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🔬Modern Optics

🔬modern optics review

12.1 Quantization of the electromagnetic field

3 min readLast Updated on July 22, 2024

Electromagnetic fields, once thought to be continuous, are actually made up of tiny packets of energy called photons. This quantum view explains how light interacts with matter and forms the basis for many modern technologies.

Quantizing the electromagnetic field involves treating it like a quantum harmonic oscillator. This approach allows us to describe light using creation and annihilation operators, opening up new ways to manipulate and understand electromagnetic radiation.

Quantization of the Electromagnetic Field

Field quantization concept and implications

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  • Treats classical fields as composed of discrete quanta instead of continuous values
    • Allows fields to take on only specific, discrete values (quantization)
    • Contrasts with classical fields (electromagnetic field) which are treated as continuous and can have any value
  • Quantizes the electromagnetic field into photons, the fundamental quanta of light
    • Photons have specific properties (energy, momentum) determined by the frequency and wavelength of the electromagnetic wave
    • Governs the interaction between matter and the electromagnetic field through the absorption and emission of photons

Quantized electromagnetic field Hamiltonian

  • Derived using the analogy with the quantum harmonic oscillator
    • Treats electric and magnetic field components as conjugate variables, similar to position and momentum in the harmonic oscillator
    • Expresses the Hamiltonian in terms of creation and annihilation operators for each mode of the field
  • Takes the form: H=k,λωk(ak,λak,λ+12)H = \sum_{\mathbf{k},\lambda} \hbar \omega_{\mathbf{k}} (a_{\mathbf{k},\lambda}^{\dagger} a_{\mathbf{k},\lambda} + \frac{1}{2})
    • k\mathbf{k}: wave vector, determines the direction and wavelength of the mode
    • λ\lambda: polarization of the mode (horizontal, vertical)
    • ωk\omega_{\mathbf{k}}: angular frequency of the mode, given by ωk=ck\omega_{\mathbf{k}} = c|\mathbf{k}|, where cc is the speed of light
    • ak,λa_{\mathbf{k},\lambda}^{\dagger} and ak,λa_{\mathbf{k},\lambda}: creation and annihilation operators for the mode (k,λ)(\mathbf{k},\lambda)
  • Represents the total energy of the quantized electromagnetic field
    • Each mode contributes an energy ωk\hbar \omega_{\mathbf{k}} for each photon in that mode, plus a ground state energy of 12ωk\frac{1}{2}\hbar \omega_{\mathbf{k}}
    • Creation and annihilation operators add or remove photons from each mode, changing the energy of the field

Properties of photons as quanta

  • Massless particles with zero rest mass
  • Always travel at the speed of light in vacuum (cc)
  • Energy: E=ωE = \hbar \omega, where ω\omega is the angular frequency of the electromagnetic wave
  • Momentum: p=kp = \hbar k, where kk is the wave number (magnitude of the wave vector)
  • Spin-1 particles with an intrinsic angular momentum of \hbar
    • Spin is related to the polarization of the electromagnetic wave
  • Exhibit wave-particle duality, displaying both wave-like and particle-like properties depending on the context

Problem-solving with quantized fields

  • Photon number states: Describe the state of the electromagnetic field using nk,λ|n_{\mathbf{k},\lambda}\rangle, where nn represents the number of photons in the mode (k,λ)(\mathbf{k},\lambda)
    • Creation operator ak,λa_{\mathbf{k},\lambda}^{\dagger} adds a photon to the mode: ak,λnk,λ=n+1n+1k,λa_{\mathbf{k},\lambda}^{\dagger}|n_{\mathbf{k},\lambda}\rangle = \sqrt{n+1}|n+1_{\mathbf{k},\lambda}\rangle
    • Annihilation operator ak,λa_{\mathbf{k},\lambda} removes a photon from the mode: ak,λnk,λ=nn1k,λa_{\mathbf{k},\lambda}|n_{\mathbf{k},\lambda}\rangle = \sqrt{n}|n-1_{\mathbf{k},\lambda}\rangle
  • Coherent states α|\alpha\rangle: Eigenstates of the annihilation operator, representing classical-like electromagnetic waves
    • Defined by ak,λα=ααa_{\mathbf{k},\lambda}|\alpha\rangle = \alpha|\alpha\rangle, where α\alpha is a complex number
    • Have a well-defined amplitude and phase, and minimize the uncertainty relation between the electric and magnetic field components
  • Squeezed states: Reduce uncertainty in one quadrature (electric field) at the expense of increased uncertainty in the other quadrature (magnetic field)
    • Generated by applying the squeezing operator S(z)=exp(12(za2za2))S(z) = \exp(\frac{1}{2}(z^* a^2 - z {a^{\dagger}}^2)) to the vacuum state or a coherent state
    • Have applications in precision measurements and quantum information processing


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AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.