The Role of the Effective Range in Strongly-Interacting Few-Body Systems

R. Weiss, E. Pazy, Nir Barnea

Lucas Platter

R. E. Barfknecht

In this thesis we study the dynamics of one-dimensional strongly interacting few-body systems. We present the theory which allows us to describe such systems as a spin chain where the exchange coefficients are determined by the trapping geometry, and comment on the experimental feasibility of these mod- els. We then proceed to apply our formalism to different dynamical problems, where the strength of the interactions between the atoms in the trap plays a fundamental role. We find that interesting effects - ranging from changes in magnetic correlations to impurity oscillations and spin-charge separation - arise in this context. Additionally, we perform studies of the static corre- lations and many-body dynamics of bosonic systems away from the strongly interacting limit.

Dmitry S. Petrov

This chapter discusses the few-body problem, which plays a very important role in the physics of ultracold gases. It describes the structure of molecules and their formation in three-body collisions (three-body recombination), atom-molecule and molecule-molecule collisional properties, structure of trimers and larger clusters, numerous phenomena that go under the name of Efimov physics, and many other problems. The chapter is organized as follows. Section 3.2 discusses the effective range expansion for the two-body scattering amplitude and puts a special emphasis on the role of the resonance width. Section 3.3 applies the zero-range approach to the problem of three atoms. It considers the phenomenon of the three-body collapse, or Thomas collapse, and defines the Efimov effect. Then, it introduces a model system of one light and two heavy atoms, where we can use the Born‐Oppenheimer approximation. Section 3.4 describes the method of Skorniakov and Ter‐Martirosian (STM) for few-body problems with resonant short-range interactions. Section 3.5 considers other rapidly developing directions in theoretical few-atom physics.

R. J. Furnstahl

Mohd. Imran, M. A. H. Ahsan

We investigate a rotating system of $N$ spinless bosons confined in quasi-two-dimensional harmonic trap with repulsive finite-range Gaussian interaction potential of large $s$-wave scattering length. Exact diagonalization of the Hamiltonian matrix in subspaces of quantized total angular momenta $L_{z}$ is carried out to obtain the $N$-body ground states in slow rotating regime of $0 \le L_{z} \le 2N$. We study the finite-range effects of the Gaussian repulsion potential on the many-body ground state properties in different subspaces of quantized $L_{z}$. The study ranges from the few-body ($N=2$) system to the many-body ($N=16$), system where quantum (Bose) statistics becomes perceptible. In particular, we analyze the effect of interaction range of the Gaussian potential on the ground state energy of different $L_{z}$ states as well on the critical angular velocity $\Omega_{\bf c1}$ of first vortex ($L_{z}=N$) state. For an optimal value of the range of the Gaussian interaction potential, the nucleation of the first centered vortex may begin at a lower value of the rotational angular velocity as compared to the zero-range interaction potential. Moreover, we explore the role of interaction potential range on the quantum correlation (measured in terms of the degree of condensation and the von Neumann entropy), for small systems ($N=2$ to $8$) where an understanding of few-body effects provide valuable insight into large systems ($N=8$ to $16$) which admit the possibility of many-body correlation. The results obtained indicate that the finite-range Gaussian interaction potential enhances the degree of condensation compared to the zero-range interaction potential.