Session N3: Few Body Interactions

Efimov physics in an ultracold Bose-Fermi gas of $^{40}$K and $^{87}$Rb atoms

We present measurements of Efimov physics in an ultracold Bose-Fermi gas of $^{40}$K and $^{87}$Rb atoms near an interspecies Feshbach resonance. In particular, we measure loss rates due to inelastic collisions in the trapped gas. We find a resonance in the inelastic collisions of Feshbach molecules with $^{87}$Rb atoms, but no feature in the measured rates of inelastic collisions of three atoms.   

Efimov scaling symmetry in ultracold Li-Cs mixtures

In 1970, Vitaly Efimov predicted the existence of an infinite series of universal trimer states in a three-body system as well as a universal geometric scaling in those trimer states. After the first observation of the Cs-Cs-Cs trimer state in 2005, there have been many other observations of these homonuclear trimers in different atomic species. However, the universal scaling symmetry predicted by Efimov remains elusive in homonuclear systems due to a large scaling constant $\lambda $0 $=$ 22.7. For heteronuclear atomic systems this scaling constant can decrease dramatically for systems with large atomic mass ratios. I will report our investigation on the discrete scaling symmetry of Efimov quantum states for a heteronuclear mixture of Li-6 and Cs-133 atoms. We have identified three enhanced loss features associated with coupling to three different trimer states near an interspecies Feshbach resonance, from which we determine a geometric scaling factor. Our result represents the first model-independent experimental testing of the discrete scaling symmetry in three--body systems.   

Deterministic Control of two Fermions in a Double Well

The behavior of an ensemble of fermionic particles confined in a periodic potential is one of the richest topics of condensed matter physics. The simplest and most widely used theoretical description of such systems is provided by the Fermi-Hubbard Hamiltonian. We realize this Hamiltonian by deterministically preparing systems of two fermionic atoms trapped in a double well potential in a quantum state of our choice. We have studied the tunneling dynamics of this system as a function of the interparticle interactions and found good agreement with theoretical expectations. We have thus obtained a single-site addressable realization of the Fermi-Hubbard model where all parameters can be fully controlled and freely tuned. As a first experiment we prepared systems of one $|\uparrow\rangle$ and one $|\downarrow\rangle$ atom in the ground state of the double well, introduced repulsive (attractive) interparticle interactions and observed the crossover into a Mott-insulating (charge-density-wave) regime by measuring the occupation statistics of the individual sites. By adding a third well to the system this approach could be be used to directly observe ordered charge-density-waves and antiferromagnetic ordering.   

Two-body and Three-body Contacts for Identical Bosons near Unitarity

In a recent experiment with ultracold trapped $^{85}$Rb atoms, Makotyn et al. have 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 2-body contact $C_2$ and the 3-body contact $C_3$. We determine the contact densities for the unitary Bose gas with number density $n$ to be ${\cal C}_2 \approx 20~n^{4/3}$ and ${\cal C}_3 \approx 2~n^{5/3}$. We also show that the observed atom loss rate is compatible with that from 3-atom inelastic collisions, which gives a contribution proportional to $C_3$, but the loss rate is not compatible with that from 2-atom inelastic collisions, which gives a contribution proportional to $C_2$. We point out that the contacts $C_2$ and $C_3$ could be measured independently by using the virial theorem near and at unitarity, respectively.   

Spectroscopy for a few atoms trapped in a one-dimensional harmonic well

Spectroscopic labels for a few particles that are harmonically trapped in one-dimension and are interacting through zero-range interactions are uniquely specified by three quantum numbers that characterize the symmetries of the Hamiltonian: permutations of identical particles, parity inversion, and the separability of the center-of-mass. The exact solutions for the non-interacting and infinitely repulsive cases are reduced with respect to these symmetries. This reduction explains how states of single-component and multi-component fermions and bosons transform under adiabatic evolution from non-interacting to strong hard-core repulsion. These spectroscopic methods also clarify previous analytic and numerical results for intermediate values of interaction strength. Several examples, including adiabatic mapping for two-component fermionic states in the cases $N=3-5$, are provided.   

Quantum corrections to 3-body recombination in a dilute Bose-Einstein condensate

The rate at which the number density $n$ of a dilute Bose-Einstein condensate decreases from 3-body recombination scales like $n^3 a^4$, where $a$ is the scattering length. We calculate the corrections to the rate from quantum fluctuations in the condensate to second order in the diluteness variable $\sqrt{n a^3}$. The second order term depends on Efimov's 3-body parameter.   

Universal Dimer-Dimer and Atom-Trimer Scattering for Identical Bosons

Identical bosons with a large positive scattering length $a$ form universal bound states, including a shallow dimer, Efimov trimers, and universal tetramers. We present analytic approximations to the universal low-energy dimer-dimer and atom-trimer scattering amplitudes. The numerical parameters in the amplitude are determined using accurate 4-body results calculated by Deltuva. The coupled-channel scattering amplitudes are exactly unitary if the Efimov 3-body parameter is real. The analytic expressions for the scattering amplitudes allow the effects of deeply-bound diatomic molecules to be taken into account through the analytic continuation of the Efimov parameter.   

Long-Lived Complexes, Ergodicity and Chaos in Ultracold Molecular Collisions

Estimates for the lifetime of collision complexes formed during ultracold molecular collisions based on density-of-states arguments are shown to be consistent with similar estimate based on classical trajectory calculations. In the classical version, these collisions are shown to exhibit chaos, and their fractal dimensions are calculated versus collision energy. From these results, a picture emerges that ultracold collisions are classically ergodic, justifying the density-of-states estimates for lifetimes. These results point the way toward using the techniques of classical and quantum chaos to interpret molecular collisions in the ultracold regime.   

Efimov states and upper branch dynamics in a unitary Bose-Fermi mixture

This work explores the spectrum of a Bose-Fermi mixture and corresponding dynamics near a broad Feshbach resonance from a few-body point of view. We show that at unitarity, heteronuclear Efimov states have universal properties in scattering from the remaining atoms in the cloud. Unlike the weak interacting limit, in unitary Bose-Fermi mixtures, there is no length scale that characterizes the interactions, owing to the diverging inter-species scattering length. Moreover, the separation of energy scales of dimer states and cluster states (trimer, tetramer and beyond) causes the effective fermionic dimer-dimer scattering to have weak non-universal behavior that is from the non-adiabtic couplings to the cluster states. This allows us to utilize the upper branches of the few-body spectrums to infer the dynamics of magneto-association of fermionic dipolar molecules at relatively high temperature and at a relatively fast time scale compared to cluster formation processes. We also compare predictions of quench dynamics from a few-body perspective to time dependent mean-field calculations.   

The role of dimensionality on effective two- and three-body interactions of trapped ultracold bosons

We analyze the perturbative ground-state energies of N ultracold bosons in 1D, 2D isotropic, and 3D cylindrically symmetric harmonic potentials using renormalized perturbation theory. We assume pair-wise, zero-range (delta-function) interactions with boson-boson coupling parameter g, and in all three cases we obtain analytic expressions for the effective 2- and 3-body interaction energies to second order in g. As a function of the ratio of the transverse and longitudinal trapping frequencies, we show that the quasi-1D and quasi-2D limits of the 3D expressions agree with the ``true'' zero-range interaction 1D and 2D results. We also compare to numerical simulations using a finite-range interaction potential in a 3D trap. We anticipate that our results can be useful for experiments with anisotropically trapped ultracold atoms, when effective 3-body interactions play a significant role.