1. Introduction

Optical lattices, which are spatially periodic potentials for neutral bosonic or fermionic atoms or molecules [1–12], have a wide range of applications in quantum measurement [11,13–16], quantum computation [17], and quantum simulation [8–10,18–25]. These lattices are typically generated through laser interference [9], and the periodic potentials lead to the formation of Bloch bands and orbital effects. The orbitals in optical lattices can be utilized to construct novel interferometers [26,27], the atom-orbital qubit [17], and quantum emulators of exotic orbital physics [9,10,18,19,23,24,28]. Manipulating Bloch bands or Bloch states is a significant challenge for these studies. Several methods have been proposed for manipulating Bloch bands, such as the stimulated Raman transition method [29], band swapping technique [10,18,23,24,30], and the nonadiabatic holonomic quantum control [17]. Recently, the "shortcut" method has emerged as a fast and high-fidelity technique for manipulating Bloch states [19,31–36]. This method involves several lattice pulses, with each pulse’s time and interval optimized to manipulate Bloch states with high fidelity in tens of microseconds [19,34,36]. The shortcut method has been used for constructing new Ramsey interferometry with trapped motional quantum states [26,27], realization of the atom-orbital qubit [17], and preparation of novel quantum states [19]. However, previous shortcuts (referred to as ordinary shortcuts in this letter) have not considered robustness issues systematically, which limit their application scenarios and performance.

In this paper, we present an improved shortcut method for controlling the Bloch states of Bose-Einstein condensates (BECs) in optical lattices. Our approach combines the speed and high fidelity of ordinary shortcuts with increased robustness to external disturbances. Our method builds on the ordinary shortcut by introducing multiple optical lattice pulses, each with an adjustable hold and interval time. Unlike previous work that focused solely on maximizing fidelity, we also consider the robustness of the control process, evaluated by the response to artificially introduced lattice depth noise during the parameter optimization process. Importantly, our approach does not increase the total time or number of pulses required for control.

In the following text, we first introduce the design process of the robust shortcut in Section 2, and demonstrate its feasibility by transferring ultra-cold atoms to the D band of a one-dimensional (1D) optical lattice in experiments. We then extend our analysis to different target states and lattice depths in Section 3, verifying the universality of our method. In Section 4, we show a potential application of our approach for implementing quantum gate manipulation based on S and D bands, and provide experimental evidence of its high practical value. Finally, we analyze the robustness of our method and predict its versatility in other scenarios.

2. Designing the robust shortcut

First, we introduce the process of designing a robust shortcut. Taking a one-dimensional optical lattice as an example, it can be formed by retroreflecting a laser beam. The light beams interfere at the position of atoms, forming a periodic potential field, as shown by the black line in Fig. 1(a). The Hamiltonian of the system for BECs in the optical lattice is $H=\hat {p}^2/2m+V(x)$, where the potential part of the 1D optical lattice is $V(x)=V_0\cos ^2(kx)$ with $V_0$ as the lattice depth, and $m$ is the atomic mass. According to Bloch’s theorem, periodic potential generates Bloch bands. The bands structure of the 1D optical lattice is shown in the left panel of Fig. 1(c). From the bottom up, the Bloch bands are labeled as S, P, D, F, G bands, etc. In each band, different quasi-momentum $q$ corresponds to different Bloch states, which can be expressed as $\Psi _{n,q}(x)=u_{n,q}(x)e^{iqx}$, where $n$ is the index of the Bloch band, and $u_{n,q}(x)$ is a periodic function. The time sequence of the shortcut to manipulate Bloch states is shown in Fig. 1(b), which consists of a series of pulses. Each pulse is divided into two parts: hold and interval, with corresponding time $t^H$ and $t^I$. For the $j$-th pulse, the corresponding evolution operator is $\hat {U}_j=\hat {U}^I\hat {U}^H$, where $\hat {U}^H=e^{-i\cdot (\hat {p}^2/2m+V(x))\cdot t_j^H/\hbar }$ and $\hat {U}^I=e^{-i\cdot (\hat {p}^2/2m)\cdot t_j^I/\hbar }$. The operator $\hat {U}^H$ is for the lattice holding stage, and $\hat {U}^I$ is for the interval stage. For an initial state $|\psi _0\rangle$, after the lattice shortcut, the final state is $|\psi _f\rangle =\prod _M^1 \hat {U}_j |\psi _0\rangle$. If the target state is $|\psi _t\rangle$, the fidelity of the shortcut sequence is defined as

 

Fig. 1. (a) Experimental configuration of a 1D optical lattice where the atoms distribute in the D band. (b) The time sequence of the shortcut, consisting of a series of lattice pulses. Each pulse comprises a hold stage and an interval stage. (c) Left panel: band structure of the 1D optical lattice with lattice depth 15 $E_R$. From the bottom up, the Bloch bands are recorded as S, P, D, F, G bands, etc. In each band, different qausi-momentum $q$ corresponds different Bloch state. Center panel: the initial state is generally set to a BEC with zero momentum. Right panel: the lattice depth noise will cause jitters and shifts in lattice band structure.

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For designing an ordinary shortcut, optimizing and determining the sequence parameters $t^H_j$ and $t^I_j$ is achieved by maximizing the fidelity $F$. In our previous works, this program has been successful [17,19,26,27,36,37]. For example, the shortcut can be used to transfer BECs with zero momentum to a Bloch state with zero quasi-momentum in the D band [35–37], as shown in the left panel of Fig. 1(c). However, if there is noise during the shortcut process, the lattice band structure will change, as shown in the right panel of Fig. 1(c). This is because the evolution operator $\hat {U}^H$ can be expanded as $\hat {U}_j^H=\sum _n e^{-iE_{n,q}t_j^H/\hbar }|\Psi _{n,q}\rangle \langle \Psi _{n,q}|$, where $E_{n,q}$ is the eigenenergy of the Bloch states with $q$ quasi-momentum of the $n$th Bloch band. The fluctuations in the optical lattice will affect $E_{n,q}$, resulting in a decrease in fidelity. In this example, the atoms will be transferred into other bands, as shown in the right panel of Fig. 1(c). In experiments with high robustness requirements, the impact of noise will be even greater [17,26,27]. Therefore, it is necessary to improve the design scheme while taking into account both fidelity and robustness.

In the optimization process for the improved shortcut, we consider not only maximizing fidelity, but also the fidelity under noise conditions. We redefine a composite fidelity $F_C=\alpha |\langle \psi _f|\psi _t\rangle |^2+\beta F_{noise}$, where $F_{noise}$ is the fidelity under noise conditions, and $\alpha$ and $\beta$ are weighting factors. Reasonable selection of $F_{noise}$, $\alpha$, and $\beta$ will result in better outcomes. For high-frequency noise, which is much larger than the band gaps, it will not have a significant impact. Therefore, we set noise as the case of a deviation of depth, which is also a major error in the experiment. We calculate the fidelity for different depth offsets, and then weight and sum them to obtain the composite fidelity $F_C$. We try various construction schemes for $F_{noise}$, and find that a simple scheme can help us to search for shortcut sequences with strong robustness. We define $F_{noise}=(|\langle \psi ^a_f|\psi _t\rangle |^2+|\langle \psi ^b_f|\psi _t\rangle |^2)/2$, with $|\psi ^a_f\rangle =\prod _M^1 \hat {U}^a_j |\psi _0\rangle =\prod _M^1 e^{-i\cdot (\hat {p}^2/2m)\cdot t_j^I/\hbar } e^{-i\cdot (\hat {p}^2/2m+V_a(x))\cdot t_j^H/\hbar }|\psi _0\rangle$, where $V_a(x) = 1.3V_0\cos ^2(kx)$, and $|\psi ^b_f\rangle =\prod _M^1 \hat {U}^b_j |\psi _0\rangle =\prod _M^1 e^{-i\cdot (\hat {p}^2/2m)\cdot t_j^I/\hbar } e^{-i\cdot (\hat {p}^2/2m+V_b(x))\cdot t_j^H/\hbar }|\psi _0\rangle$, where $V_b(x) = 0.7V_0\cos ^2(kx)$. In other words, we sum the fidelity in Eq. (1) calculated when the lattice depth $V_0$ is offset by $\pm 30{\%}$ as $F_{noise}$. Then we set the other parameters as $\alpha =1$ and $\beta =0.3$ through optimizations. Finally, we calculate all possible sequences within a certain parameter space and search for the optimal $F_C$, where we set $t_j^{I,H}\leq 80$ $\mu s$, step size 0.1 $\mu s$, and the pulse number $M\leq 2$. In addition, during the optimization process, the total time will also be limited to ensure that the new sequence is fast enough.

3. Verifying the robust shortcut

We verify the feasibility and effectiveness of this robust shortcut in experiments. At the beginning of the experiment, a nearly pure BEC with $1.5\times 10^5$ $^{87}$Rb atoms at the temperature 50 nK in the $|F=2,m_F=+2\rangle$ state is prepared in a hybrid optical-magnetic trap. This trap contains a gradient magnetic field generated by an anti-Helmholtz coil pair and a dipole trap caused by a 1064 nm laser beam, with frequencies $(\omega _x,\omega _y,\omega _z)=2\pi \times (28,55,60)$ Hz. Here, the direction of the light in the dipole trap is along the x-axis, which is the same as the direction of the 1D optical lattice. The z-axis is the direction of gravity. Gravity is counteracted by the dipole trap and the gradient magnetic field. The hybrid optical-magnetic trap provides weak confinement to atoms in the perpendicular direction of the optical lattice.

At this point, the atoms are in the initial state with zero momentum, as shown at the center of Fig. 1(c). The first robust shortcut we verify is for transferring the BEC to the Bloch state $|\Psi _{3,0}\rangle$ with zero quasi-momentum on the D band, as illustrated in Fig. 1(c). We choose a excited state $|\Psi _{3,0}\rangle$ as the target state for verification because it can better reflect the characteristics of shortcut than the ground state $|\Psi _{1,0}\rangle$. The ground state can be prepared by a traditional adiabatic process. In our experiment, the 1D optical lattice is formed by a laser beam with a wavelength of $\lambda =1064$ nm and its reflected light. The lattice depth is set at 15 $E_R$, where $E_R=\hbar ^2 k^2/2m$ with $k=2\pi /\lambda$ and $m$ is the mass of the atoms. The intensity of the optical lattice light is controlled by a system with two Acousto-optical Modulators (AOMs), where the first AOM used for power feedback and the second used for generating pulse sequences [17]. The power feedback can achieve stable control of lattice depth and generate any offset to realize the deviation noise. To validate the advantages of the robust shortcut, we compare two sequences. The first one is an ordinary shortcut sequence that only considers maximizing the fidelity. For the target state $|\Psi _{3,0}\rangle$, we obtain the ordinary shortcut sequence $\{t_1^H,t_1^I,t_2^H,t_2^I\}=\{80,19,46,23\}\mu$s with a fidelity of 99.88%. To obtain the experimental fidelity, we use band-map and time-of-flight (TOF) absorption imaging methods to measure the population of the atoms in each band after the shortcut [26]. The atoms in the third Brillouin region correspond to the atoms in the D band. According to Eq. (1), the ratio of atoms in the D band is equal to the fidelity. The experimental result is demonstrated in Fig. 2(b1). It can be seen that the atoms are all at the point of $\pm 2\hbar k$, indicating that the atoms are successfully prepared in the D band. However, if there is some deviation in the lattice depth, we find that the population of other bands will rapidly increase. As shown in Fig. 2(b2), when the depth is deviated by 10%, the number of atoms near $0\hbar k$ increases, indicating a large population of the S band. For a quantitative description of the experimental results, we use a multi-mode fitting method to obtain the atomic band population. Fig. 2(c) shows the multi-mode fitting result of Fig. 2(b2), where the red circles represent the atomic column density measured experimentally, and the black dashed line represents the multi-mode fitting curves. The fitting curve includes multiple Gaussian distributions and inverse parabolic distributions at different locations, corresponding to thermal atoms and condensates, respectively. The green and yellow portions correspond to the atoms of the condensate at $0\hbar k$ and $\pm 2\hbar k$, respectively, which are the atoms in the S and D bands. From this, the fidelity of the sequence can be obtained, which is 72.34%. The experimental and theoretical results of fidelity varying with depth deviation $\delta V/V$ are shown in blue squares and solid line in Fig. 2(a), which illustrate the poor robustness of this ordinary shortcut under lattice noise. Each blue square (red circle) in Fig. 2(a) represents the average results of three repeated experiments, and the error bar corresponds to the standard deviation of the three experimental results. The program for processing the raw images is shown in Code 1 (Ref. [38]). The details of the numerical simulation for the theoretical results is shown in the Supplement 1, and the program of numerical simulation is shown in Code 2 (Ref. [39]).

 

Fig. 2. Robustness testing when target state is $|\Psi _{3,0}\rangle$ at zero qausi-momentum in the D band. (a) Fidelities for different depth deviations $\delta V/V$. The red solid line and blue solid line represent the theoretical results of robust shortcut and ordinary shortcut, respectively, while the red circles and blue squares correspond to the experimental results for robust and ordinary shortcut, respectively. Each blue square (red circle) represents the average results of three repeated experiments, and the error bar corresponds to the standard deviation of the three experimental results. (b) Raw images obtained after the shortcut, band mapping, and TOF absorption imaging process. Atoms located at $0\hbar k$ are in the S band, and those at $\pm 2\hbar k$ are in the D band. The images in (b1) to (b3) correspond to $\delta V/V$ values of 0, 0.1, and 0.3 for ordinary shortcut, while the image in (b4) corresponds to $\delta V/V=0.3$ for robust shortcut. (c) Multi-mode fitting for the raw image shown in (b2), with red circles indicating the line density (integration along the vertical axis direction in (b2)), the black dashed line representing the fitting line, the blue solid line representing the thermal atoms fitted by some Gaussian functions, and the light green and yellow areas corresponding to the condensates in the S and D band, respectively. One pixel corresponds to 6.8 $\mu m$.

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Using our improved design scheme, we can obtain a robust shortcut sequence, represented by $\{t_1^H,t_1^I,t_2^H,t_2^I\}=\{16,80,52,5\}\mu s$, with a fidelity of 99.25%. We conduct a comparative experiment and found that even with a 30% deviation in depth, the majority of atoms were still located in the D band, as shown in Fig. 2(b4). In contrast, the shortcut represented by Fig. 2(b3) resulted in a low population of atoms in the D band. These results are consistent with numerical analysis. Our robust shortcut maintains a fidelity of over 90% at deviations of $\pm$30% (as demonstrated by the red squares and dashed line in Fig. 2(a)), indicating its effectiveness.

To further illustrate the universality of robust shortcuts, we verify their efficacy for different target states. Similar to the target state $|\Psi _{3,0}\rangle$ at zero quasi momentum in the D band, we select the Bloch eigenstates $|\Psi _{1,0}\rangle$ in the S band and $|\Psi _{5,0}\rangle$ in the G band as the new target states. Fig. 3(a) shows the results for the target state $|\Psi _{1,0}\rangle$. Here, the red solid line and red circles correspond to the experimental and theoretical results of the robust shortcut, while the blue solid line and blue squares show the results of the ordinary shortcut. It can be observed that in the absence of deviation in the lattice depth, $\Delta V/V=0$, the fidelity of both shortcut sequences is similar. However, with the depth migrations, the fidelity of the ordinary shortcut rapidly decreases to less than 30%, whereas the fidelity of the robust shortcut remains above 90%. The results for the target state $|\Psi _{5,0}\rangle$ in the G band are similar, as shown in Fig. 3(b).

 

Fig. 3. Robustness testing of different target states and quantum state tomography results. (a) Target state $|\Psi _{1,0}\rangle$ in the S band. (b) Target state $|\Psi _{5,0}\rangle$ in the G band. (c) Superposition state $|\psi _t\rangle =\frac {1}{\sqrt {2}}(\Psi _{1,0}(x)+\Psi _{3,0}(x))$. (d) Raw images of the quantum state tomography process for different $t_{hold}$ times, showing the population of atoms at different momentum. (e) Fitting results for the quantum state tomography process with $\delta V/V=0$ (e1) and $\delta V/V=0.3$ (e2). The red solid line, black dashed line, and green solid line correspond to the fitting proportions of atoms in the $0\hbar k$, $\pm 2\hbar k$, and $\pm 4\hbar k$ momentum states, respectively. The red squares, black points, and green triangles correspond to the experimental proportions of atoms in the $0\hbar k$, $\pm 2\hbar k$, and $\pm 4\hbar k$ momentum states, respectively. Each experimental point in (a), (b), and (e) is obtained by averaging the results of three repeated experiments, and the error bar is the standard deviation. The error bar in (c) is obtained by a bootstrap method.

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Then we investigate its effectiveness in preparing and manipulating superposition states. Unlike previous experiments that focused on the Bloch eigenstates [9], the advantage of the shortcut method lies in its ability to efficiently handle superposition states. To this end, we set the target state as a superposition state of Bloch states with zero quasi-momentum in the S band and the D band, given by $|\psi _t\rangle =\frac {1}{\sqrt {2}}(\Psi _{1,0}(x)+\Psi _{3,0}(x))$, where the initial state is still the BEC with zero momentum. We obtain an ordinary shortcut sequence $\{t_1^H,t_1^I,t_2^H,t_2^I\}=\{59,46,13,65\}\mu s$ and a robust shortcut sequence $\{t_1^H,t_1^I,t_2^H,t_2^I\}=\{12,46,13,64\}\mu s$. As fidelity cannot be obtained through the previous band-map method due to phase information, we employ a quantum state tomography method [17] to obtain all quantum state information and then derive the experimental fidelity through Eq. (1). This porcess involves maintaining the optical lattice for different times $t_{hold}$ after the shortcut, then quickly close the lattice and perform TOF absorption imaging to obtain the momentum distribution of atoms in the optical lattice, as shown in Fig. 3(d). By fitting the momentum state proportion change curves, we can obtain all the information of the quantum state. Fig. 3(e) shows the experimental and fitting results of the momentum distribution, where the red squares and solid line represent the experimental and fitting results, respectively. Since the distribution of $+2\hbar k$ and $-2\hbar k$ (or $+4\hbar k$ and $-4\hbar k$) is equal, only the sum of the distribution $+2\hbar k$ and $-2\hbar k$ (or $+4\hbar k$ and $-4\hbar k$) is considered. From Fig. 3(e1), we obtain the fidelity of the ordinary shortcut without depth deviation, which is 98.77%. Similarly, the fidelity of the robust shortcut is 98.46%. However, when the lattice depth deviation is 30%, the experimental results of the ordinary shortcut, shown in Fig. 3(c), yield a fidelity of 65.86%, whereas the fidelity of the robust shortcut remains above 95%.

Further, we verify the effect of the robust shortcut for different lattice depths $V_0$. Fig. 4 presents the results for $V_0 = 5 E_R$. Fig. 4(a) displays the results for the target state $|\Psi _{1,0}\rangle$. We observe that the theoretical curve in red (experimental data represented by red circles) is superior to the blue curve (blue squares), indicating the advantage of the robust shortcut. Fig. 4(b) shows the results for the target state $|\Psi _{3,0}\rangle$, where the robust shortcut outperforms the ordinary one in terms of robustness.

 

Fig. 4. Robustness testing when lattice depth is 5 $E_R$. (a) For target state $|\Psi _{1,0}\rangle$ in the S band. (b) For target state $|\Psi _{3,0}\rangle$ in the D band. The red solid line and blue solid line are the theoretical results of robust shortcut and ordinary shortcut, respectively. The red circles and blue squares correspond robust and ordinary shortcut, respectively. Each experimental point is obtained by averaging the results of three repeated experiments.

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Notably, for $\delta V/V < -0.4$ in Fig. 4(a), the fidelity of the experimental results (blue squares) for the ordinary shortcut is also high, deviating from the theoretical curve. We attribute these deviations to two main reasons. First, the depths corresponding to the first two data points in Fig. 4(a) are $2 E_R$ and $3 E_R$, which are extremely shallow depths. Prior to conducting the experiments, we calibrate the lattice depth and obtain the relationship with optical intensity, such as $V=a\cdot I$ with $I$ the light intensity. However, in reality, there exists a constant offset $b$ between these two variables $V=a\cdot I + b$, and this offset has a more significant impact when the depth is low, resulting in a substantial deviation between experimental and theoretical values. The second reason is related to the data processing procedure, particularly when the S band dominates. In such cases, the calculation of the D band proportion tends to underestimate the D band’s contribution because some non-condensed atomic species cover the positions of D band atoms, preventing accurate counting of these D band atoms. These two factors lead to large differences between experiments and theories in Fig. 4(a).

In summary, our experiments demonstrate the effectiveness of the robust shortcut, which exhibits the same fidelity and speed as the ordinary shortcut while providing higher robustness.

4. Application of the robust shortcut in a quantum gate

The manipulation of orbits in optical lattices is crucial in the fields of quantum simulation and quantum information. In Ref. [17], researchers verified the feasibility of atom-orbital qubit and quantum gate control in optical lattices. It proposes an atom-orbital qubit by manipulating s and d orbitals of BECs in an optical lattice. Additionally, the letter mentions that the dynamic quantum gates’ robustness (the shortcut method) is relatively low, and thus a more robust gate, the holonomic gate, was proposed. However, dynamic quantum gates have a faster speed, and their application in atom-orbital qubits can be further promoted if their robustness improves. This difficulty can be overcome using the robust shortcut scheme described in this paper.

Here, we use the Hadamard gate as an example to demonstrate the practicality of the robust shortcut scheme. The qubit we use is based on the Bloch states with zero quasi-momentum in the S and D bands [17], denoted as $|\Psi _{1,0}\rangle$ and $|\Psi _{3,0}\rangle$. The experimental process is illustrated in Fig. 5(a) and includes BEC preparation, qubit initialization, quantum gate, and detection. The robust shortcut is utilized for qubit initialization and quantum gate. Fig. 5(b) shows the Hadamard gate process on the Bloch sphere, where the red solid line represents the Hadamard rotation axis, the north (south) pole corresponds to the Bloch states $|\Psi _{1,0}\rangle$ and $|\Psi _{3,0}\rangle$, and the red and green circles represent the initial and final states, respectively. An ideal Hadamard gate rotates the quantum state 180 degrees around the rotation axis on the Bloch sphere. In Fig. 5(b1), the initial state is $\frac {1}{\sqrt {2}}(|\Psi _{1,0}\rangle -|\Psi _{3,0}\rangle )$, and the final state after the Hadamard gate operation is $|\Psi _{3,0}\rangle$. The experimental and numerical simulation results are shown in Fig. 5(c1), where the red circles and the blue squares correspond to the robust shortcut and ordinary shortcut, respectively. It is observed that the robust shortcut exhibits stronger robustness. Fig. 5(b2) and (c2) correspond to the results with an initial state of $\frac {1}{\sqrt {2}}(|\Psi _{1,0}\rangle +|\Psi _{3,0}\rangle )$ and a final state of $|\Psi _{1,0}\rangle$, and the robust shortcut also exhibits good robustness. Therefore, the Hadamard gate based on the robust shortcut achieves a fidelity of over 80% even when the depth is offset by 20%, which is almost comparable to the robustness of the holonomic gates in Ref. [17], but the former exhibits a faster speed.

 

Fig. 5. (a) The time sequence for a quantum gate. A shortcut quantum gate is inserted between the state preparation and detection processes. (b) The demonstration of quantum gates on a Bloch sphere is shown when the initial state is (b1) $\frac {1}{\sqrt {2}}(\Psi _{1,0}(x)-\Psi _{3,0}(x))$ and (b2) $\frac {1}{\sqrt {2}}(\Psi _{1,0}(x)+\Psi _{3,0}(x))$. On the Bloch sphere, the north pole represents the state $\Psi _{1,0}(x)$ at zero quasi-momentum in the S band, while the south pole represents $\Psi _{3,0}(x)$ in the D band. The red line represents the axis of rotation for the Hadamard gate. (c) The fidelities with different depth deviations $\delta V/V$ are shown when the initial state is $\frac {1}{\sqrt {2}}(\Psi _{1,0}(x)-\Psi _{3,0}(x))$ (c1) and $\frac {1}{\sqrt {2}}(\Psi _{1,0}(x)+\Psi _{3,0}(x))$ (c2). The red solid line and blue solid line represent the theoretical results of robust shortcut and ordinary shortcut, respectively. The red circles and blue squares correspond to the experimental results of robust and ordinary shortcut, respectively. Each experimental point is obtained by averaging the results of three repeated experiments.

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5. Discussion and conclusion

In this paper, we discuss and verify the resistance of the robust shortcut to depth noise, which is the main source of noise in experiments. In the Supplement 1, we also discuss the situation with random noise and find that robust scheme still performs better. Among the random noise, the DC offset has the greatest impact. Therefore, we focus on discussing the robustness to depth deviation in this article.

Additionally, when applying orbits in optical lattices to precision measurement fields such as interferometry, the finite width of the quasi-momentum of ultracold atoms causes dephasing [26,27]. If coherent manipulations of quantum states can resist the influence of the finite width momentum distribution, the accuracy of atomic orbital interferometers will be greatly improved. We also analyze and verify the robustness of the robust shortcut against quasi-momentum noise. The shortcut is designed based on the energy difference of each Bloch eigenstate with the same quasi-momentum. The Bloch bands are curved, indicating that the energy gap of each Bloch state with different quasi-momentum is different. Consequently, a shortcut sequence can only be applied to a specific quasi-momentum. If there is an offset in quasi-momentum or finite width distribution of the quasi-momentum, the fidelity will be reduced due to the energy shift, as shown in Fig. 6(b). The robust shortcut described above is designed based on its resistance to depth noise, and its essence is to resist the noise caused by the energy difference of each Bloch eigenstate. Similarly, the influence of quasi-momentum noise on fidelity, like depth noise, comes from the change of the eigen energy difference. Therefore, the robust shortcut we designed can also resist quasi-momentum noise, as verified by our numerical simulations and experimental results in Fig. 6. In the experiment, we first give the initial BEC an initial momentum $\Delta q$ (shown in the lower part of Fig. 6(a)) and then perform a shortcut sequence and detection. We compare the previous robust and ordinary shortcut sequences for the target state $|3,0\rangle$, and the results are shown in Fig. 6(c). When the quasi-momentum shifts $0.3\hbar k$, the fidelity of the robust shortcut is more than 60%, much higher than that of the ordinary shortcut, proving that the former is noise resistant to quasi-momentum.

 

Fig. 6. Testing the robustness of the shortcut sequence under noise in quasi-momentum. (a) The initial states with zero momentum (up) and $0.3\hbar k$ (below). (b) The difference in eigen energies at different quasi-momentum values. Since each band is not flat, the corresponding eigen energies differ at different quasi-momentum values, resulting in varying fidelities for different quasi-momentum states after the same shortcut sequence. (c) The fidelities for different deviations of quasi-momentum $\delta q$. The red solid line and blue solid line represent the theoretical results of robust and ordinary shortcuts, respectively. The red circles and blue squares correspond to the experimental fidelities of robust and ordinary shortcuts, respectively. Each experimental point is obtained by averaging the results of three repeated experiments.

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It should be pointed that, in the figures presented in the paper, theoretical predictions sometimes do not fall within the experimental error bars. We attribute this to the presence of systematic biases during the measurements rather than insufficient repetitions of experimental measurements. The primary source of this systematic bias is related to deviations in lattice depth, stemming from the calibration of the relationship between depth and optical intensity before conducting the experiments. Increasing the number of repeated measurements would not bring the experimental average back in line with the theoretical predictions because there is a certain degree of deviation between the depth values used in the theoretical predictions and the actual depths in the experimental setup. This can also be observed in Fig. 3(c), where the experimental points are obtained by fitting 63 data points, yet a significant proportion of the theoretical predictions still do not fall within the experimental error bars. However, this does not affect the conclusion of this paper. Both experimental and theoretical predictions show that robust shortcut has better performance.

In conclusion, we propose an improved shortcut design scheme that retains high speed and high fidelity advantages while also ensuring extremely high robustness. We comprehensively verify the effectiveness of this new robust shortcut in experiments involving different target states, lattice depth, and quasi-momentum. The robust shortcut exhibits strong robustness during the control process. Additionally, we demonstrate an application of this scheme in quantum gate design. The robust shortcut is expected to improve the robustness of optical lattice orbit-based interferometry, quantum gates, and other processes.