  We provide another construction of the natural parametrization of
SLE$_\kappa$ for $\kappa < 4$. We construct it as the expectation of the
quantum time, which is a random measure carried by SLE in an ambient Gaussian
free field. This quantum time was built as the push forward on the SLE curve of
the Liouville boundary measure, which is a natural field-dependent measure
supported on the boundary of the domain. We moreover show that the quantum time
can be reconstructed as a chaos on any measure on the trace of SLE with the
right Markovian covariance property. This provides another proof that natural
parametrization is characterized by its Markovian covariance property.
