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Ali Mostafazadeh

Publications and source records attributed to Ali Mostafazadeh.

At least 55 records · Page 3Linked to original sources
arXiv2019

Geometric scattering of a scalar particle moving on a curved surface in the presence of point defects

A nonrelativistic scalar particle that is constrained to move on an asymptotically flat curved surface undergoes a geometric scattering that is sensitive to the mean and Gaussian curvatures of the surface. A careful study of possible realizations of this phenomenon in typical condensed matter systems requires dealing with the presence of defects. We examine the effect of delta-function point defects residing on a curved surface ${S}$. In particular, we solve the scattering problem for a multi-delta-function potential in plane, which requires a proper regularization of divergent terms entering its scattering amplitude, and include the effects of nontrivial geometry of ${S}$ by treating it as a perturbation of the plane. This allows us to obtain analytic expressions for the geometric scattering amplitude for a surface consisting of one or more Gaussian bumps. In general the presence of the delta-function defects enhances the geometric scattering effects.

quant-ph↗
arXiv2019

Potentials with Identical Scattering Properties Below a Critical Energy

A pair of scattering potentials are called $α$-equivalent if they have identical scattering properties for incident plane waves with wavenumber $k\leqα$ (energy $k^2\leqα^2$.) We use a recently developed multidimensional transfer-matrix formulation of scattering theory to obtain a simple criterion for $α$-equivalence of complex potentials in two and three dimensions.

quant-ph↗
arXiv2018

$S$-matrix pole symmetries for non-Hermitian scattering Hamiltonians

The complex eigenvalues of some non-Hermitian Hamiltonians, e.g. parity-time symmetric Hamiltonians, come in complex-conjugate pairs. We show that for non-Hermitian scattering Hamiltonians (of a structureless particle in one dimension) possesing one of four certain symmetries, the poles of the $S$-matrix eigenvalues in the complex momentum plane are symmetric about the imaginary axis, i.e. they are complex-conjugate pairs in complex-energy plane. This applies even to states which are not bounded eigenstates of the system, i.e. antibound or virtual states, resonances, and antiresonances. The four Hamiltonian symmetries are formulated as the commutation of the Hamiltonian with specific antilinear operators. Example potentials with such symmetries are constructed and their pole structures and scattering properties are calculated.

quant-ph↗
arXiv2018

Blowing up Light: A nonlinear amplification scheme for electromagnetic waves

We use blow-up solutions of nonlinear Helmholtz equations to introduce a nonlinear resonance effect that is capable of amplifying electromagnetic waves of particular intensity. To achieve this, we propose a scattering setup consisting of a Kerr slab with a negative (defocusing) Kerr constant placed to the left of a linear slab in such a way that a left-incident coherent TE wave with a specific incidence angle and intensity realizes a blow-up solution of the corresponding Helmholtz equation whenever its wavenumber $k$ takes a certain critical value, $k_\star$. For $k=k_\star$, the solution blows up at the right-hand boundary of the Kerr slab. For $k<k_\star$, the setup defines a scattering system with a transmission coefficient that diverges as $(k-k_\star)^{-4}$ for $k\to k_\star$. By tuning the distance between the slabs we can use this setup to amplify coherent waves with a wavelength in an extremely narrow spectral band. For nearby wavelengths the setup serves as a filter. Our analysis makes use of a nonlinear generalization of the transfer matrix of the scattering theory as well as properties of unidirectionally invisible potentials.

physics.optics↗
arXiv2018

Blow-up solutions of Helmholtz equation for a Kerr slab with a complex linear and nonlinear permittivity

We show that the Helmholtz equation describing the propagation of transverse electric waves in a Kerr slab with a complex linear permittivity and a complex Kerr coefficient admits blow-up solutions provided that the real part of the Kerr coefficient is negative, i.e., the slab is defocusing. This result applies to homogeneous as well as inhomogeneous Kerr slabs whose linear permittivity and Kerr coefficient are continuous functions of the transverse coordinate. For an inhomogeneous Kerr slab, blow-up solutions exist if the real part of Kerr coefficient is bounded above by a negative number.

math-ph↗
arXiv2018

Scattering due to geometry: The case of a spinless particle moving on an asymptotically flat embedded surface

A nonrelativistic quantum mechanical particle moving freely on a curved surface feels the effect of the nontrivial geometry of the surface through the kinetic part of the Hamiltonian, which is proportional to the Laplace-Beltrami operator, and a geometric potential, which is a linear combination of the mean and Gaussian curvatures of the surface. The coefficients of these terms cannot be uniquely determined by general principles of quantum mechanics but enter the calculation of various physical quantities. We examine their contribution to the geometric scattering of a scalar particle moving on an asymptotically flat embedded surface. In particular, having in mind the possibility of an experimental realization of the geometric scattering in a low density electron gas formed on a bumped surface, we determine the scattering amplitude for arbitrary choices of the curvature coefficients for a surface with global or local cylindrical symmetry. We also examine the effect of perturbations that violate this symmetry and consider surfaces involving bumps that form a lattice.

quant-ph↗
arXiv2018

Energy Observable for a Quantum System with a Dynamical Hilbert Space and a Global Geometric Extension of Quantum Theory

A non-Hermitian operator may serve as the Hamiltonian for a unitary quantum system, if we can modify the Hilbert space of state vectors of the system so that it turns into a Hermitian operator. If this operator is time-dependent, the modified Hilbert space is generally time-dependent. This in turn leads to a generic conflict between the condition that the Hamiltonian is an observable of the system and that it generates a unitary time-evolution via the standard Schrödinger equation. We propose a geometric framework for addressing this problem. In particular we show that the Hamiltonian operator consists of a geometric part, which is determined by a metric-compatible connection on an underlying Hermitian vector bundle, and a non-geometric part which we identify with the energy observable. The same quantum system can be locally described using a time-dependent Hamiltonian that acts in a time-independent state space and is the sum of a geometric part and the energy operator. The full global description of the system is achieved within the framework of a moderate geometric extension of quantum mechanics where the role of the Hilbert space of state vectors is played by a Hermitian vector bundle $\mathcal{E}$ endowed with a metric compatible connection, and observables are given by global sections of a real vector bundle that is determined by $\mathcal{E}$. We examine the utility of our proposal to describe a class of two-level systems where $\mathcal{E}$ is a Hermitian vector bundle over a two-dimensional sphere.

quant-ph↗
arXiv2018

Nonlinear Scattering and Its Transfer Matrix Formulation in One Dimension

We present a systematic formulation of scattering theory for nonlinear interactions in one dimension and develop a nonlinear generalization of the transfer matrix that has a composition property similar to its linear analog's. We offer alternative characterizations of spectral singularities, unidirectional reflectionlessness and invisibility, and nonreciprocal transmission for nonlinear scattering systems, and examine the application of our general results in addressing the scattering problem for nonlinear single- and double-$δ$-function potentials.

quant-ph↗
arXiv2018

Scattering Theory and $\mathcal{P}\mathcal{T}$-Symmetry

We outline a global approach to scattering theory in one dimension that allows for the description of a large class of scattering systems and their $\mathcal{P}$-, $\mathcal{T}$-, and $\mathcal{P}\mathcal{T}$-symmetries. In particular, we review various relevant concepts such as Jost solutions, transfer and scattering matrices, reciprocity principle, unidirectional reflection and invisibility, and spectral singularities. We discuss in some detail the mathematical conditions that imply or forbid reciprocal transmission, reciprocal reflection, and the presence of spectral singularities and their time-reversal. We also derive generalized unitarity relations for time-reversal-invariant and $\mathcal{P}\mathcal{T}$-symmetric scattering systems, and explore the consequences of breaking them. The results reported here apply to the scattering systems defined by a real or complex local potential as well as those determined by energy-dependent potentials, nonlocal potentials, and general point interactions.

quant-ph↗
arXiv2017

A Class of Exactly Solvable Scattering Potentials in Two Dimensions, Entangled State Pair Generation, and a Grazing Angle Resonance Effect

We provide an exact solution of the scattering problem for the potentials of the form $v(x,y)=χ_a(x)[v_0(x)+ v_1(x)e^{iαy}]$, where $χ_a(x):=1$ for $x\in[0,a]$, $χ_a(x):=0$ for $x\notin[0,a]$, $v_j(x)$ are real or complex-valued functions, $χ_a(x)v_0(x)$ is an exactly solvable scattering potential in one dimension, and $α$ is a positive real parameter.If $α$ exceeds the wavenumber $k$ of the incident wave, the scattered wave does not depend on the choice of $v_1(x)$. In particular, $v(x,y)$ is invisible if $v_0(x)=0$ and $k<α$. For $k>α$ and $v_1(x)\neq 0$, the scattered wave consists of a finite number of coherent plane-wave pairs $ψ_n^\pm$ with wavevector: $\mathbf{k}_n=(\pm\sqrt{k^2-(nα)^2},nα)$, where $n=0,1,2,\cdots<k/α$. This generalizes to the scattering of wavepackets and suggests means for generating quantum states with a quantized component of momentum and pairs of states with an entangled momentum. We examine a realization of these potentials in terms of certain optical slabs. If $k=Nα$ for some positive integer $N$, $ψ_N^\pm$ coalesce and their amplitude diverge. If $k$ exceeds $Nα$ slightly, $ψ_N^\pm$ have a much larger amplitude than $ψ_n^\pm$ with $n<N$. This marks a resonance effect that arises for the scattered waves whose wavevector makes a small angle with the faces of the slab.

quant-ph↗
arXiv2017

Spectral Singularities, Threshold Gain, and Output Intensity for a Slab Laser with Mirrors

We explore the consequences of the emergence of linear and nonlinear spectral singularities in TE modes of a homogeneous slab of active optical material that is placed between two mirrors. We use the results together with two basic postulates regarding the behavior of laser light emission to derive explicit expressions for the laser threshold condition and output intensity for these modes of the slab and discuss their physical implications. In particular, we reveal the details of the dependence of the threshold gain and output intensity on the position and properties of the mirrors and on the real part of the refractive index of the gain material.

physics.optics↗
arXiv2017

Unidirectional Reflection and Invisibility in Nonlinear Media with an Incoherent Nonlinearity

We give explicit criteria for the reflectionlessness, transparency, and invisibility of a finite-range potential in the presence of an incoherent (intensity-dependent) nonlinearity that is confined to the range of the potential. This allows us to conduct a systematic study of the effects of such a nonlinearity on a locally periodic class of finite-range potentials that display perturbative unidirectional invisibility. We use our general results to examine the effects of a weak Kerr nonlinearity on the behavior of these potentials and show that the presence of nonlinearity destroys the unidirectional invisibility of these potentials. If the strength of the Kerr nonlinearity is so weak that the first-order perturbation theory is reliable, the presence of nonlinearity does not affect the unidirectional reflectionlessness and transmission reciprocity of the potential. We show that the expected violation of the latter is a second order perturbative effect.

quant-ph↗
arXiv2017

Quantum Mechanics of a Photon

A first quantized free photon is a complex massless vector field $A=(A^μ)$ whose field strength satisfies Maxwell's equations in vacuum. We construct the Hilbert space $\mathscr{H}$ of the photon by endowing the vector space of the fields $A$ in the temporal-Coulomb gauge with a positive-definite and relativistically invariant inner product. We give an explicit expression for this inner product, identify the Hamiltonian for the photon with the generator of time translations in $\mathscr{H}$, determine the operators representing the momentum and the helicity of the photon, and introduce a chirality operator whose eigenfunctions correspond to fields having a definite sign of energy. We also construct a position operator for the photon whose components commute with each other and with the chirality and helicity operators. This allows for the construction of the localized states of the photon with a definite sign of energy and helicity. We derive an explicit formula for the latter and compute the corresponding electric and magnetic fields. These turn out to diverge not just at the point where the photon is localized but on a plane containing this point. We identify the axis normal to this plane with an associated symmetry axis, and show that each choice of this axis specifies a particular position operator, a corresponding position basis, and a position representation of the quantum mechanics of photon. In particular, we examine the position wave functions determined by such a position basis, elucidate their relationship with the Riemann-Silberstein and Landau-Peierls wave functions, and give an explicit formula for the probability density of the spatial localization of the photon.

quant-ph↗
arXiv2017

Exact Solution of the Two-Dimensional Scattering Problem for a Class of $δ$-Function Potentials Supported on Subsets of a Line

We use the transfer matrix formulation of scattering theory in two-dimensions to treat the scattering problem for a potential of the form $v(x,y)=ζ\,δ(ax+by)g(bx-ay)$ where $ζ,a$, and $b$ are constants, $δ(x)$ is the Dirac $δ$ function, and $g$ is a real- or complex-valued function. We map this problem to that of $v(x,y)=ζ\,δ(x)g(y)$ and give its exact and analytic solution for the following choices of $g(y)$: i) A linear combination of $δ$-functions, in which case $v(x,y)$ is a finite linear array of two-dimensional $δ$-functions; ii) A linear combination of $e^{iα_n y}$ with $α_n$ real; iii) A general periodic function that has the form of a complex Fourier series. In particular we solve the scattering problem for a potential consisting of an infinite linear periodic array of two-dimensional $δ$-functions. We also prove a general theorem that gives a sufficient condition for different choices of $g(y)$ to produce the same scattering amplitude within specific ranges of values of the wavelength $λ$. For example, we show that for arbitrary real and complex parameters, $a$ and $\mathfrak{z}$, the potentials $ \mathfrak{z} \sum_{n=-\infty}^\inftyδ(x)δ(y-an)$ and $a^{-1}\mathfrak{z}δ(x)[1+2\cos(2πy/a)]$ have the same scattering amplitude for $a< λ\leq 2a$.

quant-ph↗
arXiv2017

Perfect Broadband Invisibility in Isotropic Media with Gain and Loss

We outline a method for constructing effectively two-dimensional isotropic optical media that are perfectly and omnidirectionally invisible for both TE and TM waves provided that their wavenumber does not exceed a preassigned value $α$. This relies on the observation that a complex scattering potential $v(x,y)$ displays perfect invisibility for wavenumbers $k\leqα$ provided that its Fourier transform with respect to $y$, which we denote by $\tilde v(x,\mathfrak{K}_y)$, vanishes for $\mathfrak{K}_y\leq 2α$. We use this result to construct the permittivity profile for a realistic optical slab that is perfectly invisible in the wavenumber window $[0,α]$.

physics.optics↗
arXiv2017

Nonlinear Spectral Singularities and Laser Output Intensity

The mathematical notion of spectral singularity admits a description in terms of purely outgoing solutions of a corresponding linear wave equation. This leads to a nonlinear generalization of this notion for nonlinearities that are confined in space. We examine the nonlinear spectral singularities in arbitrary TE and TM modes of a mirrorless slab laser that involves a weak Kerr nonlinearity. This provides a computational scheme for the determination of the laser output intensity $I$ for these modes. In particular, we offer an essentially mathematical derivation of the linear-dependence of $I$ on the gain coefficient $g$ and obtain an explicit analytic expression for its slope. This shows that if the real part $η$ of the refractive index of the slab does not exceed 3, there is a lower bound on $θ$ below which lasing in both its TE and TM modes requires $η$ to be shifted by a small amount as $g$ surpasses the threshold gain. Our results suggest that lasing in the oblique TM modes of the slab is forbidden if the incidence (emission) angle of the TM mode exceeds the Brewster's angle.

physics.optics↗
arXiv2016

Spectral Singularities in the TE and TM modes of a PT-Symmetric Slab System: Optimal conditions for realizing a CPA-Laser

Among the interesting outcomes of the study of the physical applications of spectral singularities in PT-symmetric optical systems is the discovery of CPA-lasers. These are devices that act both as a threshold laser and a coherent perfect absorber (CPA) for the same values of their physical parameters. Unlike a homogeneous slab that is made to act as a CPA, a slab CPA-laser would absorb the incident waves coming from the left and right of the device provided that they have appropriate intensity and phase contrasts. We provide a comprehensive study of one of the simplest experimentally accessible examples of a CPA-laser, namely a PT-symmetric optical slab system consisting of a balanced pair of adjacent or separated gain and loss components. In particular, we give a closed form expression describing the spectral singularities of the system which correspond to its CPA-laser configurations. We determine the intensity and phase contrasts for the TE and TM waves that are emitted (absorbed) whenever the slab acts as a laser (CPA). We also investigate the behavior of the time-averaged energy density and Poynting vector for these waves. This is necessary for determining the optimal values of the physical parameters of the system that make it act as a CPA-laser. These turn out to correspond to situations where the separation distance $s$ between the gain and loss layers is an odd multiple of a characteristic length scale $s_0$. A curious by-product of our study is that, except for the cases where $s$ is an even integer multiple of $s_0$, there is a critical angle of polarization beyond which the energy of the waves emitted from the lossy layer can be larger than the energy of those emitted from the gain layer.

physics.optics↗