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AIP conference proceedings

Feshbach resonances in atomic structure and dynamics

J. H. Macek, Jacob Schach Møller, Dmitri Fedorov

Published Jan 1, 2008• 0 citations
Publisher

Abstract

Feshbach resonances are quasi‐bound states that are represented, in first approximation, as bound states in energetically closed channels. Owing to coupling with open channels they can spontaneously break up into two or more fragments. I will review those states that separate into two fragments within the framework of multichannel effective range theory. Magnetic tuning of scattering lengths is described within this context. Such tuning has opened up many opportunities for fundamental studies of few‐body systems. By using multichannel zero‐range potentials, conventionally called pseudo‐potentials, it is possible to employ the model when the few‐body systems involve three or more fragments. The emergence of Efimov states for critical values of the tuning parameters is discussed.

DOI
10.1063/1.2915640

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Feshbach resonances and the three-body problem

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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.

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

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1. Highlight “short-ranged,” “dilute,” and “low-energy” as three main features of interactions between ultracold atoms. 2. Introduce the important concept of the phase shift. 3. Introduce the $s$-wave scattering length as a universal parameter describing the low-energy interaction between ultracold atoms. 4. Discuss the relation between divergent scattering length, low-energy bound state, and jump of phase shift. 5. Discuss the relation between the scattering length and the scattering amplitude. 6. Discuss under what condition a positive scattering length describes repulsive interaction. 7. Discuss the condition when an algebraically decayed potential can be treated as a finite range one. 8. Introduce two types of zero-range single-channel potentials to capture the universal low-energy $s$-wave interaction between ultracold atoms. 9. Introduce the concept of renormalization condition and renormalizable theory. 10. Discuss how the spin rotational symmetry imposes constraints on interaction form for both alkali-metal and alkaline-earth-metal atoms. 11. Introduce Feshbach resonance as an important tool to tune scattering length. 12. Compare the two-channel Feshbach resonance with the single-channel shape resonance and compare the wide and the narrow resonances. 13. Introduce a zero-range two-channel model. 14. Introduce the confinement-induced resonance to tune interaction strength by an external potential. 15. Summarize three key conditions for a Feshbach resonance, and unify the optical Feshbach resonance, the orbital Feshbach resonance, and the confinement-induced resonance all in terms of these three conditions. 16. Introduce the Efimov effect as an important three-body effect at the vicinity of the two-body scattering resonance. 17. Highlight the symmetry aspect of the Efimov effect. 18. Discuss various connections between few-body and many-body physics. 19. Illustrate that few-body calculation can be used to determine properties of many-body systems by using high-temperature expansion as an example.

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Efimov states embedded in the three-body continuum

Nirav P. Mehta, Seth T. Rittenhouse, José Paulo D'Incao +1 more

We present analytical solutions for the three-body problem with multichannel interactions and identify a class of quasibound Efimov states that can be viewed as three-body Fano-Feshbach resonances. Our method employs a multichannel generalization of the Fermi pseudopotential to model low-energy atom-atom interactions near a magnetically tunable Fano-Feshbach resonance. We discuss the conditions under which quasibound Efimov states may be supported and identify the interaction parameters that limit the lifetimes of these states. We speculate that it may be possible to observe these states using spectroscopic methods, perhaps allowing for the measurement of multiple Efimov resonances.

arXiv:0801.2002•Physical Review A•Aug 27, 2008•20 citations
Feshbach resonances are quasi‐bound states that are represented, in first approximation, as bound states in energetically closed channels. Owing to coupling with open channels they can spontaneously break up into two or more fragments. I will review those states that separate into two fragments within the framework of multichannel effective range theory. Magnetic tuning of scattering lengths is described within this context. Such tuning has opened up many opportunities for fundamental studies of few‐body systems. By using multichannel zero‐range potentials, conventionally called pseudo‐potentials, it is possible to employ the model when the few‐body systems involve three or more fragments. The emergence of Efimov states for critical values of the tuning parameters is discussed.
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.
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.
This work investigates how the closed channel of a Feshbach resonance is characterised by experimental observables. Surprisingly, it is found that the two-body observables associated with the Feshbach resonance can be insensitive to the properties of the closed channel. In particular, it is impossible in this situation to determine the energy of the bound state causing the resonance from the usual experimental data. This is the case for all magnetic Feshbach resonances in ultracold atoms, due to their deep two-body interaction potentials. This insensitivity highlights a major difference with Feshbach resonances that involve shallow interaction potentials, such as hadron resonances. It appears however that short-range two-body correlations and three-body observables are affected by a parameter of the closed channel called the “closed-channel scattering length”. A photoassociation experiment is proposed to measure this parameter in ultracold atom systems.
1. Highlight “short-ranged,” “dilute,” and “low-energy” as three main features of interactions between ultracold atoms. 2. Introduce the important concept of the phase shift. 3. Introduce the sss-wave scattering length as a universal parameter describing the low-energy interaction between ultracold atoms. 4. Discuss the relation between divergent scattering length, low-energy bound state, and jump of phase shift. 5. Discuss the relation between the scattering length and the scattering amplitude. 6. Discuss under what condition a positive scattering length describes repulsive interaction. 7. Discuss the condition when an algebraically decayed potential can be treated as a finite range one. 8. Introduce two types of zero-range single-channel potentials to capture the universal low-energy sss-wave interaction between ultracold atoms. 9. Introduce the concept of renormalization condition and renormalizable theory. 10. Discuss how the spin rotational symmetry imposes constraints on interaction form for both alkali-metal and alkaline-earth-metal atoms. 11. Introduce Feshbach resonance as an important tool to tune scattering length. 12. Compare the two-channel Feshbach resonance with the single-channel shape resonance and compare the wide and the narrow resonances. 13. Introduce a zero-range two-channel model. 14. Introduce the confinement-induced resonance to tune interaction strength by an external potential. 15. Summarize three key conditions for a Feshbach resonance, and unify the optical Feshbach resonance, the orbital Feshbach resonance, and the confinement-induced resonance all in terms of these three conditions. 16. Introduce the Efimov effect as an important three-body effect at the vicinity of the two-body scattering resonance. 17. Highlight the symmetry aspect of the Efimov effect. 18. Discuss various connections between few-body and many-body physics. 19. Illustrate that few-body calculation can be used to determine properties of many-body systems by using high-temperature expansion as an example.
We present analytical solutions for the three-body problem with multichannel interactions and identify a class of quasibound Efimov states that can be viewed as three-body Fano-Feshbach resonances. Our method employs a multichannel generalization of the Fermi pseudopotential to model low-energy atom-atom interactions near a magnetically tunable Fano-Feshbach resonance. We discuss the conditions under which quasibound Efimov states may be supported and identify the interaction parameters that limit the lifetimes of these states. We speculate that it may be possible to observe these states using spectroscopic methods, perhaps allowing for the measurement of multiple Efimov resonances.