Colorbar for lineplot using pgfplots

Question

I am doing a lineplot with pgfplots. However, instead of the usual legend, I would prefer a colorbar, as I am basically trying to visualize a transition, and for that I chose colors in a gradient like fashion.

How can I add a colorbar below the following MWE plot with clear steps and one label on the left and one label on the right of the colorbar to indicate that my transition is from A to B:

\documentclass{article}
\usepackage[dvipsnames]{xcolor}
\usepackage{tikz,pgfplots}

\pgfplotscreateplotcyclelist{TheBlues}{%
    {color=RoyalBlue!50!black},
    {color=RoyalBlue!70!black},
    {color=RoyalBlue!90!black},
    {color=RoyalBlue!90!white},
    {color=RoyalBlue!70!white},
    {color=RoyalBlue!50!white},
}

\pgfplotsset{cycle list name=TheBlues}

\begin{document}
    \begin{tikzpicture}
        \begin{axis}[
            xmin=0,
            xmax=1,
            axis line style={
                very thick, 
                gray,
                shorten <=-0.5\pgflinewidth
            },
            tick label style ={gray},
            axis lines*=left,
        ]
            \foreach \n in {1,2,...,6}
                \addplot+[ultra thick,smooth,domain={0:1}] {x^\n};
        \end{axis}
    \end{tikzpicture}
\end{document}