Physics

General Inverse Problem Solution for Two-Level Systems and Its Application to Charge Transfer

Agostino Migliore, Hiromichi Nakazato, Alessandro Sergi, ANTONINO MESSINA

Published Sep 23, 2024• 0 citations

Abstract

Two-level quantum systems are building blocks of quantum technologies, where the qubit is the basic unit of quantum information. The ability to design driving fields that produce prespecified evolutions of relevant physical observables is crucial to the development of such technologies. Using vector algebra and recently developed strategies for generating solvable two-level Hamiltonians, we construct the general solution to the inverse problem for a spin in a time-dependent magnetic field and its extension to any two-level system associated with fictitious spin and field. We provide a general expression for the field that drives the dynamics of the system so as to realize prescribed time evolutions of the expectation values of the Pauli operators and the autocorrelation of the Pauli vector. The analysis is applied to two-state charge transfer systems, showing that the charge transfer process can be seen as a motion of the state of the associated fictitious qubit on the Bloch sphere, and that the expectation values of the related Pauli operators describe the interference between the two differently localized electronic states and their population difference. Our formulation is proposed as a basic step towards potential uses of charge transfer in quantum computing and quantum information transfer.

DOI: 10.3390/physics6030072

Related papers

Controlling the charge-transfer dynamics of two-level systems around avoided crossings

Agostino Migliore, ANTONINO MESSINA

Two-level quantum systems are fundamental physical models that continue to attract growing interest due to their crucial role as a building block of quantum technologies. The exact analytical solution of the dynamics of these systems is central to control theory and its applications, such as that to quantum computing. In this study, we reconsider the two-state charge transfer problem by extending and using a methodology developed to study (pseudo)spin systems in quantum electrodynamics contexts. This approach allows us to build a time evolution operator for the charge transfer system and to show new opportunities for the coherent control of the system dynamics, with a particular emphasis on the critical dynamic region around the transition state coordinate, where the avoided crossing of the energy levels occurs. We identify and propose possible experimental implementations of a class of rotations of the charge donor (or acceptor) that endow the electronic coupling matrix element with a time-dependent phase that can be employed to realize controllable coherent dynamics of the system across the avoided level crossing. The analogy of these rotations to reference frame rotations in generalized semiclassical Rabi models is discussed. We also show that the physical rotations in the charge-transfer systems can be performed so as to implement quantum gates relevant to quantum computing. From an exquisitely physical-mathematical viewpoint, our approach brings to light situations in which the time-dependent state of the system can be obtained without resorting to the special functions appearing in the Landau-Zener approach.

The Journal of Chemical PhysicsFeb 28, 20247 citations

On the quaternion representation of the Pauli spinor of an electron

Marc M. Cahay, George Purdy, D Morris

Abstract The solution to the time-dependent Pauli–Schrödinger equation is written using quaternion algebra, i.e. by expressing the unitary operator describing the time evolution of the Pauli spinor and the Pauli spinor itself as left-chirality quaternions. The quaternion approach is used to express the time evolution of a qubit on the Bloch sphere under the action of the commonly used quantum gates and to implement the two-qubit controlled-NOT (CNOT) gate and the three-qubit Tofolli gate. The quaternion formalism is also applied to describe: (1) a general two-qubit quaternionic gate, (2) the Larmor precession of an electron spin in an uniform magnetic field, and (3) the issue of measurements in quantum mechanics. Finally, the time-dependent Pauli–Schrödinger equation is also solved using right-chirality quaternions.

Physica ScriptaApr 2, 201919 citations

Dynamics of observables and exactly solvable quantum problems: Using time-dependent density-functional theory to control quantum systems

Mehdi Farzanehpour, I. V. Tokatly

We use analytic (current) density-potential maps of time-dependent (current) density-functional theory [TD(C)DFT] to inverse engineer analytically solvable time-dependent quantum problems. In this approach the driving potential (the control signal) and the corresponding solution of the Schr\"odinger equation are parametrized analytically in terms of the basic TD(C)DFT observables. We describe the general reconstruction strategy and illustrate it with a number of explicit examples. First we consider the real space one-particle dynamics driven by a time-dependent electromagnetic field and recover, from the general TDDFT reconstruction formulas, the known exact solution for a driven oscillator with a time-dependent frequency. Then we use analytic maps of the lattice TD(C)DFT to control quantum dynamics in a discrete space. As a first example we construct a time-dependent potential which generates prescribed dynamics on a tight-binding chain. Then our method is applied to the dynamics of spin-1/2 driven by a time-dependent magnetic field. We design an analytic control pulse that transfers the system from the ground to excited state and vice versa. This pulse generates the spin flip thus operating as a quantum not gate.

arXiv:1506.08786Physical Review AMay 24, 20162 citations

Analysis of Electron Spin States in Magnetic Fields and Hardware Implementation of Quantum Computers

Yueqian Jiang

This paper systematically reviews the theoretical progression from blackbody radiation to the foundations of quantum computing. Beginning with Planck’s energy quantization and the representation of electromagnetic wave electric field components, the wave-particle duality of photons is analogized to introduce the de Broglie relation between momentum and wave vector for matter waves, leading to the establishment of the matter wave function and the Schrödinger equation as its governing partial differential equation. The consistency between the rigorous solution for a one-dimensional infinitely deep rectangular potential well and the conclusions for a one-dimensional hydrogen atom is demonstrated. Subsequently, within the Dirac notation formalism, the wave function is expressed, and the correspondence between spatial differentiation and momentum is derived, formalizing the momentum operator (often referred to as first quantization). To lay the physical groundwork for manipulating two-qubit states in quantum computing, the potential energy function of an electron’s orbital magnetic moment in an external magnetic field is introduced, starting from the torque and work on a current-carrying rectangular coil. This is extended to the magnetic moment and potential energy of a spinning electron, where the orientation of the magnetic field determines the form of the potential and accordingly selects the three components of the Pauli operators. The properties of quantum gates corresponding to the Pauli operators are then examined for single-qubit states. A two-qubit state, constructed via the coupling of two arbitrary single-qubit states represented in a two-level system or electron spin, is shown to be uniquely expressible as a superposition of four eigen two-qubit states. The paper further investigates joint manipulations of two-qubit states, including evolution under X and Z gates conditioned on high- and low-level states, the representation of control-flow circuits (with the low-level as control), and the formulation of measurement and CNOT gates. Throughout this work, essential mathematical tools—including integration, matrix operations, complex Euler’s formula, and solutions to simple second-order ordinary differential equations—are incorporated to provide a coherent theoretical framework for state representation and gate operations in quantum computing.

International Journal of Engineering and TechnologyJan 1, 20260 citations

A generalized unitary transformation for solving time-independent and time-dependent Jaynes–Cummings models under resonant and selected off-resonance conditions

ANTONINO MESSINA, Hiromichi Nakazato, Antonio S. M. de Castro +4 more

Jaynes–Cummings (JC) models are by far the most widely used to study the dynamics of two-level systems, such as atoms or molecules under suitable conditions, coupled to quantized electromagnetic fields. Nowadays, analytical solutions of JC models are more useful than ever for understanding and controlling radiation-matter interactions relevant to quantum information processing, quantum computing, and other quantum technologies. This study presents the analysis and generalization of a method for the solution of the dynamical problem in the time-independent JC model (Carbonaro et al., 1979) that enables an exact description of the atom-field dynamics in the resonant time-dependent JC model and in a nontrivial class of off-resonance time-dependent JC models. Our analysis produces the ready-to-use forms of the evolution operator in the resonant and non-resonant cases, which are then used to derive the evolution of the atomic population inversion for some types of atom-field coupling. The central idea behind the proposed approach offers opportunities for its extension to other time-dependent cases and its generalization to JC-related models.

Annals of PhysicsJul 7, 20260 citations

Evolutionary equations for studying the dynamics of qubits in variable magnetic fields

Alexander Yu. Oshchepkov

The work develops an approach to researching the dynamics of qubits that makes it possible to monitor the behavior of the system in the process of quantum computing using computer simulation. The study deals with individual and interacting qubits based on spin ½ in a magnetic field. The goal of the work is to obtain evolutionary equations (equations of motion) for observables that adequately describe the state of the system and provide opportunities for computer modeling of processes. The dynamics of qubits are described using the formalism of the density matrix of pure states. The paper shows derivation of the equations of motion for observables being the average values of operators of physical quantities that describe the state of the system. For spin ½ in magnetic fields, the derivation of the dynamic equation for the spin polarization vector, which is a unit vector in real three-dimensional space, is shown. The shape and parameters of radiofrequency pulses that implement single-qubit operations are presented. In the case of two interacting spins ½, an expression is given for the density matrix and the Hamiltonian of the system for scalar indirect interaction of the magnetic moments of nuclei through the electron shells of atoms in the molecule. Evolution equations are derived for the average dipole moments and quadrupole moment of the system. It is shown that in the case of weak interaction the dynamics of the system can be described using a closed system of equations for the average dipole moments, which in this case coincide with the spin polarization vectors of individual spins. The presentation is accompanied by examples of numerical solutions of the resulting equations describing one-qubit and two-qubit operations. A conclusion is made about the adequacy of the created mathematical model and the possibility of its use for computer modeling of quantum computing processes, the influence of the environment on them and the stabilization of states to prevent or reduce errors in quantum computing.

ВЕСТНИК ПЕРМСКОГО УНИВЕРСИТЕТА ФИЗИКАJan 1, 20240 citations