Multifrequency Floquet Engineering of Magnon Polaritons

Technical Summary: Multifrequency Floquet Engineering of Magnon-Polaritons

Problem Statement
Floquet engineering, the technique of introducing controllable quasienergy levels into a system via periodic parameter modulation, has recently been applied to systems of cavity magnon-polaritons to manipulate their energy spectra. Traditionally, this was achieved by periodically modulating the magnon frequency using a time-varying bias magnetic field. However, the authors identify significant experimental challenges associated with this approach: generating large, uniform magnetic fields over the volume of the magnetic material with low inductance and broad frequency response is difficult. Furthermore, inhomogeneous broadening and unwanted couplings between different magnetostatic modes can occur if the field is not sufficiently homogeneous.

Methodology
To circumvent the difficulties of directly modulating the magnon frequency, the authors propose and demonstrate an alternative approach: modulating the frequency of the microwave cavity resonator. By exploiting the symmetry between magnon and photon modes in the underlying equations of the system, they achieve the same underlying physics while simultaneously gaining access to large modulation amplitudes, high frequencies, and arbitrary, precisely controllable waveforms.

The experimental platform consists of a microwave ring resonator fabricated on a low-loss printed circuit board (PCB), containing an embedded amplifier and an In-Phase/Quadrature (IQ) demodulator. A 1 mm polished yttrium iron garnet (YIG) sphere is placed within the waveguide loop to support magnon modes. The cavity frequency is tuned and modulated by applying calibrated voltages to the I and Q ports of the demodulator, which control the phase shift and amplitude of the propagating wave.

The study focuses on two distinct driving regimes:

1. Single-tone Floquet drive: The cavity frequency is modulated as $\omega(t) = \omega_c + a \sin(\Omega t)$.
2. Multifrequency Floquet drive: The cavity frequency is modulated with commensurate dual-frequency drives, $\omega(t) = \omega_c + a_1 \sin(\Omega_1 t) + a_2 \sin(\Omega_2 t + \phi)$, where $\Omega_2 = m\Omega_1$ ($m=2$ or $3$).

The system is characterized by measuring the transmission spectrum ($S_{21}$) of a feedline coupled to the resonator using a vector network analyzer (VNA). Theoretical modeling is performed using a Floquet matrix formalism for the time-periodic Hamiltonian, combined with cavity input-output theory to calculate the expected transmission spectra.

Main Results

• Single-tone modulation: The authors successfully demonstrated that modulating the cavity frequency generates sidebands at $\omega_c \pm n\Omega$. The visibility of these sidebands follows the zeros of Bessel functions, confirming the validity of calibration at high drive frequencies. When the magnon and photon frequencies are tuned to resonance (strong coupling regime), sidebands appear on the magnon-polariton branches. The authors observed anticrossings between specific sidebands (e.g., $\omega_+ - \Omega$ and $\omega_-$) when the drive frequency met certain conditions (e.g., $\Omega = 2g$). They noted that the visibility of the modes differs from previous studies where the magnon was modulated, attributing this to destructive interferences between sidebands generated by the modulation of the photon mode.
• Multifrequency modulation ($m=2$): Upon applying a second drive tone at twice the fundamental frequency ($\Omega_2 = 2\Omega_1$), the spectrum exhibited qualitatively different features compared to the single-tone drive. Most notably, new anticrossings appeared between sidebands that were previously decoupled (where the change in sideband order $\Delta n$ is even). These included anticrossings between $\omega_+ - \Omega$ and $\omega_- + \Omega$, as well as between the fundamental modes and second-order sidebands. The magnitude of these new anticrossings increased with the amplitude of the second drive tone ($a_2$).
• Phase dependence: The study showed that the relative phase ($\phi$) between the two drive tones acts as an additional control parameter. At phase differences of $\pi/2$ and $3\pi/2$, the sidebands became asymmetric, with different visibilities for sidebands of positive and negative order. This asymmetry results from interference between multiple generation paths characterized by Bessel functions of different orders.
• Multifrequency modulation ($m=3$): When the second drive frequency was set to three times the fundamental frequency ($\Omega_2 = 3\Omega_1$), the selection rule requiring an odd $\Delta n$ for finite coupling was restored. This led to level repulsions between specific sidebands (e.g., $\omega_+ - \Omega$ and $\omega_- + \Omega$) rather than anticrossings, consistent with the expansion of the coupling Hamiltonian in terms of Bessel functions.

Significance and Claims
The work claims to demonstrate a robust alternative method for Floquet engineering in hybrid magnon-photon systems. By modulating the cavity frequency instead of the magnon frequency, the authors overcome the experimental limitations associated with generating large, uniform high-frequency magnetic fields. This approach enables:

1. Large modulation depths and bandwidths.
2. The application of arbitrary and precisely controlled modulation waveforms.
3. The realization of multifrequency Floquet drives that introduce new degrees of freedom, particularly the relative phase between drive tones.

The authors conclude that their platform offers a versatile method for manipulating Floquet quasienergy levels, enabling the investigation of complex spectral features such as asymmetric sidebands and new anticrossings between previously decoupled states. The experimental results show excellent agreement with a theoretical model based on the Floquet Hamiltonian formalism and input-output theory. The work lays the foundation for future studies where cavity mode parameters, including both frequency and lifetime, can be varied temporally to explore new physics in hybrid quantum systems.