  Direct detection of vacuum fluctuations and analysis of sub-cycle quantum
properties of the electric field are explored by a paraxial quantum theory of
ultrafast electro-optic sampling. The feasibility of such experiments is
demonstrated by realistic calculations adopting a thin ZnTe electro-optic
crystal and stable few-femtosecond laser pulses. We show that nonlinear mixing
of a short near-infrared probe pulse with multi-terahertz vacuum field modes
leads to an increase of the signal variance with respect to the shot noise
level. The vacuum contribution increases significantly for appropriate length
of the nonlinear crystal, short probe pulse durations, tight focusing, and
sufficiently large number of photons per probe pulse. If the vacuum input is
squeezed, the signal variance depends on the probe delay. Temporal positions
with noise level below the pure vacuum may be traced with a sub-cycle accuracy.
