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Physical Review

Line Shapes of Resonant, Nonlinear, Paramagnetic Susceptibilities

Harro G. Andresen, H. Welling

Published Jul 5, 1965• 1 citation
Publisher

Abstract

Line shapes for both the absolute value and the real and imaginary components of the nonlinear, paramagnetic susceptibility ${\ensuremath{\chi}}_{\mathrm{NL}}(2\ensuremath{\omega})$ of a ruby crystal have been measured as a function of the magnetic field and crystal orientation at room temperature and "resonant" conditions, where the $X$-band driving frequency coincides with or is very close to the resonance frequencies of an equidistant or nearly equidistant paramagnetic three-level system. Though all measured line shapes were in good qualitative agreement with theoretical line-shape expressions, derived under the assumption of homogeneously broadened, Lorentzian-shaped, single-quantum resonances, a quantitative comparison between theory and experiment exhibited systematic discrepancies, such as too slowly decaying wings for the theoretical curves and too large theoretical amplitudes at "partly resonant" conditions for nearly equidistant three-level schemes. Line-shape measurements for single-quantum resonances of the linear susceptibility contradicted the Lorentzian line-shape assumption, but could satisfactorily be matched by a Gauss-type line shape. Although it was possible to fit the measured single-quantum resonance line shapes also by the convolution of a narrow, Lorentzian line shape with a proper combination of macroscopic inhomogeneous line-broadening effects such as Gaussian distributions of crystal $c$-axis orientations and crystal strains, the equivalent procedure for the nonlinear susceptibility resonances did not remove the discrepancy between theory and experiment. Since the approach towards thermal equilibrium within tightly coupled spin systems in solids has, in general, to be described by nonexponential relaxation processes, the line-shape expressions for the nonlinear susceptibility ${\ensuremath{\chi}}_{\mathrm{NL}}(2\ensuremath{\omega})$ were phenomenologically generalized and formulated in terms of non-Lorentzian-shaped, homogeneous, single-quantum resonances. By using these generalized line-shape expressions, it was possible to achieve a satisfactory quantitative agreement with all experimental results for both the line shapes of single-quantum, linear-susceptibility resonances as well as for the line shapes of the absolute value and the real and imaginary components of the nonlinear-susceptibility resonances.

DOI
10.1103/physrev.139.a321

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Line shapes for both the absolute value and the real and imaginary components of the nonlinear, paramagnetic susceptibility \ensuremathχNL(2\ensuremathω){\ensuremath{\chi}}_{\mathrm{NL}}(2\ensuremath{\omega})\ensuremathχNL​(2\ensuremathω) of a ruby crystal have been measured as a function of the magnetic field and crystal orientation at room temperature and "resonant" conditions, where the XXX-band driving frequency coincides with or is very close to the resonance frequencies of an equidistant or nearly equidistant paramagnetic three-level system. Though all measured line shapes were in good qualitative agreement with theoretical line-shape expressions, derived under the assumption of homogeneously broadened, Lorentzian-shaped, single-quantum resonances, a quantitative comparison between theory and experiment exhibited systematic discrepancies, such as too slowly decaying wings for the theoretical curves and too large theoretical amplitudes at "partly resonant" conditions for nearly equidistant three-level schemes. Line-shape measurements for single-quantum resonances of the linear susceptibility contradicted the Lorentzian line-shape assumption, but could satisfactorily be matched by a Gauss-type line shape. Although it was possible to fit the measured single-quantum resonance line shapes also by the convolution of a narrow, Lorentzian line shape with a proper combination of macroscopic inhomogeneous line-broadening effects such as Gaussian distributions of crystal ccc-axis orientations and crystal strains, the equivalent procedure for the nonlinear susceptibility resonances did not remove the discrepancy between theory and experiment. Since the approach towards thermal equilibrium within tightly coupled spin systems in solids has, in general, to be described by nonexponential relaxation processes, the line-shape expressions for the nonlinear susceptibility \ensuremathχNL(2\ensuremathω){\ensuremath{\chi}}_{\mathrm{NL}}(2\ensuremath{\omega})\ensuremathχNL​(2\ensuremathω) were phenomenologically generalized and formulated in terms of non-Lorentzian-shaped, homogeneous, single-quantum resonances. By using these generalized line-shape expressions, it was possible to achieve a satisfactory quantitative agreement with all experimental results for both the line shapes of single-quantum, linear-susceptibility resonances as well as for the line shapes of the absolute value and the real and imaginary components of the nonlinear-susceptibility resonances.
We explain the utility of third-order nonlinear susceptibility, χ (3) (ω 2 ; ±ω 1 , ∓ω1{\mp}\omega_{1}∓ω1​, ω 2 ), for the observation of resonances with homogeneous widths from inhomogeneously broadened distributions in three-level systems related to Raman or two-photon allowed electronic transitions. The line narrowing and high resolution capabilities are demonstrated theoretically using a model for inhomogeneous distributions of transitions in which we regard an absorption spectrum as the composition of some inhomogeneously broadened substructures. The experimental methods to separate a signal having frequency ω 2 from ingoing beam at the same frequency are outlined.
Two-dimensional (2D) line shapes derived from four types of coherent third-order nonlinear spectroscopies as a function of two independent time or frequency variables are compared. The signals scattered into two wave-vector-matching directions in the limits of high-time or high-frequency resolution are calculated for an inhomogeneously broadened ensemble of two-level systems with phenomenological dephasing and population relaxation times. While one-dimensional (1D) line shape analysis for this model is ambiguous, the contours of these 2D line shapes are shown to be characteristic of the relative time scales of the three relevant line-broadening parameters. These broadening mechanisms are also apparent by comparing 2D spectral profiles along the diagonal and antidiagonal frequency axes. For time domain experiments, in analogy with NMR correlation spectroscopy, the radiated signal is observed as a function of the two coherence periods and is 2D Fourier transformed to obtain a line shape. For experiments based on the photon echo, the ellipticity of the absolute value 2D line shape is related to the ratio of the inhomogeneous width to the dephasing rate, whereas experiments based on the transient grating method are shown to be functionally 1D. For frequency domain experiments, the contours of the 2D line shape can yield information on static and both dynamic broadening mechanisms. While the addition of spectral diffusion to such calculations will modify these results for frequency domain experiments, characterizing these dynamics in the time domain naturally leads to a 3D experiment analogous to the spin diffusion experiment in NMR.
The paramagnetic resonance properties of Cr+++ions in Al2O3(ruby) were investigated theoretically and experimentally in order to obtain information necessary for the application of this material as active material in a three-level solid-state maser (3LSSM). Numerically computed energy levels, together with their associated eigenvectors, are presented as a function of applied magnetic field for various orientations of the magnetic field with respect to the crystalline symmetry axis. A more detailed discussion is devoted to energy levels, eigenvectors and transition probabilities at angles O°, 54.74° and 90°, where certain simple relations and symmetries hold. Paramagnetic spectra for signal frequencies between 5 and 24 kmc are shown; agreement between computed and measured resonance fields is satisfactory.
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.
The optical Hanle effect (light-shift-induced zero-field level crossing) has been investigated theoretically and experimentally. In the case of a $J=0\ensuremath{\rightarrow}1\ensuremath{\rightarrow}0$ three-level system and of a $J=0\ensuremath{\rightarrow}1$ two-level system, the resonance line shape has been obtained using a nonlinear-response-function method. Approximate expressions, valid under the conditions for the observation of light shifts, are obtained and compared with exact calculations as well as with the corresponding expression for the magnetic Hanle effect. As concerns the weak laser beam which produces the coherent excitation of atomic substates, we have calculated the linear response in the limiting case of monochromatic and of broad-band excitation. The experimental investigation has been performed for the Ba resonance line using two cw dye lasers at $\ensuremath{\lambda}=5535$ \AA{} and an atomic beam. The absorption- and dispersion-shaped level crossing resonances have been observed, for narrow- and broad-band excitation, by varying the power of the strong nonresonant laser beam. We have checked that the line shapes are only dependent upon the reduced variable $\frac{{P}_{L}}{\ensuremath{\delta}}$ (PL{P}_{L}PL​; beam power; $\ensuremath{\delta}$: frequency detuning). Using the theoretical expressions, we have tried to fit the experimental curves. Provided that the Gaussian distribution of the beams is taken into account, we have obtained a good agreement between theory and experiment for the whole set of experimental data.
We analyze the absorption of microwaves by the Heisenberg-Ising chain combining exact calculations, based on the integrability of the model, with numerical calculations. Within linear response theory, the absorbed intensity is determined by the imaginary part of the dynamical susceptibility. The moments of the normalized intensity can be used to define the shift of the resonance frequency induced by the interactions and the linewidth independently of the shape of the spectral line. These moments can be calculated exactly as functions of temperature and strength of an external magnetic field, as long as the field is directed along the symmetry axis of the chain. This allows us to discuss the linewidth and the resonance shift for a given magnetic field in the full range of possible anisotropy parameters. For the interpretation of these data, we need a qualitative knowledge of the line shape, which we obtain from fully numerical calculations for finite chains. Exact analytical results on the line shape are out of reach of current theories. From our numerical work, we could extract, however, an empirical parameter-free model of the line shape at high temperatures, which is extremely accurate over a wide range of anisotropy parameters, and is exact at the free-fermion and isotropic points. Another prediction of the line shape can be made in the zero-temperature and zero magnetic field limits, where the sufficiently anisotropic model shows strong absorption. For anisotropy parameters in the massive phase, we derive the exact two-spinon contribution to the spectral line. From the intensity sum rule, it can be estimated that this contribution accounts for more than 80% of the spectral weight if the anisotropy parameter is moderately above its value at the isotropic point.