In a recent experiment with ultracold trapped Rb85 atoms, Makotyn et al. studied a quantum-degenerate Bose gas in the unitary limit where its scattering length is infinitely large. We show that the observed momentum distributions are compatible with a universal relation that expresses the high-momentum tail in terms of the two-body contact C2 and the three-body contact C3. We determine the contact densities for the unitary Bose gas with number density n to be C2 ≈ 20 n(4/3) and C3 ≈ 2n(5/3). We also show that the observed atom loss rate is compatible with that from 3-atom inelastic collisions, which gives a contribution proportional to C3, but the loss rate is not compatible with that from 2-atom inelastic collisions, which gives a contribution proportional to C2. We point out that the contacts C2 and C3 could be measured independently by using the virial theorem near and at unitarity, respectively.
In a recent experiment with ultracold trapped Rb85 atoms, Makotyn et al. studied a quantum-degenerate Bose gas in the unitary limit where its scattering length is infinitely large. We show that the observed momentum distributions are compatible with a universal relation that expresses the high-momentum tail in terms of the two-body contact C2 and the three-body contact C3. We determine the contact densities for the unitary Bose gas with number density n to be C2 ≈ 20 n(4/3) and C3 ≈ 2n(5/3). We also show that the observed atom loss rate is compatible with that from 3-atom inelastic collisions, which gives a contribution proportional to C3, but the loss rate is not compatible with that from 2-atom inelastic collisions, which gives a contribution proportional to C2. We point out that the contacts C2 and C3 could be measured independently by using the virial theorem near and at unitarity, respectively.
A fundamental quantity characterizing Fermi gases with zero-range interactions is the three-body contact C3, 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 C3 to leading order in the non-degenerate limit for the homogeneous gas with negative or infinite scattering length a. At a=∞, using a wavefunction approach, we obtain the analytical expression of C3, which has a remarkably slow 1/T0.22728 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.
We compute the virial coefficients, the contact parameters, and the momentum distribution of a strongly interacting three-dimensional Bose gas by means of a virial expansion up to third order in the fugacity, which takes into account three-body correlations exactly. Our results characterize the nondegenerate regime of the interacting Bose gas, where the thermal wavelength is smaller than the interparticle spacing but the scattering length may be arbitrarily large. We observe a rapid variation of the third virial coefficient as the scattering length is tuned across the three-atom and the atom-dimer thresholds. The momentum distribution at unitarity displays a universal high-momentum tail with a log-periodic momentum dependence, which is a direct signature of Efimov physics. We provide a quantitative description of the momentum distribution at high momentum as measured by P. Makotyn et al. [Nat. Phys. 10, 116 (2014)], and our calculations indicate that the lowest trimer state might not be occupied in the experiment. Our results allow for a spectroscopy of Efimov states in the unitary limit.
We renormalize the two-body contact interaction based on the exact solution of two interacting particles in a harmonic trap. This renormalization extends the validity of the contact interaction to large scattering lengths. We apply this renormalized interaction to a degenerate unitary Bose gas to study its stationary properties and elementary excitations using the mean-field theory and the hyperspherical method. Since the scattering length is no longer a relevant length scale at unitarity, universal properties are obtained that depend only on the average particle density. Our treatment shows that the universal relations for the total energy and for the two-body contact are $E/N=12.67{\ensuremath{\hbar}}^{2}\ensuremath{\langle}{n}^{2/3}\ensuremath{\rangle}/2m$ and ${C}_{2}/N=11.8\ensuremath{\langle}{n}^{1/3}\ensuremath{\rangle}$, respectively.
Systems consisting of identical bosons with a large scattering length satisfy universal relations determined by 2-body physics that are similar to those for fermions with two spin states. They require the momentum distribution to have a large-momentum 1/k(4) tail and the radio-frequency transition rate to have a high-frequency 1/ω(3/2) tail, both of which are proportional to the 2-body contact. Identical bosons also satisfy additional universal relations that are determined by 3-body physics and involve the 3-body contact, which measures the probability of 3 particles being very close together. The coefficients of the 3-body contact in the 1/k(5) tail of the momentum distribution and in the 1/ω(2) tail of the radio-frequency transition rate are log-periodic functions of k and ω that depend on the Efimov parameter.
We consider the resonant Fermi gas, that is, two-component fermions in three dimensions interacting by a short-range potential of large scattering length. We introduce a quantity, the three-body contact, that determines several observables. Within the zero-range model, the number of nearby fermion triplets, the large-momentum tail of the center-of-mass momentum distribution of nearby fermion pairs, as well as the large-momentum tail of the two-particle momentum distribution, are expressed in terms of the three-body contact. For a small finite interaction range, the formation rate of deeply bound dimers by three-body recombination, as well as the three-body contribution to the finite-range correction to the energy, are expressed in terms of the three-body contact and of a three-body parameter. This three-body parameter, which vanishes in the zero-range limit, is defined through the asymptotic behavior of the zero-energy scattering state at distances intermediate between the range and the two-body scattering length. In general, the three-body contact has different contributions labeled by spin and angular momentum indices, and the three-body parameter can depend on those indices. We also include the generalization to unequal masses for ↑ and ↓ particles. With respect to the relation between three-body loss rate and number of nearby triplets stated in [Petrov, Salomon and Shlyapnikov, PRL 93, 090404 (2004)], the present work adds a derivation, expresses the proportionality factor in terms of the three-body parameter, and includes the general case where there are several contributions to the three-body contact and several three-body parameters.
The aim of this thesis was to measure the three-body contact in the unitary Fermi gas for the first time. The three-body contact is a physical observable that quantifies the three-body short-range correlations. Our probe, sensitive to three fermions with spins ↑↑↓ or ↓↓↑ at short distances, is three-body loss from the optical trap, caused by recombination into a deeply bound molecule and a free atom. In our system, the two spins signify two different hyperfine states of 6Li. We calibrate that recombination process by measuring three-body loss in the motional ground state of three fermions in an optical tweezer at unitarity. The numerically determined three-body contact C_3 (3) 3 of the three fermion system and the experimentally obtained loss rate allow us to extract Im[a_3], which quantifies the coupling to the deeply bound molecular state. Depending on the initial spin configuration ↑↑↓ or ↓↓↑, we find two different values, differing by almost a factor of two. This difference can only be explained by microscopic details of the interatomic coupling. In the many-body unitary Fermi gas, we use the coupling parameter Im[a_3] to extract the zero-temperature dimensionless three-body contact density ζ3 from three-body loss measurements. We obtain a value of ζ3 = 0.22(4) and observe that the loss rate scales with a characteristic three-body constant s = 1.77, which stems from an underlying SO(2,1) symmetry. We heat the gas and determine ζ3 in the thermal regime, observing agreement with the theoretical prediction based on the virial expansion. The extracted value of ζ3 > 0, which eludes theory, indicates three-body correlations. At the same time, the experimentally confirmed constant s proves the existence of irreducible three-body correlations that cannot be explained by two-body correlations alone.