Abstract

I propose to incorporate two GaAs/InAs quantum dots in a larger circuit comprised of linear optical elements to create a spin-spin-photon polarization three-qubit Deutsch gate. Since the Deutsch gate is a universal quantum logic gate, any quantum computing task can be completed using a combination of Deutsch gates. I argue the significance of the spin-spin-photon Deutsch gate protocol, given that no Deutsch gate has been experimentally realized and the only other published Deutsch gate proposal does not use a flying qubit. The use of a flying photonic qubit facilitates quantum communication applications. In addition, I show that the versatility of my protocol can lead to two different constructions of the Toffoli gate either by fixing a parameter of the Deutsch gate or by taking a sub-circuit of the Deutsch gate. Within my Deutsch gate circuit is a smaller Toffoli gate circuit. I calculate the fidelity of the Toffoli gate circuit for different material parameters of the GaAs/InAs quantum dots. I display a schematic of the Deutsch gate circuit with removable mirrors that allow the circuit to switch into a Toffoli gate. Finally, I discuss how appropriate Toffoli gates can be adapted into Deutsch gates using a sub-circuit of the original Deutsch gate circuit.

Abstract

We give an outline of the importance and historical development of Bohmian mechanics, an ontological interpretation of quantum theory, followed by an exposition of its mathematical formulation. We then use Bohmian mechanics to obtain plots of trajectories and of the time evolution of the Bohmian mechanical quantum potential for the simple harmonic oscillator. We show how an already formed classical intuition can be relied on to help understand the behavior of the trajectories through their dynamical relationship to the quantum potential.

Nathan Stone (Senior Thesis, April 2020,
Advisor: Jean-Francois Van Huele
)

Abstract

Secure information transfer lies at the basis of today’s information age. The computation complexity of factoring a number into its prime components is the method used for information encryption. Shor’s algorithm, when implemented on a quantum computer, can factor a number with an operational complexity of approximatly O(n^3) where n is the number of qubits needed to represent the number to be factored N. Using the structure of Shor’s algorithm, I construct an algorithm that takes advantage of weak value amplification and postselection (WVAP) to construct a new "Super Prime Factorization Algorithm" which factors a number in linear computational complexity O(n) while still using the same number of qubits required in Shor’s algorithm. The results of this paper show that if a successful post selection with high probability can be obtained, then this Super Prime Factorization Algorithm, along with WVAP computation, will be instrumental for the future of quantum information and computation.

Abstract

The physics curriculum trains students in problem-solving, teamwork, and particularly in higher-level coursework, systems-thinking. Within the program, the scope of such problems is focused on physical systems. As evidenced by my experiences outside of the physics program, such a skillset proves useful in application to non-physical systems, specifically within the realm of social issues.

Abstract

The phenomenon of quantum erasure has long been a curiosity in the field of quantum mechanics because it appears to allow future entangling measurements to affect a system’s behavior in the past. A consequence of quantum erasure is the modification of the wave and particle behaviors of quantum objects. This thesis examines this effect in the context of an eraser which uses the Stern-Gerlach effect to create entanglement. We employ analytic and algebraic methods to determine how the wavefunction of a quantum object evolves over time in such a system. We quantify the wave and particle behaviors in the system through coherence and distinguishability respectively. We find that the erasure process initially destroys the coherence of the object, but each spin substate may recover a degree of coherence. The degree of coherence recovered depends on the Stern-Gerlach fields’ inhomogeneity orientations and the magnitude of the quantum object’s spin. Full coherence can always be recovered with proper inhomogeneity orientations for spin-1/2 objects. There is only one case which allows for spin-1 objects to recover full coherence. Objects with spin values greater than 1 cannot recover full coherence and the maximum recoverable coherence decreases for greater spin values. These are important considerations when it is necessary to create a quantum state with a certain degree of coherence.

Abstract

Quantum measurement theory provides a relationship between measurement error and disturbance caused in observables due to measurement. This relationship describes the lower limit of error in quantum measurement. I expand current theory from two to three observables—two disturbances and one error—to better describe three dimensional properties (like spin). I create three quantum circuits, which together, represent error and two different disturbances in a spin-1/2 system. The quantum circuits are executed on IBM’s real and simulated quantum computers. Simulate data match the two-observable relation while real data progressively gets further from the relation as the data approaches the relational boundary. For the two-observable relation, the quantum circuits are accurate on simulated quantum computers suggesting the inaccuracy in the real IBM quantum computers is due to internal IBM quantum computer workings. For three-observables, simulated and real quantum computer data suggest the constructed relation is not tight and thus not capturing the correct lower limit boundary. Tightening the three-observable relation is needed.

Rachel Kimberlyn Gardner (Senior Thesis, August 2017,
Advisor: Jean-Francois Van Huele
)

Abstract

Quantum Energy Teleportation (QET), a recent proposal, parallels with the well-established procedure Quantum Information Teleportation (QIT) on fundamental quantum principles. Analyzing Mashiro Hotta’s simple two-parameter model for QET, I expanded his model to include all eigenstates of the Hamiltonian. In attempt to deny Hotta’s claim of creating the simplest model, I created a similar one-parameter model which produced similar eigenvalues of the original model differing only by a constant shift, preventing negative values. The models’ efficiency is calculated by taking the difference between the infused and extracted energies. The significant value is the minimum of this difference, hence maximizing the efficiency. The results of these models conclude that Hotta’s original model for the ground state is most efficient.

Abstract

The dynamics of a coupled ground and coherent state are explored. The approach is focused on solving for the time evolution operator and then applying it to a tensor product of a ground and coherent state representing a physical system and environment respectively. The coherent state is then partially traced to extract the dynamics of the ground state. The time evolution operator is found by solving a series of eleven coupled differential equations. The results demonstrate that a change in coupling results in a change in the evolution of the ground state.

Abstract

Since Heisenberg introduced the relation p1 q1 ∼ h in 1927, it has been the subject of discussion and further investigation. Recent work has shown that the term ‘uncertainty’ applies to two different quantum properties. The first pertains to preparation uncertainty, the principle that one cannot prepare a quantum system such that two incompatible observables are arbitrarily well-defined. The second pertains to measurement uncertainty, the principle that the measurement with a certain degree of accuracy of one observable disturbs the subsequent measurement of a second incompatible observable. We review recent experiments showing evidence for a violation of the measurement uncertainty. We illustrate various reformulations of the Heisenberg uncertainty relation with examples using spin measurements.

Abstract

The idea of separating the universe into a system to be studied and an environment to be ignored is important to much of physics. We explore this idea in the context of quantum systems, where it is captured by the concept of a completely positive trace-preserving map, or quantum channel. We demonstrate the usefulness of the quantum channel formalism by using it to solve the problem of optimally cloning an unknown qubit, given only its purity. We find that knowing just the purity of a qubit does not give any advantage in cloning it. The simplest quantum channels are those whose input and output are single qubits. We introduce a set of these single-qubit quantum channels that are particularly simple and have straightforward physical interpretations. The question arises whether all single-qubit channels are a composition of channels from this set. We show that, in general, they are not, suggesting the complexity of the evolution of even the smallest quantum systems.

Abstract

Uncertainty relations constrain knowledge about incompatible (non-commuting) quantum observables, such as spin components of particles. I study a technique [1] to overcome these limitations, which I name the "Buddy Method", using weak measurement and particle entanglement of all three Cartesian components spin. I verify that the Buddy Method works for all four maximally entangled Bell states. I also show that the method gives better than random probabilities for partially entangled states. Finally, I generalize the method for arbitrary spin direction.

Abstract

We study the time evolution of driven quantum systems using analytic, algebraic, and numerical methods. First, we obtain analytic solutions for driven free and oscillator systems by shifting the coordinate and phase of the undriven wave function. We also factorize the quantum evolution operator using the generators of the Lie algebra comprising the Hamiltonian. We obtain coupled ODE’s for the time evolution of the Lie algebra parameters. These parameters allow us to find physical properties of oscillator dynamics. In particular we find phase-space trajectories and transition probabilities. We then search for chaotic behavior in the Lie algebra parameters as a signature for dynamical chaos in the quantum system. We plot the trajectories, transition probabilities, and Lyapunov exponents for a wide range of the following physical parameters: strength and duration of the driving force, frequency difference, and anharmonicity of the oscillator. We identify conditions for the appearance of chaos in the system.

Abstract

This work explores several formulations of inherent limitations in quantum theory. Ever since its inception with Heisenberg in 1927, uncertainty has played the lead role in imposing such limitations [1]. Yet the ubiquitous Heisenberg uncertainty principle Δx Δp ≈ h and its generalized version developed by Robertson [2] can be strengthened in many cases. I discuss and present a new derivation of a physically and mathematically extended uncertainty relation for both pure and mixed states that Schrodinger originally developed [3]. I illustrate the discrepancies between the Robertson and Schrödinger uncertainty relations by applying them to a selection of incompatible operators. I work out these uncertainties in one dimension for the infinite square well, the harmonic oscillator, and the free particle wave packet; I also expand upon my previous work [4] to calculate uncertainties for operators and states in the space of angular momentum 1/2, 1, 3/2, and 2. Incorporating the Schrödinger uncertainty relation often raises the lower bound on uncertainty with respect to incorporating the Robertson uncertainty relation. Furthermore, I connect the Schrödinger form of uncertainty to conservation law constraints and analyze some limitations on quantum computational processes. Recent work by Karasawa et al. [5] suggests that conservation laws limit the inherent accuracy of gate operations in quantum computing. One way to quantify these limitations is through a gate operation's fidelity. I extend and clarify this work on arbitrary single-qubit gate operations by incorporating the Schrödinger form of the uncertainty relation. From this, I propose an approximate computational feasibility check that suggests an upper bound on fidelity for certain quantum computations.

Abstract

I derive and apply quantum propagator techniques to atomic and condensed matter systems. I observe many interesting features by following the evolution of a wavepacket. In atomic systems, I revisit the Stern-Gerlach effect and study the spin dynamics inside an inhomogeneous magnetic field. The results I obtained are not exactly the same as the textbook description of the effect which is usually a manifestation of a perfect space and spin entanglement. This discovery can provide insight on more reliable quantum computation device designs. In condensed matter systems, the doping concentration inhomogeneity leads to the Rashba spin-orbit interaction. This makes it possible to control the spin without the external magnetic field. By propagating the wave packet in systems exhibiting Rashba spin-orbit interactions, I discover several features such as spin separation, spin accumulation, persistent spin-helix, and ripple formation.

Abstract

This work explores several formulations of inherent limitations in quantum theory. Ever since its inception with Heisenberg in 1927, uncertainty has played the lead role in imposing such limitations [1]. Yet the ubiquitous Heisenberg uncertainty principle, Δx Δp≥ h, and its generalized version developed by Robertson [2] can be strengthened in many cases. I discuss and present a new derivation of a physically and mathematically extended uncertainty relation for both pure and mixed states that Schrödinger originally developed [3]. I illustrate the discrepancies between the Robertson and Schrödinger uncertainty relations by applying them to a selection of incompatible operators. I work out these uncertainties in one dimension for the in nite square well, the harmonic oscillator, and the free particle wave packet; I also expand upon my previous work [4] to calculate uncertainties for operators and states in the space of angular momentum 1/2, 1, 3/2, and 2. Incorporating the Schrödinger uncertainty relation often raises the lower bound on uncertainty with respect to incorporating the Robertson uncertainty relation. Furthermore, I connect the Schrödinger form of uncertainty to conservation law constraints and analyze some limitations on quantum computational processes. Recent work by Karasawa et al. [5] suggests that conservation laws limit the inherent accuracy of gate operations in quantum computing. One way to quantify these limitations is through a gate operation's fidelity. I extend and clarify this work on arbitrary single-qubit gate operations by incorporating the Schrödinger form of the uncertainty relation. From this, I propose an approximate computational feasibility check that suggests an upper bound on Fidelity for certain quantum computations.

Abstract

We examine the nonrelativistic classical equations of motion of two charged massive particles in a static homogeneous magnetic eld. We discuss criteria for when the classical nonrelativistic radiationless approximation is valid for this two-particle system. We focus on the motion in the plane perpendicular to the direction of the magnetic eld. We determine conditions of boundedness in this plane for both the center-of-mass vector and the relative-position vector that describe the two-particle system. We then examine a spinor equation that describes two nonrelativistic quantum particles in a homogeneous magnetic eld. By treating some of the terms of the Hamiltonian as perturbations, we obtain analytic expressions for the energy levels of the two-particle system. We apply these analytic expression to predict the energy levels for neutral two-body systems such as hydrogen and positronium, and for the positive helium ion and any hydrogen-like ions. Finally, we explore a matrix factorization technique to derive nonrelativistic quantum wave equations which may incorporate spin, and relativistic quantum wave equations which may incorporate anti-particle wave components. From our investigation, we postulate the necessary conditions to obtain Schrodinger wave equations, Pauli wave equations, Klein-Gordon wave equations, and Dirac wave equations.

Abstract

We review some of the dfficulties in merging quantum theory with relativity. In particular, we discuss the issue of localization in quantum mechanics. We introduce the conformal group, a supergroup of the Poincare group and give its generators and corresponding algebra. We then illustrate how this allows us to construct a space-time localization operator that is consistent with special relativity and quantum theory.

Abstract

We use quantum mechanics to describe J.J. Thomson’s experiment for determining the mass-to-charge ratio m e of the electron. We review the derivation of Thomson’s classical trajectories in the electrostatic and magnetic fields. We model Thomson’s capacitor and obtain the stationary wave mechanical solutions. We construct wave packets representing the transverse probability amplitude of the beam. We allow this packet to evolve in time to observe the motion of the peak and compare it with the classical trajectory. As time progresses the packet disperses and the peak delocalizes. We discuss physical and numerical causes for this phenomenon. We also use the analytic solution in momentum space to characterize the wave packets. We consider the problem in the Heisenberg representation and confirm the correspondence between quantum expectation values and classical trajectories. The uncertainties of the Thomson trajectories are identical to those of the undeflected beam.

Johanna Sorensen (Capstone, December 2006,
Advisor: Jean-Francois Van Huele, Patrick Madden
)

Abstract

n/a

Abstract

I present a systematic study of Snyder space, the original quantized spacetime described by Hartland Snyder in Phys. Rev. 71, 38-41. I outline characteristics of representations in both an underlying de Sitter space and momentum space and discuss how information about position can be recovered. I present methods for studying systems in Snyder space, which I use to find the energy spectrum of the harmonic oscillator in one and two dimensions. I discuss the relation between Snyder space and noncommutative quantum mechanics and its place in theories of unfication and quantum gravity. Algorithms that I developed for the manipulation of noncommutative objects and for the evalu- ation of formulas from perturbation theory are included.

Abstract

The traditional explanation of the Stern-Gerlach effect carries with it several very subtle assumptions and approximations. We point out the degree to which this fact has affected the way we practice and interpret modern physics. In order to gain a more complete understanding of the Stern-Gerlach effect beyond the standard approximations and assumptions it typically carries, we introduce the inhomogeneous Stern-Gerlach effect in which the strong uniform field component is removed. This change allows us to easily identify precession as a critical concept. It also provides us with a means by which to study precession and analyze it critically. By applying and comparing several mathematical techniques to this problem we gain insight into the applicability of precession arguments and the role of standard approximations and assumptions in both analytic derivation and interpretation. This approach also allows for a more general discussion regarding the use of representations in physics and teaching.

Abstract

This thesis addresses the issue of the time variability of the fine structure constant, alpha. Recent claims of a varying alpha are set against the established standards of quantum electrodynamical theory and experiments. A study of the feasibility of extracting data on the time dependence of alpha using particles in Penning traps is compared to the results obtained by existing methods, including those using astrophysical data and those obtained in atomic clock experiments. Suggestions are made on the nature of trapped particles and the trapping fields.

Abstract

We study the feasibility of polarizing electron beams using the longitudinal Stern-Gerlach effect. After a brief historical motivation we review a semi-classical analysis for electron dynamics in the presence of an axial current ring. We derive the complete set of differential equations for the trajectories and variances in this cylindrical geometry and point out differences with expressions found in the literature. We solve a subset of the equations numerically for a particular choice of initial conditions and we provide numerical simulations of the effect. We conclude by comparing the trajectory variance with the spin separation.

Abstract

Calculating the probability current for nonrelativistic particles with spin, using the same procedure as is used for particles without spin, yields an ambiguous result. We resolve this ambiguity without appealing to relativistic quantum mechanics. A unique expression for the probability current of nonrelativistic particles with spin is derived. This expression includes an extra term that arises due to the spin of the particle. We verify that this extra spin current term is not a relativistic effect, and analyze the properties of the spin current for the case of an electron in a homogeneous magnetic field.

Abstract

In the microscopic domain, time measurements are obtained by applying quantum operators called quantum clocks. We review different types of quantum clocks, based on a time-independent Hamiltonian. Explicit microscopic models of ticking quantum clocks are constructed by using the superposition of wave functions in periodic multi-wells. Dispersion of the wave packet and accuracy of the clocks are discussed. The transition to relativistic clocks is addressed and some solutions for the relativistic double well are presented.

Abstract

In order to understand the nature of quantum computation it is useful to examine what makes a quantum algorithm different from a classical one. This work presents an in-depth analysis of Shor's algorithm, including a simulation programmed for a classical computer.

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Techniques from Supersymmetry (a theory in particle physics) are applied to nonrelativistic quantum mechanics to develop a general method of finding ladder operators for exactly solvable potentials. Bound States in the Continuum are studied, and the same techniques from Supersymmetry are used to generate potentials which permit bound states in the continuum.

Abstract

The topics of strongly couples positronium and matching of the constants as a method of deriving exactly solvable potentials are subtopics of two important problems in theoretical physics. The first topic pertaining to strongly couples positronium is that of stability from the Crater and Van Alstine two-particle wave equation for a Coulomb potential with a homogeneously charged spherical truncating potential, with regard to exactly solvable potentials in wave equations. Our contribution to this topic is to have illustrated the power of a particular method, that of matching of the constants, by deriving previously known exactly solvable potentials for the radial Schrödinger and Klein-Gordon equations using this method.

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According to classical electrodynamics, accelerating charged particles radiate. This radiation carries away some of the particles energy and momentum, and consequently affects its motion. This phenomenon is called radiation reaction.We study the equations of motion for the nonrelativistic electron including radiation reaction. In doing so, we choose to emphasize the classical quantum connection in the approach of Moniz and Sharp. We provide detailed calculations and analysis of their results. In the classical case, it is shown that the solutions to the equation of motion exhibit the well-known problems of either runaway behavior or preacceleration for a point electron. Any attempt to resolve these problems within the realms of classical electrodynamics is fruitless. Using the Heisenberg picture of quantum mechanics, we study the quantum equations of motion, and find that the quantum point electron theory does avoid some of the problems of the classical theory. Through an appropriate correspondence limit we study how the quantum theory is able to resolve some of these problems. In so doing, we gain further understanding of quantum mechanics, and its connection with classical mechanics.

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