Multiplexed microwave resonators by frequency comb spectroscopy

Coplanar waveguide resonators are central to the thriving field of circuit quantum electrodynamics. Recently, we have demonstrated the generation of a broadband microwave-frequency comb spectrum using a superconducting quantum interference device (SQUID) driven by a time-dependent magnetic field. Here, the frequency comb is used to spectroscopically probe a bank of coplanar microwave resonators, inductively coupled to a common transmission line, a standard circuit with a variety of applications. We compare the resonator line shape obtained from signals synthesized at room temperature using conventional electronics with the radiation produced in the cryogenic environment by our source, showing substantial equivalence in the estimation of the resonator quality factors. To measure non-uniformly spaced resonant frequencies, we drive the generator with a bi-chromatic tone to generate intermodulation products. Such a dense frequency comb spectrum enables simultaneous addressing of a few resonators via frequency multiplexing. Finally, we discuss the criteria for achieving effective spectroscopic coverage of a given frequency bandwidth.

💡 Research Summary

The paper presents a novel method for multiplexed spectroscopy of coplanar waveguide (CPW) microwave resonators using a cryogenic frequency‑comb generator based on a dc‑SQUID. By threading a time‑varying magnetic flux through the SQUID loop, a pump tone at frequency fₚ produces a train of voltage pulses whose Fourier spectrum consists of harmonics fₙ = n·fₚ (n = 1, 2, …). Because the comb is generated at the 60 mK stage of a dilution refrigerator, the microwave tones are delivered to the resonator chip with minimal loss and heat load, eliminating the need for many room‑temperature coaxial lines.

The experimental platform comprises three parts: (i) a cryogenic RF switch that alternately routes either a conventional room‑temperature vector network analyzer (VNA) signal or the SQUID‑generated comb to the resonator chip; (ii) a target chip containing eight quarter‑wave CPW resonators spanning 4–8 GHz, each inductively coupled to a common 50 Ω feedline; and (iii) the measurement chain (VNA for S₂₁(f) and a spectrum analyzer for the comb amplitudes).

First, the authors compare conventional VNA spectroscopy with frequency‑comb spectroscopy (FCS). For the same average photon number ⟨nₚₕ⟩≈1, the VNA shows a deeper dip (≈ −7 dB) than FCS (≈ −3 dB), a discrepancy attributed to different background‑leakage paths that modify the interference phase ϕ in the standard transmission model S₂₁(f)=1−(Qᵢ e^{−iϕ})/(Qₑ+Qᵢ+2iQₑQᵢ(f−f₀)/f₀). When the resonators are driven strongly (⟨nₚₕ⟩≈1000), both techniques yield virtually identical resonance frequencies and internal/external quality factors (Qᵢ, Qₑ), confirming that FCS can reliably extract resonator parameters even in the many‑photon regime.

A limitation of a single‑tone comb is that its harmonics are integer multiples of fₚ, making it difficult to address non‑uniformly spaced resonances. To overcome this, the authors introduce a bichromatic pump consisting of two close frequencies fₚ₁ and fₚ₂. Non‑linear mixing in the SQUID generates intermodulation products at frequencies fₙ,ₘ = n·fₚ₁ + m·fₚ₂ (n,m∈ℤ), spaced by the beating frequency |fₚ₁−fₚ₂|. In the experiment fₚ₁ = 454 MHz and fₚ₂ = 455 MHz, producing a dense comb with 1 MHz spacing. This enables simultaneous probing of resonators whose frequencies differ by as little as 1 MHz (e.g., 4.000 GHz and 4.001 GHz), which would otherwise require the 4000th and 4001st harmonics of a 1 MHz pump—an impractical requirement. The intermodulation lines also redistribute power, allowing a broader spectral coverage without increasing the pump power beyond the typical −130 dBm to −170 dBm range used in circuit QED experiments.

Finally, the authors discuss design criteria for efficient bandwidth coverage. For a target bandwidth Δf, choosing a small beat frequency Δfₚ = |fₚ₁−fₚ₂| yields a required number of lines N ≈ Δf/Δfₚ. By appropriately selecting (fₚ₁, fₚ₂) and the pump amplitudes, one can generate a comb that contains all resonator frequencies with minimal redundancy, which is especially valuable for large sensor arrays or multi‑qubit readout where wiring constraints are critical.

In summary, the work demonstrates that a low‑power, cryogenic SQUID‑based frequency comb can replace conventional room‑temperature microwave sources for resonator spectroscopy, delivering comparable accuracy in quality‑factor extraction while drastically reducing the number of required control lines. The introduction of bichromatic pumping and the resulting intermodulation spectrum provide a practical route to densely multiplexed readout of non‑uniformly spaced microwave resonators, opening new possibilities for scalable quantum‑hardware architectures and broadband microwave sensing.