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Abstract
The recent advent of quantum computing has the potential to overhaul security, communications, and scientific modeling. Superconducting qubits are a leading platform that is advancing noise-tolerant intermediate-scale quantum processors. The implementation requires scaling to large numbers of superconducting qubits, circuit depths, and gate speeds, wherein high-purity RF signal generation and effective cabling transport are desirable. Fiber photonic-enhanced RF signal generation has demonstrated the principle of addressing both signal generation and transport requirements, supporting intermediate qubit numbers and robust packaging efforts; however, fiber-based approaches to RF signal distribution are often bounded by their phase instability. Here, we present a silicon photonic integrated circuit-based version of a photonic-enhanced RF signal generator that demonstrates the requisite stability, as well as a path towards the necessary signal fidelity.
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1. Introduction
Stable and programmable RF tone generation is a cornerstone component of quantum computing architectures in order to apply qubit gate operations and transformations to perform computation, evolution, and non-demolition dispersive readouts. Traditionally, cryogenic qubits are driven using constructed RF pulses with optimized temporal-spectral characteristics and sequences designed to send the system into a superposition of states representing the information to be manipulated. These pulses are usually generated at room-temperature via direct synthesis with an arbitrary waveform generator (AWG) or in an architecture requiring fast electronic DACs mixed with a voltage-controlled oscillator (VCO) in an IQ configuration, cleaned by a bandpass filter and sent to the cryogenic chamber [1,2,3]. However, transmission of blackbody radiation down the RF lines to lower temperature stages of the chamber act as a noise source to the system and can disturb the quantum state, reducing the qubit coherence time. Often, attenuation is added to the RF lines to mitigate the impact of these thermal photons, reducing signal strength and therefore the speed of quantum operations [4–7]. Furthermore, these RF lines present heat load to every stage in the dilution refrigerator via thermal conduction and RF attenuation/filtering. In contrast, photonic methods use fiber optic cables which reduce transmission noise and provide higher bandwidth, lower phase noise, faster operation, and reduced heat load [8–15]. To leverage these advantages, we develop a photonic dispersive comb-to-tone converter (DCTC) for quantum control. A voltage-tunable optical filter (VTF) is designed which encodes RF signal information onto a carrier pulse train. The VTF additionally can be driven using slower electronic DACs and a low-jitter pulsed laser, relaxing requirements on the driving equipment compared to the electronic methods, while eliminating the VCO and broadening the tunability range of the output tone [16–19]. We first demonstrate single-tone generation for quantum control via proper control of the VTF, obtaining a spurious-free dynamic range (SFDR) of 31-dB. We show the importance of proper DCTC calibration, wherein a SFDR improvement of 14-dB is achieved. Next, after optimization of the generated tone, we generate the pulses needed to drive qubit Rabi and Ramsey oscillations as a proof-of-principle for device operation. Lastly, we measure the DCTC operation continuously and track the stability of the system over time. We find the SFDR to be stable with a 1.71-dB standard deviation, a residual frequency instability of 143.9-Hz standard deviation, and phase drift and precision instability of 0.003-rad and 0.0013-rad standard deviation respectively.
2. System architecture
Figure 1(a) shows the conventional system-level architecture used in RF quantum control generation. In this scheme, fast electronic DACs are mixed in an IQ configuration with a voltage-controlled oscillator and then recombined to create the desired waveform, subsequently sent to the cryogenic fridge. Transport through RF lines in the cryogenic chamber causes additional noise and can degrade the qubit coherence [4,6]. Figure 1(b) shows our proposed architecture: instead of fast electronic DACs, we utilize slower DACs and a low-jitter pulsed laser with a VTF to alleviate some of the noise requirements on the driving instruments. The VTF imprints characteristics of the desired RF waveform on the spectrum of the optical pulse train. This then is passed through the chromatic dispersion element to map from wavelength to time, creating the RF signal carried by the dispersed optical pulse train [14,17–19]. This signal is then photodetected at the cryogenic quantum system to extract the RF tone, only inducing heat load at the low-power photodetector. Due to the ability to integrate nearly all components at a chip-scale, the system can be scaled up to a large number of channels with optical fibers (Fig. 1(c)). The architecture is readily parallelizable to drive multiple qubits simultaneously.
Fig. 1. RF photonic drive for cryogenic systems. a, Conventional RF mixing methodology for quantum computing driver systems. One common method involves two electronic digital-to-analog converters (eDACs) driven in-phase and quadrature, and mixed to create driving RF pulses, which are then sent through a band-pass filter (BPF) to isolate only the tone of interest. This is sent into the cryogenic chamber via RF cable transport to the quantum computational system (QC), with noise introduced via the cable transport line. b, Photonic carrier methodology for quantum computing driver systems. Two slow electronic DACs and a low-jitter pulsed laser are passed to a voltage-tunable optical filter which encodes the RF onto the optical carrier. The pulse passes through chromatic dispersion element and is fiber-optically transported to the cryogenic chamber. At the quantum system, the pulse is demodulated at the photodetectors (PD), with the resulting RF filtered and sent to the quantum system. c, Concept of the dispersive-comb-to-tone converter when used to drive superconducting qubits. Due to the reduced SWAP of the system, the DCTC can be parallelized, with capabilities to drive a larger number of qubits for noise-tolerant intermediate-scale quantum computing.
The photonic integrated circuit (PIC) architecture implementing the VTF is fabricated at Advanced Micro Foundry (AMF) using the commercial silicon-on-insulator multi-project wafer platform and shown in Fig. 2(a). The VTF is a cascaded pair of Mach-Zehnder modulators (MZM) which acts as the optical encoder of the system. The first MZM is balanced which performs the amplitude modulation to assist with turning on and off the pulse train. The second MZM has an unbalanced path creating a sinusoidal interference pattern and phase modulation in this MZM leads to a phase shift of this profile, allowing for fine control of the waveform phase. Each MZM consists of an RF phase shifter from the AMF process development kit with 6-mm length [20] along with thermo-optic phase shifters to adjust the modulator bias points. After the waveform is encoded by the VTF, the optical pulse train is passed through a dispersive element which maps wavelength to time and subsequently photodetected to generate the desired RF waveform [14] with fast on/off capability for pulsed RF. The test configuration is shown in Fig. 2(b). The VTF receives optical input from a pre-amplified 40-MHz pulsed laser source before being coupled into the PIC. DC biases and electrical controls are driven by a 16-channel DAC (Texas Instruments DAC81416EVM) while RF signals are generated from a 12GSa/s AWG (Tektronix AWG7122C). Due to the low Vπ of the modulator under forward bias operating conditions, additional attenuation is added to the RF line to avoid overdriving the modulator and to utilize all 10 bits of the AWG. Once the system is optically aligned and the DC bias points of the modulators and tunable splitters are optimized and set, RF signals are sent to the modulators for waveform generation. The output of the PIC is collected and amplified to improve the signal level which then is passed through a 120-m -16ps/nm DCF which maps the envelope into time via chromatic dispersion [14,16,17].
Fig. 2. Detailed diagram of photonic integrated circuit, implementing the voltage-tunable optical filter. a, A combination of electro-optic phase shifters (EOPS) and thermo-optic phase shifters (TOPS) are used to modulate the optical pulse to generate clean signal tones. Heated delay lines (HDL) are used for fine-balancing the upper and lower arms of the VTF. MMI: multimode interferometer, EC: edge coupler. b, System-level diagram of DCTC. Optical, electrical and RF pathways are indicated in blue, orange, and green respectively. A 40-MHz laser source is sent to the PIC after EDFA amplification. In parallel, a 16-channel DAC drives the various DC connections of the PIC while the RF is generated by a 10-GSa/s AWG. After modulation of the optical pulse, the output of the PIC is amplified by an EDFA, passed through a dispersive fiber and detected by a fast photodetector. After RF amplification of the signal to help overcome the scope noise floor due to excess optical loss, the resulting RF waveform is captured on an oscilloscope for processing. A digital bandpass filter is implemented for proof-of-concept. c, Image of DCTC mounted to custom-designed PCB with fibers coupled to the input and output. DC and RF traces on the PIC are wirebonded to appropriate pads for driving the system. Inset: image of full PCB. d, Spectral transmission data of PIC. A fit to the FSR is shown in orange. e, Corresponding transmission spectrum, in linear units. The sinusoidal fit is shown in orange. f, RF symbol radius from constellation center as a function of voltage. g, DC-offset of the photodetected output as a function of voltage. h, Phase shift as a function of voltage. i, Vπ as a function of bias voltage. Vπ is measured to be less than 50-mV.
Photodetection of this signal demodulates and extracts the RF signal from the optical carrier which is subsequently amplified and sent to an oscilloscope averaging over 100 traces to improve SNR for data collection and analysis. We note that the RF signal is generated completely optically and only requires photodetection at the endpoint, eliminating long RF cables within the cryogenic chamber [17,19]. Figure 2(c) shows an image of the PIC test setup. The PIC is epoxied to a custom-designed PCB for routing of the RF and DC voltages. Wirebonds and cleaved optical fibers are used to drive electrical and optical signals to the chip. The transmission of VTF can be described as ${T_{VTF}}({V,\lambda } )= \frac{1}{2}\left( {DC(V )+ R(V )\cos \left( {\frac{{2\pi }}{{FSR}}\lambda + \theta (V )} \right)} \right)$. Figure 2(d) and 2(e) shows spectral transmission data of such a VTF on chip. A sinefit to the spectrum as a function of wavelength λ with period given by the interferometer free- spectral-range (FSR) is used to extract the RF symbol’s radius from the constellation center $R(V )$ (Fig. 2(f)), the DC-offset of the photodetected output $DC(V )$ (Fig. 2 g), and the phase shift $\theta (V )$ (Fig. 2 h), each as a function of applied voltage V to the RF phase shifter. Specifically, the phase shift is found to be relatively linear as a function of bias voltage. Additionally, a DC component of 0.5 indicates single-arm modulation and we find the strength of the generated sine wave falls off quickly after 0.8. The phase shift allows extraction of Vπ as a function of voltage as shown in Fig. 2(i), illustrating a sub-50-mV Vπ at 1-V forward bias. Due to the high linearity of the phase, acceptable loss, and low Vπ, we choose this to be our bias voltage operation point.
3. RF comb-to-tone generation
We drive the system to generate a RF tone at 3.712-GHz as a demonstration of our subsystem viability with superconducting qubits, which typically operate in the 3-5-GHz band and may not naturally align with a multiple of the frequency generator. The driving waveforms are configured to create a five-point phase-constellation such that all generated points have the same radius and exist around the origin of an IQ plot in 2π/n radian steps, where n is the desired number of points in the phase constellation. The generated RF signal is shown in Fig. 3(a). The pulse train has a clear set of interference fringes generated by the VTF. Due to the high insertion loss of the forward-biased modulators, the amplitude of the pulse and resultant RF tone is attenuated significantly – a shorter modulator design would improve this loss to more tenable levels. The digital bandpass filter extracts the clean RF tone and a sine wave is found to fit well to the data. Non-idealities such as amplitude-to-phase conversion of the modulators and nonlinearities from the forward bias operating point serve to degrade the radial and angular components of the constellation and should be compensated to achieve an optimized tone. Figure 3(b) shows an IQ plot of a constellation using an uncalibrated driving waveform. All measured constellation points have both amplitude and phase variations from their ideal counterparts, which results in a heavily distorted signal. In this uncalibrated case, angular error dominates. Figure 3(c) shows the corresponding electrical spectrum containing the tone of interest, the laser grid line of 40-MHz and any other harmonics and spurs over one Nyquist zone. With such an uncalibrated system, the SFDR is measured to be 17-dB.
Fig. 3. RF comb-to-tone generation. a, Individual pulse (blue) and extracted sinusoid (orange). The digital bandpass filter allows examination of both the modulated pulse train and resultant sinusoidal signal. b, Circle plot of generated RF tone before optimization. Ideal points are plotted in blue and measured are plotted in red. Large phase error before optimization result in a distorted RF signal. c, Electrical spectrum of the generated RF before optimization. A SFDR of 17.2-dB is measured, indicating poor optimization of the slow eDAC signals. Spurs are shown in red and the tone of interest is in blue. d, Post-calibration circle plot of generated RF tone. Theoretical points are plotted in blue and measured are plotted in red Very minor phase and amplitude error appears between measured and theoretical points, improving the generated tone purity. e, Post-calibration electrical spectrum of the generated RF. An SFDR of 31-dB is measured. Spurs are shown in red and the tone of interest is in blue. f, Temporal trace of off-on-off signal for quantum control. The duration of the on-pulse is 12-µs. Inset: Zoom-in of the trailing edge of the on-pulse. g, Temporal trace of off-on-off-on-off signal for quantum control. The duration of the on pulses are 300-ns and the center off pulse is 10.5-µs. Inset: Zoom-in of single 300-ns pulse.
Our system subsequently utilizes a synchronization circuit to reference measurement and signal generation equipment to the pulse train, allowing for a simpler iterative methodology of relative timings and driving signals for tone calibration. Figure 3(d) shows an IQ plot of the generated constellation points using such a calibrated system and driving signals using this circuit. All constellation points are calibrated to optimal locations in phase and radius with angular error significantly reduced. The corresponding RF spectrum is provided in Fig. 3(e), resulting in a SFDR of 31-dB and significantly reduced spurs. Next, we generate a series of example RF pulses as a demonstration of quantum control signals – specifically the Rabi drive tone and Ramsey oscillation signals that would drive a potential superconducting qubit in Figs. 3(f) and 3(g), as a proof-of-principle operation. Figure 3(f) has an on-time of 12-µs and Fig. 3 g has an on-time of 300-ns and an off-time of 10.5-µs between the RF pulses. Pulses are able to be quickly turned on and off such that we obtain sharp edges at both the long and short on-pulses. Therefore, the system is demonstrated to operate effectively for tone generation and pulse forming towards quantum control.
4. Long-term stability
Instability in the phase, frequency, amplitude, SDR and SFDR of the generated RF tone significantly reduce the efficacy of the comb-to-tone approach in fiber-based systems [16,19]. We examine the stability of the chip-scale interferometer and operate the system continuously to track the system stability on a short- and long-timescales. Figure 4(a) shows the measured phase over time. We find a standard deviation of 0.003-rad, with a stability within 0.03-rad over 36 hours. In comparison, the fiber version of the same architecture has an instability of 2π over one hour [16], with our chip-scale version significantly improving the stability by over three orders-of-magnitude. Short-term phase over 1, 2, and 4 hour windows have average standard deviations of 0.0013, 0.0016, and 0.0019-rad indicating long-term drift dominating the phase error rather than acquisition-to-acquisition error. Figure 4(b) shows the modified Allan deviation of the phase drift, achieving an instability at the 3 × 10−7 levels in the 10,000-sec time scales. Linear fits to various sections of the modified Allan deviation are shown, with slopes of -1.2, -0.01, and -2.03 with respective time scales up to 5,000-sec, 20,000-sec and above 30,000-sec. These are analogous to flicker phase noise, flicker frequency noise, and white phase noise in canonical RF source characterization studies. Current qubit phase coherence time (T2, on the order of ms) is shorter than the timescales measured here; this drift becomes important in optical systems with thousands of qubits, where uncorrelated drifts between drive lines would compound the error. Additionally, the improvement by a factor of almost 105 in phase stability in the realized PIC compared with a fiber-based implementation extends the applicability of this approach beyond current quantum hardware. Figure 4(c) shows the measured SFDR over time, with time bins of 1,2 and 4-hours. We find an average SFDR of 31-dB, consistent over all three time bins, and stable for the entire 35-hour measurement time. The low drift of the SFDR indicates that the chip-scale VTF is indeed stable. Figure 4(d) shows the corresponding amplitude fractional deviation of the generated RF tone over time. We measure a residual frequency instability of 143.9-Hz standard deviation, which would not significantly decrease effective decoherence time of current superconducting qubit hardware.
Fig. 4. Long-term stability of the integrated silicon chip RF comb-to-tone drive. a, Phase drift over time. The standard deviation is measured to be 0.003-radians. b, Modified Allan deviation of the phase. Linear fit slopes of -0.01, -1.2, and -2.03 are calculated for subsections and are consistent with power log models for RF sources. c, SFDR versus time. An average SFDR of 31-dB is measured, with a standard deviation of 1.71-dB. The maximum measured SFDR is 35.8-dB. d, Amplitude fractional deviation of sine versus time. Minimal fluctuation is observed in the amplitude of the generated tone. Inset: Frequency fractional deviation versus time.
5. Discussion
In this study we created a chip-scale dispersive comb-to-tone converter towards quantum system control. We generated a 3.712-GHz tone with a measured SFDR of 31-dB and demonstrated long-term stability on the order of several days in the amplitude, frequency, phase, and SFDR of the resultant tone, significantly surpassing the prior fiber-based work. Currently the SNR is limited by the high losses of the modulators and other on-chip components, and higher SNR is anticipated to enable better calibration for higher SFDR. For our system, driving superconducting qubits requires no less than -65-dBm of power at the 4 K stage of the dilution refrigerator. While the reported RF power here of -78 dBm is insufficient, several changes to the DCTC architecture and driving hardware provide an avenue to reach the required power levels. Decreasing the modulator length from 6-mm to 2-mm will provide another ≈10-dB improvement, due to the low-Vπ in forward bias. Additionally, several on-chip components for testing and individual metrology purposes add ≈ 10-dB additional loss and can be removed for more streamlined future versions. Increasing the repetition rate to 2 GHz fills in the gap between subsequent comb-to-tone pulses (100% duty cycle), creating an equivalent CW optical signal for the purpose of RF power, thereby removing an RF hit of 27-dB from the low duty cycle pulse train. Furthermore, utilizing index matching fluid can improve the cleaved-fiber coupling by 7-dB. These improvements would result in ≈ 54 dB RF power gain, providing margin to remove the two post-photodetector 20 dB RF amplifiers. The net RF power gain of 34-dB increases the generated RF power to -44 dBm, well above the driving requirement of -65 dBm.
Although a complete heating and noise analysis is beyond the scope of this paper, one can consider the heat load and shot-noise limited SNR. As discussed in the previous paragraph, the comb-to-tone generator can be designed to achieve no gap in between subsequent comb-to-tone pulses (e.g. by increasing laser repetition rate), in which case the generated microwave power matches the CW case. If one further assumes 100% RF carrier modulation depth and no internal matching resistor, the generated microwave power is:
Our current laser repetition rate of 40-MHz was chosen based on the availability of pulsed lasers and by driving the DCTC with a higher repetition rate laser, higher throughput can be achieved up to the bandwidth cutoff of the wirebonds. The modulator length can be further optimized for more efficient use of the laser power and chip size, reducing the loss of the device significantly while also reducing the overall footprint of the design. For tone generation, the phase modulation component is essential to have a low Vπ as constellation points must be able to make the full 2π-radian rotation, but the amplitude modulation portion of the VTF can be operated with a much shorter modulator in reverse bias with only small adjustments to each constellation point’s radius.
Furthermore, it is envisioned that this technology can scale to the large number of RF channels required by the current small-to-intermediate-scale cryogenic quantum processors. This would be done by integrating the rest of the DCTC onto a PIC (except the transport fiber and photodetector). The pulsed laser can be replaced by a semiconductor mode-locked laser [22] or soliton frequency combs initiated from cascaded four-wave mixing and modulation instability in high-quality factor nonlinear microresonators [23–30]. Additionally, on-chip dispersive elements such as periodic Bragg structures and crystals [31–33] can be designed and optimized for our specific dispersion profile needs and can be integrated into this PIC design.
Funding
Lawrence Livermore National Laboratory (21-ERD-033); U.S. Department of Energy (DE-AC52-07NA27344).
Acknowledgments
The authors acknowledge assistance from Kevin Chaves and the rest of the Quantum Coherence Device Physics Group at Lawrence Livermore National Laboratory for their guidance on superconducting qubit RF driving requirements. The authors also acknowledge discussions with Cody S. Fan, Murat Sarihan, Madeline Taylor, Kerry Kangdi Yu, Hangbo Yang, and Alwaleed Aldhafeeri.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request in the future.
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