  Two-mode interferometers, such as Michelson interferometer based on two
spatial optical modes, lay the foundations for quantum metrology. Instead of
exploring quantum entanglement in the two-mode interferometers, a single
bosonic mode also promises a measurement precision beyond the shot-noise limit
(SNL) by taking advantage of the infinite-dimensional Hilbert space of Fock
states. However, the experimental demonstration still remains elusive. Here, we
demonstrate a single-mode phase estimation that approaches the Heisenberg limit
(HL) unconditionally. Due to the strong dispersive nonlinearity and long
coherence time of a microwave cavity, quantum states of the form
$\left(\left|0\right\rangle +\left|N\right\rangle \right)/\sqrt{2}$ are
generated, manipulated and detected with high fidelities, leading to an
experimental phase estimation precision scaling as $\sim N^{-0.94}$. A
$9.1$~$\mathrm{dB}$ enhancement of the precision over the SNL at $N=12$, which
is only $1.7$~$\mathrm{dB}$ away from the HL, is achieved. Our experimental
architecture is hardware efficient and can be combined with the quantum error
correction techniques to fight against decoherence, thus promises the quantum
enhanced sensing in practical applications.
