1. Introduction

The accurate manipulation of quantum systems with high-fidelity is of great importance in atomic, molecular, optical physics and chemistry [1–4]. In particular, stimulated Raman adiabatic passage (STIRAP) [4,5] is a technique that can control preparation of the internal state and the dynamics of three-level quantum systems, so that an efficient and selective population transfer between quantum states is implemented. Population transfer by STIRAP has two distinct advantages: (i) it is robust against loss due to spontaneous emission from the intermediate state; (ii) it is insensitive to small variations of laser intensity, the duration, and the delay of pulses when the adiabatic condition is fulfilled. However, the adiabatic evolution in STIRAP requires a long run time for the desired parametric control, which makes the systems vulnerable to open system effects [6]. On the other hand, resonant pulses can achieve fast quantum state engineering, but they are sensitive to the parameter fluctuations that may lead to loss of fidelity. Therefore, designing an effective method that combines both the the advantages of adiabatic control and resonant interactions, which realizes fast and robust quantum state control, is a hot topic in recent years.

Shortcuts to adiabaticity (STA) [7,8], including transitionless quantum driving [9–11], Lewis-Riesenfeld (LR) invariant-based inverse engineering [12,13], fast-forward protocol [14], and time-rescaling [15], are optimal control protocols that can speed up a quantum adiabatic process through a non-adiabatic route. They offer alternative fast and accurate processes which can reproduce the same final populations, or even the same final state in a shorter time, compared to the adiabatic evolution [8]. Recently, STA has been widely used to implement rapid and robust quantum information processing in theory, such as quantum state transfer [16–27], quantum gates [28–35], generation of quantum cat state [36–38], and so on. For instance, in 2012, Chen et al. [17] used the invariant-based shortcut to accelerate the STIRAP to realize population transfers in a short time. In 2016, Li et al. [23] achieved fast and accurate population transfer in three-level quantum systems with a stimulated Raman shortcut-to-adiabatic passage (STISRAP), which is a method by employing the STA to speed up the STIRAP. In 2019, Liu et al. [31] introduced an extensible nonadiabatic holonomic quantum computation protocol, which can incorporate STA to construct a variety of extensible nonadiabatic geometric gates that are robust against several types of noises. Experimentally, efforts have also been made on investigating the applications of STA in many quantum systems [39–46]. In 2016, Du et al. [40] demonstrated a fast and high-fidelity STISRAP to speed-up the conventional "slow" STIRAP. In 2019, Kölbl et al. [46] employed the STA state transfer protocol to realize high-fidelity, reversible initialization of individual dressed states in a closed-contour, coherent driving of a single spin system.

Noises, errors, and fluctuations will decrease the accuracy and efficiency of the control of quantum states. In particular, some systematic errors cannot be avoided. For example, atoms take different Rabi frequencies induced by different positions and fields [47], and spin systems get different shifts of the amplitude of magnetic field caused by imperfections [48]. To suppress these errors, the invariant-based inverse engineering [47–52] is an effective method. In 2012, Ruschhaupt et al. [47] explored the stability of different shortcut protocols with respect to systematic errors in a generic two-level system. In 2018, Yu et al. [48] proposed a STA approach against systematic errors, to achieve robust spin flip and to generate the entangled Bell state. In 2019, Laforgue et al. [52] showed an exact and robust ultrahigh-fidelity population transfer on a three-level quantum system, using the Lewis-Riesenfeld method.

In this paper, we employ the invariant-based optimal control to realize a fast and robust three-level population transfer in a general setting. Specifically, we analytically find one solution of dynamic parameters for the optimal shortcut with zero systematic error sensitivity (SES). More importantly, we show that the SES is always very small for a family of solutions of a given functional form. Compared to the previous protocols, the optimal shortcuts can realize a perfect population transfer in a more rapid and accurate way. Notably, in contrast to the original protocol proposed by Chen and Muga [17], the time evolution of Rabi frequencies in our protocol does not diverge, even at the initial and final times, which implies that the frequencies can be easily implemented in the laboratory. The non-diverging frequencies are ensured by setting appropriate functional forms of the control signal, and our method may be applied to improve other protocols [19,28,53,54] that are based on the Lewis-Riesenfeld invariants [12] to avoid divergence (see Sec. 3.3 for detailed discussion). Moreover, in the case of negligible SES, a close relation between the amplitude of Rabi frequencies and the population of the intermediate state is formed; the smaller the population of the intermediate state is, the larger the maximal amplitudes of Rabi frequencies are. This offers alternative route for achieving high-fidelity state transfer of three-level quantum systems.

2. Stimulated Raman shortcut-to-adiabatic passage with invariant-based optimal control

We denote the three orthogonal quantum states of a three-level system by $\{\left |1\right \rangle ,\left |2\right \rangle ,\left |3\right \rangle \}$, with $\left |2\right \rangle$ being the intermediate state, and $\left |1\right \rangle$ and $\left |3\right \rangle$ being two lower-energy states which represent the initial and target states, respectively. The STIRAP is an efficient method that can realize the selective population transfer between the state $\left |1\right \rangle$ and the state $\left |3\right \rangle$ by controlling the system to evolve along the adiabatic dark state. It involves a three-state, two-photon Raman process with counterintuitive pulse orders. More specifically, the three-level quantum system first interacts with the Stokes laser, which links the state $\left |3\right \rangle$ with the state $\left |2\right \rangle$, and then it interacts with the pump laser, which leads to the transition between the state $\left |1\right \rangle$ and the state $\left |2\right \rangle$. Meanwhile, to make sure that the state vector of the system adiabatically follows the evolution of the dark state, there must be a sufficient coupling (an appropriate overlap) between the two pulses. The Hamiltonian, under the rotating wave approximation with $\hbar =1$, takes the form of

Now we are ready to apply the inverse engineering to achieve the fast quantum driving along the eigenstate of invariant $\left |\phi _{0}\left (t\right )\right \rangle$ in a given time $T$. If we want to inverse the population from $\left |1\right \rangle$ to $\left |3\right \rangle$ along this eigenstate, the boundary conditions are given by [17]

3. Main results

3.1 Optimal STISRAP based on the Dyson perturbative series and the sensitivity

Here we consider systematic errors as the “perturbing” term representing a weak disturbance to the system. Combining invariant-based shortcuts and perturbation theory, we can get approximate solutions of the Schrödinger equation and design a robust quantum state engineering against such sources of errors. The perturbing Hamiltonian is assumed to take the form

We use the technique of the Dyson series to expand the wave function up to the second-order $O\left (\lambda ^{2}\right )$,

3.2 Systematic method for the optimal control design without divergence

To begin with, let us try to simplify the boundary conditions by choosing an appropriate functional form of $\gamma$. Setting

Having assumed the functional form of $\gamma$, the original boundary conditions imply that we should find a solution of $\beta \left (t\right )$ and $\alpha \left (t\right )$ such that

Let us then consider the constraint imposed by $\sin \gamma =\tilde {b}'_i/\tilde {\alpha '}_i$ for $i=1,2$, this leads to

To summarize, we have significantly simplified the original boundary conditions by picking some useful functional forms of $\gamma$ and $\beta$. We give the final analytic expressions of $b_i$ and $r_i$ in Eqs. (37) and (38), which automatically satisfy the boundary conditions, leaving only the parameters $D$, $B$, and $T$ to be determined. For any given evolution time $T$, we can choose the population parameter $B$, and solve the phase parameter $D$ by setting $q_s$ as a very small value, i.e., $q_s\ll 1$. In general, we cannot prove that the equation $q_s=0$ for the phase parameter $D$ always has a solution, but the numerical analysis shows that $q_s$ is at order of magnitude $10^{-2}$, which means that our protocol can suppress the the noise to $1\%$. Then the parameters $\beta$ and $\gamma$ for the Hamiltonian $H$ satisfying the boundary equations Eq. (17) are well determined when the constant $D$, $B$, and $T$ are given. Once the expressions for $b_i(m)$ and $r_i(m)$ are given, we can calculate the functions $\beta (t)$ and $\gamma (t)$ easily, and the Rabi frequencies $\Omega _p$ and $\Omega _s$ can be obtained from Eqs. (10) and (11).

The above scheme of finding the parameters for any desired parameters $B$ and $T$ constitutes our main result. Notably, this scheme is not limited to the $100\%$ population transfer from the state $|{1}\rangle$ to $|{3}\rangle$; it can also be applied to arbitrary population transfer, i.e., from the state $\cos \beta _0 |{1} \rangle + \sin \beta _0 |{3}\rangle$ to $\cos \beta _1 |{1}\rangle + \sin \beta _1 |{3}\rangle$ for arbitrary $\beta _0$ and $\beta _1$, where $\beta _0$ and $\beta _1$ are probability amplitudes. To achieve this arbitrary population transfer, one should set the boundary conditions for $\beta (t)$ as $\beta (0)=\beta _0$ and $\beta (T) = \beta _1$, which can be easily satisfied by adjusting the parameters $D$ and $F$ in the ansatz function of $b_i$ in Eq. (37).

3.3 Non-diverging Rabi frequencies and numerical analysis

In the following, we further explain why the solution derived by the above scheme makes sure that the Rabi frequencies do not diverge in the time evolution. Understanding why our method guarantees the non-divergence can help to improve other optimization problems in quantum control. And we verify numerically that it can realize a perfect and robust state transfer.

First of all, let us see why the Rabi frequencies $\Omega _s$ and $\Omega _p$ shown in Eqs. (10) and (11) can diverge. In their expression, the only term that can cause divergence with smooth parameters $\beta$ and $\gamma$, is the cotangent term $\dot {\beta } \cot \gamma$. This term will diverge as $\sin \gamma \rightarrow 0$, where $\sin ^{2} \gamma$ is the population of the intermediate state $ {|{2}\rangle }$ of the quantum system. On the other hand, in STIRAP, the population of the intermediate state before and after the transfer must be vanishing, which means $\sin \gamma = 0$ at the initial and final times for the control signal. This causes the divergence of Rabi frequencies in the original protocol proposed by Chen and Muga [17] and is a frequently encountered problem in quantum control. In fact, it is a common problem faced by most protocols [19,28,53,54] based on the Lewis-Riesenfeld invariants. Conventional approach to solve this problem is to start and end up with a state whose population on the intermediate state is $\epsilon \ll 1$ instead of $0$, e.g. the parameter $\gamma (t)$ in the parametrization of the eigenvector $|\psi _0(t)\rangle$ in Eq. (5) is chosen to be the constant $\gamma \equiv \epsilon$. But this will inevtiably compromise the overall performance of the control.

By contrast, our methods provide a new scheme to completely solve this divergence problem without introducing approximations and cut-off. Here we solve this divergence problem by picking appropriate functional forms of $\beta$, such that $\dot {\beta } \cot \gamma$ no longer diverges as $\sin \gamma$ goes to $0$. This is guaranteed by first viewing the parameter $\beta$ in the Rabi frequency as a function of $\gamma$ instead of the time $t$, such that $\beta (t) = \tilde {b}(\gamma )$. Next, we pick the functional dependence of the parameter $\gamma$ as $\tilde {b}(\gamma ) \equiv f(-\cos \gamma )$, where $f$ can be any arbitrary smooth function. Applying the chain rule, we can calculate the time derivative $\dot {\beta }$ as $\dot {\beta } = f'(-\cos \gamma ) \sin \gamma$, where $f'$ is the derivative of the function $f$. Therefore, the cotangent term $\dot {\beta } \cot \gamma$ becomes

In principle, our method can lead to improvements for all such quantum control protocols [19,28,53,54] where the two equations that may cause divergence (Eqs. (10) and (11)) are the fundamental building blocks for their protocol designs. One can just try our choice of the parametrization of $\gamma (t)$ in Eq. (23) and design an appropriate piece-wise function $\beta (t)$ following a similar step by step construction in our paper. It is expected to find an appropriate $\beta (t)$ suitable for various goals of quantum control, since there is plenty freedom left for picking the functional forms $f$. In our paper, we just use linear functional forms in Eq. (37), while one could also use polynomials and trigonometric functions. We leave such possible investigations for future study.

We then analyze the value of SES. Substituting the expression for the parameter $\beta$ and $\alpha$ in Eqs. (37) and (38) into the SES expression of Eq. (35), we get the analytic form of the systematic error sensitivity, which is plotted in Fig. 1. The x-coordinate is parameter $D$, and the y-coordinate represents the systematic error sensitivity, where $B=\pi /6$ and $T=0.1 \mu s$. Notably, one can find that the SES always takes a very small value ($q_s < 0.015\ll 1$) with the variations of the phase parameter $D$ and can reach zero when $D\approx -20$. This shows that we have found a family of robust protocols parametrized by the phase parameter $D$ against such systematic errors. Then the time-dependent Rabi frequencies $\Omega _{p}$ and $\Omega _{s}$ can be obtained, shown in Fig. 2(a), with maxima $\Omega ^{m}_{p}\approx \Omega ^{m}_{s}=28.50\times 2\pi ~{\rm MHz}$. Here $D=5.85$ is given to get a relatively small values of the maximum of Rabi frequencies, while the SES is a trivial value about $0.01$. The reasons for choosing such a $D$ are three-manifold: The pulses can possess the same amplitudes and symmetrical shapes with the value of the maximum of Rabi frequencies being $\Omega ^{m}_{p}\approx \Omega ^{m}_{s}=28.50\times 2\pi ~{\rm MHz}$, so it is easy to be prepared in experiment; the SES takes a very small value below $0.015$, and thus the protocol is still robust against the systematic errors; it is also convenient for further comparisons with other shortcut protocols when the same values of maximum of Rabi frequencies are designed. In principle, one can select a larger absolute value of $D$ to make the error sensitivity zero, but this may give large values of Rabi frequencies. The performances of Rabi frequencies with respect to the phase parameters with $D=5.85$, $D=11.7$, $D=17.55$, and $D=23.40$ when $B=\pi /6$ and $T=0.1 \mu s$ are plotted in Fig. 3(a), (b), (c), (d), respectively. From the figure, we can find that the Rabi frequencies show a more complicated behavior when the phase parameter is multiplied. Also, the maxima of the Rabi frequencies are not equal for the same phase parameter, and there exists the negative values of Rabi frequencies in the certain range of parameters. This trade-off is different from the trade-off in Ref. [17] that the boundary conditions of the auxiliary parameters must be set $\epsilon$ away from zero to prevent the Rabi frequencies from diverging, which ultimately leads to reduced state transfer fidelity for some protocols. From the Eqs. (6), (10), and (11), one can see that a boundary condition of $\gamma (0)=\epsilon$ with $\epsilon =0$ will lead to $\cot \gamma (0)=\infty$, implying infinite Rabi frequencies in the initial time. Thus, in their protocol, $\epsilon$ is set to a small value to avoid the infinite Rabi frequencies at the boundary times. Consequently, the population of the intermediate state with $|\sin \gamma (t)|^{2}$ is never equal to zero in the whole evolution.

 

Fig. 1. SES in Eq. (35) with respect to the parameter $D$. Here we use dimensionless units with $\hbar =1$, $T=0.1 \mu s$, and $B=\pi /6$. One can see that the condition $q_{s}=0$ is satisfied when $D\approx -20$. Moreover, the systematic error sensitivity always takes a very small value compared to $1$ for all process durations with the maximal value being $0.0107$.

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Fig. 2. (a) The Rabi frequencies $\Omega _{s}$ (red, dashed line) and $\Omega _{p}$ (blue, solid line) in the optimal STISRAP protocol. (b) Time evolution of the populations $P_1(t)$ (red, dashed line), $P_2(t)$ (green, solid line), and $P_3(t)$ (blue, dotted dashed line) by numerically solving the Schrödinger equation in the absence of relaxation. We have used $D=5.85$, $T=0.1\mu s$, and $B=\pi /6$.

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Fig. 3. The Rabi frequencies $\Omega _{s}$ (red, dashed line) and $\Omega _{p}$ (blue, solid line) with respect to time $t$ with (a) $D=5.85$, (b) $D=11.7$, (c) $D=17.55$, and (d) $D=23.40$, respectively, in the optimal STISRAP protocol. Here $T=0.1\mu s$ and $B=\pi /6$.

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To find the specific form of $H(t)$ in Eq. (1), we solve the Schrödinger equation $H(t)|\varphi (t)\rangle =i d|\varphi (t)\rangle /dt$ numerically by a Runge-Kutta method with an adaptive step, and get the time evolution of the populations $P_k(t)$ ($k$=1,2,3) during the population transfer based on the optimal STISRAP protocol, presented in Fig. 2(b). The population of the state $|k\rangle$ is defined as $P_k(t)= \langle k|\rho (t)|k\rangle$ with $\rho (t)=|\varphi (t)\rangle \langle \varphi (t)|$ being the final density matrix after time evolution of the population transfer operation on the initial state $|1\rangle$. One can see that the time evolutions governed by the Hamiltonian $H(t)$ can achieve a $100\%$ population transfer from the state $|1\rangle$ to the state $|3\rangle$, that is, $P_1(0)= 1$, $P_3(0)= 0$, and $P_1(T)= 0$, $P_3(T)= 1$. The population of the intermediate state $|2\rangle$ is $\sin ^{2}\gamma$, and it can reach its maximum $\sin ^{2}(\pi /6)=0.25$ when $t=T/2$, as we expected for $B=\pi /6$.

4. Trade-off between the population of the intermediate state and the amplitude of Rabi frequencies

In practice, in order to achieve high-fidelity state transfer in three-level quantum systems, one also needs to minimize the interference from occupation fluctuation of the intermediate level. However, we find that this needs a bigger amplitude of Rabi frequency, meaning a lager energy cost. In other words, one cannot simultaneously suppress the population of the intermediate level and reduce the energy cost. We will show such a trade-off between these two requirements in quantum state engineering.

Here, we study the relation between the population of the intermediate state and the amplitude of Rabi frequencies in the state transfer process. These two quantities are two positive factors contributing to the fidelity of the state transfer. To minimize the overall error, one has to balance these two sources of error. Thus, understanding such relation can help us to design an effective way to realize robust state transfer in different experiments. To show this, the population of the intermediate level is decreased by choosing different smaller value of $B$ in the optimal shortcut, and then appropriate Rabi frequencies are designed to achieve state transfer with zero systematic-error sensitivity for a given $D$.

First of all, substituting $B=\pi /8$ into Eq. (23), we set a new function $\gamma$. Subsequently, by plotting the change of the systematic error sensitivity with the parameter $D$, it is straightforward to find that the SES is always a small value below $0.002$, which means the effect of the noise is suppressed to $2$‰, suggesting that these family of protocols parameterized by the phase parameter $D$ is generally robust against the systematic error. In fact, an appropriate value of $D$ can be found numerically to make the sensitivity error vanish. Here, $D=10.32$ is chosen to make sure that the maxima of the two Rabi frequencies are the same and they are both relatively small ($\Omega ^{m}_{p}\approx \Omega ^{m}_{s}=38.6\times 2\pi ~{\rm MHz}$). Figure 4(a) shows the corresponding time evolution of Rabi frequencies $\Omega _{s}$ (red, dashed line) and $\Omega _{p}$ (blue, solid line). By numerically solving the Schrödinger equation when the initial state is $|1\rangle$, the populations of the three states are depicted in Fig. 4(b).

 

Fig. 4. (a) The Rabi frequencies $\Omega _{s}$ (red, dashed line) and $\Omega _{p}$ (blue, solid line) in the optimal STISRAP protocol. (b) Time evolution of the populations $P_1(t)$ (red, dashed line), $P_2(t)$ (green, solid line), and $P_3(t)$ (blue, dotted dashed line) by numerically solving the Schrödinger equation in the absence of relaxation. We have used $D=10.32$, $T=0.1\mu s$, and $B=\pi /8$.

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Secondly, $B=\pi /16$ is chosen to get a smaller population of intermediate state $|2\rangle$. In this case, the maximum of $P_2$ is $\sin ^{2}(\pi /16)\approx 0.038$. As a result, the systematic error sensitivity of the order $10^{-5}$ with respect to the parameter $D$, which means that the systematic error plays a negligible effect in the three-level population transfer. An equivalent maxima of the Rabi frequencies requires the value of $D$ to be 40.88. This leads to $\Omega ^{m}_{p}\approx \Omega ^{m}_{s}=78.3\times 2\pi ~{\rm MHz}$. Also, the time evolutions governed by the Hamiltonian possessing these Rabi frequencies can achieve a perfect population transfer from initial state $|1\rangle$ to final state $|3\rangle$. The corresponding time evolution of Rabi frequencies and the populations of the three states are plotted in Fig. 5(a) and (b), respectively.

 

Fig. 5. (a) The Rabi frequencies $\Omega _{s}$ (red, dashed line) and $\Omega _{p}$ (blue, solid line) in the optimal STISRAP protocol. (b) Time evolution of the populations $P_1(t)$ (red, dashed line), $P_2(t)$ (green, solid line), and $P_3(t)$ (blue, dotted dashed line) by numerically solving the Schrödinger equation in the absence of relaxation. We have used $D=40.88$, $T=0.1\mu s$, and $B=\pi /16$.

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From Figs. 2, 4, and 5, the performances of Rabi frequencies and populations for the three states with variation of time $t$ can be summarized as follows: (i) The systematic error sensitivity is always small ($q_s\ll 1$) for different $D$. Thus a large range of $D$ can be prescribed to get a proper $\gamma$ and $\beta$, and then the Rabi frequencies are found from Eqs. (10) and (11); (ii) By tuning an appropriate value of $D$, the equivalent maxima of Stokes and pumping pulses can be obtained, and the time evolution of the quantum state governed by the Hamiltonian with the two pulses can achieve a complete population transfer; (iii) The smaller the population of the intermediate state is, the larger the amplitudes of Rabi frequencies are.

One can increase the value of $B$, so that the population of intermediate state $|2\rangle$ is enlarged. Moreover, for $B=\pi$, an appropriate choice of $D$ can lead to a protocol whose SES is exactly zero. Substituting the given $B$ and $D$, the $\gamma$ and $\beta$ are determined from the Eqs. (37) and (38). The corresponding Rabi frequencies also can achieve a $100\%$ state transfer. The relation between the maximal population of the intermediate state and the maximal amplitude of Rabi frequencies can be summarized in Table 1, where $P_{2({\rm peak})}$ and $\Omega _\textrm {peak}$ represent the maximal population of the intermediate state and the maximal amplitude of Rabi frequencies in the transfer process, respectively. The relation is also plotted in Fig. 6, which shows a close relation between the $P_{2({\rm peak})}$ and $\Omega _\textrm {peak}$, namely, one must decrease as the other increases. This implies that a zero value of $P_{2({\rm peak})}$ is impossible as an infinite energetic cost cannot be reached. In fact, this trade-off is closely connected with the trade-off between the phase parameter and the Rabi frequencies. If one increases the absolute value of phase parameter, and thus the maximal amplitudes of the Rabi frequencies are enlarged (the energy cost is increased). Correspondingly, the SES can be suppressed to zero. In this case, the population of the intermediate state will decrease. And vice versa.

 

Fig. 6. The relation between the maximal population of the intermediate state ($P_{2({\rm peak})}$) and the maximal amplitude of Rabi frequencies ($\Omega _\textrm {peak}$) with the same values of $D$ and $B$.

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Table 1. The relation between the maximal population of the intermediate state ($P_{2({\rm peak})}$) and the maximal amplitude of Rabi frequencies ($\Omega _\textrm {peak}$) with the variations of $D$ and $B$.

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5. Comparison between the optimal invariant-based shortcut and the original protocol

To show the robustness of our optimal protocol against the systematic error, we compare it with the protocol 2 in the original invariant-based inverse engineering proposed by Chen and Muga in 2012 [17], which shall be referred as the original protocol. Both protocols can achieve the fast and high-fidelity quantum state transfer in a three-level quantum system along the eigenstate of the invariant $\left |\phi _{0}\left (t\right )\right \rangle$, in a given time $T$. To avoid the confusion of the symbols, we define the Stokes and pumping pulses in the original invariant-based shortcut with nonzero sensitivity as $\Omega _{p(o)}$ and $\Omega _{s(o)}$, respectively, and the corresponding auxiliary angles are $\gamma _{o}$ and $\beta _{o}$. Also, to guarantee a fair comparison, we let the two protocols have the same maxima of two Rabi frequencies, same population of intermediate state, and same evolution time. To this end, we should modify some parameters of the boundary conditions in the original protocol. That is, $\gamma _{o}(0)=\gamma _{o}(T)=\epsilon$ with $\epsilon =0.035$. Here, a small value of $\epsilon =0.035$ is chosen for two-manifold: an exact zero value will give infinite Rabi frequencies from Eqs. (10) and (11); it ensures the maxima of the Rabi frequencies are $\Omega ^{m}_{p(o)}=\Omega ^{m}_{s(o)}\approx 28.50\times 2\pi ~{\rm MHz}$, as in the optimal invariant-based shortcut. The other boundary conditions are the same as the Eqs. (24) and (25). We also take $\dot {\beta }_{o}(0)=0$ and $\dot {\beta }_{o}(T)=0$ to make sure that $\Omega _{s(o)}(0)=0$ and $\Omega _{p(o)}(T)=0$. One set of possible solutions for $\gamma _{o}$ and $\beta _{o}$ can be obtained by assuming a polynomial ansatz to interpolate at intermediate times that $\gamma _{o}=\sum _{j=0}^{4}a_{j}t^{j}$ and $\beta _{o}=\sum _{j=0}^{3}b_{j}t^{j}$, and then the coefficients can be directly solved in terms of the above boundary conditions. Finally, the Rabi frequencies $\Omega _{s(o)}$ and $\Omega _{p(o)}$ in the original invariant-based shortcut can be found from the Eqs. (10) and (11). In Fig. 7, we plot the time evolution of Rabi frequencies $\Omega _{s(o)}$ (green, star line) and $\Omega _{p(o)}$ (black, dotted line) in the original invariant-based shortcut, together with the optimal ones $\Omega _{s}$ (red, dashed line) and $\Omega _{p}$ (blue, solid line) in our protocol. One can find that the four Rabi frequencies vary in a different manner, while they take the same maxima in the time evolution. Note that the final transfer fidelity in the figure appears to match very well with Fig. 2 of Ref. [57], which implements an optimal shortcut to adiabaticity for the STIRAP process using counter-diabatic driving. In the paper, the authors evaluated a cost function and then numerically optimized it by a standard optimization routine. Compared to the optimal STIRAP with counter-diabatic driving, the invariant-based shortcut allows us to actively design the dynamics first according to the boundary conditions, and the Hamiltonian is then inversely obtained. Furthermore, the optimization of invariant-based shortcut with perturbation theory can give an analytic expression of systematic error sensitivity and its value can be suppressed to zero.

 

Fig. 7. The time evolution of Rabi frequencies $\Omega _{s(o)}$ (green, star line) and $\Omega _{p(o)}$ (black, dotted line) are obtained from Eqs. (10) and (11) by solving the polynomial ans$\ddot {a}$tze $\gamma _{o}=\sum _{j=0}^{4}a_{j}t^{j}$ and $\beta _{o}=\sum _{j=0}^{3}b_{j}t^{j}$ with boundary conditions in the original invariant-based inverse engineering shortcut, together with the Rabi frequencies $\Omega _{s}$ (red, dashed line) and $\Omega _{p}$ (blue, solid line) in our optimal protocol.

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Now, we can discuss the effect of systematic errors on the two different protocols. By solving the Schrödinger equation $\left (H(t)+H'(t)\right )|\varphi (t)\rangle =i d|\varphi (t)\rangle /dt$ numerically with the initial state $|1\rangle$, in both invariant-based shortcuts, the population $P_3(\lambda )$ of state $|3\rangle$ at final time $T$ changes with respect to parameter of systematic error $\lambda$, demonstrated in Fig. 8. It turns out that the final population in our optimal protocol is always higher than the original protocol with respect to $\lambda$, showing the robustness of our protocol against systematic errors. Also, note that the population of state $|1\rangle$ in our optimal protocol at the initial time is 1, with no population at the intermediate state at all. This is in contrast to the original protocol where the population of the intermediate state takes a non-vanishing value, i.e., $\sin ^{2}\epsilon$.

 

Fig. 8. Population $P_3(\lambda )$ at the final time $T=0.1\mu s$ vs systematic error $\lambda$ by solving numerically the Schrödinger equation $\left (H(t)+H'(t)\right )|\varphi (t)\rangle =i d|\varphi (t)\rangle /dt$ when the initial state is $|1\rangle$, based on the optimal invariant-based shortcut in Fig. 2(a) (zero sensitivity, red, solid line), and the original protocol with $\epsilon =0.035$ (nonzero sensitivity, blue, dashed line).

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6. Summary

In summary, we have proposed a general solution scheme for robust stimulated Raman shortcut-to-adiabatic passage with invariant-based optimal control in three-level quantum systems. We have inversely engineered a family of optimal Hamiltonians that can suppress the systematic error sensitivity. In particular, we have studied the relation between the population of the intermediate state and the amplitude of Rabi frequencies, which shows a trade-off behavior, i.e., the smaller the amplitudes of Rabi frequencies are, the larger population of the intermediate state is. Moreover, the comparison between the optimal shortcut and the original protocol presents the robustness of the optimal one against systematic errors in the time evolution. These results suggest that the optimal stimulated Raman shortcut-to-adiabatic passage indeed provides a general route for robust state transfer in three-level quantum systems, which is helpful for realizing high-fidelity quantum information processing in different experimental platforms.