Hybridizing Ferromagnetic Magnons and Microwave Photons in the Quantum Limit

We demonstrate large normal-mode splitting between a magnetostatic mode (the Kittel mode) in a ferromagnetic sphere of yttrium iron garnet and a microwave cavity mode. Strong coupling is achieved in the quantum regime where the average number of thermally or externally excited magnons and photons is less than one. We also confirm that the coupling strength is proportional to the square root of the number of spins. A nonmonotonic temperature dependence of the Kittel-mode linewidth is observed below 1 K and is attributed to the dissipation due to the coupling with a bath of two-level systems.

Figure 1

Experimental setup. (a) An YIG sphere mounted in a rectangular cavity made of oxygen free copper. The sphere is glued to an alumina (aluminum-oxide) rod oriented to the crystal axis $⟨110⟩$. The inset is a magnified picture of a sphere with a diameter of 1.0 mm. The cavity has two connector ports for transmission spectroscopy and dimensions of $22\times 18\times 3\mathrm{mm}$ that give the fundamental-mode (${\mathrm{TE}}_{101}$) resonant frequency ${\omega }_c/2\pi$ of 10.565 GHz. (b) Measurement apparatus. The YIG sphere, cavity, and magnet are cooled to 10 mK using a dilution refrigerator. A series of attenuators and isolators prevent thermal noise from reaching the sample space. The total attenuation of the input port is 48 dB at 10 GHz. The output signal is amplified by two low-noise amplifiers at 4 K and the room temperature. We use a vector network analyzer (VNA) for transmission and reflection spectroscopy. A static magnetic field perpendicular to the microwave magnetic field and parallel to the crystal axis $⟨100⟩$ is applied by using permanent neodymium magnets and a magnetic yoke made of pure iron. We use a superconducting coil for fine tuning of the static field.

Figure 2

Normal-mode splitting between the Kittel mode and the cavity mode ${\mathrm{TE}}_{101}$. (a) Amplitude of the transmission $\mathrm{Re}(S_{21})$ through the cavity as a function of the probe microwave frequency and the static magnetic field presented in the current $I$ through the superconducting coil. The field-to-current conversion ratio obtained by fitting the FMR frequency is $1.42\mathrm{mT}/\mathrm{mA}$. The phase offset in $S_{21}$ is adjusted so that the peak has a pure absorption spectrum. The horizontal dashed line shows the cavity resonant frequency, while the diagonal dashed line shows the Kittel-mode frequency, both obtained from the fitting based on the input-output theory. We used a probe microwave power of $-123\mathrm{dBm}$, which corresponds to the average photon number of 0.9 in the cavity. (b) Cross sections at static magnetic fields corresponding to $I=-3.5$, $-2.3$, $-1.1$, and 0 mA. Solid curves are experimental data with vertical offset for clarity, and dashed white lines are the fitting curves.

Figure 3

Coupling strength of the Kittel mode to the microwave cavity mode as a function of the sample diameter $d$. The bottom and top horizontal axes show the square root of the sphere volume and the corresponding net spin numbers, respectively. The dashed line is a linear fit. The slope of the line gives the single-spin coupling strength $g_0/2\pi$ to the cavity mode, which is estimated to be 39 mHz.

Figure 4

Linewidth of the Kittel mode in the 0.5-mm sample as a function of the temperature. The red dots show the linewidth obtained by fitting $S_{21}(\omega )$ measured at each temperature. The dashed line is the fitting curve to the temperature dependence below 1 K. This is based on a model in which the linewidth is dominated both by transverse relaxation to two-level systems (TLSs) near-resonant with the Kittel-mode and by surface scattering of the Kittel-mode magnons into other degenerate modes. The blue circles show the deviation from the model. The line broadening above 1 K is ascribed to the slow-relaxation process due to impurity ions and magnon-phonon scattering.