  We consider the quantum effect of slow light dark soliton (SLDS) in a cold
atomic gas with defocuing Kerr nonlinearity via electromagnetically induced
transparency (EIT). We calculate the quantum fluctuations of the SLDS by
solving the relevant non-Hermitian eigenvalue problem describing the quantum
fluctuations, and find that only one zero mode is allowed. This is different
from the quantum fluctuations of bright solitons, where two independent zero
modes occur. We rigorously prove that the eigenmodes, which consist of
continuous modes and the zero mode, are bi-orthogonal and constitute a complete
bi-orthonormalized basis, useful for the calculation on the quantum
fluctuations of the SLDS. We demonstrate that, due to the large Kerr
nonlinearity contributed from the EIT effect, a significant quantum squeezing
of the SLDS can be realized; the squeezing efficiency can be manipulated by the
Kerr nonlinearity and the soliton's amplitude, which can be much higher than
that of bright solitons. Our work contributes to efforts for developing quantum
nonlinear optics and non-Hermitian Physics, and for possible applications in
quantum information processing and precision measurements.
