Controlling electron-electron correlation in frustrated double ionization of molecules with orthogonally polarized two-color laser fields
aa r X i v : . [ phy s i c s . a t o m - ph ] J un Controlling electron-electron correlation in frustrated double ionization of moleculeswith orthogonally polarized two-color laser fields
A. Chen, M. F. Kling,
2, 3 and A. Emmanouilidou Department of Physics and Astronomy, University College London,Gower Street, London WC1E 6BT, United Kingdom Department of Physics, Ludwig-Maximilians-Universit¨at Munich,Am Coulombwall 1, D-85748 Garching, Germany Max Planck Institute of Quantum Optics, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany (Dated: August 3, 2018)We demonstrate the control of electron-electron correlation in frustrated double ionization (FDI)of the two-electron triatomic molecule D +3 when driven by two orthogonally polarized two-color laserfields. We employ a three-dimensional semi-classical model that fully accounts for the electron andnuclear motion in strong fields. We analyze the FDI probability and the distribution of the mo-mentum of the escaping electron along the polarization direction of the longer wavelength and moreintense laser field. These observables when considered in conjunction bear clear signatures of theprevalence or absence of electron-electron correlation in FDI, depending on the time-delay betweenthe two laser pulses. We find that D +3 is a better candidate compared to H for demonstrating alsoexperimentally that electron-electron correlation indeed underlies FDI. PACS numbers: 33.80.Rv, 34.80.Gs, 42.50.Hz
Frustrated double ionization (FDI) is a major processin the nonlinear response of multi-center molecules whendriven by intense laser fields, accounting roughly for 10%of all ionization events [1, 2]. In frustrated ionizationan electron first tunnel-ionizes in the driving laser field.Then, due to the electric field of the laser pulse, it isrecaptured by the parent ion in a Rydberg state [3]. Thisprocess is a candidate for the inversion of N in free-spaceair lasing [4]. In FDI an electron escapes and another oneoccupies a Rydberg state at the end of the laser pulse.FDI has attracted considerable interest in recent years ina number of experimental studies in the context of H [1]and of the triatomic molecules D +3 and H +3 [5–7].In theoretical studies of strongly-driven two-electrondiatomic and triatomic molecules, two pathways of FDIhave been identified [2, 8]. Electron-electron correlationis important, primarily, for one of the two pathways. It iswell accepted that electron-electron correlation underliesa significant part of double ionization in strongly-drivenmolecules—a mechanism known as non-sequential doubleionization [9, 10]. However, electron-electron correlationin FDI has yet to be accessed experimentally.Here, we propose a road for future experiments to iden-tify the important role of electron-electron correlation inFDI. We identify the parameters of orthogonally polar-ized two-color (OTC) laser fields that best control therelevant pathway for electron-electron correlation in FDI.We demonstrate traces of attosecond control of electronmotion in space and time in two observables of FDI as afunction of the time-delay between the fundamental 800nm and the second harmonic 400 nm laser field. We showthat, together, the FDI probability and the momentumof the escaping electron along the fundamental laser fieldbear clear signatures of the turning on and off of electron-electron correlation.Two-color laser fields are an efficient tool for control- ling electron motion [11, 12] and for steering the outcomeof chemical reactions [13–15]. Other applications includethe field-free orientation of molecules [16–18], the genera-tion of high-harmonic spectra [19–22] and probing atomicand molecular orbital symmetry [23–25].The strongly-driven dynamics of two electrons andthree nuclei poses a challenge for fully ab-initio quan-tum mechanical calculations. The latter techniques cancurrently address one electron triatomic molecules [26].Therefore, we rely on classical and semi-classical mod-els for understanding the fragmentation dynamics in tri-atomic molecules driven by intense infrared laser pulses[8, 27]. Our work employs a three-dimensional semi-classical model. This model has provided significant in-sights into FDI for strongly-driven H [2] and D +3 [8].Our previous result for the distribution of the kinetic en-ergy release of the Coulomb exploding nuclei in FDI ofD +3 was in good agreement with experiment [7].We employ the initial state of D +3 that is accessed ex-perimentally via the reaction D + D +2 → D +3 + D [5, 7].It consists of a superposition of triangular-configurationvibrational states ν = 1 – 12 [7, 28]. We assume that mostof the D +3 ionization occurs at the outer classical turn-ing point of the vibrational levels [29, 30]. The turningpoint varies from 2.04 a.u. (v = 1) to 2.92 a.u. (v = 12)[28, 31]. We initialize the nuclei at rest for all vibrationallevels, since an initial pre-dissociation does not signifi-cantly modify the ionization dynamics [32].The combined strength of the two laser fields is withinthe below-the-barrier ionization regime. To formulatethe initial state of the two electrons, we assume thatone electron (electron 1) tunnel-ionizes at time t inthe field-lowered Coulomb potential. For this quantum-mechanical step, we compute the ionization rate using asemi-classical formula [33]. t is selected using impor-tance sampling [34] in the time interval the two-colorlaser field is present. The ionization rate is then usedas the importance sampling distribution. For electron1, the velocity component that is transverse to the OTClaser fields is given by a Gaussian [35] and the componentthat is parallel is set equal to zero. The initial state ofthe initially bound electron (electron 2) is described bya microcanonical distribution [36].Another quantum mechanical aspect of our 3D modelis tunneling of each electron during the propagation witha probability given by the Wentzel-Kramers-Brillouin ap-proximation [2, 32]. This aspect is essential to accuratelydescribe the enhanced ionization process [10, 37]. In EI,at a critical distance of the nuclei, a double potential wellis formed such that it is easier for an electron bound tothe higher potential well to tunnel to the lower potentialwell and subsequently ionize. The time propagation isclassical, starting from time t . We solve the classicalequations of motion for the Hamiltonian of the strongly-driven five-body system, while fully accounting for theCoulomb singularities [32].The OTC laser field we employ is of the form E (t, Δ t) = E ω f(t)cos( ω t)ˆz + E ω f(t + Δ t)cos[2 ω (t + Δ t)]ˆxf(t) = exp –2ln2 (cid:18) t τ FWHM (cid:19) ! , (1)with ω = 0.057 a.u. for commonly used Ti:sapphire lasersat 800 nm. T ω and T ω are the corresponding periods ofthe fundamental and second harmonic laser fields, polar-ized along the ˆz- and ˆx-axis, respectively. τ FWHM = 40 fsis the full-width-half-maximum. Δ t is the time delay be-tween the ω –2 ω pulses. We consider E ω = 0.08 a.u., sincefor this field strength pathway B of FDI, where electron-electron correlation is present, prevails over pathway A—4.8% versus 3.6% [8].In FDI of D +3 the final fragments are a neutral excitedfragment D ∗ , two D + ions and one escaping electron. Inthe neutral excited fragment D ∗ the electron occupies aRydberg state with quantum number n >
1. The differ-ence between the two FDI pathways lies in how fast theionizing electron escapes following the turn on of the laserfield [2]. In pathway A, electron 1 tunnel-ionizes and es-capes early on. Electron 2 gains energy from an EI-likeprocess and tunnel-ionizes. It does not have enough driftenergy to escape when the laser field is turned off andfinally it occupies a Rydberg state, D ∗ . In pathway B,electron 1 tunnel-ionizes and quivers in the laser field re-turning to the core. Electron 2 gains energy from both anEI-like process and the returning electron 1 and tunnel-ionizes after a few periods of the laser field. When thelaser field is turned off, electron 1 does not have enoughenergy to escape and remains bound in a Rydberg state.It follows that electron-electron correlation is more pro-nounced in pathway B [2].To compute the FDI probability as a function of the FIG. 1. (a) The FDI probability and the probabilities ofpathways A and B and (b) the distribution of V max12 are plot-ted as a function of Δ t for E ω = 0.08 a.u. and E ω = 0.05a.u.. In (a) the arrows on the right indicate the correspondingprobabilities when E ω = 0 a.u.. time delay Δ t of the ω – 2 ω pulses, we useP FDI ( Δ t) = P ν ,i P ν Γ ( Δ t, ν , i)P FDI ( Δ t, ν , i) P ν ,i P ν Γ ( Δ t, ν , i) , (2)where i refers to the different orientations of the moleculewith respect to the z-component of the laser field. Weconsider only two cases of planar alignment, that is, oneside of the equilateral, molecular triangle is either parallelor perpendicular to the ˆz–axis. Γ ( Δ t, ν , i) is given by Γ ( Δ t, ν , i) = Z t f t i Γ (t , Δ t, ν , i)dt , (3)where the integration is over the duration of the OTCfield. Γ (t , Δ t, ν , i) is the ionization rate at time t fora certain molecular orientation i, vibrational state ν andtime delay Δ t. P ν is the percentage of the vibrationalstate ν in the initial state of D +3 [28]. P FDI ( Δ t, ν , i) isthe number of FDI events out of all initiated classicaltrajectories for a certain molecular orientation i, vibra-tional state ν and time delay Δ t. Due to the challengingcomputations involved, we approximate Eq. (2) using the ν = 8 state of D +3 . This approximation is justified, sincewe find that the ν = 8 state contributes the most in thesum in Eq. (2). We obtain very similar results for the ν = 7, 9 states, which contribute to the sum in Eq. (2)less than the ν = 8 state but more than the other states.In Fig. 1(a), for E ω = 0.05 a.u., we plot the FDI prob-ability as a function of the time delay for Δ t ∈ [0, T ω ].The results are periodic with T ω /2. We find that theFDI probability changes significantly with Δ t. Thischange is mainly due to pathway B with probability thatvaries from 1.2% at Δ t =-0.2, -0.7 T ω to 6.7% at Δ t =-0.4, -0.9 T ω . In contrast, the probability of pathway Achanges significantly less varying from 2.4% to 3.7%. ForE ω < ω = 0.05 a.u..Control of electron-electron correlation in double ion-ization in atoms has been demonstrated through the freeparameters Δ t and E ω of OTC laser fields [38–43]. Thetime-delay between the laser fields can significantly affectthe time and the distance of the closest approach of thereturning electron [11]. For FDI, this is demonstrated inFig. 1(b). For each classical trajectory labelled as FDI,we compute the maximum of the Coulomb potential en-ergy 1/ | r – r | , V max12 . Then, we plot the distribution ofV max12 as a function of Δ t. The minimum values of V max12 correspond to electron 1 being at a maximum distancefrom the core, i.e. minimum electron-electron correla-tion. Comparing Fig. 1(a) with (b), we find that theseminima occur at the same Δ ts, where the FDI probabil-ity and the probability of pathway B is minimum, i.e. at Δ t =-0.2, -0.7 T ω .The probability of each FDI pathway as well as V max12 are not experimentally accessible quantities. To demon-strate the presence of electron-electron correlation inFDI, in addition to the sharp change of the FDI probabil-ity with Δ t, we need one more experimentally accessibleobservable. This observable should bare clear signaturesof the prevalence of pathway A at the Δ ts where the min-ima of the FDI probability occur, i.e. at Δ t =-0.2, -0.7T ω . We find that such an FDI-observable is the changeof the momentum of the escaping electron along the po-larization direction of the fundamental ( ω ) laser field, p z ,with Δ t. FIG. 2. The distribution of p z for FDI (a1) and for pathwaysA (a2) and B (a3) are plotted as a function of Δ t. For each Δ t, the distribution of p z for FDI is normalized to 1 whilefor pathways A and B it is normalized with respect to thetotal FDI probability. The distribution of the time electron 1tunnel-ionizes during half cycles 1 and 2 for FDI (b1) and forpathways A (b2) and B (b3) is plotted as a function of Δ t.For each Δ t, the distribution of t in (b1)-(b3) is normalizedto 1. t max is plotted with white dots (appear as white lines)in (b2) and (b3). In Fig. 2(a1) we plot the distribution of p z as a functionof Δ t for one period of the results, that is, in the interval Δ t ∈ [–0.7T ω , –0.2T ω ] in steps of Δ t = 0.1 T ω . Wefind that the distribution of p z has a V-shape. It consistsof two branches that have a maximum split at Δ t =-0.7 T ω , with peak values of p z around -0.85 a.u. and0.85 a.u.. The two branches coalesce at Δ t =-0.3 T ω ,with p z centered around zero. Moreover, FDI events withelectron 1 tunnel-ionizing during half cycles with extremaat nT ω (n/2T ω ) contribute to the upper (lower) branch of the distribution of p z . n takes both positive and negativeinteger values. We find that half cycles 1 and 2, seeFig. 3(a1) and (a2), with extrema at 0 and T/2 of theE ω laser field, respectively, contribute the most to themomentum distribution of p z . Thus, it suffices to focusour studies on half cycles 1 and 2.First, we investigate the change of the distributionof the time electron 1 tunnel-ionizes t with Δ t, seeFig. 2(b1). When the second harmonic (2 ω ) field isturned off, t is centered around the extrema of half cy-cles 1 and 2 (not shown). However, when the 2 ω -fieldis turned on, depending on Δ t, electron 1 tunnel-ionizesat times t that are shifted to the right or to the left ofthe extrema of half cycles 1 and 2, see Fig. 2(b1). More-over, we find that t shifts monotonically from the lowestvalue of the shift at Δ t = –0.3 T ω to its highest valueat Δ t = –0.7 T ω . We find that this change of t is dueto the monotonic change with Δ t of the time t max whenthe magnitude of the OTC laser field is maximum. Thatis, for each Δ t, we compute the time t max when the laserfield in Eq. (1) is maximum. t max is also the time thatthe ionization rate is maximum. We plot t max for halfcycles 1 and 2 in Fig. 2(b2) and (b3). We compare t max with the distribution of t for pathways A and B. We findt max to be closest to the distribution of t for pathwayA. Indeed, only when electron 1 is the escaping electronwill the time electron 1 tunnel-ionizes be roughly equalto the time the ionization rate is maximum. In pathwayB it is electron 2 that escapes. Thus, the time t mustbe such that both the ionization rate and the electron-electron correlation efficiently combine to ionize electron2. Next, for pathway A, we explain how the two brunchesof the distribution of p z split when t shifts to the rightof the extrema of half cycles 1 and 2 ( Δ t=-0.7 T ω ) orcoalesce when t shifts to the left ( Δ t =-0.3 T ω ). Wecompute the changes in p z of the escaping electron 1 dueto the ω -field as well as due to the interaction of electron1 with the core. These momentum changes are given by Δ p Ez ( Δ t, t ) = Z ∞ t –E ω (t)dt, Δ p Cz ( Δ t, t ) = Z ∞ t X i=1 R i – r | r – R i | + r – r | r – r | ! · ˆzdt,(4) with R i the position of the nuclei. Using the times t forthe events labeled as pathway A, we plot the probabilitydistributions of Δ p Ez and of Δ p Cz at Δ t =-0.3 T ω andat Δ t =-0.7 T ω in Fig. 3(b1) and (b2), respectively. Wefind that, for both Δ ts, the distribution of Δ p Cz peaks atpositive (negative) values of Δ p Cz when electron 1 tunnel-ionizes during half cycle 1 (2). Indeed, during half cycle1 (2), electron 1 tunnel-ionizes to the left (right) of thefield-lowered Coulomb potential. Then, the force fromthe core acts along the positive (negative) ˆz-axis resultingin the distribution Δ p Cz peaking around positive (nega-tive) values for half cycle 1 (2). We find that the con-tribution of the electron-electron repulsion term is smallcompared to the attraction from the nucleus in Δ p Cz . In -0.08-0.0400.040.08-0.25 0 0.25 0.5 0.75-1-0.500.51 00.10.20.30.4 -1.2-0.6 0 0.6 1.200.050.10.15 -1.2-0.6 0 0.6 1.2 FIG. 3. Half cycles 1 and 2 for E ω (a1) and its vectorpotential (a2). For pathway A, the distributions of Δ p Ez and Δ p Cz are plotted for half cycles 1 and 2 for Δ t =-0.3 T ω (b1)and Δ t =-0.7 T ω (b2). The distribution of p z is plotted forhalf cycles 1 and 2 for Δ t =-0.3 T ω (b3) and Δ t =-0.7 T ω (b4). contrast, the distribution of Δ p Ez peaking at positive ornegative values of Δ p Ez depends on whether t shifts tothe right or to the left of the extrema of half cycles 1and 2, i.e. it depends on Δ t. For Δ t=-0.3 T ω , whent shifts to the left of the extrema of half cycles 1 (2),the vector potential is positive (negative) resulting in thedistribution of Δ p Ez peaking at negative (positive) valuesof Δ p Ez . Similarly, for Δ t=-0.7 T ω , the distribution of Δ p Ez peaks at positive (negative) values of Δ p Ez for halfcycle 1 (2).In Fig. 2(b3) and (b4), we plot the distributions of thefinal momentum p z , which is given by Δ p Ez + Δ p Cz +p z,t .The distribution of the component of the initial momen-tum of electron 1, p z,t , has a small contribution to p z and is not shown. In Fig. 3(b3), for Δ t =-0.3 T ω , weshow that the distributions of p z for half cycles 1 and 2are similar and peak at zero. They give rise to the twobranches of the distribution p z coalescing in Fig. 2(a2)and (a1). In Fig. 3(b4), for Δ t =-0.7 T ω , we find thatthe distributions of p z for half cycles 1 and 2 are quite dif-ferent with peaks at 0.85 a.u. and -0.85 a.u., respectively.They give rise to the split of the two branches of the dis-tribution p z in Fig. 2(a2) and (a1). Unlike pathway A,for pathway B the distribution of p z as a function of Δ tin Fig. 2(a3) is very broad. The reason is that electron 2has time to interact with the core since it tunnel-ionizesafter a few cycles of the laser field.Finally, we show that a similar level of control ofelectron-electron correlation with OTC fields can not beachieved for H . We choose E ω = 0.064 a.u. so that E ω for H and D +3 has the same percentage difference fromthe field strength that corresponds to over-the-barrier ionization. We choose E ω =0.04 a.u. so that E ω /E ω is the same for both molecules. We show in Fig. 4(a)that, for all Δ ts, the FDI probability significantly reduceswhen the 2 ω -field is turned on. Indeed, its maximumvalue is 2.7% compared to 6.8% for E ω = 0 a.u.. In con-trast, in D +3 the FDI probability changes from 8.5% with- FIG. 4. (a) and (b) similar to Fig. 1(a) and Fig. 2(a1), re-spectively, for H . out 2 ω -field to a maximum value of 10.5% for E ω = 0.05a.u.. We find that the FDI probability as well as the prob-ability of pathway B do not significantly change with Δ t.In addition, the two branches of the V-shaped distribu-tion p z of the escaping electron are not as pronounced inFig. 4(b) as for D +3 . The results in Fig. 4 are obtainedwhen the inter-nuclear axis of H is parallel to E ω . Wefind similar results for a perpendicular orientation, how-ever, for E ω = 0 a.u., the FDI probability is almostzero.In conclusion, we have shown that control of electron-electron correlation in FDI can be achieved employingOTC fields in D +3 . We find that the FDI probabilitychanges sharply with the time-delay between the twolaser fields. Moreover, we identify a split in the distri-bution of the final momentum of the escaping electronthat takes place at time-delays where the FDI probabil-ity is minimum. We show this split to be a signature ofthe absence of electron-electron correlation. It then fol-lows that electron-electron correlation is present for thetime-delays, where the FDI probability is maximum. Fu-ture experiments can employ our scheme to demonstratethe importance of electron-electron correlation in FDI.A. E. acknowledges the EPSRC grant no. J0171831and the use of the computational resources of Legion atUCL. 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