Spin-dependent radiative deflection in the quantum radiation-reaction regime

A new spin-dependent deflection mechanism is revealed by considering the spin-correlated radiation-reaction force during laser-electron collision. We found that such deflection originates from the non-zero work done by the radiation-reaction force along the laser polarization direction in each half-period, which is larger/smaller for spin-anti-paralleled/spin-paralleled electrons. The resulted anti-symmetric deflection is further accumulated when the spin-projection onto the laser magnetic field is reversed in adjacent half-periods. The discovered mechanism dominates over the Stern–Gerlach deflection for electrons of several hundreds of MeV and 10 PW-level laser peak power. The results provide a new perspective to study the strong-field QED physics in quantum radiation-reaction regime and an approach to leverage the study of radiation-dominated and strong-field QED physics via particle spins.


Introduction
Spin is an intrinsic property of particles that can play a role in the motion of a moving particle, as illustrated by the famous Stern-Gerlach (SG) experiment [1]. A particle with non-zero spin can be deflected by SG effect in the inhomogeneous magnetic field. In general, the spin effect strongly depends on the gradient of the magnetic field. Optical lasers, with light intensities approaching 10 23 W cm −2 , provide large gradient of magnetic fields at the order of 10 6 T within a few microns. Therefore it is a promising driver to trigger the spin effect on particle dynamics. Previous studies have shown that by colliding ∼10 1 MeV electrons with a laser of ∼10 22 W cm −2 , electrons with spin up/ down can be deflected to an angle of ±10 −7 rad by the SG force [2]. In the regime where SG effect emerges during collision, it is expected that other exotic phenomena become significant. For instance, electrons emit high-energy gamma-photons via non-linear Compton scattering and consequently feel recoil, which is usually referred to as radiation-reaction (RR) force [3,4]. In the case where the photon energy is comparable to the electron energy, quantum behavior appears such that the radiation turns to be stochastic [5][6][7][8][9][10]. The electron motion therefore is governed by the quantum radiation-reaction effect in the strong field quantum electro-dynamics (QED) picture.
In the QED perspective, when the spin state of an electron is considered it has been found out that the spinanti-paralleled electron tends to radiate more energy than the spin-paralleled electron and that the electron may undergo a spin-flip process during photon emission [11,12]. Spin-flip effect along with the so-called 'quantumjump' process were also proposed to polarize an electron beam in collision with an elliptically polarized laser pulse [13]. However, in a linearly-polarized (LP) laser pulse the electron spin projection onto the magnetic field axis in its rest frame oscillates with the laser field and the spin-induced net contribution is averaged to zero. Considering the SG effect being too weak as compared to the RR effect [2], present scenarios thus predict no sign Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence.
Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. of significant spin-dependent dynamics for LP laser fields with symmetric field distribution, except that asymmetry is introduced to the field itself in the few-cycle regime [12,13]. The new mechanism results from the coupling between the strong radiation reaction and spin effect, which is not active in the parameters investigated in [12] In this work, we show a new mechanism that leads to significant deflection of electrons by spin-induced effect in a symmetric LP laser field. This is only possible by coupling the spin dynamics into the radiationreaction force. The mechanism, which has not been revealed before, does not rely on artificially introduced asymmetry of the field but on the intrinsic oscillating nature of the laser field. We examined the spin-dependent radiation via the classical and the QED model, which both gave consistent results. This distinctive spindependent effect dominates the SG deflection effect under the configuration considered in the current work and accessible for the next-generation 10-100 PW laser facilities [14][15][16][17][18][19].
This paper is organized as follows: in section 2, we introduce both the classical and quantum models of spindependent radiation-reaction; in section 3, the deflection of polarized electron by spin-dependent radiation and its mechanism are presented; in section 4, the spin-dependent scenario is reproduced in the unpolarized electron beam by sign of asymmetric distribution of the polarization of the scattered electrons.
where E and B are the electric and magnetic fields, p the momentum, γ the Lorentz factor, e the electron charge, m the electron mass and = »- e the anomalous magnetic moment of electron [21]. The QED-based photon emission process including electron spin is correlated to the radiation spectrum of the electron for the spin-paralleled/-anti-paralleled case, similar to the Sokolov-Ternov effect [11] (see also [22]) under the locally constant field approximation (LCFA): where F(y) is the synchrotron function, zz¢ the initial/final quantization value of spin, ρ the photon polarization, · the photon energy fraction. The theory for arbitrarily polarized electrons is further developed by D Seipt et al in [12]. For electron polarization vector of s in its rest frame, the radiation probability rate is [12] d d y , Ai(z) the Airy function, ψ the laser phase, k, ¢ k and p the four-momentum of laser, photon and electron, = . In this work, we have a>100 and χ e <1, well beyond the above criteria. Here, a 0 =eE/mcω is the invariant field strength parameter.
The spin-dependent spectrum of an electron in (anti-)parallel state given by equation (3) is consistent the results from the well-known Sokolov-Ternov equation in equation (2) [12], for convenience of numerical modeling, where the electron undergoes s s during photon emission in the spin-flip case. The QED-MC algorithm is implemented to calculate photon emission events. At each time step, the radiation probability rate  is calculated according to the local χ e value as well as the spin state. According to the QED-MC methods in [25], two uniform random numbers are generated in the [0, 1] section to sample the spectrum of the probability rate. The probability rate calculated with the first random number r 1 is multiplied with time step Δψ. If the second random number , then a photon of energy r 1 is radiated. Photon recoil Δp=r 1 ·γmc is applied to the electron via momentum conservation after photon emission. To solve the energy cut-off in the low energy part of the spectrum introduced by numerical methods, we implement the modified event generator [25] to include the low energy photon emission.
The spin-flip and non-flip events are calculated by the weights of the flip probability rate   flip at each time step. If another uniformly generated random number <   r 3 f l i p , electron undergoes s s process; otherwise the spin stays undisturbed by radiation.

Classical spin-dependent radiation-reaction model
For classical description of spin-dependent radiation, one can multiply a Gaunt factor calculated by averaging equation (2) The electron spin is parallel or anti-parallel to the chosen axis B rest . In ultra-relativistic limit, the RR force is approximately g g a F s s RR 2 2 . As we will show later, the results from the classical spin-dependent RR from equation (4) are consistent with the one from statistics of the QED calculation using equation (3). The contribution from spin-flip process is ignored here due to low flip-rate. The stochastic quantum effect is omitted in classical description. However, its averaged representation successfully interprets the spin-dependent radiative deflection. As a comparison, we also calculate the deflection from the Stern-Gerlach force

Results
In our modeling, the laser propagates along the z-axis, with x the E-field direction and ŷ the B-field direction. Electron spin is defined in this coordinate with a unit vector; e.g. (0,±1,0) indicates the spin is parallel/antiparallel to the y-axis. Head-on collision between polarized electrons and a laser is firstly investigated using the QED-MC method. We start with the simplest situation where the laser is approximated by for the electric field and = E c B y x for the magnetic field. Here E 0 is the electric field amplitude, w 0 the beam waist at e −1 , ψ=ω t−kz the phase, N the pulse length measured by wavelength (800 nm), respectively. This is valid since the electrons pass only the near-axis part of the laser beam. Electrons are initially polarized along ±y so that the spin-vector does not precess. In the head-on collision setup and the approximation of (5), the B rest is consistent with the magnetic field the laboratory frame because where β=v/c. Therefore s B rest ·ˆoscillates with laser phase. In QED-MC modeling, we find that there is no tendency of anti-symmetric deflection for a spin-free electron, i.e. the averaged deflection angle q á ñ stays near zero. However, when spin is considered, electrons of opposite spin-orientation (±) tend to be scattered to opposite directions ( figure 1(a)) via QED-MC. One finds more electrons of parallel/anti-parallel polarization in the upper (θ>0) / lower (θ<0) region, resulting in slight asymmetry of electron distribution along θ in figure 1(b). We qualify such asymmetric distribution by defining the averaged deflection angle q á ñ. The phenomenon is investigated in a large parameter regime as shown in figures 1(b)-(e). We focus on the χ e <1 region, e.g. Figure 1(b) where pair-production [28] is suppressed [29]. The averaged deflection angle is calculated by repeating the collision for sufficient number of times (10 6 times in figure 1(b) and and 10 5 times in figures 1(c)-(e)) under each set of parameters. For fixed electron energy, larger field strength can separate electrons of different initial spin-orientation to larger angles, as shown in figure 1(b), while the deflection angle of spin-free electrons stays near zero as γ 0 ?a 0 . The deflection angle is slightly reduced when spin-flip is turned on. Since every electron flips only 0.18 time in average at a 0 =150, γ 0 =1000 according to our modeling, the modification is so minor as shown in figure 1(b) that we will not include it in the following. We note that the spread of the scattered angle in figure 1(b) is a clear reflection of the stochastic nature of QED photon emission. It should be noted that when the beam waist is not included (no transverse ponderomotive scattering), as illustrated in figure 1(c), the deflection angle scales linearly with a 0 and weakly depends on γ 0 . The deflection angle peaks in the area of higher a 0 and smaller γ 0 when there is a finite laser beam waist, as shown in 1(d). By comparing figure 1(c) to 1(d) one sees the amplification of the electron deflection induced by the transverse laser ponderomotive force. The latter is more effective for lower electron energies and stronger laser fields. The SG force can also deflect electrons in a similar way, but at a much smaller level (10 −5 rad) as shown in figure 1(e).
We now focus on the ponderomotive oscillation of an electron during its collision with a plane-wave as shown in figure 2(a). When RR force is included, since F RR ∼γ 2 the electron loses its energy during oscillation such that the work done by RR force towards x-direction cannot be fully compensated by the one along the −x- Consequently the net work is non-zero in the half-period. As seen in figure 2(a), in each half-period of the trajectory the damping force induces net negative/positive momentum shift Dp x RR (Δp x is used for convenience) for the concave-down/-up curve, denoted by down/up black arrows in figure 2(a). For a spin-free electron, the momentum shifts in adjacent half-periods are nearly equal in amplitude, exhibiting near-zero deflection effect.
This symmetry is broken when spin is coupled to the RR force. One can find in equation (3) or in equation (4) that electrons radiate more energies when anti-paralleled to B rest . As a result, an electron experiences larger (anti-parallel) or smaller (parallel) F RR compared to a spin-free electron. Take the electron with spin parallel to B rest (B lab equivalently) as an example, in the first half-period the momentum shift D + p x (+ denotes initially paralleled spin) along the negative s×k direction is smaller compared to the spin-free case, as shown by the shorter red arrow in figure 2(a). However, in the next half-period, the laser magnetic field is flipped while the spin orientation remains unchanged. The spin state is switched from parallel to anti-parallel to B rest . According to equation (4), the higher radiation power for anti-parallel spin corresponds to a larger momentum shift, only that it is directed along the positive s×k axis, as shown by the longer red arrow in figure 2(a). By integrating the momentum change along the x-direction, one finds a net shift relative to the spin-free electron in the s×k direction. For electron of anti-paralleled initial spin, the process is similar but the net momentum shift is flipped due to the opposite s×k vector. . We find that when the spin±electrons are separated by the spin-dependent RR force the ponderomotive force will further increase the electrons' spacing δx. The difference of ponderomotive forces between spin±electrons is d d x pond 0 Therefore dF x pond is further increased by the spacing δx. As a result, a positive feedback between ponderomotive effect and spin-dependent RR is built up.

Discussion
Now that the electron with opposite spin-orientation tend to be scattered to opposite directions in a collision with laser pulse, the averaged spin polarization therefore becomes inhomogeneous as a function of the scattered angle. We consider an unpolarized electron bunch of γ 0 =1000 with transverse size of 4λ and 12λ at e −1 with energy spread of ∼1% and angular divergence of 10 mrad colliding with a laser pulse of a 0 =150 via QED-MC, then the polarity of electrons is measured along y-direction (á ñ s y ) for different scattering angle. The spin-flip process is omitted due to low flip-rate. We use a focused laser pulse [30] that expresses the components of the field to the fifth order of diffraction angle. The pulse length is 27fs at FWHM and beam waist is 2λ at e −1 . In a focused pulse the spin precession needs to be considered. Spin-vector precession is calculated by solving the Thomas-Bargmann-Michel-Telegdi equation [20] . The electrons distribution stays symmetric along θ as shown in figure 4(b) while á ñ s y is anti-symmetric alongas shown in figure 4(c). The bunch get polarized along y because the electrons with positive s y tend to be scattered upwards while those with negative s y downwards. Experimental measurement of this anti-symmetric phenomenon provides a new degree of freedom to study the quantum RR effect in addition to spin-free phenomena [7,10,[31][32][33]. Effects of the electron beam energy spread and angular divergence are illustrated in figure 4(c). The mechanism is robust under realistic electron beam parameters. The deflection angle is dependent on the beam waist of the driving laser, which favors smaller focal spots, a natural consequence from the ponderomotive scattering. Considering that the polarization measurement at precession of the order 0.01 is demanding for polarimeters [34], one may also measure the asymmetric electron distribution for oppositely polarized electron beams as proposed in figure 1 to identify the new effect.

Conclusion
In conclusion, we investigate the collision of initially polarized electron and laser pulse and find the deflection due to spin-dependent radiation-reaction. We successfully interpreted such deflection by studying the momentum losses during half-periods of phases for parallel and anti-parallel spin-state. It leads to nonvanishing anti-symmetric momentum shift during the collision which builds up the deflection gradually. The inhomogeneous polarity after collision is also observed for an unpolarized electron bunch. The new degree of freedom opens up the paths towards observing spin-dependent QED effects.