Universal composite pulses for efficient population inversion with an arbitrary excitation profile

We introduce a method to rotate arbitrarily the excitation profile of universal broadband composite pulse sequences for robust high-fidelity population inversion. These pulses compensate deviations in any experimental parameter (e.g., pulse amplitude, pulse duration, detuning from resonance, Stark shifts, unwanted frequency chirp, etc.) and are applicable with any pulse shape. The rotation allows one to achieve higher-order robustness to any combination of pulse area and detuning errors at no additional cost. The latter can be particularly useful, e.g., when detuning errors are due to Stark shifts that are correlated with the power of the applied field. We demonstrate the efficiency and universality of these composite pulses by experimental implementation for rephasing of atomic coherences in a ${\text{Pr}}^{3+}\text{:}{\text{Y}}_2{\text{SiO}}_5$ crystal.

Figure 1

Numerical simulation: transition probability $P_{12}^{\left(n\right)}$ vs static detuning and duration $T$ of each constituent pulse (referred to as “pulse duration”) for a single pulse (a) and CPs from the U3 group [(b)–(e)] with phases from Table 1. The particular choice of ${\varphi }_2$ allows arbitrary rotation of the excitation profile and higher-order error compensation at no additional cost. Note that the excitation profiles of U3(${90}^{\circ }$) and U3($0^{\circ }$) are symmetric around the $\mathrm{\Delta }=0$ axis. The pulses are assumed rectangular with a Rabi frequency of $\mathrm{\Omega }=50\mathrm{kHz}$ (in frequency units). Thick solid lines indicate regions of high transfer probabilities beyond 0.95, thin solid lines indicate regions with transfer probability beyond 0.7, and dashed lines indicate regions with transfer probability beyond 0.5.

Figure 2

Numerical simulation: transition probability $P_{12}^{\left(n\right)}$ vs static detuning and duration $T$ of each constituent pulse for CPs from the U5 group with phases from Table 1 and Eq. (4). As expected, the high efficiency ranges are much broader than with the U3 sequence [compare with Figs. 1–1]. As we change the free parameter ${\varphi }_2$, the excitation profile rotates and allows additional higher-order error compensation for specific pulse area and detuning errors, as well as their combinations. Note that the excitation profiles of U5(${150}^{\circ }$) and U5(${330}^{\circ }$) are symmetric around the $\mathrm{\Delta }=0$ axis. The pulses are assumed rectangular with a Rabi frequency of $\mathrm{\Omega }=50$ kHz (in frequency units). For compactness of presentation we omit the plots for U5(${270}^{\circ }$) and U5(${30}^{\circ }$).

Figure 3

Experimental data: rephasing efficiency in the EIT-driven optical memory in ${\text{Pr}}^{3+}\text{:}{\text{Y}}_2{\text{SiO}}_5$, involving either single pulses (a) or different U3 rephasing sequences [(b)–(e)], vs pulse duration and detuning. Blue color indicates low and red color indicates high rephasing efficiency. The rephasing efficiency is normalized with respect to the maximum efficiency of all investigated sequences [i.e., the maximum of the U5(${60}^{\circ }$) sequence].

Figure 4

Experimental data: rephasing efficiency in the EIT-driven optical memory in ${\text{Pr}}^{3+}\text{:}{\text{Y}}_2{\text{SiO}}_5$, involving either single pulses (a) or different U5 rephasing sequences [(b)–(e)], vs pulse duration and detuning. The rephasing efficiency is normalized with respect to the maximum efficiency of all investigated sequences [i.e., the maximum of the U5(${60}^{\circ }$) sequence].