Composite pulses for high fidelity population transfer in three-level systems

Implementation of high fidelity quantum coherent control is a top priority in quantum information processing (QIP) Salières et al. (1995); Bergmann et al. (1998); Makhlin et al. (2001); Zhao et al. (2006); McCullough et al. (2000); Gordon et al. (2002); Zeng et al. (2020); Devolder et al. (2021); Levine et al. (2018); Daems et al. (2013). Many works Allen and Eberly (1975); Vitanov et al. (2001); Guéry-Odelin et al. (2019), which design different kinds of pulse shapes, have been devoted to ensuring a remarkable quantum computing performance. Originally, the resonant pulse (RP) technique, where the frequency of radiation exactly matches the transition frequency, is regarded as a popular tool to accurately achieve coherent control, due to its fast operation and simple waveform Scully and Zubairy (1997). For example, one can achieve complete population inversion through a $\pi$-pulse, or a maximum superposition state through a $\pi/2$-pulse Gerry and Knight (2004). However, the resonant pulse is extremely susceptible to external perturbations, such as fluctuations of control fields. To overcome this difficulty, the adiabatic passage (AP) technique has been developed Shankar (1994); Král et al. (2007). The AP technique selects one eigenstate of the system Hamiltonian as an evolution path and propels the initial state to adiabatically evolve along this path. In this technique, the robustness against parameter perturbations is improved at the cost of the slow evolution rate and lower fidelity. To enjoy both the ultrahigh fidelity of RP and the robustness of AP, one can adopt the composite pulses (CPs) technique.

The CPs technique, comprised of a well-organized train of constant pulses with determined pulse areas and relative phases, was conceived in early nuclear magnetic resonance (NMR) Wimperis (1991, 1994); Levitt and Freeman (1979); Levitt (1986). To date, this technique has been used extensively in QIP Brown et al. (2004); Dridi et al. (2020); Torosov and Vitanov (2011); Genov et al. (2014); Torosov and Vitanov (2018); Jones (2013); Vitanov (2011); Torosov and Vitanov (2019a); Torosov et al. (2020a); Genov et al. (2020). One unique feature of CPs is that the pulse sequence is available to compensate for the systematic error in any physical parameters (e.g., pulse duration, pulse amplitude, detuning, Stark shift, etc.) Genov et al. (2014); Brown et al. (2004); Levitt (1986); Dridi et al. (2020); Torosov and Vitanov (2011). Previous works Torosov et al. (2015); Ichikawa et al. (2011); Cohen et al. (2016); Kyoseva et al. (2019); Wang et al. (2014); Mount et al. (2015); Demeter (2016) about precise quantum control are mostly based on two-level systems. For example, by using a narrow-band composite sequence of laser pulses, the error-tolerance region of high fidelity local addressing operation in trapped ions and atoms is improved by $20\%$ Ivanov and Vitanov (2011). Besides, an arbitrary single-qubit gate, which is robust against pulse area error, could be implemented by a composite $\theta$ pulses sequence Torosov and Vitanov (2019b).

At present, works on CPs in two-level systems are relatively complete and extensive Vandersypen and Chuang (2005). As is well known, the three-level system is another representative physical model in QIP. Some interesting phenomena, such as coherent trapping Scully and Zubairy (1997), electromagnetically induced transparency Fleischhauer et al. (2005), and lasing without inversion Scully et al. (1989) manifest in three-level systems. Recently, works on a general use of CPs scheme in three-level systems are still deficient Levitt et al. (1984); Ramamoorthy and Narasimhan (1991); Torosov et al. (2011); Torosov and Vitanov (2013); Schraft et al. (2013); Ishida et al. (2018). The main difficulty is that the analytical expression of the propagator is very complicated for a general three-level system. One way to deal with this difficulty is to reduce the three-level system into one or more two-level systems. In Ref. Torosov et al. (2020b), the closed-loop three-level system of the chiral molecule is divided into three independent two-level systems Lehmann (2018); Leibscher et al. (2019). Then, the CPs sequence using RP in these two-level systems achieves the chiral resolution necessary to overcome Rabi frequency errors and detuning errors. In Refs. Genov et al. (2011); Torosov and Vitanov (2020), two powerful tools, Morris-Shore transformation Morris and Shore (1983) and Majorana decomposition Majorana (1932), are exploited to simplify a three-level system into an effective two-level system, so as to obtain the propagator of the system. Note that some approximations or restrictions are usually adopted in this simplification process. Specifically, Morris-Shore transformation Morris and Shore (1983) is applied when the system can be mapped into two sets of states with the same energies (in the rotating-wave approximation) and two coupling coefficients must have the same time dependence, while Majorana decomposition Majorana (1932) always demands that the system have the SU(2) dynamic symmetry. The common feature of these approaches Genov et al. (2011); Torosov and Vitanov (2020); Morris and Shore (1983); Majorana (1932) is to reduce the dynamics of the three-level system to a two-level system dynamics. As a result, one ignores the dynamics of the excited (leakage) state and thus the system does not exist the population leakage. However, the dynamics of the excited state should be considered in the three-level system, and that is what makes a three-level system different from a two-level system. It is therefore necessary to conceive a general CPs sequence able to directly achieve precise and robust control in a three-level system.

In three-level systems, there are some issues that must be considered. For instance, during the construction process of qubits, the existence of the leakage state, which is beyond the computational subspace, leads to an imperfect quantum operation performance. This issue is inevitable and poses a great threat to the fidelity of qubit operations. The works on suppressing leakage in three-level systems are ongoing Ghosh et al. (2017); Shi et al. (2021). Besides, compared to the two-level system, the additional modulation parameters in the three-level system may lead to various types of systematic errors. The existence of these systematic errors has a detrimental effect on the quantum operation accuracies, to various degrees. On the other hand, as shown in Refs. Genov et al. (2014); Torosov and Vitanov (2011, 2018); Dridi et al. (2020), the fidelity improves with the increase in the number of pulses. However, considering the finite coherence time, a large number of pulses would cost long evolution time, which is tremendously detrimental for quantum computation. Therefore, one generally requires the number of pulses to be as small as possible, hence it is essential to learn how to efficiently eliminate the systematic errors within finite pulses. This is especially true in the case where the quantum systems have various types of deviations.

In this work, we achieve arbitrary population transfer by phase modulation CPs in the three-level system. We directly deduce the transition probability of each state, including the excited state. In other words, we concern the dynamics of the excited state as well. By using the Taylor expansion, the transition probability of the target (leakage) state can be sorted by different order derivatives. Then, we construct a cost function to overcome the difficulty that there are not enough variables to eliminate high-order derivatives in the short pulse sequence. Different cost functions have respective purposes for accurately controlling population transfer or effectively suppressing population leakage, or both of them. We study minutely two kinds of short pulse sequences as examples to demonstrate the pulse design process, which can easily extend to longer pulse sequences. For applications, our CPs scheme further shows the feasibility of achieving the three-atoms singlet state with ultrahigh fidelity in an atoms-cavity system. The results indicate that the final singlet state is robust against several kinds of defects, including deviations in coupling coefficients, waveform distortions, and the disturbances of the decoherence environment.

The paper is organized as follows. In Sec. II, we design the CPs sequence in the three-level system through minimizing the cost function. In Sec. III, we first employ different cost functions to analytically search for the best group of phases in the two-pulse sequence. For more than two pulses, we adopt numerical methods to find different phases of the CPs sequence. In principle, the numerical methods are suitable for arbitrary long pulse sequences. In Sec. IV, we study the feasibility of the CPs sequence through a specific application in the atom-cavity system. Finally, a conclusion is given in Sec. V.

Consider a $\Lambda$-type three-level quantum system interacting with two external fields, where the three-level system has two ground states $|g\rangle$ and $|r\rangle$, and an excited state $|e\rangle$. The ground states $|g\rangle$ and $|r\rangle$, acting as a qubit, cannot be immediately coupled to each other. Hence, we require an extra excited state $|e\rangle$ to construct the indirect coupling between two ground states. More specifically, the transition $|g\rangle\leftrightarrow|e\rangle$ ($|r\rangle\leftrightarrow|e\rangle$) is driven by the external field with the coupling strength $\Omega_{1}$ ($\Omega_{2}$), the phase $\alpha$ ($\beta$), and the detuning $\Delta$. In presence of deviations in the external fields, in the interaction picture, the system Hamiltonian is given by ($\hbar=1$)

$$H\!=\!\Delta|e\rangle\langle e|\!+\!(1+\epsilon_{1})\Omega_{1}e^{i\alpha}|g\rangle\langle e|\!+\!(1+\epsilon_{2})\Omega_{2}e^{i\beta}|r\rangle\langle e|\!+\!\mathrm{H.c.},$$

(1)

where the deviations $\epsilon_{1}$ and $\epsilon_{2}$ are random unknown constants, which would give rise to the systematic errors during quantum operations.

Here, we do not intend to specify a concrete physical system, because the three-level system studied here can be found in very different physics research fields ranging from high precision spectroscopy, to quantum information processing, NMR, and metrology. For example, this physical model is quite familiar in the quantum system of a three-level atom interacting with two laser fields Scully and Zubairy (1997). In the atomic system, the inhomogeneous distribution of the laser fields leads to different interaction coefficients between the atom and the laser fields. Also, the spatial position of the atom cannot be exactly determined due to its micro-vibration. As a result, the interaction coefficients in the system are susceptible to deviations. It is worth mentioning that similar energy-level structures can also be found in trapped ions Leibfried et al. (2003), diamond nitrogen-vacancy centers Yang et al. (2010), and superconducting circuits Xiang et al. (2013); Xue et al. (2017); Kang et al. (2016, 2017); Hong et al. (2018); Chen et al. (2020). In addition, some complicated quantum systems, such as the electrons in semiconductors Greentree et al. (2004), the spin chain Bruderer et al. (2012); Chen and Li (2016), neutral atoms interactions Li and Shao (2018); Kang et al. (2018); Shi et al. (2018); Li et al. (2020); Shao (2020); Wu et al. (2021), and massive quantum particles in an optical Lieb lattice Taie et al. (2020), can be reduced to the three-level physical model as well.

Assume that the system is initially in the ground state $|g\rangle$. If there are no deviations in the external fields (i.e., $\epsilon_{1}=0$ and $\epsilon_{2}=0$), one can easily achieve perfect population transfer for the qubit. To this end, we choose the evolution time $T=2\pi/\sqrt{\Delta^{2}+4(\Omega_{1}^{2}+\Omega_{2}^{2})}$, then the population of the ground state $|r\rangle$ becomes

$$\mathcal{P}_{r}=\frac{4\Omega_{1}^{2}\Omega_{2}^{2}\cos^{2}\frac{\Delta T}{4}}{(\Omega_{1}^{2}+\Omega_{2}^{2})^{2}}.$$

(2)

Thus, the value of $\mathcal{P}_{r}$ can be modulated by altering either the detuning or the ratio of two coupling coefficients. However, in presence of deviations in the external fields, the actual population $P_{r}^{a}$ would keep away from the desired value $\mathcal{P}_{r}$. This is verified in Figs. 1(a) and 1(c), which demonstrate that the infidelity $\mathcal{F}_{r}=|P_{r}^{a}-\mathcal{P}_{r}|$ is high even for small deviations. Note that Figs. 1(a) and 1(c) also demonstrate that the value of infidelity becomes extremely low in some regions where the deviations are large, e.g., $\mathcal{F}_{r}\approx 8\times 10^{-5}$ when $\epsilon_{1}\approx-0.46$ and $\epsilon_{2}\approx 0.42$ in Fig. 1(c). However, it is meaningless in practice, because the values of the deviations $\epsilon_{1}$ and $\epsilon_{2}$ are unknown in principle and they are usually independent of each other. On the other hand, population leakage from the ground states to the excited state invariably does happen in the presence of deviations, which can be observed in Figs. 1(b) and 1(d). Hence, the goal of this work is to precisely achieve the predefined population $\mathcal{P}_{r}$, and meanwhile, to dramatically suppress the population leakage by designing a composite $N$-pulse sequence.

First, we require to calculate the population of the ground state $|r\rangle$ in absence of the deviations. Assume that the Hamiltonian of the $n$th pulse has the following form ($n=1,\dots,N$)

	$\displaystyle H_{n}$	$\displaystyle=$	$\displaystyle\Delta_{n}|e\rangle\langle e|+(1+\epsilon_{1})\Omega_{1n}e^{i\alpha_{n}}|g\rangle\langle e|$		(3)
			$\displaystyle+(1+\epsilon_{2})\Omega_{2n}e^{i\beta_{n}}|r\rangle\langle e|+\mathrm{H.c.}$

For the pulse duration $T_{n}=2\pi/\sqrt{\Delta_{n}^{2}+4(\Omega_{1n}^{2}+\Omega_{2n}^{2})}$, the corresponding propagator, labelled as $U(\theta_{n},\gamma_{n})$, can be written as (up to a global phase)

$\displaystyle U(\theta_{n},\gamma_{n})=\left(\begin{array}[]{ccc}\cos\theta_{n}e^{i\Phi_{n}}&\sin\theta_{n}e^{-i\gamma_{n}}&0\\[4.30554pt] \!-\sin\theta_{n}e^{i\gamma_{n}}&\cos\theta_{n}e^{-i\Phi_{n}}&0\\[4.30554pt] 0&0&\!\!ie^{-i\Theta_{n}}\\ \end{array}\right),$

(7)

where

	$\displaystyle\Theta_{n}$	$\displaystyle=$	$\displaystyle\frac{\pi\Delta_{n}}{2\sqrt{\Delta_{n}^{2}+4(\Omega_{1n}^{2}+\Omega_{2n}^{2})}},$
	$\displaystyle\Phi_{n}$	$\displaystyle=$	$\displaystyle\arctan\frac{(\Omega_{1n}^{2}-\Omega_{2n}^{2})\cos\Theta_{n}}{(\Omega_{1n}^{2}+\Omega_{2n}^{2})\sin\Theta_{n}},$
	$\displaystyle\theta_{n}$	$\displaystyle=$	$\displaystyle\arcsin\frac{2\Omega_{1n}\Omega_{2n}\cos\Theta_{n}}{\Omega_{1n}^{2}+\Omega_{2n}^{2}},$
	$\displaystyle\gamma_{n}$	$\displaystyle=$	$\displaystyle\beta_{n}-\alpha_{n}-\pi/2.$

Then, the total propagator of the composite $N$-pulse sequence can be written as

$\displaystyle U(\theta_{N},\gamma_{N})U(\theta_{N-1},\gamma_{N-1})\cdots U(\theta_{2},\gamma_{2})U(\theta_{1},\gamma_{1}).$

The detailed derivation of the total propagator is given in Appendix A. After some algebraic manipulations, the population of the ground state $|r\rangle$, labelled as $P^{\scriptscriptstyle(0)}_{r}$, becomes

$\displaystyle P^{\scriptscriptstyle(0)}_{r}=\sin^{2}\vartheta_{N},$

(8)

where the expression of $\vartheta_{N}$ is also presented in Appendix A. For simplicity, we set $\Delta_{n}=0$ in the following, which means that the three-level system works in the resonant regime. Apparently, the following analysis can be easily generalized to the two-photon resonant regime. Furthermore, we choose $\theta_{n}=\pi/4$ ($n=1,\dots,N$), which means that two coupling coefficients $\Omega_{1}$ and $\Omega_{2}$ remain unchanged in the $N$-pulse sequence. Note that it is also suitable for the other values of $\theta_{n}$. As a result, we only modulate the phases $\alpha_{n}$ and $\beta_{n}$, and the form of CPs now becomes

$\displaystyle U\left(\frac{\pi}{4},\gamma_{N}\right)U\left(\frac{\pi}{4},\gamma_{N-1}\right)\cdots U\left(\frac{\pi}{4},\gamma_{2}\right)U\left(\frac{\pi}{4},\gamma_{1}\right).$

In presence of deviations in the external fields, by the Taylor expansion, the actual population $P_{r}^{a}$ of the ground state $|r\rangle$ can be written as

	$\displaystyle P_{r}^{a}$	$\displaystyle=$	$\displaystyle P^{\scriptscriptstyle(0)}_{r}+P^{\scriptscriptstyle(1)}_{r}+P^{\scriptscriptstyle(2)}_{r}+\cdots$		(9)
		$\displaystyle=$	$\displaystyle C^{\scriptscriptstyle(0)}_{\!N\!,1}+C^{\scriptscriptstyle(1)}_{\!N\!,1}\epsilon_{1}+C_{\!N\!,2}^{\scriptscriptstyle(1)}\epsilon_{2}+C^{\scriptscriptstyle(2)}_{\!N\!,1}\epsilon_{1}\epsilon_{2}+C^{\scriptscriptstyle(2)}_{\!N\!,2}\epsilon_{1}^{2}+C^{\scriptscriptstyle(2)}_{\!N\!,3}\epsilon_{2}^{2}$
			$\displaystyle+{O}(\epsilon_{1}\epsilon_{2}^{2},\epsilon_{1}^{2}\epsilon_{2},\epsilon_{1}^{3},\epsilon_{2}^{3}),$

where $C^{\scriptscriptstyle(l)}_{\!N\!,k}$ ($k=1,2,\dots$) are the coefficients of the $l$th-order term in the $N$-pulse sequence. To achieve the predefined population $\mathcal{P}_{r}$, one should design the phases $\alpha_{n}$ and $\beta_{n}$ ($n=1,\dots,N$) to guarantee the validity of the following equation

$\displaystyle P^{\scriptscriptstyle(0)}_{r}=C^{\scriptscriptstyle(0)}_{\!N\!,1}=\mathcal{P}_{r}=\sin^{2}\theta.$

(10)

That is, the zeroth-order term of the actual population $P_{r}^{a}$ should be equal to the predefined population. This is the first condition that the phases $\alpha_{n}$ and $\beta_{n}$ must satisfy. Furthermore, the global phase of the propagator is inessential for the system dynamics, and what really matters is the phase difference $\alpha_{mn}=\alpha_{m}-\alpha_{n}$ and $\beta_{mn}=\beta_{m}-\beta_{n}$ ($m,n=1,\dots,N$). Hence, one can set the phases $\alpha_{1}$ and $\beta_{1}$ of the first pulse to be arbitrary. Nevertheless, they become extremely useful when the composite pulses are designed for single qubit gates, because the values of $\alpha_{1}$ and $\beta_{1}$ can be used to modulate the relative phase between the ground states. As a result, apart from Eq. (10), there still remain $(2N-3)$ free phases (variables) in the $N$-pulse sequence, which can be designed to compensate for the systematic errors caused by the deviations $\epsilon_{1}$ and $\epsilon_{2}$.

Unlike the case of two-level systems Torosov et al. (2020a); Genov et al. (2020); Torosov et al. (2015); Ichikawa et al. (2011); Cohen et al. (2016); Kyoseva et al. (2019); Wang et al. (2014); Mount et al. (2015); Demeter (2016), there exists a population leakage from the ground states to the excited state in the three-level system. Thus, we need to suppress population leakage (i.e., the population of the excited state $|e\rangle$) as well. Similar to the treatment of $P_{r}^{a}$, the actual population $P_{e}^{a}$ of the excited state $|e\rangle$ can be expanded by the Taylor expansion, as follows

	$\displaystyle P_{e}^{a}\!$	$\displaystyle=$	$\displaystyle\!P^{\scriptscriptstyle(0)}_{e}+P^{\scriptscriptstyle(1)}_{e}+P^{\scriptscriptstyle(2)}_{e}+\cdots$		(11)
		$\displaystyle=$	$\displaystyle\!D^{\scriptscriptstyle(2)}_{\!N\!,1}\epsilon_{1}\epsilon_{2}\!+\!D^{\scriptscriptstyle(2)}_{\!N\!,2}\epsilon_{1}^{2}\!+\!D^{\scriptscriptstyle(2)}_{\!N\!,3}\epsilon_{2}^{2}\!+\!{O}(\epsilon_{1}\epsilon_{2}^{2},\epsilon_{1}^{2}\epsilon_{2},\epsilon_{1}^{3},\epsilon_{2}^{3}),~{}~{}~{}~{}~{}~{}$

where $D^{\scriptscriptstyle(l)}_{\!N\!,k}$ ($k=1,2,\dots$) are the coefficients of the $l$th-order term in the $N$-pulse sequence. Contrary to Eq. (9), Eq. (11) does not contain the first-order term.

Therefore, to obtain high fidelity population transfer in the qubit, the general method is to satisfy the following conditions: ($i$) eliminate the higher-order terms in $P_{r}^{a}$ as much as possible. Namely, $C^{\scriptscriptstyle(1)}_{\!N\!,1}=C^{\scriptscriptstyle(1)}_{\!N\!,2}=\cdots=0$; ($ii$) demand that the value of $P_{e}^{a}$ be as small as possible, i.e., $D^{\scriptscriptstyle(2)}_{\!N\!,1}=D^{\scriptscriptstyle(2)}_{\!N\!,2}=\cdots=0$. Note that the design procedure of the phases is more complicated in three-level systems than that in two-level systems Torosov et al. (2020a); Genov et al. (2020); Torosov et al. (2015); Ichikawa et al. (2011); Cohen et al. (2016); Kyoseva et al. (2019); Wang et al. (2014); Mount et al. (2015); Demeter (2016), since there are two coefficients [$C^{\scriptscriptstyle(1)}_{\!N\!,1}$ and $C^{\scriptscriptstyle(1)}_{\!N\!,2}$] in the first-order term while there are six coefficients [$C^{\scriptscriptstyle(2)}_{\!N\!,k}$ and $D^{\scriptscriptstyle(2)}_{\!N\!,k}$, $k=1,2,3$] in the second-order term, and so forth. Particularly, when considering the small number of pulses, which is often the practical case, it does not have sufficient phases to eliminate the systematic errors up to the desired order.

To solve this issue, we can proceed as follows. First, it is worth noting that the influence of the higher-order term of systematic errors on dynamical evolution would gradually diminish. Therefore, the first-order terms have the most serious effect on dynamical evolution, and we should give priority to making these terms vanish by designing suitable phases, i.e., $C^{\scriptscriptstyle(1)}_{\!N\!,1}=C^{\scriptscriptstyle(1)}_{\!N\!,2}=0$. Then, the remaining phases are designed to minimize the following cost function:

	$\displaystyle F$	$\displaystyle=$	$\displaystyle\sum_{l=2}^{\infty}\left[A_{l}F_{c}^{\scriptscriptstyle(l)}+B_{l}F_{d}^{\scriptscriptstyle(l)}\right],$		(12)
	$\displaystyle F_{c}^{\scriptscriptstyle(l)}$	$\displaystyle=$	$\displaystyle\sum\nolimits_{k=1}^{l+1}|C^{\scriptscriptstyle(l)}_{N,k}|^{2},$		(13)
	$\displaystyle F_{d}^{\scriptscriptstyle(l)}$	$\displaystyle=$	$\displaystyle\sum\nolimits_{k=1}^{l+1}|D^{\scriptscriptstyle(l)}_{N,k}|^{2},$		(14)

where $A_{l}$ and $B_{l}$ are the weighting coefficients, satisfying $0\leq A_{l-1}\leq A_{l}$ and $0\leq B_{l-1}\leq B_{l}$. Physically, the cost function $F$ represents the trade-off between the accuracy of population transfer and the leakage to the excited state $|e\rangle$. The conditions $0\leq A_{l-1}\leq A_{l}$ and $0\leq B_{l-1}\leq B_{l}$ ensure that the coefficients of the low order terms have a high weight, and thus are preferentially eliminated. Remarkably, different weighting coefficients have different effects. Setting $B_{l}=0$, we have the cost function $F=\sum_{l=2}^{\infty}A_{l}F_{c}^{\scriptscriptstyle(l)}$ aimed to keep the accuracy of the population transfer, while setting $A_{l}=0$ we have the cost function $F=\sum_{l=2}^{\infty}B_{l}F_{d}^{\scriptscriptstyle(l)}$ aimed to reduce the leakage to the excited state $|e\rangle$. Note that the minimum value of the cost function is zero, corresponding to satisfy the equations: $C^{\scriptscriptstyle(l)}_{N,k}=D^{\scriptscriptstyle(l)}_{N,k}=0$.

In this section, we present the design process of the phases $\alpha_{n}$ and $\beta_{n}$ ($n=2,...,N$) to achieve high fidelity population transfer by the $N$-pulse sequence. For the two-pulse sequence, the phases are derived analytically, while for the three-pulse sequence we combine both the analytical method and the numerical method to obtain the phases. For more than three pulses, we carry out the four(five)-pulse sequence by numerical calculations to demonstrate the feasibility of eliminating the higher-order terms of the systematic errors. It is worth mentioning that the cycle of the phases is $2\pi$, thus we restrict the values of all phases in the interval $[0,2\pi)$ following from here.

In above section, we have showed the design process of composite pulses in a general physical system, where the three-level structure can be found in atomic systems Scully and Zubairy (1997), trapped ions Leibfried et al. (2003), diamond nitrogen-vacancy centers Yang et al. (2010), and superconducting circuits Xiang et al. (2013); Xue et al. (2017); Kang et al. (2016, 2017); Hong et al. (2018); Chen et al. (2020), etc. In this section, we demonstrate that the CPs scheme can be also generalized to complicated systems for performing different quantum tasks, e.g., coherent conversion between two qubits Yang et al. (2021) or preparation of entangled states Shao et al. (2010), etc. The fundamental is to reduce complicated systems to the familiar three-level physical model. As an example, in presence of deviations, we next prepare the three-atom singlet state with ultrahigh fidelity in an atom-cavity system, and this approach is easily extended to other complicated systems.

As shown in Fig. 6(a), consider the atom-cavity system, where three identical four-level atoms are trapped in a bimodal cavity. The four-level atom has three ground states $|1\rangle$, $|2\rangle$, and $|3\rangle$, and an excited state $|e\rangle$, as shown in Fig. 6(b). The transition $|2\rangle_{k}\leftrightarrow|e\rangle_{k}$ ($|3\rangle_{k}\leftrightarrow|e\rangle_{k}$) is resonantly coupled by the cavity mode $a$ ($b$) with the coupling constant $\lambda_{k}^{a}$ ($\lambda_{k}^{b}$), where the subscript $k$ represents the $k$th atom. The transition $|1\rangle_{k}\leftrightarrow|e\rangle_{k}$ is resonantly coupled by the laser field with the coupling strength $\Omega_{k}$ and the phase $\alpha_{k}$. In the interaction picture, the Hamiltonian of the atom-cavity system reads ($\hbar=1$)

	$\displaystyle H$	$\displaystyle\!=\!$	$\displaystyle\sum_{k=1}^{3}\Big{[}(1\!+\!\epsilon_{k})\Omega_{k}\exp(i\alpha_{k})|e\rangle_{kk}\langle 1|\!+\!\lambda_{k}^{a}(1\!+\!\zeta_{k}^{a})|e\rangle_{kk}\langle 2|\hat{a}$		(31)
			$\displaystyle+\lambda_{k}^{b}(1+\zeta_{k}^{b})|e\rangle_{kk}\langle 3|\hat{b}+\mathrm{H.c.}\Big{]},$

where $\hat{a}$ and $\hat{b}$ are the annihilation operators of the cavity mode $a$ and $b$, respectively. $\epsilon_{k}$ and $\zeta_{k}^{a}$ ($\zeta_{k}^{b}$) are respectively the deviations of the laser fields and the cavity mode $a~{}(b)$ due to the inhomogeneities of spatial distribution.

Apparently, the excited number is a conserved quantity in this system since $[H,\hat{N}_{e}]=0$, where the excited number operator is defined by $\hat{N}_{e}=\sum_{k=1}^{3}(|e\rangle_{kk}\langle e|+|1\rangle_{kk}\langle 1|)+\hat{a}^{\dagger}\hat{a}+\hat{b}^{\dagger}\hat{b}$. When the condition $\Omega_{k}\ll\lambda_{k}^{a}$ ($\lambda_{k}^{b}$) is satisfied, we can restrict the system dynamics into the single-excited subspace and the Hamiltonian is approximated by Shao et al. (2010)

	$\displaystyle H^{\prime}$	$\displaystyle=$	$\displaystyle\Big{[}\frac{\sqrt{3}}{3}(1+\epsilon_{1})\Omega_{1}\exp(i\alpha_{1})|\Psi_{1}\rangle\langle\Psi_{3}|$		(32)
			$\displaystyle+\frac{\sqrt{6}}{3}(1+\epsilon_{2})\Omega_{2}\exp(i\beta_{1})|\Psi_{3}\rangle\langle\Psi_{2}|+\mathrm{H.c.}\Big{]},~{}~{}$

where we set $\Omega_{3}=\Omega_{2}$, $\alpha_{2}=\alpha_{3}=\beta_{1}$, $\epsilon_{3}=\epsilon_{2}$, and

	$\displaystyle|\Psi_{1}\rangle$	$\displaystyle=$	$\displaystyle{1}/{\sqrt{2}}\big{(}|123\rangle-|132\rangle\big{)},$
	$\displaystyle|\Psi_{2}\rangle$	$\displaystyle=$	$\displaystyle{1}/{2}\big{(}|231\rangle-|213\rangle+|312\rangle-|321\rangle\big{)},$
	$\displaystyle|\Psi_{3}\rangle$	$\displaystyle=$	$\displaystyle{1}/{\sqrt{6}}\big{(}|e23\rangle\!-\!|2e3\rangle\!-\!|32e\rangle\!+\!|23e\rangle\!+\!|3e2\rangle\!-\!|e32\rangle\big{)}.$

The state $|lmn\rangle$ represents the first atom in $|l\rangle$, the second atom in $|m\rangle$, and the third atom in $|n\rangle$. Since the cavity mode $a$ ($b$) is in vacuum state, we have ignored it in the Hamiltonian given by Eq. (32).

It is easily found from Eq. (32) that the atom-cavity system can be reduced to the three-level physical model studied in Sec. II. Hence, if the initial state of this system is $|\Psi_{1}\rangle$, by fixing the ratio of two coupling coefficients and modulating the phases according to the above composite pulses theory, one can obtain the following superposition state

$\displaystyle|\Psi\rangle=\cos\theta|\Psi_{1}\rangle+e^{-i\gamma}\sin\theta|\Psi_{2}\rangle.$

(33)

By setting $\gamma=0$ and $\theta=\arctan\sqrt{2}$, the state $|\Psi\rangle$ becomes

	$\displaystyle|\Psi_{S}\rangle$	$\displaystyle=$	$\displaystyle 1/\sqrt{3}|\Psi_{1}\rangle+\sqrt{6}/3|\Psi_{2}\rangle$
		$\displaystyle=$	$\displaystyle{1}/{\sqrt{6}}\big{(}|123\rangle\!-\!|132\rangle\!+\!|231\rangle\!-\!|213\rangle\!+\!|312\rangle\!-\!|321\rangle\big{)},$

which is actually the three-atom singlet state.

We first explore how the deviations of two coupling constants $\lambda_{k}^{a}$ and $\lambda_{k}^{b}$ affect the fidelity $F_{S}$ of the singlet state. The numerical results simulated by the three-pulse sequence scheme are shown in Fig. 7(a), where we choose $\zeta_{a},\zeta_{b}\in[0,1]$ to guarantee the validity of the effective Hamiltonian given by Eq. (32) (i.e., the condition $\Omega_{k}\ll\lambda_{k}^{a},\lambda_{k}^{b}$ is well satisfied). As a comparison, we also plot in Fig. 7(b) the fidelity $F_{S}$ as a function of the deviations $\zeta_{a}$ and $\zeta_{a}$ by the resonant pulses scheme shown in Ref. Shao et al. (2010). The results demonstrate that both schemes are robust against the deviations in two coupling constants, since the impact on the fidelity $F_{S}$ can be negligible no matter how large the deviations are, cf., $F_{S}\geq 0.9978$ in Fig. 7. The reason for this phenomenon is that both $\zeta_{a}$ and $\zeta_{b}$ are absent in the Hamiltonian given by Eq. (32). Besides, an interesting finding is that the fidelity $F_{S}$ tends to a higher value when the deviations $\zeta_{a}$ and $\zeta_{b}$ are much larger. A reasonable interpretation is given as follows. During the process of deriving the Hamiltonian given by Eq. (32), the strong coupling condition $\Omega_{k}\ll\lambda_{k}^{a}$ ($\lambda_{k}^{b}$) is exploited. The coupling constant $\lambda_{k}^{a}$ ($\lambda_{k}^{b}$) increases with the raising of the deviation $\zeta_{a}~{}(\zeta_{b})$. As a result, the dynamics of this system is more and more restricted in the single-excited subspace so that the Hamiltonian given by Eq. (32) ensures a better validity for suppressing the leakage to other subspaces. Thus, a higher fidelity of the singlet state can be achieved when a larger deviation $\zeta_{a}~{}(\zeta_{b})$ appears.

Next, we address the influence of the two deviations $\epsilon_{1}$ and $\epsilon_{2}$ on the fidelity $F_{S}$. In Fig. 8(a), we display the fidelity $F_{S}$ versus the deviations $\epsilon_{1}$ and $\epsilon_{2}$ for the three-pulse sequence. For comparison, we also plot in Fig. 8(b) the performance of the fidelity for the resonant pulses scheme shown in Ref. Shao et al. (2010). As shown by the area surrounded by the blue-solid (or pink-dotted) curve in Fig. 8(b), it is clear that the scheme shown in Ref. Shao et al. (2010) is highly susceptible to the deviations $\epsilon_{1}$ and $\epsilon_{2}$. However, the fidelity of the singlet state obtained by the three-pulse sequence remains in a high value ($F_{S}\geq 0.99$) even when a very large deviation occurs. This benefits from the fact that the second-order coefficients of systematic errors are restricted to an extremely low value by the specific set of phases. Furthermore, as the pulse number increases, the higher-order coefficients can be restricted to a low value that ensures a robust behavior against the deviations $\epsilon_{1}$ and $\epsilon_{2}$. Thus, the CPs sequence can further improve an error-tolerant performance of the singlet state.

Another experimental issue, which inevitably impacts the performance of the final fidelity, is the waveform distortion. In the ideal condition, our input pulse shape is associated with a perfect square waveform. However, in practice, the input square wave always produces a tiny smooth rising and falling edge. Let us study this issue by taking the three-pulse sequence as an example. First of all, we design a group of functions in which the waveform distortion can be arbitrarily adjustable. The functions are given by

$$\alpha(t)\!=\!\left\{\begin{aligned} \displaystyle\!&\alpha_{2}-\frac{\alpha_{2}-\alpha_{1}}{1+\exp[\chi(t\!-\!T)]},~{}~{}~{}0\leq t\leq 3T/2,\\[6.45831pt] \displaystyle\!&\alpha_{3}-\frac{\alpha_{3}-\alpha_{2}}{1+\exp[\chi(t\!-\!2T)]},~{}3T/2<t\leq 3T,\\ \end{aligned}\right.$$

(34)

$$\beta(t)\!=\!\left\{\begin{aligned} \displaystyle\!&\beta_{2}-\frac{\beta_{2}-\beta_{1}}{1+\exp[\chi(t\!-\!T)]},~{}~{}~{}0\leq t\leq 3T/2,\\[6.45831pt] \displaystyle\!&\beta_{3}-\frac{\beta_{3}-\beta_{2}}{1+\exp[\chi(t\!-\!2T)]},~{}3T/2<t\leq 3T,\\ \end{aligned}\right.$$

(35)

where $T=\pi/\sqrt{\Omega_{1}^{2}+\Omega_{2}^{2}}$. Here, $\chi$ is a dimensionless parameter that defines the magnitude of the rising and falling edge of the square wave.

As $\chi$ increases, the functions (34) and (35) gradually become a square wave. Specifically from Fig. 9(a) and 9(b), we see that the functions are almost square waveforms for $\chi$=1000 (the blue-dotted curves). For $\chi\rightarrow\infty$, they describe perfect square waves. However, when $\chi$ is small, e.g., $\chi$=10, the duration of the rising and falling edge of the square wave becomes excessively long and the waveform has a serious shape change. Then, functions (34) and (35) turn into a smooth time-dependent pulse, as shown by the red-dashed curves in Fig. 9(a) and 9(b). Obviously, this kind of waveform does not satisfy the request of our CPs scheme. As illustrated in Fig. 9(c), even though the waveform suffers from serious distortion (the red-dashed curve), the fidelity $F_{S}$ can still maintain a high value ($>$ 0.99). Moreover, as revealed by the cyan-solid and pink-dashed-dotted curves in Fig. 9(c), a slight distortion in the waveform hardly makes an impact on the final fidelity. Hence, the CPs sequence has the ability to acquire a precise and robust singlet state even with a distorted waveform. Finally, we investigate the influence of the phase shift errors on the fidelity. In presence of the deviations in the phases, the actual phases can be rewritten as

$\displaystyle\alpha^{a}(t)=(1+\epsilon_{\alpha})\alpha(t),~{}~{}~{}~{}~{}~{}\beta^{a}(t)=(1+\epsilon_{\beta})\beta(t),$

(36)

where $\epsilon_{\alpha}$ and $\epsilon_{\beta}$ represent the phase shift errors. We plot in Fig. 9(d) the relation between the fidelity and the phase shift errors, which shows that the current CPs sequence is sensitive to phase shift errors Torosov et al. (2021). The reason is that we do not consider the phase shift errors in the Hamiltonian given by Eq. (3). To tackle this problem, one requires to regard the phase shift errors as variables in Eq. (3), and then constructs a phase-error-corrected composite pulses sequence Torosov and Vitanov (2019c).

In a more realistic situation, one has to consider the influence of decoherence on the fidelity of the singlet state. In this system, the decoherence mainly comes from two aspects: ($i$) the atomic spontaneous emission from the excited state to the ground states. ($ii$) the decay of two cavity modes. Under the decoherence environment, the dynamics of this system are governed by the Lindblad master equation and the form can be written as

$\displaystyle\dot{\rho}\!=\!-i[H,\rho]\!+\!\sum_{k,l=1}^{3}\!\!\gamma_{k}^{le}\mathcal{L}(\hat{\sigma}_{k}^{le})\rho\!+\!\kappa_{a}\mathcal{L}(\hat{a})\rho\!+\!\kappa_{b}\mathcal{L}(\hat{b})\rho,~{}~{}$

(37)

where $\hat{\sigma}_{k}^{le}=|l\rangle_{kk}\langle e|$ and the Lindblad super-operator is

$\displaystyle\mathcal{L}(\hat{o})\rho=\hat{o}\rho\hat{o}^{{\dagger}}-\frac{1}{2}(\hat{o}^{{\dagger}}\hat{o}\rho+\rho\hat{o}^{{\dagger}}\hat{o}),$

(38)

where $\hat{o}$ is the standard Lindblad operator. $\gamma_{k}^{le}$ and $\kappa_{a}$ ($\kappa_{b}$) are the dissipation rate from the excited state $|e\rangle$ to the ground state $|l\rangle$ and the decay rate of the cavity mode $a$ ($b$), respectively. For simplicity, we set $\gamma_{k}^{le}=\gamma/3$ and $\kappa_{a}=\kappa_{b}=\kappa$.

Figure 10 displays the fidelity $F_{S}$ as a function of the dissipation rate $\gamma$ and the decay rate $\kappa$ by both the three-pulse sequence scheme and the resonant pulses scheme shown in Ref. Shao et al. (2010). We observe from Fig. 10(b) that the fidelity cannot maintain a relatively high value in the resonant pulses scheme even if the system does not suffer from the decoherence, cf., $F_{S}<0.956$ in Fig. 10(b). However, in the three-pulse sequence scheme, the fidelity can reach a high value ($\geq 0.99$) when the dissipation rate is small, cf., the left region of the pink-dotted line in Fig. 10(a). Another intriguing aspect in Fig. 10 is that the fidelity does not sharply drop in the resonant pulses scheme while the range of fidelity varies widely in the three-pulse sequence scheme. This is because the evolution time in the resonant pulses scheme is much shorter than that in the three-pulse sequence scheme. As a result, the influence of the decoherence on the system is relatively small by the resonant pulses scheme. From this point, we find that the longer pulse sequence is not the optimal choice in a severe decoherence environment, even though the longer pulse sequence can help to improve the robustness of the system. Furthermore, according to the pink-dotted line in Fig. 10(a), it is seen that the influence of the cavity decay is almost negligible, since $F_{S}\geq 0.99$ even when the decay rate $\kappa$ increases to 0.1. This phenomenon can be explained by the fact that the strong coupling constant $\lambda_{k}^{a}(\lambda_{k}^{b})$ ensures the decoupling of the single-photon state to minimize the loss of the cavity decay. From a physical point of view, the cavity is regarded as intermediary to realize indirect coupling between atoms, and it is almost in the vacuum state under the strong coupling condition.