The quantum Cramer–Rao bound (QCRB) provides an ultimate precision limit in parameter estimation. The sensitivity of spatial measurements can be improved by using the higher-order Hermite–Gaussian mode. However, to date, the QCRB-saturating tilt measurement has not been realized. Here, we experimentally demonstrate tilt measurements using a higher-order HG40 mode as the probe beam. Using the balanced homodyne detection with an optimal local beam, which involves the superposition of high-order HG30 and HG50 modes, we demonstrate the precision of the tilt measurement approaching the QCRB. The signal-to-noise ratio of the tilt measurement is enhanced by 9.2 dB compared with the traditional method using HG00 as the probe beam. This scheme is more practical and robust to losses, which has potential applications in areas such as laser interferometer gravitational-wave observatories and high-sensitivity atomic force microscopes.

Normal-mode splitting (NMS) is an evident signature of strong coupling interaction for observing quantum phenomena such as optomechanical squeezing and entanglement. In the previous literature, optical parametric amplifier (OPA) and coherent feedback are respectively proposed to enhance the NMS. Here, we combine OPA and coherent feedback into one optomechanical system to enhance the NMS. The controllable parameters such as input optical power, OPA gain and phase, coherent feedback strength are varied to observe the NMS variation. Especially, we consider the positive and negative feedback in terms of amplitude reflectivity of the beam splitter for coherent feedback. The NMS appears mostly with the positive coherent feedback rather than the negative. Furthermore, the largest mode separation occurs at an OPA phase of approximately −π/4 rather than zero, since the effective cavity detuning changes the effective intracavity round-trip phase, and therefore changes the OPA amplification/deamplification condition. The results indicate that the interplay between OPA and coherent feedback could enhance the NMS with more controllable parameter freedoms. This scheme provides a promising way to increase the optomechanical coupling strength, thereby has potential applications in the ground state cooling of a mechanical oscillator, preparation of optomechanical quantum states and sensitive detection of a weak force.