Strong Coupling of the Cyclotron Motion of Surface Electrons on Liquid Helium to a Microwave Cavity

The strong coupling regime is observed in a system of two-dimensional electrons whose cyclotron motion is coupled to an electromagnetic mode in a Fabry-Perot cavity resonator. Rabi splitting of eigenfrequencies of the coupled motion is observed both in the cavity reflection spectrum and ac current of the electrons, the latter probed by measuring their bolometric photoresponse. Despite the fact that similar observations of Rabi splitting in many-particle systems have been described as a quantum-mechanical effect, we show that the observed splitting can be explained completely by a model based on classical electrodynamics.

Figure 1

(a) Schematic diagram of the experimental setup. (b) Sketch of the experimental cell and the Fabry-Perot resonator. (c) Circuit for the microwave power measurements.

Figure 2

Left: results of simultaneous measurements of (a) the microwave power reflected from the cavity, $P_r$, and (b) the real part of the dc admittance of the cell, $\mathrm{Re}(Y)$, as a function of the microwave probe frequency $\omega$ and the magnetic field in units of ${\omega }_c$. The electron density is $n_e\approx 2\times 10^8{\mathrm{cm}}^{-2}$, and the microwave probe power is $P_{\mathrm{in}}\approx 180\mathrm{nW}$. The dashed lines correspond to the calculated eigenfrequencies of the cavity-field-2DES coupled motion with a coupling strength $g/2\pi \approx 77\mathrm{MHz}$. The horizontal stripes in (a) are caused by a parasitic standing wave formed in the waveguide due to a slight impedance mismatching. Right: absolute value of (c) the normalized reflected power $|S_{11}{|}^2$ and (d) the normalized electron current density $j_x/({\sigma }_0{E}_{\mathrm{in}})$ calculated using the model described in the text for $n_e=10^8{\mathrm{cm}}^{-2}$, $\gamma =4\times 10^8{\mathrm{s}}^{-1}$, and $\nu ={10}^8{\mathrm{s}}^{-1}$.

Figure 3

(a) Normalized spectra of the microwave power reflected from the cavity at several values of the input power. Measurements were performed at the magnetic field corresponding to the pure cyclotron resonance frequency ${\omega }_c=88.35\mathrm{GHz}$. The electron density is $n_e\approx 5\times 10^7{\mathrm{cm}}^{-2}$. The splitting appears only at high powers due to the power-induced narrowing of the cyclotron resonance. (b) Power dependence of the splitting $\mathrm{\Delta }$ measured as the distance between the two minima of $P_r$ near the degeneracy point. Inset: dependence of the coupling strength $g$ on the square root of the electron density, $\sqrt{n_e}$, at an input power of $P_{\mathrm{in}}\approx 70\mathrm{nW}$.