arXiv:0911.2693v1 [physics.atom-ph] 13 Nov 2009 r o eea,eat o ote rvd simple a provide they do nor exact, general, they power, not interpretive some are have fields laser IR+APT packet wave using delay to on akin [13]. dependence replicas approach invoked the An been describe also to has applied. pic- approximation” been man’s based “simple have photon the 12] time-independent, a [11, and ture 10] pic- physical field [5, electric some ture time-dependent, time-dependent gain a the both to though, try solving insight, To by described [9]. Schr¨odinger equation be course, of distribu- angular the over control 8]. has [7, been as tion has done 6], method be a [5, of such can realized control using which probability Experimental ionization field scale. the IR tens-of-attoseconds and de- a the an APT on vary pulse, the to IR ability between the an lay through thus with control pulse, overlapped permits attosecond When APT an se- APT. is which label periodic of the a Tem- each give pulses, target. to of the combine ries illuminate — harmonics to range these used ultraviolet porally, and extreme isolated har- the be of in can range typically specific a — filtering, monics selective in- con- With approximately fundamental at [4]. Letter, nm). the Ti:Sapphire 800 this of a (APT) (usually harmonics in laser train of (IR) on frared range pulse focus large will attosecond a isolated we tains an which single as latter, a The or as [3] either pulse generation, high-harmonic laser-based of field control nascent and 2]. the [1, study of physics to goals attosecond dynam- such abilities central electron These With the the are created. over scales. also control time is units for ics these avenue atomic on an of act forces, time order that relevant the forces on the quire is Since a.u. motion (1 their electrons. for scale bound of motion hl hs tepst nesadtednmc in dynamics the understand to attempts these While can, field IR+APT combined the in dynamics Electron via experimentally generated are pulses Attosecond the is physics molecular and atomic of heart the At ≈ 4a) ealdsuiso lcrndnmc re- dynamics electron of studies detailed as), 24 ASnmes 32.80.Rm,42.50.Hz numbers: PACS two-ele full-dimensional, a hypersphe with adiabatic predictions Schr¨odinger dressed equation. thes the introduce confirm illustrate and We We He of pathways. carrier-env param ionization the control multiphoton and coherent delay different a the as between of APT roles equivalent and nearly pulse e the IR reveals formulation the between This delay (APTs). exampl trains relevant pulse experimentally attosecond the on focusing molecules, epeetagnrldsrpino h neato between interaction the of description general a present We ..McoadLbrtr,Kna tt nvriy Manha University, State Kansas Laboratory, Macdonald J.R. toeodpletan smliclrchrn control coherent multi-color as trains pulse Attosecond .V enadzadB .Esry D. B. and Hern´andez V. J. Dtd oebr3 2018) 3, November (Dated: lopoieacneta rmwr ihnwihpre- simply. which made within be can framework can that Thus, conceptual dictions desired a is predictions. problem provide to the also guide cannot of simple and description a rigorous systems a provide simple to to said practice be in time-dependent exact, limited the and is general exper- of but principle, particular solution in a numerical is, Schr¨odinger of equation The outcome the iment. predicting to route ainuigtelnt ag aoi nt r used are units (atomic gauge length approxi- the dipole using the mation in Schr¨odinger equation dependent both [5] Ref. in be- observed delay numerically. recently with and APT yield analytically ion and in IR modulation tween the time- so- thus for two-electron, we with account process, combined six-dimensional, analysis the by In our full Schr¨odinger equation. helium dependent confirm the of We of ionization lutions apply- [5]. single is by fields the simple, it IR+APT to illustrate conceptually it we while ing — which, cumbersome — notationally theory our of 21]. 20, 19, [18, two-color molecules well-studied controlling the to, on syn- to, well experiments a connection equally is a applies provides APT analysis and an our harmonics, because of Moreover, thesis [18]. Brumer the by Shapiro of proposed and realizations scenario are control inter- systems coherent These such canonical of 17]. pulses advantage 16, attosecond [4, taking ferences fact, by characterized In be observable. can physical between produce given phase to interfere a relative that the processes controlling multiphoton can as different parameters of these thought of This be both system. because a follows over conclusion control delay fact, IR+APT identical the in essentially and provide and, CEP the pulses both few-cycle that demonstrates for carrier- the (CEP) have of we phase effects the one envelope understand to to related 15] [14, closely developed is formulation This work. u nltctetetbgn ihtetime- the with begins treatment analytic Our formulation general most the presenting than Rather frame- theoretical a such just present we Letter, this In lp hs ncnrligteinterference the controlling in phase elope fifae I)ple vrapdwith overlapped pulses (IR) infrared of e piil n nltclyterl fthe of role the analytically and xplicitly ia oetast i h discussion. the aid to potentials rical ut-oo ae ussadaosand atoms and pulses laser multi-color to ouino h time-dependent the of solution ctron onsb netgtn h single the investigating by points e tr u omlto loshows also formulation Our eter. tn ass66506 Kansas ttan, 2 throughout), 0 Doubly Excited States -1 ∂ 1 e + - i Ψ(t)= H0 − E~(t) · d~ Ψ(t), (1) D He (1s) + e ∂t h i -2 1Po -3 1Se where H0 is the field-free Hamiltonian and d~ is the dipole U(R) (a.u.) operator. We take into account the fundamental IR field -4 with frequency ω and two harmonics of order n1 and n2: -5 0 2 4 6 8 10 E(t)= E1(t)cos(ωt + ϕ1) R (a.u.)

+ E 1 (t)cos(n [ω(t − τ)+ ϕ ]+ ϕ 1 ) n 1 1 n FIG. 1: (Color online) Adiabatic hyperspherical potential + En2 (t)cos(n2 [ω(t − τ)+ ϕ1]+ ϕn2 ). (2) curves for He. Shown are L = 0, 1, 2 and the region of double excitation. The latter two terms make up the APT in our example. More harmonics can be added to shorten the individual pulses in the APT without any conceptual difficulty, but index l represents only the net energy lω of the photons three colors is the minimum needed to describe the most involved, and can be written as l=l1 +ln1 n1 +ln2 n2 with significant effects observed in an IR+APT experiment li the number of photons of each color. such as Ref. [5] and simplifies the discussion. Note that Specializing the above discussion to helium determines ~ for our analysis the harmonics’ pulse envelopes Eni (t), H0 and d in Eq. (4). We treat the full three-dimensional which represent the envelope for the whole pulse train, motion of both electrons using the adiabatic hyperspher- must be independent of the IR-APT delay τ but are oth- ical representation, which has proven especially useful in erwise arbitrary. Much like the derivation of CEP effects the past for understanding the dynamics of He [23, 24]. in [15], we note from Eq. (1) that Ψ is periodic in τ as More specifically, we expand each component φml as well as in the IR CEP ϕ1. We can thus expand Ψ in both φ = G (R,t)Φ (R; Ω) (5) using a discrete Fourier transform, ml X mlLν Lν Lν − Ψ(τ, ϕ ; t)= eimωτ e ilϕ1 ψ (t), (3) 1 X ml ml where R is the hyperradius and Ω the five hyperangles. The channel functions ΦLν solve the field-free fixed-R where ψml(t) does not depend on either τ or ϕ1. Substi- adiabatic equation HadΦLν=ULνΦLν for a given total or- tuting this Ψ into Eq. (1) and equating coefficients of the bital angular momentum L and thus form a complete set linearly independent exponential functions, we obtain at each R indexed by ν. We use Delves’ hyperangle and treat the system in the body frame, expanding the Euler- ∂ 1 i φml =(H0 − lω)φml − E~1 · d~(φm,l+1 + φm,l−1) angle dependence of ΦLν on Wigner D-functions (see [24] ∂t 2 for details). The adiabatic equation thus reduces to cou- 1 − ~ ~ iϕn1 iϕn1 − En1·d e φm+n1,l+n1 +e φm−n1,l−n1 pled two-dimensional partial differential equations that 2
we solve using b-splines [25]. 1 − ~ ~ iϕn2 iϕn2 − E 2·d e φ 2 2 +e φ − 2 − 2 . (4) Upon substituting φml from (5) into Eq. (4) and pro- 2 n m+n ,l+n m n ,l n
jecting out the ΦLν we find coupled, time-dependent In these equations, we have made the additional substi- equations for GmlLν (R,t) describing laser-driven motion −ilωt tution ψml(t) = e φml(t) to emphasize the connec- on the adiabatic potentials ULν . Since the experiments tion with the Floquet representation: in the limit that in [5] considered only single ionization at energies well the pulse lengths go to infinity, Eq. (4) reduces to many- below doubly-excited resonances, we can to a good ap- mode Floquet theory [22]. We stress that Eq. (4) is exact proximation consider only the lowest singlet adiabatic even for few-cycle pulses. We could further remove the potential ν=1 for each L. These potentials support the 1 harmonic phases ϕni with Fourier transforms, but that singly-excited states 1snL L, and several examples are will not be necessary for the current discussion. shown in Fig. 1. With this truncation of the ν expansion, We have rigorously obtained a representation in which we find a ground state energy of –2.895 a.u. IR+APT experiments can be understood. Further, One advantage of using the adiabatic hyperspherical Eq. (3) shows that the CEP and IR-APT delay pro- representation in the Floquet-like equations (4) is that vide similar control over the dynamics of a system. The the resulting dressed potentials — which include the ef- physical interpretation of this control as interference of fects of both Coulomb interactions and the laser field different photon pathways stems from the facts that the — give insight into the electronic dynamics in much the ilωt components φml oscillate with time-dependence e and same way that Floquet potentials have for the nuclear dy- + correlate to the photon components in the Floquet limit. namics of molecules. Applied to H2 , Floquet potentials Since we consider photons with different frequencies, the have provided natural explanations of bond-softening, 3 above-threshold dissociation, vibrational trapping, and zero-photon dissociation [26, 27]. Previous generaliza- + tions of the Floquet picture for H2 have even helped uncover unexpected phenomena such as above-threshold Coulomb explosion [28]. It would be interesting to ex- plore the analogues of these mechanisms for He in the present representation, but we will focus here on their application to the IR+APT case. Figure 2 shows the diabatic dressed potentials assum- ing ω corresponds to 800 nm and the harmonics are n1=13 and n2=15 (the two most intense components of FIG. 2: (Color online) Diabatic dressed potentials for He in a the APT in [5] by a factor of at least 5). These curves three color field (ω, 13ω, and 15ω). Only the most significant are generated by first noting that the initial state of the curves and crossing are shown, where a circle indicates a single system, 1s2 1Se, has (m,l)=(0,0). Then, per Eq. (4), large-photon crossing; a square a 1-IR photon crossing; and this state couples to the 1P o channel with one IR photon a triangle, a 3-IR photon crossing. The dashed line indicates 1 o the energy of the system, which is approximately conserved (0,±1) or with one harmonic photon (±ni,±ni). The P potential in each case is added to the figure shifted by lω. in this representation. Each of these new curves can couple to either 1Se or 1De symmetries according to dipole selection rules, shifted in and (0,0)1Se→(13,13)1P o→(13,16)1Ge. Other outcomes energy by an appropriate amount. Repeating this pro- are certainly possible and will occur with some proba- cedure, all possible multiphoton pathways combining IR bility. The goal of this kind of analysis, though, is to and harmonic photons to any particular final state can be try to identify the main routes to a given final state. generated as shown in Fig. 2. The energy in such a figure Since there exists more than one pathway to this lowest can be thought of as being approximately conserved, so above-threshold ionization (ATI) peak, interference will that the dynamics reduces to the well-understood prob- occur. Similar considerations show that there are gen- lem of multichannel dynamics. This picture, for example, erally multiple paths to ionization for each higher ATI maps ionization onto predissociation of molecules in the peak l=17,18,19,.. . opening up the possibility of control. Born-Oppenheimer approximation or, equivalently, onto In this manner, our formulation of the problem lets us atomic autoionization in the field-free adiabatic hyper- learn a considerable amount about controlling this inter- spherical representation. More specifically, transitions ference without solving the time-dependent problem. To are most likely to occur where potentials cross, and cross- see explicitly how these pathways interfere, we can derive ings of curves differing by the fewest photons will domi- an expression for any physical observable using Eq. (3). nate. In fact, diagonalizing the right-hand-side of Eq. (4) In particular, for the He IR+APT photoelectron spec- at fixed R will produce adiabatic Floquet-like potentials trum discussed above, we obtain in which these crossings become avoided, thus incorpo- rating the strength of the field in the potentials them- dP − ′ ′− selves. The remaining time dependence in E modulates ion = ei(m m )ωτ ei(l l)ϕ1 i dE X X X the strength of the coupling — or the gap at the avoided L m,m′ l,l′ ∗ crossings — as a function of time. × hEL|FmlLihEL|Fm′l′Li , (6) For the parameters above, single ionization of He re- quires the absorption of at least 16ω worth of energy, where |ELi is an energy-normalized continuum state −ilωt i.e. 16 IR photons, 1 13th harmonic plus 3 IR pho- with asymptotic kinetic energy E and FmlL=e GmlL. tons, or 1 15th harmonic plus 1 IR photons. Since All of the τ and ϕ1 dependence is displayed analytically it requires the fewest photons, we expect the latter since the amplitudes hEL|FmlLi are independent of these scenario to be most likely, leaving the system with parameters, determining only which pathways will inter- (m,l)=(15,16) and L=0 or 2 based on dipole selection fere. Equation (6) shows that the delay dependence is rules. All together, we would write these pathways as determined only by the difference in m, essentially the (0,0)1Se→(15,15)1P o→(15,16)1Se or 1De. Following the harmonic order, between two pathways. For the lowest dressed potentials in Fig. 2 suggests that only L=2 will ATI peak discussed above, the final states of two of the be populated substantially, however, since it crosses the pathways identified were (13,16)1De and (15,16)1De for (15,15)1P o curve and L=0 does not. The pathway be- the 13th and 15th harmonics, respectively. Since these ginning with the 13th harmonic would mainly proceed have the same final l (net energy absorbed from the field), via (0,0)1Se→(13,13)1P o→(13,16)1De by similar argu- they will contribute at the same final energy and inter- ments. The above two paths are highlighted in green fere, resulting in a sinusoidal dependence on ωτ with pe- in Fig. 2. Another set of likely pathways (light blue in riod π since |m–m′|=2. This π periodicity in the ion- Fig. 2) ends on 1Ge: (0, 0)1Se→(15,15)1P o→(15,16)1Ge ization yield is exactly what was observed in both the 4

0.00 0.10 0.20 0.30 0.40 with atoms and molecules upon which a relatively simple 1.5 (a) interpretive, predictive picture can be built. In the pro-

2.4 1.2 (b) cess, we introduced dressed potentials for He — again L=4 based on a rigorous derivation — that confer many of 0.9 L=2

) the benefits such potentials have provided for molecules -3 Ε/ω 0.6 1.2 on the atomic problem. We have also shown that con- (10 ion

P trol via the CEP and the delay between multicolor fields, 0.3 16ω IR+APT in particular, are conceptually equivalent, be- 0.0 L=0 ing the result of interference between different multipho- 0.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 ωτ/π ωτ/π ton pathways between the initial and final states. Ex- periments exercising this control are thus quite properly FIG. 3: (Color online) (a) Calculated photoelectron spectrum thought of as coherent control. 12 2 for an 800 nm IR pulse with I1 = 5.0 × 10 W/cm and I13 = I15 = 0.1I1. (b) Ionization probability as a function of delay for L=0, 2, and 4. This work is supported by the Chemical Sciences, Geo- science, and Biosciences Division, Office for Basic Energy Sciences,Office of Science, U.S. Department of Energy. experiment and numerical results presented in [5, 6]. In general, the periodicity in ωτ will be given by 2π/|m−m′|. Thus, if even harmonics had also been included, the pe- riodicity would have been 2π since |m–m′| would equal 1. To see the 2π periodicity, however, one would have [1] P. B. Corkum and Z. Chang, Opt. and Pho. News 19, 25 (2008). to measure a quantity that – unlike the energy spectrum 81 (6) – takes into accout interference between different final [2] F. Krausz and M. Ivanov, Rev. Mod. Phys. , 163 (2009). Ls, such as the momentum distribution. [3] M. Hentschel et al., Nature 414, 509 (2001). To confirm our analysis, we carried out a full- [4] P. M. Paul et al., Science 292, 1689 (2001). dimensional numerical solution of the time-dependent [5] P. Johnsson et al., Phys. Rev. Lett. 99, 233001 (2007). Schr¨odinger equation (1) using the adiabatic hyperspher- [6] P. Ranitovic et al., New J. Phys. (submitted). ical representation. Figure 3(a) shows the resulting pho- [7] O. Guy´etand et al., J. Phys. B: At. Mol. Opt. Phys 38, toelectron spectrum, again for parameters chosen to be L357 (2005). [8] O. Guy´etand et al., J. Phys. B: At. Mol. Opt. Phys 41, similar to the experiment in [5]. As expected, the cal- 051002 (2008). culations show ATI peaks separated by ω. The largest [9] P. Johnsson et al., Phys. Rev. Lett. 95, 013001 (2005). peak is the lowest one (indicated by the arrow), which [10] J. Mauritsson et al., Phys. Rev. Lett. 100, 073003 (2008). was discussed above, and the predicted π period of its [11] A. Maquet and R. Ta¨eib, J. Mod Opt. 54, 1847 (2007). modulation is clearly seen. A closer look at the 16ω peak [12] M. Meyer et al., Phys. Rev. A 74, 011401(R) (2006). reveals that it is split in energy with the lower-energy [13] P. Riviere et al., New J. Phys. 11, 053011 (2009). 99 peak primarily L=2 and the higher-energy peak L=4 [see [14] V. Roudnev and B. D. Esry, Phys. Rev. Lett. , 220406 Fig. 3(b)]. These details, however, only serve to bolster (2007). [15] J.J.Hua and B. D. Esry, J. Phys. B: At. Mol. Opt. Phys the value of the dressed potentials we introduced in Fig. 2 42, 085601 (2009). since they allowed us to predict that the 16ω peak should [16] J. M. Dahlstr¨om et al., Phys. Rev. A 80, 033836 (2009). be dominated by L=2 and 4. The time-dependent cal- [17] N. Dudovich et al., Nat. Phys. 2, 781 (2006). culations and the dressed potential picture complement [18] P. Brumer and M. Shapiro, Annu. Rev. of Phys. Chem. each other, providing both quantitative and qualitative 43, 257 (1992). understanding of these processes. [19] E. Charron, A. Giusti-Suzor, and F. H. Mies, Phys. Rev. Lett. 71, 692 (1993). This discussion hardly exhausts the insight that can [20] E. Charron, A. Giusti-Suzor, and F. H. Mies, Phys. Rev. be gleaned from Eq. (6). Another example is the fact Lett. 75, 2815 (1995). that the delay τ between IR and APT will not generate [21] T. T. Nguyen-Dang et al., Phys. Rev. A 71, 023403 a left-right asymmetry along the laser polarization since (2005). all of the Ls corresponding to a given ATI peak are either [22] S.-I. Chu and D. A. Telnov, Phys. Rep. 390, 1 (2004). even or odd (with only odd harmonics in the APT). An [23] X. Guan, X. M. Tong, and Shih-I Chu, Phys. Rev. A 73, exception to this statement occurs for broad bandwidth 023403 (2006). [24] C. D. Lin, Adv. At. Mol. Opt. Phy. 22, 77 (1986). pulses since the interference between even and odd Ls [25] C. de Boor, A practical guide to splines (Springer-Verlag, necessary for asymmetry can be produced in the overlap 1978). of neighboring ATI peaks. Of course, the CEP will also [26] J. H. Posthumus, Rep. Prog. Phys. 67, 623 (2004). contribute to the asymmetry in this case. [27] A. Giusti-Suzor et al., J. Phys. B: At. Mol. Opt. Phys In this Letter, we have presented a general, exact rep- 28, 309 (1995). resentation for the interaction of multicolor laser pulses [28] B. D. Esry et al., Phys. Rev. Lett. 97, 013003 (2006).