Anomalous Long-Distance Coherence in Critically Driven Cavity Magnonics

Developing quantum networks necessitates coherently connecting distant systems via remote strong coupling. Here, we demonstrate long-distance coherence in cavity magnonics operating in the linear regime. By locally setting the cavity near critical coupling with traveling photons, nonlocal magnon-photon coherence is established via strong coupling over a 2-m distance. We observe two anomalies in this long-distance coherence: first, the coupling strength oscillates twice the period of conventional photon-mediated couplings; second, clear mode splitting is observed within the cavity linewidth. Both effects cannot be explained by conventional coupled-mode theory, which reveals the tip of an iceberg of photon-mediated coupling in systems under critical driving. Our Letter shows the potential of using critical phenomena for harnessing long-distance coherence in distributed systems.

Figure 1

Setup. (a) The YIG sphere is placed on a microstrip. An external magnetic field $H$ applied parallel to the microstrip drives the magnon mode with the uniform precession. The dielectric cylinder is placed on another microstrip. The cavity mode profile in the top right resembles the TE01 mode. The lateral spacing $d$ between the centers of the cavity and the microstrip is tunable by using a step motor. The magnon and cavity modes are coupled remotely via photons traveling in coaxial cables that connect the microstrips. The traveling phase $\mathrm{\Phi }={\mathrm{\Phi }}_L+\mathrm{\Delta }\mathrm{\Phi }$ is controlled by an inserted phase shifter that tunes $\mathrm{\Delta }\mathrm{\Phi }$, while ${\mathrm{\Phi }}_L=2\pi L/\lambda$ is set by the photon propagation distance $L$. The transmission spectra $S_{21}(\omega )$ are measured by connecting the end ports of the microstrips to the vector network analyzer. (b) Theoretical model. The magnon and cavity photon modes are coupled to the traveling photons with rates ${\kappa }_{m,L(R)}$ and ${\kappa }_{c,L(R)}$, respectively, and measured by input ${\widehat{s}}_{+1}$ and output ${\widehat{s}}_{-2}$ fields.

Figure 2

Critical coupling. (a) The dielectric cylinder is calibrated by tuning its lateral spacing $d$ to the microstrip. (b) The transmission amplitude $|S_{21}|$ and (c) group delay ${\tau }_g$ reveal two critical coupling conditions marked by the green dashed lines. (d) The effective damping rate $\beta /2\pi$ and (e) the extrinsic damping rate ${\kappa }_{c,L(R)}/2\pi$ are extracted by fitting the transmission using Eq. (1). The dotted line in (d) shows the intrinsic damping rate ${\beta }_0/2\pi$. The arrows in (d) and (e) mark the cavity setting conditions for the data presented in Figs. 3 and 4.

Figure 3

Long-distance coupling: the phase dependency. The left and right panels compare the results measured at $d=4.92$ and 6.90 mm, respectively. (a),(d) Transmission amplitude $|S_{21}|$; (b),(e) group delay ${\tau }_g$ measured as a function of the frequency detuning $\mathrm{\Delta }=(\omega -{\omega }_c)/2\pi$ at different phase setting $\mathrm{\Delta }\mathrm{\Phi }$. The dashed curves in (a) and (b) are calculated by using Eq. (5) with $\eta =2$ and $\delta =0.996$. (c) The real and imaginary components of the complex coupling strength $G=J+i\mathrm{\Gamma }$ are plotted as blue and red circles, while the solid sinusoidal curves are added to guide the eye. (f) The measured inverse amplitude $1/|S_{21}({\omega }_c)|$ are plotted in comparison with the result calculated using Eq. (5) with $\eta =1$ and $\delta =1$.

Figure 4

Long-distance coupling: the field dependency. The left and the right panels compare the results measured at $d=4.92$ and 6.90 mm, respectively. (a),(c) $|S_{21}|$ mapping plotted as a function of the probing frequency $\omega$ and the field detuning ${\mathrm{\Delta }}_m$, showing level repulsion caused by remote coupling. (b),(d) Transmission spectra plotted with the same color bar as in the top panel. Hybridized modes are indicated in (b) by the red arrows at ${\mathrm{\Delta }}_m=0$. The black dotted curves and the red dashed curves are theoretical calculations using Eq. (5) with $\eta =1$, $\delta =1$ and $\eta =2$, $\delta =0.996$, respectively.