Excitation of H{}_{2}^{+} with one-cycle laser pulses: Shaped post-laser-field electronic oscillations, generation of higher- and lower-order harmonics

# Excitation of H+2 with one-cycle laser pulses: Shaped post-laser-field electronic oscillations, generation of higher- and lower-order harmonics

\nameGuennaddi K. Paramonova, Oliver Kühnb, and Andr D. Bandraukc CONTACT: G.K. Paramonov. Email: gk.parmon@gmail.com aInstitut für Chemie, Universität Potsdam, Karl-Liebknecht Strasse 24-25, 14476 Potsdam, Germany; bInstitut für Physik, Universität Rostock, Albert-Einstein-Strasse 23-24, D-18059 Rostock, Germany; cLaboratorie de Chimie Thorique, Facult des Sciences, Universit de Sherbrooke, Sherbrooke, Qubec, J1K 2R1, Canada
###### Abstract

Non Born-Oppenheimer quantum dynamics of H excited by shaped one-cycle laser pulses linearly polarized along the molecular axis have been studied by the numerical solution of the time-dependent Schrödinger equation within a three-dimensional model, including the internuclear separation, , and the electron coordinates and . Laser carrier frequencies corresponding to the wavelengths  nm through  nm were used and the amplitudes of the pulses were chosen such that the energy of H was close to its dissociation threshold at the end of any laser pulse applied. It is shown that there exists a characteristic oscillation frequency  au (corresponding to the period of  fs and the wavelength of  nm) that manifests itself as a “carrier” frequency of temporally shaped oscillations of the time-dependent expectation values and that emerge at the ends of the laser pulses and exist on a timescale of at least 50 fs. Time-dependent expectation values and of the optically-passive degree of freedom, , demonstrate post-laser-field oscillations at two basic frequencies and . Power spectra associated with the electronic motion show higher- and lower-order harmonics with respect to the driving field.

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RESEARCH ARTICLE

ne-cycle laser pulses; post-laser-field electronic oscillations; generation of higher and lower harmonics

## 1 Introduction

Coherent oscillations of bound particles (electrons or muons) that persist after laser field excitation of a molecule  [1, 2, 3] is an interesting new phenomenon, which generalizes the well-known recollision model of Corkum [4] to the case when high-order harmonics are generated due to coherent oscillations of bound electrons or muons after the end of the laser field that induced their motion. Coherent post-laser-field oscillations of bound particles were found to exist both in ordinary (electronic) and muonic molecules, as shown by numerical propagation of the respective non-Born-Oppenheimer time-dependent Schrödinger equations [1, 2, 3].

Coherent post-laser-field electronic oscillations were first found in the heavy molecular ion T after its excitation by a UV laser pulse [1]. Subsequently [2], coherent and shaped post-laser-field oscillations of a muon were found in muonic and molecules after their excitation by super-intense attosecond soft X-ray laser pulses at the wavelength of  nm. It was found, in particular, that only odd harmonics were generated in the homonuclear molecule by the optically active degree of freedom, as suggested by the concept of inversion symmetry [5]. Only even harmonics were generated in by the optically passive, transversal degree of freedom, which is excited only due to the wave properties of an electron. In contrast, both odd and even harmonics were generated in the heteronuclear molecule by the optically active degree of freedom due to inversion symmetry breaking [5], and both even and odd harmonics were generated in by the optically passive degree of freedom. It was also shown in Ref. [2] that the appearance of coherent muonic oscillations in and after the end of the laser field is a purely non-Born-Oppenheimer effect: the post-laser-field muonic oscillations did not occur if the Born-Oppenheimer approximation was employed.

In our recent work [3], shaped post-laser-field electronic oscillations were also found to exist in H excited by two-cycle laser pulses at the wavelengths and 200 nm. It was shown, in particular, that there exists a characteristic oscillation frequency  au (corresponding to the wavelength of  nm) that plays the role of a “carrier” frequency of temporally shaped post-laser-field electronic oscillations both at  nm and at  nm. In the present work, we investigate the non-Born-Oppenheimer quantum dynamics of H excited by shaped one-cycle laser pulses, i.e. the shortest periodic excitation of H is studied. The laser carrier frequencies corresponding to the wavelengths , 50, 100 150, 200, 300 and 400 nm are used such as as to cover the domains of both and . Similarly to our previous work [3], the amplitudes of one-cycle laser pulses are chosen such that the energy of H after the ends of the pulses are  au, i.e., slightly below its dissociation threshold  au.

## 2 Model, equations of motion, and numerical methods

The three-body three-dimensional model with the Coulombic interactions representing the excited by the laser field linearly polarized along the axis is shown in Figure 1. The nuclear motion is assumed to be restricted to the polarization direction of the laser electric field. The electron () moves in three dimensions with conservation of cylindrical symmetry. Therefore, only two electronic coordinates, and , measured with respect to the center of mass of the two protons () should be treated explicitly together with the internuclear separation .

The component of the dipole moment of along the axis reads [6]

 dz(z)=−ez[1+Me/(2Mp+Me)], (1)

where is the electron charge, and are the proton and the electron masses, respectively. The homonuclear molecular ion does not have a permanent dipole moment, therefore its vibrational motion is excited only indirectly due to electronic motion induced by the laser field along the axis [7, 8, 9]. Electronic motion along the transversal axis occurs only due to the wave properties of the electron.

The time-dependent non-Born-Oppenheimer Schrödinger equation that governs the quantum dynamics of in the classical laser field reads

 iℏ∂∂tΨ−ℏ22mn∂2Ψ∂R2−ℏ22me(∂2Ψ∂ρ2+1ρ∂Ψ∂ρ)
 −ℏ22me∂2Ψ∂z2+V(R,ρ,z)Ψ−dz(z)E(t)Ψ. (2)

In Equation (2), is the nuclear reduced mass, is the electron reduced mass. In the atomic units (au) used throughout the paper, we have: and and for the field amplitude and intensity  V/cm and  W/cm, respectively.

 V(R,ρ,z)=e2(1/R−1/r1−1/r2), (3)

where the electron-proton distances (see Figure 1) are

 r1(R,ρ,z)=[ρ2+(z+R/2)2]1/2 (4)

and

 r2(R,ρ,z)=[ρ2+(z−R/2)2]1/2. (5)

The time-dependent laser electric field is chosen as follows:

 E(t)=E0sin2(πt/tp)sin(ωlt),0≤t≤tp, (6)

where is the amplitude, is the pulse duration at the base, and is the laser carrier frequency. With the symmetric (the -type) envelope and the integer number of optical cycles at the base (one optical cycle in the present case), the laser electric field has a vanishing direct-current component, , and satisfies therefore Maxwell’s equations in the propagation region [10]. Note before proceeding, that at a small number of optical cycles per pulse duration, as in the present work for example, the carrier-envelope phase (CEP) may play a very important role [11]. We shall address this issue in our future work. Here, we just assume that the carrier-envelope phase is equal to zero, see Equation (6).

The numerical methods used to solve the three-dimensional Equation (2) have been described in our previous works [8, 9]. In particular, the dissociation probability has been calculated with the time- and space-integrated outgoing flux for the nuclear coordinate ; the ionization probabilities have been calculated with the respective fluxes separately for the positive and the negative direction of the axis as well as for the outer end of the axis.

The size of the -grid has been chosen such as to be substantially larger than the maximum electron excursion along the axis, , and the size of the -grid has been chosen accordingly. The maximum electron excursion, au, corresponds to the field parameters used at  nm ( au and  au) and the choice of the and grids has been based on this value. Specifically, the three-dimensional wave-function of Equation (2) was damped with the imaginary smooth optical potentials, adapted from [12], at  au, at  au and at  au for the electronic motion, and at  au for the nuclear motion.

Initially, at , the was assumed to be in its ground vibrational and ground electronic state. The wave function of the initial state was been obtained by the numerical propagation of Equation (2) in the imaginary time without the laser field ().

## 3 Results

### 3.1 Resonant properties, laser-driven dynamics, and free evolution of H+2 on a short timescale

As it was already mentioned, the amplitudes of the one-cycle laser pulses used in the present work have been chosen such that the energies of at the ends of the laser pulses were slightly below the dissociation threshold ( au) and similar,  au. The amplitudes of one-cycle laser pulses required to achieve the aforementioned energy are plotted in Figure 2 versus their wavelength by curve 1. For the sake of comparison, similar results obtained with two-cycle laser pulses are presented in Figure 2 by curve 2.

Two different domains of the laser wavelength can be clearly distinguished in Figure 2. The domain of a large change of the laser pulse amplitude required to excite to the energy of  au with the laser wavelength corresponds to  nm (or ). The domain of a relatively small change of with corresponds to  nm (). It is also seen from Figure 2 that the most efficient excitation of by one-cycle and two-cycle laser pulses, which requires the minimum laser electric-field amplitude, takes place at  nm, which plays the role of a resonant, or the optimal laser wavelength. It can be expected therefore that power spectra of electron oscillations resulting from excitation of to  au at various laser wavelengths  nm should always contain harmonics corresponding to  nm.

Indeed, it was shown in our recent work [3], where excitation of with two-cycle laser pulses at and 800 nm was studied, that the respective power spectra contain the strongest harmonics corresponding to  nm, the fourth harmonic at  nm and the first (identical) harmonic at  nm. It will be interesting to check therefore whether the excitation of with laser wavelengths  nm would result in the generation of lower-order harmonics.

An important problem of the laser-driven dynamics with few-cycle laser pulses is the electron-field following. According to the well-known recollision model of Corkum [4], electron follows the field out-of-phase: the expectation value decreases when electric-field strength increases. Previously, a perfect electron-field out-of-phase following on the level of expectation values of the laser-driven electronic degree of freedom have been found only in the infrared [8, 9] and near-infrared [13] domains of the laser carrier frequency, with a large number of optical cycles per pulse duration being involved. It was also found in our previous work [14] that electrons of the extended H-H system follow the applied laser field out-of phase at the laser carrier frequency  au, and in-phase at  au, with the number of optical cycles per pulse duration being about 33.

The laser-driven dynamics of excited by two-cycle laser pulses was studied in our recent work [3]. It was found, in particular, that at  nm, the expectation value follows the field out-of-phase only approximately (during the first optical cycle), while at  nm, the out-of-phase electron-field following does not take place even approximately. It would be moreover difficult to expect the out-of-phase electron-field following for expectation values at one-cycle excitation.

The laser-driven dynamics of excited by one-cycle laser pulses at and 300 nm is presented in Figure 3 on the timescale of 3 fs (left and right panel, respectively). It is seen from Figures 3(a), 3(b), 3(d) and 3(e) that electron approximately follows the laser field out-of-phase only during the first half-cycle, while during the second half-cycle of the laser pulse, the expectation value may change even in-phase with the laser field. Nevertheless, after the ends of one-cycle laser pulses, expectation values demonstrate rather regular oscillations with the period of  fs corresponding to the wavelength  nm and frequency  au.

The transversal electron degree of freedom, , is optically passive if the laser field is aligned along the axis (Figure 1). It can be excited therefore only due to the wave properties of electron. The time-dependent expectation values at laser wavelengths and 300 nm are presented in Figures 3(c) and (f), respectively. It is seen that expectation values start to increase at the end of the first optical half-cycle and demonstrates quite regular (yet not harmonic) oscillations during the second optical half-cycle and after the ends of the one-cycle laser pulses. Two major frequencies of -oscillations can be distinguished at a close look. Indeed, time intervals between two highest maxima of -oscillations in Figures 3(c) and (f) are  fs, corresponding to the frequency of , or the wavelength of  nm. On the other hand, time intervals between the neighboring maxima of -oscillations in Figures 3(c) and (f), are  fs, corresponding to the frequency of , or the wavelength of  nm.

While the appearance of electronic -oscillations with the frequency being very close to the frequency of electronic -oscillations could be expected, the “frequency-doubling” of electronic -oscillations occurring at needs more explanations. Such a frequency-doubling of electronic -oscillations by its -oscillations occurring at was explained in our previous works [8, 9] as follows. During electronic oscillations along the axis, the electronic density is substantially delocalized also in the transversal, direction, due to the wave properties of the electron. This takes place at every turning point of electronic -oscillations, i.e., twice per every cycle of electronic -oscillations, giving rise to excitation of along the axis in a stepwise manner. Similar frequency-doubling of muonic oscillation also takes place in muonic and molecules excited by super-intense soft X-ray laser pulses at the wavelength of  nm on the attosecond timescale [2].

The characteristic feature of the post-laser-field electronic -oscillations in excited by one-cycle laser pulses is their asymmetry with respect to [Figures 3(b) and (e)]. This is especially evident on a longer timescale in comparison with smoothly shaped oscillations of expectation values , as shown in Figure 4 on the timescale of 12 fs. The left panel presents expectation values at the laser wavelength  nm (a) and  nm (b). The right panel presents expectation values at  nm (c) and  nm (d). Post-field oscillations of occur with the same frequency,  au (corresponding to the wavelength  nm) as those of . The existence of coherent post-laser-field oscillations of and in excited to the energy of  au by two-cycle laser pulses was rationalized in detail in our recent work [3] where approximately the same characteristic oscillation frequency was calculated. Thus in the present case of one-cycle laser pulses similar arguments hold true to explain the appearance of post-laser-field electronic oscillations. Note that the characteristic oscillation frequency corresponding to  nm is in line with the results presented in Figure 2 evidencing the most efficient excitation of to  au at the laser wavelength of  nm. Also note that asymmetry of time-dependent expectation values with respect to implies the time-dependent polarization of after its excitation by one-cycle laser pulses.

### 3.2 Ionization, dissociation and post-laser-field electronic oscillations of H+2 on a long timescale

Numerical simulations of the laser-driven quantum dynamics and subsequent free evolution of were performed as long as ionization and dissociation probabilities were small enough, not more than about 2%.

The time-dependent ionization and dissociation probabilities are presented in Figure 5 on the timescale of 50 fs for the case when is excited by one-cycle laser pulse at the laser wavelength  nm. Note that ionization probabilities have been calculated with the respective time- and space-integrated outgoing fluxes separately for the negative and the positive direction of the axis (curves 2 and 3, respectively) as well as for the outer end of the axis (curve 1). The dissociation probability (curve 4) has been similarly calculated for the nuclear coordinate .

Since the optimal laser-pulse amplitude required to prepare at the energy of  au at the laser wavelength  nm is substantially larger than at larger wavelengths, both dissociation and ionization probabilities presented in Figure 5 are the largest obtained in this work. It is seen from Figure 5 that the ionization probability for the outer end of the axis is larger than those for both negative and positive direction of the axis. This can be explained by the aforementioned fact that the electronic density is delocalized in the direction twice per every cycle of electronic -oscillations, giving rise to the frequency-doubling of electronic -oscillations as compared to electronic -oscillations. It is also seen from Figure 5 that ionization starts much prior to dissociation, therefore ‘ionizative dissociation’ occurs. Indeed, the decrease of the electron density between the two protons due to ionization disturbs the initial equilibrium configuration of and thus allows the Coulombic repulsion of the protons to act more efficiently resulting in the elongation of the internuclear distance in and its subsequent dissociation.

The time-dependent expectation values of internuclear distances in are presented in Figure 6 on the timescale of 50 fs for the laser wavelengths of  nm (curve 1),  nm (curve 2),  nm (curve 3) and  nm (curve 4). It is seen from Figure 6 that at , 100 and 200 nm (curves 2, 3 and 4, respectively), the time-dependent internuclear distances behave very similar to each other, all demonstrating local maxima at  fs and at  fs (elongation of the bond) as well as local minima at  fs (contraction of the bond length). Curve 1, corresponding to the case of  nm, looks at a first glance as an exception due to much more substantial increase of the bond length caused by a comparatively strong laser field applied to prepare close to its dissociation threshold and, therefore, more efficient ionizative dissociation of resulting in a more substantial overall bond elongation. Nevertheless, both elongation and contraction of the internuclear distance can be seen in curve 1 as well. Since the nuclear motion in the symmetric molecule is activated only by the electronic motion induced by the laser field along the axis, it is interesting to find a correlation between the nuclear motion and post-laser-field electronic -oscillations.

In Figure 7 post-laser-field electronic oscillations are presented by the time-dependent expectation values for the same laser wavelengths as in Figure 6. It is clearly seen from Figure 7 that local minima of the Coulomb force correspond to local maxima of the bond length of (elongation of the bond), while the local maximum of the Coulomb force at  fs corresponds to the local minimum of the bond length (contraction of the bond length). We can conclude therefore, that periodic elongation-contraction of the bond of (Figure 6) is controlled by compressing-expanding electron acceleration along the axis (Figure 7) which takes place with the period of  fs corresponding to the frequency of shaped post-laser-field oscillations occurring with the carrier oscillation frequency  au (corresponding to the wavelength of  nm). Similar electron-nuclei correlations were found in our recent work [3] where two-cycle laser pulses were used to excite and a more detailed explanation is given for post-laser-field electron-nuclei correlations in terms of below-resonance vibrational frequency [9, 15] and for the existence of characteristic oscillation frequency (corresponding to  nm) in terms of a continuum state prepared by the laser pulses.

Finally, to complete this section, we present in Figure 8 the time-dependent expectation values [Figure 8(a)] and [Figure 8(b)] calculated on the long timescale of 50 fs at the laser wavelengths of , 50, 100 and 200 nm.

It is seen from Figures 8(a) and 8(b) that expectation values and first demonstrate fast oscillations on the time interval of about  fs. Again, two major frequencies of electronic -oscillations can be distinguished at a close look: ( nm) and ( nm). The frequency-doubling of electronic -oscillations has been already described in Section 3 and presented in Figures 3(c) and 3(f) therein.

Afterwards, at  fs, expectation values demonstrate a rather smooth behavior, with the electron excursion along the transversal coordinate being quite strongly dependent of the laser pulse amplitude used to excite initially along the axis. It is also seen from Figure 8(b) that there exists a nice correlation between the time-dependent values and the internuclear distances of Figure 6. Indeed, at , 100 and 200 nm [curves 2, 3 and 4, respectively, in both Figure 6 and Figure 8(b)], local maxima of both and occur at  fs and at  fs, while their local minima occur at  fs. Since no frequency-doubling of post-laser-field electronic -oscillations (as compared to -oscillations) takes place, we conclude that the low-frequency oscillations of at  fs are induced by the periodic elongation-contraction of the bond length in (Figure 6), rather than by post-laser-field electronic -oscillations (Figure 7). Again, since the periodic elongation-contraction of the bond length in is very small at  nm (curve 1 in Figure 6), local maxima and minima of are not seen at  nm as well [curve 1 in Figure 8(b)].

### 3.3 Power spectra generated by post-laser-field electronic oscillations in H+2

In this section we present the power spectra generated by post-laser-field electronic motion calculated in the acceleration form, and , for the electron coordinates and , respectively.

It is straightforward to show with the Ehrenfest’s theorem, that the acceleration of the expectation value can be written in the following form:

 d2dt2⟨z⟩=−1me[⟨∂V/∂z⟩+E(t)]. (7)

Since the applied laser field aligned along the axis does not excite the transversal degree of freedom directly, its excitation can occur only due to the wave properties of electron. Therefore, the electric-field term does not appear in the equation for the acceleration of the expectation value at all. It can also be shown, by making use of Ehrenfest’s theorem, that the acceleration of the expectation value reads

 d2dt2⟨ρ⟩=−1me⟨∂V/∂ρ⟩. (8)

The power spectrum of any time-dependent expectation value is defined by the squared modulus of the Fourier transform:

 (9)

where

 ⟨S(t)⟩=⟨Ψ(t)|S|Ψ(t)⟩. (10)

In the case under consideration, the time-dependent expectation value in Equation (9) will stand accordingly for defined by Equation (7), or for defined by Equation (8). In the given above definitions we took into account that the power spectra defined by Equations (9) and (10) do not depend on the sign of .

#### 3.3.1 Generation of lower-order harmonics at λl<200 nm

As it was discussed earlier (Section 3.1), power spectra resulting from excitation of close to its dissociation threshold at various laser wavelengths  nm might always generate harmonics corresponding to  nm due to the most efficient excitation of at this wavelength (Figure 2). Therefore, at  nm, generation of lower-order harmonics with respect to electronic -motion can be expected.

In Figure 9, power spectra in the acceleration form, and , generated due to the laser-initiated electron motion along the coordinate, are presented for the case when is excited by one-cycle laser pulse at the laser wavelength of  nm.

It is seen from Figure 9(a) that the strongest lower-order harmonic of the spectrum corresponds, as it was expected, to the wavelength of  nm, while a weaker lower-order harmonic corresponds to  nm. In the power spectrum generated by the transversal degree of freedom, Figure 9(b), the strongest lower-order harmonic at  nm corresponds to the doubled frequency of -oscillations (where corresponds to the wavelength of  nm). It is also seen from Figure 9(b) that the lower order harmonic corresponding to  nm occurs in the power spectrum as well.

In Figure 10, power spectra and , are presented for the case when is excited by one-cycle laser pulse at the laser wavelength of  nm. Again, as it is seen from Figure 10(a), the strongest lower-order harmonic of the spectrum corresponds to the wavelength of  nm, while the second, much weaker, lower-order harmonic corresponds to  nm (i.e., it is very close to  nm). In the power spectrum of the transversal degree of freedom, Figure 10(b), the strongest lower-order harmonic at  nm corresponds to the doubled frequency of -oscillations , while a weaker lower order harmonic at  nm corresponds to .

Power spectra and for the case of the laser wavelength  nm are shown in Figures 11(a) and (b), respectively. The strongest lower-order harmonic of the spectrum generated by the optically active degree of freedom corresponds to the wavelength of  nm, while a comparatively very weak harmonic corresponds to  nm. We can conclude therefore that at the laser wavelength  nm one lower-order and one “identical” harmonic are generated by the optically active degree of freedom of .

In the power spectrum generated by the optically passive degree of freedom, Figure 11(b), a new feature can be observed as well. Indeed, while the strongest lower-order harmonic at  nm corresponds to , and a weaker lower order harmonic at  nm corresponds to , the other harmonic corresponding to , at  nm, is the higher-order harmonic with respect to the laser wavelength  nm used to excite . We can conclude therefore, that the laser wavelength of  nm manifests itself as a beginning of the appearance of the higher-order harmonics, at least in the power spectra generated by the optically passive, transversal degree of freedom. The physical reason behind this feature is the above described frequency-doubling of the electronic -oscillations, with respect to the laser-initiated electronic -oscillations, caused by the wave-properties of an electron.

The situation is developing further at a larger wavelength,  nm, approaching the most efficient for the excitation of close to its dissociation threshold laser wavelength of  nm, as depicted in Figure 2.

Power spectra and are presented in Figure 12 for the case when is excited by one-cycle laser pulse at the laser wavelength of  nm. It is clearly seen from Figure 12(a) for the spectrum that the strongest harmonic at  nm is still the lower-order harmonic with respect to the laser wavelength  nm used to excite , while the other, a weak harmonic at  nm, is the higher-order harmonic.

In the power spectrum of Figure 12(b) for the optically passive degree of freedom, the lower-order harmonic at  nm corresponding to is not the strongest one anymore. The strongest harmonic at  nm is a higher-order harmonic, as well as a weaker one at  nm, both corresponding to the accordance with the frequency-doubling of the electronic -oscillations.

We can conclude therefore from the results presented in this section (Figures 9 through 12) that when the laser wavelength increases from small values of about 25 nm and approaches 200 nm, generation of higher-order harmonics starts, due to the frequency-doubling of the electronic -oscillations, from the optically passive degree of freedom at  nm [Figure 11(b)] and develops such that both and power spectra have higher-order harmonics at  nm (Figure 12).

#### 3.3.2 Generation of higher-order harmonics at λl≥200 nm

As it was already discussed in Section 3.1 and depicted in Figure 2 therein, there are two different domains of the laser wavelength characterizing the efficiency of excitation of to the energy close to its dissociation threshold by one-cycle and two-cycle laser pulses. The domain of  nm to be considered in this section corresponds to a very small change of the optimal laser pulse amplitude required to excite from its ground state to the energy of  au. Nevertheless, since  nm is still the most efficient laser wavelength (see Figure 2), we can expect that power spectra resulting from excitation of at various laser wavelengths  nm would contain harmonics corresponding to  nm.

In Figure 13, power spectra in the acceleration form, and , generated due to the laser-initiated electron motion along the coordinate, are presented for the case when is excited by one-cycle laser pulse at the laser wavelength of  nm.

It is seen from Figure 13(a) that the strongest harmonic in the spectrum generated by the optically active degree of freedom is the first, or identical harmonic corresponding to the wavelength of  nm, while a much weaker, the second-order harmonic corresponds to  nm. The appearance of the second-order harmonic in the power spectrum of the symmetric molecule is the specific feature of one-cycle and two-cycle [3] laser pulses, because if laser pulses with many optical cycles are used to excite a symmetric molecule, even harmonics should not appear in the power spectra generated by optically active degrees of freedom at all, as suggested by the concept of inversion symmetry [5].

In the power spectrum generated by the optically passive degree of freedom, Figure 13(b), the strongest second-order harmonic at  nm corresponds to the doubled frequency of -oscillations , while a weaker, the first-order harmonic at  nm, corresponds to .

In Figure 14, power spectra and are presented for the case when at the laser wavelength of  nm. Again, as it is seen from Figure 14(a), the strongest harmonic in the spectrum generated by the optically active degree of freedom corresponds to the wavelength of  nm. The other higher-order harmonic in the power spectrum , that corresponding to  nm, is (formally) the third-order harmonic with respect to the laser wavelength of  nm used to excite . On the other hand, the higher-order harmonic at  nm is the second-order harmonic with respect to the smallest-frequency harmonic at  nm appeared in the power spectrum. Such a new nomenclature of the harmonic order may often be very suitable, because the smallest-frequency harmonics in all power spectra generated by the optically-active degree of freedom analyzed in this section above [see Figures 9(a) through 14(a)] are those corresponding to  nm. The physical reason behind this feature is that  nm is the most efficient laser wavelength with respect to excitation of from the ground state to the energy close to its dissociation threshold by one-cycle and two-cycle laser pulses, see Figure 2.

Since the smallest-frequency harmonics in all power spectra generated by the optically-passive degree of freedom [Figures 9(b) through 14(b)] also correspond to  nm, a new nomenclature of the harmonic order described above for power spectra can be used to analyze power spectra as well. Indeed, it is seen from Figure 14(b) that the higher-order harmonic with the smallest-frequency is that at  nm, while the higher-order harmonic at  nm is approximately the second-order harmonic with respect to the smallest-frequency one (at  nm) and the third-order harmonic with respect to the laser wavelength of  nm used to excite . Needless to add that, as usual, harmonic at  nm corresponds to the frequency of , while harmonic at  nm corresponds to the doubled frequency of -oscillations , where the oscillation frequency corresponds to the wavelength  nm.

Finally, in Figure 15, power spectra and are presented for the case when is excited by one-cycle laser pulse at the laser wavelength of  nm. In the usual nomenclature of the harmonic order, three sequential peaks appeared in the power spectrum , Figure 15(a), correspond respectively to the second, the fourth and the fifth harmonics with respect to the laser wavelength of  nm used to excite . In a suggested new nomenclature, the second and the third peaks (those at 98 and 82 nm) approximately correspond to the second harmonic of the smallest-frequency peak at 200 nm.

In the power spectrum , Figure 15(b), the peak at 190 nm corresponds to the second and that at 110 nm approximately corresponds to the fourth harmonic with respect to the laser wavelength of  nm used. On the other hand, the peak at 110 nm approximately corresponds to the second harmonic of the smallest-frequency peak at 190 nm. Again, the peak at 190 nm corresponds to the frequency of , while the peak at 110 nm corresponds to the doubled frequency of -oscillations .

## 4 Conclusion

In the present work, the non-Born-Oppenheimer quantum dynamics of excited by linearly polarized along the molecular axis shaped one-cycle laser pulses has been numerically studied at different laser-carrier frequencies corresponding to the laser wavelengths of 25, 50, 100, 200, 300 and 400 nm. The amplitudes of the one-cycles laser pulses have been optimized such that the energy of at the end of each pulse,  au, was close from the below to its dissociation threshold. For the sake of completeness, some results were obtained by making use of two-cycle laser pulses as well. The present work provides a detailed and in-depth extension of our previous study [3] where shaped two-cycle laser pulses at the laser wavelengths of 800 and 200 nm were used to excite close to its dissociation threshold.

The basic results obtained in the present work are summarized as follows.

(I) The most efficient excitation of by one-cycle shaped laser pulses, which requires the minimum laser electric-field amplitude to achieve the required energy of about -0.515 au at the end of the pulse, takes place at the laser wavelength of  nm corresponding to the laser-carrier frequency of  au. This frequency plays the role of a characteristic oscillation frequency  au and manifests itself as the carrier frequency of temporally shaped post-laser-field oscillations of the time-dependent expectation values and corresponding to the optically active degree of freedom, which exist on a long timescale of at least 50 fs after the ends of the pulses used to excite initially. The corresponding values for excited by two-cycle shaped laser pulses [3] are almost identical.

(II) The optically passive, transversal degree of freedom, which is excited only due to the wave properties of the electron, also demonstrates post-laser-field oscillations of the time-dependent expectation values and which occur at two basic frequencies and . The latter frequency corresponds to the frequency-doubling of electronic -oscillations as compared to electronic -oscillations described previously [8, 9, 3].

(III) A characteristic feature of power spectra generated by and degrees of freedom when is excited by one-cycle laser pulses is a small number of generated harmonics: not more than three relatively strong harmonics can be observed in the power spectra presented in Section 3.3. A similar feature was observed in the case when was excited by two-cycle laser pulses [3]. In a more general case, the number of generated harmonics correlates with the number of optical cycles used to excite a molecule, with the overall trend being “the smaller optical cycles used to excite a molecule is, the smaller number of harmonics is generated”. Such a correlation was discussed in more detail in our previous work [3] for both ordinary (electronic) and muonic molecules.

(IV) The positions of many peaks in power spectra presented in Section 3.3 are not equal to integer multiples of the laser carrier frequency , especially in the case of lower-order harmonic generation at  nm described in Section 3.3.1. It is known [5] that such harmonics may be be generated by single atoms and molecules due to resonance effects. In the current case of one-cycle laser pulses used to excite close to its dissociation threshold and the resonant properties of this process depicted in Figure 2, the situation can be rationalized by introducing the suggested above new nomenclature of the harmonic order as follows. Since the smallest-frequency harmonics in all power spectra and presented in Section 3.3 correspond to the optimal wavelength  nm, a new harmonic order can be defined as follows:

 (ωωl)′=(ωωl)λoptlλl. (11)

It is easy to check that Equation (11) works very well for all power spectra and is also quite suitable for power spectra where the frequencies of strongest harmonics are given by and .

## Acknowledgement

This work has been financially supported by the Deutsche Forschungsgemeinschaft through the Sfb 652 (O.K.), which is gratefully acknowledged.

## References

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