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Abstract

The Belinski-Khalatnikov-Lifshitz (BKL) conjecture predicts a chaotic
alternation of Kasner epochs in the evolution of generic classical spacetimes
towards a spacelike singularity. As a first step towards understanding the full
quantum BKL scenario, we analyze a vacuum Bianchi II model with local
rotational symmetry, which presents just one Kasner transition. During the
Kasner epochs, the quantum state is coherent and it is thus characterized by
constant values of the different quantum fluctuations, correlations and
higher-order moments. By computing the constants of motion of the system we
provide, for any peaked semiclassical state, the explicit analytical transition
rules that relate the parametrization of the asymptotic coherent state before
and after the transition. In particular, we obtain the modification of the
transition rules for the classical variables due to quantum back-reaction
effects. This analysis is performed by considering a high-order truncation in
moments (the full computations are performed up to fifth-order, which
corresponds to neglecting terms of an order $\hbar^3$), providing a solid
estimate about the quantum modifications to the classical model. Finally, in
order to understand the dynamics of the state during the transition, we perform
some numerical simulations for an initial Gaussian state, that show that the
initial and final equilibrium values of the quantum variables are connected by
strong and rapid oscillations.