High-order Harmonic Generation Signals Research Articles

Revealing Berry curvature of the unoccupied band in high harmonic generation

Experimental determinations of the net Berry curvature have been made for the occupied band because they are basically available on the basis of electronic and magnetic responses of carriers occupying the band, for instance, through the measurement of the anomalous Hall effect or the magneto-optical Kerr effect. Here, we investigate the circular dichroism of the high-order harmonic generation (HHG) in undoped monolayer ${\mathrm{MoS}}_{2}$. Breaking the threefold rotational symmetry of the pristine monolayer ${\mathrm{MoS}}_{2}$ by applying uniaxial strain and an external bias field, we find that the third-order HHG signals deliver nonvanishing dichroic signatures. These signatures are essentially identical as the net Berry curvatures obtained from the electron-doped monolayer ${\mathrm{MoS}}_{2}$ with the aforementioned symmetry broken. This finding implies an experimental challenge to observe the net Berry curvature of the unoccupied empty band, which should open another stage toward the advancement of valleytronics.

Correlation effects in high-order harmonic generation from finite systems

Using the Hubbard model we study how the process of high-order harmonic generation (HHG) is modified by beyond mean-field electron-electron correlation for both finite and bulk systems. A finite-size enhancement of the HHG signal is found and attributed to electrons backscattering off the lattice edges. Additionally, with increasing strength of the electron-electron correlation an enhancement of the high-frequency regime of the HHG spectrum is found. This is attributed to the on-site Coulomb repulsion between electrons giving rise to a localized quiver motion of the electrons. The finite-lattice enhancement dominates the HHG spectra from a few harmonic orders until a threshold from which the correlational enhancement dominates. This threshold is determined by the degree of correlation and decreases into the low-frequency regime for increasing electron-electron correlation. This infers that as the Mott insulator limit of high electron-electron correlation is approached, the finite-size effect on the electron dynamics becomes negligible.

Simple model of dephasing for high-order harmonic generation in solids

We analyze high-order harmonic generation (HHG) in a disordered semiconductor within the context of the Anderson model of disorder. Employing the theoretical methods pioneered for the study of disordered metals, we show that disorder is a source of ultrafast dephasing of the HHG signal in semiconductors. Furthermore, it is shown that the dephasing effect induced by disorder on HHG spectra depends on both strength and correlation length of the disorder and very weakly on the frequency and intensity of the laser. Our results suggest that HHG has the potential to be a new spectroscopic tool for the analysis of disordered solids.

High-order harmonic generation in solid C60

High-order harmonic generation (HHG) has unleashed the power of strong laser physics in solids. Here we investigate HHG from a large system, solid ${\mathrm{C}}_{60}$, with 240 valence electrons engaging harmonic generation at each crystal momentum. We employ the density functional theory and the time-dependent Liouville equation of the density matrix to compute HHG signals. We find that under a moderately strong laser pulse, HHG signals reach 15th order, consistent with the experimental results from ${\mathrm{C}}_{60}$ plasma. The helicity dependence in solid ${\mathrm{C}}_{60}$ is weak, due to the high symmetry. In contrast to the general belief, HHG is unsuitable for band structure mapping in ${\mathrm{C}}_{60}$. However, we find a window of opportunity using a long wavelength, where harmonics are generated through multiple-photon excitation. In particular, the fifth-order harmonic energies closely follow the transition energy dispersion between the valence and conduction bands. This finding is expected to motivate future experimental investigations.

All-optical measurement of high-order fractional molecular echoes by high-order harmonic generation.

An all-optical measurement of high-order fractional molecular echoes is demonstrated by using high-order harmonic generation (HHG). Excited by a pair of time-delayed short laser pulses, the signatures of full and high order fractional (1/2 and 1/3) alignment echoes are observed in the HHG signals measured from CO 2 molecules at various time delays of the probe pulse. By increasing the time delay of the pump pulses, much higher order fractional (1/4) alignment echo is also observed in N 2O molecules. With an analytic model based on the impulsive approximation, the spatiotemporal dynamics of the echo process are retrieved from the experiment. Compared to the typical molecular alignment revivals, high-order fractional molecular echoes are demonstrated to dephase more rapidly, which will open a new route towards the ultrashort-time measurement. The proposed HHG method paves an efficient way for accessing the high-order fractional echoes in molecules.

High-order harmonic generation in imperfect crystals

High-order harmonic generation (HHG) in imperfect crystals, where the disorder is modeled by random shifts of the ionic positions, is studied using time-dependent density-functional theory. When irradiated by midinfrared laser pulses, the disorder-free system produces HHG spectra with two plateaus. Compared with the disorder-free system, disordered systems are found to emit suppressed harmonics in the first plateau region and enhanced harmonics in the second plateau region. The suppression of harmonics in the first plateau becomes less pronounced when decreasing the displacement of the nuclei, while the enhancement in the second plateau region is insensitive to the range of the ionic displacement. We have confirmed these findings for many different disordered sample systems and for different laser field strengths. The increase of the HHG signals in the second plateau region is proposed to stem from a change of the dynamics in the system, evidenced by the transition matrix elements between the field-free Kohn-Sham orbitals. In addition, a time-frequency profile of HHG spectra shows that the emission of harmonics is less regular in the time domain for a disordered system than for the disorder-free system.

Generating high-order optical and spin harmonics from ferromagnetic monolayers

High-order harmonic generation (HHG) in solids has entered a new phase of intensive research, with envisioned band-structure mapping on an ultrashort time scale. This partly benefits from a flurry of new HHG materials discovered, but so far has missed an important group. HHG in magnetic materials should have profound impact on future magnetic storage technology advances. Here we introduce and demonstrate HHG in ferromagnetic monolayers. We find that HHG carries spin information and sensitively depends on the relativistic spin–orbit coupling; and if they are dispersed into the crystal momentum k space, harmonics originating from real transitions can be k-resolved and carry the band structure information. Geometrically, the HHG signal is sensitive to spatial orientations of monolayers. Different from the optical counterpart, the spin HHG, though probably weak, only appears at even orders, a consequence of SU(2) symmetry. Our findings open an unexplored frontier—magneto-high-order harmonic generation.

Sturmian–Floquet approach to high-order harmonic generation

We show that the Floquet approach with a Sturmian basis means an efficient description of high-order harmonic generation (HHG) with monochromatic excitation. This method, although it involves numerical calculations, is close to analytic approaches with a corresponding deeper insight into the dynamics. As a first application, we investigate the role of atomic coherence during the process of HHG: as shown, different coherent superpositions of initial atomic states produce observably different HHG spectra. For linearly polarized excitation, we demonstrate that the question whether the constituents of the initial superpositions are dipole coupled or not strongly influences the dynamics. By investigating time-dependent HHG signals, we also show that the preparation of the initial atomic state can be used for the control of the high-harmonic radiation.

Sub-femtosecond nuclear dynamics and high-harmonic generation: Can muonated species be used as a probe of isotope effects?

Sub-femtosecond nuclear dynamics and high-order harmonic generation (HHG) studies are reported for the X̃2B1 and Ã2A1 states of Mu2O+. The photoelectron spectra and autocorrelation functions are calculated by solving the time-dependent Schrödinger equation, and the HHG signals from the autocorrelation functions for the two cationic states. Good agreement is observed with our earlier studies, with the autocorrelation function ratios revealing maxima as a function of time. Expectation values of bond lengths and bond angle show quasiperiodic oscillations that reflect repeated passages of the wavepacket at minima of the potential surfaces, thence being responsible for the HHG peaks.

Effect of Initial Vibrational-State Excitation on Subfemtosecond Photodynamics of Water.

We discuss the effect of initial vibrational-state excitation on the subfemtosecond photodynamics of water. Photoelectron spectra of Franck-Condon ionization to the (2)B1 state of the H2O(+) (D2O(+)) from the ground and several vibrationally excited states of the neutral are reported. Also calculated are ratios of the high-order harmonic generation (HHG) signals as a function of time for each initial vibrational state of the neutral molecule as predicted from the ratios of the square of the autocorrelation functions for D2O(+) and H2O(+). They reveal maxima as a function of time for each vibrational state of the neutral molecule. In turn, the HHG signals are found to be enhanced with vibrational excitation, with the calculated expectation values of the bond lengths and bond angle revealing quasiperiodic oscillations in time for all initial vibrational states of the neutral species. Although the bond lengths show only a marginal increase, the bond angle is found to be enhanced markedly by vibrational excitation, this being therefore responsible for the observed rise in the HHG signal.

Subfemtosecond quantum nuclear dynamics in water isotopomers.

Subfemtosecond quantum dynamics studies of all water isotopomers in the X̃ (2)B1 and à (2)A1 electronic states of the cation formed by Franck-Condon ionization of the neutral ground electronic state are reported. Using the ratio of the autocorrelation functions for the isotopomers as obtained from the solution of the time-dependent Schrödinger equation in a grid representation, high-order harmonic generation signals are calculated as a function of time. The results are found to be in agreement with the available experimental findings and with our earlier study for D2O(+)/H2O(+). Maxima are predicted in the autocorrelation function ratio at various times. Their origin and occurrence is explained by calculating expectation values of the bond lengths and bond angle of the water isotopomers as a function of time. The values so calculated for the (2)B1 and (2)A1 electronic states of the cation show quasiperiodic oscillations that can be associated with the time at which the nuclear wave packet reaches the minima of the potential energy surface, there being responsible for the peaks in the HHG signals.

Above- and below-threshold high-order-harmonic generation ofH2+in intense elliptically polarized laser fields

We present an ab initio three-dimensional precision calculation and analysis of high-order-harmonic generation (HHG) of the hydrogen molecular ion subject to intense elliptically polarized laser pulses by means of the time-dependent generalized pseudospectral method in two-center prolate spheroidal coordinates. The calculations are performed for the ground and first excited electronic states of ${\mathrm{H}}_{2}{}^{+}$ at the equilibrium internuclear separation $R=2$ a.u. as well as for the stretched molecule at $R=7$ a.u. The spectral and temporal structures of the HHG signal are explored by means of the wavelet time-frequency analysis. Several aspects of ellipticity-dependent dynamical behaviors are uncovered. We found that the production of above-threshold harmonics for nonzero ellipticity is generally reduced, as compared with linearly polarized fields. However, below-threshold harmonics still appear quite strong except when the polarization plane is perpendicular to the molecular axis. Weak even harmonics are detected in the HHG spectra of stretched molecules. This effect can be explained by the broken inversion symmetry due to dynamic localization of the electron density near one of the nuclei. Multiphoton resonance and two-center interference effects are analyzed for the exploration of the quantum origin of the predicted HHG spectral and dynamical behavior.

High-harmonic generation from a coherent superposition of electronic states: Controlling interference patterns via short and long quantum orbits

Laser pump-probe schemes are explored numerically from a molecular time-dependent Schr\"odinger equation for monitoring attosecond electron motion by high-order-harmonic generation (HHG) from a coherent superposition of electronic states. Varying the time delay between the probe and pump pulse on an attosecond time scale alters the HHG signal, leading to universal interference patterns. By using an extended three-step recollision model, we show that various regular interference patterns in the HHG spectra, including continuous harmonic frequency redshifts as a function of pump-probe delay, are related to interference between specific pairs of short or long quantum orbits. For small excited-state population (regime A), interferences are controlled by electron-tunneling times, whereas for equal populations (regime B), recollision time control dominates thus allowing for control of the HHG process by the pump-probe delay time and by electron-state populations. We show that each specific pattern is closely related to the interference between pairs of specific (short or long) quantum orbits originating from a specific electron bound state. One can generate a specific pattern by varying the populations in the coherent superposition, the laser intensity, wavelength, and ionization potential of the electronic states.

High-order harmonic generation from graphene: Strong attosecond pulses with arbitrary polarization

We explore high-order harmonic generation (HHG) from a graphene sheet exposed to intense femtosecond laser pulses based on the Lewenstein model. It is demonstrated that the HHG cutoff frequency increases with graphene size up to the classical limit for distant diatomic systems. In contrast to two-center systems, the cutoff frequency remains constant with increasing power of the harmonics as the graphene diameter extends beyond maximal electron excursion. It is shown that the extended nature of the graphene sheet allows for strong HHG signals at maximum cutoff for linearly as well as circularly polarized laser pulses, the latter opening for generation of strong circularly polarized attosecond pulses.

Comparison of high-order harmonic generation in uracil and thymine ablation plumes

We present studies of high-order harmonic generation (HHG) in laser ablation plumes of the ribonucleic acid nucleobase uracil and its deoxyribonucleic acid counterpart thymine. Harmonics were generated using 780 nm, 30 fs and 1300 nm, 40 fs radiation upon ablation with 1064 nm, 10 ns or 780 nm, 160 ps pulses. Strong HHG signals were observed from uracil plumes with harmonics emitted with photon energies >55 eV. Results obtained in uracil plumes were compared with those from thymine, which did not yield signs of harmonic generation. The ablation plumes of the two compounds were examined by collection of the ablation debris on a silicon substrate placed in close proximity to the target and by time-of-flight mass spectrometry. From this evidence we conclude that the differences in HHG signal are due to the different fragmentation dynamics of the molecules in the plasma plume. These studies constitute the first attempt to analyse differences in structural properties of complex molecules through plasma ablation-induced HHG spectroscopy.

PROPERTIES OF HIGH-ORDER HARMONIC GENERATION SIGNALS FROM SWITCHED-OFF ALIGNED MOLECULES

A quantum theory of intense-field pump-probe experiments proposed recently, which successfully described the highorder harmonic generation (HHG) signals of adiabatic and non-adiabatic aligned molecules, is applied here to investigate the HHG signals from molecules aligned by using switched-off schema. The results are presented in the two externally available control parameters: the delay time, td, between the pump and the probe pulse, and the relative angle, α, between their polarizations. To provide more succinct and clearer information of the beats involved in the signal, the signal is also presented in the frequency domain. The results show that the HHG signals of switched-off aligned molecules mimic those of the adiabatic and non adiabatic aligned molecules, and therefore, it can be used to determine some molecular properties, as powerful as non-adiabatic signal. In addition, the HHG signals of switched-off aligned molecules also have benefit, which is it gives a greater modulation depth.

Theory of intense-field dynamic alignment and high-order harmonic generation from coherently rotating molecules and interpretation of intense-field ultrafast pump-probe experiments

A quantum theory of intense-field pump-probe experiments proposed by us recently [F. H. M. Faisal et al., Phys. Rev. Lett. 98, 143001 (2007) and F. H. M. Faisal and A. Abdurrouf, Phys. Rev. Lett. 100, 123005 (2008)] is derived here fully and applied to investigate the phenomena of dynamic alignment and high-order harmonic generation (HHG) from coherently rotating linear molecules. The theory is developed from the basic quantum transition amplitude for the HHG and used to relate the Fourier transform (FT) of the expectation value of the dipole operator to the rate of emission of the HHG photons. It permits us to give analytical expressions for the HHG signals and their simultaneous dependence on the two externally available control parameters---the delay-time, ${t}_{d}$, between the pump and the probe pulse, and the relative angle, $\ensuremath{\alpha}$, between their polarizations. A relation between the basic ``one-molecule'' and the macroscopic ``many-molecule'' HHG signals is obtained from the phase-matching condition for HHG in an ideal medium. The requirement for the coherent HHG signal and the ``elastic'' molecular transition, in contrast to the ``inelastic'' transitions and the ``hyper-Raman'' emission, is discussed. The effect of the ``delayed'' probe pulse on the dynamic alignment induced by the pump-pulse, the mean rotational energy of the molecule during the period between the pump and the probe pulse, as well as a method of estimating the effective temperature of the molecules are analyzed. A ``revival theorem'' on the number of fractional ``revivals,'' equal to the lowest power of the ``cosine operator'' in the Hamiltonian of the system, times the maximum powers of the ``cosine-moments'' present in the signal, is derived and used to interpret the observed fractional revivals and their relative phases. A ``magic'' polarization angle ${\ensuremath{\alpha}}_{c}=\mathrm{arctan}\phantom{\rule{0.2em}{0ex}}\sqrt{2}\ensuremath{\approx}55\ifmmode^\circ\else\textdegree\fi{}$, at which the signals for all ${t}_{d}$ approach each other closely, is identified as a generic signature of a ${\ensuremath{\sigma}}_{g}$ symmetry of the active orbital. Similarly, the presence of a ``crossing neighborhood'' near ${\ensuremath{\alpha}}_{c}$ is shown to be a generic signature of an active ${\ensuremath{\pi}}_{g}$ orbital. At an operational angle ${\ensuremath{\alpha}}_{c}\ensuremath{\approx}55\ifmmode^\circ\else\textdegree\fi{}$ in the laboratory, a steady emission of high-order harmonic radiation from coherently rotating molecules with ${\ensuremath{\sigma}}_{g}$ orbital symmetry (e.g., ${\mathrm{N}}_{2}$) can be obtained. Finally, explicit numerical calculations are performed at specific experimental parameter values in the time domain as well as in the frequency domain. The results well reproduce all the salient features of the experimental observations for ${\mathrm{N}}_{2}$ and ${\mathrm{O}}_{2}$, and provide a unified theoretical interpretation of the same.

Alignment dependence of high-order harmonic generation from CO2

High-order harmonic generation (HHG) from aligned CO2 is studied within the framework of the strong-field approximation (SFA). Our results are in qualitative agreements with recent pump–probe experiments. The experimentally observed inverted modulation in HHG signals as a function of pump–probe delay time has previously been attributed to the quantum interference from the two oxygen centres. Our results, however, indicate that this is not necessary for the inverted modulation. The angular dependence of the HHG and the evolution of the HHG yield as functions of delay time are influenced strongly by the depletion of the ground state and, therefore, are sensitive to the probe laser intensity.

High-Order Harmonic Generation by Two Non-CollinearFemtosecond Laser Pulses in CO

We have experimentally studied the high order harmonic generation (HHG)in CO molecules by two femtosecond laser pulses using a pump–probetechnique. The delay time between two pulses is longer than the pulseduration, and the pump intensity is about 6.2×1014 W/cm2.It is found that the HHG signal is independent of the time delay in thepicosecond region, but it is dependent on the distance between thecentres of focuses of the two beams. The phase modulation induced bythe pump pulse is regarded to be responsible for this.