Resonator Characterization via the Pound-Drever-Hall Method

August 21, 2023 by Jim Phillips

This blog discusses methods for measuring the frequency fluctuations and quality factor, 𝑄, of a resonator. Resonators are physical systems that naturally oscillate at a well-defined frequency. At this resonant frequency, energy exchanges periodically between two forms, for example, kinetic and spring energy as in a scanning probe microscope’s cantilever or electric and magnetic field as in a microwave resonator. Variations in the resonant frequency can be an extremely sensitive indicator of physical parameters affecting the resonator such as the environment of an SPM cantilever or the length of an optical cavity. In addition to the resonant frequency 𝑓₀, another important quantity is the sharpness of the resonance, usually parameterized by quality factor 𝑄=𝑓₀/𝐹𝑊𝐻𝑀, where 𝐹𝑊𝐻𝑀 is the resonance full-width at half-maximum, or linewidth. Resonators are widely used because they enhance signal strength by a factor 𝑄. Also, for a given resonant frequency, a higher quality factor means a narrower linewidth, making the resonator more sensitive to a physical quantity of interest.

The extremely high sensitivity achievable with resonators is used in many applications including scanning probe microscopes, gyroscopes, and quantum computing. We focus here on the case of a superconducting resonator used in quantum technologies, but the techniques can be applied in numerous other fields. In a superconducting quantum computing circuit, the qubit is commonly part of a mm-scale superconducting resonator with resonant frequency in the 4-8 GHz range. The qubit is often coupled to a readout resonator in the same frequency range. Shifts in the readout resonator’s frequency of the order of one linewidth indicate changes in the quantum state of the qubit. Loss mechanisms make the resonator measurement more difficult by adding noise and reducing 𝑄. Measuring and understanding loss mechanisms in superconducting microwave resonators will help us to improve the lifetimes of superconducting quantum circuits for quantum information technologies.

When developing the fabrication process for such resonators, it is often helpful to measure the frequency fluctuations and 𝑄 of an isolated resonator that is not coupled to a qubit (Fig. 1). Measuring a pure resonator is simpler than measuring a qubit and can exclude confounding factors resulting from the qubit’s nonlinearity. To probe noise mechanisms important in quantum measurements, the measurements need to extend down to extremely low power, with the mean number of microwave photons in the resonator about 10⁻². A resonator can be operated smoothly over such a power range; a qubit cannot be [1,2].

The most common approach to characterize the resonator is to perform a frequency sweep, traditionally using a Vector Network Analyzer (VNA). However, sweeping frequency and fitting the measured curve to estimate the center frequency and 𝑄 requires time and the process can be cumbersome,

Figure 1. Measurements of resonator frequency fluctuations. The quantity that is fluctuating is the center frequency of the resonator, 𝑓₀. The square of the fluctuations with respect to the mean is referred to as the “power” and has a magnitude in Hz². The fluctuation itself has a frequency, and we often study the distribution of fluctuations across the frequency spectrum. The power spectral density (PSD) has units of Hz²/Hz. The root power spectral density is plotted, and it has units of Hz/√Hz.

By contrast, the Pound-Drever-Hall (PDH) method requires only a measurement at a single frequency. It is faster, though for a state-of-the-art VNA, PDH may be faster only by a modest factor. The PDH signal is linearly related to the frequency shift when the shift is small, so unlike a VNA, PDH provides a real-time feedback signal. Further, the Zurich Instruments digital lock-in amplifiers that we recommend for PDH are equipped both for generating the FM signal and for feedback to hold the oscillator frequency centered on the resonator, capabilities not normally included in a VNA.

Other measurement methods include ringdown spectroscopy, a phase-locked loop (PLL), and dual-frequency resonance tracking (DFRT). These can all be done with Zurich Instruments lock-in amplifiers. However, compared with PDH, ringdown spectroscopy is slower, the PLL is more sensitive to phase noise and DFRT has lower bandwidth. Therefore, we concentrate on PDH.

Pound-Drever-Hall sensing

PDH is ideal for measuring resonator frequency fluctuations and 𝑄. It has been used in locking microwave and optical oscillators with the highest precision for many decades, including in work that has won several Nobel prizes. It is named for pioneers of precision measurement in the microwave and optical domains. In a resonator, it measures rapid center frequency fluctuations. It also measures 𝑄 directly.

Basis for the PDH signal

The resonator center frequency, 𝑓₀, fluctuates in response to the noise processes of interest, and we wish to make our oscillator frequency, 𝑓, follow those fluctuations.

Then, the fluctuations of 𝑓 are easily recorded and form a record of the fluctuations of 𝑓₀. To make 𝑓 follow 𝑓₀, we use a feedback loop. For the feedback loop, we need a measure of the frequency offset, 𝑓−𝑓₀.

Figure 2. PDH block diagram.

The parts of the feedback loop are listed below, marked A, B, C, … to correspond to the letters in Fig. 2.

A. The resonator center frequency is 𝑓₀, which is typically 4-8 GHz.

B. The oscillator frequency is 𝑓, generated by Osc1. 𝑓 should match 𝑓₀.

C. We require a signal at 𝑓 with sinusoidal frequency modulation (FM) at a modulation frequency, 𝑓_𝑚. Here, 𝑓_𝑚= 50 MHz (although, on a superconducting resonator with 𝑄~500,000, the linewidth would be ~10 kHz and we might choose 𝑓_𝑚= 100 kHz. The signal’s Fourier spectrum has a carrier at 𝑓 and sidebands at 𝑓 ± 𝑝 𝑓_𝑚with𝑝 = 1,2,… . In practice, it is sufficient to supply just the first upper and lower sidebands, 𝑝 = 1. The lock-in amplifier supplies the FM signal by adding signals from Osc1, Osc2, and Osc3.

D. The composite signal thus prepared is converted to an analog signal and upconverted to 6 GHz in a mixer. Since the addition described in C is digital, the analog signal after conversion can cover a wide range of power levels. This is necessary for testing loss mechanisms down to the single quantum level.

E. We thus have a pure FM signal like the one shown.

F. The FM signal interacts with the resonator at A. When 𝑓 ≠ 𝑓₀, some of the FM is converted to amplitude modulation (AM). This AM forms the basis for our measure of 𝑓 − 𝑓₀.

G. We need to remove the microwaves and measure just the amplitude. To do this, the signal passes through a power detector such as a diode and a low-pass filter. This gives us the AM as a signal at 𝑓_𝑚. When the feedback is working and 𝑓 is very close to 𝑓₀, this signal is very small. PID controllers by themselves use a DC input signal. By integrating a PID controller with a lock-in amplifier as we do here, it gains the ability to take this very small AC signal as an input.

H. The lock-in amplifier provides a demodulator to take the tiny AC signal at G and convert it to a tiny but precise DC signal. Since this signal is digital, the errors normally associated with DC signals do not arise.

I. Without the feedback loop running, we can sweep the oscillator frequency 𝑓 around 𝑓₀ to study the input to the PID controller, the “error signal.” Shown is a typical PDH error signal. We will work at the center of the red ellipse. For small departures of 𝑓 from 𝑓₀, at the center of the ellipse, the demodulated PDH signal is linearly proportional to the frequency departure.

J. The PID loop filter filters and scales the PDH error signal to form the control signal according to the filter’s P and I coefficients. These set the PDH feedback loop bandwidth, typically ranging from a few Hz to a few kHz. The loop filter controls the frequency, 𝑓, of the “carrier” (the central tone) of the FM signal. As it tunes the carrier to follow 𝑓₀, the lock-in amplifier’s MOD unit keeps the upper and lower sideband signals, made by Osc2 and Osc3, at a spacing of 𝑓_𝑚 from the carrier. The carrier and sidebands form the tunable signal described in C above. This completes the feedback loop.

K. Another demodulator operating at 2𝑓_𝑚 provides a measure of 𝑄, discussed below.

To frequency modulate the oscillator signal, we could have used an external analog frequency modulator. However, working over a wide range of power requires frequent recalibration. Instead, we simply use the Zurich Instruments MOD option, which harnesses oscillators Osc2 and Osc3 (C in figure) to create the needed sidebands at 𝑓 − 𝑓_𝑚, and 𝑓 + 𝑓_𝑚.

With the PDH loop locked, we record 𝑓(𝑡) as a direct measure of resonator frequency fluctuations. The Fourier transform and Allan Deviation of this quantity are key indicators of resonator loss mechanisms, with different loss mechanisms appearing in different ranges of frequency or averaging time. These studies can be done as a function of carrier power. At a power corresponding to as little as 0.01 microwave photon in the resonator, the full contribution to loss of Two-Level System (TLS) loss mechanisms is revealed.

Determining Quality Factor, Q

When the PDH loop is locked and 𝑓 = 𝑓₀, the AM signal at 𝑓_𝑚 is zero. However, there is a strong AM signal at 2𝑓_𝑚. The signal demodulated at 2𝑓_𝑚, the “second order signal,” (K in Fig. 2) provides a direct measure of 𝑄. It is plotted against 𝑓_𝑚 in Fig. 3 [3]. For values of 𝑓_𝑚 near the resonator linewidth, 𝑓₀/𝑄, this second-order signal provides a measure of 𝑄. To determine 𝑄, one first varies 𝑓_𝑚, finding a dependence like that in the figure. Then one measures at just one value of 𝑓_𝑚, near 𝑓₀/𝑄. This provides a fast measure of 𝑄 as a function of time.

Figure 3. Q Signal as a function of modulating frequency. Data in blue, model in orange.

Conclusion

We have seen how the PDH method improves upon materials measurements for quantum technology and yields results not otherwise obtainable. PDH can be used over the full range of power needed for sensitive materials studies. A Zurich Instruments lock-in amplifier performs key functions, making PDH quick and easy to set up.

References

[1] T. Lindström; J. Burnett; M. Oxborrow; A Ya. Tzalenchuk, “Pound-locking for characterization of superconducting microresonators,” Rev Sci Instrum 82, 104706 (2011).

[2] C.R.H. McRae, H. Wang, J. Gao, M.R. Vissers, T. Brecht, A. Dunsworth, D.P. Pappas, and J. Mutus, “Materials loss measurements using superconducting microwave resonators,” Rev Sci Instrum 91, 091101 (2020).

[3] John Pitten, Jim Phillips, Brandon Boiko, Josh Mutus, and Corey Rae McRae, “Rapid characterization of superconducting microwave resonators using the Pound-Drever-Hall technique,” APS March Meeting 2023.