We implement a general imaging method by measuring the complex degree of coherence using linear optics and photon number resolving detectors. In the absence of collective or entanglementassisted measurements, our method is optimal over a large range of practically relevant values of the complex degree of coherence. We measure the size and position of a small distant source of pseudo-thermal light, and show that our method outperforms the traditional imaging method by an order of magnitude in precision. Finally, we show that a lack of photon number resolution in the detectors has only a modest detrimental effect on measurement precision and simulate imaging using the new and traditional methods with an array of detectors; showing that the new method improves both image clarity and contrast.

Exploring the quantum behaviour of macroscopic objects provides an intriguing avenue to study the foundations of physics and to develop a suite of quantum-enhanced technologies. One prominent path of study is provided by quantum optomechanics which utilizes the tools of quantum optics to control the motion of macroscopic mechanical resonators. Despite excellent recent progress, the preparation of mechanical quantum superposition states remains outstanding due to weak coupling and thermal decoherence. Here we present a novel optomechanical scheme that significantly relaxes these requirements allowing the preparation of quantum superposition states of motion of a mechanical resonator by exploiting the nonlinearity of multi-photon quantum measurements. Our method is capable of generating non-classical mechanical states without the need for strong singlephoton coupling, is resilient against optical loss, and offers more favourable scaling against initial mechanical thermal occupation than existing schemes. Moreover, our approach allows the generation of larger superposition states by projecting the optical field onto NOON states. We experimentally demonstrate this multi-photon-counting technique on a mechanical thermal state in the classical limit and observe interference fringes in the mechanical position distribution that show phase superresolution. This opens a feasible route to explore and exploit quantum phenomena at a macroscopic scale.

We introduce quantum hypercube states, a class of continuous-variable quantum states that are generated as orthographic projections of hypercubes onto the quadrature phase-space of a bosonic mode. In addition to their interesting geometry, hypercube states display phase-space features much smaller than Planck's constant, and a large volume of Wigner-negativity. We theoretically show that these features make hypercube states sensitive to displacements at extremely small scales in a way that is surprisingly robust to initial thermal occupation and to small separation of the superposed state-components. In a high-temperature proof-of-principle optomechanics experiment we observe, and match to theory, the signature outer-edge vertex structure of hypercube states.

We implement an optimal imaging method using number-resolving photon detectors. Measuring the complex degree of coherence, we obtain the size and position of a distant pseudothermal light source and show this method outperforms traditional imaging.

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