Transmission-Mode RF-SET for Rapid Charge Readout

• Transmission-mode RF-SET is a charge readout system that integrates a single electron transistor with a superconducting inductor for rapid measurement of spin and charge states.
• Its design eliminates the directional coupler used in reflection-mode, simplifying circuitry and enhancing impedance matching for efficient multiplexing.
• Experimental benchmarks in Si/SiGe heterostructures show integration times as low as 100–300 ns, matching performance of state-of-the-art RF reflectometry systems.

A transmission-mode RF-SET (radio-frequency single electron transistor) is an advanced spectroscopic and charge readout system optimized for rapid, high-fidelity single-shot measurements of spin and charge states in semiconductor quantum devices. Utilizing a monolithically integrated SET, this architecture capacitively couples the SET to a 50 Ω coplanar feedline via a superconducting niobium inductor, providing an impedance-transforming network suited for transmission-mode (S_21) rather than reflection-mode (S_11) measurements. This configuration enables rapid, multiplexable, and experimentally simplified readout of double quantum dot structures, especially when implemented in Si/SiGe heterostructures. Its distinguishing feature is the elimination of the directional coupler required in reflection-mode, reducing circuit complexity without sacrificing single-shot spin-readout speed, and matching the integration timescales achieved in state-of-the-art RF reflectometry (Fattal et al., 7 Apr 2025).

1. Circuit Topology and Lumped-Element Model

The transmission-mode RF-SET employs a 50 Ω coplanar waveguide (feedline), capacitively coupled ($C_C \approx 100$ pF) to a planar-spiral superconducting niobium inductor ($L_C$). The inner pad of the inductor is connected to the coupling capacitor on the PCB, while the outer pad is wire-bonded to the source lead of the monolithic Si/SiGe SET. The SET is positioned adjacent to a double quantum dot, facilitating proximity-based charge readout. In the lumped-element model, neglecting small series resistances, the total input impedance is

$$Z_\text{tot}(\omega) = \frac{1}{j \omega C_C} + j \omega L_C + \frac{R_S}{1 + j \omega R_S C_P},$$

where $R_S$ is the differential resistance of the SET, and $C_P$ is the parasitic capacitance, including the accumulation capacitance of the 2DEG under the gates. The transmission S-parameter,

$$S_{21}(\omega) = \frac{2}{2 + Z_0 / Z_\text{tot}(\omega)},$$

with characteristic impedance $Z_0 = 50~\Omega$, captures the insertion-type S-parameter relevant for this configuration. In contrast, the reflection coefficient is

$$S_{11}(\omega) = \frac{Z_\text{tot}(\omega) - Z_0}{Z_\text{tot}(\omega) + Z_0}.$$

This architecture enables direct monitoring of charge transitions via changes in $R_S$, affecting both impedance and transmitted signal amplitude.

2. Theoretical Framework and Resonator Characteristics

2.1 Transmission and Power Coefficient

The transmission coefficient is defined by the voltage divider formed by $Z_0$ and $L_C$0,

$L_C$1

with transmitted power $L_C$2.

2.2 Resonance Conditions and Quality Factors

Resonance occurs where $L_C$3. In the limit $L_C$4, the resonance frequency is approximated as

$L_C$5

or more generally,

$L_C$6

The unloaded and coupling quality factors are

$L_C$7

where $L_C$8 includes parasitic RF losses $L_C$9. The total capacitance is $$Z_\text{tot}(\omega) = \frac{1}{j \omega C_C} + j \omega L_C + \frac{R_S}{1 + j \omega R_S C_P},$$0, the sum of parasitic and 2DEG-related contributions. The loaded quality factor obeys

$$Z_\text{tot}(\omega) = \frac{1}{j \omega C_C} + j \omega L_C + \frac{R_S}{1 + j \omega R_S C_P},$$1

2.3 SNR Scaling

For small perturbations of $$Z_\text{tot}(\omega) = \frac{1}{j \omega C_C} + j \omega L_C + \frac{R_S}{1 + j \omega R_S C_P},$$2, such as those induced by charge transitions, the corresponding change in transmitted voltage is $$Z_\text{tot}(\omega) = \frac{1}{j \omega C_C} + j \omega L_C + \frac{R_S}{1 + j \omega R_S C_P},$$3. Under white amplifier noise $$Z_\text{tot}(\omega) = \frac{1}{j \omega C_C} + j \omega L_C + \frac{R_S}{1 + j \omega R_S C_P},$$4, the single-shot SNR is

$$Z_\text{tot}(\omega) = \frac{1}{j \omega C_C} + j \omega L_C + \frac{R_S}{1 + j \omega R_S C_P},$$5

where $$Z_\text{tot}(\omega) = \frac{1}{j \omega C_C} + j \omega L_C + \frac{R_S}{1 + j \omega R_S C_P},$$6 is the integration time. Empirically, $$Z_\text{tot}(\omega) = \frac{1}{j \omega C_C} + j \omega L_C + \frac{R_S}{1 + j \omega R_S C_P},$$7 for $$Z_\text{tot}(\omega) = \frac{1}{j \omega C_C} + j \omega L_C + \frac{R_S}{1 + j \omega R_S C_P},$$8, transitioning to SNR saturation at longer timescales due to $$Z_\text{tot}(\omega) = \frac{1}{j \omega C_C} + j \omega L_C + \frac{R_S}{1 + j \omega R_S C_P},$$9 noise. In practice, SNR is quantified by the separation of IQ-distribution means for distinct charge states.

3. Experimental Implementation and Performance Benchmarks

Device realization occurs in a Si/SiGe heterostructure, with the superconducting niobium inductor ($R_S$0H) fabricated on a Si die, wire-bonded to both the coupling capacitor and SET source ohmic. The cryogenic chain includes $R_S$1 dB input attenuation and 33 dB amplification at 4 K (Caltech CITLF3). RF tones at $R_S$2 MHz are synthesized and I/Q demodulated using Zurich Instruments UHFLI.

The charge readout protocol proceeds by stepping gate voltages along a compensated path in the double quantum dot stability diagram, recording $R_S$3 single-shot $R_S$4 samples at varying $R_S$5 and RF power. Each dataset is fit with a Gaussian to obtain state means and variances, from which SNR is computed.

Measured benchmarks for minimum integration time to achieve $R_S$6 are:

Transition Type	$R_S$7 for $R_S$8
Interdot charge transition (ICT)	100 ns
Dot-reservoir transition (DRT)	300 ns

These results are consistent with leading RF reflectometry systems reporting $R_S$9–$C_P$0 ns.

4. Turn-On Behavior, Capacitive Shifts, and RF Losses

A global turn-on experiment, where all SET gates are swept from $C_P$1 V, reveals:

1. Unaccumulated 2DEG ($C_P$2 V): $C_P$3 MHz, high $C_P$4.
2. Partial 2DEG ($C_P$5 V): $C_P$6 MHz ($C_P$7 fF), moderate $C_P$8 drop.
3. Full 2DEG ($C_P$9 V): $$S_{21}(\omega) = \frac{2}{2 + Z_0 / Z_\text{tot}(\omega)},$$0 MHz ($$S_{21}(\omega) = \frac{2}{2 + Z_0 / Z_\text{tot}(\omega)},$$1 fF), strong $$S_{21}(\omega) = \frac{2}{2 + Z_0 / Z_\text{tot}(\omega)},$$2 collapse as dissipative RF channels form.

At $$S_{21}(\omega) = \frac{2}{2 + Z_0 / Z_\text{tot}(\omega)},$$3 V, the SET switches to DC conduction and resonance nearly vanishes.

RF losses are modeled by introducing a gate-dependent $$S_{21}(\omega) = \frac{2}{2 + Z_0 / Z_\text{tot}(\omega)},$$4 in parallel with $$S_{21}(\omega) = \frac{2}{2 + Z_0 / Z_\text{tot}(\omega)},$$5 and including accumulated $$S_{21}(\omega) = \frac{2}{2 + Z_0 / Z_\text{tot}(\omega)},$$6. The equivalent resistance is $$S_{21}(\omega) = \frac{2}{2 + Z_0 / Z_\text{tot}(\omega)},$$7, further degrading $$S_{21}(\omega) = \frac{2}{2 + Z_0 / Z_\text{tot}(\omega)},$$8:

$$S_{21}(\omega) = \frac{2}{2 + Z_0 / Z_\text{tot}(\omega)},$$9

Fitting measured $Z_0 = 50~\Omega$0 lineshapes with the general notch-type line formula,

$Z_0 = 50~\Omega$1

allows direct extraction of $Z_0 = 50~\Omega$2, $Z_0 = 50~\Omega$3, $Z_0 = 50~\Omega$4, and hence $Z_0 = 50~\Omega$5 and $Z_0 = 50~\Omega$6 as functions of gate bias. Experimental results confirm $Z_0 = 50~\Omega$7 decreases from $Z_0 = 50~\Omega$8 kΩ to $Z_0 = 50~\Omega$9 kΩ as the 2DEG forms, accounting for observed resonance quality factor collapse.

5. Design Guidelines and Optimization Strategies

Key parameters for optimal performance are:

• Parasitic $$S_{11}(\omega) = \frac{Z_\text{tot}(\omega) - Z_0}{Z_\text{tot}(\omega) + Z_0}.$$0: Minimize by utilizing high-quality, low-loss dielectrics, reducing 2DEG extent under gates, and engineering ohmic contacts for ultra-low resistance.
• $$S_{11}(\omega) = \frac{Z_\text{tot}(\omega) - Z_0}{Z_\text{tot}(\omega) + Z_0}.$$1 Maximization: Select superconductors with low kinetic inductance and fabricate narrow-linewidth Nb spirals; optimize Nb–dielectric interfaces.
• Impedance Matching: For transmission mode, target $$S_{11}(\omega) = \frac{Z_\text{tot}(\omega) - Z_0}{Z_\text{tot}(\omega) + Z_0}.$$2 (as opposed to $$S_{11}(\omega) = \frac{Z_\text{tot}(\omega) - Z_0}{Z_\text{tot}(\omega) + Z_0}.$$3 in reflection) to maximize $$S_{11}(\omega) = \frac{Z_\text{tot}(\omega) - Z_0}{Z_\text{tot}(\omega) + Z_0}.$$4. This leads to design choices of $$S_{11}(\omega) = \frac{Z_\text{tot}(\omega) - Z_0}{Z_\text{tot}(\omega) + Z_0}.$$5 and $$S_{11}(\omega) = \frac{Z_\text{tot}(\omega) - Z_0}{Z_\text{tot}(\omega) + Z_0}.$$6.
• Multiplexing: Multiple $$S_{11}(\omega) = \frac{Z_\text{tot}(\omega) - Z_0}{Z_\text{tot}(\omega) + Z_0}.$$7–$$S_{11}(\omega) = \frac{Z_\text{tot}(\omega) - Z_0}{Z_\text{tot}(\omega) + Z_0}.$$8–SET branches may be placed along the feedline, each with distinct $$S_{11}(\omega) = \frac{Z_\text{tot}(\omega) - Z_0}{Z_\text{tot}(\omega) + Z_0}.$$9. The flat and monotonic $R_S$0 response simplifies channel separation and digitization compared to reflection.
• Minimizing $R_S$1: Achievable via increasing $R_S$2 (within SET stability), lowering line and amplifier noise, and further enhancing $R_S$3 through optimized impedance matching and balanced $R_S$4.

6. Comparative Analysis and Prospective Applications

Transmission-mode RF-SET achieves competitive performance with leading RF reflectometry systems, attaining $R_S$5 in the $R_S$6–$R_S$7 ns regime for charge transitions in Si/SiGe quantum dots. The simplicity of the microwave assembly—owing to the omission of a directional coupler—and the straightforward frequency multiplexing capability position this architecture for scalable, rapid, and parallel spin-qubit readout and detailed studies of fast charge dynamics in quantum dot systems (Fattal et al., 7 Apr 2025). A plausible implication is the facilitation of large-scale quantum computing readout hardware by leveraging these architectural simplifications and multiplexing flexibility.