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European Journal of Physics

Potential and Feshbach s-wave resonances in coupled atomic collision channels

Grover Andrade-Sánchez, Vı́ctor Romero-Rochı́n

Published Sep 7, 2023• 0 citations
Publisher

Abstract

Abstract We discuss s-wave scattering in an atomic binary collision with two coupled channels, tunable by an external magnetic field, one channel open and the other closed for the incident energies considered. The analysis is performed with a stylized model of square-well potentials. This simplification allows for a pedagogically thorough discussion of the different scattering resonances that appear in coupled channels. One of them, the potential resonances at vanishing energy, occurs as a bound state of the coupled system emerges, in turn, tuned at a very precise value of the external field. The other resonances, described by Feshbach theory, occur when the incident energy is near a bound state of the closed channel, as if it were decoupled from the open channel. These resonances exist for values of the external field above a particular threshold value. Besides the potential intrinsic value of this study in a quantum mechanics course, as the analysis can be performed with minor numerical calculations, it is also an aid for the understanding of current research advances in the exciting field of ultracold gases.

DOI
10.1088/1361-6404/acf799

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Abstract We discuss s-wave scattering in an atomic binary collision with two coupled channels, tunable by an external magnetic field, one channel open and the other closed for the incident energies considered. The analysis is performed with a stylized model of square-well potentials. This simplification allows for a pedagogically thorough discussion of the different scattering resonances that appear in coupled channels. One of them, the potential resonances at vanishing energy, occurs as a bound state of the coupled system emerges, in turn, tuned at a very precise value of the external field. The other resonances, described by Feshbach theory, occur when the incident energy is near a bound state of the closed channel, as if it were decoupled from the open channel. These resonances exist for values of the external field above a particular threshold value. Besides the potential intrinsic value of this study in a quantum mechanics course, as the analysis can be performed with minor numerical calculations, it is also an aid for the understanding of current research advances in the exciting field of ultracold gases.
We discuss sss-wave scattering in an atomic binary collision with two coupled channels, tunable by an external magnetic field, one channel open and the other closed for the incident energies considered. The analysis is performed with a stylized model of square-well potentials. This simplification allows for a pedagogically thorough discussion of the different scattering resonances that appear in coupled channels. One of the them, the potential resonances at vanishing energy, occur as a bound state of the coupled system emerges, in turned tuned at a very precise value of the external field. The other resonances, described by Feshbach theory, occur when the incident energy is near a bound state of the closed channel, as if it were decoupled from the open channel. These resonances exist for values of the external field above a particular threshold value. Besides the potential intrinsic value of this study in a quantum mechanics course, as the analysis can be performed with minor numerical calculations, it is also an aid for the understanding of current research advances in the exciting field of ultracold gases.
The physics of Feshbach resonance is analyzed using an analytic expression for the s-wave scattering phase shift and the scattering length a which we derive within a two-channel tight-binding model. Employing a unified treatment of bound states and resonances in terms of the Jost function, it is shown that, for strong interchannel coupling, Feshbach resonance can occur even when the closed channel does not have a bound state. This may extend the range of ultracold atomic systems that can be manipulated by Feshbach resonance. The dependence of the sign of a on the coupling strength in the unitary limit is elucidated. As a by-product, analytic expressions are derived for the background scattering length, the external magnetic field at which resonance occurs, and the energy shift ε-ε(B), where ε is the scattering energy and ε(B) is the bound-state energy in the closed channel (when there is one).
By applying a magnetic field across a trapped ultracold gas of alkali metal atoms, it is possible to alter the binding energies of near threshold molecular states. The proximity of molecular states to the scattering threshold has a strong effect on the threshold scattering characteristics. Because of this the magnetic field strength can be used to control low energy scattering within the gas. This resonant phenomena is referred to as a magnetically tuneable Feshbach resonance. This thesis looks at these and related phenomena in the context of trapped potassium atoms. First, we perform coupled-channels calculations in order to characterise a group of previously unreported Feshbach resonances, occurring in a host of different collision channels. Next, we compare these characterisations to empirical data, and more simplified models. Second, a detailed investigation into the photodissociation of weakly bound Feshbach molecules is carried out. This allows us to gain new insights into the structure of the exit channel interaction. We make semi-classical arguments which enable experimentalists to directly measure the height of potential barriers. We also perform detailed numerical calculations which, in conjunction with experimental data, allow us to develop a fine-tuned potential model for the exit state. Third, we model weakly bound triatomic states and their impact on near threshold scattering. Then, we discuss the use of three body recombination as a tool for observing Efimov states. Furthermore, we model the three body recombination rate using a simplified two channel model. Finally we suggest a method intended to provide the basis for numerical calculations using realistic Born-Oppenheimer potentials. It could also be used to analytically study near threshold three body physics.
A Feshbach resonance in the s-wave scattering length occurs if the energy of the two atoms in the incoming open channel is close to the energy of a bound state in a coupled closed channel. Starting from the microscopic Hamiltonian that describes this situation, we derive the effective atom–molecule theory for a Bose gas near a Feshbach resonance. In order to take into account all two-body processes, we have to dress the bare couplings of the atom–molecule model with ladder diagrams. This results in a quantum field theory that exactly reproduces the scattering amplitude of the atoms and the bound-state energy of the molecules. Since these properties are incorporated at the quantum level, the theory can be applied both above and below the critical temperature of the gas. Moreover, making use of the true interatomic potentials ensures that no divergences are encountered at any stage of the calculation. We also present the mean-field theory for the Bose–Einstein condensed phase of the gas.
Magnetically tunable Feshbach resonances exist in ultracold collisions between atoms in 2S^{2}\mathrm{S}2S and 3P0{}^{3}{\mathrm{P}}_{0}3P0​ states, such as an alkali-metal atom colliding with Yb or Sr in a clock state. We investigate the mechanisms of these resonances and identify the terms in the collision Hamiltonian responsible for them. They involve indirect coupling between the open and closed channels, via intermediate channels involving atoms in 3P1{}^{3}{\mathrm{P}}_{1}3P1​ states. The resonance widths are generally proportional to the square of the magnetic field and are strongly enhanced when the magnitude of the background scattering length is large. For any given pair of atoms, the scattering length can be tuned discretely by choosing different isotopes of the 3P0{}^{3}{\mathrm{P}}_{0}3P0​ atom. For each combination of an alkali-metal atom and either Yb or Sr, we consider the prospects of finding an isotopic combination that has both a large background scattering length and resonances at a high but experimentally accessible field. We conclude that 87Rb+Yb^{87}\mathrm{Rb}+\mathrm{Yb}87Rb+Yb, Cs+Yb\mathrm{Cs}+\mathrm{Yb}Cs+Yb, and 85Rb+Sr^{85}\mathrm{Rb}+\mathrm{Sr}85Rb+Sr are particularly promising combinations.
Abstract Taking the ultracold 85 Rb– 87 Rb collision system as an example, we investigated the Feshbach resonances of nonzero partial waves above the threshold. The self-energy at the threshold, which represents the coupling strength between open and closed channels, is considered a critical parameter to quantitatively describe the properties of Feshbach resonances. The total elastic and inelastic cross sections are calculated as functions of the magnetic field B and collision energy E col , ranging from 0.1 to 600 μ K. For a large absolute value of the self-energy at the threshold, the resonance decays rapidly with increasing collision energy, and narrow resonances of nonzero partial waves can be clearly resolved in the contour plot of the inelastic cross section versus the collision energy and magnetic field. It was found that the resonance tail appeared at the given magnetic field when the cross section decreased from the maximal value of the resonance peak to the minimum value, where a long resonance tail indicates an appreciable resonance in a relatively large region of collision energy. This relationship between the self-energy and the properties of Feshbach resonances still exists in the thermally averaged inelastic rate coefficient. The bound-state energies for nonzero partial waves split owing to the spin–spin interaction, which results in multiple nearly-overlapping resonances. Both the spin–spin and second-order spin–orbit effects are included. However, multiple nearly-overlapping resonances for nonzero partial waves are difficult to resolve in thermally averaged rate coefficients.