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🔬Modern Optics

🔬modern optics review

12.2 Photon statistics and coherent states

4 min readLast Updated on July 22, 2024

Light behaves in fascinating ways at the quantum level. Photon statistics reveal how light particles interact and arrive at detectors. Understanding these patterns helps us grasp the quantum nature of light and its unique properties.

Coherent states, like laser light, show Poissonian statistics with random photon arrivals. Other quantum states can exhibit sub-Poissonian or super-Poissonian behavior, leading to antibunching or bunching effects. These phenomena are crucial for quantum optics applications.

Photon Statistics

Types of photon statistics

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  • Photon statistics describe the probability distribution of photon numbers in a light field
    • Determined by the second-order correlation function, g(2)(τ)g^{(2)}(\tau) measures the temporal correlations between photons at different times
  • Poissonian statistics exhibit a Poisson distribution of photon numbers
    • Characteristic of coherent states (laser light) where the variance in photon number equals the mean photon number, (Δn)2=n\langle(\Delta n)^2\rangle = \langle n\rangle
    • Photon arrivals are independent and randomly distributed in time
  • Sub-Poissonian statistics have a narrower photon number distribution than Poissonian
    • Variance in photon number is less than the mean photon number, (Δn)2<n\langle(\Delta n)^2\rangle < \langle n\rangle indicating reduced photon number fluctuations
    • Found in non-classical light sources (single photon emitters, squeezed states) with antibunching and regularized photon arrivals
  • Super-Poissonian statistics have a broader photon number distribution than Poissonian
    • Variance in photon number is greater than the mean photon number, (Δn)2>n\langle(\Delta n)^2\rangle > \langle n\rangle indicating increased photon number fluctuations
    • Characteristic of thermal light sources (incandescent bulbs) and chaotic light with bunching and clustered photon arrivals

Properties of coherent states

  • Coherent states are quantum states that most closely resemble classical light
    • Eigenstates of the annihilation operator, a^α=αα\hat{a}|\alpha\rangle = \alpha|\alpha\rangle where α\alpha is a complex amplitude, α=αeiϕ\alpha = |\alpha|e^{i\phi}
    • Generated by displacing the vacuum state, α=D^(α)0|\alpha\rangle = \hat{D}(\alpha)|0\rangle using the displacement operator, D^(α)=exp(αa^αa^)\hat{D}(\alpha) = \exp(\alpha \hat{a}^\dagger - \alpha^* \hat{a})
  • Coherent states are minimum uncertainty states, satisfying ΔxΔp=/2\Delta x \Delta p = \hbar/2
    • Have equal uncertainties in the quadrature components (amplitude and phase)
    • Represented by a circular Gaussian distribution in phase space (Wigner function)
  • Coherent states exhibit a Poissonian photon number distribution, P(n)=eα2α2n/n!P(n) = e^{-|\alpha|^2}|\alpha|^{2n}/n!
    • Mean photon number is given by n=α2\langle n\rangle = |\alpha|^2
    • Photon number variance equals the mean, (Δn)2=n\langle(\Delta n)^2\rangle = \langle n\rangle
  • Coherent states have a constant phase and amplitude, leading to a well-defined classical limit
    • Expectation values of the electric field operator oscillate with a stable phase, E^(t)αcos(ωt+ϕ)\langle\hat{E}(t)\rangle \propto |\alpha|\cos(\omega t + \phi)
    • Coherent states provide a bridge between classical and quantum descriptions of light, serving as a reference for other quantum states

Calculations for quantum light states

  • Photon number distribution, P(n)P(n), gives the probability of measuring nn photons in a given state
    1. Fock states (number states): P(n)=δn,mP(n) = \delta_{n,m} for m|m\rangle where δn,m\delta_{n,m} is the Kronecker delta
    2. Coherent states: P(n)=eα2α2n/n!P(n) = e^{-|\alpha|^2}|\alpha|^{2n}/n! with mean photon number n=α2\langle n\rangle = |\alpha|^2
    3. Thermal states: P(n)=nn/(1+n)n+1P(n) = \langle n\rangle^n/(1+\langle n\rangle)^{n+1} with mean photon number n=(eω/kBT1)1\langle n\rangle = (e^{\hbar\omega/k_BT} - 1)^{-1}
  • Second-order correlation function, g(2)(τ)g^{(2)}(\tau), characterizes the temporal correlations between photons
    • Defined as g(2)(τ)=:I^(t)I^(t+τ):/I^(t)2g^{(2)}(\tau) = \langle: \hat{I}(t)\hat{I}(t+\tau) :\rangle / \langle \hat{I}(t) \rangle^2 where I^(t)\hat{I}(t) is the intensity operator and ::: : denotes normal ordering
    1. For coherent states, g(2)(τ)=1g^{(2)}(\tau) = 1 indicating no correlation between photons
    2. For thermal states, g(2)(0)=2g^{(2)}(0) = 2 and g(2)(τ)=1+e2τ/τcg^{(2)}(\tau) = 1 + e^{-2|\tau|/\tau_c} where τc\tau_c is the coherence time, showing bunching at short timescales
    3. For single-photon states, g(2)(0)=0g^{(2)}(0) = 0 indicating perfect antibunching and the impossibility of detecting two photons simultaneously

Experiments demonstrating quantum light

  • Photon antibunching experiments demonstrate the non-classical nature of light
    • Characterized by g(2)(0)<g(2)(τ)g^{(2)}(0) < g^{(2)}(\tau) and g(2)(0)<1g^{(2)}(0) < 1 indicating a reduced probability of detecting two photons simultaneously compared to classical light
    • Observed in resonance fluorescence from single atoms (quantum dots) where the emission of a photon is followed by a dead time before the next emission
    • Requires a Hanbury Brown and Twiss (HBT) setup with a beamsplitter and two detectors to measure the second-order correlation function
  • Photon bunching experiments reveal the temporal clustering of photons
    • Characterized by g(2)(0)>g(2)(τ)g^{(2)}(0) > g^{(2)}(\tau) and g(2)(0)>1g^{(2)}(0) > 1 indicating an increased probability of detecting two photons simultaneously compared to classical light
    • Observed in thermal light sources (black body radiation) and some nonlinear optical processes (parametric down-conversion)
    • Bunching arises from the constructive interference of multiple photon paths leading to clustered arrivals
  • HBT experiment uses a beamsplitter and two detectors to measure the second-order correlation function
    • Correlates the photon detection events between the two detectors at different time delays
    • Enables the study of non-classical light sources (single photon emitters, entangled photon pairs) and quantum optical phenomena (antibunching, bunching, entanglement)
    • Played a crucial role in the development of quantum optics and the understanding of the quantum nature of light


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