Part I, Chapter 2

Non-Classical Light States

Photon statistics reveal the quantum nature of light: bunching, antibunching, and correlations beyond classical physics.

2.1 Photon Statistics Classification

The statistical properties of photon number fluctuations distinguish three fundamental classes of light. The key parameter is the Mandel Q parameter:

$$Q = \frac{\langle (\Delta n)^2 \rangle - \langle n \rangle}{\langle n \rangle} = \frac{\langle n^2 \rangle - \langle n \rangle^2 - \langle n \rangle}{\langle n \rangle}$$

Three Classes of Photon Statistics

Poissonian (Q = 0) — Coherent Light

Δn² = 〈n〉. The variance equals the mean, as in laser light. This is the boundary between classical and quantum statistics.

Super-Poissonian (Q > 0) — Thermal/Chaotic Light

Δn² > 〈n〉. Fluctuations exceed the Poisson level. Thermal light from blackbody sources exhibits Q = 〈n〉, giving Δn² = 〈n〉 + 〈n〉².

Sub-Poissonian (Q < 0) — Non-Classical Light

Δn² < 〈n〉. Fluctuations are suppressed below the shot noise limit. This has no classical analog and is a hallmark of quantum light (e.g., squeezed states, Fock states).

Derivation 1: Thermal (Bose-Einstein) Photon Statistics

Step 1. A single mode in thermal equilibrium at temperature T has the density operator ρ = (1 - e^-βℏω) ∑_n e^-nβℏω |n〉〈n|.

Step 2. The probability of finding n photons is:

$$P(n) = \frac{\bar{n}^n}{(1 + \bar{n})^{n+1}}, \qquad \bar{n} = \frac{1}{e^{\hbar\omega/k_BT} - 1}$$

Step 3. Computing the variance:

$$\langle (\Delta n)^2 \rangle = \bar{n} + \bar{n}^2 = \bar{n}(1 + \bar{n})$$

The extra 〈n〉² term (compared to Poisson) reflects photon bunching — the tendency of thermal photons to arrive in clusters.

2.2 Second-Order Correlation Function g²(τ)

The normalized second-order correlation function measures intensity-intensity correlations and is the central quantity for characterizing photon statistics:

$$g^{(2)}(\tau) = \frac{\langle \hat{a}^\dagger(t) \hat{a}^\dagger(t+\tau) \hat{a}(t+\tau) \hat{a}(t) \rangle}{\langle \hat{a}^\dagger \hat{a} \rangle^2}$$

At zero time delay, this simplifies to:

$$g^{(2)}(0) = \frac{\langle \hat{n}(\hat{n}-1) \rangle}{\langle \hat{n} \rangle^2} = 1 + \frac{\langle (\Delta n)^2 \rangle - \langle n \rangle}{\langle n \rangle^2}$$

Derivation 2: g²(0) for Coherent, Thermal, and Fock States

Coherent state |α〉: Using 〈â†â†ââ〉 = |α|&sup4; and 〈â†â〉 = |α|²:

$$g^{(2)}_{\text{coh}}(0) = \frac{|\alpha|^4}{|\alpha|^4} = 1$$

Thermal state: Using 〈n(n-1)〉 = 2〈n〉²:

$$g^{(2)}_{\text{th}}(0) = \frac{2\bar{n}^2}{\bar{n}^2} = 2$$

Fock state |n〉: Using 〈n|&hat;n;(&hat;n;-1)|n〉 = n(n-1):

$$g^{(2)}_{\text{Fock}}(0) = \frac{n(n-1)}{n^2} = 1 - \frac{1}{n}$$

For n = 1: g²(0) = 0 (perfect antibunching). For n = 0: g²(0) is undefined. The condition g²(0) < 1 is a definitive signature of non-classical light.

Summary of g²(0) Values

g²(0) = 2: Thermal/chaotic light (bunching)

g²(0) = 1: Coherent light (Poissonian, random arrivals)

g²(0) < 1: Non-classical light (antibunching)

g²(0) = 0: Single-photon Fock state (perfect antibunching)

2.3 Hanbury Brown-Twiss Experiment

The Hanbury Brown-Twiss (HBT) experiment measures photon intensity correlations using a beam splitter and two detectors. It was originally developed in 1956 to measure the angular diameters of stars.

Experimental Setup

Light is split by a 50:50 beam splitter. Two single-photon detectors record arrival times. The coincidence rate R(τ) as a function of time delay τ directly measures g²(τ).

• Thermal light: g²(τ) = 1 + |g¹(τ)|², peaked at τ = 0 (bunching)
• Coherent light: g²(τ) = 1 for all τ (no correlation)
• Single photons: g²(0) = 0, rising to 1 at large τ (antibunching)

Derivation 3: Beam Splitter Transformation for HBT

Step 1. A lossless 50:50 beam splitter transforms input modes â, &bcirc; into output modes &ccirc;, &dcirc;:

$$\hat{c} = \frac{1}{\sqrt{2}}(\hat{a} + i\hat{b}), \qquad \hat{d} = \frac{1}{\sqrt{2}}(i\hat{a} + \hat{b})$$

Step 2. For a single photon input |1〉_a|0〉_b, the output is:

$$|1\rangle_a |0\rangle_b \to \frac{1}{\sqrt{2}} (|1\rangle_c |0\rangle_d + i|0\rangle_c |1\rangle_d)$$

Step 3. The coincidence probability (both detectors fire simultaneously) is:

$$P_{\text{coinc}} = \langle \hat{n}_c \hat{n}_d \rangle = 0$$

A single photon cannot be split — it goes to one detector or the other, never both. This is the essence of photon antibunching and confirms the particle nature of light.

2.4 Photon Antibunching

Photon antibunching (g²(0) < g²(τ)) is a purely quantum phenomenon with no classical counterpart. Classically, g²(0) ≥ g²(τ) always holds (the Cauchy-Schwarz inequality for intensities). Violating this proves the quantum nature of the light source.

Derivation 4: Classical Cauchy-Schwarz Inequality for g²

Step 1. For a classical field with intensity I(t), the correlation function is:

$$g^{(2)}_{\text{cl}}(\tau) = \frac{\langle I(t) I(t+\tau) \rangle}{\langle I(t) \rangle^2}$$

Step 2. The Cauchy-Schwarz inequality for real random variables states:

$$\langle I(t) I(t+\tau) \rangle^2 \leq \langle I(t)^2 \rangle \langle I(t+\tau)^2 \rangle = \langle I^2 \rangle^2$$

Step 3. Since 〈I²〉 ≥ 〈I〉², we get g²(0) ≥ 1 and g²(0) ≥ g²(τ) for all classical light.

Conclusion: Any measurement of g²(0) < 1 violates the classical inequality and constitutes proof of the quantum (particle) nature of the detected light. First observed by Kimble, Dagenais, and Mandel (1977) using fluorescence from single sodium atoms.

2.4.1 Single-Photon Sources

Modern single-photon sources that exhibit antibunching include:

• Single atoms/ions in optical cavities (cavity QED)
• Quantum dots (InAs/GaAs) — widely used in integrated photonics
• Nitrogen-vacancy centers in diamond — room temperature operation
• Single molecules — first demonstrated with pentacene in p-terphenyl
• Carbon nanotubes — telecom wavelength emission

2.5 Squeezed Light and Sub-Poissonian Statistics

Amplitude-squeezed states exhibit sub-Poissonian photon statistics. While the squeezed vacuum has super-Poissonian statistics (photons come in pairs), a displaced squeezed state with squeezing in the amplitude quadrature has reduced number fluctuations.

Derivation 5: Photon Number Variance for Amplitude-Squeezed Light

Step 1. For a coherent displaced squeezed state |α, ξ〉 with real α and amplitude squeezing (squeeze the X&sub1; quadrature), the photon number variance is:

$$\langle (\Delta n)^2 \rangle = |\alpha|^2 e^{-2r} + 2\sinh^2 r \cosh^2 r$$

Step 2. For large |α| (strong coherent amplitude), the first term dominates and the mean photon number is approximately 〈n〉 ≈ |α|².

Step 3. The Mandel Q parameter becomes:

$$Q \approx |\alpha|^2 (e^{-2r} - 1) < 0 \quad \text{for } r > 0$$

The negative Q confirms sub-Poissonian statistics. The photon number is more precisely defined than in a coherent state — a signature of quantum noise reduction.

2.6 Higher-Order Correlations and Quantum State Verification

Beyond g²(τ), higher-order correlation functions g&sup(n)⊃ provide increasingly stringent tests of non-classicality and reveal the full counting statistics of photon sources.

2.6.1 Third-Order Correlations g³(0)

The third-order correlation function measures three-photon coincidences:

$$g^{(3)}(0) = \frac{\langle \hat{n}(\hat{n}-1)(\hat{n}-2) \rangle}{\langle \hat{n} \rangle^3}$$

g³(0) Values

Coherent: g³(0) = 1 (Poissonian statistics at all orders)

Thermal: g³(0) = 6 = 3! (super-bunching)

Fock |n〉: g³(0) = n(n-1)(n-2)/n³ = (1-1/n)(1-2/n)

Single photon: g³(0) = 0 (no three-photon events possible)

2.6.2 Photon Number Distributions and Mandel Q

The full photon number distribution P(n) provides more information than g²(0) alone. Different states with the same mean photon number can have vastly different P(n) shapes:

Coherent State: Poisson Distribution

P(n) = e^-&bar;n;&bar;n;ⁿ/n!. Bell-shaped curve centered at &bar;n; with width σ = √&bar;n;. The distribution is fully characterized by &bar;n; alone. The relative fluctuations decrease as 1/√&bar;n; for large &bar;n; — the shot noise limit.

Thermal State: Geometric (Bose-Einstein) Distribution

P(n) = &bar;n;ⁿ/(1+&bar;n;)ⁿ⁺¹. Monotonically decreasing — the most probable outcome is always n = 0 (no photons). This reflects the wave-like nature of thermal radiation where intensity fluctuations dominate.

Number-Squeezed Light

P(n) is narrower than Poissonian with Q < 0. In the ideal limit (Fock state), P(n) = δ_n,n₀ — zero fluctuations. Experimentally, Q values as negative as -0.6 have been achieved using regulated single-electron tunneling (Coulomb blockade) in quantum dots.

2.6.3 Degree of Non-Classicality

Several measures quantify how non-classical a light state is:

• Wigner function negativity: Volume of negative regions measures non-classicality
• Non-classical depth: Amount of noise that must be added to make P(α) non-negative
• Entanglement potential: Maximum entanglement achievable by sending the state through a beam splitter
• Klyshko criterion: P(n) oscillations that violate classical bounds

2.7 Photon Bunching in Detail

The bunching of thermal photons (g²(0) = 2) can be understood from two complementary perspectives: the wave picture and the particle picture.

2.7.1 Wave Picture: Intensity Fluctuations

Thermal light is a superposition of many randomly phased waves. The total field amplitude is a Gaussian random variable (central limit theorem). The intensity I = |E|² follows an exponential distribution:

$$P(I) = \frac{1}{\langle I \rangle}\exp\!\left(-\frac{I}{\langle I \rangle}\right)$$

The variance of intensity is 〈(ΔI)²〉 = 〈I〉², giving g²(0) = 〈I²〉/〈I〉² = 2. Photons are more likely to be detected in pairs because the intensity spends more time at high values (bunching).

2.7.2 Particle Picture: Boson Statistics

Photons are bosons and obey Bose-Einstein statistics. The probability of detecting two photons in the same mode is enhanced by a factor of 2! = 2 compared to distinguishable particles (the permanent vs determinant of the mode overlap matrix). This is the same boson enhancement that leads to stimulated emission and Bose-Einstein condensation.

2.7.3 Temporal and Spatial Coherence

The g²(τ) function reveals the coherence time of the source:

$$g^{(2)}(\tau) = 1 + |g^{(1)}(\tau)|^2$$

where g¹(τ) is the first-order (field) correlation function. For a Lorentzian spectral line with width Δν, g¹(τ) = exp(-πΔν|τ|), giving:

$$g^{(2)}(\tau) = 1 + e^{-2\pi\Delta\nu|\tau|}$$

The bunching peak has width τ_c = 1/(πΔν) — the coherence time. For spatially extended sources, the spatial coherence area (Van Cittert-Zernike theorem) determines the transverse scale over which bunching is observed.

2.8 Modern Experimental Techniques

Measuring photon statistics requires detectors and techniques matched to the quantum nature of light.

Superconducting Nanowire Single-Photon Detectors (SNSPDs)

Efficiency >98%, timing jitter <20 ps, dark count rate <1 Hz. The workhorse detector for modern quantum optics experiments. Cooled to ~2 K, a thin superconducting nanowire absorbs a photon, creating a resistive hotspot that disrupts the supercurrent.

Transition-Edge Sensors (TES)

Photon-number-resolving capability up to ~20 photons. Operated at the superconducting transition temperature (~100 mK), a small temperature rise from photon absorption produces a large resistance change. Used for direct P(n) measurement and boson sampling verification.

Homodyne and Heterodyne Detection

Balanced homodyne detection measures a single quadrature with shot-noise-limited sensitivity. By varying the local oscillator phase, complete tomographic reconstruction of the quantum state is possible. Heterodyne detection simultaneously measures both quadratures at the cost of one unit of added vacuum noise.

Time-Correlated Single-Photon Counting (TCSPC)

Records the arrival time of each photon with picosecond resolution. Building up the histogram of time differences between consecutive detections yields g²(τ) directly. Essential for characterizing single-photon sources and measuring fluorescence lifetimes.

Applications

Quantum Key Distribution

Single-photon sources with g²(0) ≈ 0 are essential for secure QKD. Any multiphoton emission allows an eavesdropper to split off one photon (photon number splitting attack) without disturbing the signal. True single-photon sources eliminate this vulnerability.

Stellar Intensity Interferometry

The original HBT experiment measured angular diameters of stars (Sirius: 0.0068 arcsec) by correlating intensity fluctuations at two separated telescopes. Modern versions (e.g., VERITAS array) achieve sub-milliarcsecond resolution using Cherenkov telescopes.

Sub-Shot-Noise Measurements

Amplitude-squeezed light enables spectroscopy and absorption measurements with signal-to-noise ratios exceeding the shot noise limit. This is used in biological imaging and precision measurements of weak optical signals.

Quantum Dot Single-Photon Sources

Self-assembled InAs/GaAs quantum dots embedded in photonic crystal cavities achieve g²(0) < 0.01 with >90% indistinguishability. These are leading candidates for scalable photonic quantum computing and quantum networking.

2.9 Photon Pair Sources

Correlated photon pairs are essential for testing quantum mechanics and for quantum information applications. The primary sources are:

Spontaneous Parametric Down-Conversion (SPDC)

A pump photon at frequency ω_p splits into signal and idler photons (ω_s + ω_i = ω_p) in a χ&sup(2); crystal. Energy and momentum conservation (phase matching) determine the output wavelengths and directions. Type-I produces identically polarized pairs; Type-II produces orthogonally polarized pairs, directly creating polarization-entangled states.

Spontaneous Four-Wave Mixing (SFWM)

Two pump photons annihilate to create a signal-idler pair in a χ&sup(3); medium (optical fiber, silicon waveguide, or atomic vapor). Compatible with integrated photonics and telecommunications fiber networks.

Cascade Emission in Atoms

Two-photon cascade emission (e.g., calcium 4p² → 4s4p → 4s²) produces time-energy entangled pairs. Used in the original Aspect experiments (1982) testing Bell's inequality.

Key Equations Summary

Mandel Q Parameter:

$$Q = \frac{\langle (\Delta n)^2 \rangle - \langle n \rangle}{\langle n \rangle}$$

Second-Order Correlation:

$$g^{(2)}(0) = \frac{\langle \hat{n}(\hat{n}-1) \rangle}{\langle \hat{n} \rangle^2}$$

Thermal Photon Variance:

$$\langle (\Delta n)^2 \rangle = \bar{n} + \bar{n}^2 = \bar{n}(1 + \bar{n})$$

Siegert Relation (thermal light):

$$g^{(2)}(\tau) = 1 + |g^{(1)}(\tau)|^2$$

Classical Inequality:

$$g^{(2)}(0) \geq 1 \text{ (classical)}, \qquad g^{(2)}(0) < 1 \text{ (quantum antibunching)}$$

Historical Context

1909 — Einstein: Derived the photon number fluctuation formula showing both wave and particle contributions, foreshadowing quantum optics.

1956 — Hanbury Brown & Twiss: Demonstrated intensity correlations in light from a mercury arc lamp, sparking a fierce debate about whether photons could be "bunched."

1963 — Glauber: Developed the full quantum theory of optical coherence, introducing g¹ and g² correlation functions. Nobel Prize 2005.

1977 — Kimble, Dagenais & Mandel: First observation of photon antibunching from single sodium atoms, proving the quantum nature of light emission.

1985 — Slusher et al.: First generation of squeezed light using four-wave mixing in sodium vapor, achieving noise reduction 0.3 dB below the vacuum level.

2000 — Michler et al.: First demonstration of single-photon emission from a semiconductor quantum dot at room temperature.

Interactive Simulation

This simulation computes and visualizes g²(τ) for thermal, coherent, and single-photon sources, and shows the photon counting histograms for each class.

Photon Statistics: g2(tau) and Counting Distributions

Python

script.py96 lines

import numpy as np
import matplotlib.pyplot as plt
from math import factorial

# --- Parameters ---
n_bar = 5.0       # mean photon number
tau_c = 1.0       # coherence time (arb. units)
n_max = 20        # max photon number

n_vals = np.arange(0, n_max + 1)
tau_vals = np.linspace(-3, 3, 500)

fig, axes = plt.subplots(2, 2, figsize=(12, 10))
fig.patch.set_facecolor("#0a0a0a")
for ax in axes.flat:
    ax.set_facecolor("#111111")

# Plot 1: g2(tau) for different light sources
ax1 = axes[0, 0]
# Thermal: g2(tau) = 1 + exp(-2|tau|/tau_c)
g2_thermal = 1 + np.exp(-2 * np.abs(tau_vals) / tau_c)
g2_coherent = np.ones_like(tau_vals)
# Single photon: g2(tau) = 1 - exp(-|tau|/tau_r) with tau_r = 0.5
tau_r = 0.5
g2_single = 1 - np.exp(-np.abs(tau_vals) / tau_r)

ax1.plot(tau_vals, g2_thermal, color="#f97316", linewidth=2, label="Thermal")
ax1.plot(tau_vals, g2_coherent, color="#a78bfa", linewidth=2, label="Coherent")
ax1.plot(tau_vals, g2_single, color="#c084fc", linewidth=2, label="Single photon")
ax1.axhline(y=1, color="gray", linestyle="--", alpha=0.5)
ax1.set_xlabel("Time delay (tau/tau_c)", color="white")
ax1.set_ylabel("g^(2)(tau)", color="white")
ax1.set_title("Second-Order Correlation Function", color="#c4b5fd", fontsize=14)
ax1.legend(facecolor="#1a1a2e", edgecolor="#7c3aed", labelcolor="white")
ax1.set_ylim(-0.1, 2.5)
ax1.tick_params(colors="white")
for spine in ax1.spines.values():
    spine.set_color("#7c3aed")

# Plot 2: Photon counting distributions
ax2 = axes[0, 1]
P_coherent = np.array([n_bar**n / factorial(n) * np.exp(-n_bar) for n in n_vals])
P_thermal = np.array([n_bar**n / (1 + n_bar)**(n + 1) for n in n_vals])
P_fock = np.zeros(n_max + 1)
P_fock[5] = 1.0  # Fock state |5>

width = 0.25
ax2.bar(n_vals - width, P_coherent, width=width, color="#a78bfa", alpha=0.8, label="Coherent")
ax2.bar(n_vals, P_thermal, width=width, color="#f97316", alpha=0.8, label="Thermal")
ax2.bar(n_vals + width, P_fock, width=width, color="#c084fc", alpha=0.8, label="Fock |5>")
ax2.set_xlabel("Photon number n", color="white")
ax2.set_ylabel("P(n)", color="white")
ax2.set_title("Photon Number Distributions", color="#c4b5fd", fontsize=14)
ax2.legend(facecolor="#1a1a2e", edgecolor="#7c3aed", labelcolor="white")
ax2.tick_params(colors="white")
for spine in ax2.spines.values():
    spine.set_color("#7c3aed")

# Plot 3: Mandel Q parameter vs squeezing
ax3 = axes[1, 0]
r_vals = np.linspace(0, 2, 100)
alpha_vals = [2, 3, 5]
colors = ["#a78bfa", "#c084fc", "#f97316"]
for alpha_val, col in zip(alpha_vals, colors):
    Q_vals = alpha_val**2 * (np.exp(-2 * r_vals) - 1)
    ax3.plot(r_vals, Q_vals, color=col, linewidth=2,
             label="alpha = " + str(alpha_val))
ax3.axhline(y=0, color="gray", linestyle="--", alpha=0.5)
ax3.set_xlabel("Squeezing parameter r", color="white")
ax3.set_ylabel("Mandel Q", color="white")
ax3.set_title("Mandel Q for Amplitude-Squeezed Light", color="#c4b5fd", fontsize=14)
ax3.legend(facecolor="#1a1a2e", edgecolor="#7c3aed", labelcolor="white")
ax3.tick_params(colors="white")
for spine in ax3.spines.values():
    spine.set_color("#7c3aed")

# Plot 4: g2(0) as function of mean photon number for Fock states
ax4 = axes[1, 1]
n_fock = np.arange(1, 21)
g2_fock = 1 - 1.0 / n_fock
ax4.plot(n_fock, g2_fock, "o-", color="#a78bfa", linewidth=2, markersize=6)
ax4.axhline(y=1, color="#f97316", linestyle="--", alpha=0.7, label="Coherent (g2=1)")
ax4.axhline(y=2, color="red", linestyle="--", alpha=0.5, label="Thermal (g2=2)")
ax4.set_xlabel("Fock state photon number n", color="white")
ax4.set_ylabel("g^(2)(0)", color="white")
ax4.set_title("g^(2)(0) for Fock States |n>", color="#c4b5fd", fontsize=14)
ax4.legend(facecolor="#1a1a2e", edgecolor="#7c3aed", labelcolor="white")
ax4.tick_params(colors="white")
for spine in ax4.spines.values():
    spine.set_color("#7c3aed")

plt.tight_layout()
plt.savefig("output.png", dpi=150, bbox_inches="tight", facecolor="#0a0a0a")
plt.show()
print("Plot saved: Photon statistics analysis for different quantum light states.")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Further Topics

Connections to Other Fields

• Condensed matter: HBT-type correlations (noise correlations) in ultracold atoms reveal antibunching of fermions and bunching of bosons released from optical lattices, directly imaging quantum statistics in real space (Felling et al., 2005).
• Nuclear physics: Intensity correlations in pion emission from heavy-ion collisions (HBT interferometry) measure the spatial extent of the nuclear fireball.
• Quantum information: Photon statistics characterization is essential for certifying quantum random number generators, where deviation from Poissonian statistics could compromise security.
• Astrophysics: Modern stellar intensity interferometry (Dravins et al.) uses SPAD arrays on Cherenkov telescopes to measure stellar diameters with sub-milliarcsecond resolution, reviving Hanbury Brown's original astronomical technique.