Attosecond control of spin polarization in electron-ion recollision driven by intense tailored fields
David Ayuso, ?lvaro Jiménez-Galán, Felipe Morales, Misha Ivanov, Olga Smirnova
AAttosecond control of spin polarization in electron-ion recollisiondriven by intense tailored fields
David Ayuso , Alvaro Jim´enez-Gal´an , FelipeMorales , Misha Ivanov , , and Olga Smirnova , ∗ Max-Born Institute for Nonlinear Optics and Short Pulse Spectroscopy,Max-Born-Straße 2A, D-12489 Berlin, Germany Department of Physics, Imperial College London,South Kensington Campus, SW72AZ London, UK Institute f¨ur Physik, Humboldt-Universitt zu Berlin,Newtonstraße 15, D-12489 Berlin, Germany and Technische Universit¨at Berlin, Ernst-Ruska-Geb¨aude,Hardenbergstraße 36A, 10623 Berlin, Germany
Abstract
We show that electrons recolliding with the ionic core upon tunnel ionization of noble gas atomsdriven by a strong circularly polarized laser field in combination with a counter-rotating secondharmonic are spin polarized and that their degree of polarization depends strongly on the recollisiontime. Spin polarization arises as a consequence of (1) entanglement between the recolliding electronand the ion, and (2) sensitivity of ionization to the sense of electron rotation in the initial state. Wedemonstrate that one can engineer the degree of spin polarization as a function of time by tuningthe relative intensities of the counter-rotating fields, opening the door for attosecond control ofspin-resolved dynamics. ∗ [email protected] a r X i v : . [ phy s i c s . a t o m - ph ] D ec . INTRODUCTION The Stern-Gerlach experiment [1–3] revealed, in 1922, that an electron possesses an in-trinsic angular momentum that is quantized and that is independent of its orbital angularmomentum: the spin. Electron spin governs the behavior of matter, arranging the elec-tronic shells of the elements in the periodic table through the Pauli exclusion principle [4]and giving rise to magnetism [5]. Ever since its discovery, finding ways of producing spinpolarized electrons has attracted the interest of physicists [6]. In 1969, Fano demonstratedthat one-photon ionization of atoms with circularly polarized light in the energy region ofCooper minima can lead to the generation of electrons with a high degree of spin polar-ization [7]. Another way of producing polarized currents is via ionization from a selectedstate of an atom or a molecule presenting fine structure splitting [8]. This investigationhas been extended to the multiphoton case in the perturbative regime [9–11]. However, de-spite its importance, spin polarization with strong laser fields has received no attention untilvery recently [12–15]. The first theoretical predictions of spin polarization in noble gasesupon strong field ionization with circularly polarized light [12] have just been experimentallyconfirmed [14].Spin polarization in the strong field regime is a consequence of electron-ion entanglementand the sensitivity of the ionization yield to the sense of electron rotation in the initial state[12]: electrons that counter-rotate with the field ionize more easily than the co-rotatingelectrons, yielding different ionization rates for p − and p + electrons in noble gases [16–21] and diatomic molecules [15]. The possibility of inducing recollision of spin-polarizedelectrons with the parent ion can open new directions in attosecond spectroscopy [13, 14].Not surprisingly, the degree of spin polarization is higher for higher ellipticity of the ionizingfield. The flip side of the coin, however, is that high ellipticity of the ionizing field reduces thechance of electron return to the parent ion. In this context, the use of an intense circularlypolarized laser field in combination with its counter-rotating second harmonic, known asa bi-circular field, constitutes a powerful tool for introducing the spin degree of freedominto attosecond science, due to the opportunity to combine circular polarization with theefficient recollision offered by these fields [13, 22–26]. The application of bi-circular fields canlead to the production of ultrashort circularly and elliptically polarized laser pulses in theXUV domain [25–31]. Their chiral nature offers unique possibilities for probing molecular2hirality [32] or symmetry breaking [33] at their natural time scales via high harmonicgeneration spectroscopy. Recent theoretical work [13] has indicated that electrons producedupon strong field ionization with bi-circular fields are spin polarized.Here we present a detailed theoretical study of spin polarization in electron-core recollisiondriven by bi-circular fields, emphasizing the possibilities for, and the physical mechanismsof controlling the degree of spin-polarization by changing the parameters of the bi-circularfield. The paper is organized as follows. Section II describes the theoretical approach, whichis based on the strong field approximation (SFA). Section III describes our results, focusingon the analytical analysis of how the properties of the quantum electron trajectories definethe spin polarization. This allows us to establish the origin of spin polarization in bi-circularfields (section III A) and show how to achieve its attosecond control by tailoring the laserfields (section III B). Section IV concludes the paper. II. METHOD
Consider ionization, followed by electron-parent ion recollision, of xenon atoms drivenby a strong right circularly polarized (RCP) field in combination with the counter-rotatingsecond harmonic. The resulting electric field can be written, in the dipole approximation,as: F ( t ) = (cid:104) F ,ω cos ( ωt ) + F , ω cos (2 ωt ) (cid:105) ˆx + (cid:104) F ,ω sin ( ωt ) − F , ω sin (2 ωt ) (cid:105) ˆy (1)where F ,ω and F , ω are the amplitudes of the right and left circularly polarized fields,respectively, with frequencies ω and 2 ω . Within the strong-field approximation (SFA), thecontinuum electron wave function at time t is given by [34]: | Ψ( t ) (cid:105) = i (cid:90) tt dt (cid:48) e i IP( t (cid:48) − t ) F ( t (cid:48) ) (cid:90) d p d ( p + A ( t (cid:48) )) | p + A ( t ) (cid:105) V (2)where IP is the ionization potential, p is the drift (canonical) momentum, related to the thekinetic momentum k ( t ) by k ( t ) = p + A ( t ), d ( p + A ( t )) = (cid:104) p + A ( t ) | ˆ d | Ψ (cid:105) is the transitiondipole matrix element from the initial ground state | Ψ (cid:105) (the system is assumed to be in theground state at t = t ) to a Volkov state | p + A ( t ) (cid:105) V , given by | p + A ( t ) (cid:105) V = 1(2 π ) / e − iS V ( t,t (cid:48) , p ) e i [ p + A ( t )] · r (3)3here S V ( t, t (cid:48) , p ) is the Volkov phase: S V ( t, t (cid:48) , p ) = 12 (cid:90) tt (cid:48) dτ (cid:2) p + A ( τ ) (cid:3) (4)Eq. 2 can be used to calculate different observables, such as photoelectron yields, inducedpolarization and harmonic spectra [34]. Here we are interested in analyzing the degree ofspin-polarization of the electrons that are driven back to the ionic core. This requires ameasure of the recollision probability, resolved on the state of the ion and on the spin ofthe returning electron. The latter is determined by the initial magnetic quantum numberof the state from which the electron tunnels and the state of the ion that has been createdupon ionization, as described in [12]. As for the recollision probability, given that the sizeof the returning wave packet far exceeds the size of the atom, an excellent measure of therecollision amplitude is the projection of the continuum wave function (eq. 2) | Ψ( t ) (cid:105) onany compact object at the origin; the recollision current will scale with the object area. Toobtain the required recollision probability density at the origin, we simply project | Ψ( t ) (cid:105) onthe delta-function at the origin, yielding a rec ( t ) = i (cid:90) tt dt (cid:48) F ( t (cid:48) ) (cid:90) d p d ( p + A ( t (cid:48) )) e − i [ S V ( t,t (cid:48) , p )+IP( t − t (cid:48) )] (5)The degree of spin polarization of the recolliding electrons as a function of the recollisiontime t is given by the normalized difference between the recollision probability densities forelectrons recolliding with spin up ( w ↑ ( t ) = | a ↑ ( t ) | ) and spin down ( w ↓ ( t ) = | a ↓ ( t ) | ) [12]:SP( t ) = w ↑ ( t ) − w ↓ ( t ) w ↑ ( t ) + w ↓ ( t ) (6)The densities w ↑ ( t ) and w ↓ ( t ) are obtained from the recollision densities w p + ,p − IP P / , / ( t ) = (cid:12)(cid:12) a p + ,p − IP P / , / ( t ) (cid:12)(cid:12) correlated to ionization from the p + and p − orbitals, resolved on the ionicstates P / and P / , and the corresponding Clebsch-Gordan coefficients [12]: w ↑ ( t ) = w p + IP P / ( t ) + 23 w p − IP P / ( t ) + 13 w p − IP P / ( t ) (7) w ↓ ( t ) = w p − IP P / ( t ) + 23 w p + IP P / ( t ) + 13 w p + IP P / ( t ) (8)The contribution of the p orbital is negligible [16, 18]. The key quantities in these expres-sions are the recollision densities resolved on the initial orbital and the final ionic state, w p − IP P / = (cid:12)(cid:12) a p − IP P / (cid:12)(cid:12) , etc. Application of the saddle-point method (see e.g. [34]) to the4ntegral eq. 5 allows us to perform the semi-classical analysis of this expression in termsof electron trajectories, getting insight into the physical origin of spin polarization duringrecollision. The saddle points are calculated by solving the following set of equations [34]:[ p + A ( t i )] (cid:90) t r t i dτ [ p + A ( τ )] = 0 (10)where IP is the ionization potential, t i and t r are the complex ionization and recollisiontimes, respectively. Eq. 9 describes tunneling and eq. 10 requires that the electron returnsto the core.Fig. 1 shows a schematic representation of the process on the complex time plane. Theelectron enters the barrier at complex time t i = t (cid:48) i + it (cid:48)(cid:48) i . The motion in the classically forbid-den region occurs along the imaginary time axis and the electron is born in the continuumat a real time t (cid:48) i . As a result, the recollision time t r and the canonical momentum p are, ingeneral, complex. To further simplify the analysis, we can take into account that for mostof the relevant trajectories the imaginary part of their recollision time is rather small. Thisallows one to keep the recollision time on the real time axis, also simplifying the treatmentof the usual divergences near the cutoff region, see [34].The recollision densities correlated to ionization from p + and p − orbitals are proportionalto: w p m IP ∝ (cid:12)(cid:12)(cid:12) e − i [ S V ( t r ,t i , p )+IP( t r − t i )]+ imφ k ( ti ) (cid:12)(cid:12)(cid:12) (cid:39) e (cid:61){ S V ( t (cid:48) i ,t i , p ) }− t (cid:48)(cid:48) i e − m (cid:61){ φ k ( ti ) } (11)In this expression, the first key quantity that determines the magnitude of w p m IP is theimaginary part of action. It is mostly accumulated between the times t i = t (cid:48) i + it (cid:48)(cid:48) i and t (cid:48) i , i.e. in the classically forbidden region. The second key quantity, which depends on theprojection m of the angular momentum, is the complex-valued ionization angle φ k ( t i ) . It isgiven by the following expression: φ k ( t i ) = atan (cid:32) − k (cid:48) x ( t i ) k (cid:48) y ( t i ) (cid:33) + i atanh (cid:32) k (cid:48) x ( t i ) k (cid:48)(cid:48) y ( t i ) (cid:33) (12)with k x ( t i ) = k (cid:48) x ( t i ) + ik (cid:48)(cid:48) x ( t i ) and k y ( t i ) = k (cid:48) y ( t i ) + ik (cid:48)(cid:48) y ( t i ) being the complex velocities alongx and y directions, respectively. Note that the difference between the recollision densitiesfrom p + and p − orbitals depends solely on the imaginary part of the ionization angle.5inally, the electron recollision energy is calculated as E rec = [ p + A ( t r )] t r on the real time axis. Real time I m a g i n a r y t i m e t i " t r 't r "0 Recollisiont r = t r ' + it r "EXACT SOLUTIONSTunnel entranceti = t i ' + it i "t i 'Tunnel exit Real time I m a g i n a r y t i m e t i "t r "0 Recollisiont r = t r ' APPROXIMATE SOLUTIONSTunnel entranceti = t i ' + it i "t i 'Tunnel exitPropagation I o n i z a t i o n I o n i z a t i o n Propagation
FIG. 1. Schematic representation of the contour time integration of the action. Ionization starts ata complex time t i = t (cid:48) i + it (cid:48)(cid:48) i , the electron tunnels out of the potential barrier at the real time t (cid:48) i , andreturns to the ionic core at t r = t (cid:48) r + it (cid:48)(cid:48) r (left panel). If the imaginary part of the recollision time issufficiently small, one can keep the recollision time on the real time axis (right panel), simplifyingthe treatment of the cutoff region. III. RESULTS
The Lissajous curves of the electric field considered here (see eq. 1) and of the corre-sponding vector potential A ( t ), given by F ( t ) = − d A ( t ) /dt , are shown in fig. 2, as wellas the ionization and the recollision time windows (the field parameters are given in thefig. 2 caption). The resulting electric field has a three-fold symmetry, with 3 peaks percycle oriented at angles 0, 2 π/ π/ xy plane. Ionization is more likely tooccur near the maxima of the electric field, where the tunneling barrier is thinner. Electronsliberated just before these maxima are unlikely to return to the core, those released afterthe maximum can recollide.Consider strong field ionization of a xenon atom from the outermost 5p shell. The spin-orbit interaction splits the energy levels of the ion into P / and P / , with ionizationpotentials IP P / = 12 .
13 eV and IP P / = 13 .
43 eV. Our calculations considered bothionic states, as needed for calculating spin polarization. The saddle point equations (eqs. 96nd 10) have been solved numerically, allowing the ionization and return times to be complex(exact solutions), and also by keeping the return time on the real time axis (approximatesolutions), as represented in fig. 1. The real and imaginary parts of the ionization time,the complex part of the recollision time and the recollision energy (evaluated using eq. 13)are shown in fig. 3, as functions of the real part of the return time. Our exact solutionsagree with those reported previously in [24] and the approximate solutions agree well withthe exact ones. We can see that the imaginary part of the recollision time (fig. 3c) is rathersmall, except near the cutoff, where the saddle point method diverges. The main advantageof using approximate solutions and keeping the recollision time on the real time axis is thatthe ionization time and the recollision energy behave smoothly in the vicinity of the cutoff,while being very similar to the exact solutions outside this region.Let us compare now the results for the states P / and P / of the ion. As expected, thereal part of the ionization time (fig. 3a) and the recollision energy (fig. 3c) are almost iden-tical in both cases. The imaginary part of the ionization time, however (fig. 3b), is slightlysmaller for the P / state, with the lower IP, resulting in higher ionization amplitudes. F x (t) (atomic units) F y ( t ) ( a t o m i c un i t s ) -0.05 0 0.05 0.1-0.1-0.08-0.06-0.04-0.0200.020.040.060.080.1 recollision ionizationshort trajectorieslong trajectories -1.5 -1 -0.5 0 0.5 1 1.5-1.5-1-0.500.51 recollisionionization A y ( t ) ( a t o m i c un i t s ) A x (t) (atomic units)short trajectorieslong trajectories FIG. 2. Electric field (left panel) and vector potential (right panel) resulting from combining aRCP field of frequency ω = 0 .
05 a.u. and intensity I = 10 W cm − with a LCP field of frequency2 ω and equal intensity. The ionization and recollision time-windows are indicated in the figuresfor short (green) and long (yellow) trajectories for one of the three ionization bursts. .5 0.6 0.7 0.8 0.9 1 1.1 1.20.040.060.080.10.120.140.160.18 short trajectories long trajectories Approx. solutionsExact solutionsApprox. solutionsExact solutionsReal recollision phase (units of π ) I m a g i n a r y r e c o lli s i o n p h a s e ( un i t s o f π ) Real recollision time (asec)
800 1000 1200 1400 1600 1800
Real recollision time (asec)
800 1000 1200 1400 1600 1800
Real recollision phase (units of π ) R e a l i o n i z a t i o n p h a s e ( un i t s o f π ) I m a g i n a r y i o n i z a t i o n p h a s e ( un i t s o f π ) R e c o lli s i o n e n e r g y ( e V ) short trajectories long trajectorieslong trajectories long trajectoriesshort trajectories short trajectoriesApprox. solutionsExact solutionsApprox. solutionsExact solutions R e a l i o n i z a t i o n t i m e ( a s e c ) I m a g i n a r y i o n i z a t i o n t i m e ( a s e c ) I m a g i n a r y r e c o lli s i o n t i m e ( a s e c ) A BC D
FIG. 3. Saddle point solutions for the bi-circular field represented in fig. 2 as functions of the realpart of the recollision time: real (A) and imaginary (B) parts of the ionization time, imaginarypart of the recollision time (C), and recollision energy (D). Full saddle points (dashed lines) havebeen calculated allowing both ionization and recollision times to be complex, whereas approximatesolutions (full lines) have been obtained by keeping the time of return on the real time axis (seefig. 1). Results are shown for the ionic states of xenon P / (red lines) and P / (blue lines),with ionization potentials IP P / = 12 .
13 eV and IP P / = 13 .
43 eV.
We have evaluated the degree of spin polarization in recollision (eq. 6) using the saddlepoint solutions shown in fig. 3. Total spin polarization is shown in fig. 4 as a function ofthe recollision time, together with the degree of polarization resolved in the P / and P / states of the core. It is clear from the figure that recolliding electrons are spin-polarized andthat their degree of polarization depends strongly on the recollision time. Electrons thatreturn to the core at earlier (later) times are more likely to have spin up (down). Note alsothat spin polarization resolved in the ionic states P / and P / has opposite sign. Both8pin polarization resolved on the states of the ion and the total spin polarization changesign at the recollision phase (time) of 0 . π rad (1 .
11 fsec). Each return time is associatedwith a given recollision energy, which is the well-known time-energy mapping [34] (see fig.3d). Fig. 4 shows spin polarization as a function of the recollision energy for short and longtrajectories. Whereas for the short trajectories spin polarization changes dramatically as afunction of the recollision energy, for the long trajectories the variation is rather smooth. Sp i n p o l a r i z a t i o n Recollision time (asec)Recollision phase (units of π )Approx. solutionsExact solutionsshort trajectories long trajectoriestotal FIG. 4. Total spin polarization (black lines) and spin polarization resolved in the P / (red lines)and in the P / (blue lines) states of the core as a function of the recollision time. Spin polarizationhas been calculated using the exact (full lines) and the approximate (dashed lines) saddle pointssolutions shown in fig. 3.
10 15 20 25 30 35 40 45-0.6-0.4-0.200.20.40.60.81 15 20 25 30 35 40 45 Sp i n p o l a r i z a t i o n Recollision energy (eV)Approx. solutionsExact solutions short trajectories long trajectories totaltotal Recollision energy (eV) -0.6-0.4-0.2010.20.40.60.8
FIG. 5. Total spin polarization (black lines) and spin polarization resolved in the P / (red lines)and in the P / (blue lines) states of the core as a function of the recollision energy for short (leftpanel) and long (right panel) trajectories. Spin polarization has been calculated using the exact(full lines) and the approximate (dashed lines) saddle points solutions shown in fig. 3. A. Origin of spin polarization
To better understand the physical origin of spin polarization in recollision, let us analyzethe recollision densities for different ionic channels. These are presented in fig. 6 as a functionof the recollision time, as well as the total recollision densities corresponding to electronswith spin up and spin down (eqs. 7 and 8). There are three important things worth notinghere. First, the recollision densities correlated to the P / state of the core ( w p − IP P / and w p + IP P / ) are higher than those for the P / state ( w p − IP P / and w p + IP P / ) because the lowerionization potential of this ionic state leads to smaller imaginary ionization times (see fig.3b) - the tunneling barrier is thinner. Second, all recollision densities exhibit a maximumvalue that arises at lower recollision times in the case of the p + orbital ( w p + IP P / and w p + IP P / ).Third, the densities resolved on the P / and P / states of the core cross at φ r = 0 . π rad ( t r = 1044 asec) and φ r = 0 .
70 rad ( t r = 1061 asec), respectively, leading to changes ofsign in spin polarization (see fig. 4).In order to understand these features, we have examined the saddle point solutions at t = t i , when the electron enters the classically forbidden region. The ionization velocity andthe ionization angle are shown in fig. 7 as a function of the recollision time. We can see that,for a recollision phase (time) of 0 . π rad (1 .
11 fsec), the real part of the ionization angle10resents a jump of π and its imaginary component becomes zero. A purely real ionizationangle leads to equal tunnelling probabilities for p + and p − orbitals (see eq. 11) and thus nospin polarization. Recollision time (asec)Recollision phase (units of π) long trajectoriesshort trajectories R e c o lli s i o n d e n s i t y ( a r b . un i t s ) FIG. 6. Recollision densities for p + and p − electrons correlated to the states of the ion P / and P / as a function of the recollision time (full lines) and total recollision densities for electrons withspin up and spin down (dashed lines), calculated using the approximate quantum orbits resultingfrom keeping the time of return on the real time axis. k x ( t i ) ( a t o m i c un i t s ) I o n i z a t i o n a n g l e ( un i t s o f π ) k y ( t i ) ( a t o m i c un i t s ) Real recollision phase (units of π ) Real recollision phase (units of π ) Real recollision phase (units of π ) Real recollision time (asec)
800 1000 1200 1400 1600 1800
Real recollision time (asec)
800 1000 1200 1400 1600 1800
Real recollision time (asec)
800 1000 1200 1400 1600 1800
Real partImaginary part Real partImaginary part Real partImaginary part long trajectoriesshort trajectories long trajectoriesshort trajectories long trajectoriesshort trajectories
FIG. 7. Real (full lines) and imaginary (dashed lines) parts of the electron velocity at t = t i (when it enters the classically forbidden region) along the x (left panel) and the y (central panel)directions, and ionization angle (right panel) calculated using eq. 12, as a function of the recollisiontime. The time-dependent sensitivity of the recollision densities to the sense of rotation of theelectron in its initial state can be understood by examining different quantum trajectories.11ig. 8 contains a representation of the values of the electric field and the ionization velocityat t = t i of three quantum orbits that recollide with the P / state of the ion at differenttimes: φ r = 0 . π rad (positive spin polarization), φ r = 0 . π rad (no spin polarization)and φ r = 0 . π rad (negative spin polarization), calculated by keeping the time of returnon the real time axis. We will refer to them as trajectories A, B, and C, respectively. Thethree trajectories have similar values of k (cid:48)(cid:48) ( t i ) and F ( t i ). However, their values of k (cid:48) ( t i ) arevery different. Let us analyze the motion of the electron through the classically forbiddenregion, which occurs in imaginary time (see fig. 1) and along the complex plane of spatialcoordinates ( r = r (cid:48) + i r (cid:48)(cid:48) ). The real part of the trajectory depends on k (cid:48)(cid:48) and F (cid:48) according to k (cid:48)(cid:48) = d r (cid:48) /dτ and F (cid:48) = d k (cid:48)(cid:48) /dτ , with τ being the complex time variable. Under the barrier, dτ = − dt (cid:48)(cid:48) (see fig. 1). Equivalently, the motion in the plane of imaginary coordinates isdictated by k (cid:48) = − d r (cid:48)(cid:48) /dτ and F (cid:48)(cid:48) = − d k (cid:48) /dτ . Trajectories A, B and C are depicted infig. 9. Their real parts in the classically forbidden region are almost identical because theypresent similar values of k (cid:48)(cid:48) ( t i ) and F (cid:48) ( t i ). The motion in the imaginary plane, however, isdifferent due to the very distinct values of k (cid:48)(cid:48) ( t i ). Trajectory B presents k (cid:48) ( t i ) = 0 and thusits motion in the complex plane is solely dictated by the imaginary value of the electric field,which barely changes its direction during tunneling. Thus, the motion in the imaginaryplane occurs along a straight line. The initial values of k (cid:48) for trajectories A and C are nonzero and point in opposite directions (see fig. 8). During tunneling, they are modified by F (cid:48)(cid:48) ,giving rise to clockwise motion in trajectory A and to anti-clockwise motion in trajectoryB along the plane of imaginary coordinates (see fig. 9). Because of its initial angularmomentum, p + ( p − ) electrons can be driven more easily along trajectory A (B) than p − ( p + ) electrons, which leads to different recollision densities and leads to the time-dependentspin-polarization in recollision. 12 F x (t), F x (t i ), k x (t i ) (atomic units) F y ( t ) , F y ( t i ) , k y ( t i ) ( a t o m i c un i t s ) F (t) F' (t i ) F' (t i ) F' (t i ) k' (t i )=0 k' (t i ) k' (t i ) F" (t i ) F" (t i ) F" (t i ) k" (t i )/ k" (t i )/ k" (t i )/ Փ r = 0.65 π (A) Փ r = 0.75 π (C) Փ r = 0.69 π (B) FIG. 8. Real and imaginary parts of the kinetic momentum k ( t i ) and the electric field F ( t i ) at thesaddle point of ionization t = t i ; k ( t i ) = k (cid:48) ( t i ) + i k (cid:48)(cid:48) ( t i ) and F ( t i ) = F (cid:48) ( t i ) + i F (cid:48)(cid:48) ( t i ). Solutions areshown for one ionization burst. The electric field considered here, resulting from combining a RCPfield of frequency ω = 0 .
05 a.u. and intensity I = 10 W cm − with a counter-rotating secondharmonic of equal intensity, is represented in the figure. -8 -6 -4 -2 0012345678 -2.5 -2 -1.5 -1 -0.5 0-2-1.5-1-0.500.5 Փ r = 0.69 π (B) Under-the-barrier motionExcursion in the continuumUnder-the-barrier motionExcursion in the continuum R {x(t)} (atomic units) I {x(t)} (atomic units) I { y ( t ) } ( a t o m i c un i t s ) R { y ( t ) } ( a t o m i c un i t s ) Փ r = 0.65 π (A) Փ r = 0.75 π (C) Փ r = 0.65 π (A) Փ r = 0.75 π (C) Փ r = 0.69 π (B) p - p + FIG. 9. Real (left panel) and imaginary (right panel) components of the quantum orbits thatrecollide with the core at φ r = 0 . π (traj. A, red lines), 0 . π (traj. B, black lines) and 0 . π (traj. C, green lines). The corresponding recollision times ( t r = φ r /ω ) are t r = 314, 334 and 363asec. For illustration purposes, the sense of rotation of electrons in p + and p − orbitals is depictedin the right panel. . Attosecond control of spin polarization In this section we discuss how modifying the parameters of the driving fields can affectthe degree of spin polarization of the recolliding electrons. In particular, we analyze theeffect of varying the relative intensities of the two counter-rotating fields. Fig. 10 containsa representation of the electric fields resulting from making the intensity of the secondharmonic half and twice the intensity of the fundamental field (see parameters of the fieldsin fig. 10 and in its footnote). Increasing the relative intensity of the fundamental fieldshrinks the width of the field lobes. Enhancing the relative intensity of the second harmonichas the opposite effect. The corresponding recollision energy and spin polarization, obtainedwith these fields, are shown in 10, as a function of the recollision time, for one optical cycleof the fundamental field. For comparison purposes, the results obtained for equal intensitiesof the counter-rotating fields (already discussed in the previous section), are included in fig.10.Spin polarization is presented in fig. 10 (lower panels), also as a function of the recollisiontime. We can see that relatively small modifications of the fields intensities lead to dramaticchanges in the degree of polarization, allowing to achieve a high degree of control. Inparticular, by tuning the relative intensities of the fields, it is possible to select the instantat which spin polarization changes it sign: increasing the intensity of the fundamental fieldshifts the change of sign towards earlier times, whereas increasing the intensity of its secondharmonic has the opposite effect. 14 .5 0.75 1 1.25 1.5 1.75 2 0.5 0.75 1 1.25 1.5 1.75 20.5 0.75 1 1.25 1.5 1.75 20.5 0.75 1 1.25 1.5 1.75 20.5 0.75 1 1.25 1.5 1.75 2 -0.8-0.6-0.4-0.200.20.40.60.81 ++ + Sp i n p o l a r i z a t i o n Recollision phase (units of π ) R e c o lli s i o n e n e r g y ( e V ) totalRecollision time (asec) Recollision phase (units of π ) Recollision phase (units of π )totaltotal Recollision time (asec) Recollision time (asec) FIG. 10. Attosecond control of spin polarization. Upper figures: Lissajous curves representing theelectric fields resulting from combining a RCP field with frequency ω = 0 .
05 a.u. and a LCP fieldwith frequency 2 ω with different relative intensities: I ω = I ω / I ω = I ω (centralcolumn) and I ω = 2 I ω / I ω and I ω considered in each case areindicated in the figure. Middle panels: recollision energy as a function of the recollision time.Lower panels: spin polarization as a function of the recollision time. Results have been calculatedby keeping the time of return on the real time axis. IV. CONCLUSIONS
The possibility of inducing recollision with spin-polarized electrons can open new direc-tions in attosecond spectroscopy. Electron spin and orbital angular momentum can play animportant role in well-established recollision-driven techniques such as photoelectron diffrac-15ion and holography [35–40] or high harmonic generation [24, 26, 29, 32, 41–45]. We haveshown that the use of intense two-color counter-rotating bi-circular fields can drive electron-core recollision with a degree of spin polarization that depends on the recollision time andtherefore on the recollision energy. Electron spin polarization upon tunnel ionization is in-trinsically related to the generation of spin-polarized currents in the ionic core [46]. In thiscontext, the potential of inducing recollision within one optical cycle of the driving field canallow for probing spin-polarized currents in atoms and molecules with sub-femtosecond andsub-Angstrom resolution. The time-dependence of spin polarization could be exploited toreconstruct information of the recollision process itself from spin-resolved measurements ofdiffracted electrons. Furthermore, our work shows that the degree of spin-polarization canbe modified as desired by tailoring the driving fields. In particular, we have found that smallvariations in the relative intensities of the counter-rotating fields can change dramaticallythe level of polarization of the recolliding currents, opening the way for attosecond controlof spin-resolved dynamics in atoms and molecules.
V. ACKNOWLEDGEMENTS
The authors acknowledge support from the DFG grant SM 292/2-3 and from the DFGSPP 1840 “Quantum Dynamics in Tailored Intense Fields”. [1] W. Gerlach and O. Stern, Zeitschrift f¨ur Physik , 110 (1922).[2] W. Gerlach and O. Stern, Zeitschrift f¨ur Physik , 349 (1922).[3] W. Gerlach and O. Stern, Zeitschrift f¨ur Physik , 353 (1922).[4] W. Pauli, Zeitschrift f¨ur Physik , 765 (1925).[5] J. M. D. Coey, Magnetism and Magnetic Materials (Cambridge University Press, 2010).[6] J. Kessler,
Polarized Electrons (Springer, 1976).[7] U. Fano, Physical Review , 131 (1969).[8] N. A. Cherepkov, Journal of Physics B: Atomic and Molecular Physics , 2165 (1981).[9] P. Lambropoulos, Advances in Atomic and Molecular Physics, , 87 (1976).[10] S. N. Dixit, P. Lambropoulos, and P. Zoller, Physical Review A , 318 (1981).
11] T. Nakajima and P. Lambropoulos, EPL (Europhysics Letters) , 25 (2002).[12] I. Barth and O. Smirnova, Physical Review A , 013401 (2013).[13] D. B. Miloˇsevi´c, Physical Review A , 051402 (2016).[14] A. Hartung, F. Morales, M. Kunitski, K. Henrichs, A. Laucke, M. Richter, T. Jahnke,A. Kalinin, M. Sch¨offler, L. P. H. Schmidt, M. Ivanov, O. Smirnova, and R. D¨orner, Na-ture Photonics , 526 (2016), letter.[15] K. Liu and I. Barth, Physical Review A , 043402 (2016).[16] I. Barth and O. Smirnova, Physical Review A , 063415 (2011).[17] T. Herath, L. Yan, S. K. Lee, and W. Li, Physical Review Letters , 043004 (2012).[18] I. Barth and O. Smirnova, Physical Review A , 013433 (2013).[19] I. Barth and O. Smirnova, Physical Review A , 065401 (2013).[20] I. Barth and M. Lein, Journal of Physics B: Atomic, Molecular and Optical Physics , 204016(2014).[21] J. Kaushal, F. Morales, and O. Smirnova, Physical Review A , 063405 (2015).[22] H. Eichmann, A. Egbert, S. Nolte, C. Momma, B. Wellegehausen, W. Becker, S. Long, andJ. K. McIver, Physical Review A , R3414 (1995).[23] S. Long, W. Becker, and J. K. McIver, Physical Review A , 2262 (1995).[24] D. B. Miloˇsevi´c, W. Becker, and R. Kopold, Physical Review A , 063403 (2000).[25] D. B. Miloˇsevi´c, Opt. Lett. , 2381 (2015).[26] L. Mediˇsauskas, J. Wragg, H. van der Hart, and M. Y. Ivanov, Physical Review Letters ,153001 (2015).[27] T. Zuo and A. D. Bandrauk, J. Nonlinear Optic. Phys. Mat.6 , 533 (1995).[28] M. Ivanov and E. Pisanty, Nature Photonics , 501 (2014), news and Views.[29] O. Kfir, P. Grychtol, E. Turgut, R. Knut, D. Zusin, D. Popmintchev, T. Popmintchev, H. Nem-bach, J. M. Shaw, A. Fleischer, H. Kapteyn, M. Murnane, and O. Cohen, Nature Photonics , 99 (2015), article.[30] F. Mauger, A. D. Bandrauk, and T. Uzer, Journal of Physics B: Atomic, Molecular andOptical Physics , 10LT01 (2016).[31] A. D. Bandrauk, F. Mauger, and K.-J. Yuan, Journal of Physics B: Atomic, Molecular andOptical Physics , 23LT01 (2016).
32] O. Smirnova, Y. Mairesse, and S. Patchkovskii, Journal of Physics B: Atomic, Molecular andOptical Physics , 234005 (2015).[33] D. Baykusheva, M. S. Ahsan, N. Lin, and H. J. W¨orner, Physical Review Letters , 123001(2016).[34] O. Smirnova and M. Ivanov, “Multielectron high harmonic generation: Simple man on acomplex plane,” in Attosecond and XUV Physics (Wiley-VCH Verlag GmbH & Co. KGaA,2014) pp. 201–256.[35] M. Spanner, O. Smirnova, P. B. Corkum, and M. Y. Ivanov, Journal of Physics B: Atomic,Molecular and Optical Physics , L243 (2004).[36] M. Meckel, D. Comtois, D. Zeidler, A. Staudte, D. Paviˇci´c, H. C. Bandulet, H. P´epin,J. C. Kieffer, R. D¨orner, D. M. Villeneuve, and P. B. Corkum, Science , 1478 (2008),http://science.sciencemag.org/content/320/5882/1478.full.pdf.[37] Y. Huismans, A. Rouz´ee, A. Gijsbertsen, J. H. Jungmann, A. S. Smolkowska, P. S. W. M.Logman, F. L´epine, C. Cauchy, S. Zamith, T. Marchenko, J. M. Bakker, G. Berden, B. Redlich,A. F. G. van der Meer, H. G. Muller, W. Vermin, K. J. Schafer, M. Spanner, M. Y. Ivanov,O. Smirnova, D. Bauer, S. V. Popruzhenko, and M. J. J. Vrakking, Science , 61 (2011),http://science.sciencemag.org/content/331/6013/61.full.pdf.[38] C. I. Blaga, J. Xu, A. D. DiChiara, E. Sistrunk, K. Zhang, P. Agostini, T. A. Miller, L. F.DiMauro, and C. D. Lin, Nature , 194 (2012).[39] M. Meckel, S. Staudte, A.and Patchkovskii, D. M. Villeneuve, P. B. Corkum, R. Dorner, andM. Spanner, Nature Physics , 594 (2014), article.[40] M. G. Pullen, B. Wolter, A.-T. Le, M. Baudisch, M. Hemmer, A. Senftleben, C. D. Schroter,J. Ullrich, R. Moshammer, C. D. Lin, and J. Biegert, Nat Commun (2015), article.[41] D. B. Miloˇsevi´c and W. Becker, Physical Review A , 011403 (2000).[42] H. J. W¨orner, J. B. Bertrand, B. Fabre, J. Higuet, H. Ruf, A. Dubrouil,S. Patchkovskii, M. Spanner, Y. Mairesse, V. Blanchet, E. M´evel, E. Con-stant, P. B. Corkum, and D. M. Villeneuve, Science , 208 (2011),http://science.sciencemag.org/content/334/6053/208.full.pdf.[43] A. Fleischer, O. Kfir, T. Diskin, P. Sidorenko, and O. Cohen, Nature Photonics , 543 (2014),article.[44] E. Pisanty, S. Sukiasyan, and M. Ivanov, Physical Review A , 043829 (2014).
45] R. Cireasa, A. E. Boguslavskiy, B. Pons, M. C. H. Wong, D. Descamps, S. Petit, H. Ruf,N. Thire, A. Ferre, J. Suarez, J. Higuet, B. E. Schmidt, A. F. Alharbi, F. Legare, V. Blanchet,B. Fabre, S. Patchkovskii, O. Smirnova, Y. Mairesse, and V. R. Bhardwaj, Nature Physics , 654 (2015), letter.[46] I. Barth and O. Smirnova, Journal of Physics B: Atomic, Molecular and Optical Physics ,204020 (2014).,204020 (2014).