letters to nature focused laser beam and some electrons pass through more intense Observation of the Kapitza-Dirac regions,a numerical solution of the Schrodinger equation gives acceptable agreement with the experimental data (Fig.2).The effect parameters used in the numerical simulation (laser focus, 125 um;laser intensity in the standing wave,5X 10Wm2; Daniel L Freimund",Kayvan Aflatooni&Herman Batelaan electron velocity,1.1 X 10ms;optics transmission,70%;over- lap,45 um)are consistent with the experimental parameters.An Department of Physics and Astronomy,University of Nebraska-Lincoln, overlap of 45 um indicates the FWHM of the height of the standing 116 Brace Laboratory,PO Box 880111,Lincoln,Nebraska 68588-0111,USA wave.We calculate that with perfect overlap(standing wave FWHM of 125 um)between the two counter-propagating laser beams,a In their famous 1927 experiment,Davisson and Germer observed' laser light intensity ten times lower would yield a comparable the diffraction of electrons by a periodic material structure,so diffraction pattern.The small asymmetry in the diffraction pattern showing that electrons can behave like waves.Shortly afterwards, (somewhat larger in the experiment than in the simulation)is Kapitza'and Dirac'predicted that electrons should also be attributed to a misalignment of the electron beam of approximately diffracted by a standing light wave'.This Kapitza-Dirac effect is 1 mrad with respect to the laser and is indicative of the onset of analogous to the diffraction of light by a grating,but with the Bragg scattering. roles of the wave and matter reversed.The electron and the light In some early experiments-15 attempts were made to measure grating interact extremely weakly,via the ponderomotive the deflection of free electrons due to a light wave.Two experiments potential",so attempts to measure the Kapitza-Dirac effect had reported an effect while two others did noRegardless of this to wait for the development of the laser.The ideas that the controversy no diffraction peaks were observed.Indeed,recent underlying interaction with light is resonantly enhanced for reviews state that the Kapitza-Dirac effect has not been observed electrons in an atom led to the observation?that atoms could be for electrons310.Explanations were offered to account for the diffracted by a standing wave of light.Deflection of electrons by controversy of the early experiments.Schwartz6 has suggested high-intensity laser light,which is also a consequence of the that in two experiments the interaction strength was accidentally Kapitza-Dirac effect,has also been demonstrated.But the such that the height of the first-order diffraction peak was at a coherent interference that characterizes wave diffraction has not minimum.Considering the experimental difficulty of obtaining hitherto been observed".Here we report the diffraction of free uniform laser intensity,this explanation seems unlikely.Fedorov7 electrons from a standing light wave-a realization of the on the other hand,has suggested that a slow adiabatic turn-on is the Kapitza-Dirac effect as originally proposed. main reason for the previous failure to observe the deflection owing In our experiment,an electron beam crosses two counter-propa- to the 'ponderomotive potential.In agreement with Fedorov,our gating laser beams which form the standing wave light grating simulation also shows that increasing the laser beam spatial width (Fig.1).To reach sufficiently high laser intensities,we used a causes the Kapitza-Dirac effect to vanish for finite-sized electron Nd:YAG laser with 10-ns pulses and an energy of 0.2]per pulse beams.We have kept Fedorov's suggestion in mind while designing focused to a beam waist 125 um in diameter.Each counter-propa- this experiment.Additionally,the greater stability and reliability of gating laser beam travels an equal distance not differing by more modern lasers and the improved performance of electronics have than 1 mm.This is well within the coherence length of the laser aided this experiment compared to earlier attempts to observe the beam(5 mm)where the standing wave is formed.A 380-eVelectron Kapitza-Dirac effect. beam is collimated by two 10-um-wide molybdenum slits separated Our results demonstrate that no fundamental problems stood by 24 cm.A third slit cuts the height of the electron beam to the size in the way of observing the effect.At much higher laser of the laser beam waist.Subsequently,the electron beam crosses the intensities the important 1988 experiment by Bucksbaum et al. standing wave about 1 cm after the third slit.A fourth 10-um slit, showed that electrons could be deflected by the ponderomotive 24 cm downstream from the interaction region,is used to scan the potential.Bucksbaum observed two classical rainbow scattering electron beam profile.The measured spatial width(full-width at peaks separated by about 1,000 photon recoils.We observe half-maximum,FWHM)of the electron beam is 25 um.This is a quantum mechanical diffraction peaks separated by two photon considerably narrower width than the expected distance between recoils.An important difference between these experiments is that the zero and first diffraction order,55 um=2A/(X 24 cm), the rainbow peaks are not coherent,whereas diffraction peaks are where Aag is the de Broglie wavelength of the electrons and Aomt is the coherent. wavelength of the laser light,532 nm.We may thus expect the diffraction peaks to be resolved.The factor of two takes into account the ratio between the light grating periodicity and the light wavelength.The electrons are detected as a function of time with Lens an electron multiplier.Each laser pulse is used as a start signal,and the detection of electrons is used as the stop signal for a time to amplitude converter.A multi-channel scaler records the pulses from Electron gun ctron detector the converter into coincidence time spectra.From the time spectra taken at various positions,the diffraction pattern is obtained directly. The diffraction pattern is shown in Fig.2.The diffraction orders are clearly resolved and fall at their expected positions(n 55 pm, ht grating n=0,±l,±2，..).The heights of the diffraction peaks might be expected to be given by the analytic solution of the Schrodinger equation in the diffractive limit.However,this is not the case. Laser beam Given that some electrons pass through less intense regions of the Figure 1 Schematic of our apparatus.Electrons are collimated by four molybdenum slits and diffract from a standing wave of light formed by two counter-propagating laser Present addres Physics,Fort Hayes sity. 600 Park Street,Hays,Kansa beams.The electrons must be described by a quantum mechanical wave while the 7601-4099.U5A standing light wave acts as a grating. 142 2001 Macmillan Magazines Ltd NATURE|VOL 413|13 SEPTEMBER 2001 www.nature.com
................................................................. Observation of the Kapitza±Dirac effect Daniel L. Freimund*, Kayvan A¯atooni*² & Herman Batelaan* * Department of Physics and Astronomy, University of NebraskaÐLincoln, 116 Brace Laboratory, PO Box 880111, Lincoln, Nebraska 68588-0111, USA .............................................................................................................................................. In their famous 1927 experiment, Davisson and Germer observed1 the diffraction of electrons by a periodic material structure, so showing that electrons can behave like waves. Shortly afterwards, Kapitza2 and Dirac3 predicted that electrons should also be diffracted by a standing light wave4 . This Kapitza±Dirac effect is analogous to the diffraction of light by a grating, but with the roles of the wave and matter reversed. The electron and the light grating interact extremely weakly, via the `ponderomotive potential'5 , so attempts to measure the Kapitza±Dirac effect had to wait for the development of the laser. The idea6 that the underlying interaction with light is resonantly enhanced for electrons in an atom led to the observation7 that atoms could be diffracted by a standing wave of light. De¯ection of electrons by high-intensity laser light, which is also a consequence of the Kapitza±Dirac effect, has also been demonstrated8 . But the coherent interference that characterizes wave diffraction has not hitherto been observed9,10. Here we report the diffraction of free electrons from a standing light waveÐa realization of the Kapitza±Dirac effect as originally proposed. In our experiment, an electron beam crosses two counter-propagating laser beams which form the standing wave light grating (Fig. 1). To reach suf®ciently high laser intensities, we used a Nd:YAG laser with 10-ns pulses and an energy of 0.2 J per pulse focused to a beam waist 125 mm in diameter. Each counter-propagating laser beam travels an equal distance not differing by more than 1 mm. This is well within the coherence length of the laser beam (5 mm) where the standing wave is formed. A 380-eVelectron beam is collimated by two 10-mm-wide molybdenum slits separated by 24 cm. A third slit cuts the height of the electron beam to the size of the laser beam waist. Subsequently, the electron beam crosses the standing wave about 1 cm after the third slit. A fourth 10-mm slit, 24 cm downstream from the interaction region, is used to scan the electron beam pro®le. The measured spatial width (full-width at half-maximum, FWHM) of the electron beam is 25 mm. This is a considerably narrower width than the expected distance between the zero and ®rst diffraction order, 55 mm 2l dB=l opt 3 24 cm, where ldB is the de Broglie wavelength of the electrons and lopt is the wavelength of the laser light, 532 nm. We may thus expect the diffraction peaks to be resolved. The factor of two takes into account the ratio between the light grating periodicity and the light wavelength. The electrons are detected as a function of time with an electron multiplier. Each laser pulse is used as a start signal, and the detection of electrons is used as the stop signal for a time to amplitude converter. A multi-channel scaler records the pulses from the converter into coincidence time spectra. From the time spectra taken at various positions, the diffraction pattern is obtained directly. The diffraction pattern is shown in Fig. 2. The diffraction orders are clearly resolved and fall at their expected positions (n 3 55 mm; n 0; 6 1; 6 2;¼). The heights of the diffraction peaks might be expected to be given by the analytic solution of the SchroÈdinger equation in the diffractive limit11. However, this is not the case. Given that some electrons pass through less intense regions of the focused laser beam and some electrons pass through more intense regions, a numerical solution of the SchroÈdinger equation gives acceptable agreement with the experimental data (Fig. 2). The parameters used in the numerical simulation (laser focus, 125mm; laser intensity in the standing wave, 5 3 1014 W m22 ; electron velocity, 1:1 3 107 m s21 ; optics transmission, 70%; overlap, 45 mm) are consistent with the experimental parameters. An overlap of 45 mm indicates the FWHM of the height of the standing wave. We calculate that with perfect overlap (standing wave FWHM of 125mm) between the two counter-propagating laser beams, a laser light intensity ten times lower would yield a comparable diffraction pattern. The small asymmetry in the diffraction pattern (somewhat larger in the experiment than in the simulation) is attributed to a misalignment of the electron beam of approximately 1 mrad with respect to the laser and is indicative of the onset of Bragg scattering. In some early experiments12±15 attempts were made to measure the de¯ection of free electrons due to a light wave. Two experiments reported an effect12,13, while two others did not14,15. Regardless of this controversy no diffraction peaks were observed. Indeed, recent reviews state that the Kapitza±Dirac effect has not been observed for electrons9,10. Explanations were offered to account for the controversy of the early experiments. Schwartz16 has suggested that in two experiments the interaction strength was accidentally such that the height of the ®rst-order diffraction peak was at a minimum. Considering the experimental dif®culty of obtaining uniform laser intensity, this explanation seems unlikely. Fedorov17, on the other hand, has suggested that a slow adiabatic turn-on is the main reason for the previous failure to observe the de¯ection owing to the `ponderomotive potential'. In agreement with Fedorov, our simulation also shows that increasing the laser beam spatial width causes the Kapitza±Dirac effect to vanish for ®nite-sized electron beams. We have kept Fedorov's suggestion in mind while designing this experiment. Additionally, the greater stability and reliability of modern lasers and the improved performance of electronics have aided this experiment compared to earlier attempts to observe the Kapitza±Dirac effect. Our results demonstrate that no fundamental problems stood in the way of observing the effect. At much higher laser intensities the important 1988 experiment8 by Bucksbaum et al. showed that electrons could be de¯ected by the ponderomotive potential. Bucksbaum observed two classical rainbow scattering peaks separated by about 1,000 photon recoils. We observe quantum mechanical diffraction peaks separated by two photon recoils. An important difference between these experiments is that the rainbow peaks are not coherent, whereas diffraction peaks are coherent. letters to nature 142 NATURE | VOL 413 | 13 SEPTEMBER 2001 | www.nature.com ² Present address: Department of Physics, Fort Hayes State University, 600 Park Street, Hays, Kansas 67601-4099, USA. Apertures Lens Laser beam Light grating Electron gun Electron detector Figure 1 Schematic of our apparatus. Electrons are collimated by four molybdenum slits and diffract from a standing wave of light formed by two counter-propagating laser beams. The electrons must be described by a quantum mechanical wave while the standing light wave acts as a grating. © 2001 Macmillan Magazines Ltd
letters to nature The observation of the Kapitza-Dirac effect opens the door to achieved in 100-ps pulse Nd:YAG lasers2)will raise the strength of various new experiments.Because the diffracted electron beams the magnetic field of the laser beam to the extent that the electron are coherent with each other,the Kapitza-Dirac effect constitutes spin would rotate by 180 in such a field.The question thus arises of a coherent beam splitter.Just as for atoms,the combination of whether the electron spin in the diffraction process could flip. three such beam splitters can be used to construct a Mach- Although classical arguments for a circularly polarized travelling Zehnder interferometer's.Compared to biprism electron interfe- wave seem to rule out this possibility2,this question,in general,and rometers,this new type of electron interferometer would operate in particular for standing waves,is to our knowledge unanswered. at very low electron energies and seems to be well suited to study, The atom optics counterpart of this effect is the "optical Stern- for example,forward electron-atom scattering phase shifts Gerlach effect"and has been observed.However,this result cannot Instead of using three consecutive beam splitters,it may also be easily be extended to free electrons owing to the half-integer value of possible to use the coherence of the diffraction pattern itself. the spin.A spin flip in combination with diffraction would When I2 molecules are placed in a YAG laser beam (with experi- constitute a polarizing beam splitter for free electrons or,in other memt)theegn lom the eou words,a microscopic Stern-Gerlach magnet.We have to keep in mind that Stern-Gerlach magnets for free electrons do not exist" only at the antinodes of the standing wave.The result is that the By increasing the laser intensity further to 108Wcm(for a laser periodically aligned I2 molecules will write a sinusoidal phase shift wavelength of 1 um),it is interesting to note that electrons are so on the incoming electron waves.This shift will modify the light that relativistic speeds can be reached24.Thus the study of diffraction pattern and could be used to monitor the I,alignment the interaction of free electrons with laser light can probably as it is influenced by,for example,molecular dissociation or be extended from quantum mechanics to include spin,chaotic ionization. behaviour and relativistic mechanics. ▣ Apart from the use of the Kapitza-Dirac effect as a tool,it is interesting to study in itself.It has been shown experimentally that Received 18 May:accepted 19 July 2001. atoms moving through a standing light wave represent an example 1.Davisson,C.Germer,L.H.The scattering ofelectrons by a single crystal of nickel.Nature 119,558- of classical and quantum chaos.The largest angles to which atoms 56019271. can be deflected are determined by the boundary between regular 2.Mott,N.Man of courage.Nature 350,31 (1991). and chaotic motion,and shaking the standing wave back and forth 3.Aitchison,I.Spin doctors.Natre 392,771-773(1998). 4.Kapitza,P.L&Dirac,P.A.M.The reflection of electrons from standing light waves.Proc.Camb.Phil. leads to the observation of Anderson localization2.Our experiment Se.29,297-300(1933). shows that the same experimental regime can be reached for 5.Boot.H.A.H.Harvie.R.B.R-S.Charged particles in a non -uniform radio-frequency field.Nature electrons.The charge of the electron affords a convenient means 180,1187(1957). 6.Altshuler,S.,Frantz,L.M.Braun s from standing light waves.Phy's.Rev. of studying the effect of external interactions on quantum chaotic Let.17,231-232(1966. behaviour. 7.Gould,P.L.Ruff,G.A.Pritchard,D.E.Diffraction of atoms by light:the near-resonant Kapitza- Increasing the laser intensity to 1015 Wcm2(which is readily Dirac effect.Phys.Rev.Lett.56,827-830(1986) 8.Bucksbaum,P.H..Schumacher,D.W.Bashkansky,M.High intensity Kapitza-Dirac effect.Phys. R.Let.6L,1182-1185(988). 9.Adams.C.S..Sigel,M.Mlynek,I.Atom optics.Plrys Rep.240,143-210(1994). 10.Fedorv,M.V.in Laser Science and Tecmology A Handbook No.13,1-77 (Harwood Academic,New York,1991). 11.Batelaan,H.The Kapitza-Dirac effect.Contemp.Phys 41,369 -381(20001. 12.Bartell.L.S.Roskos.R.R.Thor on,H.B.Reflection of clectrons by standing light waves: experimental study.Phys.Rev.166, 49 1504(1968 0.05 13.Schwartz,H.,Tourtelotte,H.A.Gaertner,W.W.Direct observation of nonlinear scattering of electrons by laser beam.Phys.Lett.19,202-203(1965) 14.Takeda,Y.Matsui,I.Electron reflection by standing wave of giant laser pulse.I.Plrys.Soc./pn 25, 1202(1968). 15.Pfeiffer,H.-Chr.Experim prufung nen beim Kapitza- Dirac-effekt.Phys.Lett.A 26,362-363 (1968). 16.Schwarz.H.The Kapitza-Dirac effect at high laser intensities.Phy .LctA43.457-478(1973). 17.Fedorov,M.V.Stimulated scattering ofelectrons by photons and adiabatic switching on hypothesis. 0p.Commumn.12,205-209(1974). 18.Rasel,E,Oberthaler,M.K.,Batelaan,H..Schmeidmayer,I.Zeilinger,A.Atom wave interferometry with diffraction gratings of light.Phys Rev.Lett.75.2633-2637(1995). 10 55 55 1i0 19.Forrey.R.C..Dalgarno,A.Schmiedmayer.I.Determining the electron forward-scattering Position (um) amplitude using electron interferometry.Phys.Rev.A 59,R942-R945(1999). 20.Meshulach,D.Silverberg.Y.Coherent quantum control of two-photon transitions by a femto- econd laser pulse.Nature 396,239-242(1998). 21.Larsen.L 1..Wendt-Larsen,I.Stapelfeldt,H.Controlling the branching ratio of photodissociation 0.10- using aligned molecules.Phrys Rev.Lett.83,1123-1126(1999). 22.Robinson,I.C.etal.Can a single-pulse standing wave induce chaos in atomic motion?Phys.Rev.Lett. 76,3304-3307(1996) 23.Steck,D.A..Milner,V.,Oskay.W.H.Raizen,M.G.Quantitative study of amplitude noise effects on 0.05 dynamical localization.Phys Rev.E62,3461-3475(2000). 24.Bucksbaum,P.H.Atoms in Strong Fields (ed.Nicolaides,C.A.)381-405 (Plenum,New York, 1990). 25.Sleator,T.Pfau,T.,Balykin.V,Carnal.O.Mlynek.I.Experimental demonstration of the optical Stern-Gerlach effect.Phys.Rev.Lett.68,1996-1999 (1992) Q 26.Batelaan,H.Gay.T.1.Schwendiman,L )Stern-Gerlach effect for electron beams.Plrys.Rev.Lett. 79,4517-4521(1997. -110 -55 0 55 110 Position (um) Acknowledgements Figure 2 Experimental data.The electron detection rate is presented as a function of We thank P.Burrow,G.Gallup and T.Gay for discussions.This work was supported by the detector position.Our data (black points)agree reasonably well with a numerical solution Research Corporation,the NRI and the NSF Experimental Program to Stimulate Competitive Research. of the Schrodinger equation(described in the text)and clearly show diffraction peaks, which is the signature of the Kapitza-Dirac effect.The bottom figure shows the electron Correspondence and requests for materials should be addressed to H.B. beam profile with the laser beams turned off. (e-mail:hbatelaan2@unl.edu). NATURE|VOL 41313 SEPTEMBER 2001www.nature.com ©2001 Macmillan Magazines Ltd 143
The observation of the Kapitza±Dirac effect opens the door to various new experiments. Because the diffracted electron beams are coherent with each other, the Kapitza±Dirac effect constitutes a coherent beam splitter. Just as for atoms, the combination of three such beam splitters can be used to construct a Mach± Zehnder interferometer18. Compared to biprism electron interferometers, this new type of electron interferometer would operate at very low electron energies and seems to be well suited to study, for example, forward electron±atom scattering phase shifts19. Instead of using three consecutive beam splitters, it may also be possible to use the coherence of the diffraction pattern itself. When I2 molecules are placed in a YAG laser beam (with experimental parameters almost identical to those used in our experiment) they will be aligned along the laser polarization axis20,21, but only at the antinodes of the standing wave. The result is that the periodically aligned I2 molecules will write a sinusoidal phase shift on the incoming electron waves. This shift will modify the diffraction pattern and could be used to monitor the I2 alignment as it is in¯uenced by, for example, molecular dissociation or ionization. Apart from the use of the Kapitza±Dirac effect as a tool, it is interesting to study in itself. It has been shown experimentally that atoms moving through a standing light wave represent an example of classical and quantum chaos. The largest angles to which atoms can be de¯ected are determined by the boundary between regular and chaotic motion22, and shaking the standing wave back and forth leads to the observation of Anderson localization23. Our experiment shows that the same experimental regime can be reached for electrons. The charge of the electron affords a convenient means of studying the effect of external interactions on quantum chaotic behaviour. Increasing the laser intensity to 1015 W cm-2 (which is readily achieved in 100-ps pulse Nd:YAG lasers24) will raise the strength of the magnetic ®eld of the laser beam to the extent that the electron spin would rotate by 1808 in such a ®eld. The question thus arises of whether the electron spin in the diffraction process could ¯ip. Although classical arguments for a circularly polarized travelling wave seem to rule out this possibility24, this question, in general, and in particular for standing waves, is to our knowledge unanswered. The atom optics counterpart of this effect is the ``optical Stern± Gerlach effect'' and has been observed25. However, this result cannot easily be extended to free electrons owing to the half-integer value of the spin. A spin ¯ip in combination with diffraction would constitute a polarizing beam splitter for free electrons or, in other words, a microscopic Stern±Gerlach magnet. We have to keep in mind that Stern±Gerlach magnets for free electrons do not exist26. By increasing the laser intensity further to 1018 W cm-2 (for a laser wavelength of 1mm), it is interesting to note that electrons are so light that relativistic speeds can be reached24. Thus the study of the interaction of free electrons with laser light can probably be extended from quantum mechanics to include spin, chaotic behaviour and relativistic mechanics. M Received 18 May; accepted 19 July 2001. 1. Davisson, C. & Germer, L. H. The scattering of electrons by a single crystal of nickel. Nature 119, 558± 560 (1927). 2. Mott, N. Man of courage. Nature 350, 31 (1991). 3. Aitchison, I. Spin doctors. Nature 392, 771±773 (1998). 4. Kapitza, P. L. & Dirac, P. A. M. The re¯ection of electrons from standing light waves. Proc. Camb. Phil. Soc. 29, 297±300 (1933). 5. Boot, H. A. H. & Harvie, R. B. R.-S. Charged particles in a non-uniform radio-frequency ®eld. Nature 180, 1187 (1957). 6. Altshuler, S., Frantz, L. M. & Braunstein, R. Re¯ection of atoms from standing light waves. Phys. Rev. Lett. 17, 231±232 (1966). 7. Gould, P. L., Ruff, G. A. & Pritchard, D. E. Diffraction of atoms by light: the near-resonant Kapitza± Dirac effect. Phys. Rev. Lett. 56, 827±830 (1986). 8. Bucksbaum, P. H., Schumacher, D. W. & Bashkansky, M. High intensity Kapitza±Dirac effect. Phys. Rev. Lett. 61, 1182±1185 (1988). 9. Adams, C. S., Sigel, M. & Mlynek, J. Atom optics. Phys. Rep. 240, 143±210 (1994). 10. Fedorov, M. V. in Laser Science and Technology; An International Handbook No. 13, 1±77 (Harwood Academic, New York, 1991). 11. Batelaan, H. The Kapitza±Dirac effect. Contemp. Phys. 41, 369±381 (2000). 12. Bartell, L. S., Roskos, R. R. & Thompson, H. B. Re¯ection of electrons by standing light waves: experimental study. Phys. Rev. 166, 1494±1504 (1968). 13. Schwartz, H., Tourtelotte, H. A. & Gaertner, W. W. Direct observation of nonlinear scattering of electrons by laser beam. Phys. Lett. 19, 202±203 (1965). 14. Takeda, Y. & Matsui, I. Electron re¯ection by standing wave of giant laser pulse. J. Phys. Soc. Jpn 25, 1202 (1968). 15. Pfeiffer, H.-Chr. Experimentelle pruÈfung der streuwahrscheinlichkeit fuÈr elektronen beim Kapitza± Dirac-effekt. Phys. Lett. A 26, 362±363 (1968). 16. Schwarz, H. The Kapitza±Dirac effect at high laser intensities. Phys. Lett. A 43, 457±478 (1973). 17. Fedorov, M. V. Stimulated scattering of electrons by photons and adiabatic switching on hypothesis. Opt. Commun. 12, 205±209 (1974). 18. Rasel, E., Oberthaler, M. K., Batelaan, H., Schmeidmayer, J. & Zeilinger, A. Atom wave interferometry with diffraction gratings of light. Phys. Rev. Lett. 75, 2633±2637 (1995). 19. Forrey, R. C., Dalgarno, A. & Schmiedmayer, J. Determining the electron forward-scattering amplitude using electron interferometry. Phys. Rev. A 59, R942±R945 (1999). 20. Meshulach, D. & Silverberg, Y. Coherent quantum control of two-photon transitions by a femtosecond laser pulse. Nature 396, 239±242 (1998). 21. Larsen, J. J., Wendt-Larsen, I. & Stapelfeldt, H. Controlling the branching ratio of photodissociation using aligned molecules. Phys. Rev. Lett. 83, 1123±1126 (1999). 22. Robinson, J. C. et al. Can a single-pulse standing wave induce chaos in atomic motion? Phys. Rev. Lett. 76, 3304±3307 (1996). 23. Steck, D. A., Milner, V., Oskay, W. H. & Raizen, M. G. Quantitative study of amplitude noise effects on dynamical localization. Phys. Rev. E 62, 3461±3475 (2000). 24. Bucksbaum, P. H. Atoms in Strong Fields (ed. Nicolaides, C. A.) 381±405 (Plenum, New York, 1990). 25. Sleator, T., Pfau, T., Balykin, V., Carnal, O. & Mlynek, J. Experimental demonstration of the optical Stern±Gerlach effect. Phys. Rev. Lett. 68, 1996±1999 (1992). 26. Batelaan, H., Gay, T. J. & Schwendiman, J. J. Stern±Gerlach effect for electron beams. Phys. Rev. Lett. 79, 4517±4521 (1997). Acknowledgements We thank P. Burrow, G. Gallup and T. Gay for discussions. This work was supported by the Research Corporation, the NRI and the NSF Experimental Program to Stimulate Competitive Research. Correspondence and requests for materials should be addressed to H.B. (e-mail: hbatelaan2@unl.edu). letters to nature NATURE | VOL 413 | 13 SEPTEMBER 2001 | www.nature.com 143 –110 –55 0 55 110 0 0.05 Position (µm) –110 –55 0 55 110 0 0.05 0.10 Count rate (per second per channel) Count rate (per second per channel) Position (µm) Figure 2 Experimental data. The electron detection rate is presented as a function of detector position. Our data (black points) agree reasonably well with a numerical solution of the SchroÈdinger equation (described in the text) and clearly show diffraction peaks, which is the signature of the Kapitza±Dirac effect. The bottom ®gure shows the electron beam pro®le with the laser beams turned off. © 2001 Macmillan Magazines Ltd