2.1 Fock states and coherent states

Fock states and coherent states 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 photon antibunching, 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 ($\hat{n} = \hat{a}^\dagger \hat{a}$)
• Fock states are orthogonal to each other, meaning that the inner product of two different Fock states is zero ($\langle n | m \rangle = \delta_{nm}$)
• The vacuum state is a special Fock state with zero photons ($|0\rangle$)

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 quantum interference 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 ($\hat{a}|\alpha\rangle = \alpha|\alpha\rangle$)
  • The complex amplitude ($\alpha$) determines the average number of photons ($|\alpha|^2$) and the phase of the coherent state
• Coherent states have a Poissonian photon number distribution, with the variance equal to the mean photon number ($\langle (\Delta \hat{n})^2 \rangle = |\alpha|^2$)
• Coherent states maintain their shape and properties under the action of the annihilation operator ($\hat{a}|\alpha\rangle = \alpha|\alpha\rangle$)

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 ($|\alpha\rangle = \hat{D}(\alpha)|0\rangle$)
  • The displacement operator is defined as $\hat{D}(\alpha) = \exp(\alpha \hat{a}^\dagger - \alpha^* \hat{a})$
  • Applying the displacement operator to the vacuum state shifts the state in phase space by the complex amplitude $\alpha$

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 Poissonian distribution
• Fock states are eigenstates of the photon number operator ($\hat{n}|n\rangle = n|n\rangle$), while coherent states are eigenstates of the annihilation operator ($\hat{a}|\alpha\rangle = \alpha|\alpha\rangle$)

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 ($\langle n | m \rangle = \delta_{nm}$), while coherent states are not orthogonal and have a non-zero overlap ($\langle \alpha | \beta \rangle = \exp(-|\alpha - \beta|^2/2)$)

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 $|n\rangle$ results in a transition to the state $|n-1\rangle$
• 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 $|\alpha\rangle$ results in a transition to a coherent state with a slightly reduced amplitude $|\alpha'\rangle$, where $|\alpha'|^2 = |\alpha|^2 - 1$

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 $|n\rangle$, 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 ($P(m) = \delta_{mn}$)

Coherent state photon number distribution

• Coherent states have a Poissonian photon number distribution, characterized by the mean photon number $|\alpha|^2$, where $\alpha$ is the complex amplitude of the coherent state
• The probability of measuring $n$ photons in a coherent state $|\alpha\rangle$ is given by the Poisson distribution: $P(n) = (|\alpha|^{2n} e^{-|\alpha|^2}) / n!$
• The variance of the photon number distribution for a coherent state is equal to the mean photon number, $\sigma^2 = |\alpha|^2$

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 $|\alpha|^2$, the Poisson distribution can be approximated by a Gaussian distribution with mean $|\alpha|^2$ and variance $|\alpha|^2$
• 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