Quasi-two-dimensional dipolar scattering

We study two-body dipolar scattering with one dimension of confinement. We include the effects of confinement by expanding this degree of freedom in harmonic oscillator states. We then study the properties of the resulting multichannel system. We study the adiabatic curves as a function of $D/l$, the ratio of the dipolar and confinement length scales. There is no dipolar barrier for this system when $D/l<0.34$. We also study the WKB tunneling probability as a function of $D/l$ and scattering energy. This can be used to estimate the character of the scattering.

Figure 1

(a) The adiabatic potential energy curves, $V_0(\rho /l)$, for $\mathrm{D̄}=0.5$ (solid black), $\mathrm{D̄}=1$ (red circles), $\mathrm{D̄}=2$ (blue $\times$ ’s), and $\mathrm{D̄}=5$ (green squares) in harmonic oscillator units. (b) The height of the dipolar barrier as a function of $\mathrm{D̄}$. The first excited threshold, $2{\mathrm{D̄}}^2$ (solid black), and ${\mathrm{D̄}}^{5/2}$ (blue dashed) are plotted in dipolar energy units.

Figure 2

(a) The lowest adiabatic curve on a log-log scale for various values of $\mathrm{D̄}$, which are 0.5 (black triangles), 1 (red circles), 2 (blue $\times$ ’s), 5 (green squares), and 10 (violet diamonds) compared with the pure 2D scattering potential (solid black). (b) The lowest diabatic curve for $\mathrm{D̄}=0.1$ (blue ×’s), 0.34 (black triangles), and 1 (red circles) as a function of $\mathrm{\rho ̃}$. Also shown is pure dipolar repulsion (solid black) and the absolute value of the attractive 2D centrifugal term (solid blue).

Figure 3

(a) The scattering rate, $4\sin {}^2(\delta )$, as a function of the inner boundary phase for $\mathrm{D̄}=1$ at different energies: 0.001 (solid red), 0.013 (black dashed), and 0.17 (blue with circles). (b) The background scattering rate as a function of the scattering energy for $\mathrm{D̄}$ equal to 1 (red circles), 2 (open blue squares), and 5 (black triangles). The pure 2D scattering is shown as a solid blue curve.

Figure 4

(a) The WKB tunneling probability as a function of energy for $\mathrm{D̄}$ equal to 1 (black line), 2 (red circles), 3 (blue triangles), 4 (maroon squares), and 5 (violet diamonds). The fitted resonance widths from the scattering calculations are shown for $\mathrm{D̄}$ $=$ 1 (black circles), 2 (red open circles), and 3 (blue open triangles). The green $\times$’s in each plot are the fitted $P_T$ from Eq. (4). (b) The WKB tunneling probability as a function of energy for $\mathrm{D̄}$ equal to 1 (black line), 2 (red circles), 5 (violet diamonds), 10 (open blue circles), 20 (open black triangles), and 30 (open green squares). (c) $P_T$ as a function of $\mathrm{D̄}$ for the energies of 1 (black), 3.2 (red with circles), 10 (blue $\times$’s), 32 (green squares), and 100 (violet diamonds). The open green circles are the fit $139.3(E/E_D){}^2{e}^{-5.86{\mathrm{D̄}}^{2/5}}$ for $E/E_D=10$ and 100 and show very good agreement.

Figure 5

The character of the quasi-2D scattering as a function of $E/E_D$ vs $\mathrm{D̄}$. The excited threshold (solid blue) and the height of the adiabatic barrier (solid red) are shown. The WKB tunneling probability contour for $P_T=$ ${10}^{-2}$ (dashed black), ${10}^{-3}$ (solid black), and ${10}^{-4}$ (dash dot black) are plotted. The green $\times$ ’s in each plot are the fitted $P_T$ from Eq. (4) and the greens circles are the fit for the semiclassical energy regime from Eq. (5).