Overcoming intrinsic material limitations through cavity feedback

Abstract

Magnons, the quanta of spin waves, have significant potential for use in modern technologies, especially when strongly coupled to another mode for read-out and control. However, while magnons strongly interact with microwave photons via the magnetic-dipole interaction to form hybrid cavity-magnon polariton modes, the weak magnetostrictive magnon-phonon interaction, together with large polariton linewidths dominated by magnon dissipation, has so far restricted magnonic-spheres to the weak-coupling regime. The material-limited magnon dissipation rate in particular has been regarded as an unavoidable limitation in these systems. Here, we surpass this long-standing limitation by implementing an active microwave feedback loop to suppress the linewidth of cavity-magnon polaritons and strongly suppress their effective decay rate below the magnon-limited linewidth, thereby enhancing the polariton-phonon cooperativity from C=1 to C=150. As a key milestone, we achieve normal-mode splitting between a cavity-magnon polariton and a mechanical mode, providing direct evidence of three-mode hybridization among photons, magnons, and phonons. Our results establish feedback as a general route to accessing strong-coupling regimes in systems previously thought to be limited by material properties and hence open new opportunities for coherent control in hybrid quantum systems.

Figures (5)

• Figure 1: Feedback reduces the hybridized polariton dissipation below the magnon linewidth. (a) Schematic of the cavity magnomechanical system and microwave feedback loop. The copper cavity is surrounded by a tunable magnet and probed by a vector network analyzer (VNA) through port 1. A fraction of the cavity output (port 2) is directed to the VNA for measurement, while the remaining signal is routed through the feedback loop. In the feedback loop, the signal is phase shifted (PS), amplified (Amp), and subsequently fed back into the cavity through port 1. A pump tone can be added to the circuit using an independent microwave source (MWS). (b) Polariton spectra demonstrating strong coupling between magnons and cavity photons as the bias field is tuned. (c) Transmission amplitude of the cavity--magnon polariton modes with increasing feedback gain $g_{\mathrm{fb}}$. Each trace is normalized and vertically offset for clarity. For each spectrum, the bias magnetic field was adjusted to maintain symmetric polariton modes. In the absence of feedback (NO-FB), the system exhibits broad hybrid modes, while increasing feedback gain progressively narrows the polariton linewidths. (d) Extracted linewidths of the upper ($\tilde{\kappa}_{+}$) and lower ($\tilde{\kappa}_{-}$) polariton modes, together with the intrinsic magnon dissipation rate $\kappa_m$, as functions of the feedback gain $g_{\mathrm{fb}}$. The feedback-induced linewidth reduction suppresses the polariton dissipation more than an order of magnitude below the magnon-limited linewidth.
• Figure 2: Feedback-induced strong magnomechanical coupling. (a) Frequency schematic of the measurement showing the pump tone, $\omega_d$, approximately one mechanical frequency, $\omega_b$, red-detuned from the upper polariton mode, $\omega_+$. (b) Transmission amplitude, $|S_{21}|$, of the upper cavity--magnon polariton as a function of the probe detuning $\Delta_{sd}=\omega_s-\omega_d$ in the absence of feedback. As the red-detuned pump tone is increased, a transparency peak appears, reflecting interference due to the mechanical spectrum, due to the cavity enhanced coupling. Despite the increasing drive power, the transparency feature remains small and no normal-mode splitting is observed. (c) When feedback gain is applied, there is a clear transparency peak that evolves into a well-defined normal mode splitting as the gain is increased (with constant pump tone power, $P_d=18\,\mathrm{dBm}$). (d) Contour plots of the measured (left) and predicted (right) transmission as a function of $g_{\mathrm{fb}}$, highlighting the emergence and evolution of the normal-mode splitting with increasing feedback gain.
• Figure 3: Extracted parameters characterizing the feedback-induced strong-coupling regime. (a) Coupling and dissipation rates extracted from fits to the transmission amplitude shown in Fig. \ref{['Figure_2']}(d) as functions of the feedback gain $g_{\mathrm{fb}}$. The effective polariton--mechanical coupling strength $G_+$ and the mechanical dissipation rate $\kappa_b$ remain approximately constant, while the effective upper-polariton linewidth $\tilde{\kappa}_+$ is strongly suppressed with increasing feedback gain. The blue shaded region highlights the parameter range in which the strong-coupling condition $G_+ > \tilde{\kappa}_+, \kappa_b$ is satisfied. (b) Polariton--mechanical cooperativity $C = G_+^2/(\tilde{\kappa}_+ \kappa_b)$ calculated from the extracted rates as a function of the feedback gain $g_{\mathrm{fb}}$, showing a significant enhancement driven by feedback-induced linewidth suppression.
• Figure 4: Normal-mode (avoided level crossing) spectrum of the upper polariton--mechanical system enabled by feedback-induced linewidth suppression. (a) Schematic diagrams illustrating the relative positions of the lower and upper polariton modes ($\omega_{-}$, $\omega_{+}$), the drive/pump tone $\omega_d$, and the mechanical frequency $\omega_b$ for three representative drive frequencies (blue, orange, and red). (b) Corresponding line cuts of the transmission amplitude $|S_{21}|$ (left column) and phase (right column) measured at the three drive frequencies indicated in panel (a), plotted as functions of the probe detuning $\delta_s=\omega_s-\omega_0$, where $\omega_0/2\pi$ denotes the center frequency of each measurement sweep (roughly $\omega_+$). The evolution from a single resonance into two well-resolved peaks, accompanied by the characteristic phase response, reveals normal-mode splitting in the strong-coupling regime. (c) Experimental color map of the transmission amplitude $|S_{21}|$ as a function of the probe and drive detunings, $\delta_s=\omega_s-\omega_0$ and $\delta_d=\omega_0-\omega_d$, measured at a fixed probe power of $P_d=15\,\mathrm{dBm}$. A microwave feedback loop with gain $g_{\mathrm{fb}}=29.75\,\mathrm{dB}$ is applied, and the feedback phase is tuned to $\phi\simeq0$. The clear avoided crossing demonstrates coherent hybridization between the upper polariton mode and the mechanical resonance. Colored arrows indicate the three drive frequencies corresponding to the line cuts shown in panel (b). (d) Theoretical simulation of the transmission amplitude $|S_{21}|$ plotted in the same detuning coordinates $(\delta_s,\delta_d)$, showing excellent agreement with the experimentally observed avoided-crossing spectrum.
• Figure 5: Transmission spectra of the cavity–magnon system measured inside the microwave feedback loop as a function of increasing drive power (bottom to top), shown for direct comparison with the no-feedback measurements in Fig. \ref{['Figure_2']}(c). The feedback gain is fixed at $g_{\mathrm{fb}} = 27~\mathrm{dB}$ and the phase is tuned to $\phi\simeq 0$, corresponding to the regime of maximal polariton linewidth suppression. In contrast to the no-feedback configuration, where increasing drive power alone does not produce resolvable normal-mode splitting due to the large intrinsic polariton linewidth, the feedback-enabled reduction of the linewidth allows the drive-enhanced magnomechanical interaction to overcome the dissipation rates, leading to a clear emergence of normal-mode splitting and entry into the strong-coupling regime. Dashed lines represent fits to the theoretical model.