From Ladder Operators to Photons: The Harmonic Oscillator → Quantum Optics Bridge

Introduction

One of the most elegant connections in quantum physics is the bridge between the quantum harmonic oscillator (QHO) and quantum optics. The mathematical framework we use to describe a simple mass on a spring reappears, almost unchanged, when we describe light at the quantum level.

TL;DR — The same operators that raise and lower energy levels in the quantum harmonic oscillator become photon creation and annihilation operators when you quantize a mode of the electromagnetic field. The mathematics stays identical—only the physical interpretation changes.

1. The Quantum Harmonic Oscillator and Ladder Operators

The Starting Point

The one-dimensional quantum harmonic oscillator (QHO) describes a particle bound by a quadratic potential. Its Hamiltonian is:

\hat H=\frac{\hat p^2}{2m}+\frac12 m\omega^2 \hat x^2

This looks messy, but we can simplify it dramatically by introducing ladder operators.

Defining the Ladder Operators

We define two special operators, $\hat a$ (lowering) and $\hat a^\dagger$ (raising):

\hat a=\sqrt{\frac{m\omega}{2\hbar}}\!\left(\hat x+\frac{i}{m\omega}\hat p\right),\qquad \hat a^\dagger=\sqrt{\frac{m\omega}{2\hbar}}\!\left(\hat x-\frac{i}{m\omega}\hat p\right)

These operators satisfy the fundamental canonical commutation relation:

[\hat a,\hat a^\dagger]=1

The Elegant Result

With these operators, the Hamiltonian takes a remarkably simple form:

\hat H=\hbar\omega\!\left(\hat a^\dagger \hat a+\tfrac12\right)

The operator $\hat n=\hat a^\dagger\hat a$ is called the number operator, and it has eigenstates $|n\rangle$ with eigenvalues $n$ :

\hat n|n\rangle=n|n\rangle

We can build all energy eigenstates from the ground state $|0\rangle$ by repeatedly applying the creation operator:

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

How the Ladder Operators Act

These operators "climb" the energy ladder by adding or removing quanta:

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

The names "lowering" and "raising" operators come from this behavior—they literally lower or raise the quantum number by one.

2. Quantized Light: Oscillators Become Photons

The Same Mathematics, New Physics

Here's where the magic happens: when we quantize the electromagnetic field, a single mode (with frequency $\omega$ ) has the exact same Hamiltonian as the quantum harmonic oscillator:

\hat H_{\text{field}}=\hbar\omega\!\left(\hat a^\dagger \hat a+\tfrac12\right)

The mathematics is identical, but the physical interpretation transforms completely:

Operator	QHO Interpretation	Quantum Optics Interpretation
$\hat a$	Lowers energy by $\hbar\omega$	Annihilates a photon
$\hat a^\dagger$	Raises energy by $\hbar\omega$	Creates a photon
$\vert n\rangle$	Energy eigenstate	Fock state with $n$ photons
$[\hat a,\hat a^\dagger]=1$	Canonical commutation	Same algebra, different meaning

Why This Works

In quantum field theory, we decompose the electromagnetic field into independent modes (like Fourier modes). Each mode oscillates at a specific frequency, just like a harmonic oscillator. When we quantize the field, these oscillations become quantized—and the quanta are photons.

The ladder operators of the QHO become the photon creation and annihilation operators. The "particle" that gets created or destroyed is no longer a material particle on a spring—it's a quantum of the electromagnetic field itself.

3. The Complete Electromagnetic Field

Many Modes, Many Oscillators

A single mode is just the beginning. The full electromagnetic field contains infinitely many modes, each characterized by:

A wavevector $\mathbf{k}$ (direction and wavelength)
A polarization state
A frequency $\omega_k = c|\mathbf{k}|$

The complete Hamiltonian is a sum over all these independent modes:

\hat H=\sum_k \hbar\omega_k\!\left(\hat a_k^\dagger \hat a_k+\tfrac12\right)

Each mode has its own pair of creation and annihilation operators, and they satisfy:

[\hat a_k,\hat a_{k'}^\dagger]=\delta_{kk'}

This means operators from different modes commute—each mode behaves like an independent quantum harmonic oscillator. The electromagnetic field is essentially an infinite collection of independent oscillators, one for each possible mode.

4. Quadratures, Coherent States, and Quantum Measurements

Field Quadratures: Phase Space for Light

Just as the harmonic oscillator has position and momentum, the electromagnetic field has analogous observables called quadratures:

\hat X=\frac{1}{\sqrt{2}}(\hat a+\hat a^\dagger),\qquad \hat P=\frac{1}{i\sqrt{2}}(\hat a-\hat a^\dagger)

These satisfy the canonical commutation relation:

[\hat X,\hat P]=i

Think of $\hat X$ and $\hat P$ as the "position" and "momentum" of the field mode—they define a phase space for light, with the same quantum uncertainty as mechanical systems.

Coherent States: The Quantum Description of Laser Light

Coherent states $|\alpha\rangle$ are eigenstates of the annihilation operator:

\hat a|\alpha\rangle=\alpha|\alpha\rangle

These special states have remarkable properties:

They minimize uncertainty: $\Delta X \cdot \Delta P = 1/2$ (the quantum limit)
They're the closest quantum analog to classical electromagnetic waves
They describe laser light in quantum optics
The photon number follows a Poissonian distribution with mean $\bar n=|\alpha|^2$

Coherent states are essentially the ground state of the oscillator, displaced in phase space to some complex amplitude $\alpha$ .

Squeezed States: Redistributing Quantum Noise

Squeezed states are even more exotic. They redistribute uncertainty between the two quadratures:

Reduce noise in $\hat X$ at the expense of increased noise in $\hat P$ (or vice versa)
Still respect the Heisenberg uncertainty principle: $\Delta X \cdot \Delta P \geq 1/2$
Enable quantum-enhanced measurements beyond the standard quantum limit
Used in gravitational wave detectors (LIGO) and precision metrology

All these concepts—quadratures, coherent states, squeezed states—are just relabelings of the QHO's phase-space picture, now applied to the quantum electromagnetic field.

5. The Big Picture: Same Mathematics, Different Physics

A Side-by-Side Comparison

The table below summarizes the beautiful correspondence between the QHO and quantum optics:

Mathematical Object	QHO Interpretation	Quantum Optics Interpretation
$[\hat a,\hat a^\dagger]=1$	Canonical ladder algebra	Photon creation/annihilation algebra
$\hat n=\hat a^\dagger\hat a$	Energy quantum number	Photon number operator
$\hat X,\hat P$	Scaled position/momentum	Electric and magnetic field quadratures
$\\|n\rangle$	$n$ -th energy eigenstate	Fock state with exactly $n$ photons
$\\|\alpha\rangle$	Displaced ground state	Coherent state (laser light)

The Key Insight

Quantization maps each electromagnetic field mode to a harmonic oscillator. The mathematical structure is preserved completely:

The ladder operators remain ladder operators
The commutation relations stay the same
The energy spectrum has the same structure

What changes is what the operators act on:

In the QHO: a particle in a potential
In quantum optics: the electromagnetic field itself

The bridge between these theories is purely algebraic. The physics—the actual observable phenomena—is in the interpretation of these operators, not in their mathematical properties.

This is one of the most profound examples of mathematical unification in physics: the same abstract structure describes both mechanical oscillations and the quantum nature of light.

Appendix: Mathematical Reference

Number (Fock) States

The number basis forms the foundation of quantum optics:

\langle n|\hat a^\dagger \hat a|n\rangle=n,\qquad \hat a|0\rangle=0

The vacuum state $|0\rangle$ is annihilated by $\hat a$ , and has zero photons but non-zero energy $E_0 = \hbar\omega/2$ (vacuum energy).

Coherent State Expansion

A coherent state can be expanded in the Fock basis:

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

The expected photon number in this state is:

\langle \hat n \rangle_{|\alpha\rangle}=|\alpha|^2

This shows that the amplitude $\alpha$ directly relates to the intensity of the light (since intensity is proportional to photon number).

Squeezing Operator

The single-mode squeezing operator is defined as:

\hat S(\zeta)=\exp\!\left[\tfrac12(\zeta^\ast \hat a^2-\zeta\,\hat a^{\dagger 2})\right],\quad \zeta=re^{i\phi}

where $r$ is the squeezing parameter and $\phi$ determines the squeezing angle. Applying this operator to the vacuum state produces a squeezed vacuum state, which has reduced noise in one quadrature at the expense of increased noise in the conjugate quadrature.