If the inhomogeneous linewidth is larger than the available Rabi frequency, the first step of preparing and initialising a set of qubits is to create a narrow ensemble for each qubit using spectral holeburning. This can be achieved by burning a trench in the line, and burning a narrow antihole back into the trench [148], as illustrated in Figure 8.7. This method was first suggested for quantum computing using rare
Number of qubits Q u b it co n ce n tr at io n 4 MHz 10 MHz 100 MHz 1 GHz 2 4 6 8 10 12 14 16 18 20 10−20 10−15 10−10 10−5 100
Figure 8.8: Concentration of ions in an ensemble qubit after distillation as a function of the number of prepared qubits. The dashed line shows the limiting qubit concentration of 10−12 qubit ions per Eu3+ site.
earth doped crystals [33].
This spectral holeburning step can place a limit on the allowable Rabi frequency, as the width of the trench is limited by the smallest hyperfine splitting in the ma- terial. If 151Eu3+ is present, for instance, this is 27 MHz in EuCl
3·6D2O. The Rabi
frequency must be substantially smaller than this number to avoid pulses applied to the prepared qubit exciting the edges of the trench. As the width of the pre- pared feature must be smaller than the Rabi frequency, qubits prepared by spectral holeburning are limited to hundreds of kilohertz linewidths.
After the spectral holeburning step, the qubit preparation sequence continues as described in the previous section, with a qubit distillation step. However, the problem with using spectral holeburning to prepare a feature comes in this step: the concentration of ions in the final antihole is very low. If a 1 MHz wide fea- ture is prepared in a 100 MHz wide line, only 1% of the ions in each line remain in the prepared qubit. Note that it is the total width of the line, including any isotope or other discrete shifts, that is important, not the inhomogeneous width. In EuCl3·6H2O, isotope shifts mean the total width is 600 MHz while the inhomoge-
neous width is 100 MHz. When qubit distillation is applied to a material in which the qubits were created with spectral holeburning, the qubit distillation process drastically reduces the concentration of ions in each qubit. This is demonstrated by Figure 8.8, which shows the concentration of ions in a distilled qubits for a dopant
concentration C0 = 0.5% and a prepared qubit width of 1 MHz. The number of
qubits that can be prepared without reducing the ensemble size below its detection limit is 4 for a linewidth of 1 GHz, 5 for 100 MHz, 10 for 10 MHz and 17 for 4 MHz. Current EuCl3·6D2O crystals have a total linewidth of 600 MHz, suggesting that,
using spectral holeburning, four interacting qubits could be prepared.
Spectral holeburning to prepare narrow ensembles followed by qubit distillation is unavoidable in the rare earth doped quantum computing schemes of Ohlsson et al. [4] and Longdell and Sellars [161,30]. However, distillation works slightly differently in those schemes and it is worthwhile elucidating the difference. In a doped crystal, the ions in one qubit have a random spatial distribution relative to ions in another qubit so the interaction between two qubits is very inhomogeneous. The distillation process, therefore, is required not only to select out ensemble qubits that can interact with each other, but ones that interact with a particular strength. The two schemes do this in different ways. The Ohlsson scheme uses spectral holeburning to select ions that have shifted more than the width of the qubit, similar to the method here. The disadvantage of this method, as explained by Longdell [161] is that almost none of the ions have such large interaction frequencies because the average interaction strength is much smaller than the qubit width. The scheme of Longdell et al. uses a coherent technique to select out ions that have shifted by less than the qubit width. While this is a much larger proportion of the ions in the ensemble, the necessity of using a small interaction means that gates are very slow. Even with a larger ensemble in each qubit than in the Ohlsson scheme, the fidelity of the two-qubit conditional phase shift gate demonstrated by Longdell et al. [30] was limited by the size of the ensemble.
In both schemes, the qubit distillation process removes a much larger proportion of the ensemble than would occur in a stoichiometric crystal with a similar dopant concentration and inhomogeneous linewidth. Additionally, the inhomogeneity re- maining in the interaction strengths between qubits is of the same size as the qubit width, whereas in a stoichiometric crystal it is likely to be less than 1 kHz. The reason these early schemes concentrated on doped crystals is that it was possible to use crystals that were very well characterised, such as Y2SiO5 and YAlO3. Be-
cause suitable stoichiometric crystals, such as EuCl3·6H2O, have not been intensively
studied, any quantum computing demonstrations must be preceded by an in-depth characterisation, such as that presented in this thesis.
(a) (b) (c)
Figure 8.9: A single qubit NOT gate operated using the optical transitions. The “atoms” in these images are included to show where coherence is transferred. (a) A π
pulse transfers the coherence from the |0i state to the |ei state.(b) Anotherπ pulse, this
time on the|1i → |ei transition, exchanges the coherence between these two levels. For a general qubit rotation gate, theπpulse is replaced by a pulse of the appropriate angle and
phase. (c) The coherence in|ei is transferred back down into|0i to complete the gate.