Quantum Optics Fundamentals

Build your intuition for the quantum world that powers the next technological revolution — from wave-particle duality to photon entanglement.

What Is Quantum Optics?

Quantum optics applies quantum mechanics to light and its interaction with matter at the single-photon level. Unlike classical optics, it acknowledges that light is composed of discrete energy packets — photons — that exhibit fundamentally quantum behaviour.

"The most beautiful thing we can experience is the mysterious. It is the source of all true art and science."
— Albert Einstein

At its core, quantum optics explores how photons exhibit superposition, entanglement, and interference — phenomena with no classical analogue — and how these properties enable quantum computation, communication and sensing.

photon λ

Building Blocks of Quantum Optics

Click any concept to expand a full explanation.

Light behaves as both a wave and a particle depending on how it is measured. In Young's double-slit experiment, a single photon passes through both slits simultaneously (wave) yet always arrives at a single location (particle).

The wave function Ψ(x,t) gives the probability amplitude. |Ψ|² yields the probability of detecting a photon at position x. The de Broglie relation λ = h/p connects wavelength and momentum.

This duality is a fundamental property of nature, not a measurement limitation. It underpins all quantum interference phenomena that power quantum computers and sensors.

Double-Slit de Broglie Wave Function

The electromagnetic field is quantized into discrete photons. Fock states |n⟩ are eigenstates of the photon number operator n̂ = â†â — they describe light with exactly n photons.

Creation and annihilation operators: â†|n⟩ = √(n+1)|n+1⟩. The vacuum state |0⟩ carries zero-point energy ½ℏω per mode — the origin of vacuum fluctuations driving spontaneous emission.

Coherent states (laser light) are superpositions with Poisson photon number distributions, fundamentally different from classical waves.

Fock States |n⟩ ↠Operator Zero-Point Energy

Two photons are entangled when their joint state cannot be factored into a product of individual states. Measuring one instantly defines the state of the other, regardless of distance.

The four Bell states: |Φ±⟩ = (|HH⟩ ± |VV⟩)/√2 and |Ψ±⟩ = (|HV⟩ ± |VH⟩)/√2 are maximally entangled. Violations of Bell inequalities confirm this is real, not a hidden-variable effect.

SPDC (Spontaneous Parametric Down-Conversion) in nonlinear crystals is the standard method for generating entangled photon pairs in the lab.

Bell States EPR Paradox SPDC

A quantum system exists in a superposition of states: |ψ⟩ = α|0⟩ + β|1⟩, where |α|² + |β|² = 1. Only measurement collapses it to a definite state.

Quantum algorithms work by manipulating amplitudes — constructive interference amplifies correct answers; destructive interference cancels wrong ones.

The Mach-Zehnder interferometer splits a photon into superposition, lets paths acquire different phases, then recombines them — output probabilities are determined entirely by interference.

|ψ⟩=α|0⟩+β|1⟩ Mach-Zehnder Amplitude Amplification

ΔxΔp ≥ ℏ/2 — position and momentum cannot simultaneously have zero uncertainty. For photons: ΔnΔφ ≥ ½ between photon number and phase.

This is not a measurement limit — it is built into the structure of quantum mechanics. Any state that reduces one uncertainty necessarily increases the other.

Squeezed states exploit this: reduce phase noise below the standard quantum limit (at the cost of more number uncertainty) — enabling LIGO's sub-shot-noise sensitivity.

ΔxΔp ≥ ℏ/2 Squeezed States Shot Noise Limit

QKD distributes cryptographic keys with information-theoretic security. Any eavesdropping disturbs the quantum states, revealing the interceptor. The BB84 protocol sends single photons in random polarisation bases.

Security proof: the no-cloning theorem prevents Eve from copying quantum states; any measurement she performs introduces detectable errors in the Quantum Bit Error Rate (QBER > 11% = eavesdropping detected).

CV-QKD (continuous-variable) uses coherent states and homodyne detection, making it compatible with standard telecom fibre at speeds exceeding 1 Gbit/s.

BB84 No-Cloning Theorem CV-QKD QBER

Interaction with the environment causes decoherence — quantum superpositions collapse into classical mixtures. Maintaining coherence is the central engineering challenge of quantum technology.

Quantum error correction encodes one logical qubit into many physical qubits. The surface code detects and corrects errors without measuring (and collapsing) the logical state.

Photons are naturally decoherence-resistant — they rarely interact with room-temperature environments — making optical channels ideal for quantum communication.

Surface Code Logical Qubit T₂ Coherence Time

Lasers produce coherent light — photons sharing identical frequency, phase and direction. Coherent states |α⟩ are eigenstates of the annihilation operator: â|α⟩ = α|α⟩.

Stimulated emission: an excited atom releases a photon identical to an incoming one. Population inversion (more atoms excited than ground state) sustains amplification.

Modern quantum optics requires going beyond lasers — single-photon emitters and photon-number-resolving detectors probe the deepest nonclassical states of light.

Coherent State |α⟩ Stimulated Emission Population Inversion

The Math of Quantum Light

E = hν

Planck–Einstein Relation

Energy of a photon equals Planck's constant times its frequency. The founding equation of quantum theory.

p = ℏk

Photon Momentum

Photon momentum equals reduced Planck constant times wave number. Links quantum and electromagnetic theories.

ΔxΔp ≥ ℏ/2

Heisenberg Uncertainty

Position and momentum cannot simultaneously be exactly known. Fundamental limit in all quantum systems.

|Φ+⟩=(|HH⟩+|VV⟩)/√2

Bell State

Maximally entangled two-photon state. Foundation of quantum cryptography and teleportation.

â†|n⟩=√(n+1)|n+1⟩

Creation Operator

Adds one photon to a Fock state. The algebraic engine of quantum field theory.

ρ=Σpᵢ|ψᵢ⟩⟨ψᵢ|

Density Matrix

Describes mixed quantum states — essential for modelling decoherence and open systems.

Quick Quantum Quiz

Question 1 / 4

What is a photon?

Question 2 / 4

What does quantum entanglement mean for two photons?

Question 3 / 4

Which protocol uses photon polarisation for quantum key distribution?

Question 4 / 4

What does the Heisenberg Uncertainty Principle say about photon number and phase?

Quantum Terminology

Qubit

Quantum bit — basic unit of quantum information. Can exist in superposition of |0⟩ and |1⟩ simultaneously.

Photon

Elementary particle of light. Massless, travels at c, carries energy E = hν.

Entanglement

Non-classical correlation between particles. Measuring one instantly determines the outcome of the other.

Decoherence

Loss of quantum properties through environmental interaction. The central obstacle in quantum computing.

Squeezed Light

Quantum light state with noise in one quadrature reduced below vacuum level.

Beam Splitter

Optical device splitting a photon into superposition of transmitted and reflected paths — the fundamental quantum gate.

SPDC

Spontaneous Parametric Down-Conversion: one pump photon → two entangled daughter photons in a nonlinear crystal.

No-Cloning

Theorem: an unknown quantum state cannot be perfectly copied. Guarantees security of quantum cryptography.

Fock State

Quantum state of exactly n photons. Written |n⟩. Eigenstate of the photon number operator.

QKD

Quantum Key Distribution — uses quantum mechanics to distribute cryptographic keys with provable security.

Bell State

One of four maximally entangled two-qubit states. Foundation of quantum information protocols.

Coherence

Fixed phase relationship enabling quantum interference. Coherence time T₂ limits qubit lifetime.