  We construct a transport map from Poisson point processes onto
ultra-log-concave measures over the natural numbers, and show that this map is
a contraction. Our approach overcomes the known obstacles to transferring
functional inequalities using transport maps in discrete settings, and allows
us to deduce a number of functional inequalities for ultra-log-concave
measures. In particular, we provide the currently best known constant in
modified logarithmic Sobolev inequalities for ultra-log-concave measures.
