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Journal of Physics Conference Series

Disentangling the Time-Evolution Operator of a Single Qubit

Marco Enríquez, Sara Cruz y Cruz

Published May 26, 2017• 7 citations
Publisher

Abstract

We show that the time-evolution equation for a two-level system can be turned into a system of non-linear differential equations when it is written in the disentangled form. The solutions are determined by solving a parametric-oscillator-like equation with a certain time-dependent frequency. The formalism is used to deal with the problem of a two-level atom interacting with a circularly polarized field. The possibility of generating exactly solvable Hamiltonians is briefly discussed.

DOI
10.1088/1742-6596/839/1/012015

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We show that the time-evolution equation for a two-level system can be turned into a system of non-linear differential equations when it is written in the disentangled form. The solutions are determined by solving a parametric-oscillator-like equation with a certain time-dependent frequency. The formalism is used to deal with the problem of a two-level atom interacting with a circularly polarized field. The possibility of generating exactly solvable Hamiltonians is briefly discussed.
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.
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.
Time evolution operator for a {h(1)⊕h(1)}⊎u(2)\left\{ h(1) \oplus h(1) \right\} \uplus u(2){h(1)⊕h(1)}⊎u(2) time-dependent quantum Hamiltonian; a self-consistent resolution method based on Feynman's disentangling rules
In this article the time evolution operator of two interacting quantum oscillators, whose Hamiltonian is an element of the complex {h(1)⊕h(1)}⊎u(2)\left\{ h(1) \oplus h(1) \right\} \uplus u(2){h(1)⊕h(1)}⊎u(2) algebra, is analyzed using the Feynman time ordering operator techniques. This method is consistently used to generate the conditions and to formally find explicit disentangled expressions for such operator. In this way, it is shown that all the problem reduces to solve a complex Riccati-type differential equation. Some closed solutions to this differential equation are found and then concrete disentangling expressions for the time-ordered evolution operator are given. Finally, the time evolution of the coherent states linked to the isotropic 2D quantum oscillator are analyzed under alternative time-independent an time-dependent Hamiltonian systems.
We use the evolution operator method to describe time-dependent quadratic quantum systems in the framework of nonrelativistic quantum mechanics. For simplicity, we consider a free particle with a variable mass M(t), a particle with a variable mass M(t) in an alternating homogeneous field, and a harmonic oscillator with a variable mass M(t) and frequency ω(t) subject to a variable force F(t). To construct the evolution operators for these systems in an explicit disentangled form, we use a simple technique to find the general solution of a certain class of differential and finite-difference nonstationary Schrödinger-type equations of motion and also the operator identities of the Baker–Campbell–Hausdorff type. With known evolution operators, we can easily find the most general form of the propagators, invariants of any order, and wave functions and establish a unitary relation between systems. Results known in the literature follow from the obtained general results as particular cases.
In this paper we introduce an alternative approach to studying the evolution of a quantum harmonic oscillator subject to an arbitrary time dependent force. With the purpose of finding the evolution operator, certain unitary transformations are applied successively to Schrödinger's equation reducing it to its simplest form. Therefore, instead of solving the original Schrödinger's partial differential equation in time and space the problem is replaced by a system of ordinary differential equations. From the obtained evolution operator we workout the propagator. Even though we illustrate the use of unitary transformations on the solution of a forced harmonic oscillator, the method presented here might be used to solve more complex systems. The present work addresses many aspects regarding unitary transformations and the dynamics of a forced quantum harmonic oscillator that should be useful for students and tutors of the quantum mechanics courses at the senior undergraduate and graduate level.