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Three body contact in the unitary Fermi gas

Carl Heintze

Published Jan 1, 2026• 0 citations
Publisher

Abstract

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 $\zeta_3$ from three-body loss measurements. We obtain a value of $\zeta_3$ = 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 $\zeta_3$ in the thermal regime, observing agreement with the theoretical prediction based on the virial expansion. The extracted value of $\zeta_3$ > 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.

DOI
10.11588/heidok.00038186

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•Jan 1, 2013•1 citation
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\zeta_3ζ3​ from three-body loss measurements. We obtain a value of ζ3\zeta_3ζ3​ = 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\zeta_3ζ3​ in the thermal regime, observing agreement with the theoretical prediction based on the virial expansion. The extracted value of ζ3\zeta_3ζ3​ > 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.
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 ↑\uparrow↑ and ↓\downarrow↓ 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.
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.
A fundamental quantity characterizing Fermi gases with zero-range interactions is the three-body contact C3C_3C3​, 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 C3C_3C3​ to leading order in the non-degenerate limit for the homogeneous gas with negative or infinite scattering length aaa. At a=∞a=\inftya=∞, using a wavefunction approach, we obtain the analytical expression of C3C_3C3​, which has a remarkably slow 1/T0.227281/T^{0.22728}1/T0.22728 dependence on the temperature TTT. In the Feynman diagram technique, which we use for a<0a<0a<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.
Abstract We revisit the properties of the two-component Fermi gas with short-range interactions in three dimensions, in the limit where the s -wave scattering length diverges. Such a unitary Fermi gas possesses universal thermodynamic and dynamical observables that are independent of any interaction length scale. Focusing on trapped systems of N fermions, where N ≤ 10 , we investigate how well we can determine the zero-temperature behavior of the many-body system from published few-body data on the ground-state energy and the contact. For the unpolarized case, we find that the Bertsch parameters extracted from trapped few-body systems all lie within 15% of the established value. Furthermore, the few-body values for the contact are well within the range of values determined in the literature for the many-body system. In the limit of large spin polarization, we obtain a similar accuracy for the polaron energy, and we estimate the polaron’s effective mass from the dependence of its energy on N . We also compute an upper bound for the squared wave-function overlap between the unitary Fermi system and the non-interacting ground state, both for the trapped and uniform cases. This allows us to prove that the trapped unpolarized ground state at unitarity has zero overlap with its non-interacting counterpart in the many-body limit N → ∞ .
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.
Ultracold gases have become versatile systems to study quantum many-body effects. Their high degree of control and the tunability of the atomic interactions led to important advances in the understanding of strongly correlated matter. In the regime of strong interactions, unlike fermions, the study of bosons is hampered by three-particle recombination that leads to atom losses and keeps the system from reaching a real equilibrium state. In this thesis, we have performed the first quantitative comparison between theory and experiment on the three-particle loss coefficient L_3(a,T), for arbitrary scattering length a and temperature T. For unitary two-particle scattering (|a| → ∞), we show that the three-particle loss coefficient follows the law L_3(∞, T ) = λ_3/T^2, that we have tested using a trapped non-degenerate ^{7}Li gas maintained at constant temperature. The measured value of λ_3 = 2.5(3)_{stat}(6)_{cyst} × 10^{−20}(μK)^2cm^6s^{−1} is, within the error bars in good agreement with the theory prediction λ_3^{th} = 1.52 × 10^{−20}(μK)^2cm^6s^{−1}. We have extended our measurements to arbitrary values of the scattering length a. For a < 0, the theory predicts a smooth connection between the previously derived zero-temperature model for L_3(a, 0) and the unitarity limited loss coefficient L_3(∞, T ). We also show that a second Efimov resonance in 7Li should be observable near a = − 500 a0 for a temperature of 1 μK. Finally, we compare our theoretical prediction with measurements performed at Innsbruck with ^{133}Cs and at Cambridge with ^{39}K. The theory also shows remarkable quantitative agreement with these measurements.