A compact electron matter wave interferometer for sensor technology
Andreas Pooch, Michael Seidling, Moritz Layer, Alexander Rembold, Alexander Stibor
AA compact electron matter wave interferometer for sensor technology
A. Pooch , M. Seidling , M. Layer, A. Rembold and A. Stibor , + Institute of Physics and Center for Collective Quantum Phenomena in LISA + ,University of T¨ubingen, Auf der Morgenstelle 15, 72076 T¨ubingen, Germany (Dated: February 20, 2018)Remarkable progress can be observed in recent years in the controlled emission, guiding and detectionof coherent, free electrons. Those methods were applied in matter wave interferometers leading to highphase sensitivities and novel sensor technologies for dephasing influences such as mechanical vibrationsor electromagnetic frequencies. However, the previous devices have been large laboratory setups. Forfuture sensor applications or tests of the coherence properties of an electron source, small, portableinterferometers are required. Here, we demonstrate a compact biprism electron interferometer that canbe used for mobile applications. The design was optimized for small dimensions by beam path simulations.The interferometer has a length between the tip and the superposition plane before magnification of only47 mm and provides electron interference pattern with a contrast up to 42.7 %. The detection of twodephasing frequencies at 50 and 150 Hz was demonstrated applying second order correlation and Fourieranalysis of the interference data. I. INTRODUCTION
Matter wave interferometers for electrons [1–4] have sig-nificantly improved in the last decades. They are appliedto measure the rotational phase shift due to the Sagnac ef-fect [5], to study Coulomb-induced quantum decoherence[6, 7], the magnetic Aharonov-Bohm effect [8–11] or theTalbot-Lau effect for magnetic field sensing [12]. The topicis influenced by recent technical innovations and improve-ments concerning the beam source [13–15], the precise elec-tron guiding [3, 16], the coherent beam path separation[4, 14, 17, 18] and the development of spatial and temporalsingle-particle detection methods [19–22]. The progress haspotential novel applications in electron microscopy [23, 24]and sensor technology for inertial forces [25], mechanicalvibrations [21] and electromagnetic frequencies [19, 20].Small deviations of the partial waves in the two sepa-rated beam paths in a matter wave interferometer lead to aclear phase shift on the detector after they get superposed.This simple feature makes interferometric measurements ex-tremely sensitive towards external perturbations. In contrastto neutral atoms, the phase of electron matter waves canbe shifted not only by mechanical vibrations, temperaturedrifts or rotations of the setup but also by external elec-tromagnetic frequencies. Usually, these perturbations leadto a time dependent dephasing, causing a “wash-out” ofthe temporally integrated interference pattern that can beobserved in a reduced interference contrast. This is par-ticularly a challenge for sensitive long-time phase measure-ments such as proposed for the measurement of the electricAharonov-Bohm effect [26], for decoherence measurements[7] or the interferometry of ions [27, 28].We recently demonstrated in a biprism electron inter-ferometer [4] that such dephasing effects can on the onehand be corrected and on the other hand used for an accu-rate measurement of the perturbation frequencies [20, 21].Thereby, the dephasing was detected and reduced with thehigh spatial and temporal single-particle resolution of a de-lay line detector [22]. A second-order correlation analysis incombination with a Fourier analysis was performed on the detection events after the interference was recorded. It canreveal multifrequency electromagnetic oscillations and me-chanical vibrations. The spectrum of the unknown externalfrequencies, their amplitudes, the interference contrast andthe pattern periodicity can be extracted from a spatially“washed-out” pattern [19–21]. For that reason electronmatter wave interferometers have a high potential in sensortechnology. However, due to their large dimensions currentexperimental setups are not suitable for portable sensor ap-plications [2–4]. To apply an electron interferometer as asensor for electromagnetic and vibrational frequencies or forthe mobile analysis of the coherence of a beam source, it isnecessary to construct a small and transportable device.In this article we present a compact biprism matter waveinterferometer for free electrons with minimized distancesbetween all parts and a high mechanical stability. The dis-tance between the tip and the superposition plane beforemagnification is only 47 mm including all components forbeam guiding and diffraction. In combination with the re-cently developed tools for spectrum analysis by correlationtheory [19–21], the compact setup is an important prereq-uisite for a portable, mobile sensor based on matter waveinterferometry with electrons.
II. EXPERIMENTAL SETUP
The setup of the biprism interferometer is illustrated infig. 1. The source for coherent electrons is a field emissiontip that can be prepared by pulsed etching of a polycrys-talline tungsten wire [29]. The beam gets aligned by twodeflection electrodes to coherently illuminate a biprism fiber.It acts as a beamsplitter for the electron matter waves, if asmall positive voltage is applied on the fiber. The electricforce bends the separated beam paths towards each otherand superimposes them in front of a magnifying quadrupolelens. The key feature of the electron biprism, analogue tothe optical biprism, is that all possible beam paths get de-flected by the same angle, leading to a common angle ofsuperposition at the entrance of the quadrupole [2]. The a r X i v : . [ phy s i c s . i n s - d e t ] M a r FIG. 1. Sketch of the compact biprism electron interferome-ter (not to scale). The coherent electron beam is field emittedby a tungsten tip and guided by double deflectors. A biprismfiber separates and combines the partial matter waves thatinterfere at the entrance of a magnifying quadrupole. Animage rotating coil can rotate the resulting pattern for align-ment. The magnified interferogram is amplified by two multi-channel plates and detected by a delay line detector. superposition of the two partial beams leads to an interfer-ence pattern parallel to the biprism fiber. The interferogramhas a typical pattern periodicity of several hundred nanome-ters and needs to be magnified by the quadrupole lens tofit the spatial resolution of the multi-channel plate (MCP)detector. Such lenses are optimal suited for biprism interfer-ometry since it is only necessary to magnify the pattern inthe direction normal to the interferences. A small misalign-ment of the biprism fiber towards the magnifying axis, canbe corrected by an image rotating coil. In our experiment, adrift distance of 169.8 mm between the exit aperture of thequadrupole lens and the MCPs of the detector was chosen.The electron signal gets amplified by the MCP and detectedby a hexagonal delay line [30] with a spatial and temporalresolution of ∼ µ m and ∼ a = 19 mm and to the entrance of thequadrupole lens b = 28 mm. The tip emits electrons with anacceleration voltage of U e = 2250 V into a double deflectormodule, consisting of four pairs of flat deflection electrodeswith a length of 5 mm and oriented around the beam with adistance of 5.5 mm between them. The positive and nega-tive voltage applied between two opposing electrode pairs isequal to keep a zero potential on the beam axis. The biprismfiber consists of a glass fiber with a diameter of 400 nm thatwas manufactured by a special procedure described else-where [4, 32, 33]. It is coated with a gold-palladium alloy FIG. 2. (a) Image of the components in the beam pathfrom the tip to the quadrupole with the upper copper shellremoved. (b) Interferometer parts from the tip to thequadrupole mounted together within both half-shells. to ensure a smooth, conductive surface and is glued on aholder isolated by a non-conductive foil. The fiber is posi-tioned between two grounded titanium electrodes that are4 mm apart from each other. The rotating coil is windedaround a 7 mm-diameter tube. The quadrupole lens is madeout of four opposing cylindrical electrodes with a length of10 mm, a diameter of 7.6 mm and a distance normal to thebeam path towards each other of 6.7 mm. As for the dou-ble deflectors, the electrodes are mounted within isolatingholders made out of
Macor . An aperture with a diame-ter of 4 mm is positioned at the entrance and exit of thequadrupole lens to decrease the amount of secondary strayelectrons on the detector. The interferometer is magneti-cally shielded by a mu-metal tube and in a vacuum cham-ber at ∼ × − mbar to ensure a long lifetime and stableemission of the tip [34]. III. EXPERIMENTAL RESULTS
With this device it was possible to observe high contrastelectron interference pattern such as shown in fig. 3 (a)for 5 × detected particles. The temporal and spatialinformation of every detection event was recorded fora correlation analysis after data acquisition. The smallbend in the middle of the pattern in fig. 3 (a) is dueto a deviation of the beam from the optical axis in thequadrupole lens. It can be corrected by a fourth degreepolynomial fit on the fringes and subsequent straightening.The resulting pattern in the marked red rectangle (7 mm ×
32 mm) is exhibited in fig. 3 (b). Thus, nine fringesare visible in the image. The color bar indicates theintensity distribution of the incoming particles. In fig. 3 (c)the average intensity along the y -direction is plottedagainst the distance x on the screen. The distributionis enfolded by a sinc -function due to the similarity tothe double slit experiment analysis [35]. Therefore, thedata in fig. 3 (c) was fitted with the model function I ( x ) = I · (cid:16) K m · cos( πxs m + φ ) (cid:17) · sinc ( πxs + φ ) revealing a contrast K m = (37.2 ±
5) % and the fringedistance s m = (0.85 ± φ , φ , theaverage intensity I and the width of the interference pat- (a) (b)(c)-16 -8 0-16-80816 x [mm] y [ mm ] -16-80816 0246810N cts -6 -4 -2 0 12 x [mm] i n t e n s i t y [ a . u .] FIG. 3. (a) Image of the interference pattern on the detec-tor screen with the set experimental values U e = 2250 V, U f = (0.559+0.69) V and quadrupole voltages +2960 V be-tween the electrodes normal to the fringes and -2910 V paral-lel to them. (b) Focus of the region within the red rectanglewhere the fringes were straightened by a fourth degree polyno-mial. (c) Blue curve: average intensity along the y -directionon the screen. Red curve: numerical fit of the intensity. tern s are additional fitting parameters. The magnificationof the interferogram in fig. 3 (a) is 2517 ±
6. It is givenby the determined fringe distance s m on the detector aftermagnification divided by the theoretical fringe distance s = 338 nm at the entrance of the quadrupole, which iscalculated with s = λ dB θ , and θ = 2 γ · aa + b . Thereby, λ dB is the de Broglie wavelength of the electrons, θ thesuperposition angle at the entrance of the quadrupole, γ = π ln ( R/r ) U f U e the deviation angle of the biprism fiberwith an applied voltage of U f , r is the radius of the fiberand R its distance to the grounded electrodes [36].We additionally performed beam path simulations withthe program Simion [37]. The superposition angle wasextracted from the simulation by a method described in[26]. It is known that contact potentials between thegold/palladium coating of the fiber and the titanium elec-trodes influence the effective potential interacting with theseparated beams [38, 39]. For that reason, we adapted thebiprism voltage until nine interference stripes fit in the su-perposition area. This is achieved by adding 0.69 V in thesimulation to the experimentally applied voltage of 0.559 V.This extra voltage is considered to be the contact poten-tial and agrees well comparing the literature values for thework functions of the averaged 80:20 % gold/palladium al-loy and the one of titanium [40]. Their difference amountsto 0.71 eV. Our simulations revealed a superposition angleof 7 × − rad and a pattern periodicity of 369 nm beforemagnification. The quadrupole magnification was simulatedto be 2886, which can be considered as an upper bound forperfect beam alignment on the optical axis. The reason- (a) (b)(c) 0.8 1 1.2 1.4-3.5-2023.5 g (2) ( u,
0) [a.u.] u [ mm ] τ [s] 0 . . . g (2) ( u, τ )
50 100 150 · − ω/ π [Hz] a m p li t ud e [ a . u .] FIG. 4. (a) Blue line: the second-order correlation function g (2) ( u,
0) at the correlation time τ = 0. Red curve: fit func-tion to reveal the contrast and the pattern periodicity. (b)The second-order correlation function g (2) ( u, τ ) for τ rangingbetween 0 and 5 seconds. g (2) ( u, τ ) is extracted from the datain fig. 3 (b). The periodic fringe pattern is clearly observable.(c) Plot of the amplitude spectrum calculated via a numericalFourier transformation of g (2) ( u = M u · s g / , τ ) , M u ∈ N andsubsequent averaging. Periodic perturbations of the interfer-ence fringes in time become visible and identify two charac-teristic frequencies at 50 Hz and 150 Hz. able small variations to the theoretical fringe distance s and magnification are possibly due to the neglected beamadjustment voltages and small deviations between the sim-ulated and experimental setup distances.As recently demonstrated [20, 21] and discussed above,contrast reducing dephasing effects, such as electromag-netic oscillations or mechanical vibrations can be isolatedand corrected by a second-order correlation analysis of themeasured interference pattern. We applied this method,described in detail elsewhere [20, 21, 41], to the interfer-ence data in fig. 3 (b). The analysis provides the second-order correlation function g (2) ( u, τ ) with the correlationlength u and correlation time τ between the detected par-ticles. Consequentially, we can determine the fringe dis-tance s g and the contrast K g of the unperturbed interfer-ence pattern at the temporal position τ = 0 , g (2) ( u, ,as shown in fig. 4 (a). The red curve is a fit function g (2) ( u,
0) = 1 + K g · cos ( πus g + φ g ) + O with the fit pa-rameters contrast K g , the fringe distance s g , the phase φ g and the offset O [41]. We obtain s g = (0.84 ± K g = (42.5 ±
7) %. The contrast K g is higher than K m suggesting that disturbing effects may wash-out theinterference pattern. The significant high standard devia-tions for both values are due to a limited number of de- . . . . biprism voltage [V] f r i n g e d i s t a n ce ( s m , s g ) [ mm ] c o n t r a s t( K m , K g ) [ % ] s m s g K m K g FIG. 5. Fringe distances s m and s g together with the inter-ference contrasts K m and K g as a function of the biprismvoltage. The data is evaluated by spatial integration and cor-relation analysis. The error bars for s m and s g are smallerthan the dot size. K m is in most cases lower than K g due tothe influence of the detected dephasing perturbations. tected particles and therefore a large noise. In fig. 4 (b) g (2) ( u, τ ) is shown for correlation times up to five seconds.Along the u -axis the unperturbed fringe pattern with a dis-tance s g can be observed. The amplitude spectrum of thecorrelation function, |F ( g (2) ( u, τ ))( u, ω ) | , reveals the per-turbation characteristics. It is determined using a numeri-cal temporal Fourier transformation at the spatial positions ( u = M u · s g / , τ, M u ∈ N ) , where the correlation functionhas its maximum signal [21, 41]. In fig. 4 (c) the averageover all amplitude spectra calculated at the spatial positions u = M u · s g / is plotted. For optimal settings of the spatialand temporal discretization step size of the numerical cor-relation function [41], two clear peaks at 50 Hz and 150 Hzevolve. These frequencies affect the electron waves and leadto unwanted contrast loss. Their spatial perturbation ampli-tude before the magnification can be determined accordingto the description in [21] to 14.3 nm at 50 Hz and 14.8 nmat 150 Hz. Thereby, it is assumed that the perturbationtakes place before magnification, which is reasonable sincethe amplitudes are low. They probably originate from theutility frequency of the electrical power supplies. Their de-tection demonstrates the usability of our device as a sensorfor external perturbation frequencies.For further characterization of the compact interferome-ter, we present a series of measurements with variable volt-ages U f at the biprism fiber, going from (0.051+0.69) Vto (0.900+0.69) V in ∼
50 mV steps. With those sixteendata files, the same analysis is executed as described above.The data is shown in fig. 5, whereas U f is plotted againstboth the fringe distance and the contrast. The values for s g and K g refer to the unperturbed case originating from thesecond-order correlation analysis. As expected from theory[36], the fringe distance decreases with increasing biprismvoltage, since the superposition angle of the partial waves increases. This is observed for both s m and s g leading tonearly the same results. The contrast K m increases up to ∼
38 % at ∼ K m decreases due to diffraction effects at the edge of the fiber.For higher voltages a reduced contrast is expected due tocoherence considerations with a beam source of finite extent[28, 36]. K g reveals a similar curve progression with a max-imum of 42.7 %. As expected, the contrast K g is at mostvoltages higher than the contrast K m disturbed by the twodephasing frequencies. The first four measurements of K g can not be evaluated with the g (2) analysis because of theinfluence of diffraction. It is not included in the theory ofthe g (2) analysis and significant for small biprism voltages. IV. CONCLUSION
We demonstrated a compact biprism matter wave inter-ferometer for free electrons providing interference fringeswith a contrast up to 42.7 %. The dimension and the ro-bustness of the setup is sufficient to be integrated in amobile device. The interference data was recorded by adelay line detector with a high spatial and temporal single-particle resolution. This allowed a second-order correlationand Fourier analysis revealing the undisturbed contrast andthe pattern periodicity of the interferogram. It was com-pared to the values obtained by pure spatial signal inte-gration. Our device allowed the identification of dephasingfrequencies at 50 Hz and 150 Hz and therefore demonstratedits applicability for the detection of external perturbationswith the recently developed second-order correlation dataanalysis [19–21, 41]. Identifying dephasing oscillations fromthe lab environment is a very helpful tool to improve ex-periments for sensitive phase measurements. The setup canalso be applied to test the coherence of novel beam sources[14, 15]. Especially the simple design and the small dimen-sions make the interferometer easy to handle and usable invarious environments. The sensitivity towards vibrational orelectromagnetic dephasing and inertial forces such as rota-tion and acceleration could be increased significantly with alarger beam path separation. This would require an inter-ferometer scheme with two or three biprism fibers in com-bination with a quadrupole or an einzel-lens [5, 25, 26].
V. ACKNOWLEDGEMENTS
This work was supported by the Vector Stiftung andthe Deutsche Forschungsgemeinschaft through the EmmyNoether program STI 615/1-1 and the research grantSTI 615/3-1. A.R. acknowledges support from the Evan-gelisches Studienwerk e.V. Villigst. The authors thankN. Kerker, R. R¨opke and G. Sch¨utz for helpful discussions.
VI. REFERENCES [1] F. Hasselbach, Rep. Prog. Phys. , 016101 (2010)[2] G. M¨ollenstedt and H. D¨uker, Z. Phys. A - Hadron Nucl. , 377 (1956)[3] F. Hasselbach, Z. Phys. B , 443 (1988)[4] G. Sch¨utz, A. Rembold, A. Pooch, S. Meier, P. Schneeweiss,A. Rauschenbeutel, A. G¨unther, W.T. Chang, I.S. Hwangand A. Stibor, Ultramicroscopy , 9 (2014)[5] F. Hasselbach and M. Nicklaus, Phys. Rev. A , 143 (1993)[6] W. H. Zurek, Rev. Mod. Phys.
715 (2003)[7] P. Sonnentag and F. Hasselbach, Phys. Rev. Lett. ,200402 (2007)[8] Y. Aharonov, and D. Bohm, Phys. Rev. , 485 (1959)[9] G. M¨ollenstedt, and W. Bayh, Physikalische Bl¨atter , 299(1962)[10] A. Tonomura, N. Osakabe, T. Matsuda et al., Phys. Rev.Lett. , 792 (1986)[11] R.G. Chambers, Phys. Rev. Lett. , 3 (1960)[12] R. Bach, G. Gronniger and H. Batelaan, Appl. Phys. Lett. , 254102 (2013)[13] H. S. Kuo, I. S. Hwang, T. Y. Fu, Y. C. Lin, C. C. Chang,and T. T. Tsong, Japanese J. Appl. Phys. , 227601 (2015)[15] P. Hommelhoff, Y. Sortais, A. Aghajani-Talesh et al., Phys.Rev. Lett. , 077401 (2006)[16] J. Hammer, J. Hoffrogge, S. Heinrich et al., Phys. Rev.Appl. , 044015 (2014)[17] C.C. Chang, H.S. Kuo, I.S. Hwang and T.T. Tsong, Nan-otechnology , 115401 (2009)[18] B. Cho, T. Ichimura, R. Shimizu and C. Oshima, Phys. Rev.Lett. , 246103 (2004)[19] A. Rembold, G. Sch¨utz, W. T. Chang, A. Stefanov, A.Pooch, I. S. Hwang, A. G¨unther and A. Stibor, Phys. Rev.A , 033009 (2017)[22] O. Jagutzki, V. Mergel, K. Ullmann-Pfleger, L. Spielberger,U. Spillmann, R. D¨orner, H. Schmidt-B¨ocking, Nucl. Instr.Meth. Phys. Research A , 244-249 (2002)[23] W.P. Putnam, and M.F. Yanik, Phys. Rev. A , 040902(R)(2009)[24] P. Kruit, R.G. Hobbs, C-S. Kim et al., Ultramicroscopy ,31 (2016)[25] J.F. Clauser, Physica B , 262 (1988)[26] G. Sch¨utz, A. Rembold, A. Pooch, H. Prochel and A. Stibor,Ultramicroscopy , 65 (2015)[27] F. Hasselbach and U. Maier, Quantum Coherence and De-coherence - Proc. ISQM-Tokyo 98 ed. by Y.A. Ono and K.Fujikawa (Amsterdam: Elsevier), 299 (1999)[28] U. Maier, doctoral thesis, University of T¨ubingen (1997)[29] W.T. Chang, I.S. Hwang, M.T. Chang et al., Rev. Sci. Instr. , 083704 (2012)[30] Roentdek, model DLD40X[31] Private communicaton with the company Roentdek[32] F. Warken, E. Vetsch, D. Meschede, M. Sokolowski and A.Rauschenbeutel, Optics Express , 11952 (2007)[33] F. Warken, A. Rauschenbeutel and T. Bartholom¨aus, Pho-tonics Spectra , 73 (2008)[34] K. S. Yeong and J. T. L. Thong, J. Appl. Phys. , 104903(2006)[35] E. Hecht, Optik. 6. Aufl. , Berlin, New York: Walter DeGruyter Incorporated (2014)[36] F. Lenz, and G. Wohland, Optik , 315 (1984)[37] Simion, Version 8.1, Scientific Instrument Services Inc.,USA[38] E. Krimmel, C. M¨ollenstedt, and W. Rothemund, Appl.Phys. Lett. , 209 (1964)[39] W. Br¨unger, Z. Physik250