We study the quantum three-body free space problem of two same-spin-state fermions of mass
m interacting with a different particle of mass
M, on an infinitely narrow Feshbach resonance with infinite
s-wave scattering length. This problem is made interesting by the existence of a tunable parameter, the mass ratio
α=m/M. By a combination of analytical and numerical techniques, we obtain a detailed picture of the spectrum of three-body bound states, within {\sl each} sector of fixed total angular momentum
l. For
α increasing from
0, we find that the trimer states first appear at the
l-dependent Efimovian threshold
αc(l), where the Efimov exponent
s vanishes, and that the {\sl entire} trimer spectrum (starting from the ground trimer state) is geometric for
α tending to
αc(l) from above, with a global energy scale that has a finite and non-zero limit. For further increasing values of
α, the least bound trimer states still form a geometric spectrum, with an energy ratio
exp(2π/∣s∣) that becomes closer and closer to unity, but the most bound trimer states deviate more and more from that geometric spectrum and eventually form a hydrogenoid spectrum.
We study the quantum three-body free space problem of two same-spin-state fermions of mass
m interacting with a different particle of mass
M, on a narrow Feshbach resonance with infinite
s-wave scattering length. This problem is made interesting by the existence of a tunable parameter, the mass ratio
α=m/M. By a combination of analytical and numerical techniques, we obtain a detailed picture of the spectrum of three-body bound states, within {\sl each} sector of fixed total angular momentum
l. For
α increasing from
0, we find that the trimer states first appear at the Efimovian threshold
αc, where the Efimov exponent
s vanishes, and that the {\sl entire} trimer spectrum (starting from the ground trimer state) is geometric for
α→αc+, with a global energy scale that has a finite and non-zero limit. For further increasing values of
α, the least bound trimer states still form a geometric spectrum, with an energy ratio
exp(2π/∣s∣) that becomes closer and closer to unity, but the most bound trimer states deviate more and more from that geometric spectrum and eventually form a hydrogenoid spectrum.
In the free three-dimensional space, we consider a pair of identical $\ensuremath{\uparrow}$ fermions of some species or in some internal state and a pair of identical $\ensuremath{\downarrow}$ fermions of another species or in another state. There is a resonant
s-wave interaction (that is, of zero range and infinite scattering length) between fermions in different pairs and no interaction within the same pair. We study whether this
2+2 fermionic system can exhibit (as the
3+1 fermionic system) a four-body Efimov effect in the absence of three-body Efimov effect, that is, the mass ratio $\ensuremath{\alpha}$ between $\ensuremath{\uparrow}$ and $\ensuremath{\downarrow}$ fermions and its inverse are both smaller than 13.6069\dots{}. For this purpose, we investigate scale invariant zero-energy solutions of the four-body Schr\"odinger equation, that is, positively homogeneous functions of the coordinates of degree $s\ensuremath{-}7/2$, where
s is a generalized Efimov exponent that becomes purely imaginary in the presence of a four-body Efimov effect. Using rotational invariance in momentum space, it is found that the allowed values of
s are such that
M(s) has a zero eigenvalue; here the operator
M(s), that depends on the total angular momentum $\ensuremath{\ell}$, acts on functions of two real variables (the cosine of the angle between two wave vectors and the logarithm of the ratio of their moduli), and we write it explicitly in terms of an integral matrix kernel. We have performed a spectral analysis of
M(s), analytical and for an arbitrary imaginary
s for the continuous spectrum and numerical and limited to
s=0 and $\ensuremath{\ell}\ensuremath{\le}12$ for the discrete spectrum. We conclude that no eigenvalue of
M(0) crosses zero over the mass ratio interval $\ensuremath{\alpha}\ensuremath{\in}[1;13.6069\ensuremath{\cdots}]$, even if, in the parity sector ${(\ensuremath{-}1)}^{\ensuremath{\ell}}$, the continuous spectrum of
M(s) has everywhere a zero lower border. As a consequence, there is no possibility of a four-body Efimov effect for the
2+2 fermions. We also enunciated a conjecture for the fourth virial coefficient of the unitary spin-
1/2 Fermi gas, inspired from the known analytical form of the third cluster coefficient and involving the integral over the imaginary
s axis of
s times the logarithmic derivative of the determinant of
M(s) summed over all angular momenta. The conjectured value is in contradiction with the experimental results.
We calculate the spectrum of three-body Efimov bound states near a Feshbach resonance within a model which accounts both for the finite range of interactions and the presence of background scattering. The latter may be due to direct interactions in an open channel or a second overlapping Feshbach resonance. It is found that background scattering gives rise to substantial changes in the trimer spectrum as a function of the detuning away from a Feshbach resonance, in particular in the regime where the background channel supports Efimov states on its own. Compared to the situation with negligible background scattering, the regime where van der Waals universality applies is shifted to larger values of the resonance strength if the background scattering length is positive. For negative background scattering lengths, in turn, van der Waals universality extends to even small values of the resonance strength parameter, consistent with experimental results on Efimov states in
39K. Within a simple model, we show that short-range three-body forces do not affect van der Waals universality significantly. Repulsive three-body forces may, however, explain the observed variation between around $\ensuremath{-}8$ and $\ensuremath{-}10$ of the ratio between the scattering length where the first Efimov trimer appears and the van der Waals length.
The Efimov states are three-body bound states, which appear universally when the s-wave scattering length between the particles is resonantly large.Since the Efimov states and their characteristic discrete scale invariance have recently been observed in ultracold atoms, the Efimov states and their associated few-body phenomena have attracted widening interest both theoretically and experimentally.In this thesis, I theoretically study the Efimov physics and related phenomena, focusing on the following three topics: (i) Universal three-body physics for a mass-imbalanced Fermi system (Chapter 3).For a system of two identical spinless (i.e., spin-polarized) fermions and one distinguishable particle which interact via a short-range potential with a large s-wave scattering length, two classes of universal three-body bound states have been known to appear in different regimes of the mass ratio: the Efimov trimers and the Kartavtsev-Malykh trimers, which feature the discrete and continuous scale invariance, respectively.I have found the third class of universal three-body bound states, which I call the "crossover trimers".The crossover trimers show neither the discrete nor continuous scale invariance.I have identified the regions of these three classes of trimers as a function of the mass ratio and the s-wave scattering length, and shown that the Kartavtsev-Malykh trimers and the Efimov trimers can continuously transform into each other via the crossover trimers as the mass ratio and the s-wave scattering length are varied.Owing to the presence of the crossover trimers, the Kartavtsev-Malykh trimers dissociate into a particle and a dimer when the s-wave scattering length is varied, inducing resonances in the particle-dimer scattering.I have calculated the elastic particle-dimer scattering lengths in arbitrary angular-momentum channels, and shown that the particle-dimer resonances indeed occur.From the resonance positions, I have found accurate values of the critical mass ratios at which the Kartavtsev-Malykh trimers in the higher angular-momentum channels appear. (ii) Universal three-body parameter (Chapter 4).In the Efimov physics, the three-body parameter fixes the short-range three-body phase and hence the scale of the energy spectrum.Until quite recently, it has been widely held that the threebody parameter should depend on the short-range part of the inter-atomic potential, and hence vary almost randomly between different atomic species and hyperfine states.However, mounting evidence in recent experiments suggests otherwise.While some theoretical studies have succeeded in reproducing the universal behavior in the three-body parameter, the underlying physical mechanism has remained unclear and controversial.In this thesis, I elucidate the physical origin of the universal three-body parameter.I propose that a non-adiabatic deformation of the three-body wave function induced by a universal two-body correlation results in a universal three-body repulsion, which prevents three particles from coming close and renders the three-body parameter universal.This mechanism is verified by reproducing the universal three-body repulsion with a simple model wave function.The mechanism found here suggests that the three-body parameter would also be universal for non-van der Waals types of potentials.For various classes of deep two-body potentials, it is shown that the effective range is the relevant length scale to characterize the two-body correlation and sets the three-body parameter.Depending on the shape of the two-body correlation, there
Abstract An unusual class of equal mass p-wave universal trimers with symmetry L Π = 1 ± is identified, for both a two-component fermionic trimer with s- and p-wave scattering length close to unitarity and for a one-component fermionic trimer at p-wave unitarity. Moreover, fermionic trimers made of atoms with two internal spin components are found for L Π = 1 ± , when the p-wave interaction between spin-up and spin-down fermions is close to unitarity and/or when the interaction between two spin-up fermions is close to the p-wave unitary limit. The universality of these p-wave universal trimers is tested here by considering van der Waals interactions in a Lennard–Jones potential with different numbers of two-body bound states; our calculations also determine the value of the scattering volume or length where the trimer state hits zero energy and can be observed as a recombination resonance. The faux-Efimov effect appears with trimer symmetry L Π = 1 − when the two fermion interactions are close to p-wave unitarity and the lowest 1 / R 2 coefficient gets modified, thereby altering the usual Wigner threshold law for inelastic processes involving three-body continuum channels.