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Scattering resonances in the extreme quantum limit

Jesse S. Hersch

Published Jan 1, 1999• 3 citations
OpenAlex

Abstract

This thesis addresses topics in low energy scattering in quantum mechanics, in particular, resonance phenomena. Hence the title: the phrase \\extreme quantum limit" refers to the situation when the wavelengths of the particles in the system are larger than every other scale, so that the behavior is far into the quantum regime. A powerful tool in the problems of low energy scattering is the point scatterer model, and will be used extensively throughout the thesis. Therefore, we begin with a thorough introduction to this model in Chapter 2. As a rst application of the point scatterer model, we will investigate the phenomenon of the proximity resonance, which is one example of strange quantum behavior appearing at low energy. Proximity resonances will be addressed theoretically in Chapter 3, and experimentally in Chapter 4. Threshold resonances, another type of low energy scattering resonance, are considered in Chapter 5, along with their connection to the E mov and Thomas eects, and scattering in the presence of an external con ning potential. Although the point scatterer model will serve us well in the work presented here, it does have its limitations. These limitations will be removed in Chapter 6, where we describe how to extend the model to include higher partial waves. In Chapter 7, we extend the model one step further, and illustrate how to treat vector wave scattering with the model. Finally, in Chapter 8 we will depart from the topic of low energy scattering and investigate the inuence of diraction on an open quantum mechanical system, again both experimentally and theoretically.

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This thesis addresses topics in low energy scattering in quantum mechanics, in particular, resonance phenomena. Hence the title: the phrase \extreme quantum limit" refers to the situation when the wavelengths of the particles in the system are larger than every other scale, so that the behavior is far into the quantum regime. A powerful tool in the problems of low energy scattering is the point scatterer model, and will be used extensively throughout the thesis. Therefore, we begin with a thorough introduction to this model in Chapter 2. As a rst application of the point scatterer model, we will investigate the phenomenon of the proximity resonance, which is one example of strange quantum behavior appearing at low energy. Proximity resonances will be addressed theoretically in Chapter 3, and experimentally in Chapter 4. Threshold resonances, another type of low energy scattering resonance, are considered in Chapter 5, along with their connection to the E mov and Thomas eects, and scattering in the presence of an external con ning potential. Although the point scatterer model will serve us well in the work presented here, it does have its limitations. These limitations will be removed in Chapter 6, where we describe how to extend the model to include higher partial waves. In Chapter 7, we extend the model one step further, and illustrate how to treat vector wave scattering with the model. Finally, in Chapter 8 we will depart from the topic of low energy scattering and investigate the inuence of diraction on an open quantum mechanical system, again both experimentally and theoretically.
It is well known that at long wavelengths $\ensuremath{\lambda}$ an s-wave scatterer can have a scattering cross section $\ensuremath{\sigma}$ on the order of ${\ensuremath{\lambda}}^{2}$, much larger than its physical size, as measured by the range of its potential. Very interesting phenomena can arise when two or more identical scatterers are placed close together, well within one wavelength. We show that, for a pair of identical scatterers, an extremely narrow p-wave ``proximity'' resonance develops from a broader s-wave resonance of the individual scatterers. A new s-wave resonance of the pair also appears. The relation of these proximity resonances (so called because they appear when the scatterers are close together) to the Thomas and Efimov effects is discussed.
Scattering theory studies the behaviour of (quantum) systems over time and length scales which are very large, compared with the time or length that are characteristic of the interaction which affects them. The chapter begins with an outline of the basic problems of scattering theory, i.e. the existence of scattering states, their uniqueness, weak asymptotic completeness the existence of the scattering transformation, S , the reduction of S due to symmetries, the analyticity of S , asymptotic completeness and the existence of wave operators. After considering the Schrödinger equation for stationary states, the integral equation for scattering problems is derived and studied. The Born approximation and the Born series are presented, and the conditions which ensure convergence of the Born series are also discussed. Further topics are partial wave expansion, its convergence and the uniqueness of the solution satisfying the asymptotic condition, the Levinson theorem. Lastly, in the first part of the chapter, scattering theory from singular potentials is introduced, with emphasis on the polydromy of the wave function, following early work in the literature, and the general problems in the theory of resonances are studied. In the second part, we examine in detail a separable potential model, the occurrence of bound states in the completeness relationship, an excitable potential model and the unitarity of the Möller wave operator. Lastly, we study the survival amplitude associated with quantum decay transitions. Aims and problems of scattering theory Scattering theory is the branch of physics that is concerned with interacting systems on a scale of time and/or distance that is large compared with the scale of the interaction itself, and it provides the most powerful tool for studying the microscopic nature of the world.
The purpose of these lectures is to give an accessible and self contained introduction to quantum scattering theory in one dimension. Part A defines the theoretical playground, and develops basic concepts of scattering theory in the time domain (Asymptotic Condition, in- and out- states, scattering operator S^\hat{S}S^). The aim of Part B is then to build up, in a step-by-step fashion, the time independent scattering theory in energy domain. This amounts to introduce the Lippmann-Schwinger equation for the stationary scattering states (denoted as ∣ψE(±1)±⟩| ψ_{E(\pm 1)}^\pm \rangle∣ψE(±1)±​⟩), to discuss fundamental properties of ∣ψE(±1)±⟩| ψ_{E(\pm 1)}^\pm \rangle∣ψE(±1)±​⟩, and subsequently to construct S^\hat{S}S^ and T^\hat{T}T^ operators in terms of ∣ψE(±1)±⟩| ψ_{E(\pm 1)}^\pm \rangle∣ψE(±1)±​⟩. Physical contents of the S^\hat{S}S^ and T^\hat{T}T^ operators is then illuminated by deriving explicit formulas for the probability of transmission/reflection of our quantum particle through/from the interaction region of the potential. An illustrative numerical example is given, which also highlights an existence of scattering resonances. Finally, Part C elaborates the nonhermitian scattering theory (Siegert pseudostate formalism), which offers an extremely powerful tool suitable for clear cut understanding of the resonance phenomena.
Scattering processes are a fundamental way of experimentally probing distributions and properties of systems in several areas of physics. Considering two-body scattering at low energies, when the de Broglie wavelength is larger than the range of the potential, partial waves with high angular momentum are typically unimportant. The dominant contribution comes from l=0l=0l=0 partial waves, commonly known as sss-wave scattering. This situation is very relevant in atomic physics, e.g. cold atomic gases, and nuclear physics, e.g. nuclear structure and matter. This manuscript is intended as a pedagogical introduction to the topic while covering a numerical approach to compute the desired quantities. We introduce low-energy scattering with particular attention to the concepts of scattering length and effective range. These two quantities appear in the effective-range approximation, which universally describes low-energy processes. We outline a numerical procedure for calculating the scattering length and effective range of spherically symmetric two-body potentials. As examples, we apply the method to the spherical well, modified Pöschl-Teller, Gaussian, and Lennard-Jones potentials. We hope to provide the tools so students can implement similar calculations and extend them to other potentials.
Most of our knowledge about the structure of matter and interaction between particles is derived from scattering experiments. From a theoretical point of view, scattering problems are concerned with the continuous (and positive) energy eigenvalues and unbound eigenfunctions of the Schro¨dinger equation. We have already encountered one-dimensional examples of scattering in Chapter 4, in the discussion of reflection and transmission of an incident particle with definite momentum from a potential step or barrier. In this chapter we consider scattering from a more formal point of view. We will confine our discussion to elastic scattering (scattering without loss or gain of energy by the projectile) from central potentials although many of the concepts and results are applicable to inelastic scattering.
This report on progress explores recent advances in our theoretical and experimental understanding of the physics of open quantum systems (OQSs). The study of such systems represents a core problem in modern physics that has evolved to assume an unprecedented interdisciplinary character. OQSs consist of some localized, microscopic, region that is coupled to an external environment by means of an appropriate interaction. Examples of such systems may be found in numerous areas of physics, including atomic and nuclear physics, photonics, biophysics, and mesoscopic physics. It is the latter area that provides the main focus of this review, an emphasis that is driven by the capacity that exists to subject mesoscopic devices to unprecedented control. We thus provide a detailed discussion of the behavior of mesoscopic devices (and other OQSs) in terms of the projection-operator formalism, according to which the system under study is considered to be comprised of a localized region (Q), embedded into a well-defined environment (P) of scattering wavefunctions (with Q + P = 1). The Q subspace must be treated using the concepts of non-Hermitian physics, and of particular interest here is: the capacity of the environment to mediate a coupling between the different states of Q; the role played by the presence of exceptional points (EPs) in the spectra of OQSs; the influence of EPs on the rigidity of the wavefunction phases, and; the ability of EPs to initiate a dynamical phase transition (DPT). EPs are singular points in the continuum, at which two resonance states coalesce, that is where they exhibit a non-avoided crossing. DPTs occur when the quantum dynamics of the open system causes transitions between non-analytically connected states, as a function of some external control parameter. Much like conventional phase transitions, the behavior of the system on one side of the DPT does not serve as a reliable indicator of that on the other. In addition to discussing experiments on mesoscopic quantum point contacts that provide evidence of the environmentally-mediated coupling of quantum states, we also review manifestations of DPTs in mesoscopic devices and other systems. These experiments include observations of resonance-trapping behavior in microwave cavities and open quantum dots, phase lapses in tunneling through single-electron transistors, and spin swapping in atomic ensembles. Other possible manifestations of this phenomenon are presented, including various superradiant phenomena in low-dimensional semiconductors. From these discussions a generic picture of OQSs emerges in which the environmentally-mediated coupling between different quantum states plays a critical role in governing the system behavior. The ability to control or manipulate this interaction may even lead to new applications in photonics and electronics.