Open Access
Add to BibSonomy
Abstract
Shortening the operation time of implementing scheme and reducing the influence of harmful factors have always been the research objectives pursued by people. Based on invariant-based reverse engineering, we present a general scheme for implementing robust population transfer in a three-level system via optimal shortcut to adiabatic passage. The systematic error sensitivity is introduced to measure the robustness of the process. The smooth Rabi frequencies are expressed with some coefficients, which are also related to the systematic error sensitivity and the population of intermediate state. When the amplitude of control field is given, the transfer can be optimized within as small systematic error sensitivity as possible, i.e., the robustness against systematic errors is further improved by choosing suitable correlation coefficient. Additionally, we apply the technique to achieve robust excitation fluctuation transfer between two membranes in an optomechanical system. The relation between the fidelity of excitation fluctuation transfer and variation of effective optomechanical coupling strengths is analysed. Numerical result shows that the fidelity keeps over 0.95 even if the coupling strengths deviates from 20% of the theoretical value. Moreover, comparison with existing literature [Opt. Express 29, 7998 (2021) [CrossRef] ], the proposed scheme possesses stronger robustness against variations of effective optomechanical coupling strengths and lower population of unwanted states. The idea may provide a promising approach for quantum information processing.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Stimulated Raman adiabatic passage (STIRAP) [1–3] which controls quantum state in a three-level quantum system is a relatively mature technique. Based on STIRAP, quantum state engineer is robust against small variations of experimental parameters, and thus the method is further explored [4] and applied to various systems [5–8]. However, STIRAP requires long evolution time to make sure the system without transition, which may result in more devastation from decoherence or decay. To overcome this problem, the shortcuts-to-adiabaticity (STA) is put forward [9,10]. The so-called STA is that it speeds up a quantum adiabatic process through a non-adiabatic route, namely, it does not need to satisfy adiabatic condition. Up to the present, the STA technologies based on different methods, such as Lewis-Riesenfeld invariants [11–15], transitionless quantum driving (TQD) [16–22], fast-forward approach [23,24], iterative interaction pictures [25–27], and dressed states [28–31], have been developed and applied [32,33]. Additionally, reverse engineering has been widely researched and used for constructing the STA [34] because this strategy means that the parameters just need to satisfy the initial and final conditions and have multiple set of solutions. Therefore, it is necessary to optimize these solutions. Based on the theory of reverse engineering, one can optimize experimental parameters for various purpose, such as weakening the influence of decoherence [35–37], cutting down energy consumption [37–40], optimizing duration [41], reducing the leakage error [42], enhancing the robustness against amplitude noise or systematic errors [43–49].
In practice, there are some factors impairing the ideal consequences, such as noises, errors, and fluctuations. Hence, how to decrease the effect of devastating factors is a hot topic. As one of the error sources, systematic errors greatly decrease the fidelity of target state. To suppress the impact of systematic errors, the strategy of combining optimal control with STA is an excellent choice. In Refs. [43,44], systematic error sensitivity (SES) was introduced to measure the robustness against systematic errors, and the optimally robust shortcuts to population inversion in two-level quantum systems were presented. Subsequently, Yu et al. extended this idea to a two-interaction-spin system and achieved fast spin flip [45]. In 2020, Kang et al. proposed a protocol to generate singlet states of spin qubits which was optimized with respect to the errors of control fields [46]. Besides, optimization of STA with respect to the error and fluctuation has been further applied to different quantum tasks [47,48]. On the other hand, quantum state transfer is an important building block for modern quantum information processing and has been extensively researched in various systems, such as optomechanical systems [50–52], quantum dots [53–55], spin chains [56–58], etc. Recently, Song et al. proposed a general optimal-STA to achieve population transfer in a three-level system, where the Rabi frequencies of control fields were derived based on the inverse function theorem and further optimized to enhance the robustness against the small shift of control fields [49].
Optomechanical system which studies the interaction between an optical mode and a mechanical mode has achieved significant advancements in both theory and experiment [59–65]. These progresses enable optomechanical system to be an ideal platform to observe the quantum behaviors of macroscopic matter. In particular, the quantum state transfer in an optomechanical system may provide a new way for quantum communication between two truly macroscopic systems. In recent years, numerous protocols have been proposed to achieve quantum state transfer between two mechanical oscillators [66–70]. In Ref. [69], akin to STIRAP, adiabatic transfer of energy fluctuations between membranes is presented. Then the adiabatic process is accelerated via shortcuts to adiabaticity [70].
Inspired by above works, for the purpose of further increasing the robustness with respect to systematic errors, we now exploit an alternative optimal-STA to realize a fast and robust population transfer in a three-level system. The smooth control fields are inversely engineered based on Lewis-Riesenfeld invariant. The population of the intermediate state, systematic error sensitivity, and the amplitude of Rabi frequencies are expressed with some common time-independent coefficients. Consequently, when the amplitude of Rabi frequencies is given, there exists an optimal route toward the target state with the property of strong-robustness against systematic error. Simultaneously, as an example, we apply this idea to implement the excitation fluctuation transfer between two membranes in an optomechanical system. Numerical simulations show that the proposed scheme has strong robustness against variations of effective optomechanical coupling strengths and low population of unwanted states.
2. Optimization of population transfer via invariant-based reverse engineering
Consider a $\Lambda -$type system with three bare states denoted by $|1\rangle$, $|2\rangle$, $|3\rangle$, where the transitions $|1\rangle \leftrightarrow |2\rangle$ and $|2\rangle \leftrightarrow |3\rangle$ are resonantly driven by two external pulses with Rabi frequencies $\Omega _{p}$ and $\Omega _{s}$, respectively. The corresponding Hamiltonian is [1–3]
According to the theory of Lewis and Riesenfeld [11], the dynamical invariant $I(t)$ for the Hamiltonian, which satisfies
is given by [12,13]By solving the Eq. (2), the Rabi frequencies $\Omega _{p}$ and $\Omega _{s}$ can be reversely designed as
Thus, the solution of the wave function can be formed as [11]
where $C_{n}$ denotes a time-independent amplitude and the Lewis-Riesenfeld phase $\theta _{n}$ is in the form ofNext, the design of the experimentally feasible Rabi frequencies to realize a robust and fast population transfer will be shown. Assuming that the system is initially in the state $|1\rangle$ and evolves along $|\xi _0(t)\rangle$ into the target state $|3\rangle$, the boundary conditions are given by
with $T$ being the operation time for shortcuts to adiabaticity. However, from a practice point of view, the Rabi frequencies of control fields may have a deviation from ideal value because of inhomogeneous broadening or calibration imperfections. Hence, robustness against these errors is one of the criteria to measure the quality of a protocol. When Rabi frequencies of control fields have a systematic error rate $\delta$, i.e., $\Omega _{p}\rightarrow (1+\delta )\Omega _{p}, \Omega _{s}\rightarrow (1+\delta )\Omega _{s}$, the perturbed Hamiltonian is expressed as $H_{\delta }=\frac {1}{2}\delta \Omega _{p}|2\rangle \langle 1| +\frac {1}{2}\delta \Omega _{s}|2\rangle \langle 3|+H.c.$. By using time-dependent perturbation theory, the state of system $|\Psi (T)\rangle$ takes the form ofThus, SES in Eq. (11) is approximate to
where $\theta =\theta _+=-\theta _{-}$.Noticeably, we aim to design smooth Rabi frequencies which drive a three-state system to target state with great robustness against systematic error. On the one hand, $\beta (t)$ is set as
where $s=t/T$, and $A,B$ are dimensionless parameters which can be optimized. $B$ is related to $Q_{S}$ and energy cost which is quantized by pulse peak $max(\Omega _{p},\Omega _{s})=\Omega _{max}$ [38,49], and $B$ is set as $B\geq 2$ to automatically satisfy the boundary conditions in Eqs. (9a) and (9b) and simultaneously make sure that $\dot {\beta }$ doesn’t increase dramatically. Besides, $A$ is related to $Q_{S}$, $\Omega _{max}$, and the maximal value $P_{2m}=\sin ^2 (A)$ of population of the intermediate state $|2\rangle$. For the sake of convenience, it is set as $0<A<\frac {\pi }{2}$ in the following discussion. On the other hand, Eq. (9e) meansSo, $\dot {\alpha }(t)$ is chosen as
to automatically satisfy the boundary conditions in Eq. (9d) and Eq. (9e). Here, $D$ is a dimensionless parameter related to $Q_{S}$ and $\Omega _{max}$, and $C$ is a dimensionless parameter satisfying Eq. (9c). To reversely design $\alpha$ according to Eq. (9c), $D$ is set as a positive integer. Table 1 shows the values of $C$ corresponding to $D=3,4,5,6,7,8,9,10$. Next, we will explore the optimal value of $A,B,D$ when $\Omega _{max}$, $Q_{S}$, $P_{2m}$ are under overall consideration.
Table 1. $C$ values corresponding to different $D$.
While in the case that $D$ is given, $Q_{S}$ or $\Omega _{max}$ depends on $A,B$, while $P_{2m}$ only depends on $A$ and small $A$ means small $P_{2m}$. As an example, we take $D=8$ and plot $Q_{S}$ versus $A$ and $B$ in Fig. 1. It can be seen from Fig. 1 (a) that $Q_{S}$ is close to zero only if $A<0.1$ or $A\simeq 0.26$. Figure 1 (b) and Fig. 1 (c) show that $B$ has a relatively small effect on $Q_{S}$ and one can choose optimal $B$ to make $Q_{S}$ as small as possible when $A$ is fixed, for example, setting $A=0.3$ and $D=8$, $Q_{S}$ can be decrease to 0.064 if $B=6.2$. The population $P_{3}$ of target state $|3\rangle$ versus $\delta$ for different $A$ and $B$ is illustrated in Fig. 2. Apparently, $P_{3}$ decreases with the increasing of $|\delta |$. The robustness against the systematic errors is enhanced with optimal $A$ ($B$) when $B$ ($A$) is fixed, which agrees well with Fig. 1. $\Omega _{max}$ as a function of $A$ or $B$ is illustrated in Fig. 3. It shows that $\Omega _{max}$ drops with the increasing of $A$ when $A<0.95$ while it increases with the increasing of $B$. However, $\Omega _{max}$ slowly increases with the increasing of $A$ when $A>0.95$, as shown in the inset of Fig. 3 (a). This is because the variation trend of $\Omega _{max}$ is mainly contributed from the second item on the right of Eq. (6) when $A$ is a small value, while it is mainly dominated by the first term on the right of Eq. (6) when A is greater than a certain value such as 0.95. Therefore, when integer $D$ is given, there is a balance point between population of the intermediate state and energy cost, as well as robustness against systematic errors and energy cost.
Fig. 1. (a) $Q_{S}$ versus $A$ when $B=2$ and $D=8$, (b) $Q_{S}$ versus $B$ when $A=\pi /6$ and $D=8$, (c) $Q_{S}$ versus $B$ when $A=0.3$ and $D=8$.
Fig. 2. $P_{3}$ versus $\delta$ for (a) $A=\frac {\pi }{12},\frac {\pi }{6},\frac {\pi }{2}$ when $B=2,D=8$, (b) $B=2,6,8$ when $A=\frac {\pi }{6},D=8$, (c) $B=2,6,8$ when $A=0.3,D=8$.
Fig. 3. (a) $\Omega _{max}$ versus $A$ when $B=2$ and $D=8$, (b) $\Omega _{max}$ versus $B$ when $A=\pi /6$ and $D=8$.
We notice that $Q_{S}$ and $\Omega _{max}$ are also relevant to $D$. The values of $Q_{S}$ and $\Omega _{max}$ for different $D$ are shown in Table 2. Apparently, when $A$ and $B$ are fixed, for a smaller $D$, both $Q_{S}$ and $\Omega _{max}$ are smaller. It is worth mentioning that the influences of $D$ on $Q_{S}$ and $\Omega _{max}$ are much slighter than that of $A$. Figure 4 displays $P_{3}$ versus $\delta$ for $D=3,6,10$ when $B=2,A=\frac {\pi }{6}$. It shows that the robustness against the systematic errors is slightly enhanced with the decreasing of $D$. In a word, when population of the intermediate state is fixed, one can enhances the robustness against the systematic errors and meanwhile consumes less energy of control field.
Table 2. $Q_{S}$ and $\Omega _{max}$ for different $D$ when $A=\frac {\pi }{6}$ and $B=2$.
In practice, the parameters can be optimized according to the experimental conditions of different systems. In general, the coupling coefficient between two states is experimentally limited. When $\Omega _{max}$ is given, one can optimize $A,B,D$ to achieve as small as possible $P_{2m}$ and $Q_{S}$. Next, we take $\Omega _{max}=15$ and $\Omega _{max}=20$ (in unit of $1/T$) as an example and set $D=8$ since the impact of $D$ on $Q_{S}$ is mild. The contour lines of $Q_{S}$ (solid lines) and $\Omega _{max}$ (dotted lines) versus $A$ and $B$ are presented in Fig. 5. The result reveals that $Q_{S}$ can be minimized to about 3 if $A=0.56$ and $B=5.6$ when $\Omega _{max}=15$, while $Q_{S}$ can be minimized to about 1.6 if $A=0.44$ and $B=5.8$ when $\Omega _{max}=20$. In other words, when the maximum value of coupling coefficient is given, one can make systematic error sensitivity and population of the intermediate state as small as possible via optimal control.
Fig. 5. The contour lines of $Q_{S}$ (solid lines) and $\Omega _{max}$ (dotted lines) versus $A$ and $B$ when $D=8$.
3. Application of optimal invariant-based reverse engineering in an optomechanical system
In this section, we apply the above optimal invariant-based reverse engineering to achieve fast and robust excitation fluctuation transfer between two membranes. The considered model is composed of two membranes ($B_{1}$, $B_{2}$) with the frequency $\omega _{mj} (j=1,2)$ placed in an optical cavity which is divided into three independent subcavities ($L$, $M$, $R$) with frequency $\omega _{cf}$ ($f=L,M,R$). The laser fields with Rabi frequency $\Omega _{f}$ and frequency $\omega _{f}$ are added to drive the modes $A_{L}$, $A_{M}$, and $A_{R}$, respectively, as shown in Fig. 6. The Hamiltonian of the optomechanical system takes the form of
Here,
Now, we will apply the proposed idea in Sec. 2 (labeled OS-scheme) to implement excitation fluctuation transfer between two membranes, i.e., the first membrane $B_{1}$ is initially in a thermal state with one phonon number and finally transfers excitation fluctuation to the second membrane $B_{2}$. Assuming that the amplitudes of coupling strengths are taken as $G_{max}=\frac {2.85\times 2\pi }{T}$, which is chosen in Ref. [49], the minimum values of $Q_{s}$ and the corresponding $A,B$ for different D are shown in Table 3. From Table 3, the dimensionless parameters $(A,B,C,D)$ are set as $(0.338,2,\frac {3\pi ^{2}}{8T},3)$ to achieve both low population of the intermediate state and strong robustness against systematic errors. In this case, time-dependent parameters $\alpha ^{OS}$ and $\beta ^{OS}$ can be deduced as
Table 3. The minimum values of $Q_{s}$ and the corresponding $A,B$ for different D.
In Ref. [49] (labeled IS-scheme), the time-dependent parameters $\beta ^{IS}$ has the same form as $\beta ^{OS}$, i.e.,
where $A^{IS}=\pi /6$, while $\alpha ^{IS}$ is expressed as [49]Fig. 6. Schematic diagram of the system. Two membranes acting as two mechanical oscillators are placed within a three-mode optical cavity, which is driven by three amplitude-modulated lasers.
For a specified amplitude of coupling strength $G_{max}=\frac {2.85\times 2\pi }{T}$, the coupling strengths of both schemes with time are plotted in Fig. 7. Obviously, the curves of the coupling strengths in OS-scheme are simpler than those in IS-scheme. According Eq. (19), $G_{1}$ and $G_{2}$ are directly proportional to $\Omega _{L}$ and $\Omega _{R}$ which are generally expressed as superposition of multiple Gaussian functions. Both coupling strengths in IS-scheme need to be zero simultaneously when $t=T/2$ from Fig. 7(b). These characteristics have the potential to result in OS-scheme experimentally easier to be implemented than IS-scheme. With the chosen parameters, the fidelity $F=\langle \delta b_{2}^{\dagger}\delta b_{2}\rangle$ versus $\delta$ is displayed in Fig. 8. It shows that OS-scheme is more robust against systemic errors originating from coupling strengths fluctuations than IS-scheme, where the fidelity of OS-scheme keeps over 0.95 even if the coupling strength deviates from 20% of the theoretical value. In addition, excitation fluctuations of $A_{L}$ or $A_{R}$ are worthy of note because larger population of excitation fluctuations of $A_{L}$ or $A_{R}$ will make the scheme suffer from decay. Figure 9 shows the variation of excitation fluctuations of $A_{L}$ or $A_{R}$ with time based on two schemes. Apparently, for the same coupling strengths, the OS-scheme possesses less excitation fluctuations of $A_{L}$ and $A_{R}$ than those in IS-scheme.
Fig. 7. The coupling strengths $G_{1}$ (red line) and $G_{2}$ (green line) versus $t/T$ (a) with Eq. (23) where $A=0.338$, (b) with Eqs. (24) and (25) where $A^{IS}=\pi /6, R=5.85$ [49].
4. Conclusion
In conclusion, we have proposed a feasible and robust scheme for quantum state transfer in a general three-level system with optimal shortcuts to adiabatic passage by invariant-based reverse engineering. SES is introduced to measure the robustness with respect to the systematic errors. When the amplitude of Rabi frequency is given, one can optimize parameters to minimize SES and the population of the intermediate state by choosing suitable correlation coefficient. The proposal has strong universality and can be applied to various systems. As an example, we study the average excitation fluctuation transfer between two membranes in an optomechanical system. Numerical results indicate that OS-scheme possesses greater robustness against systematic error and less population of unwanted state than Ref. [49] when the amplitude of coupling strength is fixed, where the fidelity of the excitation fluctuations transfer in OS-scheme keeps over 0.95 even if the coupling strength deviates from 20% of the theoretical value. By the way, the idea is not limited to robust quantum state transfer, it may also be applied to generate high-fidelity quantum entanglement or quantum gates in different experimental platforms. Hence, the work may open a new avenue for quantum information processing.
Funding
National Natural Science Foundation of China (No. 11674059, No. 12074067); Natural Science Foundation of Fujian Province (No. 2019J01431, No. 2020J01191); Department of Education, Fujian Province (No. JAT210585).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. K. Bergmann, H. Theuer, and B. W. Shore, “Coherent population transfer among quantum states of atoms and molecules,” Rev. Mod. Phys. 70(3), 1003–1025 (1998). [CrossRef]
2. N. V. Vitanov, T. Halfmann, B. W. Shore, and K. Bergmann, “Laser-induced population transfer by adiabatic passage techniques,” Annu. Rev. Phys. Chem. 52(1), 763–809 (2001). [CrossRef]
3. N. V. Vitanov, A. A. Rangelov, B. W. Shore, and K. Bergmann, “Stimulated Raman adiabatic passage in physics, chemistry, and beyond,” Rev. Mod. Phys. 89(1), 015006 (2017). [CrossRef]
4. F. Ahmadinouri, M. Hosseini, and F. Sarreshtedari, “Stimulated Raman adiabatic passage: Effects of system parameters on population transfer Stimulated Raman population transfer,” Chem. Phys. 539, 110960 (2020). [CrossRef]
5. U. Gaubatz, P. Rudecki, M. Becker, S. Schiemann, M. Kölz, and K. Bergmann, “Population switching between vibrational levels in molecular beams,” Chem. Phys. Lett. 149(5-6), 463–468 (1988). [CrossRef]
6. K. S. Kumar, A. Vepsälaïnen, S. Danilin, and G. S. Paraoanu, “Stimulated Raman adiabatic passage in a three-level super-conducting circuit,” Nat. Commun. 7(1), 10628 (2016). [CrossRef]
7. D. A. Golter and H. Wang, “Optically driven Rabi oscillations and adiabatic passage of single electron spins in diamond,” Phys. Rev. Lett. 112(11), 116403 (2014). [CrossRef]
8. V. Fedoseev, F. Luna, I. Hedgepeth, W. Löffler, and D. Bouwmeester, “Stimulated Raman adiabatic passage in optomechanics,” Phys. Rev. Lett. 126(11), 113601 (2021). [CrossRef]
9. X. Chen, A. Ruschhaupt, S. Schmidt, A. del Campo, D. Guéry-Odelin, and J. G. Muga, “Fast optimal frictionless atom cooling in Harmonic traps: shortcut to adiabaticity,” Phys. Rev. Lett. 104(6), 063002 (2010). [CrossRef]
10. X. Chen, I. Lizuain, A. Ruschhaupt, D. Guéry-Odelin, and J. G. Muga, “Shortcut to adiabatic passage in two- and three-level atoms,” Phys. Rev. Lett. 105(12), 123003 (2010). [CrossRef]
11. H. R. Lewis and W. B. Riesenfeld, “An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field,” J. Math. Phys. 10(8), 1458–1473 (1969). [CrossRef]
12. X. Chen, E. Torrontegui, and J. G. Muga, “Lewis-Riesenfeld invariants and transitionless quantum driving,” Phys. Rev. A 83(6), 062116 (2011). [CrossRef]
13. X. Chen and J. G. Muga, “Engineering of fast population transfer in three-level systems,” Phys. Rev. A 86(3), 033405 (2012). [CrossRef]
14. M. Lu, Y. Xia, L. T. Shen, J. Song, and N. B. An, “Shortcuts to adiabatic passage for population transfer and maximum entanglement creation between two atoms in a cavity,” Phys. Rev. A 89(1), 012326 (2014). [CrossRef]
15. Y. H. Chen, Y. Xia, Q. Q. Chen, and J. Song, “Efficient shortcuts to adiabatic passage for fast population transfer in multiparticle systems,” Phys. Rev. A 89(3), 033856 (2014). [CrossRef]
16. M. V. Berry, “Transitionless quantum driving,” J. Phys. A: Math. Theor. 42(36), 365303 (2009). [CrossRef]
17. A. del Campo, “Shortcuts to adiabaticity by counterdiabatic driving,” Phys. Rev. Lett. 111(10), 100502 (2013). [CrossRef]
18. Y. Liang, Q. C. Wu, S. L. Su, X. Ji, and S. Zhang, “Shortcuts to adiabatic passage for multiqubit controlled-phase gate,” Phys. Rev. A 91(3), 032304 (2015). [CrossRef]
19. Y. X. Du, Z. T. Liang, Y. C. Li, X. X. Yue, Q. X. Lü, W. Huang, X. Chen, H. Yan, and S. L. Zhu, “Experimental realization of stimulated Raman shortcut-to-adiabatic passage with cold atoms,” Nat. Commun. 7(1), 12479 (2016). [CrossRef]
20. B. B. Zhou, A. Baksic, H. Ribeiro, C. G. Yale, F. Joseph Heremans, P. C. Jerger, A. Auer, G. Burkard, A. A. Clerk, and D. D. Awschalom, “Accelerated quantum control using superadiabatic dynamics in a solid-state lambda system,” Nat. Phys. 13(4), 330–334 (2017). [CrossRef]
21. Y. C. Li and X. Chen, “Shortcut to adiabatic population transfer in quantum three-level systems:Effective two-level problems and feasible counterdiabatic driving,” Phys. Rev. A 94(6), 063411 (2016). [CrossRef]
22. N. V. Vitanov, “High-fidelity multistate stimulated Raman adiabatic passage assisted by shortcut fields,” Phys. Rev. A 102(2), 023515 (2020). [CrossRef]
23. S. Masuda and K. Nakamura, “Acceleration of adiabatic quantum dynamics in electromagnetic fields,” Phys. Rev. A 84(4), 043434 (2011). [CrossRef]
24. S. Masuda and S. A. Rice, “Fast-forward assisted STIRAP,” J. Phys. Chem. A 119(14), 3479–3487 (2015). [CrossRef]
25. S. Ibá nez, X. Chen, E. Torrontegui, J. G. Muga, and A. Ruschhaupt, “Multiple Schrödinger pictures and dynamics in shortcuts to adiabaticity,” Phys. Rev. Lett. 109(10), 100403 (2012). [CrossRef]
26. S. Ibá nez, X Chen, and J. G. Muga, “Improving shortcuts to adiabaticity by iterative interaction pictures,” Phys. Rev. A 87(4), 043402 (2013). [CrossRef]
27. X. K. Song, Q. Ai, J. Qiu, and F. G. Deng, “Physically feasible three-level transitionless quantum driving with multiple Schr$\ddot {o}$dinger dynamics,” Phys. Rev. A 93(5), 052324 (2016). [CrossRef]
28. A. Baksic, H. Ribeiro, and A. A. Clerk, “Speeding up adiabatic quantum state transfer by using dressed states,” Phys. Rev. Lett. 116(23), 230503 (2016). [CrossRef]
29. Y. H. Kang, Y. H. Chen, Z. C. Shi, J. Song, and Y. Xia, “Fast preparation of W states with superconducting quantum interference devices by using dressed states,” Phys. Rev. A 94(5), 052311 (2016). [CrossRef]
30. A. Baksic, R. Belyansky, H. Ribeiro, and A. A. Clerk, “Shortcuts to adiabaticity in the presence of a continuum: applications to itinerant quantum state transfer,” Phys. Rev. A 96(2), 021801 (2017). [CrossRef]
31. F. Y. Zhang and C. P. Yang, “Tunable coupling of spin ensembles,” Opt. Lett. 43(3), 466 (2018). [CrossRef]
32. A. del Campo and K. Kim, “Focus on Shortcuts to Adiabaticity,” New J. Phys. 21(5), 050201 (2019). [CrossRef]
33. D. Guéry-Odelin, A. Ruschhaupt, A. Kiely, E. Torrontegui, S. Martínez-Garaot, and J. G. Muga, “Shortcuts to adiabaticity: Conceptsmethods, and applications,” Rev. Mod. Phys. 91(4), 045001 (2019). [CrossRef]
34. Y. H. Kang, Y. H. Chen, Q. C. Wu, B. H. Huang, Y. Xia, and J. Song, “Reverse engineering of a Hamiltonian by designing the evolution operators,” Sci. Rep. 6(1), 30151 (2016). [CrossRef]
35. A. Levy, A. Kiely, J. G. Muga, R. Kosloff, and E. Torrontegui, “Noise resistant quantum control using dynamical invariants,” New J. Phys. 20(2), 025006 (2018). [CrossRef]
36. C. K. Hu, J. M. Cui, A. C. Santos, Y. F. Huang, M. S. Sarandy, C. F. Li, and G. C. Guo, “Experimental implementation of generalized transitionless quantum driving,” Opt. Lett. 43(13), 3136 (2018). [CrossRef]
37. A. C. Santos and M. S Sarandy, “Generalized shortcuts to adiabaticity and enhanced robustness against decoherence,” J. Phys. A: Math. Theor. 51(2), 025301 (2018). [CrossRef]
38. H. L. H. Mortensen, J. J. W. Sørensen, K. Mølmer, and J. F. Sherson, “Fast state transfer in a Λ-system: a shortcut-to-adiabaticity approach to robust and resource optimized control,” New J. Phys. 20(2), 025009 (2018). [CrossRef]
39. A. Tobalina, J. Alonso, and J. G. Muga, “Energy consumption for ion-transport in a segmented Paul trap,” New J. Phys. 20(6), 065002 (2018). [CrossRef]
40. A. C. Santos, A. Nicotina, A. M. Souza, R. S. Sarthour, I. S. Oliveira, and M. S.Sarandy, “Optimizing NMR quantum information processing via generalized transitionless quantum driving,” Europhys. Lett. 129(3), 30008 (2020). [CrossRef]
41. D. Stefanatos and E. Paspalakis, “Efficient generation of the triplet Bell state between coupled spins using transitionless quantum driving and optimal control,” Phys. Rev. A 99(2), 022327 (2019). [CrossRef]
42. B. J. Liu and M. H. Yung, “Leakage suppression for Holonomic quantum gates,” Phys. Rev. Appl. 14(3), 034003 (2020). [CrossRef]
43. A. Ruschhaupt, X. Chen, D. Alonso, and J. G. Muga, “Optimally robust shortcuts to population inversion in two-level quantum systems,” New J. Phys. 14(9), 093040 (2012). [CrossRef]
44. X. J. Lu, X. Chen, A. Ruschhaupt, D. Alonso, S. Guérin, and J. G. Muga, “Fast and robust population transfer in two-level quantum systems with dephasing noise and/or systematic frequency errors,” Phys. Rev. A 88(3), 033406 (2013). [CrossRef]
45. X. T. Yu, Q. Zhang, Y. Ban, and X. Chen, “Fast and robust control of two interacting spins,” Phys. Rev. A 97(6), 062317 (2018). [CrossRef]
46. Y. H. Kang, Z. C. Shi, Y. Xia, and J. Song, “Robust generation of logical qubit singlet states with reverse engineering and optimal control with spin qubits,” Adv. Quantum Technol. 3(12), 2000113 (2020). [CrossRef]
47. R. H. Zheng, Y. H. Kang, S. L. Su, J. Song, and Y. Xia, “Robust and high-fidelity nondestructive Rydberg parity meter,” Phys. Rev. A 102(1), 012609 (2020). [CrossRef]
48. S. Liu, J. H. Shen, R. H. Zheng, Y. H. Kang, Z. C. Shi, J. Song, and Y. Xia, “Optimized nonadiabatic holonomic quantum computation based on Förster resonance in Rydberg atoms,” Front. Phys. 17(2), 21502 (2022). [CrossRef]
49. X. K. Song, F. Meng, B. J. Liu, D. Wang, L. Ye, and M. H. Yung, “Robust stimulated Raman shortcut-to-adiabatic passage with invariant-based optimal control,” Opt. Express 29(6), 7998 (2021). [CrossRef]
50. L. Tian, “Adiabatic state conversion and pulse transmission in optomechanical systems,” Phys. Rev. Lett. 108(15), 153604 (2012). [CrossRef]
51. Y. D. Wang and A. A. Clerk, “Using interference for high fidelity quantum wtate transfer in optomechanics,” Phys. Rev. Lett. 108(15), 153603 (2012). [CrossRef]
52. G. D. de Moraes Neto, F. M. Andrade, V. Montenegro, and S. Bose, “Quantum state transfer in optomechanical arrays,” Phys. Rev. A 93(6), 062339 (2016). [CrossRef]
53. U. Hohenester, G. Panzarini, F. Troiani, E. Molinari, and C. Macchiavello, “Coherent population transfer in coupled semiconductor quantum dots,” Appl. Phys. Lett. 77(12), 1864 (2000). [CrossRef]
54. B. Chen, Q. H. Shen, W. Fan, and Y. Xu, “Long-range adiabatic quantum state transfer through a linear array of quantum dots,” Sci. China Phys. Mech. 55(9), 1635–1640 (2012). [CrossRef]
55. A. D. Greentree, J. H. Cole, A. R. Hamilton, and L. C. L. Hollenberg, “Coherent electronic transfer in quantum dot systems using adiabatic passage,” Phys. Rev. B 70(23), 235317 (2004). [CrossRef]
56. M. Christandl, N. Datta, A. Ekert, and A. J. Landahl, “Perfect state transfer in quantum spin networks,” Phys. Rev. Lett. 92(18), 187902 (2004). [CrossRef]
57. K. Eckert, O. Romero-Isart, and A. Sanpera, “Efficient quantum state transfer in spin chains via adiabatic passage,” New J. Phys. 9(5), 155 (2007). [CrossRef]
58. C. Di Franco, M. Paternostro, and M. S. Kim, “Perfect state transfer on a spin chain without state initialization,” Phys. Rev. Lett. 101(23), 230502 (2008). [CrossRef]
59. J. Y. Yang, D. Y. Wang, C. H. Bai, S. Y. Guan, X. Y. Gao, A. D. Zhu, and H. F. Wang, “Ground-state cooling of mechanical oscillator via quadratic optomechanical coupling with two coupled optical cavities,” Opt. Express 27(16), 22855 (2019). [CrossRef]
60. X. Y. Zhang, Y. H. Zhou, Y. Q. Guo, and X. X. Yi, “Simultaneous cooling of two mechanical oscillators in dissipatively coupled optomechanical systems,” Phys. Rev. A 100(2), 023807 (2019). [CrossRef]
61. H. K. Lau and A. A. Clerk, “Ground-state cooling and high-fidelity quantum transduction via parametrically driven bad-cavity optomechanics,” Phys. Rev. Lett. 124(10), 103602 (2020). [CrossRef]
62. J. Q. Liao and L. Tian, “Macroscopic quantum superposition in cavity optomechanics,” Phys. Rev. Lett. 116(16), 163602 (2016). [CrossRef]
63. D. G. Lai, X. Wang, W. Qin, B. P. Hou, Franco Nori, and J. Q. Liao, “Tunable optomechanically induced transparency by controlling the dark-mode effect,” Phys. Rev. A 102(2), 023707 (2020). [CrossRef]
64. Y. Zhang, Z. Zhu, Y. Cui, H. Yu, C. Jiang, and X. Cao, “Optomechanically induced transparency, amplification, and Fano resonance in a multimode optomechanical system with quadratic coupling,” EPJ Quantum Technol. 8(1), 7 (2021). [CrossRef]
65. A. Vochezer, T. Kampschulte, K. Hammerer, and P. Treutlein, “Light-mediated collective atomic motion in an optical lattice coupled to a membrane,” Phys. Rev. Lett. 120(7), 073602 (2018). [CrossRef]
66. S. Felicetti, S. Fedortchenko, R. Rossi, S. Ducci, I. Favero, T. Coudreau, and P. Milman, “Quantum communication between remote mechanical resonators,” Phys. Rev. A 95(2), 022322 (2017). [CrossRef]
67. J. Cheng, X. T. Liang, W. Z. Zhang, and X. Duan, “Optomechanical state transfer between two distant membranes in the presence of non-Markovian environments,” Chin. Phys. B 27(12), 120302 (2018). [CrossRef]
68. Y. Wang, Y. H. Kang, C. S. Hu, B. H. Huang, J. Song, and Y. Xia, “Quantum control with Lyapunov function and bang-bang solution in the optomechanics system,” Front. Phy. 17(3), 32501 (2022). [CrossRef]
69. D. Garg, A. K. Chauhan, and A. Biswas, “Adiabatic transfer of energy fluctuations between membranes inside an optical cavity,” Phys. Rev. A 96(2), 023837 (2017). [CrossRef]
70. Y. H. Chen, Z. C. Shi, J. Song, and Y. Xia, “Invariant-based inverse engineering for fluctuation transfer between membranes in an optomechanical cavity system,” Phys. Rev. A 97(2), 023841 (2018). [CrossRef]
71. M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014). [CrossRef]
72. P. Facchi, V. Gorini, G. Marmo, S. Pascazio, and E. C. G. Sudarshan, “Quantum Zeno dynamics,” Phys. Lett. A 275(1-2), 12–19 (2000). [CrossRef]
73. P. Facchi and S. Pascazio, “Quantum Zeno subspaces,” Phys. Rev. Lett. 89(8), 080401 (2002). [CrossRef]
