We analyze the stationary current of Bose particles across the Bose-Hubbard chain connected to a battery, focusing on the effect of interparticle interactions. It is shown that the current magnitude drastically decreases as the strength of interparticle interactions exceeds the critical value which marks the transition to quantum chaos in the Bose-Hubbard Hamiltonian. We found that this transition is well reflected in the nonequilibrium many-body density matrix of the system. Namely, the level-spacing distribution for eigenvalues of the density matrix changes from Poisson to Wigner-Dyson distributions. With the further increase of the interaction strength, the Wigner-Dyson spectrum statistics change back to the Poisson statistics which now marks fermionization of the Bose particles. With respect to the stationary current, this leads to the counter-intuitive dependence of the current magnitude on the particle number.
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