# Introduction to Ion Trap Quantum Computing

### 1. Confining the ions

#### 1.1 Producing the ions

`Ions' are just atoms which have lost or gained one or more electrons, thus aquiring an electrical charge. In our case the ions are calcium with one electron removed. They are therefore positively charged. To get hold of some ions, we heat up a small quantity of calcium metal to around 800 degrees Celcius in high vacuum (a hundred thousand millionth of the pressure of normal air). At this temperature, the metal evaporates enough to produce a little puff of calcium vapour. We strip electrons from the atoms in this vapour by firing a beam of fast-moving electrons through the vapour. Some of the resulting ions will, by chance, happen to lie inside our ion trap. Those that do will be caught.

#### 1.2 Trap structure and electric potentials

Here is a picture of the first ion trap electrode structure to be used in our lab (it dates from 1999 - a similar structure has since been donated to the Science Museum, London). The four trapping electrodes are the central ones, they are coloured blue here. They are approximately 1 mm in diameter and 3 cm long, arranged at the corners of a square, with two short pin-like electrodes at each end, which provide confinement along the axis. The rest of the structure consists of auxilliary electrodes, threaded steel rods to support everything, and connecting wires.

If we define the axis of the trap (parallel to the long electrodes) to be the z direction, then the electric potential seen by the ions, in the xy plane, is shown in the plot below. The round flat regions are the electrodes.

At the centre of this picture you see a saddle shape: that is, a U shape in one direction (say x)) and an inverted U in the other (y). This means the ions are pushed back to the middle, i.e. confined, along x, and expelled along y. The trap works by rapidly oscillating the potential, so the confinement is alternately along x and y. Averaged over a cycle of this fast (6 MHz) oscillation, an ion at any fixed point has zero average potential energy, so it is neither expelled not confined, to first approximation. However, it does experience an oscillating force, which causes it to jiggle very slightly too and fro. This jiggling or "micromotion" has an associated kinetic energy, which is larger when the ion experiences a larger oscillating force. This oscillating force is zero at the centre of the trap, and grows larger as the ions moves off in any direction. Therefore the net result is confinement in all directions, because the ion seeks the place of minimum micromotion. In short, the oscillating field provides a minimum in total energy of the ion, at the trap centre.

We use this oscillating potential to stop the ions from escaping radially. They are confined in the z direction, that is to say, prevented from escaping along the axis, by putting a fixed positive voltage on the pin-like electrodes at either end of the trap. This completes the essential business of confining the ions. However, this is only the beginning of the story of the ion trap quantum information processor.

### 2. Controlling the motion of trapped ions

#### 2.1 Requirements

We need our ions to be approximately stationary. To be more precise, we need the thermal (ie random) kinetic energy of the ions to be small compared to the controlled kinetic energies which will be involved in the operation of the processor. How large are these controlled kinetic energies? We control the motion of the ions by using laser beams. The ions pick up and lose kinetic energy when they absorb and emit single photons. The significant energy is therefore the so-called recoil energy. This is the kinetic energy gained by a stationary ion when it absorbs or emits one photon of the relevant wavelength. In our case this energy is

0.00000000000000000000000000002 Joules.

Obviously such a small number is difficult to grasp. It is better to express it as a frequency by dividing by Planck's constant h. We thus find that the recoil energy is h R where R = 32,000 Hertz (a Hertz is a cycle per second). To have a thermal energy small compared to this, we need to cool the ions to a temperature of about a micro-Kelvin, that is, a millionth of a degree celsius above absolute zero.

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To work out the recoil energy: |

The calcium ion has atomic weight 40 and mass M=6.6×10^{-26} kg. The relevant atomic transition has wavelength 397 nm. The momentum of a photon of this wavelength is p=h/(397 nm)=1.7×10-27 kg m/s. An ion emitting such a photon recoils with this momentum and therefore with velocity 2.5 cm/s. Therefore the recoil energy is (1/2)mv^{2}=2.1×10^{-29}Joules=(32000 Hertz)h as above |

It is remarkable that one can even contemplate obtaining such low temperatures in the lab, and indeed this is the major difficulty in our experiment. We achieved it for the first time in our lab in 2004 (we were not the first world-wide). It is based on using laser light to provide cooling.

#### 2.2 Laser cooling, resonance and the Doppler effect

It might come as a surprise that a laser can cool something, but once you think about it the principle is not hard to understand. Suppose you are running along and someone throws a big football which bounces off your chest. This will slow you down! Even though the football itself carries a lot of energy, its effect was to reduce your motional energy, not increase it. The significant quantity here is not just the energy, but the momentum (the potential directional 'push'). The football carries a 'push' in one direction, which opposes your 'push', so you push it and it pushes you, slowing you both down.

The argument applies similarly to light hitting an atom. The light carries both energy and momentum. If the light is travelling opposite to the direction of motion of the atom, its momentum will oppose that of the atom, and the atom will slow down. The significant thing about laser light is that it is highly regular light. This means that even though it carries a lot of energy, it is, in a valid sense, extremely `cold', because the random component of its energy is so small, and heat is measurement of how random the energy really is: imagine an ice cube moving at a thousand miles per hour, with a lot of energy, but nevertheless very cold.

In order to make sure our ions only ever absorb light which is moving against them (to slow them down), rather than along with them (which would accelerate them), we make use of two basic physical principles: resonance, and the Doppler effect.

**Resonance:**

This the ubiquitous phenomenon in which a relatively weak oscillating disturbance has a large effect on a system, when the oscillation has just the right frequency.

For example, the strings of a piano all resonate at different frequecies. If you press the foot pedal to lift all the dampers on a piano, and then sing one note towards the piano strings, you can notice the following: when you stop singing, one of the strings will have been set in motion much more than all the others, and you will hear it vibrating. This is the string which was resonant at the frequency of the note you sang.

The charged electron cloud in any atom or ion has similar resonant properties, only now the interaction is with light rather than sound, and the resonant frequencies are much higher. As a result, when our calcium ions are illuminated by a laser beam, they only absorb (and re-emit) the light when the latter has exactly the right frequency (the frequency of oscillation of light waves is what determines their colour). The precision of this resonant phenomenon is remarkable: a frequency change of only one part in ten thousand million would prevent the atom absorbing the light. In our case the resonant light is violet and has a frequency of eight hundred million million oscillations per second.

**The Doppler effect:**

The second ingredient we make use of is the Doppler effect. This is the effect whereby if you move towards a source of sound (or it moves towards you, for example the engine note of an approaching car), you will hear the sound at a higher note, while if you move away the pitch of the note will fall (children playing at cars or aeroplanes know this when they sing "neee-oww" as they run along and zoom past one another). The same phenonemon applies to frequencies of light (although, instead of musical pitch, "frequency" as a word used to describe light corresponds to colour).

Therefore, as any one of our ions moves randomly in the trap, it will experience the incident laser light first at one frequency, then at another, depending on the direction of motion of the ion. We cunningly tune the laser so that its frequency is just below the resonant frequency of the ion. In this way, the ion will only `see' the light (ie interact resonantly with it) when its motion causes the light frequency to be Doppler-shifted upwards. But this happens only when the ion moves towards (not away from) the laser, which is exactly what we need to ensure the ion is always slowed down (not speeded up) by the photons!

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The details of the cooling process are rather subtle. An ion moving at the recoil velocity produces a Doppler shift of 63 kHz of the 397 nm radiation. However the natural linewidth of the strong optical transitions in calcium is 20 MHz. This is much too broad a resonance to permit the cooling we need. Therefore we use stimulated Raman transitions between Zeeman sublevels of the ground state. These levels have essentially zero spontaneous decay rate so the effective linewidth is given by the Fourier transform of the duration of the laser pulse used to drive the transition. In our case this is typically 100 micro-seconds so the linewidth is γ = 10 kHz. |

The situation is best understood by taking into consideration the quantization of the motion of the ion. The ion trap provides a harmonic potential well. The energy of vibration in such a well is quantized: The neighbouring energy levels are separated by hv_{z} where the vibration frequency v_{z} is determined by the electric field parameters of the trap and is chosen to be a few times the recoil frequency (i.e. v_{z} = 120 kHz). By using 100 micro-second laser pulses we ensure that these vibrational energy levels are well resolved. The laser is tuned to excite transitions from vibrational quantum number n to n-1 thus optically pumping the population towards n=0. The residual population of higher levels at the end of the cooling is determined by the relative rate of off-resonant transitions n to n+1. A thorough treatment leads to the conclusion that the mean vibrational quantum number when steady state is reached is given by n-bar = (γ/v_{z})^{2} = 0.01. |

This corresponds to a temperature of 1.25 micro Kelvin. In practice noise sources in the apparatus limit the cooling to somewhat higher temperatures. |

### 3. Quantum information processing

*(Note: for much of this discussion, we will be using computing terminology to describe the ions as "quantum bits" or "qubits." For more information on how ions can be the quantum analogy of computer zero/one bits, please visit the Oxford Centre for Quantum Computation website.)*

#### 3.1 Ions as quantum bits - controlling storage, bit by bit

The ion trap has a single quantum register, consisting of a number of qubits. The number of qubits in the register is equal (in the simplest case, which is all we will consider) to the number of ions trapped. In initial experiments this number will be three to ten; in the long term it might extend to hundreds. To process information in the ions, we need two ingredients:

1. change any individual qubit

2. perform the logical XOR operation between any pair of qubits

Since each qubit is stored in a separate ion, to change any individual qubit (the first requirement), we simply illuminate the relevant ion with a pulse of light. This is possible because the ions are separated by twenty microns (thousandths of a millimetre) or so, and we can focus a laser beam down to a width of a few microns.

#### 3.2 Ions as magnets - pointing up for 0, down for 1... or somewhere in between?

The ion can be thought of as a tiny magnet, which can have its "north pole" pointing up to represent a stored zero, or down to represent a stored one. So far this is similar to the use of magnetic storage in conventional computers, except our magnets are much smaller. However, the significant difference is that our magnets can point not only up or down, but also in other directions such as sideways, and the magnets are so weak that no other system is affected by them.

What does this mean? Well, quantum-mechanically, you represent a magnet pointing off to one side (i.e. neither exactly up or exactly down) as a superposition of the up and down states, that is: every time you look at an atomic "magnet" which is a simple, up/down system (generally called a single "spin"), then whatever axis you choose to look at it along, it will appear to either be a 1 or a 0 in this representation: it CANNOT be pointing off to one side. All the superposition gives you is the relative probabilities of seeing the magnet in 1 or 0.

This is a completely different situation to normal magnets. If you chose an axis, you might ask "by how much does the magnet project itself along this axis?" and you would get an answer as a fraction dependent on the angle between the magnet axis and the one you chose. But in a quantum system, every spin is pointed either parallel to the axis, or opposite to parallel (antiparallel) and rather than giving you the percentage of pointing along the axis for one magnet, the only probabilities you have a handle on are the percentage of ions, all in the same magnetic state, that point parallel to the axis as opposed to antiparallel.

It is this quantization of spin-pointing that allows us to manipulate the atoms as quantum bits, allowing the register to store quantum superpositions of zero and one. Light carries with it an electromagnetic field: in fact, light waves can be considered as just being an electromagnetic field. So:

CLASSICAL

A pulse of laser light changes the orientation of the single-ion "magnet" because the fields present in the light twist the magnet round. The magnet has frequency-dependent response and the duration of the pulse twists the magnet by a certain amount.

QUANTUM

The laser light fields cause a mixing of the two possible magnetic states of the ion (0 or 1). If we know the initial state, then for a given frequency and duration of pulse, we can calculate the final superposition or mix accurately.

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We apply a fixed external magnetic field of around 100 Gauss (0.01 Tesla) to the whole ion trap. The magnetic moment of each ion is provided by its single valence electron (calcium-40 has zero nuclear spin) resulting in a Zeeman effect which splits the spin up/down states by 140 MHz. A pair of laser beams of wavelength around 397 nm and frequency difference 140 MHz generated by acousto-optic modulation drive stimulated Raman transitions between the two states. The Rabi frequency is R1 R2/Delta: R1 and R2 are the Rabi frequencies for the S-P transitions driven by each laser beam alone and Delta is the detuning from resonance of these single-photon transitions. The Rabi frequency has to be chosen small compared to the smallest energy level separations in the problem (i.e. 120 kHz = the separation of the vibrational levels). |

#### 3.3 Ions as an interacting system - using the whole ion string

To achieve the second requirement (the XOR between two ions) a further subtlety is needed. First, let us recall what the exclusive-or (XOR) operation is:

* If two bits are the same, their XOR is zero.

* If two bits are different, their XOR is one.

The way to carry this out on a pair of quantum bits is well known: leave the first bit unaffected, and flip the second bit if and only if the first bit is a one (sometimes called a "controlled not" operation). Our problem is, how do we flip ion B if and only if ion A is pointing down? (remember that these ions might not be next to one another, and in any case their magnets are so weak that they do not influence one another). The subtle method is to use the motion of the ions, and the fact that they repel one another through their electric charge.

We first shine a laser on ion A in such a way that the ion absorbs and remits a photon only if its magnet is pointing down. Such a selective pulse of light is possible through the phenomenon of resonance, mentioned previously in the context of laser cooling: take it as read that the ion with magnet down resonates at a slightly different frequency to the same ion with magnet up.

However, we give the ion a little too much energy (high-frequency), by a carefully determined amount. What is this amount? Well, all the ions are able to rattle to and fro in their positions in the trap, but only in a plane perpendicular to the string length. A vibration in this direction is transmitted via the strong ion-ion electrical repulsion to all the ions, and they all vibrate backwards and forwards, the quantum-mechanical equivalent of Newton's Cradle, with all the little pendulums moving together.

Because this is the realm of quantum mechanics, this whole-string vibration is quantized, and we must therefore give it an exact amount of energy to be certain of exciting it. This is the excess energy we give to ion A. So by exciting ion A, we have also set the whole ion string in motion, conditional on ion A being excited (if ion A is in the wrong magnetic state, it doesn't absorb the photon and the ion string doesn't start vibrating).

Now we give ion B a light pulse with slightly too little energy, (low-frequency). So it will only flip if the string is vibrating and thus able to give up its energy, to counteract the deficit between what we give B and what frequency it is resonant at.

To complete the operation a final pulse is applied to ion A so as to exactly cancel the vibration of the string, and return A to its initial state. Overall, a controlled-not between any pair of ions A and B can thus be achieved, in which the motion of the string of ions has been used as a kind of communal one-qubit conveyor belt.

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The first Raman pulse is at frequency ω_{0}-ω_{z} where ω_{0} is the Zeeman splitting and ω_{z} is the vibrational level separation. It drives a π-pulse on the transition |down; n=0> --- |up; n=1>. |

The second pulse is at frequency ω_{0} and duration so as to drive a π-pulse on the transition |up; n=1> --- |down; n=1> and simultaneously a 2π-pulse on the transition |up; n=0> --- |down; n=0>. |

The third pulse is identical to the first. |

### 4. Further reading

A more thorough introduction to ion trap QC is given in:

A. M. Steane, "The ion trap quantum information processor,"

Appl. Phys. B. 64 , 623-642 (1997).

See here for information about our Quantum Error Correction work, including accessible introductions, papers, and other links.