Heisenberg-limit spin squeezing with the spin Bogoliubov Hamiltonian

It is well established that the optimal spin squeezing under a one-axis-twisting Hamiltonian follows a scaling law of J-2/3 for J interacting atoms after a quench dynamics. Here, we prove analytically and numerically that the spin squeezing of the ground state of the one-axis-twisting Hamiltonian actually reaches the Heisenberg limit J-1. By constructing a bilinear Bogoliubov Hamiltonian with the raising and lowering spin operators, we exactly diagonalize the spin Bogoliubov Hamiltonian, which includes the one-axis twisting Hamiltonian as a limiting case. The ground state of the spin Bogoliubov Hamiltonian exhibits excellent spin squeezing, which approaches the Heisenberg limit in the case of the one-axis-twisting Hamiltonian. It is possible to realize experimentally the spin-squeezed ground state of the one-axis-twisting Hamiltonian in dipolar spinor condensates, ultracold atoms in optical lattices, spins in a cavity, or alkali atoms in a vapor cell.