Matter-wave interferometry: towards antimatter interferometers
Simone Sala, Fabrizio Castelli, Marco Giammarchi, Stefano Siccardi, Stefano Olivares
MMatter-wave interferometry:towards antimatter interferometers
Simone Sala , , Fabrizio Castelli , , Marco Giammarchi ,Stefano Siccardi and Stefano Olivares , Dipartimento di Fisica, Universit`a degli Studi di Milano, I-20133 Milano, Italy INFN Sezione di Milano, I-20133 Milano, ItalyE-mail: [email protected]
Abstract.
Starting from an elementary model and refining it to take into accountmore realistic effects, we discuss the limitations and advantages of matter-waveinterferometry in different configurations. We focus on the possibility to apply thisapproach to scenarios involving antimatter, such as positrons and positronium atoms.In particular, we investigate the Talbot-Lau interferometer with material gratings anddiscuss in details the results in view of the possible experimental verification.
1. Introduction
Matter-wave interference is at the heart of the quantum mechanical nature of particles.While this phenomenon has been observed for electrons [1, 2, 3], neutrons [4, 5], atomsand molecules [6, 7, 8] using a variety of different experimental tools, no experimentaltests exist on elementary antimatter particles, or matter-antimatter systems. However,beams of antiparticles at low energy are becoming increasingly available, as in the caseof antiprotons at the CERN Antiproton Decelerator [9] or in the case of positrons (andthe associated positronium production) in Na source systems coupled with Surko traps[10]. In this paper we discuss the optical analogy and the main principles of Fraunhoferand Talbot matter-wave interference regimes, considering material gratings, in order tointroduce the issues and the problems of antimatter interferometry. Positrons ( e + ) areproposed as our first antimatter system to study and positronium (Ps) is the atom thatwe will be considering as a matter-antimatter symmetric system. The antiproton ( p )case will also be shortly discussed.The paper is structured as follows. In section 2 we review the basic elements ofquantum diffraction theory of particles from a grating and describe the build-up ofthe statistical interference pattern. Section 3 focuses on the incoherence due to thesource, such as the effect of the particle velocity spectrum and the source geometricalextension. In section 4 we address the interaction between particles and a grating a r X i v : . [ qu a n t - ph ] S e p atter-wave interferometry: towards antimatter interferometers L G = hmv x y s c r ee n Figure 1. (Color online) A single particle of momentum p = mv impinging on an N -slit grating ( G ); detection will take place on a screen, placed at a distance ( L ). Thegrating has a period D and the width of each slit is a . The z -axis is orthogonal to the x – y plane. considering both neutral particles and charged particles. Section 5 is devoted to Talbot-Lau interferometry: we describe the geometry, and its advantages with respect to singlegrating setups. Furthermore, we numerically show how the fringe visibility is affected bythe particle velocity spread, when realistic parameters are used to carry out Monte Carlosimulated experiments. Finally, we close the paper drawing some concluding remarksin section 6.
2. Basic quantum model of diffraction
In this section we review the basics of matter-wave interferometry. We assume that aparticle moving along the y -axis with de Broglie wavelength λ = h/ ( mv ), m and v beingrespectively its mass and its velocity along the y axis, interacts with an N -slit gratinglaying in the x – z plane (see Fig. 1). Upon assuming that the slits are sufficiently largealong the z -axis, so that diffraction is negligible along that direction, we can representthe state just after the grating at time t = 0 and y = 0, being t = y/v , as the followingsuperposition state [11, 12, 13]: ψ ( N ) ( x, t = 0) ∝ N (cid:88) n =1 ψ n ( x, t = 0) , (1)where ψ n ( x, t = 0), n = 1 , . . . , N , are the wave functions describing the particle passedthrough the n -th slit (we assume, as usual, that the slits are independent). For a systemof identical slits with period D , we can write ψ n ( x,
0) = ψ ( x − nD ). Indeed, the actualexpression of ψ n ( x, t = 0) is dictated by the characteristics of the diffraction gratingand its interaction with the incoming particle. As the grating prepares the system inthe state of Eq. (1), we can assume that the motion along the x -axis is governed by thefree Hamiltonian: H eff = p x m . (2)Therefore, the evolved state ψ ( x, t ) is obtained by solving the Schr¨odinger equationwith the Hamiltonian (2). In particular, the particle probability density distribution atter-wave interferometry: towards antimatter interferometers x -axis on the screen at position y = L (the interference pattern), is given by I ( x ) = | ψ ( N ) ( x, t = L/v ) | , where: ψ ( N ) ( x, t ) = 1 √ λL (cid:90) + ∞−∞ exp (cid:104) i πλL ( x − x (cid:48) ) (cid:105) ψ ( N ) ( x (cid:48) ,
0) d x (cid:48) , (3)that is formally identical to the Fresnel integral of classical optics [14]. The most commonapproach found in literature [12, 13, 15, 16] is to adopt an “effective” point of view andpostulate a convenient form for the initial single-slit wave function, for example: ψ n ( x,
0) = a − χ [ − a + nD, a + nD ] ( x ) (4)where a is the slit width and χ Ω ( x ) = (cid:40) x ∈ Ω , . This is the quantum mechanical analog of assuming uniform illumination in thetreatment of light diffraction. Another useful choice for the initial single-slit wavefunction is a Gaussian function centered on the slit interval with a suitable variance σ , namely (we drop the overall normalization constants); ψ n ( x,
0) = exp (cid:20) − ( x − nD ) σ (cid:21) . (5)This choice is more convenient, as many calculations can be easily carried outanalytically on Gaussian functions. In this case, the parameter σ is usually set to σ = a/ (2 √ π ). Upon introducing the rescaled variables:ˆ x = xσ , ˆ D = Dσ , and ˆ L = (cid:126) t mσ = Lλ πσ , (6)and considering a two-slit setup, the time evolved wave function outgoing from a doubleslit setup reads ψ (ˆ x, ˆ L ) = (cid:88) n =1 , C n exp − (ˆ x − ˆ x n ) (cid:16) L (cid:17) (cid:16) − i ˆ L (cid:17) where ˆ x = − ˆ D/ x = + ˆ D/
2. The generalization for a set of N equally separatedslits is straightforward. We have introduced the relative normalization constants C n ,to account for a possible asymmetry in the beam preparation [13]. In the following weassume perfect symmetry C = C = 1. After simple algebraic manipulations, defining F ± = exp (cid:34) − (ˆ x ± ˆ D/ L ) (cid:35) the intensity reads I (ˆ x, ˆ L ) = F + + F − + 2 (cid:112) F + F − cos (cid:34) ˆ L ˆ x ˆ D L ) (cid:35) (7) atter-wave interferometry: towards antimatter interferometers Figure 2.
The evolution of the interference pattern (7) shown with its dependence onthe screen distance L . which clearly shows the appearance of an interference pattern due to the oscillating term(see Fig. 2). It is worth noting that the condition for observing the interference maximain far field turns out to be the usual relation of classical optics, thus in the limit ˆ L (cid:29) x ˆ D L = 2 nπ → xDπLλ = nπ which is indeed the expected classical relation. The formal analogy with classical optics[see Eq. (3)] also ensures that the choice of the initial single slit profile impacts onlythe envelope of the intensity pattern and not its oscillatory behavior. The classicalFraunhofer field outgoing a double slit setup reads I class ( x, L ) ∝ sinc (cid:18) πa xLλ (cid:19) (cid:20) (cid:18) πD xLλ (cid:19)(cid:21) while starting from (7) it easy to recover a Fraunhofer-like expression by taking the farfield limit in the form ˆ L (cid:29) x (cid:29) ˆ D , so that F + = F − (cid:39) exp (cid:18) − ˆ x L (cid:19) and finally, in order to highlight the similarity with the classical expression in theFraunhofer limit, we use Eqs. (6) in Eq. (7), obtaining: I ( x, L ) = 2 exp (cid:20) − (cid:16) π σ xλL (cid:17) (cid:21) (cid:20) (cid:18) πD xLλ (cid:19)(cid:21) . (8)So, much alike the classical case, in a quantum treatment based on the free evolutionof single-slit wave functions, the latter factorizes and determines the envelope of thepattern.
3. Incoherence due to the source
In order to describe a real experiment, the model introduced so far is not enough, sincemany relevant departures from the ideal situation arise. For instance, the particles’speeds vary according to a given distribution, the particle source has a finite size and thecollimation stage unavoidably introduces transverse momenta. Furthermore, focusing atter-wave interferometry: towards antimatter interferometers y s L G s o u r ce d e t ec t o r x y Figure 3.
Sketch of an interferometer operating in the far field, where an incoherentextended source (transverse size σ s ) illuminates an N -slit grating G (period D and slitwidth a ) from a distance y s . on the scenario we are interested in, unstable antimatter atoms like Ps can decay inflight. All of these issues lead to incoherence effects. In general, if q = ( q , q , . . . ) is thevector of the physical parameters q k which can classically fluctuate in a real experiment,we can describe the overall incoherence effect by averaging the ideal intensity I ( x, t | q )given a suitable distribution p ( q ), that is:¯ I ( x, t ) = (cid:90) I ( x, t | q ) p ( q ) d q (9)There are two relevant examples of incoherence: the one due to a finite transversecoherence length, the other due to the presence of a non-monochromatic beam. In thissection we focus on the first one, whereas the effects of a non-monochromatic beam willbe considered in section 5 in the context of the Talbot-Lau interferometry.The experimental results show that the patterns of matter-wave experiments withmulti-slit gratings can be described by considering a limited number of slits [15, 17].We define an experimental parameter l , the coherence length , as the typical transverselength scale on the plane of the grating that sets how many slits can coherently take partto the interference process. From the physical point of view, a finite coherence length is aresult of both the spatial extension and the intrinsic incoherence of the sources typicallyemployed in matter-wave experiments. In order to take into account this effect, we adda (common) random transverse momentum k x along x -direction to the wave function ψ n ( x, t = 0) associated with each slit, namely, ψ n ( x, t = 0) exp (i xk x ). If we assumethat k x is distributed according to a Gaussian distribution with zero mean and variance1 /l , one obtains the following analytical result for the intensity, valid in the Fraunhoferlimit [17]: I (cid:48) ( x, L ) = 2 πNλL (cid:12)(cid:12)(cid:12) ˆ ψ ( x ) (cid:12)(cid:12)(cid:12) (cid:40) N − (cid:88) n =1 N − nN exp (cid:20) − ( nD ) l (cid:21) cos (cid:18) πnDxλL (cid:19)(cid:41) , (10)where ˆ ψ ( x ) = (cid:90) ψ ( x (cid:48) ) exp (cid:18) i 2 πxλL x (cid:48) (cid:19) d x (cid:48) , (11) atter-wave interferometry: towards antimatter interferometers ψ ( x ) = ψ ( x, t = 0). Since for nD (cid:29) l , the corresponding exponential termsuppresses the interference, l can be regarded as the coherence length predicted inRef. [17], which is inversely proportional to the transverse momentum spread. Thiscould allow to determine the coherence length a priori , without resorting to a fit ofthe model to experimental data. However the transverse momentum distribution is notalways easily guessed. For example, if two successive slits are used as collimators, abound on the maximum transverse momentum could be established with a geometricalconstruction [17]. Nevertheless, since we are interested in more compact geometries asin the presence of in-flight decay of unstable antimatter, limiting the dimensions of theapparatus will be of the utmost importance. So we would need an estimate on thecoherence length for an apparatus of the kind of Fig. 3.We can obtain the intensity within the framework of the model given in Eq. (9) asfollows. We assume that at random time a particle is emitted with a speed v from a point x s of the source (located at the distance y s from the grating), following a distribution p ( x s , v ) that is determined by the nature of the source itself. After its emission theparticle crosses the grating and produces an interference intensity pattern that dependsparametrically on these quantities. Under the same assumption there discussed, theoverall intensity at the screen is thus given by Eq. (9), that now reads:¯ I ( x, L ) = (cid:90) I ( x, L | x s , v ) p ( x s , v ) d x s d v. (12)The integration can be performed via Monte Carlo (MC) method, as it scales wellwith the dimension of the parameter space. Moreover, we can refine our analysis,e.g., taking into account the instability of the particles and their lifetime. As a firstapproximation, we could simply discard the particles that do not reach the detectorplane. This corresponds to employ a detector able to discriminate between a true eventand the background noise induced by the decay in flight.In order to obtain the same results as in Eq. (10) for a suitable choice of l and in the Fraunhofer approximation, we should consider a monochromatic beam andaverage only over the source dimension x s , assuming a uniform probability density p ( x s ) = σ − s χ [ − σ s / ,σ s / ( x ), σ s being the source dimension. It is worth noting that inour simulations the average intensities are computed retaining the full accuracy of theFresnel integral, i.e., without the Fraunhofer approximation. In a setup like the oneshown in Fig. 3 a comparison between Eqs. (10) and (12) shows that the coherencelength l can be estimated as [18]: l ≈ y s λ σ s . (13)The dependence on the physical parameters is in agreement with the naive estimateassociating the coherent illumination region in this kind of setup with the width of thecentral diffraction peak for a slit of size σ s , where σ s is the transverse extension of thesource [18]. In Fig. 4 we show the Monte Carlo simulations of the far field interferencepattern for different values of the source dimension σ s and a particular choice of the atter-wave interferometry: towards antimatter interferometers x position [m]-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -3 × I n t en s i t y [ a . u .] l = 1 µ m l = 10 µ m l = 100 µ m Figure 4.
Monte Carlo simulation of the far field setup (Fig. 3) from Eq. (12) for amonochromatic beam of Ps atoms with λ = 3 . m Ps = 2 m e , v = 10 m / s) anddifferent values of the source dimension σ s = 900 ,
90 and 9 µ m, corresponding to thecoherence length l = 1 ,
10 and 100 µ m (as shown in the plot). We also set D = 10 µ m, a = 3 µ m, N = 10, y s = 0 . L = 1 m. Interference disappears when l < D ,and the contrast starts to decrease when l ≈ D . The dashed curves refers to thecorresponding single-slit diffraction envelopes. other involved parameters. For the simulations we considered a typical Ps velocity v = 10 m / s [19], which leads to λ = 3 . D , we observe a decrease in the contrast or visibility of thepattern (see Fig. 4): C = I max − I min I max + I min (14)defined as the difference between the intensity of a maximum and its adjacent minimum.We will reconsider the implication for the design of an experiment in sect. 5.
4. Interaction with material gratings
In the previous section we addressed the interferometry problem assuming that theparticle did not interact with the grating. As physical quantum mechanical objects theparticles interact in various ways with the walls of the material grating. The formalismwe developed so far is sufficiently general to account for this effect by modifying theinitial wave function accordingly.In the previous sections we have seen that the fundamental building block forquantum models of diffraction from a grating is the single slit outgoing wave function ψ n ( ξ, V ( ξ, y ) acting on the particle is known in the region within one slit, we can atter-wave interferometry: towards antimatter interferometers n ( ⇠ ) y⇠ a m a M n ( ⇠ ) e i' ( ⇠ ) Figure 5.
Trapezoidal bars cross section with the narrower side of the slit facing thebeam, displayed for a wedge angle β and nominal maximum and minimum widths a M and a m = a M − δ tan β , respectively. account for the interaction by treating the slit as a phase mask , producing a transmissionfunction of the form t A ( ξ ) = e i ϕ ( ξ ) [20, 21, 22], implying the substitution (see Fig. 5) ψ n ( ξ, → ψ n ( ξ,
0) e i ϕ ( ξ ) . (15)The standard approach [20, 22, 23] is now to determine the phase shift ϕ ( ξ ) via thesemiclassical eikonal approximation. Denoting with v the particle speed we can write: ϕ ( ξ ) = − (cid:126) v (cid:90) V ( ξ, y ) d y. (16)As discussed, the Fourier transform ˆ ψ of the single slit wave function sets the envelopeof the diffraction pattern [see Eq. (11)], which in the absence of interactions dependson the nominal (geometrical) width of the slit a . We will show that as a first orderapproximation the effect of a potential is a reduction of the effective slit width; this hasbeen observed in various situations [20], a notable example being the C experiments[6]. We find: ˆ ψ ( x ) = (cid:90) slit ψ ( ξ ) exp (cid:18) i 2 πξxλL (cid:19) d ξ (17)where x denotes the transversal coordinate on the screen plane, consistently with ournotation. As suggested by Ref. [20], we approximate the above Fourier transform usinga cumulants expansion [24] [a normalization of ψ ( ξ ) ensuring (cid:82) slit ψ ( ξ ) d ξ = 1 is implied]log ˆ ψ ( x ) = ∞ (cid:88) n =1 κ n (i ζ ) n n ! with ζ = 2 πxλL (18)where the cumulants κ n are defined in terms of the raw moments µ k of ψ ( ξ ), namely, µ k = (cid:90) ψ ( ξ ) e i ϕ ( ξ ) ξ k d ξ, and we are interested in the first two terms only κ = µ , and κ = µ − µ , (19) atter-wave interferometry: towards antimatter interferometers ψ ( x ) ≈ exp (cid:18) i κ ζ − κ ζ (cid:19) (20) | ˆ ψ ( x ) | ≈ exp (cid:8) − ζ (cid:61) m[ κ ] − (cid:60) e[ κ ] ζ (cid:9) (21)In order to give a physical meaning to the expansion parameters, we analyze the caseof no interaction, with a rectangular wave function ψ ( ξ ) = a − χ [ − a / ,a / . The firstraw moment (the mean) of this distribution vanishes due to parity ψ ( ξ ) = ψ ( − ξ ), andin general we expect this to be true in any realistic situation, as both a reasonableinteraction potential and the wall geometry will be symmetric with respect to ξ → − ξ .The second moment is simply evaluated κ = µ = 1 a (cid:90) a / − a / ξ d ξ = a | ˆ ψ ( x ) | ≈ exp (cid:20) − (cid:16) πa xλL (cid:17) (cid:21) . As one may expect, the exact expression is the well known ˆ ψ ( x ) = sinc[ πax/ ( λL )], thecentral peak of the sinc function coincides with the Gaussian approximation, suggestingto identify an effective slit width as follows: a eff = (cid:112) (cid:60) e[ κ ] . (22)We will now distinguish the two cases of neutral and charged particles in term of thepotential. The nonretarded van der Waals atom-surface potential [20], which affects all types ofneutral polarizable particles, is expressed in terms of the distance from the surface andthe coefficient C as V vdW ( r ) = − C r , ( r >
10 ˚A) . (23)It has been shown with direct electron microscope imaging, that diffraction gratingsof the type commonly used for matter-wave experiments can have a trapezoidal (seeFig. 5) slit profile [20, 22, 23, 25, 26], as a result of the fabrication process. Thereforeit is useful to study this immediate generalization, from which the trivial parallel-planeprofile is recovered in the limit β →
0. Calculations also show that the introductionof even a small wedge angle has a significant impact on the effective width of theslits compared to parallel-planes approximation at the same thickness and interactionstrength.The potential can be written in terms of the distance of the generic point ( ξ, y )from the right and left grating walls, respectively d and d . The symmetry ofthe system implies that these quantities are related by d ( ξ, y ) = d ( − ξ, y ). Since atter-wave interferometry: towards antimatter interferometers d ( ξ, y ) = a m / − ξ + y tan β , as it is evident from Fig. 5, the projection of the distanceon the normal vector to the side wall is simply obtained by multiplying d ( ξ, y ) for thecosine of the wedge angle β . The integration in Eq. (16) is straightforward, and yieldsfor the phase shift of the potential (23) ϕ ( ξ, β ) = C v (cid:126) cos β tan β ) − [ a m − ξ − y tan β )] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y = δy =0 . (24)Taking both surfaces into account we obtain: ϕ ( ξ, β ) = ϕ ( ξ, β ) + ϕ ( − ξ, β ) . (25)We note that at fixed geometry the ratio R = C /v sets the overall scale of theinteraction strength, as ϕ ( ξ, β ) ∝ C /v . Numerical estimates of Eq. (22) with thephase shift (25) are in good agreement with the experimental and theoretical results in[20]. Therefore, this kind of interactions can be accounted for by a reduction in effectiveslit width, at least up to the highest interaction strength tested experimentally, that is R max = C /v ≈ . · − meV · nm (this is obtained for Kr atoms at v = 400 m / swith a SiN x grating, and a measured C = 1 . · nm ). Hoinkes’ empirical rule [27],which has been confirmed experimentally [20], states that for a given material C is linearin the particle static polarizability α . The static polarizability of Ps atoms, α (Ps) , inthe ground state (estimated treating it as an hydrogen-like atom with the appropriatereduced mass) is α (Ps) ≡ α (H) ≈ .
33 ˚A , roughly twice that of the Kr atoms. Inturn, Ps atoms have an interaction scale R < R max down to speeds of v (Ps)min ≈
800 m / s,corresponding to a very low energy E ≈ . · − meV, while the lower speed limit for anexperiment with antihydrogen atoms (assuming α ( H ) = α ( H ) ) would be v (H)min ≈
100 m / s.Therefore, we can safely conclude that a treatment of the van der Waals interaction interms of Eq. (22) is fully adequate to describe experiments involving current Ps sources[28, 29]. The above procedure is easily generalized to all potentials depending on the distancefrom the surface as ∝ r − n . Relevant examples are the retarded van der Waals interaction( n = 4) and the electrostatic potential ( n = 1). In the AEgIS experiment at CERN[30], the production of antihydrogen and Ps atoms also involves, as an intermediatestep, the realization of a steady beam of charged antimatter, specifically e + and p .It will be interesting to carry out interferometry experiments on these objects as well,because no successful demonstration of interference has been obtained for these systems,yet. Moreover, to the best of our knowledge, this statement also applies to any kind ofcharged system heavier than an electron.It is a standard result in electrostatics that the potential acting on a point charge q sitting at an orthogonal distance r from a dielectric surface (relative permittivity (cid:15) )is that of a point charge q (cid:48) = q (1 − (cid:15) ) (1 + (cid:15) ) − located on the axis of symmetry with atter-wave interferometry: towards antimatter interferometers V el ( r ) = 1 − (cid:15) (cid:15) q π(cid:15) r (26)where r the distance from the grating wall. By using Eq. (16) we find: ϕ el1 ( ξ, β ) = q (1 − (cid:15) )(1 + (cid:15) ) − π(cid:15) (cid:126) v sin β log (cid:20) a m − ξ − δ tan β ) a m − ξ (cid:21) , (27)where a m , δ and β are the same as in Fig. 5. Once the total phase shift ϕ ( ξ, β ) = ϕ el1 ( ξ, β ) + ϕ el1 ( − ξ, β ) is obtained, as in Eq. (25), we can follow the same procedure andaccount for the electrostatic interaction introducing an effective slit width, given by(22). However, the orders of magnitude involved might be very different. If we comparethe potential strengths at the center of a slit of width a = a m = a M (for the sake ofsimplicity we set β = 0), then we obtain the following expression for the ratio: V el V vdW = q π(cid:15) − (cid:15) (cid:15) a C , which shows a quadratic dependence on the slit width, descending from the differentpower-law scaling of the potentials. Assuming that q = e , the electron charge, we alsohave e (4 π(cid:15) ) − = 1439 .
964 meV · nm. Consistently, we recall from the previous section,that C is of the order of a few meV · nm , in turn we have V el /V vdW ∼ a [nm].Though the difference seems very large, this is only a pointwise estimate, while theeffective slit width is determined by the behavior of the phase shift over the whole rangeof ξ . Nevertheless, we can see that there are realistic situations where the calculatedimpact of electrostatic interactions is indeed very high, as discussed in table 1. We seethat in view of the higher typical potential strength in comparison with the van derWaals interaction, for low energy antiprotons Eq. (22) predicts sizeable reduction ineffective slit width, i.e., (cid:38) a eff e + [ nm] a eff p [ nm]0.1 401.3 148.11 477.2 285.810 497.1 397.4100 499.7 460.0 Table 1.
Calculated effective width for realistic parameters (order of magnitude)applicable to possible experiments, a = 0 . µ m, β = 5 ◦ , (cid:15) = 4, for e + and p of varyingenergy. The grating thickness is set to δ = 500 nm and δ = 160 nm respectively,reflecting the typical scale necessary to absorb the particles completely outside theslits for silicon at 1 keV reference energy. Exact numerical calculation of the envelope function [still using the eikonalapproximation for the phase shift (16)] shows that for low energy particles there is indeed atter-wave interferometry: towards antimatter interferometers
12a stark departure from the sinc shape expected for weak interactions. Experimental datawith electrons in the 0 . ÷ | a eff − a | /a (cid:46)
10% for that energy range.
5. Talbot-Lau interferometry
In section 3 we have discussed how the contrast of interference patterns is affected by theratio between the coherence length l and the grating period D . This imposes technicalconstraints on the design of an interferometer using the geometry shown in Fig. 3. Firstof all the finite resolution of the detector has to be taken into account. This parametergreatly depends on the kind of detector and on the particles involved. For example, inthe case of anti-hydrogen, e + and p emulsion detectors could be employed, which arecapable of a spatial accuracy up to 0 . ÷ µ m [30, 31]. Being L the grating to detectordistance and D the grating period, the Fraunhofer diffraction orders are separated by∆ x = LλD . If δx is the experimental sensitivity, in order to resolve each maximum of the diffractionpattern within at least an interval M δx , with M integer, we should have LλD ≥ M δx ⇒ L ≥ M D δxλ , which imposes a constraint on L . It is clear that for a fixed wavelength and geometryboth increasing M and reducing the distance L , which is of utmost importance withdecaying particles, requires a decrease in the grating period D .Moreover, starting from Eq. (13) and requiring that the coherence length l is atleast ˜ M times the grating spacing D , we obtain the following condition on the source-grating distance: y s ≥ ˜ M D σ s λ . Therefore, to obtain a good coherence either the distance y s has to be increased or thesource dimension σ s reduced as much as possible. Note that, apparently, by reducing theperiod D we can satisfy both conditions on L and y s , however a reasonable small valuefor D is fixed by the grating construction constraints. This poses technical challengesdue to the particle decay in the first case (for ortho-Ps atoms with a lifetime τ = 142 ns[32] and realistic thermal speeds v ≈ m / s [19], y s should be in the range of a fewcentimeters), and to difficulties in manipulating the beam size in the latter. In viewof these consideration, we suggest that a different kind of interferometer would be bestsuited for experiments with antimatter, namely a Talbot-Lau setup [33, 34], which issketched in Fig. 6. atter-wave interferometry: towards antimatter interferometers L d e t ec t o r x y L G G b e a m Figure 6.
Sketch of a Talbot-Lau interferometer. The first grating ( G ) is illuminatedby an incoherent beam of mean wavelength λ , and acts as an intensity mask providingthe necessary coherence for illuminating the second grating ( G ). Their separation L is set to the observation distance from G and the gratings have the same period D and slit width a . If L matches the Talbot Length T L = D /λ , this setup produces onthe detector plane high contrast fringes with period D . It is worth noting that the usual Talbot-Lau configuration involves a third gratingas a scanning mask [33, 35], that is not necessary in our case since we assume thatthe high resolution of the detector will allow to directly resolve the diffraction pattern.The fundamental property of this geometry is that it produces high contrast fringesregardless of the coherence and spatial extension of the illuminating beam. There aretwo physical phenomena governing this apparatus: the Talbot self-imaging effect [34],stating that in the Fresnel region of a coherently illuminated periodic grating self-imagesof the grating transmission function will appear at L = nT L = nD /λ (as well as rescaledsub-images with a fractional period for half-integer multiples ), and the so called Laueffect [36]. This effect can be understood as arising from an incoherent superposition ofpatterns produced by laterally displaced, mutually independent, point sources, the roleof which in the apparatus of Fig. 6 is played by the slits of the first grating [34]. Theperiodic images thus produced can overlap “constructively” if the first grating has asuitable periodicity; under these conditions the elementary displacement on the screenplane produced by moving between adjacent sources equals the Talbot image periodor an arbitrary integer multiple of the latter. In particular, this “resonance” conditionis met in the configuration of Fig. 6 when L = T L . Geometrically, this setup bears astrong similarity to a classical moir´e deflectometer [37]. What discriminates betweenthe purely classical and the quantum interference regime is the condition for diffractionto be negligible, namely [37]: LλD (cid:28) D. (28)A moir`e deflectometer and a Talbot-Lau interferometer as defined in Fig. 6 havein common that they produce a fringe pattern with period D . The question that nownaturally arises is: how can the experimental results prove that the observed fringes are atrue interference effect and not simple classical geometrical shadow patterns produced byballistic particles? As mentioned, high contrast fringes are expected only if the gratingseparation is an integer multiple of the Talbot length, while for “classical projectiles” atter-wave interferometry: towards antimatter interferometers L L/T0.1 0.2 0.3 0.4 0.5 0.6 1 2 3 C on t r a s t [ a . u .] Figure 7.
Monte Carlo simulation of the fringe visibility modulation (contrast) as afunction of
L/T L (Log scale), for a well defined particle velocity. For definiteness thefollowing parameters (realistic for an experiment with 1 keV antiprotons) were chosen D = D = D = 265 nm, N = 40 and a = 90 nm. The calculated Talbot length for thisperiod and energy is T L = 77 . L/T L (cid:28)
1: this corresponds to a classical behavior of the particles. the contrast does not depend on this condition. This property ultimately descends fromthe longitudinal periodicity of the so-called
Talbot carpet , which is a distinctive featureof diffraction in the Fresnel region. Therefore, the observation of this kind of pattern isa proof of the wave character of the interfering particles. From the experimental point ofview, this can be done by continuously adjusting the grating separation or changing theparticle energy (hence the Talbot Length) in a monochromatic beam, and measuring themodulation in contrast as a function of the parameter
L/T L . If the apparatus is trulyoperating as an interferometer and not as a classical device, distinct peaks in contrastshould be detected, as shown in Fig. 7 [38]. Recalling (28), we see that the classicallimit corresponds to L (cid:28) T L , as it is confirmed by numerical calculations showing aweak dependence of the contrast on L in this region. This is clear from Fig. 7: asthe grating distance L falls below T L = 77 . F on the charged particles:this force will be negligible if the corresponding deviation from a straight trajectory issmaller than the typical size of the finest structure in the observable pattern. Let uscall this quantity ∆, and introduce the flight time of the particle τ = L/v . Using theidentity ∆ = F crit τ /m , we obtain the following relation: F crit = h DmL λ , atter-wave interferometry: towards antimatter interferometers D . If we set L = T L = D /λ , we have: F crit = h mD , (29)that is independent of the particle energy. Furthermore, from the Lorentz force F = q ( E + vB ) one can deduce the critical values of the involved electric and magneticfields by the simple relations E crit = F crit /q and B crit = F crit /qv . It is worth noting thatin the case of Fraunhofer interferometry (far field) a similar calculation with ∆ = Lλ/D leads to a critical force F ( F )crit = h / ( mLDλ ) ∝ v which is thus affected by the particleenergy.Overall, the Talbot-Lau geometry has several advantages which are especiallyrelevant for anti-matter interferometry. In particular, it allows to minimize the totallength of the apparatus, a crucial requirement for decaying Ps, and to employ a largersource with weak coherence requirements, significantly increasing the particle flux. We now present a realistic estimate of the expected contrast signal in an e + experiment,following the experimental methodology outlined in this section (see Figs. 6 and7). We assume that the particle mean energy E can be tuned between 5 keV and20 keV (reasonable for current continuous e + beams) with a narrow Gaussian energydistribution ( σ/E (cid:46) L/T L , where theTalbot length reads: T L = D √ mE h . (30)A contrast peak is expected around L ≈ T L . The above formula makes it clear that toscan this region there are specific complications related to each choice of which parameterto vary: the energy is bounded by technical constraints and furthermore provides a sub-linear scaling, whereas the grating distance can be varied arbitrarily. There are howevertechnical complications in physically moving an apparatus sensitive to alignment overconsiderable lengths.We recall that in this configuration D sets the periodicity of the interference pattern,which should be larger than the detector resolution. However as evident from (30), theTalbot length scales quadratically with D . Thus it rapidly becomes very large, and toolong an apparatus poses additional challenges related both to grating alignment andshielding of a larger region from stray fields. We set D = 2 µ m corresponding to a TalbotLength T L = 0 .
326 m at the median energy of the considered energy range, namely E = 10 keV. Using Eq. (29) we obtain E crit ≈ . / m and B crit ≈ . E = 5 keV,which corresponds to the worst scenario.The maximum magnetic field is particularly critical, as the requirement is smallerthan the natural magnetic field of the Earth, however considering that experiments withelectrons and similar length scales involved have been successfully carried out [39], we atter-wave interferometry: towards antimatter interferometers Figure 8.
Monte Carlo simulations of the expected contrast modulation as a functionof E , for the monochromatic case and with a Gaussian distribution of increasing σ centered on E . The total number of slits is set to N = 40, sufficiently large to providerealistic results. See text for more details on the parameters. believe that an appropriate mu-metal based shielding will be enough to circumvent theproblem. Moreover an uniform constant magnetic field will only rigidly shift the patternin space [40], in fact the above limits indicate the maximum allowed fluctuations (eitherin time or in space) of the E and B .Another problem our theoretical analysis allows to account for is the electrostaticinteraction with the grating walls. First of all we have to set the slit width, and thus theopen fraction of the grating; this is best set at a/D ≈ a = 0 . µ m. Highervalues could improve the total particle flux minimizing the losses inside the materialgrating, but will also reduce the contrast due the overlapping diffraction peaks. Weremark that for the gratings to work as true intensity masks, their thickness must besufficient to stop all the positrons outside the slits. For such low energies few microns ofSiN x will be sufficient. Therefore, assuming these parameters and a small wedge angle β = 10 ◦ , a grating thickness δ = 800 nm and E = 5 keV, equation (22) predicts aneffective slit width a eff = 0 . µ m. This deviation is very small ( ≈ . σ = 0 .
25 keV or σ = 0 . atter-wave interferometry: towards antimatter interferometers
6. Conclusion and outlook
In this paper we reviewed the basic elements of diffraction theory applied to matter-wave interferometry. In particular, we focus our analysis on the possible issues arisingfrom the use of charged or neutral antimatter particles. To sustain our investigationwe also performed Monte Carlo simulated experiments based on realistic parameters.In particular, we have considered the effect due to a realistic source (extended andnon-monochromatic) and the interaction with the grating as well as the influence ofstray electromagnetic fields. We also found that van der Waals interactions with thematerial grating become critical for highly polarizable particle systems. In this scenarioa possible solution could be resorting to light gratings [41, 42]. We have shown that thebetter configuration to carry out matter-wave interferometry with decaying particles isgiven by the Talbot-Lau setup also in the presence of a Gaussian distribution of theparticle energy, which realistically describes the actual e + and p beams. Furthermore,exploiting the high resolution capabilities of the antiparticle detectors, such as thenuclear emulsions, we have shown that the typical Talbot-Lau setup involving threegratings can be reduced to a two-grating configuration which indeed simplifies theexperimental implementation. Our analysis paves the way to further investigations inorder to design an experiment to demonstrate antimatter-wave interference also in viewof possible applications in the emerging field of gravity experiments using antimatter[41, 43, 44]. Acknwledgments
The authors would like to thank R. Ferragut and C. Pistillo for useful discussions andsuggestions. SO acknowledges financial support by MIUR (project FIRB “LiCHIS”- RBFR10YQ3H) and by EU through the Collaborative Project QuProCS (GrantAgreement 641277).
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