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We present a model that provides a description of the microwave dielectric loss in oxides. The dielectric loss ͑tan ␦͒ in single crystal and polycrystalline MgO and Al 2 O 3 is measured over the temperature range 70-300 K. We are able to model the dielectric loss in terms of a two-phonon difference model. There are two key parameters in this model: The third derivative, 3 , of the lattice potential and the linewidth, ␥, of the thermal phonons. In polycrystalline samples, rather than considering the different mechanisms of extrinsic loss, it is assumed that the main effect of extrinsic factors is a modification of the linewidth of the thermal phonons. By varying ␥(T), it is shown that the model can describe the loss in both single crystals and polycrystallines materials. In single crystal and polycrystalline MgO, we use ␥ as a fitting parameter. In single crystal and polycrystalline Al 2 O 3 , we obtain ␥(T) by Raman spectroscopy. The theory gives the right order of magnitude of the measured loss.

Polycrystalline alumina with extremely low microwave dielectric loss is reported with properties analogous to a theoretical ensemble of randomly oriented, single crystal sapphire grains. By avoiding deleterious impurities and by careful control of microstructure, we show that grain boundaries in aluminum oxide have only a limited influence on the dielectric loss. A method of measuring the electric permittivity and loss tangent of low-loss microwave ceramic dielectrics is reported. The electrical parameters such as relative permittivity and loss tangent are extracted using the radial mode matching technique. Ϫ5 at 10 GHz. 1 Polycrystalline analogs of these materials usually have similar permittivity, but losses that are generally a factor of at least two higher. The measurement of low loss, low permittivity microwave dielectric ceramics using the TE 01␦ is limited in accuracy by the electric filling factor and geometric factor of the conducting shield. By carefully choosing the dimensions of the conducting shield for optimum electric filling factor P e and geometric factor G, it is possible to measure the tan ␦ with 10% accuracy for tan ␦Ͼ10 Ϫ5 .2 The preferred method of measurement for tan ␦Ͻ10Ϫ6 utilizes high order whispering gallery modes, 3,4 which have electric filling factors approaching unity and very high geometric factors. Unfortunately, the modal number density is usually very high in the frequency range in which whispering gallery modes exist, and careful choice of cavity dimensions is necessary to avoid spurious modes.5 This letter seeks to extend the usefulness of the TE 01␦ mode measurement method by eliminating a great source of error: the support. The relative permittivity of the dielectric resonator can be calculated using the radial mode matching method. 6 Instead of solving for frequency, as is usually done, one can instead solve for r . After calculation of the electric filling factor and geometric factors, the tan ␦ can be obtained from the expressionwhere P e is the electric filling factor, Q 0 is the unloaded quality factor, R s is the surface resistivity, and G is the geometric factor. An alumina dielectric resonator was constructed as shown in Fig. 1 7 This material is high purity and fine grained, both of which are necessary in order to achieve very low loss. 8,9 The dimensions of the conducting shield are diameter 36.00 mm, height 23.48 mm. The alumina puck has diameter 10.69 mm and height 4.34 mm. The alumina spacer has diameter 4.13 mm and height 6.77 mm.The resonant frequency of the quasi-TE 011 (TE 01␦ ) mode was measured at room temperature ͑300 K͒ and found to be 8.95 557 GHz. The measurement was performed in transmission (S 21 ) using an Agilent network analyzer ͑HP8722͒. Input and output coupling to the resonator was achieved using coaxial transmission lines with small loops formed by soldering the central conductors to the outer shield of the coaxial cable. The coupling was Ϫ40.5 dB, measured from the insertion loss. The loaded quality factor Q L of the mode was measured and f...

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ABSTRACT:One of the most important problems in developing devices for quantum computation is the coupling and dissipation of states by thermal noise. We present a study of a two-state electric dipole in a crystal coupling to noise from a reservoir. As a realization of such an energy-dissipating dipole, we report and analyze dielectric loss measurements in single crystal and polycrystalline Al 2 O 3 over the temperature range 70 -300 K. We are able to model the dielectric loss in terms of a quasi-classical model that uses the fluctuation-dissipation theorem. Two key parameters in this model are the crystal oscillator energy and reservoir-lattice coupling constant. In polycrystalline samples, it is assumed that the main effect of structural disorder is a modification of the spectrum of the thermal phonons, so that acoustical vibrations acquire some optical mode character. The temperature dependence of the linewidth of the high dielectric strength infrared (IR) mode at 438 cm Ϫ1 and the quasi-degenerate Raman mode of the k ϭ 0 (418 cm Ϫ1) transition are also investigated and are shown to be related simply to the dielectric loss. The model reproduces the unusual temperature dependence of the dielectric loss observed experimentally. The implications for the coupling of quantum mechanical objects to noise and quantum information processing are discussed.