# Second Quantization¶

*The references that inspired this chapter are all mentioned in the*
References
*section*.

## Quantization of the electromagnetic field¶

The Maxwell equations for the electric and magnetic fields \(\vec{E}\) and \(\vec{B}\) describe the evolution of electromagnetic fields. They can also be expressed in terms of the vector and scalar potentials \(\vec{A}\) and \(\Phi\) from which \(\vec{E}\) and \(\vec{B}\) can be derived [T5] (see also wikipedia).

Assuming the Coulomb gauge condition and introducing the current density \(J\),
the evolution of the fields is governed by following equations for
the *transverse* (subscript \(_T\)) and the *longitudinal* (subscript \(_L\)) components

For free electromagnetic waves, \(\vec{J_T} = 0\), resulting in

By expanding the vector potential as a sum of contributions from modes in a virtual cavity,
we can obtain [T5] following equation for the **contribution of an individual mode**
(characterized by their wave vector \(\vec{k}\) and their spatial coordinate index \(\lambda\))
to the **total radiative energy**

with \(\epsilon_0\) the permittivity of free space and \(V\) the volume.

This equation strikingly resembles the algebraic formulation of the harmonic oscillator. By virtue of the correspondence principle we make following associations

and we define the corresponding Hamiltonian as

## Quantized atom-field interaction¶

The Hamiltonian of an atom interacting with an electromagnetic field can be expressed as the sum of the atomic, radiative and coupling Hamiltonians \(\hatsubsup{H}{A}{}, \, \hatsubsup{H}{R}{}, \, \hatsubsup{H}{AR}{}\).

The atomic Hamiltonian takes into account the atom’s electronic energy levels. It can be formulated in terms of its eigenvectors \(\ket{i}\) with eigenvalues \(\hbar \omega_i\):

In case of a two-level system, e.g. by considering only two energy levels of the atom,
the **atomic Hamiltonian** involves the
Pauli operator
\(\pauliZ\) and can be rewritten in terms of the raising and lowering operators
\(\pauliPM = \frac{1}{2} \left( \pauliX \pm i \pauliY \right)\)

This form of the Hamiltonian is known as **second quantization** [T5].

The **field Hamiltonian** involves the creation \(\hatsup{a}{\dagger}\) and annihilation \(\hat{a}\)
as described above (or the number operator \(\hat{n}\)) i.e. for one mode

Finally the **coupling Hamiltonian** describes the atom’s **electric dipole** coupling
to the electromagnetic field.
By using the Rotating Wave Approximation, this Hamiltonian reduces to

with the *vacuum Rabi frequency* \(\Omega_0\) [B2].

The complete Hamiltonian as described in this section is called the **Jaynes-Cummings Hamiltonian**,
named after Jaynes and Cummings who introduced it as an idealization of the matter-field coupling
in free space.

The eigenstates of this interaction Hamiltonian are generally entangled states
denoted by \(\ket{+, n}\) and \(\ket{-, n}\) and called the **dressed states**
of the atom-field system.
The dressed states are clearly distinguished from the uncoupled states at atom-field resonance.