Time evolution of gluon coherent state and its von Neumann entropy in heavy-ion collisions

We propose a new prescription for evaluating a von Neumann entropy in the initial stage of high-energy heavy-ion collisions utilizing the time evolution of classical Yang-Mills (CYM) field: The von Neumann entropy is computed for the quantum coherent states constructed so as to give the classical gluon fields as the expectation values. The entropy is to be liberated when the complete decoherence is achieved. As a demonstration, the time evolution of the CYM dynamics is solved with an initial condition which mimics the Glasma state, though in a non-expanding geometry; the Glasma state is characterized by the longitudinal color-electric and -magnetic fields with gluon fields' fluctuations around it. We find that the initial longitudinal fluctuations of the fields play essential roles for the entropy production in two ways: First, the field fluctuations at $t=0$ themselves act as a source of the von Neumann entropy prepared before the time evolution. Second, the initial fluctuations triggers field instabilities, and hence the larger the strength of them, the more the entropy production at later time.