Electron-Spin Dynamics in Elliptically Polarized Light Waves

Electron-spin dynamics in elliptically polarized light waves

1, 1, 2 1, 2 Heiko Bauke, ∗ Sven Ahrens, and Rainer Grobe 1Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany 2Intense Laser Physics Theory Unit and Department of Physics, Illinois State University, Normal, Illinois 61790-4560 USA (Dated: November 4, 2014) We investigate the coupling of the spin angular momentum of light beams with elliptical polarization to the spin degree of freedom of free electrons. It is shown that this coupling, which is of similar origin as the well-known spin-orbit coupling, can lead to spin precession. The spin-precession frequency is proportional to the product of the laser-ﬁeld’s intensity and its spin density. The electron-spin dynamics is analyzed by employing exact numerical methods as well as time-dependent perturbation theory based on the fully relativistic Dirac equation and on the nonrelativistic Pauli equation that is amended by a relativistic correction that accounts for the light’s spin density.

PACS numbers: 03.65.Pm, 31.15.aj, 31.30.J–

1. Introduction quantum mechanical equations of motion, relativistic and nonrelativistic. Subsequently, we solve these equations via time- dependent perturbation theory in Sec. 3 and by numerical meth- Novel light sources such as the ELI-Ultra High Field Facility, ods in Sec. 4. In Sec. 5 we estimate under which experimental for example, envisage to provide field intensities in excess of 20 2 conditions the predicted spin precession may be realized before 10 W/cm and field frequencies in the x-ray domain [1–6]. we summarize our results in Sec. 6. Such facilities open the possibility of probing light-matter interaction in the ultra-relativistic regime [7–9]. In this regime, the quantum dynamical interaction of electrons, ions, or atoms can be modified due to relativistic corrections of the Dirac equation 2. Electrons in standing laser fields to the nonrelativistic Schrödinger equation. In certain setups, however, relativistic effects may change the dynamics not only quantitatively, but also qualitatively. As an example, spin-orbit 2.1. Laser fields with elliptical polarization interaction, that is, the interaction of a particle’s spin with its orbital angular momentum, can lead to a splitting of atomic We examine electrons in standing light waves formed by two energy levels, which are degenerated within the framework of counterpropagating laser beams with circular or elliptic polar- nonrelativistic quantum theory. At ultra-high laser intensities ization. The electric and magnetic field components of two also, spin effects can set in [10–12]. elliptically polarized laser fields counterpropagating in the di- In our recent publication [13], we demonstrated that relativis- rection of the x axis are given by tic effects may also modify the spin dynamics of an electron ! ! 2π(x ct) 2π(x ct) in external electromagnetic waves with elliptical polarization. E1,2(r, t) = Eˆ cos ∓ ey + cos ∓ η ez , This effect can be attributed to a coupling of the spin density λ λ ± that is associated with the external laser field to the electron’s (1a) ! ! spin. The coupling between the electronic spin and the pho- Eˆ 2π(x ct) 2π(x ct) B (r, t) = cos ∓ η e cos ∓ ez . tonic spin is of similar origin as the well-known spin-orbit 1,2 c ∓ λ ± y ± λ effect. Both effects have in common that two angular momen- (1b)

arXiv:1409.2647v2 [quant-ph] 3 Nov 2014 tum degrees of freedom are coupled that are of different origin. Other examples for the interaction between different kinds of Here, we have introduced the speed of light c, the position T angular momentum have been studied in very different contexts, vector r = (x, y, z) , the time t, and ex, ey, and ez denoting e. g., the angular momentum of phonons in a magnetic crystal unit vectors in the direction of the coordinate axes. The two and the angular momentum of the crystal [14] or the orbital electromagnetic waves (1) feature the same wavelength λ, the angular momentum of light and the internal degrees of motion same electric field amplitude Eˆ, and the same intensity in molecules [15], to mention two recent examples. 2 In this contribution, we elaborate and extend the theoreti- I = ε0cEˆ (2) cal description of the electron’s spin dynamics as studied in but have opposite helicity. The parameter η ( π, π] deter- Ref. [13]. The article is organized as follows. In Sec. 2 we ∈ − describe the considered laser setup and derive the electron’s mines the degree of ellipticity with η = 0 and η = π correspond- ing to linear polarization and η = π/2 to circular polarization. Introducing the wave number k =±2π/λ and the lasers’ angular frequency ω = kc, the total electric and magnetic fields of the ∗ [email protected] standing laser wave that is formed by superimposing the two 2

Note, that for plane wave fields the photonic spin density may be expressed solely by the electric field and the magnetic z y B vector potential, see (5), and many works utilize the expression ε E A to define the photonic spin density, which is not equiv- x 0 × λ alent to ε0(E A + cB C)/2 for none-plane-wave fields, e. g., standing waves.× The total× amount of spin angular momentum, E however, does not depend on which of both densities is chosen [22], this means Z Z ε FIG. 1: (Color online) Schematic illustration of the electric (solid red ε E A d3r = 0 (E A + cB C) d3r , (7) line) and magnetic (dotted blue line) field components of the standing 0 × 2 × × laser wave (3) with η = π/2 that is a superposition of two counterpropagating circularly polarized laser waves of opposite helicity and equal where the integration goes over all space regions with non- wavelength λ. The electric and magnetic components E and B are vanishing electromagnetic fields. parallel to each other and rotate around the propagation direction. Furthermore, note that the given expressions for the photonic spin density require that the electromagnetic vector potentials A and C are given in Coulomb gauge. If another gauge is electric and magnetic field components (1a) and (1b) follow as applied then A and C in the definition of the spin density have to be replaced by A⊥ and C⊥, which are defined by splitting the vector potentials A and C such that A = A⊥ + Ak and ∇ A⊥ = 0 E(r, t) = 2Eˆ cos kx (cos ωt ey + cos(ωt η) ez) , (3a) · − and ∇ Ak = 0 and similarly for the electric vector potential C 2Eˆ [23]. Thus× the Coulomb gauge is the most convenient gauge to B(r, t) = sin kx ( sin(ωt η) e + sin ωt ez) . (3b) c − − y study effects of the photonic spin and it is therefore employed here. All predicted effects of this contribution result from For circular fields (η = π/2), the electric and the magnetic gauge invariant quantum mechanical wave equations and are, components (3) are parallel to each other and rotate around the therefore, independent of the choice of the gauge. propagation direction as sketched in Fig. 1. The elliptically polarized fields (1) carry spin angular momentum [16, 17]. The partition of a general light beam’s total angular momentum into orbital angular momentum and spin 2.2. Quantum dynamics in momentum angular momentum was debated in the literature. A definition space that seems to be well established now [18–21] utilizes Coulomb gauge vector potentials for both the magnetic as well as the Introducing the window function, which allows for a smooth electric field component. The Coulomb gauge vector potentials turn-on and turn-off of the laser field, A1,2 and C1,2 of the elliptically polarized fields (1) are  sin2 πt if 0 t ∆T,  2∆T ≤ ≤ Eˆ w(t) = 1 if ∆T t T ∆T, (8) C1,2(r, t) = ( sin(kx ωt η) ey sin(kx ωt) ez) ,   2 π(T t) ≤ ≤ − −ω ± ∓ ± − ∓ sin − if T ∆T t T, (4a) 2∆T − ≤ ≤ Eˆ the magnetic vector potential of the combined laser fields (3) A1,2(r, t) = ( sin(kx ωt) ey sin(kx ωt η) ez) . is given by −ω ∓ ∓ ∓ ∓ ± (4b) 2w(t)Eˆ A(r, t) = cos kx (sin ωt e + sin(ωt η) ez) . (9) These potentials are related to the electromagnetic fields via − ω y − E = A˙ = c∇ C , B = ∇ A = C˙ /c and 1,2 1,2 1,2 1,2 1,2 1,2 The parameters T and ∆T denote the total interaction time and the potentials− comply− × with the Coulomb× gauge as−∇ A = 1,2 the turn-on and turn-off intervals. The quantum mechanical ∇ C = 0. With these quantities the spin density of each· laser 1,2 evolution of an electron of mass m and charge q = e in the field· is given by the expression ε (E A + cB C )/2, 0 1,2 1,2 1,2 1,2 laser field with the vector potential (9) is governed by− the Dirac which reduces in the case of the plane× wave fields (1)× to equation ˆ2 ε0 ε0E λ sin η 2 (E A +cB C ) = ε E A = ex . i~Ψ(˙ r, t) = cα i~∇ qA(r, t) + mc β Ψ(r, t) (10) 2 1,2 × 1,2 1,2 × 1,2 0 1,2 × 1,2 2πc · − − (5) T The total photonic spin density of the setup is with the Dirac matrices α = (αx, αy, αz) and β [24, 25]. The quasi one-dimensional sinusoidal structure of the vector poten- 2 ε0 ε0Eˆ λ sin η tial (9) allows us to cast the partial differential equation (10) (E A + cB C) = ex , (6) 2 × × πc into a set of coupled ordinary differential equations [12]. For this purpose, we make the ansatz with E = E1 +E2, B = B1 +B2, A = A1 + A2, and C = C1 +C2. X γ γ Thus, the ellipticity parameter η determines also the light’s spin Ψ(r, t) = cn (t)ψn (r) (11) density. n,γ 3 with integer n, γ + , , + , , and the four-component for the now two-component wave function Ψ(r, t) with A(r, t) basis functions ∈ { ↑ −↑ ↓ −↓} given by (9), B(r, t) = ∇ A(r, t), E(r, t) = A˙ (r, t), and the × T − r vector of Pauli matrices σ = (σx, σy, σz) . In leading order γ k γ inkx Eq. (19) features four terms that may cause spin dynamics. The ψn (r) = un e , (12) 2π so-called Zeeman term σ B(r, t) mediates the coupling of ∼ · with uγ defined as the electron’s spin to the magnetic field and the anticommuta- n tor expression σ B(r, t), ( i~∇ qA(r, t))2 is the lowest s   ∼ { · − − } 2 χ / order relativistic correction to the Zeeman coupling. It is well- + /+ n + mc  ↑ ↓  u ↑ ↓ = E   , (13a) known that the term σ E(r, t) i~∇ in (19) leads to the n  nck~σx /  2 n 2 χ↑ ↓ ∼ · × E n+mc so-called spin-orbit interaction, i. e., the coupling between the s E 2  nck~σx /  electron’s spin and its orbital angular momentum. The term n + mc  2 χ↑ ↓ u / =  n+mc  , (13b) σ (E(r, t) A(r, t)), however, may be interpreted as a cou- n−↑ −↓ E − E /  2 n χ ∼ · × E ↑ ↓ pling of the spin density of the external electromagnetic wave to the particle’s spin. In Sec. 4 we will show that the relativistic χ = (1, 0)T, and χ = (0, 1)T, and the relativistic energy- ↑ ↓ electron motion in the setup of Sec. 2 can be described in lead- momentum relation ing order by the nonrelativistic Pauli equation complemented p 2 2 2 n = (mc ) + (nck~) . (14) by the relativistic correction due to the electromagnetic wave’s E spin density as the only relativistic correction. This leads us to Introducing the four-tuples the relativistic Pauli equation + + T cn(t) = (cn ↑(t), cn ↓(t), cn−↑(t), cn−↓(t)) (15) 1 q~ i~Ψ(˙ r, t) = ( i~∇ qA(r, t))2 σ B(r, t) and employing the expansion (11) the Dirac equation becomes 2m − − − 2m · ! in momentum space q2~ + σ (E(r, t) A(r, t)) Ψ(r, t) . (20) X 4m2c2 · × i~c˙n(t) = nβcn(t) + Vn n (t)cn (t) . (16) E , 0 0 n0 We also formulate the relativistic Pauli equation (20) in momentum space by utilizing the ansatz (11) as for the Dirac Here, Vn,n0 (t) denotes the interaction Hamiltonian. Its components are given by equation but now with γ , and the two-component basis functions ∈ {↑ ↓} r t qEˆ γ,γ0 w( ) γ k γ inkx Vn,n (t) = δn,n0 1 + δn,n0+1 ψn (r) = χ e . (21) 0 k − × 2π γ γ0 γ γ0 un †αyu sin ωt + un †αzu sin(ωt η) , (17) Using the expansion (11) with the functions (21) and introduc- n0 n0 − ing the pair with γ, γ0 + , , + , and δn,n denoting the Kronecker ∈ { ↑ −↑ ↓ −↓} 0 T delta. As the basis functions (13) are common eigenfunctions of cn(t) = (cn↑(t), cn↓(t)) , (22) the free Dirac Hamiltonian, the canonical momentum operator, the relativistic Pauli (20) is given in momentum space by as well as of the z component of the Foldy-Wouthuysen spin operator [26], the spin expectation value (with the z axis as the n2k2~2 quantization axis) is conveniently given by the absolute squared i~c˙n(t) = cn(t) γ 2m values of the coefficients cn (t) via q2Eˆ2w(t)2 t c t c t c t + 2 2 (1 cos η cos(2ω η))( n 2( ) + 2 n( ) + n+2( )) ~ X + 2 2 + 2 2 2k mc − − − s (t) = c ↑(t) + c−↑(t) c ↓(t) c−↓(t) . (18) z n n n n ~qEˆw t 2 n | | | | − | | − | | i ( ) + ( σy sin(ωt η) + σz sin ωt)(cn 1(t) cn+1(t)) 2mc − − − − In order to interpret the electron’s relativistic quatum dynam- ~q2Eˆ2w(t)2 sin η ics it will be beneficial to consider the weakly relativistic limit c t c t c t + 2 3 σx( n 2( ) + 2 n( ) + n+2( )) . (23) of the Dirac equation (10). In this limit, this equation reduces 4km c − via a Foldy-Wouthuysen transformation [26–28] to Neglecting also the relativistic corrections due to the term σ (E(r, t) A(r, t)) in (20), Eq. (23) reduces to the nonrelativistic∼ ·

( i~∇ qA(r, t))2 q~ Pauli equation× in momentum space i~Ψ(˙ r, t) = − − σ B(r, t) + qφ(r, t) 2m − 2m · n2k2~2 ( i~∇ qA(r, t))4 q2~2 i~c˙n(t) = cn(t) − − (c2 B(r, t)2 E(r, t)2) 2m − 8m3c2 − 8m3c4 − q2Eˆ2w(t)2 q~ q~2 + (1 cos η cos(2ωt η))(cn 2(t) + 2cn(t) + cn+2(t)) σ (E(r, t) ( i~∇ qA(r, t))) ∇ E(r, t) 2k2mc2 − − − − 4m2c2 · × − − − 8m2c2 · ˆ ! i~qEw(t) q~ n o + ( σy sin(ωt η) + σz sin ωt)(cn 1(t) cn+1(t)) . + σ B(r, t), ( i~∇ qA(r, t))2 Ψ(r, t) (19) 2mc − − − − 8m3c2 · − − (24) 4

The two-component basis functions (21) are common eigen- matrices functions of the free Pauli Hamiltonian, the canonical momen-    ζ, † ζ0, ζ, † ζ0,  ~ ζ,ζ0 un ↑ αyu ↑ un ↑ αyu ↓ tum operator, and the nonrelativistic spin operator σz/2. Thus,  n0 n0  Vy;n,n =   , (29a) the spin expectation value (with the z axis as the quantization 0  ζ, † ζ0, ζ, † ζ0,  un ↓ αyun ↑ un ↓ αyun ↓ axis) is given by the absolute squared values of the coefficients  0 0  γ  ζ, † ζ0, ζ, † ζ0,  c (t) via ζ,ζ0 un ↑ αzu ↑ un ↑ αzu ↓ n  n0 n0  Vz;n,n =   , (29b) 0  ζ, † ζ0, ζ, † ζ0,  un ↓ αzu ↑ un ↓ αzu ↓ ~ X n0 n0 s (t) = c (t) 2 c (t) 2 . (25) z n↑ n↓ and 2 n | | − | | qEˆ Equations (16), (23), and (24) are the basic equations that ζ,ζ0 Vn,n (t) = δn,n0 1 + δn,n0+1 we are going to solve in the next sections numerically and 0 k − × ζ,ζ0 ζ,ζ0 analytically via time-dependent perturbation theory. In the Vy;n,n sin ωt Vz;n,n cos ωt , (29c) following, we assume that the electron is initially at rest with 0 − 0 spin aligned to the z axis, which corresponds to the initial with ζ, ζ0 +, . We further introduce the 4 4 matrix + ∈ { −} × condition c0 ↑(0) = 1 (or c0↑(0) = 1 in case of the Pauli equation) ++ + ! γ Vn,n (t) Vn,n− (t) and cn (0) = 0 else. Vn,n (t) = +0 0 (30) 0 V − (t) V −− (t) n,n0 n,n0 that comprises 16 matrix elements (17) of the interaction Hamil- 3. Time-dependent perturbation tonian V(t). In this notation, the free Hamiltonian H0 reads ! n1 0 theory H0;n,n = E δn,n . (31) 0 0 n1 0 −E The short-time evolution of the electron dynamics may be cal- Consequently, matrix elements of the free propagator (27) are culated via time-dependent perturbation theory [29] that allows given by us to devise an analytic expression for the Rabi frequency of ++ ! the spin precession. In this scheme the Hamiltonian is split into U0;n,n (t) 0 U0;n,n (t) = 0 (32) a free part and a possibly time-dependent interaction part, viz., 0 0 U−− (t) 0;n,n0 H = H0 + V(t). Then the time evolution operator that maps a quantum state from time 0 to time t is given by with ++ 1 U0;n,n (t) = exp( i nt/~)δn,n0 , (33a) U(t) = U0(t) + U1(t) + U2(t) + ... (26) 0 − E U−− (t) = exp(i nt/~)δn,n 1 . (33b) 0;n,n0 E 0 with the free propagator As our numerical simulations indicate that spin changing quan- ! + i tum transitions occur from the state with coefficient c0 ↑ to the U0(t) = exp H0t (27) + −~ state with c0 ↓, we do not need to calculate the full time evolution matrix, knowing and the corrections due to the interaction Hamiltonian V(t) ++ ++ ++ ++ U0,0 (t) = U0;0,0(t) + U1;0,0(t) + U2;0,0(t) Z t Z t Z t 1 3 2 ++ ++ + U3;0,0(t) + U4;0,0(t) + ... (34) Un(t) = n dtn ... dt2 dt1 (i~) 0 0 0 will be sufficient. A short calculation shows that U++ (t) is U (t tn)V(tn) ... U (t t )V(t )U (t ) (28) n;0,0 0 − 0 2 − 1 1 0 1 a zero matrix for odd n. In the follwing we will calculate for n > 0. To simplify the following calculations we neglect the the propagators in second and fourth order time-dependent turn-on and turn-off times; i. e., we set ∆T = 0 for the reminder perturbation theory. of this section. Furthermore, we specialize to the case of laser First, we calculate the second-order propagator, that is, ac- waves with circular polarization, i. e., η = π/2. cording to (28), ++ U2;0,0(t) = Z t Z t 1 2 3.1. Dirac equation t t U++ t t V ++ t U++ t t V ++ t 2 d 2 d 1 0;0,0( 2) 0,1 ( 2) 0;1,1( 2 1) 1,0 ( 1) − ~ 0 0 − − + + + V −(t )U−− (t t )V − (t ) For the Dirac equation in momentum space (16) the operators 0,1 2 0;1,1 2 − 1 1,0 1 ++ ++ ++ H0, V(t), and U0(t, 0) are represented by matrices of infinite + V0, 1(t2)U0; 1, 1(t2 t1)V 1,0(t1) − − − − − dimension that we will conveniently write in terms of sub- + + ++ + V0,−1(t2)U0;−−1, 1(t2 t1)V −1,0(t1) U0;0,0(t1) . (35) matrices of size 2 2 and 4 4. First, we introduce the 2 2 − − − − − × × × 5 s Note that in the last equation we took advantage of the fact that 2 + 1 mc the interaction Hamiltonian and the free particle propagator dn = 1 + , (40a) √ n are banded matrices. Rewriting the matrix elements (29c) of 2 s E the interaction Hamiltonian by expanding the trigonometric sgn n mc2 functions into exponentials dn− = 1 (40b) √2 − n E ˆ the integral (35) evaluates to [30] ζ,ζ0 qE Vn,n (t) = δn,n0 1 + δn,n0+1 0 2k − × 2 ˆ2 ++ q E i t/~ ζ,ζ0 ζ,ζ0 iωt ζ,ζ0 ζ,ζ0 iωt U t 0 t iV V e + iV V e− , (36) 2;0,0( ) = i 2 e− E − y;n,n0 − z;n,n0 y;n,n0 − z;n,n0 2k ×

r0,1r1,0( 1 + σx) r0,1r1,0( 1 σx) r0, 1r 1,0( 1 + σx) − + − − + − − − and utilizing the identity 1 + 0 ~ω 1 + 0 + ~ω 1 + 0 ~ω −E E − −E E −E− E − r0, 1r 1,0( 1 σx) t0,1t1,0(1 σx) t0,1t1,0(1 + σx) + − − − − + − + ++ ++ + + ! 1 + 0 + ~ω 1 + 0 ~ω 1 + 0 + ~ω iV V iV − V − y;n,n0 z;n,n0 y;n,n0 z;n,n0 −E− E E E − E E ± + − + ± − = 1 1 ! iV − V − iV −− V −− t0, 1t 1,0( σx) t0, 1t 1,0( + σx) ± y;n,n0 − z;n,n0 ± y;n,n0 − z;n,n0 + − − − + − − + O(1) . (41) ! 1 + 0 ~ω 1 + 0 + ~ω rn n ( σz + iσ ) tn n ( iσ σz) , 0 ∓ y , 0 ± y − (37) E− E − E− E tn n ( iσ σz) rn n ( σz iσ ) , 0 ± y − , 0 ± − y Note, that we give in (41) only those terms explicitly that grow linear in time t. Terms that oscillate but remain bounded over with the coeﬃcients time are omitted. Summing the various terms in (41) ﬁnally yields

+ + 2 2 tn,n = dn dn + dn−dn− , (38) q Eˆ 0 0 0 U++ t i 0t/~t + + 2;0,0( ) = i 2 2 e− E . (42) rn,n = d−d d d− , (39) − 2k mc 0 n n0 − n n0 ++ Thus, U2;0,0(t) is diagonal and therefore no spin ﬂip is predicted and in second order time-dependent perturbation theory.

The fourth order time-dependent perturbation theory propagator is explicitly given by

Z t Z t Z t Z t 1 4 3 2 U++ t t t t t 4;0,0( ) = 4 d 4 d 3 d 2 d 1 ~ 0 0 0 0 X X ++ +,ζ3 ζ3,ζ3 ζ3,ζ2 ζ2,ζ2 ζ2,ζ1 ζ1,ζ1 ζ1+ ++ U (t t4)V (t4)U (t4 t3)Vn ,n (t3)U (t3 t2)Vn ,n (t2)U (t2 t1)V (t1)U (t1) , (43) 0;0,0 − 0,n3 0;n3,n3 − 3 2 0;n2,n2 − 2 1 0;n1,n1 − n1,0 0;0,0 (n1,n2,n3) ζ1,ζ2,ζ3 +, ∈{ −} where the multi-indices (n1, n2, n3) and (ζ1, ζ2, ζ3) can take six and eight diﬀerent values, respectively. Thus the sum in (43) runs over 48 terms. All possible values of the tuple (n1, n2, n3) are shown in Tab. 1. Expanding the trigonometric functions of the interaction Hamiltonian into exponentials and reordering the terms of (43) we get

++ U4;0,0(t) = ˆ !4 X X X qE +,ζ3 +,ζ3 ζ3,ζ2 ζ3,ζ2 ζ2,ζ1 ζ2,ζ1 ζ1,0 ζ1,+ η4iV V η3iVy,n ,n Vz,n ,n η2iVy,n ,n Vz,n ,n η1iV V ν(t) (44) 2k~ − y,0,n3 − z,0,n3 − 3 2 − 3 2 − 2 1 − 2 1 − y,n1,0 − z,n1,0 (n1,n2,n3) ζ1,ζ2,ζ3 η1,η2,η3,η4 +, +1, 1 ∈{ −} ∈{ − } with the phase integral

Z t Z t4 Z t3 Z t2 ν(t) = dt4 dt3 dt2 dt1 0 0 0 0 ! i exp (t t ) + ζ n (t t ) + ζ n (t t ) + ζ n (t t ) + t exp (iω (η t + η t + η t + η t )) . (45) −~ E0 − 4 3E 3 4 − 3 2E 2 3 − 2 1E 1 2 − 1 E0 1 4 4 3 3 2 2 1 1

For those terms where the indices η1, η2, η3, η4 sum to zero the phase integral grows linearly in time, viz. 6

3.2. Nonrelativistic Pauli equation TABLE 1: All six possible values of the multi-indices (n1, n2, n3) that appear in the sums of (43) and (44) and in the integral (45). In order to calculate the precession frequency in the framework number n1 n2 n3 of the nonrelativistic Pauli equation (24) we deﬁne the “free” 1 1 2 1 Hamiltonian 2 1 0 1 ! n2k2~2 q2Eˆ2 3 1 0 -1 H = + δ 1 . (53) 0;n,n0 2m k2mc2 n,n0 4 -1 -2 -1 5 -1 0 -1 Then the interaction Hamiltonian becomes 6 -1 0 1 Vn,n0 (t) = V1;n,n0 (t) + V2;n,n0 (t) (54a)

i i with ν(t) = ( ζ )/~ η ω ( ζ )/~ (η + η )ω 0 1 n1 1 0 2 n2 1 2 i~qEˆ E − E − E − E − V (t) = (σ cos ωt σ sin ωt)(δ δ ) , i i t/~ 1;n,n0 y z n,n0+1 n,n0 1 e− E0 t + O(1) , (46) 2mc − − − × ( ζ n )/~ (η + η + η )ω (54b) E0 − 3E 3 − 1 2 3 2 2 where O(1) indicates some integration constant. If η1 + η2 + q Eˆ V2;n,n (t) = (δn,n 2 + δn,n +2)1 . (54c) η3 + η4 , 0, the function ν(t) shows oscillatory behavior, and, 0 2k2mc2 0− 0 therefore, these cases can be neglected. 2 ˆ2 2 2 A Taylor expansion of the propagator (44) up to the fourth Neglecting the constant term q E /(k mc ) in (53), which order in the laser’s wave number k yields finally causes just a global phase rotation that may be removed by ! a suitable gauge, yields the free propagator ++ Ω i t/~ 4 U (t) = i Ω 1 + σ e− E0 t + O(k ) , (47) 2 2 ! 4;0,0 ϕ 2 x n k ~ U n n (t) = exp i t δn n 1 . (55) 0; , 0 − 2m , 0 with the two angular frequencies The second order propagator for the coefficient c (t) q4Eˆ4λ6 0 Ω = (48) ϕ (2π)6~3mc4 U2;0,0(t) = Z t Z t and 1 2 dt dt U (t t ) V (t )U (t t )V (t ) 4 ˆ4 5 2 2 1 0;0,0 2 2;0,2 2 0;2,2 2 1 2;2,0 1 q E λ − ~ 0 0 − − Ω = 5 2 2 5 . (49) (2π) ~ m c + V2;0, 2(t2)U0; 2, 2(t2 t1)V2; 2,0(t1) − − − − − In contrast to the second order perturbation theory we have + V1;0,1(t2)U0;1,1(t2 t1)V1;1,0(t1) found now a contribution in the propagator that causes a spin − + V1;0, 1(t2)U0; 1, 1(t2 t1)V1; 1,0(t1) U0;0,0(t1) (56) precession around the x axis due to the σx term in (47). The − − − − − fourth order short-time propagator yields for the initial condi- becomes after time integration + γ tion c0 ↑(0) = 1 and cn (0) = 0 else the spin-flip probability q2Eˆ2 k2~2/(2m)1 + ω~σ 2 2 U t x t O Eˆ4 , + 2 Ω t 2;0,0( ) = i 2 2 2 2 2 2 + ( ) (57) c ↓(t) = . (50) − 2m c (k ~ /(2m)) (ω~) | 0 | 4 − Anticipating the oscillatory behavior of the long-time spin dy- which reduces in the limit k < mc~ to ! namics (see Sec. 4 and Ref. [13]), which means ΩP~k ΩP U2;0,0(t) = i 1 + σx t , (58) + 2 2 Ωt 4mc 2 c0 ↓(t) = sin , (51) | | 2 where we identify the frequency Ω as the precession frequency (49) q2Eˆ2λ of the electron’s spin [12] by comparing the Taylor expansion ΩP = (59) of (51) with (50). The spin precession frequency (49) is pro- 2πm2c3 2 portional to the photonic spin density %σ = ε0Eˆ λ/(πc), the denotes the spin-precession angular frequency for the nonrel- laser field’s intensity I given in (2), and the fourth power of the ativistic Pauli equation. Note that in the case of the nonrel- wavelength ativistic Pauli equation already second order time-dependent α2 perturbation theory predicts spin precession in contrast to the I 4 el fully relativistic Dirac equation. Furthermore, spin-precession Ω = %σ λ 2 2 3 , (52) 2π m c angular frequencies for the nonrelativistic Pauli equation (59) 2 2 2 3 where the proportionality factor αel/(2π m c ) is independent and the Dirac equation (49) feature qualitatively different de- of the laser’s electromagnetic field and αel denotes the fine- pendencies on the electric field strength Eˆ: quadratic scaling structure constant. in the former case, quartic scaling in the latter case. 7

I (W/cm2) Dirac eq. rel. Pauli eq. Pauli eq. 1021 1022 1023 0.50 1016

0.25 1015 ~ / ) . T 0 00

( 14 z 10 s s) /

0.25 (1 Dirac eq. (theory) − Ω 13 10 Dirac eq. 0.50 − 0 5 10 15 20 25 rel. Pauli eq. T (1000 laser periods) 1012 Pauli eq. (theory) Pauli eq. 1011 FIG. 2: Expectation value of the quantum mechanical electron spin 1014 1015 in z direction sz(T) as a function of the total interaction time T by Eˆ (V/m) using three different equations of motion (the Dirac equation (16), the relativistic Pauli equation (23), and the nonrelativistic Pauli equation FIG. 3: Angular frequency Ω of the spin precession as a function of (24)) to model the quatum dynamics. The wavelength and the ampli- the laser’s electric field strength Eˆ and its intensity I for lasers of the tude of the applied circularly polarized laser fields are λ = 0.159 nm wavelength λ = 0.159 nm. Depending on the applied theory (the Dirac and Eˆ = 2.057 1014 V/m, respectively, and the field’s switch-on-off × equation (10), the relativistic Pauli equation (20), or the nonrelativistic interval ∆T corresponds to five laser cycles. Pauli equation) the spin of an electron in two counterpropagating circularly polarized light waves scales with the second or the fourth power of Eˆ. Theoretical predictions for the spin-precession frequency 4. Numerical results are given by (49) and (59), respectively.

In order to corroborate our analytical results of Sec. 3, we carried out some numerical simulations solving the time- dependent Dirac equation as well as the time-dependent rel- dependent perturbation theory for the Dirac equation there ativistic and nonrelativistic Pauli equations. Figure 2 shows is no spin precession. In order to derive the spin-precession the expectation value of the quantum mechanical electron spin frequency we had to calculate the dynamics in fourth order in z direction sz(T) as a function of the total interaction time perturbation theory. In second order perturbation theory for T. Time-dependent perturbation theory predicts different pre- the relativistic Pauli theory the spin forces due to the magnetic cession frequencies (49) and (59) for the nonrelativistic Pauli field and the spin forces due to the light’s spin density just can- equation and for the Dirac equation, respectively. This is con- cel each other. As shown in Ref. [13], the cancellation of the sistent with our numerical simulations, which yield oscillatory spin forces can also be understood by quasi-classical considera- behavior of the spin with different angular frequencies for the tions if one assumes that the electron is evenly spread over the nonrelativistic Pauli equation and for the Dirac equation; see range of a wavelength. The observed spin precession results Fig. 2. Note that the relativistic Pauli equation (23) and the because the electron’s wave function actually accumulates at Dirac equation (16) yield spin evolutions that are virtually indis- the maxima of the magnetic field of the standing light wave tinguishable. Thus, the relativistic Pauli equation is an excellent at x = λ/4. In order to demonstrate this effect let us utilize approximation to the fully relativistic Dirac equation for the the relativistic± Pauli equation (20). A snapshot of the electron setup and the parameters that we consider here. density %(x, t) = Ψ (x, t) 2 + Ψ (x, t) 2 during the electron’s In Fig. 3 we compare the angular frequencies of the spin interaction with the| laser↑ field| is| shown↓ | in the left part of Fig. 4; precession that result from the nonrelativistic Pauli equation, here Ψ and Ψ denote the upper and lower components of the the relativistic Pauli equation, and the fully relativistic Dirac wave function↑ ↓Ψ. The shape of the wave function can be char- equation for the case of circular polarization (η = π/2). Our acterized by calculating the density at x = λ/4. If the density numerical results reflect the different scaling of the angular %(x, t) is evenly distributed, then λ%(λ/4, t) = 1. Thus, larger frequency as a function of the peak electric field strength for values indicate deviations from a flat distribution. As shown in the nonrelativistic Pauli equation and the Dirac equation. The the right part of Fig. 4, the electron density at x = λ/4 oscillates numerical results for the spin-precession frequency are in an at frequencies much larger than the spin-precession frequency. excellent agreement with the theoretical predictions (49) and The quantity λ%(λ/4, t) oscillates around some mean value that (59), respectively. Note that the spin-precession frequency (49) is larger than one and that depends on the electric field am- for the Dirac equation also describes the spin-precession fre- plitude Eˆ. Our numerical simulations indicate that the larger quency that results from the relativistic Pauli equation very Eˆ the more the electron density tends to concentrate around well. As demonstrated in Ref. [13] the spin precession angular x = λ/4. Furthermore, the wave packet shows a beating be- frequency becomes proportional to sin η if the light has ellip- havior± as displayed in the right most part of Fig. 4. Note that tical polarization. This means that the spin precession effect the beat’s envelope does not have the same period as the spin ceases to exist in the case of linear polarization (η = 0). precession. Thus, the electron’s spatial density and its spin In Sec. 3.1 we demonstrated that in second order time- oscillate in an asynchronous manner. 8

% = Ψ 2 + Ψ 2 Ψ 2 Ψ 2 | ↑| | ↓| | ↑| | ↓| 2.0 0.50 0.5 1.8 . 0.25

0 4 ) t , 1.6 ~ 4 /

0.3 ) 0.00 t λ/ ( z ( 0.2 1.4 s λ% 0.25 0.1 1.2 − 0.0 1.0 0.50 0.50 0.25 0.00 0.25 0.50 0 50 100 150 200 0 5 10 15 20 25− − − x/λ t (laser periods) t (1000 laser periods)

FIG. 4: Left display: Electron density % = Ψ 2 + Ψ 2 and densities of the individual spinor components Ψ 2 and Ψ 2, respectively, after t = 4400 laser periods. The densities accumulate| ↑| in| regions↓| of high magnetic ﬁelds near x = λ/4. Right displays:| ↑| The| scaled↓| electron density ± λ%(x, t) at x = λ/4 (solid gray line, left axis) and the expectation value of the spin sz(t) (dashed black line, right axis) as a function of time t. All laser parameters as in Fig. 2. Note that oscillations of %(λ/4, t) are too fast to be resolved on the scale of the right most display.

5. Considerations on the electric and from the third inequality of (60) follows with (61a) field strength q Eˆ | | < 1 . (61b) 2kmc2 The sinusoidal oscillatory motion of the electron’s spin in the For laser wavelengths that are longer than half of the Compton standing laser field of two elliptically polarized laser fields is wavelength the condition (61b) is always fulfilled provided that of perturbative character. It starts to breakdown if the laser the condition (61a) is met. In Fig. 5 we show the spin dynamics field becomes too strong resulting in an anharmonic motion for three different peak electric field strengths, for the cases of the spin. Criteria for the breakdown of the harmonic spin q Eˆ/(k2~c) < 1 with Eˆ = 2.06 1014 V/m, q Eˆ/(k2~c) 1 precession may be devised from the relativistic Pauli equation with| | Eˆ = 3.09 1014 V/m, and× q Eˆ/(k2~c) |>| 1 with Eˆ≈ = (23). The external laser field can be treated as a small perturba- 4.11 1014 V/m×, respectively. The| | figure clearly shows how tion if the kinetic-energy coefficient k2~2/(2m) in (23) is large the inharmonicities× start to set in at q Eˆ/(k2~c) 1. compared to the others. This means the conditions Generally speaking the observed| electron-spin| ≈ dynamics is not a strong-field effect. Spin flips may be observed also for q2Eˆ2 k2~2 ~ q Eˆ k2~2 ~q2Eˆ2 k2~2 weak laser fields but the spin-precession frequency becomes < , | | < , < (60) 2k2mc2 2m 2mc2 2m 4km2c3 2m small in this regime. In this case, the electron must be trapped for many laser cycles in the laser field’s focus to observe a full have to be met. The first two inequalities are both equivalent spin flip. If one assumes that the electron stays for cycles to in the laser’s focus, then the required electric field strengthN is bounded as q Eˆ !1/4 !1/4 | | < 1 (61a) ω6~2m2 (2π)6c6~2m2 ω2~ (2π)2c~ k2~c = < Eˆ < = . 2 q4 2 q4λ6 q c q λ2 N N | | | | (62) Eˆ = 2.06 1014 V/m Eˆ = 4.11 1014 V/m × × Eˆ = 3.09 1014 V/m λ (nm) 0.50 × 1 0.5 0.3 0.2 0.15 3.5

0.25 3.0

~ 2.5 / m) ) . / T 0 00 V ( 2.0 z s 14 1.5 0.25 (10 ˆ − E 1.0 0.50 0.5 − 0 2 4 6 8 10 12 14 16 T (1000 laser periods) 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 ω (1019 1/s) FIG. 5: Expectation value of the quantum mechanical electron spin in z direction sz(T) as a function of the total interaction time T for FIG. 6: Laser parameters for which a full spin flip is feasible (gray different peak electric field strengths Eˆ. Other parameters as in Fig. 2. shaded area). The upper and lower bounds are given by (62) assuming The quantum dynamics was simulated by solving the time-dependent that the electron can stay for = 5000 laser cycles in the electromag- Dirac equation. netic field’s focus. N 9

The lower bound follows from > ω/(2Ω) and the upper spin precession around the laser waves’ propagation axis. The bound from (61a). For the lowerN limit we made the rather pes- spin precession is caused by the coupling of the light’s spin den- simistic assumption that a full spin flip would be required to sity to the electron’s spin degree of freedom. This mechanism detect a change of the spin orientation. More sensitive electron- is of similar origin as spin-orbit coupling. The spin-precession spin measurements would allow to employ shorter laser pulses frequency is directly proportional to the product of the laser’s and/or lower intensities to verify the predicted effect. Further- intensity and the electromagnetic field’s spin density. The ob- more, the inequalities in (62) implicate that an increase of the servation of the predicted spin precession may become feasible laser’s wavelength may also require an increase of interaction by employing near-future high-intensity x-ray laser facilities. time in terms of laser cycles . This is because when the Furthermore, it might be interesting to generalize the setup con- wavelength λ is increased but allN other parameters remain fixed, sidering electron-vortex beams [31] and/or light-vortex beams then the lower bound will finally exceed the upper bound of and to study a possible interaction of the light’s spin or its (62) and then the effect becomes unverifiable. Thus, there is a orbital angular momentum with the electron beam’s orbital specific range of wavelengths and electric field strengths that angular momentum. render a full spin flip possible that depends on the parameter as illustrated in Fig. 6. N Acknowledgments 6. Conclusions S. A. acknowledges the nice hospitality during his visit at Illi- We investigated electron motion in two counterpropagating el- nois State University. This work was supported by the NSF liptically polarized laser waves of opposite helicity and found (USA).

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