Reduced Dynamics from the Unitary Group to Some Flag Manifolds : Interacting Matrix Riccati Equations

TL;DR

This paper explores the reduction of unitary dynamics in quantum systems to flag manifolds, deriving interacting matrix Riccati equations and analyzing their nonlinear superposition properties, extending previous work in quantum control.

Contribution

It introduces a new framework for reduced quantum dynamics on flag manifolds using matrix Riccati equations, generalizing prior models.

Findings

Derived a set of interacting Riccati differential equations.

Identified a nonlinear superposition formula for these Riccati equations.

Extended previous models to more general flag manifold settings.

Abstract

In this paper we treat the time evolution of unitary elements in the N level system and consider the reduced dynamics from the unitary group U(N) to flag manifolds of the second type (in our terminology). Then we derive a set of differential equations of matrix Riccati types interacting with one another and present an important problem on a nonlinear superposition formula that the Riccati equation satisfies. Our result is a natural generalization of the paper {\bf Chaturvedi et al} (arXiv : 0706.0964 [quant-ph]).