Scattering Matrix Formalism Research Articles

Setting benchmarks for modelling gas\u2013surface interactions using coherent control of rotational orientation states

The coherent evolution of a molecular quantum state during a molecule-surface collision is a detailed descriptor of the interaction potential which was so far inaccessible to measurements. Here we use a magnetically controlled molecular beam technique to study the collision of rotationally oriented ground state hydrogen molecules with a lithium fluoride surface. The coherent control nature of the technique allows us to measure the changes in the complex amplitudes of the rotational projection quantum states, and express them using a scattering matrix formalism. The quantum state-to-state transition probabilities we extract reveal a strong dependency of the molecule-surface interaction on the rotational orientation of the molecules, and a remarkably high probability of the collision flipping the rotational orientation. The scattering matrix we obtain from the experimental data delivers an ultra-sensitive benchmark for theory to reproduce, guiding the development of accurate theoretical models for the interaction of H2 with a solid surface.

Modeling long imperfect SNS junctions and Andreev bound states using two impurities and the T -matrix formalism

We provide a new analytical tool to calculate the energies of Andreev bound states (ABS) in long imperfect SNS junctions, at present these can only be described by numerical tools. We model an NS junction as a delta-function "Andreev" impurity, i.e., a localized potential which scatters an electron into a hole with opposite spin. We show using the scattering matrix formalism that, quite surprisingly, an "Andreev" impurity is equivalent to an NS junction characterized by both Andreev reflection and a finite amount of normal scattering. The ABS energies are then calculated using the T-matrix formalism applied to a system with two Andreev impurities. Our results lie between those for a perfect long SNS junction limit described by the Andreev approximation (ABS energies depend linearly on the phase and are independent of the chemical potential) and the particle-in-the-box limit (bound state energies are independent of the phase and have a linear dependence on the chemical potential). Moreover, we recover a closed-form expression for the ABS energies by expanding around the particle-in-the-box limit.

Hitting a ball on a spring: a simple model for understanding decoherence with wavefunctions

The typical approach to studying decoherence begins by examining a combined system and interacting environment (termed bath), and then deriving a master equation to examine the behaviour of the density matrix after tracing over the bath, typically by invoking the Markovian approximation. This approach is quite successful for a large variety of systems however is frequently a non-intuitive picture that can mask the fundamental behaviour of the system. We examine here a simple model that captures the essence of decoherence and study it using the wavefunction picture, invoking the analytic concepts of the correspondence principle, the method of images, the phase jitter concept, and the scattering matrix formalism. We complement this with a numerical study using the split-operator FFT method, the forced harmonic oscillator method, and the thawed Gaussian approximation.

Unidirectional reflectionless transmission for two-dimensional PT -symmetric periodic structures

Unidirectional reflectionless propagation (or transmission) is an interesting wave phenomenon observed in many $\mathcal{PT}$-symmetric optical structures. Theoretical studies on unidirectional reflectionless transmission often use simple coupled-mode models. The coupled-mode theory can reveal the most important physical mechanism for this wave phenomenon, but it is only an approximate theory, and it does not provide accurate quantitative predictions with respect to geometric and material parameters of the structure. In this paper, we rigorously study unidirectional reflectionless transmission for two-dimensional (2D) $\mathcal{PT}$-symmetric periodic structures sandwiched between two homogeneous media. Using a scattering matrix formalism and a perturbation method, we show that real zero-reflection frequencies are robust under $\mathcal{PT}$-symmetric perturbations, and unidirectional reflectionless transmission is guaranteed to occur if the perturbation (of the dielectric function) satisfies a simple condition. Numerical examples are presented to validate the analytical results, and to demonstrate unidirectional invisibility by tuning the amplitude of balanced gain and loss.

Secondary Quantization of the Electromagnetic Field in Inhomogeneous Structures Based on Formalism of Scattering Matrix

Secondary Quantization of the Electromagnetic Field in Inhomogeneous Structures Based on Formalism of Scattering Matrix

Spin transport and Spin Tunnelling Magneto-Resistance (STMR) of F[formula omitted]NCSC[formula omitted]F spin valve

In this work, we study the spin transport at the Ferromagnet|Noncentrosymmetric Superconductor (F|NCSC) junction of a Ferromagnet|Noncentrosymmetric Superconductor|Ferromagnet (F|NCSC|F) spin valve. We investigate the Tunnelling Spin-Conductance (TSC), spin current and Spin Tunnelling Magneto-Resistance (STMR), and their dependence on various important parameters like Rashba Spin-Orbit Coupling (RSOC), strength and orientation of magnetization, an external in-plane magnetic field, barrier strength and a significant Fermi Wavevector Mismatch (FWM) at the ferromagnetic and superconducting regions. The study has been carried out for different singlet-triplet mixing of the NCSC gap parameter. We develop Bogoliubov-de Gennes (BdG) Hamiltonian and use the extended Blonder–Tinkham–Klapwijk (BTK) approach along with the scattering matrix formalism to calculate the scattering coefficients. Our results strongly suggest that the TSC is highly dependent on RSOC, magnetization strength and its orientation, and singlet-triplet mixing of the gap parameter. It is observed that NCSC with moderate RSOC shows maximum conductance for a partially opaque barrier in presence of low external magnetic field. For a strongly opaque barrier and a nearly transparent barrier a moderate value and a low value of field respectively are found to be suitable. Moreover, NCSC with large singlet component is appeared to be useful. In addition, for NCSC with large RSOC and low magnetization strength, a giant STMR (%) is observed. We have also seen that the spin current is strongly magnetization orientation dependent. With the increase in bias voltage spin current increases in transverse direction, but the component along the direction of flow is almost independent.

Analytical design of flat-top transmission filters composed of several resonant structures.

Resonant properties of composite structures consisting of several identical resonant structures (e.g. multilayer thin-film structures or guided-mode resonance gratings) separated by phase-shift layers are investigated theoretically. Using the scattering matrix formalism, we analytically demonstrate that, at properly chosen thicknesses of the phase-shift layers, the composite structures comprising two or four resonant diffractive structures with a Lorentzian transmittance profile optically implement the Butterworth filters of the order two or three, respectively, and enable achieving flat-top transmission spectra with steep slopes and low sidebands. In addition, we show that the composite structures consisting of three or four second-order Butterworth filters can accurately approximate the fourth- or fifth-order Butterworth filters, respectively. The presented theoretical results are confirmed by rigorous numerical simulations of composite structures consisting of the so-called W-structures (simple three-layer resonant structures comprising a high-index core layer and two low-index cladding layers in a high-index dielectric environment). The simulation results confirm the formation of flat-top transmittance peaks, the shape of which fully agrees with the derived theoretical description. Moreover, we demonstrate an exceptionally simple mechanism of controlling the transmittance peak width, which consists in changing the thicknesses of the cladding layers of the initial W-structure and enables generating flat-top transmission peaks with a significantly subnanometer width.

Resonant properties of composite structures consisting of several resonant diffraction gratings.

We theoretically and numerically investigate resonant optical properties of composite structures consisting of several subwavelength resonant diffraction gratings separated by homogeneous layers. Using the scattering matrix formalism, we demonstrate that the composite structure comprising N gratings has a multiple transmittance zero of the order N. We show that at the distance between the gratings satisfying the Fabry-Pérot resonance condition, an (N - 1)-degenerate bound state in the continuum (BIC) is formed. The results of rigorous numerical simulations fully confirm the theoretically predicted formation of multiple zeros and BICs in the composite structures. Near the BICs, an effect very similar to the electromagnetically induced transparency is observed. We show that by making the proper choice of the thicknesses of the layers separating the gratings, nearly rectangular reflectance or transmittance peaks with steep slopes and virtually no sidelobes can be obtained. In particular, one of the presented examples demonstrates the possibility of obtaining an approximately rectangular transmittance peak with a significantly subnanometer width. The presented results may find application in the design of optical filters, sensors and devices for optical differentiation and transformation of optical signals.

Effect of Rashba spin–orbit coupling, magnetization, and mixing of gap parameter on tunneling conductance in F∣NCSC junction of an F∣NCSC∣F spin valve

In this paper, we study the quantum transport at the Ferromagnet∣Noncentrosymmetric Superconductor (F∣NCSC) interface of an F∣NCSC∣F spin valve. In this context, we investigate the tunneling conductance and its dependence on Rashba spin–orbit coupling (RSOC) considering different barrier strength and a significant Fermi wavevector mismatch (FWM) at the ferromagnetic and superconducting regions. The study is carried out for different magnetization orientations and strengths. We develop a Bogoliubov–de Gennes (BdG) Hamiltonian, introducing RSOC and exchange interaction for such a hybrid structure. To study charge conductance, we use an extended Blonder–Tinkham–Klapwijk approach along with scattering matrix formalism to calculate the scattering coefficients. Our results strongly suggest that the tunneling conductance is strongly dependent on RSOC, magnetization strength, its orientation, and the FWM. This work has also been done for different singlet–triplet mixings of the gap parameter. We have observed that with the rise of singlet–triplet mixing ratio, the conductance decreases. It is also observed that a transparent barrier with moderate RSOC and magnetization strength with arbitrary orientation is highly suitable for maximum conductance.

Photon Blockade in Weakly Driven Cavity Quantum Electrodynamics Systems with Many Emitters.

We use the scattering matrix formalism to analyze photon blockade in coherently driven cavity quantum electrodynamics systems with a weak drive. By approximating the weak coherent drive by an input single- and two-photon Fock state, we reduce the computational complexity of the transmission and the two-photon correlation function from exponential to polynomial in the number of emitters. This enables us to easily analyze cavity-based systems containing ∼50 quantum emitters with modest computational resources. Using this approach we study the coherence statistics of photon blockade while increasing the number of emitters for resonant and detuned multiemitter cavity quantum electrodynamics systems-we find that increasing the number of emitters worsens photon blockade in resonant systems, and improves it in detuned systems. We also analyze the impact of inhomogeneous broadening in the emitter frequencies on the photon blockade through this system.

Nonlinear chiral refrigerators

We investigate a mesoscopic refrigerator based on chiral quantum Hall edge channels. We discuss a three-terminal cooling device in which charge transport occurs between a pair of voltage-biased terminals only. The third terminal, which is to be cooled, is set as a voltage probe with vanishing particle flux. This largely prevents the generation of direct Joule heating which ensures a high coefficient of performance. Cooling operation is based on energy-dependent quantum transmissions. The latter are implemented with the aid of two tunable scattering resonances (quantum dots). To find the optimal performance point and the largest temperature difference created with our refrigerator, it is crucial to address the nonlinear regime of transport, accounting for electron-electron interaction effects. Our numerical simulations show that the maximal cooling power can be tuned with the quantum dot couplings and energy levels. Further, we provide analytical expressions within a weakly nonlinear scattering-matrix formalism which allow us to discuss the conditions for optimal cooling in terms of generalized thermopowers. Our results are important for the assessment of chiral conductors as promising candidates for efficient quantum refrigerators with low dissipation.

Electron waiting times in hybrid junctions with topological superconductors

We investigate the waiting time distributions (WTDs) of superconducting hybrid junctions, considering both conventional and topologically nontrivial superconductors hosting Majorana bound states at their edges. To this end, we employ a scattering matrix formalism that allows us to evaluate the waiting times between the transmissions and reflections of electrons or holes. Specifically, we analyze normal-metal–superconductor (NIS) junctions and NISIN junctions, where Cooper pairs are spatially split into different leads. The distribution of waiting times is sensitive to the simultaneous reflection of electrons and holes, which is enhanced by the zero-energy state in topological superconductors. For the NISIN junctions, the WTDs of trivial superconductors feature a sharp dependence on the applied voltage, while for topological ones they are mostly independent of it. This particular voltage dependence is again connected to the presence of topological edge states, showing that WTDs are a promising tool for identifying topological superconductivity.

Memory Efficient Method for Electromagnetic Multi-Region Models Using Scattering Matrices

This paper describes the scattering matrix approach to obtain the solution to electromagnetic field quantities in harmonic multi-layer models. Using this approach, the boundary conditions are solved in such way that the maximum size of any matrix used during the computations is independent of the number of regions defined in the problem. As a result, the method is more memory efficient than classical methods used to solve the boundary conditions. Because electromagnetic sources can be located inside the regions of a configuration, the scattering matrix formulation is developed to incorporate these sources into the solving process. The method is applied to a 3D electromagnetic configuration for verification.

Majorana Fermions in One-Dimensional Structures at LaAlO3/SrTiO3 Oxide Interfaces

We study one-dimensional structures that may be formed at the LaAlO 3 /SrTiO 3 oxide interface by suitable top gating. These structures are modeled via a single-band model with Rashba spin-orbit coupling, superconductivity and a magnetic field along the one-dimensional chain. We first discuss the conditions for the occurrence of a topological superconducting phase and the related formation of Majorana fermions at the chain endpoints, highlighting a close similarity between this model and the Kitaev model, which also reflects in a similar condition the formation of a topological phase. Solving the model in real space, we also study the spatial extension of the wave function of the Majorana fermions and how this increases with approaching the limit condition for the topological state. Using a scattering matrix formalism, we investigate the stability of the Majorana fermions in the presence of disorder and discuss the evolution of the topological phase with increasing disorder.

Evolution of the transmission phase through a Coulomb-blockaded Majorana wire

We present a study of the transmission of electrons through a semiconductor quantum wire with strong spin-orbit coupling in proximity to an s-wave superconductor, which is Coulomb-blockaded. Such a system supports Majorana zero modes in the presence of an external magnetic field. Without superconductivity, phase lapses are expected to occur in the transmission phase, and we find that they disappear when a topological superconducting phase is stabilized. We express tunneling through the nanowire with the help of effective matrix elements, which depend on both the fermion parity of the wire and the overlap with Bogoliubov-de-Gennes wave functions. Using a modified scattering matrix formalism, that allows for including electron-electron interactions, we study the transmission phase in different regimes.

Optical properties of one dimensional metal-air graded system

Using the scattering matrix formalism, in this work it is presented the numerical study of the optical properties of 1D photonic graded systems, constructed by multiple bi-layers of Drude-type metal and air, where the total amount of material, as well as the length proportion of them in each bi-layer are kept constant. A graded variation of the bi-layers length, following an arithmetic progression defined in terms of a variable δ is considered. It is shown the formation of a transmission band that grows with the number of bi-layers and how the increase of the metallic plasma frequency or the size of the system modifies the transmission spectra, resembling the response of different photonic crystals. For the transverse electric (TE) and magnetic (TM) modes, it is presented the redistribution of the transmission peaks to higher frequencies as a function of the incident angle and δ. For TM mode, it is observed the appearance of maximum transmission peaks below the metallic plasma frequency, in an equal number as the air layers of the system, whose widths decrease with the augment of the incident angle. It is observed the variation of the transmission peaks with the increase of δ, which are shifted to higher frequencies, due to the material distribution which is mainly located on one side of the system. Additionally an analytical approach for graded structures is developed.

Quantization of the Electromagnetic Field in Three-Dimensional Photonic Structures on the Basis of the Scattering Matrix Formalism (S Quantization)

A technique for quantization of the electromagnetic field in photonic nanostructures with three-dimensional modulation of the dielectric constant is developed on the basis of the scattering matrix formalism (S quantization in the three-dimensional case). Quantization is based on equating the eigenvalues of the scattering matrix to unity, which is equivalent to equating each other the sets of Fourier expansions for the fields of the waves incident on the structure and propagating away from the structure. The spatial distribution of electromagnetic fields of the modes in a photonic nanostructure is calculated on the basis of the R and T matrices describing the reflection and transmission of the Fourier components by the structure. To calculate the reflection and transmission coefficients of individual real-space and Fourier-space components, the structure is divided into parallel layers within which the dielectric constant varies as a function of two-dimensional coordinates. Using the Fourier transform, Maxwell’s equations are written in the form of a matrix connecting the Fourier components of the electric field at the boundaries of neighboring layers. Based on the calculated reflection and transmission vectors for all polarizations and Fourier components, the scattering matrix for the entire structure is formed and quantization is carried out by equating the eigenvalues of the scattering matrix to unity. The developed method makes it possible to obtain the spatial profiles of eigenmodes without solving a system of nonlinear integro-differential equations and significantly reduces the computational resources required for calculating the probability of spontaneous emission in three-dimensional systems.

Numerical emulation of Thru-Reflection-Line calibration for the de-embedding of Surface Acoustic Wave devices

In this contribution, a rigorous numerical calibration is proposed to characterize the excitation of propagating mechanical waves by interdigitated transducers (IDTs). The transition from IDT terminals to phonon waveguides is modeled by means of a general circuit representation that makes use of Scattering Matrix (SM) formalism. In particular, the three-step calibration approach called the Thru-Reflection-Line (TRL), that is a well-established technique in microwave engineering, has been successfully applied to emulate typical experimental conditions. The proposed procedure is suitable for the synthesis/optimization of surface-acoustic-wave (SAW) based devices: the TRL calibration allows to extract/de-embed the acoustic component, namely resonator or filter, from the outer IDT structure, regardless of complexity and size of the letter. We report, as a result, the hybrid scattering parameters of the IDT transition to a mechanical waveguide formed by a phononic crystal patterned on a piezoelectric AlN membrane, where the effect of a discontinuity from periodic to uniform mechanical waveguide is also characterized. In addition, to ensure the correctness of our numerical calculations, the proposed method has been validated by independent calculations.

Tunable χ / P T $\chi /\mathcal {P}\mathcal {T}$ Symmetry in Noisy Graphene

We investigate the resonant regime of a mesoscopic cavity made of graphene or a doped beam splitter. Using Non-Hermitian Quantum Mechanics, we consider the Bender-Boettcher assumption that a system must obey parity and time reversal symmetry. Therefore, we describe such system by coupling chirality, parity, and time reversal symmetries through the scattering matrix formalism and apply it in the shot noise functions, also derived here. Finally, we show how to achieve the resonant regime only by setting properly the parameters concerning the chirality and the PT symmetry.

MESH: A free electromagnetic solver for far-field and near-field radiative heat transfer for layered periodic structures

We describe MESH (M ultilayer E lectromagnetic S olver for H eat transfer), a free software that combines rigorous coupled wave analysis (RCWA) and scattering matrix formalism to simulate the radiative heat transfer both in the near-field and far-field regimes for layered three-dimensional structures made of planar layers. Each layer can have in-plane one-dimensional or two-dimensional periodicity. In this paper, we provide detailed discussions of the algorithms of MESH, which enables it to be a flexible tool for different types of radiative heat transfer simulations. We also discuss aspects of the codes related to parallelization and user scripting. Program summaryProgram Title: MESHProgram Files doi:http://dx.doi.org/10.17632/zx9v3bf3hf.1Licensing provisions: GNU General Public License 3 (GPL)Programming language: C, C++.External routines: Lua[1], Python[2] and LAPACK and BLAS linear-algebra software[3], and optionally MPI message-passing interface[4]. Armadillo[5] is already contained in MESH.Nature of problem: Far-field and near-field radiative heat transfer in layered periodic structures.Solution method: Fourier modal method (rigorous coupled wave analysis) and the scattering matrix method.[1] R. Ierusalimschy, L.H. de Figueiredo, W.C. Filho, Lua an extensible extension language, Software: Practice and Experience 26 (1996) 635652. http://www.lua.org.[2] Python Software Foundation. Available at http://www.python.org[3] MKL: https://software.intel.com/en-us/intel-mkl[4] T.M. Forum, MPI: A Message Passing Interface, in: Supercomputing 93, Portland, OR, 878883, 1993[5] Conrad Sanderson and Ryan Curtin. Armadillo: a template-based C++ library for linear algebra. Journal of Open Source Software, Vol. 1, pp. 26, 2016. http://dx.doi.org/10.21105/joss.00026