The Hanbury Brown and Twiss experiment is a game-changer in quantum optics. It shows how light behaves in ways we can't explain with classical physics. By measuring photon correlations, it reveals the quantum nature of light.

This experiment connects to the broader study of photon statistics and correlation functions. It gives us a way to measure and understand how light particles interact and behave, opening up new frontiers in quantum optics research.

The HBT Experiment

Experimental Setup and Principles

• The Hanbury Brown and Twiss (HBT) experiment is a seminal experiment in quantum optics that demonstrates the correlation between photons emitted by a thermal light source
• The setup consists of a light source (usually a mercury vapor lamp or a laser), a beam splitter, two photodetectors, and a coincidence counter
  • The light from the source is split into two paths by the beam splitter
  • Each path is directed towards a photodetector
• The photodetectors are connected to a coincidence counter, which measures the correlation between the arrival times of photons at the two detectors
• The key principle of the HBT experiment is the measurement of the second-order correlation function, $g^{(2)}(\tau)$, which quantifies the probability of detecting a photon at time $t+\tau$ given that a photon was detected at time $t$

Measuring the Second-Order Correlation Function

• The second-order correlation function, $g^{(2)}(\tau)$, is defined as:
  • $g^{(2)}(\tau) = \frac{\langle I(t)I(t+\tau) \rangle}{\langle I(t) \rangle^2}$
  • Where $I(t)$ is the intensity of the light at time $t$, and $\langle \cdot \rangle$ denotes the time average
• The value of $g^{(2)}(\tau)$ provides information about the statistical properties of the light source:
  • For a coherent light source (laser), $g^{(2)}(\tau) = 1$ for all $\tau$
  • For a thermal light source (mercury vapor lamp), $g^{(2)}(\tau) > 1$ for small $\tau$ and approaches 1 for large $\tau$
  • For a non-classical light source (single-photon emitter), $g^{(2)}(\tau) < 1$ for small $\tau$ and approaches 1 for large $\tau$
• The HBT experiment measures $g^{(2)}(\tau)$ by recording the coincidence counts between the two detectors as a function of the time delay $\tau$ between the detection events
  • The coincidence counts are normalized by the product of the individual detector count rates to obtain $g^{(2)}(\tau)$

Quantum Nature of Light

Evidence for the Quantum Nature of Light

• The HBT experiment provided the first direct evidence of the quantum nature of light by showing that photons from a thermal source exhibit bunching
  • Bunching refers to the tendency of photons to arrive at the detectors in groups, resulting in a higher probability of detecting coincident photons at small time delays
• Classical wave theory predicts that the intensity fluctuations at the two detectors should be uncorrelated, resulting in a flat second-order correlation function, $g^{(2)}(\tau) = 1$
  • However, the HBT experiment demonstrated that $g^{(2)}(\tau) > 1$ for small time delays ($\tau \approx 0$), indicating photon bunching
• The bunching effect cannot be explained by classical wave theory and requires a quantum description of light in terms of photons
  • In the quantum picture, photons from a thermal source are emitted randomly and independently, but they tend to arrive at the detectors in bunches due to the Bose-Einstein statistics of the photon field

Impact on Quantum Optics

• The HBT experiment laid the foundation for the field of quantum optics and inspired further investigations into the quantum properties of light
• The observation of photon bunching in the HBT experiment led to the development of the quantum theory of optical coherence, which describes the statistical properties of light in terms of the correlation functions
• The HBT technique has become a powerful tool for characterizing the statistical properties of various light sources and investigating quantum phenomena such as entanglement and squeezing
  • For example, the HBT setup can be used to measure the degree of entanglement between photon pairs generated by spontaneous parametric down-conversion (SPDC)
  • It can also be used to study the photon statistics of non-classical light sources, such as single-photon emitters and squeezed states of light

Photon Bunching vs Antibunching

Photon Bunching

• Photon bunching occurs when the second-order correlation function, $g^{(2)}(\tau)$, is greater than 1 for small time delays ($\tau \approx 0$), indicating that photons tend to arrive at the detectors in groups
• Bunching is a characteristic of thermal light sources, such as the mercury vapor lamp used in the original HBT experiment
  • In a thermal light source, photons are emitted randomly and independently, but they tend to arrive at the detectors in bunches due to the Bose-Einstein statistics of the photon field
• The degree of bunching can be quantified by the value of $g^{(2)}(0)$, with $g^{(2)}(0) > 1$ for bunched light
  • For a thermal light source, $g^{(2)}(0) = 2$, indicating strong bunching
  • For a coherent light source (laser), $g^{(2)}(0) = 1$, indicating no bunching

Photon Antibunching

• Photon antibunching occurs when $g^{(2)}(\tau) < 1$ for small time delays, indicating that photons tend to arrive at the detectors with a minimum time separation
• Antibunching is a purely quantum effect and cannot be observed with classical light sources
  • It requires a non-classical light source, such as a single-photon emitter or a quantum dot
• The observation of photon antibunching provides strong evidence for the particle-like nature of light and the existence of single-photon states
  • In a single-photon emitter, photons are emitted one at a time, with a minimum time separation between successive photons
  • This results in a reduced probability of detecting coincident photons at small time delays, leading to antibunching
• The degree of antibunching can be quantified by the value of $g^{(2)}(0)$, with $g^{(2)}(0) < 1$ for antibunched light
  • For an ideal single-photon emitter, $g^{(2)}(0) = 0$, indicating perfect antibunching
  • In practice, non-ideal single-photon emitters have $0 < g^{(2)}(0) < 1$, indicating partial antibunching

Applications of HBT Technique

Quantum Optics Applications

• The HBT technique has become a powerful tool for characterizing the statistical properties of light sources and investigating the quantum nature of light
• In quantum optics, the HBT setup is used to study the photon statistics of various light sources, such as:
  • Single-photon emitters (quantum dots, nitrogen-vacancy centers in diamond)
  • Atomic ensembles (cold atoms, trapped ions)
  • Nonlinear optical processes (spontaneous parametric down-conversion, four-wave mixing)
• Measuring the second-order correlation function, $g^{(2)}(\tau)$, allows researchers to distinguish between classical and non-classical light sources and to investigate quantum properties of light, such as:
  • Entanglement: The HBT technique can be used to measure the degree of entanglement between photon pairs generated by SPDC or other nonlinear optical processes
  • Squeezing: The HBT setup can be used to study the photon statistics of squeezed states of light, which have reduced quantum fluctuations in one quadrature at the expense of increased fluctuations in the other quadrature

Astronomical Applications

• In astronomy, the HBT technique has been applied to measure the angular diameter of stars through a method called intensity interferometry
  • By measuring the correlation between the intensity fluctuations of starlight at two telescopes separated by a distance, astronomers can determine the angular size of the star
  • This technique works even for stars too small to be resolved by conventional imaging techniques, such as the Hubble Space Telescope
• The HBT technique has also been used to study the coherence properties of astronomical sources, such as:
  • Pulsars: The HBT setup can be used to measure the temporal coherence of pulsar emission, providing insights into the emission mechanism and the properties of the neutron star
  • Active galactic nuclei (AGN): The HBT technique can be used to study the spatial coherence of the emission from AGN, helping to constrain the size and structure of the emitting region
• Intensity interferometry using the HBT technique has the potential to revolutionize astronomical imaging by enabling the construction of large arrays of telescopes that can achieve unprecedented angular resolution
  • Projects such as the Cherenkov Telescope Array (CTA) and the Stellar Intensity Interferometry (SII) initiative aim to use the HBT technique to study a wide range of astrophysical phenomena, from exoplanets to black holes