The high-frequency limit of spectroscopy

We consider a quantum-mechanical system, initially in its ground-state, exposed to a time-6 dependent potential pulse, with a slowly varying envelope and a carrier frequency ω 0 . By working 7 out a rigorous solution of the time-dependent Schr¨odinger equation in the high-ω 0 limit, we show 8 that the linear response is completely suppressed after the switch-oﬀ of the pulse. We show, at the 9 same time, that to the lowest order in ω − 1 0 , observables are given in terms of the linear density 10 response function χ ( r , r ′ , ω ), despite the problem’s inherent nonlinearity. We propose a new spec-11 troscopic technique based on these ﬁndings, which we name the Nonlinear High-Frequency Pulsed 12 Spectroscopy (NLHFPS). An analysis of the jellium slab and jellium sphere models reveals very high 13 surface sensitivity of NLHFPS, which produces a richer excitation spectrum than accessible within 14 the linear regime. Combining the advantages of the extraordinary surface sensitivity, the absence 15 of constraints by the conventional dipole selection rules, and the clarity of theoretical interpreta-16 tion utilizing the linear response time-dependent density functional theory, NLHFPS emerges as a 17 powerful characterization method in nanoscience and nanotechnology. 18

In optical spectroscopy, we use light-based probes to study the investigated systems' structure and composition.A significant part of spectroscopy involves linear effects, such as when light is absorbed or scattered off material targets, allowing their imaging and characterization, teaching us almost solely about dipole-allowed transitions.Nonlinear spectroscopy methods go beyond this limitation, studying otherwise hidden or dark transitions applicable to a large variety of systems and processes [1].Examples of nonlinear spectroscopy include the second-order harmonic generation (SHG) approach, used to study interfaces and adsorbed molecules and serves as high-resolution optical microscopy in biological systems [2], multiphoton excitation fluorescence (MPEF) used for imaging in biological systems, as well as various Raman scattering methods [3].This Article introduces what can arguably be considered the ultimate nonlinear spectroscopy: a rigorous high-frequency limit of molecular spectroscopy leading to new spectroscopic techniques applicable to molecules and materials.The new spectroscopy relies on the second-order derivatives of the system's nuclear electric potential.As a result, the spectroscopy is sensitive to surfaces and film structure.It can also lead to new types of spectra in molecules.
Nonlinear optical (NLO) methods form a group of powerful experimental techniques for characterizing molecular, biomolecular, condensed matter systems, including materials of reduced dimensionality.The use of nonlinear spectroscopies, especially in surface science and nanoscale systems, is growing due to their high interfacial sensitivity.However, nonlinear spectroscopy is often more challenging to interpret since its descriptions involve much more sophisticated theoretical techniques as compared to the linear case [4].
where the unperturbed Hamiltonian is N and v ext (r) being the number of electrons and the ex- W (r i , t). (3) We assume that the time-dependence in the pulse poten- where n(r) = where χ(r, r ′ , ω) is the density-density response function 111 of the interacting electron system.To leading order in 112 ω −1 0 we find the density oscillations proportional to ω −2 0 113 while the total energy absorbed by the system is propor-114 tional to ω −4 0 , as can be seen in the final expression: In the dipole case equations ( 5), ( 8) and ( 9) are replaced with 117 and where and Ê0 is the unit vector along E 0 .
121 Notably, the pairs of equations, Eq. ( 8) and Eq. ( 12) for the density response and Eq. ( 9) and Eq. ( 13) for where ω is a frequency in the optical range and σ ≫ ω −1 , 135 we expect to see a large response or energy absorption at 136 frequencies ω that correspond to twice the excitation en-137 ergies.On the other hand, when the pulse is extremely  Hydrogen atom.We now investigate how the highfrequency limit is approached as the frequency increases, by a calculation for an exactly solvable system, namely the hydrogen atom.Assuming the atom, initially in its ground state, is subjected to the doubly modulated Gaussian pulse with a spherically symmetric quadruple potential W (r, t) cos ω 0 t = W 0 r 2 e −(t/σ) 2 cos ωt cos ω 0 t, we numerically time-propagate the Scrödinger equation (1).Upon the end of the pulse, we look at the populations of the excited states, plot them in Fig. 1 versus the enveloping function frequency ω (the second frequency), and compare with the asymptotic limit.The latter, according to Eqs. ( 5), (7), and ( 16) is given by where φ n,s (r) are the hydrogenic s-orbitals and ǫ n are the corresponding eigenenergies, and we have restricted the comparison to the transitions to the s-states only.
Similarly, in the dipole case, we propagate the system   16), from the ground-state of the hydrogen atom to some of its excited s-states.The black thick line is the asymptotic limit of Eq. ( 17).Spectra at finite frequencies are obtained by the numerical propagation of the TD Schrödinger equation.
The parameters of the pulse used were σ = 50 a.u. and W0 = 0.125 a.u.
under the potential For hydrogen atom where Y lm (θ, φ) are spherical harmonics.Therefore, from the ground state, transitions to s-and d-states are possible only, with the corresponding amplitudes where, in Eq. ( 20), we can further simplify with account of the pulse of Eq. ( 18).We observe the principal difference between the change of the occupancies of the s-and p-levels: while the latter gets much more (approximately three orders of magnitude) populated in the middle of the pulse duration, it gives the electron away upon the pulse end, while the former keeps the accepted electron with finite probability.This type of behavior is characteristic for spherically symmetric systems in the high-frequency regime, which is in accordance with our asymptotic theory.This is the linear response that dominates the s → p transition at the time of the pulse duration, which is gone upon the pulse's extinction.In particular, we conclude that the usual dipole selection rules do not hold in this process.With the use of Eqs. ( 20) and (21), in Fig. 4 we compare the excitation and ionization processes' probabilities for hydrogen atom initially in its ground-state and exposed to the Gaussian pulse.We conclude that the ionization is dominant for short pulses, in which case electron is stripped off by a sudden impact, while for longer pulses transitions to excited bound states become preferential.We also note that transitions to the d-states play insignificant role compared with those to the s-states.
Jellium slab.We proceed by considering a slab of the thickness d with the positive constant background charge density n + = ( 4 3 πr 3 s ) −1 , where r s is the 3D density parameter.Within the Kohn-Sham (KS) density-functional theory (DFT) [15] and using the local density approximation (LDA), we calculate the ground-state KS band structure and electron density.To this system, we apply the doubly modulated dipole pulse of Eq. ( 18), and we use our theory to determine the total energy transfer from the pulse to the slab in the high carrier frequency regime.The problem being one-dimensional, the operator in Eq. ( 14) reduces to the Laplacian, and we have by where Θ(x) is the Heaviside's step-function.Resulting  15), ( 18) (σ = 500 a.u.) at asymptotically large frequency ω0 as a function of 2ω, as obtained through Eq. ( 13).Right: absorption per unit time from the monochromatic field of the frequency ω in the linear response regime.Two slabs of the thicknesses d = 25 and 40 a.u. and the density parameter rs = 5 are considered.x-axes are scaled to the bulk plasma energy ωp = 4.2 eV.Parameters used correspond to the jellium model of solid potassium.

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We have demonstrated that, to the leading order in the z-z 8. Comparison of the standard linear-response energy absorption spectrum of the ethylene molecule to that of the highfrequency response in the dipole approximation (Eq.13).x − x refers to the linear response auto-correlation function with the electric field along the x-axis, while ∇xvext − ∇xvext stands for the auto-correlation function in the high-frequency nonlinear regime, and the similarly for two other directions.
Using Eq. (5) together with the spectral representation of the many-body interacting density response function where η is a positive infinitesimal, we obtain Eq. ( 8) for the density oscillations in the system upon the end of the pulse.
For the total energy absorbed by the system from the pulse we can write which, with the use of the completeness of the basis set and then, by Eqs. ( 5) and (34), finally written in the form 516 of Eq. ( 9).
where E(t) is the uniform in space electric field, then [∇W (r, t)] 2 = E 2 (t), and Eq. ( 33) integrates to zero identically.With the use of Eqs. ( 27), (29), and the commutator relations and noting that in Eq. ( 27) the sum of the 4th, 5th, and 519 6th terms on the RHS evaluates to zero, as it can be 520 directly verified, we arrive at the dipole counterpart of 521 Eq. ( 33) For the linearly polarized field 523 we then immediately arrive at Eqs. ( 11)- (13).
Excitation probability, upon the end of the pulse of Eq. ( 16), from the ground-state of the hydrogen atom to some of its excited s-states.The black thick line is the asymptotic limit of Eq. ( 17).Spectra at nite frequencies are obtained by the numerical propagation of the TD Schrodinger equation.The parameters of the pulse used were σ = 50 a.u. and W0 = 0:125 a.u.
Excitation probability, upon the end of the pulse of Eq. ( 18), from the ground-state of the hydrogen atom to some of its excited s-states.The black thick line is the asymptotic limit of Eq. ( 20).Spectra at nite frequencies are obtained by the numerical propagation of the TD Schrodinger equation.The parameters of the pulse used were σ = 50 a.u. and E0 = 0:125 a.u.
Comparison of the standard linear-response energy absorption spectrum of the ethylene molecule to that of the highfrequency response in the dipole approximation (Eq.13).x -x refers to the linear response auto-correlation function with the electric eld along the x-axis, while rxvext -rxvext stands for the autocorrelation function in the high-frequency nonlinear regime, and the similarly for two other directions.

N 107 F
i=1 δ(r i − r) is the electron density opiωt C(t) 2 dt (6) is the pulse envelop frequency-auto-correlation function, 106 and nd (r) = [∇W (r)] 2 .(7) Further development then shows (see Methods), that 108 the time-dependent oscillation in the electron density of 109 the system after the pulse ends are given by 110

123
the total energy absorption are identical except for: the 124 power of ω −2 0 , and the difference between F nd and F d .125 Furthermore, in these two pairs of equations we witness 126 a hybridization of linear and quadratic response quanti-127 ties: the linear density-density response function is mul-128 tiplied by the quadratic frequency envelop C 2 (ω).Such 129 hybridization may lead to interesting effects, although in-130 terpretation of the spectrum may pose a challenge.How-131 ever, in two extreme cases it is still simple to interpret the 132 spectrum.When the pulse envelop is nearly monochro-133 matic, i.e. of the form 134 3

Figures 1
FIG. 2. Excitation probability, upon the end of the pulse of183

FIG. 6 .
FIG.6.Same as Fig.5, but for slabs of the density parameter rs = 2 and the corresponding bulk plasma energy ωp = 16.7 eV (jellium model of solid aluminum).
nonlinear technique an ideal tool to study this otherwise 280 subtle type of excitation.It is instructive to note that 281 F d (z) of Eq. (22) provides, effectively, the impact mode 282 of the complementary linear response problem, which is 283 known to be favorable for MP excitation[17].In Fig.6284 (r s = 2), left panel, we also see a prominent broad peak 285 at 2ω below the BP frequency, while MP is indistinguish-286 able in the linear response spectrum in the right panel.

287
We, therefore, conclude that the corresponding excitation 288 exists at the surface of metallic aluminum, and the high-289 frequency nonlinear technique provides a unique way to 290 detect it, while the traditional method of the electron en-291 ergy loss spectroscopy (EELS) does not possess sufficient 292 sensitivity [16].The oscillating structures at 2ω > ω p 293 in Figs. 5 and on both sides from ω p in Fig. 6 differ 294 for different slab thicknesses, and they can, therefore, be 295 attributed to the interference effect between the two sur-296 faces of the slabs.Finally, the absence of the conventional 297 (dipole) surface plasmon (SP) peak at ω s = ω p / √ 2 is due 298 to the strictly normal to the surface direction of the ex-299 citing field (q = 0), in which case the amplitude of the 300 SP vanishes.301 Jellium sphere.In contrast to a slab, for a sphere, 302 the second derivative in the RHS of Eq. (14) does not 303 reduce to Laplacian and, consequently, F d (r) is not given 304 by the positive background density only.Instead, we 305 have 306 307 ground.Due to the symmetry, the density-response 308 function χ(r, r ′ , ω) splits in angular momentum into 309 χ lm (r, r ′ , ω), the latter acting separately on each har-310 monic of the externally applied potential.The problem 311 becoming one-dimensional again, we calculate χ 00 and 312 χ 20 , apply them to Eq. (23), and plug the result into 313

319
Right: absorption per unit time from the monochromatic field 320 of the frequency ω0 in the linear response regime.Positions 321 of classical Mie plasmons ω l are shown by vertical lines.Two 322 spheres of the radii R = 30 and 40 a.u. and the density 323 parameter rs = 5 are considered.results of calculations for two spheres, with 327 radii R = 30 and 40 a.u., and the density parameter 328 r s = 5, are presented, for the nonlinear ω 0 → ∞ and the 329 linear-response regimes, in the left and right panels, re-330 spectively.Within the classical electrodynamics, a sphere 331 of the Drude metal supports an infinite series of Mie 6 plasmons ω l = l/(2l + 1)ω p , l = 1, 2, . . .[18].In the 333 monochromatic linear-response (right panel of Fig. 7), we 334 observe the p-mode only of this series, red-shifted by the 335 quantum size effect.336 According to Eq. (23), energy absorption in the nonlin-337 ear ω 0 → ∞ regime (left panel of Fig. 7) originates from 338 the superposition of the s-and d-modes.As plotted ver-339 sus the second modulation frequency ω, it reveals a rich 340 spectrum of the underlying excitations.The leftmost fea-341 ture near 0.57ω p comes from the d-mode Mie plasmon ω 2 , 342 red-shifted in the quantum calculation.The dominating 343 broad peak with the maximum near 0.80ω p does not have 344 an analogue within the classical electrodynamics, and, 345 similar to the multipole plasmon modes in the case of a 346 slab, it becomes accessible with the use of the high-ω 0 347 nonlinear regime.A signature of the bulk plasmon on 348 the right shoulder of this peak can be also observed, in-349 dicating the possibility of the direct recognition of the 350 constituents of nano-particles by their bulk plasmon fre-351 quencies ω p with the use of laser pulses.The latter is, ob-352 viously, impossible in the linear-response regime.We also 353 note structures above ω p , which are due to the (dressed) 354 single-particle excitations, affected by the quantum in-355 terference.356 Finally, we consider molecular electronic spectroscopy.357 Referring back to the above-discussed very short pulse 358 spectroscopy, in the dipole interaction case, we used time-359 dependent local density approximation calculations to 360 produce linear response estimates of the high-frequency 361 energy absorption (using Eq. 13) and, in Fig. 8, com-362 pare to standard low-frequency energy absorption for 363 the ethylene molecule.The spectra's differences in the 364 two regimes are due to the dipole versus F d selection 365 rules, which further emphasizes the high-frequency spec-366 troscopy's aptitude for probing the excitations forbidden 367 in the linear regime.See Methods for details concerning 368 this calculation.369 Discussion.We have considered the excitation of an 370 arbitrary quantum-mechanical system by an externally 371 applied electric field of high-frequency ω 0 and finite du-372 ration in time.After the end of the pulse, the state of 373 the system being a superposition of the eigenstates of 374 the unperturbed Hamiltonian, the expansion of the cor-375 responding transition amplitudes in the power series in 376 ω −1 0 has been performed, with the leading terms found of 377 the order ω −4 0 for the uniform applied field (dipole case) 378 and of ω −2 0 , otherwise.

524
Derivation in the Kramers-Henneberger's ac-525 celeration frame.For an arbitrary u(t), if a function

Figure 7
Figure 7 Probability of the excitation and ionization of hydrogen atom, initially in its ground state, to s-(left) and d-(right) states, relative to the total excitation plus ionization probability, plotted versus the pulse width σ.The pulse shape is purely Gaussian C(t) = e −(t/σ) 2 .