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arXiv:2202.10516 · Applied Physics Reviews

Probing quantum devices with radio-frequency reflectometry

Florian Vigneau, Federico Fedele, Anasua Chatterjee, D. J. Reilly, Ferdinand Kuemmeth, M. Fernando González-Zalba, Edward A. Laird, Natalia Ares

Published Feb 24, 2023• 115 citations
arXivHTMLPDF

Abstract

Many important phenomena in quantum devices are dynamic, meaning that they cannot be studied using time-averaged measurements alone. Experiments that measure such transient effects are collectively known as fast readout. One of the most useful techniques in fast electrical readout is radio-frequency reflectometry, which can measure changes in impedance (both resistive and reactive) even when their duration is extremely short, down to a microsecond or less. Examples of reflectometry experiments, some of which have been realized and others so far only proposed, include projective measurements of qubits and Majorana devices for quantum computing, real-time measurements of mechanical motion, and detection of non-equilibrium temperature fluctuations. However, all of these experiments must overcome the central challenge of fast readout: the large mismatch between the typical impedance of quantum devices (set by the resistance quantum) and of transmission lines (set by the impedance of free space). Here, we review the physical principles of radio-frequency reflectometry and its close cousins, measurements of radio-frequency transmission and emission. We explain how to optimize the speed and sensitivity of a radio-frequency measurement and how to incorporate new tools, such as superconducting circuit elements and quantum-limited amplifiers into advanced radio-frequency experiments. Our aim is threefold: to introduce the readers to the technique, to review the advances to date, and to motivate new experiments in fast quantum device dynamics. Our intended audience includes experimentalists in the field of quantum electronics who want to implement radio-frequency experiments or improve them, together with physicists in related fields who want to understand how the most important radio-frequency measurements work.

DOI
10.1063/5.0088229

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Many important phenomena in quantum devices are dynamic, meaning that they cannot be studied using time-averaged measurements alone. Experiments that measure such transient effects are collectively known as fast readout. One of the most useful techniques in fast electrical readout is radio-frequency reflectometry, which can measure changes in impedance (both resistive and reactive) even when their duration is extremely short, down to a microsecond or less. Examples of reflectometry experiments, some of which have been realized and others so far only proposed, include projective measurements of qubits and Majorana devices for quantum computing, real-time measurements of mechanical motion, and detection of non-equilibrium temperature fluctuations. However, all of these experiments must overcome the central challenge of fast readout: the large mismatch between the typical impedance of quantum devices (set by the resistance quantum) and of transmission lines (set by the impedance of free space). Here, we review the physical principles of radio-frequency reflectometry and its close cousins, measurements of radio-frequency transmission and emission. We explain how to optimize the speed and sensitivity of a radio-frequency measurement and how to incorporate new tools, such as superconducting circuit elements and quantum-limited amplifiers into advanced radio-frequency experiments. Our aim is threefold: to introduce the readers to the technique, to review the advances to date, and to motivate new experiments in fast quantum device dynamics. Our intended audience includes experimentalists in the field of quantum electronics who want to implement radio-frequency experiments or improve them, together with physicists in related fields who want to understand how the most important radio-frequency measurements work.
Radio frequency (RF) reflectometry is a well understood measurement technique in common use by RF engineers today. This technique has been applied at low temperature to probe the impedance of nanoscale devices for over two decades. These measurements typically employ basic resonant impedance transformers to bring the device impedance close to the characteristic impedance of the measurement apparatus. In 2013 it was discovered by Colless et al. that reflectometry could be used to measure extremely small impedance changes in a gate coupled to a quantum dot to directly probe the charge state of the dot. This has simplified measurements of certain nanoscale devices including single electron boxes (SEBs) and semiconductor qubits. Despite the maturity of RF reflectometry measurements in general, researchers working at low temperature continue to make basic mistakes in their measurements. The root cause of most of these mistakes is the failure to properly calibrate the measurement apparatus. In some cases this leads researchers to unrealistic conclusions when trying to match measurements to theory. This dissertation demonstrates that a low temperature RF calibration can be very easy and straightforward using inexpensive, off the shelf SMA terminators as calibration standards. A proper calibration allows for direct quantitative comparisons between data measured on different reflectometers or between data and simulations. The consistency between measurements and simulations then allows for straightforward design and implementation of improved impedance transformers, increasing the SNR of extremely sensitive measurements, with simple data-based simulations. A new type of single electron device, the capacitively coupled RF single electron transistor (CSET) is proposed and demonstrated which can rival the traditional RF SET in sensitivity but integrates more effectively into arrays and may offer a level of resistance to electrostatic discharge damage.
Fast measurement of small currents and even smaller fluctuations in devices with large impedance is not trivial. Standard low frequency lock-in measurements suffer from stray capacitance of the order of nF reducing the bandwidth to merely kHz and a time resolution of ms. One approach of increasing the bandwidth is to connect the high-impedance device next to a low input impedance amplifier, though, the signal is decreased by the ratio of the impedances of the device and the amplifier, easily a factor of a thousand. Therefore, long averaging times are required to detect the small signal. Moreover, 1/f noise still affects the measurements. The issues of small bandwidth, impedance mismatch and 1/f noise can be simultaneously addressed using resonant LC circuits. The success of this impedance transformer lies in replacing the slow operation of applying voltages and measuring currents by applying fast radio frequency signals and measuring the reflectance. LC circuits remain appealing because of a rather simple assembly. Nonetheless, the challenge to reproducibly achieve the resonance frequencies in gigahertz (GHz) range while matching to high impedance still remains due to the large parallel stray capacitance of the measurement setup. Resonance frequency in GHz range is desirable for two reasons. Firstly, measurements are faster and 1/f noise smaller. Yet more importantly, one can profit from the wide range of measurement techniques developed in the context of circuit quantum electrodynamics, such as FPGA based hardware and quantum limited Josephson parametric amplifiers. \n \nIn this thesis, a matching circuit namely stub tuner based on coplanar transmission lines is designed, fabricated and integrated with mesoscopic devices. Owing to the simple design, we could predict the frequency response of a stub tuner in terms of the standard circuit model (Chapter 3). We initiated certain necessary fabrication adaptations to achieve integration of niobium circuits with devices. In particular, a reliable stamping of carbon nanotubes from the growth substrate to the target substrate was accomplished, improving both the device quality and yield (Chapter 4). RF reflectometry is employed to perform admittance measurements on CNT quantum dots. We could reliably deduce all circuit and device properties using the resonance response. The measurement bandwidth is shown to be in MHz range which is three orders of magnitude larger than that for DC measurements. In particular, using the phase response of the circuit, double-dot qubit energy and dephasing rate are studied (Chapter 5). We could even get rid of conventional contacts that could degrade the device properties, and still extract quantum capacitance and dissipation in graphene p-n junctions. The residual doping is shown to affect not only the mean free path but also the Fermi velocity. We further see that at small doping, electron-electron interactions could modify the Fermi velocity and that the dispersion relation could deviate from linearity (Chapter 6). \n \nIn the second half of the scientific results, real strength of stub tuner is tested in transmission measurements. The large bandwidth and signal to noise ratio is utilized to measure extremely clean noise power densities at high speed and draw comparisons with the corresponding average current. Clear changes in the shot noise spectrum are seen when a transport channel is added or removed from the bias window of the quantum dot. We especially observe noise enhancement outside the Coulomb diamonds due to blocking states (Chapter 7). Finally, we investigate noise properties of quantum dots when leads are superconducting. Distinct charge transfers due to multiple Andreev reflections are visible in the shot noise. The main finding, however, was the thermal noise at zero bias. We observed its periodic suppression and emergence across the quantum dot resonance. In another device, shot noise due to the Kondo effect is found to be enhanced in the superconducting state when the Kondo temperature exceeds the value of the superconducting gap (Chapter 8). Simple design and reliable calibration of the GHz matching circuit in both reflectance and transmission pave a way for fast and non-invasive measurements of mesoscopic conductors. \n \n \n \n \n
Radio-frequency (rf) reflectometry offers a fast and sensitive method for charge sensing and spin readout in gated quantum dots. We focus in this work on the implementation of rf readout in accumulation-mode gate-defined quantum dots, where the large parasitic capacitance poses a challenge. We describe and test two methods for mitigating the effect of the parasitic capacitance, one by on-chip modifications and a second by off-chip changes. We demonstrate that on-chip modifications enable high-performance charge readout in Si\mathrm{Si}Si/Six{\mathrm{Si}}_{x}Six​${\mathrm{Ge}}_{1\ensuremath{-}x}$ quantum dots, achieving a fidelity of 99.9% for a measurement time of $1\phantom{\rule{0.2em}{0ex}}\ensuremath{\mu}\mathrm{s}$.
Radio-frequency reflectometry allows for fast and sensitive electrical readout of charge and spin qubits hosted in quantum dot devices coupled to resonant circuits. Optimizing readout, however, requires frequency tuning of the resonators and impedance matching. This is difficult to achieve using conventional semiconductor or ferroelectric-based varactors in the detection circuit as their performance degrades significantly in the mK temperature range relevant for solid-state quantum devices. Here we explore a different type of material, strontium titanate, a quantum paraelectric with exceptionally large field-tunable permittivity at low temperatures. Using strontium titanate varactors we demonstrate perfect impedance matching and resonator frequency tuning at 6 mK and characterize the varactors at this temperature in terms of their capacitance tunability, dissipative losses and magnetic field sensitivity. We show that this allows us to optimize the radio-frequency readout signal-to-noise ratio of carbon nanotube quantum dot devices to achieve a charge sensitivity of 4.8 μμμe/Hz1/2^{1/2}1/2 and capacitance sensitivity of 0.04 aF/Hz1/2^{1/2}1/2.
This thesis focuses on implementing radio frequency (rf) reflectometry techniques for dispersive detection of charge and spin dynamics in nanoscale devices. I have investigated three aspects of rf reflectometry using state-of-the-art silicon (Si) complementary metal-oxide-semiconductor (CMOS) nanowire field effect transistors (NWFETs). First, a high-sensitivity capacitive gate-based charge sensor is developed by optimising the external matching circuit to detect capacitive changes in the high frequency resonator. A new circuit topology is used where superconducting niobium nitride (NbN) inductor is connected in parallel with a single-gate Si NWFET resulting in resonators with loaded Q-factors in the 400-800 range. For a resonator operating at 330 MHz, I have achieved a charge sensitivity of 7.7 μe/Hz\mu e/\sqrt{\text{Hz}}μe/Hz​ and, when operating at 616 MHz, I get 1.3 μe/Hz\mu e/\sqrt{\text{Hz}}μe/Hz​. This gate-based sensor can be used for fast, accurate and scalable techniques for quantum state readout in Si CMOS based quantum computing. Second, this new circuit topology for the resonator is used with a dual-gate Si NWFET. This dual-gate device geometry provides access to a double quantum dot (DQD) system in few electron regime. The spin-state of the two-electron DQD system is detected dispersively using Pauli spin blockade between joint singlet S(2,0) and triplet T−_-−​(1,1) states in a finite magnetic field BBB. The singlet-triplet relaxation time T1T_1T1​ at B=4.5B=4.5B=4.5~T is measured to be $\sim$1 ms using standard homodyne detection technique. Third, I expand the range of applications of gate-based sensing to accurate temperature measurements. I have experimentally demonstrated a primary thermometer by embedding a single-gate Si NWFET with the rf capacitive gate-based sensor. The thermometer, termed as gate-based electron thermometer (GET), relies on cyclic electron tunneling between discrete energy levels of a quantum dot and a single electron reservoir in the NWFET. I have found that the full-width-half-maximum (FWHM) of the resonator phase response depends linearly with temperature via well known physical law by using the ratio kB/ek_\text{B}/ekB​/e between the Boltzmann constant and the electron charge. The GET is also found to be magnetic field independent like other primary thermometers such as Coulomb blockade and shot noise thermometers.