Enhancing and controlling single-atom high-harmonic generation spectra: a time-dependent density-functional scheme
EEnhancing and controlling single-atomhigh-harmonic generation spectra: atime-dependent density-functionalscheme
Alberto Castro , , ∗ , Angel Rubio , and E. K. U. Gross ARAID Foundation, Edificio CEEI, Mar´ıa de Luna s/n, 50018 Zaragoza Spain Institute for Biocomputation and Physics of Complex Systems (BIFI), and Zaragoza Centerfor Advanced Modelling (ZCAM), University of Zaragoza, 50018 Zaragoza, Spain Nano-Bio Spectroscopy Group and ETSF Scientific Development Centre, Departamento deF´ısica de Materiales, Centro de F´ısica de Materiales CSIC-UPV/EHU-MPC and DIPC,Universidad del Pa´ıs Vasco UPV/EHU, E-20018 San Sebasti´an, Spain Max-Planck Institut f¨ur Mikrostrukturphysik, Weinberg 2, D-06120 Halle, Germany ∗ acastro@bifi.es Abstract:
High harmonic generation (HHG) provides a flexible frameworkfor the development of coherent light sources in the extreme-ultravioletand soft x-ray regimes. However it suffers from low conversion efficienciesas the control of the HHG spectral and temporal characteristics requiresmanipulating electron trajectories on attosecond time scale. The phasematching mechanism has been employed to selectively enhance specificquantum paths leading to HHG. A few important fundamental questionsremain open, among those how much of the enhancement can be achievedby the single-emitter and what is the role of correlations (or the electronicstructure) in the selectivity and control of HHG generation. Here we addressthose questions by examining computationally the possibility of optimizingthe HHG spectrum of isolated Hydrogen and Helium atoms by shapingthe slowly varying envelope of a 800 nm, 200-cycles long laser pulse.The spectra are computed with a fully quantum mechanical description,by explicitly computing the time-dependent dipole moment of the systemsusing a first-principles time-dependent density-functional approach (exactfor the case of H). The sought optimization corresponds to the selectiveenhancement of single harmonics, which we find to be significant. Thisselectivity is entirely due to the single atom response, and not due to anypropagation or phase-matching effect. In fact, this single-emitter enhance-ment adds to the phase-matching techniques to achieving even larger HHGenhancement factors. Moreover, we see that the electronic correlation playsa role in the determining the degree of optimization that can be obtained. © 2018 Optical Society of America
OCIS codes: (000.0000) General.
References and links
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1. Introduction
At sufficiently high intensities, matter no longer reacts linearly to light, and may re-emit atinteger multiples ( harmonics ) of the frequency of the incoming source [1]. According to per-turbation theory, the intensity of the harmonics decreases exponentially with their order. How-ever, the spectrum of atoms and molecules exposed to very intense, typically infrared, laserpulses was found to present unexpectedly high harmonics [2, 3], and its shape was observed tohave a plateau extending non-perturbatively over many orders of magnitude – a process knownas high harmonic generation (HHG) [4, 5]. The light emitted in this manner is coherent andmay reach the extreme ultraviolet and soft X-ray frequency regime opening the path towardsthe coherent manipulation and control of matter at its natural time scale. These properties canbe of paramount importance for many technological and scientific purposes in ultrafast sci-nce – most notably, the generation of attosecond pulse trains or single isolated attosecondpulses, or the external seeding of free-electron lasers [6, 7, 8]. These advances allow to followthe electron dynamics [9]. Unsurprisingly, a big effort has been devoted to first understand-ing the underlying physics, and then to controlling and fine-tuning the efficiency and spectralcharacteristics of the harmonic radiation. The latter can be done by modifying the non-linearmedium, or by post-processing the signal with filters, gratings, etc. However, one advantageousalternative is to modify the characteristics of the parent pulse, which obviously will modifythe spectral outcome. The most obvious manner of doing this is by systematically varying thedefining parameters of this parent pulse [10, 11, 12]. However, the current availability of ad-vanced pulse shaping tools [13], together with the development of closed-loop quantum controltechniques [14], provides a superior optimization alternative [15]. In this manner, the success-ful selective enhancement of harmonic orders could be achieved when using a hollow fibercontainer for the generating medium [16, 17, 18]. Gas jet ( free focusing ) geometries were alsoemployed [19, 20, 21], but although some degree of control could be achieved (for example, theextension of the cut-off frequency), the very selective order enhancement or depletion obtainedwith the hollow fibers was not observed. This fact seems to imply that this type of selectiveenhancement cannot be explained from the single-atom response only; instead, the propaga-tion effects present in the capillary set-up apparently play a fundamental role. In this context.quasiphase matching (QPM) approach is commonly used to achieve independent phase controlbetween multiple high-harmonics [22].However, a full interpretation of the optimisation mechanisms can only be achieved withtheoretical input, for which purpose one may utilise quantum simulations in combination withthe theoretical branch of quantum optimal control [23, 14] (QOCT). Recently, Schaefer andKosloff [24, 25] have addressed this task, showing the possibility of enhancing the emissionat desired frequencies for simple few level systems and one-dimensional one.electron system.Here we address, by first principles simulations based on time-dependent density functionaltheory, the role of many electron interactions in the high harmonic generation, and provide com-pelling evidence that a single-atom HHG emission can be enhanced by few orders of magnitudein a controlled manner, with standard laser shaping techniques available in many experimentallabs.The three-step model successfully describes the key features of HHG [26, 27, 28], at leastqualitatively. It combines a quantum description of the ionisation and recombination of theelectrons, with a classical description of the intermediate electronic propagation. Lewenstein etal [29] developed an approximate, mostly analytical, quantum description based on the strongfield approximation (SFA): it neglects the contribution of excited bound states, the depletion ofthe ground state, and considers the continuum electrons to be free of the influence of the parention. A more precise approach consists of propagating Schr¨odinger’s equation [30, 31, 32], anexpensive method that quickly becomes prohibitive as we increase the number of electrons.For one-electron problems the approach is perfectly feasible, and this fact has encouraged theuse of the single active electron approximation (SAE), which assumes that only one electron issignificantly disturbed by the field, and its evolution may be computed on the combination ofthe laser field and the potential originating by the parent ion.This single electron picture is commonly used to describe recollision processes and HHGin atoms and relies on the fact that under HHG conditions there is one electron being ionised.However this doesn’t imply the other electrons do not play a role. There is indeed no formaljustification for the use of the SAE and in fact, many-body effects have been shown recently toplay an important role in HHG providing an explanation of why heavier atoms emit strongerHHG than lighter ones [33] and the giant enhancement of He HHG at 100 ev [34]. However, theSAE has been successful in explaining a few features of the HHG spectra such as the spectralutoff, the phase structure of the spectrum and the prediction of the generation of attosecondpulses.In spite of all those experimental and theoretical efforts, it is clear that the topic of selec-tive HHG generation deserves further microscopical analysis, and in this work, we explorethe optimisation possibilities of one and two electron systems (the Hydrogen and the Heliumatoms), isolating the single atom response, so that we can learn how much selectivity in theHHG spectrum can be obtained from isolated atoms that can nicely complement QPM schemesin enhancing further the HHG selective emission. For this purpose, we employ a global optimi-sation scheme that acts on the envelope of the generating pulse, maintaining the fundamentalfrequency and minimising undesired ionisation (and for molecules also dissociation) processes.For the case of Helium, we report results obtained both with the single active electron approxi-mation, and with time-dependent density-functional theory (TDDFT) [35, 36], in order to assessthe influence of the electron-electron interaction in the optimisation process. As many-electroneffects may be relevant, TDDFT appears as the ideal framework to capture them in the HHGspectra (see for example Ref. [37]) as it combines a very good compromise between accuracyand computational efficiency. The present optimisation scheme has been implemented in thefirst-principles code octopus [38, 39], that allows the treatment of more complex molecularand extended systems. However for the purposes of the present work, it is better to stay at thesimplest level of one and two electron systems. Larger electronic systems would offer a widerrange of possibilities for HHG enhancement.
2. Theory
Within the dipole approximation and in the length gauge, the experimentally measured har-monic spectrum can be theoretically approximated by the following formula: H ( ω ) = | (cid:90) T d t d d t (cid:104) ˆ (cid:126) µ (cid:105) ( t ) e − i ω t | , (1)i.e. the power spectrum of the second derivative of the expectaction value of the dipole momentˆ (cid:126) µ = − ∑ Ni = ˆ (cid:126) r i (although see Ref. [40] for a discussion on the pertinence of using, alternatively,the first derivative or the dipole moment itself). This object is given by:d d t (cid:104) ˆ (cid:126) µ (cid:105) ( t ) = (cid:104) N ∑ i = ∇ v ( ˆ (cid:126) r i ) (cid:105) + N ε ( t ) (cid:126) π , (2)where v is the (static) ionic potential, N is the number of electrons, ε ( t ) is the laser pulse electricfield, and (cid:126) π is the polarization vector (see the Methods appendix below for some extra detailsabout the theory). Note that this expression can be read as both the acceleration of the electronicsystem, and as the corresponding back-reaction of the nucleus (or nuclear center of mass, if weare dealing with a molecule). This is not surprising since the electromagnetic emission mustbe related with a charge acceleration. The expression corresponds, except for the mass factor,with the classical force acting on the nucleus, considered as a point particle. We will thereforerewrite Eq. (1) as: H ( ω ) = | (cid:126) f ( ω ) | . (3)where (cid:126) f ( ω ) = (cid:82) T (cid:126) f ( t ) e − i ω t ) is the Fourier transform of: (cid:126) f ( t ) = (cid:104) N ∑ i = ∇ v ( ˆ (cid:126) r i ) (cid:105) + N ε ( t ) (cid:126) π . (4)rom a TDDFT perspective, the use of this force functional is convenient since it can beexplicitly written as a density functional: (cid:126) f ( t ) = (cid:90) d r n ( (cid:126) r , t ) ∇ v ( (cid:126) r ) + N ε ( t ) (cid:126) π . (5)Usually, the electric field ε ( t ) is factorised into a sinusoidal function determining the funda-mental frequency ω , and an envelope function f that determines the overall laser-pulse shape: ε ( t ) = f ( t ) sin ( ω t ) . (6)This factorisation – and the concept of a fundamental frequency – is meaningful for long andquasi-monochromatic pulses, but as the technology has reached the optical period limit, it hasstarted to lose its relevance. Nevertheless, the existence of a fundamental frequency is implicitwhen speaking of harmonics, which are defined as radiation at integer multiples of preciselythat frequency. These will only be well defined if the envelope function is smooth compared tothe sinusoidal term, i.e. its frequencies are much lower than ω .Therefore, in this work, we investigate the possibility of manipulating the envelope function f , leaving the sinusoidal factor sin ( ω t ) unchanged, in order to influence the shape of the HHGspectrum. This manipulation cannot be unconstrained, as the envelope must be composed offrequencies much lower than ω . Moreover, we have searched for solutions that preserve thefluence or total integrated energy of the pulse: I = (cid:90) d t ε ( t ) . (7)This type of requirement of a specific structure for the solution field (in terms of frequencies,fluence, etc.) can be respected following essentially two routes: by imposing penalties on unde-sired features of the pulses in the definition of the optimising function, or by constraining fromthe beginning the search space. This latter option can be achieved by establishing a parametri-sation of the control field (in this case, the envelope) that enforces the required condition, andis the route that we have chosen for this work. The search for the optimum is in this mannerperformed in the space of parameters that determine the control field; the remaining necessaryingredient is the definition of a merit function that encodes the physical requirements. More-over, the assumption of low frequencies for f implies that the spectrum of ε is concentratedaround ω . Therefore, the N ε ( t ) (cid:126) π term in Eqs. (2), (4), and (5) does not contribute to the HHGspectrum in the region we are interested in and in the following we will safely ignore it.We have shown how the HHG spectrum may be explicitly computed solely in terms of thesystem electronic density n ( (cid:126) r , t ) . For systems with more than one electron, this fact is conve-nient since it allows to use time-dependent density-functional theory [35, 36] (TDDFT) (seeMethods). One can substitute the propagation of the real interacting system by the propagationof a system of fictitious non-interacting electrons whose density is however identical to that ofthe real one, despite the fact that its wave function is a single Slater determinant.Hereafter, we will restrict the discussion to one and two-electron systems, the extension tosystems with larger number of electrons is straightforward in the TDDFT framework. The one-electron case obviously does not need a TDDFT treatment, although it may be treated as suchby considering one single occupied orbital. For such one-orbital problem, the exchange andcorrelation potential must cancel the Hartree term: v xc [ n ]( (cid:126) r , t ) = − v H [ n ]( (cid:126) r , t ) , (8)so that the resulting equation reduces to the initial Schr¨odinger equation. For two-electron sys-tems, we use the exact-exchange approximation (EXX) to the xc term, which for this two-lectron case amounts to setting: v xc [ n ]( (cid:126) r , t ) = − v H [ n ]( (cid:126) r , t ) , (9)Note that in this form TDDFT is identical to time-dependent Hartree-Fock that provides a fooddescription of the non-linear properties of two-electron systems except for the description ofcharge-transfer excitations (see for example Ref. [41]).The electric field amplitude will be determined by the specification of a set of M parame-ters u , . . . , u M ≡ u : ε ( t ) = ε [ u ]( t ) . The evolution of the TDKS system is in consequence alsogoverned by the choice of parameters u , i.e. the orbitals and density are functionals of theparameters: u → ϕ [ u ] , u → n [ u ] . We may then use the tools of QOCT to find the set u that max-imizes a given target function G , defined in terms of a functional of the density of the system,i.e.: G [ u ] = ˜ F [ n [ u ]] . (10)This functional ˜ F is designed to favour the desired behaviour of the system (in this case, acertain form of the HHG spectrum, to be detailed below). Note that it is defined in terms of thesystem density, and not in terms of the full many-body wave function. This definition ensuresthat the substitution of the real by the Kohn-Sham system in the optimization entails no furtherapproximation. The theory must however be developed in terms of a functional of the Kohn-Sham orbitals, which can be easily defined as: F [ ϕ ] = ˜ F [ µϕ ∗ ϕ ] , (11)where µ is the occupation of the orbital, i.e. one or two for one- or two-electron calculations,respectively.We must now choose a form for F in such a way that its maximization leads to the desiredHHG optimization, namely the selective increase of one harmonic peak – that should leavethe neighboring ones as low as possible. There is substantial liberty to design F , and it is notevident what functional form should lead to better results. F [ ϕ ] = ∑ k α k H [ ϕ ]( k ω ) , (12)where α k takes a positive value for the harmonic to be enhanced, and negative values for theones that we wish to reduce. However, this choice proved to be problematic, since the modu-lation of the source signal with the envelope function leads to displacements, sometimes sub-stantial, of the harmonic peaks with respect to the precise integer multiples k ω . A generaldefinition that solves this problem (and that includes the previous one as a particular case), is: F [ ϕ ] = (cid:90) d ωα ( ω ) H [ ϕ ]( ω ) = (cid:90) d ωα ( ω ) | (cid:126) f [ ϕ ]( ω ) | , (13)where we have made explicit the fact that both H and (cid:126) f , defined in Eqs. (1) and (4) are function-als of the time-dependent evolution for the system. The function α permits to establish somefinite window around each harmonic peak k ω , that will be positive for the harmonic orders thatwe want to enhance, and negative for the ones that we want to reduce. Finally, a third optionis to seek for the maximum of the spectrum in these frequency windows around the harmonicorders, i.e.: F [ ϕ ] = ∑ k α k max ω ∈ [ k ω − β , k ω + β ] { log H [ ϕ ]( ω ) } , (14)where the real number β determines the size of the window.nce the function G has been defined (through the definition of the target functional F ),it remains to use some optimization algorithm to find the optimal u set. There are numerousoptions, and we may divide them on two groups, depending on whether or not they requirethe computation of the gradient of G – in addition of the computation of the function itself.The methods that employ the gradient are of course more efficient, as long as this gradientcan itself be computed efficiently. The simplest scheme is steepest descents, but one can alsouse conjugate gradients or, in our case, the Broyden-Fletcher-Goldfarb-Shanno (GFBS) quasi-Newton method.For the function G , the gradient is given by [42]: ∇ G [ u ] = (cid:90) T d t ∇ ε [ u ]( t ) Im (cid:104) χ [ u ]( t ) | ˆ (cid:126) r · ˆ π | ϕ [ u ]( t ) (cid:105) . (15)This expression uses an auxiliary orbital χ [ u ] defined by the following equations of motion:i ∂∂ t χ [ u ]( (cid:126) r , t ) = − ∇ χ [ u ]( (cid:126) r , t ) + v ∗ KS [ n [ u ]]( (cid:126) r , t ) χ [ u ]( (cid:126) r , t )+ ˆ K [ ϕ [ u ]( t )] χ [ u ]( (cid:126) r , t ) − i δ F [ ϕ [ u ]] δ ϕ ∗ [ u ]( (cid:126) r , t ) , (16) χ [ u ]( (cid:126) r , T ) = . (17)The operator ˆ K [ ϕ [ u ][ t ]] is defined as:ˆ K [ ϕ [ u ]( t )] χ [ u ]( (cid:126) r , t ) = − ϕ [ u ]( (cid:126) r , t ) Im (cid:90) d r (cid:48) χ ∗ [ u ]( (cid:126) r (cid:48) , t ) f Hxc [ n [ u ]( t )]( (cid:126) r ,(cid:126) r (cid:48) ) ϕ [ u ]( (cid:126) r (cid:48) , t ) , (18)where f Hxc is the so-called kernel of the Kohn-Sham Hamiltonian, which, for our two-electroncase treated within the EXX approximation, is given by: f Hxc [ n ]( (cid:126) r ,(cid:126) r (cid:48) ) =
12 1 | (cid:126) r − (cid:126) r (cid:48) | , and is null forthe one-electron case (zeroing the full ˆ K operator).The functional derivative of F , needed in Eq. (16), for the HHG target defined in Eq. (13), is: δ F δ ϕ ∗ ( (cid:126) r , t ) = (cid:126) g [ ϕ ]( t ) · ∇ v ( (cid:126) r ) ϕ ( (cid:126) r , t ) , (19)where (cid:126) g [ ϕ ]( t ) = µ (cid:82) d ω α ( ω ) Re (cid:104) (cid:126) f [ ϕ ]( ω ) e − i ω t (cid:105) .However, we cannot compute this functional derivative for the target defined in Eq. (14) dueto the presence of the “max” function, at least in a simple and efficient manner. In consequence,when using this target definition we could not make use of any of the optimization algorithmsthat make use of the gradient, and turned to the gradient-free NEWUOA algorithm [43], whichis a very efficient scheme for optimization problems with a moderate number of degrees offreedom, such as the ones treated here.In fact, for the optimizations attempted in this work, we observed numerically that the targetof Eq. (14) provided much better results than the target of Eq. (13), and therefore we will onlyshow below gradient-free optimizations; in a forthcoming publication, where the target is theHHG cut-off extension, we will present gradient-based optimizations based on a target of thetype given in Eq. (13).Therefore, it remains to specify the set of parameters u that determine the envelope of theelectric fields. The requirements are: (i) the envelope should have a given cut-off frequency; (ii)the field should smoothly approach zero at the end points of the propagation time interval; (iii)the total integral of the field should be zero, and (iv) the fluence or total integrated intensity ofhe pulse should have a constant pre-defined value. This last condition is merely a choice, andnot a physical constraint that experimentalists face.The first step to parametrize the applied time-dependent electroc field ε ( t ) in order to enforceall these constraints is to expand the envelope in a Fourier series: f ( t ) = L ∑ i = f i g i ( t ) , (20)where g i ( t ) = (cid:113) T cos (cid:0) π T it (cid:1) ( i = , . . . , L ) (cid:113) T sin (cid:0) π T ( i − L ) t (cid:1) ( i = L + , . . . , L ) (21)This series fixes the maximum possible ( cut-off ) frequency to π T L . Note that it explicitly omitsthe zero-frequency term, which is a desired restriction, in order to fulfill: (cid:82) T d t f ( t ) = f i coefficients is not yet, however, our parameter space, sincewe still want to enforce the conditions f ( ) = f ( T ) =
0, and fix the fluence: I = (cid:82) d t ε ( t ) = I .As discussed in Ref. [44], these conditions reduce the degrees of freedom from 2 L to 2 L − u , . . . u L − are finally the hyperspherical angles that characterize a sphereof constant fluence, determining the Fourier coeffiencients: f i = f i [ u ] .In all the OCT calculations to be shown below we have fixed the wavelength of the fun-damental frequency ω to 800 nm, a very common value used in laboratories equipped witha Ti:sapphire source. The total pulse duration is fixed to 200 cycles, T =
200 2 π / ω , whichcorresponds to 533 fs approximately. The envelope function f ( t ) is then restricted to have fre-quencies no larger than ω /
60. The fluence [Eq. (7)] is then fixed to a value (around 5.0 a.u.)that ensures a sufficiently non-linear response of both the Hydrogen and Helium atoms, whilenot causing a substantial ionization. Fixing the fluence does not imply fixing the peak intensity;however the simultaneous existence of a maximum frequency puts a limit on it; in practice, thepeak intensities observed in the optimal pulses are in the range of 5 10 - 10 W/cm .The optimization are started from randomly generated sets of parameters u . Since the pro-cedure finds local maxima, we have performed several searches for each case, choosing after-wards the best among them. In order to have some “reference” to compare the optimal run to,we define a reference pulse as: ε ref ( t ) = ε cos (cid:18) π t − TT (cid:19) cos ( ω t ) , (22)i.e. a cosinoidal envelope that peaks at t = T / ε , chosen to fulfill the fluencecondition.
3. Results
The calculated HHG spectrum emitted by the Hydrogen and Helium atoms, irradiated by thereference pulse, is depicted in Fig. 1. Note that there is a range of harmonics with comparableintensities forming a plateau (9 to 19 in H and 15 to 21 in He). Because of this, we have selectedthat range (shaded in the plot) to perform the selective optimisations. The range is displayed,this time with a linear y axis scale, in the inset. For the case of He we show the EXX andSAE results. As in He two electrons populate the only orbital in a spin-compensated configu-ration, the SAE approximation, in this case, consists in neglecting the interaction between theelectrons during the action of the field, freezing the potential to its ground state shape. In thisadiabatic-DFT context, it amounts to ignoring the time-evolution of the Hartree, exchange and ig. 1. HHG spectrum of the Hydrogen (top) and He (bottom) atoms, with the referencepulse of Eqn. (22. For the case of the He HHG spectra we show two results: one solvingthe TDDFT equations using the EXX functional (green) and the other solving the single-active-electron (SAE) (red) equation, commonly used by the strong-field community. Tomake more clear the comparison between EXX and SAE we shifted the SAE spectra by0.5 ω in the x-axis. The shaded area contains the harmonics of interest. This area is alsodisplayed in the inset, with a linear y -axis scale. orrelation potentials, and is useful to gauge the relevance that correlations may have on theHHG optimisation.Let’s discuss first the case of optimising the HHG spectra of H. We used the target givenby Eq. (14) to optimise the odd orders from the 9th to 19th. To enhance the 9th harmonic, forexample, we set α =
5, and α = α = α = α = α = − α k are zero). Inthis manner, the sum of all coefficients is zero, avoiding any improvement of the merit functiondue to a mere overall reduction or increase of the spectrum. The results are displayed in Fig. 2.From top to botton, in the left panels, the spectra produced by the optimal fields for the 19th,17th, . . . , 9th harmonic. In the right panels, the optimal fields themselves; their envelopes inreal time, as well as their power spectrum.The resulting fields produce considerably higher harmonic outputs than the unshaped, refer-ence field. To quantify this point we introduced an enhancement factor that is displayed in eachplot, defined as: κ j = max ω ∈ [ k ω − β , k ω + β ] { H [ ϕ ]( ω ) } H ref ( j ω ) , (23)where H ref is the spectrum obtained with the reference field, and H the one obtained with theoptimal field (the computation of the max function is not needed for the former, because due tothe regularity of its envelope function, H ref always peaks at the precise integer multiples j ω ).This enhancement factor greatly vary from case to case (i.e. it is 6 for j =
13, and 208 for j = y -scale; they are scaled in each case to thevalue of the maximum of the plot.We turn now our attention to the case of the Helium atom, that contains two electrons. Theinteraction between these is treated here with TDDFT, within the EXX approximation. As inthe previous case we performed optimisations based on the target given by Eq. (14), now for theorders 15th to 21st, fixing the coefficients α k in an analogous manner. The results are displayedin Fig. 3. From top to bottom, in the left panels, the spectra produced by the optimal fields forthe 21st, 19th, 17th, and 15th harmonic. In the right panels, the optimal fields themselves.The enhancement factors achieved are quite large, and as in the case of Hydrogen, rather dif-ferent from case to case. This rather large enhancement of the wanted harmonic is not accompa-nied by a full depletion of the neighbouring ones – in fact, they are also increased. This partialselectivity is also similar to the Hydrogen results. To quantify the role of electron-electroninteractions we show in the same Fig. 3 the SAE results (red curve). Qualitatively, the SAEresults are not very different to the ones obtained with the EXX functional, in terms of intensityenhancements and selectivity. The fact that the calculated optimal fields and the spectra are dif-ferent for both EXX and SAE illustrate not only the intrinsic non-linearity of the optimisationalgorithms and the rather large number of possible local maxima, but also the fact that electroninteraction does play a role in the generation and optimisation of harmonics. Indeed, by look-ing in more detail to the results shown in Fig. 3 for the 15th and 19th harmonic optimisation,we see that EXX with respect to SAE provides a better selectivity and harmonic enhancement,measured by the height of the desired harmonic and the quenched of the neighbouring ones.Therefore electron correlation seems to play a role in the optimisation of harmonics. This fact,together with the common knowledge that heavier noble gases emit stronger HHG radiationthat light ones (whereas the SAE that predicts similar spectra) [45, 46, 47] support our findingsabout the limitations of the SAE approximation and the role of electron interactions. In factwe can expect larger enhancement factors to be reached by applying the present optimisationtechniques to heavier atomic/molecular systems. ig. 2. Optimized HHG spectra (left panels), and corresponding optimal fields (right pan-els), for the Hydrogen atom case. The optimal fields are plotted in the time domain (onlythe envelope function f ( t ) is shown), and in the frequency domain. The HHG spectra areshown in a linear scale, normalized in each case up the value of the maximun value. Theenhancement factor defined in Eq. (23) is also shown.ig. 3. Optimized HHG spectra (left panels), and corresponding optimal fields (right pan-els), for the Helium atom case. As in Fig. (1) we show in green the results within TDDFTusing the EXX functional and in red the ones using the SAE approximation. The optimalfields are plotted in the time domain (only the envelope function f ( t ) is shown), and inthe frequency domain. The HHG spectra are shown in a linear scale, normalized in eachcase up the value of the maximun value. The enhancement factor defined in Eq. (23) is alsoshown. . Conclusion In conclusion, we have investigated, by theoretical means, the possibility of tuning the shape ofthe HHG spectrum of the Hydrogen and Helium atoms by shaping the slowly varying envelopeof a 800 nm, 200-cycles long laser pulses. For this purpose, we have optimised a functionaldesigned to enhance selected harmonics. The allowed modifications of the pulse are very con-strained, since we enforce a maximum envelope frequency no larger than 1/60 of the fundamen-tal frequency. This means very slowly varying envelopes. However, the picture that emerges ofour analysis is that these relatively small modifications produce strong variations of the spec-tra, allowing for significative increases of the harmonic intensities. These enhancements arenot fully selective, since the neighbouring harmonics also increase, but to a lesser extent. Theoutcome depends of the precise definition of the target functional, which is a topic to be inves-tigated further. There is ample freedom to choose this object, and a different option may yieldbetter selectivity – while perhaps reducing the total enhancement, or vice-versa.The spectra have been computed with a fully quantum mechanical description, by explicitlycomputing the time-dependent dipole moment of the systems. The results presented here corre-spond to the single-atom response – we have not propagated Maxwell’s equations in a atomicgaseous medium. Therefore, this work demonstrates the relevance of the single atom responsefor HHG and how this single-atom response is significantly altered by the envelope of the laserpulse, even for the small modifications allowed in our scheme. We have shown that few ordersof magnitude HHG enhancement factor can be reached at the single-atom level. Thus, if thisfact is combined with the phase matching method used for HHG generation we would be ableto reach much higher global harmonic enhancement in atomic and molecular gases. Moreover,our results illustrate the role of electron-electron interactions in this optimisation and controlof HHG. This can be qualitatively rationalised in terms of the larger Hilbert space spanned bythe interacting system as compare to the simplest single-active electron scheme (or any othernon-interacting electron approach).
Appendix: Methods