Strong Coupling Between RF Photons and Plasmons of Electrons on Liquid Helium

Technical Summary: Strong Coupling Between RF Photons and Plasmons of Electrons on Liquid Helium

Problem and Motivation
Plasmons, arising from the collective motion of electrons, are fundamental excitations in diverse electronic systems. While localized plasmons in metallic nanoparticles and 2D semiconductor systems have been exploited for applications ranging from biosensing to photocatalysis, their utility in quantum information processing is often limited by high dissipation rates. Achieving the strong-coupling regime—where the coherent coupling rate exceeds dissipation rates—is challenging in these conventional systems due to disorder and impurity scattering. Electrons floating on liquid helium offer an exceptionally clean, disorder-free environment where electron scattering is strongly suppressed. While strong coupling between single-electron orbital states and microwave photons has been demonstrated, establishing a hybrid platform for collective plasmon modes coupled to photons remains a critical step toward using plasmons as carriers of quantum information.

Methodology
The authors realized a tunable plasmon–photon hybrid platform using a Corbino geometry device. The system consists of 2D electrons floating approximately 10 nm above a liquid helium surface, confined within a circular area by concentric electrodes on top and bottom plates. A nanofabricated inductor connected to the top center electrode forms an LC resonator, which couples to the electron layer via a radial RF electric field.

The experimental approach involved two primary measurement regimes:

1. Frequency-Domain Measurements: The reflection coefficient of an RF signal applied to the resonator was measured while tuning the plasmon frequency ($\omega_p$). Tuning was achieved by adjusting DC voltages ($V_{BC}, V_{BM}, V_{BO}$) to control the electron density ($n_0$) and the effective radius of the electron cloud, thereby altering the wave vector $k_{0,\mu}$ of the plasmon modes.
2. Time-Domain Measurements: To directly observe coherent energy exchange, 20 ns RF pulses were injected into the resonator. The reflected signal was demodulated into in-phase (I) and quadrature (Q) components to track the ring-down dynamics and energy storage in the resonator.

Theoretical modeling was performed using a self-consistent charge-density equation derived from the continuity equation and Coulomb interactions. This model calculated plasmon frequencies and impedances ($Z_p$) based on simulated electron density profiles, allowing for the prediction of reflection coefficients and coupling strengths without additional free parameters.

Key Contributions and Results

• Demonstration of Strong Coupling: The authors established strong coupling between the plasmon mode of electrons on liquid helium and RF photons confined in an LC resonator. In the frequency domain, an avoided level crossing was observed when the plasmon frequency was tuned to the resonator frequency ($\omega_p = \omega_0$). Fitting the spectrum yielded a coupling strength of $g/2\pi \approx 4.55$ MHz and a plasmon decay rate of $\gamma_p/2\pi \approx 5.10$ MHz.
• Coherent Energy Exchange: Time-resolved measurements revealed Rabi-like oscillations (classical normal-mode beating) in the energy stored in the resonator after the excitation pulse was switched off. This provided direct evidence of coherent coupling. Analysis of the ring-down dynamics yielded a coupling strength of $g/2\pi = 4.906 \pm 0.009$ MHz. The observation of real-valued mode splitting ($2\Lambda_0$) confirmed the system operates in the strong-coupling regime.
• Tunability and Mode Identification: The platform demonstrated high tunability, allowing the excitation of fundamental and higher-order radial plasmon modes ($\mu = 1, 2, \dots$). The authors successfully identified avoided crossings corresponding to the first and second radial modes by comparing experimental data with numerical simulations of electron density profiles.
• Temperature-Dependent Decay and Wigner Crystal Transition: The LC resonator served as a sensitive probe of electron-on-helium physics.
  • Decay Mechanism: The plasmon decay rate ($\gamma_p$) was found to increase with temperature, consistent with scattering mechanisms dominated by electron-ripplon interactions below 700 mK and electron-helium vapor scattering above this threshold. The experimental data aligned well with theoretical calculations neglecting many-electron effects.
  • Wigner Crystallization: A distinct change in the detuning ($\Delta = \omega_0 - \omega_p$) was observed around 250 mK, attributed to the formation of a Wigner crystal. As electrons crystallize, the coupling to surface dimples (dimple sublattice) increases, shifting the plasmon frequency. Notably, the plasmon linewidth ($\gamma_p$) remained insensitive to this transition, consistent with theoretical predictions that scattering rates in the optical branch of coupled plasmon-ripplon modes do not significantly differ between the liquid and crystal phases at these frequencies.

Significance and Claims
The paper claims that this work establishes electrons on liquid helium as a tunable plasmon–photon hybrid platform. The primary significance lies in demonstrating strong coupling between collective electron excitations (plasmons) and photons in a low-dissipation environment, a regime difficult to access in conventional semiconductor 2D materials.

The authors position this result as a step toward cavity quantum electrodynamics (cQED) with floating-electron plasmons. They note that while the current operating frequency (120 MHz) is below the thermal energy at the experimental temperature (180 mK), rendering the system unsuitable for quantum information storage in its current state, the platform provides a pathway for future development. The authors suggest that confining plasmons to smaller regions to achieve gigahertz frequencies would enable operation in the quantum regime, allowing for coherent exchange of quantum information between plasmons and microwave photons in superconducting resonators. Additionally, the study validates the LC resonator as a quantitative tool for investigating fundamental electron-on-helium phenomena, such as the Wigner crystal transition and temperature-dependent scattering processes.