Dispersive detection of a charge qubit with a broadband high-impedance quantum-Hall plasmon resonator

Domain: neuroscience
Published: 2026-02-10
Editorial outcome: minor_revision

Abstract
Cavity quantum electrodynamics (cQED) provides strong light-matter interactions that can be used for manipulating and detecting quantum states. The interaction can be enhanced by increasing the resonator’s impedance, while approaching the quantum impedance (h/e2) remains challenging. Edge plasmons emergent as chiral bosonic modes in the quantum Hall channels provide high quantized impedance of h/νe2 that can exceed 10 kΩ for the Landau-level filling factor ν ≤ 2, well beyond the impedance of free space. Here, we apply such a high-impedance plasmon mode in a quantum-Hall plasmon resonator to demonstrate dispersive detection of an nearby charge qubit formed in a double quantum dot. The phase shift in microwave transmission through the plasmon resonator follows the dispersive shift associated with the qubit state, in agreement with the cQED theory. The high impedance allows us to perform dispersive detection of qubit spectroscopy with a plasmon resonator having a broad bandwidth. Leveraging these topological edge modes, our results establish two-dimensional topological insulators as a new platform of cQED.

Introduction
Quantum Hall (QH) states are well-understood two-dimensional topological insulators showing highly quantized Hall conductance in the units of e2/h1,2. Although the non-interacting picture is conveniently used to describe the edge transport, the low-energy excitation should be described by collective plasmon modes, particularly when highly nonequilibrium or high-frequency transport is concerned3–6. Interestingly, the impedance of the plasmon mode is also quantized as h/νe2 at the Landau-level filling factor ν5,7. Thus, the plasmons with this fixed impedance should show perfect impedance matching over the whole channel, even in the presence of a non-uniform electrostatic environment. Because the plasmon mode in the QH regime is theoretically constructed from electrons in terms of the bosonization scheme, the interaction-induced emergent quasiparticles should follow bosonic characteristics8,9. Actually, a recent experiment demonstrates plasmon-assisted tunneling in a double quantum dot (DQD) coupled to a plasmon resonator, where integer multiples of plasmons are absorbed by or emitted from the DQD10. This implies possible applications for cavity quantum electrodynamics (cQED)11,12. Since the vacuum fluctuation of the electric potential in the resonator increases with the resonator’s impedance, strong cQED effects are expected with high impedance (Z=13 kΩ at ν=2), as compared to conventional 3D cavities (Z=377 Ω) and sophisticated Josephson-junction arrays (a few kΩ) for photons13–17. While strong, ultra-strong, and deep strong coupling regimes are expected in plasmonic cQED systems10,12, their experimental realizations remain challenging. One of the applications of cQED is the dispersive detection of a qubit, where the transmission amplitude and phase of photons (or plasmons in our case) are sensitive to the qubit state. Because the signal is enhanced by the quality factor Q of the resonator or cavity, high-Q resonators are usually appreciated at the cost of long measurement time due to their narrow bands18–20. However, one can choose a high-Z and low-Q resonator to achieve broadband dispersive detection.

Here, we study the plasmon transmission characteristics of a plasmonic resonator with a high Z of 13 kΩ and a low Q of 4 coupled to a semiconductor charge qubit in a DQD. A clear dispersive response is observed by changing the qubit parameters, i.e., the energy detuning and tunneling coupling of the DQD. Moreover, the broadband resonator allows us to demonstrate qubit spectroscopy over a wide energy range. These characteristics agree well with the dispersive shift calculated by the standard Jaynes-Cummings model. The demonstrated dispersive shift is obtained without significantly populating the qubit excited state. Some outlooks for strong coupling regimes are discussed.

Results
We consider a cQED system schematically shown in Fig. 1a, where the charge qubit in the DQD is coupled to the closed edge channel forming a plasmon resonator with the perimeter of one plasmon wavelength. We measure the plasmon transmission coefficient S21 by sending a microwave at probe frequency fp from port 1 and receiving the signal at port 2 to investigate the qubit-resonator system. For a broadband resonator with low Q, we can vary fp around the bare resonant frequency fr of the resonator over a wide frequency width of about fr/Q. Here, the amplitude of electric potential for a given plasmon power in the resonator is enhanced by the large impedance Z. This potential can induce qubit transition near the resonant condition (fp = fq), where fq is the transition frequency of the qubit, and a dispersive shift is expected at large qubit-plasmon detuning Δq-r = hfq - hfr in energy. The dynamics is characterized by the coupling strength g = fr η sinθq sqrt(Ze2/h) in frequency with a dimensionless factor η sinθq (<1) described below. Under the dispersive regime with large Δq-r ≫ hg, the plasmon resonant frequency f0 = fr + Δf is shifted by Δf ≃ -hg2/Δq-r, if the qubit is prepared in the ground state. This dispersive shift can be measured by the phase ϕ = arg(S21) ≃ 2QΔf/fr of S21 at fp ≃ fr. Because ϕ is proportional to QZ from the above formulas, a reasonable signal amplitude is expected with a high-Z and low-Q resonator.

The plasmon modes are confirmed by measuring S21 at various B and fp. For convenience, we evaluate the relative change Δ|S21| = |S21/S21^0| from the reference S21^0 measured when the resonator is disabled by opening the lower constrictions GNL to connect the edge channel to the lower ohmic contact S in Fig. 1a (see Supplementary Note 3 for details). The representative data in Fig. 1c show a distinct peak labeled n = 1 and a faint peak labeled n = 2, both of which follow the clear 1/B dependences (the solid lines). These dependences agree well with the edge magnetoplasmon modes7,21. Excess charge density ρc in the channel induces voltage Vch = ρc/CΣ for the effective channel capacitance CΣ (≃ 0.28 nF/m), and resulting Hall current I = σxy Vch flows along the channel at the plasmon velocity vp = σxy/CΣ for the Hall conductance σxy. For the perimeter L, the resonant frequencies appear at fr = n vp/L with integer n. Therefore, fr can be changed by tuning σxy with B, as shown in Fig. 1c, and by tuning CΣ with the gate voltages VGres, as shown in Supplementary Fig. 3c. The overall fr ∝ 1/B dependence in Fig. 1c can be understood with the classical Hall conductivity σxy = en e/B, and the pronounced peaks with narrow width are seen at around the integer filling factors ν = 1, 2, and 3 (see Supplementary Note 4 for details). In the following experiments, we focus on the robust QH state at ν ≃ 2 (B = 3.6 T). The frequency dependences of the amplitude Δ|S21| in Fig. 1d and the phase shift Δϕ = arg(S21/S21^0) in Fig. 1e show the resonant frequency fr = 2.45 GHz, plasmon decay rate κ ≃ 0.6 GHz, and Q = f/κ ≃ 4 for n = 1. We shall show below that dispersive detection is successful for this low-Q and high-Z resonator.

We now activate the DQD with quantum dots QD1 and QD2, which are located close to the resonator. Instead of measuring the current through the DQD (see Supplementary Note 5), we probe the DQD by measuring S21. Tunneling between the two dots and between the dot and the lead induce finite change in S21 when the tunneling rate is comparable to or greater than the plasmon frequency fp, as presented in Supplementary Note 6. In Fig. 2c, we plot Δϕ = arg(S21/S21^0) measured at fp = 2.5 GHz as a function of the two gate voltages Vg1 and Vg2, which tune the electron numbers (N1, N2) in QD1 and QD2, respectively, under appropriate gate voltage Vc for reasonable tunnel coupling. The negative peaks (blue for Δϕ < 0) elongated to the upper-right direction can be understood as due to interdot charge transitions between (N1, N2+1) and (N1+1, N2). The hexagonal charge stability diagram for (N1, N2) is outlined by the black dashed lines. An excess electron occupies QD1 and QD2 on the left and right sides of the transition signal, respectively, as shown in the inset to Fig. 2c. The energy difference ε between the localized states in QD1 and QD2 can be swept by changing Vg1 and Vg2 in the opposite directions, as shown by the red arrow labeled ε. The average qubit energy δ, measured from the chemical potential of the leads, can be changed by sweeping the gate voltages in the orthogonal direction labeled δ. Around the center of the peak (δ = 0) in the δ direction, the qubit is effectively isolated from the leads due to the Coulomb blockade with the inter-dot charging energy of Δ12 ≃ 74 μeV (see Supplementary Note 5). The dimensionless coupling strength η ≃ dε/d(eVch) measures how much energy in ε is modulated by the plasmon potential eVch. The appearance of the signal suggests a finite η (≃ 0.032 estimated from the following analysis), which is not optimized yet but sufficient to study the dispersive shift.

The qubit excitation energy hfq = sqrt(ε2+4tc2) is given by the tunnel coupling tc and the energy difference ε22. Therefore, hfq takes the minimum value of 2tc at the center of the peak (ε = 0). Based on the dispersive shift Δf ≃ -hg2/Δq-r discussed above, the negative phase shift (Δϕ < 0) is associated with the negative frequency shift (Δf < 0) under the positive qubit-resonator detuning Δq-r = hfq - hfr > 0. The simple negative peaks seen in Fig. 2c suggest that 2tc > hfr. We can reach zero detuning (Δq-r = 0) by reducing the interdot tunnel coupling tc via gate voltage Vc, as shown in Fig. 2d. While a single negative peak in Δϕ is seen in panel (i) at Vc = -413 mV, the single negative peak splits into two with a ‘w’ shaped profile in (ii)-(iv). Δϕ at ε = 0 turns out to be positive in (iv) at Vc = -415 mV, which indicates that 2tc is reduced below hfr. Zero detuning (Δq-r = 0) is reached at the boundary of positive and negative Δϕ regions.

We calculate the dispersive shift by using the standard cQED analysis20, as summarized in Methods: Quantum Master Equation. The Jaynes-Cummings Hamiltonian describes the coupling between the resonator and the qubit with coupling strength g = fr η sinθq sqrt(Ze2/h). Here, η (< 1) is the electrostatic coupling coefficient, and sinθq = 2tc/sqrt(ε2+4tc2) describes the relative amplitude of the transition dipole moment of the qubit22. Because sinθq depends on ε and is maximized to sinθq = 1 at ε = 0, we shall describe g = gc sinθq with maximal coupling gc = fr η sqrt(Ze2/h) at ε = 0. The coupling to the environment is considered by using the Markovian quantum master equation with resonator decay rate κ, qubit relaxation rate γ1, and qubit dephasing rate γϕ. Here, we consider the thermal plasmon number npl,th = (e^(hfr/kBTB)-1)^-1 ≃ 0.45 in the resonator. We assumed that the qubit is coupled to the phonon environment with the thermal phonon number nph,th = (e^(hfq/kBTB)-1)^-1 ranging from ≃ 0 at ε ≫ kBTB to 0.48 at ε = 0 and 2tc/h = 2.4 GHz. The input-output theory for the probe frequency fp is used to calculate S21 (see Methods: Input-Output Theory for details).

We analyze the dispersive shift, i.e., how Δϕ changes with ε, for two representative data shown in the bottom panels (the circles) of Fig. 3a for 2tc < hfr and Fig. 3b for 2tc > hfr. The simulations (the solid lines) reproduce the experimental data very nicely by choosing the parameters tc, γ1, γϕ, gc (shown in the figure caption). Because γ1 and γϕ cannot be separated with high accuracy, we evaluate the decoherence rate γ = γ1/2 + γϕ in the following. The fitting suggests the maximal coupling gc ≃ 55 MHz for both cases, while γ depends on tc (γ ≃ 0.34 GHz at tc ≃ 1.2 GHz in Fig. 3a and 0.07 GHz at tc ≃ 1.43 GHz in Fig. 3b). Whereas our particular device was in the weak coupling regime (gc < γ, κ), the experiment was successful in demonstrating the dispersive detection of the qubit. The expected coupling dependency of gc = fr η sqrt(Ze2/h) implies a relatively small η ≃ 0.032, which can be improved, as discussed below. The upper panels of Figs. 3a and 3b show the estimated eigen-energies with gc = 55 MHz (the solid lines) and the uncoupled bare energies with gc = 0 (the dashed lines). The evolution of Δϕ reflects the eigenstates of the system, and the conditions for Δϕ = 0 (the black circles in Fig. 3a) coincide with the resonance at Δq-r = hfq - hfr = 0 (the crossing points of the dashed lines at ε = ±ε0).

In Fig. 3c, we show the measurements at various Vc, as summarized in Supplementary Note 7. The single negative peak observed at Vc > -413 mV changes to the ‘w’ shaped profile at Vc ≤ -413 mV. The feature is reproduced well with the simulations (the red solid lines). The fitting parameters tc and γ are plotted as a function of Vc in Fig. 3d and 3e, respectively. The qubit parameter 2tc can be tuned smoothly across hfr (the dashed line in Fig. 3d) with Vc. γ in Fig. 3e decreases with increasing Vc and thus with increasing tc. This dependency implies that low-frequency fluctuation in ε is the dominant decoherence source23–25, and thus intrinsic γ must be lower. Furthermore, the obtained 2tc is plotted as a function of the resonant condition ε0 in Fig. 3f. The dependence is consistent with the qubit energy hfq = sqrt(ε0^2 + 4tc^2) at fq = 2.5 GHz (the red solid line). In this way, the plasmon resonator successfully measures the qubit characteristics.

When the input RF power P is increased, the average plasmon number npl = ⟨a+a⟩ = npl,th + npl,drv increases with the driven plasmon number npl,drv ≃ (2e/h Vd/κ)^2 from the thermal one npl,th ≃ 0.45, where Vd is the drive voltage on the channel (See Methods: Quantum Master Equation). Under strong plasmon drive, the electron in the qubit is no longer in the ground state (Fig. 4a). As a result, the occupation probability Pg in the ground state decreases, and Pe in the excited state increases with increasing P, as shown in Fig. 4b, calculated for the experimental conditions for Fig. 3a. The qubit polarization Pz = Pe - Pg = ⟨σz⟩ is fully depolarized (Pz = 0) in the large limit of Vd (npl,drv ≫ 1), as shown in the upper scale of Fig. 4b. This depolarization reduced by half at around Vd ≃ 10 μV (blue circle), which is used to evaluate Vd and npl in our resonator.

For this purpose, we choose the condition close to the resonance at ε = 0 and 2tc ≃ 2.4 GHz close to fr = 2.5 GHz, which is identical to that in Fig. 3a. As P is increased, the phase Δϕ and amplitude ΔA at ε = 0 changes, as shown in Fig. 4c and 4d (see Supplementary Note 8 for details). Here, Δϕ and ΔA are normalized by the average values taken in the low-power region (P < -100 dBm). Both phase and amplitude decrease with increasing P and are halved at P ≃ -85 dBm for Δϕ and P ≃ -76 dBm for ΔA. These dependences are reproduced with simulations of Δϕ and ΔA by assuming a different proportionality factor k in the relation Vd = k sqrt(P) (see Methods: Input-Output theory), as shown by the blue line for the Vd scale in each figure. The different k might be regarded as uncertainty of the estimate, because the theory suggests the same dependency for non-zero Δq-p. Nevertheless, the high-Z and low-Q resonator is sensitive to Vd and the corresponding plasmon number npl,drv down to 10 or even lower. This analysis ensures that the dispersive detection in Fig. 3 with P = -87.5 dBm, marked by the vertical gray dashed line in Fig. 4c and 4d, was performed without significantly populating the qubit excited state.

We have successfully demonstrated plasmon measurements to evaluate the cQED of a charge qubit, which is expected in the strong coupling regime (gc > κ, γ)26. Several strategies can be considered to achieve this goal. First, both γ and κ can be reduced by optimizing device structure and material quality. The qubit decay rate γ of the present device might be raised by external electrical noise or background charge fluctuations27,28, and thus γ can be reduced significantly, as reported in recent Ref. 16 with γ < 5 MHz. The Q factor of the present resonator remains low, possibly due to the nearby charge puddles (see Supplementary Fig. 4), and thus higher Q is anticipated with high-quality materials, as reported in Ref. 29 with Q ≃ 18. We should have significant room for improving the cQED environment. Second, the coupling strength, gc = fr η sqrt(Ze2/h), can be enhanced by increasing Z and η. This is encouraging for entering the ultra-strong coupling regime (gc/fr > 0.1). The impedance Z = h/(νe²) can be increased by choosing smaller ν = 2, 1, 1/3, … towards the fractional QH regimes. The coupling efficiency η (≃ 0.032 in the present device) can also be increased by optimizing the capacitive coupling conditions in the present device (see Supplementary Note 6) or by introducing screening electrodes so that one dot couples more strongly to the resonator than the other, as demonstrated in some papers with large η ≃ 0.316, 30. Fig. 5 shows how the figure of merit gc/fr increases with Z and η in comparison with reported values for DQD-based cQED systems (see Supplementary Note 9)15–17, 30–35. The demonstrated scheme provides unique opportunities to explore strong plasmon–matter interactions and enable novel approaches to quantum state manipulation.

Alternatively, the high impedances of 2D topological edge channels are seen in various materials such as QH states in graphene36–38, quantum spin Hall states in HgTe/CdTe and InAs/GaSb39–42, and quantum anomalous Hall states in magnetically doped tetradymite semiconductors43,44. There might be suitable materials to integrate a plasmon resonator and a qubit in the strong coupling regime.

In addition to high-Z resonators, QH resonators can provide novel cQED functionalities. For example, the extremely slow plasmon velocity45, as compared to the light speed, is advantageous for integrating small multiple resonators on a chip. Chirality can be used for non-reciprocal coupling between multiple qubits annularly attached to a chiral plasmon resonator, which can be elaborated from a microwave circulator46. Furthermore, it would be intriguing to explore the coupling to topological quasiparticles like anyons in fractional QH states47.

Methods
Device fabrication
The device is fabricated using a GaAs/AlGaAs heterostructure with a two-dimensional electron gas (2DEG) 100 nm beneath the surface, having an electron density of 1.8 × 10^11 cm^-2. The fabrication process involves several lithographic and metallization steps. Initially, the mesa structure of the device is defined by optical lithography and wet chemical etching. Then, the ohmic contacts are fabricated by using photolithography, followed by metal deposition and alloying. The fine gate electrodes in the plasmon resonator and double quantum dots (DQDs) are fabricated by using electron-beam lithography, followed by electron-beam evaporation of Ti/Au (10/20 nm) metal films. Finally, large gate electrodes, including those for coplanar microwave transmission lines and bonding pads, are fabricated using photolithography and deposition of Ti/Au (10/250 nm) metal films.

The plasmon resonator is defined in a disk-shaped 2DEG region with a radius of ~18 μm. Two arc-shaped gate electrodes located on the left and right of the resonator are patterned with 36 μm in length and 5 μm in width, which are used as input and output ports for the plasmon signal. Adjacent to the plasmon resonator, several narrow gates with 120 nm in width are patterned for defining the DQD with each dot in a size of approximately 100 × 100 nm^2. The input and output ports of the resonator are connected to the center conductor of the coplanar microwave transmission lines, which has 32 μm in width and is spaced 18 μm from the ground planes, providing a characteristic impedance of 50 Ω. The coplanar transmission lines are connected to external coaxial cables to enable the measurement of the plasmon transmission characteristics.

Spectrum measurement
The transmission characteristics of the plasmon resonator (Fig. 1c, 1d, and 1e) are investigated using a vector network analyzer (VNA). The microwave signal generated from Port 1 (P1) of the VNA first goes through the coaxial cables in a dilution refrigerator, and then reaches the coplanar transmission line in the sample. The coplanar transmission line is connected to the input electrode (Ginj) of the resonator, which allows the microwave signal excite the plasmon mode in the channel. The transmission signal through the edge channel is detected using the output electrode (Gdet). The returning signal through the coaxial cables is subsequently amplified by a low-temperature amplifier (26 dB gain) at the 4 K stage in the dilution refrigerator and a room-temperature amplifier (38 dB gain) at 300 K. Finally, the amplified signal reaches Port 2 (P2) of the VNA, and the amplitude and phase of the signal are measured.

We employed a homodyne detection scheme (Fig. 2b) to measure the plasmon transmission spectra for qubit detection. To improve the detection signal, a lock-in technique is utilized to modulate the microwave signal at a frequency f ~ 133 Hz. In this detection scheme, the microwave signal is generated by an external microwave source instead of a VNA, and the VNA detection is replaced with an IQ mixer for signal demodulation, as seen in Fig. 2b. The demodulated signal is filtered before the measurement of the in-phase (I) and quadrature (Q) components. From the measured signal, we calculate the amplitude based on A = sqrt(I^2+Q^2) and phase based on ϕ = arg(Q/I).

Quantum Master equation
Theoretically, we model the DQD charge qubit as an effective two-level system interacting with a single plasmon mode in the resonator. The coupled system is described by the Jaynes–Cummings type model26

H = hfr a+a + hfq/2 σz + hg (a+σ- + aσ+) + Hp (1)

where a (a+) is the plasmon annihilation (creation) operator, σ- (σ+) the qubit lowering (rising) operator, and σz the Pauli matrix. The qubit frequency is given by hfq = sqrt(ε^2+4tc^2) with the effective plasmon-qubit coupling strength g = gc sinθq accounting for the mixing of plasmon and qubit state, where gc is the bare coupling strength and sinθq = 2tc/hfq is the qubit transition dipole moment. A microwave of frequency fp probing the transmission through the resonator can be modeled by Hp = eVd (e^-i2πfpt a+ + ei2πfpt a), where Vd is a measure of the drive amplitude in the resonator. This drive causes rotation of the qubit that depends on the drive amplitude.

The dissipations in the system are described with a density matrix ρ through the Markovian master equation

dρ/dt = -i/h [H, ρ] + κ D[a]ρ + γ1/2 D[σ-]ρ + γ1/2 nph,th D[σ-]ρ + γ1/2 nph,th D[σ+]ρ + κ npl,th D[a+]ρ + γϕ D[σz]ρ (2)

where D[A]ρ = AρA+ - 1/2 {A+A, ρ} is the Lindblad operator, and γϕ is the dephasing rate. The finite base temperature TB is considered with the thermal plasmon number npl,th = (e^(hfr/kBTB)-1)^-1 in the resonator, and thermal phonon number nph,th = (e^(hfq/kBTB)-1)^-1 in the qubit environment. The total qubit decoherence rate is γ = γ1/2 + γϕ. In the frame rotating at the frequency of the external probe fp, we obtain

H0 = Δr-p a+a + Δq-p/2 σz + hg (a+σ- + aσ+) + eVd (a+ + a) (3)

where Δr-p = hfr - hfp denotes the energy detuning between the resonator and the probe and Δq-p = hfq - hfp denotes the energy detuning between the qubit and the probe. To obtain the average value of the qubit polarization ⟨σz⟩, we derive the equations for the evolution of the field and the qubit polarization as

d⟨a⟩/dt = -(κ/2 + iΔr-p) ⟨a⟩ - i eVd/h - ig ⟨σ-⟩ (4)

d⟨σ-⟩/dt = -(γ1/2 (2nph,th + 1) + γϕ + iΔq-p) ⟨σ-⟩ + ig ⟨aσz⟩ (5)

d⟨σz⟩/dt = -2ig (⟨aσ+⟩ - ⟨a+σ-⟩) - γ1 (2nph,th + 1) ⟨σz⟩ + 1 (6)

d⟨a+a⟩/dt = ig (⟨aσ+⟩ - ⟨a+σ-⟩) + i eVd/h (⟨a+⟩ - ⟨a⟩) - κ ⟨a+a⟩ + κ npl,th (7)

To solve these equations [Eqs. (4–7)], we approximate ⟨aσz⟩ = ⟨a⟩ ⟨σz⟩, ⟨aσ+⟩ = ⟨a⟩ ⟨σ+⟩, and ⟨a+σ-⟩ = ⟨a+⟩ ⟨σ-⟩. These approximations provide steady state solutions,

⟨a⟩ = -i eVd/h / (κ/2 + iΔr-p + iχ ⟨σz⟩) (8)

⟨σ-⟩ = g χ ⟨a⟩ ⟨σz⟩ (9)

⟨σz⟩ = -1 / (2nph,th + 1 + 4|⟨a⟩|^2 Im(χ)) (10)

⟨a+a⟩ = npl,th + κ/2 Im(χ) ⟨σz⟩ - hκ/2 Im(⟨a⟩) (11)

where χ = hg^2 / (Δq-p - ihγth) is the electric susceptibility associated with the decoherence γth = γ1/2 (2nph,th + 1) + γϕ and qubit-probe energy detuning Δq-p. In the dispersive regime with |Δq-p| ≫ g, γth, the electric susceptibility induces a dispersive shift of the plasmon frequency48

Δfr = χ ⟨σz⟩ = Δq-p hg^2 / (Δq-p^2 + h^2γth^2) ⟨σz⟩ (12)

Input–Output Theory
To model the transmission, we use the input-output theory. With the input field αin,1 = sqrt(κ) ⟨a⟩, where we use the input-output relation. As the left Port 1 is used as the plasmon injection, we obtain the transmission coefficients

S21 = -i κ / (κ/2 + iΔr-p + iχ ⟨σz⟩) (13)

where we used a_in,1 = sqrt(κ) and a_in,2 = 0. As the experimental probe frequency fp is set close to the resonance frequency fr, we can use Δr-p = 0 in Eq. (13), and substituting χ = hg^2 / (Δq-p - ihγth) yields the amplitude of the transmitted signal

A = |S21| = |κ / (κ + 2i hg^2 ⟨σz⟩ / (Δq-p - ihγth))| (14)

and the phase

ϕ = -arg(S21) = tan^-1(4 h^2g^2 Δq-p ⟨σz⟩ / (κ (Δq-p^2 + h^2γth^2) - 2h^3g^2 γth ⟨σz⟩)) (15)

In the limit of weak coupling g ≪ κ, γth, Eq. (14) can be further simplified as,

A = |S21| ≃ |κ / (κ + 2h^2g^2 ⟨σz⟩ γth / (Δq-p^2 + h^2γth^2))| (16)

and Eq. (15) can be simplified as

ϕ = -arg(S21) ≃ tan^-1(4 h^2g^2 Δq-p ⟨σz⟩ / (κ (Δq-p^2 + h^2γth^2))) (17)

where we used an approximation of tan^-1(x) ≃ x. Next, we calculate the spectral shift of A and ϕ with respect to the reference amplitude A0 and phase ϕ0 at large qubit detuning ε ≫ 0 (or Δq-p ≫ 0). By taking the limit Δq-p ≫ 0 for Eqs. (16) and (17), we obtain

A0 = 1 (18)

ϕ0 = 0 (19)

Therefore, the spectral shift ΔA = A - A0 and Δϕ = ϕ - ϕ0 can be written as

ΔA = A - A0 ≃ -2 h^2g^2 γth ⟨σz⟩ / (κ (Δq-p^2 + h^2γth^2)) (20)

Δϕ = ϕ - ϕ0 ≃ 2 hg^2 Δq-p ⟨σz⟩ / (κ (Δq-p^2 + h^2γth^2)) (21)

Equation (21) can be expressed as a function of impedance Z and quality factor Q = fr/κ as

Δϕ ≃ 2 fr (η sinθq)^2 Δq-p ⟨σz⟩ ZQ / (Δq-p^2 + h^2γth^2) (22)

The experimental data in Fig. 3 in the main text and Fig. 7 in the Supplementary Note 7 are fitted by using Eq. (21), where ⟨σz⟩ is obtained by self-consistently solving Eqs. (8–10) with the plasmon voltage Vd = 7.5 μV (for P = -87.5 dBm). Under this drive, the qubit remains mostly in the ground state with population Pg ≃ 0.7 and small excited state population Pe ≃ 0.3. Fitting the data in Figs. 3(a) and 3(b) suggests the maximal coupling strength gc ≃ 55 MHz. In fitting the data of Fig. 3(c), gc is fixed to extract the parameters of interdot coupling tc, qubit relaxation rate γ1, dephasing rate γϕ. As the γ1 and γϕ cannot be determined with high accuracy, the total decoherence rate γ = γ1/2 + γϕ is studied in the main text.

The Eqs. (20) and (21) are also used to simulate the drive dependence of ΔA and Δϕ shown in Figs. 4c and 4d, respectively, where the normalized values of ΔA/ΔA0 = ⟨σz⟩/⟨σz⟩0 and Δϕ/Δϕ0 = ⟨σz⟩/⟨σz⟩0 are compared with the experimental data. The values of ΔA0 and Δϕ0 are calculated at zero drive limit (Vd = 0) with qubit polarization ⟨σz⟩0 = -1 / (2nph,th + 1).

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The manuscript by Lin et al. presents compelling experimental evidence of coupling between quantum Hall (QH) plasmons and a double quantum dot system. This is a novel and intriguing result with potential implications for hybrid quantum systems, especially in contexts where topologically protected modes are desirable. The topic is timely and of high relevance for the quantum technologies community. I am inclined to support publication, but I have several concerns and requests for clarification that I believe must be addressed before proceeding:\n1) One of the core features of quantum Hall edge modes is their chirality. It is not entirely clear how this feature is impacting the results of this work. Specifically:\na. In the Methods section, Eq. (3) describes a potential that appears to be that of a standing wave. This seems inconsistent with the absence of counter-propagating modes in a chiral system. Maybe is it possible that phi_0 is itself time-dependent and encodes the chiral propagation? If so, this should be made more explicit. If not, what defines phi_0 and how does the formulation account for unidirectional propagation?\nb. More generally, how does chirality manifest in the coupling between the edge and the double quantum dots? I can imagine that this could impact coupling of more that one double dot\n2) Could the authors clarify whether parasitic capacitances have been included in the model and how they might affect the plasmon propagation? As shown in Mahoney et al., Phys. Rev. X 7, 011007 (2017), such capacitances can even reverse the effective chirality of the signal propagation. A brief discussion or estimate of their effect in this geometry would we helpful.\n3) Is it possible to tune the plasmon velocity at fixed magnetic field via gate voltages on the coupling gates? Intuitively, one could imagine that adjusting the electrostatic environment could alter the local capacitance per unit length or the confinement profile, and thereby modify the plasmon dispersion. Have the authors explored this experimentally?\n4) The capacitance to the gates and ground planes is likely non-uniform along the edge. How does this spatial variation affect the plasmon-dot coupling? Could this be used to optimize the placement of the quantum dot for enhanced coupling?\n5) While demonstrating plasmon-double dot coupling is a significant step, the coupling strength reported here is relatively modest compared to state-of-the-art charge-photon coupling rates, which can be substantially above the 100 MHz range (e.g., Samkharadze et al., Science 359, 1123 (2018); de Palma et al., Nat. Commun. 15, 2703 (2024); Yu et al., Nat. Nanotechnol. 18, 1360 (2023)). This is somewhat surprising, especially given the high impedance of QH edge modes. In the Outlook section, the authors suggest possible improvements to increase the coupling, but I believe the paper would benefit significantly from a more quantitative analysis of these prospects. Can the authors provide estimates for how large g could realistically become, and how it compares to coupling in superconducting resonator systems? A more explicit comparison with these technologies could greatly strengthen the relevance of this work.\n6) The observed Q factor is quite low. Do the authors attribute this to dielectric losses, residual bulk conduction, edge disorder due to etching, or other mechanisms? The authors mention that this can be solved by higher mobility materials, but could improvements be expected also by defining edges electrostatically rather than through etching?\n7) In the Introduction, the authors assert that backscattering is absent in QH systems. While this is true in the DC limit, backscattering can occur at finite frequency, especially at low magnetic fields.\n8) The reported dissipation rate gamma is higher than what has been demonstrated in other GaAs samples (e.g., a few MHz as in reference 15). Could the authors comment on whether this elevated dissipation is an artifact of the device geometry, the fabrication method (e.g., etched edges), or other limitations?\n9) The coupling strength g is sometimes provided in frequency but its equations are in energy. Good to specify if the extracted values of g are divided by h or hbar.", "reviewer_2": "In the article entitled “Dispersive detection of a charge qubit with a broadband high-impedance quantum-Hall plasmon resonator”, Lin and cowokers explore circuit quantum electrodynamics with plasmonic resonators formed by the edge states of a closed quantum Hall droplet. The authors couple a plasmonic resonator to a double quantum dot operated as a charge qubit. Analyzing the phase response of the resonator within the standard framework of the Jaynes-Cumming model (dispersive regime), the authors extract the qubit properties (decoherence rate, spectrum) for different gate voltages. The authors also investigate how the power sent in the input probe tone affects the resonator response.\nOverall, the paper reads easily. The analysis of the results is sound and the experimental data are in rather good agreement with the theoretical models derived. In my opinion, the data and the analysis support the main claim of the manuscript that is the dispersive detection of a semiconductor charge qubit with a quantum Hall resonator. This work provides a new alternative platform for circuit quantum electrodynamics. To my knowledge, this represents the first demonstration of the use of a quantum Hall plasmonic resonator in such experiments and this is the main novelty of this work. This work could interest a broad community of researchers in the field of mesoscopic quantum physics, in particular those working on high-mobility 2D electronic systems. I think it may stimulate theoretical and experimental works on the use quantum Hall resonators as probing tools, in particular in the context of fractional quantum (anomalous) Hall effect.\nDespite this, I think the authors should further motivate their work and further highlight the perspectives it offers. I also have few concerns regarding the future of this platform. Additionally, I have questions regarding some parts of the data analysis and the interpretation of the data. Once these points have been addressed, I could recommend the publication of this manuscript in Nature Communications.\nMajor questions:\n(1) I think standard transport characterizations of the device/heterostructure are missing. Measures of the longitudinal Rxx and transverse Rxy resistances in a Hall bar made on the same sample/heterostructure would allow to evaluate the quality of the quantum Hall states forming at the magnetic field at which the device is operated. In particular, a measure of Rxx at ν = 2 would provide insights on the residual bulk conductivity and thus on the dissipation in the resonator. Do the authors have such a measurement? Can they measure Rxx in a configuration where the central part is open (no resonator formed like in Fig. S1)?\nSimilarly, the mobility of the GaAs/AlGaAs heterostructure should be reported. This quantity is relevant especially as higher mobility samples should enable the formation of resonators with higher Q as mentioned by the authors.\n(2) In my opinion, a more in-depth discussion about the advantages and prospects offered by quantum Hall plasmonic resonators should be carried out. In particular the use of plasmonic resonators instead of high-impedance superconducting resonators (Z0 = 7.9 kOhms in Janik et al., Nat. Comm., 16, 2103 (2025)) should be further motivated. Is the high characteristic impedance the only advantage of quantum Hall plasmonic resonators?\n(3) The current quality factor is low Q = 4 and this limits the possibilities offered by the resonator despite offering a large detection bandwidth. As acknowledged by the authors, achieving the strong coupling regime is desirable and one solution is to increase the Q factor. I think the discussion regarding solutions to improve Q needs to be extended and clarified. More particularly:\n- The sentence “Because the plasmon velocity also decreases with decreasing ν, the resonator size can be reduced to concentrate the plasmon power in the resonator” is not clear to me. Can the authors provide more detailed explanations? I think the effects of the resonator length L and of the plasmon velocity v should be discussed separately. The authors could discuss how the resonator linewidth κ scales with L and v.\n- It would be insightful to visualize how the quality factor evolves as function of the magnetic field B and filling factor ν based on data of Fig. 1c. A general comment on the separate effects of B and ν on Q will also be welcome considering potential extensions of this work to materials where the electron density can be tuned such as graphene.\n- I suggest that the authors extend their comment on improving the device structures. In particular, recommendations on the design of the resonators (gated/ungated, edges defined by etching or electrostatically, etc.) would be useful for future works on the topic.\n(4) In Fig. 2c., the authors should explain why only some charge transitions are visible. It would be also interesting to discuss whether a plasmonic resonator can be used to detect the last electron transition or not. Similarly, do the authors have an estimate of the number of electrons in the quantum dots?\n(5) For their analysis of the plasmon number in the resonator (Fig. 4), the authors assume that at low power the qubit is in its ground state at the charge degeneracy point. I am wandering if this assumption holds and more precisely if the thermal population of the excited state can be neglected. Do the authors have a measure or an estimate of the electron temperature in the system? Does the thermal population of the excited state influence their analysis (both for the data presented in Fig. 3 and Fig. 4)?\n(6) In Fig. 4b and Fig. 4c, the amplitude and the phase response seem to drop at different external input powers. In particular, the phase response does not follow the simulation. Can the authors comment on that and explain this discrepancy?\n(7) From the derivation presented in Methods (equation 17), it is not clear to me how the authors are able to extract separately γ1 and γϕ. It seems that only γ = γ1/2 + γϕ can be extracted. Can the authors explain it?\n(8) In Supplementary section 1, equation S9 is simplified to equation S10 assuming that GsL << σxy = 77 µS. With L = 113 µm, Gs = 0.55 S/m, we have GsL = 62 µS ~ σxy. Thus the validity of the simplification can be questioned. Can the authors discuss this?\nOther points:\n(9) In Fig. 1c, the second mode nR = 2 is very faint. Is it because the Ginj, Gdet electrodes have a length comparable to the wavelength of this mode? Would it be interesting to couple to higher plasmon modes (nR ≥ 2) in order to achieve resonators with higher Q?\n(10) In Fig. S3c, the qubit dephasing rate increases from 30 MHz to 60 MHz with the probe frequency. Do the authors have an explanation for this behavior? Also I suggest to put the y-axis of this figure in linear scale, as it is done Fig. 3e.\n(11) I suggest the authors to report the value of the electrostatic coupling coefficient η = 0.023 earlier in the manuscript and not at the very end. How large can this coefficient be in realistic devices?\n(12) If I am not mistaken, the formula for the number of plasmons in the resonator np=(2εd/κ) should be replaced np= (2eεd/hκ) considering that εd as unit of a voltage.\n(13) Fig. 2.c and Fig. 2.d: The definitions of ε and δ in terms of gate voltages Vg1 and Vg2 should be specified and chosen to be consistent with the charge labels. The orientations of corresponding axes should also be checked. For example, the schematics in the inset of Fig. 2.c seem to contradict the charge labeling.\nAbout the references:\nFor the sentence “While strong, ultra-strong, and deep strong coupling regimes are expected in plasmonic cQED systems”\nCan the authors provide references that support this statement?\nFor the sentence “Because the plasmon velocity also decreases with decreasing ν”, I suggest to add Kumada et al., “Edge magnetoplasmon transport in gated and ungated quantum Hall systems”, Phys. Rev. B (2011) as reference or any other appropriate work.\nI also suggest the authors to add two more recent references regarding the coupling of superconducting resonators to charge qubits in 2D system (germanium heterostructures)\n- Janik et al., “Strong charge-photon coupling in planar germanium enabled by granular aluminium superinductors”, Nat. Comm., (2025)\n- De Palma et al., “Strong hole-photon coupling in planar Ge for probing charge degree and strongly correlated states”, Nat. Comm., (2024)\nI suggest to mention another recent work on quantum-Hall resonators and potentially make some comparisons: Frigerio et al., Gate tunable edge magnetoplasmon resonators”, Comm. Phys., 7, 314 (2024).", "reviewer_3": "The authors report of a system where a charge q-bit (tunnel coupled double quantum dot) is coupled to the plasmon excitation of a Hall edge channel. This system realizes the standard Jaynes -Cumming QED Hamiltonian describing a two-level system coupled with a single boson. The authors perform several experiments which allow to gain information on the q-bit characteristics, by a heterodyne measurement on the transmission through the plasmon channel. In the dispersive regime, they extract the energy levels and coupling strength of the double quantum dots (Fig. 3), and they reconstruct its charge state diagram (Fig. 2). For a nearly resonant condition and by varying the input drive on the plasmon channel, they also estimate the average number of plasmons by measuring the polarization saturation of the double dot.\nThese types of measurements have been reported before (and a pretty common) form qubits coupled with transmission line resonators; the novelty here is the fact that the boson resonator is provided by the plasmon of an edge Hall state. While such plasmon excitations have been recently reported https://www.nature.com/articles/s42005-024-01803-6, to my knowledge, here the authors report their coupling with a qbit for the first time. Also, the authors work in a regime of high impedance of the plasmon mode, which results in higher coupling strength (although the quality factor is low). These novelty grant publication in Nature communications.\nThe paper can benefit from minor improvements. For instance, in the second paragraph of page 3 there is a confusion between the plasmon velocity and the Landau filling factor that are noted with the same greek letter “nu”. In general, it is not obvious for an unspecialized reader why the plasmon frequency should scale as the inverse of the magnetic field, while it is independent on the Landau level filling factor. The authors might provide a more detailed explanation on that point. In Figure 1c the transitions between filling factors seems to be correlated to changes in the transmission signal, can the authors comment on that? Finally, do they control the positions of the anti-nodes of of the plamon wave-voltage?"}, "author_response": "We sincerely thank the referees for taking the time to review our manuscript and providing valuable comments and suggestions to improve it. In the following, we provide the detailed responses to the questions and comments that were raised by the referees. The changes in the revised manuscript have been highlighted in red.\nREVIEWER COMMENTS\nReply to Reviewer #1’ comments\nThe manuscript by Lin et al. presents compelling experimental evidence of coupling between quantum Hall (QH) plasmons and a double quantum dot system. This is a novel and intriguing result with potential implications for hybrid quantum systems, especially in contexts where topologically protected modes are desirable. The topic is timely and of high relevance for the quantum technologies community. I am inclined to support publication, but I have several concerns and requests for clarification that I believe must be addressed before proceeding:\nReply: We thank the reviewer for the recognition of the novelty of this work. We also thank the reviewer for careful reading and providing the valuable comments. Below is our point-by-point response. We hope that these responses satisfactorily address the reviewer's concerns and lead to a recommendation for publication.\n1) One of the core features of quantum Hall edge modes is their chirality. It is not entirely clear how this feature is impacting the results of this work. Specifically:\na. In the Methods section, Eq. (3) describes a potential that appears to be that of a standing wave. This seems inconsistent with the absence of counter-propagating modes in a chiral system. Maybe is it possible that phi_0 is itself time-dependent and encodes the chiral propagation? If so, this should be made more explicit. If not, what defines phi_0 and how does the formulation account for unidirectional propagation?\nReply: We thank the reviewer for pointing out this error. Equation (3) is not correct for describing the chiral plasmon wave in the channel. As this equation is not used in the main text, we have removed it and its derivation in the revised manuscript. We provide the detailed calculation of the channel potential using a circuit model in the Supplementary Note 2, where the potential 𝑉 (𝑥,𝑡)= 𝑉(cid:3560) (𝑥)e(cid:3036)(cid:2985)(cid:3047)~𝑒(cid:2879)(cid:3036)((cid:3038)(cid:3051)(cid:2879)(cid:3104)(cid:3047)) [see Eqs. (S5-S8) for 𝑉(cid:3560) (𝑥)] describes the chiral propagation of plasmons in the resonator.\nb. More generally, how does chirality manifest in the coupling between the edge and the double quantum dots? I can imagine that this could impact coupling of more that one double dot\nReply: In contrast to usual standing waves in non-chiral resonators, the chirality provides circulating resonant modes. Otherwise, the chirality does not play significant roles in this study. However, the chirality can be used for non-reciprocal coupling of multiple qubits. This advantage is discussed as a future prospect in Lines 217-222 in the Prospects section.\n2) Could the reviewer for pointing out the effect of parasitic capacitance between the RF ports and the related reference. The parasitic capacitance in our device was evaluated using a setup shown in Supplementary Fig. 2f, and the measured result is shown in Supplementary Fig. 3a. The role of the parasitic capacitance (estimated to be ~ 1 fF) is negligible, because the signal |S p| only through the parasitic capacitance is 24 dB smaller than the signals |S | and |S 0| through the edge channel. Therefore, this small parasitic capacitance is not included in the circuit model.\n3) Is it possible to tune the plasmon velocity at fixed magnetic field via gate voltages on the coupling gates? Intuitively, one could imagine that adjusting the electrostatic environment could alter the local capacitance per unit length or the confinement profile, and thereby modify the plasmon dispersion. Have the authors explored this experimentally?\nReply: We thank the reviewer for this comment. The plasmon velocity can be tuned via gate voltage V applied on G and G . The experimental results are provided in Supplementary Fig. 3c. The characteristics of edge plasmons are described with the sentence “f can be changed by tuning  with B, as shown in Fig. 1c, and by tuning C with the gate voltages V , as shown in Supplementary Fig. 3c’ in Lines 93-94 in the Device section.\n4) The capacitance to the gates and ground planes is likely non-uniform along the edge. How does this spatial variation affect the plasmon-dot coupling? Could this be used to optimize the placement of the quantum dot for enhanced coupling?\nReply: In our device, the edge channel is predominantly defined by gate electrodes, including the microwave input, output ports, and the coupling region to the DQD, and only a small portion is defined by etching, as shown in Fig. 1a and the optical picture in Fig. 1d. We considered such non-uniformity in the simulation, as shown in Supplementary Note 2. Because the plasmon potential V in ch Supplementary Fig. 2c is continuous over the non-uniform channel, the double dot can be placed at any point. In other words, a larger coupling capacitance between the dot and the plasmon channel is essential for enhancing the coupling.\n5) While demonstrating plasmon-double dot coupling is a significant step, the coupling strength reported here is relatively modest compared to state-of-the-art charge-photon coupling rates, which can be substantially above the 100 MHz range (e.g., Samkharadze et al., Science 359, 1123 (2018); de Palma et al., Nat. Commun. 15, 2703 (2024); Yu et al., Nat. Nanotechnol. 18, 1360 (2023)). This is somewhat surprising, especially given the high impedance of QH edge modes. In the Outlook section, the authors suggest possible improvements to increase the coupling, but I believe the paper would benefit significantly from a more quantitative analysis of these prospects. Can the authors provide estimates for how large g could realistically become, and how it compares to coupling in superconducting resonator systems? A more explicit comparison with these technologies could greatly strengthen the relevance of this work.\nReply: We thank the reviewer for introducing the relevant studies on charge–photon systems. These papers are cited in the revised manuscript. Following the reviewer’s suggestion, we have significantly revised the discussion and outlook sections by providing a more quantitative analysis on the limitations in our system and relevant strategies for improvements. In specific, we have compared the coupling coefficient of our system to the state-of-the-art superconducting systems in the new added Fig. 5 in the main text. The detailed parameters for the resonator, qubit and coupling strength are summarized in the Supplementary Table 1. Based on the reported values and technologies, we discuss the prospect of the strong coupling regime in our system (Lines 197-212).\n6) The observed Q factor is quite low. Do the authors attribute this to dielectric losses, residual bulk conduction, edge disorder due to etching, or other mechanisms? The authors mention that this can be solved by higher mobility materials, but could improvements be expected also by defining edges electrostatically rather than through etching?\nReply: We attribute the low Q factor to disorders around the channel. Such disorders induce charge puddles and lead to plasmon dissipation. To clarify this point, we have further analyzed the field dependent resonance spectra of Fig. 1c in Supplementary Note 4, We find that at integer filling factors (ν = 1, 2, 3), the Q factor remains nearly constant, whereas it drops rapidly when ν deviates from integer values. These results indicate significant impact of the QH states and therefore are expected to appear for both gate-defined and etching channels. Nevertheless, we should be able to increase Q by improving the materials and device structure, as described in Lines 200-202.\n7) In the Introduction, the authors assert that backscattering is absent in QH systems. While this is true in the DC limit, backscattering can occur at finite frequency, especially at low magnetic fields.\nReply: In the revised manuscript, this assert is revised. The original sentence has been modified as “Thus, the plasmons with this fixed impedance should show perfect impedance matching over the whole channel even in the presence of non-uniform electrostatic environment” in Lines 27-28.\n8) The reported dissipation rate gamma is higher than what has been demonstrated in other GaAs samples (e.g., a few MHz as in reference 15). Could the authors comment on whether this elevated dissipation is an artifact of the device geometry, the fabrication method (e.g., etched edges), or other limitations?\nReply: We thank the reviewer for this comment. We suspect that the relatively large γ is related to the charge noise in the device and/or external noise coming from the measurement apparatus. This is discussed in Lines 197-199 in the revised manuscript.\n9) The coupling strength g is sometimes provided in frequency but its equations are in energy. Good to specify if the extracted values of g are divided by h or hbar.\nReply: We thank the reviewer for careful reading. In the revised manuscript, we have clarified the unit of 𝑔 and unified it to GHz.\nReply to Reviewer #2’ comments\nIn the article entitled “Dispersive detection of a charge qubit with a broadband high-impedance quantum-Hall plasmon resonator”, Lin and cowokers explore circuit quantum electrodynamics with plasmonic resonators formed by the edge states of a closed quantum Hall droplet. The authors couple a plasmonic resonator to a double quantum dot operated as a charge qubit. Analyzing the phase response of the resonator within the standard framework of the Jaynes-Cumming model (dispersive regime), the authors extract the qubit properties (decoherence rate, spectrum) for different gate voltages. The authors also investigate how the power sent in the input probe tone affects the resonator response.\nOverall, the paper reads easily. The analysis of the results is sound and the experimental data are in rather good agreement with the theoretical models derived. In my opinion, the data and the analysis support the main claim of the manuscript that is the dispersive detection of a semiconductor charge qubit with a quantum Hall resonator. This work provides a new alternative platform for circuit quantum electrodynamics. To my knowledge, this represents the first demonstration of the use of a quantum Hall plasmonic resonator in such experiments and this is the main novelty of this work. This work could interest a broad community of researchers in the field of mesoscopic quantum physics, in particular those working on high-mobility 2D electronic systems. I think it may stimulate theoretical and experimental works on the use quantum Hall resonators as probing tools, in particular in the context of fractional quantum (anomalous) Hall effect.\nDespite this, I think the authors should further motivate their work and further highlight the perspectives it offers. I also have few concerns regarding the future of this platform. Additionally, I have questions regarding some parts of the data analysis and the interpretation of the data. Once these points have been addressed, I could recommend the publication of this manuscript in Nature Communications。\nReply: We thank the reviewer for recognizing the novelty and significance of our work. We also thank the reviewer for careful reading and providing the valuable comments. Below we provide a point-by-point response, which we hope can satisfactorily address the reviewer’s concerns and support the recommendation for publication.\nMajor questions:\n(1) I think standard transport characterizations of the device/heterostructure are missing. Measures of the longitudinal Rxx and transverse Rxy resistances in a Hall bar made on the same sample/heterostructure would allow to evaluate the quality of the quantum Hall states forming at the magnetic field at which the device is operated. In particular, a measure of Rxx at ν = 2 would provide insights on the residual bulk conductivity and thus on the dissipation in the resonator. Do the authors have such a measurement? Can they measure Rxx in a configuration where the central part is open (no resonator formed like in Fig. S1)?\nReply: We thank the reviewer for this comment. In the revised manuscript, we have provided the low-frequency R and R data in the Supplementary Fig. 1, measured with the resonator device by opening constrictions under the gates G and G . For filling factor ν = 2 studied in this work, the observed vanishing longitudinal resistance and quantized Hall plateau confirm the well-developed quantum Hall state with negligible bulk scattering.\nSimilarly, the mobility of the GaAs/AlGaAs heterostructure should be reported. This quantity is relevant especially as higher mobility samples should enable the formation of resonators with higher Q as mentioned by the authors.\nReply: We have followed the reviewer’s suggestion. In the revised manuscript, we have added the as-grown sample mobility (μ = 460 cm2/Vs) in Line 71 in the Device description section.\n(2) In my opinion, a more in-depth discussion about the advantages and prospects offered by quantum Hall plasmonic resonators should be carried out. In particular the use of plasmonic resonators instead of high-impedance superconducting resonators (Z0 = 7.9 kOhms in Janik et al., Nat. Comm., 16, 2103 (2025)) should be further motivated. Is the high characteristic impedance the only advantage of quantum Hall plasmonic resonators?\nReply: We thank the reviewer for this suggestion and for introducing the related literature on high-impedance superconducting resonator. In the revised manuscript, we have substantially expanded the Discussion section to highlight the advantages and prospects of the plasmon resonator. A new figure (Fig. 5) is added for comparing the coupling coefficient between the plasmon resonator and the state-of-the-art superconducting resonators. In addition to the high impedance, the QH plasmon resonator is expected to provide several advantages. The slow plasmon velocity provides compact resonator designs for higher integration. The chirality of plasmons can be used to design non-reciprocal qubit couplings. The system might be extended to topological quantum computing in the future. These points are discussed in Lines 217-222 in the Prospects section.\n(3) The current quality factor is low Q = 4 and this limits the possibilities offered by the resonator despite offering a large detection bandwidth. As acknowledged by the authors, achieving the strong coupling regime is desirable and one solution is to increase the Q factor. I think the discussion regarding solutions to improve Q needs to be extended and clarified. More particularly:\n- The sentence “Because the plasmon velocity also decreases with decreasing ν, the resonator size can be reduced to concentrate the plasmon power in the resonator” is not clear to me. Can the authors provide more detailed explanations? I think the effects of the resonator length L and of the plasmon velocity v should be discussed separately. The authors could discuss how the resonator linewidth κ scales with L and v.\nReply: We thank the reviewer for this comment. Because this sentence is not directly related to enhance 𝑔, we have removed it from this paragraph. Instead, we focus on the resonator length (for a given resonant frequency) and described its benefit as ‘The extremely slow plasmon velocity, as compared to the light speed, is advantageous for integrating small multiple resonators on a chip’ in Lines 218-219 in the Prospects section.\n- It would be insightful to visualize how the quality factor evolves as function of the magnetic field B and filling factor ν based on data of Fig. 1c. A general comment on the separate effects of B and ν on Q will also be welcome considering potential extensions of this work to materials where the electron density can be tuned such as graphene.\nReply: The extracted Q-factor is plotted as a function of magnetic field B in Supplementary Fig. 4d in the revised manuscript. We found that Q takes nearly a constant value of ~4 at integer fillings (e.g.,  = 1, 2, 3), while it rapidly decreases when deviating from integer . These results indicate that the development of better materials plays a crucial role in increasing Q, as we described in Lines 200-202 in the Prospects section.\n- I suggest that the authors extend their comment on improving the device structures. In particular, recommendations on the design of the resonators (gated/ungated, edges defined by etching or electrostatically, etc.) would be useful for future works on the topic.\nReply: We thank the reviewer for this suggestion. In the revised manuscript, we have extended the discussion to include the device design and optimization to obtain larger geometrical coupling in Lines 203-209 in the Prospects section.\n(4) In Fig. 2c., the authors should explain why only some charge transitions are visible. It would be also interesting to discuss whether a plasmonic resonator can be used to detect the last electron transition or not. Similarly, do the authors have an estimate of the number of electrons in the quantum dots?\nReply: We thank the reviewer for this comment. To support Fig. 2c, we have provided the phase-response data over a wider gate voltage range in Supplementary Fig. 6. In addition to the charge degeneracy lines, we also observed other signals including the charging lines of the QD1 and QD2. The signals are expected to be visible when the transition has a tunneling rate close to the resonance frequency. We have added a sentence in Lines 104-106 to provide a further explanation for Fig. 2c. As in this work, we focused on the plasmon-qubit coupling, the possibility to detect the last electron has not been evaluated, we would like to leave this research as a future work.\n(5) For their analysis of the plasmon number in the resonator (Fig. 4), the authors assume that at low power the qubit is in its ground state at the charge degeneracy point. I am wandering if this assumption holds and more precisely if the thermal population of the excited state can be neglected. Do the authors have a measure or an estimate of the electron temperature in the system? Does the thermal population of the excited state influence their analysis (both for the data presented in Fig. 3 and Fig. 4)?\nReply: We thank the reviewer for this important comment. Because our dilution refrigerator shows the base lattice temperature T ~ 100 mK during the measurement, this thermal energy cannot be neglected. We reanalyzed all data by considering the finite-temperature effects with the thermal plasmon number n = ~ 0.45 in the resonator. In addition, we considered the phonon bath as a major dissipation environment for the DQD qubit. The corresponding thermal phonon number n = changes from ~ 0 at  >> k T to ~ 0.48 for the data in Fig. 3a (See our theoretical model in the Methods section). Using this model, we have refitted the data in Fig. 3 and Fig. 4 in the main text, and Fig. 7 and Fig. 8 in the Supplementary materials. The simulation also suggests that the thermal population of the excited state P = in the qubit is finite (~ 0.24) even in the absence of plasmon drive (See Fig. 4b). The simulation reproduces the experimental data quite well with the extracted parameters (𝑔 , t , ) and shows that the dispersive readout at P = -87.5 dBm in Fig. 3 was performed without significantly populating the qubit excited state.\n(6) In Fig. 4b and Fig. 4c, the amplitude and the phase response seem to drop at different external input powers. In particular, the phase response does not follow the simulation. Can the authors comment on that and explain this discrepancy?\nReply: The difference between the phase and amplitude response actually cannot be explained within our QED framework. Further investigations on theory and experiment are needed. In the revised manuscript,  and A are fitted by a slightly different \"proportionality factor (not shown)\" between V and √𝑃. The difference can be regarded as finite uncertainty, as marked by the hatched regions in Figs. 4c and 4d.\n(7) From the derivation presented in Methods (equation 17), it is not clear to me how the authors are able to extract separately γ1 and γϕ. It seems that only γ = γ1/2 + γϕ can be extracted. Can the authors explain it?\nReply: We thank the reviewer for this comment. Because γ and γ appear in different ways in Eq. (5.3), it should be possibly to separate them in principle. However, as the reviewer suggested, separation of γ and γ, and thus γ , is not always satisfactory. Therefore, we plotted only γ in Fig. 3e and Fig. S7c. For the simulation, we use the representative values of γ and γ shown in the figure caption to Fig. 3. We have added a sentence in Lines 148-149 in the main text. In addition, we apologize for our mistake in the second equality in Eq. (17), which may be misleading. The equation is corrected and shown as Eq. (12) in the revised manuscript.\n(8) In Supplementary section 1, equation S9 is simplified to equation S10 assuming that GsL << σxy = 77 µS. With L = 113 µm, Gs = 0.55 S/m, we have GsL = 62 µS ~ σxy. Thus the validity of the simplification can be questioned. Can the authors discuss this?\nReply: We thank the reviewer for pointing out this question. We simplify Eq. S9 to Eq. S10 to obtain the standard resonance formulism. As the reviewer pointed out, the plasmon dissipation is too large for further simplification of Eq. S9. In the revised manuscript, we have removed the Eq. S10 and the corresponding description.\nOther points:\n(9) In Fig. 1c, the second mode nR = 2 is very faint. Is it because the Ginj, Gdet electrodes have a length comparable to the wavelength of this mode? Would it be interesting to couple to higher plasmon modes (nR ≥ 2) in order to achieve resonators with higher Q?\nReply: Yes, for n = 2, the faint signal is because the electrodes (G , G ) have a length comparable to the wavelength. Our device is designed to maximize the signal injection and detection for the first mode. It would be also interesting to use higher-order mode for possible larger coupling and higher Q but we have not yet investigated that.\n(10) In Fig. S3c, the qubit dephasing rate increases from 30 MHz to 60 MHz with the probe frequency. Do the authors have an explanation for this behavior? Also I suggest to put the y-axis of this figure in linear scale, as it is done Fig. 3e.\nReply: We have followed the reviewer’s suggestion and put the y-axis of Fig. S3c (now Supplementary Fig. 7c in the revised manuscript) in a linear scale. As the referee suggested, the qubit dephasing rate  should not change with the probe frequency f . Actually, the variation of  in the f dependency of Fig. 7c is smaller than that in the V dependency of Fig. 3e. The small increase of qubit dephasing rate with increasing the probe frequency may originate from unwanted variations, such as the transmission power and the electrical noises in the measurement circuits. However, we have not reached conclusive explanations for this change.\n(11) I suggest the authors to report the value of the electrostatic coupling coefficient η = 0.023 earlier in the manuscript and not at the very end. How large can this coefficient be in realistic devices?\nReply: We have adopted the reviewer's suggestion and described the value of (= 0.032 in the revised manuscript) on Lines 118-121 in the Device section and Lines 152-154 in the Dispersive shift section, where we explained how = 0.032 is estimated from the obtained 𝑔 = 55 MHz and the expected estimation 𝑔 =𝑓η(cid:3493)𝑍𝑒(cid:2870)/ℎ. In addition, we describe how  can be increased further on Lines 205-209 in the Prospects section.\n(12) If I am not mistaken, the formula for the number of plasmons in the resonator np=(2εd/κ) should be replaced np=(2eεd/hκ) considering that εd as unit of a voltage.\nReply: We thank the reviewer for pointing out this mistake. Yes, the unit of  is volt. In the revised manuscript, we have corrected this formula as “n = (2eV /hκ)”, where we have changed  to V to unify the symbols in the paper for better readability.\n(13) Fig. 2.c and Fig. 2.d: The definitions of ε and δ in terms of gate voltages Vg1 and Vg2 should be specified and chosen to be consistent with the charge labels. The orientations of corresponding axes should also be checked. For example, the schematics in the inset of Fig. 2.c seem to contradict the charge labeling.\nReply: We thank the reviewer for careful reading. In the revised manuscript, we have corrected the orientations of the ε and -axes in the inset of Fig. 2c to make them consistent with the charge labeling in Fig. 2d.\nAbout the references:\nFor the sentence “While strong, ultra-strong, and deep strong coupling regimes are expected in plasmonic cQED systems” Can the authors provide references that support this statement?\nReply: Two references (Refs. 10 and 12) have been added to support this statement.\nFor the sentence “Because the plasmon velocity also decreases with decreasing ν”, I suggest to add Kumada et al., “Edge magnetoplasmon transport in gated and ungated quantum Hall systems”, Phys. Rev. B (2011) as reference or any other appropriate work.\nReply: We thank the reviewer for this suggestion. The paper [Kumada et al., Phys. Rev. B (2011)] has been cited in the revised manuscript.\nI also suggest the authors to add two more recent references regarding the coupling of superconducting resonators to charge qubits in 2D system (germanium heterostructures)\n- Janik et al., “Strong charge-photon coupling in planar germanium enabled by granular aluminium superinductors”, Nat. Comm., (2025)\n- De Palma et al., “Strong hole-photon coupling in planar Ge for probing charge degree and strongly correlated states”, Nat. Comm., (2024)\nReply: We thank the reviewer for introducing the related papers. These two papers have been cited in the Prospects section in the revised manuscript.\nI suggest to mention another recent work on quantum-Hall resonators and potentially make some comparisons: Frigerio et al., Gate tunable edge magnetoplasmon resonators”, Comm. Phys., 7, 314 (2024).\nReply: We thank the reviewer for introducing the related paper on plasmon resonator. This paper justifies the feasibility of large Q factors (8–18 in the GHz range). In the revised manuscript, we have cited this paper in the Prospects section.\nReply to Reviewer #3’ comments\nThe authors report of a system where a charge q-bit (tunnel coupled double quantum dot) is coupled to the plasmon excitation of a Hall edge channel. This system realizes the standard Jaynes -Cumming QED Hamiltonian describing a two-level system coupled with a single boson. The authors perform several experiments which allow to gain information on the q-bit characteristics, by a heterodyne measurement on the transmission through the plasmon channel. In the dispersive regime, they extract the energy levels and coupling strength of the double quantum dots (Fig. 3), and they reconstruct its charge state diagram (Fig. 2). For a nearly resonant condition and by varying the input drive on the plasmon channel, they also estimate the average number of plasmons by measuring the polarization saturation of the double dot.\nThese types of measurements have been reported before (and a pretty common) form qubits coupled with transmission line resonators; the novelty here is the fact that the boson resonator is provided by the plasmon of an edge Hall state. While such plasmon excitations have been recently reported https://www.nature.com/articles/s42005-024-01803-6, to my knowledge, here the authors report their coupling with a qbit for the first time. Also, the authors work in a regime of high impedance of the plasmon mode, which results in higher coupling strength (although the quality factor is low). These novelty grant publication in Nature communications.\nReply: We thank the reviewer for the positive evaluation and for recognizing the novelty of our work. We also thank the reviewer for the valuable comments and the recommendation for publication. Our point-by-point responses are provided below.\nThe paper can benefit from minor improvements. For instance, in the second paragraph of page 3 there is a confusion between the plasmon velocity and the Landau filling factor that are noted with the same greek letter “nu”.\nReply: We thank the reviewer for careful reading. We apologize for using similar letters: Greek letter  for the filling factor and italic Time-Roman letter v for the plasmon velocity. To avoid possible confusion, we use v for the plasmon velocity in the revised manuscript.\n-In general, it is not obvious for an unspecialized reader why the plasmon frequency should scale as the inverse of the magnetic field, while it is independent on the Landau level filling factor. The authors might provide a more detailed explanation on that point.\nReply: We thank the reviewer for this comment. In the revised manuscript, we have revised the sentences in Lines 89-97 to detailed explain why the plasmon frequency scale as the inverse of the magnetic field. The basic characteristics of edge plasmons are described.\n-In Figure 1c the transitions between filling factors seems to be correlated to changes in the transmission signal, can the authors comment on that?\nReply: We thank the reviewer for raising this point. We have further analyzed the resonance spectra in Fig. 1c, and presented the extracted resonance frequency f, bandwidth , and quality factor Q in Supplementary Fig. 4. Larger signal and quality factors are obtained at integer filling factors (e.g.,  = 1, 2, 3), indicating a close correlation between the plasmon transport and the formation of QH states. These correlations have been discussed in Supplementary Note 4 in the revised manuscript.\n-Finally, do they control the positions of the anti-nodes of of the plamon wave-voltage?\nReply: We thank the reviewer for this important comment. Because the plasmon wave circulates in the resonator, the resonant mode has no anti-nodes. Therefore, the DQD can be placed at any point near the resonator. This is discussed with chirality on Lines 219-221 in the revised manuscript."}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have addressed my concerns satisfactorily. I particularly appreciate the new Fig. 5 with the prospects of the field. I believe the updated manuscript is ready for publication.", "reviewer_2": "The authors have addressed my previous questions and have made substantial improvements to their analysis. In particular, they have modified their theoretical models to include thermal effects. Also, they have provided an insightful figure (Fig. 5) enabling to compare their work to similar experiments performed with superconducting resonators and to assess the room for improvement in the coupling strength.\nYet, I think some clarifications and modifications are still needed before publication.\n(1) Errors bars on the parameters fitted should be reported.\n(2) I find the modified text related to Fig. 4c-d confusing.\nAccording to the authors, the qubit polarization starts dropping at n = 2. This plasmon number should correspond to pl,drv fixed values of the drive voltage Vd and of the power applied P considering that n = (2eVd/hκ)² and that P is proportional pl,drv to Vd². Yet, in the experiments presented, Δϕ and ΔA start to drop at different powers P. To take this into account, the authors fit the data separately using different proportionality factors between P and Vd². After the rescaling, I would expect that the threshold powers obtained for Δϕ and ΔA correspond to the same Vd but the authors actually report two different values (Vd ≅ 2 µV and 3 µV).\nCan the authors clarify this? Likewise, can they clarify the meaning of the n = 3 and n = 6 values reported? Also, pl,drv pl,drv\ncan they provide more details on how the related simulations shown in Sup. Fig. 8 are performed?\n(3) The authors should draw a guideline in Fig. 4c-d to mark the drive voltage Vd corresponding to the power P = -87.5 dBm they used in the experiments presented in Figure 3.\n(4) Input-Output Theory: I suggest the authors to provide more details on how they go from equations 8 and 9 to equations 10 and 11.\n(5) I suggest to remove the diagonal black dashed line in Fig 5. This line could be misleading considering that the x-axis of the graph is broken.\n(6) The following typos and sentences have to be corrected or clarified:\n- Fig 1.d-e: It is written “Experiemnt” instead of “Experiment”\n- Bibliography item 48: It is written “Nonabelian anyons” instead of “Non-abelian anyons”\n- Sup. Note 3, page 5: It is written “small transmission amplitude |S21p| ~ 56 dB”. I guess it should be “-56dB”\n- Sup. Note 4, page 6 It is written “Whereas the overall bandwidth becomes broader with increasing the field…”. In Sup. Fig. 4, the bandwitdh κ rather seems to decrease with B."}, "author_response": "RESPONSES TO THE REFEREES\nReply to Reviewer #1’ comments\nThe authors have addressed my concerns satisfactorily. I particularly appreciate the new Fig. 5 with the prospects of the field. I believe the updated manuscript is ready for publication.\nReply: We thank the referee again for reviewing our manuscript and recommending it for publication.\nReply to Reviewer #2’ comments\nThe authors have addressed my previous questions and have made substantial improvements to their analysis. In particular, they have modified their theoretical models to include thermal effects. Also, they have provided an insightful figure (Fig. 5) enabling to compare their work to similar experiments performed with superconducting resonators and to assess the room for improvement in the coupling strength.\nReply: We thank the referee again for careful reading and providing the valuable comments. Based on these comments, we have revised our manuscript. At below, we provide the point-by-point response.\nYet, I think some clarifications and modifications are still needed before publication.\n(1) Errors bars on the parameters fitted should be reported.\nReply: The errors of the fitting parameters (𝑔 , t , ) are reported in the caption of Fig. 3 for the representative data in Fig. 3a and 3b.\n(2) I find the modified text related to Fig. 4c-d confusing.\nAccording to the authors, the qubit polarization starts dropping at n = 2. This plasmon number should correspond to pl,drv fixed values of the drive voltage Vd and of the power applied P considering that n = (2eVd/hκ)² and that P is proportional pl,drv to Vd². Yet, in the experiments presented, Δϕ and ΔA start to drop at different powers P. To take this into account, the authors fit the data separately using different proportionality factors between P and Vd². After the rescaling, I would expect that the threshold powers obtained for Δϕ and ΔA correspond to the same Vd but the authors actually report two different values (Vd ≅ 2 µV and 3 µV).\nCan the authors clarify this? Likewise, can they clarify the meaning of the n = 3 and n = 6 values reported? Also, pl,drv pl,drv\ncan they provide more details on how the related simulations shown in Sup. Fig. 8 are performed?\nReply: We thank the reviewer for pointing out this unclear point. As we understand the specific values of n = 2, 3 and 6 are not physically important, we removed all of them. In the revised paper (Lines 181-182 and 187-195), we focused on the power P (blue circle in Fig. 4c-4d) at which the signals  and A are reduced by half in the experiment, and the voltage amplitude V (blue circle in Fig. 4b) at d 1 which the polarization P is reduced by half in the simulation. By comparing the experiment and the z simulation, we introduced a proportionality factor k in 𝑉 =𝑘√𝑃, as shown in the inserted V scale (cid:2914) d with the k value in Fig. 4c and 4d. We believe this revision significantly improves the readability.\nThe power dependent data in Supplementary Fig. 8 are simulated based on Eqs. (14-15) and Eqs. (5.1-5.4) in the methods section. The details are provided in the Supplementary Note 8.\n(3) The authors should draw a guideline in Fig. 4c-d to mark the drive voltage Vd corresponding to the power P = -87.5 dBm they used in the experiments presented in Figure 3.\nReply: A guideline (vertical gray dashed line) has been added in Fig. 4c and 4d to mark the drive voltage V for P = -87.5 dBm.\nd\n(4) Input-Output Theory: I suggest the authors to provide more details on how they go from equations 8 and 9 to equations 10 and 11.\nReply: The details on this derivation are provided in Lines 328-338 in the methods section: Input-Output Theory.\n(5) I suggest to remove the diagonal black dashed line in Fig 5. This line could be misleading considering that the x-axis of the graph is broken.\nReply: We believe the diagonal line is important, as both  and Z should be improved for increasing 𝑔 . Because the Referee worries possible misleading of the x-axis, we have broken the diagonal and (cid:2913) horizontal dashed lines in the revised Figure 5. The broken x-axis is now clearly visible with the broken diagonal line.\n(6) The following typos and sentences have to be corrected or clarified:\n- Fig 1.d-e: It is written “Experiemnt” instead of “Experiment”\nReply: We thank the reviewer for careful reading. This typo has been corrected in Fig. 1d and 1e.\n- Bibliography item 48: It is written “Nonabelian anyons” instead of “Non-abelian anyons”\nReply: This typo has been corrected in Bibliography item 48.\n- Sup. Note 3, page 5: It is written “small transmission amplitude |S21p| ~ 56 dB”. I guess it should be “-56dB”\nReply: This error has been corrected in Supplementary Note 3.\n- Sup. Note 4, page 6 It is written “Whereas the overall bandwidth becomes broader with increasing the field…”. In Sup. Fig. 4, the bandwitdh κ rather seems to decrease with B."}]

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