Wigner Function Tomography via Optical Parametric Amplification

Abstract: Wigner function tomography is indispensable for characterizing quantum states, but its commonly used version, balanced homodyne detection, suffers from several weaknesses. First, it requires efficient detection, which is critical for measuring fragile non-Gaussian states, especially bright ones. Second, it needs a local oscillator, tailored to match the spatiotemporal properties of the state under test, and fails for multimode and broadband states. Here we propose Wigner function tomography based on optical parametric amplification followed by direct detection. The method is immune to detection inefficiency and loss, and suitable for broadband, spatially and temporally multimode quantum states. To prove the principle, we experimentally reconstruct the Wigner function of squeezed vacuum occupying a single mode of a strongly multimode state. We obtain a squeezing of $-7.5\pm 0.4$ dB and a purity of $0.91^{+0.09}_{-0.08}$ despite more than $97\%$ loss caused mainly by filtering. Theoretically, we also consider the reconstruction of a squeezed single photon - a bright non-Gaussian state. Due to strong multimode parametric amplification, the method allows for the simultaneous tomography of multiple modes. This makes it a powerful tool for optical quantum information processing.