 Bosonic annihilation operator

A bosonic annihilation operator removes one excitation from a specified bosonic mode:Its product with its adjoint operator defines the number operator $N=a^{†}a$, and the canonical commutation relation $[a,a^{†}]=I$ fixes the ladder factors. Acting on the vacuum gives the zero vector, not a normalized state with negative occupation.

A coherent state is an eigenstate of this operator, although the annihilation operator is not a self-adjoint operator. In conditional photon subtraction, applying $a$ describes the successful-event state change only after normalization; the norm also contains information about the event's probability. Algebraic annihilation should therefore be distinguished from a deterministic device that always removes one quantum.