In Situ Generation of High-Energy Spin-Polarized Electrons in a Beam-Driven Plasma Wakefield Accelerator
Zan Nie, Fei Li, Felipe Morales, Serguei Patchkovskii, Olga Smirnova, Weiming An, Noa Nambu, Daniel Matteo, Kenneth A. Marsh, Frank Tsung, Warren B. Mori, Chan Joshi
IIn Situ
Generation of High-Energy Spin-Polarized Electrons in a Beam-DrivenPlasma Wakefield Accelerator
Zan Nie, ∗ Fei Li, † Felipe Morales, Serguei Patchkovskii, Olga Smirnova, Weiming An, NoaNambu, Daniel Matteo, Kenneth A. Marsh, Frank Tsung, Warren B. Mori,
1, 4 and Chan Joshi ‡ Department of Electrical and Computer Engineering,University of California Los Angeles, Los Angeles, California 90095, USA Max Born Institute, Max-Born-Str. 2A, D-12489 Berlin, Germany Department of Astronomy, Beijing Normal University, Beijing 100875, China Department of Physics and Astronomy, University of California Los Angeles, Los Angeles, California 90095, USA (Dated: January 27, 2021)
In situ generation of a high-energy, high-current, spin-polarized electron beam is an outstandingscientific challenge to the development of plasma-based accelerators for high-energy colliders. Inthis Letter we show how such a spin-polarized relativistic beam can be produced by ionizationinjection of electrons of certain atoms with a circularly polarized laser field into a beam-driven plasmawakefield accelerator, providing a much desired one-step solution to this challenge. Using time-dependent Schr¨odinger equation (TDSE) simulations, we show the propensity rule of spin-dependentionization of xenon atoms can be reversed in the strong-field multi-photon regime compared withthe non-adiabatic tunneling regime, leading to high total spin-polarization. Furthermore, three-dimensional particle-in-cell (PIC) simulations are incorporated with TDSE simulations, providingstart-to-end simulations of spin-dependent strong-field ionization of xenon atoms and subsequenttrapping, acceleration, and preservation of electron spin-polarization in lithium plasma. We show thegeneration of a high-current (0.8 kA), ultra-low-normalized-emittance ( ∼
37 nm), and high-energy(2.7 GeV) electron beam within just 11 cm distance, with up to ∼
31% net spin polarization. Highercurrent, energy, and net spin-polarization beams are possible by optimizing this concept, thus solvinga long-standing problem facing the development of plasma accelerators.
In high-energy lepton colliders, collisions between spin-polarized electron and positron beams are preferred [1].Spin-polarized relativistic particles are chiral and there-fore ideally suited for selectively enhancing or suppressingspecific reaction channels and thereby better character-izing the quantum numbers and chiral couplings of thenew particles. To enable science at the ever-increasingenergy frontier of elementary particle physics while simul-taneously shrinking the size and cost of future colliders,development of advanced accelerator technologies is con-sidered essential. While plasma-based accelerator (PBA)schemes have made impressive progress in the past threedecades, a concept for in situ generation of spin-polarizedbeams has thus far proven elusive. The most commonspin-polarized electron sources are based on photoemis-sion from a Gallium Arsenide (GaAs) cathode [2]. Spin-polarized positron beams may be obtained from pair pro-duction by polarized bremsstrahlung photons, the latterproduced by passing a spin-polarized relativistic electronbeam through a high-Z target [3]. Unfortunately, noneof the above methods can generate ultra-short (few mi-crons long) and precisely (fs) synchronized spin-polarizedelectron beams necessary for injection into PBAs.The only previous proposal for producing spin-polarized electron beams from PBA [4–7] involves inject-ing spin-polarized electrons into a wake excited by a mod-erate intensity laser pulse or a moderate charged electronbeam in a density down-ramp. However, this proposal isa two-step scheme. The first step requires the generationof spin-polarized electrons outside of the PBA set-up by employing a complicated combination (involving multi-ple lasers) of molecular alignment, photodissociation andphotoionization of hydrogen halides [8, 9]. Even thoughthe spin polarization of the hydrogen atoms can be high,the overall net spin polarization of electrons ionized fromboth hydrogen and halide atoms is expected to be low[5]. The second step involves the injection of these spin-polarized electrons crossing the strong electromagneticfields of the plasma wake. To avoid severe spin depo-larization due to these strong electromagnetic fields, thewakefield should be moderately strong, which limits boththe accelerating gradient and charge of the injected elec-trons.In the one-step solution we propose here, the genera-tion and subsequent acceleration of spin-polarized elec-trons is integrated within the wake itself. Using a combi-nation of TDSE [10–12] and 3D-PIC [13–15] simulations,we show that spin-polarized electrons can be produced insitu directly inside a beam-driven plasma wakefield ac-celerator and rapidly accelerated to multi GeV energiesby the wakefield without significant depolarization. Elec-trons are injected and simultaneously spin-polarized viaionization of the outermost p-orbital of a selected noblegas (no need for pre-alignment) using a circularly po-larized laser [16]. The mitigation of depolarization is an-other benefit of laser-induced ionization injection [17, 18]:the electrons can be produced inside the wake close tothe wake axis, where the transverse magnetic and elec-tric fields of the wake are near zero [19], minimizing boththe beam emittance and depolarization due to spin pre- a r X i v : . [ phy s i c s . p l a s m - ph ] J a n cession. A third advantage of our scheme is that thewake can be in the highly nonlinear or bubble regimewhere electrons are rapidly accelerated to c minimizingthe emittance growth and accelerating the electrons athigher gradients.The proposed experimental layout of our scheme isshown in Supplementary Materials. A relativistic driveelectron beam traverses a column of gas containing a mix-ture of lithium (Li) and xenon (Xe) atoms. The ioniza-tion potentials of the 2 s electron of Li atoms and the out-ermost 5 p electron of Xe atoms are 5.4 eV and 12.13 eV,respectively. The electron beam fully ionizes Li atomsand produces the wake while keeping Xe atoms union-ized. If the driving electron beam is ultra-relativistic( γ (cid:29)
1) and sufficiently dense ( n b > n p , k p σ r,z < s electrons of the Li atoms are ionized during therisetime of the beam current and blown out by the trans-verse electric field of the beam to form a bubble-like wakecavity [19, 20] that contains only the Li ions and theneutral Xe atoms. Now an appropriately delayed circu-larly polarized ultra-short laser pulse copropagating withthe electron beam is focused at the entrance of the Liplasma to strong-field ionize the 5 p electron of the Xeatoms, producing spin-polarized electron beam close tothe center (both transversely and longitudinally) of thefirst bucket of the wake. The injected electrons are sub-sequently trapped by the wake potential and acceleratedto ∼ ∼
11 cm without significant depo-larization.It is known that strong field ionization rate of a fixedorbital in circularly polarized fields depends on the senseof electron rotation (i.e. the magnetic quantum num-ber m l ) in the initial state [21–23]. Based on this phe-nomenon and spin-orbit interaction in the ionic core,spin-polarized electrons can be produced by strong-fieldionization [16]. Here we use Xe atoms as an example, butthere are many other possibilities. Xe has six p -electronsin its outermost shell, with m l ≡ l z = 0 , ±
1. Strong-fieldionization from the p orbital ( m l = 0) in circularly po-larized laser fields is negligible in the strong–field regime[22, 23]. Consider first ionization from the p + orbital (co-rotating with the laser field) into the two lowest states ofXe + , P / and P / , see the left half of the ionizationpathways in Fig. 1(a). Removal of a spin-up p + electron( s z = 1 / l z = 1) would create a hole with j z = +3 / P / . Re-moval of a spin-down p + electron ( s z = − / l z = 1)would create a hole with j z = +1 / P / and P / states, with the Clebsch-Gordan coefficients squared splitting the two pathwaysas 1/3 for P / and 2/3 for P / . Repeating the sameanalysis for the p − electron (right half of ionization path-ways in Fig. 1(a)), one obtains the following expressionsfor the ionization rates W ↑ and W ↓ of spin-up and spin- (a) (b)(c)(d) p p p p (e) Co-rotatinglaser field Counter-rotatinglaser field P P S Xe p - (m l = -1)p + (m l = +1) m j = 1/2 m j = -1/2m j = 3/2 m j = 1/2 m j = -1/2 m j = -3/2 p p p p Up Down
Energy [eV] i on . p r abab ili t y [ a r b . u .] -3 -2 R a t e ( f s - ) Spin-upSpin-down Intensity (W/cm ) -2 -1 Y i e l d Spin-upSpin-down -1 Intensity (W/cm ) S p i n po l a r i z a t i on w/o averagingw/ averaging FIG. 1: (a) Schematic of spin-dependent photoionizationshowing possible ionization pathways from Xe to Xe + . (b)TDSE simulation results of the multi-photon ionization pho-toelectron spectra for the final ionic state, Xe + ( P / ) orXe + ( P / ), the energy and the initial quantum number m l = ± λ = 260 nmlaser pulse with peak intensity I = 2 . × W/cm . (c,d)Log-log plot of the simulated ionization rates and yields ofspin-up and spin-down electrons as a function of laser peak in-tensity of a 260 nm, 10 fs (FWHM), circularly polarized laser.(e) Spin polarization as a function of peak laser intensity with-out and with focal-volume averaging. down electrons [16]: W ↑ = W p + + 23 W p − + 13 W p − (1) W ↓ = W p − + 23 W p + + 13 W p + (2)where W p + , W p − , W p + , and W p − denote ionizationrates of a p + electron into the P / state, a p − electroninto the P / state, a p + electron into the P / state,and a p − electron into the P / state, respectively. Netspin polarization arises under two conditions: (i) either p + ionization dominates p − or vice versa and (ii) one ofthe two ionic states is more likely to be populated.In the adiabatic tunneling regime of strong-field ion-ization (Keldysh parameter [24] γ K (cid:28) p + and p − electrons are the same and ioniza-tion is not spin-selective. In the non-adiabatic tunnel-ing regime ( γ K ∼
1) [25], the p − electrons are morelikely to be ionized [22, 23, 26, 27], and the popula-tion of Xe + ( P / ) is suppressed due to its higher ion-ization potential ( I p ( P / ) = 13 .
44 eV compared toI p ( P / ) = 12 .
13 eV), satisfying both conditions for gen-erating spin-polarized electrons. Both the m l -dependentionization rates and the resulting spin polarization havebeen experimentally verified [26–30]. However, the ob-served spin polarization generated by ionization of Xe at800 nm and 400 nm changes sign both between the twoionization channels and across the photoelectron spec-trum [16, 28–30], reducing the net spin polarization uponintegrating over all photoelectron energies and both ionicstates.Theory and simulations show that propensity rules forionization can be reversed in the multi-photon regime( γ K (cid:29)
1) [31–33]. From our TDSE simulations, ion-ization of Xe by the third harmonic ( λ =260 nm) of aTi:Sapphire laser is strongly dominated by the removalof a p + electron at all laser intensities, until saturation,and for all photoelectron energies, with ionization intoXe + ( P / ) strongly suppressed (Fig. 1(b)), which leadsto high total spin-polarization. We have performed simu-lations for a range of intensities from 3 . × W/cm to6 . × W/cm , by solving the TDSE for each intensityfor four ionization pathways: p + , p + , p − , and p − ,and calculated the corresponding spin-up and spin-downelectron ionization rates and yields (Fig. 1(c,d)) accord-ing to Eq. (1)(2). The net spin-polarization with integra-tion over temporal and spatial intensity distribution, allphotoelectron energies, and final ionic states (see Supple-mentary Material of Ref. [34]) is shown in Fig. 1(e). Forthe laser intensity we used in the following PIC simula-tions ( I = 2 . × W/cm ), the net spin-polarizationreached 32% after focal-volume averaging.We have incorporated the spin-dependent ionizationresults into our wakefield acceleration simulations. Bytightly focusing a 260 nm circularly polarized laser pulseat the appropriate position in the wake bubble wherethe longitudinal and transverse electric fields are zero(Fig. 2(a)), electrons with a net spin polarization are gen-erated and injected into the wakefield. The trapping con-dition is given by [18] ∆Ψ ≡ Ψ − Ψ init (cid:46) −
1, whereΨ ≡ e ( φ − A z ) mc is the normalized pseudo potential of thewake, and Ψ init is the pseudo potential at the positionwhere the electron is born (injected). The pseudo poten-tial is maximum at the center of the bubble and minimumclose to the rear. For this reason, we choose to inject elec-trons where Ψ init is maximum so that the injected elec-trons are most easily trapped by the wake (Fig. 2(a,b)).Previous studies have shown that spin dynamics due tothe Stern-Gerlach force, the Sokolov-Ternov effect (spinflip), and radiation reaction force are negligible in ourcase [4–7]. Therefore, only spin precession needs to beconsidered. We have implemented the spin precessionmodule into the 3D-PIC code OSIRIS [13, 14] followingthe Thomas-Bargmann-Michel-Telegdi (T-BMT) equa-tion [35] d s /dt = Ω × s (3)where Ω = em ( γ B − γ +1 v c × E )+ a e em [ B − γγ +1 v c ( v · B ) − v c × E ] . Here, E , B are the electric and magnetic field, v is the electron velocity, γ = √ − v /c is the relativisticfactor, and a e ≈ . × − is the anomalous magneticmoment of the electron.
160 200z ( μ m)n (×10 cm ) driver 17 -3 (a) 1 Ψ Xe,wake 18 -3
240 28040 360 400 z ( μ m)440 480 x ( μ m ) x Xe,laser 18 -3 n (×10 cm )
Li 17 -3 S p i n v e c t o r den s i t y ( a r b . u . ) ΔΨ Ionizing laserDrive beamInjected electrons (b)(c) (d)
FIG. 2: (a),(b) Two snapshots show the charge density distri-bution of driving electron beam (grey), beam ionized Li elec-trons (green), laser ionized Xe electrons (brown), and wake-field ionized Xe electrons (blue) at (a) z = 210 µ m (end of ion-ization injection) and (b) z = 425 µ m (after being trapped).The dashed lines in (b) show the on-axis wake pseudo po-tential. The wakefield ionized Xe electrons (blue) are onlygenerated at the tail of the bubble and cannot be trapped bythe wake. (c),(d) The spin vector density distribution of Xeelectrons ionized by the UV laser at the same moment of (a)and (b). As shown in Fig. 2(c) and (d), the spin vector distribu-tion is at first concentrated around the top and bottompoints of s z = ± ◦ , which is negligible compared tothe spread due to spin precession induced by the wake-field at later times (Fig. 2(d)).Figure 3 describes start-to-end simulations incorporat-ing both the TDSE and PIC components. The wholesimulation consists of two stages: the injection and trap-ping stage (0-0.74 mm) and acceleration stage (0.74-110 mm). The injection and trapping stage was simu-lated using the OSIRIS code [13, 14] with high tempo-ral resolution and the acceleration stage was simulatedusing the QPAD code [15, 36] with lower temporal res-olution. The density profiles of Xe and Li gases areshown in Fig. 3(a). The Xe gas column, with a density of n Xe = 8 . × cm − , is 420 µ m long. The exact lengthof the Xe region is not important as long as Xe is notionized by the electron beam. The Li gas, with a densityof n Li = 8 . × cm − , extends across the whole inter-action region and provides background plasma electronswhen ionized by the drive electron beam. The drivingbeam electron energy is 10 GeV with a Gaussian profile n b = N (2 π ) / σ r σ z exp( − r σ r − ξ σ z ), where N = 4 . × (658 pC), and σ r = σ z = 11 . µ m are the transverse andlongitudinal beam sizes, respectively. Such a beam hasa maximum electric field of 16 GV/m, which is far largerthan that required to fully ionize the Li atoms, but notthe Xe atoms. It forms the plasma and blows out theplasma electrons to create the wake cavity. The 260 nmionization laser is delayed by 148 fs (44.5 µ m) from thepeak current position of the drive electron beam. Thelaser pulse has Gaussian envelope with pulse duration(FWHM) of 30 fs and focal spot size of w = 1 . µ m.The peak laser intensity is 2 . × W/cm (the sameintensity as in Fig. 1(b)) to make a tradeoff between netspin polarization and ionization yield. At this peak laserintensity, the 5 p (outermost) electron of Xe is partiallyionized ( ∼
32% at focus) while the 5 p (second) electronof Xe is not ionized at all ( < − ).Evolution of injected beam parameters includingcharge, peak current, normalized emittance, and spinvector distribution as a function of propagation distancein the plasma are shown in Fig. 3(b)-(e). All photoion-ized electrons with charge of 3 pC (Fig. 3(b) left axis) areinjected, trapped and accelerated to 2.7 GeV (Fig. 3(c))within 11 cm to give a peak current of I = 0 . (cid:15) n = 36 . x and z directions are shown in Fig. 3(d) and (e), respec-tively. The spin spread in x (or y ) direction is symmetricso that (cid:104) s x (cid:105) ≈ (cid:104) s y (cid:105) ≈
0) as shown in Fig. 3(d).Therefore, the net spin polarization P = P z = (cid:104) s z (cid:105) only depends on the spin distribution in the z direction.The spin depolarization mainly occurs during the first500 µ m distance as electrons are injected into the wakeuntil they become ultra-relativistic ( γ ∼ (cid:104) s z (cid:105) = 30 .
7% (Fig. 3(e)), corresponding to 96%of the initial spin polarization at birth. This result iscomparable to the first-generation GaAs polarized elec-tron sources, that are most commonly used in conven-tional rf accelerators. The reason why depolarization issmall in our case is that the injected electrons are alwaysclose to the axis of the wake so that the transverse mag-netic and electric fields they feel are close to zero. Ina nonlinear wake bucket, the transverse magnetic field B φ scales linearly with distance from the center of thewake ( B φ ∝ r ) [19]. From Eq. (3), the spin precessionfrequency Ω ≈ − eB φ /mγ when γ ∼
1. Therefore, if theelectrons are close to the axis ( r ≈ ≈
0. In addition, once the electron energy isincreased to ultra-relativistic level ( γ (cid:29)
1) by the longi-tudinal wakefield, the spin precession effect is negligible P ea k c u rr en t ( k A ) n ( n m ) s x s z (a) D en s i t y ( × c m ) - Xe Li (b)
Injection and trapping stage Acceleration stage d N / d s ( a r b . u . ) x d N / d s ( a r b . u . ) z d N / d ( a r b . u . )
024 0.81.6 C ha r ge ( p C ) (c)(d)(e) FIG. 3: (a) The density profiles of the Xe and Li gases usedin the simulations. (b) Evolution of beam charge (left blueaxis), peak current (right red axis) and normalized emittance (cid:15) n (right green axis). (c) Evolution of Lorentz factor γ . Thedashed line presents mean energy (cid:104) γ (cid:105) . d, Evolution of spinvector in the x direction: s x . The dashed line represents (cid:104) s x (cid:105) .e, Evolution of spin vector in the z direction: s z . The topbox plots the s z distribution in the range of 0.8 and 1. Thecentral box plots (cid:104) s z (cid:105) (net spin polarization) in the range of0.2 and 0.4. The bottom box plots the s z distribution inthe range of − − .
8. The long vertical dashed blackline marks the focal position ( z = 0 .
18 mm) of the ionizationlaser. The plots in the range of 0.74-2.5 mm and 2.5-110 mmare shown in two temporal scales to clearly present the wholeevolution dynamics but the actual simulation was run withone temporal resolution in the whole acceleration stage. [4].We have investigated how the variation of injectedbeam charge (by either varying the Xe density or the spotsize of the ionization laser) affects the final spin polariza-tion of the injected electrons. The parameter scanningresults are summarized in Fig. 4(a). The spin polariza-tion drops slowly and linearly with the increase of thebeam charge. This indicates that the space charge forceis the probable cause of spin depolarization in our case,which is confirmed by analyzing the tracks of the ion-ized electrons (see Supplementary Material for details,which includes Ref. [38]). Considering practical issues in S p i n po l a r i z a t i on ( % ) Xe density scanningSpot size scanning S p i n po l a r i z a t i on ( % ) n ( n m ) xy (a) (b)Laser offset in x direction ( m)Injected charge (pC) FIG. 4: (a) Spin polarization v.s. injected beam charge byeither varying the Xe density (blue) or the spot size of the ion-ization laser (red). The five data points of Xe density scanningcorrespond to Xe density of 8 . × cm − , 8 . × cm − ,1 . × cm − , 3 . × cm − , and 7 . × cm − whilekeeping the spot size of 1.5 µ m. The four data points of spotsize scanning correspond to ionization laser spot size of 1 µ m,1.5 µ m, 2 µ m, and 2.5 µ m while keeping the Xe density of8 . × cm − . (b) Spin polarization (left) and normalizedemittance (right) after propagation distance of 0.74 mm v.s.laser transverse displacement in x direction. experiments, we have investigated how the laser trans-verse offset relative to the drive electron beam affectsthe spin polarization and normalized emittance as shownin Fig. 4(b). The spin polarization is essentially not af-fected by the transverse displacement in ± µ m range.The normalized emittance in x direction grows with thelaser offset in x direction and the normalized emittance in y direction remains almost the same. These emittancesare within values envisioned for future plasma-based col-liders. Another possible issue in experiments might bethe synchronization between the drive electron beam andthe ionizing laser pulse. To make sure the ionized elec-trons are trapped by the wake (meet the trapping con-dition ∆Φ (cid:46) − ±
80 fs in our simulation case. This requirementcan be further relaxed if using higher drive beam chargeand lower plasma density.Here we have used a single collinearly (to the elec-tron beam) propagating laser pulse for ionizing the Xeatoms. To obtain even lower emittance ( <
10 nm) beams,one could use two transverse [39] or longitudinal [40] col-liding laser pulses instead. We also note that the beamcharge, peak current, and the maximum spin polariza-tion observed here are not limited by theory. The firsttwo can be increased by optimizing the ionizing laser pa-rameters, drive beam parameters, and the beam loadingwithin the wake. The latter may be increased by usingelectrons in the d or f orbitals instead of p orbitals –for instance by using Yb III [41, 42]. A modified versionof this scheme may also be useful for generating a spin-polarized electron beam in a laser wakefield accelerator(LWFA) [43].We thank Nuno Lemos and Christopher E. Claytonfor useful discussions regarding this work. 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