Stationary and dynamical properties of two harmonically trapped bosons in the crossover from two dimensions to one

Abstract

We unravel the stationary properties and the interaction quench dynamics of two bosons, confined in a two-dimensional anisotropic harmonic trap. A transcendental equation is derived giving access to the energy spectrum and revealing the dependence of the energy gaps on the anisotropy parameter. The relation between the two and the one dimensional scattering lengths as well as the Tan contacts is established. The contact, capturing the two-body short range correlations, shows an increasing tendency for a larger anisotropy. Subsequently, the interaction quench dynamics from attractive to repulsive values and vice versa is investigated for various anisotropies. A closed analytical form of the expansion coefficients of the two-body wavefunction, during the time evolution is constructed. The response of the system is studied by means of the time-averaged fidelity, the spectra of the spatial extent of the cloud in each direction and the one-body density. It is found that as the anisotropy increases, the system becomes less perturbed independently of the interactions while for fixed anisotropy quenches towards the non-interacting regime perturb the system in the most efficient manner. Furthermore, we identify that in the tightly confined direction more frequencies are involved in the dynamics stemming from higher-lying excited states.

Two-body quench dynamics of harmonically trapped interacting particles

“…6, which plots r(t = π/2ω) as a function of the maximum number of terms (N max ) in the sum in Eq. (34). As can be seen r(t = π/2ω) diverges logarithmically.…”

“…There have been experimental measures of the contrast [26] in the regime of impurity atoms residing in a Fermi sea of other atoms and there have been experimental measures of the separation between 6 Li atoms in a quenched anisotropic trap [27]. This experimental work has been complimented by theoretical work [28][29][30][31][32][33][34][35][36][37][38][39][40]. Our work is the trapped two-body limit of such systems.…”

“…Much work has been done regarding the contrast (and other quantities) in one-and two-dimensional systems of two bosons in recent years [34][35][36][37].…”

Analytical solution for the spectrum of two ultracold atoms in a completely anisotropic confinement

“…Note added: When we finished this work, we realized that recently there is a related work [53]. The authors derived the expression of J 2D (E) for E < E 0 , and a recurrence relation of J 2D (E) for arbitrary E. With this recurrence relation they also obtained the complete energy spectrum of two atoms in a 2D anisotropic confinement, as well as the eigen-states.…”

“…II.D, one can also derive the corresponding eigen-state |Ψ with eigen-energy E, which satisfies Eq. ( 51) and the boundary conditions (52,53), as well as the normalization condition Ψ|Ψ = 1. With Eq.…”