ence
If the accumulated phase shifts due to the interactions are the same in both periods of free evolution of the photon echo sequence then the effect of the interaction should be rephased. This is illustrated by the experiments where the perturbing pulse was applied at either the middle or the end of a photon
5.6 Characterisation of dipole-dipole interactions 147
Control
Target
π/2
π
echo
τ δ
τ δ
τ
π
π
Figure 5.21:Pulse sequence used to demonstrate the recovery of the photon echoes when the two periods of free evolution in the photon echo sequence are equivalent. A series of shots were taken as the value of delta was increased from zero to one. The shots alternated between having no perturbing pulses applied, having just the later perturbing pulse applied and having both perturbing pulses applied. The delay between the two driving pulses of the photon echo,τ, was 400µs. The results are presented in Fig. 5.22.
echo sequence (see Fig. 5.17). With the command that we had over the state of the control anti-hole it was possible to demonstrate this rephasing when the effect of the interaction wasn’t constant during the entire pulse sequence. These experiments were useful because they provided a measure of how well the single qubit operations could be applied to the wide control anti-hole. Furthermore they were arrived at by looking at the effect on the target anti-hole. Thus, the measurements were sensitive to the imperfections in the control pulses in a manner similar to potential quantum computation demonstrations.
Experiments with a similar theme have been carried out by Altner and Mitsunaga [154] but instead of coherent control over the perturbing ensemble they used a co-doped sample. Ions with long coherence times were used for the target ions and ions with a much shorter lifetime were used for the control. The symmetry between the two periods of free evolution of the photon echo was achieved by having the lifetime of the perturbing ions short compared to the length of these periods of evolution.
The pulse sequence that was used is shown in Fig. 5.21 and the results in Fig. 5.22.
In the case of one perturbing pulse (red dots of Fig. 5.22) the behaviour is easy to understand. As delta is increased the length of time for which the control ions are excited reduces. The interaction therefore causes less
0 0.2 0.4 0.6 0.8 1 20 40 60 80 100 120 140 160 180
δ
echo heightFigure 5.22: Echo recovery. The results obtained using the pulse sequence of Fig. 5.21. The dots (red) correspond to applying only the latter perturbing pulse and the open circles (green) to applying both perturbing pulses. The blue crosses correspond to when no perturbing pulses were applied. The blue line shows the mean echo height in the absence of perturbing pulses. The other two lines are least squares fits of an arbitrary quadratic to the data.
5.6 Characterisation of dipole-dipole interactions 149
decoherence.
The experimental points represented by open circles (green) in Fig. 5.22 correspond to the application of both of the perturbation pulses, depicted in Fig. 5.21.
If the first perturbing pulses were an ideal π pulse for all of the control ensemble, then the first pulse would put all of the control ensemble into the excited state, then the second would put them all back in the ground state. As a result, each of the free evolution periods in the photon echo sequence would have the control ensemble excited for a fraction δ of the time. This should result in no echo demolition because the effect of the interactions should be rephased. The phase shift due to the interaction accumulated by the target ions during each half of the photon echo sequence would be the same.
In practice, the echoes from the target ions when the control ions had been addressed twice (open circles Fig. 5.22) were smaller than those when the control ions were left untouched (blue crosses). The main contribution to the asymmetry that caused the reduction in the echo heights was the imperfection in the π pulses. The 210 kHz Rabi frequency used was about the same as the spectral width of the control anti-hole, rather than being much bigger.
For the smaller values of delta, the primary reason for this asymmetry was the fact that all control ions were not placed perfectly into the ground state by the second pulse. As delta was increased, the length of time after the second pulse decreased and this resulted in the echo heights represented by open circles (green) getting larger. Neglecting population decay, the periods of each photon echo sequence during which the ions were excited were the same even with imperfect π pulses on the control.
However for larger values of delta, when the fraction of time spent with the excited control ions excited was largest, population decay became more important. This means that atδ = 1, the echo heights represented by open circles (green) do not reach those of the unperturbed echoes unlike the singly perturbed echoes, which are shown as dots (red).
How much closer the open circles (green) in Fig. 5.22 are to the crosses (blue) than to the dots (red) is a measure of the echo recovery. In this case, the echo recovery was much smaller than in the experiments depicted in the lower graph in Fig. 5.20 (Sec. 5.6.5). In those experiments, a perturbing
π pulse caused significant echo demolition whereas the echo height after a 2π perturbation was almost the same as the unperturbed height. This is primarily a consequence of how the ensembles were prepared. The ‘2π’ pulses
that were applied to the anti-holes are intended to select out ions based on their Rabi frequencies. For ensembles with spectral width small compared to the Rabi frequency this would occur. However, for ensembles where the spectral width of the ensemble was comparable to the Rabi frequency one must remember that the selection of ions was based on theirgeneralised Rabi frequency. Keeping this in mind it is easy to see why the 2π pulses did a better job of putting ions back in the ground state than two separated π
pulses did. We chose our ensembles so they would consist of ions that ended up back in the ground state at the end of a ‘2π’ pulse. Another contributing factor to the smaller recovery in Fig. 5.22 was the use of ensembles for the control anti-hole that were wider in frequency. This can be seen in the larger amount of echo demolition that occurred when aπ pulse was applied to the control ensemble half way through an echo sequence on the control.