Ref [1] makes a point of explaining why we have quantization of light by introducing Maxwell’s equations in a quantized way, for a one-dimensional (1D) cavity, but we will see that the result can be simply extended to higher dimensions.

Starting from Maxwell's equations, the vector potential is decomposed into normal modes and each mode is treated as a quantum harmonic oscillator. Promoting the mode amplitudes to operators gives the annihilation and creation operators $\hat{a}$ and $\hat{a}^\dagger,$ and the key result is the single-mode Hamiltonian:

$\hat{H} = \hbar\omega\left(\hat{a}^\dagger\hat{a} + \frac{1}{2}\right)$

The $\frac{1}{2}$is the zero-point energy — even with no photons ($⟨\hat{a}^\dagger\hat{a}⟩ = 0$), the field carries energy $\frac{1}{2}\hbar\omega$. This is physically real and gives rise to vacuum fluctuations, which set the vacuum electric field strength: $E_\text{zpf} = \sqrt{\frac{\hbar\omega}{2\varepsilon_0 V}}$ where $V$ is the mode volume.

You can store magnetic flux and charge in both a qubit and in a resonator, and so we're going to quantize those two elements using the variables of charge and magnetic flux.

How is it achieved?

• Confine atom and photon in a cavity (cavity QED)
• Goal: control the interaction somehow, engineer matter/light interactions, e.g. in solid-state circuits (circuit QED)

Figure 2.1 from [1]

Cavity QED - Controlling Light-Matter Interactions:

We use Chapter 2 of Ref [1] for a basic theoretical discussion about the light-matter interaction and preliminary theory for measurement and characterization of superconducting circuits.

What is the aim of cavity QED?

• CQED allows us to work with individual quanta, so that a single photon can coherently interact with an atom
• CQED allows us to suppress, enhance, and even make an excitation coherent

We consider the simplest example of the light-matter interaction where a two-level quantum system (a qubit) interacts only with a single mode of light. In practice, this situation can be achieved by placing the qubit inside a cavity that supports a discrete set of modes. By a proper choice of qubit and cavity frequencies, cavity mode geometry, qubit placement and orientation, the qubit can effectively interact with only one of the modes of the cavity.

Figure 2.1 from [2]

Why is this challenging on the level of single (artificial) atoms and single photons?

• Single-photon fields $E_0$ are small in 3D

• Dipole moment d is small

  → photon/dipole interaction $\hbar g \sim dE_0$ is equally small

• Spatial and temporal mode-matching in order to control the absorption probability is difficult

How is cavity QED useful for quantum information processing?

• Isolating qubits from their electromagnetic environment, thus maintaining their addressability
• Reading out the state of qubits
• Coupling qubits to each other
• Converting stationary qubits to flying qubits