High-harmonic generation in diatomic molecules: a quantum-orbit analysis of the interference patterns
aa r X i v : . [ phy s i c s . a t o m - ph ] J u l High-harmonic generation in diatomic molecules: a quantum-orbit analysis of theinterference patterns
C. Figueira de Morisson Faria
Department of Physics and Astronomy, University College London,Gower Street, London WC1E 6BT, United Kingdom (Dated: November 23, 2018)We perform a detailed analysis of high-order harmonic generation in diatomic molecules within thestrong-field approximation, with emphasis on quantum-interference effects. Specifically, we investi-gate how the different types of electron orbits, involving one or two centers, affect the interferencepatterns in the spectra. We also briefly address the influence of the choice of gauge, and of theinitial and final electronic bound states on such patterns. For the length-gauge SFA and undressedbound states, there exist additional terms, which can be interpreted as potential energy shifts. If,on the one hand, such shifts alter the potential barriers through which the electron initially tun-nels, and may lead to a questionable physical interpretation of the features encountered, on theother hand they seem to be necessary in order to reproduce the overall maxima and minima in thespectra. Indeed, for dressed electronic bound states in the length gauge, or undressed bound statesin the velocity gauge, for which such shifts are absent, there is a breakdown of the interferencepatterns. In order to avoid such a problem, we provide an alternative pathway for the electron toreach the continuum, by means of an additional attosecond-pulse train. A comparison of the purelymonochromatic case with the situation for which the attosecond pulses are present suggests thatthe patterns are due to the interference between the electron orbits which finish at different centers,regardless of whether one or two centers are involved.
I. INTRODUCTION
In the past few years, high-order harmonic genera-tion (HHG) and above-threshold ionization (ATI) fromaligned molecules in strong laser fields of femtosecondduration have proven to be a powerful tool for resolving,or even controlling, processes in the subfemtosecond andsubangstrom scale. For instance, one may employ HHGand ATI in the tomographic reconstruction of molecularorbitals [1], and in the attosecond probing of dynamicchanges in molecules [2].This is possible due to the fact that the physical mecha-nisms governing both phenomena take place in a fractionof the laser period, i.e., within hundreds of attoseconds[3], and involve the recombination or the elastic scatter-ing of an electron with its parent molecule [4]. Thereby,high-order harmonics or high-energy photoelectrons, re-spectively, are generated. Thus, the spectral features arehighly dependent on the spatial configuration of the ionswith which the electron rescatters or recombines, andyield patterns which are characteristic of the molecule.Furthermore, they also depend on the alignment angle ofthe molecule, with respect to the laser-field polarization.Due to their simplicity, in particular diatomicmolecules have been investigated, and minima and max-ima have been encountered in their HHG and ATI spec-tra. Such patterns have been observed both theoreti-cally [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]and experimentally [19], and have been attributed tothe interference between high- order harmonics or photo-electrons generated at different centers in the molecule.They are, in a sense, the microscopic counterpart ofthose obtained in a double-slit experiment. Further-more, the energy positions of the maxima and min- ima depend on the alignment angle and on the inter-nuclear distance and, additionally, reflect the bondingor antibonding nature of the highest occupied molecu-lar orbitals in question [14, 15, 18]. Such features havebeen studied either numerically, by solving the time-dependent Schr¨odinger equation (TDSE) [14, 15, 18], orsemi-analytically, by employing the strong-field approxi-mation (SFA) [5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 17]. In par-ticular, in [15], the contributions to the yield from eachmolecular center have been singled out within a TDSEcomputation. Therein, it has been explicitly shown thatthe maxima and minima in the spectra are obtaineddue to the interference between contributions from dif-ferent centers, in agreement with the double-slit modelin [15, 18].Specifically in the SFA framework, the transition am-plitude can be written as a multiple integral, with a semi-classical action and slowly-varying prefactors. The struc-ture of the molecule (and thus its double-slit character)can be either incorporated in the prefactors or in theaction. The latter approach also takes into account pro-cesses in which an electron rescatters or recombines witha center in the molecule different from the site of its re-lease [5, 9, 10, 11, 13].An open question is, however, which types of electronorbits are responsible for specific interference features.For instance, are the dips and maxima originated by theinterference between orbits in which the electron leavesand returns to the same center (regardless of which), orbetween those in which the electron leaves one atom andrecombines with the other? On the other hand, it couldalso be that the interference patterns result from the com-bined effect of all such orbits, and one is not able to at-tribute them to specific sets.Since the strong-field approximation is a semi-analytical method, and allows an immediate associationwith the classical orbits of an electron returning to itsparent molecule, it appears to be an ideal tool for tack-ling this problem. This approximation, however, pos-sesses several drawbacks. First, the SFA is gauge de-pendent, which leads to different pre-factors and action,depending on whether the velocity gauge or the lengthgauge is taken. Second, even for a specific gauge, the pre-cise expressions for the pre-factors are not agreed upon,and some of them lead to interference patterns which arein disagreement with the experiments, and with resultsfrom other methods [15, 20].The main objective of this work is to make a detailedassessment of the contributions and relevance of the dif-ferent types of recombination scenarios to the above-mentioned interference patterns, within the Strong-FieldApproximation, for a diatomic molecule. Throughoutthe paper, we approximate the strong, infra-red fieldby a linearly polarized monochromatic wave E ( t ) = E sin( ωt ) e x , and consider a linear combination of atomicorbitals (LCAO approximation). In general, we considerthat the electron reaches the continuum by tunnelingionization. As an exception, however, we also take anattosecond-pulse train superposed to the monochromaticwave [21, 22, 23, 24]. The attosecond-pulse train pro-vides an additional pathway for the electron to reach thecontinuum, and, recently, has proven to be a powerfultool in order to control high-order harmonic generation[21, 23, 24] and above-threshold ionization [22, 23]. Inthe context of the present work, it is a convenient wayto avoid problems related to spurious potential-energyshifts. These shifts are present in the length gauge andartificially modify the potential barrier through which theelectron tunnels. Hence, they may lead to a questionablephysical interpretation, what the relevance of the differ-ent sets of orbits to the patterns concerns.This paper is organized as follows. In Sec. II, wediscuss the strong-field approximation transition ampli-tude for high-order harmonic generation, starting fromthe general expressions (Sec. II A), and, subsequently,addressing the specific situation of a diatomic moleculein the LCAO approximation (Sec. II B). Thereby, weconsider the situation for which the structure of themolecule is either incorporated in the pre-factor, or inthe semiclassical action, in the presence and absence ofthe attosecond-pulse train. When discussing the formercase, we emphasize the role of the overlap integrals, inwhich the dipole moment and the atomic wave func-tions are localized at different centers in the molecule.In the latter case, we follow the model in Ref. [11], fora purely monochromatic driving field, and our previouswork [23] when the attosecond pulses are present, veryclosely. Subsequently (Sec. III), we investigate the inter-ference patterns. Finally, in Sec. IV we summarize thepaper and state our main conclusions. II. TRANSITION AMPLITUDESA. General expressions
In general, the strong-field approximation (SFA) con-sists in neglecting the influence of the laser field when theelectron is bound, the atomic or molecular binding poten-tial when the electron is in the continuum and the inter-nal structure of the system in question, i.e., contributionsfrom its excited bound states. The SFA transition am-plitude for high-order harmonic generation reads, in thespecific formulation of Ref. [26] and in atomic units, b Ω = i Z ∞−∞ dt Z t −∞ dt ′ Z d pd ∗ rec ( ˜p ( t )) d ion ( ˜p ( t ′ ))exp[ iS ( t, t ′ , Ω , k )] + c.c., (1)with the action S ( t, t ′ , Ω , p ) = − Z tt ′ [ p + A ( τ )] dτ − I p ( t − t ′ ) + Ω t (2)and the prefactors d rec ( ˜p ( t )) = h ˜p ( t ) | O dip . e x | ψ i and d ion ( ˜p ( t ′ )) = h ˜p ( t ′ ) | H int ( t ′ ) | ψ i . In the above equations, O dip , e x , H int ( t ′ ) , I p , and Ω denote the dipole oper-ator, the laser-polarization vector, the interaction withthe field, the ionization potential, and the harmonic fre-quency, respectively. The explicit expressions for ˜p ( t ) aregauge dependent, and will be given below. The above-stated equation describes a physical process in which anelectron, initially in a field-free bound-state | ψ i , is cou-pled to a Volkov state | ˜p ( t ′ ) i by the interaction H int ( t ′ )of the system with the field. Subsequently, it propagatesin the continuum and is driven back towards its parention, with which it recombines at a time t , emitting high-harmonic radiation of frequency Ω . In Eq. (1), addition-ally to the above-mentioned assumptions, the further ap-proximation of considering only transitions from a boundstate to a Volkov state in the dipole moment has beenmade. In the single atom case, this has been justified bythe fact that the remaining contributions, from the so-called “continuum-to-continuum” transitions, were verysmall. For a discussion of the various formulations of theSFA see, e.g., [25, 26, 27] and in particular [28].Due to the fact that it can be carried out almostentirely analytically, the SFA is a very powerful ap-proach. It possesses, however, the main drawback ofbeing gauge-dependent (for general discussions see, e.g.,Ref. [30], and for the specific case of molecules, Ref.[11]). Apart from the obvious fact that the interactionHamiltonians H int ( t ′ ), which are present in d ion ( ˜p ( t ′ )),are different in the length and velocity gauges [31], inmost computations where the SFA is employed, field-free bound states are taken, which are not gauge equiva-lent. Indeed, a field-free bound state (cid:12)(cid:12)(cid:12) ψ ( L )0 E in the lengthgauge would be gauge equivalent to the field-dressed state (cid:12)(cid:12)(cid:12) ψ ( V )0 E = χ v ← l (cid:12)(cid:12)(cid:12) ψ ( L )0 E in the velocity gauge, with χ v ← l =exp[ i A ( t ) · r ] . Such a phase shift causes a translation p → p − A ( t ) on a momentum eigenstate | p i . Hence, forfield-free bound states in both gauges it leads to differentdipole matrix elements d rec ( ˜p ( t )) and d ion ( ˜p ( t )). Explic-itly, in the length gauge ˜p ( t ) = p + A ( t ), while in thevelocity gauge ˜p ( t ) = p . For computations involving a single atom, the latter ar-tifact can be avoided by placing the system at the originof the coordinate system. For systems composed of sev-eral centers, such as molecules, however, this ambiguitywill always be present. Indeed, in the literature, differentresults have been reported for molecular SFA computa-tions in the velocity and in the length gauge [7, 8, 11, 17].In recent, modified versions of the length-gauge SFA, thisproblem has been eliminated for ATI by considering theinitial bound state (cid:12)(cid:12)(cid:12) ˜ ψ ( L )0 E = exp[ − i A ( t ′ ) · r ] (cid:12)(cid:12)(cid:12) ψ ( L )0 E . Sucha state is gauge-equivalent to a field-free bound state inthe velocity gauge, and, physically, may be interpretedas a field-dressed state, in which the laser-field polariza-tion is included [17, 32]. For HHG, one may proceed ina similar way, with the main difference that the dress-ing should also be included in the final state, with whichthe electron recombines. In the dressed modified SFA, ˜p ( t ) = p . B. Diatomic molecules
We will now apply the SFA to a diatomic molecule.For this purpose, we will consider the simplest scenario,namely a one-electron system and frozen nuclei. Fur-thermore, we will assume that the molecular orbitalfrom which the electron is released and with which itmay recombine is a linear combination of atomic or-bitals (LCAO approximation). Explicitly, the molecularbound-state wave function reads ψ ( r ) = C ψ ( ψ (1)0 ( r ) + ǫψ (2)0 ( r )) , (3)where ǫ = ± , r = r − R / , and r = r + R / C and C , respectively,and C ψ is a normalization constant. For homonuclearmolecules, which we will consider here, ψ (1)0 = ψ (2)0 = ϕ .The positive and negative signs for ǫ correspond to bond-ing and antibonding orbitals, respectively. Within thiscontext, there exist two main approaches for comput-ing high-order harmonic spectra, which will be discussednext.
1. Prefactors
The simplest and most widely used [6, 7, 8, 12, 15,17, 20] approach is to incorporate the structure of themolecules in the prefactors d rec ( ˜p ( t )) and d ion ( ˜p ( t )) , and to employ the same action S ( t, t ′ , Ω , p ) as in thesingle-atom case. The multiple integral in the transi-tion amplitude (1) can then either be solved numeri- cally, or using saddle point methods [29]. The latterprocedure can be applied for high enough intensitiesand low enough frequencies, and consists in approximat-ing (1) by its asymptotic expansion around the coordi-nates ( t s , t ′ s , p s ) for which S ( t, t ′ , Ω , p ) is stationary. Thisimplies that ∂ t S ( t, t ′ , Ω , p ) = ∂ t ′ S ( t, t ′ , Ω , p ) = 0 and ∂ p S ( t, t ′ , Ω , p ) = . In this paper, we employ the specificsaddle-point approximations discussed in Ref. [33].For a single atom placed at the origin of the coordinatesystem, this leads to the equations[ p + A ( t ′ )] = − I p , (4) Z tt ′ dτ [ p + A ( τ )] = 0 , (5)and 2(Ω − I p ) = [ p + A ( t )] . (6)Eq. (4) expresses the conservation of energy at the time t ′ at which the electron reaches the continuum by tun-neling ionization. This equation possesses no real solu-tion, which reflects the fact that tunneling has no clas-sical counterpart. In the limit I p →
0, one obtains sucha condition for a classical particle reaching the contin-uum with vanishing drift velocity. Eq. (5) constrains theintermediate momentum of the electron, so that it re-turns to its parent ion, and, finally, Eq. (6) describes theconservation of energy at a later time t , when the elec-tron recombines with its parent ion and a high-frequencyphoton of frequency Ω is generated.The matrix element d rec ( ˜p ) = h ˜p | O dip · e x | ψ i thenreads d rec ( ˜p ) = C ψ (2 π ) / h e i ˜p · R / I ( r ) + ǫe − i ˜p · R / I ( r ) i , (7)where I ( r j ) = Z O dip ( r + r · e x exp[ i ˜p · r j ] ϕ ( r j ) d r j , (8)and O dip ( r + r ) is the dipole moment. In the lengthgauge, which we are mostly adopting in this paper, ˜p ( t ) = p + A ( t ) . Unless strictly necessary, in order tosimplify the notation, we do not include the time depen-dence in ˜p . The dipole moment can be written in severalforms. If one considers the length form a natural choice is O ( l )dip ( r ) = − e r + e r + e r = e r . Other possibilities are toconsider the operator O dip in its velocity and accelerationforms [20, 34].Inserting O dip ( r ) in Eq. (8) yields I ( r j ) ∝ I j ( r j ) + I jν ( r j ) , with j = 1 , ν = j, (9)where I j ( r j ) = Z O dip ( r j ) · e x exp[ i ˜p · r j ] ϕ ( r j ) d r j (10)and I jν ( r j ) = Z O dip ( r ν ) · e x exp[ i ˜p · r j ] ϕ ( r j ) d r j , ν = j. (11)Specifically, if the dipole is in the length form, the above-stated integrals read I j ( r j ) = − i ∂ ˜ p x φ ( ˜p ) (12)and I jν ( r j ) = 12 ( − i∂ p x φ ( ˜p ) + ǫ j R x φ ( ˜p )) , (13)where φ ( ˜p ) = Z exp[ i ˜p · r j ] ϕ ( r j ) d r j . (14)In I jν , ǫ j = +1 for j = 1 and ǫ j = − j = 2. Eq.(7) is then explicitly written as d ( b )rec ( ˜p ) = 2 iC ψ (2 π ) / (cid:20) − cos( ϑ ) ∂ p x φ ( ˜p ) + R x ϑ ) φ ( ˜p ) (cid:21) , (15)for bonding molecular orbitals (i.e., ǫ > , or d ( a )rec ( ˜p ) = 2 C ψ (2 π ) / (cid:20) sin( ϑ ) ∂ p x φ ( ˜p ) − R x ϑ ) φ ( ˜p ) (cid:21) , (16)in the antibonding case (i.e., ǫ < , with ϑ = ˜p · R / . In Eqs. (15) and (16), the terms with a purely trigono-metric dependence on the internuclear distance yield thedouble-slit condition in [18]. The maxima and minimain the spectra which are caused by this condition are ex-pected to occur for ˜p · R = 2 nπ and ˜p · R = (2 n + 1) π, (17)respectively, for bonding molecular orbitals (i.e., ǫ > . For antibonding orbitals, the conditions are reversed, i.e.,the maxima occur for the odd multiples of π and theminima for the even multiples. If the velocity gauge istaken, the above stated conditions hold for ˜p ( t ) = p ,instead of ˜p ( t ) = p + A ( t ). This is due to the factthat the initial and final free-field sttes are not gaugeequivalent, as discussed in Sec. IIA.The remaining terms grow linearly with the projec-tion R x of the internuclear distance along the directionof the laser-field polarization, and may lead to unphysi-cal results [11, 20]. For this reason, they are sometimesneglected in the integrals I jν ( r j ) [16]. There exists, how-ever, no rigorous justification for such a procedure. In-deed, only recently, it has been shown that such termscan be eliminated by considering an additional interac-tion which depends on the nuclear coordinate. This in-teraction is present in a modified molecular SFA, in itsdressed and undressed versions [17], and leads to contri-butions which cancel out the linear term in R x . In the length gauge, if the length form of O dip is taken, d rec ( ˜p ( t )) = d ion ( ˜p ( t ′ )), with ˜p ( t ) = p + A ( t ) , while inthe velocity gauge, d ( b )ion ( ˜p ) = C ψ [ p + A ( t ′ )] (2 π ) / cos( ϑ ) φ ( ˜p ) , (18)or d ( a )ion ( ˜p ) = − i C ψ [ p + A ( t ′ )] (2 π ) / sin( ϑ ) φ ( ˜p ) , (19)with ˜p ( t ) = p , for bonding and antibonding molecularorbitals, respectively.
2. Modified saddle-point equations
Physically, if one employs the pre-factors (15) and (16),this means that one is not modifying the saddle-pointequations (4)-(6). Therefore, the orbits along which theelectron is moving in the continuum remain the same asin the single-center case. This approach is questionablein several ways. From the technical viewpoint, there isno guarantee that such pre-factors are slowly varying, ascompared to the semiclassical action, especially for largeinternuclear distances [9, 11]. Furthermore, in general,they do not incorporate processes in which the electronleaves one center of the molecule and recombines withthe other, which, physically, are expected to be presentin molecular HHG [11].A slightly more sophisticated approach is to exponen-tialize the pre-factors obtained in the two-center case andincorporate the terms in ˜p · R / d ( b )rec ( ˜p ) = − iC ψ (2 π ) / (cid:20) cos (cid:18) ˜p · R (cid:19) ∂ ˜ p x φ ( ˜p ) (cid:21) , (20)for which the second term in Eq. (13) is absent. Eq. (20)is also very similar to the dipole matrix element in thevelocity form, apart from the fact that, in the latter case, ∂ ˜ p x φ ( ˜p ) is replaced by ˜p φ ( ˜p ) [20]. In the expression forthe antibonding case, the cosine term in (20) should bereplaced by sin( ˜p · R / M = X j =1 2 X ν =1 M jν (21)of the transition amplitudes M jν = C ψ (2 π ) / Z t dt ′ Z dt Z d pη ( p , t, t ′ ) × exp[ iS jν ( p , Ω , t, t ′ )] , (22)with η ( p , t, t ′ ) = [ ∂ ˜ p x φ ( ˜p ( t ))] ∗ ∂ ˜ p x φ ( ˜p ( t ′ )) . The terms S jν correspond to a modified action, which incorporates thestructure of the molecule.Explicitly, for the undressed length-gauge SFA, S jj = S ( p , Ω , t, t ′ ) + ( − j +1 ξ ( R, t, t ′ ) (23)and S jν = S ( p , Ω , t, t ′ ) + ( − ν +1 ξ ( R, t, t ′ ) , ν = j, (24)where ξ ( R, t, t ′ ) = [ A ( t ) − A ( t ′ )] · R / ξ ( R, t, t ′ ) = p · R +[ A ( t ) + A ( t ′ )] · R / . Eq. (23) and (24) may be di-rectly related to physical processes involving one or twocenters in the molecule, respectively, as it will be dis-cussed next.For this purpose, we will solve the multiple integralsin (22) employing saddle-point methods. The condi-tions ∂ p S jν ( p , Ω , t, t ′ ) = , ∂ t S jν ( p , Ω , t, t ′ ) = 0 and ∂ t ′ S jν ( p , Ω , t, t ′ ) = 0 upon the derivative of the actionyield saddle-point equations, which, as in the previoussection, can be related to the orbits of an electron re-combining with the molecule. Explicitly, for the modifiedaction S jj , [Eq. (23)], the saddle-point equations read[ p + A ( t ′ )] − I p − E ( t ′ ) · R / , (25)[ p + A ( t )] − I p + E ( t ) · R / , (26)for j = 1 , or[ p + A ( t ′ )] − I p + E ( t ′ ) · R / , (27)[ p + A ( t )] − I p − E ( t ) · R / , (28)for j = 2 . The saddle point equations (25) and (27)correspond to the tunnel ionization process from cen-ter C and C , respectively. Curiously, both equationscontain extra terms, if compared to equation (4) for thesingle-atom case. Such terms are dependent on the inter-nuclear distance R and the external laser field E ( t ′ ) atthe time the electron is freed, and may be interpreted aspotential-energy shifts in the barrier through which theelectron tunnels out. Similar terms are also observed inEqs. (26) and (28) for the energy conservation at the timethe electron recombines, as compared to the single-atomexpression (6).The remaining saddle point equation is given by thesame expression as in the single-atom case, i.e., Eq. (5),and means that, for M and M , the electron is ejectedand returns to the same center. If on the other hand, weconsider the modified action (24), this yields Z tt ′ [ p + A ( s )] ds ± R = 0 , (29)which, physically, mean that the electron is leaving fromone center and recombining with the other. The negative and positive signs refer to the transition amplitudes M (center C to center C ) and M (center C to center C ), respectively. In the former case, the remaining sad-dle points are given by (25) and (28), i.e., the electrontunnels from C and recombines with C , whereas in thelatter case they are given by (27) and (26), which physi-cally, expresses the fact that the electron is ejected at C and recombines with C . The energy shifts ± E ( τ ) · R / , τ = t, t ′ in (25)-(28) areabsent in the velocity gauge SFA [11], and in a modifiedlength-gauge SFA, in which the electric field polarizationis incorporated in the initial and final electronic boundstates. In both cases, the action S jj , associated to orbitsinvolving only one center, is given by the single-atomexpression (2), which leads to the saddle-point equations(4)-(6). The action S jν related to two-center orbits reads˜ S jν = S ( p , Ω , t, t ′ ) + ( − ν +1 p · R , ν = j. (30)The above-stated expression leads to the single-atom equations (4), (6) for tunneling and rescatter-ing, together with the two-center return condition(29). The prefactors, however, are different in bothcases. In the dressed modified length-gauge SFA, η D ( p , t, t ′ ) = [ ∂ p x φ ( p )] ∗ ∂ p x φ ( p ) , while in the velocitygauge η V ( p , t, t ′ ) = [ ∂ p x φ ( p )] ∗ φ ( p )[ p + A ( t ′ )] / . One should note that, within the specific model em-ployed here, the physical process in which the electronmoves directly from one center to the other, withoutreaching the continuum, is not being considered. Sucha process leads to a strong set of harmonics in the low-energy range of the spectra. Since, however, we are fo-cusing on the plateau harmonics, the contributions fromthis extremely short set of orbits are not of interest tothe present discussion. For a detailed study of this case,see, e.g., [5, 10].
3. Additional attosecond pulses
The role of the energy shifts observed in (25)-(28)is notwell understood. A way of eliminating such terms is tomodify the length-gauge SFA and include the influenceof the laser field in the initial and final states. How-ever, even without such modifications, it is possible toprovide an additional pathway for the electron to reachthe continuum, so that, at least in the context of tunnel-ing ionization, these shifts can be avoided. For instance,if the electron is ejected by a high-frequency photon, itdoes not have to tunnel through potential barriers withenergy shifts whose physical meaning is not clear. Such apathway can be provided by a time-delayed attosecond-pulse train E h ( t ) superposed to a strong, near infra-redfield E l ( t ) = E sin ωt e x . Indeed, it has been recentlyshown that such pulses can be used to control the elec-tron ejection in the continuum, and thus high-harmonicgeneration and above-threshold ionization [21, 22, 23].In [23, 24], we employed such a scheme to controlquantum-interference effects for high-harmonic genera-tion and above-threshold ionization for the single-atomcase, within the SFA framework. Our previous findingssuggest that the probability of the electron reaching thecontinuum, in case it is ejected by the attosecond pulses,is roughly the same for all sets of orbits. Indeed, it ap-pears that the sole, or at least main factor determiningthe intensities in the spectra is the excursion time of theelectron in the continuum. In the specific case studiedin [23], there was a set of very short orbits, which led toparticularly strong harmonics. Therefore, an attosecondpulse train superposed to a strong laser field is an idealsetup to avoid any artifacts due to modified tunnelingconditions.In [23], we have approximated the attosecond pulsetrain by a sum of Dirac-Delta functions in the time do-main. This yields E h ( t ) = E h π ∞ X n =0 ( − n σ ( t ) δ ( t − nπω − t d ) ǫ x , (31)where ω , E h , and σ ( t ) denote the laser field frequency,the attosecond-pulse strength and the train temporal en-velope, respectively. This approximation is the limitingcase for a train composed of an infinite set of harmon-ics, and has the main advantage of allowing an analytictreatment of the transition amplitudes involved up to atmost one numerical integration. Furthermore, it is a rea-sonable asymptotic limit for pulses composed by a largehigh-harmonic set. We consider here σ = const, which,physically, corresponds to an infinitely long attosecond-pulse train. Clearly, the total field is given by E ( t ) = E l ( t ) + E h ( t ) . We will now assume that the attosecond pulses are theonly cause of ionization and that the subsequent propaga-tion of the electron in the continuum is determined onlyby the monochromatic field. Hence, E ( t ′ ) ≃ E h ( t ′ ) and A ( t ) ≃ A l ( t ) in Eqs. (1) and (22). This eliminates theintegral over the ionization time in the transition ampli-tudes. Hence, the values for which the single atom-action S ( t, t ′ , Ω , k ) , or the modified action S jν is stationary mustbe determined only with respect to the variables t and p . Physically this means that the recombination andreturn conditions remain the same, with regard to thepurely monochromatic case, but that there is no longera saddle-point equation constraining the initial electronmomentum. In fact, the electron is being ejected in thecontinuum with any of the energies N ω − I p , since allharmonics composing the train are equivalent. For theother extreme limit, namely a high-frequency monochro-matic wave, we refer to [24], where we provide a detaileddiscussion within the SFA.Explicitly, if the action is not modified, these assump-tions lead to the transition amplitude M ( D )h = iπC ψ E h σ (2 π ) / ∞ X n =0 ( − n Z + ∞−∞ dt Z d p exp [ iS ( p , Ω , t, t ′ )] × d ∗ rec ( ˜p ( t )) d ion ( ˜p ( t ′ )) . (32) In case one considers the transition amplitudes (22), thisyields M jν = iπC ψ E h σ (2 π ) / ∞ X n =0 ( − n Z + ∞−∞ dt Z d pη ( ˜p , t, t ′ )(33) × exp[ iS jν ( p , Ω , t, t ′ )] . (34)In both equations, t ′ = t d + nπ/ω. The saddle-point equa-tions (5) and (6) for the single atom case can then becombined assin ωt − ( − n sin ωt d =[ ω ( t − t d ) − nπ ] cos ωt ∓ s Ω − I p U p ! , (35)which will give the return times t. If, on the other hand,these equations are modified, it is possible to distinguishfour main scenarios. Specifically, for the processes inwhich the electron leaves and returns to the same center,the saddle-point equations differ from (35) only in a shift I p → I p ± E l ( t ) · R / . The negative and positive signscorrespond to M and M , respectively. For the sce-narios involving two centers, the saddle-point equationsread sin ωt − ( − n sin ωt d + ǫ R x ω U p =[ ω ( t − t d ) − nπ ] cos ωt ∓ s Ω − ˜ I p U p , (36)with ˜ I p = I p + ǫ E l ( t ) · R / . The case ǫ = − ǫ = +1corresponds to M and M , respectively. III. HARMONIC SPECTRA
We will now present high-harmonic spectra, in thepresence and absence of the attosecond pulses. We re-strict the electron ionization times to the first half cycleof the driving field. We also consider the six shortest pairsof orbits for the returning electron. Due to wave-packetspreading, the contributions from the remaining pairs arenegligible [35]. For simplicity, we employ a bonding com-bination of 1s states, for which φ ( ˜p ) ∼ ˜p + 2 I p ] , (37)and assume that the molecule is aligned parallel to thelaser-field polarization, so that R x = R. A. Prefactors
We will commence by considering the single-center ac-tion (2) and the prefactors discussed in the previous sec-
FIG. 1: (Color Online) High-harmonic spectra computed em-ploying the single-atom orbits and two center prefactors, us-ing a bonding ( ǫ = +1), linear combination of atomic orbitals,for internuclear distances 1 a.u. ≤ R ≤ H +2 , which wasapproximated by the linear combination of 1 s atomic orbitalswith I p = 0 . I = 1 × W / cm , and ω = 0 .
057 a.u.,respectively. tion. In Fig. 1, we depict high-harmonic spectra com-puted for a wide range of internuclear distances, employ-ing the prefactor (15) or the modified expression (20), forwhich the linear term in R x is absent [upper and lowerpanel, respectively]. The figure illustrates how the linearterm masks the interference patterns. In fact, for a.u.1 ≤ R ≤ ς = R | cos θφ ( ˜p ) / (2 ∂ p x φ ( ˜p )) | , where θ is the alignment angleand φ ( ˜p ) is given by Eq. (14). If ς ≃
1, the maxima willpossess the same order of magnitude and there will be
20 40 60 80 100 120 140-30-28-26-24-22-20-18-16 n = 1 n = 5n = 3
Log H a r m on i c Y i e l d ( a r b . un i t s ) Harmonic Order N Modified prefactor Modified action
FIG. 2: (Color online) Spectra computed employing thesingle-atom orbits and two center prefactors (black thin lines),as compared to those obtained employing modified saddle-point equations (green thick lines). We consider here themodified length form (20) of the dipole operator, which ex-cludes the term with a linear dependence on R x . The atomicsystem was chosen as H +2 , which was approximated by thelinear combination of 1 s atomic orbitals with I p = 0 . R = 5a.u., and θ = 0 , respectively. The driving field intensity andfrequency are given by I = 1 × W / cm , and ω = 0 . no noticeable modulation, while if ς < ς > , how-ever, one expects that the linear term in R will prevail.Hence, the critical value for the internuclear distance is R c = 2 | sec θ∂ p x φ ( ˜p ) /φ ( ˜p ) | . This expression depends onthe bound states with which the electron recombines, andalso on the harmonic energy. For instance, specifically for1 s states, R c ∼ p x sec θ/ ( ˜p + 2 I p ) . Above the ioniza-tion threshold, according to Eq. (6), R c ∼ p x sec θ/ Ω.Hence, one expects the linear term to be more promi-nent as the harmonic energy increases, leading to clearer,though incorrect, patterns. In order to avoid such prob-lems, we will employ the prefactor (20) throughout.In Fig. 2, we compare spectra computed using eitherthe pre-factor (20) or modified saddle-point equations.Both spectra are very similar, with maxima and min-ima at harmonic frequencies Ω = I p + n π / (2 R x ), asexpected from the double-slit condition. This similarityholds not only for the gross features, but, additionally, forthe substructure caused by other types of quantum inter-ference. Close to the minima, however, the yield from thelatter case is larger. Nevertheless, the very good overallagreement shows that, in fact, the patterns obtained canalso be interpreted as the result of the quantum interfer-ence between different types of electron orbits. B. Interference effects
In order, however, to identify which sets of orbits causethe dips and the maxima, we will analyze the interferencebetween their individual contributions. Such results areshown in Fig. 3. In the upper panels, we present thespectra computed from topologically similar sets of or-bits, i.e., from processes involving only one, or two cen-ters [panel 3.(a)]. In this case, the main interference pat-terns are absent. This strongly suggests that they aredue to the quantum interference of topologically differentsets of orbits: the orbits along which an electron leavesand returns to the same center, and those along which itreaches the continuum at one center and recombines withthe other. Physically, this could be attributed to the factthat, in this case, there would be an appreciable phasedifference between the two sets of orbits, since the latterare much longer than the former. This phase differencewould cause the overall modulation.Hence, there are two remaining possibilities. Con-cretely, the modulation can be due to the quantum inter-ference either between processes in which the electronleaves from different centers and recombines with thesame center (i.e., between the orbits which start at C j and end at C ν , with ν = j, and those starting and end-ing at C ν , and j = 1 , C j andrecombines at C ν , ν = j, or it is freed and recombines at C j , with j = 1 , | M + M | , from the orbits starting at C . In order to understand this better, one must have acloser look at the individual contributions from differentsets of orbits, and, in particular their orders of magni-tude. Such contributions, depicted in Panels (a) and (b)of Fig. 4, show that the transition probabilities | M | and | M | , which correspond to the orbits starting fromthe center C , are roughly four orders of magnitude largerthan | M | and | M | , i.e., than those from the orbitsstarting at C . Therefore, it is not surprising that theyield is dominated by | M + M | in the previous fig-ure. Furthermore, these results exhibit no maxima andminima. Hence, they support the assumption that suchfeatures are due to the interference of different types oforbits. Finally, the contributions from orbits startingat the same center possess the same order of magnitude.This suggests that tunneling ionization is the main mech-anism determining the relevance of a particular type oforbits to the spectra, and that are local differences in thebarrier through which the electron must tunnel, depend-
20 40 60 80 100 120 140-30-28-26-24-22-20-18-16-28-26-24-22-20-18-16-28-26-24-22-20-18-16 (c)
Harmonic Order N |M + M | |M + M | (a) |M + M | |M + M | (b) Log H a r m on i c Y i e l d ( a r b . un i t s ) |M + M | |M + M | FIG. 3: (Color Online) Contributions to the high-harmonicyield from the quantum interference between different typesof orbits, for internuclear distance R = 5 a.u. The remainingparameters are the same as in Fig. 3. Panel (a): Orbitsinvolving similar scattering scenarios, i. e., | M + M | , and | M + M | . Panel (b): Orbits ending at the same center,i.e., | M + M | and | M + M | . Panel (e): Orbits starting at the same center, i.e., | M + M | and | M + M | . Forcomparison, the full contributions | M + M + M + M | are displayed as the light gray circles in the picture. ing on the center it starts from.An inspection of the imaginary parts Im[ t ′ ] of the starttimes provides additional insight into this problem. Dueto the fact that tunneling ionization is a process whichhas no classical counterpart, this quantity is always non-vanishing, even if the energy range in question is lowerthan the maximal harmonic energy. The larger Im[ t ′ ] is,the less probable it will be that tunneling ionization takesplace. Such an interpretation has been successfully em-ployed in [36] in order to determine the dominant pairs oforbits, in the context of nonsequential double ionizationwith few-cycle laser pulses, and will be also consideredin this work. For that purpose, we will take the shortest
20 40 60 80 100 120 140-30-28-26-24-22-20-18-16-30-28-26-24-22-20-18-16 (b)
Log H a r m on i c Y i e l d ( a r b . un i t s ) Harmonic Order N |M | |M | (a) |M | |M | FIG. 4: (Color online) Contributions to the high-harmonicyield from specific types of orbits, for internuclear distance R = 5 a.u. The remaining parameters are the same as inFig. 3. Panel (a): Yield from the orbits starting and endingat the same center, i. e., transition probabilities | M | and | M | . Panel (b): Yield from the orbits starting and endingat different centers, i.e., transition probabilities | M | and | M | . pairs of orbits utilized in the computation of the tran-sition probabilities in Fig. 4 and, for each case, displayIm[ t ′ ]. These are the dominant pairs of orbits. The longerpairs have a less significant influence on the spectra, dueto the spreading of the electronic wave packet [35].Such results are depicted in Fig. 5. Clearly, Im[ t ′ ]is around four times larger for the orbits starting fromthe center C , as compared to those starting from C . This means that, in order to reach the continuum, theelectron must overcome a larger barrier if it comes from C . Since, roughly speaking, the ionization probabilityper unit time decreases exponentially with Im[ t ′ ], oneexpects the contributions from the orbits starting from C to be around four orders of magnitude smaller thanthose from the orbits starting at C . An inspection ofFig. 4 shows that this is indeed the case.The above-stated effect could, however, be an artifactof the strong-field approximation in the length gauge.Indeed, the terms ± E ( t ′ ) · R in the saddle-point equa-tions (25) and (27) can be interpreted as potential en- Harmonic Order N I m [t ’ ] C C C C (b)(a) I m [t ’ ] C C C C FIG. 5: (Color online) Imaginary parts of the start times t ′ of the shortest pairs of orbits, which contribute to the matrixelements M , and M , [panel (a)], and to M , and M [panel (b)], for the same field and atomic parameters as inthe previous figure. ergy shifts, due to the fact that the electron is displacedfrom the origin [11, 17]. Such terms increase or sink thepotential barrier for C or C , respectively, and, conse-quently, change the orders of magnitude in the contribu-tions starting from different centers. Even though, as awhole, the results match those obtained by other means,their physical interpretation is controversial.One may, however, avoid this problem by providingan additional pathway for the electron to reach the con-tinuum. For that purpose, we shall superpose a time-delayed attosecond pulse train to the strong laser field,employing the model discussed in Sec. II B 3 and in ourprevious work [23, 24]. The maximal harmonic ener-gies for this specific model are strongly dependent onthe time delay t d between the attosecond-pulse train theinfra-red field, extending from the ionization potential,for φ = 0 . π to I p + 1 . U p , for φ = nπ . Furthermore,there exist many intermediate delays, for which a doubleplateau is present. This substructure may be detrimen-tal to the identification and physical interpretation of theinterference patterns. In order to avoid such problems,we will consider here vanishing time delay, i.e., φ = 0 . In Fig. 6, we depict the spectra obtained for a diatomicmolecule subjected to such a field, assuming either a two-center prefactor and the single-center saddle-point equa-tion (35), or the modified saddle point equations (36)0
FIG. 6: (Color Online) High-harmonic spectra for a moleculealigned parallel to the laser-field polarization, and internu-clear distance R = 3 . φ = 0) with respect to it. We con-sider a bonding combination of 1 s atomic orbitals, and takethe intensity of the attosecond-pulse train to be I h = I l / | M jν | , with j = 1 , ν = 1 ,
2. The interferenceminima are indicated by arrows in the figure. [Fig. 6.(a)]. Both computations exhibit a minimum nearthe harmonic frequency Ω = 71 ω , in agreement with Eq.(17). If only the contributions | M jν | from the individualscattering scenarios are taken, such a minimum is absent[Fig. 6.(b)]. Therefore, it is due to interference effectsbetween different sets of orbits. One should note that, incontrast to the purely monochromatic case, all contribu-tions exhibit the same order of magnitude. This is dueto the fact that, if the attosecond pulses are present, theelectron is being ejected in the continuum with roughlythe same probability, regardless of the center it left from.The precise role of the various recombination scenar-ios is illustrated in Fig. 7. For clarity, we concentrateon the plateau region around the interference minimum.The main difference observed, with regard to the purelymonochromatic case, is that the overall shape of the spec-trum, and consequently its minimum, is due to the collec- FIG. 7: (Color Online) Contributions to the high-harmonicyield from the quantum interference between different typesof orbits, for the same field and molecular parameters as inFig. 7. Panel (a): Contributions from topologically similarscattering scenarios, i. e., contributions from | M + M | and | M + M | . Panel (b): Contributions from orbits start-ing at the same center. Panel (c): Contributions from orbitsending at the same center. For comparison, the overall spectraare displayed as the blue symbols in the figure. The interfer-ence minimum is indicated by the vertical lines near Ω = 71 ω . tive interference of several types of orbits. This is in con-trast to the previous results, for which they were causedby the processes starting at a center C j and ending atdifferent centers, i.e., | M jv + M jj | , with ν = j and ν, j = (1 , | M + M | involving only one-center sce-narios. This is shown in Fig. 7.(a), and contradicts thepreviously made assumption that such features are dueto the interference between topologically different sets oforbits. In fact, it seems that the absence of overall max-ima and minima for the one-center contributions, in thepurely monochromatic case [Fig. 3.(a)], is due to the dif-ferent orders of magnitude for M and M [c.f. Fig.4.(a)].Furthermore, one needs several different processes inorder to obtain the correct position of the minimum. Forinstance, in Fig. 7.(a), the overall spectrum closely fol-lows | M + M | in the low-energy region. In the vicin-ity of the minimum and for higher energies, however, itfollows neither such contrbutions nor | M + M | , fromtwo-center processes. In Fig. 7.(b), where we display the1contributions | M jj + M jν | , with ν = j and ν, j = (1 , , from the orbits starting from the same center, the spec-trum closely follows | M + M | before the minimum,and | M + M | after the minimum.The remaining panel [Fig. 7.(c)] depicts the contribu-tions from | M νν + M jν | , with ν = j and ν, j = (1 , finishing at the same center. In thiscase, the interference minimum is absent. This is a strongevidence that the relevant condition for the presence ofsuch features is that the orbits taken into account end atdifferent centers, instead of being topologically different(which is the case for both Fig. 7.(b) and Fig. 7.(c)).Therefore, these results agree with the double-slit pic-ture, which has been put across in [18].The additional attosecond pulses have the advantageof not introducing changes in the standard length-gaugeSFA formulation. They modify, however, the physics ofthe problem, since they provide a different mechanismfor the electron to reach the continuum. Clearly, there isalso the possibility of eliminating the spurious potentialenergy shifts, by considering a different version of theSFA, such as the field-dressed length-gauge formulationproposed in [17], or the velocity-gauge formulation.In Fig. 8, we display high-order harmonic spectra forthe field-dressed SFA in the length gauge, and for thefield-undressed SFA in the velocity gauge [panels (a) and(b), respectively]. For simplicity, we exhibit the resultsobtained with the modified prefactor (20), instead of amodified action. We also provide curves for which onlythe cosine term has been set to one, in order to facilitatethe identification of interference effects. As an overallfeature, we do not observe the interference patterns ex-hibited in the previous figures. Indeed, the curves withand without the cosine term are very similar. There is,however, a minimum near Ω = 115 ω for the former case.For the dressed length-gauge SFA, the above-statedfeatures can be attributed to the fact that the conditionfor maxima and minima is now given by (17), with p instead of ˜p = p + A ( t ) . The harmonic frequencies forwhich they occur can be easily obtained from condition(6), and are given byΩ = I p + (cid:2) n π /R x + 2 nπA ( t ) /R x + A ( t ) (cid:3) / . (38)An upper bound for Ω can be estimated as follows. Atthe electron return times, the vector potential is roughly A ( t ) . p U p . This yields, for the parameters in Fig. 8,Ω ∼ ω , which is slightly larger than the minimumencountered.A breakdown of the interference patterns also occursin the velocity gauge, for the very same reasons. In-deed, the interference condition for the SFA in the ve-locity gauge and for the field-dressed SFA in the lengthgauge are identical. This is a direct consequence ofthe fact that field-free initial and final electron statesin the velocity gauge are gauge-equivalent to the field-dressed states considered in this paper. This gauge equiv-alence will lead to identical recombination form factors d rec ( p ) = h p | O dip . e x | ψ i . Since the interference condi-
20 40 60 80 100 120 140-23-22-21-20-19-18-17-16-28-27-26-25-24-23-22-21 = 115(a) Velocity gauge, undressed n = 1
Log H a r m on i c Y i e l d ( a r b . un i t s ) Harmonic order N cos( p . R /2) = 1 with cos( p . R /2) (a) Length gauge, dressed = 115 n = 1 cos( p . R /2) = 1 with cos( p . R /2) FIG. 8: (Color Online) High-order harmonic spectra for thesame field and molecular parameters as in Fig. 2, but com-puted with the field-dressed modified length-gauge formula-tion of the strong-field approximation [Panel (a)], comparedto its field-undressed velocity-gauge counterpart [Panel (b)].The arrows in the figure indicate the harmonic order for whicha minimum is observed, and the vertical line marks the roughestimate for such a minimum. The solid and dotted lines cor-respond to the prefactor (19) and to the situation for whichcos p · R /2 has been set to one, respectively. tions (17) are mainly determined by d rec ( p ), they willbe the same. Hence, the harmonic orders for whichthe maxima and minima occur are given by Eq. (38),and therefore are unrealistically high. The discrepan-cies between both yields stem from the form factors d ion ( p ) = h p | H int ( t ′ ) | ψ i , which are gauge-dependent.One may, however, consider field-dressed states in thevelocity gauge, which are gauge-equivalent to field-freestates in the length gauge. This is achieved by apply-ing the transformation χ v ← l = exp[ i A ( t ) · r ] in the initialand final length-gauge electronic bound states. This leadsto a shift p → p + A ( t ) on a momentum eigenstate | p i ,which is exactly the opposite shift induced in the field-dressed length-gauge SFA. This shift has the main con-sequence that the interference condition (17) now holdsfor ˜p = p + A ( t ), even in the velocity gauge. The results2
20 40 60 80 100 120 140-30-28-26-24-22-20-18-16 n = 5n = 3n = 1
Log H a r m on i c Y i e l d ( a r b . un i t s ) Harmonic Order N Length gauge (undressed) Velocity gauge (dressed)
FIG. 9: (Color Online) High-order harmonic spectra for thesame field and molecular parameters as in Fig. 2, but com-puted with the field-dressed modified velocity-gauge formula-tion of the strong-field approximation. In order to facilitatethe comparison, the undressed length-gauge version of theSFA is provided, and the velocity-gauge yield has been nor-malized in approximately 2 orders of magnitude, to match thelength-gauge results. In both cases, for simplicity, the mod-ified prefactor (18) has been employed, instead of modifiedsaddle-point equations. obtained employing such dressed states in the velocity-gauge SFA, depicted in Fig. 9, are indeed very similar tothose obtained using its field undressed, length gauge for-mulation. In fact, we have observed mainly quantitativedifferences, due to different prefactors d ion ( ˜p ) [37].One should note, however, that the transformation χ v ← l introduces the same additional potential energyshifts in the modified action as in the undressed length-gauge case. Thus, the price one pays for recovering thecorrect interference conditions is the loss of a direct con-nection to simple classical models. IV. CONCLUSIONS
The present results support the viewpoint that themaxima and minima in the high-order harmonic spectraof diatomic molecules are due to the interference of elec-tron orbits finishing at different centers in the molecule.This seems to hold regardless of whether the electron hasbeen released in one center at the molecule and recom-bines with a different center, or whether it is ejected andreturns to the same center. Such conclusions have beenreached by employing modified saddle-point equations,within the strong-field approximation. These modifica-tions lead to orbits involving different centers, and are aslightly more refined approach than the standard proce-dure of considering single-center saddle-point equations and modified prefactors.In this framework, we considered that the electron hasbeen ejected by tunneling ionization and by an additionalattosecond-pulse train, and compared the similarities anddifferences from both physical situations. In the formercase, depending from which center the electron is leav-ing, it must overcome unequal potential barriers to reachthe continuum, whereas in the latter case it is ejectedwith roughly equal probability. Especially in the purelymonochromatic case, we observed that the contributionsfrom orbits starting at one of the centers, namely C ,were much larger than those from C , due to a narrowerpotential barrier. In particular, there exist potential-energy shifts which are proportional to the electric field atthe ionization time and the internuclear distance, whichcause such differences. It is however noteworthy that theelectron excursion times have been confined to the firsthalf-cycle of the laser field. If the other half-cycle hadbeen taken, the potential barrier would reverse and thecontributions starting from C would be more prominent.The above-mentioned potential-energy shifts, however,do not possess a clear-cut physical interpretation. In fact,they are only present in the standard length-gauge formu-lation of the strong-field approximation, i.e., if field-freebound states are taken when the electron is ejected andrecombines, and are the source of several problems. Forinstance, they reflect the fact that the SFA is translation-dependent. Moreover, due to their existence, it is diffi-cult to establish an immediate connection between thisapproach and the classical equations of motion of an elec-tron in the laser field.On the other hand, it seems that such shifts are nec-essary in order to obtain the correct energy position ofthe maxima and minima in the HHG spectra. In fact, animproved formulation of the SFA, in which the influenceof the laser field is included in the electron bound states,restores its translation invariance, provides an unprob-lematic classical limit [17], but yields incorrect energypositions for the interference patterns. This discrepancyis related to the fact that the field dressing alters thedouble-slit interference conditions (17). A similar ab-sence of interference features has been reported very re-cently in Ref. [38], for HHG computations using a field-dressed version of the SFA in the length gauge.Finally, when employing different gauges, from our re-sults it is clear that the dressing of the initial and thefinal states plays a far more important role than the dif-ferent interaction Hamiltonians H int ( t ′ ). If the dressingis applied consistently so that the the electronic boundstates are gauge-equivalent, the interference patterns willremain the same. This is due to the fact that the inter-ference condition (17) will then remain invariant. Forinstance, the spectra obtained in the velocity-gauge SFAwith undressed states are very similar to those computedin the length gauge with field dressed states. The sameholds for the undressed length-gauge, and the dressedvelocity-gauge SFA spectra. In the two former cases,there is a breakdown of the interference patterns, as com-3pared to the field-undressed length gauge SFA. However,such patterns can be restored in the velocity gauge, bydressing the electronic bound states appropriately. Acknowledgments
We would like to thank L. E. Chipperfield, R. Torres,J. P. Marangos, and H. Schomerus for useful discussions, and W. Becker for calling Ref. [17] to our attention.We are also grateful to the Imperial College and to theUniversity of Stellenbosch for their kind hospitality. Thiswork has been financed by the UK EPSRC (AdvancedFellowship, Grant no. EP/D07309X/1). [1] J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. P´epin,J. C. Kieffer, P. B. Corkum and D. M. Villeneuve, Nature , 867 (2004).[2] H. Niikura, F. L´egar´e, R. Hasbani, A. D. Bandrauk, M.Yu. Ivanov, D. M. Villeneuve and P. B. Corkum, Nature , 917 (2002); H. Niikura, F. L´egar´e, R. Hasbani, M.Yu. Ivanov, D. M. Villeneuve and P. B. Corkum, Nature , 826 (2003); S. Baker, J. S. Robinson, C. A. Haworth,H. Teng, R. A. Smith, C. C. Chiril˘a, M. Lein, J. W. G.Tisch, J. P. Marangos, Science , 424 (2006).[3] A. Scrinzi, M. Y. Ivanov, R. Kienberger, and D. M. Vil-leneuve, J. Phys. B , R1 (2006).[4] P. B. Corkum, Phys. Rev. Lett. , 1994 (1993); K. C.Kulander, K. J. Schafer, and J. L. Krause in: B. Piraux etal. eds., Proceedings of the SILAP conference , (Plenum,New York, 1993).[5] R. Kopold, W. Becker and M. Kleber, Phys. Rev. A ,4022 (1998).[6] J. Muth-B¨ohm, A. Becker, and F. H. M. Faisal, Phys.Rev. Lett. , 2280 (2000); A. Jar´on-Becker, A. Becker,and F. H. M. Faisal, Phys. Rev. A , 023410 (2004); A.Requate, A. Becker and F. H. M. Faisal, Phys. Rev. A , 033406 (2006).[7] T. K. Kjeldsen and L. B. Madsen, J. Phys. B , 2033(2004); Phys. Rev. A , 023411 (2005); Phys. Rev. Lett. , 073004 (2005); T. K. Kjeldsen, C. Z. Bisgaard, L. B.Madsen, H. Stapelfeld, Phys. Rev. A , 013418 (2005);C. B. Madsen and L. B. Madsen, Phys. Rev. A , 023403(2006).[8] V. I. Usachenko, and S. I. Chu, Phys. Rev. A , 063410(2005).[9] H. Hetzheim, M. Sc. thesis (Humboldt University Berlin,2005).[10] V. I. Usachenko, P. E. Pyak, and Shih-I Chu, Laser Phys. , 1326 (2006).[11] C. C. Chiril˘a and M. Lein, Phys. Rev. A , 023410(2006).[12] M. Lein, Phys. Rev. Lett. , 053004 (2005); C. C.Chiril˘a and M. Lein, J. Phys. B , S437 (2006).[13] H. Hetzheim, C. Figueira de Morisson Faria, and W.Becker Phys. Rev. A, in press (arXiv:0704.0712).[14] S. X. Hu and L. A. Collins, Phys. Rev. Lett. , 073004(2005); D. A. Telnov and Shih-I Chu, Phys. Rev. A ,013408 (2005).[15] G. Lagmago Kamta and A. D. Bandrauk, Phys. Rev. A , 011404 (2004); ibid. , 053407 (2005).[16] X. Zhou, X. M. Tong, Z. X. Zhao and C. D. Lin, Phys.Rev. A , 061801(R) (2005); ibid. , 033412 (2005).[17] D. B. Miloˇsevi´c, Phys. Rev. A , 063404 (2006).[18] M. Lein, N. Hay, R. Velotta, J. P. Marangos, and P. L. Knight, Phys. Rev. Lett. , 183903 (2002); Phys. Rev.A , 023805 (2002); M. Spanner, O. Smirnova, P. B.Corkum and M. Y. Ivanov, J. Phys. B , L243 (2004).[19] B. Shan, X. M. Tong, Z. Zhao, Z. Chang, and C. D. Lin,Phys. Rev. A , 061401(R) (2002); F. Grasbon, G. G.Paulus, S. L. Chin, H. Walther, J. Muth-B¨ohm, A. Beckerand F. H. M. Faisal, Phys. Rev. A , 041402(R)(2001);C. Altucci, R. Velotta, J. P. Marangos, E. Heesel, E.Springate, M. Pascolini, L. Poletto, P. Villoresi, C. Vozzi,G. Sansone, M. Anscombe, J. P. Caumes, S. Stagira, andM. Nisoli, Phys. Rev. A , 013409 (2005); T. Kanai, S.Minemoto and H. Sakai, Nature , 470 (2005).[20] C. C. Chiril˘a and M. Lein, J. Mod. Opt. , 1039 (2007).[21] K. J. Schafer, M. B. Gaarde, A. Heinrich, J. Biegert,and U. Keller, Phys. Rev. Lett. , 023003 (2004); M.B. Gaarde, K. J. Schafer, A. Heinrich, J. Biegert, andU. Keller, Phys. Rev. A , 013411 (2005); J. Biegert,A. Heinrich, C. P. Hauri, W. Kornelis, P. Schlup, M. P.Anscombe, M. B. Gaarde, K. J. Schafer and U. Keller,J. Mod. Opt. , 87 (2006); J. Biegert, A. Heinrich, C.P. Hauri, W. Kornelis, P. Schlup, M. P. Anscombe, K. J.Schafer, M. B. Gaarde and U. Keller, Laser Physics ,899 (2005).[22] P. Johnsson, R. L´opez-Martens, S. Kazamias, J. Maurits-son, C. Valentin, T. Remetter, K. Varj´u, M. B. Gaarde,Y. Mairesse, H. Wabnitz, P. Sali`eres, Ph. Balcou, K. J.Schafer, and A. L’Huillier, Phys. Rev. Lett. , 013001(2005); P. Johnsson, K. Varj´u, T. Remetter, E. Gustafs-son, J. Mauritsson, R. Lopez-Martens, S. Kazamias, C.Valentin, Ph. Balcou, M. B. Gaarde, K. J. Schafer, andA. L’Huillier, J. Mod. Opt. , 233 (2006).[23] C. Figueira de Morisson Faria, P. Sali`eres, P. Villain andM. Lewenstein, Phys. Rev. A , 053416 (2006).[24] C. Figueira de Morisson Faria and P. Sali`eres, Laser Phys. , 390 (2007).[25] W. Becker, S. Long, and J. K. McIver Phys. Rev. A ,1540 (1994).[26] M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huillierand P. B. Corkum, Phys. Rev. A , 2117 (1994).[27] W. Becker, S. Long, and J. K. McIver, Phys. Rev. A ,4112 (1990); ibid. , 1540 (1994); M. Lewenstein, K. C.Kulander, K. J. Schafer and Ph. Bucksbaum, Phys. Rev.A , 1495 (1995).[28] W. Becker, A. Lohr, M. Kleber, and M. Lewenstein,Phys. Rev. A , 645 (1997).[29] P. Sali`eres, B. Carr´e, L. LeD´eroff, F. Grasbon, G.G. Paulus, H. Walther, R. Kopold, W. Becker, D. B.Miloˇsevi´c, A. Sanpera and M. Lewenstein, Science ,902 (2001).[30] A. Fring, V. Kostrykin and R. Schrader, J. Phys. B. , , 023415 (2005).[31] In the length or velocity gauge, this interaction is givenby H LI ( t ′ ) = r . E ( t ′ ) or H VI ( t ′ ) = [ p + A ( t ′ )] /
2, respec-tively.[32] O. Smirnova, M. Spanner and M. Ivanov, J. Phys. B ,S307 (2006).[33] C. Figueira de Morisson Faria, H. Schomerus and W.Becker, Phys. Rev. A , 043413 (2002).[34] K. Burnett, V. C. Reed, J. Cooper and P. L. Knight,Phys. Rev. A , 3347 (1992); J. L. Krause, K. Schaferand K. Kulander, Phys. Rev. A , 3347 (1992).[35] Since the bound state with which the returning electronrecombines is highly localized, a broader wave packetmeans a less pronounced overlap upon recombination and, consequently, weaker harmonics.[36] C. Figueira de Morisson Faria, X. Liu, A. Sanpera andM. Lewenstein, Phys. Rev. A , 043406 (2004).[37] In these SFA formulations, for exponentially decayingstates, φ ( ˜p ) and ∂ p x φ ( ˜p ) exhibit a singularity, which canbe eliminated by being incorporated in the action. Thishas not been included in our model, and may lead tofurther differences between both yields. We expect, how-ever, these discrepancies to be minimal, as this additionalterm is slowly varying. For a discussion in the length-gauge context, c.f., C. Figueira de Morisson Faria andM. Lewenstein, J. Phys. B , 3251 (2005).[38] O. Smirnova, M. Spanner and M. Ivanov, J. Mod. Opt.54