We investigate the three-body properties of two identical ↑ fermions and one distinguishable ↓ atom interacting in a strongly confined two-dimensional geometry.We compute exactly the atom-dimer scattering properties and the three-body recombination rate as a function of collision energy and mass ratio m ↑ /m ↓ .We find that the recombination rate for fermions is strongly energy dependent, with significant contributions from higher partial waves at low energies.For m ↑ m ↓ , the s-wave atom-dimer scattering below threshold is completely described by the scattering length.Furthermore, we examine the ↑↑↓ bound states (trimers) appearing at large m ↑ /m ↓ and find that the energy spectrum for the deepest bound trimers resembles that of a hydrogen atom confined to two dimensions.
We consider the problem of three distinguishable fermions confined to a quasi-two-dimensional (quasi-2D) geometry, where there is a strong harmonic potential in one direction. We go beyond previous theoretical work and investigate the three-body bound states (trimers) for the case where the two-body short-range interactions between fermions are unequal. Using the scattering parameters from experiments on ultracold
6Li atoms, we calculate the trimer spectrum throughout the crossover from two to three dimensions. We find that the deepest Efimov trimer in the
6Li system is unaffected by realistic quasi-2D confinements, while the first excited trimer smoothly evolves from a three-dimensional-like Efimov trimer to an extended 2D-like trimer as the attractive interactions are decreased. We furthermore compute the excited trimer wave function and quantify the stability of the trimer against decay into a dimer and an atom by determining the probability that three fermions approach each other at short distances. Our results indicate that the lifetime of the trimer can be enhanced by at least an order of magnitude in the quasi-2D geometry, thus opening the door to realizing long-lived trimers in three-component Fermi gases.
This thesis concerns quasi-two-dimensional (quasi-2D) ultracold Fermi systems with short range interactions. Here, quasi-2D geometries are those that lie between two- and three-dimensions. We study few- and many-body systems composed of two or three distinguishable species of fermions. Of principle interest are the bound states that form in these systems and how these bound states are affected by the quasi-2D geometries. In the first research chapter, we focus on the two-dimensional (2D) regime of quasi- 2D. We begin by developing an effective 2D model of this 2D-like geometry. We use this model to analyse a Fermi system consisting of three fermionic species with equal interactions between all distinguishable pairs. Starting with the few-body problem, we show that there is always a three-body bound state (trimer) in the effective 2D model and that these 2D trimers are stable. This is notably different from the extremely unstable ‘Efimov’ trimers found in three-dimensional (3D) systems. Using a low-density expansion and a variational approach, we investigate the fate of the 2D trimers as the system’s density is increased from the three-body limit to the many-body limit. We find that remnants of trimers can persist in the form of strong three-body correlations in the many-body system. In the second research chapter, we analyse a realistic experimental system of three distinguishable fermions confined to quasi-2D. We develop a model that goes beyond the previous chapter in two regards. Firstly, we allow for unequal interactions between the pairs of fermions. Secondly, we use a full quasi-2D model that is not specific to the 2D-like regime. With this general model, we use the scattering and confinement parameters from experiments on ultracold 6Li atoms to calculate the trimer spectrum across the dimensional crossover from 2D to 3D. We show that one state smoothly evolves from an unstable 3D-like trimer to a 2D-like trimer as the attractive interactions are decreased. We furthermore compute the trimer wave function and quantify the stability of the trimer with respect to three-body recombination by determining the probability that three fermions approach each other at short distances. Our results indicate that the lifetime of the trimer can be enhanced by at least an order of magnitude in the quasi-2D geometry. This opens the door to realising long-lifetime trimers in ultracold gas experiments and furthermore, the many-body state of trimers predicted in the preceding chapter. vii In the final research chapter, we examine an alternative approach to accessing quasi-2D geometries by using a system of coupled 2D planes. We study the two-component Fermi gas in this geometry, as a preliminary to studying the three-component gas. We find that the interplanar coupling can be tuned to drive a dimensional crossover from 2D to 3D. We study the effect of this dimensional crossover on the two- and many-body systems. In general, we find that two-body correlations are strengthened by weakening the coupling between 2D planes. In the many-body system, we use mean-field theory to find that the critical temperature of pairing is increased by weakening the coupling between 2D planes. This represents a first step towards a more general representation of quasi-2D geometries, namely: A system of coupled quasi-2D planes.
We study the zero-energy collision of three fermions, two of which are in the spin-down ($\ensuremath{\downarrow}$) state and one of which is in the spin-up ($\ensuremath{\uparrow}$) state. Assuming that the two-body and three-body interactions have a finite range, we find a parameter
D called the three-body scattering hypervolume. We study the three-body wave function asymptotically when three fermions are far apart or one spin-$\ensuremath{\uparrow}$ (spin-$\ensuremath{\downarrow}$) fermion and one pair, formed by the other two fermions, are far apart and derive three asymptotic expansions of the wave function. The three-body scattering hypervolume
D appears in the coefficients of such expansions at the order of ${B}^{\ensuremath{-}5}$, where
B=[(s12+s22+s32)/2]1/2 is the hyperradius of the triangle formed by the three fermions (we assume that the three fermions have the same mass), and
s1,s2,s3 are the sides of the triangle. We compute the
T-matrix element for three such fermions colliding at low energy in terms of
D in the absence of two-body interactions. When the interactions are weak, we calculate
D approximately using the Born expansion. We also analyze the energy shift of three two-component fermions in a large periodic cube due to
D and generalize this result to the many-fermion system.
D also determines the three-body recombination rates in two-component Fermi gases, and we calculate the three-body recombination rates in terms of
D and the density and temperature of the gas.
We solve the three-body problem of a quasi-one-dimensional ultracold Fermi gas with parabolic confinement length a (perpendicular) and 3D scattering length a. On the two-body level, there is a Feshbach-type resonance at a (perpendicular)/a approximately 1.46, and a dimer state for arbitrary a (perpendicular)/a. The three-body problem is shown to be universal, and described by the atom-dimer scattering length a(ad) and a range parameter b(ad). In the dimer limit a (perpendicular)/a>>1, we find a repulsive zero-range atom-dimer interaction. For a (perpendicular)/a<<-1, however, the potential has long range, with a(ad)>0 and b(ad)>>a(ad). There is no trimer state, and despite a(ad)=0 at a( perpendicular)/a approximately 2.6, there is no resonance enhancement of the interaction.
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
aad and
bad. In the tightly bound ``dimer limit'' ${a}_{\ensuremath{\perp}}∕a\ensuremath{\rightarrow}\ensuremath{\infty}$, we find
bad=0 and
aad 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:
aad and
bad 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.