Complex temperature dependence of coupling and dissipation of cavity magnon polaritons from millikelvin to room temperature

Hybridized magnonic-photonic systems are key components for future information processing technologies such as storage, manipulation, or conversion of data both in the classical (mostly at room temperature) and quantum (cryogenic) regime. In this work, we investigate a yttrium-iron-garnet sphere coupled strongly to a microwave cavity over the full temperature range from $290\mathrm{K}$ to $30\mathrm{mK}$. The cavity-magnon polaritons are studied from the classical to the quantum regimes where the thermal energy is less than one resonant microwave quanta, i.e., at temperatures below $1\mathrm{K}$. We compare the temperature dependence of the coupling strength $g_{\mathrm{eff}}\left(T\right)$, describing the strength of coherent energy exchange between spin ensemble and cavity photon, to the temperature behavior of the saturation magnetization evolution $M_s\left(T\right)$ and find strong deviations at low temperatures. The temperature dependence of magnonic disspation is governed at intermediate temperatures by rare-earth impurity scattering leading to a strong peak at $40\mathrm{K}$. The linewidth ${\kappa }_m$ decreases to $1.2\mathrm{MHz}$ at $30\mathrm{mK}$, making this system suitable as a building block for quantum electrodynamics experiments. We achieve an electromagnonic cooperativity in excess of 20 over the entire temperature range, with values beyond 100 in the millikelvin regime as well as at room temperature. With our measurements, spectroscopy on strongly coupled magnon-photon systems is demonstrated as versatile tool for spin material studies over large temperature ranges. Key parameters are provided in a single measurement, thus simplifying investigations significantly.

Figure 1

Experimental setup and position of the YIG sphere in cavity resonator. (a) Simulated magnetic field distribution of the “bright” mode resulting in an enhancement between the posts at resonance. (b) Schematics of the experimental setup. For reflection and transmission measurements one or two loops with one winding are used for inductively coupling the microwave signal from the VNA into the resonator. The YIG sphere is placed in the magnetic field's antinode in the middle between the posts for enhanced coupling in the bright mode. (c) Photograph of the assembled YIG-cavity resonator system.

Figure 2

Transmission (${\mathcal{S}}_{21}$) measurement of the dispersion for the cavity-magnon polariton for $T=50\mathrm{K}$. Resonant coupling appears at $227\mathrm{mT}$ with a coupling strength of $g_{\mathrm{eff}}\left(T\right)/2\pi =29\mathrm{MHz}$. The raw data (not shown) had magnetic field independent oscillations due to standing waves in the cables; for analysis of the data they were removed as described in the Appendix. As an example, the inset shows the background-corrected resonant linear amplitudes of the coupled system with a fit of Eq. (5) to the data. The red line shows the background corrected data and the other one (dotted) the corresponding fit. The dotted lines in the main plot illustrate the uncoupled dispersion of the cavity (field independent) and Kittel magnon (field dependent).

Figure 3

Temperature dependence of the saturation magnetization ${\mu }_0{M}_s$ (SQUID measurements) and the effective coupling strength $g_{\mathrm{eff}}\left(T\right)$ of the Kittel mode measured with the single port cavity resonator (reflection). The dotted lines in the main figure correspond to fits of the data to Bloch's $T^{3/2}$ law. The left inset shows a zoom into the region around resonance of the magnon polariton for a temperature of 100 K. The dotted lines indicate additional couplings to parasitic spin wave modes close to the Kittel mode. For lower temperatures the fields for the couplings start overlapping and the coupling strength of the Kittel mode decreases. The left shows exemplarily a value of the subKelvin temperature measurement which is consistent to the higher temperature data ($T\ge 5\mathrm{K}$). The millikelvin regime is discussed in detail in a separate publication [37].

Figure 4

Magnon linewidth ${\kappa }_m$ as derived from a fit to input-output theory via Eqs. (4) and (5) for reflection (red, circles) and transmission (blue, triangles) data (measured with our cavity resonator equipped with two ports). The error bars are obtained from the fit's covariance matrix. Higher error bars for the transmission data result from the influence of additional avoided crossings in the vicinity of the main splitting. The data taken in the millikelvin setup with $-135\mathrm{dBm}$ input power at the inductive loop coupler of the cavity resonator is shown on the left side of the figure. We observe a total change of the magnon linewidth by a factor of six in the range between $30\mathrm{mK}$ to $290\mathrm{K}$ with a peak around $40\mathrm{K}$. This behavior is goverened by impurity rare-earth ion relaxation [16, 32, 47, 48]. The green curve models the temperature dependence, taking a possible contribution from rare-earth impurity scattering at different temperatures into account. The sum of them results in the black line (stars). The millikelvin data are analyzed closer in a separate publication [37].

Figure 5

Temperature dependence of the cooperativity $C$ (squares) and the total cavity resonator losses ${\kappa }_l\left(T\right)$ (triangles) from the measurement with the single port cavity resonator. The cooperativity decreases with decreasing temperature but is much larger than one for all temperatures. In contrast to the magnon linewidth ${\kappa }_m$ the total cavity resonator loss ${\kappa }_l$ does not change significantly as a function of temperature. Thus, the temperature dependence of the cooperativity inversely reflects the changes of the magnon linewidth.

Figure 6

(a) Temperature dependence of the applied static field in the resonance condition for the Kittel mode. The external static field values for resonant coupling increases by $4\%$ from 5 to $290\mathrm{K}$. The increase is attributed to the decrease of the saturation magnetization $M_s$ as a function of temperature (cf. Fig. 3) since demagnetizing fields have to be taken into account, as well. Thus, $H_{\mathrm{res}}=H_{\mathrm{ext}}-H_{\mathrm{demag}}$. The error bars are in the order of $0.1\mathrm{mT}$, and are covered by the data symbols. (b) Temperature dependence of the gyromagnetic ratio. The values are calculated from ${\omega }_{\mathrm{res}}=\gamma ·{\mathrm{H}}_{\mathrm{ext}}$. It decreases towards higher temperatures due to the changes in $H_{\mathrm{demag}}$. The total decrease is less than $4\%$ and thus small compared to the temperature dependence of the coupling strength $g_{\mathrm{eff}}\left(T\right)$. (c) Temperature dependence of the cavity frequency. It decreases towards higher temperatures. The total decrease is less than $0.4\%$ and thus very small compared to the temperature dependence of the coupling strength $g\left(T\right)$. The error bars are of the order of $0.1\mathrm{mT}$.

Figure 7

Typical field dependence of the linewidth parameters at $T=50\mathrm{K}$. The datapoints are taken from a reflection measurement. The red curve (circles) shows values for the magnon linewidth ${\kappa }_m$, the blue curve (squares) the linewidth related to the resonator, labeled with ${\kappa }_l$, and the green one (stars), labeled with ${\kappa }_e$, the field dependence of the coupling of the resonator to the microwave feedline. Since the magnon linewidth is dominated by scattering within the sphere, it does not depend on the resonance condition. Indeed, both ${\kappa }_m$ and ${\kappa }_e$ are constant within the error bars for the displayed range of applied external field. Approaching resonance opens a new loss channel due to the energy transfer to the magnons in the YIG sphere. The total cavity losses ${\kappa }_l$ increase by $50\%$ when reaching the resonance condition.

Figure 8

Difference between the applied field for resonant coupling of the cavity photon to the Kittel mode (${\mathrm{H}}_{\mathrm{res}}^{\mathrm{Kittel}}$) and to other magnetostatic modes (${\mathrm{H}}_{\mathrm{res}.}^{\mathrm{MS}}$) in the specimen, respectively. As shown exemplary in the left inset of Fig. 3, the difference is estimated from the distance of the resonant field for coupling to the Kittel mode to the field for the additional coupling to other magnetostatic modes. For lower temperatures, the probability for parasitic coupling to other modes increases and the coupling strength of the Kittel mode is lowered (cf. Fig. 3). Above $200\mathrm{K}$, the distance is large and thus the parasitic coupling rate decreases below the noise level. For temperatures below $50\mathrm{K}$, the resonance frequencies are almost identical. At the same time, the overall signal-to-noise ratio is reduced and the field difference cannot be resolved.

Figure 9

Ratio between the coupling strength $g_{\mathrm{eff}}\left(T\right)$ and the square root of the magnetization $M_s$. Over the whole temperature range the value of that relative quantity changes by $6\%$.