Part I, Chapter 1

Quantum Description of Light

Quantizing the electromagnetic field reveals that light consists of discrete photons, each carrying energy ℏω.

1.1 Quantization of the Electromagnetic Field

The classical electromagnetic field in a cavity of volume V can be decomposed into normal modes. Each mode behaves as a harmonic oscillator with generalized coordinate q and momentum p. The quantization procedure promotes these to operators, yielding the quantum theory of radiation.

1.1.1 Classical Mode Expansion

Consider a cubic cavity with periodic boundary conditions. The vector potential is expanded as:

$$\mathbf{A}(\mathbf{r}, t) = \sum_{\mathbf{k}, \lambda} \sqrt{\frac{\hbar}{2\epsilon_0 \omega_k V}} \left[ a_{\mathbf{k}\lambda} \hat{\boldsymbol{\epsilon}}_{\mathbf{k}\lambda} e^{i\mathbf{k}\cdot\mathbf{r}} + a_{\mathbf{k}\lambda}^* \hat{\boldsymbol{\epsilon}}_{\mathbf{k}\lambda}^* e^{-i\mathbf{k}\cdot\mathbf{r}} \right]$$

where λ labels the two polarization directions and ω_k = c|k|. The Hamiltonian becomes a sum of independent harmonic oscillators.

Derivation 1: Quantized Field Hamiltonian

Step 1. The classical energy of the electromagnetic field is:

$$H = \frac{1}{2} \int_V \left( \epsilon_0 |\mathbf{E}|^2 + \frac{1}{\mu_0} |\mathbf{B}|^2 \right) d^3r$$

Step 2. Substituting the mode expansion for E = -∂A/∂t and B = ∇ × A, orthogonality of the mode functions gives:

$$H = \sum_{\mathbf{k}, \lambda} \hbar \omega_k \left( a_{\mathbf{k}\lambda}^* a_{\mathbf{k}\lambda} + \frac{1}{2} \right)$$

Step 3. Promote amplitudes to operators: a → â, a* → â† with [â_kλ, â_k'λ'†] = δ_kk'δ_λλ'.

Result:

$$\hat{H} = \sum_{\mathbf{k}, \lambda} \hbar \omega_k \left( \hat{a}_{\mathbf{k}\lambda}^\dagger \hat{a}_{\mathbf{k}\lambda} + \frac{1}{2} \right)$$

Each mode has equally spaced energy levels E_n = ℏω(n + 1/2), with the ground state containing 1/2 ℏω of zero-point energy per mode.

1.2 Photon Number States (Fock States)

The eigenstates of the number operator &hat;n; = â†â are the Fock states |n〉, containing exactly n photons. They form a complete orthonormal basis for each field mode.

Properties of Fock States

$$\hat{a}^\dagger |n\rangle = \sqrt{n+1}\, |n+1\rangle, \qquad \hat{a} |n\rangle = \sqrt{n}\, |n-1\rangle$$

$$\hat{n} |n\rangle = n |n\rangle, \qquad \langle n | m \rangle = \delta_{nm}$$

$$|n\rangle = \frac{(\hat{a}^\dagger)^n}{\sqrt{n!}} |0\rangle$$

Figure. Photon number distribution for a Fock state |n=3⟩. The distribution is a single bar at n=3: the photon number is perfectly defined with zero variance (Δn = 0).

Derivation 2: Electric Field Fluctuations in Fock States

Step 1. For a single mode, the electric field operator is:

$$\hat{E} = i\mathcal{E}_0 \left( \hat{a} e^{i\mathbf{k}\cdot\mathbf{r} - i\omega t} - \hat{a}^\dagger e^{-i\mathbf{k}\cdot\mathbf{r} + i\omega t} \right)$$

where the single-photon field amplitude is:

$$\mathcal{E}_0 = \sqrt{\frac{\hbar \omega}{2 \epsilon_0 V}}$$

Step 2. The expectation value of E in a Fock state vanishes: 〈n|Ê|n〉 = 0 (since 〈n|â|n〉 = 0 for all n).

Step 3. The variance is:

$$\langle n | \hat{E}^2 | n \rangle = \mathcal{E}_0^2 (2n + 1)$$

Even the vacuum state |0〉 has nonzero field fluctuations ΔE = &Escr;₀, reflecting the quantum nature of the electromagnetic field.

1.3 Coherent States

Coherent states |α〉 are the quantum states most closely resembling classical electromagnetic waves. They are eigenstates of the annihilation operator: â|α〉 = α|α〉. Laser light is well-described by a coherent state.

Derivation 3: Coherent State Expansion in the Fock Basis

Step 1. Write |α〉 = ∑_n c_n|n〉 and apply â|α〉 = α|α〉:

$$\sum_n c_n \sqrt{n}\, |n-1\rangle = \alpha \sum_n c_n |n\rangle$$

Step 2. Matching coefficients gives the recursion c_n = (α/√n) c_n-1, yielding c_n = (αⁿ/√(n!)) c₀.

Step 3. Normalization 〈α|α〉 = 1 requires |c₀|² e^|α|² = 1.

Result:

$$|\alpha\rangle = e^{-|\alpha|^2/2} \sum_{n=0}^{\infty} \frac{\alpha^n}{\sqrt{n!}} |n\rangle$$

The probability of finding n photons follows a Poisson distribution:

$$P(n) = |\langle n | \alpha \rangle|^2 = \frac{|\alpha|^{2n}}{n!} e^{-|\alpha|^2} = \frac{\bar{n}^n}{n!} e^{-\bar{n}}$$

where the mean photon number is &bar;n; = |α|² and the variance equals the mean: Δn² = &bar;n;.

1.3.1 Displacement Operator

A coherent state can also be generated from the vacuum by the displacement operator:

$$|\alpha\rangle = \hat{D}(\alpha)|0\rangle, \qquad \hat{D}(\alpha) = e^{\alpha \hat{a}^\dagger - \alpha^* \hat{a}}$$

This displaces the vacuum state in phase space by the complex amplitude α, where Re(α) corresponds to the field quadrature X&sub1; and Im(α) to X&sub2;.

1.3.2 Minimum Uncertainty

Coherent states are minimum uncertainty states. Defining quadrature operators:

$$\hat{X}_1 = \frac{1}{2}(\hat{a} + \hat{a}^\dagger), \qquad \hat{X}_2 = \frac{1}{2i}(\hat{a} - \hat{a}^\dagger)$$

we find ΔX&sub1; = ΔX&sub2; = 1/2, so that ΔX&sub1;ΔX&sub2; = 1/4, saturating the Heisenberg uncertainty relation.

1.4 Vacuum Fluctuations

The vacuum state |0〉 is the ground state of the quantized field. Although it contains no photons, it possesses zero-point energy ℏω/2 per mode and nonzero field fluctuations. These vacuum fluctuations have measurable consequences.

Physical Consequences of Vacuum Fluctuations

• Spontaneous emission: An excited atom decays even without incident photons
• Lamb shift: Vacuum fluctuations shift atomic energy levels by ~1 GHz in hydrogen
• Casimir effect: Two conducting plates attract due to modified vacuum modes
• Anomalous magnetic moment: Vacuum fluctuations contribute to g-2 of the electron

Derivation 4: Casimir Force Between Parallel Plates

Step 1. Between two perfectly conducting plates separated by distance d, the allowed wave vectors are quantized: k_z = nπ/d. The vacuum energy per unit area is:

$$\frac{E_{\text{vac}}}{A} = \sum_n \int \frac{d^2 k_\perp}{(2\pi)^2} \frac{\hbar c}{2} \sqrt{k_\perp^2 + \left(\frac{n\pi}{d}\right)^2}$$

Step 2. This sum diverges and must be regularized. Using zeta-function regularization (or an exponential cutoff), the difference between the energy with and without plates yields a finite, d-dependent result.

Step 3. The Euler-Maclaurin formula converts the sum-minus-integral into a Bernoulli number expression. The leading term gives:

Result:

$$\frac{F}{A} = -\frac{\pi^2 \hbar c}{240\, d^4}$$

The negative sign indicates an attractive force. At d = 1 μm, this gives F/A ≈ 1.3 × 10&sup-3; N/m². First precisely measured by Lamoreaux in 1997.

1.5 Squeezed States

While coherent states have equal uncertainty in both quadratures, squeezed states reduce the fluctuations in one quadrature below the vacuum level at the cost of increased fluctuations in the conjugate quadrature, still satisfying the uncertainty principle.

$$|\alpha, \xi\rangle = \hat{D}(\alpha)\hat{S}(\xi)|0\rangle, \qquad \hat{S}(\xi) = \exp\!\left[\frac{1}{2}(\xi^* \hat{a}^2 - \xi (\hat{a}^\dagger)^2)\right]$$

Writing ξ = r e^iθ, the quadrature variances become:

$$\Delta X_1^2 = \frac{1}{4} e^{-2r}, \qquad \Delta X_2^2 = \frac{1}{4} e^{2r}$$

For r > 0, X&sub1; is squeezed below the vacuum level while X&sub2; is anti-squeezed. The uncertainty product remains at the minimum: ΔX&sub1;ΔX&sub2; = 1/4.

Derivation 5: Mean Photon Number in a Squeezed Vacuum

Step 1. The squeezed vacuum is |ξ〉 = &Scirc;(ξ)|0〉. Using the Bogoliubov transformation:

$$\hat{S}^\dagger(\xi) \hat{a} \hat{S}(\xi) = \hat{a} \cosh r - \hat{a}^\dagger e^{i\theta} \sinh r$$

Step 2. The mean photon number is:

$$\bar{n} = \langle \xi | \hat{a}^\dagger \hat{a} | \xi \rangle = \langle 0 | \hat{S}^\dagger \hat{a}^\dagger \hat{a} \hat{S} | 0 \rangle = \sinh^2 r$$

Step 3. The photon number distribution contains only even photon numbers (photons are created in pairs):

$$P(2m) = \frac{(2m)!}{(m!)^2 2^{2m}} \frac{(\tanh r)^{2m}}{\cosh r}, \qquad P(2m+1) = 0$$

This is a signature of the pairwise photon creation by the squeezing operator.

1.6 Phase Space Representations

Quantum states of light can be visualized using quasi-probability distributions in phase space. The most common are the Wigner function W(α), the Husimi Q-function, and the Glauber-Sudarshan P-representation.

1.6.1 Wigner Function

The Wigner function provides a phase-space picture that is the closest quantum analog of a classical probability distribution. For a single mode with density operator ρ:

$$W(\alpha) = \frac{2}{\pi}\text{Tr}\left[\hat{D}^\dagger(\alpha)\hat{\rho}\hat{D}(\alpha)(-1)^{\hat{n}}\right]$$

Key properties of the Wigner function:

• It is real-valued and normalized: ∫ W(α) d²α = 1
• Marginals give correct probability distributions for any quadrature
• It can take negative values — a signature of non-classical states
• Coherent states have a Gaussian Wigner function with width 1/2 in each quadrature
• Fock states have concentric ring structures with alternating positive and negative regions

1.6.2 Wigner Functions of Key States

Vacuum State |0〉

$$W_0(\alpha) = \frac{2}{\pi}e^{-2|\alpha|^2}$$

A symmetric Gaussian centered at the origin with variance 1/4 in each quadrature.

Coherent State |β〉

$$W_\beta(\alpha) = \frac{2}{\pi}e^{-2|\alpha - \beta|^2}$$

Same Gaussian shape as vacuum but displaced to the point β in phase space.

Single-Photon Fock State |1〉

$$W_1(\alpha) = \frac{2}{\pi}(4|\alpha|^2 - 1)e^{-2|\alpha|^2}$$

Negative at the origin: W₁(0) = -2/π. This negativity is a hallmark of non-classicality and has been experimentally measured using homodyne tomography.

Cat State (|α〉 + |-α〉)/√2

The Wigner function shows two Gaussian peaks at ±α plus an interference fringe pattern between them with rapid oscillations. The fringes are spaced by 1/(2|α|) and extend to negative values, encoding the quantum coherence between the two components.

1.6.3 Husimi Q-Function

The Q-function is always non-negative and represents the probability of obtaining outcome α in a heterodyne measurement:

$$Q(\alpha) = \frac{1}{\pi}\langle\alpha|\hat{\rho}|\alpha\rangle \geq 0$$

The Q-function is the convolution of the Wigner function with a vacuum Gaussian, smoothing out the non-classical features. It is always non-negative, unlike the Wigner function.

1.6.4 Glauber-Sudarshan P-Representation

Any density operator can be written as a mixture of coherent states:

$$\hat{\rho} = \int P(\alpha)|\alpha\rangle\langle\alpha|\, d^2\alpha$$

The P-function is more singular than the Wigner function. For thermal light, P(α) is a well-behaved Gaussian. For coherent states, P(α) = δ²(α - α₀). For non-classical states (Fock, squeezed), P(α) becomes negative or more singular than a delta function — the definition of non-classical light in quantum optics.

1.7 Quantum State Tomography

To fully characterize a quantum state of light, one performs quantum state tomography— reconstructing the density matrix or Wigner function from a set of measurements.

1.7.1 Homodyne Detection

Balanced homodyne detection measures a single quadrature X_θ = X&sub1; cosθ + X&sub2; sinθ by interfering the signal with a strong local oscillator at phase θ. The photocurrent difference is proportional to X_θ.

By sweeping θ from 0 to π and collecting the quadrature histograms, one obtains the marginal distributions of the Wigner function. The inverse Radon transform then reconstructs the full Wigner function — this is optical homodyne tomography, first demonstrated by Smithey et al. (1993).

1.7.2 Photon Number Resolving Detection

Transition-edge sensors (TES) and superconducting nanowire detectors can resolve the exact number of photons in a pulse. This enables direct measurement of the photon number distribution P(n) and extraction of g²(0) without relying on correlation measurements.

Modern Quantum State Characterization

• Maximum likelihood estimation: The standard method for density matrix reconstruction from incomplete tomographic data
• Machine learning approaches: Neural networks trained to reconstruct quantum states from measurement data with fewer measurements
• Direct Wigner function measurement: Using photon parity measurements at displaced points in phase space (Lutterbach & Davidovich scheme)
• Quantum process tomography: Characterizing quantum channels by reconstructing the process matrix from input-output state pairs

1.8 Generating Squeezed Light

Squeezed states are produced experimentally through nonlinear optical processes that generate photon pairs correlated in time and frequency.

1.8.1 Optical Parametric Amplification (OPA)

A χ&sup(2); nonlinear crystal pumped by a strong laser at frequency 2ω amplifies vacuum fluctuations at ω, producing squeezed vacuum. The OPA Hamiltonian is:

$$\hat{H}_{\text{OPA}} = i\hbar\chi(\hat{a}^{\dagger 2} - \hat{a}^2)$$

where χ ∝ pump amplitude × χ&sup(2);. This is exactly the squeezing operator S(ξ) with ξ = χt. Modern OPA sources (GEO600, LIGO) achieve >15 dB of squeezing using periodically poled KTP (PPKTP) crystals.

1.8.2 Four-Wave Mixing

In χ&sup(3); media (optical fibers, atomic vapors), the Kerr nonlinearity generates squeezing via self-phase modulation. The first squeezing experiment (Slusher et al., 1985) used four-wave mixing in sodium vapor. Fiber-based squeezers are compact and broadly tunable.

1.8 Photon Number Engineering

Modern quantum optics laboratories routinely create and manipulate specific photon number states. Key techniques include:

Conditional Preparation via SPDC

Spontaneous parametric down-conversion (SPDC) produces photon pairs. Detecting one photon (the herald) conditionally prepares the other in a single-photon state |1〉. By using multiple SPDC sources and photon-number-resolving detection, multi-photon Fock states up to |3〉 have been prepared.

Photon Subtraction and Addition

Tapping off a small fraction of a beam with a beam splitter and detecting a photon implements the annihilation operator â on the remaining field. Applied to a squeezed state, photon subtraction creates highly non-Gaussian states useful for continuous-variable quantum information processing.

Cavity QED Preparation

Single atoms passing through a high-Q microwave cavity can deterministically prepare Fock states. The Haroche group at ENS Paris has prepared states up to |7〉 and observed the progressive decoherence of Schrödinger cat states in real time.

Superconducting Circuit QED

Superconducting qubits coupled to microwave resonators prepare and measure arbitrary Fock states and cat states with exquisite control. The Schoelkopf group at Yale has demonstrated quantum error correction using photon number states in a cavity.

Applications

Gravitational Wave Detection (LIGO)

LIGO uses squeezed vacuum states injected into its dark port to reduce quantum shot noise, improving sensitivity below the standard quantum limit. Since 2019, squeezed light has been routinely used in the O3 and O4 observing runs, increasing the detection range by ~40%.

Quantum Cryptography

Single-photon Fock states and weak coherent pulses form the basis of quantum key distribution (BB84 protocol). The no-cloning theorem guarantees that any eavesdropping attempt introduces detectable errors, enabling information-theoretically secure communication.

Quantum Computing with Photons

Linear optical quantum computing (KLM scheme) uses single photons, beam splitters, and photodetectors to implement universal quantum gates. Boson sampling experiments have demonstrated quantum computational advantage with 50+ photons.

Casimir Effect Engineering

The Casimir force becomes significant in MEMS/NEMS devices at submicron separations. It must be accounted for in nanoscale device design and has been proposed as a mechanism for frictionless bearings and actuators in nanomachinery.

Historical Context

1900 — Planck: Introduced energy quantization E = nhν to resolve the blackbody radiation problem, unknowingly laying the foundation for quantum optics.

1905 — Einstein: Proposed the photon concept to explain the photoelectric effect, establishing that light itself is quantized, not just the emitters.

1927 — Dirac: Developed the first quantum theory of the electromagnetic field, introducing creation and annihilation operators for photons.

1948 — Casimir: Predicted the attractive force between conducting plates due to vacuum fluctuations, later confirmed experimentally by Lamoreaux (1997).

1963 — Glauber: Introduced coherent states and the quantum theory of optical coherence, earning the 2005 Nobel Prize in Physics.

1981 — Caves: Showed that squeezed states could improve interferometric sensitivity beyond the shot noise limit.

Interactive Simulation

This simulation compares the photon number distributions for coherent, thermal, and squeezed vacuum states, and visualizes their Wigner functions in phase space.

Quantum Light: Photon Statistics & Wigner Functions

Python

script.py92 lines

import numpy as np
import matplotlib.pyplot as plt
from math import factorial, cosh, sinh, tanh

# --- Parameters ---
n_bar = 5.0        # mean photon number for coherent and thermal states
r = 1.0            # squeezing parameter
n_max = 20         # max photon number to plot

n_vals = np.arange(0, n_max + 1)

# --- Photon number distributions ---
# Coherent state: Poisson distribution
P_coherent = np.array([n_bar**n / factorial(n) * np.exp(-n_bar) for n in n_vals])

# Thermal state: Bose-Einstein distribution
P_thermal = np.array([n_bar**n / (1 + n_bar)**(n + 1) for n in n_vals])

# Squeezed vacuum: only even photon numbers
P_squeezed = np.zeros(n_max + 1)
for n in n_vals:
    if n % 2 == 0:
        m = n // 2
        P_squeezed[n] = (factorial(2 * m) / (factorial(m)**2 * 2**(2*m))) * (tanh(r))**(2*m) / cosh(r)

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: Photon number distributions
ax1 = axes[0, 0]
width = 0.25
ax1.bar(n_vals - width, P_coherent, width=width, color="#a78bfa", alpha=0.8, label="Coherent")
ax1.bar(n_vals, P_thermal, width=width, color="#f97316", alpha=0.8, label="Thermal")
ax1.bar(n_vals + width, P_squeezed, width=width, color="#c084fc", alpha=0.8, label="Squeezed vac")
ax1.set_xlabel("Photon number n", color="white")
ax1.set_ylabel("P(n)", color="white")
ax1.set_title("Photon Number Distributions", color="#c4b5fd", fontsize=14)
ax1.legend(facecolor="#1a1a2e", edgecolor="#7c3aed", labelcolor="white")
ax1.tick_params(colors="white")
for spine in ax1.spines.values():
    spine.set_color("#7c3aed")

# Plot 2: Wigner function of coherent state
ax2 = axes[0, 1]
x = np.linspace(-5, 5, 200)
p = np.linspace(-5, 5, 200)
X, P = np.meshgrid(x, p)
alpha_re = np.sqrt(n_bar)
W_coherent = (2 / np.pi) * np.exp(-2 * ((X - alpha_re)**2 + P**2))
im2 = ax2.contourf(X, P, W_coherent, levels=30, cmap="magma")
ax2.set_xlabel("X₁", color="white")
ax2.set_ylabel("X₂", color="white")
ax2.set_title("Wigner: Coherent State", color="#c4b5fd", fontsize=14)
ax2.tick_params(colors="white")
for spine in ax2.spines.values():
    spine.set_color("#7c3aed")

# Plot 3: Wigner function of squeezed vacuum
ax3 = axes[1, 0]
W_squeezed_wig = (2 / np.pi) * np.exp(-2 * (X**2 * np.exp(2*r) + P**2 * np.exp(-2*r)))
im3 = ax3.contourf(X, P, W_squeezed_wig, levels=30, cmap="magma")
ax3.set_xlabel("X₁ (squeezed)", color="white")
ax3.set_ylabel("X₂ (anti-squeezed)", color="white")
ax3.set_title("Wigner: Squeezed Vacuum (r=1.0)", color="#c4b5fd", fontsize=14)
ax3.tick_params(colors="white")
for spine in ax3.spines.values():
    spine.set_color("#7c3aed")

# Plot 4: Variance comparison
ax4 = axes[1, 1]
r_vals = np.linspace(0, 2, 100)
var_squeezed = 0.25 * np.exp(-2 * r_vals)
var_anti = 0.25 * np.exp(2 * r_vals)
var_vacuum = 0.25 * np.ones_like(r_vals)
ax4.semilogy(r_vals, var_squeezed, color="#a78bfa", linewidth=2, label="Squeezed quadrature")
ax4.semilogy(r_vals, var_anti, color="#f97316", linewidth=2, label="Anti-squeezed quadrature")
ax4.semilogy(r_vals, var_vacuum, color="gray", linewidth=1, linestyle="--", label="Vacuum level")
ax4.set_xlabel("Squeezing parameter r", color="white")
ax4.set_ylabel("Quadrature variance", color="white")
ax4.set_title("Squeezing vs. Anti-Squeezing", 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 and Wigner functions for quantum light states.")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Key Equations Summary

Quantized Field Hamiltonian:

$$\hat{H} = \sum_{\mathbf{k},\lambda} \hbar\omega_k \left(\hat{a}_{\mathbf{k}\lambda}^\dagger \hat{a}_{\mathbf{k}\lambda} + \frac{1}{2}\right)$$

Coherent State:

$$|\alpha\rangle = e^{-|\alpha|^2/2} \sum_{n=0}^{\infty} \frac{\alpha^n}{\sqrt{n!}} |n\rangle$$

Squeezed Quadratures:

$$\Delta X_1^2 = \frac{1}{4}e^{-2r}, \qquad \Delta X_2^2 = \frac{1}{4}e^{2r}$$

Casimir Force:

$$\frac{F}{A} = -\frac{\pi^2 \hbar c}{240\, d^4}$$

Single-Photon Field Amplitude:

$$\mathcal{E}_0 = \sqrt{\frac{\hbar\omega}{2\epsilon_0 V}}$$

Wigner Function (vacuum):

$$W_0(\alpha) = \frac{2}{\pi}e^{-2|\alpha|^2}$$

Quantum Description of Light

1.1 Quantization of the Electromagnetic Field

1.1.1 Classical Mode Expansion

1.2 Photon Number States (Fock States)

Properties of Fock States

1.3 Coherent States

1.3.1 Displacement Operator

1.3.2 Minimum Uncertainty

1.4 Vacuum Fluctuations

Physical Consequences of Vacuum Fluctuations

1.5 Squeezed States

1.6 Phase Space Representations

1.6.1 Wigner Function

1.6.2 Wigner Functions of Key States

1.6.3 Husimi Q-Function

1.6.4 Glauber-Sudarshan P-Representation

1.7 Quantum State Tomography

1.7.1 Homodyne Detection

1.7.2 Photon Number Resolving Detection

Modern Quantum State Characterization

1.8 Generating Squeezed Light

1.8.1 Optical Parametric Amplification (OPA)

1.8.2 Four-Wave Mixing

1.8 Photon Number Engineering

Applications

Gravitational Wave Detection (LIGO)

Quantum Cryptography

Quantum Computing with Photons

Casimir Effect Engineering

Historical Context

Interactive Simulation

Quantum Light: Photon Statistics & Wigner Functions

Key Equations Summary

Recommended Reading