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.
While Efimov physics in ultracold atoms is usually modeled with an isolated Feshbach resonance, many real world resonances appear in close vicinity to each other and are therefore overlapping. Here we derive a realistic model based on the mutual coupling of an open channel and two closed molecular channels while neglecting short-range physics as permitted by the narrow character of the considered resonances. The model is applied to three distinct scenarios with experimental relevance. We show that the effect of overlapping resonances is manifested most strikingly at a narrow resonance in whose vicinity there is a slightly narrower one. In this system the Efimov ground state extends not only over the scattering length zero crossing between the two resonances but also over the pole of the second resonance to finally meet the dissociation threshold below it. In the opposite scenario, when a narrow resonance is considered in the vicinity of a slightly broader one, we observe that the Efimov features are pushed to lower binding energies and smaller scattering lengths by a significant factor facilitating their experimental investigation. Both scenarios are compared with the case of two narrow resonances which are far enough away from each other to be effectively decoupled. In this case the two-channel model results are recovered. Finally, we analyze the rich excitation spectrum of the system and construct and explain its nodal pattern.
It is well-known that cold atoms near
s-wave Feshbach resonances have universal properties that are insensitive to the short-range details of the interaction. What is less known is that atoms near higher partial-wave Feshbach resonances also have remarkable universal properties. We illustrate this with a single atom interacting resonantly with two fixed static centers. At a Feshbach resonance point with orbital angular momentum $L\ensuremath{\ge}1$, we find
2L+1 shallow bound states whose energies behave like
1/R2L+1 when the distance
R between the two centers is large. We then compute corrections to the binding energies due to other parameters in the effective range expansions. For completeness we also compute the binding energies near
s-wave Feshbach resonances, taking into account the corrections. Afterwards we turn to the bound states at large but finite scattering volumes. For
p-wave and higher partial-wave resonances we derive a simple formula for the energies in terms of a parameter called ``proximity parameter.'' These results are applicable to a free atom interacting resonantly with two atoms that are localized to two lattice sites of an optical lattice, and to one light atom interacting with two heavy ones in free space. Modifications of the low energy physics due to the long range van der Waals potential are also discussed.
For a system of three identical bosons interacting via short-range forces, when two of the atoms are about to form a two-body
s-wave dimer, the Efimov effect takes place leading to the formation of an infinite number of three-body (Efimov) states. The lowest Efimov state crosses the three-body break-up threshold when the
s-wave two-body scattering length is $a\ensuremath{\approx}\ensuremath{-}9.73{r}_{\mathrm{vdW}}$,
rvdW being the van der Waals length. This article focuses on a generalized version of this Efimov scenario, where two of the atoms are about to form a two-body
d-wave dimer, resulting in strong
d-wave interactions. In his paper [Phys. Rev. A 62, 050702(R) (2000)], Bo Gao predicted that for broad resonances the
d-wave dimer is always formed when $a\ensuremath{\approx}0.956{r}_{\mathrm{vdW}}$. Here we find that a single universal three-body state associated with the
d-wave dimer is also formed near the three-body breakup threshold at $a\ensuremath{\approx}1.09{r}_{\mathrm{vdW}}$, or alternatively
a2=0.902rvdW, where
a2 is the two-body
d-wave scattering length. The signature of such a universal three-body state is signaled experimentally by an enhancement of the three-body recombination rate. The three-body effective potential curves that are crucial for understanding the recombination dynamics are also calculated and analyzed. An improved method to calculate the couplings, effective potential curves, and recombination rate coefficients is presented.
The interaction of two atoms at energies in the microkelvin range is mainly characterized by one parameter, namely, the s-wave scattering length. Efimov showed that when the scattering length was effectively infinite, i.e. much larger than the natural length scale of the atoms, a large number of three-body bound states could form even when the two-body system supported only one weakly bound state, or possibly none at all. The states are bound in an effective 1/ R 2 potential and therefore form a series with a characteristic energy spacing. These states are called Efimov states and are conceptually important since they indicate a new way to form aggregates of particles. They bear some relation to vibrationally bound states of three atoms. The theory of these states and their relation to vibrational binding is reviewed in light of recent experimental measurements.
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=0 partial waves, commonly known as
s-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.