Composite pulses for optimal robust control in two-level systems

In this work, we put forward an approach of constructing the optimal composite pulse sequence for robust population transfer in two-level systems. This approach is quite universal and applicable to a variety of systems, because the modulation parameters of composite pulses are obtained by minimizing the homemade cost function rather than nullifying the error coefficients of the transition probability. Specifically, we design different forms of the cost function for implementing optimal robustness with respect to the single or multiple errors. When slightly adjusting the constraints of the cost function, this approach can be easily extended to achieve arbitrary population transfer. The numerical results demonstrate that the optimized sequences are immune from various systematic errors, allowing us to produce a broadening excitation range of the transition probability. Therefore, this work offers an optimal robust design for controlling quantum states in error-prone environments.

Figure 1

The width $W_k$ of the high-excitation range vs the weighting factor $r$, where the modulation parameters of the optimal CP sequences are obtained by minimizing the cost function ${\mathcal{J}}_5$.

Figure 2

(a) Excitation profiles for the $\mathrm{E}{2}_{{\epsilon }_A}^{\left[l\right]}$ sequence ($l=1,\cdots ,5$), where $r=0.0001$ and the modulation parameters come from Table 1. The $\mathrm{E}{2}_{{\epsilon }_A}^{\left[1\right]}$ sequence is the optimal one, while the other sequences are obtained by the extreme values of the cost function ${\mathcal{J}}_5$. (b) Transition probability error $1-P_{{\epsilon }_A}^{\left(2\right)}$ vs the pulse area error ${\epsilon }_A$ in a logarithmic scale.

Figure 3

(a) Excitation profiles for different optimal CPs by choosing $r={10}^{-k}$ in the cost function ${\mathcal{J}}_4$, where $k=0,\cdots ,4$. (b) Transition probability error $1-P_{{\epsilon }_A}^{\left(2\right)}$ vs the error ${\epsilon }_A$ in a logarithmic scale.

Figure 4

(a) Excitation profiles for different optimal CPs by choosing $\mathcal{M}=1,\cdots ,6$ in the cost function ${\mathcal{J}}_{\mathcal{M}}$, where $r=0.01$. (b) Transition probability error $1-P_{{\epsilon }_A}^{\left(2\right)}$ vs the error ${\epsilon }_A$ in a logarithmic scale.

Figure 5

(a) Excitation profiles for the single resonant $\pi$ pulse (${\pi }_{{\epsilon }_A}, {\pi }_{{\epsilon }_{\mathrm{\Omega }}}$, and ${\pi }_{{\epsilon }_{\mathrm{\Delta }}}$) and the two-pulse sequence ($\mathrm{O}{2}_{{\epsilon }_A}, \mathrm{O}{2}_{{\epsilon }_{\mathrm{\Omega }}}$, and $\mathrm{O}{2}_{{\epsilon }_{\mathrm{\Delta }}}$), where the modulation parameters come from Table 2. (b) Transition probability error $1-P_{\epsilon }^{\left(2\right)}$ vs the error $\epsilon$ in a logarithmic scale.

Figure 6

Excitation profiles for the two-pulse sequence by (a) all-parameter modulation and (b) phase modulation, where the modulation parameters are given in Table 3. The curves represent the transition probability $\gamma$ locked at the level $0.1,0.2,...$, and 1. (c) The high-excitation range (i.e., $P_{{\epsilon }_A}^{\left(2\right)}\ge 0.999\gamma$) for the two types of modulation.

Figure 7

Transition probability $P_{{\epsilon }_{\mathrm{\Delta }}{\epsilon }_A}^{\left(2\right)}$ vs the errors ${\epsilon }_{\mathrm{\Delta }}$ and ${\epsilon }_A$ in (a) the $\mathrm{O}{2}_{{\epsilon }_{\mathrm{\Delta }}{\epsilon }_A}^{\left[1\right]}$ sequence and (b) the $\mathrm{O}{2}_{{\epsilon }_{\mathrm{\Delta }}{\epsilon }_A}^{\left[2\right]}$ sequence, where the modulation parameters are given in Table 2. The white, black, and green curves correspond to $P_{{\epsilon }_{\mathrm{\Delta }}{\epsilon }_A}^{\left(2\right)}=0.9$, 0.99, and 0.999, respectively (from outside in).

Figure 8

(a) Excitation profiles for the $\mathrm{O}{3}_{{\epsilon }_A}, \mathrm{D}{3}_{{\epsilon }_A}^{\left[1\right]}, \mathrm{D}{3}_{{\epsilon }_A}^{\left[2\right]}, \mathrm{T}{3}_{{\epsilon }_A}^{\left[1\right]}, \mathrm{T}{3}_{{\epsilon }_A}^{\left[2\right]}$, and BB1 sequences, where the modulation parameters of the first five sequences are given in Table 5. (b) Transition probability error $1-P_{{\epsilon }_A}^{\left(3\right)}$ vs the pulse area error ${\epsilon }_A$ in a logarithmic scale.

Figure 9

Transition probability $P_{{\epsilon }_{\mathrm{\Delta }}{\epsilon }_A}^{\left(3\right)}$ vs the errors ${\epsilon }_{\mathrm{\Delta }}$ and ${\epsilon }_A$ in (a) the $\mathrm{O}{3}_{{\epsilon }_{\mathrm{\Delta }}{\epsilon }_A}^{\left[1\right]}$ sequence, (b) the $\mathrm{O}{3}_{{\epsilon }_{\mathrm{\Delta }}{\epsilon }_A}^{\left[2\right]}$ sequence, (c) the $\mathrm{D}{3}_{{\epsilon }_A}^{\left[1\right]}$ sequence, (d) the $\mathrm{D}{3}_{{\epsilon }_A}^{\left[2\right]}$ sequence, (e) the $\mathrm{U}{3}_{{\epsilon }_{\mathrm{\Delta }}{\epsilon }_A}^{\left[1\right]}$ sequence, (f) the $\mathrm{U}{3}_{{\epsilon }_{\mathrm{\Delta }}{\epsilon }_A}^{\left[2\right]}$ sequence, (g) the short CORPSE sequence, (h) the CORPSE sequence, (i) the reduced CinBB sequence, and (j) the CinBB sequence, where the modulation parameters of the first eight sequences are given in Table 5. The white, black, green, blue, and red curves correspond to $P_{{\epsilon }_{\mathrm{\Delta }}{\epsilon }_A}^{\left(3\right)}=0.9$, 0.99, 0.999, 0.9999, and 0.99999, respectively (from outside in).