5.2 Creation and annihilation operators

Creation and annihilation operators are key tools in quantum optics. They describe how energy quanta are added or removed from a system, like photons in an electromagnetic field. These operators help us understand the behavior of quantum states and their interactions.

The operators' properties, including their commutation relations, are crucial for building quantum field theories. They allow us to construct Fock states, representing specific numbers of energy quanta, and coherent states, which are important for describing laser light and other quantum optical phenomena.

Creation and annihilation operators

Definition and properties

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• Creation (###â†_0###) and annihilation (â) operators are mathematical tools used to describe the behavior of quantum harmonic oscillators and quantum field theories
• The creation operator adds a quantum of energy to the system, increasing the occupation number of a particular quantum state by one
  • It is the adjoint (conjugate transpose) of the annihilation operator
• The annihilation operator removes a quantum of energy from the system, decreasing the occupation number of a particular quantum state by one
• The creation and annihilation operators are non-Hermitian, meaning they are not equal to their adjoints
• The creation and annihilation operators are unbounded, meaning they can act on states with arbitrarily large occupation numbers (infinite-dimensional Hilbert space)

Commutator and its significance

• The commutator of the creation and annihilation operators is equal to the identity operator, [[â, â†] = 1](https://www.fiveableKeyTerm:[â,_â†]_=_1)
• The non-zero commutator indicates that the creation and annihilation operators do not commute, which is a fundamental property of quantum mechanics
• The commutator plays a crucial role in deriving the properties of the harmonic oscillator Hamiltonian and the number operator
• The commutator also ensures that the creation and annihilation operators satisfy the canonical commutation relations, which are essential for quantizing the electromagnetic field

Commutation relations of operators

Commutators of creation and annihilation operators

• The commutator of two creation operators or two annihilation operators is always zero: $[â, â] = [â†, â†] = 0$
  • This implies that the order of applying two creation or two annihilation operators does not matter
• The commutator of a creation operator and an annihilation operator is equal to the identity operator: $[â, â†] = 1$
  • This non-zero commutator is a fundamental property of quantum mechanics and leads to the uncertainty principle

Applications of commutation relations

• The commutation relations can be used to derive the properties of the harmonic oscillator Hamiltonian and the number operator
• The number operator, defined as $[N̂](https://www.fiveableKeyTerm:n̂) = â†â$, has eigenstates that are the Fock states, and its eigenvalues are the occupation numbers
  • The number operator commutes with the creation and annihilation operators: $[N̂, â†] = â†$ and $[N̂, â] = -â$
• The harmonic oscillator Hamiltonian can be expressed in terms of the creation and annihilation operators as $Ĥ = ℏω(â†â + 1/2)$, where $ω$ is the angular frequency of the oscillator
  • The ground state energy of the harmonic oscillator is $E_0 = ℏω/2$, which is a consequence of the non-zero commutator of the creation and annihilation operators

Construction of Fock states

Definition and properties of Fock states

• Fock states, also known as number states, are eigenstates of the number operator $N̂$ and represent states with a well-defined number of quanta
• The ground state, or vacuum state, $|0⟩$, is the state with no quanta and is annihilated by the annihilation operator: $â|0⟩ = 0$
• Fock states form a complete orthonormal basis for the Hilbert space of the harmonic oscillator
  • Completeness means that any state in the Hilbert space can be expressed as a linear combination of Fock states
  • Orthonormality means that the inner product of two different Fock states is zero, and the inner product of a Fock state with itself is one: $⟨m|n⟩ = δ_{mn}$, where $δ_{mn}$ is the Kronecker delta

Constructing Fock states using creation operators

• Fock states can be constructed by applying the creation operator to the ground state: $|n⟩ = \frac{(â†)^n}{\sqrt{n!}} |0⟩$, where $n$ is the number of quanta in the state
• The normalization factor $1/\sqrt{n!}$ ensures that the Fock states are orthonormal
• The creation operator acts as a "ladder operator," increasing the number of quanta in a state by one with each application
• Example: The first excited state (single-quantum state) can be obtained by applying the creation operator once to the ground state: $|1⟩ = â†|0⟩$

Operator action on Fock states

Creation operator action on Fock states

• The creation operator acting on a Fock state increases the number of quanta by one: $â†|n⟩ = \sqrt{n+1} |n+1⟩$
  • The resulting state is proportional to the Fock state with one additional quantum
• The proportionality factor $\sqrt{n+1}$ ensures that the resulting state is properly normalized
• Example: $â†|2⟩ = \sqrt{3} |3⟩$, meaning that applying the creation operator to a two-quantum state results in a three-quantum state with an amplitude of $\sqrt{3}$

Annihilation operator action on Fock states

• The annihilation operator acting on a Fock state decreases the number of quanta by one: $â|n⟩ = \sqrt{n} |n-1⟩$
  • The resulting state is proportional to the Fock state with one less quantum
• The proportionality factor $\sqrt{n}$ ensures that the resulting state is properly normalized
• The annihilation operator acting on the ground state gives zero: $â|0⟩ = 0$, as there are no quanta to remove
• Example: $â|3⟩ = \sqrt{3} |2⟩$, meaning that applying the annihilation operator to a three-quantum state results in a two-quantum state with an amplitude of $\sqrt{3}$

Number operator action on Fock states

• The number operator acting on a Fock state returns the number of quanta in that state: $N̂|n⟩ = n|n⟩$
  • This demonstrates that Fock states are eigenstates of the number operator, with the number of quanta as the eigenvalue
• The expectation value of the number operator in a Fock state is equal to the number of quanta in that state: $⟨n|N̂|n⟩ = n$
• Example: $N̂|5⟩ = 5|5⟩$, meaning that the number operator acting on a five-quantum state returns the same state multiplied by the number of quanta (5)

Coherent states and their significance

• The creation and annihilation operators can be used to construct coherent states, which are eigenstates of the annihilation operator and have a well-defined phase
• Coherent states are important in quantum optics and can be used to describe laser light
  • They exhibit Poissonian photon number statistics and minimize the uncertainty relation between the quadrature operators (position and momentum)
• Coherent states can be generated by applying the displacement operator, which is a linear combination of creation and annihilation operators, to the ground state: $|α⟩ = e^{α↠- α^*â}|0⟩$, where $α$ is a complex number representing the amplitude and phase of the coherent state