More than 65 years ago, Jost and Kohn [R. Jost and W. Kohn, Phys. Rev. 87, 977 (1952)] derived an explicit expression for a class of short-range model potentials from a given effective range expansion with the s-wave scattering length as being negative. For as > 0, they calculated another class of short-range model potentials [R. Jost and W. Kohn, Dan. Mat. Fys. Medd 27, 1 (1953)] using a method based on an adaptation from Gelfand-Levitan theory [I. M. Gel'fand and B. M. Levitan, Dokl. Akad. Nauk. USSR 77, 557-560 (1951)] of inverse scattering. We here revisit the methods of Jost and Kohn in order to explore the possibility of modeling resonant finite-range interactions at low energy. We show that the Jost-Kohn potentials can account for zero-energy resonances. The s-wave phase shift for positive scattering length is expressed in an analytical form as a function of the binding energy of a bound state. We show that, for small binding energy, both the scattering length and the effective range are strongly influenced by the binding energy; and below a critical binding energy the effective range becomes negative provided the scattering length is large. As a consistency check, we carry out some simple calculations to show that Jost-Kohn potentials can reproduce the standard results of contact interaction in the limit of the effective range going to zero.

We theoretically study the effects of trap-confinement and interatomic interactions on Josephson oscillations (JO) and macroscopic quantum self-trapping (MQST) for a Bose-Einstein condensate (BEC) confined in a trap which has a symmetric double-well (DW) potential along zaxis and 2D harmonic potentials along x-and y-axis. We consider three types of model interaction potentials: contact, long-range dipolar and finite-range potentials. Our results show that by changing the aspect ratio between the axial and radial trap sizes, one can induce a transition from JO to MQST for contact interactions with a small scattering length. For long-range dipolar interatomic interactions, we analyze transition from Rabi to Josephson regime and Josephson to MQST regime by changing the aspect ratio of the trap for a particular dipolar orientation. For a finite-range interaction, we study the effects of relatively large scattering length and effective range on JO and MQST. We show that JO and MQST are possible even if scattering length is relatively large, particularly near a narrow Feshbach resonance due to the finite-range effects.

We study the dynamics of a nonlinear dissipative bosonic Josephson junction (BJJ) with a time-dependent sinusoidal perturbation in interaction term. We demonstrate parametric resonance where the system undergoes sustained periodic oscillations even in the presence of dissipation. This happens when the frequency of the perturbation is close to twice the frequency of the unperturbed Josephson oscillations and the strength of perturbation exceeds a critical threshold. We have formulated the threshold conditions for parametric oscillations. To explore the nature of the oscillations, we carry out a multiple time scale analysis of the stability boundaries in terms of the V-shaped Arnold's tongue in the parameter space. Full numerical simulations have been performed for the zero-, running-and π-phase modes of nonlinear Josephson effect. Our results demonstrate that in π-phase mode, the system is capable of making a transition from regular parametric to chaotic parametric oscillations as one crosses the stability boundary. Also, the phase difference undergoes phase slip before executing sustained parametric oscillations. PACS numbers: 74.50.+r,67.85.Hj,95.10.Fh,52.35.Mw

We study the dynamics of one-dimensional bosons trapped in a box potential, in the presence of a barrier creating a tunable weak-link, thus realizing a one dimensional Bose-Josephson junction. By varying the initial population imbalance and the barrier height we evidence different dynamical regimes. In particular we show that at large barriers a two mode model captures accurately the dynamics, while for low barriers the dynamics involves dispersive shock waves and solitons. We study a quench protocol that can be readily implemented in experiments and show that self-trapping resonances can occur. This phenomenon can be understood qualitatively within the two-mode model.