Concepts

This page explains the signal processing and algorithms at a level useful for operators. See Yu et al. 2022 for derivations and performance measurements.

Resonator Readout

Superconducting microwave resonators appear as dips in forward transmission (S21) on a shared RF feedline. For umux, an RF SQUID couples each TES to a unique resonator – detector current shifts the resonance frequency. A flux ramp (sawtooth) drives all SQUIDs through multiple flux quanta, linearizing the periodic SQUID response. The detector signal appears as a phase shift in this modulation.

Flux ramp phase modulation. Top: resonance frequency vs. time. Middle: flux ramp sawtooth. Bottom: slow detector signal appears as phase shift of the periodic response.

Firmware Signal Chain

Firmware block diagram. Blue: polyphase filter banks. Green: baseband processor. Yellow: timing. Pink: streaming. Purple: DC generation.

Polyphase filter bank – splits 614.4 MHz baseband into 512 time-multiplexed 2.4 MHz channels (2x oversampled, 100 dB rejection, ~6 us delay). Only 416 channels per band are used (center 500 MHz).

Conceptual single-channel downconversion: multiply by channel frequency, lowpass filter, decimate by 256.

Baseband processor – per-channel processing at 2.4 MHz:

Per-channel baseband processor. Red: external inputs. Green: FPGA computation. Blue: filter bank interface. Pink: data outputs.

Eta Calibration

The eta parameter converts IQ response into a frequency error estimate. It rotates and scales the resonator circle so small frequency shifts map to a single quadrature:

\[\eta = \frac{2\Delta f}{(I_+ - I_-) + j(Q_+ - Q_-)}\]

\[\hat{\Delta f} = \mathrm{Im}\left[S_{21} \times \eta\right]\]

Left: uncalibrated resonator in complex plane. Right: after eta rotation, frequency shifts project onto one axis.

Eta is measured once per resonator (S.eta_scan()) and is stable typically stable across cooldowns. It is stored in firmware BRAM as etaMag + etaPhase.

Tone Tracking

SMuRF’s tracking loop minimizes the frequency error by updating probe tone frequencies in real time. The algorithm parameterizes the flux-ramp-modulated resonance as a truncated Fourier series (DC + 3 harmonics = 7 coefficients per channel):

\[\hat{f}[n] = \sum_{i=1}^{3}\left(a_i \sin\omega_i n + b_i \cos\omega_i n\right) + C\]

Updated each sample via stochastic gradient descent:

\[\vec{\alpha}[n+1] = \vec{\alpha}[n] + \mu\,\hat{\Delta f}[n]\,\vec{h}[n]^T\]

where mu is the tracking gain (lmsGain in firmware). The demodulated detector signal is the phase of the first harmonic:

\[\theta = \arctan(b_1 / a_1)\]

averaged over each flux ramp frame and streamed at the frame rate.

Why tone tracking matters: by keeping tones at S21_min, power on the cryogenic amplifier is reduced 5–10 dB per tone. This suppresses intermodulation products and enables 1000+ channel multiplexing without linearity degradation.

Flux Ramp

• Reset rate: typically 2–10 kHz (= detector sample rate)

• Phi0 rate: reset_rate x phi0_per_ramp (typ. 4–6); should be >10 kHz to exceed resonator TLS 1/f

• Blanking: configurable dead time around sawtooth resets excluded from tracking to avoid transients

• Carrier frequency (f1): phi0_rate, the fundamental tracked by the loop filter

Data Output Levels

The firmware supports multiple tap points:

1. Raw IQ (2.4 MHz/ch) – take_debug_data()

2. Frequency + delta-f – after eta, before tracking

3. Demodulated phase (frame rate) – take_stream_data()

The software SMuRF Processor further applies phase unwrapping, a 4th order Butterworth lowpass (default 63 Hz at 4 kHz), downsampling, and file writing.