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arXiv:quant-ph/0410223 · Physical Review A

Matter diffraction at oblique incidence: Higher resolution and the He34 Efimov state

Gerhard C. Hegerfeldt, Martin Stoll

Published Mar 3, 2005• 3 citations

Abstract

We study the diffraction of atoms and weakly bound three-atomic molecules from a transmission grating at non-normal incidence. Due to the thickness of the grating bars, the slits are partially shadowed. Therefore, the projected slit width decreases more strongly with the angle of incidence than the projected period, increasing, in principle, the experimental resolution. The shadowing, however, requires a revision of the theory of atom diffraction. We derive an expression in the style of the Kirchhoff integral of optics and show that the diffraction pattern exhibits a characteristic asymmetry which must be accounted for when comparing with experimental data. We then analyze the diffraction of weakly bound trimers and show that their finite size manifests itself in a further reduction of the slit width by $(3∕4)⟨r⟩$ where $⟨r⟩$ is the average bond length. The improved resolution at non-normal incidence may in particular allow us to discern, by means of their bond lengths, between the small ground state of the helium trimer [$⟨r⟩\ensuremath{\approx}1\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$, Barletta and Kievsky, Phys. Rev. A 64, 042514 (2001)] and its predicted Efimov-type excited state ($⟨r⟩\ensuremath{\approx}8\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$, ibid.), and in this way to experimentally prove the existence of this long-sought Efimov state.

DOI: 10.1103/physreva.71.033606

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We study the diffraction of atoms and weakly bound three-atomic molecules from a transmission grating at non-normal incidence. Due to the thickness of the grating bars, the slits are partially shadowed. Therefore, the projected slit width decreases more strongly with the angle of incidence than the projected period, increasing, in principle, the experimental resolution. The shadowing, however, requires a revision of the theory of atom diffraction. We derive an expression in the style of the Kirchhoff integral of optics and show that the diffraction pattern exhibits a characteristic asymmetry which must be accounted for when comparing with experimental data. We then analyze the diffraction of weakly bound trimers and show that their finite size manifests itself in a further reduction of the slit width by

(3∕4)⟨r⟩

where

⟨r⟩

is the average bond length. The improved resolution at non-normal incidence may in particular allow us to discern, by means of their bond lengths, between the small ground state of the helium trimer [

⟨r⟩\ensuremath{\approx}1\phantom{\rule{0.3em}{0ex}}\mathrm{nm}

, Barletta and Kievsky, Phys. Rev. A 64, 042514 (2001)] and its predicted Efimov-type excited state (

⟨r⟩\ensuremath{\approx}8\phantom{\rule{0.3em}{0ex}}\mathrm{nm}

, ibid.), and in this way to experimentally prove the existence of this long-sought Efimov state.

The size of the helium trimer is determined by diffracting a beam of 4He clusters from a 100 nm period grating inclined by 21 degrees. 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(+0.4)(-0.5) 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.

Excited states and excitation energies of weakly bound systems, e.g., atomic few-body systems and clusters, are difficult to study experimentally. For this purpose we propose a new and very general atom-optical method which is based on inelastic diffraction from transmission gratings. The technique is applicable to the recently found helium trimer molecule 4He3, allowing for the first time an investigation of the possible existence of an excited trimer state and determination of its excitation energy. This would be of fundamental importance for the famous Efimov effect.

Cold helium atoms are a unique system in which a single excited three-body Efimov bound state occurs, naturally, without the need for an external magnetic field. While three-body bound-state energies of cold helium atoms have previously been investigated, recent experimental techniques have allowed their structure to also be studied. The weak interaction between helium atoms leads to a helium-helium (dimer) scattering length

a

much larger than the helium-helium effective range of interaction

r

. This feature is exploited in a theory that systematically expands observables in powers of

r/a

, known as short-range effective field theory (SREFT), which has been used successfully to investigate properties of cold-atom systems. Using SREFT, we investigate the average bond length of atoms in the three-body ground state and excited Efimov bound state of cold helium atoms. At leading-order (next-to-leading order) in SREFT, we find the average bond length of the

^{4}\mathrm{He}

trimer ground state is 8.35(33)

\AA{}

(10.29(2)

\AA{}

) and the average bond length of the excited

^{4}\mathrm{He}

trimer Efimov bound state is 103(4)

\AA{}

(105.3(2)

\AA{}

).

In atom optics, optical wave theory has been very successful in describing the diffraction of atoms by microfabricated structures. For a molecule, however, with a binding energy smaller than the kinetic energy and with a non-negligible diameter, wave optics may no longer be adequate due to the effects of finite size and breakups. To study the role of the two effects we consider the diffraction of weakly bound diatomic systems such as the helium dimer. We find striking deviations from classical optics, in particular from Babinet's principle, and use the results to clarify the role of finite size and breakups, respectively, in the diffraction from strips, slits, and transmission gratings. As an application the determination of the

{\mathrm{He}}_{2}

size by transmission through holes and by diffraction experiments is investigated.

Atomic diffraction through double slits and transmission gratings is well described in terms of the associated de Broglie waves and classical wave optics. However, for weakly bound and relatively large systems, such as the He

{}_{2}

dimer, this might no longer hold true due to the possibility of breakup processes and finite-size effects. We therefore study the diffraction of weakly bound two-particle systems. If the bar and slit widths of the grating are much larger than the diameter of the two-particle system, we recover the usual optics results. For smaller widths, however, deviations therefrom occur. We find that the location of possible diffraction peaks is indeed still governed by the usual grating function from optics, but the peaks may have a lower intensity. This is not unexpected when breakup processes are allowed. More unusually though, diffraction peaks that would be absent for de Broglie waves may reappear. The results are illustrated for diffraction of He

{}_{2}

.

Exciting experiments in the field of atom and molecule optics have lately drawn much attention to the effects involved in the coherent diffraction of particle beams. We review the influence of the finite size of the particles and of their energy level spectrum on the diffraction pattern. In turn, we demonstrate how experimental diffraction measurements allow to determine these quantities of weakly bound molecules by considering the diffraction of the dimer and trimer of helium, and the deuterium molecule dimer.