Cooperative polariton dynamics in feedback-coupled cavities

The emerging field of cavity spintronics utilizes the cavity magnon polariton (CMP) induced by magnon Rabi oscillations. In contrast to a single-spin quantum system, such a cooperative spin dynamics in the linear regime is governed by the classical physics of harmonic oscillators. It makes the magnon Rabi frequency independent of the photon Fock state occupation, and thereby restricts the quantum application of CMP. Here we show that a feedback cavity architecture breaks the harmonic-oscillator restriction. By increasing the feedback photon number, we observe an increase in the Rabi frequency, accompanied with the evolution of CMP to a cavity magnon triplet and a cavity magnon quintuplet. We present a theory that explains these features. Our results reveal the physics of cooperative polariton dynamics in feedback-coupled cavities, and open up new avenues for exploiting the light–matter interactions.

Notes 2 and 3, measurement results for the A cavity and the A-P coupled cavities are discussed in detail, respectively.

Supplementary Note 2: Characterization of the A resonator
The resonant nature of the feedback resonator A is verified by direct measurement of its resonance frequency and quality factors. For the purpose of such a characterization, we have fabricated a stand-alone resonator that is not coupled to the resonator P, which has an identical geometrical design of the one that is coupled to the P resonator.
where κ A > 0 is the intrinsic damping parameter of the resonator A at zero voltage, g A < 0 is the voltage dependent gain parameter of the resonator A, and κ 1 = κ 2 is the coupling rate at port 1 and 2, respectively. From the fit, we determine the resonance frequency ω c /2π = 3 GHz and the quality factor Q = 17 at V = 0 V, as shown in Supplementary Fig.   2a. The voltage dependences of the resonance frequency and quality factor are measured and displayed in Supplementary Figs. 2c and 2d, respectively. By the tuning of the voltage, is increased up to 4 × 10 4 while the resonance frequency remains stable with a deviation of less than 1%.

Supplementary Note 3: Quantifying the coupling between A and P resonators
The design requirements of resonator A and P are described in Supplementary Note 1.
We set the two resonators to work at the weak coupling regime, which allows their dissipation to be coupled while operating at the nearly same frequency. The measured |S 21 | spectra of the resonators before and after coupling are shown in Supplementary Fig. 3. The effect of the coupling can be determined by using the input-output theory for two coupled oscillators: where κ P and κ A are the intrinsic damping parameters of the resonator P and A, respectively, and κ 1 = κ 2 is the energy loss rate at port 1 and port 2, respectively. δω accounts for a small mode frequency shift due to the change of effective dielectric constant when the two cavities are brought together. By using 2 )| to fit the spectra measured on the two resonators separately as shown in Supplementary Figs. 3a and 3b, we determine the intrinsic damping parameters κ P = 126 MHz and κ A = 180 MHz. By using Supplementary Equation (2) to fit the measured spectrum of the A-P coupled cavity as shown in Supplementary Fig. 3c, we determined κ 1 /2π=κ 2 /2π = 25 MHz, δω = 160 MHz, and the coupling parameter is estimated as g/2π 1 MHz. Such a weak coupling does not lift the mode degeneracy, but leads to a coupled mode with a mixed damping rate that depends on both κ P and κ A . An YIG sphere is usually described as a many-body system with N spins S j , (j = 1, · · · N ). The total spin of the sphere, S = N j=1 S j , is the spin observable that couples to the magnetic field of a cavity, which is approximately homogeneous across a 1 mm diameter YIG sphere. Such a collective coupling of two-level systems to a cavity mode resembles Dicke's model of superradiance [1].
Agarwal [2] discovered that in the evaluation of the absorption spectra of such a coupled system in the linear dynamic regime, one only needs the eigenfunctions of a reduced space defined by the combined states |G; 0 = |( N 2 , − N 2 ); 0 , |G; 1 = |( N 2 , − N 2 ); 1 and |E; 0 = |( N 2 , − N 2 + 1); 0 , where |(S, M ) represents the collective spin eigenstate ofŜ 2 andŜ z , and |n represents the photon Fock states. Using these states and by setting the energy of |G; 0 as zero, we obtain the Hamiltonian for the A-P-M devices: where the creation (annihilation) operatorsp † (p) describe the P-cavity mode,â Here, g 0 is the vacuum Rabi frequency of the single spin as defined in the main text.
Note that in Supplementary Equation (4) which describes the magnon-photon coupling in the P-M device. It creates the cavity magnon polariton (CMP). Note that in the linear dynamic regime where the number of CMP m N , the many-spin system is far from being saturated by the photon excitation, so that adding photons may increase m but does not enhance the coupling strength Ω P M . Therefore, the P-M coupling in the reduced space of Agarwal can be simply described by the elementary process of creating one CMP (m =1) from the ground state |G; 0 , and we get from Supplementary Equation (4) From Supplementary Equation (5) at the detuning ∆ = ω r − ω c , we get the CMP eigen- with the eigenfrequencies where c ± = (Ω ± ∆/2)/2Ω are the state amplitudes and Ω ≡ Ω 2 0 + (∆/2) 2 . Our result is consistent with Agarwal's theory for N atoms [2], and it is also in agreement with the CMP theory set on the footing of coupled harmonic oscillators [3].
Supplementary Equation (10) consistently treats the P-M and A-P-M coupling in the reduced space of CMP, which describes two coherently linked cooperative dynamics: the collective excitation of N spins via the magnon-photon coupling, and the collective deexcitation of m CMPs via the polariton-feedback-photon coupling. It leads to the dressed states of each CMP mode, whose Rabi frequency Ω ± can be calculated by considering the coupling between the combined CMP-photon states |± |n and |G; 0 |n + 1 . We find Ω ± = (Ω ± ∆/2) 2 + 2(f Ω 0 ) 2 (Ω ± ∆/2)/Ω, where in addition to the Dicke factor of √ N that appears in Ω 0 , a new feedback factor emerges with f = n m .
Physically, f denotes the de-excitation ratio of the CMPs, and it is the signature of cooperative polariton dynamics involving collective coupling of m CMPs with n feedback photons.
Experimentally by using the A-P-M device, f is determined by the gain of the device as explained in the main text, which can be tuned by changing the voltage.