Spectroscopy of Wigner molecules on superfluid helium using a superconducting resonator

TL;DR

This paper demonstrates the integration of Wigner molecules on superfluid helium with a superconducting resonator, enabling microwave spectroscopy and strong coupling, which advances quantum simulation and control of electron correlations.

Contribution

It introduces a novel cQED platform for Wigner molecules on helium, allowing for the first time their spectral characterization and strong photon coupling in the microwave regime.

Findings

Successfully prepared 1-4 electron Wigner molecules on a resonator

Observed their unique microwave spectra for the first time

Achieved a single-electron-photon coupling strength of 4.8 MHz

Abstract

Electrons on helium form a unique two-dimensional electron system on the interface of liquid helium and vacuum. On liquid helium, trapped electrons can arrange into strongly correlated states known as Wigner molecules, which can be used to study electron interactions in the absence of disorder, or as a highly promising resource for quantum computation. Wigner molecules have orbital frequencies in the microwave regime and can therefore be integrated with circuit quantum electrodynamics (cQED), which studies light-matter interactions using microwave photons. Here, we experimentally realize a cQED platform with the orbital state of Wigner molecules on helium. We deterministically prepare one to four-electron Wigner molecules on top of a microwave resonator, which allows us to observe their unique spectra for the first time. Furthermore, we find a single-electron-photon coupling strength of g/2$\pi$ = 4.8$\pm$0.3 MHz, greatly exceeding the resonator linewidth $\kappa$/2$\pi$ = 0.5 MHz. These results pave the way towards microwave studies of strongly correlated electron states and coherent control of the orbital and spin state of Wigner molecules on helium.}

Tables

Table 1. Supplementary Table 1: Modeled unloading voltages compared with experimental jump locations.

Unloading voltage	Model (V)	Experiment (V)	Comment
$V_{\mathrm{tg}}^{\left(1\right)}$	-0.305	-0.305	Taken from experiment and used as parameter in the model
$V_{\mathrm{tg}}^{\left(2\right)}$	-0.248	-0.246	Obtained from fit with ${\omega }_e/2\pi =26$ GHz
$V_{\mathrm{tg}}^{\left(3\right)}$	-0.205	-0.202	Obtained from fit with ${\omega }_e/2\pi =26$ GHz
$V_{\mathrm{tg}}^{\left(4\right)}$	-0.165	-0.168	Obtained from fit with ${\omega }_e/2\pi =26$ GHz
$V_{\mathrm{tg}}^{\left(5\right)}$	-0.127		Not observed in experiment
$V_{\mathrm{tg}}^{\left(6\right)}$	-0.092		Not observed in experiment

Table 2. Supplementary Table 2: Summary of the contributions to the linewidth of a single electron from different types of noise or decay expected in our device.

Type	Mechanism	Magnitude
Dephasing	Voltage noise from the gates	0.5 MHz
Dephasing	Helium vibrations in the dot	110 MHz
Dephasing	Reservoir electrons on the resonator	7 MHz
Transverse	Microwave leakage through gates	1 MHz

Table 3. Supplementary Table 3: Simulated effect of each electrode on the single electron mode frequency f e subscript 𝑓 𝑒 f_{e} and the resulting contribution to the single electron linewidth, assuming a voltage noise on each electrode of 5 μ 𝜇 \mu V.

Electrode $i$	Simulated slope (GHz/V)	${\gamma }_{\phi }^i/2\pi$ (MHz)
Resonator	-48	0.2
Trap	95	0.5
Resonator guard	-7	0.1
Trap guard	-11	0.1
Total		0.5

Equations

$$n\approx\frac{\varepsilon_{0}\varepsilon_{\mathrm{He}}}{et_{\mathrm{He}}}a_{\mathrm{res}}V_{\mathrm{res}}^{\mathrm{th}}=6\times 10^{12}\,\mathrm{m}^{-2},\$$

$$V_{\mathrm{unload}}^{(1)}=-\frac{V_{\mathrm{bar}}}{\beta }=-0.305\mathrm{V}$$

$$V_{\mathrm{unload}}^{(2)}=V_{\mathrm{unload}}^{(1)}+\frac{3}{4}\frac{E_0}{\beta e}$$

$$V_{\mathrm{unload}}^{(3)}=V_{\mathrm{unload}}^{(1)}+1.31037\frac{E_0}{\beta e}$$

$$V_{\mathrm{unload}}^{(4)}=V_{\mathrm{unload}}^{(1)}+1.83545\frac{E_0}{\beta e}$$

$$E_0={\left(\frac{m_e{\omega }_e^2{e}^4}{2{\left(4\pi \right)}^2{\epsilon }_0^2{\epsilon }_{\mathrm{He}}^2}\right)}^{\frac{1}{3}}$$

$$E/e={\alpha }_0(V_{\mathrm{trap}})x^2+{\alpha }_1(V_{\mathrm{trap}})y^2+{\alpha }_2(V_{\mathrm{trap}})y^4.$$

$$\frac{A}{A_0}=\left|\frac{\sqrt{{\kappa }_1{\kappa }_2}}{i({\kappa }_1+{\kappa }_2+{\kappa }_{\mathrm{int}})/2-\chi ({\omega }_0)}\right|,$$

$$\chi ({\omega }_0)=\frac{g^2}{({\omega }_0-{\omega }_e)+i\gamma }.$$

$$\frac{\alpha }{2\pi }=\frac{1}{2\pi }\frac{3e{\alpha }_2\hslash }{m_e^2{\omega }_e^2}\approx 85\mathrm{MHz}.$$

$$\mathcal{L}=\left(\begin{array}{cc}{L}_{11}& L_{12}\\ L_{21}& L_{22}\end{array}\right)\text{and}\mathcal{C}=\left(\begin{array}{cc}{C}_{11}& C_{12}\\ C_{21}& C_{22}\end{array}\right).$$

$$Z_{\text{diff}}=2\sqrt{\frac{L_{11}-L_{12}}{C_{11}+|C_{12}|}}.$$

$$f_0=\frac{1}{4\ell }\frac{1}{\sqrt{(L_{11}-L_{12})(C_{11}+|C_{12}|)}},$$

$$\hslash g=\mathbf{d}\cdot \mathbf{E}.$$

$$\frac{1}{4}\hslash {\omega }_0=\frac{1}{2}\frac{1}{T}{\int }_0^T\mathrm{d}t{\cos }^2({\omega }_{0}t)\int {\mathrm{d}}^{3}r\epsilon E^2(r)=\frac{1}{4}C{V}_{\mathrm{zpf}}^2=\frac{1}{2}C{V}_{\mathrm{rms}}^2.$$

$$g/2\pi =\mathbf{d}\cdot \mathbf{E}=\frac{1}{2}e{E}_y{f}_0\sqrt{\frac{Z}{m_e{\omega }_e}}.$$

$$S_{ij}=\frac{\sqrt{{\kappa }_i{\kappa }_j}}{(\omega -{\omega }_0)+i{\kappa }_{\mathrm{tot}}/2},$$

$$\frac{{\kappa }_3}{{\kappa }_c}=\frac{|S_{31}(\omega ={\omega }_0){|}^2}{|S_{21}(\omega ={\omega }_0){|}^2}.$$

$$\rho gh=\frac{\sigma }{R},$$

$$z(x,h)=d_0+\frac{\rho gh}{2\sigma }\left(x^2-\frac{w^2}{4}\right),$$

$$z(0,h)=t_{\mathrm{He}}=\max \left(0,d_0-\frac{\rho gh}{2\sigma }\frac{w^2}{4}\right).$$

$$g_{0i}/2\pi =\frac{e}{2\pi }\iint ⟨0|\left(x\frac{\partial V_{\mathrm{RF}}}{\partial x}+y\frac{\partial V_{\mathrm{RF}}}{\partial y}\right)|i⟩dxdy,$$

$$\gamma =\frac{{\gamma }_1}{2}+{\gamma }_{\phi }.$$

$$t_{\mathrm{He}}=d_0-\frac{\rho gh{w}^2}{8\sigma },$$

$$\Delta t_{\mathrm{He}}=\frac{\partial t_{\mathrm{He}}}{\partial h}\Delta h=\frac{\rho g{w}^2}{8\sigma }\Delta h$$

$$\frac{{\gamma }_{\phi }^{\mathrm{He}}}{2\pi }=\frac{\partial f_e}{\partial t_{\mathrm{He}}}\Delta t_{\mathrm{He}},$$

$$\frac{{\gamma }_{\phi }^{\mathrm{noise}}}{2\pi }=\sqrt{\sum _i{\left(\frac{\partial f_e}{\partial V_i}\right)}^2\Delta V_i^2}.$$

$$V=\frac{Q}{C}=\frac{N}{A}\frac{e{t}_{\mathrm{He}}}{{\epsilon }_0{\epsilon }_{\mathrm{He}}}=\frac{ne{t}_{\mathrm{He}}}{{\epsilon }_0{\epsilon }_{\mathrm{He}}},$$

$$\Delta V_{\mathrm{res}}=\frac{\partial V}{\partial t_{\mathrm{He}}}\Delta t_{\mathrm{He}}=\frac{ne}{{\epsilon }_0{\epsilon }_{\mathrm{He}}}\Delta t_{\mathrm{He}}$$

$$\frac{{\gamma }_{\phi }^{\mathrm{res}}}{2\pi }=\left|\frac{\partial f_e}{\partial V_{\mathrm{res}}}\right|\frac{ne}{{\epsilon }_0{\epsilon }_{\mathrm{He}}}\Delta t_{\mathrm{He}}$$

$$\gamma /2\pi =2\sqrt{2\ln 2}\frac{\partial f_e}{\partial V_{\mathrm{trap}}}\Delta V_{\mathrm{trap}}=67\mathrm{MHz},$$

Full text

\newcites

mainMain text \newcitesmethMethods \newcitessmSupplementary Materials

Spectroscopy of Wigner molecules on superfluid helium

using a superconducting resonator

G. Koolstra

gkoolstra@uchicago.edu

The James Franck Institute and Department of Physics, University of Chicago, Chicago, Illinois 60637, USA

  

Ge Yang

The James Franck Institute and Department of Physics, University of Chicago, Chicago, Illinois 60637, USA

  

David I. Schuster

David.Schuster@uchicago.edu

The James Franck Institute and Department of Physics, University of Chicago, Chicago, Illinois 60637, USA

Electrons on helium form a unique two-dimensional electron system on the interface of liquid helium and vacuum \citemainMonarkhaKono2004. On liquid helium, trapped electrons can arrange into strongly correlated states known as Wigner molecules \citemainRousseauPRB2009, which can be used to study electron interactions in the absence of disorder, or as a highly promising resource for quantum computation \citemainPlatzman1999, DykmanPRB2003, Lyon2006, Schuster2010. Wigner molecules have orbital frequencies in the microwave regime and can therefore be integrated with circuit quantum electrodynamics (cQED), which studies light-matter interactions using microwave photons \citemainWallraffNature2004. Here, we experimentally realize a cQED platform with the orbital state of Wigner molecules on helium. We deterministically prepare one to four-electron Wigner molecules on top of a microwave resonator, which allows us to observe their unique spectra for the first time. Furthermore, we find a single-electron-photon coupling strength of $g/2\pi=4.8\pm 0.3$ MHz, greatly exceeding the resonator linewidth $\kappa/2\pi=0.5$ MHz. These results pave the way towards microwave studies of strongly correlated electron states and coherent control of the orbital and spin state of Wigner molecules on helium.

The orbital state of electrons on helium consists of the lateral motion of strongly correlated electrons. Since the electron-phonon coupling in helium is small compared with semiconductors, this motion is expected to have low dissipation, making the orbital state an attractive candidate for a long-lived quantum bit \citemainSchuster2010, DaniilidisNJOP2013. In addition, by adding a magnetic field gradient from a micro-magnet \citemainPioroNatPhys2008, the orbital state offers a path towards the electron spin state \citemainViennot2015, Mi2018, LandigNature2018, Samkharadze2018Science, PengPRA2017. Since the orbital frequency of electrons on helium is in the microwave regime, and electrons can couple strongly to microwave photons \citemainSchuster2010, ShlomiPRA2017, Mi2017, Stockklauser2017, cQED can play a unique role in the detection and manipulation of the orbital state.

A small ensemble of electrons on helium behaves differently from other confined electron systems, such as semiconductors or atoms, where the electron wavefunctions are delocalized and overlap. On the surface of liquid helium electron interactions dominate \citemainRees2011PRL, ReesPRL2012 and are largely unscreened, which results in strongly correlated electron configurations known as Wigner molecules. The symmetry of these molecules changes for each additional electron, which has been observed in charging diagrams of small islands of liquid helium \citemainPapageorgiou2005, RousseauPRB2009. Additionally, theory has predicted Wigner molecule configurations and orbital frequencies in various trapping potentials \citemainBoltonSandM1993, BedanovPRB1994, SchweigertPRB1995, KongPRE2002. Spectroscopy of Wigner molecules on helium could provide insight into both the internal molecular structure and the molecule’s environment, but the lack of a microwave interface has prevented this to date.

Here we realize the coupling of Wigner molecules on helium to a microwave cavity that serves as an electron detector and harbors an electron reservoir. We transfer electrons from the reservoir to a small island where we control the charge with single electron resolution and perform spectroscopy of a single electron and few-electron Wigner molecules. We observe unique spectra which serve as a fingerprint for the molecule’s internal structure, and a large electron-photon coupling. These results open the door to coherent control of the orbital and spin state of Wigner molecules on helium.

At the heart of our cQED device lies a superconducting microwave resonator with an integrated electron-on-helium quantum dot (Fig. 1a). Our coplanar stripline resonator consists of two niobium center pins, which are joined at one end (Fig. 1b, c) and are situated below the ground plane at the bottom of a micro-channel (width $w=3.5$ µm, and depth $d_{0}\approx$ 1.2 µm). The microwave mode with resonance frequency $f_{0}=6.399$ GHz and linewidth $\kappa_{\mathrm{tot}}/2\pi=0.4$ MHz has an RF electric field that is concentrated between the center pins. As liquid 4He fills the channel, its surface is stabilized due to helium’s surface tension, after which the liquid helium can serve as a defect-free substrate for electrons (Fig. 1d).

After depositing electrons over the resonator, we detect a dispersive resonance frequency shift that depends strongly on the resonator bias voltage $V_{\mathrm{res}}$ (Fig. 2a) and the number of electrons on the resonator \citemainGeYang2016. For the experiments presented hereafter, we fix $V_{\mathrm{res}}$ at 0.6 V such that electrons on the resonator can be treated as a reservoir with constant electron density. Furthermore, our measurements are performed at $T=25$ mK and low incident microwave power ($n_{\mathrm{ph}}\approx 5$) such that electrons respond linearly to the resonator’s driving force.

We use the dot in Fig. 1c to isolate individual electrons from the reservoir, which requires fine control over the electrostatic potential. We achieve this using three sets of electrodes near the tip of the resonator where the microwave electric field is strongest. The size of the electrodes near the dot is much larger than in semiconducting quantum dots, because the unscreened electron interaction results in inter-electron distances exceeding 200 nm. With appropriate voltages applied to the electrodes, the smooth electrostatic potential (Fig. 2d,e) allows for trapping of electrons. Furthermore, due to the dot’s oblong shape, the lateral motion of trapped electrons is primarily in the $y$-direction (see Fig. 1d), such that it couples to the transverse microwave field of the resonator.

To load the dot we use the trap electrode (Fig. 1c, green) to attract reservoir electrons towards the dot, and the resonator guard (blue) to create a barrier between the dot and reservoir. Only if the trap voltage is sufficiently positive, and the resonator guard is sufficiently negative can electrons be loaded and contained in the dot, respectively. When monitoring the resonance frequency shift $\Delta f_{0}$ in response to these two voltages, we only see significant signal in an area that is marked by two converging dashed lines in Fig. 2b. The dashed lines are obtained from simulation of the electrostatic potential near the dot (see Methods), and indicate the presence of a barrier between reservoir electrons and electrons in the dot. Well within the predicted trapping region, we observe resonance frequency shifts that depend sensitively on $V_{\mathrm{trap}}$ and $V_{\mathrm{rg}}$, indicating that trapped electrons in the dot interact with the resonator. The observed shift depends on the number of trapped electrons, which increases for a larger trap voltage, as well as the shape of the electrostatic potential.

To deterministically prepare few-electron Wigner molecules, we partially unload the dot using the trap guard electrode (orange in Fig. 1c). A partial unload consists of briefly sweeping the trap guard voltage to $V_{\mathrm{unload}}<0$, which decreases the trap depth (see Fig. 3a), followed by a measurement of the resonator transmission at $(V_{\mathrm{trap}},V_{\mathrm{tg}})=(0.175,0.0)$ V. The plateaus in resonator transmission shown in Fig. 3b are reproduced after reloading the dot, but are absent when the dot is initially empty. Therefore, each plateau is associated with a constant number of trapped electrons, and the final change in transmission at $V_{\mathrm{unload}}=-0.305$ V leaves the dot empty.

The sudden changes in transmission are consistent with single electrons leaving the dot. We show this by modeling the trap as an axially symmetric harmonic well in which the electron configurations can be calculated analytically \citemainBedanovPRB1994, SchweigertPRB1995. From the voltage at which the last electron escapes, we estimate unloading voltages for two, three and four electrons, using the effective trap curvature as the only free parameter (see Methods). Red arrows in Fig. 3b indicate these estimates, and agree within 3 mV with the plateau edges. This unloading method therefore allows for deterministic preparation of one to four-electron Wigner molecules.

The increasing length of transmission plateaus $\Delta V_{N}$ with decreasing $N$ is a telltale sign of strongly interacting electron ensembles, such as Wigner molecules \citemainRousseauPRB2009, and is in stark contrast to an equally-spaced charging diagram typically seen in metallic islands or semiconducting quantum dots. For an electron on helium dot, an unscreened interaction results in a significantly different charge configuration each time an electron is removed from the dot (Fig. 3d), resulting in the characteristic irregular spacing.

While a Wigner molecule is trapped in the dot, we use the resonator to observe it’s unique spectrum, which provides insight in the electron configurations and orbital frequencies. We perform spectroscopy by monitoring the resonator’s transmission while varying the trap voltage, which deforms the trap and therefore controls the orbital frequencies. For this measurement, a Wigner molecule can be trapped and studied for hours, since the trap depth is large compared to the zero-point energy and thermal energy. Fig. 3c shows five different spectroscopy traces, each corresponding to the different-sized Wigner molecules from Fig. 3b. To retrieve electron configurations and orbital frequencies, we numerically minimize the total energy of the ensemble and solve the coupled equations of motion \citemainGeYang2016. The electron configurations (Fig. 3d) change significantly as electrons are added or removed from the dot, and show correlated electron motion, originating from strong electron interactions. The largest signal in Fig. 3c occurs for a single electron at $V_{\mathrm{trap}}=0.175$ V when its orbital frequency is resonant with the resonator. In our model, the orbital frequency of larger Wigner molecules remains detuned for all $V_{\mathrm{trap}}$, which is due to a strong anharmonic component in the electrostatic potential. From the quartic term in this potential, we estimate a single-electron anharmonicity of 85 MHz, which holds promise for creating an electron-on-helium orbital state qubit.

We now focus on a single trapped electron and investigate its properties by tuning the orbital frequency into resonance with the resonator. Fig. 4a shows a crossing of the orbital frequency with the resonator around $V_{\mathrm{trap}}=0.184$ V, which is accompanied by a rapid change in $\Delta f_{0}$ (Fig. 4c). By fitting the measured frequency shift to a model, which takes into account one orbital mode coupled to a single resonator mode \citemainCottet2017, we obtain a single-electron-photon coupling strength $g=2\pi\times(4.8\pm 0.3)$ MHz and electron linewidth $\gamma=\gamma_{1}/2+\gamma_{\varphi}=2\pi\times(77\pm 19)$ MHz. The coupling strength is large compared to the resonator linewidth ($\kappa/2\pi\approx 0.5$ MHz), indicating that each photon measures the presence of the electron, and the coupling is similar to that measured in semiconducting quantum dot cQED architectures \citemainMi2017. In addition, our estimate of the anharmonicity (see Supplementary Figure 7) is similar to that in superconducting qubits, indicating that with a reduced linewidth the orbital state of a single electron on helium can be used as a qubit.

The total linewidth $\gamma$ is three orders of magnitude larger than expected from the electron-phonon coupling in 4He and charge noise from the bias electrodes, respectively ($\gamma/2\pi<0.1$ MHz) \citemainSchuster2010. We identify the dominant source of excess noise as classical helium fluctuations in the dot, caused by the pulse tube refrigerator. This is corroborated by a measurement of the crossing voltage as function of time, which shows spectral features of the pulse tube refrigerator. To estimate the dephasing rate due to helium fluctuations, we estimate an electron’s sensitivity to helium fluctuations from electrostatic simulations ($\partial f_{e}/\partial t_{\mathrm{He}}\approx 80$ MHz/nm) and independently measure helium fluctuations ($\Delta t_{\mathrm{He}}\approx 1.4$ nm), yielding $\gamma_{\varphi}/2\pi\approx 110$ MHz. Therefore, we expect the single electron linewidth to be limited by dephasing due to helium level fluctuations.

Reducing the linewidth and increasing the coupling strength offers a path towards coherent control of a single electron on helium and may enable more accurate spectroscopic studies of Wigner molecules, through direct measurement of the electron orbital frequencies using two-tone spectroscopy \citemainSchusterPRL2005. In the next generation of electron on helium dots, one can passively or actively reduce the vibrations that excite the helium surface \citemainPelliccioneRSI2013, or engineer a dot geometry that has a reduced sensitivity to classical helium vibrations. In addition, microwave resonators made of high kinetic inductance superconductors can enhance the coupling strength in our device via an increased characteristic impedance \citemainSamkharadze2018Science, Shearrow2018.

In conclusion, we have integrated an electron-on-helium dot with a superconducting microwave resonator and observed distinct spectra of Wigner molecules consisting of up to four electrons. The large anharmonicity and coupling strength of a single electron on helium hold promise for creating an electron-on-helium qubit, which can be readily integrated with superconducting qubits while leveraging established protocols. Finally, when combined with a magnetic field gradient, the orbital state offers a clear path towards control of single electron and Wigner molecule spin states.

Acknowledgements

Acknowledgements.

This research was supported by the DOE, Office of Basic Energy Sciences, Materials Sciences and Engineering Division. We thank K.W. Lehnert for the parametric amplifier used in this work, S. Chakram, D.D. Awschalom, D.J. van Woerkom, E. Kawakami, and other members of the Schuster lab for insightful discussions and P.J. Duda and D.C. Czaplewski for assistance and advice during fabrication of the device. This work made use of the Pritzker Nanofabrication Facility of the Institute for Molecular Engineering at the University of Chicago, which receives support from Soft and Hybrid Nanotechnology Experimental (SHyNE) Resource (NSF ECCS-1542205), a node of the National Science Foundation’s National Nanotechnology Coordinated Infrastructure.

Author contributions

The experiment was conceived by D.I.S. All authors contributed to the design of the experiment. G.Y. and G.K. fabricated the samples. G.K. performed the measurements and analyzed the data. G.K. and D.I.S. wrote the manuscript and all authors commented on the manuscript.

Competing interests

The authors declare no competing interests.

Methods

Fabrication. First an 80 nm thick Nb ground plane was evaporated onto a high-resistivity ($>10$ k$\Omega$cm) Si $\langle 100\rangle$ wafer, followed by deposition of a 100 nm thick silicon oxide sacrificial layer, which was used to protect the Nb ground plane during the following etch steps. The micro-channels were defined using a Raith EBPG-5000+ electron beam lithography system and etched using a CHF3/SF6 chemistry, immediately followed by an HBr/O2 etch. In the second step the resonator center pins were defined using e-beam lithography. After development, evaporation of a 150 nm thick Nb layer and lift-off, the center pins remained on the bottom of the micro-channel. To improve robustness of the device and avoid electrical breakdown at low temperatures, we etched away an additional $\sim$400 nm of Si substrate in between the resonator center pins. To this end, another layer of 80 nm thick silicon oxide was deposited, after which the additional Si was etched with the previously described etch chemistry. The silicon oxide layer was removed using buffered HF and a DI water rinse.

Measurements. All measurements were performed in an Oxford Triton 200 dilution refrigerator with a base temperature of 25 mK. The chip was mounted in a custom-designed hermetic sample cell and sealed with indium to prevent superfluid helium leaks. Helium was supplied to the sample cell from a high purity 4He gas cylinder and, using a control volume ($V\approx 25$ cm3) in a gas handling system, we were able to introduce a controlled amount of helium to the sample cell. The experiment was performed in a regime where the channel was almost full and the liquid helium film was stabilized due to surface tension \citemethMMartyJPC1986.

Electrons were captured on the helium surface by thermal emission from a tungsten filament situated above the chip, while applying a positive voltage to the resonator DC bias electrode ($V_{\mathrm{res}}=3.0$ V) and a negative bias voltage to the filament. We assume electrons in the reservoir were distributed uniformly across the resonator and estimate the electron density from the resonator voltage at which electrons can no longer be contained on the resonator, as depicted by the sudden increase in $\Delta f_{0}$ in Fig. 2a. At $V_{\mathrm{res}}^{\mathrm{th}}=0.18$ V we estimate the density

[TABLE]

where $a_{\mathrm{res}}$ is the resonator electrode lever arm, $t_{\mathrm{He}}$ is the helium thickness, $\varepsilon_{\mathrm{He}}=1.056$ is the dielectric constant of helium and $e$ is the elementary charge. This density corresponds to approximately $10^{5}$ reservoir electrons, whose orbital frequency stayed far detuned from $f_{0}$ during experiments with electrons in the dot.

The pulse tube refrigerator is a continuous source of mechanical vibrations which excites the liquid helium surface. These vibrations were detected by the microwave resonator as a slowly varying resonance frequency jitter, with a standard deviation of approximately $6.8$ kHz in the absence of reservoir electrons. This jitter complicated the measurement of small resonance frequency shifts due to trapped electrons, which were typically of the same order as the jitter. However, since the dominant frequency components in the mechanical noise spectrum were below 10 Hz, we circumvented this issue by sweeping electrode voltages faster than 1/10 Hz*-1*, such that signatures of trapped electrons became visible after averaging.

Electrostatic simulations of the dot. The electrostatic potential near the dot was obtained by solving Poisson’s equation using the finite element method with ansys maxwell. We separately solve the potential for each electrode that contributes to the dot potential by applying 1 V on a single electrode while keeping all other electrodes grounded. We minimize numerical noise in the potential by increasing the vertex density in the center of the dot and imposing strong convergence criteria. For post-processing the potential values are cast to a regular Cartesian grid using interpolation.

The two converging dashed line segments in Fig. 2b are obtained by considering both the potential along the channel and the reservoir density. The reservoir density $n$ sets the chemical potential of the reservoir via (approximately) $e^{2}nt_{\mathrm{He}}/\varepsilon_{0}\varepsilon_{\mathrm{He}}$, and for larger $n$, $V_{\mathrm{rg}}$ must be more negative to maintain a barrier between reservoir and dot (Fig. 2d). For our device, this non-zero barrier condition is captured by a line segment with slope 1.15. The reservoir density $n$ determines the offset of this line segment, and was measured by increasing $V_{\mathrm{trap}}$ until electron transport occurred onto the trap electrode. From an equation similar to Eq. (1) we find $n\approx 4\times 10^{12}$ m*-2*. The horizontal line segment was found by finding the minimum $V_{\mathrm{trap}}$ for which the reservoir extends left of $x=1.5$ µm at $V_{\mathrm{rg}}=0$. Fig. 2c shows a situation above this threshold, for which the loading operation should result in trapped electrons.

Unloading the dot. The dot was unloaded by sweeping the trap guard to $V_{\mathrm{tg}}=V_{\mathrm{unload}}<0$ while keeping all other electrodes constant at $(V_{\mathrm{res}},V_{\mathrm{trap}},V_{\mathrm{rg}})=(0.6,0.15,-0.4)$ V. The electrodes were then ramped back to $(V_{\mathrm{trap}},V_{\mathrm{tg}})=(0.175,0)$ V in order to probe the resonator transmission. The speed of the ramp did not change the charging diagram of Fig. 3b.

To confirm that changes between transmission plateaus in Fig. 3b are associated with single electron transport, we simulated unloading using a combination of electrostatic simulations and analytical calculations. Even though the electrode geometry in the dot produced a complex and anharmonic trapping potential on the scale of the dot ($8\times 4$ µm), the small extent of the electron ensemble ($0.5\times 0.5$ µm) allowed us to simulate the unloading with an axially symmetric harmonic well. The unloading voltage $V_{\mathrm{unload}}$ decreased the trap depth and resulted in unloading of the dot. We modeled this process as a linear decrease in barrier height: $V_{b}=V_{\mathrm{bar}}+\beta V_{\mathrm{unload}}$, where $V_{\mathrm{bar}}=22$ meV was obtained from electrostatic simulations and $\beta$ was determined from the final jump $(A/A_{0})^{2}$ in Fig. 3b. The energies of the Wigner molecules were calculated analytically \citemethMKongPRE2002, which resulted in the unloading voltages $V_{\mathrm{unload}}^{(N)}$:

[TABLE]

where

[TABLE]

and depends only on the trap curvature at the unloading point ($\omega_{e}$), electron mass ($m_{e}$) and other physical constants. Best agreement between model and experiment was found with an effective trap curvature $\omega_{e}/2\pi=26$ GHz, which produces the red arrows in Fig. 3b.

If the dot had initially contained five electrons, our model would have predicted an additional plateau starting at $V_{\mathrm{unload}}^{(5)}$ = -0.127 V. Since we did not observe this plateau we concluded the trap was initially loaded with $N=4$ electrons.

Modeling of Wigner molecule spectra. To accurately model Wigner molecule spectra, we needed a more sophisticated model of the electrostatic potential than a axially symmetric harmonic well. Instead, the electrostatic potential was approximated by

[TABLE]

Without a quartic term, the method described below predicts crossings for all Wigner molecules at equal $V_{\mathrm{trap}}$, which is inconsistent with experiment. Eq. (3) represents a model that reproduces the observed spectroscopy traces. The coefficients $\alpha_{i}$ were obtained by first fitting Eq. (3) to the electrostatic potential obtained via finite element modeling, and were then slightly adjusted to reproduce the spectroscopy traces from experiment, using the following method.

For a particular trap voltage the Wigner molecule configurations were found through numerical minimization of the total energy, which included a small screening correction to the interaction energy due to the metal electrodes under the electrons. In addition, we neglected the kinetic term in the total energy, since at $T=25$ mK the kinetic energy is approximately three orders of magnitude smaller than the interaction energy. Next, using the electron positions as input, the cavity frequency shift and orbital frequencies were determined by solving the linearized equations of motion of the coupled cavity-electron system. We then took the strongest-coupled orbital frequency $\omega_{e}$ and calculated its effect on the resonator via

[TABLE]

where $\kappa_{1,2,\mathrm{int}}$ represents the coupling through port 1 and 2 of the resonator and the internal loss rate, respectively. In addition, the susceptibility is given by

[TABLE]

$g/2\pi$ was fixed at 5 MHz (estimated from the resonator geometry, see Supplementary Information) and $\gamma/2\pi$ was adjusted to get good agreement for $N=1$. $\gamma$ was not further adjusted for $N>1$ Wigner molecules, since for those molecules all orbital modes stayed far detuned and the modeled traces only weakly depended on $\gamma$. With this method we obtained the resonator responses shown as solid black traces in Fig. 3c.

We obtained better agreement between the data and model for one and two electrons, compared with three and four electrons. This can be attributed to the larger size of the three and four-electron Wigner molecules, since the approximation of the electrostatic potential in Eq. (3) only holds for small $x,y$. In addition, each Wigner molecule spectrum was averaged about 2000 times which blurs sharp features, such as the one in the modeled three-electron trace.

The anharmonicity of a single electron was estimated by treating the $y^{4}$ term in Eq. (3) as a perturbation to the harmonic oscillator Hamiltonian. We define the anharmonicity $\alpha$ as $\hbar\alpha=(E_{2}-E_{1})-(E_{1}-E_{0})$, where $E_{n}$ are the perturbed eigenenergies. Near the crossing with the resonator we find $\alpha_{2}\approx 0.014$ $\mu$m*-4*, leading to

[TABLE]

Extracting single electron properties. To extract $g$ and $\gamma$ from the data in Fig. 4c, we used the same model for the resonator transmission as in Eq. (4), which was based on input-output theory and assumed that one orbital mode coupled to the resonator. To fit the frequency shift vs. trap voltage, we needed to know $\omega_{e}$ as function of $V_{\mathrm{trap}}$. We used quadratic fits to a finite element model of the electrostatic potential, which accurately predicted the single-electron crossing voltage, to find the dependence of $\omega_{e}$ on $V_{\mathrm{trap}}$. For the data in Fig. 4c, this method predicted a sensitivity near the crossing of $\partial f_{e}/\partial V_{\mathrm{trap}}=95$ GHz/V and also gives the top horizontal axis in Fig. 4c.

Since the measured frequency shift remained less than a linewidth, the phase ($\Delta\varphi$) was a direct measure of the cavity frequency shift and the conversion was made via $\Delta\varphi=\arctan\left(\Delta f_{0}/\kappa_{\mathrm{tot}}\right)\approx\Delta f_{0}/\kappa_{\mathrm{tot}}$, where $\kappa_{\mathrm{tot}}=\kappa_{1}+\kappa_{2}+\kappa_{\mathrm{int}}$. Using the simulated $\omega_{e}$ vs. $V_{\mathrm{trap}}$, we fit the measured cavity frequency shift to $\Delta f_{0}=\Delta\varphi\kappa_{\mathrm{tot}}$, which gave the values listed in the main text. Quoted uncertainties were fit uncertainties.