Transient absorption and reshaping of ultrafast XUV light by laser-dressed helium
Mette B. Gaarde, Christian Buth, Jennifer L. Tate, Kenneth J. Schafer
aa r X i v : . [ phy s i c s . a t o m - p h ] F e b Transient absorption and reshaping of ultrafast xuv light by laser-dressed helium
Mette B. Gaarde , , ∗ Christian Buth , , , † Jennifer L. Tate , and Kenneth J. Schafer , Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803, USA The PULSE Institute for Ultrafast Energy Science,SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA and Max-Planck-Institut f¨ur Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany (Dated: May 29, 2018)We present a theoretical study of transient absorption and reshaping of extreme ultraviolet ( xuv )pulses by helium atoms dressed with a moderately strong infrared ( ir ) laser field. We formulate theatomic response using both the frequency-dependent absorption cross section and a time-frequencyapproach based on the time-dependent dipole induced by the light fields. The latter approach canbe used in cases when an ultrafast dressing pulse induces transient effects, and/or when the atomexchanges energy with multiple frequency components of the xuv field. We first characterize thedressed atom response by calculating the frequency-dependent absorption cross section for xuv energies between 20 and 24 eV for several dressing wavelengths between 400 and 2000 nm andintensities up to 10 W/cm . We find that for dressing wavelengths near 1600 nm, there is anAutler-Townes splitting of the 1 s → p transition that can potentially lead to transparency forabsorption of xuv light tuned to this transition. We study the effect of this xuv transparency in amacroscopic helium gas by incorporating the time-frequency approach into a solution of the coupledMaxwell-Schr¨odinger equations. We find rich temporal reshaping dynamics when a 61 fs xuv pulseresonant with the 1 s → p transition propagates through a helium gas dressed by an 11 fs, 1600 nmlaser pulse. PACS numbers: 32.80.Fb, 32.80.Qk, 32.80.Rm, 42.50.Hz
I. INTRODUCTION
The advent of ultrafast xuv and even x-ray lightsources that can be synchronized to optical or ir laserpulses has given rise to several recent studies of the tran-sient absorption of such radiation by laser-dressed atoms,both experimental [1–5] and theoretical [6–8]. For exam-ple, many of the experiments done in attosecond physicsinvolve the transient absorption of attosecond xuv radi-ation by atoms interacting with an ir laser field. This isbecause the strong-field process of high harmonic gen-eration ( hhg ), which is used to produce the attosec-ond xuv radiation as either single pulses or trains ofpulses, results in the xuv field being precisely synchro-nized with the driving ir field [1, 9]. It is then possi-ble to perform experiments using the xuv field and areplica of the original ir field with attosecond precision[10]. Glover et al. also showed that it is possible to over-lap pulses of synchrotron-produced soft x-ray radiationwith an 800 nm dressing laser in a study of laser-inducedtransparency in neon [2].In this paper we explore how an ultrafast xuv pulseinteracts with a simple atom, helium, in the presence ofa moderately strong ir field which may be either shorteror longer in duration than the xuv pulse. We have as ourgoal formulating theoretical methods that can be used tocalculate the absorption and emission of xuv radiationby strongly dressed atoms even when the xuv pulses are ∗ Electronic address: [email protected] † Electronic address: [email protected] on the femtosecond time scale, and may include multiplefrequencies in a comb. In addition we want to be able tostudy the transient absorption and reshaping of radiationas it propagates through a macroscopic amount of gas.We will restrict ourselves in this study to cases where the ir laser dresses the atom without appreciably exciting it,leaving higher ir intensities for a future paper. We willalso restrict ourselves to xuv wavelengths and intensitieswhere single excitations below the first ionization thresh-old at 24.6 eV dominate the xuv absorption. Even giventhese restrictions, the ir laser has a substantial impacton the Rydberg and continuum states of the atom and,in this way, enables profound control over resonant xuv absorption [7, 11–14].In the calculations we present we will consider the sim-plest case, where there are just two radiation fields, onewhich dresses the atom and one which is absorbed andpossibly reshaped. The fundamental problem of a two-color field like this has been studied before in the con-text of x-ray absorption by neon [2, 7, 15–17], argon [18],and krypton [19] atoms. Specifically, the examination oflaser-dressed atoms led to the discovery of electromag-netically induced transparency ( eit ) for x-rays [7], bet-ter characterized as Autler-Townes splitting [12] becausethe transparency is not predominantly caused by destruc-tive interference. There have also been several studies ofhelium in the context of the two-color problem we arediscussing. It was investigated with an optical laser andthe xuv free electron laser in Hamburg [20–22], and theimpact of laser-dressing helium on the production of xuv radiation via high harmonic generation ( hhg ) was stud-ied theoretically in [23].We begin by characterizing the single atom responsein terms of the cross section for absorption of xuv ra-diation of frequency ω X . First we calculate the linear,frequency-dependent xuv absorption cross section usinga Floquet-like method (non-Hermitian perturbation the-ory – nhpt ) that treats the xuv field as a monochromaticsource. This method has been extensively tested in thecontext of x-ray absorption on the 1 s → p resonanceof laser-dressed neon, alluded to above [7, 19]. Next, weoutline a method using direct integration of the time-dependent Schr¨odinger equation ( tdse ) which achievesessentially the same goal using pulses of finite duration.The cross section is extracted by projecting out the initialstate from the final state wave function. The two meth-ods, nhpt and the tdse -projection method, are shownto agree when the xuv pulse bandwidth is very small.The tdse -projection approach is, however, potentiallymore flexible in dealing with situations where the dress-ing laser couples many states of the atom. We find thatthe tdse -projection method can be used with a reason-able amount of effort to study laser-dressed absorptionover a wide range of xuv frequencies and dressing wave-lengths. As an example of the usefulness of the method,we show representative results for several dressing wave-lengths between 0.4 and 2.0 µm .We next extend the treatment of the xuv interactionto deal with cases where the atomic response varies as afunction of time or frequency in a non-trivial way. Thiscould for instance be because the ir dressing pulse is soshort that non-adiabatic effects cause the cross sectionto vary substantially over the bandwidth of the pulse.Another interesting situation is when the ir pulse is sostrong that multiphoton processes cause the atom to ex-change energy with the light field over a large range offrequencies in many different orders of nonlinearity, sothat it is no longer practical to (artificially) separate thelinear/non-linear absorption from the driven linear/non-linear emission. We therefore develop a time-frequencyapproach to the atomic response, based on the time-dependent energy exchange between the atom and thelight fields. Our method is similar in spirit to the treat-ment provided by for instance Tannor [24] and Pollardand Mathies [25] with the important difference that weare not separating the atomic response into different lin-ear and non-linear orders but keeping everything in onefrequency dependent response function. We find thatwhen we use long ( ∼
30 fs) xuv pulses we get good agree-ment between linear absorption cross sections calculatedusing the time-frequency and the tdse -projection crosssection. Having obtained this good agreement over arange of frequencies and dressing laser intensities gives usconfidence that we can calculate the full time-frequencyresponse of the dressed atom.Finally we show how this time-frequency approach isconsistent with our solution of the coupled Maxwell waveequation ( mwe ) and the tdse . This allows for a gener-alized, ab-initio description of linear and non-linear ab-sorption, emission, and phase matching in a macroscopicmedium. We apply this formalism to studying the prop- agation of an xuv pulse in a macroscopic helium gasdressed by a moderately intense 11 fs, 1600 nm laserpulse. We find that the xuv pulse, which is resonant withthe 1 s → p transition in the undressed atom, undergoesrich temporal absorption and reshaping dynamics.The paper is structured as follows. In Sec. II, wefirst discuss the three formalisms for calculating absorp-tion cross sections of laser-dressed atoms. In Sec. III wepresent our framework for the macroscopic calculations.Then we use the methods to study laser-dressed helium;computational details are given in Sec. IV and resultsare presented in Sec. V. We end the paper with a briefconclusion in Sec. VI. II. SINGLE ATOM RESPONSE
This section contains three derivations of the one-photon absorption cross section for xuv light interact-ing with an atom in the presence of a long wavelengthdressing field. All three formalisms are based on the sin-gle active electron ( sae ) approximation, and in all caseswe use linearly polarized fields where the ir and xuv polarization vectors are parallel. We use atomic unitsthrough-out this section [26]. A. Non-Hermitian Rayleigh-Schr¨odingerperturbation theory
Our nhpt treatment of dressed xuv absorption is dis-cussed in detail in references [18, 19, 27]. Here we pro-vide a brief account to highlight the essential steps in thederivation and to facilitate a discussion of the other twoformalisms.In the nhpt formalism, the one-photon xuv absorptioncross section follows from σ = 2 Γ I J X , (1)where J X is the constant xuv photon flux of a continuouswave xuv light source [19], and Γ I is the transition ratefrom the initial state to Rydberg orbitals or the contin-uum. The factor of 2 accounts for the number of electronsin the atomic orbital which is used as the initial state | I i .To determine Γ I with nhpt the full Hamiltonian of anatom in two-color light ˆ H = ˆ H + ˆ H is decomposed intoa strongly interacting part ˆ H = ˆ H AT + ˆ H EM , L + ˆ H I , L +ˆ H EM , X that contains the atomic electronic structure ˆ H AT in Hartree-Fock-Slater approximation [28, 29]. The inter-action with light is expressed in terms of nonrelativisticquantum electrodynamics [18, 19, 27]; the free ir laserand xuv fields are ˆ H EM , L and ˆ H EM , X , respectively, andthe interaction of the atomic electrons with the laser fieldis ˆ H I , L . The weak interaction with the xuv light is rep-resented by ˆ H = ˆ H I , X [19].Next we represent ˆ H in a complex-symmetric directproduct basis of electronic states—without the initialstate | I i —and photonic number states. In doing so, weassume that the initial state and its energy E I are notnoticeably influenced by the laser dressing. The matrixrepresentation of ˆ H , H ( m )0 ~c ( m ) F = E ( m ) F ~c ( m ) F , (2)is diagonalized, yielding eigenvectors ~c ( m ) F which rep-resent the expansion coefficients of new laser-dressedstates | F ( m ) i for eigenvalues E ( m ) F [19]. Here, m is themagnetic quantum number that is conserved for linearlypolarized light.When the Hamiltonian ˆ H is represented in the newbasis of laser-dressed states [Eq. (2)], the excitationor ionization of a ground-state electron of an atomdue to xuv photoabsorption is described as a reso-nance in the spectrum of the non-Hermitian, complex-symmetric representation of the Hamiltonian in the basis {| I i , | F ( m ) i | ∀ F, m } [30–32]. The complex energy of theresonance state that | I i becomes due to the coupling toexcited states and the continuum via xuv light is usuallyrefered to as the Siegert energy [30, 33] and satisfies E res = E R − i Γ I / . (3)The real part of the resonance energy is E R , and Γ I stands for the transition rate from the ground state toa laser-dressed Rydberg orbital or the laser-dressed con-tinuum [Eq. (2)] via photoabsorption. We determinethe Siegert energy [Eq. (3)] of the initial state | I i insecond-order non-Hermitian perturbation theory. Thetotal transition rate out of | I i is given byΓ I = 2 Im (cid:20)X m,F h I | ˆ H | F ( m ) i h F ( m ) | ˆ H | I i E ( m ) F − E I (cid:21) (4)and the absorption cross section is finally obtained fromEqs. (4) and (1) as: σ ( ω X ) = 8 π α ω X Im (cid:20)X m,F ( D ( m ) F ) E ( m ) F − E I − ω X (cid:21) . (5)Here α denotes the fine-structure constant and D ( m ) F is acomplex-scaled transition dipole matrix element betweenthe initial state | I i and the F th laser-dressed atomicstate with projection quantum number m [19]. B. Projection treatment of xuv absorption
As an alternative to the treatment above, we can ob-tain the linear absorption cross section by a direct so-lution of the tdse in the sae approximation [34]. Thecross section is extracted by projecting the final statewave function obtained at the end of a finite pulse ontothe initial wave function. As such, we avoid calculatingthe dressed states directly, making explicit use of onlythe laser-free initial and final states. To simplify the treatment of finite duration pulseswhen using the projection method, we replace thequantum electrodynamic treatment of xuv radiation inSec. II A, by a semiclassical description of light [35]. Webegin by choosing the vector potential of the xuv lightof carrier frequency ω X to be ~ A X ( t ) = − E X ( t ) ω X sin( ω X t ) ~e x . (6)The electric field of the xuv light field is then given byderivative with respect to time, ~ E X ( t ) = − ∂ ~ A X ( t ) /∂t : ~ E X ( t ) = (cid:20) E X ( t ) cos( ω X t ) + 1 ω X ∂ E X ( t ) ∂t sin( ω X t ) (cid:21) ~e x . (7)Here, E X ( t ) = p παI X ( t ) is the envelope of the xuv pulse and I X ( t ) is its cycle-averaged intensity. Our spec-ification of the vector potential in Eq. 6 ensures that theintegrated electric field and the vector potential at theend of the pulse, A ( t f ), are zero when E X ( t ) is zero atthe initial and final times. It leads to the second term onthe right hand side of Eq. 7 which is a small correctionof order ∆ ω X /ω X near the center of the pulse, for pulseswith a bandwidth of ∆ ω X . By ensuring that A ( t f ) = 0we obtain results which are independent of the electro-magnetic gauge.For a Gaussian envelope pulse with a full width athalf maximum ( fwhm ) duration of τ X , the bandwidth ofthe pulse is given by ∆ ω X = 4 ln 2 /τ X . In our calcula-tions, we first specify ∆ ω X and this dictates the value of τ X . The Gaussian envelope is then approximated by atrigonometric pulse [36]: I X ( t ) = I X , cos n (cid:16) πtT n (cid:17) θ (cid:16) T n − | t | (cid:17) ≡ I X , g n ( t ) , (8)with an integer n > θ function [37].The total pulse duration is defined as T n = π τ X − n , (9)The envelope (8) converges rapidly to a Gaussian func-tion in the limit lim n →∞ g n ( t ) → exp (cid:16) − tτ X ) (cid:17) . Usingthe approximative function (8) instead of a true Gaus-sian function has the advantage that it goes to zero on afinite support, which allows us to satisfy the requirement A ( t f ) = 0 exactly.In the tdse -projection formalism we also need to dressthe atom with a laser field with frequency ω L . We do thisby using a laser field of the form ~ E L ( t ) = E L ( t ) ~e L sin( ω L t ) . (10)The envelope function E L ( t ) is now a trapezoidal pulsewith a linear ramp of one optical cycle at each end anda flat section that completely spans over the xuv pulse.The pulse contains an integer number of laser cycles, sowe again obtain zero vector potential at the end of thedressing pulse. We assume this field is too weak to exciteor ionize the atom on its own, an assumption that we canexplicitly check by running the calculation once withoutthe xuv field.To calculate the cross section for absorption we beginwith the atom in its ground state | ψ I i at time t anduse the grid-based methods of reference [34] to propagatethe wave function forward in time until the end of thecombined xuv and dressing pulse at time t f . At this timewe calculate the probability that the atom has remainedin its ground state P I ( t f ) by projecting the final wavepacket | ψ ( t f ) i onto the initial wave packet: P I ( t f ) = |h ψ I | ψ ( t f ) i| . (11)Given P I ( ∞ ) = P I ( t f ) from the tdse calculation, weobtain the probability that the atom is excited or ion-ized from 1 − P I ( t f ). Because we are dealing with aone-photon absorption process where we assume the in-tensity is well below saturation, a linear relation be-tween xuv absorption rate and xuv photon flux holds:Γ( t ) = σ ( ω X ) J X ( ω X , t ). We use this assumption totransform the probability to absorb an xuv photon intoan expression for the cross section:2[1 − P I ( ∞ )] = σ ( ω X ) ∞ Z −∞ J X ( ω X , t ) d t . (12)This is equivalent to the steady state expression inEq. (1): The factor 2 again stems from the two electronsin the spatial orbital I which contribute equally.The underlying assumption in Eq. (12) is that we cancalculate the absorption cross section σ ( ω X ) for a smallrange of frequencies ∆ ω X around ω X by calculating theresponse of the atom to a pulse of bandwidth ∆ ω X . ForEq. (12) to be meaningful, the cross section needs to beapproximately constant over the bandwidth ω X of thepulse. For a low-bandwidth pulse, we can further use therelation I X ( t ) ≈ ω X J X ( ω X , t ) between photon flux andintensity. Then, the time-integral on the left-hand sidecan be solved analytically for the pulse shape (7). In thisway, we find the xuv absorption cross sections σ ( ω X )from Eq. (12) by dividing the probability to excite anatom out of the ground state 1 − P I ( t f ) by the integralover the xuv flux.As we stated in the introduction, though we expectthat the two methods for calculating the frequency-dependent absorption cross section should agree, the tdse projection approach is potentially more flexible indealing with situations where the dressing laser couplesmany states of the atom which forces the Hamiltonianmatrix in Eq. 2 to be very large. C. Time-frequency treatment of ultrafast xuvabsorption
In this section we extend the treatment of the xuv in-teraction to deal with cases where the atomic responsevaries as a function of time or frequency in a non-trivialway. This could be because the dressing ir pulse issubstantially shorter than the xuv pulse, or when non-linear interactions would cause the atom to exchange en-ergy with multiple xuv frequency components in differ-ent non-linear orders.We start by deriving a frequency-dependent responsefunction e S ( ω ) from the time-dependent energy exchangebetween the atom and the light field. e S ( ω ) is defined sothat when integrated over all frequencies, it yields thetotal excitation probability. This includes excitation tocontinuum states, i.e. , ionization. We can then expressthe total energy gained by the atom from the light fields,∆ E , as the sum over the frequency dependent excitationprobability e S ( ω ) times the photon energy:∆ E = Z ∞−∞ ω e S ( ω ) dω. (13)To calculate the response function we use that the totalatomic energy gain can also be expressed as a sum overthe rate at which energy is gained:∆ E = Z ∞−∞ ω e S ( ω ) dω = Z ∞−∞ dEdt dt. (14)We calculate this rate directly from our one electronHamiltonian, H = H A + E ( t ) z , as: dEdt = ddt h ψ | H | ψ i = h ψ | ∂H∂t | ψ i = h z i ∂ E ∂t . (15)We note that E ( t ) is the full electric field consisting ofthe sum of the dressing laser and the xuv fields. Thismeans that we are simultaneously treating the exchangeof energy between the atom and all frequencies of thelight field. In the following we will denote h z i ( t ) by z ( t ).The time-dependent dipole moment is related to z ( t ) by d ( t ) = − z ( t ) for a single electron. We now calculate ∆ E :∆ E = Z ∞−∞ z ( t ) ∂ E ∂t dt (16)= − Z ∞ ω n ˜ z ( ω ) e E ∗ ( ω ) o dω . (17)In this derivation we have used that both z ( t ) and E ( t )are real functions of time so that ˜ z ( − ω ) = ˜ z ∗ ( ω ) and e E ( − ω ) = e E ∗ ( ω ). Using Eq. (14) we then have an expres-sion for the response function: e S + ( ω ) = − n ˜ z ( ω ) e E ∗ ( ω ) o ω > , (18)where the + subscript on e S + ( ω ) explicitly indicates thatwe are only integrating over positive frequencies.We calculate the dipole spectrum in the sae approx-imation ˜ d SAE ( ω ) via the time-dependent acceleration a ( t ): a ( t ) = d zdt = −h ψ ( t ) | [ H, [ H, z ]] | ψ ( t ) i , (19)The dipole spectrum is then given by ˜ d SAE ( ω ) =˜ a ( ω ) /ω , where ˜ a ( ω ) denotes the Fourier transform of a ( t ). The full (two-electron) dipole moment is ˜ d ( ω ) =2 ˜ d SAE ( ω ).In the weak- ir limit where it is meaningful to talkabout an absorption cross section, we can write thefrequency-dependent energy exchange function ω e S ( ω ) bymeans of a generalized cross section ˜ σ ( ω ) and the spectralenergy density of the electric field, ω e J ( ω ). The spectralflux e J ( ω ) is defined as [53]: e J ( ω ) = 14 παω (cid:12)(cid:12)(cid:12) e E ( ω ) (cid:12)(cid:12)(cid:12) , (20)This means that once we calculate the response function e S ( ω ), the generalized cross section is given by:˜ σ ( ω ) = 4 παω e S ( ω ) | e E ( ω ) | . (21)Inserting the response function from Eq. (18) we obtainthe cross section, now defined for both positive and neg-ative frequency components: σ ( ω ) = 8 παω Im ( ˜ d SAE ( ω ) e E ( ω ) ) . (22)This equation is the generalized, time-frequency, multi-mode equivalent of Eq. (1) which was derived for thesteady-state case.To calculate the generalized cross section in Eq. (22),and the macroscopic polarization field described in thefollowing sub-section, we multiply the time-dependentacceleration in Eq. (19) with a window function W ( t ), a W ( t ) = a ( t ) W ( t ) and calculate ˜ d SAE ( ω ) from theFourier transform of a W ( t ). In Eq. (22) we also cal-culate e E ( ω ) from W ( t ) E ( t ) for normalization purposes.The window function on the time-dependent accelera-tion is necessary in particular in those cases where the xuv light is resonant with an atomic transition. The xuv light then induces a strong coherence between theground state and the excited state which in the numer-ical calculation will go on “ringing” until long after the xuv pulse is over. This ringing does not correspond tostimulated emission or absorption of xuv radiation. Thewindow function we use is a trigonometric function asgiven in Eq. (8) and is in general chosen to have the same fwhm duration as the longer of the ir and xuv pulses.The choice of window function has some influence on thevalue of the cross section for the un-dressed atom aroundthe field-free resonances. When the atom is laser-dressed so that the xuv light is no longer absorbed as strongly,the ringing is strongly suppressed by the laser field andthe influence of the window function is very small.It is interesting to note here that for intense or few-cycle ir fields, and/or for multi-mode xuv fields, the signof the response function e S + ( ω ) (and therefore the sign ofthe generalized cross section) for a particular frequency ω in Eq. (18) can be positive or negative. When e S + ( ω )is positive the atom will predominantly absorb light ofthat frequency, and when e S + ( ω ) is negative the atom willpredominantly emit light of that frequency. This makesthe response function a powerful tool for studying thedynamics of the light-atom energy exchange, in particularin combination with a sliding time-window on the time-dependent acceleration. This would in principle allowfor the time-resolution of when different frequencies areabsorbed or emitted during a dynamical process. We willdiscuss a simple application of this in connection with themacroscopic reshaping of an xuv pulse presented in theResults section. III. MACROSCOPIC RESPONSE, INCLUDINGABSORPTION
As we will show at the end of this section, the relation-ship derived in the previous section, between the dipolespectrum driven by an arbitrary pulse and the absorp-tion cross section for the frequencies contained in thatpulse, is consistent with our general framework for theinteraction between an ultrafast, multi-color pulse anda macroscopic medium. This framework consists of thecoupled solutions of the mwe and the tdse for all fre-quencies ω of the electric field ˜ E ( ω ) of the multi-colorpulse. We will express all quantities in SI units in thissection. In a frame that moves at the speed of light, andin the slowly evolving wave approximation which workswell even for few-femtosecond pulses [38], the mwe takesthe following form: ∇ ⊥ ˜ E ( ω ) + 2 iωc ∂ ˜ E ( ω ) ∂z = − ω ǫ c ( ˜ P ( ω ) + ˜ P ion ( ω )) . (23)The electric field ˜ E ( ω ) and the source terms ˜ P ( ω ) and˜ P ion ( ω ) are also functions of the cylindrical coordinates r and z . We solve this equation by space-marching throughthe helium gas, at each plane z in the propagation direc-tion calculating the response terms ˜ P ( ω ) and ˜ P ion ( ω ) vianumerical integration of the tdse , and then using themto propagate to the next plane in z . The macroscopicpolarization field ˜ P ( ω ) is calculated from two times theone-electron single atom dipole moment ˜ d SAE ( ω ):˜ P ( ω ) = 2 ρ ˜ d SAE ( ω ) = 2 ρeω √ π Z ∞−∞ a ( t ) W ( t )e iωt d t, (24)where ρ is the atomic density, a ( t ) is the time-dependentacceleration calculated as described in Sec. II C. As thedriving field for the tdse calculation we use the evolvingelectric field E ( t ) at the plane z . This means that ˜ P ( ω ) ingeneral includes both the linear and nonlinear responseof the atom to the multicolor field. The term ˜ P ion ( ω ) isdue to the space- and time-dependent free-electron con-tribution to the refractive index and is also calculatedwithin the sae-tdse , see [39]. This term is very small inthe cases considered in this paper and we will ignore ithereafter.By calculating the source terms in each z -plane andusing them to propagate to the next z -plane, we are cou-pling both the linear and non-linear response generatedin one step back into the full electric field so that it cancontribute to the driving electric field in the next step. Inmuch of the work described in the literature, see for in-stance [38–42], the non-linear response is separated fromthe linear response, and the propagation of the newlygenerated radiation (via nonlinear processes) is separatedfrom the propagation of the driving field. Absorptionand dispersion of different frequency components of thelight fields are then added separately, typically using tab-ulated, frequency-dependent values. It has been shownin a number of papers that such an approach offers avery complete description of both the generation of newfrequencies via nonlinear processes, and the macroscopiceffects of phase matching and ionization-driven reshap-ing of the ultrafast propagating pulse [38–42]. However,it cannot describe ultrafast or dynamical reshaping ofthe xuv pulses driven by for instance absorption, dis-persion, or laser-induced transparency. More generally,processes that are due to the combined response to thestrong dressing or driving laser field and the weaker xuv fields are not described in a self-consistent manner be-cause the generated radiation is not included into thedriving field.In the following we will argue that the approach pre-sented in this paper, which allows us to calculate thenon-linear response of the dressed atom, also allows usto describe the absorption and dispersion of the ultra-fast pulses in a self-consistent manner, to within the sae approximation. Let us first rewrite the macroscopic po-larization field as:˜ P ( ω ) = ρ ˜ d ( ω ) = ρ [Re( ˜ d ( ω )˜ E ( ω ) ) + i Im( ˜ d ( ω )˜ E ( ω ) )] ˜ E ( ω ) (25)The last term on the right is proportional to the general-ized cross section in Eq. (22). By inserting this expressioninto the MWE in Eq. (23) we get: ∇ ⊥ ˜ E ( ω )+2 i ∂ ˜ E ( ω ) ∂z = − ωǫ c ρ Re( ˜ d ( ω )˜ E ( ω ) ) ˜ E ( ω ) − iρ ˜ σ ( ω ) ˜ E ( ω ) . (26)The second term on the right hand side clearly will leadto absorption at frequency ω with absorption coefficient ρ ˜ σ ( ω ) when ˜ σ ( ω ) is positive, which is the case in theweak field limit when the atomic response is linear. Inthis linear case, the first term on the right hand sidecan likewise be interpreted as a generalized expression for the dispersion experienced in the gas medium, withthe frequency dependent correction to the refractive in-dex given by ∆˜ n ( ω ) = ρ ǫ Re( ˜ d ( ω )˜ E ( ω ) ). The strength of ourtime-dependent approach is that even when the drivingfield is strong enough to induce non-linear processes, weare able to treat all of the linear and non-linear processeswithin one time-dependent calculation, rather than ar-tificially separating processes of different non-linearitiesand assigning them a frequency- and intensity-dependentweight. IV. COMPUTATIONAL DETAILS
Computations with the time-independent theory ofSec. II A were carried out with the dreyd computer pro-gram from the fella suite [43]. The computational pa-rameters are specified in analogy to Ref. [19]. However,in this work, we do not rely on the Hartree-Fock-Slatermean-field approximation [28, 29] to describe the atomicelectronic structure. Instead, we use a pseudopotentialfor helium, constructed from the ground state Hartree-Fock potential, calculated on a very fine radial grid bystandard iterative methods [44]. We set the K edge ofhelium to the value of E = − . a us-ing 3001 finite-element functions. From its eigenfunctionswe choose, for each orbital angular momentum l , the100 functions which are lowest in energy to form atomicorbitals [19]. In doing so, we consider spherical harmonicswith up to l = 7 [37, 45]. Continuum electrons are treatedwith a smooth exterior complex scaling complex absorb-ing potential [46–48] which is parametrized with the com-plex scaling angle θ = 0 .
13 rad, a smoothness of the pathof λ = 5 a − , and an exteriority of r = 10 a [19]. Thereis only radiative decay of singly excited states of heliumwith comparatively long lifetimes to all other time scalesin the problem; therefore, we set the linewidth of a K va-cancy in helium to zero. Finally, we diagonalize the in-volved Floquet-type matrices to obtain the cross section.Without the laser field this is done exactly; when thelaser is present we use 4000 Lanczos iterations [18].Computations with the tdse -projection method ofSec. II B were carried out with a one-electron tdse solver code which is based on the algorithms describedin Ref. [34]. The same potential used above is trans-ferred to a radial grid with spacing of 0 . a and used forthe TDSE-projection and fully time-dependent computa-tions (see below). The interpolation of the Hartree-Fockpotential onto the coarse grid used for the TDSE propa-gation introduces a small error in the helium 1s ionizationpotential which we correct by slightly changing the po-tential at the first grid point [34]. The pulse shape isgiven by Eq. (7-9) with n = 6. This means that the to-tal propagation time is 4.67 τ X . Typically we use a boxof 200 au in size, with a 50 au absorbing boundary atthe outer edge [34]. The maximum angular momentumand time step size are adjusted to achieve convergence.We use ℓ m ax = 8 and 1500 steps per dressing laser cycle.In some cases where the xuv pulse was very long or the xuv wavelength was very close to the ionization thresh-old, the box size was increased to 1000 au ensure that nowave function amplitude that might reflect from the ab-sorbing boundary could interfere with amplitude excitedat a later time. We specify the bandwidth ∆ ω X of the xuv pulse instead of the fwhm duration τ X as in Eq. (8).For our (approximately) Gaussian pulse, we use the time-bandwidth product τ X ∆ ω X = 4 ln 2 to convert betweenthe two quantities [49] with ∆ ω X = 0 .
05 eV which corre-sponds to a duration of τ X = 36 . xuv light; to a very good approximation, wefind a linear relationship as should hold for a one-photonabsorption process (1). An xuv intensity of 10
10 Wcm isemployed in Figs.1-4.The calculations with the time-frequency method ofSec. II C were performed with the TDSE solver describedabove. For the calculations in Figs. 1(b) and (c) and theinset in Fig. 2 we have used ℓ m ax = 8 and approximately4000 steps per cycle of the dressing laser field. The sizeof the radial grid was 150 a (using 750 points) with a 250point absorbing boundary. The intensity envelope of the ir pulse is cos ( βt/τ IR ), where τ IR is the fwhm durationof the ir pulse and β = 2 arccos(0 . / ). The intensityenvelope of the xuv is usually chosen to be the fourthpower of the ir envelope (to be consistent with the xuv being a high order harmonic produced by the ir pulse).This gives a fwhm pulse duration for the xuv pulse ofabout half that of the ir pulse. The window functiondiscussed in Sec. II C is a Hann window with a fwhm duration very close to that of the ir pulse. The windowfunction was chosen such that the long-pulse calculationin Fig. 1(b) can be compared to those in Fig. 1(a): the fwhm bandwidth of the windowed acceleration spectrum˜ a W ( ω ) has the same 0.05 eV bandwidth as the TDSE-projection approach.For the mwe-tdse calculations in Fig. 6 we employ twotime scales. One time scale defines the spectral resolu-tion of the macroscopic, propagating, electric fields. Thistime scale typically extends to ± fwhm of thelongest of the ir and xuv pulses and contains approxi-mately 5500 time points. The other time scale is usedfor the tdse solution and extends only over the finiteduration of the longest pulse, and typically contains 6000points per ir laser cycle. The macroscopic length scalescover 160 µ m in the radial direction, with 200 grid points,40 of which contain an absorber that prevents reflectionsfrom the edge of the grid, and 1 mm in the propagationdirection, with 600 grid points. In the propagation direc-tion we only evaluate the dipole moment every 20 steps,and rescale the response to the appropriate density andphase in between, see [39] for details. The initial spatialdistribution of both the xuv and the dressing laser beamis Gaussian. The xuv beam has a confocal parameter of 10 cm and a corresponding focal diameter of 60 µ m. The1600 nm dressing pulse has a confocal parameter of 2 cmand a focal diameter of 140 µ m. This means that in thespatial dimension, the xuv beam is always overlappedwith the ir beam. The ir beam changes only marginallyduring the propagation in the helium gas. V. RESULTS AND DISCUSSIONA. Single atom absorption cross sections
The helium absorption cross section for linearly po-larized xuv light, in the absence of laser light, is dis-played in Fig. 1. In part (a), we compare results from dreyd [43] with results of the tdse -projection method[Eq. (12)]. To be able to compare these two results, wehave convoluted the dreyd cross sections with a Gaus-sian with the same bandwidth of ∆ ω X = 0 .
05 eV thatwas used in the TDSE calculation. This leads to goodagreement between the two results. We note that thepresence of a spectral bandwidth in both calculationsmeans that we are only able to resolve spectral featuresto within 0.05 eV. The peaks at 21 . . . . s → snp tran-sitions with n ∈ { , , , , . . . } . In parts (b) and (c) weshow cross sections calculated using the time-frequencyapproach leading to Eq. (22), around the 2 p and the 6 p and 7 p states. These calculations were done using an ex-tremely weak 764 nm ir pulse and harmonics 13 (b) or15 (c) of the ir frequency [54]. Harmonic 13 is resonantwith the 2 p state and harmonic 15 is in between the 6 p and the 7 p states. We show the results of using three dif-ferent xuv pulse durations (30 fs, 15 fs, and 7.5 fs). The ir pulse has twice the duration of the xuv pulse and anintensity of 10 (low enough that it does not influ-ence the cross sections). The 30 fs calculation leads to a0.05 eV bandwidth of the dipole moment around the 2 p state, after applying the time-domain window functiondiscussed in Sec. IV. The calculated cross section is inreasonably good agreement with the results in (a). Theshorter xuv pulses lead to broader absorption cross sec-tions. For the 15 fs xuv pulse the 6 p and 7 p states canstill be distinguished as separate features in the absorp-tion spectrum. Using 7.5 fs xuv pulse the cross sectioncan be calculated over a much larger frequency range,spanning both below and above the ionization threshold,and as a consequence one can no longer distinguish the6 p and 7 p states. The value of the cross section in thiscalculation is in good agreement with the value in Henke et al. [50] of 7.5 Mbarn just above threshold, as we expectwhen using pulses that span the ionization threshold.In Fig. 2 we show how the xuv cross section changeswhen the helium atom is exposed to an infrared laser fieldwith an intensity of 10
12 Wcm and a wavelength of ∼ dreyd [43] and the tdse -projection method (which haveagain both been calculated/convoluted with a 0.05 eV A b s o r p t i on c r o ss s e c t i on [ M ba r n ]
02 0.20.150.10.050468 Photon energy (eV) 24Photon energy (eV)Photon energy (eV)
FIG. 1: (Color online) The xuv absorption cross section ofa helium atom. (a) The dashed red lines were obtained with dreyd [43] the solid black lines were obtained using the tdse -projection method. (b) and (c) show close-ups of the crosssection around the 2 p (b), 6 p , and 7 p (c) states calculatedwith the time-frequency method [Eq. (22] for different xuv pulse durations. The results obtained using 30 fs, 15 fs, and7.5 fs xuv pulses are shown in black (circles), red (squares),and green (open diamonds), respectively. bandwidth), and they are found to be in good agreementover a broad energy range.The absorption of the dressed atom in Fig. 2 changessignificantly from the undressed case, although many ofthe field-free resonances can still be recognized. Thepeak due to the 2 p state has broadened and shifted tolower energy, whereas the higher np peaks are shifted tohigher energies. In addition, several new absorbing fea-tures have appeared between 21 eV and 22 eV. The insetin Fig. 2 shows cross sections for the dressed helium atomcalculated using the time-frequency approach of Eq. (22),for an xuv pulse duration of 7.5 fs. The 764 nm ir pulseduration is 15 fs and the ir peak intensity varies between10 (undressed, as shown in Fig. 1(b)) and 10
12 Wcm .The inset details the shift and broadening of the 2 p res-onance as the dressing laser intensity is increased. Wehave chosen to use the 7.5 fs xuv pulses for these cal-culations in order to be able to cover the shift of the 2 p resonance within the bandwidth that can be addressedwithin Eq. (22).The extra peaks in the 800 nm dressed-atom cross sec-tion shown in Fig. 2 result from complex multiphotoneffects and do not have a straight forward interpreta-tion. Other dressing-laser wavelengths offer more insightinto the non-linear optics driven by the two-color field.We first show two figures exploring the impact of thedressing laser wavelength on the xuv absorption crosssection. The wavelengths we have used are listed in Ta-ble I together with the corresponding photon energies.All of these wavelengths can be produced from standard
200 021234567 21 22 23 24 25Photon energy (eV) Photon energy (eV) C r o ss s e c t i on [ M ba r n ] Cross sect. [100 Mbarn]20.6 21 21.4
FIG. 2: (Color online) The xuv absorption cross section ofa helium atom dressed by an intense 800 nm ir laser pulsewith a peak intensity of 10
12 Wcm . The dashed red lines wereobtained with dreyd [43]; the cross section obtained fromthe tdse -projection is plotted with dashed black lines. Theinset shows the cross section calculated using Eq. (22), usingan xuv pulse duration of 7.5 fs ( xuv pulse duration 7.5 fs),and an 764 nm ir pulse with a duration of 15 fs and a peakintensity of 10 (solid black curve), 10
11 Wcm (dashed red),5 ×
11 Wcm (dot-dashed green), or 10
12 Wcm (dotted blue),respectively.Wavelength [nm] 400 500 620 800 1400 1600 2000Photon energy [eV] 3.10 2.48 2.00 1.55 0.89 0.78 0.62TABLE I: Correspondance between wavelenght and photonenergy for the involved laser light. Ti:Sapphire high power, short pulse laser systems via fre-quency mixing in nonlinear materials.The laser-dressed xuv absorption cross section, calcu-lated using the TDSE-projection method are displayedin Figs. 2, 3, and 4 for several laser wavelengths at anintensity of 10
12 Wcm . For 400 nm and 500 nm light, wesee only a moderate impact of the laser dressing. Theimpact is mostly on the 2 p state, as the largest dipolecoupling exists to other close-by Rydberg states. 620 nmand 800 nm light exhibit complex multiphoton effectswhich manifest in complicated multi-peak structures inthe cross sections. The dressing pulses with longer wave-lengths all induce systematic behavior. In all three cases,the single 1 s → s p transition in Fig. 1(a) (withoutdressing) is split into two lines by the laser in Fig. 3;the transitions from the 1 s orbital into higher Rydbergorbitals are replaced by a continuous, weak absorptionfeature.We would like to elucidate the origin of the double-peak feature around 21 eV in the long-wavelength seriesshown in Fig. 4. It is much simpler than the correspond-ing feature for the wavelengths in Figs. 2 and 3. To thisend, we make a Λ-type model for helium which is shownin Fig. 5. It comprises the ground state of helium andthe 1 s − p , 1 s − s excited states. The laser photonenergy is denoted by ω L whereas Γ s − s and Γ s − p are parameters for the laser-induced decay widths of therespective excited states. The overall agreement of the FIG. 3: (Color online) Laser-dressed xuv absorption crosssection of helium for 400, 500, and 620 nm laser wavelengthsat a laser intensity of 10
12 Wcm .FIG. 4: (Color online) Laser-dressed xuv absorption crosssection of helium for 1400, 1600, and 2000 nm laser wave-lengths at a laser intensity of 10
12 Wcm (red solid curves).The results of the simple three-level model of Fig. 5 are in-dicated by the dashed black curves for Γ s − p = 0 . s − s = 0 .
05 eV. model curves with the ab initio data in Fig. 4 is sat-isfactory. The reason for the success of the three-levelmodel is—as in Ref. [7]—the fact that the splitting be-tween the 2 s and 2 p Rydberg orbitals in helium is 0 .
84 eV, i.e. , the laser is almost in resonance with this transi-tion, within the laser-induced line widths, for midinfraredwavelengths [Table I]. Furthermore, the other levels ofhelium couple only weakly.The Λ-type model explains the double peaked struc-ture in Fig. 4 in terms of a splitting of the 1 s − p and1 s − s states into an Autler-Townes doublet. This fea- FIG. 5: (Color online) Λ-type three-level model for helium.The laser photon energy is ω L and the xuv photon energyis ω X . The laser-induced decay widths of the 1 s − p and1 s − s excited states are denoted by Γ s − p and Γ s − s ,respectively. ture raises the possibility that the dressing laser could beused to induce transparency to the xuv radiation tunedto the 1 s → s p transition. A similar mechanism wasfound for the suppression of resonant absorption of x raysin neon [7, 15–17], argon [18], and krypton [19] atoms andcalled eit for x rays [7]. In the next section we study theanalogous effect in helium. B. xuv pulse shaping in a macroscopic medium
In this section we present an application of the time-frequency approach to absorption in a macroscopic non-linear medium. We study how the laser induced trans-parency discussed above may be used to temporally con-trol the xuv pulse shape in a helium gas, in analogy withthe eit for x-rays discussed in [18]. In that x-ray study,the absorption was exclusively described in terms of anintensity-dependent absorption cross section, which thenin turn enforces a one-to-one mapping of the absorptionto time through the intensity. This does not allow fortruly dynamical effects. In addition, the intensities weexplore here are much lower that those used in the x-raystudy, which means that ionization of the Rydberg statesdoes not play a large role.We calculate the electric field of a combined two-color xuv - ir pulse after propagation through a 1 mm long he-lium gas jet with a density of 1 . × cm − (6 mbarat room temperature). We solve the coupled mwe-tdse in the form of Eq. (23) as described in section III, seealso [39]. The initial xuv pulse has a wavelength of58.7 nm (21.1 eV, resonant with the 1 s → s p tran-sition), a pulse duration of 61 fs and a peak intensity of10 W/cm . The 1600 nm dressing pulse has a peak in-tensity of 10 W/cm . We have used different ir pulsedurations between 122 fs and 11 fs. We have checkedthat the reshaping discussed below is no different whenwe use a higher xuv intensity of 10 W/cm .Fig. 6 shows the radially integrated spectrum (a) and0 (a)(b) R ad i a ll y i n t eg r a t ed s pe c t r u m ( a r b . un i t s ) -100 -50 0 50 100Time (fs) Initial Final, un-dressed R ad i a ll y i n t eg r a t ed y i e l d ( a r b . un i t s ) Final, IR 10 W/cm
12 2
Initial Final, un-dressedFinal, IR 10 W/cm
12 2
Final, 2 x
10 W/cm
12 2
FIG. 6: (Color online). ir -assisted xuv absorption in 1 mmlong macroscopic helium gas with a density of 4 × cm − .The initial 61 fs xuv pulse is resonant with the 2 p state ofthe undressed helium atom. We show the xuv spectrum in(a) and time profile in (b), both before (solid lines) and af-ter propagation. Final profiles at the end of the un-dressedmedium are shown with dotted red lines, final profiles at theend of the medium dressed by an 11 fs, 1600 nm ir pulse witha peak intensity of 10 W/cm are shown with dashed bluelines. In (b) we also show the final time profile when the in-tensity of the dressing pulse is 2 × W/cm (thin greenline). time profile (b) of the xuv pulse before and after prop-agation through the helium gas. When the atoms areundressed the xuv radiation is strongly depleted via theresonant absorption, as is shown by the dotted red lines[55]. The absorption length at this atomic density is lessthan 0.1 mm. During the first few absorption lengthsthe xuv yield decreases exponentially. The large disper-sion across the resonance, and to a lesser extent the fre-quency dependence of the absorption cross section, sub-sequently leads to reshaping of the depleted beam uponfurther propagation in the gas. This causes the double-peaked shape of the spectrum emerging at the end of themedium. The time profile of the final xuv field is cor-respondingly irregular, as seen by the red dotted line inFig. 6(b). We then apply a 1600 nm, 10 W/cm dressing pulsewhich is much longer than the xuv pulse (123 fs vs 61 fs).This means that the xuv pulse encounters a sample ofstrongly dressed atoms which are no longer resonant withthe xuv energy, see Fig. 4, and the gas is therefore trans-parent to the xuv light. The spectrum of the final xuv pulse is nearly indistinguishable from the initial spectrumand is not shown in Fig. 6(a). The final xuv pulse shapeis also nearly identical to the initial pulse shape exceptfor a 1.6 fs delay caused by the different group veloci-ties of the ir and the xuv pulses (also not shown in thefigure).Next, we apply an 11 fs dressing ir pulse which is sub-stantially shorter than the xuv pulse. This means thatthe dressing pulse turns on and off within the fwhm du-ration of the xuv pulse, thereby strongly coupling the2 s and 2 p states in a dynamical manner. The final spec-tral and temporal xuv profiles at the end of the mediumare shown with dashed blue lines in Fig. 6(a) and (b).The xuv time profile at the end of the medium is dom-inated by an approximately 10 fs pulse, superimposedon a much weaker longer pulse, and the corresponding xuv spectrum has broad shoulders at frequencies sub-stantially beyond the initial xuv bandwidth. We notethat the final xuv pulse is not symmetric around timezero and that in particular, the short sub-pulse is delayedby approximately 4 fs from the center of the dressing ir pulse. We attribute this to the complicated absorp-tion and emission dynamics driven by the two-color pulseas explored in more detail next.The time-dependent acceleration driven by the initialtwo-color pulse is shown in Fig. 7, solid black line. Weare showing the envelope of the acceleration to avoid thefast oscillations at the resonance frequency. On the ris-ing edge of the xuv pulse, before the ir pulse turns on,this acceleration represents the absorption of the xuv light via population transfer to the 2p state. This meansthat when the ir pulse arrives there is already popula-tion in the 2p state which will couple strongly to the 2sstate. This causes the suppression of the accelerationwhich starts around t = − t = − t = +4 fs, we find that they have opposite signs. Thismeans that whereas the dipole response early in the pulseand up until t ≈ − xuv light, the peak in Fig. 7 around t = +4 fs correspondsto emission of xuv radiation. We interpret this behavioras coming from Rabi-like oscillations of the excited statepopulation between the 2s and the 2p states, driven bythe ir field. The emission happens when the populationreturns to the 2p state while the strong ir field is stillon. This interpretation would predict that at higher ir intensity the Rabi cycling should be faster. We indeedfind that if we increase the ir intensity to 3 × W/cm ,the time-dependent acceleration has two revivals withinthe ir pulse duration, both corresponding to emission1 | a ( t ) | ( a r b . un i t s ) FIG. 7: (Color online). Single atom time-dependent accelera-tion driven by the initial xuv - ir pulse in Fig. 6. The durationof the two pulses are 61 fs and 11 fs, respectively, and the xuv intensity is 10 W/cm . In solid black line we show the resultof using an ir intensity of 1 × W/cm , in dashed red linethe ir intensity is 3 × W/cm . (dashed red line in Fig. 7). The Rabi oscillation periodfor resonant population transfer between the 2s and the2p states is approximately 10 fs (6 fs) for a constant in-tensity of 1 × W/cm (3 × W/cm ). This is ingood agreement with the time scale of the oscillations inthe acceleration seen in Fig. 7, especially considering thatthe ir intensity is changing rapidly between t = −
10 fsand t = 10 fs.Finally, returning to Fig. 6(b) and the xuv pulse thatemerges from the helium gas dressed by the 11 fs ir pulse,we can now attribute the delay of the short xuv pulse tothe excited state dynamics in the strongly dressed atomicgas. Fig. 8 shows the evolution of the xuv time profileshown in Fig. 6, as a function of propagation distance.The complicated atomic response shown in Fig. 7, whichincludes absorption at early times and emission around t = 4 fs, is reflected in the propagating xuv electric field.After the first few hundred microns of propagation the xuv time profile has been substantially depleted on therising edge and is dominated by a much shorter pulsepeaking shortly after t = 0. We note that this in turn willchange the atomic response from that plotted in Fig. 7since the dressing ir pulse and the xuv pulse are thenmore comparable in duration. VI. CONCLUSION
In this paper, we have investigated the response oflaser-dressed helium atoms to xuv radiation, within the sae approximation. In particular, we have focused onthe calculation of absorption cross sections and their ap-plication to absorption in a macroscopic medium.First, we introduced a time-independent method basedon nhpt . The interaction with the xuv light was treatedin terms of a one-photon process while a Floquet-like ap-proximation was used to describe the impact of the dress- -60 -40 -20 6040200Time (fs)00.51.01.52.02.53.0 X U V t i m e p r o f il e ( a r b . un i t s ) FIG. 8: (Color online). Evolution of xuv time profile duringpropagation through the macroscopic helium gas. The timeprofiles at different propagation distances have been displacedvertically, starting with z = 0 at the bottom to z = 1 . z = 0 .
01 mm. ing laser. Second, we devised a time-dependent methodto compute the cross section with using a direct inte-gration of the tdse and projection of the final wavepacket onto the initial atomic state. We showed thatthe projection-based approach, which was implementedusing finite pulses, yields the same results compared withthe time-independent results. Third, we presented a ver-satile time-frequency approach to evaluating an atomicresponse function which can be used even when the dress-ing laser pulse is so short that it introduces transient ef-fects, or in cases where the atom exchanges energy withmultiple frequency components of the multi-color lightfield. We showed that this method, when used to calcu-late linear absorption cross sections, agrees with the firsttwo. Finally, we showed that this third approach can beimplemented in a combined mwe-tdse solver to describeabsorption and ultrafast pulse reshaping in a macroscopicmedium.We used the TSDE-projection method to investigatethe dependence of the xuv absorption cross section onthe wavelength of the laser dressing at 10
12 Wcm laserintensity. We found complex multiphoton physics for800 nm light and shorter wavelengths. For longer, midin-frared wavelengths, however, we showed that the impactof the laser dressing in the 1 s → p transition in heliumcan be described in terms of a Λ-type three-level model2previously used to describe eit for x-rays [7]. As in theearlier study, the transparency in helium is caused pre-dominantly by Autler-Townes splitting brought about bythe strong one photon coupling induced by the dressinglaser, in this case between the 2 p and 2 s states. We in-vestigated the macroscopic reshaping of an ultrafast xuv pulse resonant with this transition, for the case when thetransparency is induced by an ir pulse which is substan-tially shorter than the xuv pulse. This means that theabsorption properties of the helium atom change dynami-cally during its interaction with the xuv light. We foundrich temporal reshaping dynamics in which the atomsboth absorb and subsequently emit the xuv radiationin a process strongly influenced by Rabi oscillations be-tween the 2 s and 2 p states. This leads to an xuv pulseemerging from the macroscopic medium which has beenshortened from 60 fs to 10 fs and whose peak intensity hasincreased by approximately a factor of two. The increasein the peak intensity, which results from the coherentpopulation pumped into the 2 p state before the dressingpulse arrives, is a truly dynamical effect which cannotbe described in terms of a single absorption cross sectiononly.Our results open up several possibilities for futureresearch on ultrafast quantum optics. The control of xuv absorption by laser dressing of helium enables,for example, the possibility for post-generation ultrafastshaping of xuv pulses [7]. And although we have con-fined ourselves in this work to the case of one xuv fieldwith a dressing laser, it is a straightforward extensionof the method to treat multiple xuv frequencies some ofwhich some could be resonant with dressed atomic tran- sitions, and the complex interferences that would resultfrom this [1, 3]. Also, we have used moderately strongIR fields that do not cause any excitation on their own,but the time-dependent treatment is not limited to theseintensities. Using higher IR intensities will lead to gen-eration of harmonics in the non-linear medium. Har-monics with energies below and slightly above the ion-ization threshold, for which absorption dynamics playsthe largest role, have recently attracted a lot of atten-tion, for instance as a source of vuv and xuv frequencycombs [51] or as a seed for free electron lasers [52]. Thetreatment of these processes necessitates using methodswe have developed in this paper, since the atomic ab-sorption and emission properties will be changing on anultrafast time scale. Acknowledgments
This work was supported by the National ScienceFoundation under grant Nos. PHY-0449235 (CB andJT), PHY-0701372 (KS), and PHY-1019071 (MG). Wealso acknowledge support from the PULSE Instituteat Stanford University (MG and KS) and from theMarie Curie International Reintegration Grant (CB)within the 7 th [1] P. Johnsson, J. Mauritsson, T. Remetter, A. L’Huillier,and K. J. Schafer, Phys. Rev. Lett. , 233001 (2007).[2] T. E. Glover, M. P. Hertlein, S. H. Southworth, T. K.Allison, J. van Tilborg, E. P. Kanter, B. Kr¨assig, H. R.Varma, B. Rude, R. Santra, et al., Nature Phys. , 69(2009).[3] P. Ranitovic, X. M. Tong, B. Gramkow, S. De, B. De-Paola, K. P. Singh, W. Cao, M. Magrakvelidze, D. Ray,I. Bocharova, et al., New Journal of Physics , 013008(2010).[4] E. Goulielmakis, Z.-H. Loh, A. Wirth, R. Santra,N. Rohringer, V. S. Yakovlev, S. Zherebtsov, T. Pfeifer,A. M. Azzeer, M. F. Kling, et al., Nature , 739 (2010).[5] M. J., R. T., S. M., K. Kluender, A. L’Huillier, K. J.Schafer, O. Ghafur, F. Kelkensberg, W. Siu, P. Johnsson,et al., Physical Review Letters , 053001 (2010).[6] M. Wickenhauser, J. Burgd¨orfer, F. Krausz, andM. Drescher, Phys. Rev. Lett. , 023002 (2005).[7] C. Buth, R. Santra, and L. Young, Phys. Rev. Lett. ,253001 (2007), arXiv:0705.3615.[8] T. Pfeifer, M. J. Abel, P. M. Nagel, A. Jullien, Z.-H. Loh,M. J. Bell, D. M. Neumark, and S. R. Leone, ChemicalPhysics Letters , 11 (2008).[9] M. Drescher, M. Hentschel, R. Kienberger, M. Uiber- acker, V. S. Yakovlev, A. Scrinzi, T. Westerwalbesloh,U. Kleineberg, U. Heinzmann, and F. Krausz, Nature , 803 (2002).[10] T. Remetter, P. Johnsson, J. Mauritsson, K. Varj, Y. Ni,F. L´epine, E. Gustafsson, M. Kling, J. Khan, R. L´opez-Martens, et al., Nature Physics , 323 (2006).[11] K.-J. Boller, A. Imamoˇglu, and S. E. Harris, Phys. Rev.Lett. , 2593 (1991).[12] M. Fleischhauer, A. Imamoˇglu, and J. P. Marangos, Rev.Mod. Phys. , 633 (2005).[13] C. Buth and K. J. Schafer, Phys. Rev. A , 033410(2009), arXiv:0905.3756.[14] M. Swoboda, T. Fordell, K. Kl¨under, J. M. Dahlstr¨om,M. Miranda, C. Buth, K. J. Schafer, J. Mauritsson,A. L’Huillier, and M. Gisselbrecht, Phys. Rev. Lett. ,103003 (2010), arXiv:1002.2550.[15] R. Santra, C. Buth, E. R. Peterson, R. W. Dunford, E. P.Kanter, B. Kr¨assig, S. H. Southworth, and L. Young, J.Phys.: Conf. Ser. , 012052 (2007), arXiv:0712.2556.[16] C. Buth, R. Santra, and L. Young, Rev. Mex. F´ıs. S ,59 (2010), arXiv:0805.2619.[17] L. Young, C. Buth, R. W. Dunford, P. J. Ho, E. P. Kan-ter, B. Kr¨assig, E. R. Peterson, N. Rohringer, R. Santra,and S. H. Southworth, Rev. Mex. F´ıs. S , 11 (2010), arXiv:0809.3537.[18] C. Buth and R. Santra, Phys. Rev. A , 043409 (2008),arXiv:0809.3249.[19] C. Buth and R. Santra, Phys. Rev. A , 033412 (2007),arXiv:physics/0611122.[20] T. Laarmann, A. R. B. de Castro, P. G¨urtler, W. Laasch,J. Schulz, H. Wabnitz, and T. M¨oller, Phys. Rev. A ,023409 (2005).[21] D. Charalambidis, P. Tzallas, N. A. Papadogiannis,L. A. A. Nikolopoulos, E. P. Benis, and G. D. Tsakiris,Phys. Rev. A , 037401 (2006).[22] T. Laarmann, A. R. B. de Castro, P. G¨urtler, W. Laasch,J. Schulz, H. Wabnitz, and T. M¨oller, Phys. Rev. A ,037402 (2006).[23] E. Cormier and P. Lambropoulos, J. Phys. B , 3095(1997).[24] D. J. Tannor, Introduction to Quantum Mechanics: Atime-dependent perspective (University Science Books,Sausalito, CA, 2007).[25] W. T. Pollard and M. R. A., Annual Reviews of PhysicalChemistry , 497 (1992).[26] A. Szabo and N. S. Ostlund, Modern quantum chem-istry: Introduction to advanced electronic structure the-ory (McGraw-Hill, New York, 1989), 1st, revised ed.,ISBN 0-486-69186-1.[27] C. Buth, R. Santra, and L. S. Cederbaum, Phys. Rev. A , 032505 (2004), arXiv:physics/0401081.[28] J. C. Slater, Phys. Rev. , 385 (1951).[29] J. C. Slater and K. H. Johnson, Phys. Rev. B , 844(1972).[30] V. I. Kukulin, V. M. Krasnopol’sky, and J. Hor´aˇcek, The-ory of resonances (Kluwer, Dordrecht, 1989), ISBN 90-277-2364-8.[31] N. Moiseyev, Phys. Rep. , 211 (1998).[32] R. Santra and L. S. Cederbaum, Phys. Rep. , 1(2002).[33] A. J. F. Siegert, Phys. Rev. , 750 (1939).[34] K. J. Schafer, in Strong field laser physics , edited byT. Brabec (Springer, New York, 2008), vol. 134 of
Springer series in optical sciences , pp. 111–145, ISBN978-0-378-40077-8.[35] D. P. Craig and T. Thirunamachandran,
Molecular quan-tum electrodynamics (Academic Press, London, 1984),ISBN 0-486-40214-2.[36] I. Barth and C. Lasser, J. Phys. B , 235101 (2009).[37] G. B. Arfken and H. J. Weber, Mathematical methods forphysicists (Elsevier Academic Press, New York, 2005),sixth ed.[38] T. Brabec and F. Krausz, Rev. Mod. Phys. , 545 (2000).[39] M. B. Gaarde, J. L. Tate, and K. J. Schafer, J. Phys. B , 132001 (2008).[40] A. L’Huillier, P. Balcou, S. Candel, K. J. Schafer, andK. C. Kulander, Phys. Rev. A , 2778 (1992).[41] E. Priori, G. Cerullo, M. Nisoli, S. Stagira, S. D. Sil-vestri, P. Villoresi, L. Poletto, P. Ceccherini, C. Altucci,R. Bruzzese, et al., Phys. Rev. A , 063801 (2000).[42] V. S. Yakovlev, M. Y. Ivanov, and F. Krausz, Opt. Ex-press , 15351 (2007).[43] C. Buth and R. Santra, fella – the free electronlaser atomic, molecular, and optical physics programpackage , Argonne National Laboratory, Argonne,Illinois, USA (2008), version 1.3.0, with contribu-tions by Mark Baertschy, Kevin Christ, Chris H.Greene, Hans-Dieter Meyer, and Thomas Sommerfeld,chemistry.anl.gov/Fundamental Interactions/FELLA main.shtml.[44] S. E. Koonin, Computational Physics (Westview Press,Boulder, CO, 1998).[45] E. Merzbacher,
Quantum mechanics (John Wiley &Sons, New York, 1998), 3rd ed., ISBN 0-471-88702-1.[46] N. Moiseyev, J. Phys. B , 1431 (1998).[47] U. V. Riss and H.-D. Meyer, J. Phys. B , 2279 (1998).[48] H. O. Karlsson, J. Chem. Phys. , 9366 (1998).[49] J.-C. Diels and W. Rudolph, Ultrashort laser pulse phe-nomena , Optics and photonics series (Academic Press,Amsterdam, 2006), 2nd ed., ISBN 0-12-215493-2.[50] B. Henke, E. Gullikson, and J. Davis, Atomic Data andNuclear Data Tables , 181 (1993).[51] D. C. Yost, T. R. Schibli, J. Ye, J. L. Tate, J. Hostet-ter, K. J. Schafer, and M. B. Gaarde, Nat. Phys. , 815(2009).[52] X. He, M. Miranda, J. Schwenke, O. Guilbaud, T. Ru-chon, C. Heyl, E. Georgiadou, R. Rakowski, A. Persson,M. B. Gaarde, et al., Phys. Rev. A , 063829 (2009).[53] Our Fourier transformation convention is E ( t ) = √ π ∞ R −∞ e E ( ω ) e − iωt d ω and e E ( ω ) = √ π ∞ R −∞ E ( t ) e iωt d t .[54] The ir field is included as a technical convenience for per-forming the low- ir intensity and high- ir intensity calcula-tions consistently, both for the time-frequency approachand for the mwe-tdse approach.[55] We note that a weak 61 fs ir pulse with a peak intensityof 10 W/cm2