  The generation of an entangled coherent state is one of the most important
ingredients of quantum information processing using coherent states. Recently,
numerous schemes to achieve this task have been proposed. In order to generate
travelling-wave entangled coherent states, cross phase modulation, optimized by
optical Kerr effect enhancement in a dense medium in an electromagnetically
induced transparency (EIT) regime, seems to be very promising. In this
scenario, we propose a fully quantized model of a double-EIT scheme recently
proposed [D. Petrosyan and G. Kurizki, {\sl Phys. Rev. A} {\bf 65}, 33833
(2002)]: the quantization step is performed adopting a fully Hamiltonian
approach. This allows us to write effective equations of motion for two
interacting quantum fields of light that show how the dynamics of one field
depends on the photon-number operator of the other. The preparation of a
Schr\"odinger cat state, which is a superposition of two distinct coherent
states, is briefly exposed. This is based on non-linear interaction via
double-EIT of two light fields (initially prepared in coherent states) and on a
detection step performed using a $50:50$ beam splitter and two photodetectors.
In order to show the entanglement of a generated entangled coherent state, we
suggest to measure the joint quadrature variance of the field. We show that the
entangled coherent states satisfy the sufficient condition for entanglement
based on quadrature variance measurement. We also show how robust our scheme is
against a low detection efficiency of homodyne detectors.
