Chapter 27 Key Takeaways

The Big Picture

The electromagnetic field is a quantum system. Each mode of the field is a quantum harmonic oscillator, and photons are the excitation quanta of these oscillators. Three families of states — Fock states, coherent states, and squeezed states — capture qualitatively different quantum properties of light. The beam splitter, when fed with non-classical inputs, produces genuinely quantum phenomena (entanglement, Hong-Ou-Mandel bunching) that have no classical analogues and form the foundation of photonic quantum technologies.

Key Equations

Field Quantization

The single-mode Hamiltonian: $$\hat{H} = \hbar\omega\left(\hat{a}^\dagger\hat{a} + \frac{1}{2}\right), \qquad [\hat{a}, \hat{a}^\dagger] = 1$$

The electric field operator: $$\hat{E} = \mathcal{E}_0\left(\hat{a}e^{-i\omega t} + \hat{a}^\dagger e^{i\omega t}\right), \qquad \mathcal{E}_0 = \sqrt{\frac{\hbar\omega}{2\epsilon_0 V}}$$

Fock States $|n\rangle$

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

• Mean electric field: $\langle n|\hat{E}|n\rangle = 0$
• Field fluctuations: $\langle n|\hat{E}^2|n\rangle = \mathcal{E}_0^2(2n+1)$
• Photon number uncertainty: $\Delta n = 0$

Coherent States $|\alpha\rangle$

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

• Mean photon number: $\bar{n} = |\alpha|^2$
• Photon number variance: $(\Delta n)^2 = |\alpha|^2 = \bar{n}$ (Poissonian)
• Mean electric field: $\langle\hat{E}\rangle \propto 2|\alpha|\cos(\omega t - \phi)$ (oscillates classically)
• Quadrature uncertainties: $\Delta X_1 = \Delta X_2 = 1/2$ (standard quantum limit)
• Displacement operator: $|\alpha\rangle = \hat{D}(\alpha)|0\rangle = e^{\alpha\hat{a}^\dagger - \alpha^*\hat{a}}|0\rangle$
• Time evolution: $|\alpha(t)\rangle = |\alpha e^{-i\omega t}\rangle$ (rotates in phase space)
• Overlap: $|\langle\beta|\alpha\rangle|^2 = e^{-|\alpha-\beta|^2}$ (not orthogonal, overcomplete)

Squeezed States

Squeezing operator: $\hat{S}(\xi) = \exp[\frac{1}{2}(\xi^*\hat{a}^2 - \xi(\hat{a}^\dagger)^2)]$, where $\xi = re^{i\theta}$

• Squeezed quadrature: $\Delta X_1 = \frac{1}{2}e^{-r}$
• Anti-squeezed quadrature: $\Delta X_2 = \frac{1}{2}e^{r}$
• Product: $\Delta X_1 \cdot \Delta X_2 = 1/4$ (minimum uncertainty preserved)
• Mean photon number (squeezed vacuum): $\bar{n} = \sinh^2 r$
• Photon number parity: only even $n$ for squeezed vacuum

Beam Splitter (50:50)

$$\hat{a}_3 = \frac{1}{\sqrt{2}}(\hat{a}_1 + i\hat{a}_2), \qquad \hat{a}_4 = \frac{1}{\sqrt{2}}(i\hat{a}_1 + \hat{a}_2)$$

General beam splitter ($\theta$ = mixing angle): $$\hat{U}_{\text{BS}}(\theta) = \begin{pmatrix}\cos\theta & i\sin\theta \\ i\sin\theta & \cos\theta\end{pmatrix}$$

• Single photon in: $|1,0\rangle \to \frac{1}{\sqrt{2}}(|1,0\rangle - i|0,1\rangle)$ (entangled)
• Coherent state in: $|\alpha,0\rangle \to |\alpha/\sqrt{2},\; i\alpha/\sqrt{2}\rangle$ (product state)

Hong-Ou-Mandel Effect

$$|1,1\rangle \xrightarrow{\text{50:50 BS}} \frac{-i}{\sqrt{2}}(|2,0\rangle + |0,2\rangle)$$

• Coincidence probability: $P_{1,1} = 0$ (for identical photons)
• HOM dip: $P_{\text{coinc}}(\tau) = \frac{1}{2}(1 - e^{-\tau^2/\tau_c^2})$
• Visibility: $\mathcal{V} = 1$ for perfectly indistinguishable photons

Second-Order Correlation Function

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

Classical bound: $g^{(2)}_{\text{classical}}(0) \geq 1$. Violation proves nonclassical light.

Comparison Table: Three State Families

Conceptual Hierarchy

1. Classical EM field = collection of independent oscillator modes
2. Quantization = promote each mode to a QHO with $[\hat{a}, \hat{a}^\dagger] = 1$
3. Fock states = energy eigenstates = definite photon number
4. Coherent states = eigenstates of $\hat{a}$ = closest to classical waves
5. Squeezed states = minimum uncertainty with asymmetric quadratures = beat the SQL
6. Beam splitter = unitary mode mixing = creates entanglement from non-classical inputs
7. HOM effect = two-photon interference = direct proof of boson statistics
8. Photon statistics ($g^{(2)}$) = classifies light as bunched/random/antibunched

Common Mistakes to Avoid

1. "A Fock state is a single photon." Not necessarily — $|n\rangle$ is a Fock state for any $n \geq 0$. $|1\rangle$ is a single-photon Fock state; $|0\rangle$ is the vacuum Fock state.

2. "A coherent state has a definite number of photons." No — it has a Poisson distribution over all $n$. For large $\bar{n}$, the relative fluctuation $\Delta n/\bar{n} = 1/\sqrt{\bar{n}}$ is small, making it appear definite.

3. "The beam splitter always creates entanglement." Only for non-classical inputs (Fock states). Coherent state inputs produce product-state outputs.

4. "The HOM effect is due to photon-photon interaction." Photons do not interact (in vacuum). The effect arises from destructive interference of quantum amplitudes, enabled by bosonic indistinguishability.

5. "Squeezed states violate the uncertainty principle." They do not — they saturate it ($\Delta X_1 \cdot \Delta X_2 = 1/4$). They redistribute uncertainty between quadratures while respecting the bound.

6. "$g^{(2)}(0) < 1$ is merely unusual." It is impossible for any classical field. It is the definitive proof that the electromagnetic field must be quantized.

What Connects Forward

• Ch 28 (Measurement Problem): Delayed-choice experiments with single photons challenge our understanding of measurement and reality.
• Ch 31 (Path Integrals): Feynman's approach: sum over all photon paths through an interferometer.
• Ch 34 (Second Quantization): The creation/annihilation operators of this chapter are the foundation of quantum field theory for all particles.
• Ch 37 (QFT Preview): Quantum electrodynamics — the full relativistic theory of photons interacting with charged matter.
• Ch 39 (Capstone: Bell Test): Photon polarization entanglement enables the most precise tests of Bell inequalities.