Cascaded Charge-Sensing Protocol

• Cascaded charge-sensing protocols are defined as non-destructive sampling methods that combine multiple amplifier readings to reduce noise without sacrificing throughput.
• By employing sequential, statistically independent measurements, the approach achieves order-of-magnitude improvements in signal fidelity for CCD imaging and quantum dot spin readout.
• Its scalability and efficient design enable high-speed, low-noise operation crucial for advanced astronomical imaging and quantum information processing applications.

A cascaded charge-sensing protocol is a measurement architecture in which the same charge packet or configuration is non-destructively sampled at a series of sites—amplifiers, sense nodes, or sensors—arranged in a chain or distributed structure. This approach enables the reduction of measurement noise through the statistical combination of largely independent samplings, without incurring the per-measurement time penalty that would apply if each sample were fully destructive. In charge-coupled device (CCD) and quantum-dot qubit contexts, such cascaded protocols grant order-of-magnitude improvements in noise and fidelity, unlocking new regimes in low-signal detection, photon-limited imaging, and quantum information readout (Lin et al., 10 Jun 2024, Lapi et al., 29 May 2024, Diepen et al., 2020).

1. Fundamental Principles of Cascaded Charge Sensing

In traditional CCDs, charge packets are measured destructively at a single floating-diffusion sense node, with a noise floor determined by amplifier properties, typically $\sigma_1 \sim$ 2–3 electrons rms. The Skipper CCD innovation enabled repeated non-destructive sampling of each packet via toggling between a sense node and summing well, averaging out amplifier noise by the factor $1 / \sqrt{N}$, but at a time cost proportional to $N$.

The cascaded or multi-amplifier protocol (often realized as MAS-CCD) replaces a single read node with a series of $M$ floating-gate amplifiers arranged in sequence along the serial register. As each packet traverses these amplifiers, it is non-destructively measured at each stage, resulting in $M$ statistically independent measurements per pixel in a single serial shift operation. The resultant noise for simple averaging is:

$$\sigma_{\text{total}} = \frac{\sigma_{1}}{\sqrt{M}}$$

This reduction incurs no per-pixel readout-rate penalty, preserving throughput essential for astronomical and photon-counting applications (Lin et al., 10 Jun 2024, Lapi et al., 29 May 2024). If each amplifier additionally performs $N$ Skipper-style samples, the reduction generalizes to $\sigma_{\text{total}} = \sigma_{1} / \sqrt{MN}$.

In quantum dot arrays, cascaded charge sensing enables readout of spin or charge information located many sites removed from the sensor, relaying the signal via controlled, Coulomb-mediated charge transitions in a domino-like cascade. The total signal measured at the remote sensor can then reflect events at the distant qubit, resulting in amplification of the measurement signal and enhanced signal-to-noise ratio (SNR) (Diepen et al., 2020).

2. Architectural and Operational Implementations

The MAS-CCD exemplifies cascaded charge sensing using a 16-channel linear output chain. The device architecture comprises:

• Pixel matrix: $1024 \times 512$ pixels at $15~\mu$m pitch, $650~\mu$m thickness, with 40 V substrate bias.
• Serial register: Incorporates 256 extended pixels, feeding 16 in-line floating-gate output stages. Each output stage sits after $\sim$15 inter-amplifier pixels.
• Amplifier stage gates: Each amplifier features a summing well (SW), output gate (OG), floating-gate sense node (FG), pixel separation (PS) barrier, and three-phase horizontal clocks (H1–H3). The final stage replaces PS with a dump gate.
• Clocking protocol per sample: For each amplifier, the packet is transferred from H3 into SW/FG, undergoes correlated double sampling (CDS) on the FG, and is returned to the H1/H2/H3 path for propagation to the next amplifier.
• Readout electronics: Multiple synchronized digital CDS channels (e.g., the “Hydra” system: 4 DESI FEEs in leader-follower mode for 16-channel readout at 38.3 kHz, 26 $\mu$s/pix) (Lin et al., 10 Jun 2024, Lapi et al., 29 May 2024).

Pseudocode for one pixel readout cycle:

for i in range(1, M+1): # M = number of amplifiers, e.g. 16 charge = move_charge_to_amplifier(i) for j in range(1, N+1): # N = number of samples per amplifier (Skipper-style) measurements[i][j] = sample_charge(charge) # Shift packet to next amplifier combined_value = weighted_average(measurements)

In quantum dot arrays, the protocol initializes electrons in a chain of dots with gate voltages tuned such that a charge transition in one dot, triggered by, e.g., a Pauli spin blockade event, sequentially drives neighboring dots across their Coulomb thresholds. The charge transfer sequence propagates to a sensor at the edge, coupling the initial event to a measurable signal even for dots remote from the sensor (Diepen et al., 2020).

3. Quantitative Noise and Signal Models

For MAS-CCDs, assuming identical, statistically independent amplifier noise $\sigma_1$, simple averaging yields:

$$\sigma_{\text{total}} = \frac{\sigma_{1}}{\sqrt{M}}$$

$$\text{For}~N~\text{samples per amplifier:}~\sigma_{\text{total}} = \frac{\sigma_{1}}{\sqrt{M N}}$$

Covariances between amplifiers or timing electronics are incorporated via:

$$\text{Var}(\bar{X}) = \frac{1}{M^2} \sum_{i} \sigma_i^2 + \frac{2}{M^2} \sum_{i<j} \text{Cov}(i,j)$$

Charge transfer inefficiency between amplifiers (amplifier-to-amplifier charge transfer efficiency, ACTE) is modeled as:

$\eta_i = \frac{S_{i+1}}{S_i}$

$S_k = S_0 \prod_{i=0}^{k-1} \eta_i$

Low charge transfer inefficiencies degrade cumulative SNR, emphasizing the requirement for $\eta_i \to 1$ for all $i$.

Common-mode noise (from digitization, shared power supplies, etc.) is often suppressed via real-time subtraction or optimal weighting of channels. The optimal estimator variance, given amplifier-specific noise $\sigma_i$ and a common-mode component $\sigma_{\text{CM}}$, is

$$\sigma_{\text{total}}^2 = \frac{1}{A N_s} \sum_{i=1}^A \sigma_{amp,i}^2 + \sigma_{CM}^2$$

where $A$ is the number of amplifiers and $N_s$ samples per amplifier (Lapi et al., 29 May 2024).

4. Performance Benchmarks and Scalability

Key metrics for the cascaded protocol as implemented in 16-stage MAS-CCDs include:

• Read noise: With $N = 1$ and $M = 16$ at 26 $\mu$s/pix, $\sigma_\text{total} = 1.03~e^-$ rms/pix (vs single-stage $\sigma_1\sim 4.1~e^-$ rms) (Lin et al., 10 Jun 2024, Lapi et al., 29 May 2024). Sub-electron noise ($\sim 0.5~e^-$ rms) is accessible with $N > 1$.
• Throughput: No slowdown per pixel relative to single-amplifier CCD; pixel rates up to 72 kHz achieved.
• ACTE: $\eta_i > 0.9999$ for most stages under $2~\text{ke}^-$; one stage measured at $\eta \sim 0.997$. Above $50~\text{ke}^-$ per packet, ACTE drops sharply and defines effective full-well (Lin et al., 10 Jun 2024).
• Linearity: $\pm 2.5\%$ over $330$–$35,000~e^-$ (Lin et al., 10 Jun 2024).
• Noise/common-mode decorrelation: At slow speeds, decorrelation can yield up to 25% further noise reduction; marginal at high pixel rates (Lapi et al., 29 May 2024).
• Quantum dot spin readout: Cascade-based protocols reach $>99.9\%$ fidelity in $1.7~\mu$s, with SNR enhancements $\times3.5$ compared to conventional methods (Diepen et al., 2020).

Scaling $M$ further (e.g., $M = 32$ or $64$) yields further reductions in noise, with hybrid cascaded-plus-Skipper sampling targeting $\ll 0.1~e^-$ rms for photon-starved applications (Lin et al., 10 Jun 2024).

5. Comparison with Other Charge Readout Architectures

Protocol	Noise Reduction	Sampling Penalty	Scalability
Standard CCD	None	None	Single amplifier
Skipper CCD	$1/\sqrt{N}$	$N\times$ slowdown	Single amplifier
MAS-CCD	$1/\sqrt{M}$ (or $1/\sqrt{M N}$)	No slowdown (single sample per amp)	Scalable $M$
Quantum dot cascade	SNR gain $\times 3.5$	None (relayed measurement)	Array/fan-out

In conventional architectures, enhanced SNR is only possible at the expense of throughput or by multiplying amplifier count at the device boundary. In the MAS-CCD protocol, the vertical gain comes from internal reuse of a single charge packet with negligible added time per sample (Lin et al., 10 Jun 2024, Lapi et al., 29 May 2024). In quantum dot arrays, the protocol circumvents the limitations imposed by short-range capacitive coupling, enabling sensors at system peripheries to conduct high-fidelity central measurements (Diepen et al., 2020).

6. Optimization and Practical Implementation Strategies

Performance is sensitive to numerous parameters:

• Clock amplitudes: Lower swings reduce clock-induced charge and noise, but too low impairs ACTE. Empirically, $H_1$ low must be $\sim 1$ V below $PS_\text{low}$.
• Temperature: Operation at 143 K suppresses dark and leakage currents.
• PS/Node removal efficiency: Must be tuned ($\epsilon\ll 10^{-5}$) via negative $PS/H$ clock levels to minimize charge trapping (Lapi et al., 29 May 2024).
• Common-mode noise handling: When present, adjacent empty pixels can be used for subtraction.
• Sampling strategy: Region-of-interest sampling restricts $N > 1$ to critical regions, balancing total readout time.
• Gain equalization and optimal weighting: Combined pixel value is calculated as a gain-equalized, optimally weighted sum across amplifiers (Lapi et al., 29 May 2024).
• Grounding and electronic design: Board/interface grounding is essential for minimizing groupwise correlated noise.

For quantum dot cascades, step tuning of each participating dot near its charge transition is required for reliable domino operation. Multi-stage pulsing may be employed to extend chain length and improve adiabaticity (Diepen et al., 2020).

7. Scientific Applications and Future Perspectives

The adoption of cascaded charge-sensing protocols directly targets demanding regimes in astronomical and quantum information science:

• Faint object spectroscopy: Noise floors $\lesssim1~e^-$ dramatically enhance sensitivity for ground-based mapping at blue-visible wavelengths (e.g., DESI/S5 science drivers), enabling 30–50% S/N gain at 400 nm, shorter exposures, and higher mapping speeds (Lin et al., 10 Jun 2024).
• Space-based coronagraphy and faint imaging (HWO): Requirements $<0.1~e^-$ rms for count rates $<1~$photon hr$^{-1}$ pix$^{-1}$ are accessible with cascaded+Skipper hybrid reading (Lin et al., 10 Jun 2024).
• Time-domain astronomy: Fast, low-noise full-frame readout ($\lesssim1$ min) is feasible, reducing cosmic-ray occupancy in survey data (Lin et al., 10 Jun 2024).
• Quantum computation: In quantum dot arrays, cascade-based remote spin readout enables dense two-dimensional arrays with edge-only sensing and measurement fidelities exceeding $99.9\%$ within $\sim1.7~\mu$s (Diepen et al., 2020).

Planned scaling includes ASIC-based readout for 64–128 channel MAS-CCDs at sub-$10~\mu$s/pix rates (Lin et al., 10 Jun 2024), and further exploration of clock-induced charge artifacts and dynamic common-mode adjustment. In quantum dot arrays, improved pulse engineering and on-chip amplification are under investigation to extend cascade length and robustness.

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Cascaded charge-sensing protocols thus provide a path to near-quantum-limited sensitivity in imaging and quantum system readout, with architectural flexibility and throughput scalable for next-generation instrumentation and processors (Lin et al., 10 Jun 2024, Lapi et al., 29 May 2024, Diepen et al., 2020).