Flux ramp modulation based MHz frequency-division dc-SQUID multiplexer

We present a MHz frequency-division dc-SQUID multiplexer that is based on flux ramp modulation and a series array of $N$ identical current-sensing dc-SQUIDs with tightly coupled input coil. By running a periodic, sawtooth-shaped current signal through an additional modulation coil being tightly, but non-uniformly coupled to the individual SQUIDs, the voltage drop across the array changes according to the superposition of the flux-to-voltage characteristics of the individual SQUIDs within each cycle of the modulation signal. In this mode of operation, an input signal injected in the input coil of one of the SQUIDs and being quasi-static within a time frame adds a constant flux offset and leads to a phase shift of the associated SQUID characteristics. The latter is inherently proportional to the input signal and can be inferred by channelizing and down-converting the sampled array output voltage. Using a prototype multiplexer as well as a self-developed high-speed readout electronics for real-time phase determination, we demonstrate the simultaneous readout of four signal sources with MHz bandwidth per channel.

(Dated: 19 January 2021) We present a MHz frequency-division dc-SQUID multiplexer that is based on flux ramp modulation and a series array of N identical current-sensing dc-SQUIDs with tightly coupled input coil. By running a periodic, sawtooth-shaped current signal through an additional modulation coil being tightly, but non-uniformly coupled to the individual SQUIDs, the voltage drop across the array changes according to the superposition of the flux-to-voltage characteristics of the individual SQUIDs within each cycle of the modulation signal. In this mode of operation, an input signal injected in the input coil of one of the SQUIDs and being quasi-static within a time frame adds a constant flux offset and leads to a phase shift of the associated SQUID characteristics. The latter is inherently proportional to the input signal and can be inferred by channelizing and down-converting the sampled array output voltage. Using a prototype multiplexer as well as a self-developed high-speed readout electronics for real-time phase determination, we demonstrate the simultaneous readout of four signal sources with MHz bandwidth per channel.
Direct-current superconducting quantum interference devices (dc-SQUIDs) are presently one of the most sensitive wideband devices for measuring any physical quantity that can be naturally converted into magnetic flux. For this reason, dc-SQUIDs are nowadays routinely used for applications ranging from investigations of magnetic nanoparticles to diagnostics in health care or the exploration of mineral deposits 1 . The intrinsic compatibility with mK operation temperatures as well as the excellent noise performance make SQUIDs also key components for the readout of cryogenic particle detectors 2 .
The maturity of fabrication technology allows building SQUID systems with hundreds or thousands of identical sensors. Moreover, the size of present-day SQUID systems is not limited by fabrication technology but other system constraints such as cooling power or system (e.g. wiring) complexity. This applies in particular to SQUID systems operating at mK-temperatures as used, for example, to read out cryogenic particle detectors. For this reason, suitable multiplexing techniques are required to realize multi-channel SQUID systems providing ultra-low power dissipation at cryogenic temperatures, a readout bandwidth of several 10 kHz up to some MHz, a large dynamic range as well as a linear relation between the input and output signal.
Existing SQUID-based multiplexing techniques include time-division multiplexing 3 , frequency-division multiplexing using MHz 4 and GHz 5 carriers, codedivision multiplexing 6 as well as hybrid multiplexing schemes 7,8 . But despite of the great success and their numerous advantages, they suffer from minor drawbacks a) sebastian.kempf@kit.edu that practically limit their application: State-of-the-art MHz frequency-division multiplexers, for example, employ large on-chip passive filter circuits limiting the overall channel count per given chip area. Furthermore, the frame rate of time-division SQUID multiplexers is too low to acquire wideband signals without signal deterioration. Though GHz frequency multiplexing can easily resolve these issues, it comes at the expense of an elaborated cryogenic microwave setup as well as a complex readout electronics.
In view of this, we present a novel MHz frequencydivision SQUID multiplexing technique being a suitable alternative to multiplex several tens to a about a hundred signal sources. It is based on flux ramp modulation, a modulation technique that had been originally developed for linearizing the output signal of a microwave SQUID multiplexer 9 . It relies on injecting a periodic, sawtoothshaped modulation current signal I ramp (t) into the modulation coil of a current-sensing SQUID to induce a linearly rising flux ramp with an amplitude of several flux quanta inside the SQUID loop causing the SQUID output voltage to vary according to its flux-to-voltage characteristic. The flux ramp repetition rate sets the effective sampling rate and hence the signal bandwidth. It is chosen such that the input signal appears to be quasi-static within a time frame of the flux ramp. In this configuration, the input signal leads to a constant magnetic flux offset causing a phase-shift of the flux-to-voltage characteristics that depends linearly on the actual amplitude of the input signal. Determining this phase shift, e.g. using Software-defined radio (SDR) based readout electronics, provides an intrinsically linearized measure of the input signal. A phase-shift of 2π corresponds to a magnetic flux change of one magnetic flux quantum. channel multiplexer that is based on our multiplexing approach. Four SQUIDs (one SQUID for each readout channel) are connected in series and dc-current biased such that the voltage V SQ across the array is the superposition of the output voltages V SQ,i of the individual SQUIDs. Each SQUID is equipped with a tightly coupled input coil with mutual inductance M in that is connected to the actual signal source, e.g. a superconducting pick-up coil or a cryogenic detector. Furthermore, a common modulation coil is inductively coupled to each SQUID, the mutual inductance M mod,i being chosen to have a unique value (the actual choice of the mutual inductance values is discussed below). By injecting a sawtooth-shaped current signal I ramp (t) into the modulation coil, a linearly rising flux ramp is induced inside each SQUID loop. The actual amplitude of the flux ramp depends on the mutual inductance M mod,i and the amplitude of the modulation current, the latter being adjusted such that multiple flux quanta are induced inside each SQUID loop. For this reason, the flux ramp causes the output voltage V SQ,i of the i-th SQUID to vary according to its flux-to-voltage characteristics, where the number of periods depends on the height of the flux ramp (see Fig. 1b). The periodic oscillation of the output voltage V SQ,i (t) of the i-th SQUID hence acts as a carrier signal which is phase-modulated by the signal source connected to the SQUID input. In case that the mutual inductances M mod,i are properly chosen, the carrier frequencies f c,i = I ramp,pp M mod,i f ramp /Φ 0 are unique and can be set by the amplitude I ramp,pp and repetition rate f ramp of the modulation signal. Since the latter simultaneously defines the effective sampling rate and hence the bandwidth per channel specified by the applications, the carrier frequencies are in practice mainly set by the amplitude I ramp,pp of the modulation signal. The series connection of the individual SQUIDs allows to superpose the carriers into the output voltage V SQ across the entire array (see Fig. 1c and d). By channelizing this overall output voltage signal V SQ (t) for each cycle of the flux ramp using, e.g. using digital down converters combined with subsequent low-pass filters, the phase of the individual carriers can be continuously monitored and acquired in real-time. In this sense, N signals can be simultaneously read out using only two bias lines as well as two lines connected to the common modulation coil.
To prove our novel multiplexing approach, we designed, fabricated and characterized a four-channel prototype multiplexer. We used our well-established fabrication process for Nb/Al-AlOx/Nb-Josephson tunnel junctions 10 as well as a SQUID design that we had previously developed. The SQUID layout is hence not optimized with respect to this multiplexing application. More precisely, the SQUID impedance is not matched to the line impedance, the flux-to-voltage transfer coefficient is not maximized and the on-chip wiring is not optimized for transmitting high-frequency signals. We therefore had to expect a reduced signal to noise level. Fig. 2a shows an optical microscope photograph of one of our fabricated multiplexers. The common modulation coil and the bias lines are colored in blue and orange, whereas the input coils are colored in red. Besides the electrical contact pads that are required for multiplexer operation and that are marked with colored dots, additional pads for initial diagnostics are placed on the left side of the chip. The latter allows, for example, to tap the individual SQUID voltages V SQ,i or to modulate the flux of only a subset of the entire SQUID array. The SQUIDs are washer-type parallel gradiometers consisting of four planar, single-turn coils that are connected in parallel (see Fig. 2b). Each coil is built by two superconducting washers of different size that are connected in series. The bigger washer is used to tightly couple the input coil, while the smaller washer is used to couple the modulation coil. This arrangement allows spatially and thus inductively separating the input and modulation coil to avoid parasitic coupling of the flux ramp into a potential superconducting input circuit as formed, for example, when connecting a superconducting pickup coil to the SQUID input. The Josephson tunnel junctions are located in the lower part of the SQUID and are resistively shunted to ensure a non-hysteretic behavior of the SQUID. In addition, the SQUID is equipped with washer shunts to damp the fundamental SQUID resonance. For our prototype device, we aimed to equally space the mutual inductances M mod,i in the range M mod,min ≤ M mod,i < 2M mod,min with i = 1, . . . , 4 and M mod,min being the mutual inductance of the weakest coupled SQUID. In general, this prevents higher harmonics of the carrier signals to appear in the target carrier frequency range and ensures that an integer number of flux quanta is induced in each SQUID loop when injecting a proper modulation signal into the common modulation coil. The latter is essential to avoid the occurrence of transients in the flux-to-voltage characteristics during the ramp reset after each cycle. We set the mutual inductance M mod,i by adjusting the overlap of the modulation coil and the underlying SQUID washer. For the tightest coupled SQUID, i.e. the SQUID with the largest mutual inductance M mod,i , the modulation coil runs directly on top of the SQUID washer (see Fig. 2c) to yield the highest possible magnetic coupling factor. For the other SQUIDs supposed to be weaker coupled, the diameter of the modulation coil is reduced (see, for example, Fig. 2d for the weakest coupled SQUID). We performed numerical inductance calculations of our SQUID design by means of InductEx 11 and managed to adjust the overlap to yield almost equally spaced mutual inductance values in the range 29.3 pH ≤ M mod,i < 58.6 pH. However, the precision of the performed simulations as well as to a minor degree fabrication inaccuracies, e.g. layer thickness inhomogeneities across the wafer as well as alignment errors, led to a slightly non-uniform spacing for the final prototype device. We thus had to deal with voltage transients during the ramp resets deteriorating the phase determination and increasing the white noise level. To avoid this kind of complication for future devices, we plan to use a similar wiring strategy as presently used for code-division multiplexing 8 . In particular, we are going to build twodimensional M × N multiplexers for which the N modulation coils within a row and the M SQUIDs within a column are serially connected to each other. Assuming that the mutual inductance values within a row are virtually identical, in particular after optimizing the fabrication process, this scheme allows to inject individual flux ramp signals to each column and thus to ensure that each SQUID is modulated with an integer number of flux quanta.
We mounted the fabricated prototype multiplexer on a custom-made sample holder, electrically connected the chip by means of ultrasonic wedge bonds, and immersed the setup enclosed with a soft-magnetic and superconducting shield into a liquid helium transport dewar. We dc-biased the multiplexer using the low-noise bias current source of a commercial high-speed dc-SQUID electronics 12 and amplified the voltage drop across the array by means of a capacitively coupled, low-cost amplifier cascade 13 . For synthesizing the modulation signal, digitizing the output voltage of the multiplexer as well as real-time channelization and phase determination, we employed a prototype SDR electronics we had previously developed for operating a microwave SQUID multiplexer 14,15 . The electronics comprises a Xilinx Zynq  Fig. 1(a). The solid lines result from averaging fifty neighboring points to increase visibility. UltraScale+ FPGA board 16 as well as ADC 17 and DAC 18 evaluation boards. The latter are connected to the FPGA board via FMC connectors and are synchronized via an external 10 MHz reference. To avoid a degradation of the linear rise of the flux ramp, we replaced the inductive input transformers of the DAC board by new inductive transformers being open for the frequency span between 15 kHz and 100 MHz. This allowed synthesizing adequate flux ramps signals. However, we later figured out that the effective ac-coupling still distorted the flux ramp linearity manifesting as a change of slope of the ramp during each cycle. The resulting carrier frequency change within each time frame significantly deteriorated the phase determination and ultimately led to an increased white noise floor. For this reason, we plan for dc-coupled DACs to be used in later applications. For channelizing and demodulation of the continuously sampled output voltage signal, we used digital down-conversion with subsequent low-pass filtering for phase estimation.
We comprehensively characterized the multiplexer varying, for example, the flux ramp repetition rate f ramp or the amplitude I ramp,pp of the modulation current I ramp (t). The resulting carrier signal frequencies were in the frequency range between 10 MHz and 50 MHz and the effective sampling rate was ranging between several 200 kHz and about 1.2 MHz. This potentially allows, for example, to read out some tens of high-speed cryogenic microcalorimeters with sub-µs rise time. Fig. 3 shows as an example the four output signals when connecting test signal generators to the input coils of channels SQ2 to SQ4, running the multiplexer with a flux ramp repetition rate of f ramp = 1 MHz and adjusting the amplitude to yield carrier frequencies of about 26.6 MHz, 32.5 MHz, 38.0 MHz, and 45.5 MHz. The input signals are clearly resolved. This proves that our multiplexing technique is performing as intended. The cross-talk between different channels was measured by applying a sinusoidal test signal to one channel and measuring the spectral intensity of the associated frequency bin in the other channels. For all channels, the crosstalk is below −40 dB. The white noise level of all four channels is about 25 µΦ 0 / √ Hz and therefore seems to be rather high. However, it can be explained by the small SQUID flux-to-voltage transfer coefficient, the voltage noise of the used amplifier cascade, the noise penalty due to flux ramp modulation 9 , the remaining non-linearity of the flux ramp, the impedance mismatch as well as the non-optimized device design. By optimizing the overall setup, we expect to yield a noise level similar to state-of-the-art dc-SQUIDs.
In conclusion, we successfully demonstrated a novel MHz frequency-division dc-SQUID multiplexing technique. We showed that our approach avoids some drawbacks of other existing multiplexing techniques and allows for reading out signal sources with MHz signal bandwidth at cryogenic temperatures. In combination with the enormously large dynamic range resulting from converting the input signal into a phase shift, our multiplexing approach paves the way for realizing various applications requiring a simple setup, large signal bandwidth and dynamic range as well as low noise.