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Symmetry

Single-Qubit Driving Fields and Mathieu Functions

Marco Enríquez, Alfonso Jaimes-Nájera, Francisco Delgado

Published Sep 16, 2019• 2 citations
Publisher

Abstract

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.

DOI
10.3390/sym11091172

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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.
Geometric phase for novel analytical solutions (Barnes and Das Sarma) of time-dependent two-level quantum systems is discussed, specically for a general single-axis driving term, which is represented by a function J(t) in the Hamiltonian, and its corresponding evolution operator.It is demonstrated how general results for corresponding phases (total, dynamic and geometric) can be obtained.Using a specic case, it was found that over time in which the driving eld is appreciably dierent from zero, the corresponding geometric phase changes (in the specic example by ∆β ≈ 0.8 radians) thus enabling detection.The results are relevant to qubit control and to 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.
We consider a periodically driven quantum system described by a Hamiltonian which is the product of a slowly varying Hermitian operator $V(\mathbit{\ensuremath{\lambda}}\left(t\right))$ and a dimensionless periodic function with zero average. We demonstrate that the adiabatic evolution of the system within a fully degenerate Floquet band is accompanied by non-Abelian (noncommuting) geometric phases appearing when the slowly varying parameter $\mathbit{\ensuremath{\lambda}}=\mathbit{\ensuremath{\lambda}}\left(t\right)$ completes a closed loop. The geometric phases can have significant values even after completing a single cycle of the slow variable. Furthermore, there are no dynamical phases masking the non-Abelian Floquet geometric phases, as the former average to zero over an oscillation period. This can be used to precisely control the evolution of quantum systems, in particular for performing qubit operations. The general formalism is illustrated by analyzing a spin in an oscillating magnetic field with arbitrary strength and a slowly changing direction.