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Full-Text Articles in Physics

Volumetric Imaging Using The Pupil-Matched Remote Focusing Technique In Light-Sheet Microscopy, Sayed Hassan Dibaji Foroushani Dec 2023

Volumetric Imaging Using The Pupil-Matched Remote Focusing Technique In Light-Sheet Microscopy, Sayed Hassan Dibaji Foroushani

Optical Science and Engineering ETDs

ABSTRACT

The dissertation explores innovative techniques in light sheet microscopy, a pivotal tool in biomedical imaging, to enhance its speed, resolution, and efficiency in capturing dynamic biological processes. Light sheet microscopy allows for quick 3D imaging of biological specimens ranging from cells to organs with high spatiotemporal resolution, large field-of-view, and minimal damage, making it vital for in vivo imaging.

The first project introduces a novel optical concept designed to optimize Axially Swept Light Sheet Microscopy (ASLM). This technique is crucial for imaging specimens ranging from live cells to chemically cleared organs due to its versatility across different immersion media. …


Radio Insights Into Gamma-Ray Mysteries, Seth M. Bruzewski Dec 2023

Radio Insights Into Gamma-Ray Mysteries, Seth M. Bruzewski

Physics & Astronomy ETDs

In the time since its launch, the \textit{Fermi Gamma-Ray Space Telescope} has provided new and unparalleled views of the $\gamma$-ray sky, dramatically increasing our understanding of sources of high-energy radiation. During that same time, however, its ``unassociated'' sources have provided a consistent mystery: approximately one third of the modern gamma-ray sky remains completely unaccounted for in other electromagnetic regimes. While some of the fainter sources simply pose challenges in achieving the necessary signal-to-noise ratio, others are well constrained and have resisted traditional investigations for years, and in some cases, for over a decade. Radio astronomy has traditionally been the best …


Modeling Lithographic Quantum Dots And Donors For Quantum Computation And Simulation, Mitchell Ian Brickson Dec 2023

Modeling Lithographic Quantum Dots And Donors For Quantum Computation And Simulation, Mitchell Ian Brickson

Physics & Astronomy ETDs

Our first focus is on few-hole quantum dots in germanium. We use discontinous Galerkin methods to discretize and solve the equations of a highly detailed k·p model that describes these systems, enabling a better understanding of experimental magnetospectroscopy results. We confirm the expected anisotropy of single-hole g-factors and describe mechanisms by which different orbital states have different g-factors. Building on this, we show that the g-factors in Ge holes are suciently sensitive to details of the device electrostatics that magnetospectroscopy data can be used to make a prediction of the underlying confinement potential. The second focus is on designing quantum …


A Review On Possible Physical Meaning Of Elastic-Electromagnetic Mathematical Equivalences, Florentin Smarandache, Victor Christianto Dec 2023

A Review On Possible Physical Meaning Of Elastic-Electromagnetic Mathematical Equivalences, Florentin Smarandache, Victor Christianto

Branch Mathematics and Statistics Faculty and Staff Publications

It is known, despite special theory of relativity has been widely accepted, in our recent draft submitted to this journal it is shown that some experiments have been carried out suggesting superluminal wave propagation, which make Minkowski lightcone not valid anymore. Therefore, it seems worth to reconsider the connection between elastic wave and electromagnetic wave equations, as in their early development. In this paper we will start with Maxwell-Dirac isomorphism, then we will find its connection with elastic wave equations.


Godel, Escherian Staircase And Possibility Of Quantum Wormhole With Liquid Crystalline Phase Of Iced-Water - Part Ii: Experiment Description, Victor Christianto, T. Daniel Chandra, Florentin Smarandache Dec 2023

Godel, Escherian Staircase And Possibility Of Quantum Wormhole With Liquid Crystalline Phase Of Iced-Water - Part Ii: Experiment Description, Victor Christianto, T. Daniel Chandra, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

The present article was partly inspired by G. Pollack’s book, and also Dadoloff, Saxena & Jensen (2010). As a senior physicist colleague and our friend, Robert N. Boyd, wrote in a journal (JCFA, Vol. 1, No. 2, 2022), for example, things and Beings can travel between Universes, intentionally or unintentionally [4]. In this short remark, we revisit and offer short remark to Neil Boyd’s ideas and trying to connect them with geometry of musical chords as presented by D. Tymoczko and others, then to Escherian staircase and then to Jacob’s ladder which seems to pointto possibility to interpret Jacob’s vision …


Godel, Escherian Staircase And Possibility Of Quantum Wormhole With Liquid Crystalline Phase Of Iced-Water - Part I: Theoretical Underpinning, Victor Christianto, T. Daniel Chandra, Florentin Smarandache Dec 2023

Godel, Escherian Staircase And Possibility Of Quantum Wormhole With Liquid Crystalline Phase Of Iced-Water - Part I: Theoretical Underpinning, Victor Christianto, T. Daniel Chandra, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

As a senior physicist colleague and our friend, Robert N. Boyd, wrote in a journal (JCFA, Vol. 1,. 2, 2022), Our universe is but one page in a large book [4]. For example, things and Beings can travel between Universes, intentionally or unintentionally. In this short remark, we revisit and offer short remark to Neil’s ideas and trying to connect them with geometrization of musical chords as presented by D. Tymoczko and others, then to Escher staircase and then to Jacob’s ladder which seems to point to possibility to interpret Jacob’s vision as described in the ancient book of Genesis …


Exploring Non-Orientable Topology: Deriving The Poincaré Conjecture And Possibility Of Experimental Vindication With Liquid Crystal, Victor Christianto, Florentin Smarandache Nov 2023

Exploring Non-Orientable Topology: Deriving The Poincaré Conjecture And Possibility Of Experimental Vindication With Liquid Crystal, Victor Christianto, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

This review investigates the potential of non-orientable topology as a fundamental framework for understanding the Poincaré conjecture and its implications across various scientific disciplines. Integrating insights from Dokuchaev (2020), Rapoport, Christianto, Chandra, Smarandache (under review), and other pioneering works, this article explores the theoretical foundations linking non-orientable spaces to resolving the Poincaré conjecture and its broader implications in theoretical physics, geology, cosmology, and biology.


Understanding The Nature Of Pulsars And Characterizing Propagation Effects Using Pulsar Timing, Pratik Kumar Oct 2023

Understanding The Nature Of Pulsars And Characterizing Propagation Effects Using Pulsar Timing, Pratik Kumar

Physics & Astronomy ETDs

Pulsars are highly magnetized stellar remnants, among the densest known objects, and primarily produce radio emission in the form of lighthouse beams sweeping across the line of sight as a regular train of pulses. Apart from providing tests for matter in high-density regimes, general relativity, and plasma emission; perhaps the most notable characteristic is their applicability as precise astronomical clocks to measure various effects. Pulsar Timing Arrays (PTAs) are galactic scale detectors analogous to ground-based detectors of Gravitational Waves (GWs) like LIGO, with the aim of detecting low-frequency nano-Hz GWs from coalescing binary supermassive black holes. PTAs consist of a …


Super-Resolution Microscopy With Color Centers In Diamond, Forrest A. Hubert Aug 2023

Super-Resolution Microscopy With Color Centers In Diamond, Forrest A. Hubert

Optical Science and Engineering ETDs

This dissertation explores the development and application of diamond color centers, specifically the silicon-vacancy (SiV) and nitrogen-vacancy (NV) centers, in super-resolution microscopy and magnetic imaging techniques. It demonstrates the potential of SiV centers as photostable fluorophores in stimulated emission depletion (STED) microscopy, with a resolution of approximately 90 nm. The research also presents a method for nanoscale magnetic microscopy using NV centers by combining charge state depletion (CSD) microscopy with optically detected magnetic resonance (ODMR) to image magnetic fields produced by 30 nm iron-oxide nanoparticles. The individual magnetic feature width reaches ~100 nm while resolving magnetic field patterns from nanoparticles …


Application Of Quantum Mechanical Techniques To Optical Waveguide Structures, Stuart Ward Aug 2023

Application Of Quantum Mechanical Techniques To Optical Waveguide Structures, Stuart Ward

Physics & Astronomy ETDs

The focus of this dissertation is on the application of supersymmetric quantum mechanics to the problem of microbending in optical waveguides and on the analysis of soliton decay due solely to quantum mechanical effects.

The techniques of supersymmetric quantum mechanics are applied to the equation of motion describing light propagation in an optical waveguide which is undergoing microbending. Based on these supersymmetric techniques, given a particular refractive index profile, one may derive a new refractive index profile which results in less loss due to the microbending -- the particular example of the monomial index profile is analyzed in detail. An …


Near- And Far- Field Optical Response Of Ensembles Of Nanostructures, Lauren Zundel Aug 2023

Near- And Far- Field Optical Response Of Ensembles Of Nanostructures, Lauren Zundel

Physics & Astronomy ETDs

The ability of metallic nanostructures to support collective oscillations of their conduction electrons, known as surface plasmons, makes them attractive candidates for a wide range of applications in areas as diverse as cancer therapy, biosensing, and solar energy harvesting. These applications are especially promising for periodic arrays of nanostructures, which can support collective modes known as lattice resonances, and for nanostructures with extreme aspect ratios that give rise to enhanced light-matter interaction. In this Thesis, we employ a coupled dipole model to theoretically explore the lattice resonances supported by complex arrays of nanoparticles containing multiple nanoparticles per unit cell. We …


High-Power Laser Cooling And Temperature-Dependent Fluorescence Studies Of Ytterbium Doped Silica, Brian Topper Aug 2023

High-Power Laser Cooling And Temperature-Dependent Fluorescence Studies Of Ytterbium Doped Silica, Brian Topper

Optical Science and Engineering ETDs

Experimental observation of optical refrigeration using ytterbium doped silica glass in recent years has created a new solution for heat mitigation in high-power laser systems, nonlinear fiber experiments, integrated photonics, and precision metrology. Current efforts of different groups focus on compositional optimization, fiber fabrication, and investigating how much silica can be cooled with a laser. At the start of this work, the best effort in laser cooling ytterbium doped silica saw cooling by 6 K from room temperature. This dissertation follows the experimental efforts that culminated in the increase of this initial record by one order of magnitude. Comprehensive spectroscopic …


Modified Geometries, Clifford Algebras And Graphs: Their Impact On Discreteness, Locality And Symmetr, Roma Sverdlov Jul 2023

Modified Geometries, Clifford Algebras And Graphs: Their Impact On Discreteness, Locality And Symmetr, Roma Sverdlov

Mathematics & Statistics ETDs

In this dissertation I will explore the question whether various entities commonly used in quantum field theory can be “constructed". In particular, can spacetime be “constructed" out of building blocks, and can Berezin integral be “constructed" in terms of Riemann integrals.

As far as “constructing" spacetime out of building blocks, it has been attempted by multiple scientific communities and various models were proposed. But the common downfall is they break the principles of relativity. I will explore the ways of doing so in such a way that principles of relativity are respected. One of my approaches is to replace points …


Filaments And Their Application To Air Lasing, Spectroscopy, And Guided Discharge, Ali Rastegari May 2023

Filaments And Their Application To Air Lasing, Spectroscopy, And Guided Discharge, Ali Rastegari

Optical Science and Engineering ETDs

Laser filamentation is a fascinating phenomenon that occurs when an intense laser beam travels through transparent materials, in particular air. At sufficiently high power (TW in the near IR, GW in the UV), instead of spreading out like a regular laser beam, something remarkable happens: the laser beam becomes tightly focused, creating a thin and intense column of light called a laser filament. Laser filamentation is characterized by two main properties: (I) a high-intensity core that remains narrow over long distances beyond the Rayleigh range and (II) a low-density plasma channel within the core. In recent years, laser filamentation has …


Rigorous Analysis Of Markov Processes With Applications To Quantum Information, Samuel Edwin Slezak May 2023

Rigorous Analysis Of Markov Processes With Applications To Quantum Information, Samuel Edwin Slezak

Physics & Astronomy ETDs

We present a rigorous analysis of the rapid convergence of techniques based on Markov chains for the simulation of thermal quantum systems. We show that a classical computing algorithm called path integral Monte Carlo is capable of simulating thermal states of transverse field Ising models above a threshold temperature by demonstrating the existence of a rapidly mixing Markov chain. We then turn to quantum computing algorithms and show that an idealized version of quantum Metropolis sampling can efficiently simulate systems that satisfy the eigenstate thermalization hypothesis. In a related result, we find a class of stoquastic frustration free Hamiltonians that …


The Quantum Computational Utility Of Symmetry-Protected Topological Order: From Near-Term Advantages To Universal Measurement-Based Quantum Computing, Austin Kevin Daniel May 2023

The Quantum Computational Utility Of Symmetry-Protected Topological Order: From Near-Term Advantages To Universal Measurement-Based Quantum Computing, Austin Kevin Daniel

Physics & Astronomy ETDs

Quantum computers offer new avenues to approach difficult problems by taking advantage of the strange and often nonintuitive phenomena present in quantum physics. Though many quantum algorithms are believed or known to outperform the best known classical algorithms, the fundamental mechanism granting them their power remains elusive. In measurement-based quantum computation (MBQC), two key resources have been show to enable universal and provably nonclassical quantum computations, respectively. These are symmetry-protected topological order (SPTO), a notion describing a class of quantum magnets with hidden long-range correlations in their entanglement structure, and quantum contextuality, the fact that a quantum measurement outcome inherently …


Femtotesla Magnetometry And Nanoscale Imaging With Color Centers In Diamond, Yaser Silani Apr 2023

Femtotesla Magnetometry And Nanoscale Imaging With Color Centers In Diamond, Yaser Silani

Optical Science and Engineering ETDs

Intriguing photophysical properties of color centers in diamond make them ideal candidates for many applications from imaging and sensing to quantum networking. In the first part of this work, we have studied the silicon vacancy (SiV) centers in diamond for nanoscale imaging applications. We showed that these centers are promising fluorophores for Stimulated Emission Depletion (STED) microscopy, owing to their photostable, near-infrared emission and favorable photophysical properties. In the second part, we built a femtotesla Radio-Frequency (RF) magnetometer based on the diamond nitrogen vacancy (NV) centers and magnetic flux concentrators. We used this sensor to remotely detect Nuclear Quadrupole Resonance …


The Computational Model Of Nanofluid Considering Heat Transfer And Entropy Generation Across A Curved And Fat Surface, Sayer Obaid Alharbi, Florentin Smarandache, Awatif M.A. Elsiddieg, Aisha M. Alqahtani, M. Riaz Khan, V. Puneeth, Nidhal Becheikh Jan 2023

The Computational Model Of Nanofluid Considering Heat Transfer And Entropy Generation Across A Curved And Fat Surface, Sayer Obaid Alharbi, Florentin Smarandache, Awatif M.A. Elsiddieg, Aisha M. Alqahtani, M. Riaz Khan, V. Puneeth, Nidhal Becheikh

Branch Mathematics and Statistics Faculty and Staff Publications

The entropy generation analysis for the nanofluid flowing over a stretching/shrinking curved region is performed in the existence of the cross-diffusion effect. The surface is also subjected to second-order velocity slip under the effect of mixed convection. The Joule heating that contributes significantly to the heat transfer properties of nanofluid is incorporated along with the heat source/sink. Furthermore, the flow is assumed to be governed by an exterior magnetic field that aids in gaining control over the flow speed. With these frameworks, the mathematical model that describes the flow with such characteristics and assumptions is framed using partial differential equations …


The Fluid Margin Between Physical Causal Closure And Non-Physical Causal Closure, Extended To The Neutrosophic Causal Closure Principle, Florentin Smarandache Jan 2023

The Fluid Margin Between Physical Causal Closure And Non-Physical Causal Closure, Extended To The Neutrosophic Causal Closure Principle, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

We plead for a fluid margin, or mixed/indeterminate buffer zone, between Physical and Non-Physical Causal Closures, and for a Neutrosophic Causal Closure Principle claiming that the chances of all physical effects are determined by their prior partially physical and partially non-physical causes.


Investigation Of Gaas Double Heterostructures For Photonic Heat Engines, Nathan Giannini-Hutchin Dec 2022

Investigation Of Gaas Double Heterostructures For Photonic Heat Engines, Nathan Giannini-Hutchin

Optical Science and Engineering ETDs

The creation of a laser cooled semiconductor device has been a long sought achievement. GaAs-based devices have emerged as a promising candidate for the realization of this goal. Efforts to improve the efficiency of such devices have enabled the material to exhibit external quantum efficiencies (EQE, a measure of the probability that an excitation leads to the emission of a photon) of 99.5\%. Despite this impressive feat, a laser coolable device remains elusive.

To investigate the obstacles to such a device, the material characteristics of GaAs-based double heterostructures (DHS) are theoretically and experimentally examined. Through this study, a GaAs $\vert$ …


Investigation Of Laser And Nonlinear Properties Of Anderson Localizing Optical Fibers, Cody Ryan Bassett Nov 2022

Investigation Of Laser And Nonlinear Properties Of Anderson Localizing Optical Fibers, Cody Ryan Bassett

Optical Science and Engineering ETDs

In this dissertation, I investigate the possibility of lasing and nonlinear phenomena in completely solid-state transverse Anderson localizing optical fibers (TALOFs). I examine three areas within this range of topics. The research in nonlinear phenomena focuses on four-wave mixing (FWM). FWM is of high interest in TALOFs due to the fact that guided localized modes of the fiber each have different propagation constants, and thus unique possible FWM pairs can be generated from the same input pump beam. I demonstrate the generation of FWM in the TALOF by pumping it with 532 nm light into a localized mode and observing …


Distance Estimates To Evolved Stars Using Infrared Emission And Verification And Validation Of The Plasma Code Empire, Brandon M. Medina Aug 2022

Distance Estimates To Evolved Stars Using Infrared Emission And Verification And Validation Of The Plasma Code Empire, Brandon M. Medina

Physics & Astronomy ETDs

Gaining insight into the structure and dynamics of the Milky Way is important for understanding the universe on a large scale. Evolved stars on the Asymptotic Giant Branch are useful for studying the Milky Way because their emission is peaked in the infrared, where interstellar extinction effects are not as dominant. To further understand the physical properties of these objects like luminosity and investigate the Galaxy's structure, we need distance estimates. Obtaining distance estimates for these evolved stars via trigonometric parallax measurements is time-consuming, so infrared surveys studying Asymptotic Giant Branch stars can benefit from other distance estimate methods. In …


Ultrashort Pulse Laser Filamentation Electrical And Optical Diagnostic Comparison, James E. Wymer Aug 2022

Ultrashort Pulse Laser Filamentation Electrical And Optical Diagnostic Comparison, James E. Wymer

Optical Science and Engineering ETDs

Results presented here examine the effect of changing gas pressure on the radio frequency (RF) emissions of an ultrashort pulse laser filament plasma and how those emissions vary longitudinally in the laser focal region. We use a WR284 rectangular waveguide with a 1.5 cm hole that allows the beam through. A 3.2 GHz microwave signal is emitted in the waveguide, and signals are received through a waveguide-to-coax antenna connected to an HP8470B Schottky diode. By enabling and disabling the 3.2 GHz signal, we measure both the self-emitted RF from a USPL filament and subsequently the degree of attenuation a filament …


Atomic Gradiometry Based On The Interference Of Microwave Optical Sidebands, Kaleb L. Campbell Jul 2022

Atomic Gradiometry Based On The Interference Of Microwave Optical Sidebands, Kaleb L. Campbell

Optical Science and Engineering ETDs

We describe a novel pulsed magnetic gradiometer based on the optical interference of sidebands generated using two spatially separated alkali vapor cells. The sidebands are produced with high efficiency using parametric frequency conversion of a probe beam interacting with Rubiduim 87 atoms in a coherent superposition of magnetically sensitive hyperfine ground states. First, experimental evidence of the sideband process is described for both steady-state and pulsed operation. Then, a theoretical framework is developed that accurately models sideband generation based on density matrix formalism. The gradiometer is then constructed using two spatially separated vapor cells, and a beat-note is generated. The …


Fiesta And Shock-Driven Flows, Brian E. Romero Jul 2022

Fiesta And Shock-Driven Flows, Brian E. Romero

Mechanical Engineering ETDs

In this study, the interaction of a shock with various gas and particle interfaces is analyzed through simulations using a new, GPU capable, multi-species flow solver, FIESTA (Fast, Interface Evolution, Shocks, and Transport in the Atmosphere), de- veloped for this research. The cases studied include the interaction between a shock and i) a two-dimensional (2D), circular cloud of a dense gas; ii) a 2D curtain of a dense gas; iii) a three-dimensional (3D) cylinder of a dense gas, and iv) a 3D curtain of solid particles.

In simulations of a 2D gas curtain and a 3D gas column, the curtain …


Resource Estimation For Quantum Simulation Algorithms, Changhao Yi Jul 2022

Resource Estimation For Quantum Simulation Algorithms, Changhao Yi

Physics & Astronomy ETDs

A major application of quantum computers is simulating other quantum systems that are intractable to simulate classically. The broad family of algorithms for this problem go by the name of quantum simulation. Product formulas provide resource efficient and practical methods to simulate Hamiltonian dynamics. In this thesis, we study the resource estimation of quantum simulation by product formula from two aspects. First, we provide a detailed analysis of the algorithm itself. Using the effective Hamiltonian perspective, we successfully reduce the circuit complexity of quantum phase estimation and digital adiabatic simulation. Second, we analyze the performance of dynamical decoupling, a widely-used …


Applications Of Machine Learning Algorithms In Materials Science And Bioinformatics, Mohammed Quazi Jun 2022

Applications Of Machine Learning Algorithms In Materials Science And Bioinformatics, Mohammed Quazi

Mathematics & Statistics ETDs

The piezoelectric response has been a measure of interest in density functional theory (DFT) for micro-electromechanical systems (MEMS) since the inception of MEMS technology. Piezoelectric-based MEMS devices find wide applications in automobiles, mobile phones, healthcare devices, and silicon chips for computers, to name a few. Piezoelectric properties of doped aluminum nitride (AlN) have been under investigation in materials science for piezoelectric thin films because of its wide range of device applicability. In this research using rigorous DFT calculations, high throughput ab-initio simulations for 23 AlN alloys are generated.

This research is the first to report strong enhancements of piezoelectric properties …


An Interdisciplinary Approach To Understanding Volcanoes And Their Processes, Katherine Cosburn May 2022

An Interdisciplinary Approach To Understanding Volcanoes And Their Processes, Katherine Cosburn

Physics & Astronomy ETDs

To better understand volcanoes and their processes is important from both a fundamental science perspective and for hazard monitoring purposes. The complexity and limitations we face in pursuing such a science are numerous and this dissertation explores how an interdisciplinary approach combining physics, computer science, and volcanology can address this complexity in a straightforward and meaningful way. This is achieved through various modelling techniques across three studies: (1) a first-order analytic modelling of stratovolcano topographic shape, (2) the use of a Bayesian joint inversion on gravity and novel cosmic-ray muon measurements for imaging flat-lying subsurface density anomalies, and (3) the …


Examination Of Ionization In Cesium Diode Pumped Alkali Lasers With An Ion Chamber Diagnostic, Benjamin Oliker May 2022

Examination Of Ionization In Cesium Diode Pumped Alkali Lasers With An Ion Chamber Diagnostic, Benjamin Oliker

Optical Science and Engineering ETDs

Diode pumped alkali lasers (DPALs) are leading candidates for future high power applications, with many potential utilities for the military, aerospace, communications, and scientific diagnostics. A critical step in their development is measurement and understanding of unwanted ionization processes that occur inside the laser, which decrease efficiency, reduce the usable alkali population, and increase heat load. In this dissertation, direct measurement of the ionization rate of a cesium DPAL gain medium are made for the first time, via application of an ion chamber diagnostic. Results will demonstrate that the rate of ionization is slow compared to pump absorption, with a …


Nonequilibrium And Nonlinear Dynamics In Collective Spin Models And Implementations Using Quantum Feedback Control, Manuel H. Munoz Arias May 2022

Nonequilibrium And Nonlinear Dynamics In Collective Spin Models And Implementations Using Quantum Feedback Control, Manuel H. Munoz Arias

Physics & Astronomy ETDs

Out-of-equilibrium dynamics generalizes the study of ground states of quantum Hamiltonians at zero temperature, to that of dynamical quasi-steady states of quantum systems far from equilibrium. In this dissertation I discuss dynamical quantum phase transitions and out-of-equilibrium phases of matter in models of collective spins with multi-body interactions. These models, based on collective degrees of freedom, allow an exact description of the thermodynamic limit via the mean-field description. In this limit, the nonequilibrium dynamics of properties of quantum states is mapped to the nonlinear dynamics of classical variables, and thus it can be analyzed using tools from the theory of …