Fig. 1: Illustration of the effective light cones.

Herein, we describe the interacting bosons by the Bose-Hubbard type Hamiltonian (1). We first consider how fast boson particles move to distant regions, as shown in Fig. 2. The light cone for the boson particle transport is proved to be almost linear up to logarithmic corrections (denoted by the blue shaded line), as shown in Result 1. Conversely, if we consider the propagation of the full information (see also Fig. 3), the speed can be much faster than the particle transport. The effective light cone is proved to be polynomial with time, and the exponent is equal to the space dimension D (denoted by the orange shaded line), where the mathematical form of the Lieb-Robinson bound is given in Result 2. We can explicitly construct a protocol to achieve the light cone using dynamics with time-dependent Bose-Hubbard type Hamiltonians (see Fig. 4).