Green's operator for Hamiltonians with Coulomb plus polynomial potentials

Proefschrift--Utrecht. / Summary in English. Bibliography: p. 65-67.

Thesis (Ph. D.)--Lehigh University, 2001. / Includes vita. Includes bibliographical references (leaves 103-108).

<p> This dissertation describes the theoretial exploration of electron transfer (ET) processes at the interface between bulk and molecular or nanoscale materials. Analysis of simple model Hamiltonians, those for the two- and three-level electronic systems as well as for a single electronic level coupled to a continuum, inform an understanding of electron transfer in nontrivial systems. A new treatment of the three-level system at an undergraduate level encapsulates the hopping and superexchange mechanisms of electron transfer. The elegance of the behavior of ET from a single-level/continuum system precedes a treatment of the reverse process&mdash;quasicontinuum-to-discrete level ET. This reverse process, relevant to ET from a bulk material to a semiconductor quantum dot (QD) offers a handle for the coherent control of ET at an interface: the shape of an electronic wavepacket within the quasicontinuum. An extension of the single-level-to-continuum ET process is the injection of an electron from a QD to a wide-bandgap semiconductor nanoparticle (NP). We construct a minimal model to explain trends in ET rates at the QD/NP interface as a function of QD size. Finally, we propose a scheme to gate ET through a molecular junction via the coherent control of the torsional mode(s) of a linking molecule within the junction.</p>

<p> The scaling of controlled quantum systems to large numbers of degrees of freedom is one of the long term goals of experimental quantum information science. Trapped-ion systems are one of the most promising platforms for building a quantum information processor with enough complexity to enable novel computational power, but face serious challenges in scaling up to the necessary numbers of qubits. In this thesis, I present both technical and operational advancements in the control of trapped-ion systems and their juxtaposition with photonic modes used for quantum networking. After reviewing the basic physics behind ion trapping, I then describe in detail a new method of implementing Raman transitions in atomic systems using optical frequency combs. Several dierent experimental setups along with simple theoretical models are reviewed and the system is shown to be capable of full control of the qubit-oscillator system. Two-ion entangling operations using optical frequency combs are demonstrated along with an extension of the operation designed to suppress certain experimental errors. I then give an overview of how spatially separated ions can be entangled using a photonic interconnect. Experimental results show that pulsed excitation of trapped ions provide an excellent single photon source that can be used as a heralded entangling gate between macroscopically separated systems. This heralded entangling gate is used to show a violation of a Bell inequality while keeping the detection loophole closed and can be used a source private random numbers. Finally, the coherent Coulomb force-based gates are combined with the probabilistic photon-based gates in a proof of concept experiment that shows the feasibility of a distributed ion-photon network.</p>

<p> Randomized benchmarking is an experimental procedure intended to demonstrate control of quantum systems. The procedure extracts the average error introduced by a set of control operations. When the target set of operations is intended to be the set of Clifford operators, the randomized benchmarking algorithm is particularly easy to perform and its results have an important interpretation with respect to quantum computation. The aim of the benchmark is to provide a simple, useful parameter describing the quality of quantum control with an experiment that can be performed in a standard way on any prospective quantum computer. This parameter can be used to fairly compare different experiments or to mark improvement in a single experiment. </p><p> In this thesis I discuss first the original randomized-benchmarking procedure and the importance of the Clifford operators for its implementation. I develop the statistical analysis of the results and the physical assumptions that are required for the simplest analysis to apply. The original procedure does not extend in an obvious way to benchmarking of more than one qubit, so I introduce a standardized procedure for randomized benchmarking that applies to any number of qubits. This new procedure also enables the benchmarking of an individual control operation. I describe two randomized-benchmarking experiments I helped to design: one involved a single qubit and utilized a variation of the original procedure and the second involved two qubits and demonstrated the new procedure. I conclude with several potential extensions to the original and new procedures that give them reduced experimental overhead, the ability to describe encoded operations, and fairer comparisons between experiments.</p>

<p> In quantum mechanics, the process of measurement is intrinsically probabilistic. As a result, continuously monitoring a quantum system will randomly perturb its natural unitary evolution. An accurate measurement record documents this stochastic evolution and can be used to reconstruct the quantum trajectory of the system state in a single experimental iteration. We use weak measurements to track the individual quantum trajectories of a superconducting qubit that evolves under the competing influences of continuous weak measurement and Rabi drive. We analyze large ensembles of such trajectories to examine their characteristics and determine their statistical properties. For example, by considering only the subset of trajectories that evolve between any chosen initial and final states, we can deduce the most probable path through quantum state space. Our investigation reveals the rich interplay between measurement dynamics, typically associated with wavefunction collapse, and unitary evolution. Our results provide insight into the dynamics of open quantum systems and may enable new methods of quantum state tomography, quantum state steering through measurement, and active quantum control.</p>

<p> An approach has been developed for extracting approximate quantum state-to-state information from classical trajectory simulations which "quantizes" symmetrically both the initial and final classical actions associated with the degrees of freedom of interest using quantum number bins (or "window functions") which are significantly narrower than unit-width. This approach thus imposes a more stringent quantization condition on classical trajectory simulations than has been traditionally employed, while doing so in a manner that is time-symmetric and microscopically reversible. </p><p> To demonstrate this "symmetric quasi-classical" (SQC) approach for a simple real system, collinear <i>H + H</i>₂ reactive scattering calculations were performed [S.J. Cotton and W.H. Miller, J. Phys. Chem. A 117, 7190 (2013)] with SQC-quantization applied to the <i>H</i>₂ vibrational degree of freedom (DOF). It was seen that the use of window functions of approximately 1/2-unit width led to calculated reaction probabilities in very good agreement with quantum mechanical results over the threshold energy region, representing a significant improvement over what is obtained using the traditional quasi-classical procedure. </p><p> The SQC approach was then applied [S.J. Cotton and W.H. Miller, J. Chem. Phys. 139, 234112 (2013)] to the much more interesting and challenging problem of incorporating non-adiabatic effects into what would otherwise be standard classical trajectory simulations. To do this, the classical Meyer-Miller (MM) Hamiltonian was used to model the electronic DOFs, with SQC-quantization applied to the classical "electronic" actions of the MM model&mdash;representing the occupations of the electronic states&mdash;in order to extract the electronic state population dynamics. It was demonstrated that if one ties the zero-point energy (ZPE) of the electronic DOFs to the SQC windowing function's width parameter this very simple SQC/MM approach is capable of quantitatively reproducing quantum mechanical results for a range of standard benchmark models of electronically non-adiabatic processes, including applications where "quantum" coherence effects are significant. Notably, among these benchmarks was the well-studied "spin-boson" model of condensed phase non-adiabatic dynamics, in both its symmetric and asymmetric forms&mdash;the latter of which many classical approaches fail to treat successfully. </p><p> The SQC/MM approach to the treatment of non-adiabatic dynamics was next applied [S.J. Cotton, K. Igumenshchev, and W.H. Miller, J. Chem. Phys., 141, 084104 (2014)] to several recently proposed models of condensed phase electron transfer (ET) processes. For these problems, a flux-side correlation function framework modified for consistency with the SQC approach was developed for the calculation of thermal ET rate constants, and excellent accuracy was seen over wide ranges of non-adiabatic coupling strength and energetic bias/exothermicity. Significantly, the "inverted regime" in thermal rate constants (with increasing bias) known from Marcus Theory was reproduced quantitatively for these models&mdash;representing the successful treatment of another regime that classical approaches generally have difficulty in correctly describing. Relatedly, a model of photoinduced proton coupled electron transfer (PCET) was also addressed, and it was shown that the SQC/MM approach could reasonably model the explicit population dynamics of the photoexcited electron donor and acceptor states over the four parameter regimes considered. </p><p> The potential utility of the SQC/MM technique lies in its stunning simplicity and the ease by which it may readily be incorporated into "ordinary" molecular dynamics (MD) simulations. In short, a typical MD simulation may be augmented to take non-adiabatic effects into account simply by introducing an auxiliary pair of classical "electronic" action-angle variables for each energetically viable Born-Oppenheimer surface, and time-evolving these auxiliary variables via Hamilton's equations (using the MM electronic Hamiltonian) in the same manner that the other classical variables&mdash;i.e., the coordinates of all the nuclei&mdash;are evolved forward in time. In a complex molecular system involving many hundreds or thousands of nuclear DOFs, the propagation of these extra "electronic" variables represents a modest increase in computational effort, and yet, the examples presented herein suggest that in many instances the SQC/MM approach will describe the true non-adiabatic quantum dynamics to a reasonable and useful degree of quantitative accuracy.</p>

<p>We studied two topics: i) how much physical resources are needed to store information and ii) decoherent histories theory applied to Grover search. Given a system consisting of <i>d</i> degrees of freedom each of mass <i>m</i> to store an amount <i>S</i> of information, we find that its average energy, &lang;<i>H</i>&rang;, or size, &lang;<i>r</i>²&rang;, can be made arbitrarily small individually, but its product &lang;P&rang; = &lang;<i>H</i>&rang;&lang;<i> r</i>²&rang; is bounded below by (exp{<i>S/d</i>} &minus; 1)²<i>d</i>²/<i>m.</i> This result is obtained in a nonrelativistic, quantum mechanical setting, and it is independent of earlier thermodynamical results such as the Bekenstein bound on the entropy of black holes. </p><p> The second topic is decoherent histories applied to the Grover search problem. The theory of decoherent histories is an attempt to derive classical physics from positing only quantum laws at the fundamental level without notions of a classical apparatus or collapse of the wave-function. Searching for a marked target in a list of <i>N</i> items requires &Omega;(<i> N</i>) oracle queries when using a classical computer, while a quantum computer can accomplish the same task in <i>O</i>([special characters omitted]) queries using Grover's quantum algorithm. We study a closed quantum system executing Grover algorithm in the framework of decoherent histories and find it to be an exactly solvable model, thus yielding an alternate derivation of Grover's famous result. We also subject the Grover-executing computer to a generic external influence without needing to know the specifics of the Hamiltonian insofar as the histories decohere. Depending on the amount of decoherence, which is captured in our model by a single parameter related to the amount of information obtained by the environment, the search time can range from quantum to classical. Thus, we identify a key effect induced by the environment that can adversely affect a quantum computer's performance and demonstrate exactly how classical computing can emerge from quantum laws. </p>

<p> We study quantum theory as an example of entropic inference. Our goal is to remove conceptual difficulties that arise in quantum mechanics. Since probability is a common feature of quantum theory and of any inference problem, we briefly introduce probability theory and the entropic methods to update probabilities when new information becomes available. Nelson's stochastic mechanics and Caticha's derivation of quantum theory are discussed in the subsequent chapters. </p><p> Our first goal is to understand momentum and angular momentum within an entropic dynamics framework and to derive the corresponding uncertainty relations. In this framework momentum is an epistemic concept &ndash; it is not an attribute of the particle but of the probability distributions. We also show that the Heisenberg's uncertainty relation is an osmotic effect. Next we explore the entropic analog of angular momentum. Just like linear momentum, angular momentum is also expressed in purely informational terms. </p><p> We then extend entropic dynamics to curved spaces. An important new feature is that the displacement of a particle does not transform like a vector. It involves second order terms that account for the effects of curvature . This leads to a modified Schr&ouml;dinger equation for curved spaces that also take into account the curvature effects. We also derive Schrodinger equation for a charged particle interacting with external electromagnetic field on general Riemannian manifolds. </p><p> Finally we develop the entropic dynamics of a particle of spin 1/2. The particle is modeled as a rigid point rotator interacting with an external EM field. The configuration space of such a rotator is <i>R</i>³ &times; <i>S</i>³ (<i>S</i>³ is the 3-sphere). The model describes the regular representation of <i>SU</i>(2) which includes all the irreducible representations (spin 0, 1/2, 1, 3/2,...) including spin 1/2.</p>