High-fidelity quantum gates in the presence of dispersion

We numerically demonstrate the control of motional degrees of freedom of an ensemble of neutral atoms in an optical lattice with a shallow trapping potential. Taking into account the range of quasimomenta across different Brillouin zones results in an ensemble whose members effectively have inhomogeneous control fields as well as spectrally distinct control Hamiltonians. We present an ensemble-averaged optimal control technique that yields high-fidelity control pulses, irrespective of quasimomentum, with average fidelities above 98$\%$. The resulting controls show a broadband spectrum with gate times in the order of several free oscillations to optimize gates with up to 13.2$\%$ dispersion in the energies from the band structure. This can be seen as a model system for the prospects of robust quantum control. This result explores the limits of discretizing a continuous ensemble for control theory.

Figure 1

The first four energy bands (highlighted in gray) for a 1D optical lattice are shown in the localized basis for potential depth $r=13$ and compared to the sinusoidal trapping potential. Refer to Fig. 2.

Figure 2

The first four energy 1D band structures are shown in the momentum basis for four different potential depths $r$. The lower potential depth $r=2$ (top left) shows the largest amount of energy dispersion with the closest energy crossings. The depth $r=30$ (bottom right) shows a lesser amount of dispersion and larger energy splittings.

Figure 3

Comparison of two optimized controls for preparing an $X_{\pi }$ gate when the optimization uses only ten points in quasimomentum space. Both pulses are well behaved in the sense that they do not translate the lattice by a full lattice site or possess very large excursions from the initial value for $r$. (a) An optimized pulse for a potential depth $r=17$ with 5.4$\%$ dispersion and fidelity 99.3$\%$, which is calculated by sampling over 100 quasimomentum values. The duration of the pulse was five free oscillations (at $k=0$). (b) An optimized pulse for potential depth $r=12$ and dispersion 13.2$\%$ with a sampled fidelity of only 66.8$\%$. The time duration of the pulse was 25 free oscillations (at $k=0$).

Figure 4

Comparison of gate error for well-behaved optimized controls in Fig. 3 where the dotted blue line corresponds to control sequence (a) and the solid red line to control sequence (b). For (a) the gate error is at most 1.8$\%$ across the possible quasimomenta and the average gate error is 0.6$\%$. In (b) the gate error has been minimized for the quasimomenta points that were sampled for optimization but was 66.8$\%$ on average when sampled over 100 points.

Figure 5

The maximum fidelity for an optimized pulse over a range of times, from one to ten free oscillations (at $k=0$), preparing an $X_{\pi }$ gate as a series of different potential depths. The potential depth ranged from $r=0.25$ to $r=110$, giving a dispersion ranging from 93.8$\%$ to 0.01$\%$, respectively. Each point is an average of optimized pulses from one Rabi and ten random initial pulses.

Figure 6

Same as Fig. 5 but with times from 15 to 70 free oscillations (at $k=0$).

Figure 7

Pulses for performing $X_{\pi }$ gate at $r=2$ for three different times that were optimized specifically for $k=0.5$ and the fidelity response over quasimomenta is shown. This shows the effect of Nyquist bandwidth limits of the pulses on fidelity. Shorter pulses have larger spectral bandwidth and thus affect a larger range of quasimomenta.