Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0501407 · Physical Review A

Multichannel scattering and Feshbach resonances: Effective theory, phenomenology, and many-body effects

Georg M. Bruun, Andrew David Jackson, Evgeni E. Kolomeitsev

Published May 26, 2005• 44 citations
arXivHTMLPDF

Abstract

A low-energy effective theory based on a microscopic multichannel description of the atom-atom interaction is derived for the scattering of alkali-metal atoms in different hyperfine states. This theory describes all scattering properties, including medium effects, in terms of the singlet and triplet scattering lengths and the range of the atom-atom potential and provides a link between a microscopic description of Feshbach scattering and more phenomenological approaches. It permits the calculation of medium effects on the resonance coming from the occupation of closed channel states. The examination of such effects is demonstrated to be of particular relevance to an experimentally important Feshbach resonance for $^{40}\mathrm{K}$ atoms. We analyze a recent rethermalization rate experiment on $^{40}\mathrm{K}$ and demonstrate that a measurement of the temperature dependence of this rate can determine the magnetic moment of the Feshbach molecule. Finally, the energy dependence of the Feshbach interaction is shown to introduce a negative effective range inversely proportional to the width of the resonance. Since our theory is based on a microscopic multichannel picture, it allows the explicit calculation of corrections to commonly used approximations such as the neglect of the effective range and the treatment of the Feshbach molecule as a point boson.

DOI
10.1103/physreva.71.052713

Related papers

Prediction of Feshbach resonances from three input parameters

Thomas M. Hanna, Eite Tiesinga, Paul S. Julienne

We have developed a model of Feshbach resonances in gases of ultracold alkali-metal atoms using the ideas of multichannel quantum-defect theory. Our model requires just three parameters describing the interactions---the singlet and triplet scattering lengths, and the long-range van der Waals coefficient---in addition to known atomic properties. Without using any further details of the interactions, our approach can accurately predict the locations of resonances. It can also be used to find the singlet and triplet scattering lengths from measured resonance data. We apply our technique to $^{6}\text{L}\text{i-}^{40}\text{K}$ and $^{40}\text{K}\text{\ensuremath{-}}^{87}\text{R}\text{b}$ scattering, obtaining good agreement with experimental results, and with the more computationally intensive coupled channels technique.

arXiv:0903.0884•Physical Review A•Apr 30, 2009•39 citations

Effective-range approximations for resonant scattering of cold atoms

Caroline L. Blackley, Paul S. Julienne, Jeremy M. Hutson

Studies of cold atom collisions and few-body interactions often require the energy dependence of the scattering phase shift, which is usually expressed in terms of an effective-range expansion.We use accurate coupled-channel calculations on 6 Li, 39 K, and 133 Cs to explore the behavior of the effective range in the vicinity of both broad and narrow Feshbach resonances.We show that commonly used expressions for the effective range break down dramatically for narrow resonances and near the zero crossings of broad resonances.We present an alternative parametrization of the effective range that is accurate through both the pole and the zero crossing for both broad and narrow resonances.However, the effective-range expansion can still fail at quite low collision energies, particularly around narrow resonances.We demonstrate that an analytical form of an energy and magnetic-field-dependent phase shift, based on multichannel quantum defect theory, gives accurate results for the energy-dependent scattering length.

arXiv:1402.3272•Physical Review A•Apr 3, 2014•34 citations

Properties of cold collisions of K39 atoms and of K41 atoms in relation to Bose-Einstein condensation

H. M. J. M. Boesten, J. M. M. J. Vogels, J. G. C. Tempelaars +1 more

We have determined properties of cold $^{39}\mathrm{K}$+$^{39}\mathrm{K}$ and $^{41}\mathrm{K}$+$^{41}\mathrm{K}$ collisions by a multichannel inverted perturbation approach applied to spectroscopic data of highly excited $^{39}\mathrm{K}$ singlet and triplet bound states. We predict positive scattering lengths for the $^{39}\mathrm{K}$+$^{39}\mathrm{K}$ and $^{41}\mathrm{K}$+$^{41}\mathrm{K}$ singlet potentials, as well as for the $^{41}\mathrm{K}$+$^{41}\mathrm{K}$ triplet potential, and a negative value for $^{39}\mathrm{K}$+$^{39}\mathrm{K}$ triplet scattering. From a study of the magnetic field dependence we conclude that $^{41}\mathrm{K}$ may be a bosonic atom with a Feshbach resonance in the magnetic field range where these atoms can be trapped in the $f=1$, ${m}_{f}=\ensuremath{-}1$ hyperfine state, thus providing a possibility to switch between stable and instable condensates in a Bose-Einstein condensation experiment.

Physical Review A•Nov 1, 1996•32 citations

Ultracold scattering of alkali-metal atoms in magnetic fields

Caroline L. Blackley

This thesis reports on calculations of the scattering properties of a variety of ultracold alkali-metal mixtures. In particular, we have calculated the scattering properties of homonuclear mixtures of 85Rb, in a variety of incoming channels, and we have calculated the properties of heteronuclear mixtures of the isotopologues of Rb and Cs, and K and Cs. In general, we are interested in the location and character of Feshbach resonances in these mixtures with a view towards ultracold molecule formation. In 85Rb there is a rich Feshbach structure and potential uses for the resonances that we find, in the scattering lengths of the various incoming channels, are discussed. In 85RbCs there is a rich Feshbach structure and the prospects for ultracold molecule formation using this system are detailed. Similarly, we detail the Feshbach resonances of 87RbCs and discuss our results in the context of the successful formation of ultracold ground-state molecules. In the isotopologues of KCs each system has a rich Feshbach structure and we detail the location and width of the resonances, as well as the potential for ultracold molecule formation using each of the isotopes of potassium. In addition to scattering calculations, we have also calculated the location and character of the highest-lying bound states of each system. We have investigated the energy dependence of the scattering length using accurate coupled-channel calculations on 6Li, 39K and 133Cs to explore the behaviour of the effective range in the vicinity of both broad and narrow Feshbach resonances. We present an alternative parametrization of the effective range and further demonstrate that an analytical form of an energy and magnetic field-dependent phase shift, based on multichannel quantum defect theory, gives accurate results for the energy-dependent scattering length. Lastly, we examine the effect of additional external fields on alkali-metal collisions and discuss how external fields can be used to manipulate the interaction properties of a system.

•Jan 1, 2015•1 citation

Microscopic many-body theory of atomic Bose gases near a Feshbach resonance

R. A. Duine, Henk T. C. Stoof

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.

arXiv:cond-mat/0210544•Journal of Optics B Quantum and Semiclassical Optics•Apr 1, 2003•49 citations

Two-Body Interaction

Hui Zhai

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.

Cambridge University Press eBooks•Jan 30, 2021•0 citations
A low-energy effective theory based on a microscopic multichannel description of the atom-atom interaction is derived for the scattering of alkali-metal atoms in different hyperfine states. This theory describes all scattering properties, including medium effects, in terms of the singlet and triplet scattering lengths and the range of the atom-atom potential and provides a link between a microscopic description of Feshbach scattering and more phenomenological approaches. It permits the calculation of medium effects on the resonance coming from the occupation of closed channel states. The examination of such effects is demonstrated to be of particular relevance to an experimentally important Feshbach resonance for 40K^{40}\mathrm{K}40K atoms. We analyze a recent rethermalization rate experiment on 40K^{40}\mathrm{K}40K and demonstrate that a measurement of the temperature dependence of this rate can determine the magnetic moment of the Feshbach molecule. Finally, the energy dependence of the Feshbach interaction is shown to introduce a negative effective range inversely proportional to the width of the resonance. Since our theory is based on a microscopic multichannel picture, it allows the explicit calculation of corrections to commonly used approximations such as the neglect of the effective range and the treatment of the Feshbach molecule as a point boson.
We have developed a model of Feshbach resonances in gases of ultracold alkali-metal atoms using the ideas of multichannel quantum-defect theory. Our model requires just three parameters describing the interactions---the singlet and triplet scattering lengths, and the long-range van der Waals coefficient---in addition to known atomic properties. Without using any further details of the interactions, our approach can accurately predict the locations of resonances. It can also be used to find the singlet and triplet scattering lengths from measured resonance data. We apply our technique to 6Li-40K^{6}\text{L}\text{i-}^{40}\text{K}6Li-40K and $^{40}\text{K}\text{\ensuremath{-}}^{87}\text{R}\text{b}$ scattering, obtaining good agreement with experimental results, and with the more computationally intensive coupled channels technique.
Studies of cold atom collisions and few-body interactions often require the energy dependence of the scattering phase shift, which is usually expressed in terms of an effective-range expansion.We use accurate coupled-channel calculations on 6 Li, 39 K, and 133 Cs to explore the behavior of the effective range in the vicinity of both broad and narrow Feshbach resonances.We show that commonly used expressions for the effective range break down dramatically for narrow resonances and near the zero crossings of broad resonances.We present an alternative parametrization of the effective range that is accurate through both the pole and the zero crossing for both broad and narrow resonances.However, the effective-range expansion can still fail at quite low collision energies, particularly around narrow resonances.We demonstrate that an analytical form of an energy and magnetic-field-dependent phase shift, based on multichannel quantum defect theory, gives accurate results for the energy-dependent scattering length.
We have determined properties of cold 39K^{39}\mathrm{K}39K+39K^{39}\mathrm{K}39K and 41K^{41}\mathrm{K}41K+41K^{41}\mathrm{K}41K collisions by a multichannel inverted perturbation approach applied to spectroscopic data of highly excited 39K^{39}\mathrm{K}39K singlet and triplet bound states. We predict positive scattering lengths for the 39K^{39}\mathrm{K}39K+39K^{39}\mathrm{K}39K and 41K^{41}\mathrm{K}41K+41K^{41}\mathrm{K}41K singlet potentials, as well as for the 41K^{41}\mathrm{K}41K+41K^{41}\mathrm{K}41K triplet potential, and a negative value for 39K^{39}\mathrm{K}39K+39K^{39}\mathrm{K}39K triplet scattering. From a study of the magnetic field dependence we conclude that 41K^{41}\mathrm{K}41K may be a bosonic atom with a Feshbach resonance in the magnetic field range where these atoms can be trapped in the f=1f=1f=1, ${m}_{f}=\ensuremath{-}1$ hyperfine state, thus providing a possibility to switch between stable and instable condensates in a Bose-Einstein condensation experiment.
This thesis reports on calculations of the scattering properties of a variety of ultracold alkali-metal mixtures. In particular, we have calculated the scattering properties of homonuclear mixtures of 85Rb, in a variety of incoming channels, and we have calculated the properties of heteronuclear mixtures of the isotopologues of Rb and Cs, and K and Cs. In general, we are interested in the location and character of Feshbach resonances in these mixtures with a view towards ultracold molecule formation. In 85Rb there is a rich Feshbach structure and potential uses for the resonances that we find, in the scattering lengths of the various incoming channels, are discussed. In 85RbCs there is a rich Feshbach structure and the prospects for ultracold molecule formation using this system are detailed. Similarly, we detail the Feshbach resonances of 87RbCs and discuss our results in the context of the successful formation of ultracold ground-state molecules. In the isotopologues of KCs each system has a rich Feshbach structure and we detail the location and width of the resonances, as well as the potential for ultracold molecule formation using each of the isotopes of potassium. In addition to scattering calculations, we have also calculated the location and character of the highest-lying bound states of each system. We have investigated the energy dependence of the scattering length using accurate coupled-channel calculations on 6Li, 39K and 133Cs to explore the behaviour of the effective range in the vicinity of both broad and narrow Feshbach resonances. We present an alternative parametrization of the effective range and further demonstrate that an analytical form of an energy and magnetic field-dependent phase shift, based on multichannel quantum defect theory, gives accurate results for the energy-dependent scattering length. Lastly, we examine the effect of additional external fields on alkali-metal collisions and discuss how external fields can be used to manipulate the interaction properties of a system.
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.
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.