Spin polarization of electrons by ultraintense lasers

TL;DR

This paper demonstrates that ultraintense laser interactions can rapidly induce high degrees of electron spin polarization through the Sokolov-Ternov effect, occurring in less than 10 femtoseconds at intensities above 10^{23} W/cm^2.

Contribution

It introduces the possibility of achieving ultrafast electron spin polarization using high-intensity lasers, extending the understanding of spin dynamics in extreme fields.

Findings

Spin polarization can reach 70%-90% in less than 10 fs.

High intensity lasers can induce rapid spin flip transitions.

The Sokolov-Ternov effect is effective in laser-matter interactions at extreme intensities.

Abstract

In a strong magnetic field, ultra-relativistic electrons or positrons undergo spin flip transitions as they radiate, preferentially spin polarizing in one direction -- the Sokolov-Ternov effect. Here we show that this effect could occur very rapidly (in less than 10 fs) in high intensity ($I\gtrsim10^{23}$ W/cm$^{2}$) laser-matter interactions, resulting in a high degree of electron spin polarization (70%-90%).

Equations

$$\mathbf{\zeta }=\frac{\mathbf{E}\times \mathbf{\beta }}{\Vert \mathbf{E}\times \mathbf{\beta }\Vert }={\mathbf{e}}_z.$$

$$\frac{d}{d\tau }{n}^{\downarrow }(t)=\frac{d{N}^{\uparrow \downarrow }}{d\tau }{n}^{\uparrow }(t)-\frac{d{N}^{\downarrow \uparrow }}{d\tau }{n}^{\downarrow }(t).$$

$$\frac{d{N}_{\rho }^{s{s}^{\prime }}}{d\tau }={\mathcal{P}}_{\mathrm{class}}{\int }_0^{\eta /2}d\chi \frac{dy}{d\chi }\frac{F(\eta ,\chi ,s,s^{\prime },\rho )}{\hslash \omega /\gamma },$$

$${\mathcal{P}}_{\mathrm{class}}=\frac{2m_e^2{c}^3{e}^2{\eta }^2}{3{\hslash }^2}.$$

$$\chi =\frac{\hslash \omega E}{2m_e{c}^2{E}_S}$$

$$\hslash \omega =\frac{\gamma m_e{c}^2\xi y}{1+\xi y},$$

$$y=\frac{4\chi }{3\eta (\eta -2\chi )}.$$

$$F(\eta,\chi,s,s^{\prime},\rho)=\frac{9\sqrt{3}y}{16\pi(1+\xi y)^{4}}\Bigg{\{}\frac{1+ss^{\prime}}{2}\bigg{\{}\left(1+\frac{1}{2}\xi y\right)^{2}\left[\int_{y}^{\infty}K_{5/3}(x)dx+\rho K_{2/3}(y)\right]\\ +\left[\frac{\xi^{2}y^{2}}{2}\int_{y}^{\infty}K_{1/3}(x)dx-s^{\prime}(2+\xi y)\xi yK_{1/3}(y)\right]\frac{1+\rho}{2}\bigg{\}}\\ +\frac{1-ss^{\prime}}{2}\frac{\xi^{2}y^{2}}{4}\bigg{\{}\int_{y}^{\infty}K_{5/3}(x)dx-\rho K_{2/3}(y)+\left[2\int_{y}^{\infty}K_{1/3}(x)dx-4s^{\prime}K_{1/3}(y)\right]\frac{1-\rho}{2}\bigg{\}}\Bigg{\}},$$

$${\epsilon }_{\mathrm{rad}}=\frac{4\pi e^2}{3m_e{c}^2\lambda }\approx 1.18\times 10^{-8}$$

$${\left[g(\eta ){\epsilon }_{\mathrm{rad}}{\gamma }^4\right]}^2+{\gamma }^2=a_0^2.$$

$$g(\eta )=\sum _{s{s}^{\prime }\rho }\frac{{\mathcal{P}}_{\rho }^{s{s}^{\prime }}}{{\mathcal{P}}_{\mathrm{class}}}=\sum _{s{s}^{\prime }\rho }{\int }_0^{\eta /2}d\chi \frac{dy}{d\chi }F(\eta ,\chi ,s,s^{\prime },\rho )$$

$$\frac{n^{\downarrow }-n^{\uparrow }}{n_0}=\frac{\frac{d{N}^{\uparrow \downarrow }}{dt}-\frac{d{N}^{\downarrow \uparrow }}{dt}}{\frac{d{N}^{\uparrow \downarrow }}{dt}+\frac{d{N}^{\downarrow \uparrow }}{dt}}\left(1-e^{-t/t_{\mathrm{pol}}}\right),$$

Full text

I Introduction

At the intensities which will be reached by next-generation ultraintense lasers ($\gtrsim 5\times 10^{22}-10^{24}$ W/cm2), such as several of those comprising the ‘Extreme Light Infrastructure’ Mourou et al. (2007), light-matter interactions are predicted to reach the new \colorred quantum electrodynamic (QED) plasma regime. \colorblack Matter in the laser focus is rapidly ionized creating a plasma whose behavior is characterized by the interplay of relativistic plasma and ‘strong-field’ quantum electrodynamic processes Di Piazza et al. (2012). Understanding this interplay is of fundamental interest: this regime is similar to that inferred to exist in extreme astrophysical environments such as pulsar Goldreich and Julian (1969) and active black hole magnetospheres Blandford and Znajek (1977). The QED processes can strongly modify the plasma’s behavior, for example leading to complete absorption of the laser pulse, with consequences for potential applications of these lasers, ranging from compact particle accelerators Macchi et al. (2013) to x-ray source generation Di Piazza et al. (2012). \colorred Despite its importance, the role of fermion spin in collective high-intensity laser-matter dynamics has rarely been considered Ekman et al. (2017). \colorblack

The important strong-field QED processes in laser-created QED plasmas are Erber (1966); Di Piazza et al. (2012); Ritus (1985): (i) incoherent emission of $\gtrsim$ MeV energy gamma-ray photons by electrons and positrons on acceleration by the macroscopic electromagnetic fields in the laser-produced plasma (strongly non-linear Compton scattering), with the resulting radiation-reaction strongly modifying the dynamics of the emitting electron or positron Ridgers et al. (2014); Di Piazza et al. (2010); (ii) pair creation by the emitted gamma-ray photons, in the same electromagnetic fields (the multi-photon Breit-Wheeler process Burke et al. (1997)). For example: non-linear Compton scattering and the resulting radiation can lead to almost complete laser absorption Zhang et al. (2015); Seipt et al. (2017); pair cascades (where pairs emit further gamma-ray photons, which generate even more pairs), can lead to the creation of critical density pair plasmas Grismayer et al. (2016); Nerush et al. (2011); Ji et al. (2014); Del Sorbo et al. (2017). It is therefore essential that we correctly include these QED processes in our models of the interaction of next-generation laser pulses with matter. Previous treatments of gamma-ray photon emission in QED plasmas have averaged over the spin.

In this \colorred article \colorblack we demonstrate a novel process, analogous to the Sokolov-Ternov effect in a strong magnetic field Sokolov and Ternov (1968); Belomesthnykh et al. (1984), whereby the electrons in laser-generated QED plasmas rapidly spin polarize due to asymmetry in the rate of spin flip transitions, i.e. interactions where the spin changes sign during the emission of a gamma-ray photon. We discuss several signatures of the spin polarization of the plasma. (i) Consideration of the spin of the electrons leads to a new quantum correction to the radiation reaction force: in the energetically favourable spin configuration the total power radiated and so radiation reaction force is enhanced compared to the unfavourable configuration, leading to an enhancement in total power radiated by the plasma compared the prediction assuming the spin is unpolarized. (ii) The relative energy emitted in the two possible polarizations of the gamma-ray photons is modified. (iii) This modification could decrease the rate of pair production.

\color

red

Spin polarized electron beams are important in high energy physics; the use of properly polarized beams in electron-positron colliders can suppress the Standard Model background in searches for new physics beyond the Standard Model Vauth and List (2016). Spin polarized beams are also used in electron spectroscopy for studying surface & thin-film magnetism and the electronic structure of metal, semiconductor surfaces and films Subashiev et al. (1998); Siegmann (1992). Spin polarizing plasma may enable applications of ultraintense lasers in these areas.

The paper is organized as follows. In Sec. II we demonstrate the laser-induced process of electron spin polarization by deriving a simple predictive model and applying it to the case of electrons orbiting at the magnetic node in the field of two counter-propagating lasers. In Sec. III we discuss signatures & consequences of spin polarization. Finally, in Sec. IV we draw conclusions.\colorblack

II Spin polarization by laser pulses

We focus on the case of electrons orbiting in a rotating electric field – a configuration that may be realized in the plasma created at the magnetic node of two colliding, circularly-polarized laser pulses. The electric field $\boldsymbol{E}$ at the magnetic node, say the plane $z=0$, rotates with constant amplitude Bell and Kirk (2008). Consequently, electrons subjected to $\boldsymbol{E}$ also rotate in the plane $z=0$ with (normalized) velocity $\boldsymbol{\beta}$.

To describe the spin polarization dynamics of electrons in the rotating electric field (indeed in any arbitrary electromagnetic field) we need to use a proper non-precessing spin polarization basis $\boldsymbol{\zeta}$ for which $d\boldsymbol{\zeta}/d\tau=0$ King (2015), defined in the rest frame of the electron, where $\tau=t/\gamma$ is the proper time. Spin-up and spin-down electron states, defined with respect to that basis, do not mix. According to the Bargmann-Michel-Telegdi equation Bargmann et al. (1959), which describes the precession of the expectation value of the spin polarization vector in an external electromagnetic field Vieira et al. (2011), the only non-precessing spin basis for an electron rotating in the rotating electric field is

[TABLE]

We will therefore consider spin polarization in the transverse direction where $\boldsymbol{\zeta}$ is perpendicular to $\boldsymbol{\beta}$. $\boldsymbol{\zeta}$ is analogous to the direction along the magnetic field in the discussion of the Sokolov-Ternov effect.

Each electron in the rotating electric field has a projection of its spin in the direction of the vector $\boldsymbol{\zeta}$ of $s\hbar/2$, where $s=\pm 1$ and $\hbar$ is Planck’s constant. We therefore divide the plasma electron population into two fractions $n^{s}$, characterized by their spin projection being parallel ($s=1=\uparrow$, higher energy) or antiparallel ($s=-1=\downarrow$, lower energy) to $\boldsymbol{\zeta}$. The number density of electrons with spin $s=\downarrow$, for instance, evolves according to the master equation

[TABLE]

The gamma-ray photon emission rates ${dN^{ss^{\prime}}}/{d\tau}=\sum_{\rho}{dN^{ss^{\prime}}_{\rho}}/{d\tau}$, summed over photon polarization states $\rho$, describe the transitions from one spin state to the other on emission of a gamma-ray photon, i.e. spin flip transitions. An analogous equation holds for $n^{\uparrow}$, the number density of electrons with spin $s=\uparrow$.

II.1 Rate of spin flip transitions

The expressions for the rate of spin flip transitions can be simplified when the strength parameter $a_{0}=85.5\lambda/\mu\rm m\sqrt{I/(10^{22}\;W/cm^{2})}$ of the laser electromagnetic waves is large. In that case, the formation phase interval for the emission of a photon becomes very short, $\propto 1/a_{0}$, and multiple photon emissions can be considered as incoherent. $a_{0}$ is indeed large in the cases considered here, where the intensity of each of the counter-propagating laser pulses $I$ is $\gg 10^{18}$ W/cm2 and $\lambda=1\;\mu$m – typical wavelength of ultraintense lasers and assumed in all our calculations. It is also the case that the electric field at the magnetic node is much less than the critical field of QED: the Sauter-Schwinger field $E_{S}=m_{e}^{2}c^{3}/(e\hbar)$ Heisenberg and Euler (1936), where $m_{e}$ and $e$ are the electron mass and charge respectively, and $c$ is the speed of light (all in Gaussian units). Under these two assumptions t\colorblackhe rates of gamma-ray photon emission (and pair production) are well described by the corresponding rates in constant crossed electric and magnetic fields Ritus (1985); Sokolov and Ternov (1968); Baier and Katkov (1968); Kirk et al. (2009). The rates of photon emission in this constant crossed field approximation depend only on: the electron’s initial and final spin projections $s$ and $s^{\prime}$, the emitted photon’s polarization $\rho$ and the quantum efficiency parameter $\eta=E_{RF}/E_{S}$ Baier and Katkov (1968). Here, $E_{RF}=\gamma\|\boldsymbol{E}\|$ is the electric field’s magnitude in the electron’s instantaneous rest frame. The electron spin and photon polarization dependent photon emission rates read Sokolov and Ternov (1968); Bordovitsyn (1999)

[TABLE]

with the classical dipole radiation emission power

[TABLE]

The photon quantum parameter

[TABLE]

depends on the gamma-ray photon energy

[TABLE]

with $\xi={3\eta}/{2}$ and

[TABLE]

$F(\eta,\chi,s,s^{\prime},\rho)$ is the spin- and photon polarization-dependent quantum synchrotron function. It is given by:

[TABLE]

where $K$ as the modified Bessel function of the second kind. The variable $\rho$ refers to the photon polarization, where $\rho=1=\sigma$ (polarized orthogonal to $\boldsymbol{\zeta}$) and $\rho=-1=\pi$ (polarized in the direction of $\boldsymbol{\zeta}$).

Fig. 1a shows the rates in Eq. (3), summed over photon polarizations, $dN^{ss^{\prime}}/d\tau=dN^{ss^{\prime}}_{\sigma}/d\tau+dN^{ss^{\prime}}_{\pi}/d\tau$, as functions of laser power of one of the counter-propagating pulses (the rates depends on $\eta$ which in turn depends on $a_{0}$, as explained in the next paragraph) and normalized to the unpolarized rate for a single electron ${dN}/{d\tau}=\sum_{ss^{\prime}}dN^{ss^{\prime}}/d\tau$ ***Our ${dN}/{d\tau}$ is twice that defined in Refs. Kirk et al. (2009); Ridgers et al. (2014), because two electron populations, characterized by their polarizaton, are involved. . The rate of processes resulting in a final spin aligned antiparallel to $\boldsymbol{\zeta}$ (${dN^{\downarrow\downarrow}}/{d\tau}+{dN^{\uparrow\downarrow}}/{d\tau}$) is higher than those resulting in a final spin aligned parallel (${dN^{\uparrow\uparrow}}/{d\tau}+{dN^{\downarrow\uparrow}}/{d\tau}$) due to the difference in the rates of spin flip. This means that the electron spin tends to align itself antiparallel to $\boldsymbol{\zeta}$ on a time scale which we denote as the spin polarization time $t_{\mathrm{pol}}$.

In order to calculate the rates in figure 1a we had to determine $\eta$. The electrons rotating in the electric field reach a steady state when the radiative losses due to gamma photon emission balance the acceleration due to the electric field. In this case Zhang et al. (2015) $\eta\approx 206\varepsilon_{\mathrm{rad}}\gamma^{2}$, where

[TABLE]

and the electron’s Lorentz factor is given by

[TABLE]

The factor

[TABLE]

accounts for the reduction of the power radiated due to quantum effects Ridgers et al. (2014), as compared to classical emission, where $\mathcal{P}^{ss^{\prime}}$ is the power radiated during an emission where the initial and final spins are $s$ and $s^{\prime}$ respectively.

The Sokolov-Ternov effect in a magnetic field has been been observed after one hour in storage rings Sokolov and Ternov (1968), for which the quantum efficiency parameter $\eta\sim 10^{-6}$ or less Mane et al. (2005). Next-generation ultraintense lasers are expected to reach more extreme regimes, in which $\eta\sim 0.1-1$ Bell and Kirk (2008). These high values of $\eta$ can be also achieved in the interaction of relativistic particles with strong crystalline fields Uggerhøj (2005), as it has been experimentally observed Andersen et al. (2012).

II.2 Spin polarization time

We may determine the degree of spin polarization antiparallel to $\mathbf{E\times\beta}$ (i.e. $\downarrow$) in time $t$ by solving (2) with the rates (3). We find that it evolves as

[TABLE]

with the spin polarization time $t_{\mathrm{pol}}=\gamma({dN^{\uparrow\downarrow}}/{d\tau}+{dN^{\downarrow\uparrow}}/{d\tau})^{-1}$, and assuming initially unpolarized electrons with total density $n_{0}=n^{\uparrow}+n^{\downarrow}$. By inverting the signs of $s$ and $s^{\prime}$ this equation describes the polarization of positrons as well Berestetskii et al. (1982).

In Figure 1b we have plotted the solutions of Eq. (12) as a function of time, normalized to the laser period $T=\lambda/c\approx 3.33$ fs, and of $a_{0}$. We see that for laser intensities just beyond the current limit ($a_{0}\approx 200$; $I\approx 5\times 10^{22}$ W/cm2) a significant ($\approx 10\%$) spin polarization is expected to occur rapidly, i.e. within a single laser period. $\gtrsim 50\%$ spin polarization is expected in one laser period for lasers of intensity well within the reach of next-generation laser systems ($a_{0}\gtrsim 600$; $I\gtrsim 5\times 10^{23}$ W/cm2) and a maximum spin polarization of 70% after one laser period is expected at a laser intensity $5\times 10^{24}$ W/cm2 ($a_{0}\approx 2000$).

The orbit considered here is unstable on a timescale of the order of the laser period Kirk (2016). However, as we have shown, a high degree of spin polarization can indeed occur in such a short timescale.

III Consequences of spin polarization

Let us now discuss several immediate consequences of the electron polarization on the subsequent QED plasma dynamics. Figure 2a shows the spin-dependent $g^{s}(\eta)=\sum_{s^{\prime}\rho}\mathcal{P}^{ss^{\prime}}_{\rho}/\mathcal{P}_{\mathrm{class}}$ by polarized electrons. The intensity of emission from an electron initially in the state $s=\downarrow$ is up to 30% higher than that from an electron initially in the state $s=\uparrow$. Hence, the power radiated and, consequently, the radiation reaction force depend on the spin of the electron. This is currently not included in the modeling of high intensity laser-matter interactions. As a result complete spin polarization of the plasma will increase the total power radiated by the plasma by up to $15\%$ compared to the emission from an equivalent unpolarized plasma; this could provide an observable signature of the spin polarization. Moreover, we would expect this causes an increased rate of laser absorption.

Note that the effect of spin on $\eta$ and $\gamma$ through the spin-dependent $g(\eta)$ is smaller than on the total power radiated ($<5\%$ instead of 15%, as can be numerically tested). Therefore, it was neglected in the calculation of the degree of spin polarization.

The electron spin polarization also affects polarization of the emitted gamma-ray photons. The power radiated in both $\sigma$ and $\pi$ polarizations, $P_{\sigma,\pi}=\sum_{ss^{\prime}}n^{s}\mathcal{P}^{ss^{\prime}}_{\sigma,\pi}$, at $t=T$, is plotted in Fig. 2b, normalized to the total power emitted by unpolarized electrons. Electron spin polarization causes the relative power radiated as $\sigma$ photons to increase by up to 30% compared to the unpolarized case (vice versa for $\pi$ photons). The relative yields of $\sigma$ and $\pi$ photons provides another signature of the spin polarization of the plasma.

The gamma-ray photons emitted by the electrons can decay to pairs in the electromagnetic fields of the laser pulses by the multi-photon Breit-Wheeler process. Pair cascades become important rapidly as $a_{0}$ exceeds 1200 ($I=2\times 10^{24}$ W/cm2) Bell and Kirk (2008); Fedotov et al. (2010); Bulanov et al. (2013); Grismayer et al. (2017). The polarization of the gamma-ray photons has been shown to modify the rate of pairs produced by up to 30% King et al. (2013). The modification to the polarization of the gamma-ray photons, caused by the spin polarization of the electrons, would be expected to produce a reduction to the rate of pair production of the same order and thus both electron and positron spin and photon polarization must be included in cascade simulations, yet they are currently neglected. The generated positrons will also spin polarize (parallel to $\boldsymbol{\zeta}$). Although prolific pair production by a cascade will add source terms to equations for $dn^{\uparrow}/d\tau$ & $dn^{\downarrow}/d\tau$ (plus two additional equations for positrons), we would not expect a qualitative change to the spin polarization trend in the magnetic node. Indeed, the rapid increase in the rate of pair production with laser intensity means that either the interaction is in a regime where a cascade does not occur and the number of positrons is small, or the cascade is rapidly quenched as the number density of pairs reaches the relativistic critical density and the source term shuts off.

The spin magnetization of electrons in a plasma at the magnetic node, resulting from the spin polarization, can be deduced by multiplying Eq. (12) by $\mu_{B}n_{0}$ Jackson (1999), where $\mu_{B}$ is the Bohr magneton. Assuming the plasma density is equal to the relativistic critical density (upper limit for laser propagation), the spin magnetization for $1$ $\mu$m wavelength lasers with $I>5\times 10^{22}$ W/cm2 ($a_{0}>200$) is $M\sim\rm kG$, after one laser period. This quantity may be considered another observable effect of spin polarization.

IV Conclusions

We have shown that electrons in a plasma created by two counter-propagating, ultraintense ($a_{0}>200$; \colorred $I>5\times 10^{22}$ W/cm2) \colorblack laser pulses can spin polarize (to $10\%-70\%$) on a femtosecond timescale. \colorred In this laser configuration, assuming the plasma is sufficiently rarefied that collective effects are negligible, the only experimental parameter that influence the degree of spin polarization is the laser intensity, since the wavelength is currently fixed to $\sim 1\;\mu$m by technological limitations. \colorblack

\color

red Spin polarization \colorblack can enhance the radiation reaction force on electrons and positrons in the plasma by up to 15%, compared to the prediction for unpolarized electrons. Consequently, the power radiated by the plasma is enhanced by the same percentage. The polarization of the radiated gamma-ray photons is also modified, by 30%, potentially reducing the number of pairs produced in the plasma by 30%.

The spin polarization must therefore be accounted for in the modeling of next-generation laser matter interactions. The possibility of producing spin polarized electrons and positrons with ultraintense lasers also opens up new applications. Polarized electrons are fundamental for the study of particle physics and are used in the spin polarized electron spectroscopy.

Acknowledgements

This work was funded by the UK Engineering and Physical Sciences Research Council (EP/M018156/1), by the Science and Technology Facilities Council (ST/G008248/1) \colorred and by the Knut and Alice Wallenberg Foundation. The authors are grateful to Antonino Di Piazza for useful and stimulating discussions. The data required to reproduce the results in this paper is available at DOI 10.15124/1afd25a0-a1e3-49ec-afa8-2e5f6d868124. \colorblack