Microwave SQUID Multiplexing

• Microwave SQUID multiplexing is a technique that uses rf-SQUIDs coupled with superconducting resonators to convert magnetic flux changes from cryogenic detectors into measurable frequency shifts.
• It employs a synthesized frequency comb and SDR-based digital demodulation to simultaneously read out hundreds of channels with low flux noise and minimal wiring complexity.
• This scalable architecture supports applications in CMB cosmology, x‐ray spectroscopy, and particle physics by delivering high time resolution and robust noise performance.

Microwave SQUID multiplexing is a technique that enables the simultaneous readout of large arrays of cryogenic sensors—such as transition-edge sensors (TES) and metallic magnetic calorimeters (MMCs)—by encoding information from each detector into the microwave-frequency response of high-Q resonators coupled to radio-frequency (rf) SQUIDs. This approach has become the state-of-the-art solution for scaling up the number of detectors with minimal cryogenic complexity, wiring, and noise, underpinning next-generation experiments in CMB cosmology, x-ray and gamma-ray spectroscopy, and particle physics (Kempf et al., 2013, Dober et al., 2017, Neidig et al., 9 Sep 2025).

1. Principles of Microwave SQUID Multiplexing

At the core of microwave SQUID multiplexing is the translation of small magnetic flux changes—originating from the signal in a cryogenic detector—into frequency shifts of superconducting microwave resonators coupled via non-hysteretic rf-SQUIDs. Each sensor (TES or MMC) is inductively linked to an input coil of a SQUID. The SQUID, operating in a regime with screening parameter βₗ < 1 to avoid hysteresis, acts as a flux-dependent nonlinear inductor whose effective inductance $L(\Phi)$ varies as a function of the sensed flux.

This flux-dependent inductance is placed in the load or termination of a microwave resonator (typically 4–8 GHz, with loaded quality factors $Q_l \sim 5000 - 5\times10^4$), resulting in a measurable resonance frequency shift $\Delta f$ proportional to the sensed flux change.

A frequency comb of probe tones is synthesized to interrogate many resonators simultaneously on a common feedline. Each tone is dynamically tracked or monitored using heterodyne detection and frequency multiplexed digital electronics (Henderson et al., 2018, Dober et al., 2017, Neidig et al., 9 Sep 2025). A central equation for the resonator's frequency response is frequently given as:

$$f_\mathrm{res}(\Phi) = f_0 + \Delta f \cdot \sin\left( \frac{2\pi \Phi}{\Phi_0} \right)$$

where $\Phi$ is the total flux through the SQUID and $\Phi_0$ is the flux quantum.

2. System Architecture and Readout Implementation

The typical microwave SQUID multiplexer system consists of:

• Sensor array: Each TES or MMC is inductively coupled to an input coil on a non-hysteretic rf-SQUID.
• Superconducting resonators: Each SQUID is terminated by or coupled to a quarter-wave or lumped-element resonator, yielding a unique resonance frequency within the 4–8 GHz range (Kempf et al., 2013, Dober et al., 2020, Neidig et al., 9 Sep 2025).
• Microwave feedline: All resonators are capacitively coupled to a common 50 Ω transmission line, allowing microwave excitation and multiplexed readout.
• Flux ramp modulation: Linearizes the SQUID's transfer function by periodically sweeping the SQUID flux through many $\Phi_0$; the detector signal is encoded as a phase shift in the modulated waveform (Dober et al., 2017, Neidig et al., 9 Sep 2025).
• Room temperature software-defined radio (SDR) electronics: A multichannel digital signal processing system generates probe tones, acquires transmitted signals, applies digital down-conversion and channelization (using FFTs or filterbanks), and implements flux-ramp demodulation to recover each sensor signal. Full-scale SDR hardware has demonstrated reliable readout of 400 channels, with path to much higher scaling (Neidig et al., 9 Sep 2025).

The architecture is illustrated in the table:

Component	Function	Key Parameters
TES or MMC	Cryogenic sensor; signal induces SQUID flux	$I_{\mathrm{signal}}$; $T \lesssim 100$ mK
rf-SQUID ($\beta_L < 1$)	Flux-to-inductance conversion	$L_S$, $I_c$, $M_{in}$
Microwave resonator (CPW or lumped)	Converts $L(\Phi)$ to $f_\mathrm{res}$ shift	$f_0$, $Q_l$, spacing
Feedline	Combines all channels on one line	$f \in 4-8$ GHz
SDR electronics	Tone generation, DDC, flux-ramp demodulation, DSP	BW: 4 GHz, N: 400+

A key aspect is the digital demodulation of the periodic response induced by flux-ramp modulation. In flux-ramp demodulated modes, the system achieves white flux noise levels of $\sim$1.3–1.4 μ$\Phi_0$/√Hz over many channels, indicating excellent noise matching between the microwave SQUID multiplexer and the SDR backend (Neidig et al., 9 Sep 2025).

3. Multiplexing Scalability and Noise Performance

Microwave SQUID multiplexers have demonstrated scaling from modest arrays (e.g., 14–18 channels in early MMC systems) up to targeted 400 channels per SDR electronics module, with dense channel packing set by the resonator design—resonator frequency spacings as tight as 10 MHz within the 4–8 GHz band have been realized (Neidig et al., 9 Sep 2025). The scaling is fundamentally limited by:

• Resonator linewidth (set by $Q_l$ and coupling),
• Resonator–resonator frequency collision avoidance,
• Feedline and amplification bandwidth,
• Stability and linearity of the RF path and digital electronics.

In-concurrence, the flux-ramp modulation and demodulation (via software-defined sine and cosine mixing at the ramp frequency) are designed to linearize the response across multiple $\Phi_0$, suppress harmonic distortion, and maintain a flat transfer function. The increase in flux noise with FRM is described by the factor

$$\sqrt{S_{\Phi,\mathrm{FRM}}} = \sqrt{\frac{2}{\alpha}} \sqrt{S_{\Phi,\mathrm{open}}}$$

where $\alpha$ is the fractional period used for demodulation (typically $\sim$0.8). Reported white noise levels in the system after FRM are as low as 1.26–1.4 μ$\Phi_0$/√Hz (Neidig et al., 9 Sep 2025).

4. Signal Processing and Digital Demodulation

The digital backend is built around a modular SDR architecture:

• Front-end analog mixing: Upconverts/downconverts the full 4–8 GHz input to intermediate-frequency (IF) bands, each digitized at 1 GSPS.
• Digital downconversion and channelization: Each IF band is assigned to a polyphase filterbank or FFT, splitting out 10–20 MHz-wide channels for each resonator.
• Flux-ramp demodulation: The input waveform (phase-modulated at the ramp frequency) is converted to the sensor signal via arctangent or Fourier methods, using the periodic sawtooth as a digital phase reference (Neidig et al., 9 Sep 2025).
• Dynamic reconfiguration: All mixing frequencies, demodulation phases, and channel assignments are set in software, allowing real-time retuning and adaptation to device variations or experiment reconfiguration.

This SDR-based approach not only ensures scalability and flexibility but also matches the needs of experiments where cryogenic infrastructure constrains cabling and power dissipation.

5. Noise, Linearity, and Crosstalk Considerations

Maintaining low flux noise, high linearity, and minimal crosstalk is critical for large-scale deployments:

• Noise: The system achieves FRM-demodulated noise $\lesssim 1.4$ μ$\Phi_0$/√Hz, matching the theoretical expectation with scaling factor $\sqrt{2/\alpha}$. Open-loop noise under 1 μ$\Phi_0$/√Hz is typical. System design ensures that warm electronics (SDR, amplifiers) do not dominate overall noise.
• Linearity: Cryogenic amplifiers and digital mixers are selected for high linearity. The readout remains below the intermodulation threshold even when thousands of tones are present, supported by careful gain staging.
• Crosstalk: Resonators are spaced and designed to minimize mutual coupling. In practice, operating at 10 MHz or larger spacings with careful electromagnetic design has yielded negligible crosstalk (<0.3%)—limited primarily by amplifier nonlinearity and resonator hybridization (Groh et al., 2023, Neidig et al., 9 Sep 2025).

6. Applications, Upgrades, and Future Directions

Microwave SQUID multiplexing with SDR-based readout is deployed as the preferred technology in advanced MMC-based x-ray, gamma, and neutrino experiments, and is similarly dominant in large CMB sensor arrays using TES detectors (Dober et al., 2017, Dober et al., 2020, Neidig et al., 9 Sep 2025). The architecture provides:

• Reduction in system complexity and parasitic load: Single-cable or two-cable room-temperature to cryostat interfaces for 400–1000 channels, mitigating cost and thermal conduction in large arrays (Neidig et al., 9 Sep 2025).
• High time resolution and bandwidth: The combination of GHz-band resonators and processor capabilities enables fast pulse reconstruction (rise times $\sim$10 μs), supporting high count rates essential for X-ray and nuclear physics applications.
• Scalability: The custom SDR electronics, designed for 400 channels, offer a scalable path toward multi-thousand channel systems; future upgrades targeting wider bandwidth, improved processing, and higher ADC rates are in preparation to meet the demands of next-generation imaging calorimeters and CMB experiments.

7. Summary Table: Key Parameters in Recent Wideband µMUX MMC Readout Systems

Parameter	Value	Reference
Frequency coverage	4–8 GHz	(Neidig et al., 9 Sep 2025)
Channels (per SDR module)	400 (platform design), 14–18 (tested)	(Neidig et al., 9 Sep 2025)
SQUID screening $\beta_L$	≈ 0.6	(Neidig et al., 9 Sep 2025)
Flux noise (FRM demod.)	1.26–1.4 μ$\Phi_0$/√Hz	(Neidig et al., 9 Sep 2025)
Resonator Q ($Q_l$)	∼5000	(Kempf et al., 2013)
Channel spacing	10 MHz (tested), 4–8 GHz band	(Neidig et al., 9 Sep 2025)

The demonstrated system achieves reliable, low-noise operation across the full 4–8 GHz range, providing a reliable, scalable solution for reading out large MMC arrays and serving as a blueprint for next-generation high-density cryogenic detector systems (Neidig et al., 9 Sep 2025).