Strong-field triple ionization of atoms with p^3 valence shell
Jakub S. Prauzner-Bechcicki, Dmitry K. Efimov, Micha? Mandrysz, Jakub Zakrzewski
SStrong-field triple ionization of atoms with p valence shell Jakub S. Prauzner-Bechcicki, ∗ Dmitry K. Efimov,
2, 3
Micha(cid:32)l Mandrysz, and Jakub Zakrzewski
2, 4 Instytut Fizyki imienia Mariana Smoluchowskiego,Jagiellonian University in Krakow, (cid:32)Lojasiewicza 11, 30-348 Krak´ow, Poland Institute of Theoretical Physics, Jagiellonian University in Krakow, (cid:32)Lojasiewicza 11, 30-348 Krak´ow, Poland Department of Theoretical Physics, Faculty of Fundamental Problems of Technology,Wroc(cid:32)law University of Science and Technology, 50-370 Wroc(cid:32)law, Poland Mark Kac Complex Systems Research Center, Jagiellonian University in Krakow, (cid:32)Lojasiewicza 11, 30-348 Krak´ow, Poland (Dated: February 15, 2021)The interaction of strong pulsed femtosecond laser field with atoms having three equivalent elec-trons in the outer shell ( p configuration, e.g. nitrogen) is studied via numerical integration of atime-dependent Schr¨odinger equation on a grid approach. Single, double and triple ionization yieldsoriginating from a completely antisymmetric wave function are calculated and extracted using arestricted-geometry model with the soft-core potential and three active electrons. The direct tripleionization channel is found to produce a larger yield than the channel connected with single and thendirect double ionization. Compared against earlier results investigating the ns np configuration,we propose that the differences found here might in fact be accessible through electron’s momentumdistribution. I. INTRODUCTION
The study of correlations is the study of the complex-ity of the world around us. One of the amazing mani-festations of the existence of correlations in nature is thephenomenon of non-sequential double ionization (NSDI)in strong laser fields [1, 2]. Reports from experimentsshowing the recorded double ionization yield higher byseveral orders of magnitude than expected in the sequen-tial electron escape processes [3–5], followed by the mea-surements of the ion recoil momentum and latter extrac-tion of electrons’ momenta distributions with the famousfinger-like structure [6, 7] forced researchers to acknowl-edge the fundamental role of electron-electron correla-tions played in NSDI. Along with the experimental work,there were attempts to theoretically explain the observedphenomenon. It is now recognized that the process hasa stepwise character and the rescattering is in focus. Inshort, one of the electrons tunnels and begins to moveaway from its parent ion. When the phase of the fieldchanges (we deal with short pulses, usually having thewavelength on the border of visible and infrared light),the electron is turned back, accelerated and forced to rec-ollide with the ion. As a result of the recollision, energytransfer occurs and consequently the escape of the secondelectron is allowed.Higher ionization yield is also observed in the pro-cesses involving three or more electrons, then we speak ofnon-sequential multiple ionization (NSMI) [8]. Here too,rescattering plays an important role. However, theoreti-cal analysis of events involving more than two electrons isvery difficult. This is evidenced by the fact that full-size,i.e. taking into account all spatial dimensions for eachelectron, quantum calculations even for two electrons are ∗ [email protected] still very rare [9–13]. Simplified quantum models witha reduced number of dimensions are often used to over-come the numerical difficulty [6, 14–24]. And in the casewhen three and more electrons are involved, classical orsemi-classical calculations are dominant [25–31].We have recently shown that it is possible to constructa model with a reduced geometry that enables a study oftriple ionization [20]. Importantly, the electronic configu-ration of the target atoms begins to play a significant role.In the case when two electrons are involved in the process,it is usually assumed that they have opposite spins andtherefore the spatial part of the wave function is symmet-rical. If one considers the two-electron problem with theantisymmetric spatial wave function (e.g. correspondingto the S metastable state in He), the NSDI is expectedto be strongly suppressed [32]. When three electrons areat play, there is no possibility that the spatial wave func-tion is symmetrical. The electron configuration of thetarget atoms is reflected in the symmetry of the wavefunction under consideration. And so, for alkali metalswith ns np configurations we will have a spatial wavefunction which is partially antisymmetric, while for ele-ments with p configuration (e.g. nitrogen) we will have acompletely antisymmetric function. Importantly, due tosymmetry properties of the ground state for atoms withthe ns np configuration it is not possible to reduce theproblem of double ionization to the model of two activeelectrons [22]. In contrary, such a reduction is possiblein the case of atoms with the p configuration [24].In the previous work [20] we considered the triple ion-ization events in atoms with the ns np configuration. Inthat case, the dominant triple ionization channel was thesequential escape for fields with amplitudes F = 0 . F = 0 . a r X i v : . [ phy s i c s . a t o m - ph ] F e b For the range of the analyzed field amplitudes, the pro-cess in which one electron is ionized first, and then two,plays a dominant role among the three mentioned pathsof non-sequential escape.In the present paper, we concentrate on the influence ofthe initial state symmetry on triple ionization. For thispurpose, we analyze triple ionization events for atomswith the p configuration in the outer shell and com-pare the results with the physics in the ns np configu-ration. The paper is structured as follows. Section II Adescribes briefly the dimensional reduction applied in themodel and the involved parameters, while section II B thespace-division method and extraction of fluxes allowingfor calculating channel contributions in (multi)-electronionization. In section III we present the main resultsand compare them to the results obtained in our previ-ous work with different electron configuration. SectionIV contains the conclusions. II. MODEL AND METHODSA. Model
Due to a computational complexity it is virtually im-possible nowadays to tackle the three electron problemnumerically in the full phase space. Therefore, we em-ploy a judiciously designed restricted-space model [20] inwhich each of the three electrons is allowed to move alongone-dimensional (1D) track. The chosen 1D-tracks areequivalent to the lines along which the saddles, formedby the instantaneous electric field in the potential, movewhen the field amplitude is varied (see Fig. 1(a)). Thesaddles and their motion were determined with the ap-plication of local stability analysis in the adiabatic po-tential [33, 34]. The Hamiltonian in the restricted-spacereads (in atomic units): H = (cid:88) i =1 p i V a + V int , (1)where V a is the atomic potential: V a = − (cid:88) i =1 (cid:112) r i + (cid:15) + (cid:88) i,j =1; i 02 and effective electron-electroncharges q ee = √ . V int de-scribes an interaction with the external field: V int = (cid:114) F ( t )( r + r + r ) . (3) r i and p i are the i ’th electron’s coordinate and conju-gated momentum, respectively. The field is defined via FIG. 1. The restricted-space model [24]. Panel (a): The ge-ometry of the model with respect to three-dimensional space:Electrons propagate along r , r , and r axes. The field po-larization direction, (cid:126)F , is indicated by the arrow. Panels (b)and (c) - visualization of the space division within the modelas used for calculation of probability fluxes. The space is di-vided into regions corresponding to neutral states ( A ), singlyionized states ( S ), doubly ionized states ( D ), and triply ion-ized states ( T ). The borders between respective regions aremarked with different colors as shown in each panel: on panel(b) borders A − S , A − D and A − T are depicted, whereason panel (c) borders S − D , S − T and D − T . The bor-der distances are r a = 12 a.u., r b = 7 a.u., and r c = 5 a.u.,respectively. its vector potential, F ( t ) = − ∂A/∂t , and is polarizedalong the z axis in full space: A ( t ) = F ω sin (cid:18) πtT p (cid:19) sin( ω t + ϕ ) , < t < T p . (4)Here F , ω , T = 2 πn c /ω , ϕ and n c are the field ampli-tude, the pulse frequency, the pulse length, the carrier-envelope phase and the number of cycles. In the followingwe set ω = 0 . 06 which corresponds to 760 nm of laserwavelength and the number of cycles, n c = 5. The fieldamplitude and the carrier-envelope phase are varied, al-though the behavior has been fairly consistent along thewhole domain of the carrier-envelope phase hence onlythe results for ϕ = 0 are presented. As the field is po-larized along the z axis in the full space it has to beprojected onto r i tracks. The projection imposes the ge-ometric factor (cid:112) / B. Methods To calculate ionization yields we apply a space divi-sion method as commonly used before in both classicaland quantum-mechanical calculations [21, 35] - compareFig. 1(b) and Fig. 1(c). The total space is divided intoregions corresponding to neutral states ( A ), singly ion-ized states ( S ), doubly ionized states ( D ) and triply ion-ized states ( T ). Region A extends up to r a = 12 a.u.from the origin of coordinate system in each direction es-tablishing in this way a volume capable of enclosing theneutral atom wave function. Region S is defined as sumof regions for which two electrons are close to the nucleus(less than r b = 7 a.u. from the origin) and the third oneis far away. Region D is defined as sum of regions forwhich only one electron is still close to the nucleus (lessthan r c = 5 a.u from the origin) and two other electronsare already far away. The last region, T , is defined as asum of regions for which all electrons are far away fromthe origin. The populations of A , S , D and T statesare calculated as integrated probability fluxes throughthe borders of respective regions. The assignment of theregions is to some extent arbitrary and the position ofthe borders affects quantitatively the results. However,as verified before [20–22, 24] the chosen values provideresults that reflect the correct trends in the dynamics ofthe studied system.A much simpler case of two-electron system is de-scribed in detail in [16, 21], here we just briefly mentionthe methodology behind the calculations and present theborders A − S , A − D , and A − T in Fig. 1(b) and borders S − D , S − T , and D − T Fig. 1(c). The instantaneousvalue of the population in region R is calculated via theintegral: P R ( r , t ) = P R ( r , − (cid:90) t f R ( τ )d τ, (5)where f R ( τ ) represents probability flux over border ofthe R region, i.e. f R ( τ ) = − (cid:90) (cid:90) ∂R j ( r , τ ) · d σ . (6)Here d σ is a surface element and ∂R symbolizes borderof the region R .The space-division method allows straightforwardly todistinguish between direct and sequential escapes in caseof double ionization. For instance, calculating the fluxthrough A − D border allows us to obtain ionization yieldfor direct double ionization, whereas calculations of theflux through S − D border will give the ionization yieldfor the sequential process. In the case of triple ioniza-tion, the method allows only to separate direct escapes,i.e. through A − T border, from non-direct escapes thatare calculated via fluxes through S − T and D − T bor-ders. The latter two fluxes represent processes that in-volve double ionization either as the second or the firststep in a path leading to triple ion, respectively. At thisstage, however, it is not possible to tell whether thatdouble ionization being an intermediate step in the tripleionization is direct or sequential itself. We discuss thisissue and its solution later in the text. - 7 - 6 - 5 - 4 - 3 - 2 - 1 Ionization Yield F i e l d a m p l i t u d e [ a . u . ] - 7 - 6 - 5 - 4 - 3 - 2 - 1 Contributing processes F i e l d a m p l i t u d e [ a . u . ] ( a ) ( b ) FIG. 2. Numerical ionization yields as a function of the peakelectric field amplitude in atomic units. F = 0 . . × V/m or laser intensity I = 3 . × W/cm . Panel (a): Total yields for single ionization (blacktriangles), double ionization (red circles) and triple ioniza-tion (blue squares); solid lines represent averaged data afterintegration over a Gaussian beam. Panel (b): Different con-tributions to double and triple ionization - sets of red andblue lines, respectively. Solid lines without symbols show di-rect double (red) and triple (blue) escapes. Solid lines withsymbols show sequential double (red line with circles) andsequential triple (blue line with squares) escapes. Dashedblue line presents double direct ionization followed by singleionization and dash-dotted blue line shows single ionizationfollowed by direct double ionization. III. RESULTS AND DISCUSSION Let us first consider total ionization yields as a func-tion of the peak electric field amplitude, see Fig. 2(a).Results presented are obtained for the carrier-envelopephase ϕ = 0. Single ionization (SI, black triangles) sig-nal quickly saturates, then drops down for amplitudeslarger than F = 0 . P avg ( I ) ∝ (cid:90) I P ( I ) I d I (7)The averaged SI yield is depicted with solid black linein Fig. 2 (a). As expected, once the saturation level isachieved it does not drop, because the higher intensitythe lower weight is given to the respective yield. Thesame averaging procedure is used for double ionization(DI) and triple ionization (TI) yields and indicated bythe corresponding solid lines. Analyzing both the full DIyield (red circles) and its averaged counterpart (solid red - 1 0 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 Ratio F i e l d a m p l i t u d e [ a . u . ] FIG. 3. Ratios of volume averaged numerical ionization yieldsas a function of peak electric field amplitude in atomic units:DI/SI ratio (black line), TI/SI (red line), and TI/DI (blueline). line) in Fig. 2(a) it is easy to notice the characteristicknee for amplitudes close to F = 0 . ns np electron configura-tion [20]. The ratio of double to single ionization yields(see black line in Fig. 3) is practically constant for thefield amplitude values corresponding to the characteris-tic knee in the yield curve and is of the order of 10 − asreported in experiments [5, 8]. Interestingly, similar de-pendence on the field amplitude is presented by the ratioof triple to single ionization yields (see red line in Fig. 3)- one could argue that there is a weak knee in TI yieldtoo.Multi-ionization signals may be further separated intodifferent components due to the applied method of calcu-lating yields (see Fig. 2(b)). First, the double ionizationsignal is divided into two contributions, i.e. re-collisioninduced direct ionization (solid red line) and sequentialionization (red circles). The latter signal includes alsocontribution for re-collision excitation with a subsequentionization therefore the knee as a marker of re-collisionimportance is still visible. Sequential ionization signalrapidly grows for field amplitudes larger than F = 0 . → → → → ns np electron configuration [20], namely, we estimate differ-ent contributions to triple ionization based on what welearned about double ionization in this setup. More pre-cisely, we assume that that the ratio of sequential to non-sequential double ionization, as determined by the fluxesthrough S − D and A − D , holds for double ionizationbeing the intermediate step in the three-electron process.Such an assumption allows us to extract the sequentialcontribution from the mixed paths. The results are pre-sented in Fig. 2(b) with a collection of blue lines. The sig-nal that corresponds to a direct escape of three electronsis marked with the solid blue line, signals correspondingto partially direct escapes are marked with dashed anddash-dotted blue lines, and finally, signal for the sequen-tial escape is marked with blue squares. As expected thesequential triple ionization (0 → → → 3) dominates overthe whole range of field amplitudes, however, other chan-nels give non-negligible contributions too. Especially in-teresting is the fact that a direct triple ionization yield(0 → 3) is not the lowest one. The weakest signal comesfrom the process in which single ionization is followed bya direct double ionization (0 → → → → 3) is much stronger than both(0 → 3) and (0 → → 3) pathways. The observed hierar-chy of contributions is different from that obtained forLi-like atoms as reported earlier [20] for which direct es-cape was the least important channel of ionization. It isenvisaged that such a difference in the hierarchy of contri-bution may influence electron’s momentum distributionand thus be accessible in future experiments. IV. CONCLUSIONS We have studied triple ionization of atoms with p valence shell. To this end we employed the restricted-geometry model with three active electrons solved onthe grid. 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