Probing macroscopic quantum states with a sub-Heisenberg accuracy

Significant achievements in high-sensitivity measurements will soon allow us to probe quantum behaviors of macroscopic mechanical oscillators. In a recent work [Phys. Rev. A 80, 043802 (2009)], we formulated a general framework for treating preparation of Gaussian quantum states of macroscopic oscillators through linear position measurements. To outline a complete procedure for testing macroscopic quantum mechanics, here we consider a subsequent verification stage which probes the prepared macroscopic quantum state and verifies the quantum dynamics. By adopting an optimal time-dependent homodyne detection in which the phase of the local oscillator varies in time, the conditional quantum state can be characterized below the Heisenberg limit, thereby achieving a quantum tomography. In the limiting case of no readout loss, such a scheme evades measurement-induced back action, which is identical to the variational-type measurement scheme invented by Vyatchanin et al. [JETP 77, 218 (1993)] but in the context for detecting gravitational waves. To motivate macroscopic quantum mechanics experiments with future gravitational-wave detectors, we mostly focus on the parameter regime where the characteristic measurement frequency is much higher than the oscillator frequency and the classical noises are Markovian, which captures the main feature of a broadband gravitational-wave detector. In addition, we discuss verifications of Einstein-Podolsky-Rosen-type entanglement between macroscopic test masses in future gravitational-wave detectors, which enables us to test one particular version of gravity decoherence conjectured by Diosi [Phys. Lett. A120, 377 (1987)] and Penrose [Gen. Rel. Grav. 28, 581 (1996)].