  Quantum phase transitions, which are driven by quantum fluctuations, mark a
frontier between distinct quantum phases of matter. However, our understanding
and control of such phenomena is still limited. Here we report an atomic scale
control of quantum phase transition between two different topological quantum
classes of a well-defined {\pi}-conjugated polymer controlled by their length.
We reveal that a pseudo Jahn-Teller effect is the driving mechanism of the
phase transition, being activated above a certain polymer chain length. In
addition, our theoretical calculations indicate the presence of long-time
coherent fluctuations at finite temperature between the two quantum phases of
the polymer near the phase transition. This work may pave new ways to achieve
atomic scale control of quantum phase transitions, in particular in organic
matter.
