Short-Range Interaction Impact on Two-Dimensional Dipolar Scattering

We report numerical investigation of the short range interaction influence on the two-dimensional quantum scattering of two dipoles. The model simulates two ultracold polar molecules collisions in two spatial dimensions. The used algorithm allows us to quantitatively analyse the scattering of two polarized dipoles with account for strongly anisotropic nature of dipolar interaction. The strong dependence of the scattering total cross section on the short range interaction radius was discovered for threshold collision energies. We also discuss differences of calculated scattering cross section dependencies for different polarisation axis tilt angles.


Introduction
In recent years, two-dimensional (2D) systems with anisotropic interactions have been of increasing interest.One can mention a gas of dipolar atoms or ultracold polar molecules in quasi-2D geometry of magneto-optical traps [1] and dipolar Rydberg atoms [2].Beside the gas of polar molecules long range dipolar interaction also arises in a gas of 2D dipolar excitons in 2D quantum well of semiconductor heterostructures.Dipolar exciton gas quantum dynamics [3][4][5] and Bose-Enstein Condensate [6] have been also intensively investigated in recent years.Investigation of the dipolar diatomic molecules are also actual due to the prospects of their possible applications as cubits for the quantum computing [7].
Dipole-dipole interactions are long-range and strongly anisotropic, that leads to the possibility of the creation of exotic long-range quantum systems with dipolar gases [8,9].Particularly, anisotropic superfluidity [10], 2D dipolar fermions [11] and dipolar few-body complexes [12] have been investigated.For a recent progress in the physics of few-body systems with long-range interactions we refer the reader to the comprehensive review [13].
with the scattering boundary conditions in the asymptotic region ρ → ∞: The interaction potential U (ρ, φ) reads where the isotropic short-range potential V S R (ρ) is modeled by an infinitely high potential barrier at the origin, that was used in a number of papers of other the authors (see [14]).V dd (ρ, φ) denotes the long-range and strongly anisotropic interaction potential of two arbitrarily oriented dipoles: where d i , (i = 1, 2) are dipole moments.The angles α, γ denote to the tilt of each dipole d 1 , d 2 with respect to the Z axis (that is perpendicular to the plane of the motion) and β is the angle between the dipoles polarisation planes Z d 1 , Z d 2 (see Ref. [15]).If scattering amplitude is calculated the differential scattering cross section and the total cross section are defined as where dΩ = dφ dφ q and the angle φ q defines the direction q/q of the incident wave.
This expansion reduces the 2D Schrödinger equation (1) to a system of 2M + 1 coupled ordinary differential equations of the second order: with the "left-side" boundary condition ψ j (ρ → 0) → const × √ ρ ( j = 0, 1, . . ., 2M) and with the scattering boundary condition in the asymptotic region ρ → ∞ of the form The key advantages of the algorithm are summarized below.Any local interaction in this representation is diagonal.In Ref. [15] much faster convergence over the angular grid nodes number than partial wave representation was proved.After the finite-difference approximation of the boundaryvalue problem (8)(9) the obtained block matrix can be stored in a compressed form, which allows optimal resource using.The resulting algebraic system is efficiently solved by a fast implicit matrix algorithm based on the idea of the block sweep algorithm.

Short-range interaction influence on 2D dipolar scattering
The present study evidenced the strong dependence of the total cross section (6) on the short range cut-off radius ρ S R for threshold energies of dipole-dipole collisions.When the polarization axis is aligned along the z-axis the dipole-dipole interaction is isotropic and repulsive, no resonances occurs, because repulsive dipolar term prevents particles to getting too close to each other.
For a tilted (α = γ = 0.25π) polarization axis as well as for extremal case (α = γ = 0.5π) of polarization axis lying in the plane of particles motion, we discovered strong dependence of the total cross section on the cut-off radius ρ S R (4), which is illustrated in Fig. 1.The calculations were carried out at the energy E = 10 −5 a.u. and a dipolar length D = µd 1 d 2 / 2 = 1.Note, that we study the case of distinguishable particles collisions.
The sharp increases in total cross section appear due to the virtual bound states arising as the interplay of the short-range (4) and long-range (5) potentials.It should be noted the existence of small cusps for the following values of ρ S R = 0.035 a.u., 0.0895 a.u., 0.42 a.u.For small values of ρ S R a resonance width decrease with simultaneous increase of their frequency are observed.