Enhanced dissociation of H2+ into highly excited states via laser-induced sequential resonant excitation
EEnhanced dissociation of H into highly excited states via laser-inducedsequential resonant excitation Kunlong Liu, Qianguang Li, Pengfei Lan, , ∗ and Peixiang Lu , ∗ School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China School of Physics and Electronic-information Engineering,Hubei Engineering University, Xiaogan 432000, China (Dated: September 7, 2018)We study the dissociation of H in uv laser pulses by solving the non-Born-Oppenheimer time-dependent Schr¨odinger equation as a function of the photon energy ω of the pulse. Significantenhancements of the dissociation into highly excited electronic states are observed at critical ω .This is found to be attributed to a sequential resonant excitation mechanism where the populationis firstly transferred to the first excited state by absorbing one photon and sequentially to higherstates by absorbing another one or more photons at the same internuclear distance. We havesubstantiated the underlying dynamics by separately calculating the nuclear kinetic energy spectrafor individual dissociation pathways through different electronic states. PACS numbers:
I. INTRODUCTION
Understanding the electronic excitation and nuclearmotion in laser-molecular interaction is of vital impor-tance in controlling the formation and fracture of chem-ical bonds with laser fields [1–6]. For more than twodecades, many efforts have been made to study the disso-ciation of small molecules exposed to various laser pulses[7–10]. Most of the studies are focused on the interplay ofthe population between the lowest two electronic statesof H , i.e. the ground (1 sσ g ) and first excited (2 pσ u )states, because these two states are strongly coupled andisolated from other electronic levels [11–13]. For manyfamous mechanisms, such as bond softening [14], above-threshold dissociation (ATD) [15], asymmetric electronlocalization [16], charge-resonance-enhanced ionization[17, 18], and so on, the H + +H(1 s ) dissociation channelof H plays the major role in determining the featuresof the observed fragmentation phenomena. Nevertheless,the role of highly excited states has attracted people’s at-tention since recent works [19–27] revealed more detailsof the molecular fragmentation processes (e.g. multipho-ton dissociative excitation [19], high-order ATD [20], andCoulomb explosion without double ionization [21]) andsuggested that the dissociation through highly excitedelectronic states might be a ubiquitous phenomenon inlaser-molecular interactions.In an infrared field, however, it is shown that the con-tribution from highly excited states to the total dissoci-ation is two orders smaller than that from the 1 sσ g and2 pσ u states [26]. This is, on the one hand, due to theweak multiphoton coupling between the 1 sσ g and highlyexcited states [19] as well as the ac Stark shifts of theenergy levels [28]. On the other hand, the high ioniza- ∗ Corresponding authors:[email protected]@hust.edu.cn tion rate of highly excited states would also lead to thedecrease of the population of those states [19, 29]. Thesefacts result that the effects of the population dynamicsin high-lying dissociative states would be largely weak-ened and drowned by the effect of the H + +H(1 s ) channel,making the study or control of the high-lying dissociativepopulation dynamics difficult.In this paper, we study the dissociation of H in uvpulses and report a novel fragmentation process whichleads to significantly enhancement of the dissociativepopulation in highly excited states. In contrast to theprevious study [30] where the population is transferredto higher electronic states via single-photon excitation,the high-lying population in the present study is createdthrough a sequential resonant excitation (SRE) mecha-nism: For a critical photon energy, the population wouldbe first transferred to the 2 pσ u state and sequentiallyto higher states at the same internuclear distance. Ourresults show that, by taking advantage of the strong cou-pling between the 1 sσ g and 2 pσ u states, the multiphotontransition through the SRE process exhibits much higherexcitation rate than the direct multiphoton transition.The underlying mechanism has been verified by the nu-clear kinetic energy release (KER) spectra for individualdissociative electronic states. II. THEORETICAL MODEL
For numerical simulations we have solved the non-Born-Oppenheimer time-dependent Schr¨odinger equa-tion (TDSE) for a reduced-dimensionality model of H [31, 32]. The model consists of one-dimensional mo-tion of the nuclei and one-dimensional motion of theelectron, and the electronic and nuclear motions arerestricted along the polarization direction of the lin-early polarized laser pulse. To date, this model hasbeen widely used to study and identify the molecularfragmentation in strong fields [26, 27, 33–37]. Within a r X i v : . [ phy s i c s . a t m - c l u s ] J u l FIG. 1. The probabilities of the dissociation into highly excited electronic states (solid curves) and the total probability of thelowest two states (dashed curves) as a function of photon energy of the pulse for the interaction of H ( v = 0–3) with the uvpulses. The pulse intensity is 3 × W/cm and the pulse duration is 25 optical cycle. The fixed vertical short lines indicatethe approximate locations of the dissociation enhancements. this model, the length gauge TDSE can be written as(atomic units are used throughout unless otherwise in-dicated) i ∂∂t Ψ( R, z ; t ) = [ H + ε ( t ) z ]Ψ( R, z ; t ) , where H = − m p ∂ ∂R − ∂ ∂z + R + V e ( z, R ) with V e ( z, R ) be-ing the improved soft-core potential that reproduces theexact 1 sσ g potential curve in full dimensions [32]. Here, R is the internuclear distance, z is the electron positionmeasured from the center-of-mass of the protons, and m p is the mass of the proton. The laser electric field isgiven by ε ( t ) = ε sin ( πt/T d ) sin( ωt ) (0 < t < T d ) with T d , ω , and ε being the full pulse duration, the centralfrequency, and the peak electric field amplitude, respec-tively.The TDSE is solved on a grid by using the Crank-Nicolson split-operator method with a time step of ∆ t =0 .
04 a.u.. The grid ranges from 0 to 40 a.u. for R andfrom −
200 to 200 a.u. for z , with grid spacings of ∆ R =0 .
05 a.u. and ∆ z = 0 . j th electronic state can be obtainedby projecting the final wave function to the electronicbound state Ψ je ( z ; R ) at each fixed internuclear distance,i.e. Ψ jD ( z ; R ) = (cid:104) Ψ( z ; R ) | Ψ je ( z ; R ) (cid:105) Ψ je ( z ; R ) , (1)where j equals to 0 , , , ... and indicates the ground state,the first excited state, the second excited state, and soon. Then the integration of Ψ jD ( R, z ) produces the prob-ability of the population in the j th electronic state.To verify the dynamical mechanism of the dissoci-ation, the nuclear KER spectrum for the dissociationchannel through a specific electronic state is needed. Inthe present work, we calculate the channel-specific KERspectra on the basis of the resolvent technique [38]. In de-tail, a channel-specific energy window operator is defined by ˆ W j ( E N ) = δ k / [( ˆ H jN − E N ) k + δ k ] , (2)where ˆ H jN = − m p ∂ ∂R + [ V jN ( R ) − V jN ( R = ∞ )] with V jN ( R ) being the R -dependent potential energy of the j th electronic state of H . Then, the probability distri-bution for the nuclei having a kinetic energy E N and theelectron being in the j th state is extracted from Ψ jD ( R, z )by applying the energy window operator at each z , i.e. p j ( E N ; z ) = (cid:104) Ψ jD ( R ; z ) | ˆ W j ( E N ) | Ψ jD ( R ; z ) (cid:105) . (3)Finally, the probability density at E N of the KER spec-trum is given by ρ j ( E N ) = 1 c (cid:90) p j ( E N , z ) dz (4)with c = δ πk csc( π k ) [39]. In our simulation, we use theparameters δ = 0 .
004 and k = 2. III. RESULTS AND DISCUSSION
Figure 1 shows the probabilities of the dissociation intohighly excited electronic states (solid curves) and the to-tal probability of the lowest two states (dashed curves)as a function of pulse frequency ω for the interaction ofH with the uv pulse. The first four vibrational states( v = 0–3) of H (1 sσ g ) are chosen as the initial states ofFigs. 1(a)–1(d), respectively. In the present simulation,the pulse intensity of 3 × W/cm and T d = 25(2 π/ω )are used and ω ranges from 0.114 to 0.7 a.u. (correspond-ing to the pulse wavelengths from 400 to 65 nm). Underthe pulse parameters that we use here, the ionization isfound to be negligible; thus, we will focus on the popu-lation of the electronic bound states. FIG. 2. The transition location R t ( ω ) as a function of photonenergy ω for the resonant transition channels C – C given inthe text. The dotted lines illustrate the coordinates of thecrossings. As shown in Fig. 1, the total probability of the 1 sσ g and 2 pσ u states (dashed curves) is not conserved as thephoton energy varies, indicating that the interplay in-cluding only the lowest two states would no longer ac-count for the underlying dynamics even in the case ofnegligible ionization. More surprisingly, the remarkableenhancements of the dissociation into the highly excitedstates (2 sσ g , 3 pσ u , 3 sσ g , and 4 pσ u ) are observed at somecritical photon energies. Moreover, the position of theenhancement for each high-lying state is found to be al-most independent on the initial states, whereas the max-ima of the dissociation probabilities of different excitedstates exhibit different tendencies as the vibrational staterises. For instance, the maximum probability of 2 sσ g (thin black curve) tends to decrease with the vibrationalstates, but the tendency for 3 pσ u (thin red curve) is op-posite.In order to understand the features of the anomalousdissociation probability shown in Fig. 1, here we definethe transition location R t ( ω ), which represents the inter-nuclear distance where the resonant excitation channelfrom 1 sσ g to the excited state opens, as a function of thephoton energy ω . Then, based on the R -dependent po-tential energies of the electronic states of H , we calcu-lated the transition location R t ( ω ) for the following tran-sition channels [due to parity considerations, couplingbetween 1 sσ g and excited gerade (ungerade) electronicstates only occurs in even (odd) numbers of photons]: C : 1 sσ g + ω → pσ u ,C : 1 sσ g + 2 ω → sσ g ,C : 1 sσ g + 3 ω → pσ u ,C : 1 sσ g + 2 ω → sσ g ,C : 1 sσ g + 3 ω → pσ u ,C : 1 sσ g + 4 ω → sσ g . The results have been shown in Fig. 2. One can see thatthe curve of C (dashed) and other curves intersect at dif-ferent coordinates, respectively, as indicated by the dot- FIG. 3. (a) Illustration of the sequential resonant excitationmechanism. (b)–(e) The nuclear wave packet profiles on the1 sσ g curve for the first four vibrational states of H . ted lines. Note that the crossing of two curves of R t ( ω )in Fig. 2 means that, under the photon energy of thecrossing, the two represented transition channels wouldopen at the same internuclear distance. By comparingthe results of Figs. 1 and 2, we find that the abscissa val-ues of the crossings in Fig. 2 are approximately equal tothose critical photon energies that lead to the enhanceddissociation shown in Fig. 1. Thus, we suggest that, ifthe resonant excitation channels to the 2 pσ u state andto the other highly excited state occur at the same inter-nuclear distance and under the same photon energy, thedissociative population of the corresponding state wouldbe significantly increased.Based on the above analysis, we now reveal the un-derlying physical mechanism with the diagram of themolecular potential curves. Figure 3(a) illustrates theresonant transition channels C – C that lead to the en-hanced dissociation into the highly excited states. Thecritical photon energies and transition locations for thesechannels are given by the respective crossings shown inFig. 2. It can be seen in Fig. 3(a) that the first photonabsorbtion of each channel overlaps with the correspond-ing C channel. In this situation, the direct multiphotontransition becomes a two-step transition process; that is,the molecule is firstly excited to 2 pσ u by absorbing onephoton and sequentially to higher states by absorbinganother one or more photons at the same internuclear FIG. 4. The dressed molecular potential curves (left panels) and the channel-specific KER spectra of the dissociation into thefirst five excited electronic states (right panels) in uv pulses. The pulse frequencies are given in the corresponding panels andthe other pulse parameters are the same as in Fig. 1. The solid horizontal lines in the left panels indicate the initial vibrationalenergy of H ( v = 2). distance. We call this process the sequential resonant ex-citation (SRE). For the first step of the excitation, dueto the strong coupling between the lowest two states, re-markable population would be transferred from 1 sσ g to2 pσ u . Then, for the second step, because the coupling of2 pσ u to the higher states is stronger than that of the 1 sσ g state, the excited population in 2 pσ u would be efficientlytransferred to the higher state before it dissociates tolarger internuclear distance. As a result, the multipho-ton transition through the SRE process would exhibitmuch higher excitation rate than the direct multiphotontransition.Furthermore, besides the excitation rate, the distri-bution of the initial nuclear wave packet would affectthe yield of the excited population [27]. In Figs. 3(b)–3(e), we illustrate the nuclear wave packet profiles on the1 sσ g curve for the first four vibrational states of H .For v = 0, the wave packet concentrates around R = 2a.u., so only the dissociative population through the C channel is significantly enhanced [see Fig. 1(a)]. Forhigher vibrational state, the wave packet distribution ex-pands to a wider distribution in R dimension. Thus theenhancements of other SRE channels gradually becomepronounced. In contrast, as the vibrational state rises,the wave packet distribution around the C channel isdecreased and becomes modulated; thus, the maximumdissociation probability of 2 sσ g decreases and the mod-ulation structure appears in the probability curve (thinblack) of 2 sσ g [see Fig. 1(d)].Next, in order to substantiate the SRE mechanism proposed above, we calculated the channel-specific KERspectra of the dissociation into the first five excited elec-tronic states (2 pσ u , 2 sσ g , 3 pσ u , 3 sσ g , and 4 pσ u ) by usingthe resolvent technique. The results for the interactionof H ( v = 2) with the uv pulses of four critical photonenergies have been shown in the right panels of Figs. 4(a)–4(d), including the KER spectra of the 2 pσ u state (thickgray curves) obtained from the two-state model. Thechosen photon energies, i.e. ω = 0 . C , C , C , and C channels, respectively, as shown in Fig. 3(a). Forcomparison, the molecular potential curves dressed bythe corresponding photon energies are depicted in the leftpanels of Figs. 4(a)–(d). Note that only the responsibledressed potentials are plotted.As shown in Fig. 4, if only the lowest two statesare considered, the peak positions of the KER spectraof 2 pσ u (thick gray curves) are in good agreement withthe predictions of the 2 pσ u − ω curves. However, aslong as the higher excited states are taken into account,the KER spectra of 2 pσ u (thick dash-dotted curves) de-viate from those of the two-state model. Compared tothe two-state model, the dissociation yields of 2 pσ u arelower and the peak positions are shifted in the TDSE cal-culation. Such deviations indicate that the dissociativepopulation, which was supposed to dissociate along the2 pσ u curve after the single-photon transition, have beenpartially transferred to higher states before it begins todissociate. As a result, in addition to the spectra of 2 pσ u ,the KER spectra peaks of other excited states can alsobe observed in Fig. 4. Moreover, the KER spectra of thehighly excited states are found to be in good agreementwith the predictions of the responsible dressed potentialcurves. These results demonstrate that the enhanced dis-sociation into highly excited states arises from the SREprocess.Additionally, we notice that there is an enhancement inthe KER spectrum for the 2 sσ g state (thin black curve)in Fig. 4(c). According to the dressed potential curvesin the left panel, the 2 sσ g − ω and 2 pσ u − ω curvesoverlap around R = 8 . pσ u passes the overlapped region, it wouldbe partially excited to 2 sσ g via absorbing one photon,resulting in the enhancement of the dissociation yield of2 sσ g . However, due to the delay of the second excitationstep, fewer electric field is left to trigger the coupling;thus, the enhancement for 2 sσ g is weaker than that inFig. 4(b). IV. CONCLUSION
In conclusion, we have studied the dissociation of H in uv pulses as a function of the photon energy of thepulse. Our results show that the dissociation into highlyexcited electronic states provides a significant contribu-tion at some critical photon energies. This anomalous phenomenon is attributed to the SRE process in whichconsiderable population is firstly transferred to the firstexcited state and sequentially to higher states at the sameinternuclear distance. The underlying mechanism hasbeen verified by the nuclear KER spectra of the dis-sociation pathways through different electronic states.Though our present study focuses on the simplified modelof H , the essential dynamics of SRE should generalizedto more complicated molecular systems. In future stud-ies, the effect of the high-lying population dynamics onthe molecular fragmentation could be amplified via theSRE process. Moreover, the SRE process would opena feasible access to achieve efficient control of electronlocalization in highly excited states. ACKNOWLEDGMENT
This work was supported by the National Natural Sci-ence Foundation of China under Grants No. 11234004and No. 61275126, the 973 Program of China underGrant No. 2011CB808103, and the China PostdoctoralScience Foundation under Grant No. 2014M552028. Nu-merical simulations presented in this paper were par-tially carried out using the High Performance Comput-ing Center experimental testbed in SCTS/CGCL (seehttp://grid.hust.edu.cn/hpcc). [1] A. H. Zewail, Science , 1645 (1988).[2] A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer,V. Seyfried, M. Strehle, and G. Gerber, Science , 919(1998).[3] M. F. Kling, C. Siedschlag, A. J. Verhoef, J. I. Khan, M.Schultze, T. Uphues, Y. Ni, M. Uiberacker, M. Drescher,F. Krausz, and M. J. J. Vrakking, Science , 246(2006).[4] P. Lan, P. Lu, W. Cao, Y. Li, and X. Wang, Phys. Rev.A , 011402 (2007).[5] F. He, C. Ruiz, and A. Becker, Phys. Rev. Lett. ,083002 (2007).[6] X. Zhu, M. Qin, Y. Li, Q. Zhang, Z. Xu, and P. Lu, Phys.Rev. A , 045402 (2013).[7] J. H. Posthumus, Rep. Prog. Phys. , 623 (2004).[8] F. Mart´ın, J. Fern´adez, T. Havermeier, L. Foucar, Th.Weber, K. Kreidi, M. Sch¨ofler, L. Schmidt, T. Jahnke,O. Jagutzki, A. Czasch, E. P. Benis, T. Osipov, A. L.Landers, A. Belkacem, M. H. Prior, H. Schmidt-B¨oking,C. L. Cocke, R. D¨oner, Science , 629 (2007).[9] J.-Ph. Karr, J. Mol. Spectrosc. , 37 (2014).[10] S. Schiller, D. Bakalov, and V. I. Korobov, Phys. Rev.Lett. , 023004 (2014).[11] B. Moser and G. N. Gibson, Phys. Rev. A , 041402(R)(2009).[12] M. Kremer, B. Fischer, B. Feuerstein, V. L. B. de Jesus,V. Sharma, C. Hofrichter, A. Rudenko, U. Thumm, C.D. Schr¨oter, R. Moshammer, and J. Ullrich, Phys. Rev.Lett. , 213003 (2009). [13] K. Liu, Q. Zhang, and P. Lu, Phys. Rev. A , 033410(2012); K. Liu, Q. Zhang, P. Lan, and P. Lu, Opt. Ex-press , 5107 (2013).[14] P. H. Bucksbaum, A. Zavriyev, H. G. Muller, and D. W.Schumacher, Phys. Rev. Lett. , 1883 (1990).[15] A. Giusti-Suzor, X. He, O. Atabek, and F. H. Mies, Phys.Rev. Lett. , 515 (1990).[16] K. P. Singh, Pramana , 87 (2014).[17] T. Zuo and A. D. Bandrauk, Phys. Rev. A , R2511(1995).[18] A. Staudte, D. Paviˇci´c, S. Chelkowski, D. Zeidler, M.Meckel, H. Niikura, M. Sch¨offler, S. Sch¨ossler, B. Ulrich,P.P. Rajeev, T. Weber, T. Jahnke, D. M. Villeneuve, A.D. Bandrauk, C. L. Cocke, P. B. Corkum, and R. D¨orner,Phys. Rev. Lett. , 073003 (2007).[19] G. N. Gibson, L. Fang, and B. Moser, Phys. Rev. A ,041401(R) (2006).[20] J. McKenna, A. M. Sayler, F. Anis, B. Gaire, Nora G.Johnson, E. Parke, J. J. Hua, H. Mashiko, C. M. Naka-mura, E. Moon, Z. Chang, K. D. Carnes, B. D. Esry, andI. Ben-Itzhak, Phys. Rev. Lett. , 133001 (2008).[21] B. Manschwetus, T. Nubbemeyer, K. Gorling, G. Stein-meyer, U. Eichmann, H. Rottke, and W. Sandner, Phys.Rev. Lett. , 113002 (2009).[22] M. Førre, S. Barmaki, and H. Bachau, Phys. Rev. Lett. , 123001 (2009).[23] Y. Zhou, C. Huang, Q. Liao, and P. Lu, Phys. Rev. Lett. , 053004 (2012).[24] E. L¨otstedt, T. Kato, and K. Yamanouchi, J. Chem.Phys. , 104304 (2013). [25] H. Li, A. S. Alnaser1, X. M. Tong, K. J. Betsch, M.K¨ubel, T. Pischke, B. F¨org, J. Sch¨otz, F. S¨ußmann, S.Zherebtsov, B. Bergues, A. Kessel, S. A. Trushin, A. M.Azzeer and M. F. Kling, J. Phys. B: At. Mol. Opt. Phys. , 063420(2013).[27] K. Liu, P. Lan, C. Huang, Q. Zhang, and P. Lu, Phys.Rev. A , 053423 (2014).[28] X. Zhu, M. Qin, Q. Zhang, Y. Li, Z. Xu, and P. Lu, Opt.Express , 5255 (2013).[29] Y. Li, W. Hong, Q. Zhang, S. Wang, and P. Lu, Opt.Express , 24376 (2011).[30] F. He, Phys. Rev. A , 063415 (2012).[31] K. C. Kulander, F. H. Mies, and K. J. Schafer, Phys.Rev. A , 2562 (1996). [32] B. Feuerstein and U. Thumm, Phys. Rev. A , 043405(2003).[33] C. B. Madsen, F. Anis, L. B. Madsen, and B. D. Esry,Phys. Rev. Lett. , 163003 (2012).[34] R. E. F. Silva, F. Catoire, P. Rivi`ere, H. Bachau, and F.Mart´ın, Phys. Rev. Lett. , 113001 (2013).[35] K. Liu, W. Hong, Q. Zhang, and P. Lu, Opt. Express ,26359 (2011).[36] A. Pic´on, A. Jaron-Becker, and A. Becker, Phys. Rev.Lett. , 163002 (2012).[37] N. Takemoto and A. Becker, Phys. Rev. Lett. ,203004 (2010).[38] K. J. Schafer and K. C. Kulander, Phys. Rev. A , 5794(1990).[39] F. Catoire and H. Bachau, Phys. Rev. A85