Non-sequential double ionization with near-single cycle laser pulses

Transcription

1 Receved: 5 February 07 Accepted: 30 June 07 Publshed: xx xx xxxx OPEN Non-sequental double onzaton wth near-sngle cycle laser pulses A. Chen, M. Kübel,4, B. Bergues 3,4, M. F. Klng 3,4 & A. Emmanouldou A three-dmensonal semclasscal model s used to study double onzaton of Ar when drven by a near-nfrared and near-sngle-cycle laser pulse for ntenstes rangng from W/cm to W/cm. Asymmetry parameters, dstrbutons of the sum of the two electron momentum components along the drecton of the polarzaton of the laser feld and correlated electron momenta are computed as a functon of the ntensty and of the carrer envelope phase. A very good agreement s found wth recently obtaned results n knematcally complete experments employng near-snglecycle laser pulses. Moreover, the contrbuton of the drect and delayed pathways of double onzaton s nvestgated for the above observables. Fnally, an expermentally obtaned ant-correlaton momentum pattern at hgher ntenstes s reproduced wth the three-dmensonal semclasscal model and shown to be due to a transton from strong to soft recollsons wth ncreasng ntensty. Non-sequental double onzaton (NSDI) n ntense near-nfrared laser felds s a fundamental process wth electron-electron correlaton playng a key role 3. Consderable nformaton regardng NSDI has been obtaned from knematcally complete experments,.e., the momenta of the escapng electrons and ons are measured n concdence 4. Most of these experments employ mult-cycle laser pulses allowng for multple recollsons to occur before both electrons onze. Multple recollsons complcate the electron dynamcs and render the comparson wth theory dffcult. Recently, however, knematcally complete experments succeeded n confnng NSDI to a sngle laser cycle by usng carrer-envelope phase (CEP)-controlled few- and near-sngle-cycle pulses 5, 6. These experments wth near-sngle-cycle pulses allow for an easer comparson between theory and experment. To nterpret the double onzaton spectra of drven Ar measured usng near-sngle-cycle laser pulses, a smple one-dmensonal (D) classcal model was put forth 6 8. Ths model reles on the assumpton that the domnant pathways of double onzaton are, for small and ntermedate ntenstes, delayed non-sequental onzaton and, for hgher ntenstes, sequental onzaton. For strongly-drven Ar, ntermedate ntenstes refer to the ntensty range from 0 4 W/cm to W/cm. Ths model neglects the contrbuton of another major pathway of double onzaton, namely, drect onzaton as well as the Coulomb potental. Ths D model dd not acheve a quanttatve agreement wth the complete set of avalable expermental data over the whole ntensty range. Delayed onzaton also referred to as recollson-nduced exctaton wth subsequent feld onzaton, RESI 9, 0, and drect onzaton are two man pathways of NSDI. An nterestng fndng of these near-sngle cycle experments was that the correlated momenta components of the two escapng electrons along the drecton of the laser feld have a cross-shaped pattern for an ntensty around 0 4 W/cm 6 8. A cross-shaped correlated electron momenta pattern due to the delayed double onzaton mechansm was prevously dentfed n the context of strongly-drven He at an ntensty of W/cm and for a wavelength of 400 nm. In a cross-shaped correlated electron momenta pattern the double onzaton probablty s the hghest when the component of the momentum along the drecton of the laser feld s very small for one electron whle t takes a wde range of values for the other electron, see the expermental correlated electron momenta at 0 4 W/cm n Fg. 5. In the context of strongly-drven Ar, the above descrbed D model attrbuted the cross-shaped pattern of the correlated electron momenta to the delayed pathway of double onzaton 7, 8. A quantum mechancal calculaton, whch neglects the Coulomb potental, was used to refne the contrbuton of the delayed pathway of double onzaton to the cross-shaped correlated electron momenta pattern. Ths calculaton dentfed the key role that the symmetry of the excted state plays n the fnal shape of the correlated momenta. Department of Physcs and Astronomy, Unversty College London, Gower Street, London, WCE 6BT, Unted Kngdom. Jont Laboratory for Attosecond Scence, Unversty of Ottawa and Natonal Research Councl, 00 Sussex Drve, Ottawa, Ontaro, KA 0R6, Canada. 3 Max-Planck-Insttut für Quantenoptk, D-85748, Garchng, Germany. 4 Department für Physk, Ludwg-Maxmlans-Unverstät, D-85748, Garchng, Germany. Correspondence and requests for materals should be addressed to A.E. (emal: a.emmanouldou@ucl.ac.uk) Scentfc REPOrts 7: 7488 DOI:0.038/s

2 Fgure. Rato of double to sngle onzaton probablty as a functon of ntensty. Expermental results 8 are denoted by dark blue squares and lght blue crosses and computed results are presented by a sold lne wth black crcles when the focal volume effect s not accounted for and by a dashed-lne wth trangles when the focal volume effect s accounted for. The dfference n the two expermental sets results from slghtly dfferent averagng over the focal volume 8. In ths work, usng a three-dmensonal (3D) semclasscal model, NSDI of Ar s studed when Ar s drven by 750 nm near-sngle-cycle laser pulses at ntenstes rangng from W/cm to W/cm. In ths 3D model the only approxmaton s n the ntal state. There s no approxmaton durng the tme propagaton. That s, all Coulomb forces and the nteracton of each electron wth the laser feld are fully accounted for. Moreover, when analyzng the numercally obtaned doubly-onzed events, no assumptons are made regardng the prevalng mechansm of double onzaton and we use no free parameter. Ths s not the case for the D model 8. In addton, the Coulomb sngularty s fully accounted for usng regularzed coordnates 3. Ths s an advantage over models whch soften the Coulomb potental 4. Prevous successes of ths 3D model nclude dentfyng the mechansm responsble for the fngerlke structure n the correlated electron momenta 5, whch was predcted theoretcally 6 and was observed expermentally for He drven by 800 nm laser felds 7, 8. Moreover, ths model was used to nvestgate drect versus delayed pathways of NSDI for He drven by a 400 nm laser feld whle achevng excellent agreement wth fully ab-nto quantum mechancal calculatons 9. Usng ths model, n ths work, several observables are computed for dfferent ntenstes of strongly-drven Ar. These observables are the sum of the two electron momentum components along the drecton of the polarzaton of the laser feld and the double dfferental probablty of the two electron momentum components along the polarzaton drecton of the laser feld,.e. the correlated electron momenta. Furthermore, the ampltude and the phase of the asymmetry parameter that determnes the dfference of the ons escapng wth postve versus negatve momentum along the polarzaton drecton of the laser feld are computed as a functon of the carrer envelope phase (CEP) and the ntensty. Prevously obtaned expermental results over the whole ntensty range 6 8 are n better agreement wth the computed results obtaned usng the 3D semclasscal model rather than wth the computed results obtaned wth the D model n ref. 8. Throughout ths work the computed results are compared wth the expermental results that were recently publshed n and adopted from ref. 8 where the data acquston and analyss s descrbed n detal. Brefly, CEP stable laser pulses wth a full-wdth-half-maxmum pulse duraton of 4 fs and a center wavelength of 750 nm are focused onto a cold-gas jet of argon atoms nsde a reacton mcroscope. There, the momenta of ons and electrons generated n the laser focus va strong feld onzaton are recorded n concdence as a functon of the ntensty and of the CEP of the laser pulse. The CEP s measured wth a precson of roughly 00 mrad. Motvated by the good agreement we fnd n ths work between theory and experment, the strength of the 3D semclasscal model n fully accountng for the electron dynamcs s utlzed to dentfy the prevalng pathway of double onzaton as a functon of ntensty. In addton, for a small ntensty around 0 4 W/cm, the dependence of the double onzaton pathways on CEP s computed usng the 3D semclasscal model. Fnally, the transton from strong to soft recollsons s dentfed as the man reason for the expermentally observed escape of the two electrons wth opposte momenta at hgher ntenstes 0. Method For the current studes, a 3D semclasscal model s employed that s formulated n the framework of the dpole approxmaton 5. The ntal state n the 3D model entals one electron tunnelng through the feld-lowered Coulomb potental wth the Ammosov-Delone-Kranov (ADK) formula,. To obtan the tunnel onzaton rate for Ar, n the ADK formula the frst onzaton energy of Ar,.e. Ip = au.. and the effectve charge Z = are used. The ext pont of the tunnel-onzed electron s along the drecton of the laser feld and s computed usng parabolc coordnates 3. The momentum along the drecton of the electrc feld s zero whle the transverse one s gven by a Gaussan dstrbuton,. Thus, the formulaton of the ntal state of the electron that tunnel-onzes s a quantum mechancal one. The rest of the formulaton n the 3D model,.e the ntal state of the remanng electron and the tme propagaton of both electrons, s classcal. The remanng electron s ntally descrbed by a mcrocanoncal dstrbuton 4. In what follows, the ntally tunnelng and bound electrons are denoted as electrons and, respectvely. The weght of each classcal trajectory that we propagate n tme s gven by Scentfc REPOrts 7: 7488 DOI:0.038/s

3 Fgure. Percentage contrbuton of the drect (blue crcles) and the delayed (red trangles) pathways of double onzaton as a functon of the laser ntensty for t dff = /0 T (sold lnes), t dff = /0 T (dashed lnes) and t dff = /40 T (dotted lnes). The focal volume effect s not accounted for. where W = W W, () W E(t ) 0 n* 3 κ exp 3E(t ) () s the ADK onzaton rate, at the tme t 0 of tunnel-onzaton. n * s the effectve prncpal quantum number gven by Ip = Z n* and κ = I p. W s the weght for electron to have a transverse velocty equal to v at the tme t 0 : W v E(t ) 0 0 κ v exp E(t ). (3) t 0 s the tme electron tunnel-onzes through the feld-lowered Coulomb potental. In our computaton, for each classcal trajectory, the tunnel-onzaton tme s selected randomly n the tme nterval that the laser feld s on. In ths work the laser feld s lnearly polarzed and s gven by 0 E( t) = E e τ cos ( ωt + φ)z, ^ (4) 0 ( ln ( t ) ) where τ = 4 fs s the full-wdth-half-maxmum pulse duraton, ω = 0.06 a.u. (750 nm) s the frequency, E 0 s the strength and φ s the CEP of the laser feld. For ths laser feld the tunnel-onzaton tme s selected randomly effectvely n the tme nterval ( τ, τ). Once the ntal condtons for each electron are specfed, the poston and momentum of each electron are propagated forward n tme, startng at tme t 0, by solvng the classcal equatons of moton. The tme propagaton s determned by the three-body Hamltonan of the two electrons wth the nucleus kept fxed. All Coulomb forces are accounted for: the nteracton of each electron wth the nucleus and the laser feld and the electron-electron nteracton are all ncluded n the tme propagaton. Durng the tme propagaton each electron s nteractng wth the nucleus wth charge Z =. A trajectory s labeled as a doubly-onzed event f asymptotcally,.e. t, the energes of both electrons are postve. The double and sngle onzaton probabltes are gven by P DI N DI SI W = = W, P W N SI N W (5) where N DI, N SI and N are the numbers of doubly-onzed, sngly-onzed and all events, respectvely. For the results presented n ths work, the ntenstes consdered range from W/cm to W/ cm. At W/cm, CEPs are consdered rangng from φ = 5 to φ = 345 n steps of 30 ; for each φ, doubly-onzed events are obtaned as a result of runnng 400, -hour jobs; one job corresponds to CPU. For all other ntenstes, 4 CEPs are consdered rangng from φ = 0 to φ = 360 n steps of 5. For each φ, at 0 4 W/cm, W/cm, 0 4 W/cm, W/cm, W/cm and W/cm the doubly-onzed events thus obtaned are.5 0 4, 5 0 4, 0 5,.8 0 5, and 6 0 5, respectvely as a result of runnng 00, -hour jobs. For the results presented regardng total double onzaton the average has been taken over all CEPs for each ntensty. From the above, t s clear that the computatons requred, partcularly for the lower ntenstes, are challengng, snce, t s tme-consumng to obtan enough doubly-onzed events that N Scentfc REPOrts 7: 7488 DOI:0.038/s

4 Fgure 3. Probablty dstrbuton of the sum of the two electron momentum components parallel to the polarzaton of the laser feld (black sold lnes) for laser feld ntenstes from W/cm to W/ cm. For each ntensty, the probablty dstrbuton of the sum of the momenta of the delayed pathway (red dotted lnes) and of the drect pathway (blue dashed lnes) are also plotted. The focal volume effect s not accounted for. Each probablty dstrbuton s dvded by ts maxmum value. render the statstcal error very small for each ntensty and for each of the or 4 CEPs. The statstcal error s 4 proportonal to N DI. The ntense computatons requred s the reason results are obtaned for seven ntenstes n the range from W/cm to W/cm. Usng the results obtaned at these seven ntenstes an average over the focal volume s performed 5 to drectly compare wth experment. It s, however, noted that computatons at a larger number of ntenstes are needed to account more accurately for the focal volume effect (FVE). For the results presented, t s stated explctly when focal volume averagng s ncluded and when t s not. Results Double onzaton and pathways. In Fg., the rato of double to sngle onzaton probablty s computed as a functon of the laser ntensty and compared to the expermental results 8. It s found that the computed rato of double to sngle onzaton probablty reproduces well the overall pattern of the observed rato. The computed rato s found to be at most a factor of two smaller than the observed rato and by a factor of 3.5 when the focal volume effect s accounted for. Ths dfference possbly suggests that the effectve charge of Z = used to model the attractve Coulomb potental n the 3D semclasscal model durng tme propagaton overestmates the Coulomb attracton. Once the doubly-onzed events are obtaned usng the 3D semclasscal model, an analyss of the classcal trajectores s performed n tme n order to dentfy the contrbuton of the drect and the delayed pathway of NSDI as a functon of the laser ntensty. The man two double onzaton energy transfer pathways are dentfed by usng the tme dfference between the recollson tme t rec and the onzaton tme of each electron t on, wth =,, for each doubly-onzed classcal trajectory. The recollson tme s defned as the tme of mnmum approach of the two electrons and s dentfed by the maxmum n the electron par potental energy. The onzaton tme for each electron s defned as the tme when the sum of the electron s knetc energy (usng the canoncal momentum) and the potental energy due to the electron s nteracton wth the nucleus becomes postve and remans postve thereafter. The canoncal momentum of an electron s gven by p A, wth A the vector potental. The onzaton tme of electron s, thus, not necessarly the tme t 0 ths electron tunnel-onzes at the start of the tme propagaton. Ths energy s referred to as compensated energy and was ntroduced n ref. 6. A doubly-onzed trajectory s labeled as delayed or drect dependng on the tme dfferences ton t rec and t on t rec. Specfcally, Scentfc REPOrts 7: 7488 DOI:0.038/s

5 Fgure 4. Probablty dstrbuton of the sum of the two electron momentum components parallel to the polarzaton of the laser feld for laser feld ntenstes from W/cm to W/cm. The computed results wth the focal volume effect not accounted for are denoted by black sold lnes and when t s accounted for by black dashed lnes; the blue crosses denote the expermental results 8. Each probablty dstrbuton s dvded by ts maxmum value. on on rec dff on rec dff on on on t t < t & t < t (a) t t < t & t < t (b) Drect (6) on on rec dff on rec dff on t t > t & t t < t (a) t t > t & t t < t (b) rec rec dff dff Delayed (7) on rec dff on t t > t & t t > t Double Delayed (8) rec dff where t dff s a postve arbtrary parameter. The percentage of doubly-onzed events labeled as delayed or drect, out of all doubly-onzed events, depends on our choce of the tme dfference t dff. These percentages are gven by α R α DI = α N DI NDI W W, (9) where N DI s the number of α labelled doubly-onzed events, wth α denotng the drect or delayed events. Thus, α α the probablty of doubly-onzed events labeled as delayed or drect, out of all events, s gven by PDI = RDIPDI. t dff should not be chosen nether very large, such as /4 T, or very small such as /40 T. Choces n between are reasonable and lead to smlar trends of the two prevalng pathways of double onzaton. Ths s shown n Fg. where the percentages of drect and delayed doubly-onzed events are plotted for t dff equal to /0 T, /0 T and /40 T as a functon of the ntensty of the laser feld. It s found that the contrbuton of the drect and the delayed pathways to double onzaton as a functon of ntensty dsplays general trends that do not sgnfcantly depend on the choce of t dff. Both the drect and the delayed pathways of double onzaton sgnfcantly contrbute at all ntenstes. Thus, the drect pathway can not be neglected as was done n prevous models. The drect pathway contrbutes the most for ntermedate ntenstes. In Fg., at a hgh ntensty above W/cm, t s shown that the contrbuton of the drect pathway of double onzaton starts decreasng. At ths hgh ntensty a transton from strong to soft recollsons takes place, as dscussed n the followng. It s found that double delayed events contrbute no more than 5% for the smallest ntensty even when the tme dfference s chosen small and equal to /40 T. t dff = /0 T s chosen for the results presented n ths work. We fnd that wth ths choce of t dff the Scentfc REPOrts 7: 7488 DOI:0.038/s

6 Fgure 5. Frst row: measured correlated electron momenta 8. Second row: computed correlated electron momenta for all double onzaton events wth the focal volume effect accounted for. Thrd row: computed correlated electron momenta for all double onzaton events wth the focal volume effect not accounted for. Fourth row: correlated electron momenta for the drect pathway of double onzaton wth the focal volume effect not accounted for. Ffth row: correlated electron momenta for the delayed pathway of double onzaton wth the focal volume effect not accounted for. Sxth row: correlated electron momenta for the double delayed pathway of double onzaton wth the focal volume effect not accounted for. The sum of the fourth row plus the ffth row plus the sxth row s equal to the thrd row. Each double dfferental dstrbuton s dvded by ts maxmum value. dstrbutons of the sum of the two electron momentum components along the polarzaton drecton of the laser feld for the drect and the delayed pathways of double onzaton are the closest to what s expected from ref. 0. That s, the former dstrbuton dps whle the latter one peaks around zero. We fnd that dfferent results are obtaned f nstead of the compensated energy the energy of each electron s used to dentfy the onzaton tme. Namely, one fnds that at an ntensty of W/cm almost all classcal trajectores are dentfed as double delayed. Ths was the concluson n ref. 5. Usng the actual energy to dentfy the onzaton tme at an ntensty of W/cm results n the drect pathway of double onzaton stll only contrbutng 0%. However, ths s not a reasonable result. At W/cm 3.7 U p s equal to 50 ev whch s much hgher than the second onzaton energy of Ar. Moreover, the recollson at ths ntensty s strong, whch s dscussed n the secton for the correlated electron momenta as a functon of ntensty, and so the drect pathway of double onzaton should contrbute sgnfcantly. Thus, the compensated energy s employed to dentfy the onzaton tme n ths work whch leads to both the drect and delayed pathway beng the man pathways of double onzaton n agreement wth ref. 4 for the smallest ntensty. Dstrbuton of the sum of the electron momenta. In Fg. 3, the probablty dstrbutons of the sum of the two electron momentum components along the polarzaton drecton of the laser feld are presented for ntenstes from W/cm to W/cm. In Fg. 3, the contrbuton of the drect and the delayed pathways of double onzaton to the probablty dstrbuton of the sum of the momenta s also shown; the focal volume effect s not accounted for. It s found that the delayed pathway s contrbuton s a dstrbuton concentrated Scentfc REPOrts 7: 7488 DOI:0.038/s

7 around zero whle the drect pathway s contrbuton s a doubly-peaked dstrbuton, as expected from ref. 0. The drect pathway s probablty dstrbuton of the sum of the momenta s the broadest one. Therefore, ncludng only the delayed pathway of double onzaton would result n a narrower dstrbuton of the sum of the momenta than the observed one. Indeed, the D model descrbed n ref. 8 whch accounts only for the delayed pathway of double onzaton results n a narrower probablty dstrbuton of the sum of the momenta than the observed one. In Fg. 4, the expermental results for the probablty dstrbuton of the sum of the momenta n ref. 8 are compared wth one set of computed results that account for the focal volume effect (black dashed lnes) and one that does not (black sold lnes). It s found that the computed results are n good agreement wth the observed ones. Specfcally, t s found that, for each ntensty, the computed sum of the electron momenta extends over a range that s very smlar to the expermental one. For nstance, for an ntensty of W/cm, the computed sum of the momenta extends over a range from roughly a.u. to a.u., whle, for an ntensty of W/cm, t extends from 4 a.u. to 4 a.u.; for both ntenstes these ranges are n agreement wth the expermental results 8. It s noted that a dfference of the computed probablty dstrbutons of the sum of the electron momenta wth the expermental ones s that the computed ones have smaller values around zero. Ths s more so the case for the computed results that account for the focal volume effect. Ths dfference suggests that the current 3D model underestmates the contrbuton of the delayed pathway of double onzaton. Correlated electron momenta as a functon of ntensty: transton from strong to soft recollsons. For ntenstes rangng from W/cm to W/cm, the computed correlated electron momenta wth the focal volume effect accounted for are plotted and compared to the measured ones n Fg. 5. We fnd that at all ntenstes, but partcularly at smaller ones, there are fewer doubly-onzed events wth both momenta beng close to zero than n the expermentally obtaned correlated electron momenta 6, 8. In Fg. 5, we also plot the computed correlated electron momenta for all doubly-onzed events and for the drect and the delayed double onzaton pathways wthout accountng for the focal volume effect. At ntermedate ntenstes of W/cm, as n the observed correlated electron momenta n ref. 8, the computed correlated electron momenta transton to a well-known pattern 0, 7. Ths pattern nvolves both electrons escapng n the same drecton ether parallel or antparallel to the laser feld, thus, gvng rse to a much hgher probablty densty n the frst and thrd quadrants of the correlated electron momenta, rather than the second and fourth ones. We fnd that ths pattern s due to the drect pathway of double onzaton whch s the prevalng one at ntermedate ntenstes of W/cm. Ths pattern s due to strong recollsons where the two electron momentum components along the drecton of the laser feld are both determned from the vector potental at tmes just larger than the recollson tme. Thus, both electrons escape wth smlar momenta n the drecton along the polarzaton of the laser feld. We also fnd that at these ntermedate ntenstes the pattern of the correlated electron momenta for the delayed pathway of double onzaton s more spread out over all four quadrants and has a sgnfcant number of doubly-onzed events wth both electron momenta close to zero, as expected from ref. 0. At smaller ntenstes of W/cm and 0 4 W/cm, the computed correlated electron momenta resemble but do not qute have the cross-shaped pattern of the measured results 6, 8, see Fg. 5. The man dfference s that the computed correlated electron momenta have fewer doubly-onzed events wth both electron momenta beng close to zero. At these small ntenstes, we fnd that the drect pathway of double onzaton nvolves manly events wth both electrons escapng n the same drecton ether parallel or antparallel to the laser feld, however, the electron momenta are not as equal as at ntermedate ntenstes. At small ntenstes, we also fnd that the delayed pathway nvolves manly doubly-onzed events wth both electrons escapng n the opposte drecton, wth magntudes of the electron momenta that are more asymmetrc than for the drect pathway of double onzaton. The cross-shaped pattern s better reproduced by the delayed double onzaton pathway, see Fg. 5. Indeed, ths pathway does not have as many doubly-onzed events wth equal magntude of the components of the electron momenta along the drecton of the laser feld as the drect pathway does. In addton, we fnd that 63% of the events labelled as delayed doubly-onzed satsfy the condtons n eq. 7(b) whle 37% satsfy the condtons n eq. 7(a). That s, for the majorty of delayed doubly-onzed events the ntally bound electron s the one that onzes last followng recollson. We fnd that the correlated electron momenta when the tunnelng electron onzes second n the delayed doubly-onzed events (37%) resembles more a cross-shaped pattern than the correlated electron momenta when the ntally bound electron onzes second (63%), see Fg. 6. A less known pattern s that observed expermentally and retreved computatonally wth the 3D semclasscal model for ntenstes above W/cm, see Fg. 5. For these hgher ntenstes, t s found that the two electrons escape mostly wth opposte momenta for a sgnfcant number of doubly-onzed events. To dentfy the reason for ths shft n the correlated electron momenta, n Fg. 7, the tme electron tunnel-onzes, t 0, and the recollson tme, t rec, are plotted for three dfferent ntenstes, namely, 0 4 W/cm, W/cm and W/ cm and for two dfferent CEP s, namely, φ = 5 and φ = 05 for each ntensty. The tunnelng tme of electron s found to be close to the tmes correspondng to the extrema of the laser feld for all three ntenstes. However, the dstrbuton of the recollson tme s found to shft from tmes correspondng roughly to zeros of the laser feld for an ntensty of 0 4 W/cm to tmes correspondng to the extrema of the laser feld for an ntensty of W/cm. The transfer of energy from electron to electron s much smaller for the soft recollsons. For these hgher ntenstes, where soft recollsons preval, the momentum of electron s mostly determned from the vector potental at the tunnelng tme. The momentum of electron s determned by the vector potental shortly after recollson takes place whch s roughly half a laser cycle after electron tunnel-onzes. As a result the two electrons escape mostly wth opposte momenta. Ths mechansm of soft recollsons for hgher ntenstes was frst dentfed n a theoretcal study of strongly-drven N wth fxed nucle 0. For the delayed pathway of double onzaton, ths opposte momenta pattern, demonstrated wth much hgher probablty densty n the second and fourth quadrants of the correlated electron momenta, sets n at lower ntenstes of W/cm, Scentfc REPOrts 7: 7488 DOI:0.038/s

8 Fgure 6. Correlated electron momenta at an ntensty of W/cm for the delayed pathway of double onzaton for the case when the electron that onzes second s electron (a) and electron (b). Each double dfferental dstrbuton s dvded by ts maxmum value. Fgure 7. Probablty dstrbuton of the tunnelng tme t 0 of electron (blue lne) and of the recollson tme t rec (red lne) for ntenstes 0 4 W/cm, W/cm and W/cm and for two dfferent CEPs, φ = 5 and φ = 05, for each ntensty. Smlar results hold for all other CEPs. Grey lne denotes the laser feld. The lght blue arrows ndcate the mappng of a tunnelng-tme peak to a recollson-tme peak. see Fg. 5. For the drect pathway ths opposte momenta pattern sets n at hgher ntenstes of W/cm. Ths s consstent wth a smaller transfer of energy takng place from electron to electron at the recollson tme n the delayed pathway compared to the energy transfer n the drect pathway. Asymmetry parameter. The asymmetry parameter + R DI(I, φ) R DI(I, φ) A(I, φ) = + R (I, φ) + R (I, φ) (0) DI s computed as a functon of the ntensty I and the CEP (φ). R DI (I, φ) and R DI (I, φ) denote the percentage of doubly-onzed events wth ons escapng wth postve and negatve momentum, respectvely, along the drecton + of the polarzaton of the laser feld. Snce n the 3D semclasscal model the nucleus s fxed, R DI (I, φ) and R DI (I, φ) correspond to the percentage of double onzaton events where the sum of the two electrons momentum components along the drecton of the laser feld polarzaton are negatve and postve, respectvely. For each ntensty, A(I, φ) s ftted wth the snusodal functon DI + Scentfc REPOrts 7: 7488 DOI:0.038/s

9 Fgure 8. The computed results (blue crcles) for the asymmetry parameter A(I, φ) at I = W/cm as a functon of φ. The snusodal functon that s used to ft the computed results s denoted wth a red sold lne. Fgure 9. Asymmetry parameters A 0 (a) and φ 0 (b) as a functon of ntensty. The computed results for all doubly-onzed events when the focal volume effect s not accounted for are denoted by black sold lnes wth crcles and when t s accounted for by black dashed lnes wth trangles. The delayed pathway of double onzaton s denoted by red sold lnes wth trangles and the drect pathway by blue sold lnes wth crcles. For the drect and delayed pathways of double onzaton the focal volume effect s not accounted for. Expermental results 8 are denoted by lght blue crosses and dark blue squares. A(I, φ) = A (I) sn ( φ + φ (I)). () 0 0 In Fg. 8, we llustrate at W/cm how the snusodal functon n Eq. () fts our computed results for A(I, φ) wth A 0 = 0.4 and φ 0 = 46. The computed results show that for a gven ntensty the percentage of doubly-onzed events wth ons escapng wth postve versus negatve momentum changes as a functon of the CEP. In a manner smlar to the one llustrated n Fg. 8, we obtan the computed asymmetry ampltude A 0 (I) and offset phase φ 0 (I) at other ntenstes. The computed A 0 (I) and offset phase φ 0 (I) are plotted n Fg. 9(a,b), respectvely, and compared wth two sets of expermentally obtaned asymmetry parameters 8. The comparson shows that the 3D semclasscal model reproduces well the decreasng pattern of A 0 and the ncreasng pattern of φ 0 wth ncreasng ntensty. However, the computed values for these asymmetry parameters are hgher than the ones obtaned from the expermental results. Smaller values of A 0 correspond to a more spread out pattern of the correlated electron momenta. Thus, the larger values of A 0 of the computed results are consstent wth the computed correlated electron momenta havng less doubly-onzed events wth sum electron momenta close to zero compared to the measured ones. Moreover, n Fg. 9(a,b) the asymmetry parameters are plotted for each of the man two pathways of NSDI. It s shown that for both pathways the asymmetry parameter φ 0 (I) has a smlar pattern. The asymmetry parameter A 0 (I) for the delayed pathway s generally smaller. Ths s consstent wth the correlated electron momenta of the delayed pathway havng a more spread out pattern n all four quadrants than the correlated electron momenta of the drect double onzaton pathway, as we have seen n the prevous secton. Snce A 0 (I) s smaller for the delayed pathway of double onzaton, the fact that the computed values of A 0 (I) are larger than the measured ones could be due to the fact that the 3D semclasscal model underestmates the contrbuton of the delayed pathway. We have reached a smlar concluson when dscussng the dstrbuton of the sum of the electron momenta. Scentfc REPOrts 7: 7488 DOI:0.038/s

10 Fgure 0. Correlated electron momenta at an ntensty of W/cm for φ rangng from 5 to 65 wth a step of 30. For φ rangng from 5 to 65 the expermental results from ref. 8 are compared wth the computed results. Each double dfferental dstrbuton s dvded by ts maxmum value. Fgure. Percentage contrbuton of the drect and the delayed pathways of double onzaton as a functon of the CEP at an ntensty of W/cm. Correlated momenta and double onzaton pathways as a functon of CEP. In what follows, the dependence of the correlated momenta on the CEP s nvestgated at an ntensty of W/cm. In Fg. 0, the correlated momenta are plotted for φ rangng from 5 to 65 wth a step of 30. The bn sze of the CEP s chosen to be larger than 00 mrad, whch s the expermental precson of the CEP, and large enough n order to get good statstcs. For data analyss, all events are selected for whch one electron has been detected n concdence wth an Ar + on. The momentum of the second electron, whch s not detected for most events, s calculated from conservaton of momentum. Both the data and the computed results are symmetrzed wth respect to the bottom-left-to-top-rght dagonal n order to account for the two electrons beng ndstngushable. Moreover, due to the symmetry of the Hamltonan, when φ φ + 80 then p p. Ths symmetry s respected by the computed results. In the expermental results there s a small devaton from ths symmetry. Ths devaton arses from artfacts of the electron spectrometer and false concdences. For each CEP, n the top rght half of the correlated electron momenta plot the mpact of false concdences s stronger than n the bottom left half. For CEP rangng from 95 to 345 the correlated electron momenta plots have more doubly-onzed events n the top rght half. Thus, n Fg. 0, we compare the computed results wth the measured correlated electron momenta whch are more accurate,.e. for CEP rangng from 5 to 65 where the correlated electron momenta have more doubly-onzed events n the bottom left half. A good agreement s found between the computed and the expermental results for CEP rangng from 5 to 65 gven the expermental uncertanty of 00 mrad n the CEP. Specfcally, the computed correlated momenta correctly reproduce the overall observed pattern for each ndvdual CEP. A dfference between the computed and the expermental results s that the former results have less doubly-onzed events wth both electron momenta close to zero suggestng that the computatons overestmate the contrbuton of the drect pathway. To better llustrate the change of the correlated momenta pattern as a functon of the CEP plotted n Fg. 0, n Fg. the percentage of the drect and delayed pathways of double onzaton are plotted as a functon of the CEP. At an ntensty of W/cm the delayed pathway has the largest contrbuton for CEPs φ = 45 and φ = 5, Scentfc REPOrts 7: 7488 DOI:0.038/s

11 whle t has the smallest for CEPs φ = 65 and φ = 345. In Fg. 0, a comparson of the correlated momenta between φ = 45 and φ = 65 shows that there s a hgher probablty densty for both electrons to onze wth the same large momentum, wth both electrons escapng n the drecton that s opposte to the electrc feld, for φ = 65 than for φ = 45. Ths s ndeed consstent wth the drect onzaton pathway havng a larger contrbuton for φ = 65 than for φ = 45 as shown n Fg.. From Fg., t s found that the contrbuton of each of the two man pathways of double onzaton vares roughly by 0% as a functon of the CEP for the smallest ntensty of W/cm. Conclusons Usng a 3D semclasscal model we nvestgate the dependence of double onzaton observables on the ntensty and on the carrer envelope phase of a near-sngle-cycle near-nfrared laser feld employed to drve Ar. The good agreement of the computed results wth recent experments employng near-sngle-cycle laser pulses 6, 8, 8, adds to prevous successes of ths 3D model n dentfyng features of non-sequental double onzaton of two-electron atoms when drven by many-cycle laser pulses, 5, 9. A dfference between the computed and the expermental results was found to be a lower value of the dstrbuton of the sum of the two electron momentum components along the drecton of the polarzaton of the laser feld and of the correlated electron momenta around zero. Ths seems to suggest that the current 3D model overestmates the Coulomb attracton of each electron from the nucleus. Future studes can mprove on the 3D model for many electron atoms such as Ar by usng more accurate effectve potentals for the tme propagaton. Moreover, t was demonstrated that the man pathways of double onzaton, that s, the drect and the delayed pathways, both sgnfcantly contrbute at all ntenstes currently under consderaton. Furthermore, the prevalence of the drect versus the delayed pathway was nvestgated as a functon of the CEP for an ntensty of W/cm and t was shown that the results obtaned are consstent wth features of the observed correlated electron momenta 8. Fnally, a prevously-predcted n the context of a strongly-drven fxed-nucle N unexpected ant-correlaton momentum pattern at hgher ntenstes 0, s observed expermentally n the context of strongly-drven Ar 8 and also reproduced n the current work for strongly-drven Ar by a near-sngle-cycle laser feld. It s shown that ths ant-correlaton pattern s due to soft recollsons wth recollson tmes close to the extrema of the laser feld. References. Corkum, P. B. Plasma perspectve on strong feld multphoton onzaton. Phys. Rev. Lett. 7, 994 (993).. Taylor, K. T., Parker, J. S., Dundas, D. & Meharg, K. J. Theory of laser-drven double-onzaton of atoms at T:sapphre laser wavelengths. J. Mod. Opt. 54, 959 (007). 3. Becker, A., Dörner, R. & Moshammer, R. Multple fragmentaton of atoms n femtosecond laser pulses. J. Phys. B: At. Mol. Opt. Phys. 38, S753 (006). 4. Ullrch, J. et al. Recol-on and electron momentum spectroscopy: reacton-mcroscopes. Rep. Prog. Phys. 66, 463 (003). 5. Camus, N. et al. Attosecond correlated dynamcs of two electrons passng through a transton state. Phys. Rev. Lett. 08, (0). 6. Bergues, B. et al. Attosecond tracng of correlated electron-emsson n non-sequental double onzaton. Nat. Commun. 3, 83 (0). 7. Johnson, N. G. et al. Sngle-shot carrer-envelope-phase-tagged on-momentum magng of nonsequental double onzaton of argon n ntense 4-fs laser felds. Phys. Rev. A 83, 034 (0). 8. Kübel, M. et al. Complete characterzaton of sngle-cycle double onzaton of argon from the nonsequental to the sequental onzaton regme. Phys. Rev. A 93, 0534 (06). 9. Kopold, R., Becker, W., Rottke, H. & Sandner, W. Routes to nonsequental double onzaton. Phys. Rev. Lett. 85, 378 (000). 0. Feuersten, B. et al. Separaton of recollson mechansms n nonsequental strong feld double onzaton of Ar: the role of exctaton tunnelng. Phys. Rev. Lett. 87, (00).. Emmanouldou, A. Prevalence of dfferent double onzaton pathways and traces of three-body nteractons n strongly drven helum. Phys. Rev. A 83, (0).. Maxwell, A. S. & Fguera de Morsson Fara, C. Controllng below-threshold nonsequental double onzaton va quantum nterference. Phys. Rev. Lett. 6, 4300 (06). 3. Kustaanhemo, P. & Stefel, E. Perturbaton theory of Kepler moton based on spnor regularzaton. J. Rene Angew. Math. 8, 04 (965). 4. Huang, C., Zhou, Y., Zhang, Q. & Lu, P. Contrbuton of recollson onzaton to the cross-shaped structure n nonsequental double onzaton. Optcs Express, 390 (03). 5. Emmanouldou, A. Recol collsons as a portal to feld-asssted onzaton at near-uv frequences n the strong-feld double onzaton of helum. Phys. Rev. A 78, 034 (008). 6. Parker, J. S. et al. Hgh-energy cutoff n the spectrum of strong-feld nonsequental double onzaton. 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Plateau n above-threshold-onzaton spectra and chaotc behavor n rescatterng processes. Phys. Lett. A 36, 533 (997). 4. Abrnes, R. & Percval, I. C. Classcal theory of charge transfer and onzaton of hydrogen atoms by protons. Proc. Phys. Soc. 88, 86 (966). 5. Wang, P., Sayler, A. M., Carnes, K. D., Esry, B. D. & Ben-Itzhak, I. Dsentanglng the volume effect through ntensty-dfference spectra: applcaton to laser-nduced dssocaton of H +. Opt. Lett. 30, 664 (005). Scentfc REPOrts 7: 7488 DOI:0.038/s

12 6. Leopold, J. G. & Percval, I. C. Ionsaton of hghly excted atoms by electrc felds. III. Mcrowave onsaton and exctaton. J. Phys. B, 709 (979). 7. Weber, Th et al. Correlated electron emsson n multphoton double onzaton. Nature 405, 658 (000). 8. Bergues, B., Kübel, M., Klng, N. G., Burger, C. & Klng, M. F. Sngle-cycle non-sequental double onzaton. IEEE J. Sel. Top. Quantum Electron., (05). Acknowledgements A.E. acknowledges the EPSRC grant no. J0783 and the use of the computatonal resources of Legon at UCL as well as the UCL Global Engagement Fundng. M.K., M.F.K., and B.B. acknowledge support from the Max Planck Socety and the DFG cluster of excellence Munch Centre for Advanced Photoncs (MAP). Author Contrbutons A.C. performed the analyss of the computatons nvolved and contrbuted to the deas nvolved. A.E. provded the codes used for the computatons and the analyss and was responsble for the man deas nvolved n the theoretcal analyss. M.K., M.F.K., and B.B. provded the expermental results and contrbuted to the dscusson n the paper concernng the comparson of the computatons wth the expermental results. All authors revewed the manuscrpt. Addtonal Informaton Competng Interests: The authors declare that they have no competng nterests. Publsher's note: Sprnger Nature remans neutral wth regard to jursdctonal clams n publshed maps and nsttutonal afflatons. Open Access Ths artcle s lcensed under a Creatve Commons Attrbuton 4.0 Internatonal Lcense, whch permts use, sharng, adaptaton, dstrbuton and reproducton n any medum or format, as long as you gve approprate credt to the orgnal author(s) and the source, provde a lnk to the Creatve Commons lcense, and ndcate f changes were made. The mages or other thrd party materal n ths artcle are ncluded n the artcle s Creatve Commons lcense, unless ndcated otherwse n a credt lne to the materal. If materal s not ncluded n the artcle s Creatve Commons lcense and your ntended use s not permtted by statutory regulaton or exceeds the permtted use, you wll need to obtan permsson drectly from the copyrght holder. To vew a copy of ths lcense, vst The Author(s) 07 Scentfc REPOrts 7: 7488 DOI:0.038/s