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) 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⟩
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⟩ 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⟩ 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⟩=α∣α⟩ where α is a complex amplitude, α=∣α∣eiϕ
Generated by displacing the vacuum state, ∣α⟩=D^(α)∣0⟩ using the displacement operator, D^(α)=exp(αa^†−α∗a^)
Coherent states are minimum uncertainty states, satisfying ΔxΔp=ℏ/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)=e−∣α∣2∣α∣2n/n!
Mean photon number is given by ⟨n⟩=∣α∣2
Photon number variance equals the mean, ⟨(Δn)2⟩=⟨n⟩
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)⟩∝∣α∣cos(ωt+ϕ)
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), gives the probability of measuring n photons in a given state
Fock states (number states): P(n)=δn,m for ∣m⟩ where δn,m is the Kronecker delta
Coherent states: P(n)=e−∣α∣2∣α∣2n/n! with mean photon number ⟨n⟩=∣α∣2
Thermal states: P(n)=⟨n⟩n/(1+⟨n⟩)n+1 with mean photon number ⟨n⟩=(eℏω/kBT−1)−1
Second-order correlation function, g(2)(τ), characterizes the temporal correlations between photons
Defined as g(2)(τ)=⟨:I^(t)I^(t+τ):⟩/⟨I^(t)⟩2 where I^(t) is the intensity operator and :: denotes normal ordering
For coherent states, g(2)(τ)=1 indicating no correlation between photons
For thermal states, g(2)(0)=2 and g(2)(τ)=1+e−2∣τ∣/τc where τc is the coherence time, showing bunching at short timescales
For single-photon states, g(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) and g(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) and g(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