Technical White Paper

On digital synthesis and detection of microwave signalsfor quantum technology

Recent progress with integrated circuits for real-time digital signal processing enables a direct digitalapproach to synchronous drive and measurement of continuous or pulsed microwaves. High-speed dataconverters give direct access to microwave bands without analog IQ mixers, removing local-oscillatorleakage, nonlinear distortion, calibration and drift when controling and measuring quantum circuits. Multiplechannels of drive and readout are processed on one chip, replacing racks of expensive equipment andeliminating problems of synchronization and software control of multiple instruments.

RFSoC Technology5G radio telecommunication has driven the recentdevelopment of Radio-Frequency System-on-a-Chip(RFSoC) technology. This acronym designates thechip-level integration of high-speed analog-to-digitalconverters (ADC), digital-to-analog converters(DAC), a field-programmable gate array (FPGA) andmultiple CPU cores. The high density of componentsand reduced signal-propagation delay enable theinterleaving of several slower ADCs to achievesampling rates up to 5 Gs/s, 10 GS/s for DACs, on16 parallel channels, all phase coherent andsynchronized to one master clock. This extremelyhigh data throughput and processing power createsan ideal platform for a fully programmable, directdigital control and measurement system formicrowave quantum technology.

One immediate advantage of RFSoC is theelimination of analog mixers used to up-convert anddown-convert signals to microwave frequencies.Analog mixers are nonlinear components withharmonic and intermodulation distortion. They

require a separate Local Oscillator (LO) which leaksthrough, giving unwanted frequency content.Frequency-dependent gain and phase imbalance inthe IQ mixer makes it difficult to select a singlesideband [1]. This white paper describes directdigital methods which eliminate these problems.

Digital Down- and Up-ConversionTraditional digital methods limit the maximumfrequency in a waveform to the Nyquist frequency, orhalf the sampling frequency, fN=½ fs. To accessfrequencies above fN a local oscillator and analogmixer are used to down-convert or up-convert thewaveform to another frequency band. Digitalmethods perform this conversion without mixers byworking in a higher Nyquist Zone (NZ) of the ADC orDAC, where frequency content appears at so-calledalias frequencies (see fig. 1).

When a high frequency signal is down-convertedfrom a higher Nyquist zone, it is under-sampled bythe ADC and it will appear at an alias frequency.Traditional Digital Signal Processing (DSP) removes

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Figure 1: Aliasing and Nyquist Zones: Discrete periodic samples at frequency fs represent a sinusoidal signal at frequency f (blue curve). The same set of samples represents under-sampled sinusoids at frequencies in higher Nyquist zones (NZ). Aliasing of a signal at frequency f occurs at frequencies mirrored about integer multiples of the sampling frequency, nfs ±  f (lower panel).

Technical White Paper

high frequency signals with a low-pass anti-aliasingfilter having a sharp roll-off above fN. If instead aband-pass filter is used to select one Nyquist zoneof interest, we can map the high frequency signal toits alias in the 1st Nyquist zone. With Digital Down-Conversion (DDC) the filter and sampling frequencyshould be carefully chosen, so as to be absolutelysure of the zone from which the signal actuallyoriginates.

Digital Up-Conversion (DUC) is achieved byboosting output power in a higher Nyquist zone.DACs switch between discrete levels at the risingedge of a square-wave clock (see fig. 2). This modeof operation is called non-return to zero (NRZ), or"normal mode". The rapid step generates frequencycontent at image frequencies: the original signalfrequency is mirrored around each multiple of thesampling-clock frequency. Typically the higher

frequencies are removed by a reconstruction filterwith a rapid roll-off above the Nyquist frequency fN .

If instead the DAC is constructed to flip the signalabout its mid-point at each falling edge of the clock,it will boost the power in the 2nd and 3rd Nyquistzones, while reducing the power in the 1st and 4th

zones. This mode of operation is called return tocomplement (RTC), or "mixed mode". A band-passreconstruction filter is then used to select the 2nd

zone. Here again, the sampling frequency and filtershould be carefully chosen to avoid spurious tonesoutside the zone of interest. Such spurious tonescan be quite strong when the tone and its image areclose to a zone boundary.

Direct Digital Synthesis (DDS)Thus, digital circuits can directly reach microwavebands with data converters capable of 10 GS/s. But

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Figure 3: Flow chart showing the conversion steps for digital generation (upper path) and measurement (lower path) of high frequency signals. We follow a 100 MHz signal synthesized by programmable logic at each stage in the conversion path.

ProgrammableLogic (FPGA)

4 x 500 MS/s

100 MHz

2 GS/sInterpolate X 5

100 MHz

10 GS/s

NCO 3 GHz

3 ± 0.1 GHz 10 GS/s

DAC 3 ± 0.1 GHz (1st NZ)mix mode: 7 ± 0.1 GHz (2nd NZ)

2 GS/s

100 MHz100 MHz

4 GS/s

100 MHz100 MHz

1 ± 0.1 GHz1 ± 0.1 GHz

Decimate ÷ 2

NCO 1 GHz

4 GS/s ADC

3 ± 0.1 GHz (2nd NZ)7 ± 0.1 GHz (4th NZ)

Figure 2: The DAC performs rapid transitions between discrete levels (blue signal) at each edge of a clock (dotted line) running at the sampling frequency fs . The dominant frequency content of this analog waveform is in the 1st Nyquist zone, but significant frequency content exists at image frequencies. If the DAC is capable of flipping the signal to its complement at each falling edge of the clock, moving frequency content moves to higher Nyquist zones and efficiently boosting output power in the 2nd Nyquist zone.

Technical White Paper

state-of-the-art FPGA logic runs at only 500 MHz.Parallel processing of samples bridges this gapbetween the slow logic and the high-speedconverters. Additional flexibility is provided by digitalmultiplication with a Numerically ControlledOscillator (NCO). Digital mixing is mathematicallyperfect so it introduces no distortion or leakage, instark contrast to analog mixing.

Figure 3 shows a scheme to synchronouslygenerate and measure arbitrary waveforms in a2 GHz band around a user-selected centerfrequency. The waveform is first synthesized in theFPGA using 4 parallel data paths, resulting in aneffective data rate of 2 GS/s. As an example weconsider a 100 MHz signal, but any waveform withfrequency content below 1 GHz is possible. Thewaveform is interpolated by a factor of 5 to aneffective date rate of 10 GS/s. We then digitallymultiply the signal with a 3 GHz NCO. Any frequencybelow 5 GHz can be selected for the NCO, with20 μHz precision. The multiplication results in twoside-bands at 3 ± 0.1 GHz, but either side-band canbe chosen if both I and Q quadratures of the originalwaveform are synthesized. Digital mixing results inperfect suppression of unwanted sideband. Thus wecan position our signal anywhere in a 2 GHz bandcentered at the NCO frequency.

In the final output stage the samples are sent to a10 GS/s DAC which outputs the high-frequency

analog waveform. The sharp steps give frequencycontent at f = 3 ± 0.1 GHz with an image in the 2nd

Nyquist zone at fs  -  f = 7 ± 0.1 GHz. The mixedmode of the DAC boosts power in the 2nd NZ whilereducing power in the 1st NZ (see fig. 2). An externalanalog reconstruction filter removes the unwantedband.

The down-conversion of a high-frequency analogsignal occurs in a similar manner, but with a slowerADC at 4 GS/s. At this sampling frequency, signalsat 3 ± 0.1 GHz or 7 ± 0.1 GHz, in the 2nd and 4th NZrespectively, alias down to the same frequency,1 ± 0.1 GHz. These two signals are thereforeindistinguishable to the ADC.

Our example demonstrates an important precautionwith direct digital methods: one should use anexternal analog filter to remove signals outside theNyquist Zone of interest. At first, this might feel likean added burden. In reality, these filters are usuallyembedded into traditional designs and thusinaccessible to the end user. With DDS-capableconverters, the power and flexibility of choosing theanalog filters is given to the end user.

Down-conversion proceeds with multiplication by anNCO at 1 GHz, resulting in a 100 MHz signal whichis highly over-sampled at 4 GS/s. With decimationby a factor of 2 we arrive at 2 GS/s, to be analyzedby the FPGA running at 500 MHz with 4 parallel datapaths.

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Figure 4: The front and back panel showing Vivace's connections: 8 input and 8 output signal ports (14 bit resolution), 4 input and for 4 output ports for digital triggers or markers, clock reference for synchronization to external instruments, and ethernet connection for data transfer. A specification sheet is found here.

Technical White Paper

Vivace and PrestoIntermodulation Products has developed two newmeasurement platforms for readout and control ofquantum experiments. Based on the latest RFSoCchips from Xilinx, Vivace (fast and lively) andPresto (very, very fast) are named with termsnotating tempo in music, an art form with manyanalogs in quantum technology.

Intermodulation Products firmware gives exceptionaldigital signal processing power through anApplication Programming Interface (API) in Python.Numerous features are available at run-time viaPython scripting, eliminating the complex and time-consuming task of developing and re-compilingFPGA code.

A pulse-sequencing firmware is designed for phase-coherent pulse generation and readout on multipleports. Signal generation works with templates thatare pre-stored in memory. By looping, concatenatingand stretching, waveforms are modified in real-time.Vivace or Presto can perform rapid parametersearches, calibrations, or execute algorithms bystepping through waveforms and pulse shapes.There is no need to generate giga samples of dataon a lab computer and wait for transfer to on-chipmemory.

Signal analysis on-chip performs real-timesummation (averaging), or multiplication with atemplate and summation (template matching) forquantum state discrimination. Sinusodial templatesgive averaged quadrate data. One cansimultaneously match with multiple templates andthe results of template matches are compared withthreshold values. Boolean logic is then used to armor trigger subsequent events in the measurementsequence, thus enabling low-latency feedback andfeed-forward for quantum error correction. Withtemplate matching Vivace and Presto achieve nearconverter-limited latency of 200 ns.

Signal generation and analysis on all channels iscoordinated by a master sequencer with up to 16382events placed on a temporal axis with 2 ns

resolution. 500 ps time resolution is achieved byadjusting samples in a template.

Vivace and Presto also come with a continuous-wave firmware, designed for modulating anddemodulating at multiple frequencies simultaneously.192 frequencies can be distributed to the outputports (8 or 16, depending on the chip used). A keyinnovation is the ability to tune all 192 frequencies,forcing them to be integer multiples of the resolutionbandwidth. This feature completely eliminatesFourier leakage between tones separated by onlyone bandwidth. It also enables lock-in detection atmixing frequencies, to reveal quantum correlationsin the frequency domain [2]. The continuous-wavefirmware is ideal for many different kinds offrequency-domain measurement, with built-inmethods for multifrequency sweeping, for exampleto rapidly find very high-Q resonances.

Glossary of AcronymsADC – Analog to Digital ConverterAPI – Application Programming InterfaceDAC – Digital to Analog ConverterDDC – Digital Down-ConversionDDS – Direct Digital SynthesisDSP – Digital Signal ProcessingFPGA – Field Programmable Gate ArrayGS/s – Giga Samples per secondI and Q – In-phase and QuadratureLO – Local Oscillator NCO – Numerically Controlled OscillatorNZ – Nyquist ZoneRFSoC – Radio Frequency System on a Chip

References[1] Calibration of mixer amplitude and phaseimbalance in superconducting circuits. S.W. Jolin,R. Borgani, M.O. Tholén, D. Forchheimer and D.B.Haviland. Rev. Sci. Instrum. 91, 124707 (2020).

[2] Squeezing and correlations of multiple modes ina parametric acoustic cavity. G. Andersson. S.W.Jolin, M. Scigliuzzo, R. Borgani, M.O.Tholén, D.B.Haviland and P. Delsing. ArXiv:2007.05826v1

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