Sensitive detection of the Rydberg transition in trapped electrons on liquid helium using radio-frequency reflectometry

TL;DR

This work tackles the challenge of detecting Rydberg transitions in a many-electron system trapped on liquid helium by employing rf reflectometry to monitor small impedance changes in a lumped-element tank circuit. The authors combine experimental rf measurements with Green's-function simulations and an independent image-charge readout to dissect the origin of the rf response, showing that the signal is strongly enhanced near plasmon resonances and is dominated by lateral collective motion rather than vertical displacements of individual electrons. A master-equation analysis of a driven two-level system dressed by microwave excitation is presented to contrast with the observed capacitive response, arguing that vertical-displacement-based quantum-capacitance cannot account for the data. The results demonstrate high-sensitivity, fast readout of Rydberg dynamics in electrons on helium and point toward collective-electron effects as the primary mechanism, with potential implications for quantum-state readout and studies of many-electron dynamics in this platform.

Abstract

Radio-frequency reflectometry, which probes small changes in the electrical impedance of a device, provides a useful method for sensitive and fast detection of dynamic processes in quantum systems. We use this method to detect excitation of the quantized motional (Rydberg) states of trapped electrons on liquid helium. The Rydberg transition in an ensemble of electrons is detected by a change in the impedance of an rf circuit coupled to the microwave-excited electrons. To elucidate the origin of the observed response, the result is compared with an independent impedance measurement on the same electron system modulated by an electrostatic potential and with a numerical simulation using the Green's function method. Additionally, it is found that the rf response to the Rydberg resonance can be strongly enhanced by a resonant mode of the electron collective motion. Our results suggest that the observed response to the Rydberg resonance must be attributed to the lateral motion of the many-electron system rather than the vertical displacement of the individually excited electrons, as was explicate earlier. A theoretical analysis of the expected response due to the vertical displacement is given.

Figures (18)

• Figure 1: (color on line) Experimental setup. (a) 3D rendering of the experimental cell and PCB (electrical shielding is not shown) comprising an rf device for gate-based sensing of electrons on liquid helium. (b) Circuit model of the device and measurement setup. The electrical impedance of the cell is represented by a parallel combination of capacitance $C_\textrm{p}$ and resistance $R_\textrm{p}$. $R_\textrm{L}$ and $C_\textrm{L}$ model parasitic contributions to the impedance of the wire coil having inductance $L$, while $C_\textrm{par}$ model the parasitic capacitance of PCB and electrical connections in the cell. The effective resistance $R$ represent other losses in the circuit. The amplitude-modulated reflection from the device is amplifier by a cryogenic amplifier followed by a room-temperature amplifier (Fairview Microwaves FMAM3311) and low-pass filter (Mini-Circuits LSP-250+) and demodulated by a mixer (Mini-Circuits ZEM-2B+). Alternatively, the signal at the output of the cryogenic amplifier can be measured by a signal or vector analyzer. (c) In-phase and quadrature components (left panel) and amplitude and phase (right panel) of the reflection signal measured by an rf lock-in amplifier at $T=100$ mK. Solid lines show fitting as described in the text.
• Figure 2: (color on line) Exemplary reflection spectra (solid lines) taken using VNA before (blue) and after (red) charging the surface of liquid helium with electrons. The dashed line is the reflection spectrum taken with electrons confined by negative guard potentials.
• Figure 3: (color on line) The distribution of areal density of surface electrons $n_s$ for different sets of bias voltages applied to the electrodes (a) $V_\textrm{BCBM}=20$ V, $V_\textrm{BG}=V_\textrm{TC}=V_\textrm{TM}=V_\textrm{TG}=0$ (corresponds to surface charging), (b) $V_\textrm{BCBM}=30$ V, $V_\textrm{TC}=V_\textrm{TM}=18$ V, $V_\textrm{BG}=V_\textrm{TG}=-60$ V, (c) $V_\textrm{BCBM}=30$ V, $V_\textrm{TC}=18$ V, $V_\textrm{BM}=V_\textrm{BG}=V_\textrm{TM}=V_\textrm{TG}=-60$ V, and (d) $V_\textrm{BC}=V_\textrm{TC}=0$, $V_\textrm{BM}=V_\textrm{BG}=30$ V, $V_\textrm{TM}=V_\textrm{TG}=18$ V. Note that for the latter configuration the electrons form a ring-shaped distribution and are completely expelled from the central region, as described in Sec. \ref{['sec:config']}. For reference, the dashed lines indicate the radii of the central, middle and guard electrodes, $R_\textrm{central}=5.9$ mm, $R_\textrm{middle}=8.4$ mm and $R_\textrm{guard}=10.4$ mm, respectively.
• Figure 4: (color on line) Color map of the measured in-phase component of the demodulated reflection signal versus the mm-wave frequency $f_\textrm{mm}$ and the rf carrier frequency $f_\textrm{c}$ obtained for PM mm-wave excitation at the modulation frequency $f_\textrm{m}=1$ kH and confining voltages $V_\textrm{BCBM}=30$ V, $V_\textrm{TC}=V_\textrm{TM}=18$ V and $V_\textrm{BG}=V_\textrm{TG}=-60$ V corresponding to the electron density profile given by the dashed line (b) in Fig. \ref{['fig:3']}.
• Figure 5: (color on line) (a) The in-phase component of the demodulated reflection signal measured with the resonant ($f_\textrm{mm}=166.5$ GHz) PM excitation at the modulation frequency $f_\textrm{m}=1$ kHz as a function of the rf carrier frequency $f_\textrm{c}$. (b) The in-phase component of the demodulated reflection signal measured without mm-wave excitation and applying an ac voltage with the peak-to-peak amplitude of 3 V and the frequency $f_\textrm{m}=1$ kHz to the guard electrodes of the top and bottom plates. Dashed line is a simulated response from the circuit model shown in Fig. \ref{['fig:1']}(b) to the capacitive and resistive changes of the cell impedance of $\delta C_p = 170$ aF and $\delta R_p= -0.7$ M$\Omega$, respectively.