Matter Diffraction at Oblique Incidence: Higher Resolution and the Helium Trimer Efimov State

$Matter Diffraction at Oblique Incidence: Higher Resolution and the Helium Trimer Efimov State$

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Efimov long-sought this ntttfu hoeicePyi,Universität Götting Physik, für Theoretische Institut ff 0 mhdbogta mrvmn,btfi- but improvement, an brought had nm 200 3 esuytedi the study We ato.W eiea xrsini h tl fteKirchho the of style the in expression an derive We raction. steol oeuepeitdt oss an possess to predicted molecule only the is .INTRODUCTION I. ff ato tolqeicdne ihrrslto n the and resolution Higher incidence: oblique at raction ff ff 4 ato faoi n molecular and atomic of raction ato xeiet iha unpar- an with experiments raction He d 2 ff = ff ato faosadwal-on he-tmcmlclsf molecules three-atomic weakly-bound and atoms of raction n trimer and ato ekitniisi a in intensities peak raction 0 m[1 ae h way the paved [11] nm 100 ff ff ato o sodium a for raction ato fwal on rmr n hwta hi nt siz finite their that show and trimers bound weakly of raction ehr .Hgred n atnStoll Martin and Hegerfeldt C. 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It is there- per atom is a few tens of meV, much less than electronic ex- fore essential to provide sufficient experimental resolution for citation energies of the atom. Therefore, we treat atoms as the small ground state in order to evaluate diffraction data point particles and neglect their electronic degrees of freedom from a mixed beam. A limitation in the resolution is posed throughout this article. The de Broglie wavelength λdB as- by the period of the grating, d = 100nm, and the slit width, sociated with the atomic motion is typically of the order of typically s0 = 60nm, which are both large compared to the 0.1nm whereas a typical length scale of the scattering object ground state size. Transmission gratings with smaller periods is d = 100 nm. We shall, in the following, refer to this relation and slit widths are, however, presently not available. as the diffraction condition: To address this issue we consider diffraction from a custom transmission grating at oblique (non-normal) incidence, λdB d . (1) ≪ i.e. the grating is rotated by an angle θ′ about an axis parallel The free Hamilton operator for an atom of mass m is H0 = to its bars. The grating typically consists of a t 120nm thick 2 layer of silicon nitride into which the slits have≈ been etched in pˆ /2m. The interaction between the diffracting object and the a lithographic production process [11]. As the grating is ro- atom will be described by a Lennard-Jones [36] type surface tated the upstream edges of its bars cut into the incident beam, potential W(x) where x is the position of the atom. This inter- partially shadowing the slits. By this effect the projected action exhibits a strongly repulsive core at a distance l from slit s width (perpendicular to the incident beam) decreases the diffracting object of the order of the atomic diameter, and ⊥ it passes into a weak attractive C /l3 van der Waals potential more strongly with θ′ than the projected period d . Since − 3 the diffraction pattern is roughly governed by the squared⊥ slit at l & 1nm [12, 37]. Due to the low kinetic energy of the function [33] atoms in the beam it is sufficient for the purposes of this work to model the repulsive part of the interaction by a hard core. sin(nπs /d ) 2 The attractive part will be omitted for the moment and will be ⊥ ⊥ " (nπs /d ) # be included later in Sec. III. ⊥ ⊥ Generally, the scattering state p′, + for an atom with inci- where n denotes the diffraction order, one expects that at non- | i 2 dent momentum p′ and positive energy E′ = p′ /2m satisfies normal incidence more atoms or molecules are diffracted into the Lippmann-Schwinger equation [38] | | higher diffraction orders. For example, at t = 120nm and an angle of incidence (rotation angle) of θ′ = 21◦ the ratio d /s p′, + = p′ + G0(E′ + i0)W p′, + (2) ⊥ ⊥ | i | i | i is twice as large as d/s0 (normal incidence) while the diffrac- 1 tion angles, to good approximation, only increase by the ratio where G0(z) = [z H0]− is the free Green’s function, or re- d/d . This effect increases the experimental resolution. Due solvent. Denoting− the atom transition amplitude associated to the⊥ shadowing, however, the fundamental results obtained with the potential W by earlier for atom diffraction at normal incidence cannot be car- at t (p; p′) = p W p′, + , (3) ried over unchanged. h | | i While we illustrate our results using the experimentally most interesting case of helium trimer diffraction, the gen- where p = p′ + ∆p is the outgoing momentum and E = p 2/2m, the S matrix element has the usual decomposition eral findings of this work equally apply to other weakly-bound | | trimers, possibly consisting of non-identical atoms. The arti- [38] cle is structured as follows. In Section II we derive, from (3) at p S p′ = δ (p p′) 2πiδ(E E′) t (p; p′). (4) quantum mechanical scattering theory, the transition ampli- h | | i − − − tude for an atom diffracted from a bar of finite thickness. In In many applications the diffraction object may be effec- Section III we construct a periodic transmission grating from tively regarded as translationally invariant along one direction many bars and introduce the notion of a slit at non-normal whence the diffraction process can be treated in two space di- incidence. We show that if the slits are not aligned with mensions. This is the case, for instance, for diffraction from a the direction of periodicity a characteristic asymmetry of the slit if the vertical spread of the focussed atom beam is much diffraction pattern arises which went unnoticed in a previous less than the physical height of the slit. In this article we experiment [35]. The asymmetry is relevant for the precise shall always assume the scattering object to be translation- evaluation of experimental data. In Section IV we outline the ally invariant along the x -axis. To adapt the notation, we de- general scattering theory approach to trimer diffraction and 3 note the two-dimensional projections into the (x1, x2) plane work it out in Section V for non-normal trimer diffraction of all three-dimensional vectors, such as p, by their corre- from a grating. In Section VI we provide the link between sponding italic letters, such as p. As scattering does not occur diffraction data and the trimer bond length and discuss aspects along the x3-axis a delta-function δ(∆p3) expressing momen- of a helium trimer diffraction experiment. tum conservationcan be extracted from Eq. (3), leaving a two- at(2) dimensional transition amplitude t (p; p′) which satisfies

II. ATOM DIFFRACTION FROM A DEEP BAR at at(2) t (p; p′) = δ(∆p3) t (p; p′). (5)

at(2) In a typical beam diﬀraction experiment with an average In order to derive an expression for t (p; p′) we project beam velocity of the order of v = 500m/s the kinetic energy Eq. (2) into conﬁguration space. The full wave function 3

ψ(k′, x) = 2π~ x p′, + , where k′ = p′/~, is a sum of the shadow line, and p′ is the parallel component (cf. Fig. 1). In h | i a2 incident part ψinc(k′, x) = 2π~ x p′ and the scattered part accordance with the Babinet principle, the transition ampli- h | i tude shows the characteristic behavior of an optical slit func- (2) (2) ψscatt(k′, x) = 2π~ x G E′ + i0 W p′, + . (6) tion. The Babinet principle of wave optics states that two h | 0 | i complementary objects, such as a slit and a bar of the same (2) 2 Here, E′ = p′ /2m, and the two-dimensionalGreen’s func- width, cause the same diffraction pattern outside the direction | | (2) 2 1 tion, or resolvent, is G0 (z) = [z pˆ /2m]− . of illumination (forward scattering) [33]. If the scale of an object is large− compared to the wavelength of visible light it is well known that the diffraction about the forward direction depends only on its (two-dimensional) silhouette as seen from the direction of the illuminating light, a2 e.g. as for a disk and a ball. In two dimensions the silhou- A ette of a diffraction object is simply a straight line, here called + a 2 1 shadow line (line in Fig. 1) and, as in optics, due to the A diffraction condition (1) the scattered part ψscatt of the wave function can be approximated at small scattering angles about A the forward direction [39]. Neglecting the attractive part of the potential W(x) the repulsive hard core imposes Dirichlet boundary conditions on the circumference of the diffracting A object. Denoting the Green’s function in configuration space 2 by p′ − ~2 (2) (2) (2) G ( k′ ; x, x′) = x G E′ + i0 x′ (7) 0 2m 0 | | − h | | i and using the Green theorem one finds, after some algebra, FIG. 1: Two-dimensional geometry of atom diffraction from a diffraction object (bar) of finite thickness. The incident momentum ∂ of the atom is denoted by p′. The straight shadow line , which (2) A ψscatt(k′, x) da2 G0 ( k′ ; x, a) ψinc(k′, a) plays the role of the silhouette, divides the bar into an “illuminated” ≃ Z " | | ∂a1 A part and a “shadowed” part. Also drawn is the adapted coordinate ∂ (2) system (a , a ) with the a -axis centered along the shadow line. ψ (k′, a) G ( k′ ; x, a) . (8) 1 2 2 − inc ∂a 0 | | 1 #a1=0

Here, da2 is the infinitesimal line element along , and ∂/∂a1 denotes the normal derivative (cf. Fig. 1). The Green’sA function (7) can be expressed in terms of a Hankel function [40, III. ATOM DIFFRACTION FROM A DEEP GRATING chap. 3.10]: A. The atom slit function of the deep grating (2) i (1) G ( k′ ; x, x′) = H k′ x x′ . (9) 0 | | 4 0 | || − | In diffraction experiments one often employs transmission Using the asymptotic expansion [41, chap. 9.2] gratings to enhance the measurable diffraction peak intensities by a factor of N2, where N is the number of coherently illu- i( k′ x π/4) (1) x 2 i k x x/ x e | || |− | |→∞ ′ ′ minated bars. One grating bar is simply a special case of the H0 k′ x x′ e− | | · | | | || − | ∼ rπ k′ √ x general scattering object considered in the previous section. | | | | (10) In the following we arrange N identical bars to create a reg- the far field (Fraunhofer) limit of ψscatt(k′, x) can readily be ularly spaced periodic transmission grating. While the sim- calculated. Inserting the expansion into Eq. (8) shows that the plest and most familiar situation in which the bars are aligned vector k = k′ x/ x in the first exponential should be iden- along their common shadow line (a -axis, cf. Fig. 1) is treated | | | | 2 tified with the outgoing wave vector, and p = ~k with the in virtually every textbook on optics (e.g. Ref. [33]), we are outgoing momentum. Comparing this expression with the far not aware of a more general treatment where the individual field limit of Eq. (6) one arrives at the two-dimensional tran- shadow lines are not parallel to the alignment axis. This situ- sition amplitude ation arises naturally, however, for diffraction from a grating with bars of finite thickness at non-zero angle of incidence p + p A/2 at(2) i a1 a′ 1 i∆p a /~ p p = a2 2 (cf. Fig. 2). We note that apart from the period an additional t ( ; ′) 2 da2 e− (11a) −2 (2π) m~ Z A/2 length scale of the grating is given by the distance between − p + p ~ the bars. Depending on the angle of incidence the (projected) i a1 a′ 1 sin(∆pa2 A/2 ) = , (11b) distance can become small or even zero. The diffraction con- −2 (2π)2m~ ∆p /2~ a2 dition (1), which must hold for all length scales, therefore also where A denotes the length of the shadow line . Further- imposes a limit on the maximal angle of incidence. more, p is the momentum component of p normalA to the Under the diffraction condition the transition amplitude of a a′ 1 ′ 4 grating of N bars with period d along the x2-axis can be written as the coherent sum of the spatially translated amplitudes of each bar: N 1 at(2) − i(n 1 (N 1))∆p d/~ at(2) s t (p; p ) = e 2 2 t (p; p ) (12a) 2 gra ′ − − − ′ x2 Xn=0 at(2) +S 0/2 = HN (∆p2) t (p; p′). (12b) In the second line, the sum has been carried out and replaced by the grating function [33] θ sin(∆p dN/2~) H (∆p ) = 2 (13) θ′ α x1 N 2 sin(∆p d/2~) 2 d whose argument is the momentum transfer along the direc- p′ S tion of periodicity x2. Eq. (12) yields, in principle, a satisfac- S 0/2 tory description of the diffraction problem in terms of atom − s1 scattering from a bar. The literature on optics, however, commonly adapts a complementary point of view by focusing on A the apertures (slits) between the bars rather than on the aper- ture stops (bars) themselves. This is expressed, for example, by the Kirchhoff integral of optics [33]. In recent work this viewpoint has also proven very useful in the field of atom and molecule diffraction from a transmission grating: small quan- FIG. 2: Geometry of atom diffraction from a grating of identical tum mechanical effects such as the van der Waals interaction deep bars with period d along the x2 direction. The slit line , which connects the shadow lines of adjacent bars, provides aS general- [12, 17] between the atoms in the beam and the grating as A well as the finite size of the helium dimer [3] manifest them- ization of the notion of a slit at normal incidence. It lends itself to selves as an apparent reduction of the slit width of the grating. the application of the Huygens principle. Also drawn is the adapted coordinate system (s , s ) with the s -axis centered along the slit line In these articles the respective quantities could be determined 1 2 2 and the angle α by which the coordinate systems (x1, x2) and (s1, s2) quite precisely by comparison of the reduced slit width with are rotated with respect to each other. the true geometrical slit width. Unlike for grating diffraction at normal incidence, however, it is initially not evident how to define a slit in the case of non-normal incidence: the cor- rect choice may depend on the angle of incidence. We pro- bar Eq. (11a) becomes vide, therefore, a mathematical prescription which converts ff Eq. (12) into an expression in the style of the Kirchho inte- p + p D/2 at(2) i s1 ′s1 i∆p d/2~ i∆p s /~ gral. This is achieved by introducing, between every pair of t (p; p′) = e 2 ds e− s2 2 −2 (2π)2m~ 2 adjacent bars, a new coordinate system (s1, s2) as depicted in ( ZS 0/2 S /2 Fig. 2 such that the s2-axis meets the boundaries of the two 0 i∆p2d/2~ − i∆ps s2/~ bars at their respective shadow lines. The length of the re- + e− ds2 e− 2 . (15) Z D/2 ) sulting straight “slit line” will be denoted by S . We now − S 0 substitute the integration variable a2 of Eq. (11a) by Keeping in mind that the integration variable s2 was substi- ∆p A S s = a2 a 0 as a ≶ 0 tuted for a2 for a single bar the following geometrical interpre- 2 ∆ 2 2 ps2 ± 2 ± 2 tation is possible: in the integral from S 0/2 to D/2 in Eq. (15) the variable s represents the position along the upper half of where ∆p denotes the momentum transfer parallel to the s - 2 s2 2 the slit line on one side of the bar; similarly, in the inte- axis. Similarly, we denote the components of the incident S gral running from D/2 to S 0/2, s2 is the position along the and outgoing momenta with respect to the s1-axis by p′ and − − s1 lower half of the next slit line at the other side the bar. Insert- p , respectively. (An explicit expression for these momen- s1 ing Eq. (15) into Eq. (12a) the half slit lines of the N adjacent tum components in terms of the geometry of the grating will bars can be joined to yield N 1 slits between them. Collect- be given below in Eqs. (27) and (28) for the transmission grating all terms and introducing− a “slit function”, ing of Fig. 4.) Using the identities

∆pa A + ∆ps S 0 = ∆p2d , (14a) D/2 2 2 at at a (p′; ∆p ) = ds exp i∆p s /~ τ (p′; s ), (16) pa + p′ ps + p′ p1 + p′ s2 2 s2 2 2 1 a1 1 s1 1 Z D/2 − = = , (14b) − ∆pa2 ∆ps2 ∆p2 at which hold because of energy conservation, and the abbrevi- where the transmission function τ (p′; s2) inserted here is ation D = d∆p /∆p , the transition amplitude of the single unity inside the slit and zero otherwise, the transition am- 2 s2 S 5 plitude of the grating can be written, for N 1, as 1 ≫ ps + p′ at(2) i 1 s1 − tgra (p; p′) (17) 0 0.8 n= 1. 2 ~ /I ≃−2 (2π) m n

I n=+1. sin (∆p Nd/2~) 2 H (∆p ) aat(p ; ∆p ) . ~ N 1 2 ′ s2 0.6 × ( ∆ps2 /2 − − ) The first term in curly brackets is sharply peaked about the 0.4 forward direction, and in the limit N , using Eq. (14b), it +3.−3. +5. −5. p1 →∞ simply reduces to 2π~ δ(∆p2). The second term, which is a ps1 product of the grating function (13) and the slit function (16), Diffraction intensity 0.2 generates the familiar diffraction pattern of a grating [34]. The n-th order principle diffraction maximum appears at the mo- 0 mentum transfer 0 ±0.02 ±0.04 ±0.06 ±0.08 -1 Momentum transfer ∆p / h- 10 nm n2π~ s2 [ ] ∆p = . 2 d FIG. 3: Asymmetric probing of the envelope function by the grating function at non-normal incidence. Both curves show the product Introducing the angle of incidence θ′ such that p1′ = p′ cos θ′ | | 2 2 at 2 = ff (p + p′ ) /(2p′ S ) a p′; ∆p as it appears in Eq. (19) ver- and p2′ p′ sin θ′ (cf. Fig. 2), and equivalently the di raction s1 s1 s1 0 s2 | | | ∆ | angle θ such that p1 = p cos θ and p2 = p sin θ, the n-th sus the momentum transfer ps2 . The solid curve refers to the pos- | | | | itive values of ∆p on the horizontal axis. The dashed curve refers order is located at the angle θ = θn satisfying s2 to the negative values of ∆ps2 which have been mirrored onto the n2π~ positive axis for comparison. The circles and triangles on top of sin θn = sin θ′ + . (18) the solid and the dashed curves, respectively, mark those values of p′ d | | ∆ps2 where the grating function HN (∆p2) probes the slit function, i.e. where ∆p = n2π~/d is satisfied for n = 0, 1, 2,... . Their Generally, the diffraction intensities are proportional to the 2 ± ± 2 intensities correspond to the measurable diffraction peaks. For the scattering matrix element p S p′ where the components of |h | | i| calculation of this figure the grating cross section of Fig. 4 was used the outgoing momentum p must be evaluated at the angle θn. 1 with the beam parameters θ′ = 21◦ and p′ /~ = 10 nm− . Inserting Eq. (17) into Eq. (4) one finds, after some algebra, | |

+ 2 at 2 ps1 p′s1 a p′; ∆ps2 In = I0 | | , (19) 2p ! aat (p ;0) 2 ′s1 | ′ | where the intensity I0 of thezeroth diffraction order serves as a the leading non-linear term is seen to vanish like ∆p2/ p′ rel- normalization constant depending on the experimental count- ative to the linear term. Accordingly, the asymmetry| is| less at 2 2 ing rate and where, from Eq. (16), a (p′;0) = S (in the pronounced for smaller diffraction orders, for faster beams | | 0 absence of the van der Waals interaction considered below). and, in the case of molecules, for heavier molecules. While 1 4 We now discuss the intensity formula Eq. (19) to explain at p′ /~ = 10nm− (corresponding to a He beam at v the origin of the asymmetry of the diffraction pattern. The 160m| |/s), as seen in Fig. 3, the quadratic term in Eq. (20)≈ at slit function a p′; ∆ps2 is an even function of the mo- is responsible for a 8% deviation of the positive and neg- ∆ ± mentum transfer component ps2 . The geometrical factor ative fifth diffraction orders, respectively, its contribution re- (p + p )2/(2p S )2 (the component p depends on ∆p s1 ′s1 ′s1 0 s1 s2 duces to 0.7% at v = 1800m/s. This smallness explains through conservation of energy and momentum)can be shown why the asymmetry± has previously been missed (cf. Fig. 5 in to introduce, for positive incident angle θ′, a slight attenuation Ref. [35]). Clearly, in the thin grating limit α 0 the mirror → of the slit function at positive ∆ps2 and likewise an intensifi- symmetry is recovered in Eq. (20). cation at negative ∆ps2 . The product of the slit function and the geometrical factor serves in Eq. (19) as an envelope func- In the derivation so far no comment has been made about tion which is probed by the grating function at the momen- the inclusion of the attractive van der Waals interaction be- tum transfer ∆p rather than at ∆p . Since ∆p and ∆p 2 s2 s2 2 tween the atom and the grating. It can be accounted for are not proportional to each other this probing is not symmet- through the transmission function τat(p ; s ) in the slit func- ff ′ 2 ric for positive and negative di raction angles. This is de- tion (16) as outlined in Appendix A and in Refs. [12, 15]. picted in Fig. 3, and it leads to the characteristic asymmetry Unlike the case of normal incidence, at non-normal incidence ff of the di raction pattern of a deep grating at non-normal inci- the influence of the van der Waals interaction may be differ- dence. Expanding ∆p , using Eq. (14b), into a power series s2 ent on each side of the slit, introducing, in principle, an addi- ∆ in ( p2/ cos θ′) through second order, tional source of asymmetry to the diffraction pattern. Numer- 2 ical comparisons using the explicit expression of Appendix A ∆ps ∆p tan θ tan(α + θ ) ∆p 2 2 + ′ − ′ 2 , (20) for the transmission function demonstrate, however, that this cos(α + θ′) ≃ cos θ′ 2 p′ cos θ′ ! ff | | e ect is minor. 6

B. Theatomdiﬀraction pattern of the deep grating (23). One ﬁnds

S S 0/2 S The quantitative evaluation of experimental diﬀraction data 0 at 0 R1± = dξ τ p′; ξ (24) requires to determine a set of parameters describing the geom- 2 − Z0 ± 2 − etry of the particular grating (cf. Fig. 4) as well as the van der and Waals interaction coeﬃcient C3 (cf. appendix A). Previous 2 S 0/2 S 0 2 at S 0 R2± = R1± 2 dξ ξ τ p′; ξ . 2 − − Z0 ± 2 − (25)

d Using the explicit form of the transmission function derived in the appendix the length scale of the cumulants can be shown β to be set by the parameter √C3/(~v). For helium and a SiNx 3 grating C3 0.1meV nm [12]. Therefore, for the purposes of this work≈ it is sufficient to truncate the expansion Eq. (23) x2 after the second order. Inserting the first two terms into the s slit function Eq. (21) and introducing the four quantities s2 0 S + + 0 S ff = S Re (R + R−), ∆= Im(R + R−), α e 0 − 1 1 1 1 + 1 + Γ= Im(R R−), Σ= Re (R + R ), t 1 − 1 r2 2 2− the n-th order diffraction intensity relative to the zeroth order FIG. 4: Geometrical cross section of a custom diffraction grating is given, within this approximation, by as it has been used in matter diffraction experiments [3, 12]. The parameters and their typical values are: period d = 100 nm, slit width 2 s = 60 nm, thickness t = 120 nm, and wedge angle β = 6 . The 0 ◦ I ps + p′ n 1 s1 2 ~2 ~ angle by which the coordinate systems (x1, x2) and (s1, s2) are rotated =   exp (∆ps2 Σ) / exp[ Γ∆ps2 / ] = + = I0  2 2  − − with respect to each other satisfies cot α tan β s0/t and S 0 sin α 2p′ S + ∆  h i  s1 eff  t. At the above parameters α 58◦. The characteristic shape of the  q  ≈  2  2 bars is reminiscent of the lithographic production process [11].  sin [∆p S ff/2~] + sinh [∆p ∆/2~] s2 e s2 . (26) × ~ 2 ∆ps2 /2 work has shown that an immediate numerical fit of Eq. (19) to experimental data does not reliably determine these param- Here, the momentum components are to be taken explicitly at eters. Analogous to the procedure developed in Ref. [12] for the angles diffraction at normal incidence we therefore introduce a two- p = p′ cos(α + θ ) (27a) term cumulant approximation of the slit function. To this end s1 | | n we rewrite Eq. (16) as p′ = p′ cos(α + θ′) (27b) s1 | | ~ at and the momentum transfer is given by a (p′; ∆ps2 ) = i∆ps2 ∆p = p′ sin(α + θ ) sin(α + θ′) (28) s2 | | n − i∆p S /2~ ∆ps2 i∆p S /2~ + ∆ps2 e s2 0 Φ− e− s2 0 Φ (21) × ( ~ ! − ~ !) where α denotes the angle shown in Fig. 4 by which the coordinate system (s1, s2) is rotated with respect to (x1, x2). + where the functions Φ (κ) and Φ−(κ) are defined by The diffraction intensity formula (26) is, though more complicated, reminiscent of the result for normal incidence de- S 0/2 1 iκξ at S 0 rived in Ref. [12]: The long fraction involving the sine of S Φ = ′ p eff ±(κ) at ∓ dξ e± τ ′; ξ . (22) τ (p′;0) Z0 ± 2 − resembles the Kirchhoff slit function for a slit of “effective width” S eff whose intensity zeros, though, are removed by the at Here, τ ′(p′; s2) denotes the derivative of the transmission hyperbolic sine function involving ∆. The Gaussian exponen- function with respect to its position argument. As Φ±(0) = 1 tial reflects the suppression of higher diffraction orders. The the logarithm of Φ±(κ) can be expanded into a power series of asymmetry In , I n of the diffraction pattern is now embod- the form − ied by the asymmetry in ∆ps2 as a function of n in Eq. (28). j The second exponential in Eq. (26) accounts for the minor ∞ ( iκ) ff ln Φ±(κ) = ± R±. (23) additional asymmetry in the di raction pattern due to the dif- j! j Xj=1 ferent influence of the van der Waals interaction on both sides of the bar. A comparison of the diffraction intensities calcu-

The complex numbers R1± and R2±, which are known as the lated from Eq. (19) and the approximation (26) is displayed in ﬁrst two cumulants, are uniquely determined by Eqs. (22) and Fig. 5. 7

1 m2 0

/I m1 n I ρ(1) 0.1 r(23)

0.01 Diffraction Intensity m3

FIG. 6: One of three possible sets of Jacobi coordinates. The vector -8 -6 -4 -2 0 2 4 6 8 ρ(1) points from the center of mass of the subsystem (23) to atom (23) Diffraction order n 1. The vector r is the relative coordinate of the subsystem. The coordinate R (not shown) corresponds to the center of mass position FIG. 5: Diffraction intensities of a Helium atom beam at v = 500 m/s and is, therefore, identical for all three sets. from a d = 100 nm transmission grating as displayed in Fig. 4 at θ′ = 21◦ angle of incidence. The solid curve was calculated using Eq. (19), the dashed curve shows the two-term cumulant approxima- form as tion (26). To guide the eye these functions are shown continuously. mi m j mk The circles on top of the solid curve at integer n mark the experimen- R M 1 M 1 M 1 ri (i) m j mk tally accessible diffraction orders In/I0. ρ = 1 1 1 r j , (30)    m j+mk m j+mk     ( jk)   − −     r   0 1 1   rk     −          By a numerical fit of Eq. (26) to experimental diffraction where 1 and 0 denote the 3 3 unit and zero matrix, respec- × data the effective slit width S eff, amongst the other param- tively, and M = m1 + m2 + m3 is the total mass. It is sufficient eters, can be determined accurately and allows for further to restrict the combinations of indices to the ascending permu- comparison between theory and experiment along the lines of tations Ref. [12]. For completeness, we note that (i jk) = (123), (231), (312). (31) S 0/2 at S eff = S 0 Re ds2 1 τ (p′; s2) . (29) The transformation between different sets of Jacobi coordi- − Z S /2 − − 0 h i nates can be derived from Eq. (30). It takes the form This means that the geometrical slit width S appears to be re- 0 R R duced by the average deviation from unity of the transmission ( j) ( ji) (i) at  ρ  =  ρ  , (32) function τ (p′; s2). (ki) J ( jk)  r   r          where the block matrix ( ji) is given by IV. TRIMER DIFFRACTION THEORY J 10 0 m m M A trimer, in the scope of this article, is a three-atomic ( ji) = 0 i 1 k 1 . (33)  − mi+mk (m j+mk )(mk+mi)  molecule which is weakly bound by pair interactions [42]. J  m j   0 1 m +m 1  Again, the atoms themselves are treated as point particles.  − − j k    An additional three-body interaction between the atoms is as-   The three matrices ( ji) satisfy the relations det ( ji) = 1 sumed to be negligible. Central to later applications will be and ( ji) (ik) (k j) =J1. Expressed in Jacobi coordinatesJ the the helium trimer 4He in which case at least these assump- 3 HamiltonJ J operatorJ for a free trimer is given by H + V where tions are expected to be valid [30]. 0

1 2 M (i) 2 m j + mk ( jk) 2 H0 = Pˆ + qˆ + pˆ 2M 2mi(m j + mk) 2m jmk A. Three-body bound states (i) mk ( jk) ( jk) V = vi j ρˆ rˆ + v jk rˆ − m j + mk ! The masses of the three atoms at positions ri, for i = 1, 2, 3, will be denoted by mi and are assumed to be of the same order (i) m j ( jk) + v ki ρˆ + rˆ . (34) of magnitude. The interaction between atom j and k is mod- m j + mk ! ( jk) ( jk) = eled by a potential v jk( r ) where r r j rk is the relative coordinate. We introduce| | the Jacobi coordinates− R, ρ(i), r( jk) Here, P, q(i), and p ( jk) are the conjugate momenta associated sketched in Fig. 6, which can be expressed in block matrix with R, ρ(i), and r( jk), respectively. Denoting the eigenstates 8 of the center of mass momentum Pˆ by P the full trimer states Since the discovery of the Efimov effect[31] in 1970the he- | i 4 can be written in product form P,φγ P φγ satisfying lium trimer He3 has received much attention [4, 5, 6, 20, 21, | i≡| i| i 22, 23,24, 25, 26,27, 28, 29, 30,48, 49, 50, 51]. Thistrimeris [H0 + V] P,φγ = E P,φγ (35) predicted to possess, apart from its quite tightly bound ground | i | i state (Eg = 126mK), a weakly bound Efimov-type excited with energy eigenvalues state (E = −2.3mK) [4]. As both states have zero total angu- e − 2 lar momentum the corresponding wave functions only depend P ρ = ρ , = E = | | + Eγ (36) on three coordinates which may be taken as r r , 2M and the angle between ρ and r. A common way to| | visualize| | where E is the negative binding energy of the trimer bound trimer wave functions is to draw the hyperspherical probabil- γ ity density P(R): The hyperradius R, which is independent state φ γ. The| i representation of a trimer state by its wave function of the choice of the set of Jacobi coordinates, is defined as µ R(ρ, r)2 = 2 mρ2 + 1 mr2 and the corresponding probability depends on the particular set of Jacobi coordinates. Denoting 0 3 2 the common eigenstates of the relative momentum operators density can be calculated according to (i) ( jk) qˆ and pˆ by q, p i, jk, where q and p are the corresponding | i 3 3 2 eigenvalues, we introduce momentum space wave functions P(R) = d ρd r φγ(ρ, r) δ(R(ρ, r) R). (39) by Z | | −

(i, jk) The purpose of the “mass” parameter µ0 is to ensure that the φγ (q, p) = i, jk q, p φγ . (37) h | i unit of R is length. The numerical value of µ0 is, in principle, arbitrary as it simply scales R. Fig. 7 displays the hyperradial Because of the transformation Eq. (32) wave functions with probability densities for the two helium trimer states, using respect to diﬀerent sets of Jacobi coordinates can be chosen to µ /m = 1 , which were calculated numerically from the mo- satisfy the transformation relation 0 2 mentum space Faddeev equations [44, 45] in the unitary pole (i, jk) (i) ( jk) ( j,ki) ( j) (ki) approximation [46] based on the TTY potential [47]. φγ (q , p ) = φγ (q , p ). (38)

The corresponding configuration space wave functions (i, jk) B. Scattering theory approach to trimer diffraction φγ (ρ, r) are defined analogously. In order not to overload the notation we will omit in the following, where possible, the indices of the relative coordinates: if not denoted otherwise We now proceed to the diffraction of a trimer from an ex- we implicitly use (i jk) = (123) whence ρ ρ(1), r r(23) and ternal potential q q(1), p p(23). ≡ ≡ ≡ ≡ W(r1, r2, r3) = W1(r1) + W2(r2) + W3(r3) (40) 0.8 4 He3 ground state where the Wi(ri) are the interactions of the individual atoms 4 He3 excited state with the diffraction object. The full Hamilton operatoris given 0.6 by ]

-1 H = H0 + V + W. (41) nm

[ 0.4 By virtue of this structure, which is formally identical to that

P(R) of dimer diﬀraction [16], we may carry over the fundamen- × 10 tal algebraic relations from previous work. Introducing, as in 0.2 Ref. [16], the resolvents

1 G0(z) = [z H0]− (42a) 0 − 1 1 10 100 GV (z) = [z H0 V]− (42b) Hyperradius R [nm] − − 1 G (z) = [z H W]− (42c) W − 0 − FIG. 7: Hyperradial probability densities P(R) according to Eq. (39) and the two-body T matrices in three-body space, of the two theoretically predicted bound states of the helium trimer 4 He3. The states were calculated numerically using the momen- T (z) = V + VG (z)V (43) tum space Faddeev equations [44, 45] in the unitary pole approxi- V V mation [46] based on the TTY potential [47] for the helium-helium TW (z) = W + WGW (z)W (44) interaction. Clearly, the excited state, with its expectation value of the hyperradius of µ /m R = 10.1 nm, is spatially more ex- for the potentials V and W, an AGS type [52] transition oper- 0 h ie tended by almost onep order of magnitude than the ground state with ator UVV can be derived which satisﬁes the equation µ /m R = 1.1 nm. The scaling of the horizontal axis is logarith- 0 h ig mic.p UVV = TW + TWG0TVG0UVV . (45) 9

In particular, the transition amplitude is given by the matrix W(r1, r2, r3). Splitting W according to Eq. (40) into the in- element of this transition operator [16], dividual potentials Wi(ri) and using the Lippmann-Schwinger Eq. (2) the matrix element (52) can be expressed by the known t(P,φγ; P′,φγ ) = P,φγ U (E′ + i0) P′,φγ , (46) ′ h | VV | ′ i atom transition amplitudes (3): at (3) and it determines the S matrix associated with H as p , p , p W p′ , p′ , p′ , + = t (p ; p′ ) δ (p p′ ) h 1 2 3| | 1 2 3 i 1 1 1 2 − 2 (3) (3) at (3) at P,φγ S P′,φγ = δ (P P′)δγγ δ (p3 p3′ ) + t1 (p1; p1′ ) δ (p2 p2′ ) t3 (p3; p3′ ) h | VV| ′ i − ′ × − − 1 1 2πiδ(E E′)t(P,φγ; P′,φγ′ ). (47) at at − − + + t1 (p1; p1′ ) t2 (p2; p2′ ) × " E′ E3 + i0 E′ E1 + i0# We shall in the following impose the condition 3 − 1 − at 1 1 2 2 t3 (p3; p3′ ) + cycl. perm. (53) P P′ × E2′ E2 + i0 E3′ E3 + i0 Eγ , φγ V φγ | | , | | (48) − − | | |h | | i| ≪ 2M 2M Here, “cycl. perm.” indicates that all explicitly shown terms which ensures that the internal energies of the trimer (both on the right hand side of Eq. (53) should be repeatedwith their binding energy and potential energy) are much smaller than indices permuted once and twice, in ascending order. Apply- the external energy associated with the center of mass mo- ing again the condition of the weak binding energy (48) the tion. For a helium trimer beam at an incident beam velocity complex energy denominators can be approximated. Firstly, 1 1 of v = 500m/s, for example, the center of mass kinetic en- using the principal value formula (x + i0)− = iπδ(x) + x− 2 − P ergy P′ /(2M) 16meV (corresponding to 180K) exceeds it is possible to approximate the trimer| | ground≈ state energy by more than three orders of (3) 1 1 magnitude. Using the Schrödinger equation for bound trimer δ (p p′ ) + 2 − 2 " E E + i0 E E + i0# wave functions φγ(q, p) the condition (48) can be shown to 3′ − 3 1′ − 1 (3) entail the relations δ (p p′ ) 2πiδ E E′ (54) ≃ 2 − 2 − 1 − 1 q P , P′ and p P , P′ . (49) where a small correction term (E /E) was neglected. Sec- | |≪| | | | | |≪| | | | γ ondly, for three variables x, y, z Owith x + y + z = 0 the distribu- These state that the wave functions of the trimer are concen- tion identity trated in momentum space at relative momenta far smaller 1 1 1 1 1 1 than the center of mass momentum. + + Under the conditions (48) and (49) an approximation of the x + i0 y + i0 y + i0 z + i0 z + i0 x + i0 equation for UVV (45) to lowest order is possible and sufficient (2π)2 [53, chap. 3.4] whence the transition amplitude becomes = δ(x)δ(y) + δ(y)δ(z) + δ(z)δ(x) (55) − 3 t(P,φγ; P′,φγ ) P,φγ T (E′ + i0) P′,φγ . (50) can be shown to hold. If x + y + z , 0, such as in Eq. (53) ′ ≃ h | W | ′ i for x = E1′ E1, y = E2′ E2, and z = E3′ E3, Eq. (55) is We note that within this approximation the trimer binding po- still applicable− within the− same range of validity− as Eq. (54). tential V is only implicitly contained through the bound states Combining the steps one arrives at the following approximate φ and φ . The evaluationof the right handside of Eq. (50) γ γ′ expression for the matrix element of TW (E′ + E′ + E′ + i0): is| i nontrivial.| i A series of approximations, all accurate within 1 2 3 at the condition (48), may however be applied to simplify the p , p , p T (E′ + E′ + E′ + i0) p′ , p′ , p′ t (p ; p′ ) h 1 2 3| W 1 2 3 | 1 2 3i≃ 1 1 1 transition amplitude. As the first step, the matrix element of (3) (3) at δ (p2 p2′ ) δ (p3 p3′ ) 2πi δ E1 E1′ t1 (p1; p1′ ) TW (E′ + i0) can be shown to vary slowly under a variation × − − − − π 2 of E′ on the scale of the binding energies Eγ . This allows (3) at (2 ) ′ δ (p2 p2′ ) t3 (p3; p3′ ) δ E2 E2′ δ E3 E3′ to replace the energy argument of TW in Eq. (50) by the sum × − − 3 − − 2 at at at E1′ + E2′ + E3′ where Ei′ = pi′ /2mi are the energies of the t1 (p1; p1′ ) t2 (p2; p2′ ) t3 (p3; p3′ ) + cycl.perm. (56) free atoms. Introducing two| complete| sets of states Eq. (50) × becomes Upon insertion of Eq. (56) into the trimer transition amplitude (51) several momentum integrals can be carried out by 3 3 3 3 virtue of the momentum delta-functions. Moreover, the en- t(P,φγ; P′,φγ ) d qd pd q′d p′ φ∗ (q, p)φγ (q′, p′) ′ ≃ Z γ ′ ergy delta-functions allow the integration of a further momen- p , p , p T (E′ + E′ + E′ + i0) p′ , p′ , p′ . (51) tum component each. As an example, we consider the delta- × h 1 2 3| W 1 2 3 | 1 2 3i function δ(E1 E1′ ) in the second term of Eq. (56). Switching Using Eqs. (42) and (44) the algebraic relation TW (z)G0(z) = − (3) back to Jacobi coordinates, after integrating over δ (p2 p2′ ) WGW (z) can be shown to hold. Inserting this relation the ma- it becomes − trix element of T (E + E + E + i0) is replaced by W 1′ 2′ 3′ 2 m1 δ(E1 E1′ ) = δ P + q p , p , p W p′ , p′ , p′ , + (52) − M h 1 2 3| | 1 2 3 i + + 2 where p′ , p′ , p′ , + p′ , + p′ , + p′ , + is the scattering m2 m3 m2 m3 | 1 2 3 i≡| 1 i| 2 i| 3 i P′ P + q ∆p . (57) state of three independent atoms associated with the potential − " − M − m2 #    10

To proceed we decompose the momentum vectors into their where again use has been made of condition (49) to simplify components parallel and perpendicular to the incident center the functional arguments of the atom transition amplitudes of mass momentum P′: the parallel componentof q, for exam- where possible. ple, is denoted by q and the two-dimensional perpendicular V. TRIMER DIFFRACTION FROM A DEEP GRATING vector is denoted byk q . Then, by condition (49) and using ⊥ P′ = 0 by deﬁnition, the delta-function (57) can be approximated⊥ by

δ ∆P + m2+m3 ∆p + ξ ( m1 P + q ) A. The trimer slit function of the deep grating k m2 k M k k δ(E1 E1′ ) − ≃ 2 m2+m3 m1 P + q m2 M k k where the term involving the factor ξ 6(λ /d)2 is a cor- ≈ dB rection which is small by the diffraction condition (1). Inte- In this section the general trimer transition amplitude will grating, according to Eq. (51), the second term of Eq. (56) be evaluated for diffraction from a deep grating. Inserting, to over dp′, the correction involving ξ is pushed into the func- this end, the expression (17) for the atom transition amplitude tional argumentsk of the atom transition amplitudes as well as into Eq. (60) yields a very long sum containing a total of 42 the trimer wave functions. In both cases, however, these func- terms which we do not spell out explicitly. These 42 terms can tions vary slowly on the scale of ξP such that the correction be classified by the number (zero through three) of references k can be shown to be negligible altogether. Similar simplifica- to the slit function in each. In the context of a many-body tions are readily derived for the other energy delta-functions multiple scattering series expansion [53] they may respec- in Eq. (56). tively be interpreted as forward (9 terms), single (18 terms), Combining all steps the general expression for the trimer double (12 terms), and triple-scattering terms (3 terms). A transition amplitude subject to the diffraction condition and cumbersome and little elucidating calculation involving the the condition of weak binding energy is obtained. Before- transformation properties of the form factor (59) with respect hand, however, it is helpful to introduce two abbreviations. to different sets of Jacobi coordinates reveals that the eigh- Firstly, we express the atom transition amplitude in terms of teen single-scattering terms interfere almost destructively and 2 the momentum transfer ∆p = p p′ as that their net contribution is small by a factor (λ /d) com- i i − i dB pared to the forward and the triple-scattering terms.O They may ti(pi; ∆pi) = ti(pi; pi′). (58) thus be neglected by the diffraction condition (1). Similarly, the twelve double-scattering terms contribute only to order Secondly, we introducee a molecular “form factor” (λ /d) and may also be neglected. O dB 3 3 Fγγ (q; p) = d q′d p′φ∗ q′, p′ φγ q + q′, p + p′ . (59) ′ Z γ ′ The forward and triple-scattering terms, respectively, can With this notation the trimer transition amplitude assumes the be combined. As in the atom case in Eq. (5) a delta-function form δ(∆P3) can be extracted from the trimer transition amplitude m + m m leaving 2 3 ∆ at 1 ∆ t(P,φγ; P′,φγ′ ) Fγγ′ 0, P ;0 t1 P; P ≃ ( − M ⊥ M e M(m2 + m3) 2 at m3 m2 + m3 2πi d p t3 P; ∆P , p − m2P Z ⊥ M k − m2 ⊥! k m + m m e 2 3 2 ∆ + Fγγ′ 0, P p ;0, p × − m2 M ⊥ ⊥ − ⊥! (2) t(P,φ ; P′,φ ) = δ(∆P )t (P ,φ ; P ′,φ ) (61) m m + m 4π2 M2 γ γ′ 3 γ γ′ at 1 , ∆ + 2 3 2 2 t1 P;0 P p 2 d q d p × M ⊥ m2 ⊥! − 3 P Z ⊥ ⊥ e m m k at 1 ∆ 1 ∆ + Fγγ′ 0, q ;0, p t1 P; P , P q × − ⊥ − ⊥ M k M ⊥ ⊥ at m2 m2 e m2 t2 P;0, ∆P q + p × M M ⊥ − m2 + m3 ⊥ ⊥! e where bold italic letters again denote the two-dimensional at m3 m3 m3 t3 P;0, ∆P q p + cycl. perm. projections of vectors into the plane perpendicular to the x3- × M M ⊥ − m2 + m3 ⊥ − ⊥!) axis. The two-dimensional trimer transition amplitude for e (60) diffraction from a transmission grating becomes 11

P + P P2 (2) i s1 ′s1 1 P1 1 s1 t (P ,φ ; P ′,φ ) 2π~ δ(∆P )δ dq dp F (0, q , 0;0, p , 0) gra γ γ′ 2 ~ 2 γγ′ ~ 2 2 γγ′ ≃−2 (2π) M 3( Ps1 − (2π ) P Z ⊥ ⊥ − ⊥ − ⊥ k m1 at m1 m1 HN ∆P sin θ′ + ∆P + q cos θ′ a1 P ′; ∆P sin(α + θ′) + ∆P + q cos(α + θ′) × k M ⊥ ⊥ M k M ⊥ ⊥ m2 m2 at m2 m2 m2 HN ∆P q + p cos θ′ a2 P ′; ∆P q + p cos(α + θ′) × M ⊥ − m2 + m3 ⊥ ⊥! ! M M ⊥ − m2 + m3 ⊥ ⊥! !

m3 m3 at m3 m3 m3 HN ∆P q p cos θ′ a3 P ′; ∆P q p cos(α + θ′) × M ⊥ − m2 + m3 ⊥ − ⊥! ! M M ⊥ − m2 + m3 ⊥ − ⊥! !) + cycl. perm. (62)

Inserting now for each atom the slit functions (16) and rewrit- incorporates the complicated internal configuration of the ing the form factor (59) as a configuration space integral trimer molecule through the bound state wave functions. In particular, the notation = 3 3 i(q ρ+p r)/~ Fγγ′ (q; p) d ρ d re− · · φγ∗(ρ, r) φγ′ (ρ, r) m2 + m3 Z r1 = S 2 cos(α + θ′) + ρ (66a) ⊥ M ⊥ the integrations over dq and dp in Eq. (62) can be carried m1 m3 ⊥ ⊥ r2 = S 2 cos(α + θ′) ρ + r (66b) out. The three grating functions HN give rise to a triple sum ⊥ − M ⊥ m2 + m3 ⊥ of which only the on-diagonal terms contribute significantly: m1 m2 r3 = S 2 cos(α + θ′) ρ r (66c) they represent diffraction of all three atoms from the same ⊥ − M ⊥ − m2 + m3 ⊥ bar; the off-diagonal terms, which correspond to diffraction of atoms from different bars, are negligible since the proba- has been chosen to emphasize the geometrical meaning of the bility for atoms to be spatially separated as far as the distance position arguments of the atom transmission functions: The between two adjacent bars (100nm) is strongly suppressed by quantities ri / cos(α + θ′) can be interpreted as the positions of the individual⊥ atoms projected onto the slit line while the bound state wave functions of the trimer. Collecting the S remaining terms the trimer transition amplitude can be cast the integration variable S 2 represents the projected center of into the form mass position R / cos(α + θ′). The trimer transmission function (65) is, therefore,⊥ simply the product of the three atomic P + P (2) i s1 ′s1 transmission functions averaged over the wave functions of t (P ,φγ; P ′,φγ ) (63) gra ′ ≃−2 (2π)2M~ the incident and the outgoing bound trimer state. This intuitive result is a straightforward extension of the case of dimer P1 tri 2π~ δ(∆P )δ H (∆P )a (P ′; ∆P ) . diffraction [54]. 2 γγ′ N 2 γγ′ s2 × ( Ps1 − ) Here, we introduced a trimer slit function by B. Thetrimerdiffraction pattern of the deep grating D/2 tri tri a (P ′; ∆P ) = dS exp i∆P S /~ τ (P ′; S ) γγ′ s2 2 s2 2 γγ′ 2 Thanksto the formal coincidenceof Eqs. (63) and (64) with Z D/2 − − ff (64) their counterparts Eqs. (17) and (16) of atom di raction the γγ′ derivation of the n-th order relative diffraction intensity In where S 2 can be interpreted geometrically as the center of for the incident bound state φγ′ and the outgoing bound state mass position of the trimer along the slit line (cf. Fig. 2) φγ can be carried over immediately. Therefore, we write the ff and where, analog to the atom case, D = d∆P2/∆Ps2 . Both trimer di raction intensities in the form Eqs. (63) and (64) exhibit the same structure as their atom 2 tri 2 Ps + P′ aγγ P ′; ∆Ps2 counterparts. Only the new trimer transmission function, Iγγ′ = Iγγ′ 1 s1 | ′ | . (67) n 0 tri 2 which appears in the trimer slit function (64), and which turns 2P′s ! a (P ′;0) 1 | γγ′ | out as Contrary to the atom case, however, the trimer slit function tri 3 3 depends on the spatially extended trimer bound states φγ and τ (P ′; S ) = d ρ d r φ∗ (ρ, r) φ (ρ, r) γγ′ 2 γ γ′ tri 2 2 Z φγ and, therefore, in general a (P ′;0) < S . ′ γγ′ 0 Eq. (67) determines the diff| raction intensities| of both elas- at m1 r1 at m2 r2 τ1 P ′; ⊥ τ2 P ′; ⊥ tic (φ = φ ) and inelastic (φ , φ ) processes. Earlier works × M cos(α + θ )! M cos(α + θ )! γ γ′ γ γ′ ′ ′ on the helium trimer [51] as well as on van der Waals dimers at m3 r3 [54] have shown that diffraction orders corresponding to in- τ3 P ′; ⊥ , (65) × M cos(α + θ′)! elastic processes are typically less intense by five to six orders 12 of magnitude than those of elastic processes. They are, there- 1 fore, experimentally less relevant. In the following we focus gg on elastic processes. Analogous to the procedure in Section 0

γγ /I

III, In can be approximated by a two-term cumulant expan- gg n I sion. The cumulants Rγ,± j now depend on the trimer state φγ. Moreover, because of the threefold van der Waals interaction 1 + 0.1 an additional term Ωγ = Im(R R ) should be retained in 2 2,γ− 2−,γ the expansion for suﬃcient numerical accuracy. Taking these generalizations into account the n-th order relative intensity becomes

2 Diffraction Intensity 0.01 I Ps + P′ n =  1 s1  exp (∆P Σ )2/~2 ∆P Γ /~ I   s2 γ s2 γ 0 2P S 2 + ∆2  h− − i -8 -6 -4 -2 0 2 4 6 8  ′s1 eff,γ γ   q  Diffraction order n 2 ∆ / ~ + 2 ∆ ∆ /~ + ∆ 2 Ω /~2 / sin [ Ps2 S eff,γ 2 ] sinh [( Ps2 γ Ps2 γ ) 2] 4 × ~ 2 FIG. 8: Diffraction intensities of a pure beam of ground state He3 ∆Ps2 /2 at v = 500 m/s from a d = 100 nm transmission grating at θ′ = 21◦ (68) angle of incidence for the grating geometry of Fig. 4. The solid curve was calculated using Eq. (67), the dashed curve shows the two-term where, analogous to Eqs. (27) and (28), the momentum com- cumulant approximation (68). To guide the eye these functions are ponents are to be evaluated at the incident angle θ′ and the shown continuously. The circles on top of the solid curve at integer diffraction angle θn as gg gg n mark the experimentally accessible diffraction orders In /I0 .

Ps = P ′ cos(α + θn), P′ = P ′ cos(α + θ′), 1 | | s1 | | 1 and the momentum transfer parallel to the s2-axis is given by ee 0 /I

∆Ps2 = P ′ sin(α + θn) sin(α + θ′) . ee n

| | − I Analogous to the atom case the eﬀective slit width S eﬀ,γ is 0.1 related to the trimer transmission function(65) by the equation

S 0/2 tri S eff,γ = S 0 Re dS 2 1 τγγ(P ′; S 2) . (69) 0.01 − Z S 0/2 − − h i Diffraction Intensity Fig. 8 shows elastic diffraction intensities for a beam of 4He in its ground state calculated according to Eqs. (67) and 3 -8 -6 -4 -2 0 2 4 6 8 (68). The asymmetry of this diffraction pattern is not as pronounced as in the atom case (Fig. 5). This is due to the three- Diffraction order n fold mass of the trimer which entails a three times shorter de 4 FIG. 9: As in Fig. 8 but for a pure beam of excited state He3. Due Broglie wavelength. Similarly, Fig. 9 shows diffraction inten- to the larger pair distance of the excited state the effective slit width 4 sities for a beam of He3 in its excited state. is smaller, resulting in a considerably wider envelope function than Since inelastic diffraction processes are negligible an ex- for the ground state. 4 perimental diffraction pattern of a He3 beam will in general be well described by an incoherent superposition of the individual diffraction patterns of the two bound states weighted and is, on the other hand, experimentally accessible (through by their relative population numbers in the beam. In the fol- γγ In ), it represents a link between experiment and theory. Ear- lowing section we first derive the trimer size effect for a pure lier work on atom and dimer diffraction revealed that the dif- beam containing trimers in only one state. Hereafter the treat- ference S 0 S eff,γ carries information about both the van der ment of a mixed beam will be considered. Waals surface− interaction [12] and the size of weakly bound dimers [3]. The reduction of the slit width by the dimer size 1 was foundto be 2 r where r denotes the dimer bond length. VI. HOW TO DETERMINE THE TRIMER SIZE The subsequent evaluationh i h ofi helium dimer diffraction data 4 yielded the experimental result r = 5.2 0.4nm for He2 A. The trimer size effect [3]. h i ± In the following we derive the corresponding size effect for Since the effective slit width of the trimer (69) depends, on a trimer. To this end we explicitly insert the trimer transmis- tri the one hand, on the trimer bound state (through τγγ(P ′; S 2)) sion function (65) into Eq. (69). By definition the atom trans- 13

at mission functions τi (pi′; si2) in Eq. (65) are zero if their po- s2 sitional arguments si2 = ri / cos(α + θ′) lie outside the slit interval [ S /2, S /2]. This⊥ fact may be utilized to reduce − 0 0 the integration interval for the center of mass position S 2: at fixed relative coordinates ρ, r the interval may be limited, for r > 0, to (12) ⊥ r m2 ∆+ ∆+ r(12) ⊥ S 0 2 S 0 1 cos( α+ θ′) + < S 2 < (70a) ⊥ − 2 cos(α + θ′) 2 − cos(α + θ′) m1 r(31) (31) (23) r and, similarly, for r < 0, to r m3 ⊥ ⊥ ⊥ cos( α+ θ′) ⊥ S 0 ∆1− S 0 ∆2− + < S 2 < . (70b) − 2 cos(α + θ′) 2 − cos(α + θ′) Here, the geometrical quantities FIG. 10: Geometrical interpretation of the effective slit width for- 1 m2 + m3 m1 m3 1 (23) (31) (12) ∆1± = − ρ + r mula. Clearly, the expression 2 ( r + r + r ) may be in- ±2 ( M ⊥ m2 + m3 ⊥ terpreted as the “width” (projected| ⊥ diameter)| | ⊥ | of the| ⊥ trimer| perpen- m dicular to its incident direction. Taking the expectation values with ρ 3 r (71) ⊥ + ⊥ the bound state wave function yields the expression (76). The multi- ± − m2 m3 ) 1 plication by [cos(α + θ′)]− corresponds to an orthogonal projection and onto the slit line . Hence the slit width S 0 appears reduced by the projected width ofS the trimer. 1 m2 + m3 m1 m2 ∆2± = − ρ + r ±2 (− M ⊥ m2 + m3 ⊥ m2 + 1 ρ + r (72) Therefore, as the factor [cos(α θ′)]− corresponds to an or- ± ⊥ m + m ⊥ ) thogonal projection of the perpendicular coordinates onto the 2 3 slit line (cf. Fig. 10) S geom is simply smaller than S by the eff,γ 0 have been introduced. Neglecting, for the moment, the van der projected width of the trimer. Waals interaction, all atom transmission functions are unity In the presence of the van der Waals interaction an addi- inside the reduced domain of integration (70). In this case the vdW tional term S eff,γ accounting for the deviation from unity of effective slit width depends only on ∆1± and ∆2±. Accordingly, we call it the geometrical part of the effective slit width and the atom transmission functions arises. The entire effective denote it by slit width is the sum 1 geom 3 3 2 S = S geom + S vdW. (77) S eff,γ =S 0 Re d ρd r φγ(ρ, r) eff,γ eff,γ eff,γ − cos(α + θ′) Z | | + + ∆1 + ∆2 Θ(r ) + ∆1− + ∆2− Θ( r ) (73) × ⊥ − ⊥ As the general expression for the van der Waals part S vdW is eff,γ where Θ(r ) is the Heaviside step function. Both integrands in long and little informative we shall not give its general form ⊥ Eq. (73) can be simplified using the transformation properties explicitly. of the Jacobi coordinates (32) and the wave functions (38). Combining the results, the geometrical part of the effective slit width becomes ff r(23) + r(31) + r(12) B. Thesizee ect for three identical Bosons geom ⊥ γ ⊥ γ ⊥ γ S = S 0 D E D E D E (74) eff,γ − 2 cos( α + θ ) ′ In the remaining paragraphs of this section we focus on 4 where the expectation values in the numerator are defined as trimers of three identical Bosons, such as the He3. By con- sequence, we denote the masses by m = mi and the equal ( jk) (i, jk) 2 ( jk) r = d3ρ(i)d3r( jk) φ (ρ(i), r( jk)) r . (75) projected pair distances by r γ. The geometric part of the γ h| ⊥|i ⊥ γ Z ⊥ effective slit width (74) immediately reduces to

In Eq. (74) the symmetric term geom 3 r γ = h| ⊥|i 1 (23) (31) (12) S eﬀ,γ S 0 . (78) r + r + r (76) − 2 cos(α + θ′) 2 ⊥ γ ⊥ γ ⊥ γ D E D E D E represents the expectation value of the “width” (projected di- Moreover, if the spatial extent of the bound state wave func- ameter) of the trimer perpendicular to the incident direction. tion is small comparedto the slit width, the van der Waals part 14

vdW S eff,γ is to very good approximation given by In order to test the validity of this approximation we carried out a numericalanalysis of the error introducedby the replace- vdW 3 3 2 ment of Eq. (79) by Eq. (82): if applied to experimental data S Re d ρd r φγ(ρ, r) eff,γ ≃− Z | | the approximation entails, for the two theoretically predicted 1 4 S 0/2 bound states of He , a systematic overestimation of r by P ′ P ′ ρ r 3 dS 1 τat ; S τat ; S ⊥ − 2 | ⊥| 4 h| ⊥|i  2′  2′  2′  7% ( He3 ground state), or 3% (excited state). As seen from Z0 − 3 ! 3 − cos(α + θ′)     Fig. 11 the approximationis more reliable at high velocities as   1  1   P  r + ρ r Θ ρ r  the impact of the van der Waals interaction becomes smaller. τat ′ ; S | ⊥| ⊥ − 2 | ⊥| ⊥ − 2 | ⊥|  2′ +  ×  3 − cos(α θ′)    0  1  P ′ P ′ ρ + r + dS 1 τat ; S τat ; S + ⊥ 2 | ⊥| 120 2′  2′  2′ +  Z S 0/2 − 3 ! 3 cos(α θ′) 4 −    He ground state  1  1  3 P  r ρ + r Θ ρ r  at ′ + | ⊥|− ⊥ 2 | ⊥| − ⊥ − 2 | ⊥| τ  ; S 2′  . × 3 cos(α + θ′)  110   [nm]   γ    (79) eff,  S vdW Within the approximation (79) it is evident that S is indeed 4 eff,γ He excited state zero if the atom transmission functions are unity inside the slit, 100 3 and if the spatial extent of the trimer wave function is small vdW on the scale of the slit width. Therefore, if S eff,γ were but a small correction to the full effective slit width (77) it could be 0.2 0.3 0.4 0.5 0.6 0.7 neglected, and the projected trimer pair distance r could v [km/s] be determined using Eq. (78). Experience with dimerh| ⊥|i diffraction has shown, however, that the effect of the van der Waals FIG. 11: Effective slit widths versus the beam velocity v = P′ /M for 4 | | interaction can be of the same order as the pair distance [3] the ground state and the excited state of He3. The numerical results and must be accounted for. Since Eq. (79) depends on the using the full expression Eq. (79) are shown as solid curves and the full trimer wave function it cannot be used immediately for approximation Eq. (82) as dotted curves. The angle of incidence was the evaluation of experimental data and an approximation is taken as θ′ = 21◦. The approximation becomes more reliable at high required. The integrand in Eq. (79) is, however, slowly vary- velocities as the impact of the van der Waals interaction decreases. 2 Asymptotically, for high velocities, both pairs of curves approach ing on the scale of the variation of φ (ρ, r) . Therefore, the geom γ their respective upper limits S given by Eq. (78). positional arguments of the atom transmission| | functions can eff,γ approximately be replaced by their expectation values. This approach is in analogy to Ref. [3]. An analysis of the com- Theoretical studies of the helium trimer commonly state the expectation value of the pair distance r itself, where r = r , binations of ρ and r , using once more the transformation h i | | properties of the⊥ relative⊥ coordinates (32), shows that these ex- rather than a component such as r . To link experimental h| ⊥|i pectation values are expressible solely in terms of r . For results to these, a relation between r and r must be estab- ⊥ 4 h i h| ⊥|i example, h| |i lished. Both predicted He3 bound states are spherically symmetric (zero total angular momentum). Moreover, the two- 1 body scattering matrix corresponding to the He-He potential ρ r = r γ (80) * ⊥ − 2| ⊥| + h| ⊥|i is dominated by the shallow s-wave bound state pole of 4He , γ 2 and higher partial waves may to good approximation be ne- and glected [21]. By the Faddeev equations (e.g. Ref. [45]) for 1 1 1 the helium trimer bound state, it is then possible to derive the ρ r Θ ρ r = r γ . (81) * ⊥ − 2| ⊥|! ⊥ − 2| ⊥|!+γ 4 h| ⊥|i relation 1 Inserting these one finds as the final form of the van der Waals r = r . (83) part of the effective slit width h| ⊥|i 2 h i

S 0/2 In summary, the effective slit width (77) depends to good ap- vdW at P ′ S eff,γ Re dS 2′ 1 τ ; S 2′ proximation only on one trimer parameter, namely the expec- ≃− ( Z0 " − 3 ! tation value of the pair distance r . Consequently, r can, in h i h i P ′ r γ P ′ 5 r γ principle, be determined from trimer diffraction data. at ⊥ at ⊥ τ ; S 2′ h| |i τ ; S 2′ h| |i × 3 − cos(α + θ′)! 3 − 4 cos(α + θ′)!# 0 P ′ P ′ r γ at at ⊥ C. Experimental considerations + dS 2′ 1 τ ; S 2′ τ ; S 2′ + h| |i Z S /2 " − 3 ! 3 cos(α + θ′)! − 0 P ′ 5 r γ at ⊥ Using the results of the preceding sections the improvement τ ; S 2′ + h| |i . (82) × 3 4 cos(α + θ′)!# ) in resolution through diffraction at non-normal incidence over 15 normal incidence may be estimated. The evaluation process of VII. CONCLUSIONS experimental data involves two main steps: Firstly, values for the effective slit width S eff,γ must be obtained by numerical Motivated by the long-standing interest in the Efimov effect fits of the intensity formula (68) to experimental diffraction [31] we have studied the diffraction of weakly bound trimers patterns. Secondly, r is determined from a fit of Eq. (77) h i in a typicalmatter optics setup. As an earlier diffraction exper- to the values for S eff,γ. The stronger the dependence of this iment for the spatially extended helium dimer ( r = 5.2nm) procedure on r the more precisely r can be determined. [3] had indicated that the resolution providedh byi a custom As r changesh thei width of the Kirchhoh iff-like slit function in h i s0 = 60nm transmission grating, at normal incidence, may Eq. (68) a natural measure for this dependence is provided by be insufficient to resolve the helium trimer ground state ( r = the number of diffraction orders under the central maximum 0.96nm predicted [4]) it had suggested itself to use obliqueh i of the slit function to either side of the forward direction. This (non-normal) incidence at a rotated transmission grating for number, which we denote here by nc, and which we treat as a reducing the projected slit width. The partial shadowing of continuous variable, can be approximately written as the slits caused by the finite thickness of the etched material grating has required, however, a revision of the theory of d atom diffraction. In particular, the familiar mirror symme- n = ⊥ (84) c s try encountered in diffraction patterns from normal incidence ⊥ is lifted for non-normal incidence. This effect was visible in where d denotes the (projected) period perpendicular to the Fig. 5 of Ref. [35] but went unnoticed. It has been traced beam and⊥ s denotes the projected slit width. At the angle back to the non-alignment of the direction of periodicity of ⊥ of incidence θ′ the projected period is d = d cos θ′. Fur- the grating with the shadow lines of its bars, or, equivalently, thermore, neglecting for this estimate the⊥ van der Waals part, its slit lines. The weak attractive van der Waals surface inter- geom action, which introduces an additional but minor asymmetry, we insert s = S eff,γ cos(α + θ′) for the projected slit width ⊥ geom has been taken into account in a way similar to the case of of the deep grating where S will be taken from Eq. (78). eff,γ normal incidence. The relative variation of n with r can then be calculated and c Using atom diffraction as one building block, the multi- becomes, to leading order in r h/Si , h i 0 channel many-bodyquantum mechanical scattering theory approach of Refs. [15, 16] has been extended to derive the con- 1 dn 3 1 c . stitutive formulas of trimer diffraction. While this procedure nc d r ≃ 4 S 0 cos(α + θ ) structurally partly parallels that of dimer diffraction it is math- h i ′ ematically more complex due to the additional atom. The re- In contrast, at normal incidence the right hand side would sulting equations for the trimer diffraction pattern, however, be 3/(4s0). Inserting the parameters of Fig. 8 yields have been readily interpretable and provide intuitive physi- 2 1 (dnc/d r )/nc 2.6 10− nm− for θ′ = 21◦ as compared to cal insight into diffraction of weakly bound molecules: the h i2 ≈1 × 1.2 10− nm− for normal incidence. This roughly twofold significant measurable quantity is the quantum mechanical gain× in sensitivity is expected to halve the final error bars on expectation value of the “width” (projected diameter) of the r . trimer perpendicular to its flight direction. For identical Bo- h i Finally, since the population ratio of the two predicted 4He son trimers, such as the helium trimer, the width is related to 3 3 states in the nozzle beam is generally unknown the situation the molecular bond length by 4 r . This fact can be used, in principle, to determine r fromh ai matter diffraction experi- of a mixed beam must be considered. To analyze this we have h i summed diffraction patterns as shown in Figs. 8 and 9 for dif- ment. ferent population ratios. Hereafter, we have used Eq. (68) to If a transmission grating of the type used earlier by Grisenti determine, from the summed patterns, an average effective slit et al [12] is rotated by θ′ = 21◦ the projected slits appear approximately half as wide as the nominal slits of the grat- width S eff and from this an average bond length r . It turned out that the such determined r varies almost linearlyh i with ing. This leads to an estimated doubling of the resolution, 4 the population ratio from the groundh i state value of r (pure sufficient to determine the ground state pair distance of He3. 4 ground state beam) to the excited state value (pureh excitedi Moreover, should the He3 Efimov state, whose pair distance state beam). Therefore, three possible outcomes of an experi- is predicted to be larger by almost an order of magnitude [4], ment are to be expected. A value of r 1nm would be at- exist and should its population in the beam be significant, tributed to the ground state and indicateh i a ≈ negligible (or zero) it ought to be clearly distinguishable from the ground state population of the excited state. Equivalently, a result of about solely by its size. 8 nm would doubtlessly provide evidence for the excited state and its large pair distance. Thirdly, a value in between these two would indicate that both states are present and evidence Acknowledgments for the excited state would still be available. A controlled variation of the accessible beam parameters might then allow to We would like to thank R. Brühl, O. Kornilov, A. Kalinin, influence the population ratio in favor of either state and to T. Köhler and J.P. Toennies for stimulating discussions. This measure the pair distance for one state with less disturbance research was supported by the Deutsche Forschungsgemein- by the other. schaft. 16

APPENDIX A: SURFACE INTERACTION ditions in Sec. II) over the volume of the bars. Carrying out all four integrations for the typical wedge-shaped bars shown in Earlier work has shown that at beam velocities typically en- Fig. 4 the phase function can be calculated explicitly. Using countered in matter diffraction experiments the effective re- the abbreviations duction of the slit width due to both the van der Waals surface interaction and the finite molecular size can be of the same πC6 cos θ′ cos θ′ C3 = , d = d, s0 = s0, order of magnitude [3, 12]. A quantitative determination of 6 cos(α + θ′) cos(α + θ′) at the atom transmission function τ (p′; s2), which was inserted e e into Eq. (16), is thereforenecessary. As in Ref. [12] we use the it reads at eikonal approximation to write τ (p′; s2) = exp[iϕ(p′; s2)] for at s2 inside the slit, and τ (p′; s2) = 0 outside. The phase func- C3 ϕ(p′; s2) = tion ϕ(p′; s2) is given by [55] ~ 2 2 2 v cos θ′ cos (α + θ′) 2 2 2 2 p ξ− + (ξ11 d)− ξ− (ξ12 + d)− ~ 1 ′ 11 − − 12 − ϕ(p′; s2) = ( v)− dt Wsurf(s(t)), v = | |, (A1)  − Z m × tan θ′ + tan β  e e  2 2 2 2 where the straight path of integration s(t) must be taken to run  ξ− + (ξ21 d)− ξ− (ξ22 d)− + 21 − − 22 − − parallel to the direction of incidence, and to cross the slit line  tane θ′ tan β e  at the position s . The surface interaction W (x) at a po- −  2 surf  sitionS x between two bars is calculated from the integration  6 = S 0 = S 0 + = S 0 + of an attractive C6/l potential of Lennard-Jones type (the where ξ11 2 s2, ξ21 2 s2, ξ12 2 s2 s0 − cos((α+θ′) 2θ′) − S 0 − − repulsive part has already been modeled by the boundary con- + − S 0, and ξ22 = s2 s0. cos(α θ′) 2 − − e e

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