and are key concepts in quantum optics. Fock states have a fixed number of photons, while coherent states resemble classical light waves. These states showcase the quantum nature of light and its particle-wave duality.
Understanding these states is crucial for grasping quantum light behavior. Fock states exhibit non-classical properties like , while coherent states have Poissonian photon distributions. Their differences highlight the unique features of quantum light.
Fock states and their properties
Definition and characteristics
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Fock states, also known as number states, are quantum states with a well-defined number of photons
Fock states are eigenstates of the photon number operator, with the eigenvalue being the number of photons in the state
The photon number operator is defined as the product of the creation and annihilation operators (n^=a^†a^)
Fock states are orthogonal to each other, meaning that the inner product of two different Fock states is zero (⟨n∣m⟩=δnm)
The is a special Fock state with zero photons (∣0⟩)
Non-classical properties
Fock states are non-classical states of light, as they exhibit properties that cannot be explained by classical electromagnetism
Fock states have a well-defined photon number, which is a purely quantum mechanical concept
Fock states can exhibit sub-Poissonian photon number statistics, with a variance smaller than the mean photon number
Fock states can demonstrate photon antibunching, where the probability of detecting two photons simultaneously is lower than that of classical light sources
Fock states can be used to create entangled states (NOON states) and demonstrate effects
Coherent states: characteristics and generation
Characteristics of coherent states
Coherent states are quantum states that most closely resemble classical electromagnetic waves
Coherent states are eigenstates of the annihilation operator, with the eigenvalue being the complex amplitude of the state (a^∣α⟩=α∣α⟩)
The complex amplitude (α) determines the average number of photons (∣α∣2) and the phase of the coherent state
Coherent states have a Poissonian photon number distribution, with the variance equal to the mean photon number (⟨(Δn^)2⟩=∣α∣2)
Coherent states maintain their shape and properties under the action of the annihilation operator (a^∣α⟩=α∣α⟩)
Generation of coherent states
Coherent states can be generated by a laser operating far above its threshold, where the gain medium acts as a classical current source
The laser cavity selects a single mode of the electromagnetic field, and the gain medium amplifies this mode to create a coherent state
Displacement operators can be used to generate coherent states from the vacuum state (∣α⟩=D^(α)∣0⟩)
The displacement operator is defined as D^(α)=exp(αa^†−α∗a^)
Applying the displacement operator to the vacuum state shifts the state in phase space by the complex amplitude α
Fock states vs Coherent states
Photon number and eigenstate properties
Fock states have a well-defined photon number, while coherent states have an average photon number with a
Fock states are eigenstates of the photon number operator (n^∣n⟩=n∣n⟩), while coherent states are eigenstates of the annihilation operator (a^∣α⟩=α∣α⟩)
Classical and non-classical properties
Fock states are non-classical states of light, while coherent states closely resemble classical electromagnetic waves
Fock states are orthogonal to each other (⟨n∣m⟩=δnm), while coherent states are not orthogonal and have a non-zero overlap (⟨α∣β⟩=exp(−∣α−β∣2/2))
Sensitivity to photon loss
Fock states are more sensitive to photon loss than coherent states, as the loss of a single photon can significantly alter the state
The loss of a photon from a Fock state ∣n⟩ results in a transition to the state ∣n−1⟩
Coherent states maintain their properties under photon loss, with only a decrease in the average photon number
The loss of a photon from a coherent state ∣α⟩ results in a transition to a coherent state with a slightly reduced amplitude ∣α′⟩, where ∣α′∣2=∣α∣2−1
Photon number distribution: Fock vs Coherent
Fock state photon number distribution
The photon number distribution describes the probability of measuring a specific number of photons in a given state
For a Fock state ∣n⟩, the photon number distribution is a delta function centered at n, meaning that the probability of measuring n photons is 1, and the probability of measuring any other number of photons is 0 (P(m)=δmn)
Coherent state photon number distribution
Coherent states have a Poissonian photon number distribution, characterized by the mean photon number ∣α∣2, where α is the complex amplitude of the coherent state
The probability of measuring n photons in a coherent state ∣α⟩ is given by the Poisson distribution: P(n)=(∣α∣2ne−∣α∣2)/n!
The variance of the photon number distribution for a coherent state is equal to the mean photon number, σ2=∣α∣2
Comparison of photon number distributions
As the average photon number increases, the photon number distribution of a coherent state becomes more sharply peaked around the mean value, resembling a Gaussian distribution
For large values of ∣α∣2, the Poisson distribution can be approximated by a Gaussian distribution with mean ∣α∣2 and variance ∣α∣2
Fock states have a fixed photon number, while coherent states have a distribution of photon numbers centered around the average value
This difference in photon number distributions leads to distinct properties and applications for Fock states and coherent states in quantum optics and quantum information processing