  In this letter, we have considered an electron in a double quantum dot system
interacting with a detector represented by a point contact. We present a
dynamical model for the gradual decoherence of the density matrix due to the
interaction with the detector. The interaction of the qubit (quantum system) on
the quantum point contact (environment) leads to a discrete set of pointer
states of the double quantum dot (apparatus). The necessary time for the qubit
decoherence process (measurement time) has been calculated through this model.
The existence of a minimum time for the quantum measurement has been obtained.
