General relations for quantum gases in two and three dimensions. II. Bosons and mixtures

TL;DR

This paper establishes exact relations between observables in quantum gases of bosons in two and three dimensions, linking energy derivatives to scattering parameters and loss rates, with applications to Bose gases at unitarity.

Contribution

It derives new exact relations involving energy derivatives and three-body parameters for bosonic quantum gases, extending previous fermionic results and including effects of three-body loss.

Findings

Derived relations between energy derivatives and scattering length a

Connected three-body parameter R_t to three-particle proximity probability

Provided an analytic expression for three-body loss rate at infinite scattering length

Abstract

We derive exact general relations between various observables for N bosons with zero-range interactions, in two or three dimensions, in an arbitrary external potential. Some of our results are analogous to relations derived previously for two-component fermions, and involve derivatives of the energy with respect to the two-body s-wave scattering length a. Moreover, in the three-dimensional case, where the Efimov effect takes place, the interactions are characterized not only by a, but also by a three-body parameter R\_t. We then find additional relations which involve the derivative of the energy with respect R\_t. In short, this derivative gives the probability to find three particles close to each other. Although it is evaluated for a totally loss-less model, it remarkably also gives the three-body loss rate always present in experiments (due to three-body recombination to deeply bound diatomic molecules), at least in the limit where the so-called inelasticity parameter eta is small enough. As an application, we obtain, within the zero-range model and to first order in eta, an analytic expression for the three-body loss rate constant for a non-degenerate Bose gas with infinite scattering length. We also discuss the generalization to arbitrary mixtures of bosons and/or fermions.