Optics · Advanced Topics

Quantum Optics

Quantum optics treats the electromagnetic field itself as a quantum object. Coherent states, squeezed light, cavity QED, and photon entanglement underlie modern quantum communication, sensing, and computation.

Prerequisites

Quantum mechanics (Ch. QM) \cdot Lasers (Ch. Las) \cdot Electromagnetism (Ch. EM) \cdot Harmonic oscil

lator (Ch. QM)

Learning Goals

Quantise a single EM field mode as a harmonic oscillator and write the Hamiltonian in terms of creation and annihilation operators.
Define Fock states and coherent states, compute their photon-number distributions, and state the uncertainty relations for the quadratures.
$Calculate g^{(2)}(0) for thermal light, coherent light, and Fock state |1⟩, and interpret ant$ i-bunching as a non-classical signature.
Diagonalise the Jaynes-Cummings Hamiltonian in the one-excitation subspace and identify the vacuum Rabi splitting 2g.
Describe quantum teleportation and BB84 QKD, and explain why neither violates the no-cloning theorem or allows faster-than-light signalling.

QO.1 Quantization of the Electromagnetic Field

Each mode (k, λ) of the EM field is a quantum harmonic oscillator. The vector potential is expanded in creation and annihilation operators:

A(r,t) = \sum_{k,\lambda} \sqrt(\hbar/(2\varepsilon_{0}\omegaV)) \varepsilon_{k,\lambda} [a_{k,\lambda} e^{ik\cdotr-i\omegat} + a^\dagger_{k,\lambda} e^{-ik\cdotr+i\omegat}]

(QO.1)

Commutation: [a_(k,λ), a†_(k',λ')] = δ_(kk') δ_(λλ'). The Hamiltonian is: H = Σ_(k,λ) ℏω_k (a†_(k,λ) a_(k,λ) + ½) — infinite zero-point energy. Observable quantities involve normal-ordered products (zero-point subtracted).

Definition QO.1 — Fock States

n⟩ = (a^\dagger)^n/\sqrt(n!) |0⟩ are eigenstates of the photon number operator N̂ = a^\daggera with eigenval

e n. The vacuum |0⟩ has fluctuating electric field: \DeltaE = \sqrt(\hbar\omega/(2\varepsilon_{0}V)) ≠ 0. Fock states hav

e definite photon number but completely undefined phase.

QO.2 Coherent States

A coherent state |α⟩ is an eigenstate of the annihilation operator: a|α⟩ = α|α⟩, where α ∈ ℂ. In terms of Fock states:

|\alpha⟩ = e^{-|\alpha|^{2}/2} \sum_{n=0 to \infty} \alpha^n/\sqrt(n!) |n⟩ \qquad (coherent state)

(QO.2)

Properties: mean photon number ⟨N̂⟩ = |α|², Poissonian statistics P(n) = e^(−|α|²)|α|^(2n)/n!, phase uncertainty Δφ ≈ 1/(2|α|). For |α| ≫ 1: near-classical light. The electric field expectation value oscillates classically: ⟨E⟩ ∝ Re(α e^(−iωt)).

Theorem QO.1 — Minimum Uncertainty States

Coherent states saturate the Heisenberg uncertainty relation for the quadratures X_{1} = (a +

a^\dagger)/2 and X_{2} = (a - a^\dagger)/(2i): \DeltaX_{1} \DeltaX_{2} = \frac{1}{4} (minimum uncertainty). Both quadratures have equ

l noise: \DeltaX_{1} = \DeltaX_{2} = \frac{1}{2} (shot noise limit). Squeezed states reduce noise in one quadrature

below the shot noise limit at the expense of increased noise in the conjugate quadrature.

QO.3 Squeezed Light

A squeezed state is generated by the squeezing operator S(ξ) = exp(ξ*a²/2 − ξ(a†)²/2) with ξ = r e^(iθ). For the squeezed vacuum:

\DeltaX_{1} = \frac{1}{2} e^{-r}, \qquad \DeltaX_{2} = \frac{1}{2} e^{r} \qquad (squeezed quadratures, squeezing parameter r)

(QO.3)

Squeezing reduces phase noise (useful for interferometry) or amplitude noise.LIGO: injection of squeezed light (r ≈ 15 dB achieved by 2023) into the dark port reduces quantum noise below the standard quantum limit — essential for detecting gravitational waves from binary mergers at distances > 100 Mpc.

Generation: optical parametric oscillator (OPO) below threshold via χ(2) process. Pump photon at 2ω → signal + idler at ω (two-mode squeezing) or degenerate OPO (single-mode squeezing).

Example QO.1 — Hanbury Brown–Twiss Effect and g(2)(τ)

$Define the second-order coherence: g(2)(\tau) = ⟨a^\dagger(t) a^\dagger(t+\tau) a(t+\tau) a(t)⟩/⟨a^\daggera⟩^{2}. Compute g$ (2)(0) for (a) coherent state, (b) thermal light, (c) Fock state |1⟩.

Coherent state |α⟩:⟨

a^\daggera^\daggeraa⟩ = ⟨\alpha|a^\daggera^\daggeraa|\alpha⟩ = |\alpha|^{4}. ⟨a^\daggera⟩^{2} = |\alpha|^{4}. g(2)(0) = 1. Poissonian statistics — photons

arrive randomly, uncorrelated.

Thermal light:

Bose-Einstein distribution gives ⟨n^{2}⟩ = ⟨n⟩^{2} + ⟨n⟩^{2} (super-Poissonian). g(2)(0) = ⟨n(n-1)⟩

/⟨

n⟩^{2} = 2. Photons arrive in bunches — photon bunching. HBT effect: for chaotic light, coinc

ences at \tau=0 are twice as likely as at large \tau

Fock state |1⟩:⟨

1|a^\daggera^\daggeraa|1⟩ = 0 (a|1⟩ gives |0⟩, then a|0⟩=0). g(2)(0) = 0. Perfect photon anti-bunching —

one photon at a time, never two simultaneously. Signature of a single-photon emitter. g(2)(0) < 1 is a quantum optical effect with no classical analog.

Interpretation:

g(2)(0) classifies light: bunched (thermal, >1), coherent (=1), anti-bunched (<1). Single-

photon emitters (quantum dots, NV centers, trapped ions) show g(2)(0) \approx 0. Measured in HBT

setup: beam splitter + two detectors + coincidence counter.

QO.4 Cavity QED and the Jaynes-Cummings Model

A two-level atom (|g⟩, |e⟩) coupled to a single cavity mode:

H_{JC} = \hbar\omega_c a^\daggera + \hbar\omega_a \sigma_z/2 + \hbarg(a^\dagger\sigma_ + a\sigma_+) \qquad (Jaynes-Cummings Hamiltonian)

(QO.4)

where g is the atom-photon coupling, σ_+ = |e⟩⟨g|, σ_ = |g⟩⟨e|. On resonance (ω_c = ω_a): eigenstates are dressed states |±,n⟩ = (|e,n⟩ ± |g,n+1⟩)/√2 with energies ℏω_c(n+1) ± ℏg√(n+1) — the vacuum Rabi splitting 2g for n=0.

Strong coupling regime: g > (κ, γ) where κ is cavity decay rate and γ is atomic spontaneous emission rate. Coherent quantum information exchange between atom and photon. Vacuum Rabi oscillations: an excited atom in an empty cavity oscillates between |e,0⟩ and |g,1⟩ at frequency g.

Purcell effect: cavity modifies spontaneous emission rate. Purcell factor F_P = (3/4π²)(λ/n)³ Q/V — enhanced by high Q/V. F_P > 1: emission enhanced (useful for bright single-photon sources). Applications: quantum dot in photonic crystal cavity (Q ~ 10⁶, V ~ (λ/n)³), superconducting qubit in microwave resonator (circuit QED, g/2π ~ 100 MHz).

QO.5 Entanglement and Quantum Communication

Bell states (polarization-entangled photon pairs from SPDC):

|\Phi\pm⟩ = (|HH⟩ \pm |VV⟩)/\sqrt2, \qquad |\Psi\pm⟩ = (|HV⟩ \pm |VH⟩)/\sqrt2

(QO.5)

Quantum teleportation (Bennett et al. 1993): Alice teleports state |ψ⟩ = α|H⟩ + β|V⟩ to Bob using a shared Bell pair and two classical bits. Protocol: joint Bell measurement by Alice (4 outcomes) → Bob applies one of {I, X, Z, iY} conditional on outcome. State perfectly transferred — no cloning, no faster-than-light signaling.

Quantum key distribution(QKD): BB84 protocol uses polarization states in two conjugate bases. Security from no-cloning theorem — eavesdropping introduces detectable errors (QBER > 25% for full interception). Long-distance QKD via quantum repeaters (entanglement swapping + purification), satellite QKD (Micius, 2017, 1200 km).

Definition QO.2 — Common Traps

Coherent states are not number states: they have Poisson photon-number uncertainty.
Squeezing moves noise between quadratures: it does not violate the uncertainty principle.
Entanglement is not faster-than-light signaling: correlations need classical communication to become usable information.
Antibunching marks nonclassical light: $g^{(2)}(0) below 1 cannot be explained by classical intensity noise$

Exercises — QO.1–QO.5 Quantum Optics

Calculate g^{(2)}(0) for the Fock state |2⟩. Compare the values for thermal light, coherent s

tate, |1⟩, and |2⟩. Which are non-classical?

Straightforward

2.Diagonalize the Jaynes-Cummings Hamiltonian in the one-excitation subspace. What is the vacuum Rabi splitting? How is the dispersive regime used for qubit readout in circuit QED?

Intermediate

3.Explain the quantum teleportation protocol step by step. Why does it not violate the no-cloning theorem or allow faster-than-light communication?

Intermediate

4.Derive the phase sensitivity of a Mach-Zehnder interferometer with squeezed light input. How does LIGO use squeezed states to surpass the standard quantum limit?

Challenging

Key Takeaways

$EM field modes are quantum oscillators. Fock |n⟩: definite photon number; coherent |\alpha⟩: Po$ issonian stats, minimum uncertainty.
$g(2)(0): thermal=2 (bunching), coherent=1 (Poissonian), single-photon=0 (anti-bunching). V$ alues <1 are non-classical.
$Squeezed light: \DeltaX_{1}=\frac{1}{2}e(-r), \DeltaX_{2}=\frac{1}{2}e(r). Reduces shot noise in one quadrature. LIGO uses 15d$ B squeezing.
Jay $nes-Cummings: two-level atom + cavity mode. Vacuum Rabi splitting 2g. Strong coupling g>(\kappa$ for coherent exchange.
Purcell effect: cavity modifies spontaneous emission rate by Q/V. Used for efficient single-photon sources.
Bell states, quantum teleportation, BB84 QKD — all rely on entanglement and no-cloning theorem.