  Based on the strong-field approximation, we obtain analytical expressions for
the initial momentum at the tunnel exit and instantaneous ionization rate of
tunneling ionization in elliptically polarized laser fields with arbitrary
ellipticity. The tunneling electron reveals a nonzero offset of the initial
momentum at the tunnel exit in the elliptically polarized laser field. We find
that the transverse and longitudinal components of this momentum offset with
respect to the instantaneous field direction are directly related to the time
derivatives of the instantaneous laser electric field along the angular and
radial directions, respectively. We further show that the nonzero initial
momentum at the tunnel exit has a significant influence on the laser phase
dependence of the instantaneous ionization rate in the nonadiabatic tunneling
regime.
