Single-sideband modulator for frequency domain multiplexing of superconducting qubit readout Free

Recent advances have allowed many groups to demonstrate superb control over one or several superconducting qubits. This is seen, for example, in the creation and transmission of Fock and cat state superpositions^1,2 or the ability to continuously tune both the strength³ and axis^4,5 of a quantum non-demolition⁶ measurement. Efforts to combine multiple superconducting qubits into a (more coherent) logical qubit are progressing in parallel.^7,8 As algorithms on multiple logical qubits become experimentally realizable, the challenge of scaling readout for many-qubit systems grows in relevance.

Multiplexing, a signal processing technique in which a transmission medium is shared among multiple signals, is a natural way to combat such a scaling challenge. In this letter, we introduce a single-sideband modulator (SSBM) for cryogenic microwave applications and propose its use for frequency domain multiplexing (FDM) of superconducting qubit readout. We describe its design, theory of operation, and experimental performance.

In a circuit quantum electrodynamics architecture, qubits are readout by measuring the transmission of a cavity that has a resonance frequency that is modified (dressed) by a qubit.⁹ Specifically, the cavity resonance is dressed by an amount that depends on the qubit state. To date, multi-qubit experiments have scaled-up this readout protocol directly¹⁰ or with frequency domain multiplexing (FDM) hardwired into the measurement architecture.^7,11,12 With direct “brute-force” scaling, each qubit channel has a dedicated input and output line. Conversely, qubits in an FDM architecture can share a single input and output line, and their readout cavities are designed in advance to ensure that their dressed resonances are separated by multiple cavity linewidths.

As the number of readout channels in a multi-qubit measurement rises, direct scaling becomes untenable. Recognition of this fact has spurred new proposals for scalable architectures,^13,14 as well as a slew of devices for in-fridge signal processing, from non-reciprocal elements^15–20 to mixers^21,22 and switches.^14,23,24

The “hard-wired” FDM approach achieves scalability at the cost of flexibility. The spectral allotment for each channel is determined prior to fabrication and, unless the readout resonators are designed to be tunable,²⁵ remains fixed through the measurement process. This constraint inhibits efficient use of the available bandwidth, as densely packing the spectra allotted for each channel makes nearby channels sensitive to deviations in the design parameters. It also precludes the use of identical qubit-cavities, which may be desirable in quantum simulation applications. Finally, the “hard-wired” architecture limits the degree to which a many-qubit experiment can be reconfigured without fabrication of new devices.

There is thus a need for flexible multiplexed architectures that efficiently utilize the available measurement bandwidth. To this end, we introduce a microwave component for cryogenic analog signal processing, which could enable flexible FDM in a many-qubit measurement. This component is a single-sideband modulator (SSBM), a two-port device that converts incident signals up or down in frequency. Unlike commercial SSBMs which utilize resistive diodes, in our device the nonlinear elements are purely reactive. For such systems, the Manley-Rowe relations permit frequency-conversion without dissipation.²⁶ Moreover, and in contrast to other cryogenic microwave frequency converters,^19,27,28 the device (a) utilizes no resonant physics, endowing it with GHz of bandwidth, and (b) is controlled by RF signals with frequencies no greater than several hundred MHz, obviating the need for high-bandwidth GHz control lines. Finally, the device can be realized as a monolithic integrated circuit in a NbAlO_xNb tri-layer process,^29,30 allowing for high-yield wafer-scale production.

A proposed use for the device is shown in Fig. 1. By positioning a SSBM at the output of the readout resonator's strongly coupled port in each measurement channel, a single microwave line can be used to readout all the qubits simultaneously in an FDM scheme. Measurement bandwidth can be dynamically allocated among channels by converting each transmitted tone to its assigned band. This ability to assign spectrum in-situ facilitates efficient use of the available measurement bandwidth, as there is no risk of spectral overlap from uncertainty in fabrication parameters. Furthermore, it allows for the use of FDM even if the dressed cavity frequencies are spectrally irresolvable. Although other multiplexing proposals for time and code-domain schemes¹⁴ share these benefits, the flexible FDM architecture depicted in Fig. 1 distinguishes itself by requiring no timing coordination between channels.

The SSBM is created from a Hartley-type³¹ in-phase–quadrature (IQ) modulator with I and Q ports driven 90° out of phase. This IQ modulator is itself built from a pair of double-balanced modulators (Fig. 2(a)) realized with inductive bridge circuits (Fig. 2(b)), which we call Tunable Inductor Bridges (TIBs).¹⁴ Two pairs of tunable inductors form the bridge, with inductances l₁ and l₂. Signals couple differentially to the input port (left and right bridge nodes) and output port (top and bottom bridge nodes). Transmission between these ports is controlled by the imbalance in the bridge $l1–l2$⁠.

Manipulation of that imbalance is accomplished by arranging the two pairs of flux-tunable inductors in a figure-eight geometry. Their coordinated tuning is controlled with an off-chip magnetic coil and an on-chip bias line that bisects the figure eight (Fig. 2(c)).¹⁷ Series arrays of Superconducting Quantum Interference Devices (SQUIDs) are used to realize the flux-tunable inductors. Arrays are employed in place of individual SQUIDs to dilute the Josephson nonlinearity.¹⁷ 

To operate the TIB as a double-balanced modulator, we use the flux-control bias line to sinusoidally modulate the transmission through the bridge at angular frequency Ω. Borrowing language from the field of nonlinear-microwave elements, we denote the two microwave ports of the TIB as the local oscillator (L) and radio frequency (R) ports of the modulator, and the flux-control line as the intermediate frequency (IF) port. However, as a TIB has no galvanic connections between its IF and L port (or its IF and R port), we refer to it as a modulator rather than a mixer. Note also that in contrast to a common way of operating mixers by driving their nonlinear elements with strong signals in their L ports, the IF port of the TIB actuates the modulation process and signals at the L and R ports can be arbitrarily small.

We create an IQ modulator by dividing the power of an input tone into two double-balanced modulators and then summing their outputs with a 90° hybrid coupler (Fig. 2(d)). In the device reported in this letter, both double-balanced modulators are integrated on a single chip and connected to a commercial power splitter and 90° hybrid. Future versions of the SSBM, however, could also integrate the passive components on-chip.^24,32

Fig. 2(e) provides a phasor representation of a microwave tone as it propagates through the SSBM. This tone is first split into two arms with equal phase shift, each connected to the L port of a double-balanced modulator. Control currents are applied to the IF ports of both modulators, modulating the signal in each arm into two tones spectrally shifted from the original by $±Ω$ (blue/red sidebands, respectively). The phase of the control current in the lower arm, θ, is advanced $π/2$ radians with respect to that in the upper. Recombination at the 90° hybrid causes the blue (red) sidebands in each arm to constructively (destructively) interfere at the device's output. The input tone is thus converted up in frequency by Ω. Generally, the amplitude of the red and blue sidebands scales as $cos ((π/2±θ)/2)$⁠, allowing for selection of the red sideband when $θ=−π/2$ radians. We denote the IF port in the top arm as the I port of the IQ modulator and the IF port of the double-balanced modulator in the bottom arm as the Q port.

This angular dependence of sideband power is depicted in Fig. 3, which shows the transmitted power at various frequencies as a function of the IQ phase difference θ. For this measurement, the SSBM is driven by a microwave tone at $ω=2π×4$ GHz through its L port while we modulate the transmission across the bridges at $Ω=2π×3$ MHz. As this modulation is not purely comprised of a single spectral component at angular frequency Ω, higher harmonics of Ω at frequencies $ω±nΩ$⁠, with n an integer greater than 1, are also observed. We monitored the nearest 16 higher harmonics (⁠$2≤n≤9$⁠) and found the n = 5 harmonic to be the largest at this particular choice of ω and Ω. The plot shows the output of the SSBM at the input frequency (green), along with the first upper (blue) and lower (red) sidebands, and the largest of the higher harmonics (n = 5, dashed red and blue). More harmonics and phase sweeps at different IQ modulation frequencies are shown in Fig. S3 of the supplementary material. As θ is varied between $−π$ and π, we observe that all measured harmonics oscillate with θ at a rate commensurate with their order.

Two clear operation points are visible at $θ=±π/2$ radians, where the first blue (red) sideband is suppressed in favor of the first red (blue) sideband. The contrast between them exceeds 30 dB at these points, and the contrast between the desired sideband and the nearest harmonic (the 5th, in this case), which we call the sideband contrast, is approximately 23 dB.

We set the scale of the y-axis in units of dB relative to an unmodulated input tone, which we call modulation gain. This is done by statically biasing both TIBs to their state of maximal imbalance. We normalize the sideband power in this way to reveal inefficiencies in the modulation process and separate them from the intrinsic insertion loss of the SSBM's constituent components. In Fig. 3, the modulation gain in the desired sideband never exceeds −3 dB (dashed gray line) because the TIB in each arm of the interferometer is biased to modulate transmission sinusoidally. The device is realized as a distributed network, which means each arm is reflecting half its incident power on average—power which is ultimately dissipated in the power splitter or directed out the device's input port (return loss).

To characterize the device over a broader range of operation frequencies, sideband contrast for a variety of L and I frequencies is plotted in Fig. 4(a). In the proposed application of the SSBM for FDM of qubit readout, this is an important measure of device performance: when the measurement spectrum is densely filled, spurious sidebands are a source of cross-talk among channels.

In addition to sideband contrast, other important specifications for a SSBM are the modulation gain, L, R, and I frequency limits, instantaneous bandwidth, linearity, and insertion losses. Modulation gain is plotted in Fig. 4(b) as a function of Ω, for tones at several different input frequencies. Two major trends are visible: first, modulation gain decreases as Ω increases. This effect is a consequence of the bandwidth requirement of the pulse-shaping scheme used to sinusoidally modulate transmission through the TIBs, which begins to exceed our control hardware's bandwidth as Ω increases. The second trend is a variation in the modulation gain achieved at small modulation frequencies as the probe frequency ω changes. This is caused by a change in the phase of a TIB's transmission as its magnitude is modulated. This effect is evident in Fig. S2 of the supplementary material and may be partially alleviated in a future device by removal of a chip-mode at 5 GHz.

Figs. 4(a) and 4(b) also show the ranges of L, R, and I frequencies that can be processed by the SSBM. The device converts frequencies from/into the 4 to 8 GHz band, shifting them by as much as 120 MHz to the red or blue. For an FDM application in superconducting qubit readout, the quantity with which to compare this is the coupling strength of a readout cavity's strongly coupled port κ. Allowing 3 κ of spectrum per readout channel, and taking a representative value of several MHz for $κ/2π$⁠,^4,7,8,33,34 such an I range allows for 10 to 40 (or more³⁵) independent readout channels per input line.

Conversion of amplitude-modulated tones with the SSBM and measurements of its linearity are shown in supplementary material Figs. S4 and S5. The instantaneous bandwidth of the device exceeds 5 MHz, and its 1 dB compression point is −85 dBm, providing sufficient bandwidth and power-handling for dispersive qubit readout.

We now discuss the insertion loss of our SSBM. Although the dissipation of the TIBs is less than 0.5 dB,¹⁴ our SSBM dissipates 3 dB of the input power (the entire image sideband) in the fourth port of the 90° hybrid at the device's output. As noted previously, an additional 3 dB (or more) of power is dissipated/returned by our distributed network when the TIBs are operated as modulators—this is precisely the modulation gain plotted in Fig. 4(b). Finally, the TIBs themselves are not perfectly transmitting over the entire range of operating frequencies. To quantify this, unnormalized transmission in the 4–8 GHz band is plotted in Fig. S2 of the supplementary material, with independent measurements on both TIBs. Although this transmission approaches unity at some frequencies, it is degraded elsewhere by the presence of a chip mode around 5 GHz and impedance mismatches outside the 5–7 GHz band.

A future integrated design for the SSBM could improve on these specifications: removal of the chip mode and matching the circuit impedance across the 4–8 GHz band are well-posed problems in microwave engineering. An on-chip implementation would improve sideband contrast by suppressing path length differences in the network below a wavelength and shrink device footprint by two orders of magnitude. An additional 3 dB of conversion gain can be recovered by separating the common and differential output modes of the network's two arms and terminating the common mode in an open circuit—for example, with a balun. This eliminates the power dissipated in the image sideband. A next-generation device could thus operate over a several GHz frequency range with insertion loss approaching −3 dB. Further considerations for a future device are provided in the supplementary material.

A single-sideband modulator constructed from the repeated instancing of broadband modulation elements like TIBs is thus an appealing, general purpose way to engineer frequency conversion. The L and R bandwidths allow for conversion of tones in the 4–8 GHz band, and the device's 1 dB compression point at −85 dBm far exceeds the power typically³⁶ used for dispersive readout. In addition, low return, insertion, and modulation losses are achievable in future design iterations. Finally, the SSBM's nonlinear elements require no GHz frequency control lines and are actuated solely with radio frequency signals (several hundred MHz or less). It is therefore suitable for construction of the scalable and flexible multiplexed architectures needed in future many-qubit experiments with superconducting circuits.

See supplementary material for details on the experimental setup, bias-waveform pulse shaping, additional measurements, and future design considerations.

This work was supported by the ARO under Contract No. W911NF-14-1-0079 and the National Science Foundation under Grant No. 1125844. The authors thank Bradley A. Moores and Andrew P. Higginbotham for helpful discussions.