  It has been suggested that both quantum superpositions and nonlinear
interactions are important resources for quantum metrology. However, to date
the different roles that these two resources play in the precision enhancement
are not well understood. Here, we experimentally demonstrate a
Heisenberg-scaling metrology to measure the parameter governing the nonlinear
coupling between two different optical modes. The intense mode with n (more
than 10^6 in our work) photons manifests its effect through the nonlinear
interaction strength which is proportional to its average photon-number. The
superposition state of the weak mode, which contains only a single photon, is
responsible for both the linear Hamiltonian and the scaling of the measurement
precision. By properly preparing the initial state of single photon and making
projective photon-counting measurement, the extracted classical Fisher
information (FI) can saturate the quantum FI embedded in the combined state
after coupling, which is ~ n^2 and leads to a practical precision ~ 1.2/n. Free
from the utilization of entanglement, our work paves a way to realize
Heisenberg-scaling precision when only a linear Hamiltonian is involved.
