We consider a quantum simulator of the Heisenberg chain with ferromagnetic interactions based on the two-component 1D Bose-Hubbard model at filling equal to two in the strong coupling regime. The entanglement properties of the ground state of the two-component Bose-Hubbard model are compared to those of the effective spin model as the interspecies interaction approaches the intraspecies one. A numerical study of the entanglement properties of the two-component Bose-Hubbard model is supplemented with analytical expressions derived from the effective spin Hamiltonian. When the pure ferromagnetic Heisenberg chain is considered, the entanglement properties of the effective Hamiltonian are not properly predicted by the quantum simulator.
We propose a scheme to simulate a one-dimensional Bose–Hubbard model with a coupled cavity optomechanical system. By eliminating the cavity field under the condition of large detuning, an effective Bose–Hubbard model can be achieved. By adiabatically adjusting the classical pumping, we show that the effective Hamiltonian can fully explain transitions from the Mott-insulating phase to the superfluid phase. In addition, we show that the system allows the quantum state transfer after including the dissipation of the oscillators.
Using a similarity Hamiltonian renormalization procedure, we determine an effective spin-
21 representation of the Bose-Hubbard model at half-integer filling and at a finite onsite interaction energy
U. By means of bosonization, we are able to recast the effective Hamiltonian as that of a spin-
21 XXZ magnetic chain with pertinently renormalized coupling and anisotropy parameters. We use this mapping to provide analytical estimates of the correlation functions of the Bose-Hubbard model. We then compare such results with those based on density matrix renormalization group numerical simulations of the Bose-Hubbard model for various values of
U and for a number
L of lattice sites as low as $L\ensuremath{\sim}30$. We find an excellent agreement up to
10 between the output of analytical and numerical computations, even for relatively small values of
U. Our analysis implies that, also at finite
U, the one-dimensional Bose-Hubbard model with suitably chosen parameters may be seen as a quantum simulator of the XXZ chain.