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spin - polarized electron beams are useful for a number of applications in various fields of physics @xcite . they are utilized , for example , to study magnetic properties in condensed matter systems @xcite , electron - exchange processes in atomic collisions @xcite , and the inner structure of the nucleon in deep - inelastic scattering . spin - polarized electron ( and positron ) beams can also enhance the sensitivity of high - energy physics experiments @xcite . there are various methods to generate spin - polarized electrons @xcite . at first sight , the most straightforward way would be to split an electron beam into its spin components , just as a stern - gerlach setup does with a beam of atoms . however , for charged particles , the spin - separating mechanism in an inhomogeneous static magnetic field is hindered by the influence of the lorentz force . instead , spin - polarized electrons are produced , for instance , by elastic scattering from unpolarized high-@xmath0 atomic targets . due to spin - orbit coupling , considerable degrees of polarization are attainable this way  though at moderate intensities , because the polarization maxima lie near minima of the scattering cross section . alternatively , one may exploit the spin - orbit interaction in bound states and photoionize polarized atoms . of great practical importance is photoelectron emission from gaas cathodes after selective excitation into the conduction band with circularly polarized light .    in principle , also macroscopic laser fields can affect the electron spin . while laser - electron interactions usually are dominated by the coupling of the field to the electron charge , under suitable conditions the electron spin may play a role as well . spin effects have theoretically been predicted , for instance , in strong - field photoionization of atoms @xcite and , very recently , also observed in experiment for the first time @xcite . besides , spin - flip transitions were studied theoretically in laser - assisted mott @xcite and multiphoton compton scattering @xcite . in general , laser - induced spin effects were found to be rather small , unless the field frequency or intensity is very high . to the best of our knowledge , a spin polarizer for free electrons based on laser fields has not been designed yet .    in the present paper , we describe a new method to generate spin - polarized electron beams . our method relies on coherent electron scattering from laser fields and quantum pathway interferences . in the ideal case , the setup is capable of perfectly splitting an incident electron beam into its spin components along the laser magnetic field direction . thus , the electromagnetic field configuration acts as a stern - gerlach device for free electrons . the scheme of the spin polarizer is shown in fig . 1 .     scheme of the spin - polarizing interferometric beam splitter . an incident electron beam is first coherently bragg scattered off a bichromatic laser field with frequencies @xmath1 ( red ) and @xmath2 ( blue ) , splitting the beam into two portions . afterwards , the latter are coherently superposed via scattering from monochromatic standing laser waves . due to quantum interference , the outgoing electron beam is separated into its spin components along the laser magnetic field . ] coherent electron scattering through the kapitza - dirac effect on the periodic potential generated by laser waves resembles the diffraction of light on a grating , but with the roles of light and matter interchanged @xcite . the effect has been confirmed experimentally both in the bragg @xcite and diffraction @xcite regimes . related experiments observed the kapitza - dirac effect on atoms @xcite . kapitza - dirac scattering can be sensitive to the electron spin @xcite . in contrast to the original work @xcite , this spin - dependent kapitza - dirac effect relies on a 3-photon process , where two photons are absorbed and one photon is emitted ( or vice versa ) . the interaction may be considered as arising from an @xmath3 term in the hamiltonian , in combination with a @xmath4 term ( which , in general , has to compete with the spin - preserving @xmath5 term @xcite ) . when the incident electron momentum has no component along the field polarization direction , the three - photon process is rendered possible only by the nonzero spin of the electron . the latter thus attains a crucial role which is exploited in the current study . the quantum dynamics of electrons in the nonrelativistic regime , including their spin degree of freedom , is described by pauli s equation . in the presence of an electromagnetic field , described by a vector potential @xmath6 in radiation gauge , it reads @xmath7 where @xmath8 is the electron wave function as a pauli spinor , @xmath9 the electron mass and @xmath10 its charge . @xmath11 denotes the 3-vector of pauli matricies . the monochromatic kapitza - dirac diffraction in the so - called bragg regime can be formulated by a vector potential for a standing wave in the form of @xmath12 the standing wave amplitude is given by @xmath13 , its wave number by @xmath14 and its frequency by @xmath15 . the phase parameter @xmath16 allows to adjust the positions of the field nodes along the @xmath17-axis . the polarisation is chosen in @xmath18-direction . a slowly varying envelope function @xmath19 is introduced to model switching on and off of the laser field . in contrast , the bichromatic spin - dependent kapitza - dirac effect relies on electron scattering from two counterpropagating linearly polarized waves with frequencies @xmath1 and @xmath20 . this corresponds to a vector potential @xmath21 \ , . \label{eqn : vector_potential_bi}\ ] ] here and henceforth , the incident electron momentum @xmath22 is assumed to lie in the @xmath23-@xmath17-plane , being orthogonal to the laser polarization , with @xmath24 . we shall solve eq . for the field configuration of fig . [ fig : skizze ] both by analytical methods and direct numerical integration . having only @xmath17-dependence in the potentials , pauli s equation becomes effectively one - dimensional in space . moreover , by taking a temporal average of the pauli hamiltonian for the monochromatic vector potential ( with @xmath25 ) and neglecting constant terms , one obtains a ponderomotive potential @xcite @xmath26 it represents a periodic grating from which the electron beam diffracts . similarly , by means of a magnus expansion to third order of the pauli hamiltonian with the bichromatic vector potential , one finds a spin - dependent effective ponderomotive potential @xcite @xmath27    the electron wave function @xmath28 may be expanded into momentum eigenstates . the time evolution is encoded in the pauli spinors @xmath29 $ ] which are quantized along the @xmath17-axis .    by imposing energy conservation , the bragg condition allows only the two momentum eigenmodes with @xmath30 , lying @xmath31 apart , to interact with each other . we can therefore reduce the ansatz to @xmath32 due to the interaction with the laser fields , a characteristic rabi oscillation dynamics will occur between the two momentum modes . in the relevant four - dimensional subspace , the potentials are represented by the block matricies @xmath33 with the corresponding rabi frequencies @xmath34 and @xmath35 . since all the involved momentum eigenstates share the same kinetic energy , we can remove the latter by a gauge transformation . this corresponds to switching to the interaction picture . the interaction times with the laser potentials are chosen to yield a quarter ( or half ) rabi cycle in each stage of fig . [ fig : skizze ] . with @xmath36 , the corresponding time evolution operators thus read @xmath37 for a @xmath38-pulse with the potential @xmath39 , @xmath40 for a @xmath41-pulse and @xmath42 for a @xmath38-pulse with @xmath43 . combining these three interactions in the order sketched by fig . [ fig : skizze ] results in the time evolution operator @xmath44 for the whole process . the desired spin filtering effect is achieved when the relative phase is set to @xmath45 . in this case , the total time evolution operator becomes @xmath46 when acting on initially @xmath23-polarized electron states @xmath47 with @xmath24 , we obtain @xmath48 thus , incident electrons with spin - up along the @xmath23-axis go through the beam splitter with unchanged momentum , whereas spin - down electrons are scattered to the mirrored momentum state @xmath49 . we can also consider an incident electron with arbitrary spin polarisation , described by a density matrix with vector @xmath50 in the bloch sphere . then we find @xmath51 where @xmath52 is the projection of @xmath53 along the @xmath23-axis onto the @xmath18-@xmath17-plane . from the upper left and the lower right component we can read off , that the incident electron is deflected with exactly the same probability @xmath54 as having its spin in negative @xmath23-direction . also the spin for each separate momentum is in a correspondingly @xmath23-polarized pure state @xmath55 . for an unpolarized ensemble of incident electrons ( @xmath56 ) one obtains @xmath57 thus , the outgoing electrons are divided into one half with @xmath58 and spin - up ( along the @xmath23-axis ) , and another half with @xmath59 and spin - down . eq .   may be interpreted that the initial spin uncertainty is translated into momentum uncertainty , whereas the certainty of momentum is transformed into certainty of a specific spin state for each given momentum state . in order to corroborate our analytical considerations , we have performed a real - space simulation of an electron wave packet travelling through the electromagnetic field configuration of fig . [ fig : skizze ] . initially , the wave packet has a central longitudinal momentum of @xmath60 and a spatial width of @xmath61 ; its spin is oriented along the positive @xmath17-axis ( @xmath62 ) . note that the spin expectation value along the @xmath23-axis thus vanishes . figures [ fig : wavepacket_z ] and [ fig : wavepacket_y ] show the time evolution of the wave packet , involving different spin projections . in the first interaction , a vector potential [ see eq .  ] with @xmath63 and @xmath64 is switched on for a duration of @xmath65 . accordingly , the wave packet is partly reflected by the bichromatic kapitza - dirac effect forcing a selective spin flip ( see lower panel in fig .  [ fig : wavepacket_z ] ) on the scattered part . at the second interaction , both partial beams are reflected by a @xmath41-pulse of the type with @xmath66 , @xmath64 and duration @xmath67 without altering their spin state . after closing the diamond shape , a third kapitza - dirac diffraction ( a @xmath68-pulse of the same kind as before ) acts as a spin - insensitive beam splitter to coherently mix the two partial beams in two outgoing momentum channels .    , width @xmath61 and spin along the positive @xmath17-axis . the projections on @xmath62 in the upper and @xmath69 in the lower panel are shown . at the bottom , the interaction periods with the lasers are marked . [ fig : wavepacket_z ] ]    it is instructive to consider the same time evolution projected on the @xmath70-eigenstates . one can see in fig . [ fig : wavepacket_y ] , that the two output channels of fig . [ fig : wavepacket_z ] are subject to constructive or destructive interference depending on the spin content . because the corresponding parts of the wave function are superposed coherently , each momentum state is prepared in a specific @xmath70 spin state , in perfect agreement with eq .  .    , but with the wave function projected on the @xmath70 eigenstates @xmath71 in the upper panel , and @xmath72 in the lower panel . [ fig : wavepacket_y ] ] the results presented above have shown that the laser field configuration of fig .  [ fig : skizze ] acts as a stern - gerlach magnet for free electrons . this is an important finding because it fills a gap in the literature on generation of spin - polarized electron beams . a stern - gerlach - like device for free electrons based on macroscopic electromagnetic fields whose realizability sometimes is even questioned at all  provided by laser waves has so far not been known .    in view of an experimental implementation , we note that the single - color kapitza - dirac effect in the bragg regime , as required in the second and third stage of fig . [ fig : skizze ] , has successfully been observed @xcite . wavelength and @xmath73 duration , focused to @xmath74 height and @xmath75 width , resulting in an intensity of @xmath76 . ] the most demanding step is the spin - dependent bichromatic kapitza - dirac scattering in the first stage . nevertheless , despite the rather weak spin interaction , it can be accomplished utilizing nonrelativistic laser intensities .    in the example of figs . [ fig : wavepacket_z ] and [ fig : wavepacket_y ] , the xuv field intensities amount to @xmath77 and @xmath78 , corresponding to small values of the relativistic parameter @xmath79 . the bichromatic rabi frequency amounts to @xmath80 ( @xmath81 ) , implying a necessary interaction time of @xmath82 . the latter translates into a laser beam width of @xmath83 , where @xmath84 denotes the transverse component of the electron velocity , and also determines the minimum laser pulse duration @xmath85 . an electron energy of 30ev is assumed ( @xmath86 ) . if the laser beams have spherical cross section with @xmath87 , we find that the pulse energy is on the order of 50mj . thus , the required laser parameters for the first scattering step are challenging , but lie rather close to the performance values of present x - ray free - electron lasers such as the lcls ( stanford , california ) . currently , xuv pulses with @xmath88 duration , up to @xmath89 intensity and few mj pulse energy are available there @xcite . besides , improved x - ray focussing techniques to reach sub-@xmath90 beam waists have been proposed @xcite . in light of this , while being demanding , the first stage in fig . [ fig : skizze ] seems to be within experimental reach . the same conclusion was drawn in @xcite considering laser fields in the optical domain . the requirements for the second and third scattering stage are much more relaxed . here the required intensity in the above example is @xmath91 . it could be obtained by outcoupling a small portion from the main laser beam . we point out that laser fields of circular polarization might also appear attractive to coherently control the electron spin . in this case , however , the spin - flip transitions still compete with spin - preserving electron scattering because the @xmath5 interaction term can not be avoided . as a result , the desired spin flips are suppressed .    before passing on to the conclusion , we wish to mention that a spin - polarizing wien filter for electron vortex beams was put forward in @xcite . it relies on multipolar electric and magnetic field geometries with cylindrical symmetry and a conversion of orbital to spin angular momentum . besides , it is interesting to note that a spin - insensitive , but otherwise similar laser - based interferometric setup as in fig . [ fig : skizze ] is applied to atomic beams by the gain project for gravitational wave studies @xcite . with regard to electron beams , a spin - insensitive interferometric beam splitter exploiting the kapitza - dirac effect has been proposed in @xcite . a new way of generating spin - polarized electron beams by interaction with intense laser fields has been put forward . our scheme relies on coherent electron scattering through the kapitza - dirac effect in an interferometric setup of laser beams . it acts as a stern - gerlach device for free electrons , splitting the incident beam into spin components along the laser magnetic field direction . free - electron laser facilities appear particularly promising for an experimental test of our predictions . since the time scale for the spin - sensitive electron interaction is set by a rather slow bichromatic rabi dynamics , low electron energies are advantageous to limit the required spatial extent of the laser fields . the following strengths of the setup are noteworthy : ( i ) after an electron beam has been sent through , both partial beams are spin - polarized and separatly available for further use . ( ii ) it may equally well serve as spin filter and spin analyzer . ( iii ) by proper synchronization , the scheme offers a novel method of creating and characterizing ultrashort ( e.g. , femtosecond ) spin - polarized electron pulses @xcite . we finally note that analogies of the kapitza - dirac effect exist in other areas of physics , such as electron scattering from travelling waves in dielectric media @xcite or from optical near - fields in transmission microscopes @xcite . it is an interesting question whether the spin - polarizing electron beam splitter presented here can be transferred to these related phenomena as well . we thank s. ahrens for stimulating discussions at the onset of this project . this study was supported by sfb tr18 of the german research foundation ( dfg ) under project no . b11 .
a spin - polarizing electron beam splitter is described which relies on an arrangement of linearly polarized laser waves of nonrelativistic intensity . 
 an incident electron beam is first coherently scattered off a bichromatic laser field , splitting the beam into two portions , with electron spin and momentum being entangled . 
 afterwards , the partial beams are coherently superposed in an interferometric setup formed by standing laser waves . as a result 
 , the outgoing electron beam is separated into its spin components along the laser magnetic field , which is shown by both analytical and numerical solutions of pauli s equation . 
 the proposed laser field configuration thus exerts the same effect on free electrons like an ordinary stern - gerlach magnet does on atoms .