A fundamental quantity characterizing Fermi gases with zero-range interactions is the three-body contact $C_3$ , which determines several observables including the number of nearby triplets of fermions and the three-body loss rate in cold atom experiments, as shown in a companion article. Here, we compute $C_3$ to leading order in the non-degenerate limit for the homogeneous gas with negative or infinite scattering length $a$ . At $a=\infty$ , using a wavefunction approach, we obtain the analytical expression of $C_3$ , which has a remarkably slow $1/T^{0.22728}$ dependence on the temperature $T$ . In the Feynman diagram technique, which we use for $a<0$ , the correct three-body short-distance correlations emerge only after a non-trivial cancellation between leading order two-body and three-body correlations, and the resulting power-law scaling comes from the large-wavevector tail of the 3-body T-matrix, which we derive inspired by the analytical solution of the three-body problem at the unitary limit.A fundamental quantity characterizing Fermi gases with zero-range interactions is the three-body contact $C_3$, which determines several observables including the number of nearby triplets of fermions and the three-body loss rate in cold atom experiments, as shown in a companion article. Here, we compute $C_3$ to leading order in the non-degenerate limit for the homogeneous gas with negative or infinite scattering length $a$. At $a=\infty$, using a wavefunction approach, we obtain the analytical expression of $C_3$, which has a remarkably slow $1/T^{0.22728}$ dependence on the temperature $T$. In the Feynman diagram technique, which we use for $a<0$, the correct three-body short-distance correlations emerge only after a non-trivial cancellation between leading order two-body and three-body correlations, and the resulting power-law scaling comes from the large-wavevector tail of the 3-body T-matrix, which we derive inspired by the analytical solution of the three-body problem at the unitary limit.

arXiv:2608.15324Aug 15, 20260 citations

Three-body problem for ultracold atoms in quasi-one-dimensional traps

Christophe Mora, Reinhold Egger, Alexander O. Gogolin

We study the three-body problem for both fermionic and bosonic cold-atom gases in a parabolic transverse trap of length scale ${a}_{\ensuremath{\perp}}$. For this quasi-one-dimensional (quasi-1D) problem, there is a two-body bound state (dimer) for any sign of the 3D scattering length $a$ and a confinement-induced scattering resonance. The fermionic three-body problem is universal and characterized by two atom-dimer scattering lengths ${a}_{\mathit{ad}}$ and ${b}_{\mathit{ad}}$ . In the tightly bound ``dimer limit'' ${a}_{\ensuremath{\perp}}∕a\ensuremath{\rightarrow}\ensuremath{\infty}$, we find ${b}_{\mathit{ad}}=0$ and ${a}_{\mathit{ad}}$ is linked to the 3D atom-dimer scattering length. In the weakly bound ``BCS limit'' ${a}_{\ensuremath{\perp}}∕a\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$, a connection to the Bethe ansatz is established, which allows for exact results. The full crossover is obtained numerically. The bosonic three-body problem, however, is nonuniversal: ${a}_{\mathit{ad}}$ and ${b}_{\mathit{ad}}$ depend both on ${a}_{\ensuremath{\perp}}∕a$ and on a parameter ${R}^{*}$ related to the sharpness of the resonance. Scattering solutions are qualitatively similar to fermionic ones. We predict the existence of a single confinement-induced three-body bound state (trimer) for bosons.We study the three-body problem for both fermionic and bosonic cold-atom gases in a parabolic transverse trap of length scale ${a}_{\ensuremath{\perp}}$. For this quasi-one-dimensional (quasi-1D) problem, there is a two-body bound state (dimer) for any sign of the 3D scattering length $a$ and a confinement-induced scattering resonance. The fermionic three-body problem is universal and characterized by two atom-dimer scattering lengths ${a}_{\mathit{ad}}$ and ${b}_{\mathit{ad}}$. In the tightly bound ``dimer limit'' ${a}_{\ensuremath{\perp}}∕a\ensuremath{\rightarrow}\ensuremath{\infty}$, we find ${b}_{\mathit{ad}}=0$ and ${a}_{\mathit{ad}}$ is linked to the 3D atom-dimer scattering length. In the weakly bound ``BCS limit'' ${a}_{\ensuremath{\perp}}∕a\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$, a connection to the Bethe ansatz is established, which allows for exact results. The full crossover is obtained numerically. The bosonic three-body problem, however, is nonuniversal: ${a}_{\mathit{ad}}$ and ${b}_{\mathit{ad}}$ depend both on ${a}_{\ensuremath{\perp}}∕a$ and on a parameter ${R}^{*}$ related to the sharpness of the resonance. Scattering solutions are qualitatively similar to fermionic ones. We predict the existence of a single confinement-induced three-body bound state (trimer) for bosons.

arXiv:cond-mat/0412225Physical Review AMay 9, 200546 citations

Effective Linear Two-Body Method for Many-Body Problems in Atomic and Nuclear Physics

Y.E. Kim, Alexander L. Zubarev

Few-body systems. SupplementumJan 1, 20000 citations

The few-atom problem

Dmitry S. Petrov

This chapter discusses the few-body problem, which plays a very important role in the physics of ultracold gases. It describes the structure of molecules and their formation in three-body collisions (three-body recombination), atom-molecule and molecule-molecule collisional properties, structure of trimers and larger clusters, numerous phenomena that go under the name of Efimov physics, and many other problems. The chapter is organized as follows. Section 3.2 discusses the effective range expansion for the two-body scattering amplitude and puts a special emphasis on the role of the resonance width. Section 3.3 applies the zero-range approach to the problem of three atoms. It considers the phenomenon of the three-body collapse, or Thomas collapse, and defines the Efimov effect. Then, it introduces a model system of one light and two heavy atoms, where we can use the Born‐Oppenheimer approximation. Section 3.4 describes the method of Skorniakov and Ter‐Martirosian (STM) for few-body problems with resonant short-range interactions. Section 3.5 considers other rapidly developing directions in theoretical few-atom physics.This chapter discusses the few-body problem, which plays a very important role in the physics of ultracold gases. It describes the structure of molecules and their formation in three-body collisions (three-body recombination), atom-molecule and molecule-molecule collisional properties, structure of trimers and larger clusters, numerous phenomena that go under the name of Efimov physics, and many other problems. The chapter is organized as follows. Section 3.2 discusses the effective range expansion for the two-body scattering amplitude and puts a special emphasis on the role of the resonance width. Section 3.3 applies the zero-range approach to the problem of three atoms. It considers the phenomenon of the three-body collapse, or Thomas collapse, and defines the Efimov effect. Then, it introduces a model system of one light and two heavy atoms, where we can use the Born‐Oppenheimer approximation. Section 3.4 describes the method of Skorniakov and Ter‐Martirosian (STM) for few-body problems with resonant short-range interactions. Section 3.5 considers other rapidly developing directions in theoretical few-atom physics.

arXiv:1206.5752Oxford University Press eBooksNov 15, 201211 citations

The Equation for the Wave Function of an N-Particle System in the Many-Body Problem

V. G. Solo'Ev

SPhDDec 1, 19590 citations

A fundamental quantity characterizing Fermi gases with zero-range interactions is the three-body contact

C_3

, which determines several observables including the number of nearby triplets of fermions and the three-body loss rate in cold atom experiments, as shown in a companion article. Here, we compute

C_3

to leading order in the non-degenerate limit for the homogeneous gas with negative or infinite scattering length

a

. At

a=\infty

, using a wavefunction approach, we obtain the analytical expression of

C_3

, which has a remarkably slow

1/T^{0.22728}

dependence on the temperature

T

. In the Feynman diagram technique, which we use for

a<0

, the correct three-body short-distance correlations emerge only after a non-trivial cancellation between leading order two-body and three-body correlations, and the resulting power-law scaling comes from the large-wavevector tail of the 3-body T-matrix, which we derive inspired by the analytical solution of the three-body problem at the unitary limit.

a

and a confinement-induced scattering resonance. The fermionic three-body problem is universal and characterized by two atom-dimer scattering lengths

{a}_{\mathit{ad}}

and

{b}_{\mathit{ad}}

. In the tightly bound ``dimer limit'' ${a}_{\ensuremath{\perp}}∕a\ensuremath{\rightarrow}\ensuremath{\infty}$, we find

{b}_{\mathit{ad}}=0

and

{a}_{\mathit{ad}}

is linked to the 3D atom-dimer scattering length. In the weakly bound ``BCS limit'' ${a}_{\ensuremath{\perp}}∕a\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$, a connection to the Bethe ansatz is established, which allows for exact results. The full crossover is obtained numerically. The bosonic three-body problem, however, is nonuniversal:

{a}_{\mathit{ad}}

and

{b}_{\mathit{ad}}

depend both on ${a}_{\ensuremath{\perp}}∕a$ and on a parameter

{R}^{*}

related to the sharpness of the resonance. Scattering solutions are qualitatively similar to fermionic ones. We predict the existence of a single confinement-induced three-body bound state (trimer) for bosons.