadratic nonlinearity and gain and loss, respectively. An interesting feature of the problem is that because of the two harmonics, there exist in general two distinct gain/loss parameters, different values of which are considered herein. We find a number of traits that appear to be absent in the more standard cubic case. For instance, bifurcations of nonlinear modes from the linear solutions occur in two different ways depending on whether the first or the second harmonic amplitude is vanishing in the underlying linear eigenvector. Moreover, a host of interesting bifurcation phenomena appear to occur including saddle-center and pitchfork bifurcations which our parametric variations elucidate. The existence and stability analysis of the stationary solutions is corroborated by numerical time-evolution simulations exploring the evolution of the different configurations, when unstable.

physics.optics↗

arXiv2013

Nonlinear modes in binary bosonic condensates with the pseudo-spin-orbital coupling

We consider a binary Bose-Einstein condensate (BEC) with nonlinear repulsive interactions and linear spin-orbit (SO) and Zeeman-splitting couplings. In the presence of the trapping harmonic-oscillator (HO) potential, we report the existence of even, odd, and asymmetric spatial modes. They feature alternating domains with opposite directions of the pseudo-spin, i.e., anti-ferromagnetic structures, which is explained by the interplay of the linear couplings, HO confinement, and repulsive self-interaction. The number of the domains is determined by the strength of the SO coupling. The modes are constructed analytically in the weakly nonlinear system. The dynamical stability of the modes is investigated by means of the Bogoliubov-de Gennes equations and direct simulations. A notable result is that the multi-domain-wall (DW) structures are stable, alternating between odd and even shapes, while the simplest single-DW structure is unstable. Thus, the system features a transition to the complex ground states under the action of the SO coupling. The addition of the Zeeman splitting transforms the odd modes into asymmetric ones via spontaneous symmetry breaking. The results suggest possibilities for switching the binary system between states with opposite (pseudo) magnetization by external fields, and realization of similar stable states and dynamical effects in solid-state and nonlinear-optical settings emulated by the SO-coupled BECs.

cond-mat.quant-gas↗

arXiv2013

Parity-time symmetric optical coupler with birefringent arms

In this work, we propose a PT-symmetric coupler whose arms are birefringent waveguides as a realistic physical model which leads to a so-called quadrimer i.e., a four complex field setting. We seek stationary solutions of the resulting linear and nonlinear model, identifying its linear point of PT symmetry breaking and examining the corresponding nonlinear solutions that persist up to this point, as well as, so-called, ghost states that bifurcate from them. We obtain the relevant symmetry breaking bifurcations and numerically follow the associated dynamics which give rise to growth/decay even within the PT-symmetric phase. Our obtained stationary nonlinear solutions are found to terminate in saddle-center bifurcations which are analogous to the linear PT-phase transition.

quant-ph↗

arXiv2012

Discrete solitons in PT-symmetric lattices

We prove existence of discrete solitons in infinite parity-time (PT-) symmetric lattices by means of analytical continuation from the anticontinuum limit. The energy balance between dissipation and gain implies that in the anticontinuum limit the solitons are constructed from elementary PT-symmetric blocks such as dimers, quadrimers, or more general oligomers. We consider in detail a chain of coupled dimers, analyze bifurcations of discrete solitons from the anticontinuum limit and show that the solitons are stable in a sufficiently large region of the lattice parameters. The generalization of the approach is illustrated on two examples of networks of quadrimers, for which stable discrete solitons are also found.

nlin.PS↗