Chapter 27: Quantum Mechanics of Light — Photons, Coherent States, and Quantum Optics

"All the fifty years of conscious brooding have brought me no closer to the answer to the question: What are light quanta? Of course today every rascal thinks he knows the answer, but he is deluding himself." — Albert Einstein, 1951 (letter to Michele Besso)

Einstein introduced light quanta in 1905 and spent the remaining half-century of his life deeply uneasy about what they actually are. He was right to be uneasy. The photon — the quantum of the electromagnetic field — is one of the most subtle objects in all of physics. It has no mass, no charge, and no rest frame. It can be created and destroyed. Its very existence requires quantum field theory to make sense. And yet the photon is also the most experimentally accessible quantum system we have: we can produce single photons on demand, entangle pairs across continents, and use them to perform computations that no classical device can replicate.

This chapter takes you from the classical electromagnetic field to the quantum theory of light. The central mathematical insight is one you already know: the electromagnetic field, decomposed into modes, is nothing but a collection of quantum harmonic oscillators (Ch 4). Each mode has a ladder of energy eigenstates $|n\rangle$ — the Fock states — and $n$ is the number of photons in that mode. From this single idea, an extraordinary edifice follows: coherent states $|\alpha\rangle$ that describe laser light, squeezed states that beat the standard quantum limit, beam splitters that entangle photons, and the Hong-Ou-Mandel effect, one of the most elegant demonstrations that photons are genuinely quantum.

Learning paths: - 🏃 Streamlined path: Focus on Sections 27.1–27.4 (field quantization and the three state families), then skip to 27.6–27.7 (beam splitter and Hong-Ou-Mandel). These give you the physical core. - 🔬 Deep dive path: Work through everything sequentially. Section 27.5 on squeezed states is essential background for gravitational wave detection (LIGO). Section 27.8 (photon statistics) connects to quantum optics lab work. Section 27.9 leads into photonic quantum computing.

🔗 Connection: This chapter builds directly on the QHO ladder operator formalism from Ch 4 and Ch 8. If $\hat{a}|n\rangle = \sqrt{n}|n-1\rangle$ is not yet second nature, review those chapters first. We also use the density matrix language of Ch 23 in our treatment of photon statistics and decoherence.

27.1 Light as a Quantum Field

The Problem with Classical Light

Maxwell's equations are among the greatest achievements of 19th-century physics. They describe electromagnetic waves propagating through space, predict the speed of light, and unify electricity and magnetism into a single theory. And yet, as we saw in Ch 1, Maxwell's theory fails spectacularly when confronted with three experimental facts:

1. Blackbody radiation — the ultraviolet catastrophe (Planck, 1900)
2. The photoelectric effect — energy arrives in discrete packets (Einstein, 1905)
3. Compton scattering — photons carry momentum $p = \hbar k$ (Compton, 1923)

These phenomena forced physicists to accept that electromagnetic energy is quantized in units of $E = \hbar\omega$. But the early photon concept was essentially a particle picture grafted onto a wave theory — an uncomfortable hybrid. The proper resolution came only with the quantum theory of fields: the electromagnetic field is an operator-valued quantity, and photons emerge as excitations of this quantum field.

Why Field Quantization Is Necessary

You might wonder: is it really necessary to quantize the electromagnetic field? Could we not simply quantize the matter (atoms, electrons) and leave the field classical? This approach — called the semi-classical approximation — actually works surprisingly well for many phenomena, including the photoelectric effect and stimulated emission. But it fails for three crucial predictions:

1. Spontaneous emission — an excited atom in a vacuum decays even though there is no classical field to stimulate it. The quantum vacuum fluctuations of the field are responsible.
2. The Lamb shift — the $2S_{1/2}$ and $2P_{1/2}$ levels of hydrogen, degenerate in the Dirac theory, are split by $\sim 1057$ MHz due to the atom's interaction with vacuum fluctuations.
3. Photon antibunching — certain light sources produce photons that arrive one at a time, with suppressed two-photon coincidences. No classical or semi-classical theory can produce antibunched light.

The message is clear: the electromagnetic field is a quantum system in its own right, not merely a classical backdrop for quantum matter.

📊 By the Numbers: The Lamb shift — 1057.845(9) MHz — was measured by Willis Lamb and Robert Retherford in 1947. Its agreement with the prediction of quantum electrodynamics (QED) to 6 significant figures remains one of the most precise tests of any physical theory.

The Classical Electromagnetic Field in a Box

To quantize the electromagnetic field, we begin with a standard trick: we put the field in a cubic box of side length $L$ with periodic boundary conditions. This is not a physical restriction — we will take $L \to \infty$ at the end — but it discretizes the allowed wavevectors, making the mathematics well-defined.

The electric field inside the box can be expanded in plane-wave modes:

$$\mathbf{E}(\mathbf{r}, t) = \sum_{\mathbf{k}, \lambda} \mathcal{E}_{\mathbf{k}} \left[ \alpha_{\mathbf{k}\lambda}(t)\, \hat{\boldsymbol{\varepsilon}}_{\mathbf{k}\lambda}\, e^{i\mathbf{k}\cdot\mathbf{r}} + \alpha^*_{\mathbf{k}\lambda}(t)\, \hat{\boldsymbol{\varepsilon}}^*_{\mathbf{k}\lambda}\, e^{-i\mathbf{k}\cdot\mathbf{r}} \right]$$

where $\mathbf{k}$ labels the wavevector (with components $k_i = 2\pi n_i/L$ for integer $n_i$), $\lambda = 1, 2$ labels the two polarization directions, $\hat{\boldsymbol{\varepsilon}}_{\mathbf{k}\lambda}$ are the polarization unit vectors (orthogonal to $\mathbf{k}$), and $\mathcal{E}_{\mathbf{k}} = \sqrt{\hbar\omega_k/(2\epsilon_0 V)}$ is a normalization constant with $V = L^3$ and $\omega_k = c|\mathbf{k}|$.

The crucial observation is that each mode $(\mathbf{k}, \lambda)$ obeys the equation of motion:

$$\ddot{\alpha}_{\mathbf{k}\lambda} = -\omega_k^2 \alpha_{\mathbf{k}\lambda}$$

This is the equation of a simple harmonic oscillator with frequency $\omega_k$. The classical electromagnetic field is nothing but a collection of independent harmonic oscillators — one for each mode.

💡 Key Insight: The electromagnetic field is a system of infinitely many coupled oscillators. Each oscillator corresponds to a single mode (characterized by wavevector $\mathbf{k}$ and polarization $\lambda$), and the oscillator's "displacement" is the amplitude of that mode. This is why the QHO from Ch 4 is the skeleton key to quantum optics: quantizing the field means quantizing each oscillator.

Counting Modes

How many modes fit in the box? The allowed wavevectors have components $k_i = 2\pi n_i / L$, so the number of modes with frequencies between $\omega$ and $\omega + d\omega$ is (including both polarizations):

$$\rho(\omega)\,d\omega = \frac{V\omega^2}{\pi^2 c^3}\,d\omega$$

This is the density of states — the same function that appears in Planck's derivation of the blackbody spectrum (Ch 1). Each of these modes, when quantized, becomes an independent QHO. The total energy of the field in thermal equilibrium is:

$$U = \int_0^\infty \hbar\omega\,\bar{n}(\omega)\,\rho(\omega)\,d\omega = \int_0^\infty \frac{\hbar\omega^3}{\pi^2 c^3}\,\frac{1}{e^{\hbar\omega/k_BT}-1}\,d\omega$$

where $\bar{n}(\omega) = 1/(e^{\hbar\omega/k_BT}-1)$ is the Bose-Einstein mean occupation number — the average number of photons in a mode of frequency $\omega$ at temperature $T$. This is precisely the Planck distribution that resolved the ultraviolet catastrophe. The circle is complete: quantizing the electromagnetic field modes as harmonic oscillators, with $E_n = (n+\frac{1}{2})\hbar\omega$ for each mode, automatically produces the correct blackbody spectrum.

📊 By the Numbers: At room temperature ($T = 300\;\text{K}$), the thermal photon occupation number at optical frequencies ($\omega \sim 3\times 10^{15}\;\text{rad/s}$) is $\bar{n} \sim e^{-\hbar\omega/k_BT} \approx e^{-75} \approx 10^{-33}$. Thermal photons at optical frequencies are utterly negligible — the optical vacuum is extremely cold. At microwave frequencies ($\omega \sim 10^{10}\;\text{rad/s}$), $\bar{n} \sim k_BT/\hbar\omega \approx 600$. Microwave cavities in quantum computing must be cooled to millikelvin temperatures to suppress thermal photons.

27.2 Quantizing the Electromagnetic Field: Mode Decomposition

From Classical Amplitudes to Quantum Operators

The transition from classical to quantum field theory follows the canonical quantization recipe: we promote the classical amplitudes to operators and impose commutation relations.

The classical mode amplitude $\alpha_{\mathbf{k}\lambda}$ becomes the annihilation operator $\hat{a}_{\mathbf{k}\lambda}$, and $\alpha^*_{\mathbf{k}\lambda}$ becomes the creation operator $\hat{a}^\dagger_{\mathbf{k}\lambda}$. The canonical commutation relations are:

$$[\hat{a}_{\mathbf{k}\lambda}, \hat{a}^\dagger_{\mathbf{k}'\lambda'}] = \delta_{\mathbf{k}\mathbf{k}'}\delta_{\lambda\lambda'}$$

$$[\hat{a}_{\mathbf{k}\lambda}, \hat{a}_{\mathbf{k}'\lambda'}] = 0, \qquad [\hat{a}^\dagger_{\mathbf{k}\lambda}, \hat{a}^\dagger_{\mathbf{k}'\lambda'}] = 0$$

These are exactly the commutation relations of independent harmonic oscillators (Ch 8). The electric field operator becomes:

$$\hat{\mathbf{E}}(\mathbf{r}, t) = \sum_{\mathbf{k}, \lambda} \mathcal{E}_{\mathbf{k}} \left[ \hat{a}_{\mathbf{k}\lambda}\, \hat{\boldsymbol{\varepsilon}}_{\mathbf{k}\lambda}\, e^{i\mathbf{k}\cdot\mathbf{r}} + \hat{a}^\dagger_{\mathbf{k}\lambda}\, \hat{\boldsymbol{\varepsilon}}^*_{\mathbf{k}\lambda}\, e^{-i\mathbf{k}\cdot\mathbf{r}} \right]$$

The Hamiltonian

The Hamiltonian of the quantized field is the sum of individual oscillator Hamiltonians:

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

The operator $\hat{n}_{\mathbf{k}\lambda} = \hat{a}^\dagger_{\mathbf{k}\lambda}\hat{a}_{\mathbf{k}\lambda}$ is the number operator for mode $(\mathbf{k}, \lambda)$. Its eigenvalue $n_{\mathbf{k}\lambda}$ is a non-negative integer — the number of photons in that mode.

The $\frac{1}{2}$ term is the zero-point energy of each mode. Summing over all modes gives an infinite total vacuum energy — a famous divergence that is handled by normal ordering (subtracting the vacuum energy) or by the renormalization program of QED. For our purposes, we work with the normal-ordered Hamiltonian:

$$\hat{H} = \sum_{\mathbf{k}, \lambda} \hbar\omega_k\, \hat{a}^\dagger_{\mathbf{k}\lambda}\hat{a}_{\mathbf{k}\lambda}$$

🔴 Warning: The infinite vacuum energy is not merely a mathematical nuisance — it has physical consequences. The Casimir effect (an attractive force between parallel conducting plates in vacuum) is a measurable manifestation of vacuum fluctuations. The cosmological constant problem — the 120-order-of-magnitude discrepancy between the predicted vacuum energy and the observed dark energy density — remains one of the deepest unsolved problems in physics.

Single-Mode Simplification

For most of this chapter, we focus on a single mode and suppress the mode labels. The relevant operators and states are:

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

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

$$\hat{n}|n\rangle = n|n\rangle, \qquad n = 0, 1, 2, \ldots$$

This is exactly the QHO of Ch 4, but now $n$ counts photons rather than vibrational quanta. The ground state $|0\rangle$ is the vacuum — not "nothing," but a state with specific, measurable properties (vacuum fluctuations, zero-point energy).

🔗 Connection: Compare with Ch 4, Section 4.5: $\hat{a}|n\rangle = \sqrt{n}|n-1\rangle$, $\hat{a}^\dagger|n\rangle = \sqrt{n+1}|n+1\rangle$. The mathematics is identical. What has changed is the physical interpretation: in Ch 4, $n$ counted vibrational quanta of a mechanical oscillator. Here, $n$ counts photons — quanta of the electromagnetic field. The ladder operators have become creation and annihilation operators for particles.

✅ Checkpoint: Before proceeding, verify you can answer: (1) Why does each mode of the EM field behave as a harmonic oscillator? (2) What do the commutation relations $[\hat{a}, \hat{a}^\dagger] = 1$ physically encode? (3) What is the energy of a state with $n$ photons in a single mode of frequency $\omega$?

27.3 Photon Number States $|n\rangle$ (Fock States)

Definition and Properties

The Fock states (or photon number states) $|n\rangle$ are the energy eigenstates of the single-mode field. They form a complete orthonormal basis for the single-mode Hilbert space:

$$\langle m | n \rangle = \delta_{mn}, \qquad \sum_{n=0}^{\infty} |n\rangle\langle n| = \hat{I}$$

Any single-mode state $|\psi\rangle$ can be expanded as $|\psi\rangle = \sum_n c_n |n\rangle$.

The Fock state $|n\rangle$ is constructed by applying the creation operator $n$ times to the vacuum:

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

The $1/\sqrt{n!}$ is the normalization factor, arising from the successive square-root factors in $\hat{a}^\dagger|n\rangle = \sqrt{n+1}|n+1\rangle$.

🔵 Historical Note: Vladimir Fock introduced these states in the 1930s as part of the "second quantization" formalism for many-particle systems (Ch 34). In quantum optics, they were reintroduced by Roy Glauber in 1963 as part of his Nobel Prize-winning coherence theory. Glauber called them "number states"; the name "Fock states" is standard in both quantum optics and quantum field theory.

Properties of Fock States

Definite photon number. $|n\rangle$ is an eigenstate of $\hat{n}$ with eigenvalue $n$. The uncertainty in photon number is zero: $\Delta n = 0$.

Definite energy. $\hat{H}|n\rangle = \hbar\omega(n + \frac{1}{2})|n\rangle$, so the energy is exactly $(n + \frac{1}{2})\hbar\omega$.

Zero mean electric field. This is a crucial and initially surprising property:

$$\langle n | \hat{E} | n \rangle = 0 \qquad \text{for all } n$$

This follows because $\hat{E} \propto (\hat{a} + \hat{a}^\dagger)$, and both $\langle n|\hat{a}|n\rangle$ and $\langle n|\hat{a}^\dagger|n\rangle$ vanish by orthogonality ($\langle n|n-1\rangle = 0$ and $\langle n|n+1\rangle = 0$).

Nonzero field fluctuations. While the mean field vanishes, the variance does not:

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

where $\mathcal{E}_0 = \sqrt{\hbar\omega/(2\epsilon_0 V)}$ is the electric field per photon. Even the vacuum ($n = 0$) has nonzero field fluctuations: $\langle 0|\hat{E}^2|0\rangle = \mathcal{E}_0^2$. These vacuum fluctuations are responsible for spontaneous emission and the Lamb shift.

⚠️ Common Misconception: A single-photon state $|1\rangle$ does not have a well-defined electric field oscillating sinusoidally. The mean field is zero at all times! A Fock state has definite photon number but completely uncertain phase. This is a number-phase uncertainty relation analogous to position-momentum uncertainty. Only coherent states (Section 27.4) produce something resembling a classical electromagnetic wave.

The Fock State Is Hard to Make

Despite being mathematically simple, Fock states are experimentally challenging to produce. A thermal light source (lamp, sunlight) produces states with a broad photon-number distribution. A laser produces coherent states (Section 27.4), not Fock states. Producing a genuine single-photon state $|1\rangle$ requires specialized quantum optical techniques:

• Heralded single photons — spontaneous parametric down-conversion (SPDC) creates photon pairs; detecting one "heralds" the presence of the other.
• Quantum dots — semiconductor nanostructures that emit exactly one photon per excitation cycle.
• Single atoms/ions in cavities — controlled emission into a single cavity mode.
• Photon subtraction — subtracting a photon from a coherent state via beam splitter and detection.

📊 By the Numbers: The highest-fidelity single-photon sources (as of 2024) achieve purity exceeding 99.5% with quantum dots coupled to optical microcavities, with repetition rates of ~10 GHz. These sources are key enabling technologies for photonic quantum computing.

Matrix Elements and Selection Rules

The matrix elements of the field operators between Fock states are severely constrained. Since $\hat{a}|n\rangle = \sqrt{n}|n-1\rangle$ and $\hat{a}^\dagger|n\rangle = \sqrt{n+1}|n+1\rangle$, the only nonzero matrix elements are:

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

This means the electric field operator $\hat{E} \propto (\hat{a} + \hat{a}^\dagger)$ only connects states that differ by exactly one photon. This is the selection rule for the electromagnetic field: single-photon transitions only (in the dipole approximation). Multiphoton transitions require higher-order processes.

The matrix representation of the annihilation operator in the Fock basis is:

$$\hat{a} = \begin{pmatrix} 0 & \sqrt{1} & 0 & 0 & \cdots \\ 0 & 0 & \sqrt{2} & 0 & \cdots \\ 0 & 0 & 0 & \sqrt{3} & \cdots \\ \vdots & & & & \ddots \end{pmatrix}$$

This infinite-dimensional matrix is the "ladder" structure we met in Ch 8. In computational work, we truncate the Hilbert space at $n_{\max}$ photons, keeping the first $n_{\max}$ rows and columns.

The Number-Phase Uncertainty Relation

A Fock state has definite photon number ($\Delta n = 0$) but completely undefined phase. This suggests a number-phase uncertainty relation analogous to $\Delta x \cdot \Delta p \geq \hbar/2$. The precise formulation is subtle because there is no Hermitian operator corresponding to the optical phase (the Susskind-Glogower problem), but the physical content is clear:

$$\Delta n \cdot \Delta\phi \gtrsim 1$$

A Fock state, with $\Delta n = 0$, has $\Delta\phi = \pi$ (completely random phase — the probability is uniform on $[0, 2\pi)$). A coherent state, with $\Delta n = \sqrt{\bar{n}}$ and $\Delta\phi \approx 1/(2\sqrt{\bar{n}})$, approximately saturates this bound. This number-phase complementarity is one of the deepest features of quantum optics: you can know how many photons you have or what phase they carry, but not both simultaneously.

⚖️ Interpretation: The absence of a well-defined phase for Fock states has profound implications for the interpretation of quantum mechanics. In the Copenhagen interpretation, the phase simply "does not exist" for a number state. In the many-worlds interpretation, the phase takes all values simultaneously. In Bohmian mechanics, there is a definite but unknowable phase. All interpretations agree on the observable predictions: $\langle\hat{E}\rangle = 0$ and no interference fringes for a single-mode Fock state.

Multi-Mode Fock States

For multiple modes, the overall state is a tensor product:

$$|n_1, n_2, n_3, \ldots\rangle = |n_1\rangle \otimes |n_2\rangle \otimes |n_3\rangle \otimes \cdots$$

The multi-mode vacuum is $|0, 0, 0, \ldots\rangle = |0\rangle^{\otimes\infty}$, and the creation operator $\hat{a}^\dagger_{\mathbf{k}\lambda}$ adds a photon to mode $(\mathbf{k}, \lambda)$ only.

27.4 Coherent States $|\alpha\rangle$ (Glauber States)

Motivation: What State Describes a Laser?

Fock states are the energy eigenstates, but they are terrible models for laser light. A laser produces a nearly classical electromagnetic wave — an oscillating electric field with a well-defined amplitude and phase. A Fock state, by contrast, has zero mean electric field and completely uncertain phase. We need a different state.

The coherent state $|\alpha\rangle$, introduced by Roy Glauber in 1963, is the quantum state that most closely resembles a classical electromagnetic wave. It is an eigenstate of the annihilation operator:

$$\hat{a}|\alpha\rangle = \alpha|\alpha\rangle$$

where $\alpha$ is a complex number whose modulus gives the field amplitude and whose phase gives the oscillation phase.

💡 Key Insight: The annihilation operator $\hat{a}$ is not Hermitian, so its eigenvalues $\alpha$ can be any complex number — not just real numbers as for observables. The set of coherent states $\{|\alpha\rangle : \alpha \in \mathbb{C}\}$ forms a continuous family, in contrast to the discrete Fock states $\{|n\rangle : n \in \mathbb{Z}_{\geq 0}\}$.

Expansion in the Fock Basis

To find $|\alpha\rangle$ explicitly, expand in the Fock basis: $|\alpha\rangle = \sum_n c_n |n\rangle$. The eigenvalue equation $\hat{a}|\alpha\rangle = \alpha|\alpha\rangle$ gives:

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

Relabeling indices, this yields the recursion $c_{n+1} = \alpha c_n/\sqrt{n+1}$, solved by:

$$c_n = \frac{\alpha^n}{\sqrt{n!}}\,c_0$$

Normalization ($\sum_n |c_n|^2 = 1$) gives $|c_0|^2 e^{|\alpha|^2} = 1$, so $c_0 = e^{-|\alpha|^2/2}$ (choosing $c_0$ real and positive). The result is:

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

Properties of Coherent States

1. Photon number distribution is Poissonian. The probability of finding $n$ photons in $|\alpha\rangle$ is:

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

This is a Poisson distribution with mean $\bar{n} = |\alpha|^2$ and variance $(\Delta n)^2 = |\alpha|^2 = \bar{n}$.

The signal-to-noise ratio is $\bar{n}/\Delta n = \sqrt{\bar{n}}$. For a typical laser with $\bar{n} \sim 10^{12}$ photons per mode, $\Delta n/\bar{n} \sim 10^{-6}$ — the photon number is essentially definite, and the light appears classical. For $\bar{n} \sim 1$, the fluctuations are of order unity and the quantum nature is manifest.

2. Nonzero mean electric field. Unlike Fock states, coherent states have a nonzero expectation value for the electric field:

$$\langle\alpha|\hat{E}|\alpha\rangle \propto \alpha e^{-i\omega t} + \alpha^* e^{i\omega t} = 2|\alpha|\cos(\omega t - \phi)$$

where $\alpha = |\alpha|e^{i\phi}$. This oscillates sinusoidally — just like a classical wave.

3. Minimum uncertainty. Coherent states saturate the Heisenberg uncertainty relation for the field quadratures $\hat{X}_1 = (\hat{a} + \hat{a}^\dagger)/2$ and $\hat{X}_2 = (\hat{a} - \hat{a}^\dagger)/(2i)$:

$$\Delta X_1 = \Delta X_2 = \frac{1}{2}$$

The uncertainty is symmetric — equal in both quadratures — and takes the minimum value allowed by quantum mechanics. This is the standard quantum limit (SQL).

4. Displacement operator representation. The coherent state can be written as:

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

The displacement operator $\hat{D}(\alpha)$ shifts the vacuum state in phase space by the complex amplitude $\alpha$. This makes the physical picture transparent: a coherent state is a displaced vacuum.

5. Overcompleteness. Coherent states are not orthogonal:

$$\langle\beta|\alpha\rangle = e^{-|\alpha|^2/2 - |\beta|^2/2 + \beta^*\alpha}$$

$$|\langle\beta|\alpha\rangle|^2 = e^{-|\alpha - \beta|^2}$$

The overlap is exponentially small when $|\alpha - \beta|$ is large, but never exactly zero. Nevertheless, coherent states satisfy a completeness relation:

$$\frac{1}{\pi}\int |\alpha\rangle\langle\alpha|\,d^2\alpha = \hat{I}$$

They form an overcomplete basis — there are "too many" of them, and any coherent state can be expanded in terms of others. This overcompleteness, far from being a nuisance, is the foundation of the Glauber-Sudarshan P-representation and the entire modern theory of quantum coherence.

6. Time evolution. Under free evolution, $|\alpha\rangle$ evolves as:

$$e^{-i\hat{H}t/\hbar}|\alpha\rangle = e^{-i\omega t/2}|\alpha e^{-i\omega t}\rangle$$

Up to an overall phase, the coherent state remains coherent — $\alpha$ simply rotates in the complex plane at frequency $\omega$. The state traces out a circle in phase space, exactly as a classical oscillator does. No other quantum state has this property.

7. Coherent states as Gaussian states. In the position representation, the coherent state wavefunction is a Gaussian centered at $\langle\hat{x}\rangle = \sqrt{2\hbar/(m\omega)}\,\text{Re}(\alpha)$:

$$\psi_\alpha(x) = \left(\frac{m\omega}{\pi\hbar}\right)^{1/4}\exp\left[-\frac{m\omega}{2\hbar}(x - x_0)^2 + ip_0 x/\hbar - i\phi_0\right]$$

where $x_0 = \sqrt{2\hbar/(m\omega)}\,\text{Re}(\alpha)$ and $p_0 = \sqrt{2m\omega\hbar}\,\text{Im}(\alpha)$. This is the same Gaussian wave packet from Ch 4, but now displaced from the origin. As time evolves, the center of the Gaussian oscillates classically while the width remains constant — no spreading, no distortion. This is why the coherent state is "the most classical quantum state."

Phase-Space Picture: The Wigner Function

The coherent state has a particularly simple representation in quantum phase space. The Wigner function $W(X_1, X_2)$ is a quasi-probability distribution on the quadrature phase space $(X_1, X_2)$. For a coherent state $|\alpha\rangle$ with $\alpha = X_{10} + iX_{20}$:

$$W_{\text{coh}}(X_1, X_2) = \frac{2}{\pi}\exp\!\left[-2(X_1 - X_{10})^2 - 2(X_2 - X_{20})^2\right]$$

This is a circular Gaussian centered at $(\text{Re}\,\alpha, \text{Im}\,\alpha)$ with equal widths in both quadratures. The vacuum $|0\rangle$ is just the special case $\alpha = 0$: a circular Gaussian centered at the origin.

For comparison: - Fock state $|n\rangle$: The Wigner function has concentric rings (like a target pattern) with $n$ negative regions. The negativity of the Wigner function is a hallmark of non-classical states. - Squeezed state: The Wigner function is an elliptical Gaussian — narrow in one direction, broad in the other. - Thermal state: A broad Gaussian centered at the origin, wider than the vacuum.

The Wigner function is invaluable for visualizing quantum states because it is the closest quantum analog of a classical phase-space distribution. However, unlike classical distributions, it can take negative values — and it is precisely the states with negative Wigner functions that exhibit the most dramatic quantum effects.

🔵 Historical Note: Roy Glauber shared the 2005 Nobel Prize in Physics "for his contribution to the quantum theory of optical coherence." His 1963 papers established the coherent state formalism and the hierarchy of correlation functions ($g^{(1)}$, $g^{(2)}$, etc.) that define quantum coherence. The work was motivated by the invention of the laser in 1960 — physicists needed a quantum theory of laser light, and Glauber provided it.

✅ Checkpoint: A He-Ne laser operating at 632.8 nm has an output power of 5 mW in a single mode. Estimate the mean photon number $\bar{n}$ and the relative fluctuation $\Delta n / \bar{n}$. (Hint: Power = $\bar{n}\hbar\omega / T$, where $T$ is the observation time. For a coherent state, $\Delta n = \sqrt{\bar{n}}$.)

27.5 Squeezed States

Beyond the Standard Quantum Limit

Coherent states have symmetric uncertainties: $\Delta X_1 = \Delta X_2 = 1/2$. But the Heisenberg relation only requires $\Delta X_1 \cdot \Delta X_2 \geq 1/4$. What if we could reduce the uncertainty in one quadrature at the expense of increasing it in the other? This is exactly what squeezed states accomplish.

A squeezed vacuum state is generated by the squeezing operator:

$$\hat{S}(\xi) = \exp\!\left[\frac{1}{2}\left(\xi^* \hat{a}^2 - \xi (\hat{a}^\dagger)^2\right)\right]$$

where $\xi = r e^{i\theta}$ is the squeezing parameter. The real part $r \geq 0$ controls the degree of squeezing, and $\theta$ controls the squeezing direction in phase space.

For the squeezed vacuum $|\xi\rangle = \hat{S}(\xi)|0\rangle$, the quadrature uncertainties become:

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

(for $\theta = 0$). The uncertainty in $X_1$ is squeezed below the vacuum level by the factor $e^{-r}$, while the uncertainty in $X_2$ is amplified by $e^{r}$. The product $\Delta X_1 \cdot \Delta X_2 = 1/4$ still saturates the Heisenberg bound.

More generally, a squeezed coherent state is:

$$|\alpha, \xi\rangle = \hat{D}(\alpha)\hat{S}(\xi)|0\rangle$$

This has both a displaced center (like a coherent state) and anisotropic uncertainties (squeezing).

Photon Statistics of Squeezed States

The squeezed vacuum $|\xi\rangle$ has remarkable photon statistics:

• Mean photon number: $\bar{n} = \sinh^2 r$. Even though it is called a "vacuum" state, squeezing creates photons from the vacuum — the mean photon number grows exponentially with $r$.
• Photon number distribution: Only even photon numbers $|0\rangle$, $|2\rangle$, $|4\rangle$, ... appear (for $\theta = 0$). The squeezed vacuum contains photon pairs, reflecting its generation via parametric processes that create photons in pairs.
• Super-Poissonian statistics: The variance $(\Delta n)^2 > \bar{n}$, so the photon number fluctuations are larger than Poissonian.

The Fock-basis expansion of the squeezed vacuum is:

$$|\xi\rangle = \frac{1}{\sqrt{\cosh r}} \sum_{n=0}^{\infty} \frac{\sqrt{(2n)!}}{2^n n!} (-e^{i\theta}\tanh r)^n |2n\rangle$$

LIGO and Gravitational Wave Detection

The most spectacular application of squeezed light is in gravitational wave detection. LIGO (Laser Interferometer Gravitational-Wave Observatory) is a Michelson interferometer with 4 km arms that measures displacements of order $10^{-19}$ m — a fraction of the proton diameter.

At low laser power, the sensitivity is limited by shot noise — photon counting fluctuations that scale as $1/\sqrt{\bar{n}}$. At high laser power, the sensitivity is limited by radiation pressure noise — random momentum kicks from photon number fluctuations that shake the mirrors. These two noise sources combine to give the standard quantum limit (SQL).

Squeezed light breaks the SQL. By injecting squeezed vacuum into the unused port of the interferometer, LIGO reduces the phase-quadrature fluctuations (which determine position sensitivity) at the expense of amplitude-quadrature fluctuations (which contribute radiation pressure noise at low frequencies). Since 2019, both LIGO detectors inject 6 dB of squeezing, improving the detection rate of binary neutron star mergers by approximately 50%.

📊 By the Numbers: LIGO achieves displacement sensitivity of $\sim 3 \times 10^{-20}\;\text{m}/\sqrt{\text{Hz}}$ around 100 Hz, corresponding to strain $h \sim 10^{-23}/\sqrt{\text{Hz}}$. Frequency-dependent squeezing (implemented in the O4 observing run, 2023–2025) provides broadband quantum noise reduction by rotating the squeezing ellipse as a function of frequency using a 300 m filter cavity.

🧪 Experiment: The first observation of squeezed light was achieved by Slusher et al. in 1985 at Bell Labs using four-wave mixing in sodium vapor, achieving 0.3 dB of squeezing (7% noise reduction). Modern sources using optical parametric oscillators (OPOs) routinely achieve 15 dB of squeezing (97% noise reduction in one quadrature).

How Squeezed Light Is Generated

Squeezed light is produced by parametric processes in nonlinear optical media. The most common method uses an optical parametric amplifier (OPA) or optical parametric oscillator (OPO), in which a strong pump beam at frequency $2\omega$ passes through a nonlinear crystal (e.g., PPKTP — periodically poled potassium titanyl phosphate) and generates pairs of photons at frequency $\omega$.

The Hamiltonian for parametric down-conversion is:

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

where $\chi$ is proportional to the nonlinear susceptibility $\chi^{(2)}$ and the pump amplitude. This is precisely the generator of the squeezing operator $\hat{S}(\xi)$, with $r = \chi t$ (squeezing parameter proportional to interaction time). The physics is transparent: the nonlinear process creates photon pairs ($\hat{a}^{\dagger 2}$), correlating the photon number fluctuations and redistributing the quantum noise between quadratures.

The key experimental parameters are: - Pump power — higher pump power gives more squeezing, up to a limit set by the damage threshold of the crystal and the onset of higher-order nonlinearities. - Crystal length — longer crystals provide more interaction time but introduce phase-matching bandwidth limitations. - Cavity enhancement — placing the crystal inside an optical cavity (the OPO configuration) dramatically increases the effective interaction strength. - Loss — any optical loss degrades squeezing. With 1% loss, the maximum achievable squeezing is limited to about 20 dB regardless of the source quality.

The current world record for squeezed light generation is 15 dB of squeezing, achieved at the Max Planck Institute for Gravitational Physics (Hannover) in 2016. This corresponds to $r \approx 1.73$ and a quadrature noise reduction by a factor of $e^{-1.73} \approx 0.18$ — the squeezed quadrature has only 18% of the vacuum noise.

27.6 The Beam Splitter: A Quantum Optical Device

Classical Description

A lossless beam splitter is an optical element that splits an incident beam into two output beams. Classically, if a beam with amplitude $E_{\text{in}}$ hits a 50:50 beam splitter, the transmitted beam has amplitude $E_{\text{in}}/\sqrt{2}$ and the reflected beam has amplitude $iE_{\text{in}}/\sqrt{2}$, where the factor of $i$ accounts for the $\pi/2$ phase shift upon reflection.

For a general beam splitter with reflectivity $R$ and transmissivity $T = 1 - R$, the input-output relation for two input fields $(E_1, E_2)$ is:

$$\begin{pmatrix} E_3 \\ E_4 \end{pmatrix} = \begin{pmatrix} t & r \\ r' & t' \end{pmatrix} \begin{pmatrix} E_1 \\ E_2 \end{pmatrix}$$

where energy conservation requires $|t|^2 + |r|^2 = 1$ and the matrix must be unitary.

Quantum Description

In the quantum theory, the classical field amplitudes become operators. The beam splitter transformation for a 50:50 beam splitter is:

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

or in matrix form:

$$\begin{pmatrix} \hat{a}_3 \\ \hat{a}_4 \end{pmatrix} = \hat{U}_{\text{BS}} \begin{pmatrix} \hat{a}_1 \\ \hat{a}_2 \end{pmatrix} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & i \\ i & 1 \end{pmatrix} \begin{pmatrix} \hat{a}_1 \\ \hat{a}_2 \end{pmatrix}$$

You can verify that $\hat{U}_{\text{BS}}$ is unitary: $\hat{U}_{\text{BS}}^\dagger \hat{U}_{\text{BS}} = \hat{I}$, which guarantees that the output operators satisfy the same commutation relations as the input operators: $[\hat{a}_3, \hat{a}_3^\dagger] = [\hat{a}_4, \hat{a}_4^\dagger] = 1$ and $[\hat{a}_3, \hat{a}_4^\dagger] = 0$.

For a general beam splitter with transmissivity amplitude $t = \cos\theta$ and reflectivity amplitude $r = i\sin\theta$ (where $\theta$ is the mixing angle, with $\theta = \pi/4$ for 50:50):

$$\hat{U}_{\text{BS}}(\theta) = \begin{pmatrix} \cos\theta & i\sin\theta \\ i\sin\theta & \cos\theta \end{pmatrix}$$

Single Photon at a Beam Splitter

What happens when a single photon enters one port of a 50:50 beam splitter, with vacuum at the other port? The input state is $|1\rangle_1|0\rangle_2$ (one photon in mode 1, vacuum in mode 2). Using the inverse beam splitter transformation:

$$\hat{a}_1^\dagger = \frac{1}{\sqrt{2}}(\hat{a}_3^\dagger - i\hat{a}_4^\dagger)$$

the output state is:

$$|1\rangle_1|0\rangle_2 = \hat{a}_1^\dagger|0\rangle_1|0\rangle_2 \to \frac{1}{\sqrt{2}}(\hat{a}_3^\dagger - i\hat{a}_4^\dagger)|0\rangle_3|0\rangle_4 = \frac{1}{\sqrt{2}}(|1\rangle_3|0\rangle_4 - i|0\rangle_3|1\rangle_4)$$

The photon is in a superposition of being in output mode 3 and output mode 4. This is not classical beam splitting — the photon is not "split" into two half-photons. If you place detectors at both outputs, you will never get coincident clicks. Each detection event finds the whole photon in one output or the other, with 50% probability each. This was verified experimentally by Grangier, Roger, and Aspect in 1986.

💡 Key Insight: A beam splitter acting on a single photon creates entanglement between the two output modes. The state $\frac{1}{\sqrt{2}}(|1,0\rangle - i|0,1\rangle)$ is an entangled state of two spatial modes. This is the simplest example of path entanglement — the foundation of many quantum optical protocols.

Coherent State at a Beam Splitter

What about a coherent state? If $|\alpha\rangle$ enters port 1 with vacuum at port 2, the output is:

$$|\alpha\rangle_1|0\rangle_2 \to |\alpha/\sqrt{2}\rangle_3 |i\alpha/\sqrt{2}\rangle_4$$

The output is a product of two coherent states — not entangled! This is a dramatic contrast with the single-photon case. The beam splitter simply splits the coherent amplitude in two, preserving the product structure. Classical light (well-described by coherent states) does not become entangled at a beam splitter; single photons do.

⚠️ Common Misconception: Students sometimes believe that a beam splitter always creates entanglement. It does not. The key is the input state. Coherent state inputs produce product-state outputs. Fock state inputs (including single photons) produce entangled outputs. The beam splitter is a passive linear device — it creates entanglement only when the input has non-classical photon-number correlations.

Worked Example: Two Photons in One Port

Let us work through the case $|2\rangle_1|0\rangle_2$ in detail, as it illustrates the method and connects to the HOM calculation in the next section.

The input state is:

$$|2,0\rangle = \frac{(\hat{a}_1^\dagger)^2}{\sqrt{2}}|0,0\rangle$$

Using $\hat{a}_1^\dagger = \frac{1}{\sqrt{2}}(\hat{a}_3^\dagger - i\hat{a}_4^\dagger)$:

$$(\hat{a}_1^\dagger)^2 = \frac{1}{2}(\hat{a}_3^\dagger - i\hat{a}_4^\dagger)^2 = \frac{1}{2}\left[(\hat{a}_3^\dagger)^2 - 2i\hat{a}_3^\dagger\hat{a}_4^\dagger - (\hat{a}_4^\dagger)^2\right]$$

So:

$$|2,0\rangle \to \frac{1}{2\sqrt{2}}\left[(\hat{a}_3^\dagger)^2 - 2i\hat{a}_3^\dagger\hat{a}_4^\dagger - (\hat{a}_4^\dagger)^2\right]|0,0\rangle$$

$$= \frac{1}{2\sqrt{2}}\left[\sqrt{2}|2,0\rangle - 2i|1,1\rangle - \sqrt{2}|0,2\rangle\right]$$

$$= \frac{1}{2}|2,0\rangle - \frac{i}{\sqrt{2}}|1,1\rangle - \frac{1}{2}|0,2\rangle$$

The detection probabilities are:

Notice that the probability of a coincidence ($|1,1\rangle$) is $1/2$ — the same as the classical prediction. There is no HOM suppression! This is because the two photons entered the same port, so they are already correlated. The HOM effect requires one photon per input port.

The Beam Splitter as a Rotation

The beam splitter transformation has an elegant group-theoretic interpretation. Define the Schwinger angular momentum operators:

$$\hat{J}_x = \frac{1}{2}(\hat{a}_1^\dagger\hat{a}_2 + \hat{a}_2^\dagger\hat{a}_1), \quad \hat{J}_y = \frac{1}{2i}(\hat{a}_1^\dagger\hat{a}_2 - \hat{a}_2^\dagger\hat{a}_1), \quad \hat{J}_z = \frac{1}{2}(\hat{a}_1^\dagger\hat{a}_1 - \hat{a}_2^\dagger\hat{a}_2)$$

These satisfy the SU(2) algebra $[\hat{J}_i, \hat{J}_j] = i\epsilon_{ijk}\hat{J}_k$ (Ch 12). The beam splitter transformation is a rotation about $\hat{J}_x$:

$$\hat{U}_{\text{BS}}(\theta) = e^{-i2\theta\hat{J}_x}$$

For a 50:50 beam splitter, $\theta = \pi/4$, giving a $\pi/2$ rotation. This connection between beam splitters and rotations on the Bloch-Poincare sphere is the foundation of the SU(2) interferometer theory used in quantum metrology.

✅ Checkpoint: Verify the beam splitter output for input $|2\rangle_1|0\rangle_2$ using the worked example above. Then try $|3\rangle_1|0\rangle_2$ — you should find four terms with total photon number conserved at $N = 3$.

27.7 The Hong-Ou-Mandel Effect: Photon Bunching

Setup

The Hong-Ou-Mandel (HOM) effect is one of the most beautiful and conceptually revealing experiments in quantum optics. First demonstrated by Chung Ki Hong, Zhe Yu Ou, and Leonard Mandel in 1987, it reveals a phenomenon with absolutely no classical analogue.

The setup is simple: send one photon into each input port of a 50:50 beam splitter. The input state is $|1\rangle_1|1\rangle_2$ — one photon in mode 1 and one photon in mode 2.

Calculation

We need to express $|1\rangle_1|1\rangle_2 = \hat{a}_1^\dagger\hat{a}_2^\dagger|0,0\rangle$ in terms of the output operators. Using the inverse transformation:

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

The product is:

$$\hat{a}_1^\dagger\hat{a}_2^\dagger = \frac{1}{2}(\hat{a}_3^\dagger - i\hat{a}_4^\dagger)(-i\hat{a}_3^\dagger + \hat{a}_4^\dagger)$$

Expanding:

$$= \frac{1}{2}\left[-i(\hat{a}_3^\dagger)^2 + \hat{a}_3^\dagger\hat{a}_4^\dagger + i^2\hat{a}_4^\dagger\hat{a}_3^\dagger - i(\hat{a}_4^\dagger)^2\right]$$

Since creation operators for different modes commute ($\hat{a}_3^\dagger\hat{a}_4^\dagger = \hat{a}_4^\dagger\hat{a}_3^\dagger$), the two cross terms are:

$$\hat{a}_3^\dagger\hat{a}_4^\dagger - \hat{a}_4^\dagger\hat{a}_3^\dagger = 0$$

The cross terms cancel exactly! We are left with:

$$\hat{a}_1^\dagger\hat{a}_2^\dagger = \frac{1}{2}\left[-i(\hat{a}_3^\dagger)^2 - i(\hat{a}_4^\dagger)^2\right] = \frac{-i}{2}\left[(\hat{a}_3^\dagger)^2 + (\hat{a}_4^\dagger)^2\right]$$

Applying this to the vacuum:

$$|1,1\rangle_{\text{in}} \to \frac{-i}{2}\left[(\hat{a}_3^\dagger)^2 + (\hat{a}_4^\dagger)^2\right]|0,0\rangle = \frac{-i}{\sqrt{2}}\left[|2,0\rangle + |0,2\rangle\right]$$

The Result

$$\boxed{|1\rangle_1|1\rangle_2 \;\xrightarrow{\text{50:50 BS}}\; \frac{-i}{\sqrt{2}}\left(|2\rangle_3|0\rangle_4 + |0\rangle_3|2\rangle_4\right)}$$

Both photons exit the same port. The probability of finding one photon in each output ($|1,1\rangle$) is exactly zero. The two photons always "bunch" — they leave together through port 3 or port 4, each with probability 1/2.

Why Does This Happen?

There are two paths by which one photon could end up in each output: 1. Both photons are transmitted (neither reflects) 2. Both photons are reflected (neither transmits)

In classical probability, these are independent events with probability $1/4$ each, giving a total coincidence probability of $1/2$. But in quantum mechanics, we must add amplitudes, not probabilities. The amplitude for both transmitting is $(1/\sqrt{2})^2 = 1/2$, and the amplitude for both reflecting is $(i/\sqrt{2})^2 = -1/2$. The sum is $1/2 + (-1/2) = 0$. Destructive interference between the two quantum paths eliminates the coincidence signal entirely.

This destructive interference requires the two photons to be indistinguishable — identical in every degree of freedom: frequency, polarization, spatial mode, and arrival time. If the photons are distinguishable in any way (different frequencies, different arrival times, orthogonal polarizations), the interference is degraded and the coincidence signal reappears.

💡 Key Insight: The HOM effect is a direct consequence of photon indistinguishability — the fact that photons are bosons. For fermions (which obey anti-commutation relations), the same calculation gives the opposite result: fermions always emerge in different outputs ("antibunching"). The HOM dip is the most direct experimental test of the quantum statistics of light.

🧪 Experiment: The original 1987 HOM experiment used photon pairs from spontaneous parametric down-conversion (SPDC) in a KDP crystal. By varying the path-length difference between the two photons (introducing a relative time delay $\tau$), Hong, Ou, and Mandel measured the coincidence rate as a function of $\tau$. At $\tau = 0$ (perfect temporal overlap), the coincidence rate dropped to nearly zero — the famous "HOM dip." The width of the dip ($\sim 100$ fs) was determined by the coherence time of the down-converted photons. Visibilities exceeding 99% have been achieved in modern experiments.

The HOM Dip

The coincidence probability as a function of the time delay $\tau$ between the two photons is:

$$P_{\text{coinc}}(\tau) = \frac{1}{2}\left(1 - e^{-\tau^2/\tau_c^2}\right)$$

where $\tau_c$ is the coherence time (inversely related to the bandwidth of the photons). At $\tau = 0$, $P_{\text{coinc}} = 0$ (perfect destructive interference). For $|\tau| \gg \tau_c$, $P_{\text{coinc}} \to 1/2$ (photons distinguishable, classical behavior restored). The characteristic dip shape — the "HOM dip" — is the hallmark of two-photon quantum interference.

The visibility of the HOM dip is:

$$\mathcal{V} = \frac{P_{\text{coinc}}(\infty) - P_{\text{coinc}}(0)}{P_{\text{coinc}}(\infty)} = 1$$

for perfectly indistinguishable photons. Any reduction in visibility indicates partial distinguishability and is used as a diagnostic for photon source quality.

Fermions vs. Bosons at a Beam Splitter

The HOM effect provides a vivid contrast between bosonic and fermionic statistics. For fermions (e.g., electrons), the creation operators satisfy anti-commutation relations: $\{\hat{c}_3^\dagger, \hat{c}_4^\dagger\} = \hat{c}_3^\dagger\hat{c}_4^\dagger + \hat{c}_4^\dagger\hat{c}_3^\dagger = 0$. Repeating the HOM calculation with the sign flip, the cross terms add rather than cancel, and the $|2,0\rangle$ and $|0,2\rangle$ terms vanish (which they must, since two fermions cannot occupy the same single-particle state — the Pauli exclusion principle). The result is:

$$|1,1\rangle_{\text{fermion}} \;\xrightarrow{\text{50:50 BS}}\; |1,1\rangle \qquad \text{(fermions always separate)}$$

Fermions always emerge in different outputs — the exact opposite of photons. This fermionic anti-bunching has been observed experimentally with electrons at a quantum point contact (a solid-state analog of a beam splitter) by Henny et al. (1999) and Oliver et al. (1999).

The beam splitter thus acts as a universal sorter: bosons bunch, fermions anti-bunch, and classical particles show no correlation. This three-way comparison is one of the most beautiful pedagogical demonstrations in all of quantum mechanics.

27.8 Photon Statistics and Correlation Functions

First-Order Coherence: $g^{(1)}$

The first-order correlation function characterizes the coherence of the electric field:

$$g^{(1)}(\tau) = \frac{\langle\hat{a}^\dagger(t)\hat{a}(t+\tau)\rangle}{\langle\hat{a}^\dagger\hat{a}\rangle}$$

For a coherent state, $g^{(1)}(\tau) = e^{-i\omega\tau}$ — perfect coherence at all delays. The modulus $|g^{(1)}|$ determines fringe visibility in a Michelson interferometer (classical interference). First-order coherence does not distinguish quantum from classical light.

Second-Order Coherence: $g^{(2)}$

The second-order correlation function is where quantum optics truly departs from classical optics:

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

At zero delay:

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

This measures the probability of detecting two photons simultaneously, normalized by the product of single-photon detection rates. The value of $g^{(2)}(0)$ classifies the photon statistics:

Classical Bound

A crucial result: for any classical light source, $g^{(2)}(0) \geq 1$. This follows from the Cauchy-Schwarz inequality applied to classical intensities. Therefore:

$$g^{(2)}(0) < 1 \implies \text{nonclassical light}$$

Photon antibunching ($g^{(2)}(0) < 1$) is the definitive signature of quantum light. It means that photon detections are more evenly spaced than a random (Poisson) process would produce — photons tend to arrive one at a time, avoiding each other. No classical wave theory can produce this.

🧪 Experiment: The first observation of photon antibunching was by Kimble, Dagenais, and Mandel in 1977, using resonance fluorescence from a single sodium atom. They measured $g^{(2)}(0) < 1$, proving that the light emitted by a single atom cannot be described by any classical theory. This experiment is widely regarded as the first definitive proof that the quantum theory of radiation is necessary.

The Hanbury Brown-Twiss Effect

The experimental technique for measuring $g^{(2)}(\tau)$ was pioneered by Robert Hanbury Brown and Richard Twiss in the 1950s — originally for measuring stellar diameters! They split starlight on a beam splitter and measured intensity correlations between the two outputs.

Their observation of $g^{(2)}(0) = 2$ for thermal light (photon bunching) was initially controversial — some physicists argued it violated quantum mechanics. In fact, it is a beautiful confirmation of Bose-Einstein statistics: thermal photons tend to arrive in bunches because they are bosons.

🔵 Historical Note: The Hanbury Brown-Twiss (HBT) experiment (1956) predates the laser by four years. It was conducted using the light from Sirius, a star 8.6 light-years away. The fact that second-order coherence can be measured for starlight — revealing quantum statistical properties of photons that traveled for nearly a decade — is a stunning demonstration of the robustness of quantum correlations.

Sub-Poissonian Light and Photon Antibunching

Let us be precise about what antibunching means physically. In a Poisson process (the photon arrivals from a coherent source), detection events are random and independent — knowing when one photon arrived tells you nothing about when the next will arrive. The variance in the photon number over a fixed time interval equals the mean: $(\Delta n)^2 = \bar{n}$.

In antibunched light, the detections are more regular than random — knowing one photon arrived makes it less likely that another will arrive immediately after. The variance is less than the mean: $(\Delta n)^2 < \bar{n}$, or equivalently, $g^{(2)}(0) < 1$. This is sub-Poissonian statistics.

Why is this impossible classically? For any classical intensity distribution $P(I)$ with $I \geq 0$, the Cauchy-Schwarz inequality gives $\langle I^2\rangle \geq \langle I\rangle^2$, which translates directly to $g^{(2)}(0) \geq 1$. The only way to violate this bound is through quantum interference effects that suppress coincident detections — precisely what happens in photon antibunching.

The canonical source of antibunched light is a single quantum emitter: one atom, one quantum dot, one nitrogen-vacancy center in diamond. Such a source can emit at most one photon at a time (it must be re-excited before it can emit again), guaranteeing $g^{(2)}(0) = 0$ in the ideal case. This is the physical basis for single-photon sources used in quantum key distribution and photonic quantum computing.

Photon Bunching: A Deeper Look

Thermal (chaotic) light has $g^{(2)}(0) = 2$, meaning photons are twice as likely to arrive in pairs as a random process would predict. This photon bunching arises from the Bose-Einstein statistics of the thermal photon number distribution $P(n) = \bar{n}^n/(\bar{n}+1)^{n+1}$, which is broader than Poissonian.

The physical picture is illuminating. Thermal light consists of many randomly phased contributions (from independent atoms in a lamp, for instance). These contributions interfere constructively and destructively, producing fluctuations in the instantaneous intensity. When the intensity happens to be high, many photons arrive together (a "bunch"); when it is low, few arrive. The bunching is a statistical correlation arising from the wave nature of light and is present even in the classical wave theory — it is $g^{(2)}(0) < 1$ that requires quantization.

The timescale of bunching is the coherence time $\tau_c$: photons within $\tau_c$ of each other are correlated, while photons separated by much more than $\tau_c$ are independent. For thermal light, $g^{(2)}(\tau) = 1 + |g^{(1)}(\tau)|^2$, which decays from $g^{(2)}(0) = 2$ to $g^{(2)}(\infty) = 1$ over a time $\sim \tau_c$.

Comparison Summary

*For squeezed vacuum ($\alpha = 0$). Displaced squeezed states differ.

27.9 Connection to Quantum Computing with Photons

Why Photons?

Photons are natural candidates for quantum information processing:

1. Low decoherence — photons interact weakly with the environment. A photon can travel through kilometers of optical fiber without losing its quantum coherence.
2. High-speed communication — photons travel at the speed of light and are the natural carriers of quantum information in quantum networks.
3. Room-temperature operation — unlike superconducting qubits or trapped ions, photonic quantum systems do not require cryogenic temperatures.
4. Natural entanglement generation — SPDC sources produce entangled photon pairs routinely.

Encoding Qubits in Photons

There are several ways to encode a qubit in a single photon:

Polarization encoding: $$|0\rangle \equiv |H\rangle, \qquad |1\rangle \equiv |V\rangle$$

where $|H\rangle$ and $|V\rangle$ are horizontal and vertical polarization states. A half-wave plate rotates polarization, acting as a single-qubit gate. A polarizing beam splitter projects onto the $\{|H\rangle, |V\rangle\}$ basis.

Dual-rail encoding: $$|0\rangle \equiv |1\rangle_a|0\rangle_b, \qquad |1\rangle \equiv |0\rangle_a|1\rangle_b$$

The qubit is encoded in which spatial mode the photon occupies. A beam splitter acts as a Hadamard gate (for $\theta = \pi/4$), and a phase shifter in one arm performs a $Z$ rotation.

Time-bin encoding: $$|0\rangle \equiv |\text{early}\rangle, \qquad |1\rangle \equiv |\text{late}\rangle$$

The photon is in a superposition of arriving at an early or late time slot. This encoding is robust against polarization drift in optical fibers and is preferred for long-distance quantum communication.

Each encoding has distinct experimental advantages. Polarization encoding is simplest to manipulate (using standard waveplates and polarizing beam splitters) but degrades in optical fibers due to birefringence. Dual-rail encoding maps naturally onto integrated photonic circuits. Time-bin encoding is the most robust for fiber-based quantum networks.

Single-Qubit Gates with Linear Optics

In any encoding, arbitrary single-qubit gates can be implemented using beam splitters and phase shifters alone. This follows from the Euler decomposition: any SU(2) rotation can be written as a product of three rotations about two axes. In the dual-rail encoding:

• A beam splitter with mixing angle $\theta$ implements a rotation about the $x$-axis on the Bloch sphere (an $R_x(2\theta)$ gate).
• A phase shifter $e^{i\phi}$ in one arm implements a rotation about the $z$-axis ($R_z(\phi)$).
• The combination BS-Phase-BS achieves any desired single-qubit unitary.

This is remarkable: the complete SU(2) algebra for a single photonic qubit is realized by passive, room-temperature optical elements. The experimental fidelity of single-qubit gates on photonic qubits routinely exceeds 99.9%, limited only by the quality of the optical components.

Linear Optical Quantum Computing (LOQC)

In 2001, Knill, Laflamme, and Milburn (KLM) proved a remarkable theorem: universal quantum computing is possible using only single-photon sources, linear optical elements (beam splitters and phase shifters), and photon detectors. This was surprising because linear optics alone cannot create the deterministic two-qubit interactions needed for universal computation.

The KLM trick is to use measurement-induced nonlinearity: by performing measurements on ancilla photons and post-selecting on certain outcomes, one can implement an effective nonlinear interaction (a controlled-Z gate) between two photonic qubits. The gate succeeds probabilistically, but with teleportation-based error correction, the success probability can be boosted arbitrarily close to unity.

The KLM protocol requires enormous overhead in ancilla photons and detectors. Modern approaches to photonic quantum computing use alternative architectures:

• Fusion-based quantum computing (FBQC) — PsiQuantum's approach. Generate small entangled "resource states" and fuse them together using probabilistic measurements.
• Gaussian boson sampling — Xanadu's approach. Use squeezed states and linear optics to perform sampling problems believed to be classically intractable. Their Borealis machine (2022) demonstrated quantum computational advantage using 216 squeezed modes.
• Measurement-based quantum computing (MBQC) — Generate a large entangled cluster state, then perform the computation entirely through single-qubit measurements.

🔗 Connection: The quantum circuit model (Ch 25) and the measurement-based model are computationally equivalent — any circuit can be simulated by measurements on a cluster state, and vice versa. Photonic systems are particularly natural for the measurement-based approach because entanglement generation (via SPDC and beam splitters) and measurement (via photon detection) are the operations that photonics does best.

The Challenge: Photon Loss

The primary challenge for photonic quantum computing is photon loss — a photon absorbed by an optical element or failing to be detected is an irreversible error that cannot be fixed by standard error correction. The loss per component must be extremely small (< 1% per gate) for large-scale computation.

Current research focuses on: - Integrated photonics — fabricating beam splitters, phase shifters, and waveguides on silicon chips, reducing loss per component. - Photon-number-resolving detectors — superconducting nanowire single-photon detectors (SNSPDs) achieve >98% detection efficiency at telecom wavelengths. - Quantum error correction — topological codes (surface codes) adapted for photonic architectures, with built-in tolerance for loss errors.

📊 By the Numbers: Xanadu's X-series photonic chip (2024) integrates 300+ optical components on a single silicon nitride chip, with per-component loss of ~0.1 dB (~2.3%). PsiQuantum's silicon photonic platform targets per-component loss below 0.01 dB for fault-tolerant operation.

⚖️ Interpretation: Photonic quantum computing faces a fundamentally different challenge than superconducting or ion-trap approaches. Superconducting qubits suffer from decoherence (information leaks to the environment); photonic qubits suffer from loss (the qubit disappears entirely). Loss is in some ways easier to handle — you know when a photon is lost (it fails to arrive) — but it requires fundamentally different error correction strategies.

27.10 Summary and Progressive Project

Summary of Key Results

This chapter established the quantum theory of light by quantizing the electromagnetic field. The central insight is that each mode of the field is a quantum harmonic oscillator, and photons are the excitation quanta of these oscillators.

Three families of quantum optical states:

1. Fock states $|n\rangle$ — definite photon number, zero mean field, no defined phase. The energy eigenstates. Difficult to produce experimentally but essential for quantum information.

2. Coherent states $|\alpha\rangle$ — Poissonian photon statistics, nonzero mean field, well-defined phase. Minimum uncertainty with symmetric quadratures. Describe laser light. The closest quantum analog of a classical wave.

3. Squeezed states — minimum uncertainty with asymmetric quadratures. One quadrature below the vacuum noise level. Enable measurements beyond the standard quantum limit (LIGO). Generated by parametric processes.

Two quantum optical phenomena without classical analogues:

1. Hong-Ou-Mandel effect — two identical photons at a 50:50 beam splitter always exit the same port. The $|1,1\rangle$ output is eliminated by destructive interference between the "both transmitted" and "both reflected" amplitudes.

2. Photon antibunching — $g^{(2)}(0) < 1$, impossible for any classical field. The definitive proof that the electromagnetic field requires quantum mechanical treatment.

Connection to technology: Photons are leading candidates for quantum communication (they travel at the speed of light with low decoherence) and serious contenders for quantum computation (via linear optical quantum computing with measurement-induced nonlinearity). The photon is also the workhorse of quantum metrology: squeezed light improves gravitational wave detection (LIGO), entangled photon pairs enable quantum-enhanced imaging, and single photons provide provably secure quantum key distribution.

The unifying theme of this chapter is that classical optics is a special case of quantum optics, not the other way around. Every classical optical phenomenon has a quantum explanation, but quantum optics contains effects — antibunching, the HOM dip, entanglement from beam splitters — that have no classical counterpart whatsoever. The electromagnetic field is, at its core, a quantum system.

Key Equations Summary

Progressive Project Checkpoint: Quantum Optics Module

Toolkit version: v0.27 New module: quantum_optics.py

This chapter's project checkpoint adds three key functions to your quantum simulation toolkit:

1. coherent_state(alpha, n_max) — Construct a coherent state $|\alpha\rangle$ in the Fock basis up to $n_{\max}$ photons. Return the state vector and plot the Poisson distribution $P(n)$.

2. beam_splitter(state_in, theta) — Apply a beam splitter transformation with mixing angle $\theta$ to a two-mode input state. Return the output state in the two-mode Fock basis.

3. hong_ou_mandel(tau_range, tau_c) — Simulate the Hong-Ou-Mandel dip: compute and plot the coincidence probability as a function of relative time delay $\tau$.

See code/project-checkpoint.py for the implementation and code/example-01-optics.py for worked examples.

Looking Ahead

The quantum theory of light developed in this chapter is the simplest example of a quantum field theory — a theory where the number of particles is not fixed but can change dynamically. In Ch 34 (second quantization), we will see how the same mathematical framework extends to matter fields, leading to the creation and annihilation of electrons, quarks, and all other particles. The photon was the first quantum field; it will not be the last.

🔗 Connection: Forward references: - Ch 28 (Measurement Problem) — Delayed-choice experiments with single photons push quantum foundations to their limits. - Ch 31 (Path Integrals) — Feynman's approach to quantum optics: sum over all photon paths. - Ch 34 (Second Quantization) — The field quantization program of this chapter, extended to matter. - Ch 37 (QFT Preview) — Quantum electrodynamics: the full theory of photons interacting with charged matter. - Ch 39 (Capstone: Bell Test) — Full simulation of a photon-based Bell inequality experiment.