Zero area pulses

The zero area pulses used by Pryde et al. are shown in Fig. 5.5. As can be seen from Fig. 5.5(v) as well as the anti-hole there is a significant background also. The width of this background feature is here limited by the spectral width of the pulses used to measure it. In working toward the goal of big anti- holes with small background levels, ‘double-sinc’ zero area pulses as shown in Fig. 5.6 were investigated. The sharp cornered frequency spectrum of these pulses allow more pulses to be applied, thus reducing the background levels without eroding the anti-hole. The flat spectrum of these pulses around zero frequency should make the process of creating the anti-hole more robust with respect to drifts in laser frequency.

The envelope of the double-sinc pulses was given by2

y(t) =Aexp − t 2 2u2 (w1sinc(w1t)−w2sinc(w2t)) (5.1)

Here the width of the trench burnt is given by the larger of the wi and the width of the anti-hole in the middle is given by the smaller wi. Ais a scaling factor and the Gaussian term is used to remove any ringing that would occur if a rectangular window was used. The sequence length of the pulse used determines u and, in turn, the maximum spectral resolution achievable. If the fast sinc function is much shorter than the slower sinc function, it will have to be much more intense in order to have the same area. Therefore, the ratio between the spectral widths of the sinc functions is limited by dynamic range concerns. In view of all these considerations the following parameters were chosen

w1 = 50 kHz (5.2)

w2 = 1 MHz (5.3)

u = 24 µs (5.4)

The result of a photon echo sequence applied to an anti-hole prepared 1_{This was in no small part due to RF switches inexplicably failing at the hands of the} author.

5.3 Generating and characterising single qubits 121

Figure 5.5:Graphs describing the anti-holes created by Pryde et al. [80]. The amplitude of the zero-area pulses used to generate the anti-hole is given as a function of time in (i). Here the combined areas of the long weak pulses is the same as that of the short intense pulse, leading to a zero area pulse. Traces (ii) and (iii) show numerical estimates of absorption profiles of the resulting anti-hole after 13 and 130 pulses (a hole-burning efficiency of 5% was assumed). Trace (iv) shows the spectra of the pulses used in their experiments. Trace (v) is a Fourier transform of the experimental echo shape. This is equal to the product of the ion density versus frequency and the spectrum of the exciting pulses. The shoulders on this peak indicate a significant number of background ions. This figure was scanned from [149].

−50 −40 −30 −20 −10 0 10 20 30 40 50 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 time (µs) −600 −400 −200 0 200 400 600 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Freq (kHz)

Figure 5.6: The envelope and spectra of the double-sinc pulses used for creating anti- holes. Because the pulses used had to be of a limited temporal extent, a Gaussian window was used. This caused the rounded edges in the spectrum of the pulses.

5.3 Generating and characterising single qubits 123

with these zero area pulses is shown in Fig. 5.7. To prepare the anti-hole, 300 double-sinc pulses were applied to a broad burn-back feature. (See next section.) The sinc-like shape of the echo and its square shaped spectrum correspond well with what would be expected from the preparation pulses applied. Compared with the earlier results of Pryde et al. a great improve- ment in the background level can be seen. For Fig. 5.7 the overall phase was chosen so the integrated echo signal was all in one quadrature. There is a slight dispersive nature to the echo, as can be seen from the other quadrature. This dispersion is due to a slight frequency mismatch between laser and the anti-hole. This also appears as the spectrum shown in (b) being slightly off centre.

If the laser and the ions in the crystal were perfectly phase stable, the free induction decays from the two pulses would be exactly out of phase with the signal from the echo. This is not the case and it appears the phase between the laser and ions drifted approximately 30◦ during the 0.5 ms of the shot. The high frequency signal on the second FID is at 500 kHz and is due to the excitation of the edges of the trench in which the anti-hole lies (see spectrum in Fig. 5.6). These ions don’t contribute much signal to the echo because they are only weakly excited by the applied laser pulses and therefore do not get rephased well.

The well defined frequency of the ringing suggests sharp spectral features. Very recent simulations [153] have shown that such sharp features can be expected if the driving sinc pulses are intense enough to drive the ions far from the ground state.