Phase space is fundamentally different from real space as the two coordinates of phase space do not commute, i.e., [ x ^ ^ , p ^ ^ ] = i λ λ where λ λ is the dimensionless Planck constant. The ladder operator is defined as a ^ ^ = ( x ^ ^ + i p ^ ^ ) / 2 λ λ such that [ a ^ ^ , a ^ ^ † † ] = 1 . The Hamiltonian of a physical system H ^ ^ = H ( x ^ ^ , p ^ ^ ) can also be written in a function of ladder operators H ^ ^ = H ( a ^ ^ , a ^ ^ † † ) . By defining the rotational operator in phase space by T ^ ^ τ τ = e − − i τ τ a ^ ^ † † a ^ ^ where τ τ = 2 π π / n with n a positive integer, the system has n -fold rotational symmetry or Z n symmetry if the Hamiltonian commutates with rotational operator [ H ^ ^ , T ^ ^ τ τ ] = 0 , i.e., H ^ ^ = T ^ ^ τ τ † † H ^ ^ T ^ ^ τ τ → → H ( a ^ ^ , a ^ ^ † † ) = H ( T ^ ^ τ τ † † a ^ ^ T ^ ^ τ τ , T ^ ^ τ τ † † H ^ ^ a ^ ^ τ τ † † ) = H ( a ^ ^ e − − i τ τ , a ^ ^ † † e i τ τ ) . In this case, one can apply Bloch theorem to the n -fold symmetric Hamiltonian and calculate the band structure. The discrete rotational symmetric structure of Hamiltonian is called Z n phase space lattice and the corresponding eigenstates are called Z n phase space crystals.