Abstract This work analyzes the influence of an external confinement on the energy spectrum of two interacting ultracold atoms, considering either a regularized contact or a more realistic Born-Oppenheimer potential. For two harmonically confined particles with contact interaction, the known analytical solution exhibits universality in the sense that the energy spectra solely depend on the ratio of the confinement length to the scattering length. In this work it is shown that this type of universality breaks down for realistic interatomic interactions, leading to an explicit trap-size dependence of the energy spectrum. This effect is especially pronounced for very tight confinement. On the other hand, using first-order perturbation theory it is shown analytically that the here discussed universality persists for the contact interaction even in the case of anharmonic confinement, this result being also verified numerically beyond perturbation theory for a sextic trap potential. These findings demonstrate limitations of the popular contact-interaction model and highlight the necessity of considering more realistic finite-range interactions under tight trapping conditions.
Abstract We study the effects of the effective range of interaction on the eigenvalues and eigenstates of two particles confined in a three-dimensional (3D) isotropic as well as one- or quasi-one dimensional harmonic (1D) traps. For this we employ model potentials which mimic finite-range s -wave interactions over a wide range of s -wave scattering length a s including the unitarity limits a s → ± ∞ . Our results show that when the range is larger than the 3D or 1D harmonic oscillator length scale, the eigenvalues and eigenstates are nearly similar to those of noninteracting two particles in the 3D or 1D trap, respectively. In case of 3D, we find that when the range goes to zero, the results of contact potential as derived by Busch et al (1998 Foundations of Physics 28 549 ) are reproduced. However, in the case of 1D, such reproducibility does not occur as the range goes to zero. We have calculated the eigenvalues and eigenstates in a 1D harmonic trap taking one dimensional finite-range model potential. We have also calculated the bound state properties of two particles confined in a highly anisotropic quasi-1D trap taking three-dimensional finite-range model potential, and examined whether these quasi-1D results approach towards 1D ones as the aspect ratio η of the radial to axial frequency of the trap increases. We find that if the range is very small compared to the axial size of the trap, then one can reach 1D regime for η ≥ 10000 . However, for a large range, one can almost get 1D results for smaller values of η . This study will be important for the exploration of two-body or many body physics of trapped ultracold atoms interacting with narrow Feshbach resonance for which the effective range can be large.
A previously developed approach for the numerical treatment of two particles that are confined in a finite optical-lattice potential and interact via an arbitrary isotropic interaction potential has been extended to incorporate an additional anisotropic dipole–dipole interaction (DDI). The interplay of a model but realistic short-range Born–Oppenheimer potential and the DDI for two confined particles is investigated. A variation of the strength of the DDI leads to diverse resonance phenomena. In a harmonic confinement potential some resonances show similarities to s -wave scattering resonances while in an anharmonic trapping potential like the one of an optical lattice additional inelastic confinement-induced dipolar resonances occur. The latter are due to a coupling of the relative and center-of-mass motion caused by the anharmonicity of the external confinement.
We examine the problem of two particles confined in an isotropic harmonic trap, which interact via a finite-range Gaussian-shaped potential in two spatial dimensions. We derive an approximative transcendental equation for the energy and study the resulting spectrum as a function of the interparticle interaction strength. Both the attractive and repulsive systems are analyzed. We study the impact of the potential's range on the ground-state energy. We also explicitly verify by a variational treatment that in the zero-range limit the positive $\ensuremath{\delta}$ potential in two dimensions only reproduces the noninteracting results, if the Hilbert space in not truncated, and demonstrate that an extremely large Hilbert space is required to approach the ground state when one is to tackle the limit of zero-range interaction numerically. Finally, we establish and discuss the connection between our finite-range treatment and regularized zero-range results from the literature. The present results indicate that a finite-range interparticle potential is numerically amenable for treating the statics and the nonequilibrium dynamics of interacting many-particle systems (bosons) in two dimensions.
We consider two ultracold particles confined in a spherically symmetric harmonic trap and interacting via isotropic potential with absorbing boundary conditions at short range that models reactive scattering. First, we apply the contact pseudopotential with complex scattering length and investigate the properties of eigenenergies and eigenfunctions for different trap states as a function of the real and imaginary parts of the scattering length. In the analyzed case, the eigenenergies take complex values and their imaginary part can be interpreted in terms of decay rates. Then we introduce the model of a square well with absorbing boundary conditions that is later used to investigate the effects of the finite range of interaction on the properties of a reactive scattering in the trap. Finally, we analyze the decay rates for some reactive alkali dimer molecules assuming short-range probability of reaction equal to unity. In this case, the complex scattering length is given by the universal value related to the mean scattering length of the van der Waals interaction between molecules.
The two-body scattering amplitude and energy spectrum of confined ultracold atoms are of fundamental importance for both theoretical and experimental studies of ultracold-atom physics. For many systems, one can efficiently calculate these quantities via the zero-range Huang-Yang pseudopotential (HYP), in which the interatomic interaction is characterized by the scattering length
a. Furthermore, when the scattering length is dependent on the kinetic energy ${\ensuremath{\varepsilon}}_{\mathrm{r}}$ of two-atom relative motion, i.e., $a=a({\ensuremath{\varepsilon}}_{\mathrm{r}})$, the results are applicable for a broad energy region. However, when the free Hamiltonian of atomic internal state (e.g., the Zeeman Hamiltonian) does not commute with the interatomic interaction, or the center-of-mass (c.m.) motion is coupled to the relative motion, the generalization of this technique is still lacking. In this work we solve this problem and construct a reasonable energy-dependent multichannel HYP, which is characterized by a ``scattering length operator'' ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{\mathrm{eff}}$, for the above complicated cases. Here, ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{\mathrm{eff}}$ is an operator for atomic internal states and c.m. motion, and depends on both the total two-atom energy and the external field as well as the trapping parameter. The effects from the internal-state or c.m.-relative motion coupling can be self-consistently taken into account by ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{\mathrm{eff}}$. We further show a method based on the quantum defect theory, with which ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{\mathrm{eff}}$ can be analytically derived for systems with van der Waals interatomic interaction. To demonstrate our method, we calculate the spectrum of two ultracold fermionic alkaline-earth-like atoms [in electronic
1S0 $(|g\ensuremath{\rangle})$ and
3P0 $(|e\ensuremath{\rangle})$ states, respectively] confined in an optical lattice. By comparing our results with the recent experimental measurements for two
173Yb atoms and two
171Yb atoms, we calibrate the scattering lengths ${a}_{\ifmmode\pm\else\textpm\fi{}}$ with respect to antisymmetric and symmetric nuclear-spin states to be
a+=2012(19)a0 and ${a}_{\ensuremath{-}}=193(4){a}_{0}$ for
173Yb, and
a+=232(3)a0 and ${a}_{\ensuremath{-}}=372(1){a}_{0}$ for
171Yb.
In contrast to conventional ideal Bose and Fermi degenerate gases, atoms in experimental ultracold gases interact among themselves and are trapped by an external potential. In this work, we consider two particles of mass m trapped by a 3D harmonic potential of frequency ω, and interacting through an isotropic short range potential of intensity V0 and range b/2, with Eigenfunctions and eigenenergies are obtained and compared with those resulting (a) from an effective contact interaction and (b) in the absence of the trapping potential. Concepts like zero-resonance and binding energy, usually introduced for problems that allow acontinuum spectra, are discussed for trapped particles.