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

TL;DR

The study demonstrates strong coupling between plasmon modes of electrons floating on liquid helium and RF photons confined in an LC resonator, establishing a tunable plasmon–photon hybrid platform in a remarkably clean electronic system. By adjusting DC voltages, the plasmon frequency $\omega_p$ and the coupling rate $g$ are tuned to achieve hybridization, with observed values $g/2\pi \approx 4.6$–$4.9$ MHz and $\gamma_p/2\pi \approx 3.3$–$5.1$ MHz, as seen in both frequency- and time-domain measurements. Time-domain reflectometry reveals coherent energy exchange between modes with Rabi‑like oscillations, and a full coupled‑mode theory reproduces the observed dynamics, including the on‑resonance splitting $2\Lambda_0$. Temperature drives the system toward a Wigner crystal transition around $T\approx 250$ mK, where the detuning shifts and the optical plasmon branch is affected by the dimple frequency $\omega_d/2\pi \approx 29$ MHz. This work paves the way for cavity quantum electrodynamics with floating electrons by moving toward GHz plasmonic modes and integrating with superconducting resonators for quantum information processing.

Abstract

Plasmons, arising from the collective motion of electrons, can interact strongly with electromagnetic fields or photons; this capability has been exploited across a broad range of applications, from chemical reactivity to biosensing. Recently, there has been growing interest in plasmons for applications in quantum information processing. Electrons floating on liquid helium provide an exceptionally clean, disorder-free system and have emerged as a promising platform for this purpose. In this work, we establish this system as a tunable plasmon-photon hybrid platform. We demonstrate strong coupling between floating-electron plasmons and radio-frequency (RF) photons confined in an LC resonator. Time-resolved measurements reveal coherent oscillatory energy exchange between the plasmonic and photonic modes, providing direct evidence of their coherent coupling. These results represent a step towards cavity quantum electrodynamics with a floating-electron plasmon coupled to a resonator. Furthermore, the LC resonator serves as a sensitive probe of electron-on-helium physics, enabling the observation of the Wigner crystal transition and a quantitative study of the temperature-dependent plasmon decay arising from ripplon-induced scattering.

Figures (7)

• Figure 1: (a) Experimental setup with Corbino-geometry electrodes forming part of an LC resonator. The device consists of radially arranged top and bottom electrodes, each with an area of approximately $0.5~\mathrm{cm}^2$, which form the main capacitance $C$ of the LC resonator. The top center electrode is connected to an inductor $L$, which provides the resonant inductance, and to a coupling capacitor $C_\mathrm{C}$ that interfaces the resonator with the external circuit. The bottom electrodes are immersed in a 1-mm-thick layer of liquid helium (light blue cylinder), above whose surface electrons (light blue circles) float approximately $10~\mathrm{nm}$ in vacuum. An RF signal applied to the top center electrode generates an oscillating radial electric field $\delta E_r^\mathrm{RF}$ confined within the LC resonator and drives collective electron oscillations (plasmon mode, schematically illustrated by the pink wave) in electrons on helium. (b) Schematic representation of the experimental setup in the form of an equivalent electrical circuit. The plasmon excitation is modeled by an effective impedance $Z_\mathrm{p}$. Further details regarding the circuit elements are provided in the main text. (c) Measured magnitude and phase of the reflection coefficient of the LC resonant circuit, $|\Gamma_\mathrm{ref}|$ and $\arg(\Gamma_\mathrm{ref})$, in the absence of electrons (blue dots). The orange lines show a fit to Eq. \ref{['eq:reflection_coefficient_input_output']} with $g=0$. The data shown here are corrected for the phase offset and cable delay and are normalized, as is the case for all reflection data presented in this manuscript.
• Figure 2: Magnitude of the reflection coefficient $|\Gamma|$ as a function of $V_\mathrm{BM}$ and RF signal frequency $\omega/2\pi$ from both experiment in (a) and simulation in (c) for different $V_\mathrm{BC}=10, 9, 8 , 7.5, 6, 4$ V. $V_\mathrm{BO}$ is fixed to $-32$ V. (a) Measurement temperature $T = 180~\mathrm{mK}$; The RF excitation power is set to $-50~\mathrm{dBm}$ at the output of the signal generator. (b) Simulated electron density profiles $n_0(r)$ for different values of $V_\mathrm{BM}$, with $V_\mathrm{BC}$ fixed at $7.5~\mathrm{V}$ and $V_\mathrm{BO}$ fixed at $-32~\mathrm{V}$. For all values of $V_\mathrm{BM}$, the total number of electrons is kept constant. The thick blue and red lines correspond to the density profiles for which the fundamental and second plasmon modes, respectively, are tuned into resonance with the LC resonator frequency according to the simulation shown in (c). (c) Simulated magnitude of the reflection coefficient. Overlaid dashed lines indicate the calculated plasmon-mode frequencies: the fundamental ($\mu = 1$, blue), second ($\mu = 2$, red), third ($\mu = 3$, green), and fourth ($\mu = 4$, light blue) modes. (d) Simulated electron density profiles $n_0(r)$ for $V_\mathrm{BC}=10, 9, 8,$ and $7.5~\mathrm{V}$, evaluated at $V_\mathrm{BM}^{\mathrm{sim}}$ where the fundamental plasmon mode ($\mu=1$) is resonant with the LC resonator according to the simulation shown in (c). (e) Simulated magnitude of the density modulation, $|\delta n|$, at resonance for the fundamental ($\mu = 1$) and second ($\mu = 2$) modes at $V_\mathrm{BC} = 7.5~\mathrm{V}$, and for the third mode ($\mu = 3$) at $V_\mathrm{BC} = 6~\mathrm{V}$. The plasmon decay rates used in the simulations can be found in Supplementary Information \ref{['sec:ref_coeff']}, Fig. \ref{['fig:Fig_gamma_field']}.
• Figure 3: (a,c) Measurement temperature $T = 180~\mathrm{mK}$; The RF excitation power is set to $-20~\mathrm{dBm}$ at the output of the signal generator. (a) Right: Normalized measured reflection coefficient as a function of the bottom outer electrode voltage $V_\mathrm{BO}$ and the RF frequency applied to the LC resonator. The other electrode voltages are fixed at $V_\mathrm{BC} = V_\mathrm{BM} = 17~\mathrm{V}$. An avoided crossing is observed when the plasmon frequency is tuned into resonance with the LC circuit, indicating coherent coupling between the two modes. Left: Reflection spectrum at the resonance point $V_\mathrm{BO} = -29~\mathrm{V}$ (blue solid line), together with a fit to Eq. \ref{['eq:reflection_coefficient_input_output']} (orange dashed line). The fit yields a coupling strength $g/2\pi = 4.55 \pm 0.02~\mathrm{MHz}$ and a plasmon decay rate $\gamma_\mathrm{p}/2\pi = 5.10 \pm 0.07~\mathrm{MHz}$ . From these parameters, we obtain a mode splitting $2\Lambda_0\approx 9.0~\mathrm{MHz}$ (Eq. \ref{['eq:Lambda']}). (b) Numerically simulated reflection coefficient corresponding to panel (a). The plasmon decay rate $\gamma_{\mathrm p}/2\pi = 5.1~\mathrm{MHz}$ is used as an input parameter for the simulation. At the resonance voltage $V_\mathrm{BO} = -30.7~\mathrm{V}$, fitting the simulated spectrum yields $g/2\pi = 4.55~\mathrm{MHz}$, consistent with the experiment. The overlaid dashed blue lines indicate the calculated fundamental plasmon mode frequencies. (c) Measured $\sqrt{I^2 + Q^2}$ of the reflected signal following a $20~\mathrm{ns}$ RF pulse. Inset: Time-resolved $I^2 + Q^2$ response measured at the resonance point $V_\mathrm{BO} = -29~\mathrm{V}$ (blue dots). The orange dashed line shows a fit to Eq. \ref{['eq:a_square']}, up to an overall proportionality constant, yielding $g/2\pi = 4.906 \pm 0.009~\mathrm{MHz}$ and $\gamma_{\mathrm p}/2\pi = 3.30 \pm 0.02~\mathrm{MHz}$. (d) Analytical calculation corresponding to panel (c), performed with the resonance point set to $V_\mathrm{BO} = -29~\mathrm{V}$. The color scale represents the amplitude of the RF field stored in the LC resonator, $|a(t)|$, calculated using Eq. \ref{['eq:a_general']} with $g/2\pi = 4.9~\mathrm{MHz}$, $\gamma_{\mathrm p}/2\pi = 3.3~\mathrm{MHz}$, and $\kappa/2\pi = 0.39~\mathrm{MHz}$. (c,d) The detuning on the upper horizontal axis, $\Delta = \omega_0 - \omega_\mathrm{p}$, is obtained from the numerically calculated relation between $V_{\mathrm{BO}}$ and $\omega_\mathrm{p}$.
• Figure 4: The RF excitation power is set to $-50~\mathrm{dBm}$ at the output of the signal generator. (a) Resonance peaks measured at $V_\mathrm{BO} = -29~\mathrm{V}$ as a function of temperature; other measurement conditions are identical to those in Fig. \ref{['fig:strong_coupling_time_domain']}(a). (b,c,d) Temperature dependence of the plasmon decay rate $\gamma_{\mathrm{p}}$ in (b), the detuning $\Delta = \omega_0 - \omega_{\mathrm{p}}$ in (c), and the coupling strength $g$ in (d), obtained from fit of Eq. \ref{['eq:reflection_coefficient_input_output']} to the data in (a). Blue circles represent the extracted values, while the semi-transparent blue lines indicate the fitting uncertainties (95% confidence intervals, as used throughout this manuscript). In (b), the red dashed line shows $\gamma_{\mathrm{p}} = 1/\tau$, where $\tau$ is the momentum--relaxation time calculated using the theoretical model reported in Ref. Saitoh1978-jc for a pressing field of $92~\mathrm{V/cm}$.
• Figure 5: Extracted values of $\gamma_\mathrm{p}$/$2\pi$ as a function of pressing field measured at $T = 180~\mathrm{mK}$. The pressing field corresponds to $V_\mathrm{BC}$/$D$ (for $V_\mathrm{BC} = 10,\ 9,\ 8,\ 7.5,\ 6~\mathrm{V}$) plus a correction for the field due to the image charge induced by the electrons, which is the electric field experienced by the electrons near the center. The value at $V_\mathrm{BC} = 17~\mathrm{V}$ is extracted from Fig. \ref{['fig:strong_coupling_time_domain']}(a), while the values at $V_\mathrm{BC} = 10,\ 9,\ 8,\ 7.5,\ 6,\ 4~\mathrm{V}$ are extracted from Fig. \ref{['fig2']}(a). The value of $\gamma_\mathrm{p}$ for $V_\mathrm{BC}=4~\mathrm{V}$ in Fig. \ref{['fig2']}(a) could not be extracted \ref{['fig2']}(a) as the plasmon frequency changes rapidly with the radius at higher modes. The orange line indicates $\gamma_{\mathrm{p}} = 1/\tau$ from the model of Ref. Saitoh1978-jc at $T = 180~\mathrm{mK}$. The marker size is large than the error.