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Bogoliubov quasiparticles in Andreev bound states of aluminum nanobridge Josephson junctions

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Bogoliubov quasiparticles in Andreev bound states of aluminum nanobridge Josephson junctions

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Content Bogoliubov quasiparticles in Andreev bound states of aluminum nanobridge Josephson junctions by James Timothy Farmer A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (PHYSICS) May 2023 Copyright 2023 James Timothy Farmer This is for my father, who inspired me to think. ii Acknowledgements I’ve come to know many people during my time as a Ph.D. student at USC who I’m happy to count among my closest friends. It has not been an easy journey but each of these people has made it more enjoyable. I would especially like to thank John and Christina for many shared meals, games, and company. Pat and Jen, I appreciate all the times we’ve spent at (or adjacent to) the beach, climbing, or the brewery. Sean and Daisy, I’ve really enjoyed the time we’ve spent biking, bowling, and brunching. Brian, Haimeng, and Evangelos, Thank you for making the time off the mountain nearly as fun as the snowboarding. Building up the Levenson-Falk lab has provided a sense of satisfaction that only hard work can achieve. Eli, thank you for having the vision to create this group and drive the science ahead. You’ve been extraordinarily helpful and available for me to ask questions. Whether I’m puzzling out some concept, developing some tool, or imagining a new experiment, it’s often by bouncing the idea off you that I’m finally able to realize it. I would like to thank Prof. Leonid Glazman for helpful discussions on modeling the quasiparticle dynamics. Azarin, you were a tremendous boost to the lab and the quasiparticles project in particular. Your advice, feedback, and effort were all major contributions to realizing the work presented in this thesis. Shanto, thank you for being so diligent and for asking excellent questions. I could continue working on this project for years iii and still not feel that we’ve pried out all of the interesting physics, but I can step away with a little more ease knowing that you are the one to lead this work after me. I would like to thank my father, who inspired me from a young age to work hard and think through problems logically. I would also like to thank my sister for being there for me when I needed you. And finally, I would like to thank Darian. You were there before the start of this chapter of life and our relationship has evolved over its course. Your support extends beyond any boundaries of work and home; the main results of this thesis were made possible by your devel- opment of fabrication procedures and direct contributions to nanobridge resonator fabrication. You are an excellent scientist and a wonderful person. I am very happy that we get to live the remaining chapters of life together. To all my family and friends, thank you! iv TableofContents Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Chapter 1: Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1 BCS superconductivity and Bogoliubov quasiparticles . . . . . . . . . . . . . . . . 4 1.2 Andreev picture of Josephson junctions . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Aluminum nanobridge Josephson junctions . . . . . . . . . . . . . . . . . . . . . . 10 1.3.1 KO-1 current-phase relation . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3.2 Josephson inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3.3 Quasiparticle trapping in aluminum nanobridges . . . . . . . . . . . . . . 14 Chapter 2: Nanobridge resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1 Circuit description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.1.1 Reflection, transmission, and hanger geometries . . . . . . . . . . . . . . . 19 2.1.2 Nanobridge resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 Fitting the flux tuning curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3 Quasiparticle induced resonance shift . . . . . . . . . . . . . . . . . . . . . . . . . 26 Chapter 3: Nanobridge resonators as quasiparticle detectors . . . . . . . . . . . . . . . . . 29 3.1 Measurements of quasiparticle trapping . . . . . . . . . . . . . . . . . . . . . . . . 29 3.1.1 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.1.2 Naive Bayesian analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.1.3 Power dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.1.4 Clearing tone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.1.5 Quasiparticle transition rates . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.1.6 Trapping and Excitation Mechanisms . . . . . . . . . . . . . . . . . . . . . 43 3.1.7 Higher Number Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.1.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 v 3.2 Hidden Markov Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.2.1 A more relevant SNR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.2.2 Transition rates and lifetimes . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.2.3 Occupation probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.2.4 Mean occupation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.3 Bootstrapping through power sweeps . . . . . . . . . . . . . . . . . . . . . . . . . 52 Chapter 4: Simulations of quasiparticle behavior . . . . . . . . . . . . . . . . . . . . . . . . 54 4.1 Generation of quasiparticle time-series . . . . . . . . . . . . . . . . . . . . . . . . 54 4.2 Resonator response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.2.1 Signal-to-noise ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.3 Comparison of simulation processes . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.3.1 Poisson process trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.3.2 Non-Markovian trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.3.3 Pairwise trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.3.4 Bursts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.3.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.4 Evaluating the accuracy of HMM estimates . . . . . . . . . . . . . . . . . . . . . . 67 4.4.1 Scoring the different simulation processes . . . . . . . . . . . . . . . . . . 68 4.4.2 Testing the HMM performance at low SNR . . . . . . . . . . . . . . . . . . 69 Chapter 5: Environmental mechanisms of quasiparticle trapping . . . . . . . . . . . . . . . 73 5.1 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.2 Quasiparticle trapping via spontaneous emission . . . . . . . . . . . . . . . . . . . 77 5.2.1 Fitting the trap rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.3 Quasiparticle clearing in a thermal bath of phonons . . . . . . . . . . . . . . . . . 84 5.3.1 Derivation of phonon contribution to ABS clearing . . . . . . . . . . . . . 85 5.3.2 Fitting the release rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.4 Mean occupation from steady state . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.5 Conclussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Chapter 6: Driven excitations of Andreev bound states . . . . . . . . . . . . . . . . . . . . 94 6.1 Andreev bound state clearing via measurement photons . . . . . . . . . . . . . . . 94 6.1.1 Validating the power dependence . . . . . . . . . . . . . . . . . . . . . . . 96 6.1.1.1 Artificial degradation . . . . . . . . . . . . . . . . . . . . . . . . 96 6.1.1.2 Pulsed measurement . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.1.2 Fitting the power dependence . . . . . . . . . . . . . . . . . . . . . . . . . 100 6.1.3 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6.2 The low temperature and low power limit . . . . . . . . . . . . . . . . . . . . . . . 105 Chapter 7: Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 7.1 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 vi Appendix A: Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Appendix B: Amplifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 Appendix C: Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 vii ListofFigures 1.1 The electron density of states for BCS superconductors. The Cooper pairs condense at the Fermi energy E F , while a gap opens at± ∆ inside which no electron states are allowed. Quasiparticles must create electrons and holes beyond this gap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 The Andreev picture of Josephson junctions. (a) Two regions of superconductor (blue) with order parameter phase ϕ L and ϕ R are separated by a boundary (tan). Cooper pairs tunnel across this boundary by multiple Andreev reflections which form the Andreev bound states pictured in the boundary region. (b) The Andreev bound state energies are shown as a function of the phase bias across the junction for transmission coefficients of 0.7 (dashed) and 1 (solid). Red curves show the positive ABS while blue curves show the negative ABS. . . . . . . . . . 9 1.3 SEM image of an aluminum nanobridge Josephson junction. A thin constriction in the center bridges the gap between the left and right leads. This nanobridge is approximately 20 nm wide, 8 nm thick, and 100 nm long. The leads are approximately 800 nm wide and 80 nm thick. . . . . . . . . . . . . . . . . . . . . . 11 2.1 Images and schematics of microwave resonators. (a) A lumped element resonator. Signal is capacitively coupled (orange) to the resonator and has two paths to ground; the interdigitated capacitor (red and blue) and the meandering inductance (green). (b) A transmission line resonator. Signal is capacitively coupled (orange) to the center trace (blue) of the co-planar waveguide which is shorted on the other end, forming a λ/ 4 resonant mode. This resonant mode could be described by an equivalent circuit (like the lumped element schematic in panel (a)) with effective inductance L eff and capacitanceC eff . . . . . . . . . . 16 2.2 (a) Schematic diagram of a lumped element nanobridge resonator. (b) False-color SEM images of an aluminum nanobridge resonator; interdigitated capacitor is shown in red and blue, while the meandering inductance (green) has the inclusion of a nanobridge SQUID (purple and cyan). . . . . . . . . . . . . . . . . . 21 viii 2.3 The extracted resonant frequency as a function of flux. The solid curve is the fit to Equation 2.16. We extract a Josephson inductance participation ratio at zero flux of q 0 = 0.01341. We see hysteretic branches in the flux tuning, as expected for a SQUID with junctions which have non-sinusoidal CPRs. . . . . . . . . . . . 25 3.1 (a) Device schematic. A CPW resonator (green) is grounded via a two-junction Al nanobridge SQUID (magenta). Flux bias through the SQUID phase-biases the junctions as δ = πϕ . (b) Optical image of the device with inset SEM images of the SQUID (magenta) and a nanobridge junction (orange). (c) Simplified measurement schematic. A tone at ω d continuously drives the flux-tunable resonator. The reflected signal is amplified by a TWPA at base stage followed by a HEMT at 4K and room temperature amplifiers. Isolators between HEMT and room temperature amplifiers are not shown to conserve space. The amplified signal is homodyne demodulated and I and Q components are low-pass filtered at 15 MHz, then digitized by an Alazar ATS9371 at 300 MSa/s and down-sampled to 10 MSa/s before saving. Microwave lines are optionally filtered at base stage by K&L 12 GHz and custom Eccosorb 110 low-pass filters. . . . . . . . . . . . . . 32 3.2 (a) Ensemble measurement of resonator response at 0 flux (blue) and at ϕ =0.49 (orange). Whenϕ =0.49, distinct peaks are visible roughly 0.5 and 1 MHz below resonance, corresponding to 1 and 2 trapped QPs, respectively. (b) Histogram of continuous IQ data taken at 0 flux for 30 s. Data has 10 MHz sample rate and has been convolved with a Gaussian window with effective integration time of 3 µ s. (c) Data taken at ϕ = 0.47 with the same procedure as panel b. The darkest mode is due to the resonance with 0 trapped QPs. The second darkest, located near (I,Q) = (12 mV, 15 mV), corresponds to 1 trapped QP and the last mode corresponds to 2 or more trapped QPs (as this mode corresponds to the resonance moving far from the drive frequency). . . . . . . . . . . . . . . . . . . 34 3.3 (a) Initial clustering of 5 µ s integrated data using the scikit-learn Gaussian mixture module produces modes with 1σ (solid) and 2σ (dashed) contours for 0, 1, and 2 or more trapped QPs in light blue, dark blue, and orange, respectively. (b) Subsets of the data in which the occupation is constant for a long time (4⟨τ i ⟩) are individually fit to Gaussian distributions. Means and covariances of each mode are then fixed and the full dataset is fit with mode weights as the only free parameters. (c) Two sections of time series data with I in brown and Q in dashed magenta. The background color is light blue, dark blue, and orange for 0, 1, and 2+ trapped QPs, respectively. All data taken atϕ =0.47. . . . . . . . . . . . . . . 37 3.4 Ensemble-averaged resonator response atϕ =0.47 as a function of measurement power. At powers below -141 dBm the response is roughly power-independent. As we increase the measurement power beyond -139 dBm, the weight of the 1-QP mode begins to decrease and the weight of the 0-QP mode increases. . . . . 40 ix 3.5 Microwave response of the resonance at flux bias 0.45 as a function of clearing tone power. The legend in (a) shows the approximate power of the clearing tone at the plane of the device. The red curve is at low power and shows a significant bump on the low frequency side due to averaging over many configurations of QP occupation. At high clearing tone power (blue curve) we see the resonator response narrows, becomes more symmetric, and appears to be more over-coupled. We stress that the ratio of internal loss to external coupling is not actually changing; rather, the response is taller because we are no longer averaging over other configurations of QP occupation. (b) Histogram of time series data with the clearing tone at the same power as the red trace in (a). (c) Histogram of time series data with the clearing tone at the same power as the blue trace in (a). Note that any trapped quasiparticles are quickly excited back above the gap, greatly reducing the weights of the 1- and 2-QP modes and increasing the weight of the 0-QP mode. . . . . . . . . . . . . . . . . . . . . . . . 42 3.6 Histograms of mode lifetimes for the 0- (a), 1- (b), and 2-QP (c) modes. All lifetimes are exponentially distributed at long times, but show a distinct peak at low times which we attribute to the finite detection bandwidth τ det . We extract the apparent lifetimes τ ∗ i by fitting the behavior above a cutoff time, defined as the mean measured lifetime. These lifetimes are then adjusted to correct for the finite detector bandwidth as described in the text, giving τ 0 =728µ s,τ 1 =12.7µ s,τ 2 =4.73µ s. . . . . . . . . . . . . . . . . . . . . . . . . 43 3.7 (a) Schematic description of the ground state|g⟩, 2 degenerate first excited states|o⟩, and second excited state|e⟩ of a channel. Both|o⟩ states produce the same resonant frequency shift, while the|e⟩ state produces the same shift as 2 channels entering their|o⟩ states. (b-e) Response of the device to different probe tone frequencies. The probe tone is stepped to progressively lower frequencies, starting with the midpoint of the 0- (red) and 1-QP (blue) mode resonances in (b) and ending with the 2-QP mode (green) frequency in (e). As the probe moves close enough, a 3-QP mode (purple) becomes apparent. . . . . . . . . . . . . . . . 44 3.8 A Hidden Markov Model works by assuming the system can transition between a set of hidden states with constant probabilities given by the transition matrix, whose elements are T ij . Each element of the transition matrix describes the probability of transitioning from state i to j, as shown by the arrows. The diagonal elements of the transition matrix describe the probability of staying in the same state. The right side of this figure shows the Markov chain S(t) of hidden states in the red box. At each time step, the system emits data x according to the state dependent emission probabilitiesP(x|s). This time-series of observationsX(t) as shown in the blue box represents the only information we are able to measure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 x 3.9 (a) IQ histogram of input data with gaussian emission probabilities represented by ellipse overlays with radii equal to one standard deviation. (b) A subset of the observed time-series is plotted in the top panel. This data is at a 1 MHz sample rate and was recorded at a moderately high readout power of -134 dBm at the resonator. The lower panel shows the estimated occupation as extracted from the data using the Viterbi algorithm and the HMM. Notice the jump to and from state 0 to 1 around 10 and 110µ s. . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.10 Block diagram of bootstrapping process to analyze power sweeps. We start by manually providing estimates of the gaussian centers and covariances for the initial fit at the highest power. Afterwards, we iteratively step down in measurement power using the Gaussian modes from the current fit as the starting point for the next fit. After fitting the HMM, we check SNR IQ and increase the integration time ifSNR IQ < 3. This process ends when the lowest power data is fit or the integration time exceeds half of the shortest mode lifetime. 53 4.1 Block diagram of the simulation process. The first step is to initialize the system using the input parameters to generate a set of channels with fixed ABSs. Next, we generate a time series of ABS occupation using the selected processes – at each time step a set of Poisson processes determines if/how many QPs will trap and release from the ABSs as well as fluctuations in the bulk QP density. From this ABS occupation, we calculate the change in inductance due to the poisoning of channels and use this to generate a nanobridge resonator response to a continuous measurement tone. Noise is added to the resonator response to match the SNR to the quantum limit – adding half a photon of noise as we might achieve with a Josephson parametric amplifier. . . . . . . . . . . . . . . . . . . . . 56 4.2 Simulation results. Each row shows a different underlying process: Poisson trapping, non-Markovian, pairwise trapping, and Poisson with bursts. The leftmost column shows the IQ histogram of data along with the Gaussian modes from the fit (ellipses have radii equal to one standard deviation). The central column shows the distribution of times between trapping events. Not that the Non-Markovian and burst distributions have short time deviations from the exponential decay (straight in log plot) expected in Poisson processes. The right column shows the distribution of times between trapping events weighted by the bin value. Dashed lines show the Poisson model. Note that the hidden Markov model faithfully recreates the distributions even for non-Markovian simulations. . 66 xi 4.3 Power sweep simulation results. (a) The F1 score, precision, and recall of the estimated state time series using a static bootstrap method (constant integration time). The dashed line shows SNR = 3; the performance degrades for SNR < 3. (b) F1 score for dynamic (orange) and static (blue) bootstrap methods. The dynamic bootstrap method with changing integration time to keep SNR > 3 does not affect the performance, though it does stop the process before the F1 score drops below 0.85. (c) Mean occupation estimates (circles) and the true values from each simulation (solid line) for dynamic (orange) and static (blue) bootstrap methods. The dynamic bootstrap method stops before the mean occupation estimate drifts away from the true value. (d) Transition rates fit by the HMM; true rates used in the simulation are shown as dashed lines. The release rate (moving fromn = 1 to n = 0 occupation) is shown for static (orange) and dynamic (dark purple) bootstrapping, with each agreeing well with the rate used in the simulation (dark grey dashed line). The trap rate (moving fromn = 0 ton = 1 occupation) is shown for static (orange) and dynamic (light purple) bootstrapping and each agree well with the trapping rate used in the simulation (light grey dashed line) . 71 5.1 (a) Images of similar device: aλ/ 4 resonator is grounded through a DC SQUID with a pair of symmetric aluminum nanowire junctions. These junctions are approximately 25 nm× 8 nm× 100 nm. (b) The readout driveω d is generated at room temperature and attenuated along the path through the dilution refrigerator to the base stage. At 30 mK, pair breaking photons on all inputs and outputs are reduced by K&L 12 GHz low pass filters and Eccosorb CR110 infrared absorbers. The signal circulates to reflect off our device and pass through a 5.85 GHz low pass filter. The signal is then amplified by a travelling wave parametric amplifier (TWPA) whose pump is inserted via a directional coupler. The signal exits the dilution refrigerator receiving further amplification by a HEMT at 4K and a series of low noise amplifiers at room temperature. The signal is homodyne demodulated by an IQ mixer and the resulting quadratures are digitized after 15 MHz low pass filtering. A Keithley Sourcemeter sends DC current along the dashed path to a coil in the device package and a VNA is used to measure the resonance as a function of flux. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.2 Three characterizing quantities which describe quasiparticle trapping are presented against the trap depth∆ A (left column) and the temperatureT (right column). The first row is the mean occupation, also referred to as ¯n, which has a low temperature saturation and a dip from 100 to 150 mK. The middle row is the trap rateΓ trap which saturates around 120 mK. The bottom row shows the release rateΓ release with a low temperature saturation around 60 mK and some structure in the trap depth. All data is shown at a measurement power of -133 dBm, or ¯n≈ 25 photons in the resonator. . . . . . . . . . . . . . . . . . . . . . . . 78 5.3 The measured trap rate minus the low temperature trap rate (average of all data less than 80 mK) is shown along with the fit to Eq 5.7. We find the scaling factor β =0.873MHz/µ eV 3 and the superconducting gap∆=185 µ eV. . . . . . . . . . . 81 xii 5.4 The measured trap rate divided by the low temperature trap rate (average of all data less than 80 mK) is shown along with the fit to Eq 5.8. We find the fractional non-equilibrium quasiparticle densityx ne =8.5× 10 − 7 . . . . . . . . . . . . . . . 82 5.5 Measured trap rate (circles) and model (solid lines). The dependence on the trap depth∆ A is shown on the left, while temperature dependence is shown on the right. We note the peak in 30 mK data around 9 GHz on the left was observed as a period of significantly larger than normal mean occupation which lasted approximately 1 hour in laboratory time. The source of this peak has not been found and it is not reproducible. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.6 The release rateΓ release is shown at flux ϕ = 0.485 (trap depth∆ A = 12.4 GHz) as a function of measurement power (left) and temperature (right). We can see that the release rate saturates at low temperature, but the value depends strongly on the measurement power. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.7 (top) The measured release rate vs trap depth and temperature. The top left panel shows structure in the trap depth dependence which is attributed to the driven electron-photon interactions which dominate at low temperature. In the top right panel, the low temperature saturation is visible. The grey dashed line indicates the cutoff temperature (90 mK) for the fit. (bottom) The measured release rate minus the low temperature saturation is shown as circles, while the phonon clearing model (Eq. 5.16) is shown as solid curves. The 240 mK and 260 mK data in the top left panel show some clipping of the release rate data to the 1 MHz sample rate – A limitation of our measurement rather than a physical effect. 89 5.8 (top) The measured mean occupation (circles) and the corresponding fit (solid) are shown against temperature. Note that a different fit is performed at each value of ∆ A . (bottom) The fit parameter α M vs trap depth. Stars indicate the value ofα M for the three curves of the same color displayed in the top panel. . . 91 5.9 Two sources of estimate for the rate of readout photons clearing quasiparticles from the Andreev bound state traps. The measured low temperature release rate (blue) and the fit parameter from the mean occupation, shown as α/α M (orange), whereα = 38.51 is found from fitting the phonon contribution to the release rate as shown in Figure 5.7. We point out that these agree in shape and magnitude despite coming from different sources. . . . . . . . . . . . . . . . . . . 92 xiii 6.1 The quasiparticle clearing rateΓ release atT = 40 mK is shown as a function of trap depth∆ A (left) and measurement power at the resonator (right). There are a few observations to make of the left panel: power dependence changes with trap depth, and the low power saturation has structure in the trap depth. The power dependence appears to have 3 distinct regions (≲ 8 GHz,∼ 8 to∼ 10.5 GHz, and≳ 10.5 GHz). The low power saturation of the clearing rate increases slightly with trap depth towards a peak at∼ 10.5 GHz, then abruptly falls. A second buildup and peak may be evident in the lowest power and deepest trap (12 -13 GHz). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6.2 A power sweep atϕ =0.47 with measured data (circles) and artificially degraded duplicates of the data at -131 dBm (stars). The horizontal axes show “Effective power” because the artificially degraded data (stars) have been mapped onto the equivalent power that produces the same SNR. For the measured data (circles) this is the actual power at the device. The left panel shows that small amount of change with power present in the trap rate Γ trap can be fully explained by worsening of the hidden Markov model fit due to SNR. On the right, we see that the release rate Γ release has power dependence in excess of the small change produced by decreasing SNR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.3 An ensemble measurement to validate power dependence with no hidden Markov model. (a) The measurement scheme —a sequence of 50µ s pulses are input at 20 ms intervals, data is homodyne demodulated and we integrate over the window τ int . Ensemble results are histogrammed in IQ plane and fit to Gaussian modes. Performing this as a function of the delay time τ delay allows us to calculate a moving average for the mean occupation as a function of time since the pulse begins. (b) The mean occupation vsτ delay with an exponential fit (Equation 6.3). The point at 42.5 µ s in red was omitted due to poor quality of Gaussian mode fit. (c) The release rate Γ release from a separate experiment with continuous measurement and hidden Markov model analysis performed during the same cooldown as the pulsed ensemble measurement. The orange star marks the value of the clearing rate as estimated from the exponential fit to the mean occupation in panel (b) at the pulse power of -136 dBm. (d) The mean occupation from the continuous data fit with hidden Markov model is shown in blue. The grey dashed line marks the mean occupation atτ delay = 0 from the fit in panel (b), which can be interpreted as the low power saturation. The orange star marks the estimated mean occupation from the fit in panel (b) if the pulse were to stay on indefinitely at -136 dBm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 6.4 The release rate is plotted against trap depth∆ A and microwave measurement powerP , grouped into 3 regions of trap depth. Dashed lines in the right panel show linear fits to the log space data for each of these regions. The shallowest traps (≤ 7 GHz) in green have the steepest power dependence with a slope A ≈ 1.03. The midrange trap depth (8.5 to 10 GHz) in yellow has a slope of A≈ 0.61. The deepest traps (≥ 12 GHz) have slopeA≈ 0.44. . . . . . . . . . . . 101 xiv 6.5 The mean occupation ¯n (top), trap rateΓ trap (middle), and release rateΓ release (bottom) at constant temperature T = 37 mK is shown as a function of trap depth∆ A (left) and measurement powerP (right). In the left side panels we can see that the release rate and mean occupation saturate at the lowest powers. In the bottom left panel we see that the low power saturation of the release rate increases slowly with trap depth until it peaks at∼ 10.5 GHz; a secondary peak can be seen around 13 GHz. The trap rate is mostly power independent, and any apparent power dependence may be explained by SNR degradation affecting the hidden Markov model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 B.1 TWPA tune up results. The highest SNR (23 to 24 dB) is in this case achieved at the bright yellow points, such as pump frequency 8.042 GHz at pump power 4 dBm. 131 xv Abstract We provide a proof of principle demonstration for the use of aluminum nanobridge Josephson junctions integrated into a superconducting-qubit-like device for the continuous, real-time de- tection of quasiparticles trapping into Andreev bound states. Analysis methods are developed to determine the rates of quasiparticles trapping and clearing, as well as estimating the occupation from the time series voltage record. To test the efficacy of our analysis, we employ statistical sim- ulations and score the ability of hidden Markov models to produce meaningful information from the noisy data. We find that the hidden Markov model is an effective analysis tool even when the underlying assumption of Markovianity is invalidated. By providing a good initial guess for the modes at high power, we are able to iterate our way to very low powers by using each fit as a warm start for the next fit. Simulation results show that this method retains accurate estimates even at low SNR. The rate of quasiparticles trapping and clearing from Andreev bound states are explored over a range of environment parameters. From the dependence on the trap depth and temperature, we show that quasiparticles are relaxing from the bulk superconductor into the Andreev traps by spontaneous emission from a temperature dependent bulk population with a substantial non- equilibrium component. Similarly, for temperatures above∼ 90 mK, quasiparticles are cleared from the traps due to absorption from a thermal population of phonons with energy exceeding xvi the trap depth. While the trap rate appears to be dominated by spontaneous emission at all temperatures, the release rate saturates below∼ 60 mK to a level which depends on the power of our measurement signal. The power dependence of the rate of clearing quasiparticles can be grouped into three regions of different behavior. At the shallowest traps the rate increases ∼ linearly with power. In the mid- depth region when the trap depth is more than two photons, the dependence is sublinear, fitting a power law with exponent∼ 0.6. The last region fits a power law with exponent ∼ 0.4. A theoretical explanation is discussed which produces the linear dependence observed when the trap depth is less than two photons. Further complexity is found in the background release rate at low temperature and low measurement power. The background release rate increases slowly with trap depth to a peak around 10.5 GHz, then falls drastically. xvii Introduction From quantum bits to telescopes, a large population of quasiparticles poses a limit to the effective- ness of superconducting circuits. These quasiparticles can induce a variety of effects depending on the system. For researchers in superconducting quantum bits (qubits) these quasiparticles may exchange energy with the qubit or alter the qubit frequency [25, 69, 75] , limitingT 1 andT 2 coherence times. Even when quasiparticle populations are extraordinarily low [71], rare bursts of quasiparticles can induce correlated errors [75, 51, 83, 49] that are more difficult to address with quantum error correction algorithms. For researchers in astronomy, quasiparticles can limit the sensitivity of microwave kinetic inductance detectors which are used in some cutting-edge telescopes [81]. At 10 mK, the operating temperature of superconducting qubits, the thermal population is negligibly small. However, for decades researchers have repeatedly found that the steady state quasiparticle population is many orders of magnitude greater than the equilibrium population es- timate [4, 50, 12, 65, 82, 79, 22]. This led to the common usage of the term “non-equilibrium quasi- particles” to describe this problematic population arising from non-thermal sources. These non- equilibrium quasiparticles can come from sources including infrared light [7], cosmic rays and other high-energy radiation sources [75, 11, 51, 83], and materials defects [39]. Mitigation strate- gies such as improved light-tight shielding [7], input/output filtering with infrared absorbers [14, 1 62, 68], and device engineering [15, 67, 29, 64, 38, 57, 8] have reduced quasiparticle densities over the last decade. However, the lowest reported non-equilibrium populations in recent works [68, 75, 48] still present a problem for the long term development of superconducting quantum computers. Many works have probed quasiparticle populations by detecting single charge tunneling across Josephson junctions [4, 53, 46, 70, 72, 66, 69, 68, 38] or observing quasiparticles trapped in- side the Andreev bound states of a junction [85, 44, 55, 31, 28, 20, 27]. These Andreev bound states provide a complementary measurement of quasiparticle behavior, and can be used as qubit modes themselves [31, 26]. In many implementations the Andreev qubit relies on a non-equilibrium quasiparticle trapping in order to initialize the state; such qubits are vulnerable to additional trap- ping events and to accidental clearing of the quasiparticle from the Andreev bound state. There is thus a great need to better understand the behavior of quasiparticles in Andreev bound states and the mechanisms for quasiparticle transitions between Andreev states and bulk continuum states. Furthermore, quasiparticle traps have been proposed as a tool to mitigate quasiparticles’ effects on qubits [72, 82, 67, 49], as quasiparticles may diffuse great distances after being generated [58, 51, 83]. The trapping process itself is thus worthy of study, in addition to providing insight into bulk quasiparticle behavior. In this thesis, I present a study of quasiparticle trapping dynamics in the Andreev bound states of aluminum nanobridge Josephson junctions. Chapter 1 introduces foundational concepts cen- tral to this work such as superconductivity, quasiparticles, and Josephson junctions. In Chapter 2 I introduce microwave resonators and quasiparticle effects. Chapter 3 examines experimental de- tection of Andreev bound state occupation and analysis methods for time series data. Chapter 4 utilizes simulated results to validate our time series analysis. In Chapter 5 I discuss measurement 2 and modeling of electron-phonon interactions which mediate trapping and clearing of quasipar- ticles from the Andreev bound states of our aluminum nanobridge junctions. Chapter 6 describes the effect of driven electron-photon interactions. This thesis ends with some concluding remarks about the experimental results. There are additional appendices which detail the way we fabricate devices, practical considerations when tuning Josephson amplifiers, and provide some of the code used in this work. 3 Chapter1 Background In this chapter we discuss some of the scientific background that lead to this work. We first start with a brief introduction to superconductivity and Bogoliubov quasiparticles. Next, we introduce the Josephson junction —a critical component of this work and many others. We discuss the Andreev picture of Josephson junctions and examine the effects of Andreev bound states with large transmission coefficients. Finally, we introduce the aluminum nanobridge Josephson junction and derive the expected current phase relation and inductance. 1.1 BCSsuperconductivityandBogoliubovquasiparticles The groundwork for this thesis requires only a basic introduction to the microscopic theory of superconductivity formulated by Bardeen, Cooper, and Schriefer (BCS) [6]. Here I will give a simple introduction to key concepts in the BCS theory with some extra attention given to the ex- cited states (Bogoliubov quasiparticles, but in this work simply “quasiparticles” will always refer to these excited states). For further reading I recommend Chapter 3 of Tinkham’s Introduction to Superconductivity [30]. 4 A central tenet of BCS theory is the formation of stable pair bound states called Cooper pairs linking electrons of opposite spin σ and momentum k – or more generally electrons in time- reversed states – in some materials when cooled below a critical temperature T c . The Cooper pairs can be represented by the pairing of electron creation operators, leading to the BCS ground state |ψ BCS ⟩= Y k>k F u k +v k c † k↑ c † − k↓ |ϕ 0 ⟩ (1.1) where c † kσ creates an electron of spin σ and momentum k, |ϕ 0 ⟩ is the vacuum state with no particles, the amplitudes|u k | 2 +|v k | 2 =1, and the product is over momentums greater than the Fermi momentumk F .|v k | 2 is the probability that the Cooper pair(k↑,-k↓) is occupied while |u k | 2 is the probability that pair is unoccupied. u k andv k may differ by phase e iϕ , whereϕ will later be called the phase of the condensate wavefunction. In addition to the many body wavefunction above, a microscopic theory needs a Hamiltonian that describes the interaction of these electronic states. The BCS pairing-hamiltonian approxi- mates the interaction potential as a constantV eff =−| g eff | 2 to obtain H = X kσ ϵ k c † kσ c kσ −| g eff | 2 X k,k ′ c † k↑ c † − k↓ c k ′ ↑ c − k ′ ↓ . (1.2) We can see from Equation 1.1 that the BCS ground state has all available electrons grouped into Cooper pairs at the Fermi energy. In contrast, an excitation of the BCS superconductor, which we call a “quasiparticle,” is a single particle state with electron-like and hole-like properties. These 5 quasiparticles are represented by the Bogoliubov transformation [9, 74] which we derive here. We introduce the BCS gap parameter ∆= |g eff | 2 X k ⟨c − k↓ c k↑ ⟩ (1.3) and approximate the hamiltonian Equation 1.2 with constant mean particle number as H = X kσ (ϵ k − µ )c † kσ c kσ − X k ∆ ∗ c -k↓ c k↑ +c † k↑ c † − k ′ ↓ . (1.4) This hamiltonian is diagonalized by the Bogoliubov transformation c k↑ =u † k γ k0 +v k γ † k1 (1.5) c † − k↓ =− v † k γ k0 +u k γ † k1 (1.6) or γ k0 =u k c k↑ − v k c † − k↓ (1.7) γ † − k1 =v † k c k↑ +u † k c † − k↓ (1.8) The Bogoliubov operators γ k0 and γ † k1 obey the standard fermion creation-annihilation an- ticommutation laws and each participate in the creation of an electron with momentum− k or annihilation of an electron (creation of a hole) with momentum k. In either case, the result is a reduction of the system momentum byk. The conjugateγ k1 instead creates an electron atk or a hole at− k, thus increasing the system momentum byk. 6 +Δ −Δ E F Figure 1.1: The electron density of states for BCS superconductors. The Cooper pairs condense at the Fermi energy E F , while a gap opens at± ∆ inside which no electron states are allowed. Quasiparticles must create electrons and holes beyond this gap. The diagonalized hamiltonian is H = X k (ϵ k − E k +∆ b † k )+ X k E k (γ † k0 γ k0 +γ † k1 γ k1 ) (1.9) where E k = q ϵ 2 k +|∆ | 2 (1.10) is the excitation energy of these Bogoliubov quasiparticles, and it is now evident that ∆ is an energy gap for these quasiparticles since even forϵ k = 0 at the Fermi surface, the quasiparticle energy will be∆ . This gap leads to an electronic density of states as illustrated in Figure 1.1. In this picture, all of the cooper pairs are condensed at the Fermi energy and quasiparticle excitations are excluded in the region± ∆ about the Fermi energy. Thus, to break a Cooper pair would require the promotion of two electrons from the Fermi energy to the density of states above the gap, requiring at least2∆ energy. 7 1.2 AndreevpictureofJosephsonjunctions The Josephson junction is among the most important discoveries in superconductivity. It is used extensively in many technologies including quantum bits [35], magnetometers [41], ultra-low noise amplifiers [5], digital logic circuits [59], and photon detectors [80]. Physically, a Josephson junction is formed by linking two superconductors together through a weak boundary. This is most commonly done by creating a thin oxide layer between two superconducting films. Joseph- son junctions have the unintuitive ability to sustain a DC current in the absence of a voltage bias and produce an AC current when a DC voltage is applied. This behavior was first predicted by Brian Josephson in 1962 [33] and observed by Anderson and Rowell in 1963 [1]. In his seminal paper, Josephson shows that Cooper pair tunneling across the junction gives the current-phase relation (CPR) I(δ )=I c sin(δ ) (1.11) in response to a phase bias δ = ϕ R − ϕ L , where ϕ L (ϕ R ) is the phase of the BCS condensate wavefunction on the left(right) side andI c is the critical current which depends on the junction properties. In the Andreev picture, a Josephson junction is a boundary between two superconductors which hosts many parallel conduction channels. Each channel is formed by a pair of Andreev bound states (ABS) – a bound state formed by multiple Andreev reflections at the boundary edges in which an electron with momentum k is spectrally reflected as a hole in the time reversed state with momentum − k [2]. This reflection of an electron into a hole induces a 2e charge transfer into the superconductor, creating a Cooper pair. At the opposite boundary interface, the hole is reflected in a time reversed state as an electron and 2e is removed. This multiple 8 φ L φ R (a) (b) Figure 1.2: The Andreev picture of Josephson junctions. (a) Two regions of superconductor (blue) with order parameter phase ϕ L and ϕ R are separated by a boundary (tan). Cooper pairs tunnel across this boundary by multiple Andreev reflections which form the Andreev bound states pic- tured in the boundary region. (b) The Andreev bound state energies are shown as a function of the phase bias across the junction for transmission coefficients of 0.7 (dashed) and 1 (solid). Red curves show the positive ABS while blue curves show the negative ABS. Andreev reflection is what mediates the tunneling of Cooper pairs across the junction. This process is illustrated in Figure 1.2, where the tan region represents the junction between the two blue superconductors. The pair of ABS in thei th channel has energy E A,i± (δ )=± ∆ s 1− τ i sin 2 δ 2 , (1.12) measured from the Fermi energyE F , whereδ =ϕ R − ϕ L is the phase bias across the junction,∆ is the superconducting gap, andτ i is the transmission coefficient of the i th channel —the probability that an incident Cooper pair tunnels across the channel. The negative ABS is normally occupied 9 as it is below the Fermi energy, and is responsible for mediating the supercurrent. The individual current contribution from thei th channel is found from the derivative of the ABS energy, I i− (δ )= 1 φ 0 ∂(E A,i− ) ∂δ =+ ∆ 4φ 0 τ i sinδ q 1− τ i sin 2 δ 2 , (1.13) where φ 0 = Φ 0 2π is the reduced flux quantum and Φ 0 = h 2e is the magnetic flux quantum —the smallest unit of magnetic flux that can thread a loop of superconductor. The total supercurrent can then be found by summing overN e channels, producing I(δ )= Ne X i=1 ∆ 4φ 0 τ i sinδ q 1− τ i sin 2 δ 2 (1.14) For standard superconductor-insulator-superconductor junctions (the junction of choice for most superconducting qubits), we would work in the limit of N e >> 1, τ ≪ 1. Taking a Taylor expansion aroundτ i =0 we arrive at I(δ )= Ne X i=1 ∆ 4φ 0 τ i sinδ ≈ ∆ 4φ 0 N e τ sinδ. (1.15) Allowing I 0 ≡ ∆ 4φ 0 N e τ recovers the canonical Josephson current-phase relation given in Equa- tion 1.11. However, as I show in the next section we may get more interesting results forτ ≈ 1. 1.3 AluminumnanobridgeJosephsonjunctions Most Josephson junctions are formed by joining two superconductors across a thin insulating boundary, resulting in an SIS junction. However, a Josephson junction can be formed by any 10 200 nm Figure 1.3: SEM image of an aluminum nanobridge Josephson junction. A thin constriction in the center bridges the gap between the left and right leads. This nanobridge is approximately 20 nm wide, 8 nm thick, and 100 nm long. The leads are approximately 800 nm wide and 80 nm thick. interface which provides an appropriate phase boundary between the two superconductors. In fact, the boundary need not be a separate material at all —a simple geometric constriction can provide the appropriate phase boundary between two regions of superconductor. For an excellent review of various types of Josephson junctions, see Golubov, Kupriyanov, and Il’ichev [23]. Here I summarize the properties of nanobridge Josephson junctions which have been studied extensively in earlier work [43] which considered nanobridges formed with the same film thick- ness as bulk electrodes (referred to as 2D or Dayem bridges) and those formed with a thinner film than the electrodes (3D or variable-thickness bridges). In this work, we focus only on 3D nanobridge geometries which we will refer to simply as nanobridges or nanobridge junctions. The nanobridge is an SS’S junction formed from a single material in which the S’ portion of the junction is defined by a physical constriction providing a weak-link between the bulk electrodes, as seen in Figure 1.3. This nanobridge junction is not characterized by the canonical Josephson current-phase relation (Equation 1.11) but requires a more complex treatment. There is a simple way and an accurate way to derive the current-phase relation of a nanobridge junction. The accurate way is to solve the Usadel equations [73] for the exact geometry of the device. This was 11 done by Kulik and Omelyanchuk [37] for an ideal filament of diameter a and lengthL satisfying a ≪ L ≪ ξ connecting two large electrodes which act as rigid phase reservoirs, where ξ is the coherence length. Kulik and Omelyanchuk derived their result, known as the KO-1 current- phase relation, in the dirty limit of superconductivity (mean free path l ≪ ξ ), meaning that quasiparticle transport is diffusive. However, it has been shown [76] that for 3D nanobridges of length L ≤ 4ξ the current-phase relation approaches the KO-1 result. We’ll derive the KO-1 current-phase relation using a simpler model below. 1.3.1 KO-1current-phaserelation We model the ideal nanobridge as a parallel combination of N e effective conduction channels, each with a transmission coefficient τ i . Each channel, denoted by the subscripti, has a single pair of Andreev bound states with energies given by Equation 1.12 and current contribution given by Equation 1.13. To find the current-phase relation for the full junction, we need to integrate the current per channel (Equation 1.13) over the distribution of transmission coefficients. In the dirty limit of superconductivity (l ≪ ξ ), a zero-length contact between superconducting electrodes can be described as a parallel combination of conduction channels with transmission coefficients given by the Dorokhov distribution [18] ρ (τ )= 1 τ √ 1− τ . (1.16) 12 Integrating the current per channel (Equation 1.13) over the Dorokhov distribution of channel transmittivities (Equation 1.16), we obtain the current-phase relation I(δ )= Z 1 0 ρ (τ )I − (τ,δ )dτ = ∆ N e 4φ 0 cos δ 2 tanh − 1 sin δ 2 . (1.17) The equation above is the KO-1 current-phase relation with N e = 4φ 0 π/eR N , where R N is the normal state resistance of the junction [37]. Considering [76] verified that this current-phase relation approximately describes aluminum nanobridges of up to 4 coherence lengths, we assume that the Dorokhov distribution of channel transmittivities is a reasonable approximation as well. 1.3.2 Josephsoninductance When a Josephson junction has a DC voltage bias, it will produce AC current at a frequency proportional to the applied voltage. This is known as the AC Josephson effect and was predicted in 1962 by Brian Josephson [33]. In this section, we use the relation between applied voltage and the changing phase biasδ to derive the inductance of a Josephson junction. The voltage is related to the current by V =φ 0 ∂δ ∂t . (1.18) Consider that the standard relation between voltage and inductance isV =L ˙ I =L ∂I ∂δ ∂δ ∂t . Solving forL, we have L= V ∂I ∂δ ∂δ ∂t (1.19) 13 Comparing with Equation 1.18 we can swap V ∂δ ∂t = φ 0 and obtain the inductance for a Josephson junction as L J = φ 0 ∂I ∂δ . (1.20) In general, this inductance will depend on the current phase relationI(δ ). In our case, we use the KO-1 current phase relation (Equation 1.17) to obtain the inductance of an aluminum nanobridge Josephson junction L J (δ )= 8φ 2 0 ∆ N e 1− sin δ 2 tanh − 1 sin δ 2 − 1 . (1.21) This inductance will be useful later when characterizing nanobridge resonators (Chapter 2). 1.3.3 Quasiparticletrappinginaluminumnanobridges We can see from the Andreev bound state energies shown in Figure 1.2 that for a channel with non-zero transmission coefficient, the Andreev energy is below the gap when there is a phase difference across the junction. Quasiparticles in the electrodes must have energies above the gap, so it is energetically favorable for quasiparticles to relax into and become trapped in the unoccupied positive energy states (Equation 1.12). When a quasiparticle traps, the supercurrents (Equation 1.13) carried by each state in the channel interfere destructively. This cancellation has been demonstrated in point contact junctions [84]. In contrast to point contacts, the nanobridge geometry has a surplus of conduction channels, so the elimination of one or several channels due to quasiparticle trapping results only in a reduction of the overall current of the junction and therefore an increased Josephson inductance. 14 Chapter2 Nanobridgeresonators Microwave resonators are used extensively in many technologies such as quantum processors, wireless communications, and even in telescopes. A promising avenue towards realizing use- ful quantum computation extensively uses lumped element or transmission line microwave res- onators to couple and readout qubits. In some cases, the resonator modes can be used as qubits themselves [40]. Microwave resonators are also often used as detectors, such as microwave ki- netic inductance detectors (MKIDs) which are used in some cutting-edge telescopes [81]. In this chapter, I introduce microwave resonators, discuss a few common types, and provide some models for their behavior. Next, I introduce the effects of including a Josephson junction in a resonator and derive some of the expected behaviors that we use to characterize a device. We end with a derivation of the frequency shift expected when a quasiparticle traps in the junctions. 15 C g λ/4 C g L C L e C e (a) (b) Figure 2.1: Images and schematics of microwave resonators. (a) A lumped element resonator. Signal is capacitively coupled (orange) to the resonator and has two paths to ground; the inter- digitated capacitor (red and blue) and the meandering inductance (green). (b) A transmission line resonator. Signal is capacitively coupled (orange) to the center trace (blue) of the co-planar waveguide which is shorted on the other end, forming aλ/ 4 resonant mode. This resonant mode could be described by an equivalent circuit (like the lumped element schematic in panel (a)) with effective inductance L eff and capacitanceC eff . 2.1 Circuitdescription There are two fundamentally different types of superconducting microwave resonator that a re- searcher is likely to work with: transmission-line resonators and lumped-element resonators. An example of each is given in Figure 2.1. The microwave scattering off these resonators is essentially identical, but the physical imple- mentation differs. Transmission line resonators can be likened to the resonant modes of wind or 16 string instruments, where the length of the medium determines the resonant wavelengthλ . Most transmission-line resonators use a co-planar waveguide (CPW) geometry with the center trace capacitively broken on one or both ends, making it aλ/ 4 orλ/ 2 resonator, respectively. That is to say, the length of the center trace is a quarter or half of the resonant wavelengthλ . This is of course an oversimplification —in reality the resonant frequency and wavelength will be affected by stray inductance and capacitance arising from the resonator geometry, but the trace length provides a good approximation. Lumped-element resonators are most similar to a resonant LC circuit one might study in an introduction to electronic circuits. The resonator is physically implemented by combining mi- crometer scale capacitive and inductive elements. If one knows the inductanceL and capacitiace C of these elements, then the resonant frequency is given by the familiarω 0 = (LC) − 1/2 . How- ever, it is rarely the case that researchers are able to know the precise values ofL andC, including any stray inductances/capacitances, so it is common to iterate a few rounds of design modification and simulation using commercially available RF simulators such as Cadence Microwave Office or Ansys HFSS. All of these resonator types share some common terminology for parameters. The quality factor of a resonator, given in terms of the resonant frequencyω 0 and linewidthκ is given by Q= ω 0 κ . (2.1) 17 This is commonly called the “total” or “loaded” quality factor. It is determined by the coupling strength to the microwave feedline and the internal loss of the resonator in terms of the coupling quality factorQ c and internal quality factorQ i Q − 1 =Q − 1 c +Q − 1 i . (2.2) The scattering behavior of any resonator probed in the frequency domain can be modeled using the complex Lorentzian lineshape L(ω)= Q Qc h 1− 2iQ ω− ω 0 ω 0 i 1+4Q 2 ω− ω 0 ω 0 2 , (2.3) where ω is the probe frequency and ω 0 is the resonant frequency. It can be seen thatL(ω) ≈ 0 when far off resonance and L = Q/Q c when ω = ω 0 . It is worth pointing out here that Equation 2.2 guarantees thatQ/Q c ∈(0,1). Now it becomes obvious that to see any appreciable response from the resonator, one needs sufficient coupling strength —a weakly coupled (high Q c ) resonator will give little to no measurable signal. It is customary to define three regions of coupling: a resonator is “undercoupled” when Q/Q c < 0.5, “critically coupled” if Q/Q c = 0.5, and “overcoupled” forQ/Q c > 0.5. When high signal-to-noise ratio (SNR) is crucial, such as in the work presented in Chapters 3, 5, and 6, thenQ/Q c must be maximized. Researchers often have constraints on the resonator linewidthκ = ω 0 Q —such as ensuring an expected resonance shiftχ is an appropriate fraction of the linewidth or that the resonator response rate (equal to linewidth) is faster than system dynamics. Thus, it is usually ideal to maximize the internal quality factor 18 Q i such that we are working in the limitQ≈ Q c . This maintains optimal SNR while allowing us to set the linewidth by adjusting the coupling strength. 2.1.1 Reflection,transmission,andhangergeometries Resonators can further be categorized based on the expected scattering behavior into reflection, transmission, and hanger geometries. In many ways, reflection resonators are the simplest ge- ometry, requiring only a single microwave port which couples the incoming and outgoing waves on the same microwave feedline. For single port reflection resonators, the scattering parameter is given by S 11 (ω)=ae i(ωτ +ϕ ) [2L(ω)− 1], (2.4) wherea scales the response based on the overall attenuation and gain on the measurement line, the electrical delay τ is the time it takes a signal to travel the measurement path, and ϕ is an arbitrary phase rotation. It is often beneficial to consider what happens to signals which are near resonance and far off resonance. In the reflection resonator, signals which are far off resonance see a reflection coefficient |S 11 |≈− a while near resonant signals are have|S 11 |≈ a 1− 2 Q Qc . A transmission resonator is a two port device which will filter off-resonant signals, transmitting only those signal near resonance to the second port. When measured across the two ports, the scattering coefficient is S 21 (ω)=ae i(ωτ +ϕ ) L(ω). (2.5) Hanger type resonators are perhaps the most versatile, having an uninterrupted transmission line from the first port to the second allows unimpeded transmission of a broad range of signals. 19 The hanger type resonator is coupled by inductance or capacitance to this transmission line. The scattering coefficient across the two ports is given by S 21 (ω)=ae i(ωτ +ϕ ) [1−L (ω)]. (2.6) Hanger type resonators are the most common choice for quantum processors and most photon detectors as researchers can couple many resonators, with staggered resonant frequencies, to the same transmission line. 2.1.2 Nanobridgeresonators The description of microwave resonators given above encompasses those resonators which are purely linear. The inclusion of a nanobridge junction (or any other Josephson junction) would make the resonator nonlinear —the scattering coefficient depends on the oscillation amplitude. This is in principle how one makes a qubit from a superconducting circuit, but our linear induc- tance and capacitance are much larger so the energy potential forms a very deep well and the device behaves like a classical resonance. A common practice is to put two Josephson junctions in a small loop as shown in Figure 2.2, a geometry which is known as a DC Superconducting QUantum Interference Device (SQUID). This SQUID loop uses the principle of flux quantization —the magnetic flux threading a loop of superconductor must be an integer multiple of Φ 0 = h 2e , the magnetic flux quantum —to allow for easy tuning of the phase bias δ across the junctions. The idea here is that you apply some external magnetic field which threads the flux Φ through the loop and flux quantization requires the induction of a persistent current to bring the total flux Φ tot =Φ+Φ ind to the nearest integer 20 b) a) Figure 2.2: (a) Schematic diagram of a lumped element nanobridge resonator. (b) False-color SEM images of an aluminum nanobridge resonator; interdigitated capacitor is shown in red and blue, while the meandering inductance (green) has the inclusion of a nanobridge SQUID (purple and cyan). multiple of the flux quantum, i.e., Φ ind = nΦ 0 − Φ for some integer n. This induced current creates a phase bias across each junction equal to± δ = ∓ π Φ ind Φ 0 , where we have both positive and negative phase bias because the induced current is in an opposite direction for each junction in the loop. We typically work with small fields and assume that n=0 so we takeδ =πϕ where ϕ = Φ Φ 0 is the applied magnetic flux in units of the flux quantum. We embed a DC SQUID with a pair of symmetric nanobridge junctions in series with the center trace of aλ/ 4 CPW resonator at the voltage node, as shown in Figure 2.2. This results in 21 adding a little extra inductanceL S (δ )=L J (δ )/2 to the resonance such that we have ω 0 (δ )=[C eff (L eff +L S (δ ))] − 1/2 , (2.7) whereL eff andC eff are the effective linear inductance and capacitance of the equivalent circuit to our λ/ 4 transmission line resonator. Note that the resonant frequency of our resonator now becomes a function of the phase bias, which we can tune with the applied magnetic flux. In the following section, we derive a model for this flux tunable resonance and use it to determine the amount of shift we expect from a single electron occupying an Andreev bound state. 2.2 Fittingthefluxtuningcurve In the above section we see that embedding a DC SQUID in the resonator enables us to tune the resonant frequency in situ by applying a magnetic field. The degree to which the resonant fre- quency changes is dependent on the ratio of Josephson inductance to the total inductance, which we call the participation ratio q(δ ). This participation ratio is useful to determine the resonant frequency shift that results when a quasiparticle traps in an Andreev bound state, poisoning a conduction channel. To determine this participation ratio, we measure the resonant frequency as a function of applied field and fit the data to the flux tuning model we derive here. We assume we have two symmetric junctions which each approximately obey the KO-1 current-phase relation (Equation 1.17) of an ideal diffusive weak link. Because the junctions are in parallel and symmetric, and if we assume the geometric inductance of the SQUID loop 22 is small, then the total inductance of the SQUID will be half of the Josephson inductance derived in Equation 1.21, L S (δ )= L J (δ ) 2 . (2.8) Using the total inductance of the resonator L(δ ) = L eff + L S (δ ), we can write the resonant frequency as ω 0 (δ )= 1 p C eff (L eff +L S (δ )) . (2.9) If we define the resonant frequency at zero phase bias ( δ = 0) as Ω 0 = 1 q C eff(L eff +L S (0)) , then we can rewrite Equation 2.9 as ω 0 (δ )= 1 p C eff (L eff +L S (0))+C eff ∆ L S (δ ) = Ω 0 q 1+ ∆ L S (δ ) L eff +L S (0) , (2.10) where ∆ L S (δ )=L S (δ )− L S (0) (2.11) is the change in Josephson inductance due to phase biasδ . The inductance participation ratio of the squid is q(δ )= L S (δ ) L eff +L S (δ ) , (2.12) and we define the participation ratio at zero phase bias as q 0 = q(0). We can substitute q 0 in Equation 2.10 to obtain ω 0 (δ )= Ω 0 q 1+q 0 ∆ L S (δ ) L S (0) . (2.13) 23 Now we can use the definitions of ∆ L S (δ ) (Equation 2.11),L S (δ ) (Equation 2.8), andL J (δ ) (Equa- tion 1.21) to express the inductance ratio in terms of only trigonometric functions of the phase bias, ∆ L S (δ )= L J (δ )− L J (0) 2 =− 4Φ 2 0 ∆ N e sin δ 2 tanh − 1 sin δ 2 1− sin δ 2 tanh − 1 sin δ 2 (2.14) and L S (0)= 4Φ 2 0 ∆ N e . (2.15) All together, the model for the resonant frequency as a function of phase biasδ is ω 0 (δ )=Ω 0 " 1− q 0 sin δ 2 tanh − 1 sin δ 2 1− sin δ 2 tanh − 1 sin δ 2 #− 1 2 . (2.16) Thus, if we measure the resonant frequency while sweeping the current through a coil which couples magnetic flux through the SQUID, then we can fit Equation 2.16 with the zero flux reso- nanceΩ 0 and the Josephson inductance participation ratio at zero flux q 0 as free parameters. An example is shown in Figure 2.3. In fitting this curve, it is necessary to calibrate the conversion from the bias current to the magnetic flux ϕ = Φ Φ 0 . The easiest way to do this is to measure the resonant frequency over a broad range of currents such that multiple peaks are seen. Then you calibrate the conversion by dividing the number of peaks (which equals the number of flux quanta we’ve swept through) by the change in current between the peaks with highest and lowest current. The top and bottom axes show the flux ϕ and the bias current, respectively. 24 −4 −2 0 2 4 Bias current [mA] 4.24 4.25 4.26 4.27 4.28 4.29 4.30 Frequency [GHz] F/F 0 0.0 0.5 0.1 0.2 0.3 0.4 -0.1 q 0 = 0.01341 fit Resonant Frequency 0.6 Figure 2.3: The extracted resonant frequency as a function of flux. The solid curve is the fit to Equation 2.16. We extract a Josephson inductance participation ratio at zero flux of q 0 =0.01341. We see hysteretic branches in the flux tuning, as expected for a SQUID with junctions which have non-sinusoidal CPRs. 25 2.3 Quasiparticleinducedresonanceshift As discussed in Section 1.3.3, quasiparticles in the bulk superconductor will energetically favor relaxing from their initial energyE i ≳ ∆ to the Andreev bound states present in the aluminum nanobridge junction with E A (δ ) < ∆ for nonzero phase bias δ . These quasiparticles become trapped in the positive Andreev bound state, poisoning the channel and leading to an increased Josephson inductance. In the nanobridge resonators discussed above, this means the resonator frequency will shift each time a quasiparticle traps in or clears from the Andreev bound state. In this section, we determine how much of a shift we expect based on the Josephson participation ratio (Equation 2.12). To start with, we assume the two junctions are symmetric (meaning they each have the phase bias δ 1 = − δ 2 = δ ) and that the number of conduction channels is large compared to the mean number of poisoned channels N e ≫ ¯n. The second assumption ensures that poisoning one or a few channels does not significantly alter the phase bias and δ 1 ≈− δ 2 is maintained. Let’s consider what happens when a quasiparticle traps in the i th channel. The resonant frequency as given in Equation 2.9 becomes ω 0,i (δ )= 1 p C eff (L eff +L S (δ )+ϵ i ) , (2.17) whereϵ i is the small increase in inductance due to trapping a quasiparticle in channeli. We can rewrite this as ω 0,i (δ )= ω 0 (δ ) r 1+ ϵ i L eff +L S (δ ) . (2.18) 26 Sinceϵ i ≪ L eff +L S (δ ) for any channeli, then we can take the binomial approximation to write this as ω 0,i (δ )=ω 0 (δ ) 1− 1 2 ϵ i L eff +L S (δ ) . (2.19) Substituting the participation ratioq(δ )= L S (δ ) L eff +L S (δ ) gives ω 0,i (δ )=ω 0 (δ ) 1− q(δ ) 2 ϵ i L S (δ ) . (2.20) The problem is now to determine the change in inductanceϵ i . Consider that the total SQUID inductance is from the parallel combination of many Andreev channels, soL − 1 S (δ )= P j L − 1 j (δ ). When a quasiparticle traps in channeli, then we poison that channel and the SQUID inductance becomes L − 1 S,i (δ )= X j̸=i L − 1 j (δ )=L − 1 S (δ )− L − 1 i (δ ) (2.21) Inverting this gives L S,i (δ )=L ( S δ ) 1− L S (δ ) L i (δ ) − 1 (2.22) and allows us to Taylor expand to first order ( L S (δ ) L i (δ ) ≪ 1 when we have a large number of channels N e ) and get L S,i (δ )≈ L S (δ ) 1+ L S (δ ) L i (δ ) . (2.23) Finally, we can consider thatϵ i =L S,i (δ )− L S (δ ) so we find ϵ i L S (δ ) = L S (δ ) L i (δ ) . (2.24) 27 The resonant frequency due to trapping a quasiparticle in channeli is found by substituting Equation 2.24 into Equation 2.20, ω 0,i (δ )=ω 0 (δ ) 1− q(δ ) 2 L S (δ ) L i (δ ) , (2.25) where L i (δ ) is found from the phase derivative of the i th channel current contribution (Equa- tion 1.13), L i (δ )=φ 0 ∂I i− (δ ) ∂δ − 1 = 2φ 2 0 ∆ τ i 1− τ i sin 2 δ 2 3 2 cosδ +τ i sin 4 δ 2 . (2.26) Or, to put it more simply, the resonant frequency of our device whenn quasiparticles are trapped is given by ω n (δ )=ω 0 (δ )− n X i=1 χ i (δ ) (2.27) where χ i (δ )=ω 0 (δ ) q(δ )∆ τ i 4φ 2 0 L S (δ ) cosδ +τ i sin 4 δ 2 1− τ i sin 2 δ 2 3 2 (2.28) is the frequency shift due to trapping a single quasiparticle in channeli. In general this shift will depend on the transmission coefficient τ i of the channel, but for Boltzmann distributed trapping into a Dorokhov distribution of channels, taking τ ≈ 1 is a safe assumption and the resonant frequency becomes a simple step function of the numbern of trapped quasiparticles, ω n (δ )=ω 0 (δ )− nχ. (2.29) 28 Chapter3 Nanobridgeresonatorsasquasiparticledetectors A crucial component of any scientific work is the ability to extract meaningful information from data, often in the presence of significant noise. In this chapter I discuss experimental measure- ments which validate the use of nanobridge resonators to measure quasiparticles trapping in An- dreev bound states and the analysis methods used to study the trapping dynamics. In my earlier time as a graduate student I stumbled for a while trying to make meaningful estimations of the quasiparticle occupation time series from the noisy resonator data. Ultimately, I developed some naive bayesian approaches which I discuss in Section 3.1.2 that are decent if the SNR is large, but eventually I learned about hidden Markov models (HMM) which I introduce in Section 3.2 and which are the main analysis tool for our time series data, as they perform extraordinarily well. 3.1 Measurementsofquasiparticletrapping In this letter we characterize a device optimized for continuous, non-saturating measurements of QP trapping in Andreev states. Using microwave reflectometry we are able to continuously detect 0, 1, and 2 or more trapped quasiparticles in 5 us with SNR of 27. By altering the detector bias we 29 are also able to distinguish 3, 2, and 1 or fewer QPs. We discuss straightforward improvements that can further improve SNR and allow detection of many more trapped QPs at a single bias point. Our device provides a prototype for detectors optimized for continuous measurements of the dynamics of Andreev states coupled to resonant cavities. At the time of publication, the work presented in this section utilized a Bayesian inference scheme we devised to estimate the Andreev bound state occupation time-series from noisy IQ voltage records. While the method described here works reasonably well for data with a high SNR, it is far outperformed by more advanced algorithms. The results in this section are included for completeness, but I would recommend any researcher attempting to solve a similar classification problem to refer instead to the discussion of hidden Markov models (HMM) in Section 3.2. 3.1.1 Experiment As discussed in Section 1.2 and Section 1.3, aluminum nanobridges provide many conduction channels which each host a pair of Andreev bound states with energies (relative to the Fermi energyE F ) E A± =± ∆ r 1− τ sin 2 δ 2 (3.1) where ∆ is the superconducting gap, τ is the transmission coefficient of the channel, and δ is the phase bias across the junction. At temperatures T ≪ ∆ /k B , the upper Andreev state is unoccupied and the lower state is occupied, carrying the supercurrent across the junction. This is the channel’s ground state with total energy− E A . QPs in the bulk only exist at energies greater than∆ (or as unoccupied states below− ∆ ), so it is energetically favorable for a QP to drop into the unoccupied upper Andreev state, bringing the channel to the odd state with 0 energy. This 30 poisons the channel, eliminating it from carrying supercurrent and increasing the Josephson inductance. The lower-level QP may also be promoted to the upper Andreev level, bringing the channel to the excited state and producing twice the inductance shift of the odd state; or this QP may be cleared from the junction completely, creating another degenerate 0-energy odd state with 0 supercurrent. Researchers have demonstrated continuous monitoring of QPs trapping in point contact [31] and semiconductor nanowire [28, 27] junctions; however, these junctions have few channels and so quickly saturate as QP detectors. Three dimensional aluminum nanobridge Josephson junctions achieve good phase confine- ment and nonlinearity in an all-superconducting design [77, 76, 42]. Importantly, nanobridges comprise many conduction channels (∼ 100− 1000) with τ ∼ 1, approximately following the Dorokhov distribution (Equation 1.16) [17]. When 2 identical nanobridges are placed in a loop, forming a superconducting quantum interference device (SQUID), the junctions’ phase biasδ is simply set by the flux bias Φ asδ = πϕ , whereϕ ≡ Φ /Φ 0 andΦ 0 is the flux quantum. In an Al nanobridge SQUID near half-flux ( δ ≈ π/ 2, a channel withτ ≈ 1 hasE A+ /h≈ 29 GHz, or a trap depth of(∆ − E A )/h≈ 12 GHz, far greater than the thermal energy at 15 mK. These junctions thus function as QP traps with many deep trap states when they are phase-biased. Our device consists of a co-planar waveguide (CPW) resonator in which the center trace is terminated by a nanobridge SQUID; see Figure 3.1(a) and (b) for an illustration and refer to Section 2.1 for a more detailed circuit description. The fundamental (quarter-wavelength) mode of our resonator with zero flux through the SQUID is at ω 0 (0)=2π × 4.302 GHz and has linewidth κ =2π × 250 kHz, largely set by the coupling to the microwave feedline. This device was imaged and found to have junctions that appear nearly identical visually; past studies indicate they may thus be treated as symmetric[76, 42]. Trapping in either junction produces a similar resonant 31 w d LO RF I Q Pump 4K 1K 10 mK Flux (b) (a) C g Z L (c) F 20 dB 20 dB 20 dB 20 dB 20 dB K&L K&L Ecco Ecco TWPA HEMT 200 nm ADC Figure 3.1: (a) Device schematic. A CPW resonator (green) is grounded via a two-junction Al nanobridge SQUID (magenta). Flux bias through the SQUID phase-biases the junctions asδ =πϕ . (b) Optical image of the device with inset SEM images of the SQUID (magenta) and a nanobridge junction (orange). (c) Simplified measurement schematic. A tone at ω d continuously drives the flux-tunable resonator. The reflected signal is amplified by a TWPA at base stage followed by a HEMT at 4K and room temperature amplifiers. Isolators between HEMT and room temperature amplifiers are not shown to conserve space. The amplified signal is homodyne demodulated and I and Q components are low-pass filtered at 15 MHz, then digitized by an Alazar ATS9371 at 300 MSa/s and down-sampled to 10 MSa/s before saving. Microwave lines are optionally filtered at base stage by K&L 12 GHz and custom Eccosorb 110 low-pass filters. frequency shift, and so for QP detector applications they may be thought of as a single junction with twice the number of channels. We note that the resonator energy per photon is much smaller than the trap depth at high flux bias, so absorption of resonator photons should not appreciably affect the quasiparticle states. Our measurement setup is shown schematically in Figure 3.1(c). A signal generator provides a drive tone atω d (ϕ )≈ ω 0 (ϕ )− κ/ 2. A power splitter sends half of this power into the dilution refrigerator where it is attenuated then (in some cooldowns) filtered by K&L 12 GHz and Eccosorb low pass filters. The drive tone is circulated to reflect off our device, which is flux tunable via a DC 32 coil in the packaging, and amplified by a travelling wave parametric amplifier[47] (TWPA) which is pumped at 8.078 GHz. The reflected signal is further amplified before IQ demodulation with reference to the original signal. The in-phase (I) and quadrature (Q) components are 15 MHz low pass filtered before digitization at 300 MHz sample rate by an Alazar ATS9371 analog-to-digital converter (ADC). Raw data is down-sampled to 10 MHz before saving. We first characterize our device with ensemble-averaged vector network analyzer (VNA) mea- surements of the resonance. Figure 3.2(a) shows resonance measurements at flux biases of ϕ =0 and ϕ = 0.49, taken on a cooldown in which the K&L and Eccosorb filters were removed. The ϕ =0.49 trace in orange shows two shallow peaks at∼ 0.5 and 1 MHz below the main resonance. These are the resonance peaks with 1 and 2 trapped QPs, respectively, showing the resonance shifting due to the change in nanobridge inductance. These ensemble measurements average over all possible QP trapping configurations, and so a resonance peak amplitude corresponds to the probability of that configuration. We then move on to time-domain IQ measurements as described above. Panels (b) and (c) are log-scale histograms showing 30 seconds of continuous IQ data for ϕ = 0 (in blue) and ϕ = 0.47 (in orange), respectively. The data shown has been integrated by convolving with a Gaussian window of effective integration time √ 2πσ =3µ s. In the finite-flux data of panel (c) we can immediately see 3 distinct modes with excellent separa- tion in the log scale plots. The darkest peak (lower left), with the most counts by far, is due to the response with 0 QPs in Andreev traps. The next darkest (upper center) is from having 1 QP trapped while the lightest mode (middle right) is from 2 QPs in Andreev traps and/or excitation of a single channel into the excited state. Trapping of more than 2 QPs moves the resonance multiple linewidths and thus saturates the change in response to additional trapping, so these counts lie on top of the 2-QP distribution. We later discuss methods for avoiding this saturation. 33 0 10 20 I [mV] -10 0 10 20 Q [mV] 10 0 10 2 10 4 10 6 0 10 20 I [mV] -10 0 10 20 Q [mV] 10 0 10 2 10 4 10 6 −2 −1 0 1 Detuning [MHz] -1 0 1 Re[S11] f = 0 f = 0.49 (a) (b) (c) Figure 3.2: (a) Ensemble measurement of resonator response at 0 flux (blue) and at ϕ = 0.49 (orange). When ϕ = 0.49, distinct peaks are visible roughly 0.5 and 1 MHz below resonance, corresponding to 1 and 2 trapped QPs, respectively. (b) Histogram of continuous IQ data taken at 0 flux for 30 s. Data has 10 MHz sample rate and has been convolved with a Gaussian window with effective integration time of 3 µ s. (c) Data taken at ϕ = 0.47 with the same procedure as panel b. The darkest mode is due to the resonance with 0 trapped QPs. The second darkest, located near (I,Q) = (12 mV, 15 mV), corresponds to 1 trapped QP and the last mode corresponds to 2 or more trapped QPs (as this mode corresponds to the resonance moving far from the drive frequency). 34 In Section 3.1.7, we probe at a frequency close to the 2-QP resonant frequency and are able to observe 0, 1, 2, and 3 QP modes, confirming that we do indeed see multiple QPs trapping and not simply the excited state of a single channel. We have verified that these modes are indeed due to Andreev trapping of QPs by measuring the weights of each mode in the presence of a “clearing tone” at 17 GHz. A trapped QP may absorb a photon from this tone and so be promoted back into the bulk continuum; the frequency was chosen to be greater than the trap depth and because it happens to couple efficiently into the device. We find that the tone causes the 1- and 2-QP mode counts decrease while the 0-QP mode counts increase. We have also observed that both the sep- arations and the weights of the modes increase as a function of flux, which agrees qualitatively with a QP trapping picture. 3.1.2 NaiveBayesiananalysis We now turn to extracting the QP trap occupation as a function of time from the continuous IQ data. To optimize our analysis procedures we choose data that stresses the detector’s capabilities. We use data from a cooldown with the K&L and Eccosorb filters, which reduces QP generation by infrared radiation and thus shows a much lower QP number than that shown in Figure 3.2 [69], and use a low resonator drive power chosen to ensure the drive does not affect the QP configuration. We increase the integration time to 5 µ s to compensate for this loss of SNR. We first fit histograms using a Gaussian mixture expectation-maximization algorithm implemented by the Python module available from scikit-learn[60]. This module takes in a subset of data, assigns each point to one of the specified number of modes, then tweaks assignments and mode parameters to maximize the total likelihood for all data and all modes. The result is a set of 3 Gaussian 35 modes describing the data, shown by their 1-σ (solid) and 2-σ (dashed) contours overlaying the histogram in Figure 3.3(a). We then assign each time-series point to the mode with the highest posterior probability using a Bayesian state identification algorithm (described below) to extract a rough occupation. Since we expect the resonator to remain in the same mode for many consecutive points in the time series, we apply Bayesian inference with a memory window of 3 points. I.e., the probability that thek th data point⃗ x k belongs to thei th mode is P(M i |⃗ x k ,⃗ x k− 1 ,⃗ x k− 2 )= P(⃗ x k |M i )P(M i |⃗ x k− 1 ,⃗ x k− 2 ) P 2 j=0 P(⃗ x k |M j )P(M j |⃗ x k− 1 ,⃗ x k− 2 ) where P(M i |⃗ x k− 1 ,⃗ x k− 2 )= P(⃗ x k− 1 |M i )P(M i |⃗ x k− 2 ) P 2 j=0 P(⃗ x k− 1 |M j )P(M j |⃗ x k− 2 ) and P(M i |⃗ x k− 2 ) is the posterior probability of the Gaussian mode M i conditioned on the data, defined as P(M i |⃗ x k )= P(⃗ x k |M i )P(M i ) P 2 j=0 P(⃗ x k |M j )P(M j ) , (3.2) Finally, the quasiparticle occupation at the k th point in the time series is taken as the mode M i with maximumP(M i |⃗ x k ,⃗ x k− 1 ,⃗ x k− 2 )∀ i∈{0,1,2}. Unfortunately, the Gaussian mixture fit is unreliable in terms of quality of fit and reproducibility of Gaussian mode parameters, which can vary significantly even when refitting the same data with the same initial guess. This unreliability is apparent from the fit distributions shown in Figure 3.3(a), which do not faithfully represent the means of the 1- and 2-QP modes. 36 (a) (b) (c) 4 12 20 −2.5 5.0 12.5 Q [mV] 4 12 20 I [mV] 10 0 10 2 10 4 10 6 0 0.5 6000 Time [ms] Signal [mV] 0 5 10 15 I Q 6000.5 I [mV] Figure 3.3: (a) Initial clustering of 5 µ s integrated data using the scikit-learn Gaussian mixture module produces modes with 1σ (solid) and 2σ (dashed) contours for 0, 1, and 2 or more trapped QPs in light blue, dark blue, and orange, respectively. (b) Subsets of the data in which the occu- pation is constant for a long time (4⟨τ i ⟩) are individually fit to Gaussian distributions. Means and covariances of each mode are then fixed and the full dataset is fit with mode weights as the only free parameters. (c) Two sections of time series data with I in brown and Q in dashed magenta. The background color is light blue, dark blue, and orange for 0, 1, and 2+ trapped QPs, respec- tively. All data taken atϕ =0.47. To improve the fits we need to “initialize” the trapped QP configuration, thereby isolating each Gaussian mode for independent fitting. This is challenging, as we have no direct control of the trapped QP number. Fortunately, the Gaussian mixture procedure assigns most data points to the correct occupation. We use this initial assignment to fit the mean lifetime of each mode ⟨τ i ⟩ and then identify periods in the time series when the extracted QP occupation is stationary for at least4⟨τ i ⟩. By stitching these “quiet periods” together, we build up large distributions of data points that are pre-assigned to modes. Each distribution is then independently fit to a Gaussian to fix its mean and covariance. Finally, the full dataset is fit to a mixture of 3 Gaussians with these same means and covariances, with the weight of each mode as the only free parameters. Figure 3.3(b) shows the result, with 1-σ (solid) and 2-σ (dashed) contours overlaying the data histogram. It is immediately evident that the “quiet periods” method produces a better quality of fit. 37 We extract QP occupation from time series as before with updated Gaussian parameters from the “quiet periods” method. Figure 3.3(c) shows two 0.5 ms sections of data with a 5µ s Gaussian convolution. These sections were chosen to demonstrate switching events and are far more “ac- tive” than typical data, which mostly stays in the 0 QP mode. The background shading represents the extracted QP trap occupation (light blue for 0 QPs, dark blue for 1, and orange for 2 or more). Transitions between the 3 configurations are clearly visible and appear to be faithfully captured by the assignment algorithm. The detector’s SNR at our operating parameters (∼ 4 photon drive power, 5 µ s integration), defined as the squared separation of mode centers in the IQ plane di- vided by the product of their standard deviations, is 27 for0− 1 distinguishability, 32 for1− 2, and 30 for0− 2. The0− 1 SNR gives a detector noise floor of 6.1× 10 − 4 QPs / √ Hz, i.e., we can detect 0.00061 of the signal from a QP trapping with SNR = 1 after 0.5 s of integration, assuming a stationary occupation. We now briefly describe the QP behavior measured with our device. We see a mean trap occu- pation ofn qp =0.0185, state occupation probabilities ofP 0 =0.983,P 1 =0.0155,P 2 =0.00148, and state lifetimes ofτ 0 = 728µ s,τ 1 = 12.7µ s,τ 2 = 4.73µ s. These lifetimes are corrected for the detector bandwidth assuming Poisson switching processes [52]. The distributions of lifetimes do appear Poissonian in the long-time limit, but better SNR (discussed below) may resolve fast events which may show non-Poisson behavior. We also note that we see transitions between all three of the 0, 1, and 2 QP modes. We attribute the 0− 2 transitions to either correlated trapping of 2 QPs in less than a detector bandwidth or to direct|g⟩→|e⟩ excitation of a single channel; spectroscopic measurements of the trapped QPs should be able to distinguish between these processes. Future work will probe switching rates as a function of bias and environmental 38 parameters (e. g. flux and temperature), analyze correlations between switching events, and de- velop more sophisticated state-assignment algorithms that do not assume independent (Poisson) switching. 3.1.3 Powerdependence The approximately 4.3 GHz resonant frequency of our device means no single resonator photon has enough energy to excite a QP out of a deep trap state. However, multi-photon processes can occur, as there is a finite probability of multiple resonator photons for any strength of coherent drive. To find a measurement tone power that does not affect QP trapping, we take ensemble- averaged VNA measurements of the resonance as a function of probe power. Results atϕ =0.47 are shown in Figure 3.4. We find that at measurement powers below -141 dBm the resonator response is power-independent. As we increase power above this level, we find that the weight of the response near the 1-QP resonant frequency begins to decrease and the weight near the 0- QP resonant frequency begins to increase. We attribute these changes to clearing of QPs from the trap states. To optimize SNR while ensuring the detector is not affecting the QP configuration, we choose a measurement power just below -141 dBm. We note that at higher measurement powers the resonator response becomes nonlinear, leading to increased sensitivity per QP at the expense of bandwidth and saturation at a lower QP number. However, in our device this nonlinear regime happens at powers which are already affecting the QP configuration, and so we neglect it in our analysis. 39 −1.0 −0.5 0.0 0.5 Re[S11] 4.2715 4.2720 4.2725 4.2730 4.2735 Frequency [GHz] 4.27222 −0.73 −0.70 −0.67 4.27262 0.4 0.5 -137 dBm -135 dBm -145 dBm -143 dBm -141 dBm -139 dBm Figure 3.4: Ensemble-averaged resonator response at ϕ = 0.47 as a function of measurement power. At powers below -141 dBm the response is roughly power-independent. As we increase the measurement power beyond -139 dBm, the weight of the 1-QP mode begins to decrease and the weight of the 0-QP mode increases. 3.1.4 Clearingtone We expect the trap depth to be∆ − E A (ϕ =0.47)≈ 10.7 GHz. To test that our observed behavior is due to quasiparticles trapping in Andreev states, we attempt to clear these trapped quasiparti- cles by injecting a tone with sufficient energy to excite the trapped quasiparticles into the contin- uum of available states above the gap. Figure 3.5 shows the microwave response of the resonator as a function of a 17 GHz tone drive power. This frequency was chosen as it is higher than the trap depth and, due to details of our device and measurement setup, couples energy efficiently into the resonator. Ensemble measurements of the reflection coefficient show that, when driven strongly, the resonance curve recovers the single Lorentzian shape that it has at 0 flux. We also see that the second and third modes apparent in the continuous IQ measurements have greatly reduced amplitudes, indicating that QPs spend far less time in trap states as the 17 GHz tone clears them 40 out. Combined with the qualitative flux dependence of trapping—trapping modes move farther apart and become stronger as the flux bias becomes deeper—we take this as definitive evidence of QP trapping, similar to that shown in ensemble measurements of similar devices [44]. We note that the frequency of the clearing tone can be swept to spectroscopically measure the exact energies of trapped QPs. We plan such spectroscopic measurements in future studies. 3.1.5 Quasiparticletransitionrates Using our extracted QP state as a function of time, we are able to determine the lifetimes of the transition rates. We show data in Figure 3.6. The distributions of lifetimes for each of the 0-, 1-, and 2-QP modes show an exponential dependence at long times, but show a distinct peak at low times. We attribute this peak to the finite detection bandwidth τ det,i for detecting a switch out of theith mode, and correct for it by assuming a Poisson process and following the method developed by Aumentado and Naaman[52]. We first find the mean lifetime of all detector counts for a mode and set this as a cutoff time, fitting the distribution of lifetimes longer than this cutoff to an exponential. This gives an apparent state lifetime τ ∗ i for the ith mode, which is generally longer than the true lifetime. We also extract the apparent lifetimes outside the ith mode τ ∗ i,out and the detection time for switching back to theith modeτ det,out We then adjust the mode lifetime using the formula u= 1− u ∗ 2 − v ∗ 2 1− u ∗ − v ∗ u ∗ − u ∗ 2 (3.3) Where u = τ det /τ i , v ∗ = τ det,out /τ ∗ i,out , u ∗ = τ det /τ ∗ i . This procedure gives the lifetimes quoted in the main text, τ 0 = 728 µ s, τ 1 = 12.7 µ s, τ 2 = 4.73 µ s. We note that the finite detection SNR produces occasional “blips” from the 1 and 2 modes to the 0 mode and back, leading to a 41 4.275 4.276 4.277 4.278 4.279 4.280 Frequency [GHz] −0.8 0.0 0.8 Re[S11] 17 GHz @ -75 dBm 17 GHz @ -90 dBm −10 −5 0 5 10 I [mV] −10 −5 0 5 10 Q [mV] −10 −5 0 5 10 I [mV] −10 −5 0 5 10 Q [mV] (a) (b) (c) Figure 3.5: Microwave response of the resonance at flux bias 0.45 as a function of clearing tone power. The legend in (a) shows the approximate power of the clearing tone at the plane of the device. The red curve is at low power and shows a significant bump on the low frequency side due to averaging over many configurations of QP occupation. At high clearing tone power (blue curve) we see the resonator response narrows, becomes more symmetric, and appears to be more over-coupled. We stress that the ratio of internal loss to external coupling is not actually chang- ing; rather, the response is taller because we are no longer averaging over other configurations of QP occupation. (b) Histogram of time series data with the clearing tone at the same power as the red trace in (a). (c) Histogram of time series data with the clearing tone at the same power as the blue trace in (a). Note that any trapped quasiparticles are quickly excited back above the gap, greatly reducing the weights of the 1- and 2-QP modes and increasing the weight of the 0-QP mode. large population of short 0-mode lifetimes and likely biasing our results towards shorter 1- and 2-mode lifetimes. We are working to develop state assignment algorithms that are less sensitive 42 to such events. Future work will also develop detector bandwidth corrections that do not rely on the Poisson assumption. lifetime for mode 0 x 10 -3 lifetime for mode 1 lifetime for mode 2 0 8000 0.0 3.0 6.0 fit t * = 828.0 ± 6.2 ms cut off = 597.6 ms t det = 23.2 ms Time [ms] 0 200 0.00 0.06 0.12 Time [ms] 0 40 0.00 0.10 0.20 Time [ms] fit t * = 13.2 ± 0.1 ms cut off = 10.8 ms t det = 1.7 ms fit t * = 5.0 ± 0.2 ms cut off = 5.3 ms t det = 1.6 ms Figure 3.6: Histograms of mode lifetimes for the 0- (a), 1- (b), and 2-QP (c) modes. All lifetimes are exponentially distributed at long times, but show a distinct peak at low times which we attribute to the finite detection bandwidth τ det . We extract the apparent lifetimesτ ∗ i by fitting the behavior above a cutoff time, defined as the mean measured lifetime. These lifetimes are then adjusted to correct for the finite detector bandwidth as described in the text, giving τ 0 = 728 µ s,τ 1 = 12.7µ s,τ 2 =4.73µ s. 3.1.6 TrappingandExcitationMechanisms A conduction channel can take on 3 states, shown schematically in Figure 3.7(a). The ground state|g⟩ with energy− E A has a QP in the lower Andreev state and an empty upper state. The 2 degenerate|o⟩ states with 0 energy either have the lower QP excited out of its state or have an extra QP trapped in the higher state. The excited state|e⟩ with energy +E A has the lower QP excited into the upper state. We note that both|o⟩ states cause the same resonant frequency shift in our device, while a channel being excited to the|e⟩ state has the same effect as two channels excited to their|o⟩ states. As the gaps between Andreev states and between the lower Andreev state and the continuum are much larger than the gap between the upper state and the continuum, we anticipate that trapping of QPs in the upper states will be much more frequent than excitation 43 −5 0 5 10 15 Q [mV] 0 10 20 I [mV] (b) 10 0 10 2 10 4 10 6 0 −5 0 5 10 15 Q [mV] (d) (c) (e) 0 10 20 I [mV] -E A +E A (a) Figure 3.7: (a) Schematic description of the ground state|g⟩, 2 degenerate first excited states |o⟩, and second excited state|e⟩ of a channel. Both|o⟩ states produce the same resonant frequency shift, while the|e⟩ state produces the same shift as 2 channels entering their|o⟩ states. (b-e) Response of the device to different probe tone frequencies. The probe tone is stepped to pro- gressively lower frequencies, starting with the midpoint of the 0- (red) and 1-QP (blue) mode resonances in (b) and ending with the 2-QP mode (green) frequency in (e). As the probe moves close enough, a 3-QP mode (purple) becomes apparent. 44 from the lower state. However, we do see transitions between all 3 modes in our data. The 0-2 mode transitions are either the result of 2 QPs trapping in different channels (2 channels moving to their|o⟩ states) in less than a detector bandwidth, or a single channel being excited to its|e⟩ state. Clearing-tone measurements should be able to distinguish these processes: a clearing tone will take a single channel in|e⟩ to|o⟩, causing a frequency shift equivalent to clearing 1 QP. If 2 QPs are trapped, bringing 2 channels to|o⟩, then a clearing tone will clear both and bring both channels back to|g⟩, producing twice the frequency shift. We note that the presence of a 3-QP mode, discussed below, indicates that at least 2 and likely 3 channels (and QPs) are involved. We expect higher QP numbers to be involved in an environment with a higher background QP population, e. g. one in which there is much more thermal photon radiation. 3.1.7 HigherNumberDetection In order to confirm that we really see multiple QPs trapping and not just a single channel excited to the|e⟩ state, we tune our device to measure the 3-QP mode. Time series data in the main text is shown with the probe tone on or near the 1-QP resonant frequency. As the frequency shift per trapped QP is larger than the resonance linewidth, trapping 2 QPs fully saturates the response of the detector and the modes representing higher trap numbers will lie on top of the 2-QP mode in the IQ plane. However, we can rectify this issue straightforwardly by decreasing the frequency of the probe tone so that it lies closer to the resonances of these higher QP modes. This allows, for instance, distinguishing 2 vs 3 QPs at the expense of our ability to distinguish 0 vs 1. We show this procedure in Figure 3.7. We step the probe frequency from halfway between the 0- and 1-QP resonant frequencies (panel b), to the 1-QP resonance (panel c), to halfway between 1- and 2-QP 45 (panel d), to the 2-QP resonance (panel e). We initially see 3 modes, which we attribute to 0, 1, and 2 trapped QPs, circled in red, blue, and green respectively. As we move the probe frequency lower, a fourth mode (circled in purple) appears, breaking away from the previously-saturated 2-QP mode. We note that this data was taken with a higher measurement power which may affect the QP configuration and which caused the resonance to be slightly nonlinear. We used this higher power because the 3-QP mode was extremely rare in this cooldown (due to the low background QP density) and so was difficult see easily in a histogram if the SNR was not very high. This will not be a concern in situations where the mean trapped QP number is higher, or if the SNR is improved. 3.1.8 Conclusions We note that our device is not fully optimized for high sensitivity. While the resonant frequency was kept low in order to be less than the trap depth, raising the resonance slightly to∼ 6 GHz by shortening the waveguide would increase the participation ratio of the nanobridge inductance to total inductance while still remaining far below trap-clearing frequencies. Similarly, reducing the resonator’s characteristic impedance from 50Ω to an easily-achievable∼ 30Ω would further reduce the linear inductance, increasing sensitivity. Additionally, the TWPA used as a first-stage amplifier had a moderate ≈ 15 dB gain (due to being operated near the edge of its bandwidth) and adds noise above the quantum limit. Adding a standard parametric amplifier with a near- quantum-limited noise temperature and 20 dB of gain as a preamplifier will further improve SNR. We may also trade off some of this sensitivity and increase the resonator bandwidth, thus allowing for detection of more than 2 QPs without saturating the response. We also note the possibility 46 of detecting higher numbers of trapped QPs by probing the device with multiple probe tones simultaneously. By performing heterodyne measurement of probe tones centered on, e.g., the 1-, 3-, and 5-QP resonant frequencies, and correlating the measured outcomes, we should be able to detect up to 6 trapped QPs with a similar device. In conclusion, we have developed a device for the ultra-low-noise continuous detection of up to 2 quasiparticles trapping in Andreev bound states. Our device is capable of detecting a trapped QP with SNR of 27 in 5µ s, giving it a noise floor of 6.1× 10 − 4 QPs / √ Hz. Straightforward extensions are possible to higher sensitivity and QP saturation number. Our device can be used for QP studies including statistical analysis of trapping and untrapping rates and trap occupation, spectroscopic measurements of trapped QP energy distributions, effects of environmental variables such as temperature, and testing of QP mitigation techniques. 3.2 HiddenMarkovModels To understand what the HMM is doing, we will start with a little motivating example. Let’s pretend we have a system which can be in one of two states A or B at any point in a discrete time series. Let’s say the system has some constant probabilityT ij of transitioning from statei to j at each time step. Then we can form a “transition matrix”T from each of theT ij elements and the probability that we transition fromA (represented by the vector (1,0)) toB (represented by the vector (0,1)) at the next time step is given byP AB =B· TA. Thus, this transition matrix governs the dynamics of the system changing statess, as shown in Figure 3.8. The state of the system is “hidden”, so we cannot directly measure which states the system is in at timet, but at each point in time the system will “emit” some data ⃗ x for us to observe. This emission occurs according 47 A B T AB T BA T AA T BB A A A B B A x 1 x 2 x 3 x 4 x 5 x 6 P(x 1 |A) Hidden state transitions Time-series emission of data time S(t) X(t) P(x|s) P(x 2 |A) P(x 3 |A) P(x 4 |B) P(x 5 |B) P(x 6 |A) Figure 3.8: A Hidden Markov Model works by assuming the system can transition between a set of hidden states with constant probabilities given by the transition matrix, whose elements are T ij . Each element of the transition matrix describes the probability of transitioning from statei to j, as shown by the arrows. The diagonal elements of the transition matrix describe the probability of staying in the same state. The right side of this figure shows the Markov chain S(t) of hidden states in the red box. At each time step, the system emits datax according to the state dependent emission probabilities P(x|s). This time-series of observations X(t) as shown in the blue box represents the only information we are able to measure. to some state dependent probabilityP(⃗ x|s). Then taking the full time-series of hidden states as S(t) and observations asX(t), the HMM is fit by maximizing the likelihood of the observation sequence X(t) over all possible transition matrices T and emission probabilities P(⃗ x|s). Once the optimal estimates ˜ T and ˜ P(⃗ x|s) are found, we can use them to determine the most likely state sequence ˜ S(t) corresponding to the observationX(t) using the Viterbi algorithm [78, 63]. In general, a HMM is not limited to only two states, but can model systems with an arbitrary number of hidden states with emission probabilities which can be any valid probability distribu- tion. In our case, we may label our hidden states by the occupation number of ABS. Thus, when a quasiparticle traps(clears), our state is increased(decreased) by 1. Our observation time series is just the homodyne demodulated quadratures of the measurement tone, I and Q as measured by our ADC at the output of the demodulation mixer (see, for example, Figure 3.1(c)). If we histogram these observations in the IQ plane, the data will group into Gaussian modes whose centers are shifted along the resonant circle according to how many trapped quasiparticles are present (as seen earlier in Figure 3.2(b) and (c)). These Gaussians modes define the emission probabilities for 48 1 0 Time [μs] Occupation mV 0 1 (a) (b) Figure 3.9: (a) IQ histogram of input data with gaussian emission probabilities represented by el- lipse overlays with radii equal to one standard deviation. (b) A subset of the observed time-series is plotted in the top panel. This data is at a 1 MHz sample rate and was recorded at a moderately high readout power of -134 dBm at the resonator. The lower panel shows the estimated occupa- tion as extracted from the data using the Viterbi algorithm and the HMM. Notice the jump to and from state 0 to 1 around 10 and 110µ s. our system – each state has an associated mode with center determined by the shift along the resonant circle and covariance determined by the noise in the measurement. Figure 3.9 shows an example HMM fit to data. On the left we see the IQ histogram of input data with overlay ellipses representing the Gaussian emission probabilities (the ellipses are equiprobable curves at one standard deviation from the mode center). The right hand side shows a section of the time- series along with the estimated occupation. These jumps are due to single quasiparticle trapping events. 3.2.1 AmorerelevantSNR While it is typical to quote the RF power SNR (the signal power over the noise power after all amplification), Figure 3.9 hints at another, perhaps more relevant signal to noise ratio: The sepa- ration of these guassian modes over their standard deviation provides a more useful measure of 49 the ability to gain meaningful information. For instance, if we had an extremely broadband res- onator we would not be able to distinguish independent trapping events even at large RF power SNR because the mode separation in the IQ plane would simply be too small. We will call this mode separation signal-to-noise ratioSNR IQ and define it as SNR IQ = ||V 01 || 2 σ 2 01 (3.4) where||V 01 || is the distance between the centers of modes 0 and 1, and σ 2 01 is the product of projected standard deviations of modes 0 and 1 along the vector connecting them. We choose power units forSNR IQ to be consistent with other quoted SNR units. In Chapter 4 I show how thisSNR IQ can serve as an indicator of reliability of the HMM fit. We find that SNR IQ ≳ 3 is needed for an accurate fit. 3.2.2 Transitionratesandlifetimes Some of the more interesting quantities we can investigate are the transition rates between the different modes. These can be calculated from the off-diagonal elements of the HMM transition matrix,T ij , which gives the probability of transitioning fromi toj in one timestepδt . Then the approximate rate of transitions is simply the sampling ratef s = 1 δt times the probability, Γ ij =f s T ij . (3.5) The diagonal elementsT ii tell us the probability of staying in the modei. We typically don’t discuss the rate of staying in the same mode, but rather talk about the lifetime or average dwell 50 time in the mode. This can be found by considering that1− T ii is the probability that the system leaves the state i on the next time step, regardless of which state it moves to. Then we can similarly find the rate that the system leaves the state i as the sum of rates Equation 3.5 or more simply asΓ i =f s (1− T ii ), and invert this to obtain the lifetime of the modei as τ i = 1 Γ ii = δt 1− T ii . (3.6) 3.2.3 Occupationprobabilities Once the estimated state time seriesQ(t) has been extracted from the dataO(t) using the HMM fit and the Viterbi algorithm [78], it is very straightforward to count the number of observations N i in which the stateq =i. The probability of the mode is then P(i)= N i N (3.7) whereN is the total number of observations (i.e., how many points are in the time-series). 3.2.4 Meanoccupation In general we can find the mean number of trapped quasiparticles over a record as ⟨n⟩= X i n i P(i) (3.8) where n i is the number of trapped quasiparticles corresponding to state i. In practice, we label our states by the number of trapped quasiparticles so our estimated occupation time-series is a 51 set of integers which tell us the number of trapped quasiparticles at that time. Then taking a standard mean of this estimated occupation data directly gives us the mean occupation⟨n⟩. 3.3 Bootstrappingthroughpowersweeps Fitting the HMM is often finicky due to the large difference in counts between lower occupation states and high occupation states. In particular, a good fit requires a good initial estimate of the gaussian means. This is easiest to do if we can easily see the modes in the histogram, in which case getting a good starting value for the means is as simple as using an interactive plot to “point and click” each mode. It is typical for us to record data in segments as we sweep the measurement tone power. This allows us to perform a boot-strapping method when analyzing data with the HMM in which we manually provide the initial guess for the highest power data (with the best SNR). Then in subsequent fits at lower powers, we use the gaussian centers of the preceding fit as the initial guess for the next lower power. This method works well for processing data with the HMM down to very low powers, as is shown in Chapter 4 when I talk about validating the HMM. As will be shown, the HMM performance degrades when SNR IQ < 3, so we can dynamically adjust the integration time of processed data during this boot-strapping method to maintainSNR IQ ≳ 3. In doing so, we must be careful to not let the integration time approach the timescales of the quasiparticle dynamics, so we add a breaking condition that stops the boot- strap method if integration time exceeds half of the shortest mode lifetime from the previous fit. This process is illustrated in Figure 3.10 along with example data analyzed using this method. 52 Input estimated centers & covariances Fit HMM at highest power Use new center & covariance to t next lower power Breaking conditions met? Yes Adjust integration time if SNR IQ < 3 No Stop analysis Figure 3.10: Block diagram of bootstrapping process to analyze power sweeps. We start by man- ually providing estimates of the gaussian centers and covariances for the initial fit at the highest power. Afterwards, we iteratively step down in measurement power using the Gaussian modes from the current fit as the starting point for the next fit. After fitting the HMM, we check SNR IQ and increase the integration time ifSNR IQ <3. This process ends when the lowest power data is fit or the integration time exceeds half of the shortest mode lifetime. 53 Chapter4 Simulationsofquasiparticlebehavior In this chapter I discuss simulations used to develop and test our analysis methods. These simu- lations evolved to probe different regimes of quasiparticle behavior to determine if we can extract accurate information from our time-series data. As a first step, I assume that trapping events are Poisson distributed with a constant rate. Next, I introduce some non-Markovian processes into the simulation. I also test pairwise trapping. Lastly, I introduce bursts into the simulation – a phenomenon which has been observed in many quasiparticle experiments in which a cosmic ray or other high energy source generates a cascade of quasiparticles and leads to a burst of events localized in a short time. 4.1 Generationofquasiparticletime-series We start the simulation by initializing a set of N e Andreev channels, each with a transmission coefficient τ sampled from the Dorokhov distribution given in Equation 1.16. At each time step, we generate quasiparticle trapping and clearing events according to one of the methods discussed in detail in Section 4.3. When a quasiparticle trap event occurs, we must select which of the 54 N e channels will be poisoned. Here we assume the trapping process is thermal and sample a Boltzmann distribution P(τ )= e − E A (τ )/k B T P i e − E A (τ i )/k B T (4.1) where the sum is over all un-poisoned channels. Using Equation 2.26 we calculate the inductance of the poisoned channel and track these changes as a function of time – these will be used later to generate the resonator response. Clearing of trapped quasiparticles from the ABS is handled by a Poisson process. At each time step, we sample a Poisson distribution with rateΓ release a number of timesn equal to the number of poisoned channels. In this way, each sample represents the possibility that an individual chan- nel is cleared. When a clearing event happens, we select the channel to clear by random selection of then poisoned channels with uniform weights —this may not be the best model since we might expect shallower traps to clear faster, but this detail does not make a significant difference since we are usually working with a small number of poisoned channels. Figure 4.1 provides a block diagram showing the simulation approach. This simulated data can be used to test analysis meth- ods and verify models. In particular, it will be useful to explore the parameter regions in which our analysis is valid – for instance, we wouldn’t expect to have meaningful analysis if the QP dynamics are faster than our sampling rate, but how close can these two parameters be? In the following subsection I provide more detail on the Poisson processes used in generating the QP occupation data and later discuss the introduction of non-Markovian, pairwise, and burst effects – these too will be useful testing simulations to determine if our analysis is capable of detecting these effects. 55 Input parameters Analyze with HMM Build set of channels, {τ} Generate trapping events Generate resonator response Add noise Figure 4.1: Block diagram of the simulation process. The first step is to initialize the system using the input parameters to generate a set of channels with fixed ABSs. Next, we generate a time series of ABS occupation using the selected processes – at each time step a set of Poisson processes determines if/how many QPs will trap and release from the ABSs as well as fluctuations in the bulk QP density. From this ABS occupation, we calculate the change in inductance due to the poisoning of channels and use this to generate a nanobridge resonator response to a continuous measurement tone. Noise is added to the resonator response to match the SNR to the quantum limit – adding half a photon of noise as we might achieve with a Josephson parametric amplifier. 4.2 Resonatorresponse Once the quasiparticle occupation time series has been simulated, we may use it to generate the expected response of a nanobridge resonator using Equation 2.4. The time-series response is generated by fixing the drive frequency ω d and making the resonant frequency a function of time ω 0 (t)=Ω − n(t)δω (4.2) where Ω is the frequency of the resonator when no quasiparticles are trapped and n(t) is the number of trapped quasiparticles at time t. The resulting response for a reflection resonator driven atω d is S 11 (t)= 2 Q Qc h 1− 2iQ ω d − ω 0 (t) ω 0 (t) i 1+4Q 2 ω d − ω 0 (t) ω 0 (t) 2 . (4.3) 56 Equation 4.3 is a complex valued time-series where the real and imaginary parts correspond to the in-phase and quadrature signals as we would measure from homodyne demodulation with an IQ mixer, as detailed in later experimental chapters. Equation 4.2 gives an instantaneous change in the resonant frequency whenn(t) changes, but in reality a resonator can only respond to changes as fast as the resonant linewidth. To account for this, we low pass filter the resonant frequency ω 0 (t) with a cutoff frequency equal to the linewidthκ . This is only a real concern for high quality factor resonators which will shift much slower than the sampling rate. 4.2.1 Signal-to-noiseratio A realistic simulation needs to account for the noise that will be present in a real measurement. The Johnson-Nyquist noise spectral density [32, 54] S(ω)=k B T (4.4) is familar but does not account for quantum effects at low temperatures (k B T < ℏω). Clerk et al [13] provide a fairly exhaustive review of quantum noise and amplification, and in particular they provide the quantum version of the fluctuation-dissipation theorem originally derived by Callen and Welton [10] S(ω)= ℏω 2 coth ℏω 2k B T . (4.5) 57 While Equation 4.5 reproduces Equation 4.4 in the classical limitℏω <<k B T , we see that in the opposite limit, asT →0, the noise spectral density saturates to S(ω)= ℏω 2 . (4.6) This half photon noise spectral density arises due to the zero-point fluctuations of the electro- magnetic field and provides an unavoidable limit on the noise which accompanies any signal, even at zero temperature. The measurement signal which interacts with our resonator is much too small to measure directly, so we will need some amplification – a process which introduces additional noise ∗ . The review by Clerketal [13] also shows that the quantum limit on noise performance for a parametric amplifier in phase preserving operation is an additional half photon of noise spectral density. All together, the total noise power entering the first stage of amplification is the sum of input noise and amplifier noise P N,0 =η ℏωΓ M (4.7) where Γ M is the measurement rate (inverse of the integration time) and η = η in +η amp is the sum of input noise photons and first stage amplifier noise photons ( η ≥ 1 due to the quantum limit discussed above). We now have a model for the noise power and need to determine the signal power to calculate the SNR. The input power to the first stage amplifier can be determined by considering that the resonator linewidthκ =κ i +κ c represents the total loss rate to internal and external sources (κ i ∗ It is possible to amplify a single quadrature of a signal with zero added noise when using a parametric amplifier in phase sensitive operation, but the tradeoff is de-amplification of the other quadrature. 58 being the internal loss rate to dissipative sources andκ c being the loss rate through the coupled microwave feedline). Thus we can say that the power radiating from the resonator is P rad = ¯nℏωκ (4.8) where ¯n is the average number of photons populating the resonator. We can also relate this radiated power to the amplitude of oscillations in the resonator usingP =V 2 /R, giving V rad = p Z 0 P rad = p Z 0 ¯nℏωκ (4.9) where Z 0 is the characteristic impedance (50 Ohm is standard). For a reflection resonator, the incoming and outgoing signals interfere to produce a standing wave at the amplitude V rad . Ex- plicitly, we can relate the incoming and outgoing signal powers,P in andP out respectively, to the resonator photon numbern as √ ¯nℏωκ = p P in + p P out (4.10) On resonance,P out ∝ κ c andP in ∝ κ . We can plugP in = κ κ c P out into Equation 4.10 to find the available power at the output of the device in terms of the resonator photon number ¯n as P out = ¯nℏωκ (1+ p κ κ c ) 2 (4.11) 59 and we can similarly find the input power as P in = ¯nℏωκ (1+ p κ c κ ) 2 . (4.12) In the extremely overcoupled limit (κ c >>κ i ), then nearly all of the radiated power will exit via the microwave feedline and we haveP out = ¯nℏωκ/ 4, which is often encountered in the literature with little explanation. We can determine the RF SNR at the output of our first amplifier from Equation 4.11 and Equation 4.7 asP out /P N,0 , or SNR = ¯nκ η Γ M (1+ p κ κ c ) 2 (4.13) whereΓ M is the measurement rate (inverse of the integration time). If this first stage of amplifi- cation has high gain in addition to quantum-limited noise performance, then we may be able to maintain near quantum-limited SNR even after additional stages of amplification, each with their own added noise. The idea here is to build an amplifier chain which satisfies the condition that the added noise of a following amplifier is small when compared to the amplified noise output of the preceding amplifier. This ensures that the total signal power to noise power is approximately constant from the first stage of amplification through to the final stage. In these simulations, unless otherwise stated, we simulate the resonator response with quan- tum limited SNR by adding Gaussian noise. The SNR for a signal with root-mean-square ampli- tudeV RMS and noise with standard deviationσ is SNR = V 2 RMS σ 2 . (4.14) 60 We can useV peak = √ 2V RMS in Equation 4.14 and solve forσ to obtain σ = V peak √ 2SNR . (4.15) In our simulations the resonator response is by default normalized such thatV peak = 1, so plug- ging in Equation 4.13 we get the standard deviation of noise which is needed to achieve the target SNR based on the strength of our drive tone (determined by the average photon number in Equa- tion 4.12) σ = s η Γ M (1+ p κ κ c ) 2 2¯nκ . (4.16) 4.3 Comparisonofsimulationprocesses In the following sections I describe the simulation of different processes which could produce quasiparticle trapping and use the results to validate the use of hidden Markov models to inter- pret the data. The simplest case is a Poisson process – a very common approach to modeling the occurrence of events which are believed to be independent. In contrast to the Poisson process, which is Markovian by nature, we may explore processes in which we expect the hidden Markov model to break down, such as non-Markovian processes. Later sections explore correlated trap- ping and bursts of many events in a short time period. 4.3.1 Poissonprocesstrapping There are two primary processes we will concern ourselves with: quasiparticles trapping into the ABSs and quasiparticles releasing from these ABS traps. We expect these events to occur at 61 different rates which will depend on the bulk quasiparticle density and the number of trapped quasiparticles. It is also a decent assumption that these processes are Poissonian, though we may test some other processes later. For a Poisson process, the probability thatk events occur in the time intervalδt is given by P(k,δt )= (Γ δt ) k k! e − Γ δt (4.17) whereΓ is the effective rate of events. For the trapping process, we assume the rateΓ trap ∝x wherex is the bulk quasiparticle den- sity normalized by the cooper pair density,x≡ n QP /n CP withn QP andn CP being the volumetric density of quasiparticles and Cooper pairs, respectively. Taking the trap rate as Γ trap =βx (4.18) whereβ can be interpreted as some normalized rate to be scaled by the quasiparticle density, and plugging this in to Equation 4.17, we can sample this distribution to determine the number of trap eventsk i to occur at each time stept i . Then, at each time step, we removek i channels from the set ofN eff channels with predeterminedτ andE A by sampling with weights determined by the Boltzmann distribution P(τ )= e − E A (δ,τ )/k B T P j e − E A (δ,τ j )/k B T (4.19) where the sum in the denominator is over all unpoisoned channels at the given time. For the process of clearing quasiparticles from the ABS traps, we use a constant clearing rate Γ release in Equation 4.17. For each poisoned channel at time t i , we compare a random sample r from a uniform distribution in [0,1] to the probability that a single event occurs (Equation 4.17 62 with k = 1 and Γ release ). If r < P(1,δt ), then we clear that channel, returning it to the set of effective conduction channels while keeping track of its properties ( τ ,E A ). 4.3.2 Non-Markoviantrapping The Poisson trapping process described in the previous section is a Markovian process – meaning that the probability is determined by the state of the system at that instant with no memory of previous states. We may simulate and test non-Markovian processes by introducing some memory to the system. An easy way to accomplish this is to introduce some exponential process which increases or decreases the likelihood of trapping events occurring within a short time of each other. This can be done by scaling the Poisson rates (Equation 4.17) with an exponential decay in the delay time since the last event: P tm (k,δt,ϵ )= 1− e − ϵ τ m Γ δt k k! e − 1− e − ϵ τ m Γ δt (4.20) where ϵ is the delay since the last trap event and t m is the characteristic memory time of the system. This makes it significantly less likely that a trap event occurs within t m of the previous trap event while forϵ ≫ t m this looks like a normal Poisson process. The simulation of this process is very similar to the Poisson case, at each timestept i , I sample the probability distribution Equation 4.20 to determine how many eventsk i occur. For each trap event, the selection of which channel to poison is handled identically by sampling the Boltzman distribution Equation 4.1. 63 4.3.3 Pairwisetrapping Much like the preceding section, we can modify the trapping simulation to increase the likelihood that quasiparticles trap in pairs, or alternatively that quasiparticles are cleared in pairs – possible if the Andreev channels interact and allow for the condensation of electrons from ABS to Cooper pairs. The simplest way to simulate pairwise trapping is to use multiple Poisson processes. i.e., at each time stept i , there will be a Poisson process with rateΓ trap,1 for single quasiparticle trapping as well as a Poisson process with rate Γ trap,2 for trapping of quasiparticle pairs. For pairwise clearing, we could similarly run multiple Poisson processes. One for clearing single quasiparticles, and another for clearing pairs of quasiparticles during times when 2 or more quasiparticles are trapped. 4.3.4 Bursts It is commonly observed that high energy sources such as cosmic rays and local radioactivity will produce a cascade of many quasiparticles. This has recently been shown to produce correlated errors among qubits on the same chip [83, 49]. High energy impacts have been shown to occur at a rate of approximatelyΓ burst = 1/20 ms. Naturally, we would expect to find bursts of quasi- particle trapping events in our experiments as well. To simulate this behavior, we can introduce a new Poisson process with rateΓ burst which increases the quasiparticle densityx temporarily – x should relax exponentially towards the steady state value as quasiparticles recombine to form Cooper pairs. 64 4.3.5 Results Figure 4.2 shows the results of simulations with each of the 4 underlying processes discussed above. We see that the HMM in general catches the behavior of the simulation, despite inval- idating the underlying assumption of Markovianity. In particular, we see some smoking gun signatures that can indicate that the process is or is not Poisson and Markovian. The leftmost column shows histograms of data in the IQ plane, along with the Gaussian modes assocated with each occupation. The central column shows the log counts of dwell times between trapping events. Notice that the Poisson processes give a straight line decrease as expected. However, in our non-Markovian simulation the distribution is skewed to have fewer counts at the smaller dwell times, resulting in a rounding peak around the simulated memory time of 200µ s. For burst simulations, we see that the low dwell time counts are inflated, resulting in a concave up curva- ture near the top. The rightmost column shows the probability of a given dwell time weighted by the dwell time. For Poisson processes, this probability isP(τ ) = exp(− τ/τ M )/τ M . In each of these plots, the dashed line showsτP (τ ) whereτ M is taken as the mean value of distribution. We see good agreement between the model and data for the Poisson and pairwise processes (the pairwise process is the sum of two Poisson processes which is itself a Poisson process). However, the non-Markovian process differs significantly from the Poisson model —we do however see excellent agreement between the HMM estimate (purple) and the simulated occupation (orange) indicating that the HMM accurately predicts the behavior despite invalidating the Markovian as- sumption. We see a similar deviation from the Poisson model for the burst simulation and note that once again the HMM estimate matches the simulated occupation results. 65 −1 0 1 I 0 1 Q 0 1 2 3 0 500 1000 τ [ μs ] 1 0 1 1 0 3 Counts HMM Simulated 1 0 1 1 0 3 τ [ μs ] 0.0 0.2 0.4 τ P(τ ) HMM Simulated 0 1 I 0.0 0.5 1.0 Q 0 1 2 3 0 500 1000 1500 τ [ μs ] 1 0 0 1 0 1 1 0 2 Counts 1 0 1 1 0 2 1 0 3 τ [ μs ] 0.0 0.2 0.4 τ P(τ ) −1 0 1 I −1 0 1 Q 0 1 2 3 0 500 1000 1500 τ [ μs ] 1 0 1 1 0 3 Counts 1 0 1 1 0 3 τ [ μs ] 0.0 0.2 0.4 τ P(τ ) −1 0 1 I −1 0 1 Q 0 1 2 3 0 500 1000 τ [ μs ] 1 0 1 1 0 3 Counts 1 0 1 1 0 3 τ [ μs ] 0.0 0.2 τ P(τ ) Poisson 1 0 0 1 0 2 non-Markovian 1 0 0 1 0 2 pairwise 1 0 0 1 0 2 burst 1 0 0 1 0 2 Figure 4.2: Simulation results. Each row shows a different underlying process: Poisson trapping, non-Markovian, pairwise trapping, and Poisson with bursts. The leftmost column shows the IQ histogram of data along with the Gaussian modes from the fit (ellipses have radii equal to one standard deviation). The central column shows the distribution of times between trapping events. Not that the Non-Markovian and burst distributions have short time deviations from the exponential decay (straight in log plot) expected in Poisson processes. The right column shows the distribution of times between trapping events weighted by the bin value. Dashed lines show the Poisson model. Note that the hidden Markov model faithfully recreates the distributions even for non-Markovian simulations. 66 4.4 EvaluatingtheaccuracyofHMMestimates To quantitatively evaluate the ability of the HMM to accurately estimate the number of trapped quasiparticles, we choose to use the F1 score [56]. The most familiar metric may be the accuracy —the percentage of all points which are correctly estimated. Accuracy works well for a balanced dataset where the number of observations in each group are approximately the same. However, for unbalanced datasets where one group significantly outweighs the others, then the standard accuracy breaks down. This is easy to see if we consider an example dataset with 990 observations of group A and 10 observations of group B. A terrible model would be to assume that the group is always A, however this model would have an accuracy of 99% despite having no ability to predict behavior. In contrast, the F1 score F1= 2σ R σ P σ R +σ P (4.21) handles these unbalanced datasets naturally by considering both the precisionσ P = TP TP+FP and the recallσ R = TP TP+FN whereTP represents the true positives (in the case above this is when we correctly predict group B),FP is the false positives (predicting group B when in reality it should be group A), and FN is the false negatives (predicting group A when it should be group B). In the example, since we haveTP = 0 (we always predict group A) the F1 score will also be zero. Thus, the F1 score correctly tells us that this model has no value. In the opposite case of a perfect model, we would haveFP =FN =0 andF1=1, so this model would have great value. The standard F1 score applies to binary classification, but we may generalize this to multiclass classification by taking the average of F1 scores evaluated per class α with the conditionsTP = # of observations correctly predictingα ,FP = # of observations predictingα when the correct 67 class is¬α , andFN = # of observations predicting¬α when the correct class isα . The composite F1 score is then ⟨F1⟩= 1 N X α F1 α (4.22) whereF1 α is the standard F1 (Equation 4.21) applied to each class. In terms of the quasiparticle occupation and hidden Markov model, each class represents the number of trapped quasiparticles at the given time. If the estimated occupation is ˜ n(t) and the true occupation isn(t), then we can evaluate the true positives for modek by counting the number of observations j where both the true and estimated occupation agree that there are k trapped quasiparticles, tp k = X j δ (˜ n(t j )− k)δ (n(t j )− k), (4.23) where δ (x) = 1 if x = 0 and 0 otherwise. We can then use the true positives to find the false positives as fp k = X j δ (˜ n j − k) ! − tp k (4.24) and the false negatives as fn k = X j δ (n(t j )− k) ! − tp k . (4.25) With the above definitions for tp, fp, and fn we can evaluate the F1 score for quasiparticle simulations using Equation 4.22. 4.4.1 Scoringthedifferentsimulationprocesses The F1 scores for the four simulations from Section 4.3 are shown in Table 4.1 We see that the F1 score is quite good for all simulation processes, but that it scored a little better for the Poisson and 68 Performance metrics of HMM estimated occupation Simulated Process Poisson non-Markovian pairwise burst σ P 0.990 0.998 0.946 0.948 σ R 0.994 0.994 0.992 0.992 ⟨F1⟩ 0.992 0.996 0.966 0.968 Table 4.1: The precisionσ P , recallσ R , and F1 score for the four simulation processes in Section 4.3. Precision scores the ability to assign only the correct observations to the right mode. Recall scores the ability to assign all the correct observations to the right mode. The lower precision of pairwise and burst simulations is likely due to misattributing higher order occupations to the limited number of modes we fit (4 modes). non-Markovian simulations. If we look at the IQ histograms from Figure 4.2, we can see that the pairwise and burst simulations had significantly more occupation of higher modes. Since we only fit 4 modes in this analysis, we may attribute the decreased F1 score to misattribution of higher modes —a claim which is reinforced by the similar recallσ R in all four simulations implying that we have a similar number of false negatives across the board, but the precision suffers when high occupations are present since misattribution of higher occupation increases the number of false positives. 4.4.2 TestingtheHMMperformanceatlowSNR The hidden Markov model appears to perform extremely well, even accurately predicting pro- cesses which invalidate the Markovian assumption (see Table 4.1). However, the previous tests were at a moderately high SNR so they may not reflect on the performance at ultra-low mea- surement power —a regime we are interested in studying. To test the performance of hidden Markov model analysis at low SNR, we run some simulations and score the performance with the F1 metric. 69 Using the resonator parameters of the device measured in Section 3.1, we simulate a Pois- son trapping process as a function of measurement power (using Equation 4.12 to convert be- tween power and photon number; -148 dBm corresponds to ¯n ∼ 1 photon in the resonator) at an achievable noise performance of∼ 3 times the quantum limit (i.e., we add 1.5 photons of am- plifier noise). We first analyze this data using a bootstrap method similar to that introduced in Section 3.3, with the exception that the SNR condition is not implemented —the integration time never changes. Figure 4.3(a) shows the⟨F1⟩ score, precision σ P , and recall σ R as a function of theSNR IQ . We can see that the HMM performs very well for SNR greater than 3 (indicated by the dashed line). This motivated us to develop the full bootstrap method in which we dynami- cally increase the integration time if theSNR IQ <3. Panels (b), (c), and (d) show the results for this dynamic bootstrap method in orange, while the original data with a constant sample rate is shown in blue. In panel (b) we see that the F1 score is unchanged by our dynamic bootstrapping for a given power. In panel (c) we show the extracted mean occupation⟨n⟩ (Equation 3.8) from the HMM estimated time series (circles) alongside the true mean from the simulation (solid lines). Panel (d) shows the trap and release rates from the HMM fit along with the true rates used in the simulations (grey dashed lines). The bootstrapping method allows us to provide the HMM with a warm start estimate at high power —where we can easily see the modes associated with n(t) —and iterate down to very low powers. While the F1 score shows a decreased ability to accurately estimate the occupation time series at low power, it is clear that we are still able to provide useful estimates of the mean occupation and the transition rates. In fact, while the dynamic bootstrap method neither hurts nor improves our analysis at a given power, it does provide a nice stopping point for the bootstrap 70 −155 −150 −145 −140 −135 −130 Power [dBm] 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 F1 score matched bootstrap −155 −150 −145 −140 −135 −130 Power [dBm] 0.18 0.20 0.22 0.24 Mean matched bootstrap −155 −150 −145 −140 −135 −130 Power [dBm] 0.005 0.010 0.015 0.020 0.025 0.030 Rates [MHz] matched trap matched release bootstrap trap bootstrap release 0 5 10 15 20 25 30 35 SNR 0.6 0.7 0.8 0.9 1.0 Score F1 Prec Rec (a) (b) (c) (d) Figure 4.3: Power sweep simulation results. (a) The F1 score, precision, and recall of the estimated state time series using a static bootstrap method (constant integration time). The dashed line shows SNR = 3; the performance degrades for SNR < 3. (b) F1 score for dynamic (orange) and static (blue) bootstrap methods. The dynamic bootstrap method with changing integration time to keep SNR > 3 does not affect the performance, though it does stop the process before the F1 score drops below 0.85. (c) Mean occupation estimates (circles) and the true values from each simulation (solid line) for dynamic (orange) and static (blue) bootstrap methods. The dynamic bootstrap method stops before the mean occupation estimate drifts away from the true value. (d) Transition rates fit by the HMM; true rates used in the simulation are shown as dashed lines. The release rate (moving fromn = 1 ton = 0 occupation) is shown for static (orange) and dynamic (dark purple) bootstrapping, with each agreeing well with the rate used in the simulation (dark grey dashed line). The trap rate (moving from n = 0 to n = 1 occupation) is shown for static (orange) and dynamic (light purple) bootstrapping and each agree well with the trapping rate used in the simulation (light grey dashed line) 71 iterations before the mean occupation and transition rates estimates begin to deviate from the true values (see Figure 4.3(c) and (d) at the lowest powers). 72 Chapter5 Environmentalmechanismsofquasiparticletrapping In this chapter, I present experimental results [21] probing the excitation and relaxation mecha- nisms involved in the interaction of quasiparticles with the Andreev Bound States of Josephson junctions. This work utilizes the Hidden Markov Models of Section 3.2 to monitor quasiparticle trapping dynamics by continuous, real-time measurement of a nanobridge resonator as a func- tion of Andreev energy and environment temperature. The quasiparticle trapping events are consistent with independent spontaneous emission from a bulk quasiparticle population which is a linear combination of the non-equilibrium background and a thermal equilibrium population. Above 100 mK, clearing of quasiparticles from Andreev Bound states is dominated by coupling to a bath of thermal phonons, while it is shown that non-thermal sources dominate quasiparti- cle clearing at typical qubit operating temperatures. The mean occupation (that is, the number of trapped quasiparticles on average) provides an independent confirmation of the trapping and clearing models. 73 5.1 Experiment The goal of this experiment is to monitor quasiparticle trapping dynamics and develop a model for the background processes involved in relaxation and excitation of quasiparticles to and from trap states. We use an aluminum nanobridge resonator to monitor quasiparticle trapping dynam- ics by analyzing IQ voltage records with a hidden Markov model (see Chapter 3 for a detailed explanation). The fundamental mode f 0 (ϕ ) of the resonator is 4.301 GHz at ϕ = 0, but can be flux tuned to∼ 4.25 GHz atϕ = 0.5. The resonator has a linewidthκ = 2π × 250 kHz and experiences a shiftχ (ϕ )/2π ∈[100,400] kHz due to trapping a single quasiparticle. The resonator is thermally anchored to the base stage of a dilution refrigerator and AC coupled to a microwave feedline which routes incoming and reflected signals through the signal path shown in Figure 5.1. Our readout tone ω d is generated and split at room temperature with half of the signal entering the dilution refrigerator via a series of attenuators and filters and the other half being routed to the LO port of the demodulation mixer. At the 30 mK stage, the input signal is low-pass filtered by K&L 12 GHz filters and in-house manufactured Eccosorb filters to reduce pair breaking photons. The filtered signal is circulated to interact with our device and the reflected signal is circulated towards the output. Note that the Josephson parametric amplifier (JPA) was not used in this experiment as any attempts to tune it seemed to have a significant effect on the resonance and QP behavior, which we expect was due to insufficient isolation between the JPA and device. For this experiment, the JPA was flux biased to be a few GHz from the resonator frequency. After some isolation and a low pass filter (Mini Circuits VLF-5850+), the reflected signal is amplified by a travelling wave parametric amplifier (TWPA) whose pump is coupled in after the VLF-5850+ 74 w d LO RF I Q Pump 4K 1K 30 mK Flux a) b) 20 dB 20 dB 20 dB 20 dB 20 dB K&L K&L Ecco Ecco TWPA HEMT 200 nm ADC D.U.T. 20 dB 20 dB JPA (unused) K&L K&L Ecco Ecco 20 dB D. Att. 20 dB VNA 1 2 VLF-5850+ Amp. chain 15 MHz LPF Figure 5.1: (a) Images of similar device: aλ/ 4 resonator is grounded through a DC SQUID with a pair of symmetric aluminum nanowire junctions. These junctions are approximately 25 nm× 8 nm× 100 nm. (b) The readout driveω d is generated at room temperature and attenuated along the path through the dilution refrigerator to the base stage. At 30 mK, pair breaking photons on all inputs and outputs are reduced by K&L 12 GHz low pass filters and Eccosorb CR110 infrared absorbers. The signal circulates to reflect off our device and pass through a 5.85 GHz low pass filter. The signal is then amplified by a travelling wave parametric amplifier (TWPA) whose pump is inserted via a directional coupler. The signal exits the dilution refrigerator receiving further amplification by a HEMT at 4K and a series of low noise amplifiers at room temperature. The signal is homodyne demodulated by an IQ mixer and the resulting quadratures are digitized after 15 MHz low pass filtering. A Keithley Sourcemeter sends DC current along the dashed path to a coil in the device package and a VNA is used to measure the resonance as a function of flux. 75 low pass filter. The amplified reflected signal is then routed through the fridge output, receiving additional amplification by a low noise high-electron-mobility transistor (HEMT) amplifier at the 4K stage and a chain of isolators and amplifiers at room temperature. The amplified reflection signal, after coupling -20 dB off to the VNA, is homodyne demodulated with reference to the original signal by the IQ mixer at the top right of panel (b). The DC components of the in-phase and quadrature are low pass filtered and recorded by an AlazarTech ATS 9371 digitizer at 300 MHz sample rate. The raw data is recorded in 3 second segments and downsampled to 5 MHz sample rate prior to saving. A Keithley 2400 SourceMeter is used to drive current through a coil in the device packaging, coupling magnetic flux through the SQUID loop. As a function of this flux ϕ , VNA measurements are taken to calibrate the drive frequency so as to stay at 1 shift χ below the bare resonance, ω d (ϕ )=2πf 0 (ϕ )− χ (ϕ ). (5.1) This ensures maximal separation between the modes for 0, 1, and 2 trapped quasiparticles. Data segments are collected as described above while sweeping the dilution refrigerator tem- peratureT , the flux ϕ which equates to sweeping the trap depth∆ A , and the measurement power P . The temperature is swept from 30 mK to 300 mK by applying current to a resistive heater on the mixing chamber plate and is allowed to stabilize for two hours using a PID loop before start- ing measurements. At each temperatureT , the flux ϕ is swept from 0.3 to 0.5 in a step of 0.001 Φ 0 . This corresponds to a trap depth ∆ A from 20.2 eV to 54.2 eV (or equivalently 4.88 to 13.1 GHz). For each flux ϕ , the measurement power is swept from∼ -124 dBm to -154 dBm at the res- onator. Using Equation 4.12, we estimate the measurement power corresponding to an average photon number¯n γ =1 in the resonator is∼ -147 dBm. Thus, we are sweeping from¯n γ ≈ 200 to 76 ¯n γ ≈ 0.25 photons in the resonator. The microwave readout is generated at room temperature by a Rhode & Schwarz SMF100a signal generator at 20 dBm and reduced by a series of attenuators inside and outside of the dilution refrigerator, with the final 20 dB attenuator at the base stage. Each record in the power sweep is fit to a hidden Markov model using the bootstrap method in- troduced in Section 3.3, but the data presented in this chapter is a subset where the measurement power is -133 dBm, or¯n γ ≈ 25 photons on average in the resonator. In Chapter 6 we will explore the power dependence observed in this sweep. There are three characteristics of quasiparticle trapping dynamics presented in Figure 5.2 and modeled in the following sections. Γ trap is the rate of quasiparticles relaxing from the bulk into available Andreev bound states of the junction,Γ release is the rate of quasiparticles exciting from Andreev bound states up to the bulk, and ¯n which we call the mean occupation is the time average of the number of trapped quasiparticles.Γ trap andΓ release are found from the off-diagonal elements of the hidden Markov model transition matrix—that is, they are parameters of the model used to extract the Andreev bound state occupation time series—while¯n is found from averaging the extracted occupation over the full 3 second record. 5.2 Quasiparticletrappingviaspontaneousemission When an electron relaxes from a higher energy state E 1 to a lower energy E 0 , energy conser- vation requires that this energy is transferred somewhere, typically emission of a photon with frequency ω = E 1 − E 0 ℏ . This process occurs spontaneously in any system with a nonzero pop- ulation of excited electrons, and is thus called spontaneous emission. It is reasonable to expect quasiparticles in the superconductor to behave similarly, so we begin our modeling with the rate 77 6 8 10 12 Trap Depth [GHz] 100 200 Temperature [mK] 1e-03 1e-02 1e-01 Mean Occupation 1e-04 1e-03 1e-02 1e-01 Trap Rate [MHz] 1e-02 1e-01 1e+00 Release Rate [MHz] 30 260 Temperature [mK] 4.907 13.101 Trap Depth [GHz] Figure 5.2: Three characterizing quantities which describe quasiparticle trapping are presented against the trap depth ∆ A (left column) and the temperature T (right column). The first row is the mean occupation, also referred to as ¯n, which has a low temperature saturation and a dip from 100 to 150 mK. The middle row is the trap rateΓ trap which saturates around 120 mK. The bottom row shows the release rateΓ release with a low temperature saturation around 60 mK and some structure in the trap depth. All data is shown at a measurement power of -133 dBm, or ¯n≈ 25 photons in the resonator. of quasiparticles relaxing into the Andreev bound states,Γ trap . Spontaneous emission gives a per particle rate of relaxation with cubic dependence on the frequency of radiation A(ω)= ω 3 |U| 2 3πϵ 0 ℏc 3 , (5.2) 78 as derived in Griffith’s chapter 9 [24], where U is the matrix element connecting the two states. Assuming trapping events are independent of each other and spontaneous emission dominates the quasiparticle relaxation into Andreev bound states, each quasiparticle in the bulk has a temperature- independent trapping rateA(ω). This implies the overall trap rate is separable: Γ trap (∆ A ,T)=f(∆ A )x(T), (5.3) where x(T) is the fractional quasiparticle density and ∆ A ≡ ∆ − E A is the trap depth. We expect that most bulk quasiparticles are near the gap energy, so the frequency of radiation from spontaneous emission isω≈ ∆ A /ℏ and in Equation 5.3 we usef(∆ A )∝∆ 3 A . The fractional quasiparticle density should be the sum of a non-equilibrium backgroundx ne and a thermal population: x(T)=x ne + r 2πk B T ∆ exp − ∆ k B T . (5.4) Putting this together, we obtain a model for the rate of quasiparticles relaxing into Andreev bound states via spontaneous emission, Γ trap =β ∆ 3 A x ne + r 2πk B T ∆ exp − ∆ k B T ! , (5.5) whereβ ,∆ , andx ne are the free parameters. 79 5.2.1 Fittingthetraprate The goal in this section is to fit the trap rate model Equation 5.5 to the data, but to improve the quality of our fit we will do this in multiple parts to reduce the number of free parameters in each fit. We take advantage of the low temperature saturation of trap rate for T ≤ 120 mK as can be seen in Figure 5.2 to make the approximation Γ 0 trap ≈ β ∆ 3 A x ne , (5.6) where we have eliminated the thermal population in Equation 5.5 as it is negligibly small at the lowest temperatures. Subtracting the low temperature saturated trap rate Equation 5.6 from Equation 5.5 gives a new quantity Γ − trap (∆ A ,T)≡ Γ trap (∆ A ,T)− Γ 0 trap (∆ A ,T)=β ∆ 3 A r 2πk B T ∆ exp − ∆ k B T (5.7) which does not depend on the non-equilibrium density x ne . We first fit Γ − trap (∆ A ,T) (Equa- tion 5.7) over the full trap depth and temperature manifold with free parameters ∆ and β , as shown in Figure 5.3. The resulting fit provides β = 0.873± 0.068MHz/µ eV 3 and ∆ = 185 ± 1.5µ eV. This value for the superconducting gap ∆ aligns well with expectations for thin film aluminum, which should be slightly larger than the bulk aluminum gap of∆ bulk ≈ 170µ eV. Dividing Equation 5.5 by Equation 5.6 gives the normalized quantity ∥Γ trap ∥(∆ A ,T)≡ Γ trap (∆ A ,T)/Γ 0 trap (∆ A ,T)=1+ 1 x ne r 2πk B T ∆ exp − ∆ k B T , (5.8) 80 Figure 5.3: The measured trap rate minus the low temperature trap rate (average of all data less than 80 mK) is shown along with the fit to Eq 5.7. We find the scaling factor β =0.873MHz/µ eV 3 and the superconducting gap∆=185 µ eV. which eliminates theβ dependence. Next, with∆=185 µ eV held constant, we fit ∥Γ trap ∥(∆ A ,T) (Equation 5.8) with the fractional non-equilibrium density x ne as the only free parameter, as shown in Figure 5.4. This fit results in a non-equilibrium quasiparticle density x ne = 8.50± 0.10× 10 − 7 . 5.2.2 Results The full results of the fitting procedure are shown on the unmodified data in Figure 5.5, where the solid lines are the model (Eq. 5.5) using the combined fit parameters. The fractional non- equilibrium density x ne is quite high compared to recent works [68, 75, 48] which observe a fractional density on the order of x ne ∼ 10 − 9 . Our setup uses light-tight radiation shields on all stages of the fridge, with Berkeley Black infrared-absorbing coating [61] on the interior of 81 Figure 5.4: The measured trap rate divided by the low temperature trap rate (average of all data less than 80 mK) is shown along with the fit to Eq 5.8. We find the fractional non-equilibrium quasiparticle densityx ne =8.5× 10 − 7 . Figure 5.5: Measured trap rate (circles) and model (solid lines). The dependence on the trap depth ∆ A is shown on the left, while temperature dependence is shown on the right. We note the peak in 30 mK data around 9 GHz on the left was observed as a period of significantly larger than normal mean occupation which lasted approximately 1 hour in laboratory time. The source of this peak has not been found and it is not reproducible. 82 the 100 mK and mixing chamber shields. In addition, the sample package is mounted inside of an Amumetal 4K shield with a tin-plated copper can nested inside, also with a Berkeley Black interior coating. We use custom-made Eccosorb filters as well as K&L 12 Ghz low-pass filters on all inputs and outputs. We suspect that our device geometry may contribute to the higher- than-expected density, as large areas of superconducting aluminum are galvanically coupled to the SQUID. The left panel of Figure 5.5 shows a peak in the 30 mK data near 9 GHz. This anomaly was present in the trap rate and mean occupation, while the release rate was marginally increased. We attribute this to a temporary increase in the bulk QP density, as repeated measurement under nearly identical conditions did not show this effect. The period of increased trapping lasted for approximately one hour with no change in fridge temperature and no obvious environment fac- tors to blame. We note the duration of the effect is too long to be caused by adhesive strain [3] or a strong cosmic ray [11]. At the time of this experiment, there was ongoing remodeling work in another lab in our building, so one could speculate that some construction related vibration or release of weakly radioactive materials could have caused this anomaly (if the level of radon or another radioactive gas were to increase due to release and circulation through the building AC, it might produce an effect with this timescale). The data and model presented here suggest that quasiparticles are relaxing into Andreev bound states primarily via spontaneous emission. It is also clear that the non-equilibrium and thermal quasiparticle populations have the same trapping mechanism, evident by the shared de- pendence on the trap depth at low temperature (when only non-equilibrium quasiparticles are present) and high temperature (when thermal quasiparticles dominate the population). This may be a little surprising as there is no reason that the non-equilibrium quasiparticle population must 83 behave the same as thermal quasiparticles—it is feasible that a non-equilibrium population could via a different mechanism altogether or simply have higher energy E ne > ∆ and thus relax faster than expected. The latter would result in an inability to fit high and low temperature data with the same trap depth (low temperature trapping would have trap depth∆ A,ne = E ne − E A , while thermal quasiparticles at the gap would require∆ A =∆ − E A ). As it is, the data suggests that non-equilibrium quasiparticles are trapping from energies near the gap edge∆ via the same mechanism as thermal quasiparticles. 5.3 Quasiparticleclearinginathermalbathofphonons To promote a trapped quasiparticle from an Andreev bound state to the continuum of states above the gap, sufficient energy (at least ∆ A ) must be absorbed. In a dilution refrigerator with ample light-tight shielding, we expect this energy to come from the absorption of phonons. The rate of absorption is often derived as the product of a scattering matrix element connecting states and the energy density of the exciting field [34, 24]. For clearing quasiparticles from Andreev bound states with trap depth ∆ A , we take this exciting field as the density of phonons with energy exceeding the trap depth, resulting in the phonon clearing rate per trapped quasiparticle Γ phonon (∆ A ,T)=B ABS→bulk ρ ϵ ≥ ∆ A (∆ A ,T), (5.9) whereB ABS→bulk is the absorption rate per energy density per trapped quasiparticle which con- tains the scattering matrix element and ρ ϵ ≥ ∆ A (∆ A ,T) is the density of phonons with energy exceeding the trap depth. In the following section, we expand this to a detailed model. 84 5.3.1 DerivationofphononcontributiontoABSclearing As described in Equation 5.9, we expect the electron-phonon clearing rate to be linear in the phonon density Γ phonon (∆ A ,T)∝ρ ϵ ≥ ∆ A (T). (5.10) Our first task is to find the density of phonons with energy exceeding the trap depth, ρ ϵ ≥ ∆ A (T). Phonons obey Bose-Einstein statistics and we expect them to follow the Debye model. With ω as phonon frequency and ν as the phase velocity, we have the Debye density of states D(ω) = ω 2 /2π 2 ν 3 . We obtain the phonon density by integratingD(ω) over the Bose-Einstein distribution, ρ ϵ ≥ ∆ A (T)= 1 2π 2 ν 3 Z ω D ∆ A ℏ ω 2 exp ℏω k B T − 1 dω = 1 2π 2 k B T ℏν 3 " − ∆ A k B T 2 ln 1− e − ∆ A k B T + 2∆ A k B T Li 2 e − ∆ A k B T +2Li 3 e − ∆ A k B T +f(ω D ,T) # . (5.11) In the above, Li n (x) is the polylogarithm function of order n. Eq. 5.11 groups all terms which result from the upper limit of our integration (the Debye frequencyω D ) into one function, f(ω D ,T)=− ℏω D k B T 2 ln 1− e − ℏω D k B T + 2ℏω D k B T Li 2 e − ℏω D k B T +2Li 3 e − ℏω D k B T . (5.12) There are two materials we may be interested in: silicon and aluminum. Plugging in the appro- priateω D for each material (2π × 21.98 THz for Silicon,2π × 15.37 THz for Aluminum), we find thatf(ω D ,T) evaluates to a negligibly small offset in Eq 5.11 for T ≤ 70 K, and near identically 85 zero for our measured temperature range T ≤ 300 mK. Therefore, we are justified in dropping f(ω D ,T) from our model. Substituting the phonon density (Equation 5.11) into the phonon clearing rate (Equation 5.9) provides the detailed model Γ phonon (∆ A ,T)≡ B ABS→bulk × ρ ϵ ≥ ∆ A (∆ A ,T) = B ABS→bulk 2π 2 k B T ℏν 3 " − ∆ A k B T 2 ln 1− e − ∆ A k B T + 2∆ A k B T Li 2 e − ∆ A k B T +2Li 3 e − ∆ A k B T # . (5.13) In practice, we combine the prefactors into a single fit parameter α = k 3 B 2π 2 ℏ 3 ν 3 B ABS→bulk . (5.14) 5.3.2 Fittingthereleaserate In Figure 5.6, we observe that the release rate saturates atT ≤ 60 mK to a value which depends on the power of our microwave readout tone, suggesting that low-temperature clearing is dominated by driven electron-photon interactions. This is surprising because a single readout photon (≈ 4.27 GHz) has insufficient energy to clear the ABS trap ( ∆ A (ϕ ) > 5 GHz∀ measured ϕ ). The measurement power dependence of quasiparticle clearing will be covered in detail in Chapter 6. 86 -150 -140 -130 Power [dBm] 50 100 150 200 250 300 Temperature [mK] 0.01 0.1 Release Rate [MHz] 30 300 Temperature [mK] -154 -127 Power [dBm] Figure 5.6: The release rateΓ release is shown at flux ϕ = 0.485 (trap depth∆ A = 12.4 GHz) as a function of measurement power (left) and temperature (right). We can see that the release rate saturates at low temperature, but the value depends strongly on the measurement power. For now, we focus on the -133 dBm data presented in this chapter and account for this readout- dominated electron-photon clearing by adding an additional clearing rate toΓ release , thus we can model the total release rate as Γ release (∆ A ,T)=Γ RO (∆ A )+Γ phonon (∆ A ,T), (5.15) where the electron-photon clearing rate due to measurement (read out) isΓ RO . For now, we take advantage of the low temperature saturationΓ 0 release ≈ Γ RO to eliminate this photon contribution 87 and maintain focus on the electron-phonon clearing rate. Using Equation 5.13 and 5.14, the model is Γ release − Γ 0 release ≈ Γ phonon (∆ A ,T) ≈ αT 3 " − ∆ A k B T 2 ln 1− e − ∆ A k B T + 2∆ A k B T Li 2 e − ∆ A k B T +2Li 3 e − ∆ A k B T # . (5.16) In practice, the low temperature release rateΓ 0 release (∆ A ) is taken as the average over data 60 mK and below. 5.3.3 Results With fixed ∆ = 185 µ eV, Eq. 5.16 is fit to the full ∆ A ,T manifold of data with the scaling fac- tor α as the only free parameter. The results are shown in Figure 5.7. We obtain α = 38.51± 0.36 MHz/K 3 . The model fits remarkably well considering that the only free parameter is an over- all scaling factor. Clearly the high temperature release rate is dominated by a thermal distribution of phonons, but as shown in Figure 5.6 non-thermal sources may dominate the quasiparticle ex- citation at typical qubit operating temperatures. 88 Figure 5.7: (top) The measured release rate vs trap depth and temperature. The top left panel shows structure in the trap depth dependence which is attributed to the driven electron-photon interactions which dominate at low temperature. In the top right panel, the low temperature saturation is visible. The grey dashed line indicates the cutoff temperature (90 mK) for the fit. (bottom) The measured release rate minus the low temperature saturation is shown as circles, while the phonon clearing model (Eq. 5.16) is shown as solid curves. The 240 mK and 260 mK data in the top left panel show some clipping of the release rate data to the 1 MHz sample rate – A limitation of our measurement rather than a physical effect. 89 5.4 Meanoccupationfromsteadystate Our last feature of interest is the mean occupation ¯n, which is taken directly from the extracted time series of Andreev bound state occupations, not from hidden Markov model fit parameters. We start with a simple sum over weighted probabilities: ¯n= X i iP(i), (5.17) where P(i) is the probability of having i trapped quasiparticles. In this analysis, we are only distinguishing between 1 and 0 trapped quasiparticles, as the incidence of 2 or more trapped quasiparticles is quite rare. We can therefore assume a stationary distribution to obtain P(0)Γ trap =P(1)Γ release . (5.18) Plugging (5.18) into (5.17), we obtain the model for the mean occupation: ¯n(∆ A ,T)=P(0) Γ trap (∆ A ,T) Γ release (∆ A ,T) . (5.19) Unfortunately, we are unable to eliminate the driven electron-photon contribution as we did in Equation 5.16, so we simply leave Γ RO (∆ A ) as a free parameter and fit each line cut along 90 Figure 5.8: (top) The measured mean occupation (circles) and the corresponding fit (solid) are shown against temperature. Note that a different fit is performed at each value of ∆ A . (bottom) The fit parameter α M vs trap depth. Stars indicate the value of α M for the three curves of the same color displayed in the top panel. temperature separately. We normalize by dividing out the low-temperature saturation (T ≤ 60 mK) to obtain the model ∥¯n ∆ A (T)∥= 1+ 1 xne q 2πk B T ∆ e − ∆ k B T 1+α M T 3 − ∆ A k B T 2 ln 1− e − ∆ A k B T + 2∆ A k B T Li 2 e − ∆ A k B T +2Li 3 e − ∆ A k B T . (5.20) We fit this independently for each trap depth, while holding x ne = 8.5× 10 − 7 and∆ = 185 µ eV fixed. The only fit parameter is α M ≡ α/ Γ RO (∆ A ). The results are shown in Figure 5.8. We note the characteristic dip in mean occupation forT ∈[80,150] mK arises from an increased phonon population leading to faster clearing of Andreev bound states, while the rise forT ≥ 150 mK is due to the increased population of thermal quasiparticles. 91 Figure 5.9: Two sources of estimate for the rate of readout photons clearing quasiparticles from the Andreev bound state traps. The measured low temperature release rate (blue) and the fit parameter from the mean occupation, shown asα/α M (orange), whereα = 38.51 is found from fitting the phonon contribution to the release rate as shown in Figure 5.7. We point out that these agree in shape and magnitude despite coming from different sources. We may check for self-consistency in our description by examining the relationship between α M (∆ A ) and the driven electron-photon clearing rateΓ RO (∆ A ). We directly measureΓ RO (∆ A ) as the low-temperature saturation of the release rate and compare this with the estimate obtained from α/α M , as shown in Figure 5.9. Note that the former quantity comes entirely from the hidden Markov model fit parameters, while the latter quantity comes from direct analysis of the Andreev bound state occupation time series. These quantities agree very closely, indicating that our analysis is robust. The driven electron-photon clearing rate has significant structure in its dependence on ∆ A which is repeatable. There is additional structure when one looks at the dependence on the microwave power, which is the focus of the next chapter. 5.5 Conclussion By utilizing the many Andreev bound states of aluminum nanobridge Josephson junctions, we are able to measure and explain the behavior of quasiparticle trapping dynamics in qubit-like circuits over a range of trap depth and temperature. We show that quasiparticles relax into traps 92 primarily by spontaneous emission of a phonon. The close agreement between our data and our model suggests that most quasiparticles entering the trap are originally at or near the supercon- ducting gap∆ . This indicates that any “hot” non-equilibrium quasiparticles are first relaxing to the gap in an independent process before trapping or that the majority of non-equilibrium quasi- particles exist at the gap edge, in agreement with past results [16]. We do not see any evidence of “photon-assisted trapping” (in analogy to the photon-assisted tunneling observed in tunnel junc- tions) where an infrared photon breaks a Cooper pair, promoting a quasiparticle directly into an Andreev bound state. This process may occur at lower rates, and is the subject of future work. We also show that clearing of quasiparticles from Andreev bound state traps at temperatures above 90 mK occurs primarily through absorption of thermal phonons which are distributed accord- ing to the Debye model. Other sources, such as microwave photons, are the dominant source of Andreev bound state clearing at qubit operating temperatures. Our results further elucidate the behavior of equilibrium and non-equilibrium quasiparticles in superconducting circuits. 93 Chapter6 DrivenexcitationsofAndreevboundstates In the previous chapter, models were developed to describe the passive mechanisms which con- tribute to the quasiparticle trapping dynamics. In particular, spontaneous emission was shown to be the primary relaxation mechanism for trapping at all measured temperatures while phonon ab- sorption dominates the clearing of quasiparticles from these traps at high temperatures (T ≳100 mK). However, as can be seen in Figure 5.6 this is not true for low temperatures. It is evident that the clearing rate depends strongly on the microwave readout power, even to single photon powers (-147 dBm is a single photon on average in the resonator for this device). In this chapter, we will explore the effects of measurement on the quasiparticle trapping dynamics. 6.1 Andreevboundstateclearingviameasurementphotons It has already been shown that the rate of clearing quasiparticles from the Andreev bound state traps, Γ release , has some significant dependence on the measurement power. Figure 6.1 shows that this power dependence has some significant structure in the trap depth ∆ A . In the right panel, we clearly see that the shallowest traps are clearing faster at high power than the deepest 94 6 8 10 12 Trap Depth [GHz] -150 -140 -130 Power [dBm] 0.01 0.1 1 Release Rate [MHz] -154 -127 Power [dBm] 4.907 13.101 Trap Depth [GHz] Figure 6.1: The quasiparticle clearing rate Γ release at T = 40 mK is shown as a function of trap depth∆ A (left) and measurement power at the resonator (right). There are a few observations to make of the left panel: power dependence changes with trap depth, and the low power saturation has structure in the trap depth. The power dependence appears to have 3 distinct regions (≲ 8 GHz, ∼ 8 to ∼ 10.5 GHz, and≳ 10.5 GHz). The low power saturation of the clearing rate increases slightly with trap depth towards a peak at∼ 10.5 GHz, then abruptly falls. A second buildup and peak may be evident in the lowest power and deepest trap (12 -13 GHz). 95 traps are. Looking at the left panel, we see two plateaus in the high power release rate and a third region which falls off gradually as we move to larger trap depth. In the following sections we’ll validate this observation using a few tests and then explore the power dependence of the release rate in these regions of trap depth. 6.1.1 Validatingthepowerdependence It should be surprising to see strong power dependence at low measurement powers (∼ 1 photon) considering that the frequency of measurement photons is < 4.3 Ghz while the trap depth is 4.9 ≤ ∆ A ≤ 13.1 GHz. Nonetheless, power dependence is observed. It may be natural to question the validity of this observation as the release rateΓ release is the result of a fit (from the transmission matrix of the hidden Markov model), and changing the power of our measurement implies the SNR of the measured data is also changing. It is reasonable to suspect that the fit results of the hidden Markov model may change with SNR, leading to erroneous observations. 6.1.1.1 Artificialdegradation A good test of whether the observed power dependence is physical or a fitting artifact would make a one-to-one comparison of a measured power sweep with data where the power is held constant but the SNR is changed. We can do this by taking a single record from a power sweep and duplicating it with added gaussian noise to degrade the SNR—but the measurement record re- mains the same (and therefore the underlying quasiparticle dynamics in the record are identical). Figure 6.2 shows the results of such a test. The trap rateΓ trap can be fully explained by a degrada- tion of fit quality due to changing SNR. However, the release rate Γ release has power dependence far in excess of that which can be explained away. It appears we must accept that some physical 96 −150 −140 −130 Effective power [dBm] 1e-3 8e-4 9e-4 Trap Rate [MHz] −150 −140 −130 Effective power [dBm] 1e-2 Release Rate [MHz] 4e-2 4e-3 Figure 6.2: A power sweep at ϕ = 0.47 with measured data (circles) and artificially degraded duplicates of the data at -131 dBm (stars). The horizontal axes show “Effective power” because the artificially degraded data (stars) have been mapped onto the equivalent power that produces the same SNR. For the measured data (circles) this is the actual power at the device. The left panel shows that small amount of change with power present in the trap rateΓ trap can be fully explained by worsening of the hidden Markov model fit due to SNR. On the right, we see that the release rateΓ release has power dependence in excess of the small change produced by decreasing SNR. process is allowing measurement photons to excite quasiparticles from the Andreev bound states with trap depths in excess of the photon energy. 6.1.1.2 Pulsedmeasurement The goal of this section is to verify that the observed power dependence is a real physical phe- nomenon by checking if the measurement tone changes the mean occupation¯n of Andreev bound states without using a hidden Markov model. This can be done by taking an ensemble measure- ment where we send a short pulse of measurement signal at ω d (Equation 5.1) and record the 97 homodyne demodulated response. This setup is very similar to that in Section 5.1, with the dif- ference being how the measurement tone is sent to the device. In the former case, the measure- ment signal was always on and at a constant power. In this case, we use an arbitrary waveform generator along with an additional mixer to generate pulses of measurement signal. The signal power at the resonator when the pulse is “off” is approximately -207 dBm ( ¯n γ ≈ 10 − 6 photons), which is effectively no power making it to the resonator. When the pulse is “on”, the power at the resonator is -136 dBm (¯n γ ≈ 10 photons). The pulse length is 50 µ s. If we integrate the signal over some subset of the pulse, τ int , we get a single point per pulse per channel, I and Q. The measurement scheme here is to record an ensemble of pulses and bin the integrated data into IQ histograms which are then fit to a sum of bivariate Gaussian modes, one for each quasiparticle occupation. If the amplitude of the Gaussian mode is A i where i is the quasiparticle occupation number, then the probability of havingi quasiparticles trapped is P G (i)= A i P j A j . (6.1) From these occupation probabilities we can estimate the mean occupation ¯n as ¯n= X i iP G (i). (6.2) If we sweep the delayτ delay between measurement pulse start time and integration start time, then we get a moving average of the number of trapped quasiparticles ¯n(τ delay ). This measure- ment was performed with 1 million pulses at a 50 Hz repetition rate and the data was integrated withτ int = 5µ s. The moving average ¯n(τ delay ) is shown in Figure 6.3. The resulting mean occu- 98 0 10 20 30 40 50 0.175 0.180 0.185 0.190 0.195 0.200 0.205 0.210 Mean occupation Γ = 12.49 kHz, = ∞ <n> 0.143 Time since pulse ON [ μs] −150 −145 −140 −135 −130 −125 Power [dBm] 10 Release Rate [kHz] 40 4 −150 −145 −140 −135 −130 −125 Power [dBm] 0.050 0.075 0.100 0.125 0.150 0.175 0.200 0.225 Mean occupation 20 ms 50μs 50μs τ int τ delay τ int τ delay (a) (b) (d) (c) Figure 6.3: An ensemble measurement to validate power dependence with no hidden Markov model. (a) The measurement scheme —a sequence of 50 µ s pulses are input at 20 ms intervals, data is homodyne demodulated and we integrate over the windowτ int . Ensemble results are his- togrammed in IQ plane and fit to Gaussian modes. Performing this as a function of the delay timeτ delay allows us to calculate a moving average for the mean occupation as a function of time since the pulse begins. (b) The mean occupation vsτ delay with an exponential fit (Equation 6.3). The point at 42.5 µ s in red was omitted due to poor quality of Gaussian mode fit. (c) The re- lease rateΓ release from a separate experiment with continuous measurement and hidden Markov model analysis performed during the same cooldown as the pulsed ensemble measurement. The orange star marks the value of the clearing rate as estimated from the exponential fit to the mean occupation in panel (b) at the pulse power of -136 dBm. (d) The mean occupation from the contin- uous data fit with hidden Markov model is shown in blue. The grey dashed line marks the mean occupation at τ delay = 0 from the fit in panel (b), which can be interpreted as the low power saturation. The orange star marks the estimated mean occupation from the fit in panel (b) if the pulse were to stay on indefinitely at -136 dBm. 99 pation ¯n(τ delay ) is around 0.2 initially, then drops noticeably over the duration of the pulse. The orange curve is a fit to an exponential decay ¯n(τ delay )=Ae − Γ τ delay +⟨n⟩ ∞ (6.3) where Γ is the clearing rate and the⟨n⟩ ∞ is the long time saturation. In panels (c) and (d) the results of this pulsed measurement scheme are compared with a continuous measurement power sweep (much like those in this and the preceding chapter) analyzed by hidden Markov model bootstrapping (Section 3.3) during the same cooldown. We find close agreement between the hidden Markov model results and the values from our exponential fit to pulsed measurement data, suggesting that the observed power dependence is a real physical phenomenon and also that our analysis is quantitatively reliable. 6.1.2 Fittingthepowerdependence We may note that the power dependence as shown in Figure 6.1 appears to show a linear trend above -145 dBm or so when plotted in log-log. This invites us to fit the high power portions to a linear equation log 10 (Γ release )=Alog 10 (P)+B (6.4) We fit the release rate Γ release to this linear trend in 3 regions of trap depth and for powers - 145 dBm and greater. The results are shown in Figure 6.4. The fit parameters for all three lines are consolidated in Table 6.1 The shallowest traps (green in the figure) show a slope A ≈ 1 in log space, so it would appear that this region has a release rate Γ release which is linear in the 100 6 8 10 12 Trap Depth [GHz] -14.5 -14.0 -13.5 -13.0 Log10(Power / 1 mW) -2 -1 0 Log10(Release Rate / 1 MHz) -145 -127 Power [dBm] 4.9 - 7 GHz 8.5 - 10 GHz 12 - 13.1 GHz Figure 6.4: The release rate is plotted against trap depth∆ A and microwave measurement power P , grouped into 3 regions of trap depth. Dashed lines in the right panel show linear fits to the log space data for each of these regions. The shallowest traps (≤ 7 GHz) in green have the steepest power dependence with a slopeA≈ 1.03. The midrange trap depth (8.5 to 10 GHz) in yellow has a slope ofA≈ 0.61. The deepest traps (≥ 12 GHz) have slopeA≈ 0.44. Fit Results ∆ A Group 4.9≤ ∆ A [GHz]≤ 7 8.5≤ ∆ A [GHz]≤ 10 12≤ ∆ A [GHz]≤ 13.1 A 1.03107 0.613408 0.439832 A Std. Err. 0.0047429 0.0039978 0.0047908 B 13.0415 6.7885 4.16875 B Std. Err. 0.0645398 0.0544138 0.0652081 Table 6.1: Results of fitting the log of release rate log 10 (Γ release ) to a linear model Equation 6.4. There are three separate fits by the subset of trap depth. In each group, A is the slope andB is the intercept (atlog 10 (P) = 0, orP = 1 mW). Each fit parameter is quoted along with the standard error of the fit. microwave measurement power. The middle region (yellow) has slopeA∼ 2/3 and the deepest traps (orange) have a slopeA∼ 1/3. It is difficult to ignore that the regions with different power dependence are close in frequency to multiples of the microwave measurement signal frequency f d ∼ 4.27 GHz. It would suggest that this power dependence may arise from some multi-photon process. 101 6.1.3 Theory The work in this section is primarily that of Professor Leonid Glazman of Yale University who was kind enough to discuss our results and modeling efforts. The main idea is that quasiparticles are not being driven directly from the deepest traps (E A (δ ) with transmission coefficient τ = 1) to the band of states above the gap ∆ . Instead, given that the nanobridge junction hosts many channels with a Dorokhov distribution of channel transmission coefficients, there should be some intermediate Andreev states between those deep traps and the gap. These shallower states should be short-lived, which should broaden the uncertainty in their energy. This broadening can create an effective band in the density of states allowing for a single measurement photon to drive transitions from the deepest traps (withτ ≈ 1) to the intermediate band. Once the electron is in the intermediate band, it will be driven to the bulk much more rapidly than the initial transtion to the band since the bulk density of states at∆ is so much larger. We start with a toy model in whichE 0 are the deepest trap states (τ = 1), E j are the inter- mediate states with Dorokhov distributed mid range τ , and ∆ is the gap. Let a 0 and a j be the wave functions for the deep Andreev states and intermediate Andreev states respectively. When driven at frequencyω these states evolve according to i˙ a 0 =(E 0 +ω)a 0 +ϵd 0j a j (6.5) i˙ a j =(E j − iΓ j )a j +ϵd j0 a 0 (6.6) 102 whered ij is the dipole transition matrix element,ϵ is the drive strength,Γ j =|ϵd ib | 2 ν b , andν b is the density of states above the gap. Given that the wavefunctions have the forma j = e − iΩ t , we have (Ω − E 0 − ω)a 0 =ϵd 0j a j (6.7) (Ω − E j +iΓ j )a j =ϵd j0 a 0 . (6.8) Solving Equation 6.8 fora j and plugging this in to Equation 6.7 gives the ability to solve for the rates ˜ Ω= X j |ϵd 0j | 2 ˜ Ω+ E 0 +ω− E j +iΓ j (6.9) where ˜ Ω = Ω − E 0 − ω. In the short junction limit (nanobridge length small compared to coherence length, l ≪ χ ), Kos et al [36] find that the real part of the admittance of a weak link junction is linear in the transmission coefficient, and the rate of absorption is proportional to the real part of the admittance soΓ j = γτ j whereγ will depend on phase bias and drive frequency. We also assume thatd jk → 0 (we expect no transitions between intermediate Andreev levels of different channels). For a Dorokhov distribution of τ j , we can rewrite Equation 6.9 to integrate over the intermediate Andreev energies ˜ Ω= X j ⟨ |ϵd 0j | 2 ˜ Ω+ E 0 +ω− E j +iΓ j ⟩ Dorokhov (6.10) =|ϵd 01 | 2 Z dE j ∆ E A ⟨ − iΓ j (E j − (E 0 +ω+ ˜ Ω)) 2 +Γ 2 j ⟩ Dorokhov . (6.11) 103 Now we can approximateE 0 =∆ q 1− sin 2 δ 2 ≈ ∆(1 − δ 2 8 ) and similarlyE j =∆ q 1− τ j sin 2 δ 2 ≈ ∆(1 − τ j δ 2 8 ). Working in the case where2ℏω≈ ∆ − E 0 ≈ 2(∆ − E i ), we can use these approx- imations to obtain E j − (E 0 − ω) ≈ δ 2 16 ∆(1 − 2τ j ). Plugging this in along with the dorokhov distribution ofτ gives Im ˜ Ω= |ϵd 01 | 2 Z 1 0 dτ 2τ √ 1− τ − iγτ [ δ 2 16 ∆(1 − 2τ j )] 2 +(γτ ) 2 , (6.12) where we also use Γ i = ατ i . Evaluating this integral gives decay rate of the state under driven evolution Im ˜ Ω= − i 8 √ 2π δ 2 ∆ N e |ϵd 01 | 2 (6.13) A similar expression has been derived by Kos et al. [36]. In terms of the microwave drive power this is Im ˜ Ω= − iαP (6.14) and we have the linear power dependence observed in the shallowest trap depth group (where ℏω < ∆ A < 2ℏω) of Figure 6.4. This model (Equation 6.14) assumed the trap depth ∆ A was approximately twice the drive frequency. The other groups in Figure 6.4 which fit to sub-linear exponents are outside of the scope of this model since the trap depth is greater than twice the drive frequency. We can try to form an intuitive idea of why the model stops working for∆ A ≳ ℏω if we consider that moving to deeper trap depth means we increase the spacing between the Andreev levels. It may be the case that the intermediate Andreev band is no longer within∼ ω of the gap and so measurement photons can no longer drive this transition directly, or the spacing between 104 the intermediate states may increase beyond the energy broadening such that the intermediate band picture breaks down. 6.2 Thelowtemperatureandlowpowerlimit We see in Figure 6.1 at the lowest power that the release rate saturates to a value which depends on the trap depth. In Figure 6.5 we show all data at low temperature (T = 37 mK) as a function of trap depth∆ A and measurement power. Both the mean occupation and the release rate show some low power saturation, but we may focus only on the release rate since the mean occupation simply inherits this feature from the release rate. This low power saturation shape in the release rate is repeated in every sweep we take for all low temperatures (below∼ 80 mK where thermal phonons and QPs are not dominating the dynamics). The bottom left panel of Figure 6.5 shows that the release rate climbs slowly towards a peak near ∆ A ≈ 10.5 GHz, then falls sharply and climbs to a secondary peak at∆ A ≈ 13 GHz. This background clearing rate may be related to the higher modes of the resonator (3λ/ 4 mode is∼ 12.9 GHz) or packaging (cavity modes of the enclosure, spurious modes in the circuit board). If there is some nonzero population of higher energy modes, then these photons may be absorbed to excite quasiparticles from the traps. The observed rates are quite small which may be consistent with low temperature thermal populations of higher energy modes. In this case, the observed rise in clearing rate as the Andreev trap depth increases would be associated with the changing electron density of states at the final state energy E A (δ )+ℏω highermode ≥ ∆ . The drastic fall of clearing rate occurs when the Andreev trap depth exceeds the mode energy, and E A (δ )+ℏω highermode <∆ . 105 6 8 10 12 Trap Depth [GHz] -150 -140 -130 Power [dBm] 0.0001 0.001 0.01 0.1 Mean Occupation 0.0001 0.001 Trap Rate [MHz] 0.01 0.1 1 Release Rate [MHz] -154 -127 Power [dBm] 4.939 13.101 Trap Depth [GHz] Figure 6.5: The mean occupation¯n (top), trap rateΓ trap (middle), and release rateΓ release (bottom) at constant temperatureT =37 mK is shown as a function of trap depth∆ A (left) and measure- ment powerP (right). In the left side panels we can see that the release rate and mean occupation saturate at the lowest powers. In the bottom left panel we see that the low power saturation of the release rate increases slowly with trap depth until it peaks at∼ 10.5 GHz; a secondary peak can be seen around 13 GHz. The trap rate is mostly power independent, and any apparent power dependence may be explained by SNR degradation affecting the hidden Markov model. 106 Another possible explanation is that the low power saturation of release rate is due to inter- action with some two-level systems such as trapped charges at material interfaces. 107 Chapter7 Conclusions This body of work investigates the interactions of quasiparticle populations with sub-gap states in a superconductor. It is shown that aluminum nanobridges host a set of Andreev bound states with energies which can be dynamically tuned below the superconducting gap. These aluminum nanobridges are used as an inductive element in a microwave resonator, allowing for high mea- surement bandwidth detection of single electrons occupying these states. This experimental tech- nique relies on standard homodyne demodulation of a microwave signal to obtain the DC am- plitudes of the in phase (I) and quadrature (Q) components of the signal. Near quantum-limited amplifiers must be used to achieve sufficient SNR with large enough measurement bandwidth to monitor the trapping and clearing of quasiparticles from these Andreev bound states. The measured I(t) and Q(t) signals are fit using a hidden Markov model to determine the number of trapped quasiparticles as a function of time ¯n(t) as well as the transition rates be- tween resonator modes, Γ ij . In this case, Γ ij would be the rate of transitioning from ¯n = i trapped quasiparticles to ¯n = j trapped quasiparticles. The bulk of experimental results look at these transition rates as a function of trap depth∆ A (δ )≡ ∆ − E A (δ ), measurement power, and 108 temperature. Models are developed to explain the experimental observations in several different cases. The trap rateΓ trap = Γ 01 is found to be consistent with quasiparticle relaxation via sponta- neous emission from a temperature dependent quasiparticle density x(T) at all measured tem- peratures. The quasiparticle densityx(T) is a linear combination of a constant background pop- ulation —the non-equilibrium densityx ne —and a thermal population. The agreement with spon- taneous emission at the trap depth energy∆ A would suggest that the majority of quasiparticles, at least those participating in trapping, are “cold” quasiparticles with energy near the gap edge E∼ ∆ . The clearing rate Γ release = Γ 10 is more complex in its structure. Absorption of thermal phonons dominates the release rate for high temperatures, T ≳ 90mK. At low temperatures, the clearing rate can be dominated by driven excitation via the measurement signal —despite the insufficient energy of a single measurement photon to clear quasiparticles from the trap (ℏω drive < ∆ A ). The power dependence is likely due to broadening of intermediate states be- tween the deepest Andreev traps and the superconducting gap allowing for the low energy pho- tons to promote the electrons in deep Andreev states. At low temperature and low power, some peaks are present which may indicate that two-level systems contribute to clearing quasiparticles from sub-gap states at typical qubit operating conditions. The use of hidden Markov models as an analysis method is validated using simulations to determine the accuracy of hidden Markov model results. Hidden Markov models begin to break down when the squared ratio of mode separation to mode variance is less than 3 (SNR IQ ≲ 3). It is also shown that while changing the measurement power and therefore the RF SNR can affect 109 the estimated rates, this change is small compared to the observed changes in the clearing rate —although it can fully explain changes in trap rate. These results are useful to researchers studying methods to mitigate quasiparticle induced errors in quantum bits and other superconducting devices. In particular, these results may be interesting for researchers studying Andreev qubits —a quantum bit made out of a single Andreev bound state. These Andreev qubits are particularly susceptible to errors due to quasiparticle trapping and clearing. 7.1 Futurework Like any good project in science, it seems that I’ve discovered more new threads to follow than I’ve tidied up. Luckily, I’ve been privileged to work with a great team of scientists to help me along the way and also to continue following these new threads. In addition, some collaborators are currently working on similar trapping measurements using a different type of high transmission Josephson junction [19] One of the major experiments we would like to perform is spectroscopy of these Andreev states. The idea here is to couple in a dedicated excitation signal atω e that will clear quasiparticles from the Andreev bound state traps. If we sweep the frequencyω e over a broad range, we expect to find some peaks in the clearing rate when the excitation frequency overlaps with some Andreev states. At largeω e , we should be able to drive transitions between the upper and lower Andreev energies± E A (δ ). We may also find some unexpected behavior in this spectroscopy, similar to the trap depth regions of different behavior when sweeping the measurement power. The primary reason this experiment has not been performed is due to fabrication difficulties: these nanobridge 110 junctions have a low yield. In addition, the inclusion of a fast flux line in the design which couples excitation signals of a broad frequency range necessarily also couples to the resonator mode. This couping presents as internal loss when looking at resonator reflection, so many iterations of device design and fabrication have resulted in devices which are close to workable but fail to perform. Another option for performing this spectroscopy which may be worth pursuing is to simply include an antenna in the package that will result in free space radiation of the excitation into the enclosed volume. However, this approach may be limited by geometry and frequency dependent admittance of the excitation signal. A fun side project which spun off from this work is to use a fast arbitrary waveform generator and FPGA to run a PID loop and track the resonant frequency. This relies on very fast data processing (on the FPGA) to determine the amount the resonator has shifted and to correct for this shift by altering the drive frequency to match. This primarily relies on a pound drucker diode [45] to produce an error signal which is proportional to the difference between the resonator frequencyω 0 and the measurement frequencyω d . The presence of two-level systems may be influenced by the use of an ion beam to mill away aluminum oxide during the fabrication process. In some early fabrication work we observed that samples which were exposed to our ion beam registered having more iron and chromium when measured using an XPS at the center for nanoimaging at USC. In fabricating the device used in Chapters 5 and 6, we used this ion beam to make a galvanic connection between the resonator and SQUID. This means there are potentially magnetic contaminants (which may present two level systems at a broad range of frequencies) in localized regions within∼ 20 microns of the nanobridges. These two level fluctuators may be the source of the peaks in the low power, low temperature saturation of release rate. 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Rev. Lett. 106 (25 June 2011), p. 257003.doi: 10.1103/PhysRevLett.106.257003. 122 AppendixA Fabrication “Believe nothing you hear, and only one half that you see” Edgar Allen Poe Fabrication was a challenging aspect of this work. Much trial and error went into making working devices, so I’ve compiled a few of the recipes that work for me here. There are two approaches to making nanobridge resonators: writing the whole structure with the EBL and depositing all metal in a single step, and writing only the small structure with the EBL and using “bandaids” to galvanically connect to a resonator fabricated in another step. A.1 Singlestepresonators A.1.1 Cleaning Take a Si(111) chip from the amber shelf. Most likely this has a protective resist from previous dicing and needs to be cleaned. Gather beakers for Acetone, Methanol, IPA, and DI water at the solvent fumehood. Check that the sonicator is set to 100% power at 72 kHz. Perform the following cleaning procedure: 1. Sonicate in Acetone for 6 minutes 2. Sonicate in Methanol for 3 minutes 3. Sonicate in IPA for 3 minutes 4. Swirl in DI for 30 seconds to rinse. 5. Dry with N2 6. Place on hotplate at 110 – 180 C for 20 seconds 7. Spin resist with recipe below It is important to ensure that the chip is not allowed to dry between steps 1-4. In practice, this just means to pull the chip from one beaker and place in the next as fast as possible. It is best to do the cleaning procedure immediately before spinning so as to minimize the chance that debris/residue finds its way to the silicon surface. 123 A.1.2 Spinrecipe Check the hotplates are set to 170 and 180 C. Check or set the spinner recipe for MMA EL6 as follows: 1. Ramp 100/s, speed 500, duration 5 seconds 2. Ramp 1000/s, speed 3200, duration 60 seconds 3. Ramp 1000/s, speed 0, duration 0 Check or set spinner recipe PMMA A2 as follows: 1. Ramp 100/s, speed 500, duration 5 seconds 2. Ramp 1000/s, speed 3600, duration 60 seconds 3. Ramp 1000/s, speed 0, duration 0 Place your freshly cleaned wafer on the small chuck with a + pattern and start vacuum. Start a spin recipe. Get the MMA pipette ready by blowing with N2 to remove any dust or fibers, then pull a small amount from the secondary bottle. Wait for the dry run to finish or stop it early. Squeeze out a few drops to the side, then apply a few drops of MMA to the chip, just enough to cover, and start the MMA spin recipe. After the recipe has finished, carefully grab chip with tweezers and visually inspect the bottom for resist. Carefully clean the bottom with acetone swab if needed. Place the chip on the hot plate at 170 C for 5 minutes. Use the lift pins to make your life easier. After bake has finished, place the chip back on the spinner and run a dry recipe. While this is running, get the PMMA pipette ready. After finish/stop, squeeze a few drops out to the side and then apply a few drops of PMMA to the chip, just enough to coat the surface. Run the PMMA spin recipe, then inspect/clean the bottom of chip as before – you don’t want to bake the resist onto the bottom of the chip, it won’t sit level after that. Next, place on the bakeplate at 180 C for 5 minutes. Afterwards the chip is ready for lithography. A.1.3 Beamerpreprocessing Layout files must be processed with BEAMER software to create the gpf files that Raith EBPG requires. Here are the settings I use: 1. Import the NBR CPW design with zero width paths set to 15 nm 2. Healing per layer 3. Perform extraction of layers/regions if needed 4. 3D PEC – EDGE (a) 100 nm PMMA on Si from archive; blur = 10 nm (b) 1% accuracy, 256 dose classes, 10 nm isodose grid, automatic figure sizes 124 (c) Target dose = 1 for main layers; target dose = 0.6 for undercut layers (d) Optimal contrast / uniform clearing = 0 / 100 5. FDA multiplies bridge dose by 1.4 6. Split to extract small features and bulk 7. Export small features with 0.5 nm step size/res; 8. Export bulk features with 10 nm step size/res Once BEAMER is finished, transfer the exported gpf files to EBL using filezilla. A.1.4 RaithEBPGsetup Here is the process I use at the Raith EBPG when performing lithography: 1. Load sample and adjust table height/plane such that chip surface reads -10 um on optical microscope. Note the location of corners relative to faraday cup so you can calculate center position. 2. Load stage into EBL, wait for vacuum and transfer stage to main chamber 3. While waiting for vacuum, set up cjob: (a) 10 mm square chip (b) Exposure with default performance checks, etc. 100 kV. (c) Make a layout with 2.5 mm X vector, 3.5 mm Y vector. Make sure the layout pattern doesn’t exceed your chip size. It’s not a bad idea to set a dose update. Perhaps 20-25 uC/cm 2 (d) Drag a pattern onto the layout, choose the bulk features gpf, set the beam to 50na_300um and dose to 1650 (ideal dose 1650-1750, so aim for this range in the layout dose updates) (e) Drag another pattern onto layout, choose the small features gpf, set the beam to 1na_300um and dose to 1650 (f) Save and export the job, making sure to choose the order of beam currents that makes sense (if you are going to be at 50 nA at job start, make sure it does 50 nA, then 1 nA) 4. Move to marker, resotore beam (1 nA or 50 nA), autofocus and adjust C2 if needed. Measure current and save the beam. 5. Repeat for the other beam. 6. Start the job. 125 A.1.5 Develop After unloading from EBL, set up the -15C developer station. Fill the insulated dish with thermal beads in IPA from the freezer. Fill the MIBK:IPA beaker and a pure IPA beaker enough to easily submerge the chip, place both in the thermal beads. Grab a DI water beaker and rinse thoroughly with DI spray gun before filling with fresh DI. Monitor the thermometers; red is in the thermal bath, blue is in MIBK:IPA. As soon as the blue thermometer reads -15 C, pull out thermometer and start development: 1. 60 seconds gentle swirl in -15 C MIBK:IPA. 2. Immediately transfer to chilled IPA bath and swirl for 60 sconds. 3. Immediately transfer to DI water bath and swirl for 30 seconds. 4. Dry thoroughly with N2 – Note, it’s not possible to dry on the hotplate as this will reflow the resist and destroy your lithography. At this point, you can image the chip on microscope if you like, then load into the evaporator. A.1.6 Deposition The nanobridge deposition recipe has a critical dependence on the alignment when loading the sample onto the Angstrom evaporator stage. We make use of the shadow cast by tilting the sample to create a 3D constriction forming the nanobridge junction, so we must align the sample pattern to the crosshairs on the stage with the “bottom of the pattern” (the side with undercuts) facing the notch in the stage. Once loaded, the evaporation recipe deposits 10 nm at direct incidence (this is the bridge thickness, making it thinner will decrease the number of conduction channels giving us more sensitivity at the cost of lower yield. Making it thicker reduces sensitivity but improves yield) with a rate of 4 A/s, then tilt to -35 degrees and continue depositing to a final thickness of 70 nm (the actual target thickness in recipe will be z =(70− 10)/cos(− 35). A.1.7 Liftoff After deposition, the chip should be placed in the liftoff acetone beaker and suspended in a hot bath at 45 C. Let the chip soak for 2 hours. Towards the end of the soak, gather up the beakers for methanol, ipa, and DI water at the solvent fumehood. Fill each beaker with respective solvent. Check the sonicator settings: 40% power, 72 kHz. When the soak is complete: 1. Sonicate chip in acetone liftoff beaker for 30 seconds (more time if the liftoff has not completed, or less time if you can see that the liftoff has finished.) 2. Sonicate in methanol for 10s 3. Sonicate in IPA for 10s, 4. Sonicate in DI water for 20s. 126 5. Rinse with DI bottle. 6. Dry with N2. A.1.8 SEMimaging I use the nanoSEM 450 in “immersion mode,” which gives good resolution of the bridges with careful calibration of lens alignment and stigmation. You want to minimize the exposure of devices. I typically rotate the stage to align the samples, then move by the pattern layout spacing to image each device on the sample. It’s best to keep the beam blanked when not actively focusing or imaging, use “ctrl B” to quickly toggle beam blanking on/off. Once at a device, quickly adjust the fine focus on some feature near the squid but not on it (you should adjust stigmation/lens alignment before moving near junctions as this takes time and doesn’t need to be adjusted between images). A.1.9 Dicing If any devices are worth measuring, we need to dice them to size for packaging. The first step is to spin coat the chip with a protective resist layer —this can be PMMA A6 or photoresist, it’s only purpose is to protect the surface. That said, the spin recipe is not critical, but I typically spin at 2,500 RPM to give a thick protective layer. Bake as normal for the selected resist. We have a disco dicing saw that gives great results for cutting, follow the SOP for loading and operation. Some simple arithmetic should give the spacing between cuts required for the cutting close to the device with a little extra buffer (note: with the blades we use as of this writing, 27HEDD, the width of the cut is 60 to 80 micron. I typically set the spacing to be the pattern width +300 micron —from adding a 150 micron buffer on each side) A.1.10 Finalcleaning Once diced, we need to clean the samples in preparation for packaging and measurement. I start with a 30 minute soak in warm acetone, then sonicate for a few seconds at 40% power, 72 kHz. Longer sonication tends to flip these tiny samples which should be avoided. Finish up with a quick transfer to Methanol, then IPA, then DI water (make sure to release the sample from tweezers in each beaker to ensure good solvent flow on the chip surface which aids in removing the previous solvent). Dry (very) carefully with N2 (I’ve lost many samples by accidentally blowing it away...) Once dry, load the samples to the YES 02 plasma asher. Perform a gentle descum post-ash —10 sccm, 50 W, 50 mTorr, 10 seconds. Carefully unload and the chips are ready for packaging! A.2 multistepresonators Many of the steps for this process are similar to the above recipe for single step resonators. The biggest difference will be repeating each of the above steps with some careful attention paid to align the sample between lithography iterations. 127 We start with a Si(111) wafer which has already been patterned with gold markers for alignment. The first step is to fabricate the junctions. Clean the gold marked Si(111) wafer following Section A.1.1, then spin the resist exactly as Section A.1.2. Process the junction pattern using Beamer (steps 1-4, and 7 of Section A.1.3. Next, we perform the first aligned exposure with Raith EBPG. 1. Load sample and adjust table height/plane such that chip surface reads -10 um on optical microscope. 2. Rough align the gold markers by adjusting the table rotation. 3. Note the location of markers relative to the Faraday cup 4. Load stage into EBL, wait for vacuum and transfer stage to main chamber 5. While waiting for vacuum, set up cjob: (a) 10 mm square chip (b) Exposure with default performance checks, etc. 100 kV. (c) Select Global markers and input the location of markers in the pattern relative to center (-5000,-5000; 5000,-5000; 5000,5000; -5000,5000; for a set of four markers in a 1 cm square.) (d) Make a layout with 2.5 mm X vector, 3.5 mm Y vector. Make sure the layout pattern doesn’t exceed your chip size. It’s not a bad idea to set a dose update. Perhaps 20-25 uC/cm 2 (e) Optionally, make a sub-layout to put multiple squids at each resonator site (you can SEM to select the best looking squid, then offset each resonator in later lithography step appropriately to line up with the best squid). I typically use a vertical array of 5 sites with 80 micron spacing —you’ll want to verify that this does not cause any issues for your design (you want the unused squids to be under the ground plane, not in the center trace of resonator or FF line, etc.) (f) Drag a pattern onto layout, choose the small features gpf, set the beam to 1na_300um and dose to 1650 (g) Save and export the job, making sure to choose the order of beam currents that makes sense (if you are going to be at 50 nA at job start, make sure it does 50 nA, then 1 nA) 6. Move to marker, resotore beam (1 nA), autofocus and adjust C2 if needed. Measure current and save the beam. 7. Start the job. Develop, deposit, and liftoff exactly as above for the single step process. Optionally, you could perform SEM imaging at this point (particularly if you put multiple squids at each resonator site and need to select which one to use). 128 A.2.1 Step2: resonators After liftoff and imaging, you’ll need to spin the appropriate resist for resonator lithography. This could be photolithography or e-beam lithography. Note that we no longer require any angled deposition, so you can use the thicker resist stack we often use for SIS junctions (MMA EL13 at 4k RPM with 180C bake, then PMMA A6 at 4k RPM and bake) if you are performing e-beam lithography. Performing a dose test is a good idea, but for starters we should require ∼ 750µ C/cm 2 for room temperature development in MIBK:IPA 1:3 for e-beam lithography, and photolithography should require around 100 mJ/cm 2 when using AZ 1512 resist and AZ Developer 1:1 (NOTE: many cleanroom users develop photoresist with AZ 400K. This will etch aluminum so we cannot use it for anything where aluminum has already been deposited!) Perform the lithography with special care to align the sample (using the same alignment procedure as above for e-beam, or manually for the MJB4 photolithography tool), develop as applicable, then deposit 100 nm aluminum at direct incidence (no tilt!). Liftoff same as before. A.2.2 Step3: bandaids The final step is to make galvanic connection between squid and resonator by patterning some windows over the connection points, blasting the metal surface with an ion mill (Argon from the end-hall, gridless ion mill installed in Angstrom evaporator) to remove oxide, then deposit a thick layer of aluminum to make the connection. We start by spinning resist (A6/EL13 stack for e-beam lithography or AZ 1512 for photolithography), performing aligned lithography of bandaids (the pattern is essentially a rectangle which is 10s of microns on each side overlapping the area where squid and resonator meet —there should be atleast 100µ m 2 of overlap with resonator and squid independently.), and develop as applicable. Next, load the sample in evaporator and run an ion mill and deposit recipe. Ion milling should be setup to have 80V discharge potential, 3.5A discharge/emission current, and with± 35 degree incidence (actual VAD tilt -50.74 is direct incidence for the ion beam, so± 35 degree incidence is -15.74 or -85.74 degrees tilt). The ion mill should be run for 150 seconds at each tilt for a total of 300 seconds of mill time. Then immediately deposit 250 nm of aluminum to form the contact. Liftoff and perform final cleaning as above (Section A.1.7 and Section A.1.10). 129 AppendixB Amplifiers A critical requirement of this work is high SNR. The primary way to achieve high SNR is by use of ultra low noise parametric amplifiers near the device. Here I give some practical tips for optimizing SNR using these amplifiers. B.1 Travellingwaveparametericamplifiers The TWPA is a broadband amplifier that can get within a factor of 3 or so of the quantum limit on SNR. For tuning, there are two knobs to turn —the pump power and pump frequency. The following procedure will find the “neighborhood” of the ideal pump tone. 1. Find the dispersive feature of the TWPA using a VNA 2. Set the pump frequency to the low frequency “shoulder” of this dispersive feature 3. Increase pump power until some amplification is observed 4. Keep increasing pump power while monitoring for a sharp crash in amplification 5. Set the pump power to be a few dB below the point at which crash is observed. 6. Sweep the pump frequency to look for maximum amplification After finding this neighborhood, it is beneficial to actually optimize the SNR. First, determine the frequency you wish to amplify (the frequency of your measurement signal) and connect a signal generator at this frequency and low power (the power you will typically use in the measurement). You will want to couple some of the output signal to a spectrum analyzer —some adjustment of span and bandwidth may be necessary to see the measurement signal (smaller measurement bandwidth means lower noise floor). Perform the following to optimize the SNR 1. While recording data with a spectrum analyzer, sweep the pump power and frequency over an appropriate span about the neighborhood found previously (use your own judgement to determine what is appropriate here —large enough span to see appreciable changes, small enough step size to resolve a smooth change in behavior, balance of size and resolution so it doesn’t take all day) 130 Figure B.1: TWPA tune up results. The highest SNR (23 to 24 dB) is in this case achieved at the bright yellow points, such as pump frequency 8.042 GHz at pump power 4 dBm. 2. Calculate SNR (after averaging if you took multiple records at each point in the sweep, but be sure to average linear powers rather than decibels) by subtracting the noise power (linear average of noise floor away from peak, converted back to dBm) from the peak power (in dBm). 3. Analyze to determine the ideal pump settings to maximize SNR for your measurement. In the end, you should be able to make a plot similar to that shown in Figure B.1 from which it is easy to identify the maximum SNR. If in a hurry, you can achieve excellent results by simply iterating a few linecuts (sweep pump power at constant frequency and find best SNR, then sweep frequency at constant power and find best SNR, iterate until no change —like a manual implementation of gradient descent). The only risk here is becoming fooled by a local maximum which may be less ideal (lower SNR or perhaps more sensitive to power/frequency fluctuations) 131 B.2 Josephsonparametricamplifiers The JPA is a narrowband amplifier which can give SNR at the quantum limit. Tuning a JPA is only slightly more complex than a TWPA —it has 3 knobs to turn; pump frequency, pump power, and applied flux. The pump can be applied on the signal port or the fast flux line. If on the signal port, you are performing “4 wave mixing” and the pump frequency is equal to the amplified frequency. If on the fast flux line, we have “3 wave mixing” and the pump is twice the amplified frequency. Tuneup will be similar to the TWPA procedure; the first goal is to find the correct neighborhood. 1. Flux tune the JPA such that the low frequency “shoulder” of the phase response is at the target measurement frequency 2. Record a memory trace on VNA and display alongside new data 3. Turn on the pump (at measurement frequency plus a small (MHz) offset if pumping directly on signal port, at twice this frequency if on fast flux line) 4. Increase pump power while monitoring phase response on VNA until the JPA shifts down in frequency by a small amount 5. Increase (or decrease) flux such that JPA response overlaps with memory trace 6. iterate 4 and 5 a few times until the pump power is a few dB above where the JPA first shifted. After completion, you should have some amplification from JPA. Next will be to fine tune the SNR. We’ll follow the same procedure as the TWPA but with an extra dimension (JPA flux). After calculating SNR for the sweep, it is simple to find the point with maximum SNR. 132 AppendixC Code Python has been the language of choice for almost all simulation and analysis work. The actual code used to perform simulations, measurements, and analysis can be found in repositories on my GitHub account (https://github.com/jtfarm). The hidden Markov model source code can be found at (https://github.com/hmmlearn/hmmlearn). 133

Abstract (if available)

Abstract We provide a proof of principle demonstration for the use of aluminum nanobridge Josephson junctions integrated into a superconducting-qubit-like device for the continuous, real-time detection of quasiparticles trapping into Andreev bound states. Analysis methods are developed to determine the rates of quasiparticles trapping and clearing, as well as estimating the occupation from the time series voltage record. To test the efficacy of our analysis, we employ statistical simulations and score the ability of hidden Markov models to produce meaningful information from the noisy data. We find that the hidden Markov model is an effective analysis tool even when the underlying assumption of Markovianity is invalidated. By providing a good initial guess for the modes at high power, we are able to iterate our way to very low powers by using each fit as a warm start for the next fit. Simulation results show that this method retains accurate estimates even at low SNR.
The rate of quasiparticles trapping and clearing from Andreev bound states are explored over a range of environment parameters. From the dependence on the trap depth and temperature, we show that quasiparticles are relaxing from the bulk superconductor into the Andreev traps by spontaneous emission from a temperature dependent bulk population with a substantial nonequilibrium component. Similarly, for temperatures above ∼ 90 mK, quasiparticles are cleared from the traps due to absorption from a thermal population of phonons with energy exceeding the trap depth. While the trap rate appears to be dominated by spontaneous emission at all temperatures, the release rate saturates below ∼ 60 mK to a level which depends on the power of our measurement signal.
The power dependence of the rate of clearing quasiparticles can be grouped into three regions of different behavior. At the shallowest traps the rate increases∼linearly with power. In the middepth region when the trap depth is more than two photons, the dependence is sublinear, fitting a power law with exponent ∼ 0.6. The last region fits a power law with exponent ∼ 0.4. A theoretical explanation is discussed which produces the linear dependence observed when the trap depth is less than two photons. Further complexity is found in the background release rate at low temperature and low measurement power. The background release rate increases slowly with trap depth to a peak around 10.5 GHz, then falls drastically.

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Asset Metadata

Creator Farmer, James Timothy (author)

Core Title Bogoliubov quasiparticles in Andreev bound states of aluminum nanobridge Josephson junctions

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Physics

Degree Conferral Date 2023-05

Publication Date 04/17/2023

Defense Date 03/31/2023

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag microwave resonators,OAI-PMH Harvest,quasiparticles,superconducting circuits

Format theses (aat)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Levenson-Falk, Eli (committee chair), El-Naggar, Moh (committee member), Kresin, Vitaly (committee member), Lidar, Daniel (committee member), Takahashi, Susumu (committee member)

Creator Email jamestf6@gmail.com,jtfarmer@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-oUC113017548

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Document Type Dissertation

Format theses (aat)

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