  Electronically non-adiabatic effects play an important role in many chemical
reactions. How these effects manifest in cold and ultracold chemistry remain
largely unexplored. Here, through first principles non-adiabatic quantum
dynamics calculations of the Li + LiNa $\to$ Li$_2$ + Na chemical reaction, it
is shown that non-adiabatic dynamics induces quantum interference effects that
dramatically alter the ultracold rotationally resolved reaction rate
coefficients. The interference effect arises from a conical intersection
between the ground and an excited electronic state that is energetically
accessible even for ultracold collisions. These unique interference effects
might be exploited for quantum control applications as a quantum molecular
switch. A statistical analysis of rotational populations of the Li$_2$ product
reveals a Poisson distribution implying an underlying classically chaotic
dynamics. The Poisson distribution is robust and amenable to experimental
verification and appears to be a universal property of ultracold reactions
involving alkali metal dimers.
