Feynman diagrams and perturbative expansion of the time evolution operator

In Module 1 (Lectures 1-?), you have learned the basic properties of photon and how to describe various states of light using photons and their statistics. In Module 2, you will learn how to describe atom-light interactions as exchanges of photons. Formally, this can be seen from the perturbative expansion of the unitary time evolution operator. Feynman diagrams help visualize the photon exchanges that correspond to various mathematical terms occuring in such perturbative expansion.

Virtual photons and virtual states

In atomic physics, you often hear the term "virtual state" that is connected to the initial state via an exchange of a "virtual photon." What does the term "virtual" mean? How do virtual states and virtual photons differ from real ones? You will see that the term "virtual" arises from energy non-conservation. But it doesn't mean that virtual states can be any random state - specifically, if we consider a virtual state connected from the ground state, it has the character of the excited state the ground state was "trying" to connect to. But its quantum probability amplitude carries an energy denominator due to energy non-conservation.

Let's first consider two familiar processes: one-photon absorption and emission. Consider the case where the photon is resonant with the atomic transition. In the figures below, the resonant photon is represented by a wiggly arrow which fully connects the ground state from/to the excited state. One certainly expects these processes to be "real": for example, when an excited atom spontaneously decays and emits a photon, one can "catch" the emitted photon with a photodetector and measure its energy.

• Resonant one photon absorption.png

  (1.1) a resonant absorption process

• Resonant one photon emission.png

  (1.2) a resonant emission process

Feynman diagrams with time axis

In the previous section, you have seen how to visualize a basic two-photon interaction involving a ground state and a virtual state. In this section, you will visualize the same process but in a slightly different way, so that the the time dependence (and ordering) of the photon exchange is manifest.

Perturbative calculation of transition amplitudes

In this section, you will learn how to carry out perturbative expansion of the unitary time evolution operator $\tilde{U}(t_f, t_i)$ in the interaction picture and identify $n^{th}$ order terms with $n-$ photon exchange terms.