Bosonic Kitaev phase in a frequency-modulated optomechanical array

The bosonic system possessing the identical real-space form as a fermionic Kitaev model usually exhibits the trivial topology since bosonic operators conform to the bosonic commutation relation. However, a nontrivial bosonic Kitaev phase can also emerge under some certain conditions. Here we propose a scheme to induce a topologically nontrivial bosonic Kitaev phase based on a frequency-modulated one-dimensional optomechanical array. We find that the bosonic Kitaev model with pure imaginary tunneling amplitudes induced by effective optomechanical coupling can be mapped into a subspace expanded by two Hermitian quadrature operators in which the bosonic system possesses two isolated analogous Majorana modes. The isolated analogous modes imply that the proposed bosonic Kitaev system possesses a topologically nontrivial phase. Furthermore, we investigate the effects of the modulated partial next-nearest-neighbor (NNN) hopping on the topology of the system, in which the system exhibits distinct topological behavior under different parameter regimes. In particular, the bosonic Kitaev system experiences completely different phase-transition processes with the change of real partial NNN hopping corresponding to an even and odd number of lattice sites. Our scheme provides a flexible and controllable platform to investigate bosonic Kitaev phases and phase transitions both in theory and experiment.