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Soviet Journal of Low Temperature Physics

Low-temperature anomalies of the dynamics of a paramagnetic crystal resonator

Almas F. Sadreev

Published Aug 1, 1979• 0 citations
Publisher

Abstract

The dynamics of a paramagnetic system (S = 1/2), thermodynamically equilibrium-linked with the radiation field of a resonator is considered by the methods of the quantum theory of fields. The spectrum of the resonance frequencies of the system is found for two relations of the size of the paramagnetic sample L and the wavelengths of the radiation field k−1. It is shown that when L ≪ k−1 Fain’s result, Eq. (1), is valid for the intrinsic line width, if the Zeeman frequency ω0 ≫ck. With L ≳ k−1, (the case of a crystal resonator), and kT>ω0>ρSμB2 (ρS is the paramagnetic ion density in the crystal), the temperature behavior of the intrinsic width is described by formula (33). It is shown that the critical temperature Tc is the temperature of the Bose-Einstein condensation of photons of the resonator radiation field, which is determined only by the paramagnetic ion density in the sample. Estimates are made of the intrinsic broadening of the Zeeman line in certain paramagnetic crystal resonators with impurities of rare-earth ions.

DOI
10.1063/10.0029908

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Physical Review•Jul 5, 1965•1 citation
The dynamics of a paramagnetic system (S = 1/2), thermodynamically equilibrium-linked with the radiation field of a resonator is considered by the methods of the quantum theory of fields. The spectrum of the resonance frequencies of the system is found for two relations of the size of the paramagnetic sample L and the wavelengths of the radiation field k−1. It is shown that when L ≪ k−1 Fain’s result, Eq. (1), is valid for the intrinsic line width, if the Zeeman frequency ω0 ≫ck. With L ≳ k−1, (the case of a crystal resonator), and kT>ω0>ρSμB2 (ρS is the paramagnetic ion density in the crystal), the temperature behavior of the intrinsic width is described by formula (33). It is shown that the critical temperature Tc is the temperature of the Bose-Einstein condensation of photons of the resonator radiation field, which is determined only by the paramagnetic ion density in the sample. Estimates are made of the intrinsic broadening of the Zeeman line in certain paramagnetic crystal resonators with impurities of rare-earth ions.
It is suggested that the combined effect of a resonant magnetic field (dynamic Zeeman effect) and of the fluctuations due to spin-spin and spin-lattice interactions could give rise to inelastic components in the emission spectrum of a paramagnetic system. It is also shown that a similar effect can be expected in the second harmonic emission. It is found that if the Rabi frequency is much larger than the relaxation rates, both the resonant and the second harmonic spectra show the two side lines typical of the atomic resonant fluorescence. The results are illustrated by means of a semiclassical model. Plots of expected spectra as a function of the frequency are given for some realistic parameters of paramagnetic samples.
Studies of the resonant interaction of radiation from a liquid-nitrogen-cooled ruby laser with a ruby crystal in the temperature range 4.2°–100°K are described. The interaction was measured by observation of the change which occurs in the Cr3+ ground state EPR absorption on laser irradiation. At low radiation intensities, the temperature dependence of this change is shown to be determined by the well-known thermal tuning behavior of the R lines. Precise measurement of the relative laser and ruby-sample absorption frequencies by this technique of double-resonance spectroscopy is described. Under optical saturation conditions, additional temperature-dependent pumping effects occur at low temperatures. In particular, as the ruby temperature is increased from 4.2° to 40°K, the laser-induced change in EPR absorption is observed to undergo a large increase. This increase cannot be explained by a relative thermal shift of the laser and absorption frequencies which is known to be very small in this temperature range. The phenomena are attributed in part to the temperature-dependent homogeneous linewidth which determines the widths of the holes burnt into the inhomogeneous strain-broadened R lines. Calculations indicate that the related temperature-dependent thermalization process which redistributes excited Cr3+ among the 2E levels quantitatively accounts for most of the increase.
The subject of the review is the diffraction of an electromagnetic wave on a periodic condensed matter where the Bragg condition is satisfied at the frequency of resonance excited in this medium. The Bragg systems known as resonant photonic crystals in general have been considered, and the propagation, reflection, transmission, and diffraction of electromagnetic radiation in different objects—(I) periodic quantum-well structures near the exciton resonance, (II) optical lattices of atoms cooled in a laser field, and (III) bulk crystals and multilayers with gamma-ray resonance intranuclear transitions—have been described in a unified context. The main attention has been paid to the steady-state linear diffraction, including resonant reflection and transmission, which is the best studied and allows a comparison of the three aforementioned systems with the aim of revealing specific characteristics and common features. A characteristic common property of the considered systems is the suppression of non-radiative channels and an inhomogeneous broadening of the resonance frequency. The second fundamental property of resonant photonic crystals is that the interaction of light with resonant excitation can occur in two regimes depending on the thickness of the sample. In the superradiant regime realized for a small number of resonant layers N (fine structures), the height of the peak and half-width of the reflection spectrum monotonically increase with increasing value of N. With a further increase in N, there occurs a transition to a photonic-crystal regime where the half-width of the reflection spectrum is saturated, which manifests itself in the form of an optical stop band. The theoretical description is illustrated by experimental spectra measured for all three resonant Bragg systems.
Optical dephasing of paramagnetic ions and molecules in organic and inorganic amorphous solids at low temperatures (≲100K) is characterized by unusual temperature dependences and enhanced relaxation rates compared to crystalline systems1. For a large variety of impurity-host combinations the dephasing rate has been found to follow a Tm temperature dependence with m approximately 2. Several theories have attempted to explain this behavior by assuming interactions between the optical center, phonons, and atomic tunneling systems. However, the role of the intrinsic disorder in optical dephasing in these systems is not clearly understood.
Magnetization relaxation and the steady state response of the S = 1 Ising model with random crystal field to a time varying magnetic field with a frequency ω is modelled and studied here by a method that combines the statistical equilibrium theory with the theory of irreversible thermodynamics. The method offers information on the relaxation time (τ) of the system as well as the temperature (θ) and ω dependencies of the complex (AC or dynamical) susceptibility (i.e., χ(ω) = χ′(ω) − iχ″(ω)). The so-called low- and high-frequency regions are separated by τ because τ−1 → 0 as θ approaches the critical temperatures (θc). One can choose to keep the frequency ω fixed and observe the low-frequency behaviors followed by the high-frequency behaviors when θ → θc. It is shown that χ(ω) exhibits different behaviors in low- and high-frequency regimes that are separated by the quantity ωτ: χ′(ω) converges to static susceptibility and χ″(ω) → 0 for ωτ ≪ 1. However, in the high-frequency region where ωτ ≫ 1, χ′(ω) vanishes and χ″(ω) displays a peak at the critical temperature (θc). Besides the above, the logarithm of the susceptibility components versus log(ω) is also plotted. From these plots, one plateau (a step-like) region and a shifted peak with rising temperature is observed for the real and imaginary parts, respectively.
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 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 ${\ensuremath{\chi}}_{\mathrm{NL}}(2\ensuremath{\omega})$ 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.