= Bosonic annihilation operator
{title2=$a$}
A bosonic annihilation operator removes one excitation from a specified <bosonic> mode:
$$
a|n\rangle=\sqrt n\,|n-1\rangle\quad(n\geq1),\qquad a|0\rangle=0.
$$
Its product with its <adjoint operator> defines the <number operator> $N=a^\dagger a$, and the <canonical commutation relation> $[a,a^\dagger]=I$ 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.
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