[Paper Review] Unitary boson-boson and boson-fermion mixtures: third virial coefficient and three-body parameter on a narrow Feshbach resonance

Shimpei Endo, Yvan Castin|arXiv (Cornell University)|Aug 1, 2016

Cold Atom Physics and Bose-Einstein Condensates63 references5 citations

TL;DR

This paper presents exact integral expressions for the third virial coefficient and three-body parameter in unitary mixtures of bosons and fermions on a narrow Feshbach resonance. Using a microscopic model of narrow Feshbach resonances, it derives a new exact integral formula for the three-body parameter $ R_t $, predicts a divergence at the scale exponent $ s = 1/2 $, and applies results to realistic ultracold atomic systems.

ABSTRACT

We give exact integral expressions of the third cluster or virial coefficients of binary mixtures of ideal Bose or Fermi gases, with interspecies interactions of zero range and infinite s-wave scattering length. In general the result depends on three-body parameters Rt appearing in three-body contact conditions, because an Efimov effect is present or because the mixture is in a preefimovian regime with a mass ratio close to an Efimov effect threshold. We give a new, exact integral expression of Rt for the microscopic narrow Feshbach resonance model. A divergence of Rt in the preefimovian regime at a scaling exponent s = 1/2 is predicted and physically discussed. The analytical results are applied to typical species used in cold atom experiments.

Motivation & Objective

• To derive exact analytical expressions for the third virial coefficient $ b_3 $ in unitary binary mixtures of bosons and fermions with zero-range interactions.
• To resolve the theoretical gap in zero-range models by providing a microscopic derivation of the three-body parameter $ R_t $ for narrow Feshbach resonances.
• To address the challenge of multiple three-body parameters in boson-boson mixtures by introducing a consistent framework for $ R_t $.
• To predict a physical divergence of $ R_t $ in the pre-Efimov regime at scale exponent $ s = 1/2 $, and analyze its implications.
• To apply the formalism to experimentally relevant atomic species used in ultracold quantum gases.

Proposed method

• Adopt a zero-range model with Wigner-Bethe-Peierls contact conditions for two-body interactions, invariant under scale invariance at unitarity.
• Incorporate three-body contact conditions that include the three-body parameter $ R_t $, either as a scale-invariant power-law behavior or via explicit $ R_t $-dependence when Efimov physics is present.
• Use the hyperspherical adiabatic method to solve the three-body problem in a harmonic trap, leveraging separability in hyperspherical coordinates.
• Develop a new exact integral representation for $ R_t $ based on a microscopic model of a narrow Feshbach resonance, including the coupling to molecular states.
• Apply the formalism to compute $ b_3 $ for specific atomic mixtures (e.g., $ ^6Li-^7Li $, $ ^6Li-^40K $, $ ^7Li-^6Li $) relevant to experiments.
• Introduce a manual cutoff in the geometric spectrum to exclude unphysical deeply bound trimers, ensuring validity of the zero-range approximation.

Experimental results

Research questions

• RQ1 How can the third virial coefficient $ b_3 $ be exactly expressed in unitary boson-fermion and boson-boson mixtures with zero-range interactions?
• RQ2 What is the microscopic origin of the three-body parameter $ R_t $ in a narrow Feshbach resonance, and how can it be computed from first principles?
• RQ3 How does the three-body parameter $ R_t $ behave in the pre-Efimov regime, particularly near the scale exponent $ s = 1/2 $?
• RQ4 Why do boson-boson mixtures require two distinct three-body parameters $ R_t^{\rm BBb} $ and $ R_t^{\rm bbB} $, and how can they be consistently treated?
• RQ5 What are the quantitative predictions for $ b_3 $ and $ R_t $ in experimentally realized ultracold atomic systems?

Key findings

• A new exact integral formula is derived for the three-body parameter $ R_t $ in a narrow Feshbach resonance, enabling first-principles computation of $ R_t $ from microscopic parameters.
• The three-body parameter $ R_t $ is predicted to diverge in the pre-Efimov regime at the scale exponent $ s = 1/2 $, signaling a critical transition in three-body physics.
• For boson-boson mixtures, two distinct three-body parameters $ R_t^{\rm BBb} $ and $ R_t^{\rm bbB} $ are required, and the formalism accounts for both without ambiguity.
• The third virial coefficient $ b_3 $ is computed analytically for realistic atomic species, showing smooth dependence on mass ratio when $ R_t $ is properly included.
• The model resolves the long-standing issue of missing $ R_t $ in zero-range theories by providing a microscopic derivation, enabling quantitative predictions.
• The results are consistent with experimental observations of three-body losses and Efimov physics, particularly in systems like $ ^6Li-^7Li $ and $ ^6Li-^40K $.