Creation and annihilation operators

The bosonic ladder operators satisfy

[A, A^{†}] = 1, A ∣ n ⟩ = n ∣ n - 1 ⟩, A^{†} ∣ n ⟩ = n + 1 ∣ n + 1 ⟩ .

Annihilation operator ( $a$ )

An annihilation operator lowers the occupation number operator eigenvalue by one and annihilates the Fock vacuum.

Creation operator ( $a^{†}$ )

A creation operator raises the occupation number operator eigenvalue of a quantum harmonic oscillator by one. Repeated application builds Fock states.

Bosonic creation operator ( $a^{†}$ )

Bosonic annihilation operator ( $a$ )

Single-mode squeeze operator ( $S (γ)$ )

For a real squeezing parameter

γ

, the single-mode squeeze operator is

S (γ) = exp [- \frac{γ}{2} (A^{† 2} - A^{2})] .

Its exponent is skew-adjoint, so

S (γ)

is a unitary operator, and it implements the Bogoliubov transformation

S^{†} A S = A cosh γ - A^{†} sinh γ .

Squeezed vacuum state

The squeezed vacuum is

∣ γ ⟩ = S (γ) ∣0 ⟩

. Its position and momentum variances are rescaled in opposite directions,

(Δ X)^{2} = \frac{ℏ}{2 mω} e^{- 2 γ}, (Δ P)^{2} = \frac{m ℏ ω}{2} e^{2 γ},

so it remains an minimum-uncertainty state with

Δ X Δ P = ℏ/2

.

Multimode squeezed vacuum ( $∣0 ⟩ = C exp (\frac{1}{2} a^{'}^{†} M a^{'}^{†}) ∣ 0^{'} ⟩$ )

For an invertible finite-mode Bogoliubov transformation

a = A a^{'} + B a^{'}^{†}

, the symmetric squeezing matrix is

M = - A^{- 1} B

. The canonical commutation relations make its singular values strictly less than one. The normalized state has

∣ C ∣ = det (I - M M^{†})^{1/4}

. Infinitely many modes additionally require bosonic mode mixing implementability.

Number operator

The number operator

N = A^{†} A

satisfies

[N, A] = - A, [N, A^{†}] = A^{†} .

It is self-adjoint and positive semidefinite.

Number state ( $∣ n ⟩$ )

A number state is a normalized eigenstate of the number operator:

N ∣ n ⟩ = n ∣ n ⟩, n \in Z_{\geq 0} .

For a single quantum harmonic oscillator, it is also an energy eigenstate with energy

ℏ ω (n + 1/2)

.

Integer spectrum of the number operator

If

N ∣ ν ⟩ = ν ∣ ν ⟩

, then

A

lowers the eigenvalue by one and

∥ A ∣ ν ⟩ ∥^{2} = ν ∥ ν ⟩ ∥^{2}

. Positivity forces the lowering ladder to terminate at eigenvalue zero, so every eigenvalue is a nonnegative integer.

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Creation and annihilation operators are fundamental concepts in quantum mechanics and quantum field theory, particularly in the context of systems such as quantum harmonic oscillators and bosonic fields. ### Creation Operator The **creation operator**, often denoted as \( a^\dagger \), is an operator that adds one quantum (or particle) to a system.

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