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Chemical Physics Letters

Floquet-liouville super-matrix approach for multiphoton non-linear optical processes in intense laser fields

Tak‐San Ho, Shih‐I Chu

Published Dec 1, 1985• 9 citations
Publisher
DOI
10.1016/0009-2614(85)80231-1

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A Floquet-Liouville supermatrix (FLSM) approach is presented for nonperturbative treatment of the time development of the density-matrix operator of atoms and molecules exposed to intense polychromatic fields. By extending the many-mode Floquet theory (MMFT) recently developed, the time-dependent Liouville equation for the density matrix of quantum systems undergoing relaxations (due to radiative decays and collisional dampings, etc.) can be transformed into an equivalent time-independent non-Hermitian FLSM eigenvalue problem. This yields a numerically stable and computationally efficient approach for the unified treatment of nonresonant and resonant, one- and multiple-photon, steady-state and transient phenomena in nonlinear optical processes, much beyond the conventional rotating-wave-approximation (RWA) method. Connections of the FLSM approach to the MMFT in the limit of zero relaxations are also made, providing the understanding of the physical significance of FLSM supereigenvalues and eigenvectors. In addition to the exact FLSM formalism, we have also presented higher-order perturbative results, based on the extension of the generalized Van Vleck (GVV) nearly degenerate perturbation theory, appropriate for somewhat weaker fields and near-resonant processes, but beyond the RWA limit. The implementation of the GVV method in the time-independent Floquet-Liouvillian allows the reduction of the infinite-dimensional FLSM into a finite-dimensional GVV-Liouville matrix, from which essential analytical results are readily obtained. As an illustration of the usefulness of the new formalism, we extend both the FLSM and the GVV methods to a formal study of the multiphoton-induced resonance fluorescence spectra of two-level systems subject to purely radiation relaxations. Both the time-averaged power spectrum and the time-dependent physical spectrum are exploited in details, and novel new features in intense fields are pointed out.

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Floquet–Liouville supermatrix approach. II. Intensity-dependent generalized nonlinear optical susceptibilities

Kwanghsi Wang, Shih‐I Chu

We present a practical nonperturbative method for exact treatment of intensity-dependent generalized nonlinear optical susceptibilities χ(ω) in intense polychromatic fields, valid for arbitrary laser intensities, detunings, and relaxation. By means of the many-mode Floquet theory, the time-dependent Liouville equation can be transformed into an equivalent time-independent infinite-dimensional Floquet–Liouville supermatrix (FLSM) eigenvalue problem. It is then shown that the nonlinear optical susceptibilities χ(ω) can be completely determined simply from the supereigenvalues and eigenfunctions of the Floquet–Liouvillian L̂F. In addition to this exact FLSM approach, we have also presented higher-order perturbative results, based on the extension of the Salwen’s nearly degenerate perturbation theory, appropriate for somewhat weaker fields and near-resonant multiphoton processes, but beyond the conventional perturbative or rotating wave approximation (RWA). In the case of two-level systems, for example, the implementation of Salwen’s method in the time-independent L̂F allows the reduction of the infinite-dimensional FLSM into a 4×4 dimensional effective Hamiltonian, from which essential analytical formulas for intensity-dependent χ(ω) can be obtained. These methods are applied to a detailed study of intensity-dependent spectral line shapes (such as hole burning and extra resonance peaks at the line center, and the effects of saturation, detuning, and radiative and collisional damping, etc.) and subharmonic structures in nonlinear multiple wave mixings χ[(m+1)ω1−mω2] for two-level systems in intense linearly polarized bichromatic fields.

The Journal of Chemical Physics•Mar 15, 1987•30 citations
A Floquet-Liouville supermatrix (FLSM) approach is presented for nonperturbative treatment of the time development of the density-matrix operator of atoms and molecules exposed to intense polychromatic fields. By extending the many-mode Floquet theory (MMFT) recently developed, the time-dependent Liouville equation for the density matrix of quantum systems undergoing relaxations (due to radiative decays and collisional dampings, etc.) can be transformed into an equivalent time-independent non-Hermitian FLSM eigenvalue problem. This yields a numerically stable and computationally efficient approach for the unified treatment of nonresonant and resonant, one- and multiple-photon, steady-state and transient phenomena in nonlinear optical processes, much beyond the conventional rotating-wave-approximation (RWA) method. Connections of the FLSM approach to the MMFT in the limit of zero relaxations are also made, providing the understanding of the physical significance of FLSM supereigenvalues and eigenvectors. In addition to the exact FLSM formalism, we have also presented higher-order perturbative results, based on the extension of the generalized Van Vleck (GVV) nearly degenerate perturbation theory, appropriate for somewhat weaker fields and near-resonant processes, but beyond the RWA limit. The implementation of the GVV method in the time-independent Floquet-Liouvillian allows the reduction of the infinite-dimensional FLSM into a finite-dimensional GVV-Liouville matrix, from which essential analytical results are readily obtained. As an illustration of the usefulness of the new formalism, we extend both the FLSM and the GVV methods to a formal study of the multiphoton-induced resonance fluorescence spectra of two-level systems subject to purely radiation relaxations. Both the time-averaged power spectrum and the time-dependent physical spectrum are exploited in details, and novel new features in intense fields are pointed out.
We present a practical nonperturbative method for exact treatment of intensity-dependent generalized nonlinear optical susceptibilities χ(ω) in intense polychromatic fields, valid for arbitrary laser intensities, detunings, and relaxation. By means of the many-mode Floquet theory, the time-dependent Liouville equation can be transformed into an equivalent time-independent infinite-dimensional Floquet–Liouville supermatrix (FLSM) eigenvalue problem. It is then shown that the nonlinear optical susceptibilities χ(ω) can be completely determined simply from the supereigenvalues and eigenfunctions of the Floquet–Liouvillian L̂F. In addition to this exact FLSM approach, we have also presented higher-order perturbative results, based on the extension of the Salwen’s nearly degenerate perturbation theory, appropriate for somewhat weaker fields and near-resonant multiphoton processes, but beyond the conventional perturbative or rotating wave approximation (RWA). In the case of two-level systems, for example, the implementation of Salwen’s method in the time-independent L̂F allows the reduction of the infinite-dimensional FLSM into a 4×4 dimensional effective Hamiltonian, from which essential analytical formulas for intensity-dependent χ(ω) can be obtained. These methods are applied to a detailed study of intensity-dependent spectral line shapes (such as hole burning and extra resonance peaks at the line center, and the effects of saturation, detuning, and radiative and collisional damping, etc.) and subharmonic structures in nonlinear multiple wave mixings χ[(m+1)ω1−mω2] for two-level systems in intense linearly polarized bichromatic fields.