Single mode quantization Multimode field quantization Plane wave

① Single mode quantization ② Multimode field quantization ③ Plane wave basis

Description of light matter interaction Model Atoms Light Classical dipole Classical field (wave) satisfying Maxwell’s eq. +Lorentz force eq. Semi classical Quantum mechanical system Eigenstates and eigenenergies known Classical field described by int. term inducing transition between eigenstates Quantum mechanical Quantized field (photons)

What are we going to learn in this lecture? Classical Quantum

Why quantizing? Interaction with the electromagnetic vacuum

I- EM field as a set of quantum harmonic oscillators

1 - Single mode quantization Field quantization recipe 1) Start from the classical problem and determine it oscillating modes. 2) Express each mode in the form of a harmonic oscillator. 3) Quantize it! Step 1 Cavity of volume closed by perfectly reflecting mirrors. Free space Classical monochromatic single mode electromagnetic field : single mode field frequency : time dependent field amplitude : wave number

Step 2 The amplitude of oscillation obeys the harmonic oscillator equation Electromagnetic energy contained in the cavity momentum Step 3 Imposing canonical commutation relations to position and momentum How de we prove this? Classical observables Operators

Diagonalization of • Quantum HO with unit mass and frequency • Annihilation and creation operators Bosons Same spectrum (eigenenergies and eigenstates) as the one of the number operator

Spectrum of • is hermitian • Number state and the concept of photon eigenvalues are real are eigenstates of the number operator is the “number of photons” or quanta of excitation in the field general state in the cavity Remarks: are not eigenstates of or annihilating one photon creating one photon A photon belongs to the whole standing wave and fills the cavity of quantization Experimental and particle view: localized packet of radiation with average energy

Why annihilation and creation operators? • Action of on the state is also an eigenstate of but with lower energy has removed one quantum of energy from the system • Action of on the state is also an eigenstate of but with higher energy adds one quantum of energy to the system (Claude Cohen-Tannoudji, …)

Expression of the electric field Heisenberg picture Electric field per photon positive frequency annihilation negative frequency creation

Expectation values of the electric field and the electromagnetic vacuum • changes the number of photons orthonormality of number states • Change the number of photons! ✓ ✓ Conserve the number of photons • Averaging over the cavity volume Finite energy • Electromagnetic vacuum state with no photons ( ) Randomly fluctuating field with zero mean and energy of half a virtual photon per mode Write original Hamiltonian in normal order Substract the vacuum contribution off the energy Can’t be detected No real photon to absorb!

2 - Multimode quantization Frequency of the mode • Multimode electric field Step 1 Amplitude of mode (write the classical field) Polarization of mode Step 2 (write each mode as HO) • Each mode Step 3 (Quantize each mode) is acting as a harmonic oscillator creation and annihilation operator of mode

• Eigenstates of the multimode Hamiltonian s-th mode occupied by photons • Energies zero point Divergent!? module!! … • Vacuum energy 1. Normal ordering to get rid of the infinite zero point energy 2. High momentum modes are too energetic and don’t see the cavity and pass through it! Regularization process Hard cutoff 3. The vacuum energy depends on the value of the separation L and acts as a sort of potential energy that induces a force on the cavity Still infinite! There must be another infinity to substract! Casimir force! Euler-Mc. Laurin formula Ramanujan (1913)

3 - Orthonormal basis is general normalised mode function (Helmholtz equation) In free space, for instance plane waves polarisation Vector potential operator (radiation gauge) and magnetic field are conjugate fields and can’t be simultaneously specified.