  The key polynomials, the Demazure atoms, the Schubert polynomials, and even
the Schur functions can be defined using divided difference operator. In 2000,
Hivert introduced a quasisymmetric analog of the divided difference operator.
In particular, replacing it in a natural way in the definition of the Schur
functions gives Gessel's fundamental basis. This paper is our attempt to apply
the same methods to define the remaining bases and study the results. In
particular, we show both the key polynomials and Demazure atoms have natural
analogs using Hivert's operator and that the resulting bases occur
independently and defined by other means in the work of Assaf and Searles, as
the fundemental slide polynomials and the fundamental particle basis
respectively. We further explore properties of these two bases, including
giving the structure constants for the fundamental particle basis.
