Source: Jonathan Hui, Medium: https://medium.com/@jonathan-hui/qc-how-to-build-a-quantum-computer-with-trapped-ions-88b958b81484

Emission

Electrons can transit from one energy level to another by absorbing or emitting a photon. We can use lasers, a coherent source of photons with a specific frequency, to control the energy level of an ion. Since the energy levels are quantized, it is restricted to specific levels.

Source: http://hyperphysics.phy-astr.gsu.edu/hbase/Bohr.html

When an electron drops to a lower energy state, it releases a photon with a frequency ω corresponding to the difference in energy.

In reverse, to jump to a higher orbit, the electron needs to absorb a photon of the corresponding frequency.

When an atom is in an excited state, it may spontaneously return to one of a lower energy state after some time by emitting a photon in a random direction. This is called spontaneous emission.

There is another kind of emission called stimulated emission. By shining a photon with energy equal to the allowed energy difference, one extra photon will be emitted in the same direction as the incoming photon.

Stimulated emission can be used to control the energy of an atom to another specific level before the spontaneous emission kicks in. In quantum computers, we use both emissions to initialize the state of a qubit.

Source: http://www.optique-ingenieur.org/en/courses/OPI_ang_M01_C01/co/Contenu_05.html

Overview of Qubit with Trapped Ions

We heat up Calcium to 800°C to form a vapor.
We bombard it with electrons to strip out the outer electron.
We trap these ions with an electrical field in a vacuum.
We laser-cool these ions to the motional ground state and use this state as the |0⟩. These ions are now close to stationary. Their states, as well as their motion, can be controlled with high precision using lasers, microwaves, or radio-frequency fields, which form the foundations of the quantum gates.
To measure the qubits’ states, we shine them with another laser with a specific calculated frequency. If the qubit is in one of the states say |1⟩, it will fluoresce else it remains dark.

The diagram below shows how we trap the ions with electrodes (in skin tone color) and control them with the lasers — the two red arrows form a 2-qubit operation. We use a CCD camera to check what ions are shining or not when measured by a laser beam.

Screenshot 2022-11-03 at 17.49.33.png

Source: https://www.nature.com/articles/nature07125

More Details

We have selected the laser-cooled motional ground state as |0⟩. So, what is |1⟩? To answer that, we need to learn the energy levels of an ion and its energy transition. The diagram below is the electron configuration of a Calcium atom in different orbits.

Screenshot 2022-11-03 at 18.01.15.png

After stripping an electron from a Calcium atom, there is only one electron left in the valence shell (the outer shell) and the ground state is 4S. The ion structure will resemble a hydrogen atom (with one electron) which has relatively simple energy levels. Such a simple structure allows us to use fewer lasers to cool the ions down to the ground state and manipulate them without ambiguity. The diagram below illustrates the ground states 4S and its nearest excited states of the Calcium ion. As mentioned before, we pick 4S as |0⟩.

Screenshot 2022-11-03 at 18.02.47.png

3D and 4P are the nearest excited states respectively and are separated from 4S by optical energy (emitted or absorbed a photon with wavelength inside the visible or near visible range). This diagram also demonstrates the lifetime of the excited states and where they may de-excite to. For instance, the excited 3D (5/2) state of the Calcium ion has a lifetime τ of 1.2 seconds, while the lifetime of the 4P state is only about 7 ns (lifetime is proportional to how quickly it can be de-excited or decay.)

Electric dipole transition v.s. quadrupole transition

To understand these energy transitions, interaction models between an atom and an electromagnetic field are built. But such a model is complex and needs further simplification and approximation. For example, the 4P to 4S transition is approximated with an electric dipole approximation, i.e. the transition is contributed by the electric part of the electromagnetic field (ignoring the magnetic part) and the atom behaves like an oscillating electric dipole.

The transition between 4S to 4P is an allowed electric dipole transition. But it has a short coherence time and therefore is a bad candidate for |1⟩. But it is good for preparing the qubit to |0⟩ because of its quick response time.

3D (5/2) to 4S transition is forbidden by the selection rule of the electric dipole transition (a change of 2 units in the orbital length of momentum is not allowed). While it is called forbidden transition, it does not mean it does not happen — we just need a more complex model. Indeed, such a transition is allowed under the electric quadrupole model. But the calculated transition probability per unit of time is far much smaller for this type of transition — the transition (or coupling) is weak.

But it can be good news. 3D (5/2) to 4S transition has a longer lifetime — 1.2 seconds. Therefore, 3D (5/2) is a good candidate for |1⟩ with a coherence time appropriate enough to complete the target quantum operations (trapped-ion quantum gates are currently operating in a microsecond range). This state is called a metastable state with a lifetime in the micro or second range — much longer than the excited states with electric dipole transition.

We can use the split |↑⟩ and |↓⟩ states above as |1⟩ and |0⟩ respectively. These two hyperfine levels are chosen because it is insensitive to the magnetic field to some extent. Hyperfine structures are extremely stable and can be treated as ground states. It decays to a lower state with the magnetic dipole transition which is very weak. The superposition formed from these states has coherence in seconds or minutes. We call this hyperfine qubits.

Both types of qubits are easy to measure. And optical qubits with non-zero nuclear spin and some hyperfine levels are insensitive to magnetic fields to a certain extent. This reduces the magnetic impacts on the environment. But all the control signal routing and switching are not easy to implement in addressing individual ions. In addition, driving high-fidelity gates on a long string of ion qubits is hard. This hurts scalability — how many qubits the system has. This is a major challenge in trapped ion computers.

Optical qubits have a shorter lifetime compared with the hyperfine qubit. It requires a narrow linewidth laser.

Hyperfine qubits are extremely long-lived, longer than optical qubits. It uses lower frequency lasers that are easier to handle and it is widely available commercially and highly accurate. But hyperfine qubits have a narrower energy difference with the presence of other hyperfine neighbors. Initializing and controlling the qubits requires more complex schemes. Because the microwave has a longer wavelength, addressing ions and producing spin-dependent forces precisely is harder. To overcome the problem, we can use Raman transition. But it takes relatively high-intensity laser light. In addition, Raman scattering may occur that can behave like measurement and the superposition may collapse.

Optical and hyperfine qubits are studied by different institutes (even though hyperfine qubit seems more popular).

Choice of ion

The energy level of an atom is uneven. This is important since the laser that excites an electron from |0⟩ to |1⟩ will not mistakenly excite it to an even higher energy state. Because the spacing is uneven, we can drive particular transitions with great simplicity. Besides Calcium, we do have other choices. The positively charged Calcium ion is well suited for a quantum computer. It has only a single electron in its valence shell with a close resemblance to a hydrogen atom. But, there are other candidates that have similar behavior and mark inside the red rectangles below.

Screenshot 2022-11-03 at 18.09.10.png

Since the energy levels for each element are different, the photon absorbed or emitted is different too. The choice of the atom determines the lasers required. For example, Yb ion with hyperfine qubit is heavily studied even though it is heavier and therefore harder (or requires more power) to manipulate. But this qubit can be manipulated with lasers in the microwave range. The laser needed in the implementation is available commercially with great accuracy. Some elements, like Hg, need to deal with deep ultraviolet wavelength. This type of electromagnetic radiation can quickly damage conventional optical fibers. In addition, Yb has a 1/2 spin for the nuclei which makes the hyperfine structure (only 4 possible states) simpler than other elements, and therefore easier to manipulate. For this reason, the hyperfine qubits for Yb become a favorite in quantum computers.

Ion Trap & Laser cooling

To manipulate ions, we need them to be as stationary as possible. We trap ions using electrical fields in a vacuum and laser-cooled them to extremely low temperatures. The ions will float inside the vacuum chamber under an electrical field at a precise location (close to stationary). Once it is done, the ions will have a temperature close to nearly absolute zero even if the chamber is operated at room temperature.

Measurement — State selective fluorescence detection

To measure the state of the qubit, we use cycling optical transition. We first calculate the frequency that can excite the ion from S to P for either |0> or |1> state. By shining the laser on the ions continuously, the atom will be excited and de-excited through emission. Since we are in the range of optical transition, a photon in the visible or near visible range is emitted.

Qubit manipulation

The energy gap between the selected hyperfine state is in the microwave range. We can shine a 12 GHz microwave on the ion to rotate the qubit and create superposition. However, the relatively long wavelengths limit spatial resolution. It is harder to address individual ions. In addition, we want to couple the motion with the spin in creating a 2-qubit gate. That requires a higher gradient in the electric field than what the microwave provides. Alternatively, we reuse the Raman transition (mentioned before) to control a qubit with two laser beams. This turns into a two-photon transition that behaves like the previous microwave.