Robust not gate by single-shot-shaped pulses: Demonstration of the efficiency of the pulses in rephasing atomic coherences

We derive a versatile protocol to implement a fast and robust not quantum gate, driven by single-shot-shaped pulses with appropriate time-dependent phase and intensity profiles. The pulses are derived by combining analytic computations and numerical optimizations. We experimentally demonstrate the applicability, efficiency, and robustness of the derived pulses to rephase atomic coherences in a ${\text{Pr}}^{3+}\text{:}{\text{Y}}_2{\text{SiO}}_5$ crystal.

Figure 1

Robustness with respect to the unknown static detuning $\delta$ and relative error $\alpha$ of the Rabi frequency. Contour plot of the cost function $J$ (8) at the end of the pulse for the squared $\pi$ pulse (a) and the SSSP (b). The SSSP numerical optimization for achieving maximum $\overline{J}$ has been performed for the parameter range determined by the dashed box (b). The durations of the square $\pi$ pulse and the SSSP pulse are $\pi /{\mathrm{\Omega }}_{max}$ and $\approx 38/{\mathrm{\Omega }}_{max}$, respectively.

Figure 2

Relevant hyperfine structure in PrYSO and coupling scheme for excitation of atomic spin coherences by a radio frequency $\pi /2$ pulse and a radio frequency SSSP rephasing pulse (solid, green arrow). Atomic coherences ${\rho }_{12}$ are created between state $|1\rangle$ and state $|2\rangle$. Resulting phase evolution (i.e., dephasing and rephasing, and phase changes due to SSSP pulse) are shown. Detection pulse (solid, red arrow) is used for optical Raman heterodyne detection of the spin coherence. Schematic figure not to scale.

Figure 3

Variation of the rephasing efficiency $\eta$ vs pulse duration $T$ and peak Rabi frequency ${\mathrm{\Omega }}_{\text{max}}$ for simple pulses of rectangular shape, e.g., $\pi$ pulses (1), and a single-shot-shaped pulse (2). Rephasing efficiency $\eta$ vs pulse duration $T$ and static detuning ${\delta }_{\text{static}}$ for simple pulses (3), and an SSSP (4) for a fixed peak Rabi frequency of $\approx 2\pi 125\text{kHz}$. Experimental data shown on the left column (E) and corresponding simulations on the right column (S). All efficiencies are normalized with respect to the maximal rephasing efficiency achieved with a $\pi$ pulse.

Figure 4

Experimental data on the variation of the rephasing efficiency vs the initial phase $\varphi$ of the coherence, for a rephasing sequence of two $\pi$ pulses ($T=1.6\mu \text{s}$; $\mathrm{\Omega }=2\pi 313\text{kHz}$) (black squares) or two SSSPs ($T=48\mu \text{s}$; ${\mathrm{\Omega }}_{\text{max}}=2\pi 90\text{kHz}$) (red dots).

Figure 5

(a) Profile of the figure of merit of the composite pulse of Ref. [7] with the parameters (10). (b) Profile with SSSP pulse [Fig. 1].

Figure 6

Fidelity profiles. (a) SSSP depending on 10 parameters given in Table 2; (b) composite pulse of (10); (c) BB1inC pulse of Table 3. The white lines correspond to $J=0.9$, $J=0.99$, $J=0.999$, and $J=0.9999$ from the outer to the inner line. In each figure, ${\mathrm{\Omega }}_{max}$ is the maximum intensity reached by the corresponding pulse (see Table 4).