Multiplexed microwave resonators by frequency comb spectroscopy

TL;DR

This work demonstrates a cryogenic, SQUID-driven microwave frequency comb that can spectroscopically characterize a bank of coplanar waveguide resonators in the 4–8 GHz range. By comparing frequency comb spectroscopy (FCS) with conventional VNA measurements, the authors show that FCS yields comparable internal and external quality factors, while enabling significantly different input conditions and potential multiplexing. The introduction of bi-chromatic pumping produces intermodulation products that substantially densify the comb spectrum, enabling simultaneous addressing of multiple resonators and enabling frequency multiplexing. A theoretical framework for optimizing comb spectra via two-tone pumping is developed, including a density-of-states analysis to guide pump-frequency choices and bandwidth coverage, with practical considerations for phase coherence and future improvements.

Abstract

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.

Figures (9)

• Figure 1: (a) Simplified diagram of the experimental setup. A pump tone at frequency $f_p$ modulates the flux threading a SQUID loop. The generated frequency comb is sent to the target chip, which hosts multiple resonators, via a electromechanical switch (orange path). The resonators can alternatively be measured using a VNA (blue path). (b) Comb generator. Scanning electron micrograph showing the dc SQUID and flux line. Dark gray indicates the superconducting Al, while the exposed areas of the substrate are light gray. The two red circles mark the Josephson junctions. The device produces microwave signals (orange arrows) whose frequencies are multiple integers of the pump tone (black arrow) applied via the flux line. (c) Target chip. Optical micrograph of the resonator chip, with the meanders inductively coupled to a common feedline. (d) Comb power spectrum. Harmonics in the 4-8 GHz band generated by a pump tone of frequency 496.55 MHz and amplitude 150 mV at generator level. The reported power is measured by the spectrum analyzer at room temperature after a few stages of amplification. For details, see the complete circuit setup in Fig. \ref{['fig:complete_setup']}.
• Figure 2: Amplitude response of two exemplary resonators, A and E. Comparison of the frequency spectroscopy of resonator A obtained by standard input-output measurements by a VNA (light blue dots) or by FCS (orange dots) in the few-photon regime (a) and many-photon regime (c). The top x axes report the values of the pump frequency we swept in the FCS measurements, and the bottom x axes show the actual frequencies of the resonators spectra. The input power is calibrated to be the same for spectroscopy using the VNA and the frequency comb. (b), (d) Same plots but for resonator E. For FCS of resonator A we use the 9th harmonic, while for resonator E the 13th. The table at the bottom reports the fit parameters extracted from VNA and FCS data from panels (a) to (d). The unit of $\phi$ is rad.
• Figure 3: Spectrum of a frequency comb originated by bi-chromatic pumping with $f_{p1}=454\,$MHz and $f_{p2}=455\,$MHz with amplitude 200 mV at generator level. When one of the two tones is switched off, only the corresponding harmonics are present (blue and yellow spectra). When both tones are applied, the plot shows the intermodulation products (green data) generated around the midpoint of the 11th harmonics of the two pump tones. The subscripts indicate the pair of coefficients $n,m$ according to Eq. \ref{['eq:intermodulation_products']}.
• Figure 4: Three resonators spectroscopy by frequency multiplexing. The pump frequencies are obtained from the system \ref{['sist:multiplexing']}: $f_{p1}=404.652465\,$MHz, $f_{p2}=541.065788\,$MHz. Only $f_{p1}$ is swept; see the top x-axis. $f_{p2}$ is idle. The coefficients to address resonators A, B and C (panels (a), (b), (c) respectively) are $(n_A,m_A) = (-5, 12)$, $(n_B,m_B) = (-3, 11)$ and $(n_C,m_C) = (12, 1)$. The bottom x axes indicate the resulting frequency detuning with respect to the resonance frequencies $f^R_A$, $f^R_B$ and $f^R_C$ reported in the plots.
• Figure 5: Complete circuit diagram. The orange wires indicate the frequency comb spectroscopy setup-specific lines, and the light blue wires mark the VNA circuitry.