which is the result of applying Û ′ 0 = exp -iπâ † â unitary transformation on ĤPF , with Ĥ′ PF = Û ′ 0 ĤPF Û ′ † 0 . The role of Û ′ 0 is causing a π phase shift for the photonic DOF and flip the sign of the qc displacement from a positive one in ĤPF to a negative one in ĤPF ′ .