Large Dispersive Interaction between a CMOS Double Quantum Dot and Microwave Photons

We report fast charge-state readout of a double quantum dot in a CMOS split-gate silicon nanowire transistor via the large dispersive interaction with microwave photons in a lumped-element resonator formed by hybrid integration with a superconducting inductor. We achieve a coupling rate $g_0/(2\pi )=204\pm 2\mathrm{MHz}$ by exploiting the large interdot gate lever arm of an asymmetric split-gate device, $\alpha =0.72$, and by inductively coupling to the resonator to increase its impedance, $Z_r=560\mathrm{\Omega }$. In the dispersive regime, the large coupling strength at the double quantum-dot hybridization point produces a frequency shift comparable to the resonator linewidth, the optimal setting for maximum state visibility. We exploit this regime to demonstrate rapid dispersive readout of the charge degree of freedom, with a SNR of 3.3 in 50 ns. In the resonant regime, the fast charge decoherence rate precludes reaching the strong coupling regime, but we show a clear route to spin-photon circuit quantum electrodynamics using hybrid CMOS systems.

Popular Summary

Spins in silicon are one of the most promising qubit embodiments for the scale up of quantum computers, particularly since it has been demonstrated that silicon spin qubits can be manufactured in a transistorlike fashion using the very-large scale-integration capabilities of the semiconductor industry. However, the transition from qubits fabricated in academic laboratories to those manufactured at scale in silicon foundries requires some degree of adaptation. More particularly, standard readout techniques based on charge sensors, like the single-electron transistor, are too complex to implement with current fabrication capabilities and new methods must be developed. In this paper, we demonstrate an advanced methodology to read the state of industry-fabricated silicon devices based on a hybrid technological approach that incorporates superconducting resonant electronics.

Our approach, based on the dispersive limit of circuit quantum electrodynamics, combines an industry-fabricated silicon device and a superconducting spiral inductor on two separate chiplets, to form an inductively coupled lumped-element high-impedance microwave resonator. Keeping the superconducting circuitry separate from the silicon device enables independent and optimized fabrication strategies for both. In particular, the silicon device has been optimized to have one of the largest ever reported gate couplings, an essential ingredient for sensitive dispersive readout. We achieve a SNR of 3.3 in 50 ns, the largest at this timescale for a dispersively read silicon device. This hybrid approach is not limited to silicon devices and may potentially be used to improve the readout fidelity of other semiconductor-based systems like germanium and Majorana-based quantum devices.

Figure 1

Measurement setup, resonator geometry, and double quantum-dot characteristics. (a) Two mutually coupled inductors represent the inductive coupling between a microstrip waveguide and a spiral inductor with inductance $L$, the layout of which is shown below in the dashed box. Both elements are fabricated from the same NbN film on sapphire; the gap separating inductor and waveguide is $4\mu \mathrm{m}$ wide. The inductor is wire bonded to a NWFET on a separate substrate (aluminium wire bonds are highlighted in red). The bias tee consists of a 100-pF capacitor and a 100-$\mathrm{k}\mathrm{\Omega }$ resistor. A SEM image shows a top-down view of a similar split-gate NWFET, where the orange arrow highlights the insulating spacer between the two top gates and its misalignment from the center of the nanowire. (b) Cross-section schematic of the NWFET, cutting through the nanowire perpendicular to its length. Quantum dots are formed in the top corners of the intrinsic silicon body, indicated by white shading. (c) The phase (blue), amplitude (black), and fit (red) of a reflected microwave signal applied and measured at ${\mathrm{MW}}_{\mathrm{in}}$ when the terminals of the NWFET are grounded. (d) The DQD charge stability diagram shown in normalized reflected phase data as $V_{T1}$ and $V_{B1}$ are varied. Numbers in parentheses indicate the electron numbers on each dot; the highlighted ICTs are measured in detail later.

Figure 2

Resonator-DQD interaction in the dispersive regime. (a) Reduced phase response of the resonator $\mathrm{\Delta }\varphi /{\varphi }_0$ around the $(7,11)⟷(8,10)$ ICT. Color-coded crosses denote the positions of measurements on and off the ICT, respectively, see Fig. 3. (b) Magnetospectroscopy of the $(7,11)⟷(8,10)$ ICT, which we use to calculate the interdot gate lever arm. (c) Magnitude of the signal reflected from the cavity showing the shift in the resonant frequency as $V_{T1}$ is swept across the ICT. (d) Simulated data, using Eq. (2), to reproduce the experimental data in (c).

Figure 3

Readout performance evaluation. (a) Distribution of the reflected signal in quadrature space, collected at the two points marked in Fig. 2 on and off the ICT signal. Each point is collected with an integration time $t_{\mathrm{int}}=50$ ns. For each distribution, the black cross marks the mean and the dashed circle indicates the standard deviation of distances to the mean. (b) SNR dependence on $t_{\mathrm{int}}$ for input $\mathrm{power}=-100$ dBm. Red data points are taken with a 1-MHz low-pass filter and the green data points are taken with a 20-MHz low-pass filter. The dashed lines extrapolate the data to $\mathrm{SNR}=1$, from which we extract the minimum integration time, $t_{\min }$. (c) Decrease in $t_{\min }$ with increasing input power, which saturates due to power broadening at approximately $100$ dBm [49].

Figure 4

Response of the resonator in the resonant regime. (a) Magnitude in dB of the power reflected at the resonator, as $V_{T1}$ is varied across the $(6,9)⟷(7,8)$ ICT. (b) Simulation results based on Eq. (2).

Figure 5

Quality of model fitting. (a) Blue points plot the shift in the center frequency of the cavity, $f_c$, as the detuning is swept across the ICT in the dispersive regime. (b) Green points plot the change in FWHM of the cavity resonance across the ICT in the dispersive regime. For both (a),(b) the points are extracted from the data shown in Fig. 2, and the black dashed line represents the best-fit prediction from the model [see Fig. 2]. (c),(d) Plot of the same quantities for the ICT in the resonant regime (see Fig. 4). For all panels, errors on $f_c$ and FWHM are omitted for clarity, and errors on $\epsilon$ are too small to be visible. In (a),(b) the $f_c$ and FWHM error bars are too small to be visible, and for (c),(d), the errors are roughly the same magnitude as the random scatter in the points.

Figure 6

Photograph of the multimodule setup. The silicon chip with an array of square bond pads is seen to the left, and to the right is the NbN-on-sapphire substrate. The two modules are positioned adjacent to each other on a printed circuit board. The insert magnifies the elongated-spiral inductor adjacent to the superconducting $50\mathrm{\Omega }$ waveguide.

Figure 7

Simplified circuit model. The inductively coupled spiral inductor and microstrip waveguide can be approximated as a T circuit. All of the parasitic capacitances in the system are combined with the gate capacitance into the one capacitor shown. The total losses are accounted for by the 750-$\mathrm{k}\mathrm{\Omega }$ resistor.

Figure 8

AWR AXIEM EM simulations. We model the system, including parasitic contributions, using a three-dimensional planar MOM EM analysis simulator. The insert shows the model of the sapphire chip with the microstrip waveguide and spiral inductor. The NWFET is represented by a 800-$\mathrm{k}\mathrm{\Omega }$ resistor in parallel with a 50-fF capacitor, highlighted by a red dotted box. The MLIN component resides on the Rogers 4003C PCB.

Figure 9

Simulated resonator response. (a) Simulated reflection coefficient is plotted on a Smith chart for a range of frequencies around resonance. The orange dashed line gives the response simulated by the simplified circuit model shown in Fig. 7, which has rotated about its origin on the edge of the circle by ${48}^{\circ }$ to reproduce the asymmetry observed in the magnitude data. The blue dash-dot line gives the response simulated using the combination shown in Fig. 8. (b) We plot the absolute value of the reflection coefficient on the dB scale against frequency. The experimental data from Fig. 1 is included in black.

Figure 10

Schematic of the frequency-multiplexing experiment and results. Tones from separate microwave sources ($f_1$ and $f_2$) are combined and applied to two single-gate silicon NWFETs via a common microwave waveguide. The reflected signal is amplified at room temperature, divided, and then demodulated with the relevant reference signal ($f_1$ is demodulated into phase and gain components using an Analog Devices AD8302, whereas $f_2$ is demodulated into I and Q using a Polyphase AD0540B). Coulomb diamonds for both devices are measured simultaneously at 1.78 GHz (left plot) and 2.45 GHz (right plot).

Figure 11

Measurement of the intermodulation product, $f_1-f_2$, to characterise crosstalk. The intermodulation product is only observable in the multiplexing experiments when the input power of $f_1$ ($f_2$) is raised by approximately $30$ dB ($10$ dB) above the typical experimental level. The resulting product is at least 60 dB (40 dB) below the reflected $f_1$ ($f_2$) level.

Figure 12

Determining the electron occupancy. (a) Charge stability plot focusing on a charge transition of the $B1$ dot, where we indicate loading of electrons into the $T1$ dot with the white dashed lines. (b) Charge stability plot showing a transition of the $T1$ dot where we indicate loading of electrons into the $B1$ dot. The red-dotted and green-dashed boxes indicate the ICTs investigated in Figs. 2 and 4 respectively, similar to those in Fig. 1. An orange marker is included in both plots at the same voltage coordinates $(V_{T1},V_{B1})=(0.485,0.58)$ V to aid comparison. Due to an extended time period between some of the measurements, there is a variation in the position of $T1$-dot charge transitions due to voltage drift; black arrows are used to connect these transitions between the data sets.

Figure 13

Large dispersive interaction in a NWFET with a super-top gate (STG). (a) Cross-section schematic of the NWFET, similar to Fig. 1, but with an additional gate STG above the split gate. (b) Reduced phase response of the resonator $\mathrm{\Delta }\varphi /{\varphi }_0$ in the region of an ICT. (c) Magnitude of the signal reflected from the cavity showing the shift in the resonant frequency as $V_{T1}$ is swept across the ICT. Note the change in $V_{T1}$ axes is due to a change in the $V_{\mathrm{STG}}$ setting: $V_{\mathrm{STG}}=11.15$ V in (a) and 12.1 V in (b).