Coupling an Ensemble of Electrons on Superfluid Helium to a Superconducting Circuit

The quantized lateral motional states and the spin states of electrons trapped on the surface of superfluid helium have been proposed as basic building blocks of a scalable quantum computer. Circuit quantum electrodynamics allows strong dipole coupling between electrons and a high-$Q$ superconducting microwave resonator, enabling such sensitive detection and manipulation of electron degrees of freedom. Here, we present the first realization of a hybrid circuit in which a large number of electrons are trapped on the surface of superfluid helium inside a coplanar waveguide resonator. The high finesse of the resonator allows us to observe large dispersive shifts that are many times the linewidth and make fast and sensitive measurements on the collective vibrational modes of the electron ensemble, as well as the superfluid helium film underneath. Furthermore, a large ensemble coupling is observed in the dispersive regime during experiment, and it shows excellent agreement with our numeric model. The coupling strength of the ensemble to the cavity is found to be $\approx 1\mathrm{MHz}$ per electron, indicating the feasibility of achieving single electron strong coupling.

Popular Summary

Under appropriate conditions, electrons can be trapped on the surface of superfluid helium and act like a two-dimensional electron gas. The electrons have a complicated relationship with the helium; they are attracted to the surface but cannot stand to be in the liquid. Quantum mechanics resolves this conflict by allowing the electrons to levitate above the surface of the liquid. Because the electrons are essentially in vacuum with only a weak coupling to the helium (itself a nearly perfect liquid), these electrons are exciting candidates for quantum bits and other basic quantum physics experiments. Here, electrons are trapped within a superconducting resonator, and their dynamics are observed at high speeds.

At a temperature of 25 thousandths of a degree above absolute zero, electrons are held in place above the superfluid helium using a bias voltage that creates a parabolic trapping potential above a superconducting resonator. The electrons alter the frequency of the resonator, which allows their presence to be detected. Using these techniques, the properties of the electrons (e.g., their vibrational modes) and the superfluid helium film can be measured with high precision in a nondestructive manner; the electrons can be held and studied for days at a time. These experiments represent an important step in utilizing trapped electrons as part of a quantum processor.

Since an electron’s spin can be used as a form of quantum memory, we expect that our results will motivate additional studies leading toward the development of a viable quantum computer.

Figure 1

Device, circuit schematic, and trap geometry. (a) Optical and SEM images of a cavity-electron ensemble trap on a ($2\times 7$)-mm superconducting chip. The device is positioned 5.5 mm above the bottom of a cylindrical superfluid reservoir of radius $r=3.175\mathrm{mm}$, mounted in a hermetically sealed copper box at 25 mK in a dilution refrigerator. (b) Interdigitated gap capacitors with gap width $2\mu \mathrm{m}$ at the cavity input. (c) dc bias electrode connected directly to the center pin of the cavity at a node of the standing-wave voltage distribution of the fundamental mode. (d) Sub-$\mu \mathrm{m}$-size electron trap near voltage maximum of the fundamental mode with constriction of width 500 nm. (e) Circuit schematic showing the voltage distribution of the fundamental mode (red) and the simplified measurement and control circuit connected to the center pin (pink). The cavity is measured in transmission using a low-noise amplifier (LNA) and the trap potential is tuned through a dc source connected to the center pin (pink) through a low-pass filter. (f) Cross-sectional view of the cavity waveguide gap showing the schematic trap geometry. The ground planes (gray) form a microchannel of height $d=800\mathrm{nm}$ and width $w_G=6\mu \mathrm{m}$ filled with superfluid ${}^4\mathrm{He}$ by capillary action. A dc voltage on the submerged center pin (pink) of width $w_{\mathrm{cp}}=2\mu \mathrm{m}$ and thickness $t=80\mathrm{nm}$ creates a parabolic trapping potential for electrons above the surface that couple to the rf field in the cavity.

Figure 2

Cavity response to superfluid helium. (a) Measured resonance frequency shifts $\mathrm{\Delta }{\omega }_0/2\pi$ (blue dots, left axis) and loaded quality factor $Q_L$ (green triangles, right axis) as functions of superfluid volume supplied to the cell (bottom axis) and relative bulk helium level $H$ in the reservoir (top axis). Each data point corresponds to an increase in superfluid volume of $\mathrm{\Delta }{V}_{\mathrm{sf}}\sim 2.3{\mathrm{mm}}^3$ and reservoir level $\mathrm{\Delta }H\sim 70\mu \mathrm{m}$. (b) Different filling state corresponding to the different regimes in (a). (c) Frequency shift (blue dots, left axis) and quality factor (green triangles, right axis) as functions of center pin voltage bias $V_{\mathrm{cp}}$ at fixed helium level in the capillary action regime indicated by arrow in (a). Gray dots are frequency shifts extracted from single-shot cavity transmission measurements with $N=80$ such measurements per voltage bias point $V_{\mathrm{cp}}$. The blue data points are averages over the single-shot measurements at each point.

Figure 3

Detection of a trapped electron ensemble on superfluid helium in a cavity transmission experiment. (a) Normalized transmitted power through the cavity as a function of center pin trap voltage $V_{\mathrm{cp}}$. (b) Normalized transmission spectra at $V_{\mathrm{cp}}=+0.91$ and $+2.8\mathrm{V}$, showing a shift in resonance frequency and a reduction in transmitted power at the bias points indicated by the dashed vertical lines in (a). Solid red lines are fits to Lorentzians. (c) Resonance frequency and (d) loaded quality factor as functions of trap bias in the presence (blue) and absence (red) of an electron ensemble. In panels (a)–(d), electrons are first loaded into the cavity mode volume at an initial bias of $V_{\mathrm{cp}}=+3\mathrm{V}$ and a fixed helium level in the capillary regime with an uncharged shift of $\mathrm{\Delta }{\omega }_0/2\pi =-7.58\mathrm{MHz}$ and a reservoir level of $H\simeq 4\mathrm{mm}$. The blue line shows the trap voltage being swept from $+3$ to $-1\mathrm{V}$ and back in 4 mV steps, eventually depleting the trap region, while the red line shows the same sweep for an empty trap.

Figure 4

Measured cavity frequency shifts as a function of trap voltage bias $V_{\mathrm{cp}}$. Colors indicate consecutive cycles of the voltage sweep. In each cycle $V_{\mathrm{cp}}$ is decreased (arrows) until electrons are irreversibly lost from the trap and then increased to the initial value of 3.0 V. Gray dashed lines in the background show frequency shifts predicted by molecular dynamics simulation. For each of the four isoelectron number curves the number of electrons in the simulation is depicted to the left. A red dotted line indicates modeled electron loss with respect to a leak voltage of (530) mV. The isoelectron number curves from the experiment terminate within a small neighborhood of this loss frontier.