Coherent control of magnon-polaritons using an exceptional point

Abstract

The amplitude of resonant oscillations in a non-Hermitian environment can either decay or grow in time, corresponding to a mode with either loss or gain. When two coupled modes have a specific difference between their loss or gain, a feature termed an exceptional point emerges in the excitations' energy manifold, at which both the eigenfrequencies and eigenmodes of the system coalesce. Exceptional points have intriguing effects on the dynamics of systems due to their topological properties. They have been explored in contexts including optical, microwave, optomechanical, electronic and magnonic systems, and have been used to control systems including optical microcavities, the lasing modes of a PT-symmetric waveguide, and terahertz pulse generation. A challenging problem that remains open in all of these scenarios is the fully deterministic and direct manipulation of the systems' loss and gain on timescales relevant to coherent control of excitations. Here we demonstrate the rapid manipulation of the gain and loss balance of excitations of a magnonic hybrid system on durations much shorter than their decay rate, allowing us to exploit non-Hermitian physics for coherent control. By encircling an exceptional point, we demonstrate population transfer between coupled magnon-polariton modes, and confirm the distinctive chiral nature of exceptional point encircling. We then study the effect of driving the system directly through an exceptional point, and demonstrate that this allows the coupled system to be prepared in an equal superposition of eigenmodes. We also show that the dynamics of the system at the exceptional point are dependent on its generalised eigenvectors. These results extend the established toolbox of adiabatic transfer techniques with a new approach for coherent state preparation, and provide a new avenue for exploring the dynamical properties of non-Hermitian systems.

Figures (4)

• Figure 1: Experimental configuration and strong coupling. (a) Schematic of the coupled resonators and readout electronics, showing coupling ports 1 -- 4, amplifiers (Amp), directional couplers (DC), IQ modulators (IQ) and yttrium iron garnet spheres (YIG). (b) The position of the YIG in the coplanar waveguide of the resonator. Also shown is the direction of the field components at the location of the YIG sphere. (c) Feedline transmission $\lvert\textrm{S{21}}\rvert$ as a function of waveguide loop resonator 1 frequency and probe frequency, with the frequency of resonator 2 fixed at 2.225 and magnon modes strongly detuned. An avoided crossing betweeen microwave resonances is seen, showing that they are in the strong coupling regime. (d) Transmission as a function of applied magnetic field and probe frequency, with the frequency of resonator 1 fixed at 2.225 and resonator 2 suppressed. The real part of the frequencies of the magnon modes anticross with the microwave mode, demonstrating that the yare strongly coupled. (e) Transmission as a function of applied magnetic field and probe frequency, with the uncoupled frequencies of resonators 1 and 2 fixed at 2.225, showing the complete mode manifold. Red dotted lines show the frequencies of the microwave photon supermodes, and blue dotted lines show the uncoupled frequencies of the magnon modes. (f) Transmission as a function of damping detuning $\Delta\Gamma$ between the two resonators, with the uncoupled frequencies of resonators 1 and 2 fixed at 2.225 and the applied magnetic field fixed at 68 . An exceptional point is reached at $\Delta\Gamma/2\pi = g/2\pi = \qty{13.5}{\mega\hertz}$.
• Figure 2: The theoretical energy landscape for two non-Hermitian coupled resonators. The surface shows the real parts of the eigenfrequencies ($\textrm{Re}(\lambda$)) of the coupled system as a function of the angular frequency ($\Delta\omega$) and loss ($\Delta\Gamma$) detuning of the resonators from degeneracy, and is coloured according to the imaginary parts of the eigenfrequencies ($\textrm{Im}(\lambda$)). For $\Delta\Gamma < g$ the real part of the eigenvalues exhibits an anticrossing as a function of $\Delta\omega$, and when $\Delta\Gamma > g$ the anticrossing is in the imaginary part. The two regimes are separated by the exceptional point (EP). Also shown are trajectories on the surface and their projection on to the $\{\Delta\omega$, $\Delta\Gamma\}$ plane corresponding to: (I) an ellipse which does not enclose the EP; (II) an ellipse which does enclose the EP; (III) a trajectory from $\Delta\omega= \Delta\Gamma=0$ through the EP and back to the starting point.
• Figure 3: Population transfer by encircling an exceptional point. (a) Normalized temporal profile of applied microwave power, and magnon-polariton angular frequencies ($\omega_1$, $\omega_2$) and dampings ($\Gamma_1$, $\Gamma_2$) during the experimental sequence. (b) IF signal for a trajectory encircling the exceptional point on the low loss (upper panel) and high loss (lower panel) surfaces. (c) Example elliptical trajectories (Eq. \ref{['eqn:ellipse']}) in $\{\Delta\omega, \Delta\Gamma\}$, with $C_\omega/2\pi = \qty{-10}{\mega\hertz}, C_\Gamma/2\pi = \qty{10}{\mega\hertz}$ (blue, not enclosing EP), $C_\omega/2\pi = \qty{5}{\mega\hertz}, C_\Gamma/2\pi = \qty{30}{\mega\hertz}$ (red, enclosing EP). (d) Power spectra of ringdowns shown in panel (b) during $\tau_\textrm{meas}$. The blue curve correspond to the upper panel and orange to the lower, with fits to a double Lorentzian also shown (dashed curves). Inset: power spectrum at equilibrium, showing excess occupancy of modes above the background. (e) Relative population $\hat{E}_u$ as a function of $C_\omega$ and $C_\Gamma$ for initial excitation of $\ket{l}$ (left) and $\ket{u}$ (right). Population transfer occurs for the parameter ranges marked by the white dotted boxes, where $C_\Gamma>g$ and the trajectory has the correct chirality. (f, g) Path of eigenstates of the system on the Bloch sphere along trajectories in (c). (f) A trajectory which does not encircle the EP returns to its starting point. (g) An EP-encircling trajectory, showing orthogonal start and end points.
• Figure 4: Equalising state populations by traversing beyond and back through an EP. (a) Temporal profile of applied microwave power, and resonator angular frequencies ($\omega_1$, $\omega_2$) and loss rates ($\Gamma_1$, $\Gamma_2$) for trajectory III in Fig. \ref{['fig:Trajectories']}. (b) Final state population of coupled magnon-polaritons $\hat{E}_u$ as a function of $C_\Gamma$. Error bars show experimental data, and the solid line (shaded region) the mean (standard deviation) of the stochastic model described in the text. $\hat{E}_u$ tends towards 0.5 as $C_\Gamma$ is increased beyond the EP (located at vertical line A at which $C_\Gamma=g$). (c, upper panel) $\hat{E}_u$ as a function of trajectory timespan for $C_\Gamma/2\pi = g/2\pi = \qty{13.5}{\mega\hertz}$ (touching the EP). Populations of $\ket{u}$ and $\ket{l}$ are not equal, even for long duration trajectories. (c, lower panel) $\hat{E}_u$ as a function of trajectory timespan for $C_\Gamma/2\pi = \qty{17.5}{\mega\hertz}$ (vertical line B on panel (b)). Populations of $\ket{u}$ and $\ket{l}$ are now equalised for durations longer than $\tau_\textrm{wf}\approx\qty{80}{\nano\second}$. The effect of thermal noise is much smaller due to the higher absolute mode populations. (d) Calculated instantaneous eigenvector populations during trajectories with $C_\Gamma/2\pi = \qty{13.5}{\mega\hertz}$ (upper panel) and $C_\Gamma/2\pi = \qty{17.5}{\mega\hertz}$ (lower panel), and for $\tau_\textrm{wf} = \qty{90}{\nano\second}$ (dotted line in panel (c)). Preparation of an equal admixture of $\ket{l}$ and $\ket{u}$ relies on the contrast between loss and gain states between 30 and 60 for $C_\Gamma/2\pi = \qty{17.5}{\mega\hertz}$. (e) Temporal profile of applied microwave power, and resonator angular frequencies and loss rates for two consecutive trajectories towards the EP. (f) Energies in $\ket{l}$ and $\ket{u}$ after consecutive trajectories beyond the EP ($C_\Gamma/2\pi = \qty{17.5}{\mega\hertz}$) as a function of time $T$ between consecutive trajectories, for initial population in $\ket{u}$ (upper panel) and $\ket{l}$ (lower panel). Mode energies oscillate in-phase regardless of the initially populated state (dotted lines), demonstrating the independence of final state on initial state.