Electron-spin dynamics induced by photon spins

Strong rotating magnetic fields may cause a precession of the electron's spin around the rotation axis of the magnetic field. The superposition of two counterpropagating laser beams with circular polarization and opposite helicity features such a rotating magnetic field component but also carries spin. The laser's spin density, that can be expressed in terms of the lase's electromagnetic fields and potentials, couples to the electron's spin via a relativistic correction to the Pauli equation. We show that the quantum mechanical interaction of the electron's spin with the laser's rotating magnetic field and with the laser's spin density counteract each other in such a way that a net spin rotation remains with a precession frequency that is much smaller than the frequency one would expect from the rotating magnetic field alone. In particular, the frequency scales differently with the laser's electric field strength depending on if relativistic corrections are taken into account or not. Thus, the relativistic coupling of the electron's spin to the laser's spin density changes the dynamics not only quantitatively but also qualitatively as compared to the nonrelativistic theory. The electron's spin dynamics is a genuine quantum mechanical relativistic effect.

Introduction Driven by recent developments of novel light sources that envisage to provide field intensities in excess of 10 20 W/cm 2 and field frequencies in the x-ray domain [1-6] relativistic light matter interaction has become an active field of experimental and theoretical research in recent years [7][8][9]. The spin degree of freedom of a bound or free electron may show nontrivial dynamics or may influence the electron's motion in lasers at relativistic intensities. Spin effects for electrons in intense linearly polarized laser fields were studied in [10,11]. Distinct spin dynamics has been found in the Kaptiza-Dirac effect [12,13], in [14] collapse-and-revival dynamics for the spin evolution of laser-driven electrons was predicted at 10 18 W/cm 2 , and also the dynamics of plasmas can be modified by spin effects [15]. Furthermore, the spin orientation relative to the polarization of the driving laser field affects the ionization rate of highly charged ions [16], correlations between electron spin and photon polarization in bremsstrahlung have been measured [17] and pair production rates differ for partices with spin one half and spin zero [18,19].
Besides the electron also light carries angular momentum [20]. The (classical) densities of spin and orbital angular momentum of light can be expressed in terms of the laser's electromagnetic fields and potentials. Spin and orbital angular momentum of light are of particular interest in current research [21][22][23][24] and it appears natural to ask if the light's spin density or orbital momentum density can couple to massive particles causing observable effects. In a pioneering work by Beth [25] circularly polarized light was transformed into linear polarization removing spin angular momentum from the light beam and giving a measurable torque. Furthermore, in [26] a coupling of the orbital angular momentum of light to microscopic (classical) objects causing rotational or spinning motion was demonstrated and in [27] a direct coupling between the angular momentum of light and magnetic moments was predicted by classical considerations.
In this contribution, we examine electrons and the quantum dynamics of their spin in a standing light wave formed by two counterpropagating laser beams with circular polarization. We will show that the spin density of the electromagnetic wave couples to the electron's spin causing spin precession in the laser field with a period much larger than the laser period. This precession is identified as a genuine relativistic effect that can only be explained by treating the electron quantum mechanically and taking into account relativistic corrections to the Pauli equation.
Circularly polarized lasers We consider an electron in two counterpropagating circularly polarized lasers with the same wavelength λ and the same amplitudeÊ but having opposite helicity. The electric and magnetic field components of the individual beams are given by with the speed of light c, the position r = (x, y, z) T , and the time t, and e x , e y , and e z denoting unit-vectors along the coordinate axes. The total electric and magnetic field components of the standing laser wave follow with k = 2π/λ and ω = kc from (1) as E(r, t) = 2Ê cos kx cos ωt e y + sin ωt e z , The spin angular momentum of light can be associated with its circular or elliptical polarization. Introducing the Coulomb gauge vector potentials A 1,2 and C 1,2 of the fields (1) and ∇ · A 1,2 = ∇ · C 1,2 = 0, we find that each laser carries a spin density of [28][29][30][31] Thus, the spin density of the two lasers in (1) is proportional to E 1,2 × A 1,2 and points along the propagation axis of the two lasers. Note that the definition of the spin angular momentum of light has been discussed controversially in the literature. The definition that has been applied in (4) seems to be well established now and other definitions are incompatible only when the electromagnetic fields are more complicated than the circularly polarized lasers that we consider here.

Numerical simulations and results
Due to the lasers' high intensities we assume that the lasers act on the electron but there is no backreaction of the electron to the laser field. This allows us to treat the electromagnetic fields via the classical vector potential A(r, t). The evolution of the electron of mass m and charge q = −e in the laser field, however, will be described fully quantum mechanically via the Dirac equation with the Dirac matrices α = (α x , α y , α z ) T and β [32,33]. The combined vector potential of the two counterpropagating circularly polarized lasers (1) is given by where we have deliberately introduced the window function which allows for a smooth turn-on and turn-off of the laser field. The variables T and ∆T denote the total interaction time and the time of turn-on and turn-off.
Taking into account the quasi one-dimensional sinusoidal structure of the vector potential (6) allows us to cast the partial differential equation (5) into a set of coupled ordinary differential equations [12]. For this purpose, we make the ansatz with γ ∈ {+ ↑, − ↑, + ↓, − ↓} and the basis functions with u γ n defined as χ ↑ = (1, 0) T and χ ↓ = (0, 1) T , the Pauli matrix σ x , and the relativistic energy momentum relation E n = (mc 2 ) 2 + (nck ) 2 . The basis functions (10) are common eigenfunctions of the free Dirac Hamiltonian, the canonical momentum operator, and the Foldy-Wouthuysen spin operator [34]. Thus, ψ +↑ n (r) and ψ +↓ n (r) have positive energy E n each and positive and negative spin one half, respectively, with respect to the z axis. Introducing the four-tuples c n (t) = (c +↑ n (t), c +↓ n (t), c −↑ n (t), c −↓ n (t)) T yields from (5) and (8) the Dirac equation in momentum space i ċ n (t) = E n βc n (t) + n V n,n (t)c n (t) , with the interaction Hamiltonian V n,n (t) and its components n † α z u γ n cos ωt δ n,n −1 + δ n,n +1 .
The electron is initially at rest with spin aligned to the z axis, which corresponds to the initial condition c +↑ 0 (0) = 1 and c γ n (0) = 0 else. We solve the Dirac equation (11) numerically [12] starting from this initial condition and calculate the spin expectation value s z (T ) = 2 n |c +↑ n (T )| 2 + |c −↑ n (T )| 2 − |c +↓ n (T )| 2 − |c −↓ n (T )| 2 (13) [35] after the laser's amplitude dropped to zero at time t = T , which is shown in Fig. 2. The expectation value s z (T ) oscillates periodically with a period much larger than the laser's period, i. e., it does not just follow the laser field adiabatically. The spin's expectation value in the y direction oscillates in a similar fashion but with a π/2 phase shift. Thus, the electron's spin precesses around the x axis, i. e., the propagation direction of the laser. Considering the interaction Hamiltonian V n,n (t) as a perturbation to the free Dirac Hamiltonian, time-dependent perturbation theory [12] for the Dirac equation (11) yields the angular frequency Ω of the spin precession Ω = (qÊ) 4 λ 5 (2π) 5 2 m 2 c 5 .
A comparison of this prediction for the spin-precession angular frequency with results from the numerical solution of the Dirac equation (11) shows a good agreement between theory and simulations, see black solid line and circles in Fig. 3. In particular, the numerical results confirm that the spin-precession angular frequency Ω growths with the fourth power of the electric field strengthÊ. Electron dynamics in the weakly relativistic limit In order to understand which mechanism leads to the spin precession and theÊ 4 scaling of the spin-precession frequency we consider the weakly relativistic expansion of the Dirac equation (1/s) Dirac eq. (theory) Dirac eq. rel. Pauli eq. Pauli eq. (theory) Pauli eq. class. rel. Pauli eq.

FIG. 3:
Angular frequency Ω of the spin precession as a function of the laser's electric field strengthÊ and its intensity I for lasers of the wavelength λ = 0.992 Å. Depending on the applied theory (the Dirac equation (5), the relativistic Pauli equation (15), the nonrelativistic Pauli equation, or the classical equation (21)) the spin of an electron in two counterpropagating circularly polarized light waves scales with the second or fourth power ofÊ. Theoretical predictions for the spin-precession frequency are given by (14) and (18), respectively. into a set of ordinary differential equations by a discrete plane wave ansatz [12] similarly to (8) for the Dirac equation. The angular frequency of the spin precession that results from a numerical solution of (15) agrees with the frequency from the full Dirac equation (5) as also shown in Fig. 3. The scaling of the spin-precession angular frequency changes qualitatively, however, if the relativistic correction term in (15) that is proportional to E × A is neglected. In fact, the spin-precession angular frequency scales with the second power of the electric field strength in this case, see squares in Fig. 3.
Classical electron spin dynamics The quadratic scaling of the spin-precession angular frequency inÊ for the Pauli equation (without any relativistic corrections) and its breakdown if relativistic corrections are taken into account can be understood qualitatively by the following classical arguments. The quantum mechanical dynamics of an electron's spin that is delocalized over several laser wavelengths may be described by an ensemble of classical particles with spin angular momentum placed along the x axis. Thus, let us consider a classical electron spin s at fixed position r in the magnetic field (2b). Its dynamics is governed bẏ This equation of motion can be justified via the Pauli equation, i. e., Eq. (15) neglecting the correction term that is proportional to E × A, by calculating the quantum mechanical equation of motion for the spin observable in the Heisenberg picture [37]. For parameters such that Ω L = qÊ/(mc) ω, where Ω L denotes the oscillation frequency of a spin in a constant magnetic field with amplitudeÊ/c, the spin s(t) rotates with a position dependent angular frequency around the axis of rotation of the magnetic field. For the initial condition s(0) = (0, 0, /2) T the solution of (16) is given by s(t) = 2 0, sin(2Ω P t sin 2 kx), cos(2Ω P t sin 2 kx) T (17) with which is the angular frequency averaged over a wavelength. Note that (18) also agrees with the spin-precession angular frequency for the Pauli equation (without any relativistic corrections) which can be calculated via time dependent perturbation theory and which is also confirmed by our numerical simulations, see squares in Fig. 3. Similarly, the relativistic correction in (15) to the Pauli equation leads to the classical equationṡ For the initial condition s(0) = (0, 0, /2) T the solution of (19) is given by s(t) = 2 0, sin(−2Ω P t cos 2 kx), cos(−2Ω P t cos 2 kx) T . (20) Note that due to the fast rotation of the magnetic field (2b) the coupling between the magnetic field and the spin is significantly weakened compared to a spin in a constant field such that the net effect of the spin term proportional to B in (15) is of the same order as the relativistic correction proportional to E× A. As a consequence of (17) and (20) a classical spin under the effect of both spin terms of (15) rotates in a short time interval ∆t around the angle 2Ω P (sin 2 kx − cos 2 kx)∆t. Averaged over a laser wavelength this rotation angle vanishes. Thus, our classical argument suggests that the spin of an electron that is delocalized over several laser wavelengths should not rotate. Therefore, the observed spin precession for electrons that evolves under the effect of the Pauli equation including relativistic corrections due to the lasers spin density (15) or the full Dirac equation (5) must be the result of genuine quantum effects.
In order to confirm that the precession frequency (14) is of quantum mechanical origin we also solved the coupled classical equations of motion for the spin s, the position r, and the canonical momentum p that correspond to the quantum mechanical Hamiltonian (15), namelẏ for the initial conditions r(0) = (0, 0, 0) T , p(0) = (0, 0, 0) T , and s(0) = (0, 0, /2) T . As we see in Fig. 3 the spin-precession frequency (stars) is reduced compared to (18) (dashed line) but also scales quadratically with the electric field amplitude [38]. Classically the action of the magnetic field on the electron's spin is not canceled by the action due to the laser's spin-density because the time that the point-like electron spends in different regions of the laser field is not distributed uniformly over a laser wavelength.

Conclusions
We investigated electron motion in two counterpropagating circularly polarized laser waves of opposite helicity and found spin precession around the lasers' propagation direction. The spin precession and its frequency, that scales with the fourth power of the electric field strength, can be properly described via the fully relativistic Dirac equation or the nonrelativistic Pauli equation plus a relativistic correction proportional to E × A. Because this quantity is commonly interpreted as the spin angular momentum of circularly polarized light the relativistic correction can be interpreted as a coupling of the laser's spin density to the electron spin.
The electron-spin dynamics has the remarkable feature that it is a genuine relativistic quantum effect. The quartic scaling of the spin-precession frequency can not be found within the nonrelativistic Pauli theory; and also classical models that treat electrons as point-like particles fail to reproduce the correct spin-precession frequency even if relativistic effects due to the laser's spin density are taken into account.
We have enjoyed helpful discussions with Prof. C. Müller. R. G. acknowledges the nice hospitality during his visit in Heidelberg. This work was supported by the NSF. * heiko.bauke@mpi-hd.mpg.de