Second Quantization

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

\[% https://latex.wikia.org/wiki/List_of_LaTeX_symbols % https://www.overleaf.com/learn/latex/Main_Page % % latex commands for quantum mechanics: bra & kets \newcommand{\bra}[1]{\left<#1\right|} \newcommand{\ket}[1]{\left|#1\right>} \newcommand{\bk}[2]{\left<#1\middle|#2\right>} \newcommand{\bke}[3]{\left<#1\middle|#2\middle|#3\right>} % % general shortcuts \newcommand{\bm}[1]{\boldsymbol{#1}} % bold math \newcommand{\super}[2]{#1 {}^{#2}} % superscript \newcommand{\half}{\frac{1}{2}} % % hats together with subscripts or superscript (e.g. for angular momentum) \newcommand{\hatb}[1]{\bm{\hat{#1}}} % hat + bold \newcommand{\hatsub}[2]{\hat{{#1}_{#2}}} % hat + subscript \newcommand{\hatsup}[2]{\super{\hat{#1}}{#2}} % hat + superscript \newcommand{\hatsubsup}[3]{\super{\hat{#1}}{#3}_{#2}} % hat + sub + superscript % % Pauli operators \newcommand{\pauliX}{\hatsubsup{\sigma}{X}{}} \newcommand{\pauliY}{\hatsubsup{\sigma}{Y}{}} \newcommand{\pauliZ}{\hatsubsup{\sigma}{Z}{}} \newcommand{\pauliP}{\hatsubsup{\sigma}{+}{}} \newcommand{\pauliM}{\hatsubsup{\sigma}{-}{}} \newcommand{\pauliPM}{\hatsubsup{\sigma}{\pm}{}} % % derivates \newcommand{\odv}[2]{\frac{\textrm{d} #1}{\textrm{d} #2}} \newcommand{\pdv}[2]{\frac{\partial #1}{\partial #2}}\]

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

\[- \nabla^2 \vec{A} + \frac{1}{c^2} \frac{\partial^2 \vec{A}}{\partial t^2} = \mu_0 \vec{J_T} \quad \textrm{and} \quad \frac{1}{c^2} \frac{\partial}{\partial t} \nabla{\Phi} = \mu_0 \vec{J_L}\]

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

\[- \nabla^2 \vec{A} + \frac{1}{c^2} \frac{\partial^2 \vec{A}}{\partial t^2} = 0\]

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

\[\mathcal{E_{k \lambda}} = \epsilon_0 V \omega_k^2 \left( A_{k \lambda} A_{k \lambda}^* + A_{k \lambda}^* A_{k \lambda} \right)\]

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

\[A_{k \lambda} \rightarrow \left( \hbar / 2 \epsilon_0 V \omega_k \right) ^ {1/2} \hatsubsup{a}{k \lambda}{} \quad \textrm{and} \quad A_{k \lambda}^* \rightarrow \left( \hbar / 2 \epsilon_0 V \omega_k \right) ^ {1/2} \hatsubsup{a}{k \lambda}{\dagger}\]

and we define the corresponding Hamiltonian as

\[\hatsubsup{H}{k \lambda}{} = \frac{1}{2} \hbar \omega_k \left( \hatsubsup{a}{k \lambda}{} \hatsubsup{a}{k \lambda}{\dagger} + \hatsubsup{a}{k \lambda}{\dagger} \hatsubsup{a}{k \lambda}{} \right)\]

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 \(\hbar \omega_i\):

\[\hatsubsup{H}{A}{} = \sum\limits_i \hbar \omega_i \ket{i} \bra{i}\]

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 \(\pauliPM = \frac{1}{2} \left( \pauliX \pm i \pauliY \right)\)

\[\hatsubsup{H}{A}{} = \frac{\hbar \omega_A}{2} \pauliZ = \omega_A \left( \pauliP \pauliM - \frac{1}{2} \right)\]

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

\[\hatsubsup{H}{R}{} = \frac{1}{2} \hbar \omega_k \left( \hatsubsup{a}{k \lambda}{\dagger} \hatsubsup{a}{k \lambda}{} + \frac{1}{2} \right)\]

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

\[\hatsubsup{H}{AR}{} = - i \hbar \frac{\Omega_0}{2} \left( \hat{a} \pauliP - \hatsup{a}{\dagger} \pauliM \right)\]

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.

References

• “The field oscillator”, section 3.1, and the “Jaynes-Cummings model”, section 3.4 [B2]

• “Quantization of the radiation field”, chapter 4, sections 4.2, 4.4, 4.9 [T5]