Observation antibunching with classical light in a linear interferometer

This paper demonstrates that temporal and spatial photon antibunching, typically a nonclassical phenomenon, can be observed using classical thermal light in a Hanbury Brown-Twiss interferometer by employing photon-number-resolving detectors to perform projection measurements, thereby revealing how the interplay between thermal photon statistics and measurement techniques bridges classical and nonclassical correlations.

The Gist

The Big Question: Can "Normal" Light Act "Weird"?

Imagine you are at a crowded party. Usually, people (or in this case, particles of light called photons) tend to move in groups. If you see one person walk through a door, you expect another to follow right behind them. In the world of physics, this is called bunching. This is how "classical" light, like a light bulb or a laser, usually behaves.

However, there is a "weird" behavior called antibunching. This is like walking through a door and finding that if one person enters, the next person refuses to enter for a while. They keep their distance. In the past, scientists thought this "anti-social" behavior only happened with quantum light (light made of single, isolated particles created in a lab). They believed you couldn't see this with normal, "classical" light.

The Experiment: A New Way to Count

This paper reports a surprising discovery: You can see this "anti-social" behavior with normal, thermal light (like a light bulb), but only if you look at it in a very specific way.

The researchers used a setup called a Hanbury Brown-Twiss interferometer. Think of this as a traffic intersection with a splitter in the middle. A beam of light hits the splitter and goes down two different roads to two detectors (let's call them Camera A and Camera B).

The Twist:
Usually, these cameras just count "Did a car pass? Yes/No."
In this experiment, the researchers treated the cameras like smart counters that could say, "Did exactly one car pass?" or "Did zero cars pass?"

They didn't just look at the traffic flow; they looked for a very specific pattern:

• Camera A sees exactly one photon.
• Camera B sees exactly zero photons.

The Analogy: The Rain and the Umbrella

Imagine it's raining (this is your thermal light). Raindrops usually fall in clumps. If you hold two umbrellas (the detectors) side-by-side, they usually get wet at the same time.

• Standard View (Bunching): You look at both umbrellas. If one gets a drop, the other likely gets one too. They are "bunched."
• The New View (Antibunching): Now, imagine you are a very strict observer. You only pay attention to moments where Umbrella A gets exactly one drop, and Umbrella B stays perfectly dry.

The researchers found that when they looked for this specific "One Drop / Zero Drops" pattern, the raindrops seemed to avoid Umbrella B whenever Umbrella A got hit. It looked like the raindrops were "antibunching"—refusing to be in both places at once.

What They Found

1. It's a Team Effort: The "weird" behavior didn't come from the light itself changing. It came from the combination of two things:
   • The natural "clumpy" nature of thermal light (the rain).
   • The specific way they counted the drops (looking for "One vs. Zero").
2. It Depends on the Crowd Size: The effect is sensitive to how many photons are in the beam.
   • If the light is very dim (few photons), they see antibunching (the "One vs. Zero" pattern is strong).
   • If they make the light brighter (more photons), the pattern flips, and they start seeing bunching again.
3. Laser vs. Light Bulb: When they tried this with a laser (which is very orderly), they didn't see this effect. It only happened with the "clumpy" thermal light.

The Conclusion

The paper claims that by using photon-number-resolving detectors (detectors that can count exactly how many particles arrive, not just if any arrived), you can make normal, classical light look like it's behaving like quantum light.

They observed this "antibunching" effect in both time (when the photons arrive) and space (where the photons land).

Why does this matter?
The authors suggest this helps us understand the blurry line between the "normal" world and the "quantum" world. It shows that sometimes, the "weirdness" isn't just in the light itself, but in how we choose to measure it. They believe this could be useful for multiphoton interference (making complex patterns with light) and quantum imaging (taking pictures using light in new ways).

In short: They proved that if you count the "guests" at a party very carefully, even a rowdy crowd (thermal light) can look like it's keeping its distance, mimicking the behavior of a shy, quantum crowd.

Technical Summary

1. Problem Statement

The distinction between classical and quantum phenomena is a fundamental issue in physics. While quantum resources (like single-photon sources) offer advantages in precision, sensitivity, and security, they suffer from generation difficulties, low efficiency, and susceptibility to noise. Conversely, classical light sources are robust but typically cannot exhibit nonclassical effects like photon antibunching (where photons arrive individually rather than in bunches).

Historically, antibunching has been considered a strictly nonclassical effect, observable only in systems with non-classical light states (e.g., resonance fluorescence, entangled pairs). Previous attempts to observe antibunching with classical light relied on post-selection or asymmetric interferometer setups, which could be explained by classical theory. Recent studies suggested that using Photon-Number-Resolving Detectors (PNRDs) with classical light might yield nonclassical antibunching, but the physical origin and the boundary between classical and nonclassical interpretations in these scenarios remained unclear.

The core problem addressed: Can true antibunching be observed using purely classical light (thermal light) in a linear interferometer, and if so, what is the physical mechanism? Is the resulting effect classical or nonclassical?

2. Methodology

Theoretical Framework

The authors developed a theoretical model based on a modified Hanbury Brown-Twiss (HBT) interferometer.

• Setup: A light beam (thermal or laser) is split by a 1:1 beam splitter into two paths, detected by two single-photon detectors ($D_1$ and $D_2$).
• Key Innovation: Instead of standard coincidence counting, the detectors are treated as PNRDs capable of resolving photon numbers ($m$ and $n$). The system records photon arrival time series for offline correlation analysis.
• Mathematical Derivation:
  • The authors derived the joint probability generating function $M(x, y)$ for thermal light.
  • They calculated the joint probability $P_{mn}$ of detecting $m$ photons at $D_1$ and $n$ photons at $D_2$ using Taylor expansion coefficients.
  • They defined the normalized correlation function $g^{(2)}_{mn}$ for specific detection events (e.g., $m=1, n=0$).
  • Theoretical simulations analyzed how $g^{(2)}_{mn}$ varies with the average photon number ($\bar{n}$) and the degree of coherence ($\mu$).

Experimental Setup

• Light Sources:
  • Thermal Light: Generated by passing a 780 nm continuous-wave laser through a rotating ground glass (RG).
  • Laser Light: Used as a control (coherent state) by removing the RG.
• Detection: Two single-mode fibers coupled to single-photon detectors ($D_1, D_2$). One fiber was fixed, while the other was scanned spatially.
• Data Acquisition: A HydraHarp 400 module recorded photon detection time series. Detectors were operated to resolve photon numbers (effectively acting as PNRDs via time-tagging and binning).
• Measurement: The team measured the second-order correlation function $g^{(2)}_{m0}(\tau)$, specifically looking for cases where one detector registers 1 photon and the other registers 0 photons ($m=1, n=0$).

3. Key Contributions

1. Observation of Antibunching with Classical Light: The paper demonstrates that antibunching ($g^{(2)}_{10} < 1$) can be observed using thermal light (a classical source) in a standard linear interferometer, provided the detection scheme involves photon-number projection (specifically the $1,0$ event).
2. Mechanism Identification: The authors identify that the observed antibunching arises from the combined effect of:
   • The intrinsic photon statistics of thermal light (bunching tendency).
   • The specific photon-number projection measurement (conditioning on detecting exactly one photon in one arm and zero in the other).
3. Transition from Antibunching to Bunching: A critical finding is that the nature of the correlation is not static. By varying the average photon number ($\bar{n}$):
   • At low $\bar{n}$, $g^{(2)}_{10} < 1$ (Antibunching).
   • At high $\bar{n}$, $g^{(2)}_{10} > 1$ (Bunching).
   • This transition is driven by the changing probability of zero-photon events relative to the thermal bunching statistics.
4. Clarification of Classical vs. Nonclassical Nature: The paper argues that while standard antibunching ($g^{(2)} < 1$ for $1,1$ events) is strictly nonclassical, the antibunching observed via the $g^{(2)}_{10}$ metric with classical light sits on the boundary. It cannot be interpreted by classical wave theory alone but can be explained by semi-classical models or quantum theory with projection. It serves as a "bridge" to understand the connection between classical and nonclassical correlations.

4. Results

• Theoretical Simulations:
  • For thermal light, $g^{(2)}_{11}$ (1 photon, 1 photon) is always $\ge 1$ (bunching).
  • However, $g^{(2)}_{10}$ (1 photon, 0 photons) can drop below 1. The minimum value was found to be approximately 0.84 when $\bar{n} = 0.5$ and $\mu = 1$ (coherent thermal light).
  • $g^{(2)}_{00}$ (0 photons, 0 photons) remains $\ge 1$.
• Experimental Verification:
  • Temporal Antibunching: Measured $g^{(2)}_{m0}(\tau)$ for pseudothermal light. With an average photon number $\bar{n} \approx 0.66$, a clear dip below 1 was observed for the $m=1$ case, confirming temporal antibunching.
  • Spatial Antibunching: Scanning the detector position confirmed spatial antibunching at the same low photon flux.
  • Control Experiment: Using laser light (coherent state), $g^{(2)}_{10}$ remained constant at 1, confirming that the effect is specific to the statistics of thermal light.
  • Intensity Dependence: Increasing the light intensity (increasing $\bar{n}$ to $\approx 1.98$) caused the antibunching dip to disappear and turn into a bunching peak ($g^{(2)}_{10} > 1$), validating the theoretical prediction of the transition.

5. Significance

• Fundamental Physics: The work challenges the rigid boundary between classical and quantum optics. It shows that "nonclassical" signatures like antibunching can emerge from classical light when the measurement strategy (photon-number projection) is sufficiently sophisticated.
• Quantum Imaging and Interference: The results provide a new framework for multiphoton interference and quantum imaging (specifically ghost imaging) using classical light sources. This could lead to more robust and efficient imaging systems that do not rely on fragile single-photon sources.
• Resource Efficiency: It suggests that quantum-like advantages in correlation and imaging can be achieved with robust, high-brightness classical light sources, provided the detection post-processing is optimized.
• Theoretical Unification: The paper unifies previous fragmented observations of antibunching with classical light (via post-selection, PNRDs, or asymmetric setups) into a comprehensive theoretical model based on photon-number projection.