Recollision as a probe of magnetic field effects in non-sequential double ionization
aa r X i v : . [ phy s i c s . a t o m - ph ] N ov Recollision as a probe of magnetic field effects in non-sequential double ionization
A. Emmanouilidou and T. Meltzer Department of Physics and Astronomy, University College London,Gower Street, London WC1E 6BT, United Kingdom (Dated: March 20, 2018)Fully accounting for non-dipole effects in the electron dynamics, double ionization is studied forHe driven by a near-infrared laser field and for Xe driven by a mid-infrared laser field. Using a three-dimensional semiclassical model, the average sum of the electron momenta along the propagationdirection of the laser field is computed. This sum is found to be an order of magnitude largerthan twice the average electron momentum along the propagation direction of the laser field insingle ionization. Moreover, the average sum of the electron momenta in double ionization is foundto be maximum at intensities smaller than the intensities satisfying previously predicted criteriafor the onset of magnetic field effects. It is shown that strong recollisions are the reason for thisunexpectedly large value of the sum of the momenta along the direction of the magnetic componentof the Lorentz force.
INTRODUCTION
Non-sequential double ionization in two-electron atomsis a fundamental process that explores electron-electroncorrelation in strong fields. As such, it has attracted alot of interest in the field of light-matter interactions inrecent years [1, 2]. The majority of theoretical studieson NSDI are delivered in the framework of the dipole ap-proximation, particularly the studies involving the com-monly used near-infrared laser fields and intensities [3].In the dipole approximation the vector potential A of thelaser field does not depend on space. Therefore, mag-netic field effects are neglected, since the magnetic fieldcomponent of the laser field B = ∇ × A (t) is zero. How-ever, in the general case where A depends both on spaceand time, an electron experiences a Lorentz force whosemagnetic field component F B increases with increasingelectron velocity, since F B = q v × B . It is important toaccount for magnetic field effects, since in strong field ion-ization high velocity electrons are often produced. Crite-ria for the onset of magnetic field effects both in the rel-ativistic and the non-relativistic limit have already beenformulated [4, 5]. In the non-relativistic limit, where thiswork focuses, magnetic field effects are expected to arisewhen the amplitude of the electron motion due to themagnetic field component of the Lorentz force becomes1 a.u., i.e. β = U p / (2 ω c) ≈ p theponderomotive energy.Studies addressing magnetic field effects include usinga 3D semiclassical rescattering model that accounts for F B to successfully describe the observed ionization ofNe n+ (n ≤
8) in ultra-strong fields [6]. Moreover, non-dipole effects were addressed in theoretical studies ofhigh-order harmonic generation, for instance, by neglect-ing the Coulomb potential [7] or by using a first orderexpansion of the vector potential [8]. In recent studiesof single ionization (SI), the electron momentum distri-bution along the propagation direction of the laser fieldwas computed using different quantum mechanical ap- proaches [9–12]. For example, for H interacting with a3400 nm laser field at intensities 0.5-1 × Wcm − theaverage momentum along the propagation direction ofthe laser field was found to increase from 0.003 a.u. to0.006 a.u. [10]. Thus, for single ionization, the average ofthis momentum component increases with increasing β [10, 13]. If magnetic field effects are not accounted for,then this momentum component averages to zero. Themotivation for these theoretical studies was a recent ex-perimental observation of the average momentum in thepropagation direction of the laser field [13]. Figure 1 | Range of validity of the dipole approxima-tion and momentum in double ionization . The whitearea indicates the range of intensities and wavelengths wherethe dipole approximation is valid. β = 0 . β = 1 a.u. (solid line) and β = 2 a.u. (dash line).The arrows mark the 800 nm and the 3100 nm wavelengthsdriving He and Xe, respectively. At these wavelengths, for arange of intensities, the color bars indicate the ratio of theaverage sum of the electron momenta along the direction of F B for double ionization with twice the respective electronmomentum for single ionization (cid:10) p + p (cid:11) DI / (2 h p y i SI ). This work reveals another aspect of non-sequentialdouble ionization (NSDI) which has not been previouslyaddressed. The strong electron-electron correlation inNSDI is identified as a probe of magnetic field effects bothfor near-infrared and mid-infrared intense laser fields.Specifically, the intensities considered are around 10 Wcm − for He at 800 nm and around 10 Wcm − forXe at 3100 nm where the rescattering mechanism un-derlies double ionization [14]. For these intensities, it isfound that the average sum of the two electron momentaalong the propagation direction of the laser field is un-expectedly large. It is roughly an order of magnitudelarger than twice the average of the respective electronmomentum for single ionization. This average sum of themomenta for double ionization (DI) is shown to be maxi-mum at intensities smaller than the intensities satisfyingthe criterion for the onset of magnetic field effects β ≈ METHOD
For the current studies, a 3D semiclassical model isemployed that fully accounts for non-dipole effects dur-ing the time propagation. For simplicity this model isreferred to as 3D-SMND. It is an extension of a 3Dsemiclassical model that was previously formulated inthe framework of the dipole approximation. This lattermodel is referred to as 3D-SMD. Thus, in the 3D-SMNDmodel non-dipole effects are fully accounted for in thetwo-electron dynamics. Some of the successes of the 3D-SMD model are identifying the mechanism responsiblefor the fingerlike structure [17], which was predicted the-oretically [18] and was observed experimentally for Hedriven by 800 nm laser fields [19, 20]; investigating di-rect versus delayed pathways of NSDI for He driven bya 400 nm laser field while achieving excellent agreementwith fully ab-initio quantum mechanical calculations [21];identifying the underlying mechanisms for the carrier-envelope phase effects observed experimentally in NSDIof Ar driven by an 800 nm laser field at a range of in-tensities [22, 23]. The 3D-SMD model is extended to the3D-SMND model employed in the current work to fullyaccount for the magnetic field during time propagation.The Hamiltonian describing the interaction of the fixednucleus two-electron atom with the laser field is given byH = ( p + A (y , t)) p + A (y , t)) −− c Z | r | − c Z | r | + c | r − r | , (1) where the vector potential A is given by A (y , t) = − E ω e − ( ct − yc τ ) sin ( ω t − ky)ˆx , (2) ω , k, E are the frequency, wavenumber and strengthof the electric component of the laser field, respec-tively, and c is the velocity of light. τ = FWHM / √ ln4with FWHM the full width half maximum of the laserfield. All Coulomb forces are accounted for by settingc = c = c = 1. In this work linearly polarized laserfields are considered. To switch-off a Coulomb interac-tion we set the appropriate constant equal to zero, forexample, to switch-off the interaction of electron 1 withthe nucleus we set c = 0. For A given by Eq. (2), E and B are along the x- and z-axis, respectively, whilethe propagation direction of the laser field and the direc-tion of F B are along the y-axis. Unless otherwise stated,all Coulomb forces as well as the electric and the mag-netic field are fully accounted for during time propaga-tion. Moreover, the Coulomb singularity is addressed us-ing regularized coordinates [24] which were also employedin the 3D-SMD model [17, 21, 22].The initial state in the 3D-SMND model is taken to bethe same as in the 3D-SMD model [17, 21, 22]. It entailsone electron tunneling through the field-lowered Coulombpotential with a non-relativistic quantum tunneling rategiven by the Ammosov-Delone-Krainov (ADK) formula[25, 26]. The momentum along the direction of the elec-tric field is zero while the transverse one is given by aGaussian distribution [25, 26]. A non-relativistic ADKrate results in this Gaussian distribution being centeredaround zero. In ref. [27] non-dipole effects were ac-counted for in the ADK rate. It was shown that themost probable transverse velocity ranges from 0.33 I p /cto almost zero with increasing E / (2I p ) / , with I p theionization energy of the tunneling electron. In this work,the smallest intensities considered are 5 × Wcm − for Xe and 7 × Wcm − for He. At these intensi-ties, if non-dipole effects are accounted for in the ADKrate, the transverse velocity of the tunneling electron iscentered around 0.17 I Xep /c for Xe which is 5.5 × − a.u. (I Xep =0.446 a.u.) and 0.12 I
Hep /c for He whichis 7.9 × − a.u. (I Hep =0.904 a.u.). These values aresignificantly smaller than the values of the average mo-menta along the propagation direction of the laser field,which are presented in what follows. Thus, using thenon-relativistic ADK rate is a good approximation forthe quantities addressed in this work. The remainingelectron is initially described by a microcanonical distri-bution [28]. In what follows, the tunneling and boundelectron are denoted as electrons 1 and 2, respectively.
RESULTS p y for single ionization of Xe and H The accuracy of the 3D-SMND model is established bycomputing the momentum distribution along the propa-gation direction of the laser field, p y , for single ionization(SI) and by comparing it with available experimental andtheoretical results. In ref. [16], the peak of the p y dis-tribution was observed to shift in the direction oppositeto the magnetic field component of the Lorentz force,F B , for intensities on the order of 10 Wcm − . Thisshift was attributed to the combined effect of the mag-netic field and the Coulomb attraction of the nucleus [16].To compare with these experimental results, the shift ofthe peak of the p y distribution is computed for Xe in-teracting with a 3400 nm and a 44 fs FWHM laser fieldas the intensity increases from 3-6 × Wcm − . Theshift of the peak of the p y distribution is found to varyfrom − . − .
012 a.u.. These results are inagreement with the simulations and experimental resultspresented in ref. [16]. Moreover, to compare with the re-sults in ref. [10], the average of the momentum h p y i SI iscomputed for H driven by a 3400 nm and a 16 fs FWHMlaser field for intensities 0.5-1 × Wcm − . Using the3D-SMND model, h p y i SI is found to vary from 0.0022a.u. to 0.0046 a.u.. These values differ by 27% from theresults presented in ref. [10] and are thus in reasonableagreement. The difference may be due to non-dipole ef-fects not accounted for in the ADK rate in the 3D-SMNDmodel. In addition, the quantum calculation used in ref.[10] employs a 2D soft-core potential while a full 3D po-tential is employed by the 3D-SNMD model. The singleionization results obtained in this work were computedwith at least 4 × events and therefore the statisticalerror introduced is very small. h p y i for single ionization of He and Xe The 3D-SMND model is now employed to compute h p y i SI for He driven by an 800 nm, 12 fs FWHM laserfield and for Xe driven by a 3100 nm, 44 fs FWHM laserfield; the two laser fields have roughly the same num-ber of cycles. First an analytic expression is obtainedrelating h p y i SI with the average electron kinetic energy h E k i SI [9, 13]. When an electron interacts with an elec-tromagnetic field with all the Coulomb forces switched-off, i.e. c = c = c = 0 in Eq. (1), the equations ofmotion are ˙p y = − ( v × B ) y and ˙p x = − ( v × B ) x − E.Keeping only first order terms in 1/c, ˙p x = − E result-ing in p y − p , y = p / (2c) − p , x / (2c), with p , x / y thex/y components of the electron momentum at time t .The initial momentum of the tunneling electron alongthe electric field direction is set to zero, as in the 3D-SMND model, resulting in p y − p , y = p / (2c) = E k / c. Figure 2 | Single ionization of He and Xe . h p y i SI and h E k i SI / c are plotted as a function of intensity in ( a ) for Hedriven by an 800 nm laser field and in ( b ) for Xe driven by a3100 nm laser field. Thus, h p y i = h p , y i + h E k i / c is obtained. For this sim-ple model h E k i is the drift energy of the electron. Asdiscussed in the Method section, if non-dipole effects areaccounted for in the tunneling rate then h p , y i varies from0.33 I p /c to almost zero with increasing intensity. In the3D-SNMD model non-dipole effects are not included inthe ADK rate and therefore h p , y i = 0. Indeed, usingthe 3D-SNMD model with c = c = c = 0, it is foundthat h p y i SI = h E k i SI / c, see Table 1. SI Z=2
SI SI
Z=2c , , =1 c , , =0 c =0,c , =1I ( × Wcm − ) h p y i h E k / c i h p y i h E k / c i h p y i h E k / c i ( † )0.7 3.6 2.3 1.5 1.5 2.0 2.2He 1.3 6.1 4.4 3.4 3.4 3.3 4.44.8 31 31 19 19 19 210.05 3.2 1.6 1.3 1.3 1.8 1.6Xe 0.07 3.5 2.5 2.1 2.1 2.4 2.40.22 11 12 9.9 9.9 9.1 11 ( † ) Average momentum and kinetic energy given in × − a.u. Table 1 | Single ionization results for Xe and He.
Next, h p y i SI and h E k i SI are computed with the 3D-SMND model fully accounting for all Coulomb forces andthe presence of the initially bound electron in driven Heand Xe, i.e. c = c = c = 1 with Z = 2. The tunnel-ing electron is the one that is mostly singly ionizing. InFig. 2, we show that, for He, h p y i SI varies from 0.0036 a.u.to 0.031 a.u. at intensities 0.7-4.8 × Wcm − . For Xe, h p y i SI varies from 0.0032 a.u. to 0.011 a.u. at intensities0.5-2.2 × Wcm − . In Table 1, it is shown that h p y i SI and h E k i SI / c when obtained with the full model do notdiffer by more than a factor of 3 from the values obtainedwhen all Coulomb forces are switched-off. Thus, the sim-ple model yields the correct order of magnitude for h p y i SI .It is also shown in Table 1, that with all Coulomb forcesaccounted for, h p y i SI is no longer equal to h E k i SI / c bothfor driven He and for driven Xe. For the full model, h E k i SI is no longer just the drift kinetic energy, mainlydue to the interaction of the tunneling electron with thenucleus. Indeed, using the 3D-SMND model with this in-teraction switched off, i.e. c = 0 and c = c = 1, h p y i SI is roughly equal to h E k i SI / c, see Table 1. h p y i SI is alsoshown in Table 1 to be more sensitive than h E k i SI to theinteraction of the tunneling electron with the nucleus.Summarizing the results for single ionization, propagat-ing classical trajectories with initial times determined bythe ADK rate and all Coulomb forces switched-off yieldsthe correct order of magnitude for h p y i SI . (cid:10) p + p (cid:11) for double ionization of He and Xe For double ionization, the average of the sum of theelectron momenta along the propagation direction of thelaser field (cid:10) p + p (cid:11) DI , is computed for He driven byan 800 nm laser field and for Xe driven by a 3100 nmlaser field. The parameters of the laser fields are thesame as the ones employed in the single ionization sec-tion for He and Xe. The double ionization results ob-tained in this work were computed with at least 2 × events and therefore the statistical error introduced isvery small. The results are plotted in Fig. 3a for Heat intensities 0.7-4.8 × Wcm − and in Fig. 3b forXe at intensities 0.5-2.2 × Wcm − . The values ob-tained for (cid:10) p + p (cid:11) DI are quite unexpected. Specifi-cally, (cid:10) p + p (cid:11) DI is found to be roughly an order ofmagnitude larger than twice h p y i SI , with h p y i SI com-puted in the previous section. For comparison, both (cid:10) p + p (cid:11) DI and 2 h p y i SI are displayed in Fig. 3. It isshown that (cid:10) p + p (cid:11) DI ≈ × h p y i SI for He at 1.3 × Wcm − , while (cid:10) p + p (cid:11) DI ≈ × h p y i SI for Xe at7 × Wcm − . For 1.3 × Wcm − and 800 nm β = 0 .
18 a.u., while for 7 × Wcm − and 3100 nm β = 0 .
58 a.u.. Thus, (cid:10) p + p (cid:11) DI / h p y i SI is found tobe maximum at intensities considerably smaller than theintensities corresponding to β ≈ h p y i SI which increaseswith increasing intensity as expected [10], (cid:10) p + p (cid:11) DI after reaching a maximum decreases with increasing in-tensity for the range of intensities currently considered(Fig. 3a,b). What is the mechanism responsible for thispattern? We answer this question in what follows. Figure 3 | Double ionization of He and Xe . (cid:10) p + p (cid:11) DI and 2 h p y i SI are plotted as a function of intensity in ( a ) forHe driven by an 800 nm laser field and in ( b ) for Xe drivenby a 3100 nm laser field. (cid:10) p (cid:11) DI and (cid:10) p (cid:11) DI are plotted as afunction of intensity in ( c ) for He driven by an 800 nm laserfield and in ( d ) for Xe driven by a 3100 nm laser field. Recollision probing magnetic field effects
The average electron momentum along the propaga-tion direction is non zero when the magnetic field compo-nent of the Lorentz force F B is accounted for. This forceincreases with increasing intensity—increasing strengthof the magnetic field—and with increasing velocity alongthe direction of the electric field. (cid:10) p + p (cid:11) DI is foundto be maximum at 1.3 × Wcm − for 800 nm andat 7 × Wcm − for 3100 nm, intensities where thestrength of the magnetic field is not large. It then followsthat it must be the velocities of the two escaping electronsthat are significantly larger at these intensities than atthe higher intensities considered in this work. Large elec-tron velocities at intermediate intensities are a result ofstrong electron-electron correlation, i.e. of the rescatter-ing mechanism [14]. In the rescattering scenario afterelectron 1 tunnels in the field-lowered Coulomb potentialit accelerates in the strong laser field and can return tothe core and undergo a collision with the remaining elec-tron [14]. In what follows evidence is provided that thelarge values of (cid:10) p + p (cid:11) DI are due to recollisions. Specif-ically, it is shown that recollisions are strong resulting inoverall large electron velocities roughly at the intensitieswhere (cid:10) p + p (cid:11) DI is maximum. It is also shown thatrecollisions are soft resulting in overall smaller velocitiesat higher intensities where (cid:10) p + p (cid:11) DI is found to besmaller.This transition from strong to soft recollisions isdemonstrated in the context of He driven by an 800nm laser field at intensities 0.7 × Wcm − , 2.0 × Wcm − and 3.8 × Wcm − . To do so, an analysis ofthe doubly ionizing events is performed. In Fig. 4, thedistribution of the tunneling and recollision times is plot-ted. As expected, for the smaller intensities (Fig. 4a1,b1),electron 1 tunnel-ionizes at times around the extremaof the laser field. For 3.8 × Wcm − (Fig. 4c1) theelectric field is sufficiently strong so that electron 1 cantunnel-ionize at times other than the extrema of the field.The distribution of the recollision times is also plotted.This time is identified for each double ionizing trajec-tory as the time that the electron-electron potential en-ergy 1 / | r − r | as a function of time is maximum. Forthe smaller intensities the recollision times are centeredroughly around ± /
3, with n an integer and T the pe-riod of the laser field, as expected from the rescatteringmodel [14] (Fig. 4a2,b2). At 3.8 × Wcm − the recol-lision times shift and are centered around the extrema ofthe laser field (Fig. 4c2). This shift of the recollision timessignals a transition from strong to soft recollisions [29].This transition is further corroborated by the average ki-netic energy of each electron, D E , E , plotted in Fig. 4 asa function of time; zero time is set equal to the recollisiontime of each double ionizing trajectory. For smaller in-tensities, D E , E changes sharply at the recollision time(Fig. 4a3,b3). The change in D E , E is much smaller at3.8 × Wcm − ( Fig. 4c3). The above results showthat for the smaller intensities electron 1 tunnel-ionizesaround the extrema of the field. It then returns to thecore, roughly when the electric field is small, with largevelocity and undergoes a recollision with electron 2 trans-ferring a large amount of energy (strong recollision). Thevelocities of both electrons along the direction of the elec- tric field are determined mainly by the vector potential atthe recollision time. Thus, both electrons escape mainlyeither parallel or antiparallel to the electric field. In-deed, this is the pattern seen in the plots of the corre-lated momenta along the direction of the electric fieldin Fig. 4a5,b5 where the highest density is in the firstand third quadrants. The correlated momenta are plot-ted in units of p p , with U p the ponderomotive energyequal to E / (4 ω ). These patterns of the correlated mo-menta are consistent with direct double ionization, thatis, with both electrons ionizing shortly after recollisiontakes place [30]. Indeed, analyzing the double ionizingevents it is found that for He at 0.7 × Wcm − di-rect double ionization contributes 80%. Delayed doubleionization events contribute 20%. In delayed double ion-ization, also known as RESI [30, 31], one electron ionizessoon after recollision takes place, while the other electronionizes with a delay [21]. In contrast, at the higher inten-sity of 3.8 × Wcm − , electron 1 tunnel-ionizes afterthe extrema of the laser field. It then follows a shorttrajectory and returns to the core when the electric fieldis maximum with small velocity. Electron 1 transfers asmall amount of energy to electron 2 (soft recollision).The velocities of electron 1 and 2 along the directionof the electric field are determined mostly by the valuesof the vector potential at the tunneling and recollisiontimes, respectively. As a result, the two electrons canescape opposite to each other along the direction of theelectric field. This pattern is indeed seen in the plots ofthe correlated momenta in Fig. 4c5 with high density inthe second and fourth quadrants. This antiparallel pat-tern was predicted in the context of strongly-driven N with fixed nuclei [29]. It was also seen in the case of Ardriven by intense ultra-short laser fields [22] in agreementwith experiment [23]. A similar analysis for Xe driven at3100 nm is performed (not shown). Similar results areobtained. It is found that for driven Xe strong recollisionstransition to soft ones roughly around 22 × Wcm − . NSDI c , , =1, Z=2 NSDI c , , =0t = t rec , p = p , p = p I ( × Wcm − ) (cid:10) p (cid:11) (cid:10) E / c (cid:11) (cid:10) p (cid:11) (cid:10) E / c (cid:11) (cid:10) p (cid:11) (cid:10) E / c (cid:11) (cid:10) p (cid:11) (cid:10) E / c (cid:11) ( † )0.7 12 14 25 16 13 15 25 16He 1.3 22 22 80 21 24 24 82 204.8 25 32 73 47 33 37 74 430.05 10 11 65 9 11 11 67 8Xe 0.07 14 13 80 10 15 14 81 90.22 38 27 30 17 45 34 30 16 ( † ) Average momentum and kinetic energy given in × − a.u. Table 2 | Double ionization results for Xe and He.
For single ionization of He and Xe, it was shown that using the tunneling times as the starting point, the 3D-
Figure 4 | Recollision underlying double ionization of He driven at 800 nm . The intensities considered are 0.7 × Wcm − , denoted by ( a ), 2.0 × Wcm − , denoted by ( b ), and 3.8 × Wcm − , denoted by ( c ). The distribution oftunneling times (black line) is plotted in ( a1 ), ( b1 ) and ( c1 ). The distribution of recollision times (black line) is plotted in ( a2 ),( b2 ) and ( c2 ). The electric field is denoted as a grey line in the plots of the tunneling and recollision times. (cid:10) E (cid:11) (grey line) and (cid:10) E (cid:11) (black line) are plotted as a function of time in ( a3 ), ( b3 ) and ( c3 ) with time zero set equal to the recollision time of eachdouble ionizing event. For 2.0 × Wcm − , (cid:10) E (cid:11) (blue line) and (cid:10) E (cid:11) (red line) are also plotted in the absence of the magneticfield. (cid:10) p (cid:11) (grey line) and (cid:10) p (cid:11) (black line) are plotted as a function of time in ( a4 ), ( b4 ) and ( c4 ) with time zero set equalto the recollision time of each double ionizing event. For 2.0 × Wcm − , (cid:10) p (cid:11) (blue line) and (cid:10) p (cid:11) (red line) are also plot-ted in the absence of the magnetic field. Correlated momenta along the direction of the electric field are plotted in ( a5 ), ( b5 ) and( c5 ). SMND with all Coulomb forces switched-off yields thecorrect order of magnitude for (cid:10) p (cid:11) SI . For double ion-ization of He and Xe, using the 3D-SMND model withall Coulomb forces switched-off and with initial condi-tions taken to be the recollision times and velocities, (cid:10) p (cid:11) DI and (cid:10) p (cid:11) DI are obtained and presented in Table 2.These values of (cid:10) p (cid:11) DI and (cid:10) p (cid:11) DI agree very well withthe values obtained using the 3D-SMND model with allCoulomb forces accounted for, see Table 2. This agree-ment further supports that recollision is the main factordetermining (cid:10) p + p (cid:11) DI .Finally, in what follows, the electron that contributesthe most to the maximum value of (cid:10) p + p (cid:11) DI is iden-tified both for driven He and Xe. (cid:10) p (cid:11) DI and (cid:10) p (cid:11) DI are plotted as a function of time in Fig. 4, with timezero set equal to the recollision time of each double ion-izing trajectory. It is shown in Fig. 4a4,b4,c4 that it ismainly (cid:10) p (cid:11) DI that changes significantly at the recollisiontime. This change is more sharp for the smaller inten-sities (Fig. 4a4,b4). In Fig. 4b4, at intensity 2.0 × Wcm − , it is also illustrated that in the absence of themagnetic field both (cid:10) p (cid:11) DI and (cid:10) p (cid:11) DI tend to zero withtime, as expected. In addition, in Fig. 3c,d, for drivenHe and Xe, respectively, (cid:10) p (cid:11) DI and (cid:10) p (cid:11) DI are plot-ted as a function of intensity. It is seen that (cid:10) p (cid:11) DI and (cid:10) p + p (cid:11) DI have maxima around the same inten-sities. At these intensities (cid:10) p (cid:11) DI is significantly largerthan (cid:10) p (cid:11) DI . Thus, (cid:10) p (cid:11) DI —the average momentum ofthe tunneling electron—is the one affected the most bystrong recollisions. DISCUSSION
It was shown that the average sum of the electron mo-menta along the propagation direction of the laser fieldhas large values at intensities where strong recollisionsunderlie double ionization. This is an unexpected result.For He driven by a near-infrared laser field and for Xedriven by a mid-infrared laser field, the intensities wherethe average sum of the electron momenta along the prop-agation direction of the laser field is maximum are smallerthan the intensities where magnetic field effects are pre-dicted to be large. Thus, recollision probes magnetic fieldeffects at smaller intensities than expected. However, itcan also be stated that a magnetic field probes strongrecollisions through the measurement of the sum of theelectron momenta along the propagation direction of thelaser field. It is expected that the findings reported in thiswork will serve as a motivation for future studies. Suchstudies can identify, for instance, the effect the magneticfield has on the different mechanisms of non-sequentialdouble ionization, i.e. on direct and delayed double ion-ization and the wavelengths where the magnetic field hasthe largest effect on recollisions.
ACKNOWLEDGMENTS
A.E. is grateful for fruitful discussions with PaulCorkum. She also acknowledges the EPSRC grant no.J0171831 and the use of the computational resources ofLegion at UCL.
AUTHOR CONTRIBUTIONS
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