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

Abstract

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.