An annihilation operator lowers the number of particles in a given state by one.

The above definition is also known as an -deformed annihilation operator.

The vacuum state is quenched by the action of the annihilation operator.

Writing this out in terms of creation and annihilation operators:

Notice that since (by the commutation relations) the order in which we write the annihilation operators does not matter.

Hence, they are respectively the creation and annihilation operator for phonons.

Creation and annihilation operators can act on states of various types of particles.

In a multi-mode field each creation and annihilation operator operates on its own mode.

The eigenstates of the annihilation operator are called coherent states:

The coherent state is an eigenstate of the annihilation operator in the Heisenberg picture.

The lower bound for this noise follows from the fundamental properties of the creation and annihilation operators.

Technically, one converts the field to an operator, through combinations of creation and annihilation operators.

We will now examine the normal ordering of bosonic creation and annihilation operator products.

For obvious reason the de-excitation operator is called an annihilation operator.

Normal ordering of a product quantum fields or creation and annihilation operators can also be defined in many other ways.

We shall look in detail at four special cases where and are equal to creation and annihilation operators.

The creation and annihilation operators are introduced to add or remove a particle from the many-body system.

The normal order of any more complicated cases gives zero because there will be at least one creation or annihilation operator appearing twice.

To perceive any difference between them, we need other operators, namely the creation and annihilation operators.

Similar transformations hold for the annihilation operators.

They are eigenvectors of the annihilation operator: .

One may think of a non-linear coherent state by generalizing the annihilation operator:

When developed further, the theory often contradicts observation, so that its creation and annihilation operators can be empirically tied down.

The ground state of the corresponding Hamiltonian is annihilated by all the annihilation operators:

One may redefine the creation and the annihilation operators by a linear redefinition: