  Quantum fluctuations in low-dimensional systems and near quantum phase
transitions have significant influences on material properties. Yet, it is
difficult to experimentally gauge the strength and importance of quantum
fluctuations. Here we provide a resonant inelastic x-ray scattering study of
magnon excitations in Mott insulating cuprates. From the thin film of
SrCuO$_2$, single- and bi-magnon dispersions are derived. Using an effective
Heisenberg Hamiltonian generated from the Hubbard model, we show that the
single-magnon dispersion is only described satisfactorily when including
significant quantum corrections stemming from magnon-magnon interactions.
Comparative results on La$_2$CuO$_4$ indicate that quantum fluctuations are
much stronger in SrCuO$_2$ suggesting closer proximity to a magnetic quantum
critical point. Monte Carlo calculations reveal that other magnetic orders may
compete with the antiferromagnetic N\'eel order as the ground state. Our
results indicate that SrCuO$_2$ - due to strong quantum fluctuations - is a
unique starting point for the exploration of novel magnetic ground states.
