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and scaling

In one sense, the problem of inhomogeneity is the only problem to overcome in rare earth quantum computation. The ions have experimentally verified long coherence times and large interaction strengths and, for ensembles, mea- surements akin to NMR are available.5 _{In the approach suggested above ions}

are selected from a macroscopic collection based on their resonant frequency and interactions strength. The criterion for determining which ion groups are acceptable to be part of the ensemble of quantum computers gets in- creasingly more stringent with the number of qubits. As mentioned above this will lead to exponentially fewer ions as the size increases.

5_{It was rightly pointed out by one of the referees that the ability to make projective} measurements on one qubit while leaving others untouched is important for procedures such as error correction. It is possible that such projective measurements could be made using the large difference in decay times for different energy levels in rare earth ions.

4.6 Inhomogeneity in the interaction strength and scaling 105

In order to make rare earth quantum computing scalable a better way of overcoming this inhomogeneity is needed. The possibilities discussed here involve single ion spectroscopy, either directly or by coupling to other ions, and the use of “solid state molecules”.

4.6.1

Solid state “molecules”

In liquid state NMR based quantum computing you are dealing with an en- semble of “computers” just as for the rare earth scheme described above. The problems of inhomogeneity overcome because each computer is a molecule which is identical to all the others. It should be emphasised that rare earth quantum computing doesn’t share the initialisation problem that causes NMR quantum computing to become untenable for large numbers of qubits. A material can be imagined where there are large numbers of identical col- lections of rare earth dopants. The coherent detection method used in this thesis is sufficiently sensitive that perhaps ensembles with as few as a thou- sands of atoms could produce a measurable signal (see Sec. 4.4). With the huge research effort currently being brought to bear on problems involving nanotechnology, it may in the future be possible to make such ensembles. One advantage in trying to achieve this is that ‘defective’ members of the ensemble, for example where an atom is missing, would have a different set of optical resonant frequencies due to the huge interactions which would be achieved by having the ions so close. Indeed the situation is similar to select- ing out the right ensemble from a bulk sample but with a more favourable starting position.

All the rare earths are strongly electro-positive with their bonding to other atoms essentially ionic in nature [71] and as such incorporating them in a molecular solid would be difficult. One possibility, put forward by Sellars [141], for realising something akin to “solid state molecules” is by adding de- fects to a stoichiometric sample. Crystals containing stoichiometric amounts of europium can still exhibit relatively long coherence times, as long as the inter-europium spacing is large within the crystal [142]. The europiums close to defects would provide an ensemble of a identical groups of europium ions. The defect would shift the resonant frequencies of the members of this group to differing amounts allowing them to be individually manipulated based on their optical frequencies. The fact that they are shifted out of resonance with the bulk europiums should also increase their coherence times.

4.6.2

Single dopant detection

One way of overcoming the inhomogeneity in interaction strength is by aban- doning the use of ensembles. In such a situation the computer would consist of a single cluster of ions. Because the ions are selected based on their fre- quency rather than their precise positions and because the gate operations can be tailored to given interaction strengths no complex fabrication would be required. This leaves the problem of detection of single ions. Spectroscopic measurements of single NV centres in diamond have been demonstrated [105] and while the lower oscillator strengths for rare earth ions would push current detector technology, it may be possible to detect the approximately 1000 pho- tons/sec produced when driving an optical transition strongly. The reason for such weak fluorescence is due to the long radiative lifetimes associated with rare earth ions. How long any emission lasts depends on what rate pop- ulation gets optically pumped into other hyperfine levels. For free atoms/ions outside solids strong selection rules result in “cyclic transitions” where the optically excited state only decays into the ground state from which it is being driven. This means that the state of the hyperfine level can be read out to a high fidelity. If the atom is in the hyperfine state from which the atom is being driven from then a large amount of fluorescence is generated, whereas if it is in a different state no fluorescence will be generated.

Another option [141] is to use non-radiative transitions and detect the phonons produced. The qubit would be transfered to the optical transition and then a short lived non-radiative transition would be driven from the ground state. What would determine whether this would work would be if you detect the amount of phonons that could be produced within the (∼2 ms) life time of the long lived optically excited state. Extremely sensitive calorimeters (bolometers) exist that operate on the edge of superconducting transitions [143]. These have been highly developed due to their application as detectors for astronomy. Their sensitivity has been demonstrated in their use as part of a single photon detector that operated down to a wavelength of 4 µm [144].

Other options involve coupling to other centres that can be read out more easily. A detailed theoretical examination of such a scheme using the NV cen- tre to readout an nuclear spin has been carried out recently by Pulford et al. [145]. Another option is to treat rare earth quantum computing as some- thing of a prototype and to move on to optical centres which are easier to read out. The centre that immediately comes to mind is the NV centre in diamond. If this has a reasonable Stark shift as speculated in Sec. 4.1.1 then the quantum computing operations could be envisioned that were completely