Quantum optics begins when an electromagnetic mode is treated as a bosonic quantum oscillator rather than as a purely classical wave. The same beam splitter can then mix coherent amplitudes, split a single photon into a path superposition, or make two indistinguishable photons leave together through Hong–Ou–Mandel interference.
The optical hardware may look familiar—mirrors, phase shifters, detectors and interferometers—but the object being transformed is a quantum state in Fock space. Classical field amplitudes remain useful, especially for coherent states, yet photon counting and multiphoton interference reveal structure that cannot be captured by a deterministic classical wave amplitude alone.
This guide develops coherent states, lossless beam splitters, Mach–Zehnder interferometry, two-photon interference, photon statistics, Glauber correlation functions and balanced homodyne detection. The emphasis is on operator transformations that keep state normalisation and commutation relations explicit.
Mode operators → optical unitary → transformed state → detector model → count statistics → interference evidence.
1. One optical mode is one bosonic oscillator
Guide 37 introduced creation and annihilation operators satisfying [a,a†]=1. In quantum optics, a labels one chosen field mode: a spatial, temporal, frequency and polarisation pattern.
The number operator N=a†a counts photons in that mode. Fock state |n⟩ has exactly n photons.
The optical Hamiltonian of one ideal mode is
H=ℏω(N+1/2).
In most photodetection calculations the common zero-point term does not affect count differences and is omitted from the bookkeeping.
2. Coherent states
A coherent state |α⟩ is defined by
a|α⟩=α|α⟩.
Its Fock expansion is
|α⟩=e^{-|α|²/2}Σ_{n=0}∞ α^n|n⟩/√(n!).
The complex number α acts like a classical phase-space amplitude. In the convention [x,p]=i,
⟨x⟩=√2 Re α;⟨p⟩=√2 Im α;Var(x)=Var(p)=1/2.
Thus a coherent state is displaced vacuum: it moves the centre of the uncertainty circle without changing its shape.
3. Photon-number statistics of a coherent state
The probability of finding n photons is
P(n)=e^{-|α|²}|α|^{2n}/n!.
This is a Poisson distribution with
⟨N⟩=Var(N)=|α|².
For |α|²=4, P(0)≈0.0183, P(1)≈0.0733, P(2)≈0.1465, P(3)≈0.1954 and P(4)≈0.1954.
The relative standard deviation is √Var(N)/⟨N⟩=1/|α|. Bright coherent light therefore has small fractional shot noise even though its absolute photon-number variance grows.
4. Coherent states are not photon-number eigenstates
A laser-like coherent field can have a well-defined phase-space displacement while containing an uncertain number of photons.
A Fock state does the opposite: it has sharp photon number but zero mean field amplitude.
These are different quantum resources. “More classical-looking” does not mean “no quantum uncertainty”. Coherent states retain vacuum quadrature noise and discrete Poisson photodetection statistics.
5. Displacement operator
Define
D(α)=exp(αa†−α* a).
Then
|α⟩=D(α)|0⟩
and
D†(α)aD(α)=a+α.
Displacement is a Gaussian unitary. It changes first moments while preserving covariance and photon-counting coherence properties appropriate to coherent light.
6. Lossless two-port beam splitter
A lossless beam splitter performs a unitary mixing of two input modes a,b into output modes c,d.
For a convenient 50:50 convention, write
c=(a+b)/√2
d=(a−b)/√2.
The matrix is unitary, so the output commutators remain canonical and total photon number is preserved.
Different optical conventions place factors of i on reflected amplitudes. Physical detection probabilities are unchanged when all phases are treated consistently.
7. Coherent states pass through beam splitters classically at the amplitude level
If the inputs are product coherent states |α⟩a|β⟩b, the outputs are again product coherent states:
|(α+β)/√2⟩_c |(α−β)/√2⟩_d.
This is one reason coherent states resemble classical fields so strongly under passive linear optics. The complex amplitudes obey the same linear mixing law as classical wave amplitudes.
Quantum behaviour remains in the output shot noise and in what happens for nonclassical input states.
8. Worked coherent interference
Let α=β with equal phase and equal magnitude. Then
c=√2α and d=0.
All mean optical power exits one port under this convention.
If β=−α, the outputs reverse: c=0 and d=√2α.
The interference is controlled by relative phase, exactly as in classical wave optics at the mean-field level.
9. Single photon at a beam splitter
Input |1,0⟩ means one photon in mode a and vacuum in b.
Using
a†=(c†+d†)/√2,
we obtain
|1,0⟩_ab → (|1,0⟩_cd+|0,1⟩_cd)/√2.
The photon is not split into two half-photons. The output is a coherent superposition of the two path possibilities, and one ideal photon-counting run detects the whole photon in one port or the other.
10. Mach–Zehnder interferometer
A Mach–Zehnder interferometer uses a first beam splitter to create two path amplitudes, a relative phase shift φ between the arms, and a second beam splitter to recombine them.
With the same real 50:50 convention, the effective single-photon transformation gives output amplitudes
A_c=(e^{iφ}+1)/2
and
A_d=(e^{iφ}−1)/2.
Therefore
P_c=cos²(φ/2);P_d=sin²(φ/2).
A single photon builds an interference fringe across repeated trials because the path amplitudes recombine before detection.
11. Worked Mach–Zehnder phases
- φ=0 gives Pc=1, Pd=0.
- φ=π/2 gives Pc=Pd=1/2.
- φ=π gives Pc=0, Pd=1.
The detector output encodes phase through probability. Guide 15’s quantum-metrology framework asks how precisely φ can be estimated from such distributions and more general quantum states.
12. Two photons reveal a different interference effect
Now place one indistinguishable photon in each input port: |1,1⟩.
Using
a†=(c†+d†)/√2
and
b†=(c†−d†)/√2,
their product is
a†b†=[(c†)²−(d†)²]/2.
Acting on vacuum and accounting for Fock normalisation gives
|1,1⟩ → (|2,0⟩−|0,2⟩)/√2.
The |1,1⟩ coincidence term cancels completely.
13. Hong–Ou–Mandel interference
When two indistinguishable photons arrive simultaneously at a balanced beam splitter, the alternatives “both transmitted” and “both reflected” interfere destructively for one-photon-per-output coincidences.
The photons bunch: they leave together through one output or the other.
Scanning the relative arrival delay makes the photons gradually distinguishable. The coincidence rate then rises away from zero, producing the Hong–Ou–Mandel dip first demonstrated by Hong, Ou and Mandel. [3]
14. Indistinguishability is a mode-overlap statement
Two photons interfere perfectly only when every degree of freedom relevant to the detector is sufficiently indistinguishable: spectrum, temporal wavepacket, polarisation and spatial mode.
If their single-photon mode overlap is less than one, the coincidence suppression is incomplete.
The HOM visibility is therefore widely used as an operational test of photon indistinguishability, but its interpretation depends on multiphoton contamination, detector effects and the exact input-state model.
15. Photon counting as a POVM
An ideal number-resolving detector measures the Fock-basis POVM
Π_n=|n⟩⟨n|.
For state ρ,
P(n)=Tr(ρ|n⟩⟨n|).
A threshold detector instead distinguishes “no click” from “one or more detected photons” and therefore implements a coarser POVM.
16. Detection efficiency
A simple efficiency model treats loss before an ideal detector as a beam splitter of transmissivity η whose unused port receives vacuum.
For an n-photon Fock state, the number k actually detected follows a binomial distribution:
P(k|n)=C(n,k)η^k(1−η)^{n-k}.
For coherent input with mean photon number μ, loss thins the Poisson distribution into another Poisson distribution with mean ημ.
17. Worked inefficient coherent detection
A coherent pulse has mean photon number μ=5 and detector efficiency η=0.6.
The detected counts are Poisson with mean
ημ=3.
The probability of zero detected photons is
P(0)=e^{-3}≈0.0498.
An ideal threshold detector would therefore click with probability approximately 95.0% before dark counts.
18. Dark counts
A detector can click even when no signal photon arrived. If dark counts are approximately Poisson with mean λd per gate and independent of the signal, they add to the detected coherent-state count mean.
For threshold detection, a dark-click probability d changes
P(no click)=(1-d)P(no signal detection)
under a simple independent model.
Detector calibration belongs inside the statistical model, not after the quantum calculation.
19. Glauber correlation functions
Photon-counting coherence is described by normally ordered field correlations. For one stationary mode, the zero-delay second-order correlation is
g^(2)(0)=⟨a†a†aa⟩/⟨a†a⟩².
Glauber’s quantum theory of optical coherence established the hierarchy of such measurable correlations. [1]
20. Coherent, Fock and thermal g²
For an ideal coherent state,
g^(2)(0)=1.
For Fock state |n⟩ with n≥1,
g^(2)(0)=1−1/n.
In particular, a single-photon state has g²(0)=0: two simultaneous detections from the same ideal one-photon pulse are impossible.
For ideal single-mode thermal light, g²(0)=2, showing photon bunching relative to Poisson statistics.
21. Antibunching as a nonclassical signature
Values g²(0)<1 indicate sub-Poissonian or antibunched behaviour incompatible with a simple classical mixture of coherent intensities under standard photodetection theory.
This is one operational route for identifying nonclassical light.
No one scalar characterises all nonclassicality. Squeezing, negativity of quasiprobabilities, antibunching and entanglement diagnose different structures.
22. Balanced homodyne detection
Photon counting measures energy quanta. Homodyne detection measures a field quadrature.
Mix the weak signal mode a with a strong coherent local oscillator β=|β|eiθ on a balanced beam splitter. Measure the difference between the two output photocurrents.
In the strong-local-oscillator limit, the difference signal is proportional to
|β| x_θ
up to detector and convention factors, where xθ is the rotated signal quadrature.
The local oscillator supplies a phase reference and amplifies the quadrature signal into a measurable photocurrent difference.
23. Homodyne tomography
Measure quadrature distributions for many local-oscillator phases θ. These are projections of the phase-space quasiprobability distribution.
Inverse Radon-transform methods or maximum-likelihood reconstruction can recover the Wigner function or density matrix from sufficiently complete homodyne data.
This is the continuous-variable analogue of state tomography in Guide 21, with a different measurement geometry.
24. Shot noise is a reference level, not always an absolute constant
Experimental papers often normalise quadrature variance to a “shot-noise unit” measured using vacuum or coherent light.
Some conventions set vacuum variance to 1, others to 1/2, and some electronic instruments quote power spectral density rather than dimensionless variance.
Before comparing squeezing values or covariance matrices across sources, identify the normalisation.
25. Common misconception: a beam splitter randomly chooses reflection or transmission for each photon
For a coherent quantum input, the beam splitter applies a unitary transformation to amplitudes. Random detector outcomes appear only after measurement. Treating reflection and transmission as classical random choices misses interference between alternatives.
26. Common misconception: HOM interference is ordinary single-photon path interference
The HOM dip comes from interference between two-photon probability amplitudes leading to a coincidence. It is fourth-order optical interference and depends on two-photon indistinguishability.
27. Common misconception: coherent light has deterministic photon number
Coherent states have Poisson photon-number statistics. Their classical resemblance lies in field coherence and transformation of mean amplitudes, not in absence of quantum fluctuations.
28. Worked synthesis problem
A single photon enters a balanced Mach–Zehnder interferometer. One arm acquires phase φ=π/3.
Step 1: Output probabilities.
P_c=cos²(π/6)=3/4
and
P_d=sin²(π/6)=1/4.
Step 2: Repeated trials. In 10,000 ideal independent photons, expected counts are approximately 7500 and 2500.
Step 3: Counting uncertainty. The output at port c is binomial with standard deviation √(10000·0.75·0.25)≈43.3 counts.
Step 4: Interpretation. One photon is detected per ideal trial, but the frequency across many trials reconstructs the interference probability generated by coherent path amplitudes.
Step 5: Distinguish from HOM. Replacing the one-photon input with |1,1⟩ at one beam splitter creates a different two-photon interference problem whose coincidence amplitude cancels at perfect indistinguishability.
29. Practice set
- Define a coherent state as an eigenstate equation.
- What is its mean photon number?
- What photon-number distribution does it have?
- Write one 50:50 beam-splitter convention for output operators.
- What happens to product coherent states under a passive beam splitter?
- What state results when |1,0⟩ enters a balanced splitter?
- State the Mach–Zehnder output probabilities in this guide’s convention.
- What is the ideal |1,1⟩ output of a balanced splitter for indistinguishable photons?
- What is g²(0) for a coherent state?
- What is g²(0) for a one-photon Fock state?
- How can detector inefficiency be modelled simply?
- What quantity does balanced homodyne detection measure?
Answers
a|α⟩=α|α⟩.|α|².- Poisson.
c=(a+b)/√2,d=(a−b)/√2.- They remain product coherent states with amplitudes mixed by the same linear transformation.
(|1,0⟩+|0,1⟩)/√2.cos²(φ/2)andsin²(φ/2).(|2,0⟩−|0,2⟩)/√2up to convention-dependent phases.- 1.
- 0.
- Place a fictitious beam splitter of transmissivity η before an ideal detector.
- A phase-selected field quadrature relative to a strong local oscillator.
Sources and further study
[1] Roy J. Glauber, The Quantum Theory of Optical Coherence, Physical Review 130, 2529 (1963), and Coherent and Incoherent States of the Radiation Field, Physical Review 131, 2766 (1963). Foundational coherent-state and photodetection-correlation theory.
[2] D. N. Makarov, Theory for the beam splitter in quantum optics: quantum entanglement of photons and their statistics, HOM effect. A modern review of beam-splitter transformations and photon statistics.
[3] C. K. Hong, Z. Y. Ou and L. Mandel, Measurement of subpicosecond time intervals between two photons by interference, Physical Review Letters 59, 2044 (1987). The canonical Hong–Ou–Mandel two-photon interference experiment.
[4] Alessia Allevi and Maria Bondani, Introduction to generation, manipulation and characterization of optical quantum states, Physics Letters A 418, 127720 (2021). A tutorial treatment of optical states, beam splitters, squeezing and photodetection.
Continue through Quantum Mathematics
Guide 37: Bosonic Modes, Ladder Operators, Fock States and Quadratures supplies the Fock-space operator algebra. Guide 38: Gaussian Quantum States, Covariance Matrices, Symplectic Transformations and Squeezing organises optical Gaussian states in phase space. Guide 40: Continuous-Variable Quantum Information, Gaussian Channels, Homodyne Detection and Teleportation uses these optical primitives for information processing.