Physics and Astronomy

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This is the collection for the University of Waterloo's Department of Physics and Astronomy.

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Towards Gated Quantum Emitters from Undoped Nano-LEDs
(University of Waterloo, 2024-07-25) Sherlekar, Nachiket Sunil
Quantum light emitters have the potential to transform emerging quantum technologies and their applications, such as secure quantum communication, metrology, and quantum computing. Ideally, these light sources emit on-demand at high rates and efficiencies with high degrees of single-photon indistinguishability. Additionally, these emitters can emit entangled photon pairs, are position-controllable, and wavelength-tunable. Current state-of-the-art single- and entangled-photon sources based on spontaneous parametric down-conversion (SPDC) and on-demand (or deterministic) implementations suffer from various drawbacks that make them deviate from ideality. SPDC sources emit probabilistically, and increasing their brightness degrades their single photon purity, indistinguishability, and entanglement fidelity. Of the various deterministic sources, optically-driven semiconductor quantum dots have very high single-photon efficiencies (~ 71%), purities (> 99%) and indistinguishabilities (> 99%), are position-controllable and wavelength-tunable. However, complex synchronized optical routing between the pump laser, sources and detectors is required to scale their usage. This would occupy a large footprint, restricting them to a laboratory setting. Quantum dots may be current-injected instead, but while gigahertz (GHz) emission frequencies are possible, the electron injection number is not controllable. This inability to control the electron injection is akin to non-resonant optical excitation in which there are many charges in the environment around the quantum dot, thus making current-injected quantum dots inferior to optically-driven quantum dots. This thesis proposes a novel design for a high-rate, deterministic, electrically-driven quantum emitter that combines a gate-defined lateral planar p-n junction (or nano-light-emitting-diode or nano-LED) with a quantized charge pump along a quasi-one-dimensional channel in dopant-free GaAs/AlGaAs heterostructures. In contrast to other electrically-driven sources, our implementation allows for a precise control of the injected electron number via the quantized charge pump. In addition, by using gates to define the p-type and n-type regions of the junction instead of intentional dopants (as in conventional vertical p-n junctions), the charge carrier mobility in these heterostructures is much higher. The lack of dopants also allows p-type and n-type regions to exist simultaneously on both sides of the device (such devices are termed `ambipolar'), in turn allowing flexible operation. By operating the charge pump at GHz frequencies, this source could emit a billion photons per second. Integrating a cavity at the site of emission would boost the rate of emission and the efficiency, and could also increase the single-photon indistinguishability. The following research obstacles were identified over the course of developing our nano-LED (the prerequisite for our quantum emitter): - quenching of device electroluminescence (EL) and time-instability of emissions due to parasitic charge accumulation, necessitating thermal cycling to reset the device; - alternate current pathways (both radiative and non-radiative) through the device mesa that reduce both internal and external quantum efficiency; - delocalized emission at mesa edges due to minority currents under the topgate edges, affecting extraction efficiency and position-controllability; and - multimode emission and slow rate of spontaneous emission that reduce extraction and collection efficiencies. Descriptions of our nano-LEDs and their emissions as well as solutions to the above obstacles obtained through experiment are summarized below. The nano-LEDs discussed in this thesis are gate-induced either in GaAs rectangular quantum wells or at GaAs/AlGaAs single heterojunction interfaces. All nano-LEDs reported in literature are induced using the former and not the latter. In fact, a recent theoretical study concluded that radiative electron-hole recombination was impossible in nano-LEDs induced at single heterojunction interfaces. Our demonstration of EL from nano-LEDs induced at GaAs/AlGaAs single heterojunction interfaces is the first of its kind. Since the fabrication yield using single heterojunction wafers is higher than when using rectangular quantum wells, they offer an alternative for easier fabrication of the nano-LEDs. To understand how the EL quenches in our nano-LEDs, we propose a scenario of localized parasitic charging that results in enhanced non-radiative recombination and causes a gating of the p-n channel that suppresses the diode current. To address this issue, we have devised a gate voltage sequence that we call the `Set-Reset' protocol. This protocol clears away accumulated parasitic charge, extending the lifetime of device operation without the need for thermal cycling. Our nano-LEDs can be operated in four distinct circuit measurement configurations, depending on whether the left side is p-type or n-type (with the right side being n-type or p-type, respectively), and whether the left side is grounded or floating (with the right side being floated or grounded, respectively). EL from our nano-LEDs (induced at both quantum wells and single heterojunctions) is observed not only around the p-n junction interface, but also as far as the edges of the etched mesa, indicating the presence of unwanted radiative recombination pathways. The p-side is consistently brighter in the single heterojunction samples while the n-side was brighter for the quantum well devices. A neutral and a negatively charge exciton peak was observed in the spectra from the n-side of the nano-LEDs. Spectra from the p-side were measured only for the single heterojunction devices, and showed the neutral exciton peak as well as a lower energy peak. The narrowest neutral exciton emission linewidths (0.70 meV) from lateral p-n junctions to date were recorded from the quantum well nano-LEDs. Our nano-LEDs were also shown to be compatible with radio frequency operation, necessary for quantized charge pump integration to create a quantum emitter. To address the issue of delocalized emission and time-instability of EL, we fabricated and tested a nano-LED with a novel gate architecture that included two wide surface gates placed adjacent and perpendicular to the p-n channel. The extra gates add a degree of freedom that along with standard DC operation and the Set-Reset protocol opens up many measurement configurations. A downside is that these surface gates are prone to current leakage. Several measurement configurations were explored, with two standing out---one yielded localized emission at the junction interface while using the Set-Reset protocol; another yielded time-stability of emission in DC operation. A conceptual model has been laid out that is compatible with the results from these various operating configurations. From the time-stable measurements of EL intensity and p-n current, the internal and external quantum efficiencies were estimated to be ~ 1.19x10^(-3) and ~1.95x10^(-5), respectively. These values may be boosted in future designs by incorporating insulator-separated side gates, blocking gates, and a cavity around the emission region. The side gates and blocking gates will respectively time-stabilize and localize the EL emission during DC operation, and the cavity will increase the rate of spontaneous emission and shape the mode. A long-standing problem in the field of deterministic quantum emitters is the fact that they emit light omnidirectionally and into multiple modes. Various confining structures such as tapered nanowires, micropillar cavities, photonic crystals, solid immersion lenses and circular Bragg gratings have been proposed and implemented in literature. We identified the circular Bragg grating cavity etched into a heterostructure with a Bragg mirror grown below the rectangular quantum well as the optimum solution for our nano-LED. Through simulation, both the Bragg mirror and circular Bragg grating designs were tuned to match the quantum well emission wavelength (~ 807.5 nm). The circular Bragg cavity etched into the Bragg mirror wafer around the emission region enhances the rate of spontaneous emission via the Purcell effect, and simultaneously funnels the emission into a single elliptical Gaussian mode for efficient collection. A split was included in the circular Bragg grating to make it compatible with our proposed emitter design. Theoretically, for in-plane exciton dipoles oriented parallel to this split, the cavity enhances the spontaneous emission rate by a factor of 5.3 at a center wavelength of 807.4 nm and a bandwidth of ~ 3.7 nm or ~ 7.0 meV. The split in the cavity causes emission to be linearly polarized. This linear polarization is unfortunately incompatible with the emission of polarization entangled photon pairs. The effective collection efficiency (from simulation) is ~ 30%, which is ~ 52 times greater than that of a device without a cavity. The inclusion of our cavity also boosts the internal and external quantum efficiencies by factors of 4.5 and 89, yielding values of ~ 5.32x10^(-3) and ~ 1.74x10^(-3), respectively. Design validation of the Bragg mirror using reflection measurements yielded a Bragg stopband frequency and bandwidth that closely match simulation. Simulated and measured reflection spectra from the circular Bragg gratings indicated a linear relationship between the ring width of the grating and the cavity resonance wavelength, with a consistent wavelength offset between simulation and measurement of ~ 16.3 nm. From these results, a cavity with a ring width of ~ 94.8 nm would most closely match the emission wavelength of ~ 807.5 nm. Through our proposed and implemented solutions for the obstacles facing our nano-LEDs, we pave the way for the realization of a high-rate, electrically-driven quantum emitter.
New Experimental Observables for the QCD Axion
(University of Waterloo, 2024-07-08) Madden, Amalia
The QCD axion is one of the best motivated extensions to the Standard Model of particle physics that could also serve as the dark matter. The thesis will demonstrate new experimental observables that could be used to search for the axion. These observables are based on piezoelectric materials that spontaneously break parity symmetry, thereby enabling sensitivity to the axion's fundamental, model independent coupling to gluons. The first observable explores how axion dark matter could generate an oscillating mechanical stress in a piezoelectric crystal. We call this new phenomenon ``the piezoaxionic effect". When the frequency of axion DM matches the natural frequency of a bulk acoustic normal mode of the piezoelectric crystal, the piezoaxionic effect is resonantly enhanced and can be read out electrically via the piezoelectric effect. We also point out another, subdominant phenomenon present in all dielectrics, namely the ``electroaxionic effect". An axion background can produce an electric displacement field in a crystal which in turn will give rise to a voltage across the crystal. We find that this model independent coupling of the QCD axion may be probed through the combination of the piezoaxionic and electroaxionic effects in piezoelectric crystals with aligned nuclear spins, with near-future experimental setups applicable for axion masses between $ 10^{-11}\text{eV}$ to $10^{-7}\text{eV}$, a challenging range for most other detection concepts. The second observable, the ``piezoaxionic force" demonstrates how a piezoelectric crystal can be used to source virtual QCD axions in the laboratory, giving rise to a new axion-mediated force. The presence of parity violation in the piezoelectric crystal, combined with aligned nuclear spins, provides the necessary symmetry breaking to generate an effective in-medium scalar coupling of the axion to nucleons. We propose a detection scheme that uses the axion's model-dependent pseudoscalar coupling to nuclear spins, such that the new force can be detected by its effect on the precession of a sample of polarised nuclear spins. When the distance between the source crystal and the detector is modulated at the Larmor precession frequency of the nuclear spins, the signal is resonantly enhanced. We predict that near-future experimental setups should be sensitive to the axion in the unexplored mass range from $10^{-5} \text{eV}$ to $10^{-2} \text{eV}$.
Dark Screening of the Cosmic Microwave Background with Hidden-Sector Particles and New Dynamical Observables in First Order Phase Transitions
(University of Waterloo, 2024-07-02) Pirvu, Dalila
This thesis focuses on two research directions within the field of Cosmology. It comprises the main results of my work as a PhD student. Part~\ref{PartI} introduces new observables of false vacuum decay derived from real-time numerical simulations. Part~\ref{PartII} describes a new method to search for hidden-sector particles using information from Cosmic Microwave Background (CMB) and Large Scale Structure (LSS) data. The first part studies metastable `false' vacuum decay in relativistic first order phase transitions. It is a phenomenon with broad implications for Cosmology and is ubiquitous in theories beyond the Standard Model. Describing the dynamics of a phase transition out of a false vacuum via the nucleation of bubbles is essential for understanding vacuum decay and the full spectrum of observables. We study vacuum decay by numerically evolving stochastic ensembles of field theories in 1+1 dimensions from an initially metastable state. First, we demonstrate that bubble nucleation sites cluster by measuring correlation functions in simulations. Next, we show that bubbles form with a Gaussian spread of centre-of-mass velocities for a field with an initial thermal spectrum. Finally, we show that nucleation events are preceded by oscillons - long-lived, time-dependent, pseudo-stable field configurations. We provide theoretical tools to model and generalize our findings. In the second part, we introduce a new type of secondary CMB anisotropy: the patchy screening induced by resonant conversion of CMB photons into dark-sector massive scalar (axions) and vector (dark photons) bosons as they cross non-linear LSS. In two of the simplest low-energy extensions to the SM, CMB photons can resonantly convert into either dark photons or axions when their local plasma frequency matches the mass of the hidden sector particle. For the axion, the resonance also requires the presence of an ambient magnetic field. After the epoch of reionization, resonant conversion occurs in dark matter halos if the hidden-sector particles have masses in the range $10^{-13} {\rm \; eV} \lesssim m_{{\rm A^{\prime}}} \lesssim 10^{-11} {\rm \; eV}$. This phenomenon leads to new CMB anisotropies correlated with LSS, which we refer to as dark screening, in analogy with anisotropies from Thomson screening. Each process has a unique frequency dependence, distinguishing both from the blackbody CMB. In this thesis, we use a halo model-based approach to predict the imprint of dark screening on the CMB temperature and polarisation and their correlation with LSS. We then examine $n$-point correlation functions of the dark-screened CMB and correlation functions between CMB and LSS observables to project the sensitivity of future measurements to the dark photon and axion coupling parameters.
Variational Quantum Computing: Optimization and Geometry
(University of Waterloo, 2024-06-27) Roeland, Wiersema
Quantum computing potentially offers unprecedented computational capabilities that transcend the limitations of classical computing paradigms. Despite its conceptual inception over three decades ago, recent years have witnessed remarkable progress in the realization of physical quantum computers, spurring a surge of research activity in the field. Although fault-tolerance devices remain unrealized, modern quantum hardware is getting less noisy, which allows us to investigate quantum algorithms that require only short depth circuits. One particular class of algorithms that falls into this category are variational quantum algorithms, which treat a quantum computer as a black box with tunable parameters that can be optimized via a classical optimization routine. This thesis delves into the realm of variational quantum algorithms and explores their optimization properties, trainability and geometric properties. Through a blend of numerical experiments, geometric insights, and mathematical analysis, it provides a comprehensive exploration of variational quantum algorithms paving the way for future advancements in variational quantum computing.
Dynamics and phases of matter in open quantum many-body systems
(University of Waterloo, 2024-06-24) Sang, Shengqi
The thesis is divided into two parts, both focusing on the topic of open quantum many-body systems. The first part explores the properties of quantum circuits interspersed with measurements. Tuned by the frequency of measurements, the circuit exhibits two stable dynamical phases: a weakly-monitored phase and a strongly-monitored one. For the former case, we analyze its non-equilibrium properties and unveil that it exhibits physical length scales that grow super-linearly with time. For the latter case, we demonstrate that it can maintain non-trivial quantum order when symmetries are present. The second part addresses phases of matter for mixed many-body states. We propose a real-space renormalization group approach for mixed states and apply it to derive phase diagrams for various examples. For decohered topological codes, we establish a precise relationship between the decodability and the topological phase transitions. Lastly, we introduce the notion of 'Markov length', a length scale that measures the locality of correlation, as a diagnostic for the stability of mixed state phases.
Leveraging Polarization in the Era of Submillimeter VLBI
(University of Waterloo, 2024-06-14) Ni, Chunchong
With the advancement of technology, global very long baseline interferometry (VLBI) observations at millimeter wavelengths become possible. The Event Horizon Telescope (EHT) is the first such experiment, which makes observing accretion disk and jet launching regions near supermassive black holes and active galactic nuclei (AGN) possible, including polarimetry observations. Centaurus A (Cen A) is a nearby radio-loud AGN, with large jet structures of angular size measured in degrees. It was observed by the EHT, whose first total intensity image shows a fork-shaped edge brightening jet structure. Chapter 2 applies Bayesian imaging method to the Cen A data. We first construct the total intensity image of Cen A, which we directly compare with the previous publication. Second, the Bayesian method produces the first polarization studies of Cen A jet. Both the total intensity imaging and the polarization mapping feature a full image posteriors with access to the image uncertainty. This proves to be essential in the case of Cen A, where the data is very challenging for various reasons. With polarization image posterior of Cen A, we are able to study different regions of the jet separately, eventually producing a robust estimate of a collection of important physics quantities, including magnetic field strength, the electron number density and the jet velocity. In Chapter 3, we explore the origin and influence of the interstellar scattering on observations of Sgr A*, and propose a novel method to mitigate this scattering via EHT and next-generation EHT (ngEHT) polarimetry in the future. In EHT and other radio astronomical observations of Sgr A*, scattering contaminates the image with external small-scale structures, essentially preventing further studies of the turbulence in the accretion disk. However, for credible interstellar magnetic field strengths, the scattering is proved to be insensitive to polarization. Therefore, it is possible to distinguish intrinsic and scattered structures via the image power spectra constructed in different polarization components. Via numerical experiments, we demonstrate a method for reconstructing intrinsic structural information from the scattered power spectrum. We demonstrate that this is feasible through a series of numerical experiments with general relativistic magnetohydrodynamic (GRMHD) simulation images. Specifically, we show that the ratio of the power spectra, obtained independently for different polarization components, is independent of the scattering screen. Therefore, these power spectra ratios provide a window directly into the MHD turbulence believed to drive accretion onto black holes.
Effects of Noncommuting Charges in Quantum Information and Thermodynamics
(University of Waterloo, 2024-06-13) Majidy, Shayan
The advancement of quantum theory is rooted in challenging established assumptions. This trend persists as quantum theory extends into other fields, including thermodynamics. One such assumption in thermodynamics is that conserved quantities, known as charges, commute. Lifting this assumption has led to a new subfield, noncommuting charges, at the intersection of quantum information and quantum thermodynamics. The work presented in this thesis identifies various effects of noncommuting charges and extends the topic to many-body physics and experiments. Initially, the field’s findings were conveyed in abstract information-theoretic terms. To transition these findings to experimental practice and tie them to many-body physics, constructing relevant Hamiltonians is essential. We introduce a method for constructing Hamiltonians that globally conserve noncommuting quantities while facilitating their local transport. Having demonstrated the testability of noncommuting-charge physics, we aim to delineate its effects. To do so, we construct analogous models that differ in whether their charges commute. We find that noncommuting models exhibit higher entanglement entropies. Since entanglement accompanies thermalization, our result challenges previous assertions that charges’ noncommutation hinders thermalization. Motivated by understanding noncommuting charges’ effects on entanglement, we introduce them into monitored quantum circuits. Monitored quantum circuits typically transition from a highly entangled volume-law phase to a less entangled area-law phase as one increases the rate of measurements. This holds for monitored quantum circuits with no charges and commuting ones. We find that by introducing noncommuting charges into monitored quantum circuits, the area-law phase becomes replaced with a critical phase. Since critical phases are characterized by long-range entanglement, this result reinforces entanglement enhancement by noncommuting charges. Finally, we revisit the puzzle of whether noncommuting charges promote or hinder thermalization. Most quantum many-body systems thermalize; some don’t. In those that don’t, what effect do noncommuting charges have? One type of system that does not thermalize is a system whose Hamiltonian has so-called dynamical symmetries (or spectrum-generating algebras). We find that noncommuting charges promote thermalization by reducing the dynamical symmetries in a system.
Entangled photon source for a long-distance quantum key distribution
(University of Waterloo, 2024-06-11) Oh, Sungeun
Satellite-based Quantum Key Distribution (QKD) leverages quantum principles to offer unparalleled security and scalability for global quantum networks, making it a promising solution for next-generation secure communication systems. However, many technical challenges need to be overcome. This thesis focuses on theoretical modeling and experimental validation for long-distance QKD, as well as the development and testing of the quantum source necessary for its implementation, to take strides towards realization. While various approaches exist for demonstrating long-distance QKD, here we focus on discussing the approach of sending entangled photon pairs from an optical quantum ground station (OQGS), one through free-space on one end (uplink), and the other one through ground on the other end. In the thesis, we first discuss the considerations relevant to establishing a long-distance quantum link. Since a substantial amount of research has already been conducted on optical fiber communication through ground-based methods, our focus is specifically directed towards ground-to-space (i.e., free space) quantum links. One of the most concerning aspects in free-space quantum communication is signal attenuation caused by environmental factors. We particularly examine pointing errors that arise from satellite tracking systems. To investigate this further, we designed a tracking system employing a specific tracking algorithm and conducted tracking tests to validate its accuracy, using the International Space Station (ISS) as a test subject. Our findings illustrate the potentially significant impact of inaccurate ground station-to-satellite alignment on link attenuation, according to our theoretical model. Given that photons serve as the signals for the QKD, we also investigate the background light noise resulting from light pollution around our Optical Quantum Ground Station (OQGS), which is another concerning aspect, as it could worsen the link attenuation. Consequently, we estimate the minimum photon pair rate required for successful QKD, taking into account both the obtained values from these measurements and the expected level of link loss. Having determined the minimum photon pair rate and other requirements for the long-distance QKD, we proceed to fully elaborate on the development process of the Entangled Photon Source (EPS), which is one of the crucial devices for demonstrating entanglement-based QKD. Here, the thesis includes a detailed explanation for the customization of a crystal oven. It also explains the implementations of a beam displacer scheme and a Sagnac scheme to create a robust interferometer, responsible for creating quantum entanglement. In addition, we demonstrate a novel approach to effectively compensate for the major weaknesses of the interferometer, namely spatial and temporal walk-offs. Finally, we conduct the entanglement test and demonstrate its suitability for long-distance QKD. As a side project, we investigate the performance degradation of nonlinear crystals in response to proton radiation, exploring the potential of deploying the EPS in space for downlink QKD in the future. This thesis provides a comprehensive analysis and testing of elements required for long-distance QKD, contributing to the advancement of future global quantum networks.
Developing a robust quantum simulator with trapped ions
(University of Waterloo, 2024-05-27) Hahn, Lewis
Trapped ion systems have experienced significant growth in recent years as their potential for excelling as quantum simulators has become recognized. Ions make an excellent qubit due to their long coherence times with moderate gate times, high fidelity detection and state initialization, and their ability to create long range spin interactions. As the experimental demands of trapped ions increases, so too do the demands that sustain and control them. In this thesis, I will cover the design and implementation of robust systems for the trapped ions platform and describe the development of robust lab infrastructure, equipment, and optics required to perform high contrast entangling operations on an existing four-rod system. By redesigning the DC power distribution and grounding system I have been able to supply our quantum simulator with clean DC voltages while reducing ground loops that can introduce noise into the system. With delicate alignment of the 355 nm system, our four-rod system is able to entangle qubits together. By incorporating 3D printing and inexpensive DC motors, I was able to motorize the controls used to align our 355 nm beam in the vertical direction which has made alignment reliable and accessible. With future iterations of the motorized stage, I’ve been able to achieve a resolution of 70 nm in all three axes. We then look to the next generation ion trap in the form of a meticulously engineered blade trap. With the ability to perform simulations on systems of about 30 qubits, reaching very low vacuum pressures is essential to increase ion life times. By careful preparation of our Shapal blade holder I’ve been able to preserve the 9 × 10−13 mbar pressure of our blade trap vacuum chamber. I then discuss the design of the imaging system for the blade trap which utilizes dual 0.5 NA (numerical aperture) objectives to achieve high state detection fidelity (∼ 99.9%). By simulating the imaging system and taking into consideration the effects of the systems’ efficiencies, I find that high state detection fidelity should be achievable for detection times on the order of 20 µs. This offers potential for performing in-situ mid-circuit measurement on the blade trap system. By performing some initial tests, I compare the experimental results to the simulated performance of the imaging system and find they match reasonably well.
Random State Production for Quantum Key Distribution using Weak Coherent Pulse Source
(University of Waterloo, 2024-05-27) Piatt, Matthew
This thesis addresses two challenges involved in the culmination of the Quantum En- crYption and Science Satellite mission. This mission aims to demonstrate quantum key distribution (QKD) in space. The first part of the thesis will be dedicated to the intro- duction of a quantum random number generator. This generator will be based off of the arrival time of photons and subsequently used to produce random bits that will be turned into the random states necessary for the QKD protocol. The latter half of the thesis will address creating the random states for the protocol. This is non-trivial due to the finite resources involved in the physical apparatus that comprises the chosen weak coherent pulse source for the QEYSSat mission.
Programmatic Representation of Quantum Many Body Systems
(University of Waterloo, 2024-05-23) Luo, Xiu-Zhe
The problem of simulating quantum many-body systems is fundamental in condensed matter physics, quantum computing, and quantum chemistry. The exact simulation of quantum many-body systems is generally intractable on classical computers, and developing efficient simulation methods is crucial for understanding and utilizing quantum systems. Meanwhile, from the computer science community, the development of formal languages has dramatically improved programming and software efficiency. Thus, it is natural to ask whether we can develop and utilize such representations to simulate quantum many-body systems. We propose so-called \textit{programmatic representations} for simulating quantum many-body systems on computational devices. We begin with introducing the programmatic representations for quantum circuits, quantum operators, quantum states, pulse sequences, and more general quantum programs with control flows are discussed. We further introduce the transformation of and between these representations, which leads to the development of several software frameworks, including \texttt{Yao} and \texttt{Bloqade}, which achieved state-of-the-art performance in simulating quantum circuits and Rydberg atom array dynamics. We introduce the transformation for automatic differentiation and show that by utilizing the reversibility of the quantum circuits, only constant memory overhead is needed for the automatic differentiation of quantum circuits in simulators. As a result, we report the differentiation of 10,000-layer quantum circuits that no previous software can achieve. On top of these technical developments in exact simulation, hardware modeling, and automatic differentiation, we generalize the numerical renormalization group formulations from Wilson and White, namely Wilson's NRG and White's DMRG, which we call the \textit{operator learning renormalization group (OLRG)}. OLRG allows solving general quantum many-body problems with arbitrary operator maps in lieu of a state ansatz. We introduce a theory framework guiding the design of OLRG loss functions, providing a rigorous error bound for real-time evolution. We further show OLRG can solve the quantum many-body problems with arbitrary operator maps such as neural networks using the Operator Matrix Map (OMM), and can be used to generate control parameters for a quantum device using the Hamiltonian Expression Map (HEM). We explore different hyperparameters for both OMM and HEM for a 1D transverse field Ising model and show that our theoretical loss function correctly guides both the OMM and HEM to ground truth using differentiable programming. We conclude by discussing the future directions of applying programmatic representations to quantum many-body systems and the future directions of quantum many-body system simulation.
Trapping of Ba+ using an Autoionizing Resonance and Construction of an Academic Ba+ Testbed
(University of Waterloo, 2024-05-21) Greenberg, Noah
Trapped ions are at the forefront of the quantum information processing field. This platform has demonstrated some of the highest native gate fidelities, in addition to offering fully connected qubits with incredibly long coherence and trapping lifetimes. Among the trapped ion candidates, barium is emerging as one of the top contenders. Barium has a long lived metastable state, which can either be exploited to greatly increase the fidelity of processes like state preparation or can be used to store information in additional hyperfine levels. This attractive atomic structure, combined with the fact that many transitions fall within the visible wavelength spectrum, eases optical design and makes barium an ideal trapped ion candidate to move the technology forward. Researchers have been working with barium ions for decades, but quickly trapping long chains of this element is more of a challenge compared with other species because it reacts in the atmosphere forming oxides and salts. Furthermore, the two most promising isotopes of barium, 137Ba+ and 133Ba+, have their own added barriers, with the former being a mere 11% naturally abundant and the latter usable in only microgram quantities due to its radioactivity. Improving the loading efficiency of these isotopes will be key to reliably generating long chains of these ions, and realizing quantum information processing with barium in the most complex trapped ion platforms the technology has to offer. To this end, we demonstrate the use of an autoionizing transition to increase our loading rate of Ba+ by a factor of 7 compared with a non-resonant photoionization scheme. We utilize the largest experimentally measured photoionization cross-section in barium, which is the most power efficient scheme that exists for generating Ba+ from neutral atoms in an isotope-selective manner. This is the first reported use of an autoionizing transition to trap Ba+ and this scheme is applicable to all isotopes of barium. In order to further aid in trapping of the most elusive isotope, 133Ba+, methods for fabricating microgram quantity targets for ablation loading trapped ions are also presented. Loading an ion trap with microgram materials of any element is incredibly difficult, and even more so when it only exists as a salt, as is the case with 133Ba+. Heat-treatment of these salt ablation targets is shown to increase the consistency of neutral barium production as well as the lifetime of the ablation target, which will make trapping more predictable. The work surrounding the use of a novel photoionization scheme and testing of ablation targets was done in order to facilitate the reliable trapping of long chains of Ba+ in a surface trap. The design and initial construction of an ultra-high vacuum system and the supporting optical infrastructure for eventually trapping Ba+ in a chip trap is detailed and evaluated as well. This vacuum chamber prioritizes optical access for individual addressing ions to drive laser-based gates and allow imaging of ion chains, while still maintaining sufficient optical access for multiple global addressing zones. This chamber and experiment will serve as a platform to further probe Ba+ as a front running candidate for quantum simulation and information processing in one of the most sophisticated surface trap that exists. Finally, studies of an alternative approach for driving amplitude-modulated entangling gates between ion pairs in chains is numerically simulated. This technique leverages existing practices found in signal processing for driving spin-spin interactions in ion pairs via their collective motional modes. The purpose of this numerical project is to provide a basis for creating pulse-shapes that will entangle ions and act as a resource for when the platform has the capability to manipulate qubits.
Design and Implementation of an Experimental Setup for Entanglement Harvesting
(University of Waterloo, 2024-05-17) Hak, Lucas
Relativistic quantum information is an emerging field of study at the interface of relativity and quantum theories. One key result is the prediction of extracting entanglement from quantum fields known as “entanglement harvesting”. The Superconducting Quantum Devices (SQD) group has recently designed a qubit-coupler system, suitable for generating entanglement between two causally disconnected qubits, and developed an experimental protocol to measure entanglement harvesting. In this thesis, developments to the experimental infrastructure aimed to facilitate the entanglement harvesting experiment will be presented. The proposed experiment is operated on a superconducting flux qubit device in a dilution refrigerator. We discuss the theoretical requirements for entanglement harvesting and outline the experimental proposal. We propose cryogenic electronics setup capable of providing all the of requirements for entanglement harvesting. We model the thermal state in the transmission line, and propose attenuation that will reach a low enough thermal noise to extract entanglement. We propose electronic cables capable of supplying biasing currents between 0-26.5 GHz, able to provide low dispersion of a 33 ps Gaussian pulse and low heating of static currents up to 5 mA, required for biasing an entanglement harvesting measurement. A new superconducting magnetic coil is designed and constructed, where we consider its impact on experimental operation. The thermal considerations of heating the dilution refrigerator and properly dissipating the heat are presented.
Line-Field Spectral Domain Optical Coherence Tomography: Design and Biomedical Applications
(University of Waterloo, 2024-05-10) CHEN, KEYU
Corneal diseases such as keratoconus and Fuchs' dystrophy lead to the dysfunction of the cornea, which can result in vision loss. Early-stage detection at the cellular level provides the opportunity for treatment that slows or stops disease progression and potentially for disease cure. Optical coherence tomography (OCT), often described as the optical equivalent of ultrasound imaging, enables high-speed, non-invasive volumetric imaging at a cellular resolution. These advantages of OCT have made it a useful tool in ophthalmology and beyond. High-speed OCT data acquisition is desirable, particularly for volumetric imaging, to reduce involuntary eye and body motion and suppress motion-induced artifacts. Line-scan OCT (LS-OCT) utilizes a 2D lens, such as a cylindrical lens, as the line generator to project a line-shaped detection beam onto the sample instead of the focused pencil beam traditionally used in OCT systems. Combined with high-speed 2D cameras, LS-OCT systems allow for a data acquisition speed that is 1 to 2 orders of magnitude higher than conventional point-scanning OCT systems. The three main goals of this thesis research are: (i) to develop a novel Powell lens-based line-scan OCT system, (ii) to optimize the performance of the Powell lens-based line-scan OCT system for in vivo human studies, and (iii) to develop a line-scan OCT protocol for conducting dynamic OCT (dOCT) studies on various biological tissues. A Powell lens is used in a line-field spectral domain OCT (PL-LF-SD-OCT) system to generate a line-shaped imaging beam with an almost uniform distribution of optical power along the line direction. This design overcomes the significant sensitivity loss of approximately 10 dB that is observed along the line length direction (B-scan) in LF-OCT systems based on cylindrical lens line generators. The PL-LF-SD-OCT system offers almost isotropic spatial resolution (∆x and ∆y approximately 2 µm, ∆z approximately 1.8 µm) in free space and a sensitivity of approximately 87 dB with only about 1.6 dB loss along the line length for an imaging power of 2.5 mW at an imaging rate of 2,000 frames per second (fps). Images acquired with the PL-LF-SD-OCT system allow for the visualization of cellular and sub-cellular structures of biological tissues. Following the development of the first PL-LF-OCT system, we present a second-generation system that combines sufficiently high: spatial resolution (2.4 μm × 2.2 μm × 1.7 μm (x × y × z)) to resolve individual cells; sensitivity (approximately 90 dB) to image the semi-transparent human cornea; and image acquisition rate (≥ 2,400 fps) to suppress most involuntary eye motion artifacts. In summary, the second-generation system allows for contactless, in vivo imaging of the cellular structure of the human cornea. Volumetric images acquired in vivo from the corneas of healthy subjects show corneal epithelial, endothelial, and keratocytes cells, as well as sub-basal and stromal corneal nerves. The system's high axial resolution also allows for clear identification and morphometry of the corneal endothelium, Descemet's membrane, and the pre-Descemet’s (Dua) layer. By characterizing time-dependent signal intensity fluctuations, dOCT enhances contrast in OCT images and indirectly probes cellular metabolic processes. Almost all of the dOCT studies published so far are based on the acquisition of 2D dOCT images (B-scans or C-scans) via point-scanning spectral-domain/swept-source OCT or full-field OCT respectively, due to limitations in the image acquisition rate. Here we introduce a novel high-speed Line-Field dOCT (LF-dOCT) system and image acquisition protocols designed for volumetric dOCT imaging of biological tissues. The imaging probe is based on an exchangeable telecentric lens pair that enables a selection of transverse resolution (1.1 µm to 6.4 µm) and field of view (FOV) (250×250 µm² to 1.4×1.4 mm²) suitable for different biomedical applications. The system offers an axial resolution of 2.6 µm in free space, corresponding to approximately 1.9 µm in biological tissue assuming an average refractive index of 1.38. A maximum sensitivity of 90.5 dB is achieved for 3.5 mW optical power at the tissue surface and camera acquisition rate of 2000 fps. Volumetric dOCT images acquired with the novel LF-dOCT system from plant tissue (English cucumber) and animal tissues (mouse liver and prostate tumor spheroids) allow for volumetric visualization of the tissues’ cellular and sub-cellular structure.
Computational modeling of bacterial chromosome organization: macromolecular crowding, chain heterogeneity, and chain cross-linking
(University of Waterloo, 2024-04-30) Amir Hosein, Sadeghi Isfahani
Chromosome organization is integral to life, as it plays a pivotal role in maintaining the integrity and functionality of genomic materials, as well as facilitating the transcription, replication, and transmission of genetic information during cell division. Despite lacking membrane-based compartmentalization strategies, bacteria compactly organize their chromosomes within the nucleoid, an unpartitioned subcellular space, in a hierarchical manner. Due to this open-plan layout, macromolecular crowding (MMC) is an essential factor in bacterial chromosome organization. Indeed, it has been known that chain molecules in a crowded medium can undergo phase-separation, transitioning into collapsed states. However, the extent to which bacteria rely on MMC for organizing their chromosomes is not fully understood. Moreover, chromosomes exhibit structural heterogeneity, being decorated with various proteins such as the cross-linking protein histone-like nucleoid-structuring (H-NS) and the key transcription enzyme RNA polymerase (RNAP). The resulting chain heterogeneity can influence chromosome organization in a crowded medium, with MMC potentially modulating the effects of these proteins on their target chromosomal segments. A comprehensive understanding of chromosome organization would necessitate acquiring a fuller picture of how chain heterogeneity, MMC, and the action of DNA binding proteins (DBPs) are orchestrated in a confined space. In this thesis, using molecular dynamics (MD) simulations, we study how biomolecular crowding, confinement, chain heterogeneity, and chain cross-linking affect the spatial organization of “chromosome-like” polymers. Our modeling efforts are inspired by the way Escherichia coli (E. coli) chromosomes are organized. For this, we start with a simple model and gradually improve our modeling strategy by incorporating more biological details. These efforts yield quantitative insights into some key observations such as the clustering of transcription-active units into a transcription factory as well as H-NS and crowder synergy in condensing bacterial chromosomes. First, our homogeneous-polymer model establishes a relationship between its spatial organization and the distribution of the surrounding crowders under anisotropic (cylindrical) confinement. This effort extends the applicability of the previous findings for unconfined spaces to cell-like confined spaces: in a parameter space of biological relevance, the sum of the volume fractions of monomers and crowders, rescaled by their respective size, remains constant. We then introduce chain heterogeneity, simulating the effects of transcription on an otherwise homogeneous polymer. The resulting polymer contains large monomers dispersed along the backbone with small ones in between. This effort demonstrates that the compaction transition by crowders is well correlated with the clustering: when the large monomers are of sufficient size, chain compaction and clustering of large monomers occur concomitantly at the same narrow (biologically-relevant) range of the crowder volume fraction. It also indicates that cylindrical confinement makes MMC effects more effective. Finally, we study the action of H-NS protein and its impact on chromosome compaction. H-NS, modeled as a mobile “binder,” can bind to a chromosome-like polymer. This effort elucidates how MMC and H-NS binding each play a part in compacting a bacterial chromosome, providing a quantitative understanding of the synergistic interactions between crowders and binders. Crowders intensify the binding of H-NS to the polymer, and conversely, the presence of H-NS improves the efficiency of MMC effects, indicating a bidirectional synergy in chain compaction. Additionally, we observe that the presence of crowders facilitates the clustering of binders, where the cluster size grows as the volume fraction of crowders increases. This thesis outlines a physical framework where phase separation and clustering, driven by MMC, are identified as the principal mechanisms in bacterial chromosome organization.
Correlation Functions of Heavy Operators in AdS/CFT
(University of Waterloo, 2024-04-15) Abajian, Jacob
In this thesis we investigate the holographic dual description of correlation functions of heavy operators in conformal field theory. These heavy operators have scaling dimension that scales with the CFT central charge in the large central charge limit, and they are dual to objects propagating through the bulk whose gravitational backreaction cannot be neglected. We reproduce the expected CFT correlation functions through a gravitational path integral calculation–one that requires the introduction of special terms associated to the horizons of black holes appearing in the gravitational configurations. The results for two point functions apply in arbitrary dimensions. We also discuss heavy-heavy-light-light correlation functions in various dimensions. We find results for the three point functions in two-dimensional CFT that are consistent with the expected universal behavior of heavy operator structure constants. This is consistent with the interpretation of three-dimensional Einstein gravity as the holographic dual to an ensemble of CFT2’s.
Development of III-V Semiconductor Surface Quantum Wells for Hybrid Superconducting Device Applications
(University of Waterloo, 2024-02-20) Bergeron, Emma
This thesis concerns the materials development of both InSb/InAlSb and InAs/AlGaSb surface quantum wells: Two of the most promising platforms for the study of proximity-induced superconductivity in semiconductors with strong spin-orbit interaction. Our work covers the growth, fabrication, and measurement of Hall-bar and Josephson junction devices in both material systems. We optimize surface quantum well heterostructure growth by molecular beam epitaxy (MBE) for single subband occupation and no parasitic parallel conduction. Electronic transport measurements in magnetic fields were carried out on the resulting heterostructures and analyzed. We highlight issues with the reproducibility of modulation-doped structures in InSb quantum wells and investigate the influence of doping density, buffer choice, growth parameters, and alloy composition on observed parallel conduction in the heterostructure. We show that nominally identical growths can differ by occupation of a parallel conduction channel. We also show that the window for modulation $\delta$ doping density between growths is smaller than the observed deviation in calibrated doping densities. We report on the growth, fabrication, and transport characteristics of high-quality, gate-tunable InSb two-dimensional electron gases (2DEGs) in surface quantum wells grown on (001) SI-GaAs substrates. We demonstrate the influence of modulation doping on gating characteristics, magnetotransport behavior, and spin-orbit interaction in two heterostructures, one with and one without a modulation-doped InAlSb layer. Magnetoresistance measurements confirm that intentional dopants in InSb are compatible with high-quality and reproducible transport characteristics, without parasitic parallel conduction or unstable carrier densities. This could be further tested in a 2DEG heterostructure with a short-period InSb/InAlSb superlattice doping scheme, where only the thin layer is doped. We present the first report of a surface quantum well in the lattice-matched InAs/AlGaSb material system on GaSb substrates. Deep quantum wells in this system have demonstrated record mobilities, by an order of magnitude, over the more commonly reported InAs/InGaAs system, making it a promising platform for topological quantum computing with Majorana zero modes. The surface of the quantum well is protected by lithography techniques designed to protect the surface from unnecessary chemical exposure during fabrication. Our results show that the carrier density is greatly enhanced in a surface quantum well compared to deeper structures and is highly influenced by the choice of gate dielectric in top-gated devices, often pushing the 2DEG into the second subband. However, the gating characteristics of the 2DEG show that the device can be tuned to a single-subband occupation. Josephson junctions with ex-situ sputtered contacts to these InAs surface quantum wells are fabricated using a surface passivation technique. Our lift-off process for ex-situ sputtered Nb/Ti contacts achieves smooth edges compatible with top-gated devices. We report the observation of induced superconductivity in undoped InAs surface quantum wells using this fabrication process. Two generations of SNS samples were fabricated with ebeam lithography and surface passivation techniques. The interface transparencies of the two generations of samples were determined. We observe a dependence of the critical current on junction length, corresponding to a sensitivity to elastic scattering in the semiconductor. The temperature dependence of the critical current in the junction with arbitrary transparency is modeled by the Kulik-Omelyanchuk relation. The measured excess current, resulting from Andreev reflection processes at the normal/superconducting (SN) interfaces, confirms the presence of phase-coherent behavior in our SNS devices. The process further achieves ex-situ high-transparency superconducting contacts in league with reports of epitaxial aluminum systems to InAs surface quantum wells.
Magnetotransport experiments in GaAs 2D holes and RF-QPC readout in a lateral quantum dot device
(University of Waterloo, 2024-02-16) Cockton, Nicholas
This thesis presents a set of experiments conducted on GaAs/AlGaAs heterostructures fabricated into Hall bars and a lateral quantum dot device. The Hall bar experiments seek to further the understanding of the complexity of the GaAs valence band of a two-dimensional hole gas (2DHG). Meanwhile, the quantum dot experiments aim to improve the performance of an RF-QPC readout employed for charge detection. These experiments are significant for the development of future quantum technologies that utilize spins in semiconductors, in fields such as spintronics and quantum computing. Firstly, we study Hall bar devices formed in a 2DHG of a dopant-free GaAs/AlGaAs heterostructure. The hole sheet density is adjustable via a metal top-gate that induces and confines holes to the GaAs/AlGaAs interface. We employ magnetotransport measurements to assess both the device quality and the underlying physics of the valence band. By measuring several devices, we identify fabrication parameters that lead to parasitic parallel conduction and gate hysteresis. In our devices, featuring an asymmetric quantum well hosting a 2DHG, Rashba spin-orbit interaction (SOI) leads to a beating pattern in the Shubnikov-de Haas (SdH) oscillations of the two subbands of the heavy-hole (HH) valence band. We employ Fourier analysis to isolate the frequencies present in the SdH oscillations. The effective masses of the two spin-orbit split HH subbands are extracted via the temperature dependence of the isolated oscillations, without invoking assumptions regarding the band structure. This analysis is performed for different values of hole density and ranges of magnetic field. The HH splitting induced by Rashba SOI is evaluated and compared to modulation-doped samples in literature. Moreover, we extract the out-of-plane g factor from thermal-activation measurements in the quantum Hall regime. Secondly, we investigate non-linear magnetotransport in a Hall bar device, also formed in a 2DHG of a dopant-free GaAs/AlGaAs heterostructure. We observe the SdH phase inversion phenomenon and explore the empirical relationship between the hole temperature and applied dc current through the Hall bar. In the quantum Hall regime, comprehensive maps of the differential resistances as a function of current and magnetic field (B) are generated to show the evolution of quantum Hall breakdown reaching filling factor ν = 1. By adjusting the top-gate voltage, the hole sheet density is incremented to reveal how the quantum Hall breakdown characteristics evolve with hole density. A Zener-type tunneling model is employed to describe the size of the transport diamonds in current. We examine the magnetic field dependence of the critical current compiled for different values of hole density. The zero current anomaly (ZCA) phenomenon is observed and discussed. Hysteresis observed in the quantum Hall breakdown regime of a two-dimensional electron gas (2DEG) has been linked with dynamic nuclear polarization (DNP) through the interplay of electron and nuclear spins via the hyperfine interaction. Motivated by examining the strength of the hyperfine coupling between hole and nuclear spins, we investigate hysteresis near breakdown in our 2DHG. Finally, we present measurements on a lateral double quantum dot (DQD) device formed in a 2DEG of a modulation-doped GaAs/AlGaAs heterostructure. These devices demonstrate potential as a platform for advancing quantum information technologies, particularly when employed as single spin qubits. The fast, single-shot readout of spin states is crucial, and using spin-to-charge conversion techniques, only the charge state readout becomes essential. Thus, we design and setup an optimized radio frequency quantum point contact (RF-QPC) charge sensor for the quantum dot device. Charge sensitivity measurements are conducted using a standard QPC sensor, followed by the RF-QPC. Additionally, we characterize a superconducting quantum interference device (SQUID) amplifier for integration into the RF-QPC readout. We successfully demonstrate a carrier nulling technique, specifically to accommodate the SQUID amplifier's limited dynamic range, and enable its integration into the RF-QPC readout. After Incorporating the SQUID amplifier into the RF-QPC readout, critical issues are identified and future work to address them is discussed.
Optimization of the Optical Infrastructure and Trapping Potential for a Quantum Processor Based on Trapped Barium Ions
(University of Waterloo, 2024-02-02) Tan, XingHe
Trapped ion quantum computers represent a cutting-edge quantum computing platform, keeping the highest reported state preparation and measurement fidelity of 99.97%. Recently, barium ion has emerged as the research interest of several teams due to its exceptional characteristics, including visible and inferred state transition frequencies, long-lived metastable states, and a simple hyperfine structure. Our research group aims to demonstrate trapping 16 133Ba+ ions on a micro-fabricated surface trap and build a quantum computing platform. In the long run, we intend to open access to this platform for the academic community to use. This project is named the QuantumIon project. The project started in 2019, just before the global pandemic. During the pandemic, the QuantumION team did not have the lab space and the components to prototype and benchmark subsystems in parallel with system design. Instead, the team designed the entire system on computers during the pandemic. I joined the QuantumION team in 2022 and performed the system assembly, validation and optimization when the pandemic restrictions were lifted. In this thesis, I describe the optimization of the optical system for delivering laser beams to the ions and the validation of a 0.6 NA imaging system that will be used to observe the ion fluorescence as a means of measuring its quantum state. While benchmarking the optical systems, I found that the performance of the components deviated from their ideal specifications, such as insertion loss. Therefore, I had to modify the system design to ensure the optical systems functioned in practice. In addition, I conducted a feasibility study on delivering an ablation laser via multimode fiber and simulated a trapping potential on the surface trap that achieved approximately even spacing for 12 out of the 16 trapped ions.
Thinking outside the box; fast error estimation for next generation galaxy surveys
(University of Waterloo, 2024-01-29) Schreiner, Gregory
Since the discovery of the accelerated expansion of the universe in the late 90s, the flat $\Lambda$CDM model has reigned as the best explanation for the cosmological phenomena we have observed. In spite of over two decades of study, the identities and properties of both cold dark matter (CDM) and dark energy ($\Lambda$) remain a mystery. The goal of modern precision cosmology is to measure cosmological parameters using a multitude of probes in the pursuit of deviations from the $\Lambda$CDM framework or insights into the properties of dark matter and dark energy. If the probe being considered is the matter power spectrum measured from a galaxy redshift survey, then the precision of the derived parameters is determined by the power spectrum covariance matrix. The analytic form of the covariance matrix is difficult to estimate, so common practice is to run many simulations of the survey volume to get a brute-force estimate. Next-generation cosmological surveys are set to collect higher resolution data within a larger survey volume than ever before. The complexity and number of simulations that will be required to estimate the covariance matrix of these surveys is threatening to become too computationally expensive for even the most advanced computer clusters. Thus, there is an urgent need to develop novel techniques for reducing the computation time required to achieve such precise covariance estimates. While many proposed methods seek to reduce the number of simulations required, it is also possible to leverage the volume scaling of the covariance matrix, allowing one to reduce the size of the simulations required. Super-sample covariance (SSC) is a contribution to the covariance matrix made by modes of the power spectrum that are larger than the volume of a survey or simulation. If this volume scaling of the covariance is to be taken advantage of, then the SSC within the simulations must be accurately modeled. To this end, I review methods of running separate universe (SU) simulations to account for the effects of SSC. While these methods have all been shown to recover the SSC with reasonable accuracy, they have been largely developed and tested in isolation from one another. I present my work in directly comparing the accuracy of these methods in recovering the SSC effect using ensembles of N-body simulations. Even with SSC accurately modeled, the volume scaling of the covariance does not hold for arbitrarily small volume simulations; at some point, the analytic behaviour of the covariance is expected to break down. I push the volume scaling to its limit by running many thousands of simulations at different volumes and scaling the covariance to match that of a larger volume survey. The SSC term has a nontrivial relation to the simulation volume, preventing it from scaling in the same way as the other components of the covariance. In light of this, I present a way to include SSC such that the scaled covariance could still be recovered with good accuracy. I find the scaled covariance matches the large volume covariance to within $\sim 4\%$ or better on most scales, with higher $k$ bins being biased low due to missing a small component of the SSC. The scaled covariance at very low $k$ for very small simulations is substantially lower than the large mock covariance at those scales due to very few modes of that scale being present in the small volume simulations. This creates a skewness in the distribution of power at those scales. By computing the number of modes required to avoid this skewed distribution of power, I derive a way to estimate the minimum simulation volume that could be used to accurately model the covariance at a given scale. The accurate modeling of SSC and optimal leveraging of the volume scaling of the covariance matrix are powerful complementary tools with the potential to substantially reduce the computational cost of covariance matrix estimation for future galaxy survey data.

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