Matter Wave Diffraction from an Inclined Transmission Grating: Searching for the Elusive ${}^4\mathrm{H}\mathrm{e}$ Trimer Efimov State

The size of the helium trimer is determined by diffracting a beam of ${}^4\mathrm{H}\mathrm{e}$ clusters from a 100 nm period grating inclined by 21°. Because of the bar thickness the projected slit width is roughly halved to 27 nm, increasing the sensitivity to the trimer size. The peak intensities measured out to the eighth order are evaluated via a few-body scattering theory. The trimer pair distance is found to be $⟨r⟩={1.1}_{-0.5}^{+0.4}\mathrm{nm}$ in agreement with predictions for the ground state. No evidence for a significant amount of Efimov trimers is found. Their concentration is estimated to be under 6%, less than expected.

Figure 1

Diffraction geometry at non-normal incidence: (a) a single slit of width $s_0$ in a plate of thickness $t$ with a wedge angle $\beta$ has a projected slit width of $s_{\perp }$. Both the angle of incidence ${\Theta }_0$ and the angle $\Theta$ are measured relative to the plate normal. The hypothetical thin plate drawn along the $\eta$ direction with a slit of width $S_0=\sqrt{(s_0+t\tan \beta {)}^2+t^2}$ at an angle $\alpha =\arcsin (t/S_0)$ relative to the thick plate along the $y$ direction casts the same geometrical shadow as the thick plate. (b) Transmission grating of period $d$ along the $y$ direction.

Figure 2

(a) ${}^4\mathrm{H}\mathrm{e}$ diffraction pattern at ${\Theta }_0=21°$ angle of incidence measured for the source conditions $(T_0,P_0)=(16.5\mathrm{K},7.0\mathrm{\text{bar}})$ corresponding to a trimer de Broglie wavelength of $\lambda =0.83\AA$. The signal at negative diffraction angles has been shifted upwards by a factor 10 and mirrored onto the positive side for comparison. The trimer diffraction peaks are marked by circles, dimer peaks by stars. (b) Relative trimer peak intensities $I_n/I_0$. (c) Corresponding contrast $C_n=(I_n-I_{-n})/(I_n+I_{-n})$. The curves in (b) and (c) are best-fit calculations based on Eq. (6).

Figure 3

Projected effective slit widths $s_{\perp \mathrm{eff}}=S_{\mathrm{eff}}\cos ({\Theta }_0+\alpha )$ measured at different beam velocities at an angle of incidence ${\Theta }_0=21°$ for ${}^4\mathrm{H}\mathrm{e}$, ${{}^4\mathrm{H}\mathrm{e}}_2$, and ${{}^4\mathrm{H}\mathrm{e}}_3$. The curves represent best fits of Eq. (7) for ${{}^4\mathrm{H}\mathrm{e}}_3$ and analogous expressions for ${}^4\mathrm{H}\mathrm{e}$ and ${{}^4\mathrm{H}\mathrm{e}}_2$. Their high velocity limits are given by $s_{\perp }$ for ${}^4\mathrm{H}\mathrm{e}$, by $s_{\perp }-\frac{1}{2}⟨r⟩$ for ${{}^4\mathrm{H}\mathrm{e}}_2$, and by $s_{\perp }-\frac{3}{4}⟨r⟩$ for ${{}^4\mathrm{H}\mathrm{e}}_3$. The insets illustrate the “widths” traced out by the clusters along their flight paths (see text).