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Physical Review A

Effects of arbitrary relaxation and strong-field dressing of energy levels on nonlinear optical susceptibilities

Manoj Kumar, G. S. Agarwal

Published Mar 1, 1986• 22 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.33.1817

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Physical Review A•Jan 1, 1978•146 citations
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 \ensuremathχ(n){\ensuremath{\chi}}^{(n)}\ensuremathχ(n) which arise due to inelastic collisions. Various applications of these \ensuremathχ(n){\ensuremath{\chi}}^{(n)}\ensuremathχ(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, \ensuremathχ(n){\ensuremath{\chi}}^{(n)}\ensuremathχ(n)'s become dependent on the intensity of such fields and can be formally obtained from weak-field \ensuremathχ(n){\ensuremath{\chi}}^{(n)}\ensuremathχ(n)'s, provided proper identification of unperturbed eigenfunctions, eigenvalues, and relaxation times is made. Such intensity-dependent \ensuremathχ(n){\ensuremath{\chi}}^{(n)}\ensuremathχ(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 \ensuremathχ(1){\ensuremath{\chi}}^{(1)}\ensuremathχ(1) agrees with the work of Cohen-Tannoudji and co-workers.
Optical non-linearities and spectroscopy in fields that are strongly modulated and also intense are examined by obtaining the solution of the optical Bloch equations in modulated fields as a matrix continued fraction. Numerical results for emission and absorption in such fields are presented. The saturation optical susceptibilities and the different types of spectra show resonances at the Rabi frequency if the modulation index is small. With increase in this index, additional resonant features appear at submultiples of the Rabi frequency. These additional features become more and more prominent as the modulation index increases further. For modulated fields on resonance, a dressed-atom formulation is developed to explain the results obtained from the numerical evaluation of the continued fraction. The results are also applicable to non-linear spectroscopic measurements such as Raman gain when the probe field is modulated.
Nonlinear perturbative susceptibilities have been used successfully to analyze experiments in nonlinear optics for two decades. By its very nature, a perturbative approach is limited to a small value of the expansion parameter, and nonlinear susceptibilities are not useful for strong saturating fields. Presently, a theory is presented for the treatment of FWM which is applicable to arbitrary strength of the input fields, both on and off resonance. The field harmonic time dependences are removed by a transformation to the proper rotating frame, and the resulting eigenvalue problem is solved numerically. Several new effects are predicted. For strong fields resonance frequencies are shifted and lines are split (similar to the ac Stark effect). When such a split line is acted on by another strong field (a phenomenon not studied so far), a new effect of stirring is observed, which results in the broadening and merging of the two separate components. A new extra resonance appears which is induced by the strong field and is present even in the absence of any collisions or other dephasing processes or of laser phase fluctuation. This new extra resonance is a new member in the family of extra resonances which were discussed in various previous publications.
The propagation of a weakly modulated light beam through a nonlinear material is treated. The optical field representing such a beam consists of a strong carrier-frequency component and two weak, symmetrically displaced sidebands that combine to form a field, which may be purely amplitude modulated (AM), frequency modulated (FM), or some combination of these two modulation forms. It is shown that, for any optical nonlinearity, two modulation forms exist that have the property that the form of modulation is invariant under propagation of the beam. If a modulation form other than one of these natural modes is injected into the nonlinear medium, the modulation form will change as the beam propagates, asymptotically approaching the natural mode that experiences the lower attenuation. Explicit expressions for these natural modes are presented for the case in which the nonlinear medium can be modeled as a collection of two-level atoms. For the special case of an on-resonance pump beam, the natural modes correspond to pure amplitude modulation and pure frequency modulation. This formalism provides a general description of saturation spectroscopy for both AM and FM fields. Formulas are derived for the rate at which a pure FM beam is converted to an AM beam owing to its interaction with a two-level atomic medium. We also consider the variation that is due to propagation of the depth of modulation of an AM wave interacting resonantly with two-level atoms. The formalism predicts the existence of spectral features whose shape depends sensitively on the internal relaxation processes of the material. Experimental spectra are presented for ruby, alexandrite, and fluorescein in glass and are interpreted.
The influence of nonlinear interference processes and of changes in populations in strong electromagnetic fields, and also of the Doppler broadening on resonant four-wave frequency mixing is investigated for a system of Raman-type transitions. The case in which a quantum system interacts with one weak field and two strong fields is considered. Each of the energy levels is assumed to be perturbed by just one strong field. It is shown that the Doppler broadening alters drastically the contributions of the populations of the various levels to the investigated process and that this can alter by several orders of magnitude and nonlinear susceptibilities averaged over the velocities and influence significantly the selection of optimal configurations of the transitions and of the experimental conditions for effective resonant four-wave conversion of the radiation frequencies. A numerical analysis of the derived expressions is used to account for the experimentally observed considerable differences between the spectral dependences of the output power and the conclusions drawn from perturbation theory.
The nonlinear response of a two-level system interacting with two pump beams of different frequencies is studied with a view to examine several recent experiments on nonlinear mixing of high orders, instabilities in homogeneously broadened lasers, and holes in homogeneous lines. Such a nonlinear response is calculated with use of the continued fractions given in our earlier work [J. Opt. Soc. Am. B1, 164 (1984)]. Saturation effects lead to the appearance of many structures at submultiples of the Rabi frequency. The origin of these subharmonics in nonlinear-response functions is explained. It is shown that collisions not only lead to hole burning at the line center for weak fields but can also lead in moderate fields to the appearance of structures at subharmonics of the Rabi frequency. Existence of collision-induced coherences similar to those in four-wave mixing is shown in six-wave mixing. Exact results for the degenerate nonlinear mixing of arbitrary order are presented for arbitrary values of field strengths. Saturating fields create holes at the line center of such nonlinear mixing signals. Gain profiles of two modes symmetrically displaced from line center are also given. Numerical results, with complete account of saturation and collisions, for a variety of nonlinear-response functions are presented.
A comprehensive theory of a new type of three-wave optical mixing (${\ensuremath{\omega}}_{3}=2{\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2}$) is given for the spectroscopic purpose of determining ultra-short relaxation times (picoseconds or less) associated with excited states of condensed matter. Two incident light frequencies ${\ensuremath{\omega}}_{1}$ and ${\ensuremath{\omega}}_{2}$ are chosen so that they are both resonant with an inhomogeneously broadened optical transition and $|{\ensuremath{\omega}}_{2}\ensuremath{-}{\ensuremath{\omega}}_{1}|$ is in the vicinity of inverse relaxation times. The theory is developed for its major application to broad-band electronic transitions. The nonlinear susceptibility ${\ensuremath{\chi}}_{R}^{(3)}$ for this process (named resonant Rayleigh-type mixing) is calculated by a density-matrix formalism with the model of a two-level atomic system incorporating the distribution of resonance frequencies and longitudinal (T1{T}_{1}T1​) and transverse (T2{T}_{2}T2​) relaxation times. The result shows a frequency characteristic depending only on T1{T}_{1}T1​, T2{T}_{2}T2​, and ${\ensuremath{\omega}}_{2}\ensuremath{-}{\ensuremath{\omega}}_{1}$ in a broad-band limit, which serves for the determination of T1{T}_{1}T1​ and T2{T}_{2}T2​ in the frequency domain. The analysis is further extended to include the various effects as follows. The calculation of the saturation effect in the lowest order reveals that it modifies the shape of the frequency characteristic of the nonlinear susceptibility but does not affect seriously the determination of relaxation times. The effect of spectral cross relaxation within the inhomogeneous broadening is incorporated by a generalized density-matrix formalism. The resultant ${\ensuremath{\chi}}_{R}^{(3)}$ consists of two terms. The dominant term is the same as before except that T1{T}_{1}T1​ is replaced by a combined relaxation time ${T}_{1}^{\ensuremath{'}}={({T}_{1}^{\ensuremath{-}1}+{T}_{3}^{\ensuremath{-}1})}^{\ensuremath{-}1}$, where T3{T}_{3}T3​ is the cross-relaxation time. The other term originates from the inverse spectral-diffusion process. The effect of other energy levels located between the two levels under study is analyzed with a simple three-level model. The resultant ${\ensuremath{\chi}}_{R}^{(3)}$ also consists of two terms, the dominant one being the same as that for the two-level model except that ${T}_{1}^{\ensuremath{-}1}$ should be interpreted as total population decay rate of the upper level. For accurate derivation of ${\ensuremath{\chi}}_{R}^{(3)}$ from experiment, we discuss the interference between the resonant and nonresonant susceptibility terms and the light-wave propagation effect. Finally, the close analogy between this type of optical mixing and the photon echo is discussed.