High-Fidelity and Robust Stimulated Raman Transition with Parameter-Modulated Optimal Control

High-fidelity and robust quantum control is essential for large-scale quantum information processing. The stimulated Raman transition that utilizes second-order coupling effect is a valuable and conventional technique for manipulating states in multi-level quantum systems, but its accuracy is limited by the driving-induced Stark shift. Here, we propose a new parameter-modulated method to effectively compensate the Stark-shift effect, so that we are able to realize high-fidelity and robust stimulated Raman transition with optimal control. Additionally, its robustness against different systematic errors can be further improved via optimization its average fidelity under these specific errors. Besides, our method has potential applications for high-fidelity and robust quantum control in high-order coupling scenarios.

INTRODUCTION

High-fidelity quantum manipulation is of great importance in the field of quantum optics, quantum computing, and quantum simulation [1–3]. Quantum information transmission between two quantum states in a quantum system or among different quantum systems is of fundamental importance in quantum information science [4]. However, direct quantum state transmission is impossible in most cases, and thus one needs to consult to a intermediate quantum bus [5, 6]. When the bus has long enough coherence time, quantum information transmission can be achieved by resonant coupling. However, for quantum bus with short coherence time, quantum information transmission is implemented by the stimulated Raman transition (SRT) [7–9] technique, i.e., using a two-photon resonant process to avoid the influence of the intermediate bus by decreasing its population. But the two-photon process is a second-order coupling effect that will result in the unwanted Stark shift [10], leading to the imperfect transmission.

To overcome this Stark-shift induced imperfection, stimulated Raman adiabatic passage (STIRAP) [5, 11, 12] that slowly changes the parameters of the quantum system is developed and well applied in quantum information transmission [13, 14]. The STIRAP process needs to meet adiabatic conditions, resulting in a prolonged duration of the process and thus the decoherence will introduce more detrimental effects [13, 15, 16]. To accelerate the STIRAP process and release the adiabatic conditions [17], shortcut to adiabaticity that generates a similar quantum adiabatic process in a shorter time has been proposed to achieve the high-fidelity quantum information transmission [11, 18–20]. Despite the shortcut to adiabaticity can realize fast and high-fidelity quantum state transfer [21–24], the existence of different systematic errors still influences its performance [25–28].

Alternatively, the Stark-shift effect of the SRT process can be compensated by modifying the drive frequency following its amplitude, thus realizing high-fidelity quantum transmission [29–31]. However, the compensation of the Stark shift [32] is usually complicated especially for complex driving amplitude, which is incompatible for further improving the robustness against different systematic errors. Notably, the Stark-shift effect of the SRT process can also be effectively compensated by parameter-modulated amplitudes and phases, which shares more flexible freedoms and can be used to implement high-fidelity and robust SRT.

Here, we propose a new method that parameter-modulated amplitudes and phases are optimized to realize high-fidelity and robust SRT with optimal control. We employ optimal control algorithms to design pulse parameters of the SRT process to optimize the fidelity of the transmission, which is further confirmed by numerical simulations using optimized parameters. Additionally, to enhance the robustness of the SRT process against different systematic errors, we optimize the average fidelity under different systematic errors, which is also numerically confirmed, achieving high-fidelity and robust SRT. Furthermore, due to the generality of our model, our results can be applied to a wide range of physical systems, such as nitrogen-vacancy centers [33, 34] and neutral atoms [27, 35]. Our methods unfreeze the second-order coupling induced Stark-shift effect of the SRT process and provide a promising way for high-fidelity and robust quantum control toward high-order coupling effect.

MODEL OF THE SYSTEM

For the SRT process, as shown in Fig. 1a, the population transfer between two quantum states |0〉 and |1〉, which are not directly coupled to each other, is implemented by a two-photon resonance process, where quantum states |0〉 and |1〉 are coupled to a intermediate quantum state |2〉 with Rabi frequencies ${{\Omega }_{{a,b}}}(t)$ and same detuning $\Delta (t)$. Here we consider the case that the coherence time of the intermediate quantum state |2〉 is much shorter than the quantum states |0〉 and |1〉, thus large detuning $\Delta (t)$ compared to the Rabi frequencies ${{\Omega }_{{a,b}}}(t)$ is needed to decrease the population of the intermediate quantum state |2〉. The Hamiltonian of the system in the rotating-wave approximation is given in the basis of {|0〉, |1〉, |2〉} by ($\hbar = 1$)

$${{H}_{1}}(t) = \left( {\begin{array}{*{20}{c}} 0&0&{{{\Omega }_{a}}(t)} \\ 0&0&{{{\Omega }_{b}}(t)} \\ {{{\Omega }_{a}}(t)}&{{{\Omega }_{b}}(t)}&{\Delta (t)} \end{array}} \right).$$

(1)

Fig. 1.

(Color online) Coupling schemes of the system. (a) $\Lambda$-type three-level systems and the corresponding driving fields to perform conventional SRT. (b) $\Lambda$-type three-level systems with parameter-modulated Rabi frequencies ${{\Omega }_{{0,1}}}(t)$ and the phases ${{\phi }_{{0,1}}}(t)$.

This three-level model is generic and can be found in a variety of physical systems, such as nitrogen-vacancy centers and neutral atoms. Using this Hamiltonian and simply setting the Rabi frequencies as square wave pulses ${{\Omega }_{a}}(t) = {{\Omega }_{b}}(t) = \Omega$ and the large detuning $\Delta (t)$ as a definite constant $\Delta$, the population transfer from the quantum states |0〉 to |1〉 can be realized. However, the coupling with large detuning between two energy levels can cause unwanted Stark shift of $\frac{1}{2}\sqrt {{{\Delta }^{2}} + {{{(2\Omega )}}^{2}}} - \Delta$, resulting in the infidelity for quantum information transmission. Specifically, for fast SRT process, the Stark-shift and the decoherence of the intermediate quantum state |2〉 cause the main infidelity. Inversely, for adiabatic SRT process, the decoherence of the quantum states |0〉 and |1〉 causes mainly infidelity. Optimal SRT process is discussed in Appendix A, where the fidelity of the SRT process is limited by the Stark-shift effect and the decoherence of the system.

Here, we introduce parameter-modulated Rabi frequencies ${{\Omega }_{{0,1}}}(t)$ and the phases ${{\phi }_{{0,1}}}(t)$ to the system, as shown in Fig. 1b. In this way, it is possible to compensate the Stark-shift effect of the SRT process, because parameter-modulated Rabi frequencies and the phases can effectively generate extra phase accumulations. Moreover, this approach also shares more flexible freedoms and can be used to implement high-fidelity and robust quantum information transmission. The Hamiltonian of the system with parameter‑modulated amplitudes and phases in the rot-ating-wave approximation is written in the basis of {|0〉, |1〉, |2〉} as

$${{H}_{2}}(t) = \left( {\begin{array}{*{20}{c}} 0&0&{{{\Omega }_{0}}(t){{e}^{{i{{\phi }_{0}}(t)}}}} \\ 0&0&{{{\Omega }_{1}}(t){{e}^{{i{{\phi }_{1}}(t)}}}} \\ {{{\Omega }_{0}}(t){{e}^{{ - i{{\phi }_{0}}(t)}}}}&{{{\Omega }_{1}}(t){{e}^{{ - i{{\phi }_{1}}(t)}}}}&{\Delta (t)} \end{array}} \right).$$

(2)

Based on this time-dependent Hamiltonian, we will introduce an optimization method to realize high-fidelity and robust SRT.

OPTIMIZATION OF THE SRT PROCESS

We proceed to implement high-fidelity population transfer of the SRT process. Without loss of generality, we set detuning $\Delta (t)$ as a definite constant $\Delta$ and parameterize the Rabi frequencies ${{\Omega }_{{0,1}}}(t)$ and the phases ${{\phi }_{{0,1}}}(t)$ using Fourier series up to $N$th order as:

$$\begin{array}{*{20}{c}} {{{\Omega }_{i}}(t) = \sum\limits_{n = 1}^N {{a}_{{{{i}_{n}}}}}\sin \frac{{2n\pi t}}{T} + {{b}_{{{{i}_{n}}}}}\left( {1 - \cos \frac{{2n\pi t}}{T}} \right),} \\ {{{\phi }_{i}}(t) = \sum\limits_{n = 1}^N {{c}_{{{{i}_{n}}}}}\sin \frac{{2n\pi t}}{T} + {{d}_{{{{i}_{n}}}}}\cos \frac{{2n\pi t}}{T},} \end{array}$$

(3)

where T is the total evolution time, the expansion factors $\{ {{a}_{{{{i}_{n}}}}},{{b}_{{{{i}_{n}}}}},{{c}_{{{{i}_{n}}}}},{{d}_{{{{i}_{n}}}}}\}$ are free parameters for the parameterization of the Rabi frequencies ${{\Omega }_{{0,1}}}(t)$ and the phases ${{\phi }_{{0,1}}}(t)$, and index $i = \{ 0,1\}$, $n = 1:N$ are referred to control pulses and expansion orders, respectively. The above expansion of pulse envelopes is chosen based on the practical consideration that the drives are inactive at the time boundaries for ${{\Omega }_{i}}(0)$ = Ω_i(T) = 0, where the phases have no such restrictions. This parameterization allows us to introduce optimization algorithm to modulate pulses, thereby converting the problem of control pulse design into an optimization process.

Here, Covariance Matrix Adaptation Evolution Strategy [36–38], a stochastic and derivative-free approach, is employed for numerical optimization. It generates M arrays of candidate parameters, denoted by $\{ {{{\boldsymbol{\lambda }}}_{m}} = (a_{{{{i}_{n}}}}^{m},b_{{{{i}_{n}}}}^{m},c_{{{{i}_{n}}}}^{m},d_{{{{i}_{n}}}}^{m}),m = 1:M\}$ and selects some of them as parent parameters by the cost function values to generate new iterations. The above procedure is repeated until parameters are found, which makes the cost function converge or reaches the desired value. Thus, the choice of the cost function is crucial for the performance of the outcome control. Here, we optimize the population transfer of the SRT process by defining the cost function as the state infidelity. The infidelity of the population transfer of the SRT process is calculated by [39]

$$O({{{\boldsymbol{\lambda }}}_{m}}) = 1 - {\text{Tr}}\left[ {\sqrt {\sqrt {\rho ({{{\boldsymbol{\lambda }}}_{m}})} {{\rho }_{0}}\sqrt {\rho ({{{\boldsymbol{\lambda }}}_{m}})} } } \right],$$

(4)

where ${{\rho }_{0}}$ is the density matrix of the target state and $\rho ({{{\boldsymbol{\lambda }}}_{m}})$ corresponds to the actual final state controlled by parameters λ_m. During the optimization, the decoherence is not considered for fast iteration, and the low bound of the cost function is set as $O({{{\boldsymbol{\lambda }}}_{m}}{{) = 10}^{{ - 5}}}$ for high-fidelity population transfer of the SRT process. Here, considering the technical limitation of actual pulse generator, we chose the expansion order truncated to N = 4, which gives 16 free parameters for each pulse. All the maximum values of Rabi frequencies are restricted to be ${{\Omega }_{{{\text{max}}}}} \leqslant 0.2\Delta$ to ensure that the optimized control is attainable under current technical conditions. The operation time is chosen as $T = 20{\text{/}}\Delta$ under the restriction of Rabi frequencies. Then, the Rabi frequencies ${{\Omega }_{{0,1}}}(t)$ and the phases ${{\phi }_{{0,1}}}(t)$ of the control pulses for the optimized solution to cost function of Eq. (4) are plotted in Figs. 2a and 2b, respectively.

Fig. 2.

(Color online) Population and the optimized pulse shapes. (a) and (b) are optimal amplitude and phase of the pulse in relation to time, respectively. (c) and (d) are the population of each state of the conventional SRT process and the optimized SRT process under decoherence, respectively.

For realistic physical system, the environment of the quantum system will lead to the decoherence. Thus, we employ the Lindblad master equation [40] of

$$\dot {\rho } = - i[H(t),\rho ] + \sum\limits_k {{\Gamma }_{k}}\left[ {{{l}_{k}}\rho l_{k}^{\dag } - \frac{1}{2}(l_{k}^{\dag }{{l}_{k}}\rho + \rho l_{k}^{\dag }{{l}_{k}})} \right],$$

(5)

to evaluate the evolution of the system, where $\rho$ is the density matrix of the quantum system, the operators ${{l}_{1}} = {\text{|}}0\rangle \langle 2{\text{|}}$, ${{l}_{2}} = {\text{|}}1\rangle \langle 2{\text{|}}$ and ${{l}_{3}} = {\text{|}}0\rangle \langle 1{\text{|}}$ describe dissipations from higher states to lower ones with decay rates ${{\Gamma }_{{1,2,3}}}$, and ${{l}_{4}} = {\text{|}}1\rangle \langle 1{\text{|}}$ is the dephasing of the system with rate ${{\Gamma }_{4}}$. The decoherent rates can be chosen as Γ₁ = ${{\Gamma }_{2}} = 0.001\Delta$, ${{\Gamma }_{3}} = {{\Gamma }_{4}} = 0.00001\Delta$, which is a general setting for the coherence time of the intermediate quantum state ${\text{|}}2\rangle$ much shorter than the quantum states ${\text{|}}0\rangle$ and ${\text{|}}1\rangle$. There are many physical systems meeting this condition, such as nitrogen-vacancy centers [33, 34] and neutral atoms [27, 35]. Then, we use the Lindblad master equation to numerically simulate conventional and optimized SRT process.

As discussed in Appendix A, the highest transfer fidelity for the SRT process is 99.67% for the Rabi frequency $\Omega = 0.196\Delta$ and the evolution time $T = 7{\text{/}}\Delta$. We plot the population of the quantum states ${\text{|}}0\rangle ,{\text{|}}1\rangle$ and ${\text{|}}2\rangle$ of the conventional SRT process in Fig. 2c. The infidelity is caused by the stark-shift effect about $5 \times {{10}^{{ - 5}}}$ and the decoherence about $3.25 \times {{10}^{{ - 3}}}$. In addition, we numerically simulate the optimized SRT process and plot the population of the quantum states ${\text{|}}0\rangle ,{\text{|}}1\rangle$ and ${\text{|}}2\rangle$ with parameter-modulated optimal pulse as shown in Fig. 2d. More detailed data of the optimized Rabi frequencies ${{\Omega }_{{0,1}}}(t)$ and phases ${{\phi }_{{0,1}}}(t)$ plotted in Figs. 2a and 2b are listed in Table 1. It’s clear that the state fidelity of the optimized SRT process can realize more higher fidelity of 99.91% than the conventional SRT process. That is, the Stark-shift effect of the SRT process can be effectively compensated by our optimal pulse, and the infidelity is almost caused by the decoherence effect only.

Table 1 Extracted parameters $\{ {{a}_{{{{i}_{n}}}}},{{b}_{{{{i}_{n}}}}},{{c}_{{{{i}_{n}}}}},{{d}_{{{{i}_{n}}}}}\}$ of the optimal amplitudes and phases plotted in Figs. 2a and 2b in Section 3

OPTIMIZATION OF THE ROBUSTNESS

The parameter-modulated optimal pulse has greatly improved the population transfer, so in this section, we will further introduce the optimization of the robustness of the SRT process against different systematic errors. It is worth noting that different systematic errors possibly appear in quantum system during experiment, which will result in the imperfection of the quantum evolution process. The cost function for robust optimization is defined as average infidelity

$$\bar {O}({{{\boldsymbol{\lambda }}}_{m}}) = \int P({\boldsymbol{\gamma }})O({{{\boldsymbol{\lambda }}}_{m}},{\boldsymbol{\gamma }}){\mathbf{d}\boldsymbol{\gamma }},$$

(6)

where $P({\boldsymbol{\gamma }})$ is the probability distribution of k-dimensional error array ${\boldsymbol{\gamma }} = ({{\gamma }_{1}},...,{{\gamma }_{k}}{{)}^{T}}$ and infidelity $O({{{\boldsymbol{\lambda }}}_{m}},{\boldsymbol{\gamma }})$ varies with control parameters λ_m and errors γ. We can define each error γ as an independent stochastic variable following a Gaussian distribution

$$P(\gamma ) = \frac{1}{{\sqrt {2\pi \sigma _{\gamma }^{2}} }}{\text{exp}}\left( {\frac{{{{\gamma }^{2}}}}{{2\sigma _{\gamma }^{2}}}} \right),$$

(7)

as it effectively indicates that the errors in experiment tend to concentrate around zero with standard deviation ${{\sigma }_{\gamma }}$. The averages of infidelity are numerically simulated in the range of $[ - 6{{\sigma }_{\gamma }},6{{\sigma }_{\gamma }}]$ with step $12{{\sigma }_{\gamma }}{\text{/}}10$ for each error $\gamma$. Additionally, based on the optimized SRT pulse as described in Section 3, the robust optimisation will speed up its convergence in decoherence-free case. The robust parameter-modulated control pulse will be found when the convergence of loss function meets the optimization requirement.

In the following, we begin to illustrate the improvement of the robustness against different systematic errors with our optimal control, respectively.

Amplitude Error

Consider the fluctuation in the Rabi frequency, i.e., the amplitude error ${{\Omega }_{i}}(t) \to (1 + \epsilon ){{\Omega }_{i}}(t)$, the error Hamiltonian ${{H}_{\epsilon }}(t)$ is

$${{H}_{\epsilon }}(t) = \epsilon ({{\Omega }_{i}}(t){\text{|}}i\rangle \langle 2{\text{|}} + {\text{h}}{\text{.c}}.),$$

(8)

where $\epsilon$ denotes the error ratio. Then, the overall Hamiltonian is now $H(t) \to H(t) + {{H}_{\epsilon }}(t)$. The fidelity of the conventional SRT process and the optimized SRT process under amplitude error with decoherence is shown in Fig. 3c. The fidelity of the optimised SRT process is greater than 99.5% in the range of $\epsilon \in [ - 0.08,0.08]$, as shown by the solid blue line, indicating that our optimisation is insensitive to the fluctuation of amplitude. However, the amplitude error has a significant effect on the performance of the SRT, as shown by the dotted red line. More detailed data of the optimized Rabi frequencies ${{\Omega }_{{0,1}}}(t)$ and phases ${{\phi }_{{0,1}}}(t)$ plotted in Figs. 3a and 3b are listed in Table 2.

Fig. 3.

(Color online) Optimal parameter-modulated pulse envelopes and phases in the presence of amplitude or off-resonant error and the fidelity comparison of the optimisation scheme with SRT under decoherence. (a) and (b) are optimal amplitude and phase of the pulse in relation to time respectively, (c) comparison of the robustness of the SRT and the optimisation method with amplitude error. (d) and (e) are optimal amplitude and phase of the pulse in relation to time respectively; (f) comparison of the robustness of the SRT and the optimisation method with off-resonant error.

Table 2 Extracted parameters $\{ {{a}_{{{{i}_{n}}}}},{{b}_{{{{i}_{n}}}}},{{c}_{{{{i}_{n}}}}},{{d}_{{{{i}_{n}}}}}\}$ of the optimal amplitudes and phases plotted in Figs. 3a and 3b in Section 4

Off-Resonant Error

Another significant error in the three-level system is non-resonance, where the energy shift is caused by environmental influences and imperfect controls, leading to the infidelity for quantum information transmission. To overcome this problem, further optimisation of the pulse is required to achieve perfect population transfer. The non-resonant error Hamiltonian ${{H}_{\eta }}(t)$ is

$${{H}_{\eta }}(t) = \left( {\begin{array}{*{20}{c}} \eta &0&0 \\ 0&{ - \eta }&0 \\ 0&0&0 \end{array}} \right),$$

(9)

where $\eta$ denotes the non-resonant error with unit of $0.2\Delta$ in following numerical simulation. Then the total Hamiltonian is now $H(t) \to H(t) + {{H}_{\eta }}(t)$. Taking the same definition of average infidelity as above, the fidelity of the conventional SRT process and the optimized SRT process under off-resonant error with decoherence is shown in Fig. 3f. The fidelity of the optimised SRT process (solid blue line) is flatter than the conventional SRT process (red dashed line), which indicates that our scheme is more robust than the conventional SRT under the Off-resonant error. More detailed data of the optimized Rabi frequencies ${{\Omega }_{{0,1}}}(t)$ and phases ${{\phi }_{{0,1}}}(t)$ plotted in Figs. 3d and 3e are listed in Table 3.

Table 3 Extracted parameters $\{ {{a}_{{{{i}_{n}}}}},{{b}_{{{{i}_{n}}}}},{{c}_{{{{i}_{n}}}}},{{d}_{{{{i}_{n}}}}}\}$ of the optimal amplitudes and phases plotted in Figs. 3d and 3e in Section 4

Amplitude and Off-Resonant Error

Compositing two situations above, the total Hamiltonian will be $H(t) \to H(t) + {{H}_{\eta }}(t) + {{H}_{\epsilon }}(t)$ and the probability distribution $P(\eta ,\epsilon ) = P(\eta )P(\epsilon )$. The optimal amplitudes and phases of pulse are shown in the Figs. 4a and 4b with more detailed data in Table 4, respectively. The other parameters are the same as those mentioned above. The fidelity of the conventional and optimised SRT process under the systematic errors $- 0.04 \leqslant \epsilon \leqslant 0.04$, $- 0.04 \leqslant \eta \leqslant 0.04$ with the decoherence effect are shown in Figs. 4c and 4d, respectively. It’s clear that the fidelity of our scheme is almost always above 99.5%, and can be greater than 99.8% within a considerable margin of errors. Therefore, our scheme can achieve stronger robustness to both amplitude errors and off-resonant errors than previous schemes.

Fig. 4.

(Color online) Optimal parameter-modulated pulse envelopes and phases in the presence of composite systematic errors and the fidelity comparison of the optimisation scheme with SRT under decoherence. (a) and (b) are optimal amplitude and phase of the pulses in relation to time, respectively. (c) and (d) show the fidelity of the conventional SRT process and the optimised SRT process versus both amplitude and non-resonance errors under decoherence, respectively.

Table 4 Extracted parameters $\{ {{a}_{{{{i}_{n}}}}},{{b}_{{{{i}_{n}}}}},{{c}_{{{{i}_{n}}}}},{{d}_{{{{i}_{n}}}}}\}$ of the optimal amplitudes and phases plotted in Figs. 4a and 4b in Section 4

CONCLUSIONS

In conclusion, we proposed an optimal control method by modulating the amplitude and phase parameters to effectively improve the fidelity and the robustness of the SRT. Comparing with the conventional SRT process, our scheme can effectively compensate for the Stark-shift effect and achieve both high fidelity and robust population transfer under the realistic environments, including amplitude errors, non-resonance errors, and composite errors of them. The detail simulation results show that our parameter-modulated method can achieve precise population transfer between coupled spin states, even in the presence of amplitude and/or non-resonant errors, and maintain high fidelity despite decoherence. Therefore, our scheme is insensitive to the perturbation in all control parameters and has potential applications for high-fidelity and robust quantum control toward high-order coupling effect.

Appendices

Appendix A

Optimal conventional SRT. To obtain an optimal working point for the conventional SRT process under decoherence, we present a simple optimisation by using the Hamiltonian Eq. (1) in the main text and setting the Rabi frequencies as square wave pulses ${{\Omega }_{a}}(t) = {{\Omega }_{b}}(t) = \Omega$ for simple with large time-independent detuning $\Delta$. The 3D graph of transfer fidelity of conventional SRT under decoherence is plotted with d$\Omega$ = 0.001 and d$T$ = 0.5 as shown in Fig. 5. Then we can find that the optimal working point of conventional SRT process at $\Omega = 0.196\Delta$ and $T = 7{\text{/}}\Delta$ can achieve a maximum fidelity of 99.67%, as marked with “$+$” in Fig. 5.

Fig. 5.

(Color online) Transfer fidelity and the optimal working point of conventional SRT under decoherence.

Appendix B

Optimal data. In this appendix, we present the detailed parameters of the amplitude and phase of the optimised control pulses. Considering the parameter-modulated amplitudes and phases in Eq. (3), the factors $\{ {{a}_{{{{i}_{n}}}}},{{b}_{{{{i}_{n}}}}},{{c}_{{{{i}_{n}}}}},{{d}_{{{{i}_{n}}}}}\}$ are free parameters for the parameterization of the Rabi frequencies ${{\Omega }_{{0,1}}}(t)$ and the phases ${{\phi }_{{0,1}}}(t)$, where ${{a}_{{{{0}_{n}}}}}$, ${{b}_{{{{0}_{n}}}}}$ and ${{c}_{{{{0}_{n}}}}}$, ${{d}_{{{{0}_{n}}}}}$ are the amplitude and phase of the optimised pump field, ${{a}_{{{{1}_{n}}}}}$, ${{b}_{{{{1}_{n}}}}}$ and ${{c}_{{{{1}_{n}}}}}$, ${{d}_{{{{1}_{n}}}}}$ are the amplitude and phase of the optimised Stokes field, respectively.

To optimize the SRT process, the parameters of the optimal amplitudes and phases plotted in Figs. 2a and 2b without systematic errors are listed in Table 1. Considering the optimization of the robustness of the SRT process, the parameters of optimal pulse plotted in Figs. 3a and 3b with amplitude error are listed in Table 2, and the parameters of optimal pulse plotted in Figs. 3d and 3e with off-resonant error are listed in Table 3, while the parameters of optimal pulse plotted in Figs. 4a and 4b with both amplitude error and off-resonant error are listed in Table 4, respectively.