Electric Dipole Coupling of a Bilayer Graphene Quantum Dot to a High-Impedance Microwave Resonator

We implement circuit quantum electrodynamics (cQED) with quantum dots in bilayer graphene, a maturing material platform that can host long-lived spin and valley states. Our device combines a high-impedance (Zr ≈ 1 kΩ) superconducting microwave resonator with a double quantum dot electrostatically defined in a graphene-based van der Waals heterostructure. Electric dipole coupling between the subsystems allows the resonator to sense the electric susceptibility of the double quantum dot from which we reconstruct its charge stability diagram. We achieve sensitive and fast detection of the interdot transition with a signal-to-noise ratio of 3.5 within 1 μs integration time. The charge–photon interaction is quantified in the dispersive and resonant regimes by comparing the resonator response to input–output theory, yielding a coupling strength of g/2π = 49.7 MHz. Our results introduce cQED as a probe for quantum dots in van der Waals materials and indicate a path toward coherent charge–photon coupling with bilayer graphene quantum dots.


Device Fabrication
The sputtering process was carried out in an Orion 8 magnetron sputtering system (AJA International, Inc.) in reactive dc mode from a NbTi (Nb/Ti 70/30 wt%) target (ACI Alloys, Inc.) with 99.99% purity at 1.5 × 10 −8 mTorr base pressure.We initiate the process with a target cleaning step (5 min exposure to argon (Ar) plasma at 50 sccm flow rate and 100 W) and a conditioning phase (4 sccm nitrogen (N 2 ) for 1 min with shutter closed) .The subsequent deposition stage consists of Ar/N 2 flows at 50 sccm and 4 sccm, respectively, maintaining a power of 100 W, a pressure of 3.5 mTorr, and a working distance of 10 cm.
The parameters were chosen to optimize the growth conditions for NbTiN films, film quality, thickness uniformity, and structural properties.
We sputter a nominally 15 nm thick film of NbTiN on a two-inch wafer of intrinsic silicon (ρ > 10 kΩ cm) with 100 nm of thermally grown SiO 2 (Alineason Materials Technology GmbH).The microwave circuit is patterned using a direct-write photo-lithography system (Heidelberg Instruments DWL66+) and reactive ion etching (RIE) with SF 6 /Ar.Afterwards, we deposit gold markers and bondpads in a lift-off process and dice the wafer into 5 × 8 mm chips.The inductance per unit length L l of the coplanar waveguide resonator is dominated by the large contribution of the sheet kinetic inductance of the NbTiN film.From the resonator dimensions (width 1.6 µm, gap to ground 8.6 µm and length 900 µm) and its center frequency f r = 6.033GHz we estimate the film's sheet kinetic inductance to be L □,kin ≈ 150 pH/□.
The on-chip low-pass filters on all gate lines are formed by a shunt capacitance to ground C f ≈ 0.2 fF in series with the inductance L f ≈ 210 nH of a 2 µm wide NbTiN wire.This results in a filter cut-off frequency of The vdW material stack is fabricated on the pre-patterned chips using standard me-

Input-Output Theory
The complex transmission S 21 through the microwave feedline with a resonator coupled in a notch-type configuration can be derived from input-output theory 1 to be The resonator frequency f r and total linewidth κ = κ ext + κ int are determined from a fit to the spectrum of the unperturbed resonator. 2For this fit, the external coupling κ ext is taken as a complex-valued parameter to account for an asymmetric Fano lineshape of the resonance arising from non-idealities in the resonator-feedline coupling.
The effect of the charge qubit is described by its electric susceptibility that considers the charge qubit decoherence rate γ and probe detuning.The charge qubit energy E q (δ) = 4t 2 c + δ 2 depends on the interdot tunnel coupling t c and the DQD energy detuning δ.We calculate δ from the lever arms α L dL(R) of the left plunger gate to the left (right) dot as δ = (α L dL − α L dR )eV L = 0.024 × eV L with the lever arms determined by finite bias measurements of the DQD.The effective charge qubit coupling strength is g * = g 2tc Eq(δ) iii using the bare resonator coupling strength It is proportional to the lever arm difference of the coupling gate β = α R dR −α R dL , the resonator frequency f r , and the square root of the resonator impedance Z r .

Fitting procedure
The parameters g, γ and 2t c are determined by a least-squares fitting routine.We calculate the complex transmission S 21 (f p , δ) from input-output theory for a given set of fit parameters θ = [g, γ, 2t c ] and compute the mean squared error Q(θ) between calculation and measurement data.The parameters given in the main text for each data set are the parameters θ that minimize the error, found by sampling all three parameters over a wide range of values.We numerically calculate the hessian matrix ∂θ i ∂θ j θ around θ with N the size of S 21 .From H we calculate the variance-covariance matrix K = H −1 which contains information about uncertainties and correlations of our parameter estimates.For the dispersive case with 2t c /h ≫ f r the parameters that minimize the mean squared error are g/2π = 49.72 ± 0.27 MHz, γ/2π = 643 ± 24 MHz and 2t c /h = 10.193 ± 0.046 GHz.The strongest correlation is observed between g and 2t c as expected from the definition of the effective coupling strength g * above, while the estimate of γ is mostly uncorrelated to the other two parameters.The relative uncertainty in our estimate of γ is significantly larger compared to the uncertainty in the other two parameters.This is because the resonatorqubit detuning is large over the whole range of δ.The incoherent dispersive interaction dominates the resonator response and does not depend strongly on the decoherence rate γ, rendering the mean squared error insensitive to the estimate of the decoherence rate.
For the resonant case with 2t c /h < f r the parameters that minimize the mean squared error are g/2π = 37.49±0.19MHz, γ/2π = 1.112±0.008GHz and 2t c /h = 2.986±0.015GHz. iv We observe stronger correlations between each pair of parameters compared to the dispersive case.Furthermore, the uncertainty in γ is smaller, because the resonant interaction is limited by charge qubit decoherence.Therefore, the mean squared error is more sensitive to the estimate of the decoherence rate.
chanical exfoliation from bulk crystals and polymer-based dry transfer techniques.We first deposit the bottom hBN (∼ 25 nm thick) and graphite back-gate onto the chip and clean it from polymer residues.The top hBN (∼ 35 nm thick) and bilayer graphene are placed onto the pre-deposited bottom half in a separate deposition.Metal gates are fabricated in ii a lift-off process by electron beam lithography (EBL) and metal evaporation.For ohmic contacts to the graphene, we etch hBN before metal deposition using RIE with CHF 3 .The split-gates are made from 3/20 nm Cr/Au with a 100 nm wide channel.Plunger and barrier gates are also made from 3/20 nm Cr/Au with 25 nm in width and a gate pitch of 70 nm.They are separated from the first gate layer by 20 nm of Al 2 O 3 grown by atomic layer deposition (ALD) at 150 • C. The Al 2 O 3 layer is removed by wet etching to establish contact to the resonator below the dielectric.