Topological properties of a coupled spin-photon system induced by damping

We experimentally examine the topological nature of a strongly coupled spin-photon system induced by damping. The presence of both spin and photonic losses results in a non-Hermitian system with a variety of exotic phenomena dictated by the topological structure of the eigenvalue spectra and the presence of an exceptional point (EP), where the coupled spin-photon eigenvectors coalesce. By controlling both the spin resonance frequency and the spin-photon coupling strength we observe a resonance crossing for cooperativities above one, suggesting that the boundary between weak and strong coupling should be based on the EP location rather than the cooperativity. Furthermore, we observe dynamic mode switching when encircling the EP and identify the potential to engineer the topological structure of coupled spin-photon systems with additional modes. Our work therefore further highlights the role of damping within the strong coupling regime, and demonstrates the potential and great flexibility of spin-photon systems for studies of non-Hermitian physics.

Figure 1

The square root eigenvalue topology of the spin-photon system. (a) The dispersion and (b) linewidth are calculated from ${\tilde{\omega }}_{1,2}$ according to Eq. (2) using our experimental parameters. Blue (dark) curves at $\theta >{\theta }_{\text{EP}}$ show the characteristic strong coupling behavior; a dispersion anticrossing and linewidth crossing. However, when $\theta <{\theta }_{\text{EP}}$ the yellow (light) curves display a dispersion crossing and linewidth anticrossing, even though $C>1$. The red circle indicates the exceptional point.

Figure 2

(a) Experimental setup allowing in situ control of the coupling strength by controlling the angle $\theta$ between the local microwave and static magnetic fields as illustrated in the inset. (b) Transmission spectra of the ${\mathrm{TM}}_{011}$ mode at ${\mu }_{0}H=0$ mT. Open symbols are experimental data and solid curve is a Lorentz fit. (c) $\omega \text{−}H$ dispersion measurement at $\theta ={90}^{\circ }>{\theta }_{\text{EP}}$ and (d) $\theta ={26}^{\circ }<{\theta }_{\text{EP}}$.

Figure 3

(a) A resonance anticrossing and (c) a linewidth crossing is observed when $\theta ={90}^{\circ }>{\theta }_{\text{EP}}$. On the other hand, below the EP for $\theta ={26}^{\circ }<{\theta }_{\text{EP}}$ we instead observe (b) a resonance crossing and (d) a linewidth anticrossing. In all panels open circles are experimental data and solid curves are calculations according to Eq. (2).

Figure 4

(a) Path taken to encircle the EP. A square (star) denotes the starting (ending) positions, while the red (blue) curve indicates the path taken by mode 1 (mode 2). Note that the two paths are connected. (b) Path taken without encircling the EP. Each mode now behaves independently as shown with the red and blue curves. (c) Experimental spectra observed while encircling the EP. From top to bottom the system is tuned along paths $1\to 5$. One mode stays constant at the cavity frequency while the other mode initially shifts to high frequencies, eventually disappearing, until it reappears at lower frequencies. Therefore the modes have switched positions. (d) When the EP is not circled, an anticrossing occurs during path 3 and therefore the modes maintain their relative orientation. This key difference along path 3 is highlighted with red shading. (e) The theoretical spectra calculated according to Eq. (5) when the EP is encircled and (d) not encircled.

Figure 5

(a) The topology of a three-mode spin-photon system with two cavity modes and one FMR mode. The calculation is according to the eigenvalues of Eq. (6) using our experimental parameters. A path encircling both EPs, resembling an Archimedean spiral, is shown in red. (b) Experimental observation of a single slice in the three-mode topology at $\theta ={61}^{\circ }$. Circles are experimental data and the black curve shows the calculated eigenvalues of Eq. (6).