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Journal of Physics Conference Series

New Families of Single—Qubit Control Fields: An Algorithm

Marco Enríquez, Alfonso Jaimes-Nájera

Published Apr 1, 2020• 0 citations
Publisher

Abstract

Abstract The dynamics of a two-level system interacting with a new class of analytically solvable driving fields is analyzed. Such fields are obtained using an inverse-engineering approach, which allows to exactly factorize the correspondent time-evolution operator via the Wei-Norman theorem. This technique is presented as an algorithm and using an example we show that the free parameters involved can be used to tune some aspects of the dynamics on demand such as the atomic population inversion.

DOI
10.1088/1742-6596/1540/1/012019

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Abstract The dynamics of a two-level system interacting with a new class of analytically solvable driving fields is analyzed. Such fields are obtained using an inverse-engineering approach, which allows to exactly factorize the correspondent time-evolution operator via the Wei-Norman theorem. This technique is presented as an algorithm and using an example we show that the free parameters involved can be used to tune some aspects of the dynamics on demand such as the atomic population inversion.
We report a new family of time-dependent single-qubit radiation fields for which the correspondent evolution operator can be disentangled in an exact way via the Wei–Norman formalism. Such fields are characterized in terms of the Mathieu functions. We show that the regions of stability of the Mathieu functions determine the nature of the driving fields: For parameters in the stable region, the fields are oscillating, being able to be periodic under certain conditions. Whereas, for parameters in the instability region, the fields are pulse-like. In addition, in the stability region, this family admits solutions for evolution loops in quantum control. We obtain some prescriptions to reach such a control effect. Geometric phases in the evolution are also analyzed and discussed.
Analytical solutions to the time-dependent Schrödinger equation describing a driven two-level system are invaluable to many areas of physics, but they are also extremely rare. Here, we present a simple algorithm that generates an unlimited number of exact analytical solutions. We show that a general single-axis driving term and its corresponding evolution operator are determined by a single real function which is constrained only by a certain inequality and initial conditions. Any function satisfying these constraints yields an exact analytical solution. We demonstrate this method by presenting several new exact solutions to the time-dependent Schrödinger equation. Our general method and many of the new solutions we present are particularly relevant to qubit control in quantum computing applications.
Using the Hubbard representation for S U ( 2 ) , we write the time-evolution operator of a two-level system in the disentangled form. This allows us to map the corresponding dynamical law into a set of nonlinear coupled equations. In order to find exact solutions, we use an inverse approach and find families of time-dependent Hamiltonians whose off-diagonal elements are connected with the Ermakov equation. A physical model with the so-obtained Hamiltonians is discussed in the context of the nuclear magnetic resonance phenomenon.
Control of quantum phenomena would allow for expanding control theory from classical systems to microscopic ones whose behavior is dictated by quantum mechanics. A current goal of quantum control is to develop a systematic methodology for the manipulation of systems. The approach typically used to solve dynamic quantum systems is useful to analyze characteristics of a system represented by a defined operator. The squeeze operator's actions are characterized by finding the time evolution operator using the Wei-Norman method on the associated Hamiltonian and applying this to number (Fock) states, coherent states, and Schrodinger cat states. This specific case analyzing the Squeeze operator shows that the Wei-Norman method to find time-evolution operator can reveal the dynamics of any system with an associated Lie Algebra basis. Documenting a variety of initial states and initial parameters in a library of cases provides a foundation to achieving greater control in experimental applications as well. *I would like to thank Brigham Young University for hosting this Research Experience for Undergraduates program and the National Science Foundation for funding this research through grant #1757998. To cite this abstract, use the following reference: http://meetings.aps.org/link/BAPS.2018.4CS.L04.6
In the design of quantum computing devices of the future the basic element is the qubit. It is a two-level quantum system which may describe population transfer from one steady-state to another controlled by a coherent laser field. A four-dimensional real-variable differential equation model is constructed from the complex-valued two-level model describing the wave function of the system. The state transition matrix of the model is constructed via the Wei-Norman technique and Lie algebraic methodology. The idea of parametrization using flatness-based control, is applied to construct feasible input-output pairs of the model. This input drives the state of the system from the given initial state to the given final state in a finite time producing the corresponding output of the pair. The population transfer is obtained by nullifying part of the state vector via careful selection of the parameter functions.
In this paper we aim to explore the dynamics and overall solvability of the Jaynes-Cummings \& Jaynes-Cummings-Like models. As a lens to understand these dynamics, we focused on cases where the parameters of the system were made time-dependent. All previous work on solving the dynamics of the Jaynes-Cummings models has relied heavily on the use of differential methods and setting the parameters as time-independent constants which were zero or one. To account for this, we utilized the Wei-Norman method which allowed us to analytical solve the time-dependent Hamiltonian. Through the use of this method, we can understand the more general characteristics of the dynamics associated with the Jaynes-Cummings model. This research related back to the field of Quantum information and control as the model highlights interactions with qubits and our time-dependent parameters can force certain transitions.