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Physics-Uspekhi

Reduction of the scattering matrix array

Ivan Sadovskyy

Published Sep 7, 2015• 1 citation
Publisher

Abstract

The scattering matrix approach is widely applied in wave engineering and quantum physics. Usually, a combination of multiple scattering matrices is used. In this article, we consider arbitrary arrays of interconnected scattering matrices and present a formal result for the reduced scattering matrix. We demonstrate this approach in two well-known scattering problems.

DOI
10.3367/ufne.0185.201509c.0941

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The scattering matrix approach is widely applied in wave engineering and quantum physics. Usually, a combination of multiple scattering matrices is used. In this article, we consider arbitrary arrays of interconnected scattering matrices and present a formal result for the reduced scattering matrix. We demonstrate this approach in two well-known scattering problems.
It is well known that the scattering matrix plays an important role in quantum optics. This matrix converts the incoming characteristics of scattered radiation into output characteristics. Currently, only special cases of such a scattering matrix are known, which are determined by a specific problem. In this work, a general form of the scattering matrix is obtained, which can be applied to a wide range of problems. It is shown that previously well-known and widely used scattering matrices can be obtained from the resulting scattering matrix. The results obtained can be used to describe new quantum states, the scattering matrices of which have not yet been studied.
Scattering theory provides a convenient framework for the solution of a variety of problems. In this thesis we focus on the combination of boundary conditions and scattering potentials and the combination of non-overlapping scattering potentials within the context of scattering theory. Using a scattering t-matrix approach, we derive a useful relationship between the scattering t-matrix of the scattering potential and the Green function of the boundary, and the t-matrix of the combined system, effectively renormalizing the scattering t-matrix to account for the boundaries. In the case of the combination of scattering potentials, the combination of t-matrix operators is achieved via multiple scattering theory. We also derive methods, primarily for numerical use, for finding the Green function of arbitrarily shaped boundaries of various sorts. These methods can be applied to both open and closed systems. In this thesis, we consider single and multiple scatterers in two dimensional strips ...
Subwavelength atomic arrays are a leading platform for engineering light-matter interactions, enabling near-perfect single-photon mirrors and robust quantum memories based on long-lived dark spin waves. However, a comprehensive theory of their nonlinear, multi-excitation dynamics has remained a significant challenge. We present a unified quantum scattering theory that treats both photons and collective atomic spin waves as distinct propagating excitations interacting across different spatial dimensions. Our central result is a powerful analytical reduction: we demonstrate that the complete multi-channel S-matrix and the associated scattering cross sections are exactly determined by the effective scattering dynamics solely within the atomic spin wave subspace. This maps the complex physical problem of photon-atom interactions to a conceptually simpler one involving only atomic modes. We apply this formalism to the two-excitation case, deriving the complete analytical S-matrix and scattering cross sections for systems with two-level nonlinearities. Our work provides a versatile analytic tool for analyzing and engineering complex quantum nonlinear phenomena, including multi-excitation subradiance, in large-scale atomic systems.
This thesis concerns the theoretical investigation of interference phenomena related to elastic and inelastic scattering of quantized light. The presented work is naturally divided into two parts, the first is concerning elastic scattering while the second explores inelastic scattering. In the first part we use a scattering-matrix formalism combined with results from random-matrix theory to investigate the interference of quantum optical states on a multiple scattering medium. We investigate a single realization of a scattering medium thereby showing that it is possible to create entangled states by interference of squeezed beams. Mixing photon states on the single realization also shows that quantum interference naturally arises by interfering quantum states. We further investigate the ensemble averaged transmission properties of the quantized light and see that the induced quantum interference survives even after disorder averaging. The quantum interference manifests itself through increased photon correlations. Furthermore, the theoretical description of a measurement procedure is presented. In this work we relate the noise power spectrum of the total transmitted or reflected light to the photon correlations after ensemble averaging. This analysis enables us to describe an experimental observation of the quantum nature of light that survives the averaging over disorder. In the second part we investigate inelastic scattering. This we do by first treating the scattering of light on dipoles embedded in an arbitrary dielectric environment. By considering the two different models for dipole interaction known as the minimal-coupling and electric-dipole interaction Hamiltonians, we find exact relations between the electric field and the dipole operators in the Heisenberg picture, while keeping the model of the dipoles arbitrary. Due to the exact treatment of the electric-field operators, we obtain kernels known from classical scattering theory to describe the propagation of the field from the dipoles. Using the found electric field operators we derive the Heisenberg equations of motion for the dipoles while treating them as quantum two-level systems and using the Born–Markov and rotating-wave approximations. Postponing the rotating-wave approximation to the very end of the formal calculations allows us to identify the different physical parameters of the dipole evolution in terms of physical quantities known from optics. Finally, we use our Heisenberg picture formalism to treat a dilute cloud of two-level atoms, by simplifying the equations of motion using a single-scattering approximation for the interaction between the atoms. This enables us to derive expressions for the steady-state population and fluorescence spectrum, where we find cooperative effects in both the elastic and the inelastic spectra.
The aim of the lecture is to briefly describe the mathematical background of scattering theory for two- and three-particle quantum systems. We discuss basic objects of the theory: wave and scattering operators and the corresponding scattering matrix and illustrate them on the example of the Schrödinger equation. Our goal is to present time-dependent and stationary approaches and to describe the underlying mathematical methods. We also give a sketch of scattering theory for three interacting quantum particles including a difficult problem of the asymptotic completeness of scattering channels. Along with traditional results, we discuss new scattering channels arising for long-range pair interactions.