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Digest on Nonlinear Optics: Materials, Phenomena and Devices

Liouville Operator Theory Of Nonlinear Optical Susceptibilities

Paul L. Kelley, J. P. Taran

Published Jan 1, 1990• 0 citations
Publisher

Abstract

Expressions for the n-th order nonlinear susceptibility, x(n), are given using Liouville operators. When damping is included, the n-th order susceptibility operator is expressed as a product of propagators, the output polarization dipole operator, and n input dipole Liouville operators operating on the unperturbed density operator with a sum performed over the input frequency permutations. When damping can be neglected, the susceptibility can be reexpressed as a product of propagators, the n+1 dipole operators, and the unperturbed density operator with the sum performed over the permutations of the input frequencies and the output frequency.

DOI
10.1109/nlo.1990.695904

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Expressions for the n-th order nonlinear susceptibility, x(n), are given using Liouville operators. When damping is included, the n-th order susceptibility operator is expressed as a product of propagators, the output polarization dipole operator, and n input dipole Liouville operators operating on the unperturbed density operator with a sum performed over the input frequency permutations. When damping can be neglected, the susceptibility can be reexpressed as a product of propagators, the n+1 dipole operators, and the unperturbed density operator with the sum performed over the permutations of the input frequencies and the output frequency.
We have formulated expressions providing the number of terms (Y-G diagrams)1contributing to an n-th order susceptibility of a non-linear optical process. Obtaining an accurate count of these terms is important for diverse reasons: a) To accurately specify the number of terms needed to evaluate the nonlinear optical susceptibility either analytically or numerically, b) To help develop sequences of diagrammatic contributions which can be summed to provide analytical results for the strong interaction limit. An example is the two-level system for semiclassical and quantized fields, c) To treat two or more processes (terms) which could give rise to interference, resulting, for example, in extra resonances due to a lack of complete coherence, d) To generalize results from quantum electrodynamics to situations for which multiple photon interactions and zero point fluctuations associated with non-vacuum states need to be considered. We shall discuss the number of possible diagrams in terms of three factors, which we call a permutation factor relating to the number of possible frequencies that can be assigned to each interaction vertex, a commutation factor relating to interaction vertex assignment on the two sides of the diagram, and a state factor relating to the number of diagrams generated from an enumeration of specific material state assignments between interaction vertices. We will discuss the application of these results to several situations of interest including second harmonic generation, degenerate four-wave mixing, and resonant CARS (coherent anti-Stokes Raman scattering) in a four-level system. The relevance of the present results to experimental results which have been obtained will be discussed along with future directions which need to be addressed including the summability of diagrams under specific approximations, the calculation of expectation values, and the nonperturbative representation of processes.
1. Introduction 2. Quantum Dynamics in Hilbert Space 3. The Density Operator and Quantum Dynamics in Liouville Space 4. Quantum Electrodynamics, Optical Polarization, and Nonlinear Spectroscopy 5. Nonlinear Response Functions and Optical Susceptibilities 6. The Optical Response Functions of a Multilevel System with Relaxation 7. Semiclassical Simulation of the Optical Response Functions 8. The Cumulant Expansion and the Multimode Brownian Oscillator Model 9. Fluorescence, Spontaneous-Raman and Coherent-Raman Spectroscopy 10. Selective Elimination of Inhomogeneous Broadening Photon Echoes 11. Resonant Gratings, Pump-Probe, and Hole Burning Spectroscopy 12. Wavepacket Dynamics in Liouville Space The Wigner Representation 13. Wavepacket Analysis of Nonimpulsive Measurements 14. Off-Resonance Raman Scattering 15. Polarization Spectroscopy Birefringence and Dichroism 16. Nonlinear Response of Molecular Assemblies The Local-Field Approximation 17. Many Body and Cooperative Effects in the Nonlinear Response
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.
A quantum theory of nonlinear phenomena is considered in which account is taken of all contributions from electric and magnetic multipoles, without any multipolar approximation. Exact closed analytical expressions for the multipoles are first obtained from quantum electrodynamics by applying a generalization of the Power-Zienau transformation. The standard procedure is invoked to calculate general electromagnetic susceptibility tensors ${\ensuremath{\chi}}_{\ensuremath{\mu}\ensuremath{\nu}{\ensuremath{\alpha}}_{1}{\ensuremath{\beta}}_{1}\ensuremath{\cdots}{\ensuremath{\alpha}}_{n}{\ensuremath{\beta}}_{n}}^{[n]}$ (${\ensuremath{\omega}}_{1}, \dots{}, {\ensuremath{\omega}}_{n}$). Such a tensor is a shorthand for the electric and magnetic susceptibilities as well as mixed susceptibilities. By considering a relativistic theory based on the Dirac equation, the spin susceptibilities are automatically included. The formalism in this way allows for the computation of a wide range of electro- and magneto-optical effects to any order of perturbation and to any order of multipolar approximation.
A nonlinear response theory of a quantum-mechanical system undergoing arbitrary relaxation and interacting with fields, some of which may be strong enough to saturate optical transitions, is developed. Explicit expressions for second-order and third-order susceptibilities are obtained. If the fields are weak, then these expressions show the existence of additional resonant contributions to ${\ensuremath{\chi}}^{(n)}$ which arise due to inelastic collisions. Various applications of these ${\ensuremath{\chi}}^{(n)}$'s to modulation spectroscopy, four-wave mixing, and pump-probe experiments are discussed. The general structure suggests how the additional resonances can be used to determine inelastic rates in a Doppler-broadened medium. For saturating fields, ${\ensuremath{\chi}}^{(n)}$'s become dependent on the intensity of such fields and can be formally obtained from weak-field ${\ensuremath{\chi}}^{(n)}$'s, provided proper identification of unperturbed eigenfunctions, eigenvalues, and relaxation times is made. Such intensity-dependent ${\ensuremath{\chi}}^{(n)}$'s have resonant denominators which lead to resonances at Rabi frequencies and submultiples of these frequencies, the widths of which are also dependent on the strength of the field. In the special cases intensity-dependent ${\ensuremath{\chi}}^{(1)}$ agrees with the work of Cohen-Tannoudji and co-workers.