Realising larger conditional phase shifts

As discussed above, the small conditional phase shift was the result of the experiments being restricted to a small number of interaction strength select- ing pulse sequences and the low holeburning efficiency. The experiment was carried out using the±5/2 levels in both the ground and excited states. The

5.7 Conditional phase shifts 155

holeburning efficiency was low because ions that were left in the±5/2 excited states at the end of an operation spontaneously emitted — into the ±5/2 ground states predominantly rather than into the other hyperfine levels. One possibility for improving this situation is to use, for example, the ±1/2 hy- perfine levels for the excited state. While the holeburning efficiency would be higher, the oscillator strength for the±5/2→ ±1/2 transition would be correspondingly lower. This lower oscillator strength would result in lower Rabi frequencies which is undesirable as is discussed below. One way to im- prove the holeburning efficiency without introducing this problem would be to drive an excited state hyperfine transition with a RFπafter the unwanted ions had been left to decay. For this to be useful the Rabi frequencies obtain- able would need to be higher than the optical decay rates. Using 1 kHz/G (see Sec. 3.2) for the Zeeman splitting, one can estimate that a RF amplitude of 10 G would be required to obtain a 10 kHz Rabi frequency. With a big amplifier and a resonant coil such amplitudes are available experimentally.

The problem restricting the number of interaction strength selecting pulse sequences was the accuracy to which single qubit operations could be applied to the spectrally wide anti-holes.

Wide spectral features are required for the spatial ion density to be high, which is in turn required so the mean interaction between ions in such fea- tures is high. In order to apply pulses which achieve the same operation for all the ions of an inhomogeneously broadened ensemble, the pulse length must be short compared to the inverse of the spectral width. The Rabi frequencies provided by the laser only enabled this criterion to be achieved approximately. The two ways to increase the Rabi frequency are to either increase the laser power or reduce the spot size. Laser powers greater than the ∼200 mW used in these experiments have be generated from dye lasers. However, the easiest way to increase the intensity at the sample would be to use a smaller spot size. For the experiments described in this section the laser was focused onto the sample with a 15 cm lens; Rabi frequencies of

∼210 kHz resulted. Initially experiments were attempted with a 10 cm lens which resulted in Rabi frequencies of∼400 kHz. However the resulting higher intensities were incorrectly considered responsible for the “5 MHz problem” discussed in Sec. 5.6.3. The ability to apply definite area pulses would be helped by the square features provided by the zero area pulses mentioned in Sec. 5.3.1. The fact that the use of composite pulses only slightly increased the ability to perform the single qubit operations suggests possible problems with the background level of ions, or the presence of long tails on the anti- holes, or both. The zero area pulses are good at getting rid of such problems.

There is nothing that stopped these being used in these experiments other than the desire to keep the setup as simple as possible coupled with the lack of a small amount of extra equipment.

Composite pulse sequences can be very useful in magnetic resonance ex- periments when there is inhomogeneity in the transition frequency and driv- ing strengths. The composite pulses used in this experiment were added in an ad hoc fashion and were retained because they improved the performance slightly. A more systematic look at the use of composite pulses than was carried out here would no doubt be useful.

Another way to increase the spatial density of the ions in the anti-holes is to increase the total concentration of dopant ions. It should be noted, however, that increasing the concentration of the ions would also increase strain in the crystal and thus the inhomogeneous broadening. A packet of the same spectral width would, therefore, contain a smaller fraction of the total number of ions than in a sample with a smaller concentration. Further investigation is warranted.

Ideally, the interaction strength burning process should be only limited by the homogeneous line width optical transitions but it is also sensitive to laser jitter. For these experiments the linewidth of the laser was of the or- der of 200 Hz which is larger than the ∼100 Hz homogeneous linewidth. A more stable laser would allow smaller interaction strengths to be selected by using longer interaction strength selecting pulse sequences. This would then in turn reduce the spatial ion density required and, as a result, the spectral width of the control anti-hole as well. This would allow more accurate sin- gle qubit operations to be applied to the control anti-hole and higher two qubit fidelities would result. However, the benefit of improving the laser fre- quency stability would diminish quickly as the laser’s linewidth approached the homogeneous linewidth of the optical transition.