Two atoms scattering at low and cold energies

Bimalendu Deb

Over the last two decades cold atoms have emerged as a most useful platform for exploring quantum science and quantum technology. Cooling, trapping and manipulating atomic systems require understanding of low energy collisions physics. The purpose of this book is to bridge the gap between graduate teaching in scattering theory, particularly atomic scattering, and the emergent fields of cold atoms and cold molecules. Although this book presents basics of collisions physics as applicable to different branches of physics and chemistry, it is written with a particular emphasis on the physics of low energy quantum collisions and its applications to cold atoms. Key features • Pedagogic description of the topics will make this book suitable for graduate students taking courses on quantum scattering theory and cold atomic physics. • Each chapter will have exercises and worked examples. • Emphasises applications of theory to current research topics in several inter-related areas. • Incorporates numerical techniques. • Self-contained, with appendices summarising related topics needed by students.

Robin Kaiser

Jeff Maki, Fei Zhou

Intermediate energy scale physics plays a very important role in non-equilibrium dynamics of quasi-low dimensional cold atom systems. In this article we obtain the universal scaling relations for the generalized reflection coefficient, i.e., the fraction of atoms scattered out of the initial state, at intermediate energies scales. These intermediate energies are larger than the trap frequency, but necessarily smaller than the effective range of the inter-atomic potential. For harmonically confined one and two dimensional geometries, whenever the energy of the system is commensurate with a transverse energy level, it is shown that the system becomes non-interacting. For energies close to a transverse energy level, the reflection coefficient and scattering dynamics take on universal scaling forms, only depending on the difference between the energy of the particles and the transverse energy level. In this article, the universal power law behaviors of these scattering observables are obtained.

Robin Kaiser

Marius Lysebo

Peng Zhang, Pascal Naidon, Masahito Ueda

Most of the current theories on the $p$-wave superfluid in cold atomic gases are based on the effective-range theory for the two-body scattering, where the low-energy $p$-wave scattering amplitude ${f}_{1}(k)$ is given by ${f}_{1}(k)=\ensuremath{-}1/[\mathit{ik}+1/(\mathcal{V}{k}^{2})+1/\mathcal{R}]$. Here $k$ is the incident momentum, $\mathcal{V}$ and $\mathcal{R}$ are the $k$-independent scattering volume and effective range, respectively. However, due to the long-range nature of the van der Waals interaction between two colliding ultracold atoms, the $p$-wave scattering amplitude of the two atoms is not described by the effective-range theory [J. Math. Phys. 4, 54 (1963); Phys. Rev. A 58, 4222 (1998)]. In this paper we provide an explicit calculation for the $p$-wave scattering of two ultracold atoms near the $p$-wave magnetic Feshbach resonance. We show that in this case the low-energy $p$-wave scattering amplitude ${f}_{1}(k)=\ensuremath{-}1/[\mathit{ik}+1/({\mathcal{V}}^{\mathrm{eff}}{k}^{2})+1/({\mathcal{S}}^{\mathrm{eff}}k)+1/{\mathcal{R}}^{\mathrm{eff}}]$ where ${\mathcal{V}}^{\mathrm{eff}},$ ${\mathcal{S}}^{\mathrm{eff}}$, and ${\mathcal{R}}^{\mathrm{eff}}$ are $k$-dependent parameters. Based on this result, we identify sufficient conditions for the effective-range theory to be a good approximation of the exact scattering amplitude. Using these conditions we show that the effective-range theory is a good approximation for the $p$-wave scattering in the ultracold gases of $^{6}\mathrm{Li}$ and $^{40}\mathrm{K}$ when the scattering volume is enhanced by the resonance.