Trapping and Manipulating Ions in an Ion Trap Quantum Computer

Trapping and Manipulating Ions in an Ion Trap Quantum

Computer

Roy Garcia

November 26, 2018

Columbia University, Fall 2018

1 Ion Trap

An ion trap is a mechanism that traps ions. The typical setup involves 4 electrodes with a potentialdifference across them to confine the ions. The volume enclosed is the trapping region. How do weget ions in this region? I’ll tell you about how a group at Oxford does it. They heat up calciummetal to 800K in vacuum. The metal evaporates, producing a gas of calcium atoms. To ionizecalcium, they fire a beam of electrons to knock out a valence electron in the calcium atoms. Not allatoms are ionized, but those that do will, by chance, end up in the ion trap. We’ll consider the casewhere 4 ions are trapped inside. They are confied by the potential across the electrodes given by

V = ΦDC + ΦRF (1.1)

Where the first term is the DC potential

ΦDC =1

2kU0[z2 − (x2 + y2)] (1.2)

The z-term is just that of the simple harmonic oscillator. The classical analog to this potentialcan be given by a ball oscillating back and forth in a parabolic valley at a given frequency. In thecase of the ion trap, the ions oscillate back and forth at quantized frequencies that depend on theDC potential. However, there is a very important theorem called Earnshaw’s theorem that statescharged particles cannot be held at equilibrium by static potentials. The DC potential, while itserves to have the ions moving at known frequencies, cannot alone confine them to the trappingregion. Therefore, we must include a time-dependent potential, the RF potential:

ΦRF =1

2(V0 cos(ΩT t) + Ur)(1 +

x2 − y2

R2) (1.3)

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The most important characteristic of this potential is the cosine dependence; it results in a periodicrestoring force. During one half of the cycle, the restoring force is along the z-direction, while theyare not confined in the xy-plane. During the other half of the cycle, the restoring force pulls theions to the origin of the xy-plane, while the ions are no longer confined along the z-drection. If thefrequency is high enough, these two restoring forces can be turned on and off fast enough such thatthe ions are confined to the center of the trap. Using these two potentials, we can confine our ionswithin the trapping region and make sure that they oscillate within a harmonic oscillator potential.The ions oscillating as a group is called a phonon. The frequency at which this phonon oscillatesis called a mode. A higher frequency corresponds to a higher mode, and therefore a higher energy.This makes sense, since a higher frequency means the particles are moving faster, which means theyhave more energy. The energy levels for this potential are given by:

(1.4)

Each energy state is spaced out by ~ω. These energy states correspond to the possible states of thephonon. Note that there can be an infinite number of states, since the energy levels go on to infinity.The individual ions themselves also have their own energy states. They can be either spin up orspin down, and we can flip their spin by shining a laser on them. The individual ions therefore arethe qubits, which carry the quantum information. However, we can

(1.5)

1.1 Cooling

In order to have these ions behave as phonons, moving all together and vibrating at precise energylevels, it is first necessary to cool them down to very low temperatures, such that the characteristicthermal energy is on the same order of magnitude as the phonon energy.

kBT ≤ ~ω (1.6)

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As Richard Feynman put it, heat is just the jiggling of atoms. If we want to cool down an atom,we basically want to slow it down, and rid it of its kinetic energy. In our ion trap, we have ionsmoving along the z axis. If they move to the right, we want to kick them to the left to slow themdown. If they move to the left, we want to kick them to the right. So how do we apply this kick?We know the absorption spectrum of a calcium ion. In order for the ion to transition from a lowerenergy state to a higher energy state, it must absorb a photon that carries an energy that is equalto that difference. We call this energy difference the transition energy. If the energy of the photonis too low, then the atom does not absorb it. These photons carry momentum too, so when they areabsorbed, they will kick the atom in the opposite direction, by the conservation of momentum. Weshine a laser along the z-axis. However, when the atom moves toward the laser, the light appearsblue shifted, meaning the energy of the photons appear higher. When the atom moves away fromthe laser, light is red shifted and the energy drops. Therefore, we can set up the laser to have justthe right frequency so that when the atoms move toward the laser, the photon energy correspondsto the transition energy, so the photon is absorbed and the ion gets kicked back to the origin. Whenthe ion moves away from the laser, the photon energy is too low to be asborbed, so the ion does notexperience a kick.

We have shown how we can trap and cool ions. But how do these ions carry quantum information?Each ion has an angular momentum. Within an ion, there could be an orbital angular momentum,an electron spin angular momentum, or a nuclear angular momentum. If we want to change thestate of the qubit, the ion, we can shine light on it, such that it absorbs a photon and gains a unit ofangular momentum. However, there’s no way for us to ensure where that extra angular momentumcontribution goes: it could go to spin or orbital angular momentum. One thing we do know is thetotal angular momentum does change by a unit. Therefore, we express the state of the qubit interms of the total angular momentum. As an example, we consider the case of two spin-1/2 ions asour qubits. By the rules of the addition of angular momentu, the total anguar momentum of thesystem can either be J=0 or 1. In the case that it is 1, the z-component of the angular momentumcan Jz=-1, 0 or 1. The case of Jz=-1 corresponds to the two ions with a downward spin. TheJz=1 cacse corresponsd to the two ions with an upward spin. Jz=0 means one ion is spin up, whileanother is spin down. But since there are two possible states where this occurs, either ion 1 is spinup or ion 2 is, the total state is a superposition of these two possible states. And for the case wherethe total orbital angular momentum is 0, only Jz=0 is allowed, so again we must have one ion spinup while the other spins down.

|0, 0〉J =1√2

[|01〉 − |10〉] (1.7)

|1,−1〉J = |00〉 (1.8)

|1, 0〉J =1√2

[|01〉+ |10〉] (1.9)

|1, 1〉J = |11〉 (1.10)

Let us now take, for demonstration, the qubits to be in the |0, 0〉J state. If we measure ion 1 to bespin up, then we know the state of particle 2 without having to take its measurement–it must bespin down! The state of the other particle is always opposite to the state of the first particle. Thesestates are known as the Bell states. They have the special property that if we know the state ofone particle, then we can deduce the state of the other. We then say that the qubits are entangled.They are surprisingly natural states found in nature.We can retrieve quantum information withoutcompletely destroying the quantum state.

In practice, there are two qubit types for the ion trap. The energy states of the hyperfine structure,which depend on the angular momentum of the ion can be used as states of the ion qubit. Thephonon, also acts as a qubit, with its states being its energy modes. Usually only the lowest modes,0 and 1, are used for qubit states.

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In summary, if we manage to trap ions and cool them down such that the thermal energy is on thesame order of magnitude as the phonon energy states, then we can exploit the phonon energy statesto store quantum information. We can also store quantum information in the hyperfine structure ofions like Beryllium and excited these states with lasers.

(1.11)

2 References

https://www2.physics.ox.ac.uk/research/ion-trap-quantum-computing-group/intro-to-ion-trap-qcNielsen, Michael A., and Isaac L. Chuang. Quantum Computation and Quantum Information.Cambridge University Press, 2017.

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