Zone-plate focusing of Bose-Einstein condensates for atom optics and erasablehigh-speed lithography of quantum electronic components

T.E. Judd1,2, R.G. Scott1, G. Sinuco1, T.W.A. Montgomery1, A.M. Martin3, P. Kruger1, and T.M. Fromhold1

1Midlands Ultracold Atom Research Centre, School of Physics & Astronomy,University of Nottingham, Nottingham NG7 2RD, United Kingdom

2Physikalisches Institut, Eberhard-Karls-Universitat Tubingen,CQ Center for Collective Quantum Phenomena and their Applications,

Auf der Morgenstelle 14, D-72076 Tubingen, Germany3School of Physics, University of Melbourne, Parkville, Vic. 3010, Australia

(Dated: October 23, 2018)

We show that Fresnel zone plates, fabricated in a solid surface, can sharply focus atomic Bose-Einstein condensates that quantum reflect from the surface or pass through the etched holes. Thefocusing process compresses the condensate by orders of magnitude despite inter-atomic repulsion.Crucially, the focusing dynamics are insensitive to quantum fluctuations of the atom cloud andlargely preserve the condensates’ coherence, suggesting applications in passive atom-optical ele-ments, for example zone plate lenses that focus atomic matter waves and light at the same point tostrengthen their interaction. We explore transmission zone-plate focusing of alkali atoms as a routeto erasable and scalable lithography of quantum electronic components in two-dimensional electrongases embedded in semiconductor nanostructures. To do this, we calculate the density profile of atwo-dimensional electron gas immediately below a patch of alkali atoms deposited on the surfaceof the nanostructure by zone-plate focusing. Our results reveal that surface-induced polarizationof only a few thousand adsorbed atoms can locally deplete the electron gas. We show that, as aresult, the focused deposition of alkali atoms by existing zone plates can create quantum electroniccomponents on the 50 nm scale, comparable to that attainable by ion beam implantation but withminimal damage to either the nanostructure or electron gas.

PACS numbers: 34.50.Dy, 03.75.Kk, 42.79.Ci

I. INTRODUCTION

Cooling alkali atoms to µK temperatures and belowhas opened the field of atom optics, leading to manybreakthroughs in both fundamental physics and emerg-ing applications [1]. It provides new ways to control theatoms, by tailoring their potential landscape on a µmscale [1, 2, 3, 4, 5, 6], probe their environment, for exam-ple in high-precision matter-wave sensors [7, 8] or atomicmicroscopes [9, 10, 11], and use them for matter-wavelithography of nanostructures [12, 13, 14, 15, 16]. Fo-cusing the atomic matter waves is crucial to the develop-ment of such instruments, and for emerging applicationsinvolving the transfer of cold atoms into hollow-core op-tical fibres [17, 18], but remains a challenging task. Usu-ally, spot focusing is achieved by using electromagneticlenses [19], which requires sophisticated equipment, or byreflecting the atoms from curved optical [20, 21] or mag-netic mirrors [22, 23, 24], which are hard to make andkeep atomically clean [10, 25]. To avoid these complica-tions, matter waves can be focused by diffracting themfrom commercially-available Fresnel zone plates (ZPs)[9, 10, 11], comprising a series of concentric circular aper-tures, whose focal length is proportional to the speedof the incident atoms [26]. So far, though, ZP focusingof atomic matter waves has only been demonstrated fornon-interacting He or Ne atoms with approach speeds &400 m s−1, corresponding to long focal lengths & 30 cm[9, 10, 11].

Here, we show that such ZPs can sharply focus Bose-

Einstein condensates (BECs) that are partially trans-mitted through the plate or, if they approach it slowlyenough (∼ 1 − 10 mm s−1), quantum reflect from theattractive Casimir-Polder (CP) atom-surface potential[4, 5, 27, 28, 29]. We use numerical solutions of theGross-Pitaevskii equation, supplemented by a truncatedWigner approach [28, 30], to investigate how the focus-ing dynamics depend on the ZP geometry and on theparameters and incident velocity of the BEC. Our cal-culations reveal that the focal length is similar to thatexpected from a single-particle ray picture but, for suffi-ciently high atom densities, also exhibits resonances thatoriginate from inter-atomic interactions. At the low ap-proach speeds required for quantum reflection to occur,the BECs focus within 100 µm of the surface, where trap-ping usually occurs in atom chips [31, 32, 33, 34]. Since,in this regime, the deBroglie wavelength of the incidentBEC is comparable with that of light, a single ZP canfocus both atoms and light at the same point, therebypromoting the strong light-matter interaction requiredfor few-photon nonlinear optics [18, 35, 36]. Focusingis sharpest for dilute pancake-shaped BECs, which havea narrow distribution of approach speeds and hence ex-hibit little chromatic aberration [26], are less susceptibleto disruption by dynamical excitations created during in-teraction with a surface [27, 28], span many rings of theZP so that its resolution is intrinsically high [26], andexperience less defocusing by inter-atomic repulsion. De-spite this repulsion, ZP focusing can increase the BEC’sdensity by orders of magnitude, raising the possibility

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that quantum fluctuations will significantly reduce thecondensate fraction [28]. Surprisingly, though, we findthat such fluctuations cause little depletion of the BEC.

Finally, we introduce ZP focusing of matter waves ontoa semiconductor nanostructure as a route to erasable andscalable nm-precision lithography of quantum electroniccomponents fabricated in a two-dimensional electron gas(2DEG) on, or just below, the surface. Erasable lithog-raphy is of great interest for studying quantum trans-port and control, but has so far been achieved only in asmall number of laboratories using scanning probe tech-niques [37, 38]. Alkali atoms deposited on a semiconduc-tor surface polarize because their valence electron par-tially transfers to the surface [39, 40]. We show that thisrepels 2DEG electrons strongly enough to produce localinsulating regions with dimensions determined by the fo-cal width of the BEC, which can be made . 50 nm usingexisting ZPs. Compared with existing fabrication tech-niques, ZP lithography using matter waves offers severalkey advantages, which we identify and discuss.

The paper is organised as follows. In Section II, wedefine the system parameters, introduce our model forcalculating the BEC dynamics, and use this model tostudy quantum reflection focusing from a ZP etched ina Si surface. In Section III, we investigate focusing in-duced by transmission through, and quantum reflectionfrom, a free-standing ZP structure and determine howthe focusing process affects the quantum coherence of theatom cloud. In Section IV, we explore transmission ZPfocusing as a route to fabricating erasable quantum com-ponents in high-mobility electron gases. We summarizeour results and draw conclusions in Section V.

II. QUANTUM REFLECTION FROM A ZONEPLATE

In this section, we consider quantum reflection of a23Na BEC from a ZP etched in a Si surface. This systemproduces a single focus, which is sharp due to the narrowvelocity distribution of atoms within the BEC, and istherefore simple enough to elucidate the key features ofthe focusing dynamics, in particular the effect of inter-atomic interactions.

When an alkali atom approaches within ∼ 3 µm ofa planar surface, its potential energy decreases rapidlydue to mutual polarization of the atom and the surface[4, 5, 27, 41]. At distance x′ from a perfectly conductingsurface, the atom-surface interaction can be describedby the CP potential energy VCP (x′) = −C4/x

′3(x′ +3λa/2π2), where, for a Na atom, C4 = 9.1 × 10−56

Jm4, and λa = 590 nm is the effective atomic transi-tion wavelength [4]. This attractive potential creates noclassical turning point for an incident atom. But if theatom’s approach velocity, vx, is sufficiently low, ∼ 1− 10mm s−1, the corresponding deBroglie wavelength, λdB ,is long enough to span the rapidly decreasing CP poten-tial, causing quantum reflection to occur for both non-

FIG. 1: (a) Plan view of the ZP showing raised (black) andetched (white) rings. (b) Black shape: schematic cross-sectionin the x−r plane (axes inset) through the ZP, etched to depthd (horizontal arrow). Blue shape: initial position of the BECwhen the harmonic trap is centred at x = 0. Left (right) dot-ted parabola: harmonic trap potential before (after) displace-ment. (c) Gray surface: real part of potential energy VT (x, r),defined in the text, shown for x < ∆x+ d (i.e. above the topsurface of the ZP or within an etched ring) after displacementof the harmonic trap. The displaced trap accelerates the BEC(blue) towards the ZP (right). In (c), few ZP rings are shownfor clarity.

interacting atoms [41, 42, 43, 44] and BECs [4, 5]. ForBECs, quantum reflection probabilities up to 0.7 havebeen achieved by etching an array of ∼ 100 nm-diameterpillars into the surface to enhance the action of the CPpotential [5]. In these experiments, the spacing of the pil-lars (0.5 µm) was chosen to be � λdB to avoid diffract-ing the matter waves. By contrast, here we study matterwaves scattering from a larger, 100 µm scale, ZP struc-tures specifically designed to diffract, and hence focus,an incident BEC.

We first consider a ZP comprising 12 concentric ringsetched, by standard plasma etching techniques, for ex-ample, to a depth d = 10 µm into a flat Si surface, whichlies in the x = ∆x plane [Fig. 1(a,b)]. Note that ∆x is avariable that we change in order to control the initial po-sition of the BEC relative to the surface (details below).The ring edges are at r =

√y2 + z2 = Rn = R1n

1/2

(n = 1, 2, 3, 4, ..., 24), where R1 = 20 µm is the radius ofthe inner raised disk [black in Fig. 1(a)]. In Fig. 1(b),the black shape shows a schematic cross-section throughthe ZP, with the etched rings appearing as white inden-tations.

For a single Na atom of mass m, represented by a planewave of wavelength, λdB , travelling at velocity, vx, alongthe x-axis, simple ray analysis [26] predicts that construc-tive interference between wavefronts that quantum reflectfrom adjacent raised rings will focus the wave at a dis-tance

f = R12/λdB = R1

2mvx/h = 0.023vx (1)

from the ZP, provided f � the outer radius, R24 =98 µm, of the largest ZP ring. The wave can still fo-cus if this condition is not satisfied, but the focal lengthwill differ from that predicted by Eq. (1).

3

To investigate how a BEC interacts with the ZP, wefirst consider the geometry of the system. Since the BECapproaches the ZP along a common (x) axis of circu-lar symmetry, we describe the system using cylindrical,(x, r), co-ordinates [Fig. 1(b,c)]. For each atom, the ZPcreates an attractive CP potential energy, Vzp(x, r). Ifr < R1 or R2n ≤ r ≤ R2n+1, so that the atom is directlyabove one of the raised rings, we take this potential en-ergy to be

Vzp(x, r) ={VCP (x′), if x′ > δVCP (δ)− i(x−∆x+ δ)Vim if x′ 6 δ,

where x′ = ∆x − x is the atom-surface separation andVim = 1.6× 10−26 Jm−1 [27]. The complex form of Vzp,within distance δ = 0.15 µm of the surface, avoids thedivergence of VCP (x′) as x′ → 0 and models adsorptionof those atoms that reach the surface [27]. If R2n−1 <r < R2n, so that the atom approaches an etched ring, wetake Vzp(x, r) = 0 if x < ∆x + d [i.e. above the surfaceor within the etched ring: see Fig. 1(b)] and, to simulatethe adsorption of atoms that enter the etched ring [45],Vzp(x, r) = −i[x−(∆x+d)]Vim if x > ∆x+d (i.e. beyondthe bottom of the ring).

To study quantum reflection from the ZP, we adaptthe system used in recent experimental observations ofquantum reflection for a BEC approaching a planar Sisurface [4, 5]. We first consider a dilute condensate,henceforth called BEC A, containing N = 3 × 105 23Naatoms in a harmonic trap with cylindrical symmetryabout the x-axis and frequencies ωx = 2π×3.3 rad s−1

and ωr = 2π×1.0 rad s−1 in the longitudinal (x) and ra-dial (r) directions respectively. This creates a pancake-shaped BEC [Fig. 1(b)] with longitudinal width lx ≈ 40µm, radial diameter lr ≈ 180 µm, and peak densityn0 = 6.3 × 1017 m−3. We choose the pancake shape tolimit disruption of the BEC during quantum reflection[27, 28].

Initially, the BEC is in its equilibrium ground state,centred at x = r = 0. At time t = 0, we suddenlydisplace the harmonic trap by a distance ∆x along the xaxis, so that its centre coincides with the top surface ofthe ZP at x = ∆x [Fig. 1(b)]. This causes the BEC toapproach the ZP with velocity vx ≈ ωx∆x = vx at timeT ≈ π/2ωx. We consider ∆x values for which vx > 2 mms−1 to avoid creating dynamical excitations during thereflection process [4, 5, 27]. After the trap displacement,the total potential energy of each Na atom in the BECis VT (x, r) = Vzp(x, r) + 1

2m[ω2

x(x−∆x)2 + ω2rr

2]. As

shown in Fig. 1(c), the real part of VT (x, r) decreasesrapidly near the top surface of the ZP, but is constantwithin the etched rings [45]. At time T , we switch off theharmonic trap to prevent it influencing the subsequentfocusing process. We determine the dynamics of the BECby using the Crank-Nicolson method [27] to solve theGross-Pitaevskii equation

i~∂Ψ∂t

= − ~2

2m∇2Ψ + VT Ψ +

4π~2a

m|Ψ|2 Ψ, (2)

FIG. 2: Atom density (dark high) in the x − r plane (axesinset) at t = 0 (a), 72 ms (b), 90 ms (c), and 99 ms (d) for∆x = 240 µm (vx = 5 mm s−1). Black shapes are schematiccross-sections through the ZP. Vertical bar in (b) indicatesscale. In (d), horizontal bar shows the focal length [46] andarrows mark the narrowest ring spanned by BEC A.

where a = 2.9 nm is the s-wave scattering length forNa, ∇2 is the Laplacian in cylindrical coordinates, andΨ(x, r, t) is the wave function at time t > 0, normalizedso that |Ψ|2 is the number of atoms per unit volume.

Figure 2 shows atom density profiles at key stages dur-ing the quantum reflection and focusing of BEC A fol-lowing a trap displacement ∆x = 240 µm (vx = 5 mms−1). Immediately after the trap displacement, the BECremains centred at x = 0 [Fig. 2(a)], but accelerates to-wards the ZP. At t = 72 ms [Fig. 2(b)], the BEC’s leadingedge has reached, and undergone partial quantum reflec-tion from, the rapidly-decreasing CP potential near theraised rings. Interference between the incident and re-flected matter waves weakly modulates the atom densityprofile [vertical red and black stripes in Fig. 2(b)]. In ad-dition, some atoms have entered the etched rings. By t =90 ms [Fig. 2(c)], the reflected atoms have moved awayfrom the ZP and formed an “arrow head” density pattern.The upper and lower edges of the arrow head approachone another, moving towards r = 0 where they meet, andtransiently focus, at t = 99 ms [Fig. 2(d)] before diverg-ing again [46]. It might be possible to achieve similarcompression of a BEC by using a scanning focused laserbeam, rather than a micro-fabricated diffraction gratinglike that in Fig. 1, to imprint the ZP pattern opticallyon the atom cloud [47].

Comparison of Figs. 2(a) and (d) reveals that thewidth of the focused BEC along the x axis is similar tothat of the initial state because atoms at the front of theBEC reflect and focus before those at the back. Conse-quently, the size of the focused BEC can be reduced by

4

decreasing lx. As expected from both ray and wave anal-yses [26, 48], the radial width, lfr , of the focused cloud ap-proximately equals the width of the narrowest ring thatthe atoms enter. Atoms can only enter the jth etchedring if their incident momentum exceeds that of the low-est quantized radial mode in the ring, which requiresvx ≥ vj = h/2mwj , where wj = (R2j−R2j−1) is the ringwidth [49, 50]. Resonant injection of atoms into the nar-rowest (8th) ring spanned by BEC A, marked by arrowsin Fig. 2(d), only occurs if vx ≥ v8 = 3.4 mm s−1. In thisregime, lfr ≈ w8 = 2.5 µm [Fig. 2(d)] and so the volumeof the focused cloud is a factor ≈ (lfr /lr)2 = 1.9 × 104

smaller than the initial BEC. As vx decreases, the widthof the narrowest ring that the BEC can penetrate grad-ually increases, causing lfr to increase approximately as1/vx.

We now investigate how f , and the underlying focus-ing dynamics, vary with vx. The solid curve in Fig. 3shows f(vx) calculated from Eq. (1) for a single Na atommodeled by an incident plane wave [46]. If the atom is,instead, described by a wavepacket, using Eq. (2) witha = 0, we obtain the dotted f(vx) curve in Fig. 3 [51].This curve lies slightly below the solid line given by Eq.(1) primarily because the assumption that f � R24 = 98µm made in deriving Eq. (1) is not strictly valid [26].

In Fig. 3, the f(vx) curve calculated for BEC A(dashed curve) reveals that inter-atomic interactions fur-ther reduce f . As the BEC starts to quantum reflect,atoms accumulate near the ZP surface and their repul-sive mean field decelerates those atoms that are still ap-proaching the ZP, thus reducing the BEC’s mean incidentvelocity and, consequently, also reducing f . Resonant in-jection into the ZP’s rings reduces the build up of atomsand decelerating mean field potential near the entranceto the ring, causing the approach speed and f to in-crease abruptly with increasing vx, as indicated by thetwo vertical arrows in Fig. 3. The exact vx values atwhich these resonances occur depend on the mean fieldinter-atomic repulsion within the rings [50], which variesrapidly in space and time during the reflection process.Consequently, a simple non-interacting model can onlyestimate the positions of the resonances. However, thetwo large abrupt changes in f marked by the vertical ar-rows in Fig. 3 occur close to the vx required for resonantinjection into the single-particle modes of two rings si-multaneously. Specifically, the resonant feature indicatedby the left-hand (right-hand) arrow appears to originatefrom co-excitation of the lowest and first excited radialmodes of the 3rd and 1st (7th and 2nd) rings respectively.

The dot-dashed f(vx) curve in Fig. 3 is calculated fora denser condensate, BEC B, comprising N = 3 × 106

23Na atoms in a harmonic trap with ωx(ωr) = 2π×9.9(3.0) rad s−1 and n0 = 6.7×1018 m−3. For BEC B, inter-atomic repulsion at the entrance to the etched rings is toostrong to be overcome by the resonant injection mecha-nism described above because the mean-field energy atthe entrance to, and inside, the rings far exceeds the en-ergies of the lowest single-particle radial modes. Mean

FIG. 3: f(vx) curves calculated from Eq. (1) (solid line)and from numerical solutions of Eq. (2) for a single-atomwavepacket (dotted curve), BEC A (dashed curve) and BECB (dot-dashed curve) after quantum reflection focusing. Sym-bols show data points. Vertical arrows mark vx values wheref changes abruptly, as explained in the text. Inset: curvesshowing nf/n0 (solid) and R (dotted) calculated versus vx

for BEC A.

field repulsion therefore slows atoms that approach theZP from the trailing edge of the BEC, making their in-cident velocity significantly less than vx. Consequently,the f(vx) curve for BEC B lies below those for both thesingle atom and BEC A and reveals no resonances forthe vx values shown. As vx increases, though, the in-cident kinetic energy begins to dominate the mean fieldenergy, causing the f(vx) curves for both BECs to ap-proach those of a single atom.

We now consider how the peak density of the fo-cused cloud, nf , varies with vx for quantum reflectionof BEC A. In the inset to Fig. 3, the solid curve showsnf/n0 values determined from full numerical simulationsof BEC A. The form of this curve can be understoodby noting that nf/n0 is approximately proportional toNR(vx)/n0lx(lfr )2, where R(vx) is the fraction of atomsthat quantum reflect from the ZP (dotted curve in Fig.3 inset). In Fig. 3, nf/n0 attains a peak value of ∼ 4when vx = 3 mm s−1 ≈ v8 [52]. For higher vx, nf/n0 de-creases with increasing vx because R decreases rapidly,as expected from previous quantum reflection studies[4, 5, 27, 41]. But as vx decreases below 3 mm s−1,the atoms can no longer penetrate the narrow outer ZPrings, thus increasing lfr and reducing nf/n0. Higherdensity focused clouds could be achieved either by fabri-cating fine (nm-scale) pillars within the raised ZP rings,to increase R without affecting the diffraction process [5],or by using transmission ZPs, as we consider in the nextsection.

5

III. TRANSMISSION FOCUSING ANDDEPLETION OF THE BEC

In this section, we consider a transmission ZP, whichhas the same ring pattern as the etched plate consideredin the previous section, but is only 130 nm-thick along thex direction, as in the experiments of Ref. [11]. Since thering-shaped holes extend right through the plate, trans-mission ZPs are held together by a small number of radialstruts, which do not affect the focusing process becausetheir width is � λdB . In our calculations, we have noimaginary absorption potential in the gaps [white in Fig.4(a)] so that all (∼ N/2) atoms entering the gaps emergeon the other side of the plate. Figure 4(a) shows the re-flected (left) and transmitted (right) foci calculated forBEC A, which form at t = 104 ms [53]. The transmissionfocus contains ∼ N/2 atoms, almost 50 times the number(∼ R(vx)N/2) in the reflected cloud in Fig. 2(d). Conse-quently, transmission ZP focusing can increase the den-sity of the atom cloud passing through the plate by twoorders of magnitude to ∼ 1020 m−3. This compressionincreases the atom loss rate due to three-body scatteringby six orders of magnitude [54]. However, for BEC A, weestimate that the resulting fraction of atoms lost duringthe focusing process will be < 0.1. The radial width ofthe transmission focus in Fig. 4(a) is ∼ 5µm, suggestingthat ZP focusing could assist the injection of BECs intothe 10 µm-diameter hollow-core of a photonic crystal fi-bre [17, 18], thereby increasing the fraction of atoms thatcan be transferred from a free-space trap into the fibre.

Previous studies have shown that interactions betweenthe incident and reflected components of a BEC thatquantum reflects from a solid surface can partially de-cohere the atom cloud [28], particularly when its densityis high. We have investigated whether the density in-crease produced by ZP focusing affects the coherence ofBECs A and B by calculating the number of incoherent(i.e. non-condensed) atoms, NI , as a function of time,t, using the truncated Wigner method described in Refs.[30, 55]. For BEC A, the focusing process causes negli-gible incoherent scattering and depletion of the conden-sate, with the fraction of incoherent atoms, NI/N [solidcurve in Fig. 4(b)], remaining below 0.01 throughoutour simulation. By contrast, for BEC B the incoherentfraction [dotted curve in Fig. 4(b)] rises sharply as thecloud strikes the ZP when t = T = 76 ms. This is be-cause BEC B is ∼ 10 times denser than BEC A, henceincreasing the probability of incoherent scattering events[28]. The rate of incoherent scattering, and consequentincrease of NI/N , is highest during the reflection pro-cess, i.e. when 1 . t/T . 1.2. Thereafter, the ratedecreases but remains finite due to inter-atomic scatter-ing events that occur during the focusing process. In thisregime, the contribution to NI/N made by atoms in thereflected part of BEC B only [dashed curve in Fig. 4(b)]deviates from the decoherent fraction of the whole cloud[dotted curve in Fig. 4(b)]. However, since the deviationis very small, we conclude that quantum fluctuations de-

x

r

FIG. 4: (a) Atom density profile (dark high) showing trans-mission (right) and reflection (left) foci in the x−r plane (axesinset) for BEC A at t = 104 ms. Black shapes: schematiccross-sections through solid regions of the ZP. Vertical barindicates scale. (b) Fraction of incoherent atoms NI/N cal-culated versus t for BEC A (solid curve), BEC B (dottedcurve), and the reflected component only of BEC B (dashedcurve). Calculations are for vx = 3 mm s−1.

plete the reflected part of the condensate far more thanthe transmitted part. Physically, this is because collisionsbetween the incident and reflected matter waves give riseto incoherent scattering processes that do not affect thetransmitted wave [53]. Since transmission focusing doesnot significantly deplete the condensate, it may provide auseful tool for manipulating BECs, for example to trans-fer them into microtraps or hollow core optical fibres orto co-focus coherent light and matter waves to ensurestrong interaction between them [18].

As an alternative to using the magnetic trap displace-ment technique described in Section II to direct the BECtowards the ZP [4, 5] a moving optical lattice, formed bytwo counter-propagating laser beams with slight relativefrequency detuning, would reduce the velocity spread ofthe incident atoms and the associated chromatic aber-ration. This technique might also allow ZP focusingof 2D atom clouds [56], which can be trapped withineach lattice minimum and passed sequentially throughthe ZP. This would combine high flux, sufficiently lowdensities to reduce potentially harmful inter-atomic in-teraction effects, and good focusing properties due to thesmall transverse velocity spread given by the (single par-ticle) ground state momentum distribution within theindividual wells.

IV. ZONE-PLATE LITHOGRAPHY OFTWO-DIMENSIONAL ELECTRON GASES

A. Effect of adsorbed atoms on the electron gas

We now explore the possibility of using ZPs control-lably to deposit alkali atoms onto a semiconductor sur-face oriented parallel to the plate and in its focal plane, soenabling matter-wave lithography of quantum electroniccomponents such as quantum wires and dots within a

6

2DEG.When alkali atoms are deposited on materials with

a higher electronegativity, they polarize by the partialtransfer of their valence electron to the surface [39, 40].Stronger polarization is expected for heavier, less elec-tronegative, alkali atoms. For example, polarization of asingle Rb atom on a Si surface creates an electric dipole ofmagnitude ∼ 10−29 Cm pointing away from the surface[39]. In this section, we consider Rb atoms because theyare highly polarizable and hence have low electronega-tivity [57]. The interaction of alkali atoms with GaAssurfaces has also been extensively studied and continuesto attract considerable interest, partly because it pro-vides a way to lower the work function of GaAs, whichis important for technological applications in Schottkybarriers [58, 59, 60, 61, 62, 63, 64, 65]. Polarization ofthe adsorbed atoms creates an electric field, which, as wenow explain, can strongly affect the density and electri-cal resistance of a two-dimensional electron gas (2DEG)just below the surface.

In a 2DEG, electrons from remote ionized donors forma sheet of negative mobile charge, ∼ 15 nm thick, parallelto the surface plane [66]. Typically, the 2DEG is locateda few 10s of nm below the surface, although it can beon the surface itself [67]. A voltage, applied to Ohmiccontacts, creates an electric field along the 2DEG, thusdriving current. Since the electrons are spatially sep-arated from the parent ionized donors, their mobility isusually very high, particularly at low temperatures. Con-sequently, 2DEGs are used extensively in condensed mat-ter research and also have applications in high-frequencyelectronics: mobile telephones, for example. A 2DEG canbe located within ∼ 50 nm of the semiconductor surface[67, 68], which is close enough for the potential energydue to repulsion betwen electrons in the 2DEG and thedipoles created by the adsorbed alkali atoms to be 10sof meV. This is sufficient locally to deplete the 2DEG,whose Fermi energy is typically ∼ 10 meV, so producinga large measurable increase in the 2DEG’s resistance.

To illustrate this, we have investigated the effect of87Rb atoms deposited, by ZP focusing of a BEC, ontothe surface of a GaAs/(AlGa)As heterostructure con-taining a 2DEG at a distance l = 42 nm below thesurface [57, 67]. We expect similar results for 2DEGsin Si-based devices. Figure 5 shows a schematic di-agram of the heterostructure and ZP-lithography pro-cess. The 2DEG is formed by two δ-doping layers, lo-cated 22 and 32 nm below the surface and of density1.3 × 1016m−2 and 1016m−2 respectively, similar to val-ues used in recent experiments [72, 73]. For this sample,self-consistent solution of Poisson’s equation perpendic-ular to the surface shows that the Fermi energy of the2DEG is EF = 103π~2n2DEG/em

∗ = 2.9 meV, wheren2DEG = 8 × 1014m−2 is the sheet electron density, e isthe magnitude of the electronic charge, and the electroneffective mass, m∗, is 0.067 times the free electron mass[66, 73].

To deposit the atoms and measure their effect on the

FIG. 5: Schematic diagram of the heterostructure showingthe GaAs layers (light yellow), (AlGa)As layer (dark yellow),the two δ-doping layers (+) and the 2DEG (red). Shaded bluecone represents the focusing and surface deposition of 87Rbatoms after they have passed through the ZP (black and whitestructure on the left-hand side of the figure). Scanning theheterostructure in the directions arrowed, parallel to a singleZP or a ZP arrray [69, 70], deposits the atoms in surface pat-terns with arbitrary shapes. Polarization of atoms within thesurface pattern imprints a similarly-shaped depletion regionin the 2DEG. For illustration, here the atoms are depositedwithin the blue regions on the surface to create a square quan-tum dot in the 2DEG, like that studied in [71]. Electrons enterand leave the dot via quantum point contact openings createdby the gaps in the upper and lower edges of the blue square.The two arms emerging from the left- and right-hand edges ofthe blue square form depletion barrriers in the 2DEG, whichprevent electrons flowing around the outside of the dot [71].

2DEG, the BEC and heterostructure must be held in anultra-high vacuum system with a background pressure inthe low 10−11mbar range, as in the surface-physics exper-iments of Refs. [58, 59, 60, 63], for example. Since pre-cisely the same vacuum condition is also used in standardBEC experiments, the environments required to produceand controllably deposit the ultracold atoms are com-patible. We consider atoms deposited onto the Ga-richsurface of GaAs (001)-(4 × 2)/c(8 × 2), which producesstrong ionic bonding of the alkali-atom adsorbates dueto partial electron transfer from the atom to the sub-strate [58, 61, 64]. This Ga-rich surface reconstructioncan be prepared by encapsulating the heterostructure,after molecular beam epitaxial growth, with an As over-layer and then removing this overlayer within the vac-uum system by annealing the heterostructure at temper-atures up to 850 K [58, 60]. To heat the heterostruc-ture to such high temperatures, and then cool it to 85K or below so that the adsorbed atoms do not diffuse[59, 60, 74], requires a sample holder like that describedin Refs. [60, 75], which allows the temperature to bestabilized at any value between 85 K and 850 K.

Analysis of the BEC’s focusing dynamics, obtained by

7

solving Eq. (2) numerically, shows that, to good ap-proximation, the density profile of atoms deposited onthe surface is of the form nsurf (r) = nP exp(−r2/λ2),where the peak density nP = Natom/(2πλ2) increaseswith the total number of atoms deposited, Natom. Weconsider nP > 1017m−2 (the corresponding atom num-bers are specified below), so that the mean inter-atomicspacing (. 3 nm) is much less than the distance (42 nm)between the surface and the 2DEG. This ensures thatthe atoms can be modelled by the continuous distribu-tion nsurf (r), when calculating their effect on the 2DEG.Since Ga has a similar electronegativity to that of Si, wetake the electric dipole moment of each adsorbed atomto equal that measured previously (10−29 Cm) for 87Rbon Si [39], which is comparable to values calculated us-ing density-functional-theory for alkali atoms on GaAs[61, 64]. We note, however, that our results do not de-pend critically on the precise value of this parameter. Todetermine how the atoms influence the 2DEG, we calcu-lated the electron potential energy within and above theheterostructure by solving Poisson’s equation in cylin-drical co-ordinates. Our calculations used a relaxationmethod with a variable mesh to capture the vastly dif-ferent length scales that characterize the potential varia-tion near and away from the surface dipoles. We includedthe effects of the adsorbed surface atoms, electron sur-face states, shallow donor states, and linear (Thomas-Fermi) screening by the 2DEG [73]. Our calculationsuse a “frozen charge” model [76], in which the adsorbedatoms change the potential within the heterostructureand the electron density profile in the 2DEG, but do notalter the distribution of charge within the mid-gap sur-face states or donor layers.

We first consider atoms deposited with a radial spreadλ = 1 µm consistent with the focal width shown inFig. 4(a). Figure 6(a) shows the electron potentialenergy variation, V (r), in the plane of the 2DEG di-rectly below Natom = 106 adsorbed atoms. Repul-sive interaction with the polarized atoms increases theelectron potential energy by ∼ EF /10 when r = 0.Figure 6(b) shows the corresponding electron density,n(r) = [EF − V (r)]em∗/103π~2, which decreases to ap-proximately 90% of its bulk value as r → 0. Figure 6reveals that as Natom increases to 3 × 106 [(c) and (d)],5×106 [(e) and (f)], 8×106 [(g) and (h)], the peak valueof V (r) gradually increases to EF and, consequently,n(r) falls to zero below the centre of the surface atoms.When Natom = 107 [Fig. 6(i) and (j)], which requiresseveral different BECs to be deposited, sequentially, onthe surface, the 2DEG is fully depleted when r . 0.5µm and V (r) > EF . For a range of λ values, we findthat total depletion needs only a low density of adsorbedatoms, nP ≈ 1018m−2, corresponding to approximately0.1 monolayers, which ensures that these atoms interactfar more strongly with the surface than with one another[60, 61].

Experimental confirmation of the local electron deple-tion could be obtained by depositing the atoms immedi-

FIG. 6: Electron potential energy, V (r =py2 + z2), (left-

hand column) and density, n(r), (right-hand column) in theplane of the 2DEG directly below 106 [(a) and (b)], 3 × 106

[(c) and (d)], 5 × 106 [(e) and (f)], 8 × 106 [(g) and (h)], 107

[(i) and (j)] 87Rb atoms deposited, with a Gaussian profile ofwidth λ = 1 µm, on the surface of the heterostructure, whichlies in the focal plane of a transmission ZP. In (j), the 2DEGis fully depleted by its repulsive interaction with the polarizedsurface atoms.

ately above a quantum wire, ∼ 1 µm across and 5 µmlong, microfabricated in the 2DEG [37, 77]. As the atomsare deposited, the resistance of the quantum wire wouldrapidly increase, as observed previously when circularantidots (depletion regions) are introduced in a narrowconducting channel [37, 77, 78]. Alternatively, to deter-

8

mine the profile of the adsorbed atoms spatially and as afunction of time, quantum wires [68], each comprising aquasi one-dimensional (1D) conduction channel, could befabricated within the 2DEG by implanting ions into thesemiconductor material, for example Ga ions in GaAs,thus locally disrupting the 2DEG and transforming itfrom a conductor to an insulator [79, 80, 81, 82, 83, 84].Ion beam implantation can define a conduction networkcomprising two arrays of quantum wires, each contain-ing narrow (∼ 0.1− 1 µm) parallel conduction channels,which intersect at right angles. Monitoring the resistanceof each quantum wire, would enable the deposition of al-kali atoms on the surface of the device, or even held aboveit [78], to be mapped as a function of space and time, withsub-micron spatial resolution determined by the width ofthe wire. Overcoming the ∼ µm resolution limit of op-tical imaging is important for a wide range of ultracold-atom experiments including studies of tailored interact-ing many-body systems [85] where correlation functions[86] could be measured with unprecedented spatial reso-lution, in situ observation of soliton and vortex creationand dynamics [87], and the study of atom-surface inter-actions [88].

The behaviour of adsorbed atoms depends on the tem-perature of the semiconductor surface [59, 60, 74]. Atroom temperature, the atoms will diffuse across the sur-face at a rate that could be determined by measuringthe time evolution of the quantum wires’ resistance. Bycontrast, below 85 K, the atoms will stick where theyare deposited by the ZP [59, 60], thus producing a well-defined surface polarization pattern and electric field pro-file. The spatial resolution of such patterns is limitedby the radial width, lfr , of the focused cloud, which, asdiscussed in Section II, approximately equals the widthof the narrowest ZP ring that the matter wave passesthrough [26, 48, 89, 90]. A BEC with strong repulsiveinteractions is unable to penetrate ZP rings narrowerthan the healing length (typically a few hundred nm in atrapped BEC), which therefore limits lfr . To investigatewhether this limitation can be overcome by suppressinginter-atomic interactions, we have studied the focusingand deposition of an atom cloud that is trapped opticallyand subject to a small uniform magnetic field tuned to aFeshbach resonance so that a ≈ 0 in Eq. (2), as can beachieved for a range of alkali atoms [91, 92, 93].

When there are no inter-atomic interactions, lfr is lim-ited only by the narrowest ZP ring that the atom cloudspans [48, 89, 94], which can be etched as small as 12nm [95], with expectations that ring widths < 10 nm canbe achieved [48, 93, 95, 96, 97, 98]. Consequently, thewidth of the electron depletion region produced by theadsorbed atoms is limited either by lfr or by the distancefrom the 2DEG to the surface, whichever is the larger.In both semiconductor heterostructures and graphene,2DEGs can form on the surface itself [66, 99], suggestingthat depeletion regions a few 10s of nm across are at-tainable using existing ZPs [95]. For the heterostructureconsidered here, though, adsorbed alkali atoms will not

FIG. 7: (a) Electron potential energy, V (r =py2 + z2), and

(b) electron density, n(r), calculated in the plane of the 2DEGdirectly below 1800 87Rb atoms focused, with a Gaussian pro-file of width λ = 25 nm, onto the surface of the heterostruc-ture. The 2DEG is fully depleted by its repulsive interactionwith the polarized surface atoms.

produce depletion regions smaller than the 2DEG-surfaceseparation l = 42 nm. We therefore study atoms focusedby the ZP reported in Ref. [11], which has R1 = 10.4µm and an outer ring width of 50 nm, small enough toenable alkali-atom lithography of quantum componentsin 2DEGs. To demonstrate this, we calculated V (r) inthe plane of the 2DEG, taking λ = 25 nm, correspondingto a full focal width of ∼ 50 nm. After the depositionof only 1800 atoms, when r . 50 nm, V (r) > EF [Fig.7(a)] and the electron density is zero [Fig. 7(b)]. Numer-ical solutions of Eq. (2) reveal that this sharp focus canbe obtained for a range of approach speeds, for examplevx = 2.4 mm s−1, for which f = 57µm. Due to this longfocal length, the ZP is positioned far from the surface,thus overcoming a major problem of mask-based lithog-raphy, namely that to produce features on a 10 nm scalethe mask must be held within ∼ 1 µm of the surface [70],which is extremely challenging due to the strong Casimirattraction at such small separations.

B. Prospects for ZP-based matter-wavelithography

We now consider routes to exploiting ZP-focusing ofmatter waves for flexible, high-speed, erasable lithogra-phy of quantum electronic components.

It is not necessary for ZPs to be circular in order tofocus incident waves. For example, 1D ZPs comprisingdiffraction slits with edges at positions Rn = R1n

1/2(n =1, 2, 3, ...) can focus waves into a narrow line [100]. Con-sequently, focusing matter waves by 1D and/or circularZPs can create a wide range of surface patterns, thuscomplementing existing matter-wave lithography tech-niques, such as the use of optical standing waves to focusand deposit atoms in parallel lines [12, 13, 14, 15, 16].Alkali-atom lithography using ZPs offers comparable res-olution (a few 10s of nm) to that demonstrated with op-

9

tical lattices [16], plus the flexibility to tailor the distri-bution of adsorbed atoms using robust and well devel-oped ZP systems. Moreover, in contrast to ion implan-tation [79, 80, 81, 82, 83, 84], which damages the het-erostructure and degrades the electron mobility, alkali-atom lithography of 2DEGs is a gentle non-invasive pro-cess that requires no dedicated mask to produce eachdesired surface pattern.

By mounting the heterostructure on a scanning stageand moving it under the ZP with nm-precision (see Fig.5), arbitrary patterns of adsorbed atoms (blue areas onsurface in Fig. 5) and resulting electron depletion couldbe written in dot-matrix fashion [69, 70]. Planar arraysof transmission ZPs developed at MIT for maskless X-ray lithography [69, 70] are also suitable for depositingmany alkali atom spots in parallel on a surface. SuchZP-array lithography has major advantages over othertechniques: fast write speed, scalability, and the flexibil-ity to produce a wide range of surface patterns includingarrays of identical components, lines, gratings, and inter-connects. In the case of matter waves, cold atoms wouldbe supplied to each ZP either by outcoupling them from asingle BEC, as in an atom laser [101, 102, 103, 104, 105],or by using an array of microtraps like that created re-cently by using permanent magnets to confine the atoms[106, 107, 108, 109], with each trap supplying atoms to aparticular ZP.

The polarization of the adsorbed atoms can be con-trolled in situ by applying an electric field perpendicularto the surface [39, 110]. This suggests a way to alterthe quantum components, or temporarily erase them, byusing an electric field directed towards the surface to de-polarize the adsorbed atoms. Surface patterns, or evenindividual components, could be permanently erased byusing optical or UV radiation to either remove the atomsfrom the surface or make them diffuse away due to lo-cal heating [39, 40, 111, 112, 113, 114, 115, 116, 117,118, 119, 120], and then re-written on the same wafer.Erasable lithography of quantum components in a 2DEGis of considerable interest because it allows individualcomponents to be modified or repaired [37]. Moreover,the capacity to fabricate different devices sequentially,on the same surface, is crucial for distinguishing the ef-fects of device geometry on quantum transport from thoseoriginating from material impurities and imperfections[37, 38, 72, 121, 122]. So far, though, erasable lithogra-phy has only been achieved for single quantum electroniccomponents by using scanning probe techniques [37, 38],which produce charge patterns on the surface. Comparedwith such techniques, ZP-based alkali-atom lithographyoffers high write speed and scalability. It may also be sur-prisingly cost-effective because BECs can now be created

using off-the-shelf kits costing less than $50K [123].Polarization of dense patches of adsorbed alkali atoms

creates a strong inhomogeneous electric field above thesurface of the heterostructure as well as below it [39, 40,110]. Consequently, ultracold atoms above the surfaceand electrons within the 2DEG would move in similarly-shaped potentials. Their motion may therefore corre-late, or even couple, suggesting a route to developinghybrid electronic/atomic microchip structures made byZP lithography. Since such structures are potentially re-writable, they could be fabricated and studied in situwithout needing to break the vacuum between experi-ments on different chip geometries – thus greatly reducingthe time between device fabrication and measurement.

V. SUMMARY

In summary, BECs can be sharply focused by quan-tum reflection from, or transmission through, FresnelZPs. Optimal focusing, achieved for flat dilute BECswith lx/lr � 1 and n0 < 1018 m−3, can increase thepeak atom density by up to two orders of magnitude.Despite the increased atom-atom scattering rates that ac-company this compression, the focusing process does notsignificantly reduce the condensed fraction of the atomcloud. Focal lengths obtained from numerical simulationsof the Gross-Pitaevskii equation are similar to those ex-pected from a single-particle ray analysis, but exhibit ad-ditional resonances originating from inter-atomic interac-tions. Transmission ZP focusing of matter waves providesa powerful lithographic tool for fabricating quantum elec-tronic components by depositing well-defined, potentiallyre-writable, patterns of atoms on the surface of a semi-conductor heterojunction containing a 2DEG [124]. Thisnew type of lithography offers state-of-the-art resolution,scalability by using ZP arrays, the ability to re-write allor selected components, and a possible route to creat-ing hybrid electron/atom chips that are fabricated andstudied in situ. Since all of the individual components re-quired to realize ZP-based alkali-atom lithography exist,we hope that our results will stimulate the experimentalwork required to unite these components in a practicaldemonstration of the technique.

Acknowledgements

This work is funded by EPSRC UK and the ARC. Wethank Lucia Hackermuller and Philip Moriarty for helpfuldiscussions.

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12

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