A bosonic mode is the quantum version of a harmonic oscillator. Its Hilbert space is not two-dimensional like a qubit: it contains an infinite ladder of number states |0⟩,|1⟩,|2⟩,…, and the algebra of creation and annihilation operators makes that infinite structure manageable.
Quantum optics, microwave cavities, mechanical resonators and continuous-variable quantum information all reuse the same mathematical object. Instead of beginning with an arbitrary infinite-dimensional matrix, we organise the space using a pair of noncommuting operators a and a† satisfying a simple commutation relation. From that relation come the Fock basis, number operator, quadratures, uncertainty relations and coherent-state machinery developed in later guides.
This guide uses the convention ℏ=1 and dimensionless quadratures with [x,p]=i. Under this convention the vacuum variance of each quadrature is 1/2. Other books use quadratures differing by factors of √2; formulas must be translated consistently rather than mixed.
Canonical pair → ladder operators → number states → oscillator spectrum → quadratures → phase space.
1. Start from the harmonic oscillator
The classical harmonic oscillator has position q, momentum p and Hamiltonian
H=p²/(2m)+(mω²q²)/2.
Quantisation promotes q and p to operators obeying
[q,p]=iℏ.
After setting ℏ=1 and rescaling to dimensionless variables x and p, the algebra becomes [x,p]=i.
The oscillator is important far beyond a literal mass on a spring. Each independent normal mode of the electromagnetic field behaves mathematically as one oscillator. Quantising light therefore produces one bosonic oscillator for each optical mode.
2. Define annihilation and creation operators
With the quadrature convention above, define
a=(x+ip)/√2
and
a†=(x−ip)/√2.
Using [x,p]=i, calculate
[a,a†]=1.
This compact relation is the canonical commutation relation of one bosonic mode.
3. Derive the commutator explicitly
Expand
[a,a†]=(1/2)[x+ip,x−ip].
Terms [x,x] and [p,p] vanish. The cross terms are
(1/2)(−i[x,p]+i[p,x]).
Since [p,x]=−[x,p]=−i, the expression becomes
(1/2)(−i·i+i·(−i))=(1/2)(1+1)=1.
The entire oscillator ladder follows from this noncommutativity plus one normalisable lowest-energy state.
4. Number operator
Define the number operator
N=a†a.
It is positive because for every state |ψ⟩,
⟨ψ|N|ψ⟩=||a|ψ⟩||²≥0.
The commutators
[N,a]=−a[N,a†]=a†
show that a lowers number by one unit and a† raises it by one unit.
5. Fock states
Number states, or Fock states, satisfy
N|n⟩=n|n⟩
for n=0,1,2,….
The vacuum |0⟩ is defined by
a|0⟩=0.
Repeated creation gives
|n⟩=(a†)^n|0⟩/√(n!).
The factorial normalisation is forced by the ladder algebra.
6. Ladder action
The normalised actions are
a|n⟩=√n |n−1⟩
and
a†|n⟩=√(n+1)|n+1⟩.
The square-root factors distinguish bosonic occupation from a simple shift register. They arise because repeated creation changes normalisation.
7. Worked ladder example
Apply a² to |5⟩:
a|5⟩=√5|4⟩
and then
a²|5⟩=√5√4|3⟩=√20|3⟩=2√5|3⟩.
Likewise
(a†)²|5⟩=√6√7|7⟩=√42|7⟩.
Operator order matters when raising and lowering are mixed.
8. The oscillator Hamiltonian
Substituting the ladder definitions into the oscillator Hamiltonian gives
H=ω(N+1/2)
in ℏ=1 units.
Therefore
E_n=ω(n+1/2).
The energy levels are equally spaced. The vacuum still has nonzero energy ω/2. This zero-point energy follows from the noncommutativity of x and p; both quadratures cannot simultaneously have zero variance.
9. Why the number spectrum cannot extend below zero
Suppose N had a normalised eigenstate with negative eigenvalue n<0. Then
⟨N⟩=||a|n⟩||²=n<0,
contradicting positivity of a†a.
The ladder must terminate at a lowest state annihilated by a. Repeated raising then generates the nonnegative integer spectrum.
10. Quadratures from ladder operators
Invert the definitions:
x=(a+a†)/√2
and
p=(a−a†)/(i√2).
These Hermitian operators are continuous observables. In optical language they correspond to field-amplitude quadratures analogous to cosine and sine components of a classical wave.
11. Vacuum quadrature statistics
For |0⟩,
⟨x⟩=⟨p⟩=0.
Using aa†=a†a+1,
⟨x²⟩=⟨p²⟩=1/2.
Thus
Var(x)=Var(p)=1/2.
The Heisenberg product is ΔxΔp=1/2, saturating the uncertainty relation for [x,p]=i.
12. Number-state quadrature variance
For Fock state |n⟩, symmetry gives zero quadrature means. Direct calculation yields
Var(x)=Var(p)=n+1/2.
As number increases, the phase-space distribution spreads radially even though the mean field remains zero.
A number state therefore has sharply defined photon number but poorly defined optical phase structure compared with a coherent state.
13. Rotated quadratures
Define a quadrature at phase angle θ by
x_θ=(ae^{-iθ}+a†e^{iθ})/√2.
At θ=0 this is x. At θ=π/2 it is p.
Homodyne detection measures such rotated quadratures by comparing a signal mode with a strong phase-referenced local oscillator.
14. Phase rotations
The unitary phase rotation
R(φ)=e^{-iφN}
acts on the annihilation operator as
R†aR=e^{-iφ}a.
Consequently the quadrature vector rotates in phase space:
(x,p) → (x cosφ+p sinφ, −x sinφ+p cosφ)
under the stated convention.
15. Matrix elements of x
Because x contains one raising and one lowering term, it connects only neighbouring Fock states:
⟨m|x|n⟩=(√n δ_{m,n−1}+√(n+1)δ_{m,n+1})/√2.
This tridiagonal structure is a useful computational check when truncating an oscillator Hilbert space for numerical simulation.
16. Truncation is an approximation, not a new oscillator
Numerical software often represents a bosonic mode by the finite basis |0⟩,…,|nmax⟩. The truncated creation operator cannot satisfy [a,a†]=I exactly at the top boundary.
This is unavoidable: no finite-dimensional matrices can have commutator equal to the identity because every finite-dimensional commutator has trace zero while Tr(I)>0.
Convergence therefore requires increasing the cutoff until the populated tail near nmax is negligible for the task.
17. Worked truncation diagnostic
Suppose a simulated state has photon probabilities P(0)…P(20), and the total probability in levels 18–20 is 2×10−6.
That small tail suggests the cutoff 20 may be adequate for observables insensitive to rare high-energy events.
If the tail is 0.03, the boundary is being populated significantly. Increasing nmax is necessary before trusting commutators, energies or quadrature statistics.
18. Multimode bosonic systems
For modes j=1,…,M, introduce operators aj satisfying
[a_j,a_k†]=δ_jk
and
[a_j,a_k]=[a_j†,a_k†]=0.
A multimode Fock state is written |n₁,n₂,…,nM⟩. Its total occupation is Σnj if the number operators are summed.
Passive linear optics mixes modes while preserving total photon number. Active Gaussian operations such as squeezing can create or annihilate photons in pairs while preserving different symplectic structures.
19. Identical bosons and occupation-number notation
Fock notation does not label individual photons as particle 1, particle 2 and so on. It labels how many identical bosons occupy each mode.
For two modes, |2,1⟩ means two bosons in the first mode and one in the second. The state is automatically symmetrised because it was built with commuting creation operators.
This occupation-number language is ideal for quantum optics, where individual identical photons are not naturally tracked by persistent identity labels.
20. Thermal state
A single-mode thermal state with mean occupation n̄ has geometric Fock probabilities
P(n)=n̄^n/(n̄+1)^{n+1}.
Its quadrature means vanish and
Var(x)=Var(p)=n̄+1/2.
The vacuum is the zero-temperature limiting case n̄=0.
21. Coherent states preview
A coherent state |α⟩ is an eigenstate of the annihilation operator:
a|α⟩=α|α⟩.
Its Fock expansion is
|α⟩=e^{-|α|²/2}Σ_{n=0}∞ α^n|n⟩/√(n!).
Photon number is Poisson distributed with mean |α|². Guide 39 develops coherent states as the quantum-optical analogue of classical field amplitudes.
22. Worked coherent-number preview
If |α|²=4, then
P(n)=e^{-4}4^n/n!.
For n=0,1,2,3,4 the approximate probabilities are 0.0183, 0.0733, 0.1465, 0.1954 and 0.1954.
The most likely photon numbers sit around the mean, but number remains uncertain even though the optical phase-space displacement is well defined.
23. Quadrature uncertainty is not instrumental ignorance
The relation
ΔxΔp≥1/2
follows from operator noncommutativity. It is not merely a statement that a technician has not yet built a sufficiently precise simultaneous meter.
Squeezed states redistribute uncertainty: one quadrature can fall below vacuum variance while the conjugate quadrature expands so the uncertainty principle remains satisfied. Guide 38 develops this with covariance matrices.
24. Common misconception: a† creates a distinguishable particle with an identity
Creation operators increase occupation of a mode. The resulting bosons are identical; Fock space does not attach persistent labels to individual particles.
25. Common misconception: vacuum means every observable is zero
Vacuum has zero mean photon number and zero mean quadratures, but nonzero quadrature variances and zero-point energy in the oscillator Hamiltonian.
26. Common misconception: finite matrix truncation preserves the bosonic algebra exactly
No finite-dimensional representation satisfies [a,a†]=I exactly. A cutoff is a numerical approximation whose validity must be checked against boundary population and observable convergence.
27. Worked synthesis problem
Let
|ψ⟩=(|0⟩+√2|1⟩+|2⟩)/2.
The state is normalised because the squared coefficient sum is (1+2+1)/4=1.
Step 1: Mean number.
⟨N⟩=0·(1/4)+1·(2/4)+2·(1/4)=1.
Step 2: Apply a.
a|ψ⟩=(√2|0⟩+√2|1⟩)/2=(|0⟩+|1⟩)/√2.
The squared norm of a|ψ⟩ is one, matching ⟨N⟩.
Step 3: Mean x. Since x=(a+a†)/√2 and the state has real adjacent amplitudes, ⟨a⟩=1/(2√2)+1/2≈0.853553. Therefore ⟨x⟩=√2⟨a⟩=1/2+1/√2≈1.207107.
Step 4: Interpretation. Number is uncertain while the field quadrature has a nonzero displacement. Number-space and quadrature-space descriptions reveal different structure of the same state.
28. Practice set
- State the canonical commutator [x,p] in this guide’s convention.
- Define a and a† from x and p.
- State [a,a†].
- Define the number operator.
- What is the action of a on |n⟩?
- What is the action of a† on |n⟩?
- Write the oscillator energy E_n in ℏ=1 units.
- What are the vacuum quadrature variances?
- What is a rotated quadrature x_θ?
- Why can no finite-dimensional truncation exactly satisfy [a,a†]=I?
- What does |2,1⟩ mean in a two-mode Fock basis?
- How does a thermal state’s quadrature variance depend on mean occupation?
Answers
[x,p]=i.a=(x+ip)/√2anda†=(x−ip)/√2.- 1.
N=a†a.√n|n−1⟩.√(n+1)|n+1⟩.ω(n+1/2).- 1/2 and 1/2.
(ae^{-iθ}+a†e^{iθ})/√2.- The trace of a finite commutator is zero, while the trace of identity is nonzero.
- Two indistinguishable bosons occupy mode one and one occupies mode two.
n̄+1/2in each quadrature for an unsqueezed thermal state.
Sources and further study
[1] Christian Weedbrook and colleagues, Gaussian Quantum Information, Reviews of Modern Physics 84, 621 (2012). Sections on bosonic modes, quadratures, phase space and Gaussian states provide a standard continuous-variable convention map.
[2] Roy J. Glauber, Coherent and Incoherent States of the Radiation Field, Physical Review 131, 2766 (1963). A foundational treatment of coherent states and photon-field statistics built on oscillator/Fock-space methods.
[3] Reinhold A. Bertlmann and Nicolai Friis, Quantum States of Light in Modern Quantum Theory (Oxford University Press, 2023). A modern account connecting ladder operators, Fock space, coherent states and Gaussian phase-space methods.
Continue through Quantum Mathematics
Guide 38: Gaussian Quantum States, Covariance Matrices, Symplectic Transformations and Squeezing organises continuous-variable states by first and second moments. Guide 39: Quantum Optics, Coherent States, Beam Splitters, Interferometry and Photodetection turns mode algebra into optical experiments. Guide 40: Continuous-Variable Quantum Information, Gaussian Channels, Homodyne Detection and Teleportation develops communication and processing.