Semi-infinite Line Research Articles

On the singularities of the discrete Korteweg–deVries equation

We study the structure of singularities in the discrete Korteweg–deVries (d-KdV) equation. Four different types of singularities are identified. The first type corresponds to localised, ‘confined’, singularities, the confinement constraints for which provide the integrability conditions for generalisations of d-KdV. Two other types of singularities are of infinite extent and consist of oblique lines of infinities, possibly alternating with lines of zeros. The fourth type of singularity corresponds to horizontal strips where the product of the values on vertically adjacent points is equal to 1. (A vertical version of this singularity with product equal to −1 on horizontally adjacent sites also exists). Due to its orientation this singularity can, in fact, interact with the other types. This leads to an extremely rich structure for the singularities of d-KdV, which is studied in detail in this paper. Given the important role played by the fourth type of singularity we decided to give it a special name: taishi (the origin of which is explained in the text). The taishi do not exist for nonintegrable extensions of d-KdV, which explains the relative paucity of singularity structures in the nonintegrable case: the second and third type of singularities that correspond to oblique lines still exist and the localised singularities of the integrable case now become unconfined, leading to semi-infinite lines of infinities alternating with zeros.

Two Dualities: Markov and Schur–Weyl

Abstract We show that quantum Schur–Weyl duality leads to Markov duality for a variety of asymmetric interacting particle systems. In particular, we consider the following three cases: (1) Using a Schur–Weyl duality between a two-parameter quantum group and a two-parameter Hecke algebra from [6], we recover the Markov self-duality of multi-species ASEP previously discovered in [23] and [3]. (2) From a Schur–Weyl duality between a co-ideal subalgebra of a quantum group and a Hecke algebra of type B [2], we find a Markov duality for a multi-species open ASEP on the semi-infinite line. The duality functional has not previously appeared in the literature. (3) A “fused” Hecke algebra from [15] leads to a new process, which we call braided ASEP. In braided ASEP, up to $m$ particles may occupy a site and up to $m$ particles may jump at a time. The Schur–Weyl duality between this Hecke algebra and a quantum group lead to a Markov duality. The duality function had previously appeared as the duality function of the multi-species ASEP$(q,m/2)$ [23] and the stochastic multi-species higher spin vertex model [24].

Homogeneous open quantum walks on the line: criteria for site recurrence and absorption

In this work, we study open quantum random walks, as described by S. Attal et al.. These objects are given in terms of completely positive maps acting on trace-class operators, leading to one of the simplest open quantum versions of the recurrence problem for classical, discrete-time random walks. This work focuses on obtaining criteria for site recurrence of nearest-neighbor, homogeneous walks on the integer line, with the description presented here making use of recent results of the theory of open walks, most particularly regarding reducibility properties of the operators involved. This allows us to obtain a complete criterion for site recurrence in the case for which the internal degree of freedom of each site (coin space) is of dimension 2. We also present the analogous result for irreducible walks with an internal degree of arbitrary finite dimension and the absorption problem for walks on the semi-infinite line.

Optimization and growth in first-passage resetting

We combine the processes of resetting and first passage, resulting in first-passage resetting, where the resetting of a random walk to a fixed position is triggered by the first-passage event of the walk itself. In an infinite domain, first-passage resetting of isotropic diffusion is non-stationary, and the number of resetting events grows with time according to . We analytically calculate the resulting spatial probability distribution of the particle, and also obtain the distribution by geometric-path decomposition. In a finite interval, we define an optimization problem that is controlled by first-passage resetting; this scenario is motivated by reliability theory. The goal is to operate a system close to its maximum capacity without experiencing too many breakdowns. However, when a breakdown occurs the system is reset to its minimal operating point. We define and optimize an objective function that maximizes reward for being close to the maximum level of operation and imposes a penalty for each breakdown. We also investigate extensions of this basic model, firstly to include a delay after each reset, and also to two dimensions. Finally, we study the growth dynamics of a domain in which the domain boundary recedes by a specified amount whenever the diffusing particle reaches the boundary, after which a resetting event occurs. We determine the growth rate of the domain for a semi-infinite line and a finite interval and find a wide range of behaviors that depend on how much recession occurs when the particle hits the boundary.

Landau-Zener-Stückelberg-Majorana interferometry of a superconducting qubit in front of a mirror

We investigate the Landau-Zener-Stuckelberg-Majorana interferometry of a superconducting qubit in a semi-infinite transmission line terminated by a mirror. The transmon-type qubit is at the node of the resonant electromagnetic (EM) field, hiding from the EM field. "Mirror, mirror" briefly describes this system, because the qubit acts as another mirror. We modulate the resonant frequency of the qubit by applying a sinusoidal flux pump. We probe the spectroscopy by measuring the reflection coefficient of a weak probe in the system. Remarkable interference patterns emerge in the spectrum, which can be interpreted as multi-photon resonances in the dressed qubit. Our calculations agree well with the experiments.

Run-and-tumble particle in inhomogeneous media in one dimension

We investigate the run-and-tumble particle (RTP), also known as persistent Brownian motion, in one dimension. A telegraphic noise σ(t) drives the particle which changes between ±1 values at certain rates. Denoting the rate of flip from 1 to −1 as R1 and the converse rate as R2, we consider the position- and direction-dependent rates of the form and with α ⩾ 0. For α = 0 and 1, we solve the master equations exactly for arbitrary γ1 and γ2 at large t. From our analytical expression for the time-dependent probability distribution P(x, t) we find that for γ1 > γ2 the distribution relaxes to a steady state exponentially, whereas for γ1 ⩽ γ2 the distribution does not reach a steady state and can be described by a non-trivial scaling form. We interestingly find that these features of the probability distribution P(x, t) in the two regimes γ1 > γ2 and γ1 ⩽ γ2 also remain valid for general α > 0. In particular, for general α, we argue and numerically demonstrate that the approach to the steady state in γ1 > γ2 case is exponential. On the other hand, for γ1 ⩽ γ2, the distribution P(x, t) remains time dependent and possesses certain scaling behavior. For γ1 = γ2 we derive the scaling behavior as well as the scaling function rigorously, whereas for γ1 < γ2 we provide heuristic arguments to obtain the scaling behavior and the corresponding scaling functions. We also study the dynamics on a semi-infinite line with an absorbing barrier at the origin. For α = 0 and 1, we analytically compute the survival probabilities and the corresponding first-passage time distributions. For general α ⩾ 0, we provide approximate calculations to compute the behavior of the survival probability for t → ∞ in which limit it approaches a finite value for γ1 < γ2 but goes to zero for γ ⩾ γ2. We also study the approach to the large t value in both cases. Finally, we consider RTP in a finite interval [0, M] and compute the associated exit probabilities from that interval for all α. All our analytic results are verified with a numerical simulation.

Observation of Broadband Entanglement in Microwave Radiation from a Single Time-Varying Boundary Condition.

Entangled pairs of microwave photons are commonly produced in the narrow frequency band of a resonator, which represents a modified vacuum density of states. We generate and investigate the entanglement of a stream of photon pairs, generated in a semi-infinite broadband transmission line, terminated by a superconducting quantum interference device (SQUID). A weak pump signal modulates the SQUID inductance, resulting in a single time-varying boundary condition, and we detect all four quadratures of the microwave radiation emitted at two different frequencies separated by 0.7GHz. Power calibration is done insitu, and we find positive logarithmic negativity and two-mode squeezing below the vacuum in the observed radiation, indicating entanglement.

Semiclassical analysis of dark-state transient dynamics in waveguide circuit QED

The interaction between superconducting qubits and one-dimensional microwave transmission lines has been studied experimentally and theoretically in the past two decades. In this work, we investigate the spontaneous emission of an initially excited artificial atom which is capacitively coupled to a semi-infinite transmission line, shorted at one end. This configuration can be viewed as an atom in front of a mirror. The distance between the atom and the mirror introduces a time delay in the system, which we take into account fully. When the delay time equals an integer number of atom oscillation periods, the atom converges into a dark state after an initial decay period. The dark state is an effect of destructive interference between the reflected part of the field and the part directly emitted by the atom. Based on circuit quantization, we derive linearized equations of motion for the system and use these for a semiclassical analysis of the transient dynamics. We also make a rigorous connection to the quantum optics system-reservoir approach and compare these two methods to describe the dynamics. We find that both approaches are equivalent for transmission lines with a low characteristic impedance, while they differ when this impedance is higher than the typical impedance of the superconducting artificial atom.

On entanglement Hamiltonians of an interval in massless harmonic chains

We study the continuum limit of the entanglement Hamiltonians of a block of consecutive sites in massless harmonic chains. This block is either in the chain on the infinite line or at the beginning of a chain on the semi-infinite line with Dirichlet boundary conditions imposed at its origin. The entanglement Hamiltonians of the interval predicted by conformal field theory (CFT) for the massless scalar field are obtained in the continuum limit. We also study the corresponding entanglement spectra, and the numerical results for the ratios of the gaps are compatible with the operator content of the boundary CFT of a massless scalar field with Neumann boundary conditions imposed along the boundaries introduced around the entangling points by the regularisation procedure.

Entanglement Hamiltonians in 1D free lattice models after a global quantum quench

We study the temporal evolution of the entanglement Hamiltonian of an interval after a global quantum quench in free lattice models in one spatial dimension. In a harmonic chain we explore a quench of the frequency parameter. In a chain of free fermions at half filling we consider the evolution of the ground state of a fully dimerised chain through the homogeneous Hamiltonian. We focus on critical evolution Hamiltonians. The temporal evolutions of the gaps in the entanglement spectrum are analysed. The entanglement Hamiltonians in these models are characterised by matrices that provide also contours for the entanglement entropies. The temporal evolution of these contours for the entanglement entropy is studied, also by employing existing conformal field theory results for the semi-infinite line and the quasi-particle picture for the global quench.

Radiation from free space TEM transmission lines

This work derives exact expressions for the radiation from two conductors non-isolated transverse electromagnetic (TEM) transmission lines of any small electric size cross-section in free space. The cases of infinite, semi infinite and finite transmission lines are covered and it is shown that while an infinite transmission line does not radiate, there is a smooth transition between the radiation from a finite to a semi-infinite transmission line. The present analysis is in the frequency domain and the authors consider transmission lines carrying any combination of forward and backward waves. The analytic results are validated by successful comparison with ANSYS commercial software simulation results, and successful comparisons with other published results.

Chaos in a linear wave equation

A linear partial differential equation is shown to exhibit three properties often used to define chaotic dynamics. The system comprises a one-dimensional wave equation with gain that operates on a semi-infinite line. A boundary condition enforces that the waves remain finite. It is shown that the resulting solution set is dense with periodic orbits, contains transitive orbits, and exhibits extreme sensitivity to initial conditions. Definitions of chaos are considered in light of such linear chaos.

Equivalence between quantum backflow and classically forbidden probability flow in a diffraction-in-time problem

Quantum backflow is an interference effect in which a matter-wave packet comprised of only plane waves with non-negative momenta exhibits negative probability flux. Here we show that this effect is mathematically equivalent to the appearance of classically-forbidden probability flux when a matter-wave packet, initially confined to a semi-infinite line, expands in free space.

Beachcombing on strips and islands

A group of mobile robots (beachcombers) have to search collectively every point of a given domain. At any given moment, each robot can be in walking mode or in searching mode. It is assumed that each robot's maximum allowed searching speed is strictly smaller than its maximum allowed walking speed. A point of the domain is searched if at least one of the robots visits it in searching mode. The Beachcombers' Problem consists in developing efficient schedules (algorithms) for the robots which collectively search all the points of the given domain as fast as possible. We consider searching schedules in the following one-dimensional geometric domains: the cycle of a known circumference L, the finite straight line segment of a known length L, and the semi-infinite line [0,+∞).We first consider the online Beachcombers' Problem (i.e. the scenario when the robots do not know in advance the length of the segment to be searched), where the robots are initially collocated at the origin of a semi-infinite line. It is sought to design a schedule A with maximum speed S, defined as S=infℓ⁡ℓtA(ℓ), where tA(ℓ) denotes the time when the search of the segment [0,ℓ] is completed under A. We consider a discrete and a continuous version of the problem, depending on whether the infimum is taken over ℓ∈N⁎ or ℓ≥1. We prove that the LeapFrog algorithm, which was proposed in Czyzowicz et al. (2015) [12], is in fact optimal in the discrete case. This settles in the affirmative a conjecture from that paper. We also show how to extend this result to the more general continuous online setting.For the offline version of the Beachcombers' Problem (i.e. the scenario when the robots know in advance the length of the segment to be searched), we consider the t-source Beachcombers' Problem (i.e. all robots start from a fixed number t≥1 of starting positions) on the cycle and on the finite segment. For the t-source Beachcombers' Problem on the cycle, we show that the structure of the optimal solutions is identical to the structure of the optimal solutions to the 2t-source Beachcombers' Problem on a finite segment. In consequence, by using results from Czyzowicz et al. (2014) [13], we prove that the 1-source Beachcombers' Problem on the cycle is NP-hard, and we derive approximation algorithms for the problem. For the t-source variant of the Beachcombers' Problem on the cycle and on the finite segment, we also derive efficient approximation algorithms.One important contribution of our work is that, in all variants of the offline Beachcombers' Problem that we discuss, we allow the robots to change direction of movement and search points of the domain on both sides of their respective starting positions. This represents a significant generalization compared to the model considered in Czyzowicz et al. (2014) [13], in which each robot had a fixed direction of movement that was specified as part of the solution to the problem. We manage to prove that changes of direction do not help the robots achieve optimality.

Resonance states completeness for relativistic particle on a sphere with two semi-infinite lines attached

The paper is devoted to resonances playing an important role in direct and inverse scattering problems. A model of a relativistic particle on hybrid manifold consisting of a sphere with two semi-infinite wires attached is considered. The model is based on the theory of self-adjoint extensions of symmetric operators. Completeness of resonance states in the space of square integrable functions on the sphere is proved. The proof uses the relation between the completeness and the factorization of the characteristic function in Sz.-Nagy functional model.

Response of bimaterials with interfacial tension subjected to axisymmetric body force

This paper presents an analytical treatment of the boundary value problem of an isotropic elastic bimaterial full space with interfacial tension subjected to an axisymmetric body force. The interfacial tension is modeled as the residual stress in the pre-tensed interfacial atom membrane on the basis of Gurtin and Murdoch’s surface theory. The generality of the present model is justified with several degenerated cases (e.g. classical bimaterials with bonded and unsinkable interface, homogeneous full space with interfacial tension, and half space with surface tension). By virtue of classical Hankel or Fourier transforms, the unconventional boundary value problems are addressed in a concise and systematic manner. The final solutions are all given in the form of a semi-infinite line integral. Particularly, the solutions for the interfacial response induced by interfacial point load can be obtained in exact closed form in terms of Struve and Bessel functions. Numerical results are presented to show the influence of interfacial tension on the induced responses. It is shown that the existence of interfacial tension can significantly influence the induced elastic fields and reduce the point load singularity. The results presented in this paper are useful and meaningful to the studies of nanocomposites, soft solids, and biological tissue, where the effects of surface/interface tension could be dominant.

Coupled horizontal and rocking vibrations of a rigid circular disc on a transversely isotropic and layered half-space with imperfect interfaces

A new semi-analytical method is developed for solving the coupled horizontal and rocking vibrations of a rigid circular disc on the surface of a transversely isotropic multilayered half-space with imperfect interfaces. By virtue of the dual variable and position method, the recursive relation among different layers is established so that the final expansion coefficients of the displacement and traction vectors corresponding to the surface loading can be expressed in terms of the cylindrical system of vector functions. The physical-domain response by the surface loading (i.e., the forward solution) is obtained by carrying out the semi-infinite line integral involved. To solve for the corresponding coupled interaction problem between the surface rigid disc and layered elastic medium, the loading disc area is discretized into annular load-rings containing basic horizontal constant and vertical linear loads with unknown weights. These weights are then determined by applying an integral least-square approach with a Lagrange multiplier. Finally, the coupled time-harmonic compliances are derived by virtue of the equilibrium of the applied load on the disc and the reaction force on the half-space side. Selected numerical examples on the horizontal, moment and coupling compliances are presented to study the effect of material layering, anisotropy, interface behavior as well as input frequency. The coupling compliance is investigated in detail for the first time as compared to the horizontal and moment ones. The fully coupled, semi-coupled and decoupled schemes are also discussed with numerical examples so that one can easily access the relative accuracy among them when an actual layered structure is under both horizontal and rocking vibrations. This is important since the coupling compliance could significantly contribute to the vibrational behavior of the layered system at certain frequency.

Entanglement Hamiltonian evolution during thermalization in conformal field theory

In this work, we study the time evolution of the entanglement Hamiltonian during the process of thermalization in a (1+1)-dimensional conformal field theory (CFT) after a quantum quench from a special class of initial states. In particular, we focus on a subsystem which is a finite interval at the end of a semi-infinite line. Based on conformal mappings, the exact forms of both entanglement Hamiltonian and entanglement spectrum of the subsystem can be obtained. Aside from various interesting features, it is found that in the infinite time limit the entanglement Hamiltonian and entanglement spectrum are exactly the same as those in the thermal ensemble. The entanglement spectrum approaches the steady state spectrum exponentially in time. We also study the modular flows generated by the entanglement Hamiltonian in Minkowski spacetime, which provides us with an intuitive picture of how the entanglement propagates and how the subsystem is thermalized. Furthermore, the effect of a generic initial state is also discussed.

High thermoelectric performance of one graphene nanoribbon with line defects

We investigate the quantum transport through zigzag graphene nanoribbons with embedded “5-7-5”-edge line defects, by means of the non-equilibrium Green's function technique. It is found that when two semi-infinite line defects exist in the nanoribbon, notable Fano antiresonance takes place in the quantum transport process, which enables to drive the apparent thermoelectric effect. We propose this structure to be a promising candidate for improving the thermoelectric efficiency based on graphene nanoribbons.

Dirac quantisation condition: a comprehensive review

ABSTRACTIn most introductory courses on electrodynamics, one is taught the electric charge is quantised but no theoretical explanation related to this law of nature is offered. Such an explanation is postponed to graduate courses on electrodynamics, quantum mechanics and quantum field theory, where the famous Dirac quantisation condition is introduced, which states that a single magnetic monopole in the Universe would explain the electric charge quantisation. Even when this condition assumes the existence of a not-yet-detected magnetic monopole, it provides the most accepted explanation for the observed quantisation of the electric charge. However, the usual derivation of the Dirac quantisation condition involves the subtle concept of an ‘unobservable’ semi-infinite magnetised line, the so-called ‘Dirac string,’ which may be difficult to grasp in a first view of the subject. The purpose of this review is to survey the concepts underlying the Dirac quantisation condition, in a way that may be accessible to advanced undergraduate and graduate students. Some of the discussed concepts are gauge invariance, singular potentials, single-valuedness of the wave function, undetectability of the Dirac string and quantisation of the electromagnetic angular momentum. Five quantum-mechanical and three semi-classical derivations of the Dirac quantisation condition are reviewed. In addition, a simple derivation of this condition involving heuristic and formal arguments is presented.