Universal composite pulses for robust quantum state engineering in four-level systems

We propose a protocol for robust quantum state engineering using composite pulses (CPs) in four-level systems. The analytical expression of the propagator is derived for the implementation of universal single-qubit gates and the maximum superposition state. By carefully designing the relative phases between pulses, the CP sequences can compensate for the pulse area error to any desired order. We present two classes of CP sequences, one generating robust population inversion and the other generating robust superposition states. As applications, we employ the well-designed CP sequences to achieve the conversion of the W and Greenberger-Horne-Zeilinger states with high fidelity in a Rydberg atomic system. It is shown that the CP sequences yield excellent robustness with respect to the pulse area errors, and they possess a short evolution time.

Figure 1

Two-dimensional surfaces for Eq. (7).

Figure 2

Transition probability of the target state $|4\rangle$ vs pulse area errors for (a)–(c) the single pulse, (d)–(f) the three-pulse sequence, and (g)–(i) the five-pulse sequence. The pulse areas $A_m$ satisfy Eq. (9) and all phases are given by the analytical solutions. One can see in panels (a)–(c) that the region of high transition probability ($P>99.99\%$) hardly appears in the single pulse.

Figure 3

Fidelity of the superposition state $\left(\right|1\rangle +|4\rangle )/\sqrt{2}$ vs pulse area errors for (a) the single pulse and (b) the five-pulse sequence. The fidelity is defined by $F=1/2|(\langle 4|+\langle 2\left|\right)|\psi \left(T\right)\rangle {|}^2$. The pulse areas $A_m$ satisfy Eq. (9) and the phases are given in Table 2.

Figure 4

(a) Energy-level structure of three identical Rydberg atoms driven by four external driving fields. (b) Rydberg interaction between adjacent atoms with strength $V$.

Figure 5

GHZ-state fidelity vs pulse area errors for (a) the LRI method and (b) the ${\mathrm{S}}_{{\epsilon }_{123}}^5$ sequence. The third pulse area error ${\epsilon }_3$ is set to 5%.

Figure 6

(a) GHZ-state fidelity vs the pulse area error $\epsilon$ for the ${\mathrm{S}}_{\epsilon }^3$ sequence and the LRI method. (b) Time evolution of the GHZ-state fidelity using the ${\mathrm{S}}_{\epsilon }^3$ sequence and the LRI method. Note that the appearance of slight oscillations [e.g., see the inset in panel (b)] during the evolution process is due to the high-frequency components in the Hamiltonian (20).

Figure 7

Transition probability deviation $1-P_{41,\epsilon }^3$ vs the pulse area error $\epsilon$ for different three-pulse sequences in a logarithmic scale, where the phases are given by Eq. (D4). The two gray lines indicate the pulse area errors in the interval $[-0.05,0.05]$.

Figure 8

Transition probability deviation $1-P_{41,\epsilon }^5$ vs the pulse area error $\epsilon$ for different five-pulse sequences in a logarithmic scale, where ${\mathrm{\Theta }}_{2,3}^n$ denotes the $n\mathrm{th}$ solution in Table 3. The two gray lines indicate the pulse area errors in the interval $[-0.05,0.05]$.

Figure 9

Transition probability deviation $1-P_{41,\epsilon }^N$ of the state $|4\rangle$ vs pulse area errors for different CP sequences with the pulse areas $A_m$ satisfying Eq. (9). (a) ${\epsilon }_1={\epsilon }_2={\epsilon }_3=\epsilon$. (b) ${\epsilon }_2={\epsilon }_3=0$. (c) ${\epsilon }_1={\epsilon }_3=0$. (d) ${\epsilon }_1={\epsilon }_2=0$. All phases are given in Table 2.