The g(2) correlation function is a powerful tool in quantum optics, measuring how photons are related in time. It helps us understand if light behaves classically or shows quantum effects like antibunching, where photons tend to arrive one at a time.

Photon antibunching, indicated by g(2)(0) < 1, is a key sign of non-classical light. This phenomenon is crucial for applications like quantum cryptography and computing, where single-photon sources are needed for secure communication and information processing.

Second-order Correlation Function

Definition and Relevance

• The second-order correlation function, denoted as g(2)(τ), quantifies the correlation between photons emitted by a light source at different time intervals
• Defined as the probability of detecting a photon at time t+τ, given that a photon was detected at time t, normalized by the average photon detection rates
• Provides information about photon statistics and the nature of the light source (bunched, antibunched, or coherent)
• The value of g(2)(0) characterizes photon statistics at zero time delay, indicating the degree of photon antibunching or bunching

Experimental Measurement

• Measured experimentally using a Hanbury Brown-Twiss (HBT) interferometer setup
  • Utilizes single-photon detectors and a coincidence counter
  • Splits the light beam into two paths using a beamsplitter
  • Detects photons in each path and measures the correlation between them
• The coincidence counts are recorded as a function of the time delay τ between the two detectors
• The normalized coincidence counts provide the value of g(2)(τ) at different time delays

g(2) in terms of Operators

Quantum Optics Formalism

• g(2)(τ) can be expressed in terms of the photon creation (â†) and annihilation (â) operators using quantum optics formalism
• The photon number operator is defined as n̂ = â†â, representing the number of photons in a given mode
• The electric field operator Ê(+)(t) is proportional to the photon annihilation operator â(t), while Ê(-)(t) is proportional to the photon creation operator â†(t)

Mathematical Expression

• g(2)(τ) is defined as: g(2)(τ) = ⟨â†(t)â†(t+τ)â(t+τ)â(t)⟩ / ⟨â†(t)â(t)⟩^2
  • The numerator represents the correlation between the intensity of the light at time t and t+τ
  • The denominator normalizes the correlation by the average intensities
• The expectation values in the expression are calculated using the quantum state of the light field (Fock state, coherent state, or thermal state)
• The quantum state determines the specific values of the creation and annihilation operators

Interpretation of g(2) Values

Physical Meaning

• The value of g(2)(τ), especially at zero time delay (τ = 0), provides insights into the photon statistics and the nature of the light source
• For a coherent light source (laser), g(2)(τ) = 1 for all values of τ, indicating uncorrelated photons following Poissonian statistics
• Photon bunching occurs when g(2)(0) > 1, meaning a higher probability of detecting two photons simultaneously compared to a coherent source (observed in chaotic or thermal light sources)

Photon Antibunching

• Photon antibunching is characterized by g(2)(0) < 1, indicating a lower probability of detecting two photons simultaneously compared to a coherent source
• Signature of non-classical light, observed in single-photon emitters (atoms, molecules, quantum dots)
• Perfect photon antibunching corresponds to g(2)(0) = 0, meaning zero probability of detecting two photons at the same time (ideal case for single-photon sources)
• The temporal profile of g(2)(τ) for a single-photon emitter shows a dip at τ = 0, related to the lifetime of the excited state

g(2) and Light Source Nature

Characterizing Light Sources

• g(2)(τ) provides a powerful tool to characterize the nature of the light source and distinguish between different types of emitters
• Single-photon emitters (atoms, molecules, quantum dots) exhibit photon antibunching with g(2)(0) < 1
  • Can only emit one photon at a time and require a finite time to be re-excited before emitting another photon
  • Two-level systems (single atom, quantum dot) exhibit perfect photon antibunching with g(2)(0) = 0 in the ideal case
  • Background noise and imperfections can lead to a non-zero value of g(2)(0) in practice

Applications

• Measuring g(2)(τ) helps identify photon emission characteristics and determine suitability for various applications
  • Quantum cryptography: Requires single-photon sources with low g(2)(0) to ensure secure communication
  • Quantum computing: Utilizes single-photon sources for quantum information processing and quantum gates
  • Single-photon metrology: Relies on antibunched light for precise measurements and calibration
• The degree of photon antibunching quantifies the quality and purity of a single-photon source
  • Lower g(2)(0) value indicates a higher quality source
  • Important for applications demanding high-purity single-photon states