We use the diagrammatic $T$-matrix approach to analyze the three-body scattering problem between two identical fermions and a third particle (which could be a different species of fermion or a boson). We calculate the s-wave dimer-atom scattering length for all mass ratios, and our results exactly match the results of Petrov. In particular, we list the exact dimer-atom scattering lengths for all available two-species Fermi-Fermi and Bose-Fermi mixtures. In addition, unlike that of the equal-mass particles case where the three-body scattering $T$-matrix decays monotonically as a function of the outgoing momentum, we show that, after an initial rapid drop, this function changes sign and becomes negative at large momenta and then decays slowly to zero when the mass ratio of the fermions to the third particle is higher than a critical value (around 6.5). As the mass ratio gets higher, modulations of the $T$-matrix become more apparent with multiple sign changes, related to the "fall of a particle to the center" phenomenon and to the emergence of three-body Efimov bound states.
We consider the problem of obtaining the scattering length for a fermion colliding with a dimer, formed from a fermion identical to the incident one and another different fermion. This is done in the universal regime where the range of interactions is short enough so that the scattering length $a$ for non identical fermions is the only relevant quantity. This is the generalization to fermions with different masses of the problem solved long ago by Skorniakov and Ter-Martirosian for particles with equal masses. We solve this problem analytically in the two limiting cases where the mass of the s…
The dimer-atom-atom recombination process in the system of four identical bosons with resonant interactions is studied. The description uses the exact Alt, Grassberger and Sandhas equations for the four-particle transition operators that are solved in the momentum-space framework. The dimer-dimer and atom-trimer channel contributions to the ultracold dimer-atom-atom recombination rate are calculated. The dimer-atom-atom recombination rate greatly exceeds the three-atom recombination rate.
Using the framework of effective field theory, we present a detailed study of the Efimov effect in higher partial waves for systems of two identical particles and a third distinguishable particle. Depending on the total angular momentum $L$, the two identical particles must be bosons or fermions. We derive analytical expressions for the elastic and inelastic atom-dimer scattering cross sections as well as the atom-dimer relaxation rate at the dimer breakup threshold. For the experimentally most relevant case of P-waves, we numerically calculate the atom-dimer scattering cross sections and rela…
We consider two identical fermions interacting in the p-wave channel. Each fermion also interacts with another particle in the vicinity of an s-wave resonance. We find that in addition to the Kartavtsev-Malykh universal trimer states resulting from the s-wave particle-fermion interaction, the fermion-fermion p-wave interaction induces one or two shallow trimers in a large domain of the control parameters, including a borromean regime where the ground-state trimer exists in the absence of dimers at any mass ratio between the fermions and the particle. A generic picture of the trimer spectrum em…
We study the three-body problem of two ultracold identical bosonic atoms (denoted by $B$) and one extra atom (denoted by $X$), where the scattering length $a_{BX}$ between each bosonic atom and atom $X$ is resonantly large and positive. We calculate the scattering length $a_{\rm ad}$ between one bosonic atom and the shallow dimer formed by the other bosonic atom and atom $X$, and investigate the effect induced by the interaction between the two bosonic atoms. We find that even if this interaction is weak (i.e., the corresponding scattering length $a_{BB}$ is of the same order of the van der Wa…
The system of four identical bosons is studied using momentum-space equations for the four-particle transition operators. Positions, widths and existence limits of universal unstable tetramers are determined with high accuracy. Their effect on the atom-trimer and dimer-dimer scattering observables is discussed. We show that a universal shallow tetramer intersects the atom-trimer threshold twice leading to resonant effects in ultracold atom-trimer collisions.
We investigate the three-body properties of two identical "up" fermions and one distinguishable "down" 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_up/m_down. We find that the recombination rate for fermions is strongly energy dependent, with significant contributions from higher partial waves at low energies. For m_up < m_down, the s-wave atom-dimer scattering below threshold is completely described by the scattering length. Furt…
Correlated dynamics of three-particle bound states induced by emergent impurities in Bose-Hubbard model
Sun Yat-Sen University（中山大学）；Shenzhen University（深圳大学）；Quantum Science Center of Guangdong-Hong Kong-Macao Greater Bay Area (Guangdong)（粤港澳大湾区量子科学中心（广东））
We study two identical fermions, or two hard-core bosons, in an infinite chain and coupled to phonons by interactions that modulate their hopping as described by the Peierls/Su-Schrieffer-Heeger (SSH) model. We show that exchange of phonons generates effective nearest-neighbor repulsion between particles and also gives rise to interactions that move the pair as a whole. The two-polaron phase diagram exhibits two sharp transitions, leading to light dimers at strong coupling and the flattening of the dimer dispersion at some critical values of the parameters. This dimer (quasi)self-trapping occu…
A zero range approach is used to model resonant two-body interactions between three identical bosons. A dimensionless phase parametrizes the three-body boundary condition while the scattering length enters the Bethe-Peierls boundary condition. The model is solved exactly at zero energy for any value of the scattering length, positive or negative. From this solution, an analytical expression for the rate of three-body recombination to the universal shallow dimer is extracted.
Strongly-interacting ultra-cold atoms in tight-binding optical lattice potentials provide an ideal platform to realize the fundamental Hubbard model. Here, after outlining the elementary single particle solution, we review and expand our recent work on complete characterization of the bound and scattering states of two and three bosonic atoms in a one-dimensional optical lattice. In the case of two atoms, there is a family of interaction-bound "dimer" states of co-localized particles that exists invariantly for either attractive or repulsive on-site interaction, with the energy below or above …
Within the Bose-Hubbard model, we theoretically determine the stationary states of two distinguishable atoms in a one-dimensional optical lattice and compare with the case of two identical bosons. A heterodimer has odd-parity dissociated states that do not depend on the interactions with the atoms, and the lattice momenta of the two atomic species may have different averages even for a bound state of the dimer. We discuss two possible methods of detecting the dimer. First, the different distributions of the quasimomenta of the two species may be observed in suitable time-of-flight experiments.…
Highly-elongated quasi-one-dimensional cold atom samples have been studied extensively over the past years experimentally and theoretically. This work determines the energy spectrum of two identical fermions and a third distinguishable particle as functions of the mass ratio $κ$ and the free-space $s$-wave scattering length $a_{3\text{D}}$ between the identical fermions and the distinguishable third particle in a cylindrically symmetric waveguide whose symmetry axis is chosen to be along the $z$-axis. We focus on the regime where the mass of the identical fermions is equal to or larger than th…
The zero-energy universal properties of scattering between a particle and a dimer that involves an identical particle are investigated for arbitrary scattering angular momenta. For this purpose, we derive an integral equation that generalises the Skorniakov - Ter-Martirosian equation to the case of non-zero angular momentum. As the mass ratio between the particles is varied, we find various scattering resonances that can be attributed to the appearance of universal trimers and Efimov trimers at the collisional threshold.
We use the zero-range approximation to study a system of two identical bosons interacting resonantly with a third particle. The method is derived from effective field theory. It reduces the three-body problem to an integral equation which we then solve numerically. We also develop an alternative approach which gives analytic solutions of the integral equation in coordinate representation in the limit of vanishing total energy. The atom-dimer scattering length, the rates of atom-dimer relaxation and three-body recombination to shallow and to deep molecular states are calculated either analytica…
We study Efimov physics for three identical bosons interacting via a pairwise square-well potential, analyze the validity of the separable approximation as a function of the interaction strength, and investigate what is needed to improve this approximation. We find separable approximations to be accurate for potentials with just one (nearly) bound dimer state. For potentials with more bound or almost bound dimer states, these states need to be included for an accurate determination of the Efimov spectrum and the corresponding three-body observables. We also show that a separable approximation …
The Efimov effect in heteronuclear cold atomic systems is experimentally more easily accessible than the Efimov effect for identical atoms, because of the potentially smaller scaling factor. We focus on the case of two or three heavy identical bosons and another atom. The former case was recently observed in a mixture of 133Cs and 6Li atoms. We employ the Gaussian Expansion Method as developed by Hiyama, Kino et al.. This is a variational method that uses Gaussians that are distributed geometrically over a chosen range. Supplemental calculations are performed using the Skorniakov-Ter-Martirosi…
We investigate the problem of $N$ identical bosons that are coupled to an impurity particle with infinite mass. For non-interacting bosons, we show that a dynamical impurity-boson interaction, mediated by a closed-channel dimer, can induce an effective boson-boson repulsion which strongly modifies the bound states consisting of the impurity and $N$ bosons. In particular, we demonstrate the existence of two universal 'multi-body' resonances, where all multi-body bound states involving any $N$ emerge and disappear. The first multi-body resonance corresponds to infinite impurity-boson scattering …
We discuss the three-body properties of identical bosons exhibiting large scattering length in two spatial dimensions. Within an effective field theory for resonant interactions, we calculate the leading non-universal corrections from the two-body effective range to bound-state and scattering observables. In particular, we compute the three-body binding energies, the boson-dimer scattering properties, and the three-body recombination rate for finite energies. We find significant effective range effects in the vicinity of the unitary limit. The implications of this result for future experiments…
The effective interactions between two nuclear clusters, $d+d$, $t+t$, and $α+α$, are investigated within a cluster model using local nucleon-nucleon~($NN$) forces. It is shown that the interaction in the spin-aligned $d+d$ system is repulsive for all inter-cluster distances, whereas the $α+α$ and spin-aligned $t+t$ systems are attractive at intermediate distances. The Pauli blocking between identical-nucleon pairs is responsible for the cluster-cluster repulsion and becomes dominant in the shallow binding limit. We demonstrate that two $d$-clusters could be bound if the $NN$ force has nonzero…
