Sensitivity of high-order harmonic generation to nuclear motion

Journal of Molecular Structure: THEOCHEM 947 (2010) 119–122

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Sensitivity of high-order harmonic generation to nuclear motion Ya-Hui Guo, Hai-Xiang He, Jian-Yong Liu *, Guo-Zhong He State Key Laboratory of Molecular Reaction Dynamics, Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Dalian 116023, China

a r t i c l e

i n f o

Article history: Received 13 December 2009 Received in revised form 22 January 2010 Accepted 2 February 2010 Available online 10 February 2010 Keywords: High order harmonic generation Nuclear motion Ultrafast processes

a b s t r a c t The harmonic generations of molecular ion H2+ and D2+ exposed to an intense laser field have been studied with the collinear reduced-dimensionality model. The numerical solution of the time-dependent Schrödinger equation for H2+ and its isotopic variant D2+ shows that high-order harmonic generation is sensitive to the initial vibrational states of the molecular ion. By comparing the spectra of H2+ with D2+, it is shown more intense harmonics are generated in lighter isotopes. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction The interaction of molecules and intense laser field has been widely studied in recent years. Since it is on different time scales for the nuclei and electrons in the molecular system responding to the laser pulses, abundant dynamical behaviors such as bond softening and hardening [1,2], above-threshold dissociation and ionization [3,4], high-order harmonic generation (HHG) [5,6], attosecond pulse generation and control [7] have been observed in experimental and theoretical works. Because of its potential for the coherent light emission in the extreme ultraviolet (XUV) region and generation of attosecond pulses, lots of classical and quantum studies have been carried out to understand the underlying mechanism of HHG. At present, the physical origin of the high-harmonic spectra from atoms has been well explained by the three-step model [8–10]. The electrons are first ionized by tunneling through the potential barrier formed by the Coulomb potential and the laser field, then move back and forth driven by the periodic field, finally recombine with the parent ion and emit harmonic photons of frequencies in odd multiples of the fundamental frequency due to the symmetry. According to this model, the maximum energy of the emitted harmonic photon is Ip + 3.17Up, where Ip is the ionization potential and Up is the mean kinetic energy of a free electron in the laser field. Molecular systems exposed to laser fields behave different from atoms due to the larger number of molecular degrees of freedom. It is shown that HHG from molecules depends sensitively on the molecular structure and its alignment [11]. In particular, its spectrum is strongly influenced by the internuclear distance [12,13]. Since

* Corresponding author. E-mail address: [email protected] (J.-Y. Liu). 0166-1280/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.theochem.2010.02.004

the nuclei and electrons respond to the laser pulse differently, the exact description of the dynamical laser-driven correlated electronic and nuclear behavior by means of the time-dependent Schrödinger equation (TDSE) is a demanding work even with the state-of-the-art computers. The research into molecules in strong laser pulses has been focused to rather simple molecular systems, such as H2+ and H2. Lein proposed theoretically that the harmonic signal in molecules would be approximately relevant to the nuclei vibrational autocorrelation function and gave instantaneous information about the nuclear dynamics [14]. Based on that theoretical work, Baker et al. have done high-precision experiments and probe the ultrafast motions of the nuclei in diatomic molecules [15]. Considering the motion of the nuclei, Qu et al. have found that the ionization of H2+ in intense laser field is substantially increased and the individual harmonic peaks broaden compared to fixed nuclei [16]. Recently, Shahbaz et al. have revealed characteristic nuclear signatures in the harmonic response of strongly laser-driven muonic hydrogen and deuterium atoms. Both the harmonic cutoff and the plateau height are significantly enhanced for muonic hydrogen atom due to its small nuclear mass and size as compared with the muonic deuterium atom [17]. These studies have suggested abundant new phenomena appear when the nuclear motion is considered. In this paper, we pay our attention to the status of nuclear motions to search evidence for the correlation of HHG and nuclear dynamics. 2. Methods and discussions When the diatomic molecule ion H2+ and its isotope D2+ interacting with a linearly polarized laser field, the electron oscillation may be confined to the laser polarization direction and the molecular axis aligns quickly along the polarization direction of the laser

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pulse. Thus a one-dimensional (1D) collinear model which has been confirmed to gain quite reliable results in former works [18,19] is adopted here. We utilize the parallel compute code LZH-DICP [20], which has been approved with high efficiency and validity to investigate the behavior of atoms and small molecules in strong laser fields using split-operator approach coupled to a discrete variable representation (DVR). This method has previously been applied to study nonadiabatic quantum scattering dynamics [21] and to track nuclear and electron wavepacket in strong laser field [22]. The field-free Hamiltonian of this collinear model has an ordinary form:

H0 ðz; RÞ ¼ 

1 @2 1 @2  þ Vðz; RÞ; 2 @z2 2l @R2

ð1Þ

1 1 ffi  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi is the Coulomb potential where Vðz; RÞ ¼ 1R  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ðzR=2Þ þ1

ðzþR=2Þ þ1

which is represented by a ‘‘soft-core” parameter. z is the distance from the electron to the mass center of the two nuclei, R is the distance between the nucleus, and l is the reduced mass of the two nuclei. The TDSE with the dipole approximation is

i

@ wðz; R; tÞ ¼ ½H0 ðz; RÞ þ Hint ðz; tÞwðz; R; tÞ; @t

ð2Þ

Hint accounts for the laser–electron interaction,

Hint ðz; tÞ ¼ zE0 f ðtÞ cosðx0 tÞ;

ð3Þ

where E0, f(t) and x0 are the maximum field amplitude, the envelope function, and frequency of the laser pulse, respectively. The TDSE can be numerically solved with the second-order split-operator method [23]. The initial wave function for individual vibrational state is constructed by diagonalizing the field-free Hamiltonian. The harmonic spectra are calculated by Fourier transforming the timedependent dipole acceleration a(t) which can be obtained in terms of Ehrenfest theorem [24]:

     @ aðtÞ ¼  wðz; R; tÞ ½Vðz; RÞ þ Hint ðz; tÞwðz; R; tÞ : @z

ð4Þ

The expected value for the internuclear distance at any time is defined as

RðtÞ ¼

ZZ

ZZ Rjwðz; R; tÞj2 dzdR jwðz; R; tÞj2 dzdR:

ð5Þ

Fig. 1(a and b) shows the spectra of the HHG from H2+ and its isotope D2+ interacting with a 6 fs, 800 nm laser pulse of intensity I = 1.0  1014 W/cm2. The grid is defined by 0 < R 6 20 and

Fig. 1. The high-order harmonic spectra of H2+ and D2+ arising from interaction with the 6 fs, 800 nm, I = 1.0  1014 W/cm2 sin2 shaped laser pulse, for different initial states: (a) t = 0. (b) t = 3.

100 6 z 6 100 with the spatial step DR = 0.1 and Dz = 0.2, and the time step for electron Dte is 0.2 (4.8 attoseconds). A mask function has been used to absorb the reflecting parts of the wave function after evolving for each time step. As shown in Fig. 1(a), the spectra from the vibrational ground state of H2+ and its isotopic variant D2+ are almost the same, the harmonic intensity decreases drastically for the first three odd order harmonics, then continues to gently descend for many harmonics, and no obvious cutoff harmonic. With regard to the initial states t = 3, both of the spectral structures of H2+ and D2+ are more regular for the lower harmonics, following by the smooth harmonics in the plateau and become continuous until the cutoff region. The harmonics intensity of H2+ is about one order higher than that of D2+. The enhancement of the spectra is due to the smaller reduced mass of the H2+ system, which experiences stronger acceleration in the intense laser field. It has been demonstrated both theoretically and experimentally that for D2 and H2, the harmonic yield of the former is enhanced by a factor of about 2 as compared with the latter. Our results are different with them for that the D2 and H2 possess two-electron and the electron correlation has affected the harmonic spectra [25]. To explain the phenomena described above quantitatively, the average nuclear separations of these states as a function of time are presented in Fig. 2. In the vibrational ground state, both of H2+ and D2+ lie near their equilibrium position in the beginning, the average nuclear separation is 2.656 a0 for H2+, while 2.640 a0 for D2+ (a0 is the Bohr radius). When the laser turns off, the values become 2.812 a0 for H2+ and 2.765 a0 for D2+. Obviously, there are little differences of the nuclear motions between them, therefore the harmonics responding to strong laser pulse are almost the same. Correspondingly, the HHG power spectra are very different when the molecular ions are placed in the t = 3 state, because the nuclear separations are 2.994 a0 and 2.873 a0 of H2+ and D2+ at first, and then stretch to 3.696 a0 and 3.126 a0 after the laser switching off. The increased internuclear distance DR of H2+ is close to triple of that of D2+, indicating the faster nuclear motion boosts harmonic generation efficiency significantly [18]. We have also calculated the ionization rates [26] to research the underlying physical reason of the nuclear dynamics on the harmonic emission. As shown in Fig. 3(a and b), it is interesting to see that the ionization rate for H2+ (t = 3) is one order higher than for D2+ (t = 3) while no remarkable difference exists as they are located in the ground states. These results coincide with the HHG spectra in Fig. 1. It indicates that the fast vibration of the proton in H2+ facilitates

Fig. 2. Numerical calculation results of nuclear separation of H2+ and D2+ when interacting with the same laser pulse as Fig. 1 for unlike initial states, t = 0, 3.

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the electron ionization and leads to the significantly yields of the harmonic emission compared with the D2+. In Fig. 4(a and b), the respective HHG spectra of H2+ and D2+ for the initial states t = 0–3 are presented. The laser parameters are the same as those in Fig. 1. It can be seen that the calculated

spectra have similar patterns: a rapid decrease in intensity for the first few harmonics, a plateau with harmonics at almost constant intensity, and a cutoff where the harmonics quickly decrease. However, there are still distinct differences among different vibrational states. Contrary to the well defined harmonics which are separated by twice the laser frequency of the high vibrational state, the low-frequency part for the spectrum of the t = 0 is not well resolved. This is probably caused by the adiabatic increase in the nuclear separation. For high vibrational states, during the recollision processes, the fast oscillations of the nuclei are responsible for the reduced periodicity of the electron and nuclear recombinations and form the regular low energy part harmonics [17]. It is noticeable that the plateau region exhibits two well distinct parts for the highly vibration state. A striking transition happens in the spectrum at about the 23rd harmonic. The spectrum becomes continuous above this frequency while the 2x periodicity is preserved below the transition frequency. With increasing t, it is clear the harmonic generation efficiencies of the vibrational excited states are enhanced greatly. These results agree well with the theoretical results in previous literature [27]. This is probably caused by the faster nuclear vibration of high vibrational states. In Fig. 5, the relative velocities between the two nuclei as a function of the interaction time of the 800 nm laser pulse have been shown. It is not surprising that the higher the vibrational state is, the more quickly the nuclei move. On one hand, during the same interaction time, the molecule is more stretched in the

Fig. 4. The calculated harmonic spectra of (a) H2+ and (b) D2+ in the similar laser field as Fig. 1 for initial vibrational states, t = 0–3.

Fig. 5. The relative velocity of the two nuclei of (a) H2+ and (b) D2+ in the similar laser field as Fig. 1 for initial vibrational states, t = 0–3.

Fig. 3. Calculated ionization rates of H2+ and D2+ in the similar laser field as Fig. 1 for (a) t = 0 and (b) t = 3.

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can be seen from our calculations. It is found that when molecule ions are prepared in the higher vibrational state, the nuclear vibrates violently and the emission efficiency of the HHG is enhanced. Our simulations indicate that the fast nuclear motion leads to the enhanced harmonic generation. Acknowledgements This research was supported by NKBRSF (No. 2007CB815202), 863 (No. 2006AA01A119), and NSFC (Grant No. 20633070). The authors sincerely thank Prof. Keli Han and Dr. Ruifeng Lu for providing us LZH-DICP code. References

Fig. 6. Calculated ionization rates of H2+ in the similar laser field as Fig. 1 for initial vibrational states, t = 0–3.

higher vibrational state, resulting in the higher efficiencies of the harmonic radiation. On the other hand, since the ionization potentials of the vibrational states are lower than that of the ground state, the ionization step might also play an important role for the enhancement of the harmonic yields. We have calculated the ionization rates and the results are displayed in Fig. 6. It can be seen the ionization rate is increasing as the H2+ is located in high vibrational state. In the case of D2+, similar phenomena and conclusions are tenable. 3. Conclusion In conclusion, the effect of the nuclear motion on the harmonic generation of strongly laser-driven molecular hydrogen and deuterium ion has been discussed. In comparison with hydrogen molecular ion, deuterium moves slowly in the laser field and the harmonic efficiency is significantly suppressed due to the larger nuclear mass. The harmonic response of two isotopic molecular ions for different initial vibrational states has also been studied, and a strong vibrational-state dependence of the harmonic spectra

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