The application of b ^ ^ † † increases m z by one unit while preserving n , whereas a ^ ^ † † application simultaneously increase n and decreases m z by one unit. The analogy to quantum harmonic oscillator provides solutions H ^ ^ | n , m z ⟩ ⟩ = E n | n , m z ⟩ ⟩ , where E n = ℏ ℏ ω ω c ( n + 1 2 ) and | n , m z ⟩ ⟩ = ( b ^ ^ † † ) m z + n ( m z + n ) ! ( a ^ ^ † † ) n n ! | 0 , 0 ⟩ ⟩ .