If the linewidth of the material is below 1 MHz, but the crystal is not isotopically purified in Eu3+, or for some other reason a large number of the Eu3+ sites are
occupied by ions that cannot be used in the qubit, extra steps are required to prepare the material after those described above. The first step is to ensure that each qubit contains only qubit ions, by using spectral holeburning to discard any non-qubit ions at the qubit frequency. The next step is a distillation step, in which only those computing molecules in which each qubit site is occupied by a qubit ion are selected [148]. In the case where there are two Eu3+ isotopes, the distillation
step chooses only those computing molecules where all sites are occupied by153Eu3+.
This step is necessary because if the site associated with qubit 1 is occupied by a non-qubit ion in a particular computing molecule, there is no interaction between another qubit and qubit 1 for that molecule, introducing an error into any multi- qubit gates involving qubit 1. The distillation step uses the interactions between qubits to select out suitable computing molecules. This step is illustrated in Figure
8.6 for the case of a computing molecule with three qubits. When one qubit (the control qubit) is excited, all those ions in a second qubit (the target qubit) for which the corresponding control qubit site is occupied by an excited 153Eu3+ ion
will experience a shift in their optical transition frequency. Any ions in the target qubit that do not shift do not have a corresponding 153Eu3+ ion in the control
qubit, and can be discarded by optically pumping them into the third hyperfine ground state. The roles of the target and control qubit are then reversed. After this process, the concentration of ions in each qubit is reduced by a factor of 1
P, whereP
is the concentration of the qubit ions (52% for153Eu3+ at natural abundance). This
distillation step must be repeated for every pair of qubits, and the final concentration of computing molecules in the crystal is
(a)
(b)
(c)
(d)
Figure 8.6: Qubit distillation process. If some of the qubit sites can be occupied by Eu3+ ions that are not in the ensemble qubit, such as when some sites are occupied by 151Eu3+ instead of153Eu3+ a distillation process is needed to select out those computing molecules in which all ion sites are occupied by qubit ions. This is illustrated for a 3-qubit computing molecule. Blue indicates a site occupied by a qubit ion, grey one by a non-qubit ion. (a) There are four configurations of ions contributing to each ensemble qubit, shown for qubit 1, of which only configuration A (all sites occupied by qubit ions) is desired. (b) To remove unwanted configurations, first qubit 2 is excited. This shifts the frequency of those computing molecules in qubit 1 that also have an ion in qubit 2 – configurations A and B. Configurations C and D can then be discarded using spectral holeburning to the shelving state. (c) Repeating the process by exciting qubit 3 allows configuration B to be discarded. Qubit 1 now only contains computing molecules with an ion in every qubit. (d) The whole process must be repeated for qubits 2 and 3. Each ensemble qubit is reduced to P2 of its original size (where P is the proportion of qubit ions), but all ions in each
where C0 is the dopant concentration and N is the number of qubits. Because the
ensemble size decreases exponentially with the number of qubits, if distillation is required it places a limit on the number of qubits that can be used simultaneously. The limiting size of the ensemble is ultimately determined by the desired readout fidelity. As Section 8.3 shows, for a readout fidelity of 99.99%, approximately 2000 ions are needed in the ensemble. The concentration that will give this number of ions depends on the excitation volume of the laser. For a laser spot size of 100 µm
and an excitation volume of ≈ 1 ×10−6µm3, the largest beam size at which a
10 MHz Rabi frequency is feasible, the ensemble concentration would need to be 10−12 ensemble atoms per Eu3+ site. For a dopant concentration of 0.001%, the
limiting number of qubits in a system with a natural abundance of 153Eu3+ is 25. If
the crystal is isotopically purified, the limit increases rapidly, reaching 150 qubits at 90%153Eu3+, and more than 1000 qubits at 96%. In a low linewidth material with a
moderate level of isotopic purification, therefore, the exponential loss of ensembles due to isotopic disorder will not limit the number of qubits. In fact, this is one of the few systems in which the number of qubits will be limited by the number of lines that can be optically resolved. The reason for this is that the number of resolvable lines is limited by the requirement that the separation between lines is not smaller than the Rabi frequency, but if the Rabi frequency is reduced to resolve more lines, the gate error rate increases (see Section 8.2). If it is assumed that the minimum allowed separation between frequency qubits is 10 MHz, the maximum number of possible qubits is ≈100 in EuCl3·6D2O.
A crystal of EuCl3·6D2O that is not isotopically purified presents an additional
problem that stems from the symmetry of the crystal. All qubits due to sites off the C2 axis have two ions per computing molecule contributing to them. Gates and
readout operate in the same way for these qubits as they do for qubits with only one ion site per molecule as long as both sites are occupied by qubit ions. This, however, means that each computing molecule has nearly twice as many ions in qubits as it does qubits, meaning that it requires nearly 2N distillation steps to
prepare the N interacting qubits. The prepared ensemble size is, therefore, much
smaller than if only one of the sites had to be occupied by a qubit. To maintain a large ensemble size it is necessary to discard one of the ions in each off-C2 qubit.
This can be done by taking advantage of the slight perturbation the dopant makes to the symmetry of surrounding sites. If a field is applied that breaks the symmetry of the dopant ion itself, the frequencies of the two ions in previously C2-symmetric
sites will split due to the dopant perturbation. This splitting has already been seen in the magnetic field rotation patterns in Chapter5. Sufficiently large electric fields should also split the optical transition frequencies of off-C2 satellite lines. The Stark
(a) (b) (c)
Figure 8.7: Preparation of a spectrally narrow ensemble using spectral holeburning. The frequencies illustrated in this figure are not to scale. (a) A trench at least 1 MHz across is first burnt in the inhomogeneously broadened line. (b) A feature can be burnt back into this trench by burning at a frequency offset from the trench centre frequency by one of the hyperfine splittings. (c) The resulting feature is a narrow antihole, which can be made up to three times taller (for Eu3+) than the inhomogeneous line if the burnback is done at three combinations of the ground state hyperfine frequencies.
shift measurements of Section 7.3.1showed that the Stark shift of satellite lines can vary from that of the main line by up to a few percent, suggesting that a similar variation between the two ions in one off-C2 site may be possible. In that case, fields
of 107 V.m−1 would be required for splittings of 1 MHz.
An easier way of getting splittings between ions in off-C2satellite lines is to strain
shift the line. A stress applied to the crystal along a direction away from the C2
axis and the plane perpendicular to the C2 axis must break the C2 symmetry of the
crystal, and this strain shift can be quite large. I have made initial measurements of the strain shift in EuCl3·6H2O by measuring the shift in the position of a spectral
hole when pressure is applied. These measurements are preliminary and are only described here to give an approximate size for the strain shift. The strain shift was approximately 70 kHz/kPa. This implies that about 14 MPa could resolve a 1% difference in dipole moments between the ions in an off-C2 line, and this pressure
range is trivial to reach in the lab.
Once the two ion positions in an off-C2 line are resolved, one can be discarded
using holeburning. This does not affect the number of qubits that could be prepared because the additional ion in each qubit was not counted in Equation (8.1).