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Physical Review A

Nonlinear Optical Polarization and Relaxation Phenomena in Dilute Gases

A. Ben‐Reuven, Lewis S. Klein

Published Aug 1, 1971• 31 citations
Publisher

Abstract

The Kubo-Zwanzig-Fano relaxation method is extended to the problem of a homogeneous dilute system interacting with arbitrarily strong fields composed of several monochromatic modes, neglecting breakdown and spatial correlations. The Von Neumann equation for the density matrix is solved for classical fields, using a procedure in which the harmonics numbers are treated as the classical analog (in Liouville space) of the quantum fields occupation-number representation. This solution is adapted to the case where memory effects during the absorption or emission of a photon can be neglected. Two examples illustrate the method, which is generally applicable in such problems as saturation, double resonance, and frequency mixing involving the nonlinear response of independent molecules.

DOI
10.1103/physreva.4.753

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Journal of the Optical Society of America B•Nov 1, 1997•15 citations
The Kubo-Zwanzig-Fano relaxation method is extended to the problem of a homogeneous dilute system interacting with arbitrarily strong fields composed of several monochromatic modes, neglecting breakdown and spatial correlations. The Von Neumann equation for the density matrix is solved for classical fields, using a procedure in which the harmonics numbers are treated as the classical analog (in Liouville space) of the quantum fields occupation-number representation. This solution is adapted to the case where memory effects during the absorption or emission of a photon can be neglected. Two examples illustrate the method, which is generally applicable in such problems as saturation, double resonance, and frequency mixing involving the nonlinear response of independent molecules.
The effect of a strong perturbing radiation field on spectral lines is studied using an extension of the Kubo-Zwanzig-Fano relaxation formalism. The problem of a dilute gas interacting with a strong perturbing field and a weak probe field is solved, assuming a separation of collisional and radiative processes, by introducing a classical-field renormalization technique. The solution is formulated entirely in terms of linear-response functions. This method is applied to the problem of the three-level absorber perturbed by a strong field and probed by a weak field. Multiphoton transitions, double resonance, and dynamic Stark effects can be treated as manifestations of this three-level problem.
A new theoretical approach to the problem of non-linear polarisation of matter is suggested. The density matrix equation for a low-density gas in a strong light field has been solved without using the phenomenological relaxation constants. Use is made of the general Hamiltonian that explicitly involves, besides the interaction of atoms with the external field, also their interactions with each other and with the vacuum electromagnetic field. The equation is solved by the canonical transformation method (a variant of the Bogolyubov-Mitropolski method of averages). It is generalised and used in the more complicated case of interacting multilevel atoms (molecules) in a strong light field. An expression for the cubic susceptibility is obtained, enabling analysis of the resonant saturation in both stationary and non-stationary cases. The proposed method provides correct expressions for the dynamic Stark shifts, amplitudes of multiphoton transitions and of radiational collisions, as well as for effective interatomic interactions caused by photon exchange. The proposed theoretical approach can also be used in studying non-linear optical processes in dense gases, liquids and solids.
Liouville-space (reduced-density-operator) descriptions are developed for resonant and coherent electromagnetic interactions of quantized electronic systems, taking into account environmental decoherence and relaxation phenomena. Applications of interest include electromagnetically-induced transparency and related pump-probe optical phenomena in many-electron atomic systems (in electron-ion beam interactions, gases, and high-temperature plasmas) and semiconductor materials (bulk crystals and nanostructures). Time-domain (equation-of-motion) and frequency-domain (resolvent-operator) formulations are developed in a unified manner. The standard Born (lowest-order perturbationtheory) and Markov (short-memory-time) approximations are systematically introduced within the framework of the general non-perturbative and non-Markovian formulations. A preliminary semiclassical description of the entire electromagnetic interaction is introduced. Compact Liouville-space operator expressions are derived for the linear and the general (n'th order) non-linear electromagnetic-response tensors occurring in a perturbation-theory treatment of the semiclassical electromagnetic interaction. These expressions can be evaluated for coherent initial electronic excitations and for the full tetradic-matrix form of the Liouville-space self-energy operator representing the environmental interactions in the Markov approximation. Intense-field electromagnetic interactions are treated by means of an alternative, non-perturbative method, which is based on a Liouville-space Floquet-Fourier representation of the reduced density operator. Electron-electron quantum correlations are treated by the introduction of a cluster decomposition of the reduced density operator and a coupled hierarchy of reduced-density-operator equations.
The nonlinear interaction of light with matter is described from a quantum-statistical point of view. The phenomena of two-photon emission and two-photon absorption including both the single- and two-mode cases and the Raman effect are discussed in detail. A master equation for the density operator of the light fields alone is derived. This operator equation is converted to a c number equation and analytic solutions are obtained for the diagonal matrix elements of the density operator in the Fock representation. No linearizing approximation is introduced. These solutions allow one to compute the moments of the photon distribution for the above nonlinear processes.
The Liouville equation for the density matrix can be recast so that the external field terms which couple density matrix elements are second order in the field. This approach is shown to have a profound effect on the analysis of strong field resonant nonlinear optical problems. First, in the case where parity or appropriate field restrictions apply, there is a reduction of the number of coupled equations which must be solved by at least a factor of two. Second, this reformulation naturally leads to radiative renormalization terms which are directly related to saturation, Stark shifts and Rabi splittings. The renormalization is equivalent to a Dyson equation analysis. We use this simple formalism to obtain exact solutions to a number of nonlinear optical problems including cases where there are two or more strong, resonant fields. These problems include: 1) two levels, two strong fields, 2) three levels, two strong fields, and 3) four levels, three and four strong fields. Discussed in detail are the two level case and the three and four level cases where weak fields couple the ground state to other states. We use these results to discuss the physics of optically induced transparency, population beats, optical switching in quantum wells, and fully resonant coherent anti-Stokes Raman scattering (CARS).
To formulate a description of nonlinear polarization spectroscopy that is valid over a wide range of pump intensities, we present an approach that considers any order of the external fields. The Liouville von Neumann equation governing the dynamics of a multilevel molecular system is made tractable by expansion of the particular density matrix into powers of the carrier waves of the external fields. For a two-level system a solution of the resulting set of equations to first order of the probe field but to all orders of pump field is demonstrated. A remarkable change in the homogeneously broadened line shape is obtained at high pump intensities, caused by the optical Stark effect among others. Improvements over results derived by other authors are discussed.