In this webinar, we discussed transport measurements on materials for quantum computing, why studying resonators accelerates development of materials used for quantum computing by yielding a wealth of information, and how to measure resonators effectively with the Pound-Drever-Hall method.
The recording of the full webinar can be found here. If you would like to learn more about one of the aspects covered in this webinar, or would like to see a live demo of a presented measurement, please reach out to us.
Here are answers to the questions asked afterwards, organized according to the webinar’s three sections:
Characterizing Quantum Materials with Lock-in Amplifiers
Does RF reflectometry have good sensitivity over a broad conductivity range from insulators to superconductors?
Yes – by adjusting the properties of the matching network, a wide range of impedances can be measured.
Why Studying Resonators Accelerates Qubit Development
What other sources of loss are considered, other than TLS?
Two-Level Systems (TLS) are quantum systems with two energy levels that are coupled to the resonator. They can absorb a photon from the resonator, so they reduce the Q. TLS are often caused by impurities on the surface of the resonator.
Loss at millikelvin temperatures between one and a million average photons is broken down into power-dependent and power-independent mechanisms. Power-dependent loss is primarily due to TLS. Power-independent loss mechanisms include vortex loss, radiation, and spurious modes, among others.
How can one distinguish experimentally between TLS on surfaces and in the bulk?
By measuring many devices made from the same materials stack but with differing values of surface-to-volume ratio, we can populate a loss matrix to extract loss from interfaces and bulk dielectric substrate separately.
From one of your slides, is it correct that TLS losses can be as low with niobium as with aluminum?
The lowest interface losses to date have been from Nb devices.
We can measure at higher temperatures when measuring resonators than when measuring a full qubit. Approximately how high can those temperatures be?
We can meaningfully measure resonators at temperatures up to a third of the superconducting transition temperature. Above T_c/3, quasiparticle loss starts to become significant.
Are there good review articles on these material analyses?
Materials loss measurements using superconducting microwave resonators, McRae, C.R.H., et al., Review of Scientific Instruments, 91, 091101 (2020), review article.
Does the material from which the resonators are made affect the accuracy of TLS measurements? For example, niobium as compared with tantalum resonators.
The accuracy of determining TLS density will be affected by the Q factor, but not directly by the material of the resonator.
Studying Resonators with the Pound-Drever-Hall Method
Can you talk more about the square-law power detector used?
We use devices called envelope detectors. Important parameters are the output rise and fall times. These must both be fast enough to support the modulation frequency that you plan to use. For a 5 GHz superconducting resonator with a Q of 0.5 million and a linewidth of 10 kHz, a modulation frequency of 100 kHz may be ample. However, for room temperature resonators of lower Q, 10s of MHz may be needed, and thus a faster envelope detector.
How different is the PDH approach using the transmitted signal, as compared with using the reflected signal from the cavity?
To create the PDH signal, both carrier and sidebands must reach the power detector. As long as this happens, the several modes of coupling to the resonator have similar locking properties. For the collaboration measurements that we showed, we used “hanger mode,” in which the signal passes along a transmission line with a section having coupling to the resonator. In hanger mode, the intrinsic and coupling Q values can more readily be separated [McCrae et al. 2021].
When the modulation frequency is less than the linewidth, both carrier and sidebands always reach the detector because the sidebands can pass through the resonator. When the modulation frequency is larger than the line width, however, the sidebands cannot enter the resonator. In that case, a transmission configuration in which signal must enter the resonator and come out again does not produce a PDH signal. Reflection mode works, as does hanger mode. In hanger mode, sidebands that do not interact with the resonator traverse the coupling section unaffected, so they reach the detector.
Will digital noise affect any of these measurements?
No. In almost all cases on the input side, our instruments are limited by the noise spectral density of their input amplifier, not the quantization. The internal computations are performed with an ample number of bits. The output for PDH is a frequency, and the resolution is micro-Hertz. Studying the noise in output voltage or frequency, the almost universal experience is to see normally-distributed noise, with no sign of quantization.
Please elaborate on the precision of the determination of Q, which is given with three significant digits.
The three significant digits on the slide may not be justified. We are still completing the error analysis.
Can a superconducting resonator detect the conductivity difference between superconducting and normal metal? Can we achieve sub-micron spatial resolution by making small resonators?
Yes – by measuring a resonator, we can distinguish between superconducting and normal metal both in the resonator and in nearby metals coupled to the resonator, because there will be a large effect on the Q.
A resonator that is itself sub-micron would be very difficult to make at 4-8 GHz, where the wavelength is 7.5 to 3.75 cm. However, with normal techniques for making these resonators in a size of a few mm, we might be able to use a sub-micron-sized gap in a conductor to couple to a sub-micron-sized region.
How should one choose the modulating frequency, f_m, in a PDH measurement?
Consider two types of measurement: center frequency and Q. For center frequency, we have a wide latitude. When f_m is much less than the linewidth, sensitivity suffers. When f_m is comparable with or larger than the linewidth, sensitivity reaches a plateau so that f_m does not matter. Other considerations include staying within the bandwidth of the signal-processing chain and having f_m high enough to produce a signal after demodulation with sufficient bandwidth for the intended purposes.
For measuring Q, when validating the technique and for a single reliable measurement on a given sample, f_m should be swept from 100 times smaller than the linewidth to 100 times larger. For more routine measurements when the sample and coupling are fairly well understood, a single measurement with f_m = linewidth may suffice for measuring Q.
If the goal is to measure fast at a low photon number, what are the advantages of this method compared with just using a VNA, but adding a TWPA? Is there a simple explanation as to why PDH is faster than a VNA?
In all cases, a TWPA will help. The question is, with a given preamplifier, what method gives the fastest and most reliable answers? There are two advantages of the PDH method: it is faster, and it provides a direct measure of Q.
We are working to make a quantitative speed comparison between PDH and a VNA under comparable conditions. We anticipate a severalfold improvement with PDH. The VNA must measure at multiple frequencies in order to fit the lineshape and estimate center frequency and Q. Some of these measurements are in the wings of the resonance, so SNR is reduced. PDH needs only a single measurement at a frequency near the resonance center to determine both center frequency and Q.
The advanced modulation scheme that Jim discussed may provide a further speed improvement. If we are to secure that advantage, we must use a non-linear and quantum-limited power detector as the first stage.
How can you disentangle external and internal Q factors with this method?
For a given Q, the two Q parameters have different dependences on modulating frequency, so can be estimated separately. However, it is probably better not to estimate the coupling Q when estimating intrinsic Q at low power. Instead, use an a priori estimate made at high power, since coupling Q is independent of power. This will reduce the standard deviation of the measurement of intrinsic Q.
Can PDH be used in the under-coupled regime when the Qc is not known a priori?
Yes, it can. As explained above, both Q_i and Q_c can be estimated from the dependence of the 2nd harmonic signal vs. modulating frequency. However, the information content of low-power data is better used by taking an a priori value for Q_c to be an estimate made at higher power.
General
Can either speaker be contacted after the webinar?
Yes – please contact us at:
jim.phillips@zhinst.com
coreyrae.mcrae@colorado.edu
Can you refer to some papers related to the talk today?
- Principles of Lock-in Detection, video.
- Principles of lock-in detection and the state of the art, whitepaper.
- Resonator characterization via the Pound-Drever-Hall method; blog to be published at zhinst.com in summer 2023.
- Quantum Material Characterization for Streamlined Qubit Development, webinar.
- Materials loss measurements using superconducting microwave resonators, McRae, C.R.H., et al., Review of Scientific Instruments, 91, 091101 (2020), review article.
- Rapid characterization of superconducting microwave resonators using the Pound-Drever-Hall technique, APS March Meeting 2023.
- High Precision readout of superconducting resonators for analysis of slow noise processes, J. Barnett thesis, 2013.
- Slow noise processes in superconducting resonators, arXiv, 2013.
- An introduction to Pound–Drever–Hall laser frequency stabilization, E. Black, Am. J. Phys. 69(1), January, 2001. Abstract
