Relativistic Properties of the Kapitza-Dirac Effect

The Kapitza-Dirac effect in the relativistic regime

1 1, 1 1, 2, Sven Ahrens, Heiko Bauke, ∗ Christoph H. Keitel, and Carsten Müller † 1Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany 2Institut für Theoretische Physik I, Heinrich-Heine-Universität Düsseldorf, Universitätsstraße 1, 40225 Düsseldorf, Germany (Dated: May 24, 2013) A relativistic description of the Kapitza-Dirac effect in the so-called Bragg regime with two and three interacting photons is presented by investigating both numerical and perturbative solutions of the Dirac equation in momentum space. We demonstrate that spin-flips can be observed in the two-photon and the three-photon Kapitza-Dirac effect for certain parameters. During the interaction with the laser field the electron’s spin is rotated and we give explicit expressions for the rotation axis and the rotation angle. The off-resonant Kapitza-Dirac effect, that is when the Bragg condition is not exactly fulfilled, is described by a generalized Rabi theory. We also analyze the in-field quantum dynamics as obtained from the numerical solution of the Dirac equation.

PACS numbers: 03.65.Pm, 42.50.Ct, 42.55.Vc, 03.75.-b

1. Introduction the Kapitza-Dirac effect, but found a vanishing small spin-flip probability in the investigated parameter regime by simulating the Bargmann-Michel-Telegdi equations [31]. Another deriva- ff The di raction of electrons at a standing wave of light is re- tion which has been carried out by Rosenberg [32] solved the ff ferred to as the Kapitza-Dirac e ect [1, 2]. It is the counterpart Pauli equation perturbatively in a second quantized field in the ff process of the usual di raction of light at a material grating. diffraction regime and found only tiny spin effects, too. The ff The observation of the Kapitza-Dirac e ect seemed to be fea- impact of the electron’s spin has also been investigated with sible with the advent of the laser [3, 4] but its experimental respect to free-electron motion [33], bound-electron dynam- realization has been refuted shortly after the claim of detec- ics [34], atomic photoionization [35], and Compton and Mott ff tion [5–7]. The first observation of the Kapitza-Dirac e ect in [36] scattering in strong plane-wave laser fields. In addition the so-called Bragg regime was achieved by employing atoms callapse and revival spin dynamics has been put forward in ff [8, 9] in 1986 [10] and two years later in the so-called di rac- strongly laser driven electrons [37] and notable spin signatures ff tion regime [11]. The realization of the Kapitza-Dirac e ect have been found in laser induced ionization [38]. Our recent ff with electrons was achieved in 1988 [12] in the di raction publication on the Kapitza-Dirac effect [39] solves the Dirac regime. The scattering of electrons in the Bragg regime has equation numerically and perturbatively in the Bragg regime been demonstrated in a precise and sophisticated experimental and demonstrates pronounced spin dynamics in the Kapitza- setup in 2001 [13, 14]. Among all experiments this comes clos- Dirac effect involving three photons. In the present paper, ff est to the di raction process as proposed originally by Kapitza we elaborate our theory in detail and discuss the possibility and Dirac [1]. of spin dynamics for Kapitza-Dirac scattering with only two The latter experiment has been performed at nonrelativis- interacting photons. tic laser intensities with a laser field parameter [15] ξ0 = 6 eE0/(mcω) = 5 10− (with e denoting the electron’s charge · − This article is organized as follows. In section 2, we specify and m its mass, laser electric field amplitude E0 and angular frequency ω, and the speed of light c), with nonrelativistic the Bragg and the diffraction regime and characterize the inter- electron momenta p = 0.04 mc, and at nonrelativistic pho- action in the Bragg regime employing a classical picture. The | | 6 2 quantum equations of motions (namely the Pauli- and Dirac ton energies = 5 10− mc , raising the question [13] how the Kapitza-DiracE e·ffect might be modified in the relativistic equation) are considered in section 3 where it is shown that regime. Higher intensities [16–18] and shorter wavelengths for the setup of the Kapitza-Dirac effect they take a particular [19–21] are indeed available nowadays, demanding for a rela- simple form in momentum space. Based on these equations, tivistic theory of Kapitza-Dirac scattering within the framework we study the Kapitza-Dirac effect with two interacting photons of the Dirac theory which also accounts for the electron’s spin in section 4 numerically and analytically and demonstrate the arXiv:1305.5507v1 [physics.atom-ph] 23 May 2013 degree of freedom. Beside considerations of the Kapitza-Dirac occurrence of spin-flips at relativistic electron momenta. We effect with adiabatic pulse turn on [22], two electrons [23, 24], show, that the diffraction probability is independent of the spin perturbative solutions [25–28], relativistic investigations have orientation of the incident electron beam. The spin orientation been carried out in [29, 30] based on the Klein-Gordon equa- of the diffracted electrons is rotated as a result of the interaction and therefore neglecting the spin. Freimund and Batelaan tion with the laser field. We also discuss the resonance peak raised the question, whether the electron spin is affected in structure of the diffraction with a generalized Rabi theory. In section 5, we investigate the Kapitza-Dirac effect with three interacting photons in close analogy to the two-photon case. We analyze the in-field quantum dynamics of the Kapitza-Dirac ef- ∗ [email protected] fect and we discuss the tilt of the axis, about which the electron † [email protected] spin is rotated. 2

two-photon 2. Semi-classical considerations Kapitza-Dirac effect three-photon Depending on the parameters of the laser and the incident Kapitza-Dirac effect electron, the scattering dynamics of the Kapitza-Dirac effect happens in different regimes. Phenomenologically one refers E to the Bragg regime if the electron is scattered into a single B diffraction order and one refers to the diffraction regime if many diffraction orders can be reached. Employing an argument k based on the time-energy uncertainty relation [14, 40] one can show that if the interaction time of the electron with the ϑ0 ϑ laser is short, the electron is diffracted into several diffraction orders. Long interaction times, however, permit dynamics in the Bragg regime [25]. In the Bragg regime the electron can be λ/2 diffracted only if it fulfills the classical energy- and momentum- conservation. This article focuses on the Kapitza-Dirac effect Figure 1: (Color online.) Setup of the two-photon and the three- in the Bragg regime. photon Kapitza-Dirac effect. If the electron is diffracted, it may The electron either absorbs or emits photons from or into pick up two photon momenta or three photon momenta, respectively, the counter-propagating laser beams. We denote the number from the laser field. Directions of the electron beams are indicated by dotted lines (two-photon Kapitza-Dirac effect) and dashed lines of absorbed photons from the left (right) travelling laser beam (three-photon Kapitza-Dirac effect). The gray panels emblematize with nl (nr). Negative values of nl or nr correspond to photon screens, which display the intensity of the scattered and unscattered emission of the electron into the left or the right laser beam. electron beam (colored bars), depending on the electron momentum Following this notion, classical momentum conservation re- in laser propagation direction. In both Kapitza-Dirac effects spin-flips quires may occur, indicated by the light red bar of the diffraction pattern as compared to the dark blue bars of the unflipped electron beam p = p + n k n k . (1) out in r − l intensity. According to the considerations of sections 2 and 4, the electron enters the laser almost perpendicular to the laser’s propaga- Note that, Gaussian units are used in this article and we set tion direction for the two-photon Kapitza-Dirac effect with spin-flip ~ = 1. Furthermore, energy conservation implies dynamics, implying ϑ 90°. For the three-photon Kapitza-Dirac ≈ effect, however, the energy-momentum conservation constraint (6) (p ) = (p ) + n ck + n ck , (2) E out E in r l results in relativistic momenta of the electron in laser propagation direction such that typically ϑ0 90°. with pin and pout denoting the electron’s momentum before and after interaction with the laser, the electron’s kinetic energy (p), and the photon momentum k. For convenience, we ff separateE electron’s momentum in components in laser prop- the Kapitza-Dirac e ect in the Bragg regime requires that at least one photon is absorbed from one laser beam and at least agation direction pk, laser polarization direction pE and the direction of the laser magnetic field p (see Fig. 1). Applying one photon is emitted into the counterpropagating laser beam. B + = the nonrelativistic energy-momentum relation For nr nl 0 the electron’s energy (2) is conserved and, therefore, the scattering is elastic, otherwise it is inelastic. p2 Equation (4) as well as Eq. (6) yield pnr/r = pnr/r for nr(p) = , (3) k,out − k,in E 2m elastic electron scattering corresponding to “The reflection of electrons from standing light waves” as it has been proposed the solution of Eq. (2) with respect to p yields k,in by Kapitza and Dirac [1]. Energy-momentum conservation nr nr nl nr + nl can be illustrated in an energy-momentum diagram as shown pk,in = − k + mc , (4) − 2 nr nl in Fig. 2a for the elastic two-photon Kapitza-Dirac effect. − In the case of the three-photon Kapitza-Dirac effect, two whereas Eq. (2) with the relativistic energy-momentum relation laser photons are absorbed (nr = 2) and one is emitted q (nl = 1). The energy-momentum diagram of this process r(p) = m2c4 + c2 p2 (5) − E is sketched in Fig. 2b. For such an inelastic process, energy is transferred from the laser field to the electron. This energy yields transfer is indicated by a dashed line in Fig. 2b. Approaching the limit k 0, the dashed line in Fig. 2b is shifted down- nr nl pr = − k wards while→ preserving its slope until it becomes a tangent of k,in − 2 s the energy hyperbola (5). Consequently, energy-momentum + m2c2 + (pr )2 + (pr )2 conservation requires a non-vanishing initial momentum of the nr nl nr nl 2 B,in E,in + − k . (6) electron in laser propagation direction even for small photon nr nl 2 − nrnl | − | momenta. Furthermore, energy conservation (2) enforces at Note, that the solution (6) is real-valued and finite only if nr least one of the momenta k, pk, pE, or pB to be relativistic, and nl are non-zero and have opposite signs. This means that i. e., of the order of or larger than mc. Consequently, no non- 3

E(p) a) 3. Quantum dynamics in momentum mc2 3 space

2 In order to study spin effects and the quantum dynamics of the two-photon and the three-photon Kapitza-Dirac effect in a relativistic setting we will solve the time-dependent Dirac equation. For two-photon interactions we will also utilize the 1 nonrelativistic Pauli equation given that relativistic parameters are not mandatory in this case. As we will show in this section, pk the Pauli equation and the Dirac equation for the setup of the -1-2 0 1 2 mc Kapitza-Dirac effect can be reduced to a system of ordinary differential equations by transforming them into momentum E(p) b) space mc2 3 3.1. Laser setup 2 For the Kapitza-Dirac effect we consider two counterpropagating linearly polarized lasers of equal intensity and angular frequency ω. The vector potential of this laser setup is given 1 by

pk A(x, t) = Aˆ(t) cos(k x) sin(ωt) , (8) · -1-2 0 1 2 mc where we have introduced the temporal envelope function Figure 2: Sketch of the energy- and momentum conservation in  2 πt the Kapitza-Dirac effect. The electron must reside on its relativis- sin if 0 t ∆T,  2∆T ≤ ≤ tic energy-momentum relation (5) (solid hyperbola) before as well  ˆ ˆ 1 if ∆T t T ∆T, as after the interaction and the laser can only transfer energy and A(t) = Amax  2 π(T t) ≤ ≤ − (9) · sin − if T ∆T t T, momentum quanta, which fulfill ∆ = ck (diagonal arrows). In the  2∆T − ≤ ≤ E  two-photon Kapitza-Dirac effect (part a), one photon is absorbed and 0 else, one photon is emitted, whereas in the three-photon Kapitza-Dirac effect (part b) two photons are absorbed and one photon is emitted. In which allows for a smooth turn-on and turn-off of the laser the latter process the energy of the electron changes during diffraction field. The variables T and ∆T denote the total interaction time as indicated by the nonhorizontal dashed line. and the time of turn-on and turn-off. After turn-on and before turn-off, the electric and magnetic amplitudes of the oscillating electromagnetic fields are given by relativistic limit of (6) exists in this case. For example, the ˆ ˆ momenta (4) and (6) do not converge even for small laser pho- E = kAmax , (10) ˆ ˆ ton momenta k. Rather we find with nr = 2 and nl = 1 the B = k Amax . (11) differing limiting values − − × For convenience, we will choose our coordinate system such nr mc that the orthogonal vectors k, Bˆ , and Eˆ point along the x, y, lim pk,in = (7a) k 0 → 3 and z direction. In the following all vector quantities will be projected in the directions of k, Eˆ, and Bˆ as indicated by the and indices k, E, and B, respectively. For simplicity, we omit the s index E for the vector potential, because the vector potential 2 2 2 2 r m c + pE + pB always points in the electric field direction, that is Aˆ(t) = AÊ(t). lim pk,in = . (7b) k 0 → 8

r For pk,in to be small compared to mc the number of interact- 3.2. The Pauli equation ing photons has to go to inﬁnity such that nr nl 1 and n + n 1. In this case the slope of the energy-momentum| − | | r l| ≈ The Pauli equation that governs the quantum motion of a spin- transfer (dashed lines in Fig. 2) goes to zero. Therefore, the half particle of mass m and charge q is given by touching point of the corresponding tangent that results in the limit k 0 lies at small electron momenta in laser propagation 1 q 2 q → iψ˙(x, t) = i A(x, t) ψ(x, t) σ B(x, t) ψ(x, t) direction for Kapitza-Dirac scattering with a high number of 2m − ∇ − c − 2mc · interacting photons. (12) 4

T T ζ with the Pauli matrices σ = (σ1, σ2, σ3) = (σk, σB, σE) The basis functions ψn± (x) are simultaneous eigenfunctions of and B(x, t) = A(x, t). Taking advantage of the sinusoidal the free time-independent Dirac equation with energy eigen- spatial periodicity∇ × of the vector potential A(x, t), we expand value deﬁned as ±En the wavefunction q X 2 2 2 ζ ζ n = mc + c pn (20) ψ(x, t) = cn(t)ψn(x) (13) E n,ζ and eigenfunctions of the momentum operator with momen- into momentum eigenfunctions of the free Pauli Hamiltonian tum eigenvalue pn. Furthermore, ψn±↑(x) and ψn±↓(x) are eigen- r functions of the Foldy-Wouthuysen spin operator along the Eˆ k ζ ζ ipn x / / ψn(x) = χ e · (14) direction with eigenvalues 1 2 and 1 2 [42–45]. 2π Inserting the ansatz (13) with the− basis elements (19) in with ζ , , the two spinor basis functions the Dirac equation (12) and projecting with these from the ∈ {↑ ↓} ! ! left-hand side yields the Dirac equation in momentum space 1 0 χ↑ = , χ↓ = , (15) 0 1 q sin(ωt) i˙cn(t) = nβcn(t) Ln,n 1cn 1(t) + Ln,n+1cn+1(t) , E − 2 − − and the momentum pn = p + nk. Inserting the ansatz (13) (21) into the Pauli equation (12) and projecting onto the basis ele- with the vectors ments (14) from the left-hand side yields the Pauli equation in + + T momentum space cn(t) = (cn ↑(t), cn ↓(t), cn−↑(t), cn−↓(t)) (22)

i˙cn(t) = and the coupling matrices Ln,n0 whose elements are defined as 2 2 2 (p + nk) q Aˆ(t) γ,γ0 γ γ0 2 † ˆ cn(t) + sin (ωt) [cn 2(t) + 2cn(t) + cn+2(t)] Ln,n = un α A(t) un . (23) 2m 8mc2 − 0 · 0 q sin(ωt) X h i 2Aˆ(t)p 1 iδnkˆ Aˆ(t)σ c +δ (t) , − 4mc E − × B n n δn 1, 1 3.4. Numerical procedure ∈{ − } (16) ζ T The absolute square values of the expansion coefficients c (t) where the vectors cn(t) = (cn↑(t), cn↓(t)) and the 2 2 identity n matrix 1 have been introduced. × in (13) represent the probability of finding the electron in a particular free-particle quantum state. The state that is repre- ζ sented by cn(t) has definite spin that is encoded in the index 3.3. The Dirac equation ζ and also has a definite momentum pn = p + nk. Thus, the index n counts the number of laser photon momenta relative to the reference momentum p. For convenience, we use the term The Dirac equation is a relativistic generalization of the non- “mode n” for these different electron momenta. relativistic Pauli equation. It is given by In numerical simulations, we start from an initial quantum q state with definite momentum p and spin-up polarization. This iψ˙(x, t) = cα i A(x, t) ψ(x, t) + mc2β ψ(x, t) (17) · − ∇ − c means T  with α = (α1, α2, α3) and β denoting the Dirac matrices [41].  γ 1 if n = 0 and γ = + or γ = , We transform the Dirac equation (17) into momentum space cn(0) = ↑ ↑ (24) 0 else. in close analogy to the momentum space transformation of the Pauli equation (16). For this purpose, the basis elements (14) The case γ = + applies for the Dirac equation, while γ = of the wavefunction (13) are replaced by is for the Pauli↑ equation. The initial electron momentum r ↑p, the laser intensity and the laser frequency are chosen to γ k γ ip x ψ (x) = u e n· . (18) meet the nonrelativistic Bragg condition (4) or its relativistic n 2π n generalization (6) depending on whether the Pauli equation γ The bi-spinors un with γ + , , + , are explicitly or the Dirac equation is solved numerically. The numerical given by ∈ { ↑ − ↑ ↓ − ↓} solution of the differential Eqs. (16) and (21) is obtained by s employing a Crank-Nicholson scheme [46]. Equations (16) 2 ζ ! +ζ n + mc χ and (21) couple an infinite number of modes. In numerical un = E σ cpn ζ (19a) · simulations, however, these systems are truncated to a finite 2 n +mc2 χ E En number of modes, n n n with n large enough − max ≤ ≤ max max and such that physical results are independent of nmax. The number s of included modes depends on the laser parameters and ranges σ cpn ζ ! 2 · ζ n + mc +mc2 χ typically from one dozen to several dozens. The duration u− = E − En . (19b) n 2 χζ of turn-on and turn-off phases ∆T is ten laser periods for all En 5

1.0 qualitatively and quantitatively the same results. The quan- Pauli tum dynamics exhibits the well-known Rabi oscillations of 2 0.8 c↑ | 0| the diﬀraction probability [40] from mode 0 with momentum 2 0.6 c↑ p = kek + keE to mode 2 with momentum p = +kek + keE, in | 2| − Dirac the form 0.4 ! probability + 2 ΩRT c0 ↑ + 2 2 0.2 | | c0 ↑(T) = cos , (25a) + 2 | | 2 c ↑ | 2 | ! 0.0 ΩRT 0.0 0.5 1.0 1.5 2.0 2.5 3.0 + 2 2 c2 ↑(T) = sin . (25b) T (103 laser periods) | | 2

Here, ΩR denotes the Rabi frequency. The occupation prob- Figure 3: Occupation probabilities as a function of the total interac- ability of all other modes is vanishingly small, in particular, tion time T for the two-photon Kapitza-Dirac effect calculated by em- + 2 6 c ↓(T) < 10 . This means no spin-flip occurs during two- ploying the Dirac equation (crosses and dots) (21) and the Pauli equa- | 2 | − tion (16) (solid and dashed lines). The electron enters the laser field photon Kapitza-Dirac scattering. From a naive point of view 2 with a momentum component of p = 12.5 keV/c = 2.4 10− mc the vanishing spin-flip probability might be surprising, be- E − · along the laser electric field direction and interacts with the stand- cause one might expect a precession of the electron spin in ing laser field of peak intensity 5 1021 W/cm2 (corresponding to the magnetic field of the external laser field. The question of 3 2 · eAˆmax = 8.6 10− mc ) for each laser beam and a photon energy of electron spin precession has already been investigated based · 12.5 keV (corresponding to a laser wave length of λ = 0.1 nm). The on nonrelativistic classical equations of motion [31] but no probability that the electron is diffracted from mode 0 to mode 2 significant spin effects could be found. In section 4.4, however, oscillates in Rabi cycles as a function of the total interaction time we will show that spin-flips are possible in the two-photon T. The laser field is modulated via the envelope function (9) with a Kapitza-Dirac effect for certain relativistic parameter settings. turn-on-off time of ten laser periods. simulations presented in this article unless another turn-on-off 4.2. Perturbation theory time is indicated. Note that our approach to solve the Dirac equation is tai- In the following we will complement our numerical findings lored for interactions with monochromatic laser fields. More with analytical results obtained via time-dependent perturba- general approaches include solving the Dirac equation via a tion theory. Time-dependent perturbation theory for the Dirac Fourier transform split-operator method [47], in particular by equation (21) will allow us to calculate analytical expressions employing geometric algebra [48] and by making use of a for the Rabi frequency and to derive conditions that permit graphics processing unit [49]. Other methods employ spherical spin-flip dynamics in the two-photon Kapitza-Dirac effect. The harmonics as basis functions and Runge-Kutta integration [50] perturbative solution also allows for deduction of a rotation of or are based on the method of characteristics [51]. the electron spin during diffraction. As the initial condition is given by (24) and the electron momentum is changed by two photon momenta our aim is to approximate the time evolution , 4. Two-photon Kapitza-Dirac effect operator U2,0(t 0) that maps c0(0) to c2(t), viz. c2(t) = U2,0(t, 0)c0(0) . (26) 4.1. Numerical results Since the Dirac equation (21) couples next neighboring modes only, the lowest non-vanishing contribution to U2,0(t, 0) is of second order in time-dependent perturbation theory. The In the electron scattering dynamics as described by Kapitza general second order propagator for a time-dependent Hamil- and Dirac [1] one photon is absorbed from the laser field and tonian H(t) reads [52] one is emitted into the laser field, nr = 1 and nl = 1 in our notation. For realizing this effect the parameters of the− laser as Und(t, 0) = well as of the incident electron have to be chosen such that the Z Z 1 t t2 quantum dynamics is in the Bragg regime [40]. For nr + nl = 0 , , , . 2 dt2 dt1U0(t t2)V(t2)U0(t2 t1)V(t1)U0(t1 0) (27) the nonrelativistic Bragg condition (4) as well as the relativistic i 0 0 Bragg condition (6) require that the electron enters the laser The symbol U0(t, 0) denotes the free propagator U0(t, 0) = beam with pk = k. For the numerical simulations shown in e iH0t , with the time-independent field-free Hamiltonian − 2 − Fig. 3 we choose p = kek + keE with k = 2.4 10− mc and 3 2 − ·   eAˆ = 8.6 10− mc . .. max ·  .  Figure 3 shows the quantum state after turn-off of the laser    H0;1,1  for the two-photon Kapitza-Dirac effect for different total in-   H0 =  H0;0,0  (28) teraction times T as calculated by solving the Dirac and the    H0; 1, 1  Pauli equation numerically. As all parameters are in the nonrel-  − −   .  ativistic regime the Dirac equation and the Pauli equation give .. 6 with H = βδ and the time-dependent interaction In order to ease notations, it will be useful to split the 4 4 0;a,b Ea a,b × Hamiltonian V(t) = H(t) H0. For the Dirac equation in mo- matrices Ln,n0 and Und;2,0(t, 0) into blocks of 2 2 matrices, mentum space (21), the free− propagator reads explicitly viz. ×

1 !  ++ +   a ,b a ,b  i aβt exp( i at) 0 L L − L ↑ ↑ L ↑ ↓ U , (t, 0) = e− E δ , = δ , .  n,n0 n,n0  ab  n,n0 n,n0  0;a b a b − E 1 a b Ln,n =   with L =   (32) 0 exp(i at) 0  +  n,n0  a ,b a ,b  E Ln−,n Ln−−,n L ↓ ↑ L ↓ ↓ (29) 0 0 n,n0 n,n0 The corresponding interaction Hamiltonian reads and q sin(ωt)  ++ +  Va,b(t) = La,a 1δa,b+1 + La,a+1δa,b 1 . (30) U (t, 0) U (t, 0) − 2 − −  nd;2,0 nd;2− ,0  Und;2,0(t, 0) =  +  . (33) U− (t, 0) U−− (t, 0) Inserting these expressions into (27) yields nd;2,0 nd;2,0 ab Z t Z t Explicit expressions for the matrices L are given in the ap- 2 ω n,n0 i 2β(t t2) q sin( t2) ++ Und;2,0(t, 0) = dt2 dt1e− E − L2,1 pendix A. With these deﬁnitions the sub-propagator Und;2,0(t, 0) − 0 0 2 in the space of positive energy free particle states reads for ω i 1β(t2 t1) q sin( t1) i 0βt1 times after turn-on phase and before turn-oﬀ phase (∆T = 0 e− E − L , e− E . (31) · 2 1 0 and 0 < t < T)

2 Z t Z t2 2 Z t Z t2 ++ q ++ ++ + q + + Und;2,0(t, 0) = L2,1 L1,0 dt2 dt1 sin(ωt2) sin(ωt1)v (t, t2, t1) L2,−1 L1−,0 dt2 dt1 sin(ωt2) sin(ωt1)v−(t, t2, t1) , − 4 0 0 − 4 0 0 (34) where we have introduced the two complex phases h i v+(t, t , t ) = exp i t + ∆ ++t + ∆ ++t , (35) 2 1 − E2 E1,2 2 E0,1 1 h + + i v−(t, t , t ) = exp i t + ∆ − t + ∆ −t . (36) 2 1 − E2 E1,2 2 E0,1 1 Here, ∆ ab is an abbreviation for the energy difference ∆ ab = sign(a) sign(b) , where the signum of the upper indices is n,n0 n,n0 n n0 sign(+)E= 1 and sign( ) = 1. Performing the first integralE in (34) we findE − E − −

Z t Z t2 + dt2 dt1 sin(ωt2) sin(ωt1)v (t, t2, t1) = 0 0 Z t " 1 i ++ ++ i ++ ++ i 2t i(∆ 0,2+2ω)t2 i(∆ 1,2+ω)t2 i∆ 0,2t2 i(∆ 1,2+ω)t2 e− E dt2 ++ e− E e− E ++ e− E e− E − 4 0 ∆ 0,1 + ω − − ∆ 0,1 ω − E E − # i i∆ ++t i(∆ ++ ω)t i i(∆ ++ 2ω)t i(∆ ++ ω)t e− E0,2 2 e− E1,2− 2 + e− E0,2− 2 e− E1,2− 2 . (37) − ∆ ++ + ω − ∆ ++ ω − E0,1 E0,1 −

The integral (37) may show linear or oscillating behavior de- in t pending on the laser parameters and the initial electron momentum p. For transitions from mode 0 to mode 2 the absolute value of the coeﬃcient c2(t) must grow linearly in t within 2 ++ iq t i t + ++ ++ + + perturbation theory. The expression (37) and therefore the U (t, 0) = e− E2 l L L + l−L −L− (38) nd;2,0 − 16 2,1 1,0 2,1 1,0 propagator (34) features terms growing linearly in t if and only if at least one the exponents on the right hand side of (37) is ++ zero. Taking into account ∆ n,n < n n0 ω this leads us to | E 0 | ++| − | with the coeﬃcients the unique resonance condition ∆ 0,2 = 0 which is equivalent to the classical energy momentumE conservation conditions (1) and (2) with p = p and n = 1 and n = 1. The second in r l − integral in (34) leads to the same resonance condition. Thus, + 1 1 l = + , (39a) on resonance the propagator U++ (t, 0) reads in leading order ∆ ++ ω ∆ ++ + ω nd;2,0 E0,1 − E0,1 1 1 l− = + . (39b) ∆ + ω ∆ + + ω E0,−1 − E0,−1 7

4.3. The relativistic Rabi frequency 2.0 1.0 . Employing the explicit form of the matrices Lab in (A3) and 1.5 0 9 n,n0 the propagator (38) we calculate the diﬀraction probability 0.8 + 2 + 2 + + + T 1.0 c ↑(t) + c ↓(t) which equals with c (t) = (c ↑(t), c ↓(t)) | 2 | | 2 | n n n 0.7 0.5 .

0 6 2 ) + 2 + 2 + + , c ↑(t) + c ↓(t) = c (t)†c (t) R mc

2 2 2 2 Ω (

| | | | 0.0 0.5 / / R

+ ++ ++ + E p = c (0)†U (t, 0)†U (t, 0)c (0) . (40) Ω 2 nd;2,0 nd;2,0 2 . 0.5 0 4 ++ − Expanding Und;2,0(t, 0) from Eq. (38) in the momenta k, pB, 0.3 and p yields 1.0 E − 0.2   1.5 2 2 i 2t 2 2 − 0.1 iq t Aˆ e− E  p 5p  U++ (t, 0) = max m2c2 B E  1 nd;2,0 − 16 mc2 m2c2  − 2 − 2  2.0 0.0 − 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 # − − − − ikp i3kp pB/(mc) + B σ + E σ + (k3, p3 , p3 ) . (41) 2 E 2 B O E B Figure 4: The relativistic Rabi frequency ΩR of the two-photon Taking advantage of the normalization of the initial state, i. e., Kapitza-Dirac effect normalized to the nonrelativistic Rabi frequency + 2 + 2 Ω c0 ↑(0) + c0 ↓(0) = 1, we finally get with (40) and (41) R,2 in Eq. (44). The applied parameters correspond to a photon | | | | energy of 12.5 keV. The gray filled circles indicate momenta for which 2 2 ˆ2 !2 the condition (49) is fulfilled and full spin-flips can be observed. These + 2 + 2 t q E c ↑(t) + c ↓(t) = (42) momenta can be approximated by Eq. (50) (solid black line). | 2 | | 2 | 4 8k2mc2 in leading order in k, pB, and pE. For times of the order 1.0 2 2 2 of 8kmc /(q Eˆ ) or larger we leave the domain where our P2,max fit time-dependent perturbation theory is valid. Thus, the result 0.8 P2(T) fit (42) is correct for times much shorter than 8kmc2/(q2Eˆ2) only. . P2,max Making the ansatz 0 6 P2(T) ! 0.4 + 2 2 ΩRt probability c ↑(t) = cos , (43a) | 0 | 2 0.2 ! + 2 + 2 2 ΩRt 0.0 c ↑(t) + c ↓(t) = sin (43b) | 2 | | 2 | 2 4 3 2 1 0 1 2 3 4 − − − −(ω ω )/ω 103 − ∗ ∗ · for the long-time behavior, expanding this ansatz for short + 2 ↑ times and comparing it to (42) gives the known Rabi frequency Figure 5: The off-resonant scattering probability P2(T) = c2 (T) + + 2 | | of the two-photon Kapitza Dirac effect [2, 40] c ↓(T) at time T = π/Ω (half a Rabi cycle on resonance) and | 2 | R maximal scattering probability maximized over the total interaction 2 ˆ2 q E time P2,max = maxT 0 P2(T 0) as functions of the relative detuning Ω , = . (44) R 2 2 2 (ω ω∗)/ω∗. Dots and squares represent numerical data while lines 8k mc − are obtained via a fit to (45) and (46) yielding b = 29.2 for the fitting The expression (44) for the Rabi frequency which is based parameter. on the expansion (41) is valid only in the nonrelativistic domain. The Rabi frequency for relativistic momenta k, pB, and pE can be calculated, however, with the help of computer algebra and to a modification of the Rabi frequency and a reduction of the numerical methods in a similar fashion as (44) by evaluating maximal scattering probability. In the resonant two-photon (40) and employing the fully relativistic propagator (38). The Kapitza-Dirac effect only modes 0 and 2 are populated. Thus, relativistic Rabi frequency is shown Fig. 4. Generally the the quantum system behaves similarly to an effective two-level relativistic Rabi frequency is lower than the nonrelativistic system. This leads us to the off-resonant generalization result (44). For the parameters as applied in the setup of Fig. 3 2 ! the theoretical relativistic Rabi frequency is ΩR = 7.241 + 2 + 2 ΩR 2 ΩT 15 15 · c ↑(T) + c ↓(T) = sin (45) 10 Hz, while numerically we obtain ΩR = 7.237 10 Hz | 2 | | 2 | Ω2 2 which is in a fair agreement with the theoretical prediction.· Let k∗ denote an electron momentum and a photon mo- of (25b) with the off-resonant Rabi frequency [53] mentum that fulfill the Bragg condition (6). When the Bragg r condition (6) is not exactly fulfilled by shifting the photon (ω ω )2 Ω = Ω2 + − ∗ , (46) momentum k from k∗, electrons scatter off-resonantly leading R b2 8

ω = kc, ω∗ = k∗c, and the parameter b that accounts for the fact 1.0 that the two-photon Kapitza-Dirac effect is not a pure two-level Pauli + 2 0.8 c ↑ system. Numerical simulations indicate that the parameter b | 0 | varies with the laser frequency and the electron momentum and . + 2 0 6 c2 ↓ = . = = . | | has the value b 29 2 for pB 0 and pE 1 00012mc and Dirac ˆ 3 2 0.4 eAmax = 8.6 10− mc . The off-resonant scattering probability probability + 2 · c ↑ + 2 + 2 | 0 | P2(T) = c2 ↑(T) + c2 ↓(T) at time T = π/ΩR (half a Rabi 0.2 + 2 | | | | c ↓ cycle on resonance) and maximal scattering probability max- | 2 | 0.0 imized over the total interaction time P2,max = maxT 0 P2(T 0) 0 5 10 15 20 25 obtained via numerical simulations are shown in Fig. 5 as func- T (103 laser periods) tions of the relative detuning (ω ω )/ω . These numerical − ∗ ∗ results can be fitted to (45) and (46), respectively, leading to Figure 6: The two-photon Kapitza-Dirac effect with electron mo- the value b = 29.2. menta chosen such that the spin-flip condition is fulfilled simulated by employing the Dirac equation (21) (black solid and dashed lines) and the Pauli equation (16) (gray solid and dashed lines). The electron interacts with a standing laser field of peak intensity 3 1022 W/cm2 4.4. Spin-flips 3 2 · (corresponding to eAˆ = 8.6 10− mc ) for each beam and photon max · energy of 12.5 keV. The electron is diffracted to mode 2, similarly The propagator (41) features spin-preserving terms (propor- to Fig. 3. However, because the initial electron momentum (pB = 0, p = 1.00012mc) fulfills the spin-flip condition the electron spin tional to 1) and spin-flipping terms (proportional to σE and E changes its spin orientation during the diffraction process. σB). If the condition 1 5 m2c2 p2 p2 = 0 (47) 1.4 − 2 B − 2 E .

2 1 2 −

is met, the propagator (41) predicts that spin-preserving tran- 10 1.0 · )

sitions are totally suppressed, thus, a spin-ﬂipping dynamics 2 ,

R 0.8

may become observable in the two-photon Kapitza-Dirac effect. Ω / However, this condition corresponds to an ellipse with major R 0.6 Ω ( axis pB = √2mc and minor axis pE = √2/5mc and, therefore, 0.4 we are beyond the validity of the nonrelativistic propagator 0.2 (41). Though, Eq. (47) is not a valid condition for spin-flipping 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 − − − − p /(mc) transitions its derivation gives us a hint how to calculate the B proper condition. In analogy to (41) one can expand the rel- 1 Figure 7: Relativistic Rabi frequency of the two-photon Kapitza- ativistic propagator (38) as a superposition of the matrices , Dirac effect at momenta for which full spin-flips are possible. (that σk, σB, and σE. With the help of the Frobenius inner product means along the gray filled circles in Fig. 4) one can write ++ 1 ++ 1 1 ++ 1 Und;2,0(t, 0) = tr †Und;2,0(t, 0) 2 +tr σk†Und;2,0(t, 0) 2 σk setting pB = 0 and pE = mc fulfills the conditions for spin- ++ 1 ++ 1 flipping electron scattering under the dynamics of the Dirac + tr σ† U (t, 0) σ + tr σ† U (t, 0) σ . (48) B nd;2,0 2 B E nd;2,0 2 E equation as well as under the Pauli equation as demonstrated in Fig. 6. In contrast to the setting in Fig. 3, the Pauli equation and Thus, spin-preserving transitions are expected to be suppressed the Dirac equation yield different Rabi frequencies due to the for relativistic electron momenta, for which the Pauli equation is ++ actually not applicable. The hyperbola (50) lies approximately tr 1†U (t, 0) = 0 . (49) nd;2,0 in a local minimum of the Rabi frequency (see Fig. 4). The Again this condition can be evaluated with the help of computer Rabi frequency along the gray filled dots in Fig. 4 is shown algebra and numerical methods. The result is indicated in Fig. 4 in Fig. 7. For photon momenta k that are small compared to by gray filled circles. These points lie approximately on the mc the Rabi frequency drops down by two orders of magni- hyperbola (black solid line in Fig. 4) tude compared to ΩR,2 which is a consequence of the relative strengths of the spin-preserving and the spin-flipping terms 2 2 2 2 in (41). m c + pB pE = 0 , (50) − Because the basis functions (18) are eigenfunctions of the which can be seen as the fully relativistic version of the condi- Foldy-Wouthuysen spin operator in the laser’s polarization + + tion (47). direction the coefficients c2 ↑(t) and c2 ↓(t) give immediately Employing time-dependent perturbation theory for the Pauli the spin of the diffracted beam if one chooses the electric field 2 2 2 equation one can show that m c pE = 0 is the corresponding direction as quantization axis. Perturbation theory allows us condition for spin-flips in the two-photon− Kapitza-Dirac effect also to calculate the orientation of the spin in the diffracted under the dynamics of the Pauli equation. Thus, the parameter part of the electron beam. More precisely, we ask the question 9 what is the expectation value of the spin sout in the scattered and the rotation angle beam if the beam of incident electrons has spin sin?  q  The electron spin is determined by applying the Foldy-  2 + 2   k 9pE pB  Wouthuysen spin operator to the quantum state. In our momen- γ = 2 arctan   . (59) 2m2c2 p2 5p2  tum space representation the expectation value of the Foldy-  − B − E  Wouthuysen spin operator is Thus, for nonrelativistic momenta k, p , and p only very 1 X E B s = cn(t)†Σcn(t) , (51) small spin rotations may be observed in the two-photon 2 n Kapitza-Dirac effect.

T with Σ = (Σ1,Σ2,Σ3) and ! σi 0 5. Three-photon Kapitza-Dirac effect Σi = . (52) 0 σi The incident electron beam has well defined momentum but 5.1. Numerical results may have arbitrary spin orientation, thus it can be expressed as a superposition of two positive energy states with the same According to the semi-classical considerations in section 2, momentum. Introducing the Bloch angles θ and φ we may two photons are absorbed from the laser field and one photon write the initial state as is emitted into the laser field in the three-photon Kapitza-Dirac  +    c ↑(0) θ/ effect, corresponding to nr = 2 and nl = 1. This partic-  0   cos( 2)  −  +   iφ ular Kapitza-Dirac effect has been examined in our recent c0 ↓(0) sin(θ/2)e  c0(0) =   =   (53) c−↑(0)  0  publication [39] where we demonstated that explicit spin dy-  0    namics can be observed depending on the electron’s and the c−↓(0) 0 0 laser’s parameters. Here we will investigate the three-photon T and cn(0) = (0, 0, 0, 0) for all n , 0. The spin expectation Kapitza-Dirac effect in more detail analyzing, for example, value of the initial quantum state is the in-field dynamics, conditions for full spin-flips and the Rabi frequency at relativistic electron momenta. For the read-  θ φ  1 sin( ) cos( ) ers convenience we show in Fig. 8 the time evolution of the   ζ sin = sin(θ) sin(φ) . (54) 2 2   occupation cn(t) governed by the Dirac equation in momen- cos(θ) tum space (21)| for| the parameters that have also been consid- The spin expectation value of the diffracted part is ered in [39]. The parameters of the simulation are a photon 3 momentum of k = 6.1 10− mc, an electron momentum of 3 · 1 c2(t)†Σc2(t) pE = 2.4 10− mc in laser polarization direction, an electron sout = . (55) · 2 c2(t)†c2(t) momentum of pk = 0.347 mc in laser propagation direction and Employing the time evolution operator (38) we find 1.0 + ++ ++ + 1 c0 †(0)Und;2†,0σUnd;2,0c0 (0) s = . (56) + 2 out + ++ 0.8 c ↑ 2 ++ + 0 c0 †(0)Und;2†,0Und;2,0c0 (0) | | + 2 0.6 c ↑ The time evolution operator (38) is up to a multiplicative factor | 3 | + 2 c ↓ a unitary 2 2 matrix and, therefore, may be witten as 0.4 3

probability | | × + 2 + 2 ↑ + ↓ γ γ 0.2 c3 c3 U++ = √P eiφ cos 1 i sin n σ (57) | | | | nd;2,0 2 − 2 r · 0.0 n 0 100 200 300 400 500 600 with the real valued parameters γ, r, and P which can be T (laser periods) determined via equating coefficients in (57) and in (38). The expression in the brackets is the SU(2) representation of a Figure 8: The three-photon Kapitza-Dirac effect simulated by em- rotation around the rotation axis given by the unit vector nr and ploying the Dirac equation (21) for one half Rabi cycle. The elec- the rotation angle γ. Thus, the electron’s spin orientation after tron with initial momentum 176 keV = 0.347 mc in laser propa- diffraction sout results from a rotation of sin around the axis nr gation direction (dashed line) is diffracted to the final momentum by the angle γ. 177 keV = 0.365 mc in laser propagation direction (solid line) by its in- For nonrelativistic electron momenta the parameters of this teraction with a standing light wave with peak intensity 2 1023W/cm2 2 · rotation can be uniquely identified by equating coefficients in (corresponding to eAˆmax = 0.21mc ) for each beam and photon mo- 3 (57) and (41). We find the rotation axis mentum 3.1 keV/c = 6.1 10− mc. The electron momentum in laser · 3 polarization direction is 1.2 keV = 2.4 10− mc. The probability of   ·  0  the diffracted electron thereby splits up in a spin-flipped part (down- 1   nr = q 3pE (58) ward triangles) and a spin-preserving part (upward triangles). Data − 2 2   adopted from [39]. 9pE + pB pB 10

1.0 c 2 c 2 1 c 2 c 2 | 0| | 3| − | 0| − | 3| 0.8

0.6

0.4 probability 0.2

0.0 0 2 4 6 8 10 12 14 100 200 300 400 500 600 684 686 688 690 692 694 696 698 t (laser periods)

Figure 9: (Color online.) The in-field quantum dynamics of the three-photon Kapitza-Dirac effect. The graph shows the occupation probability of the unscattered beam c 2, of the diffracted beam c 2, and the remaining occupation probability 1 c 2 c 2 of all other modes. The | 0| | 3| − | 0| − | 3| occupation probabilities oscillate in the laser field with twice the laser frequency. The laser field is shaped by a turn-on envelope of ten laser cycles (left part of the plot) followed by a period of 678 laser cycles with constant laser intensity (center part of the plot) and turn-off envelope of ten laser cycles (right part of the plot), see (9). When the laser is at maximal intensity the probability of finding the electron at a certain momentum is distributed over the neighboring modes of the modes 0 and 3. After the final turn-off, however, only the modes 0 and 3 are occupied. The relative occupation of modes 0 and 3 depends on the total interaction time T, see Fig. 8. Here T has been chosen that c (T) 2 0 | 0 | ≈ and c (T) 2 1. | 3 | ≈

2 a field amplitude of eAˆmax = 0.21mc . Starting from the initial gets partly occupied. After turn-off, however, only the modes 0 condition (24) the quantum dynamics features Rabi oscillations and 3 have a significant occupation probability and all other in the form occupation probabilities vanish. ! + 2 2 ΩRT c ↑(T) = cos , (60a) | 0 | 2 5.2. Perturbation theory ! + 2 + 2 2 ΩRT c ↑(T) + c ↓(T) = sin , (60b) | 3 | | 3 | 2 In this section we derive a perturbative short time solution of the three-photon Kapitza-Dirac effect, for obtaining analytic similarly to the two-photon Kapitza-Dirac effect. However, a expressions for the Rabi frequency and the electron spin-flip fraction of 0.33 of the diffracted electrons have flipped their probability, similarly to the two-photon Kapitza-Dirac effect. spin with respect to the quantization axis in the laser polariza- Analogously to the two-photon Kapitza-Dirac effect, we want tion direction. to approximate the propagator U3,0(t, 0), which maps the initial In contrast to the quantum state after the interaction with quantum state c0(0) to the final quantum state by c3(t) by the laser (shown in Fig. 8), also the neighboring modes of the modes 0 and 3 are occupied during the in-field quantum c3(t) = U3,0(t, 0)c0(0) . (61) dynamics, see Fig. 9. During the turn-on phase the quantum Since the Dirac equation only contains couplings to the next state populates the neighboring modes of mode 0 and in the neighboring modes, the lowest order perturbative solution is of following interaction with the laser at constant intensity mode 3 third order and reads

Z Z Z 1 t t3 t2 , = , , , , . Urd(t 0) 3 dt3 dt2 dt1U0(t t3)V(t3)U0(t3 t2)V(t2)U0(t2 t1)V(t1)U0(t1 0) (62) i 0 0 0 Utilizing the explicit form of the free propagator (29) and the interaction Hamiltonian (30) in momentum space yields the time-dependent perturbation theory propagator

3 Z t Z t3 Z t2 iq ++ ++ ++ ++ Urd;3,0(t, 0) = L3,2 L2,1 L1,0 dt3 dt2 dt1 sin(ωt3) sin(ωt2) sin(ωt1)ν (t, t3, t2, t1) 8 0 0 0 3 Z t Z t3 Z t2 iq + + ++ + + L3,−2 L2−,1 L1,0 dt3 dt2 dt1 sin(ωt3) sin(ωt2) sin(ωt1)ν− (t, t3, t2, t1) 8 0 0 0 3 Z t Z t3 Z t2 iq ++ + + + + L3,2 L2,−1 L1−,0 dt3 dt2 dt1 sin(ωt3) sin(ωt2) sin(ωt1)ν −(t, t3, t2, t1) 8 0 0 0 3 Z t Z t3 Z t2 iq + + + L3,−2 L2−−,1 L1−,0 dt3 dt2 dt1 sin(ωt3) sin(ωt2) sin(ωt1)ν−−(t, t3, t2, t1) , (63) 8 0 0 0 11 with the phases 2.0 νab , , , = (t t3 t2 t1) 1.5 120 h a+ ba +b i exp i 3t + ∆ t3 + ∆ t2 + ∆ t1 , (64) 105 − E E2,3 E1,2 E0,1 1.0 where the upper indices a and b take the values + and . Anal- 90 − 0.5 3 )

ogously to section 4.2, the integrand of the time integral ,

75 R mc Ω (

Z t Z t Z t 0.0 / 3 2 / R ab E

p 60 dt3 dt2 dt1 sin(ωt3) sin(ωt2) sin(ωt1)ν (t, t3, t2, t1) Ω 0 0 0 0.5 (65) − 45 also contains oscillating terms which become constant, if the 1.0 − 30 resonance condition of the three-photon Kapitza-Dirac effect = ω is met. Accounting only for the terms, which 1.5 15 E3 − E0 − grow linearly in time, yields the propagator 2.0 0 − 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 − − − − ++ pB/(mc) Urd;3,0(t, 0) = 3 ! q t i t ++ ++ ++ ++ + + + ++ e− E3 l L L L + l− L −L− L Figure 10: The relativistic Rabi frequency of the three-photon − 64 3,2 2,1 1,0 3,2 2,1 1,0 Kapitza-Dirac effect normalized to the full spin-flip Rabi frequency + ++ + + + + Ω in Eq. (71). The parameters in this figure are a laser intensity l −L3,2 L2,−1 L1−,0 + l−−L3,−2 L2−−,1 L1−,0 , (66) R,3 of 2 1023 W/cm2 and a photon energy of 3.1 keV . The gray filled · with the coefficients circles mark positions, in which condition (72) is fulfilled, which is approximated by the equations (73). 1 1 1 1 lab = + ∆E+b ω ∆E+a ∆E+b + ω ∆E+a 0,1 − 0,2 0,1 0,2 1 1 + . (67) The expression (70) holds only for small k, pE and pB. How- +b +a + + ∆E + ω ∆E , + 2ω 2 2 0,1 0 2 ever, one may evaluate c3 ↑(t) + c3 ↓(t) with the exact relativistic propagator (66)| , which| is| shown| in Fig. 10. The common property of the Rabi frequencies ΩR,2 and ΩR,3 is, that 5.3. The relativistic Rabi frequency both hold at the origin pE = pB = 0 of the pE-pB plane. The difference of both is, that the two-photon Kapitza-Dirac effect An expansion of the propagator U++ (t, 0) in Eq. (66) with shows no spin-flip for pE = pB = 0 and the Rabi frequency rd;3,0 is maximal for these momenta. In the three-photon Kapitza- respect to the momenta k, pB and pE yields Dirac effect pE = pB = 0 implies a full spin-flip position (see ++ next section) and the Rabi frequency has a saddle point. For Urd;3,0(t, 0) =   the parameters which are used in Fig. 8 this Rabi frequency 3 ˆ3 i 3t √ 15 q t Amax e− E 5pE 3 2 2  evaluates to ΩR = 3.43 10 Hz whereas the Rabi frequency  1 ikσB i − k pBσk 48 m2c4 mc  √ − − mc  from the simulation Ω ·= 3.34 1015 Hz agrees well with the 2 R · 3 3 3 analytical result. + (k , pE, pB) . (68) O In analogy to the two-photon Kapitza-Dirac effect, the off- + 2 + 2 Evaluating c ↑(t) + c ↓(t) in analogy to (40) with (61) and resonant diffraction probability (45) with the off-resonant Rabi | 3 | | 3 | (68) yields the short time diffraction probability frequency (46) also applies to the three-photon Kapitza-Dirac effect. For the parameters applied in Fig. 8 we find numer- + 2 + 2 c ↑(t) + c ↓(t) = ically b = 45.7. We remark, that in the case of the three- | 3 | | 3 | !2   photon Kapitza-Dirac effect a systematic shift of the resonance t2 q3Eˆ3 25 p2   2 + 2 + √ 2 B  . peak appears in the numerical simulation, as compared to 3 5 3  pE k 3 2 2 k 2 2  (69) 4 24m c k 2 − m c the peak position which we obtain from the classical resonance condition (6). For the parameters pk = 0.3470 mc and By comparing this probability with the analogous short time 3 pE = 2.4 10− mc of Fig. 8 for example, one finds the reso- expansion of the ansatz (43), one finds the Rabi frequency · 3 nance peak at the photon momentum k = 6.1 10− mc in the s numerical simulation, whereas the resonance condition· (6) pre- p2 p2 25 E B = . 3 ΩR = ΩR,3 + 1 + 3 2 √2 , (70) dicts the photon momentum k 4 4 10− mc. This shift scales 2 k2 − m2c2 with the laser intensity, as the quantum· dynamics leaves the with the Rabi frequency perturbative regime with increasing field amplitude. Therefore, the resonance peak position of the numerical solution and the q3Eˆ3 classical condition (6) converge to the same value in the limit Ω , = . (71) R 3 24m3c5k2 of small laser intensities. 12

20 4.0 15 nonrelativistic limit 3.5 10 exact result 3.0 3 , . 5

R 2 5 pE > 0 Ω 0

/ 2.0 R pE = 0 (degrees) 5 Ω . 1 5 η − . 10 1 0 − 0.5 15 − 0.0 20 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 − 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 − − − − − − − − pB/(mc) pB/(mc)

Figure 11: Relativistic Rabi frequency of the three-photon Kapitza- Figure 12: Tilt of the electron spin rotation axis nr. The dashed line Dirac effect at momenta, for which full spin-flips are possible. (that shows the analytical result (77) of the expanded propagator (68). The means along the gray filled circles in Fig. 10) black line is obtained numerically from the exact propagator (66). The tilt of the rotation angle takes place in the plane spanned by k and Bˆ, because the E component of the vector (75) and the E component 5.4. Spin-flips of the vector from the exact computation is zero.

Similarly to section 4.4, we identify the parameter space for spin-flips in the three-photon Kapitza-Dirac effect by the con- Equation (75) means that the electron spin is rotated around dition the axis of the magnetic field, if the electron momentum pB in magnetic field direction is zero. A nonvanishing p , however 1 ++ , = . B tr †Urd;3,0(t 0) 0 (72) tilts the rotation axis of the spin rotation into the direction of The numerical evaluation of this condition yields the gray filled the laser propagation direction by the angle circles in Fig. 10 and form a line and a hyperbola. The line ! n , p and the hyperbola can be approximated by the equations η arctan r k = arctan 3 2 √2 B . (77) ≡ nr,B − mc pE = 0 , (73a) 2 2 2 2 η m c + 0.847pB 0.608pE = 0 . (73b) This tilt angle is plotted together with the exact value, ob- − tained from the propagator (66) in Fig. 12. The line (73a) can also be derived from the expansion (68), because the spin-flip condition runs through the point pE = pB = 0, and the Taylor expansion is exact in the vicinity of this point. We also plot the Rabi frequency at the regions of a full 6. Conclusions and Outlook spin-flip in Fig. 11. Similar to section 4.4, the Rabi frequency of the spin-flip regions is about two orders of magnitude lower We discussed the general relativistic and nonrelativistic Bragg than the Rabi frequency of the spin-preserving regions. conditions for a Kapitza-Dirac effect with n interacting photons In analogy to the two-photon Kapitza-Dirac effect the elec- that follow from energy-momentum conservation. We investi- tron spin in the diffracted part of the laser beam can be ex- gated the two-photon and three-photon Kapitza-Dirac effect by pressed as accounting for the Bragg conditions and simulating the Pauli + ++ ++ + equation and the Dirac equation, which have been transformed 1 c0 †(0)Urd;3†,0σUrd;3,0c0 (0) sout = . (74) into a momentum space representation. In both scenarios spin 2 + ++ ++ + c0 †(0)Urd;3†,0Urd;3,0c0 (0) effects appear at relativistic momenta of the electron. We by utilizing the time evolution operator (66). For nonrelativistic also present an analytic solution of the Kapitza-Dirac effect electron momenta pB and pE the parameters of the rotation by computing the short-time evolution with time-dependent of the spin can be uniquely identified by equating coefficients perturbation theory for the two-photon and the three-photon in (57) and (68). In contrast to the two-photon Kapitza-Dirac Kapitza-Dirac effect. By the help of our analytical and numeri- effect we find significant spin rotations for electron momenta cal methods, we were able to demonstrate full spin-flips within pE and pB. For the case of the three-photon Kapitza-Dirac the two-photon Kapitza-Dirac effect. Furthermore, we pointed effect, the rotation axis is given by out, that the spin-flip in the Kapitza-Dirac effect corresponds to   a rotation of the electron spin, when the electron is diffracted.  3 2 √2 pB The diffraction probability, however, does not depend on the 1  −  nr = q  mc  (75) spin orientation of the incident electrons. Our numerical simu- 2   3 2 √2 p2 + m2c2 0 lations indicate that the off-resonant Kapitza-Dirac effect can − B be described by a generalization of Rabi theory for two-level and the rotation angle reads systems.  s   2  The Rabi frequency of the n-photon Kapitza-Dirac effect  √2 k 2 p  γ = 2 arctan  1 + 3 2 √2 B  . (76) scales with the nth power of eEˆ/(kmc2), because the lowest or-  2 2   5 pE − m c  der contribution in time-dependent perturbation theory is of nth 13 order and contains a product of n times the interaction Hamil- A. Spin-interaction matrices tonian (30). Therefore, the Rabi frequency of the three-photon Kapitza-Dirac effect is suppressed by a factor of eEˆ/(kmc2) as compared to the Rabi frequency of the two-photon Kapitza- To express the spin-interaction matrices Lab (32), we first ˆ 2 n,n0 Dirac effect. Note, that eE/k is always smaller than mc in the introduce the coefficients Bragg regime. Thus, from this point of view, higher laser intensities are in principle required for higher photon processes. The two-photon and three-photon Kapitza-Dirac effect are s 1 + mc2 different, if both are compared against the background of spon- d+ = En (A1a) n √ taneous emission. Since spontaneous emission is proportional 2 En to the square of the electric field, the radiation power sponta- s c 1 neously emitted by the electron scales like the Rabi frequency − = , dn 2 (A1b) ff √ n( n + mc ) of the two-photon Kapitza-Dirac e ect with the square of 2 E E eEˆ/(kmc2). This implies, that the spontaneously emitted energy which is emitted in one Rabi cycle, is independent of the laser intensity for the two-photon Kapitza-Dirac effect. The which allows us to define the coefficients Rabi frequency of the three-photon Kapitza-Dirac effect, however, scales with the third power of eEˆ/(kmc2). Therefore, the + + three-photon Kapitza-Dirac effect and all higher order Kapitza- tn,n = d d + pn pn d−d− , (A2a) 0 n n0 · 0 n n0 Dirac effects may only become visible for very high intensities l l + l + s = p d−d + p d d− , (A2b) of the external laser field. n,n0 n n n0 m n n0 l l + l + Even though the above considerations favor the two-photon r = p d−d p d d− , (A2c) n,n0 n n n0 − n0 n n0 Kapitza-Dirac effect for an experimental demonstration of spin l,q l q q l w = p p d−d− + pn p d−d− , (A2d) effects, the two-photon Kapitza-Dirac effect has the drawback, n,n0 n n0 n n0 n0 n n0 l that spin effects only occur for relativistic momenta of the h = el (pn pn ) d−d− . (A2e) n,n0 · × 0 n n0 injected electron in laser polarization direction. This implies less favorable short interaction times of the electron with the laser, if the electron passes through a narrowly focused laser The upper indices denote the vector components of the coeffi- beam. cients, whereas the lower indices correspond to a mode number Our numerical solution of the quantum dynamics also shows, of the wavefunction (13) of the electron. One can show [54] that many modes are excited in the in-field dynamics, indicat- that the matrices Lab in (32) are given by n,n0 ing that the results from perturbation theory might be applicable even for interaction parameters, in which higher order X X perturbative corrections should be of relevance. The good ++ l l l q j L = L−− = Aˆ (t)s 1+i jlqr Aˆ (t)σ (A3) agreement of our numerical results with the perturbative ap- n,n0 − n,n0 n,n0 n,n0 proximation moreover suggests the applicability of our predic- l lq j tions in the parameter space of intensive, optical laser beams. Therefore we conjecture, that spin signatures in the Kapitza- Dirac effect might be realizable even for the interaction of and moderately relativistic electrons with intensive laser beams in the optical regime. + + Ln,−n = Ln−,n 0 X 0 X X l l l,q q l l l 1 = tn,n Aˆ (t)σ w Aˆ (t) σ + i Aˆ (t)h , . 0 − n,n0 n n0 Acknowledgments l l,q l (A4) S. A. would like to thank Rainer Grobe for inspiring discus- sions.

[1] P. L. Kapitza and P. A. M. Dirac, Math. Proc. Cambridge Philos. [7] L. Bartell, Physics Letters A 27, 236 (1968). Soc. 29, 297 (1933). [8] C. Adams, M. Sigel, and J. Mlynek, Phys. Rep. 240, 143 (1994). [2] H. Batelaan, Rev. Mod. Phys. 79, 929 (2007). [9] M. Freyberger, A. Herkommer, D. Krähmer, E. Mayr, and [3] H. Schwarz, H. A. Tourtellotte, and W. W. Gaertner, Phys. Lett. W. Schleich, in Adv. At., Mol., Opt. Phys., Vol. 41, edited by 19, 202 (1965). B. Bederson and H. Walther (Academic Press, New York, 1999) [4] H. Schwarz, Zeitschrift für Physik 204, 276 (1967). pp. 143–180. [5] H. C. Pfeiﬀer, Phys. Lett. A 26, 362 (1968). [10] P. L. Gould, G. A. Ruﬀ, and D. E. Pritchard, Phys. Rev. Lett. 56, [6] Y. Takeda and I. Matsui, Journal of the Physical Society of Japan 827 (1986). 25, 1202 (1968). 14

[11] P. J. Martin, B. G. Oldaker, A. H. Miklich, and D. E. Pritchard, [33] P. Krekora, Q. Su, and R. Grobe, J. Phys. B: At., Mol. Opt. Phys. Rev. Lett. 60, 515 (1988). Phys. 34, 2795 (2001); M. W. Walser, D. J. Urbach, K. Z. [12] P. H. Bucksbaum, D. W. Schumacher, and M. Bashkansky, Phys. Hatsagortsyan, S. X. Hu, and C. H. Keitel, Phys. Rev. A 65, Rev. Lett. 61, 1182 (1988). 043410 (2002). [13] D. L. Freimund, K. Aflatooni, and H. Batelaan, Nature 413, 142 [34] S. X. Hu and C. H. Keitel, Phys. Rev. Lett. 83, 4709 (1999). (2001). [35] F. H. M. Faisal and S. Bhattacharyya, Phys. Rev. Lett. 93, 053002 [14] D. L. Freimund and H. Batelaan, Phys. Rev. Lett. 89, 283602 (2004). (2002). [36] C. Szymanowski, R. Taïeb, and A. Maquet, Laser Phys. 8, 102 [15] A. Di Piazza, C. Müller, K. Z. Hatsagortsyan, and C. H. Keitel, (1998); D. Ivanov, G. Kotkin, and V. Serbo, Eur. Phys. J. C 36, Rev. Mod. Phys. 84, 1177 (2012). 127 (2004); F. Ehlotzky, K. Krajewska, and J. Z. Kaminski,´ Rep. [16] V. Yanovsky, V. Chvykov, G. Kalinchenko, P. Rousseau, T. Plan- Prog. Phys. 72, 046401 (2009). chon, T. Matsuoka, A. Maksimchuk, J. Nees, G. Cheriaux, [37] O. D. Skoromnik, I. D. Feranchuk, and C. H. Keitel, Phys. Rev. G. Mourou et al., Optics Express 16, 2109 (2008). A 87, 052107 (2013). [17] F. Amiranoff, F. Augé, H. Backe et al., Proposal [38] M. Klaiber, E. Yakaboylu, C. Müller, H. Bauke, G. G. Paulus, for a European Extreme Light Infrastructure, Tech. and K. Z. Hatsagortsyan, submitted (2013). Rep. (The extreme light infrastructure, 2010) http: [39] S. Ahrens, H. Bauke, C. H. Keitel, and C. Müller, Phys. Rev. //www.extreme-light-infrastructure.eu/pictures/ Lett. 109, 043601 (2012). ELI-scientific-case-id17.pdf. [40] H. Batelaan, Contemp. Phys. 41, 369 (2000). [18] G. Mourou and T. Tajima, Science 331, 41 (2011). [41] B. Thaller, The Dirac Equation, Texts and Monographs in [19] M. Altarelli, R. Brinkmann, M. Chergui, W. Decking, B. Dob- Physics (Springer, Heidelberg, New York, 1992). son, S. Düsterer, G. Grübel, W. Graeff, H. Graafsma, J. Hajdu [42] L. L. Foldy and S. A. Wouthuysen, Phys. Rev. 78, 29 (1950). et al., eds., The European X-Ray Free-Electron Laser Techni- [43] E. de Vries, Fortschritte der Physik 18, 149 (1970). cal design report (DESY XFEL Project Group European XFEL [44] S. S. Schweber, An introduction to relativistic quantum field Project Team Deutsches Elektronen-Synchrotron Member of the theory (Dover, Mineola, 2005). Helmholtz Association, Hamburg, 2007). [45] H. Bauke, S. Ahrens, C. H. Keitel, and R. Grobe, “What is the [20] P. Emma, R. Akre, J. Arthur, R. Bionta, C. Bostedt, J. Bozek, relativistic spin operator?” (2013), arXiv:1303.3862. A. Brachmann, P. Bucksbaum, R. Coffee, F.-J. Decker et al., [46] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flan- Nature Photon. 4, 641 (2010). nery, Numerical Recipes in C++ (Cambride University Press, [21] B. W. J. McNeil and N. R. Thompson, Nature Photon. 4, 814 Cambride, 2002). (2010). [47] J. W. Braun, Q. Su, and R. Grobe, Phys. Rev. A 59, 604 (1999). [22] M. V. Fedorov, Opt. Commun. 12, 205 (1974). [48] G. R. Mocken and C. H. Keitel, Comp. Phys. Comm. 178, 868 [23] P. Sancho, Phys. Rev. A 82, 033814 (2010). (2008). [24] P. Sancho, Journal of Physics B: Atomic, Molecular and Optical [49] H. Bauke and C. H. Keitel, Comp. Phys. Comm. 182, 2454 Physics 44, 145002 (2011). (2011). [25] R. Gush and H. P. Gush, Phys. Rev. D 3, 1712 (1971). [50] S. Selstø, E. Lindroth, and J. Bengtsson, Phys. Rev. A 79, 043418 [26] M. A. E. und M. V. Fedorov, Journal of Experimental and Theo- (2009). retical Physics 89, 460 (1999). [51] F. Fillion-Gourdeau, E. Lorin, and A. D. Bandrauk, Comp. Phys. [27] M. A. Efremov and M. V. Fedorov, J. Phys. B: At., Mol. Opt. Comm. 183, 1403 (2012). Phys. 33, 4535 (2000). [52] C. J. Joachain, N. J. Kylstra, and R. M. Potvliege, Atoms in [28] O. Smirnova, D. L. Freimund, H. Batelaan, and M. Ivanov, Phys. Intense Laser Fields (Cambridge University Press, Cambridge, Rev. Lett. 92, 223601 (2004). 2011). [29] V. M. Haroutunian and H. K. Avetissian, Phys. Lett. A 51, 320 [53] M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge (1975). University Press, 1997). [30] M. V. Federov and J. K. McIver, Opt. Commun. 32, 179 (1980). [54] S. Ahrens, Investigation of the Kapitza-Dirac effect in the [31] D. L. Freimund and H. Batelaan, Laser Phys. 13, 892 (2003). relativistic regime, Ph.D. thesis, Ruprecht-Karls Universität [32] L. Rosenberg, Phys. Rev. A 70, 023401 (2004). Heidelberg (2012), http://www.ub.uni-heidelberg.de/ archiv/14049.