3.6

# Atomic energy level

The simplest quantum system is the hydrogen atom, with a single proton for a nucleus and a single electron orbiting it. In an undergraduate quantum mechanics course, students are often asked to derive its size and behavior using Schroedinger’s equation. The position of the electron isn’t fixed, like a satellite in an orbit, but instead is a quantum probability wave in a standing wave of the kind we studied in the first week, known as an orbital. For an isolated hydrogen atom, both the ground and excited states derive from the simplest standing wave we demonstrated, and are spherical. For most atoms, the shapes are more complex, as they represent standing waves with a larger number of waves.

## Simple energy levels

An atom has a minimum energy level, which we call the ground state. If it has absorbed energy from the environment, usually by absorbing a photon, we say that the atom is in an excited state. It will eventually release that energy by emitting a photon whose wavelength is determined by the amount of energy released. The atom can only be in one of a fixed set of energy levels, so it always emits a photon of a particular wavelength as it moves from one state to another. Starting in the ground state and absorbing smallest possible amount of energy, we say it moves to the first excited state. Second and higher excited states are possible, too.

We can create a qubit using the ground state and the first excited state. Just like we used the up and down arrows when talking about spin, we can write $g$ and $e$ inside our kets to describe the physical phenomenon, then map those states to data values for our computations. We can use the ground state $|g\rangle$ as our $|0\rangle$ state, and $|e\rangle$ as our $|1\rangle$ state.

## Hyperfine levels

Ground and excited states have the advantage of being conceptually simple, but a second approach is also common. In the presence of magnetic fields, the behavior of an atom is more complex than we just described, especially when the atom has more protons in its nucleus and more electrons than a hydrogen atom. The energy level is primarily defined by the distance between the nucleus and the electron, but magnetic fields distort the shape of the nucleus and the electron’s orbital. Our single, well-defined energy level “splits” into two or more slightly different energy levels.

Because the energy levels of the hyperfine states are very close together, it takes less energy to move the qubit from one state to the other. We can use microwaves to control hyperfine states, rather than laser light.

We will discuss the strengths and weaknesses further when we discuss ion trap hardware.