# 13 Ion-based quantum computers

So far we have looked at the principles of quantum mechanics and how to transfer these principles to our mathematical description of quantum computing. While there are many different approaches on how to actually build a quantum computer, which are researched at the moment, we will only look at one approach. This approach is based on trapped ions.

## 13.1 Electron in an atom

We look at a single atom with a nucleus with a positive charge and a single electron with negative charge “orbiting” the nucleus.

The electromagnetic field generated by the nucleus is essentially a potential well for the electron, since the electron is drawn to the nucleus and the potential of the electron rises with bigger distance from the nucleus. We simplify a lot here and ignore ,e.g., the spin.

We can solve the time-independent Schroedinger equation for this setup and by solving this, we will get the wave functions that are the energy eigenstates of the Hamiltonian. These wave functions are called *orbitals*.

We can use a single atom as a qubit, where we define one of the energy eigenstates as \(\ket{0}\) and a different eigenstate as \(\ket{1}\).

In the following, we will specifically use electrically charged atoms, called ions, because they are easier to capture.

## 13.2 Setup for the ion traps

The setup for our quantum computer looks as follows:

In the previous chapter, we have learned that we need to be able to perform three different operations to build a quantum computer:

- We need to initialize (cool) our qubit.
- We need to be able to apply a unitary on the qubit.
- We need to be able to measure the qubit.

### 13.2.1 Cooling

We first look into cooling our system. For cooling, we use a useful fact: If \(E_i < E_j\) are different possible energy levels, an ion is in the energy level \(E_i\) and then hit by a photon that has the energy \(E_j-E_i\), the ion will go to energy level \(E_j\).

#### 13.2.1.1 Doppler cooling

In our initial setup, we have an ion vibrating back and forth because it has too much energy. We shine a laser on it with slightly less energy than what is need for a transition. The energy of the photon of this laser is denoted by \(E=\hbar \cdot \omega\). When the ion moves towards the photon, the photon has a higher frequency from the point of view of the ion (Doppler effect). This means that the photon has a higher energy and therefore is more likely to be absorbed.

So by shining a laser on the ion, the photons of the laser “push” the ion, when it “swings” *towards* the laser similar to a pendulum, where the pendulum gets a pushback with just enough energy so it stops. This reduces the vibrations energy down to a certain level.

#### 13.2.1.2 Sideband cooling

Using the doppler cooling, we have reduced the vibration energy, but the electrons may still be excited. We now look at another technique called sideband cooling, which will set the energy of the electrons to a specific energy level \(E_0\).

The electron can have any energy level \(E_i\). If this energy level is pretty low, the possibility of a spontaneous emission of a photon, which would reduce the energy to a lower level is also quite low. So an electron with energy level \(E_1\) or \(E_2\) will probably not change to level \(E_0\). If the energy level is big enough (we will call this energy level \(E_{\text{big}}\)), the probability of a spontaneous emission of a photon, which would reduce the energy to a lower level is quite high. So when this spontaneous emission happens, the electron will reach an energy level of ,e.g., \(E_0\), \(E_1\) or \(E_2\).

Our goal is to get the energy level to \(E_0\). We know that the energy level of the electrons with \(E_{\text{big}}\) will come down eventually, so we shine a laser with the energy per photon of \(E_{\text{big}} - E_1, E_{\text{big}} - E_2, \dots\) but not with \(E_{\text{big}} - E_0\). This will “shoot” all the low energy electrons from \(E_1,E_2,\dots\) to a higher lever where they will either fall down to \(E_1,E_2,\dots\) where they will be energized again and the process is repeated, or they fall into \(E_0\) which is our desired energy level.

Using the sideband and the doppler cooling together, we can cool the vibration and electron excitation. This means that all the ions are in the state \(\ket{0}\) and the vibrations energy is also in the state \(\ket{0}\).

### 13.2.2 Unitaries

This section will be updated later.

### 13.2.3 Measurements

We now look into performing a measurement on an ion-based quantum computer. So far we have defined that the quantum state \(\ket{0}\) is represented by the energy level \(E_0\) and the quantum state \(\ket{1}\) is represented by the energy level \(E_1\).

To perform a measurement, we also use an auxiliary energy level \(E_{\text{aux}} > E_0,E_1\). Let \(\omega\) be the frequency of a photon with energy \(E_\text{aux}-E_0 (E_\text{aux}-E_0=\Delta \omega)\)

We now shine a laser with the frequency \(\omega\) on the ion. If the state is \(\ket{0}\), the photon gets absorbed and the electron jumps to \(E_\text{aux}\) and from there spontaneously back to \(E_0\).This process then repeats over and over emitting many photons. This will create fluorescence, which can be measured by light detectors. If the state is \(\ket{1}\), no ions get absorbed and no fluorescence can be seen.

So all in all, we measure an ion by shining a photon of frequency \(\omega\) onto it and then look whether it lights up.