Direct measurement of the Husimi-Q function of the electric-field in the time-domain

We develop the theoretical tools necessary to promote electro-optic sampling to a time-domain quantum tomography technique. Our proposed framework implements detection of the time evolution of both the electric-field of a propagating electromagnetic wave and its Hilbert transform (quadrature). Direct detection of either quadrature is not strictly possible in the time-domain, detection efficiency approaching zero when an exact mode-matching to either quadrature is reached. As all real signals have a limited bandwidth, we can trace out the irrelevant sampling bandwidth to optimize the detection efficiency while preserving quantum information of the relevant signal. Through the developed understanding of the mode structure of the amplitude and Hilbert transform quadratures, we propose multiplexing and mode-matching operations on the gating function to extract full quantum information on both quantities, simultaneously. The proposed methology is poised to open a novel path toward quantum state tomography and quantum spectroscopy directly in the time domain.