Holonomic swap and controlled-swap gates of neutral atoms via selective Rydberg pumping

Abstract

Holonomic quantum computing offers a promising paradigm for quantum computation due to its error resistance and the ability to perform universal quantum computations. Here, we propose a scheme for the rapid implementation of a holonomic swap gate in neutral atomic systems, based on the selective Rydberg pumping mechanism. By employing time-dependent soft control, we effectively mitigate the impact of off-resonant terms even at higher driving intensities compared to time-independent driving. This approach accelerates the synthesis of logic gates and passively reduces the decoherence effects. Furthermore, by introducing an additional atom and applying the appropriate driving field, our scheme can be directly extended to implement a three-qubit controlled-swap gate. This advancement makes it a valuable tool for quantum state preparation, quantum switches, and a variational quantum algorithm in neutral atom systems.

1 Introduction

Quantum computers, which use the best algorithms currently known, offer the possibility of solving certain computational tasks much more effectively than any classical counterpart [1–5]. This has inspired a great deal of searches for building scalable and functional quantum computers over the past two decades. Realizing a universal set of quantum gates with high fidelities is the key to implementing quantum computation. However, errors in the control process of a quantum system inevitably affect the quantum gate, and the propagation of these inaccurate control errors may quickly spoil the practical realization. Due to the fault tolerance for some types of errors in the control process, holonomic quantum computation (HQC), first proposed by Zanardi and Rasetti [6], is one of the well-known strategies to improve gate robustness by using non-abelian geometric phases [7].

Early HQC schemes were based on adiabatic evolution [6, 8, 9]. In this case, one obstacle is the long running time required for adiabatic evolution, which makes the gates vulnerable to open system effects and parameter fluctuations that may hinder the experimental implementation. To overcome this shortcoming, nonadiabatic HQC based on nonadiabatic non-abelian geometric phases [10] was proposed [11], which has become a promising quantum computation paradigm and has attracted increasing attention recently [12–26]. Meanwhile, nonadiabatic HQC has been experimentally demonstrated on various physical platforms, such as superconducting transmons [27–29], the nuclear magnetic resonance system [30–32], and the diamond nitrogen vacancy center [33–37].

As highly promising candidates for quantum computation and simulation, Rydberg atoms exhibit remarkable attributes that make them particularly attractive for these applications. One of their key advantages is their long coherence time for ground-state atoms, combined with the exceptional properties of highly excited Rydberg states. These highly excited Rydberg states not only possess an extended lifetime proportional to the third power of the principal quantum number, but also significantly interact through strong long-range Rydberg-Rydberg interactions, manifesting as Rydberg-mediated dipole-dipole or van der Waals interactions [38–42]. The presence of these strong Rydberg-Rydberg interactions enables a phenomenon known as Rydberg blockade, where the resonant excitation of two or more atoms to the Rydberg states is hindered [43, 44]. Rydberg blockade has been proven instrumental in the effective implementation of quantum logic gates, as experimentally demonstrated with individual atom [45, 46]. Another representative phenomenon observed in neutral atom systems is the Rydberg antiblockade, allowing for a resonant two-photon transition with the energy shift of the Rydberg pair states compensated for by the two-photon detuning [47–53]. Taking advantage of the unique characteristics of Rydberg atoms, they are exceptionally well suited for qubit encoding and serve as an excellent medium in the field of quantum computing and quantum simulation [52, 54–77].

In contrast to conventional Rydberg blockade or Rydberg antiblockade, the selective Rydberg pumping (SRP) mechanism provides a novel approach to selectively exciting the target quantum states of neutral atoms while effectively freezing the evolution of nontarget quantum states [78]. The SRP mechanism capitalizes on the synergistic effects of multifrequency driving fields and strong dipole-dipole interactions, wherein the first-order Rabi coupling supersedes the traditional second-order dynamics required by Rydberg antiblockade. The advantage of adopting first-order Rabi coupling lies in its typically much larger magnitude compared to the second-order interaction. Consequently, the SRP-based scheme enables significant acceleration, while concurrently reducing the attenuation of other excited Rydberg states.

In this work, we aim to implement a holonomic swap and controlled-swap gates in neutral atomic systems, using the SRP mechanism [78]. However, the previous SRP mechanism featured a time-independent interaction strength. The coupling coefficient, which governs the system’s time evolution, must be sufficiently small to achieve improved addressing of resonant terms and efficient suppression of off-resonant interactions. To overcome these challenges, we propose employing Gaussian time-dependent soft control in the SRP mechanism [79, 80]. This approach effectively mitigates the impact of off-resonant terms even at higher driving intensities compared to time-independent driving. As a result, the synthesis of logic gates is accelerated, and the decoherence effects are greatly reduced.

2 Holonomic two-qubit swap gate

Here we consider a system of two identical ⁸⁷Rb atoms, and each atom consists of three long-lived Rydberg states $|p\rangle =|61 P_{1/2},m_{J}=1/2\rangle$, $|d\rangle =|59 D_{3/2},m_{J}=3/2\rangle$, and $|f_{1}\rangle =|57 F_{5/2},m_{J}=5/2\rangle$, and two ground states $|0\rangle =|5 S_{1/2} F=1,m_{F}=1\rangle$ and $|1\rangle =|5S_{1/2},F=2,m_{F}=2\rangle$ as shown in Fig. 1. A laser field with Rabi frequency $\Omega _{1}$ is applied to drive the transition $|0\rangle \leftrightarrow |d\rangle$ with a blue detuning Δ on the order of several hundred megahertz. At the same time, the ground state $|1\rangle$ is dispersively coupled to the excited state $|d\rangle$ by one laser field with Rabi frequency $\Omega _{2}$ (which has a red detuning Δ) and another resonant laser field with Rabi frequency Ω, simultaneously. As demonstrated in Refs. [39, 81–83], the pair states $|dd\rangle$, $|pf_{1}\rangle$, and $|f_{1}p\rangle$ exhibit nearly degenerate characteristics. Consequently, the resonant dipole-dipole (Förster resonance) interaction between the two Rydberg atoms causes a hopping transition between the Rydberg states $|dd\rangle$ and $(|pf_{1}\rangle +|f_{1}p\rangle )/\sqrt{2}$, with a coupling strength of $\sqrt{2}J$. The Hamiltonian of the two-atom system in the interaction picture reads ($\hbar = 1$)

$$ H_{\mathrm{full}} =\sum_{k=1}^{2} \bigl[\Omega _{1}e^{i\Delta t} \vert 0 \rangle _{k} \langle d \vert +\bigl(\Omega +\Omega _{2}e^{-i\Delta t}\bigr) \vert 1\rangle _{k} \langle d \vert \bigr]+J|dd \rangle \bigl(\langle pf_{1} \vert +\langle f_{1}p \vert \bigr)+ \mathrm{H.c.}, $$

(1)

where $J=C_{3}/R^{3}$ with $C_{3}/(2\pi )=\mathrm{2.54~GHz~\mu m^{3}}$ and R represents the distance between the atoms. Due to the strong dipole-dipole interaction between the two Rydberg atoms, performing a rotation with the frame defined by $\exp [-\sqrt{2}iJt(|E_{+}\rangle \langle E_{+}|-|E_{-}\rangle \langle E_{-}|)]$, where $|E_{\pm}\rangle =[\sqrt{2}|dd\rangle \pm (|pf_{1}\rangle +|f_{1}p \rangle )]/2$ are the eigenstates of the Rydberg dipole-dipole interaction with the eigenvalues $\pm \sqrt{2} J$, respectively, we obtain the following transformed Hamiltonian

$$\begin{aligned}& H_{{\mathrm{full}}}= H_{1}+H_{2}, \\& \begin{aligned} H_{1}&=\sqrt{2}\Omega _{S} \vert 00\rangle \langle T_{0} \vert e^{i\Delta t}+ \frac{1}{\sqrt{2}} \vert 01\rangle \bigl[\bigl(\langle T_{0} \vert -\langle S_{0} \vert \bigr) \bigl( \Omega +\Omega _{S}e^{-i\Delta t} \bigr)+\langle d1|\sqrt{2}\Omega _{S}e^{i \Delta t} \bigr] \\ &\quad{} +\frac{1}{\sqrt{2}} \vert 10\rangle \bigl[\bigl(\langle T_{0} \vert +\langle S_{0} \vert \bigr) \bigl( \Omega + \Omega _{S}e^{-i\Delta t}\bigr)+\langle 1d|\sqrt{2}\Omega _{S}e^{i \Delta t} \bigr] \\ &\quad{} +\sqrt{2}\bigl(\Omega +\Omega _{S}e^{-i\Delta t}\bigr) \vert 11 \rangle \langle T_{1} \vert + \mathrm{H.c.} , \end{aligned} \\& \begin{aligned} H_{2}&=\Omega _{S} \vert T_{0}\rangle \bigl[\langle E_{+} \vert e^{i(\Delta - \sqrt{2}J)t}+\langle E_{-} \vert e^{i(\Delta +\sqrt{2}J)t} \bigr]+ \vert T_{1} \rangle \bigl[\langle E_{+} \vert \bigl( \Omega e^{-i\sqrt{2}J t} \\ &\quad{} +\Omega _{S}e^{-i(\Delta +\sqrt{2}J) t}\bigr)+\langle E_{-} \vert \bigl(\Omega e^{i \sqrt{2}J t}+\Omega _{S} e^{-i(\Delta -\sqrt{2}J) t} \bigr) \bigr]+ \mathrm{H.c.}, \end{aligned} \end{aligned}$$

(2)

where we have assumed all the Rabi frequencies are real and set $\Omega _{1}=\Omega _{2}=\Omega _{S}$ for simplicity. $|T_{0}(S_{0})\rangle =(|d0\rangle \pm |0d\rangle )/\sqrt{2}$ and $|T_{1}\rangle =(|d1\rangle +|1d\rangle )/\sqrt{2}$. $H_{1}$ describes the transitions between the ground states and the Rydberg states with a single excitation, and $H_{2}$ bridges the interaction between the Rydberg states with a single excitation and two excitations. Here we consider the large detuning case, such as $\Delta \gg \{\Omega _{S},\Omega \}$ and $\Delta =\sqrt{2}J$, in this case the terms oscillating with high frequencies $\{\pm (\Delta +\sqrt{2}J),\pm \Delta ,\pm \sqrt{2}J\}$ in Eq. (2) can be safely disregarded (see Appendix A for details), then the Hamiltonian can be evaluated explicitly

$$\begin{aligned} H =&\frac{1}{\sqrt{2}}\Omega \bigl( \vert 10\rangle - \vert 01 \rangle \bigr)\langle S_{0} \vert + \frac{1}{\sqrt{2}}\Omega \bigl( \vert 10\rangle + \vert 01\rangle \bigr)\langle T_{0} \vert + \sqrt{2}\Omega \vert 11\rangle \langle T_{1} \vert +\Omega _{S} \vert T_{0} \rangle \langle E_{+} \vert \\ &{}+\Omega _{S} \vert T_{1}\rangle \langle E_{-} \vert +\mathrm{H.c.}. \end{aligned}$$

(3)

Figure 1

Schematic view of the atomic-level configuration. Three types of laser fields are applied to drive each atom. One laser field with Rabi frequency $\Omega _{1}$ is applied to drive the transition $|0\rangle \leftrightarrow |d\rangle$ with a blue detuning Δ. At the same time, the ground state $|1\rangle$ is dispersively coupled to the excited state $|d\rangle$ by one laser field with Rabi frequency $\Omega _{2}$ (which has a red detuning Δ) and another resonant laser field with Rabi frequency Ω, simultaneously

In Ref. [79] soft temporal quantum control, which enables on-resonant coupling within a desired set of target systems and efficiently avoids unwanted off-resonant contributions coming from others, has been proposed. Here, we use the technique of soft quantum control and choose the Rabi frequency Ω as a time-dependent Gaussian form, $\Omega (t)=\Omega _{m}\exp [-(t-2T)^{2}/T^{2}]$, where $\Omega _{m}$ and T are the maximum amplitude and width of the Gaussian pulse, respectively. The Hamiltonian H in Eq. (3) can be rewritten as

$$ H =H_{\mathrm{S}}+H_{\mathrm{int}}, $$

(4)

where

$$\begin{aligned}& H_{\mathrm{S}} =\Omega _{S} \vert T_{0}\rangle \langle E_{+} \vert +\Omega _{S} \vert T_{1} \rangle \langle E_{-} \vert +\mathrm{H.c.}, \\& H_{\mathrm{int}} =\frac{1}{\sqrt{2}}\Omega (t) \bigl( \vert 10\rangle - \vert 01 \rangle \bigr)\langle S_{0} \vert + \frac{1}{\sqrt{2}}\Omega (t) \bigl( \vert 10\rangle + \vert 01 \rangle \bigr)\langle T_{0} \vert +\sqrt{2}\Omega (t) \vert 11 \rangle \langle T_{1} \vert + \mathrm{H.c.}. \end{aligned}$$

Under the eigenvalues $\omega _{j}$ and the corresponding projection operators $\mathbb{P}(\omega _{j})$ of $H_{\mathrm{S}}$, $H_{\mathrm{S}}$ is reformulated in the diagonal form as $H_{\mathrm{S}}^{1}=\sum_{j}\omega _{j}\mathbb{P}(\omega _{j})$. Meanwhile, $H_{\mathrm{int}}$ under the projection operators $\mathbb{P}(\omega _{j})$ is

$$ H_{\mathrm{int}}^{1}=\sum_{j,k} \mathbb{P}(\omega _{j})H_{ \mathrm{int}}\mathbb{P}(\omega _{k}). $$

(5)

Our purpose is to suppress the terms with energy mismatches in Eq. (5) for which $\omega _{j}\neq \omega _{k}$, and to keep the energy conserving ones for which $\omega _{j}=\omega _{k}$ by adopting the time-dependent control.

We first analyze the propagator $U_{D}=\exp (-i\int _{0}^{4T}H_{D}\,dt)$, where $H_{D}=H_{{\mathrm{S}}}^{1}+\sum_{j}\mathbb{P}(\omega _{j}) H_{ \mathrm{int}}\mathbb{P}(\omega _{j})$ includes the desired resonance interactions. It is easy to verify that in the latter all $\mathbb{P}(\omega _{j})H_{\mathrm{int}}\mathbb{P}(\omega _{j})$ operators commute with $H_{{\mathrm{S}}}^{1}$, so $H_{D}$ can be diagonalized in the common eigenstates $|\psi ^{D}_{j}\rangle$ of $H_{{\mathrm{S}}}^{1}$ and $\mathbb{P}(\omega _{j})H_{\mathrm{int}}\mathbb{P}(\omega _{j})$. Therefore $U_{D}= \sum_{j} e^{-i\phi ^{D}_{j}(4T)}|\psi _{j}^{D} \rangle \langle \psi _{j}^{D} |$ is also diagonal in the basis ${|\psi _{j}^{D} \rangle}$ and the dynamic phases $\phi ^{D}_{j}(4T)$ include the effect of energy shifts from $\mathbb{P}(\omega _{j})H_{\mathrm{int}}\mathbb{P}(\omega _{j})$.

If the whole Hamiltonian in Eq. (4) under the projection operators $\mathbb{P}(\omega _{j})$, $H^{1}=H_{{\mathrm{S}}}^{1}+H_{\mathrm{int}}^{1}$, is considered, the time evolution operator $U=\mathbf{T}\exp (-i\int _{0}^{4T}H^{1}\,dt)$ with T being time ordering is generally non-diagonal in the basis ${|\psi _{j}^{D} \rangle}$ and the non-commuting terms $\mathbb{P}(\omega _{j})H_{\mathrm{int}}\mathbb{P}(\omega _{k})$ ($j \neq k$) would cause unwanted transitions between the different states $|\psi _{j}^{D} \rangle$.

However, when the soft control is included one can efficiently eliminate the unwanted interactions caused by $\mathbb{P}(\omega _{j})H_{\mathrm{int}}\mathbb{P}(\omega _{k})$ ($j \neq k$) even for long evolution times. At the boundaries of the interaction times (0 and 4T), $\Omega (t)$ has negligible values, and therefore the eigenstates of the whole system coincide with those of $H_{D}$. More precisely, under the condition of adiabatic evolution [84, 85], there are no transitions among the states $|\psi _{j}^{D} \rangle$, and the propagator at the end of the evolution is

$$ U\approx \sum_{j} e^{-i\phi _{j}(4T)} \bigl\vert \psi _{j}^{D} \bigr\rangle \bigl\langle \psi _{j}^{D} \bigr\vert \equiv \bar{U}\equiv e^{-4i\bar{H}T}, $$

(6)

where $\phi _{j}(4T)$ are the dynamic phases of $H^{1}$, while the geometric phases vanish because $\Omega (t)$ returns to its original value [86]. In this manner U takes the same form as $U_{D}$ and the adiabatic average Hamiltonian for the soft quantum control scheme is

$$ \bar{H}=\sum_{n} \frac{\phi _{j}(4T)}{4T} \bigl\vert \psi _{j}^{D} \bigr\rangle \bigl\langle \psi _{j}^{D} \bigr\vert . $$

(7)

Using this method, the Hamiltonian of our current model reduces to an average form (see Appendix B for details)

$$\begin{aligned} \bar{H} =&\frac{g}{\sqrt{2}}\bigl( \vert 10\rangle - \vert 01 \rangle \bigr)\langle S_{0} \vert + \frac{g}{\sqrt{2}}|S_{0} \rangle \bigl(\langle 10 \vert -\langle 01 \vert \bigr) \\ & +\frac{1}{4T} \int _{0}^{4T}\sqrt{\Omega (t)^{2}+\Omega _{S}^{2}} \,dt \bigl( \bigl\vert \psi ^{D}_{1}\bigr\rangle \bigl\langle \psi ^{D}_{1} \bigr\vert - \bigl\vert \psi ^{D}_{2}\bigr\rangle \bigl\langle \psi ^{D}_{2} \bigr\vert \bigr) \\ &+\frac{1}{4T} \int _{0}^{4T}\sqrt{2\Omega (t)^{2}+\Omega _{S}^{2}} \,dt \bigl( \bigl\vert \psi ^{D}_{3}\bigr\rangle \bigl\langle \psi ^{D}_{3} \bigr\vert - \bigl\vert \psi ^{D}_{4}\bigr\rangle \langle \psi ^{D}_{4} \bigr\vert \bigr), \end{aligned}$$

(8)

where $g=\sqrt{\pi}\Omega _{m}\mathrm{Erf}[2]/4$. $|\psi ^{D}_{1}\rangle =(|T_{0}\rangle +|E_{+}\rangle )/\sqrt{2}$, $|\psi ^{D}_{2}\rangle =(|T_{0}\rangle -|E_{+}\rangle )/\sqrt{2}$, $|\psi ^{D}_{3}\rangle =(|T_{1}\rangle +|E_{-}\rangle )/\sqrt{2}$, and $|\psi ^{D}_{4}\rangle =(|T_{1}\rangle -|E_{-}\rangle )/\sqrt{2}$ are the eigenstates of $H_{\mathrm{S}}$ governed by $\Omega _{S}$.

In the following, the associated propagator $U=e^{-4i\bar{H}T}$ with the evolution period $\tau =4T$ can be used to generate a high-fidelity two-qubit swap gate. The time evolution operator U in the basis $\{|S_{0}\rangle ,|00\rangle ,|01\rangle ,|10\rangle ,|11\rangle \}$ reads

$$\begin{aligned} U(\tau )= \begin{pmatrix} \cos \lambda _{\tau} & 0 & \frac{\sqrt{2}i}{2}\sin \lambda _{\tau} &- \frac{\sqrt{2}i}{2}\sin \lambda _{\tau} & 0 \\ 0& 1& 0& 0& 0 \\ \frac{\sqrt{2}i}{2}\sin \lambda _{\tau} &0 & \frac{1}{2}(1+\cos \lambda _{\tau}) &\frac{1}{2}(1-\cos \lambda _{\tau}) & 0 \\ -\frac{\sqrt{2}i}{2}\sin \lambda _{\tau} &0 & \frac{1}{2}(1-\cos \lambda _{\tau}) & \frac{1}{2}(1+\cos \lambda _{\tau}) &0 \\ 0 & 0 &0 &0 & 1 \end{pmatrix}, \end{aligned}$$

where $\lambda _{\tau}=\tau g=4Tg$. Choosing the parameters to satisfy that

$$ \lambda _{\tau}=4Tg=\pi , $$

(9)

one can derive $T=\sqrt{\pi}/(\mathrm{Erf}[2]\Omega _{m})$. The final effective evolution operator $U(\tau )$ is

$$\begin{aligned} U(\tau )= \begin{pmatrix} -1 & 0 & 0 &0 & 0 \\ 0& 1& 0& 0& 0 \\ 0 &0 & 0 &1 & 0 \\ 0 &0 & 1 & 0 &0 \\ 0 & 0 &0 &0 & 1 \end{pmatrix}, \end{aligned}$$

(10)

which is a two-qubit swap gate on the computational subspace ${\mathbf {S}}=\mathrm{Span}\{|00\rangle ,|01\rangle ,|10\rangle , |11 \rangle \}$ as follows

$$\begin{aligned} U_{\mathrm{swap}}= \begin{pmatrix} 1& 0& 0& 0 \\ 0 & 0 &1 & 0 \\ 0 & 1 & 0 &0 \\ 0 &0 &0 & 1 \end{pmatrix}. \end{aligned}$$

(11)

Next, we confirm that the effect of $U(\tau )$ on S is entirely holonomic. First, we briefly review the conditions of nonadiabatic HQC proposed in Refs. [11, 12]. Consider a N-dimensional quantum system with a Hamiltonian $H(t)$. Assume that there exists a time-dependent K-dimensional subspace $\mathbf {K}(t)$ spanned by a set of orthonormal bases $\{|\Phi _{m}(t)\rangle ,m=1,\ldots ,K\}$ at each time t. Here, $|\Phi _{m}(t)\rangle$ can be obtained by the Schrödinger equation

$$ \bigl|\Phi _{m}(t)\bigr\rangle =\mathbf{T}\exp \biggl[-i \int _{0}^{t}H\bigl(t' \bigr)\,dt'\biggr]\bigl|\Phi _{m}(0) \bigr\rangle =U(t)\bigl|\Phi _{m}(0)\bigr\rangle , $$

with T being timing ordering, and $m=1,\ldots ,K$. The unitary transformation $U(\tau )=\mathbf{T}\exp [-i\int _{0}^{\tau}H(t')\,dt']$ is a holonomy matrix acting on the K-dimensional subspace $\mathbf {K} (0)$ spanned by $\{|\Phi _{m}(0)\rangle \}_{m=1}^{K}$ if $|\Phi _{m}(t)\rangle$ satisfies the following two conditions:

$$ \begin{aligned} (&{\mathrm{I}})\quad \sum _{m=1}^{K} \bigl\vert \Phi _{m}(\tau ) \bigr\rangle \bigl\langle \Phi _{m}( \tau ) \bigr\vert =\sum _{m=1}^{K} \bigl\vert \Phi _{m}(0)\bigr\rangle \bigl\langle \Phi _{m}(0) \bigr\vert , \\ (&{\mathrm{II}})\quad \bigl\langle \Phi _{m}(t) \bigl\vert H(t) \bigr\vert \Phi _{n}(t)\bigr\rangle =0, \quad m,n=1, \ldots ,K. \end{aligned} $$

(12)

Condition $(\mathrm{I)}$ ensures that the states in the subspace complete a cyclic evolution, and condition $(\mathrm{II)}$ ensures that the cyclic evolution is purely geometric.

We check the conditions $(\mathrm{I)}$ and $(\mathrm{II)}$ for the unitary operator $U(\tau )$. Condition $(\mathrm{I)}$ is satisfied since the subspace spanned by $\{U(\tau )|00\rangle ,U(\tau )|01\rangle ,U(\tau )|10\rangle ,U( \tau )|11\rangle \}$ coincides with the subspace $\mathbf {S}=\mathrm{Span}\{|00\rangle ,|01\rangle ,|10\rangle ,|11 \rangle \}$. Furthermore, since H̄ commutes with its evolution operator $U(t)$, condition $(\mathrm{II)}$ reduces to $\langle k|\bar{H}|k'\rangle =0$, where $k,k'=\{00,01,10,11\}$. Thus, both conditions $(\mathrm{I)}$ and $(\mathrm{II)}$ are satisfied and $U(\tau )$ is a holonomic two-qubit swap gate in the subspace S. Figure 2 shows the temporal evolution of all ground states $\{|00\rangle ,|01\rangle ,|10\rangle ,|11\rangle \}$ obtained from the full Hamiltonian of Eq. (1). The left panel of Fig. 2 corresponds to the case for time-dependent control, $\Omega (t)=\Omega _{m}\exp [-(t-2T)^{2}/T^{2}]$ with $\Omega _{m}/(2\pi )=0.5\text{ MHz}$ and $T=\sqrt{\pi}/(\mathrm{Erf}[2]\Omega _{m})$, where the average value of $\Omega (t)$ is $\overline{\Omega (t)}/(2\pi ) =0.22\text{ MHz}$. Furthermore, for the case of time-independent driving, the time-independent coupling coefficient Ω, which governs the system’s time evolution, must be sufficiently small (e.g. $\Delta \gg \Omega _{S}\gg \Omega$ and $\Delta =\sqrt{2}J$) to achieve better addressing of resonant terms and efficient suppression of off-resonant interactions. Consequently, we set the time-independent driving $\Omega /(2\pi ) =0.1\text{ MHz}$ in the right panel of Fig. 2. For both cases, the other parameters take the same values, which are $\Omega _{S} /(2\pi ) =5\text{ MHz}$, $\Delta /(2\pi ) =500 \sqrt{2}\text{ MHz}$, and $J/(2\pi )= 500\text{ MHz}$. Comparing the left and right panels of Fig. 2, it is shown that time-dependent control allows us to effectively mitigate the impact of off-resonant terms even at higher driving intensities (e.g. $\overline{\Omega (t)}/(2\pi ) =0.22\text{ MHz}$ and $\Omega /(2\pi ) =0.1\text{ MHz}$), achieving this within a shorter time frame $\tau =2.267~\mu\text{s}$ compared to time-independent driving, which requires the evolution time being $\tau '=5~\mu\text{s}$.

Figure 2

The temporal evolution of populations for different ground states $\{|00\rangle ,|01\rangle ,|10\rangle ,|11\rangle \}$ corresponding to (a)-(d) governed by the full Hamiltonian Eq. (1). (Left) The time-dependent control parameter is $\Omega (t)=\Omega _{m}\exp [-(t-2T)^{2}/T^{2}]$ with $\Omega _{m}/(2\pi )=0.5\text{ MHz}$ and $T=\sqrt{\pi}/(\mathrm{Erf}[2]\Omega _{m})$, where the average value of $\Omega (t)$ is $\overline{\Omega (t)}/(2\pi ) =0.22\text{ MHz}$. (Right) The corresponding numerical simulations (e)-(f) under the time-independent driving $\Omega /(2\pi ) =0.1\text{ MHz}$. For both cases, the other parameters are the same as $\Omega _{S} /(2\pi ) =5\text{ MHz}$, $\Delta /(2\pi ) =500 \sqrt{2}\text{ MHz}$, and $J/(2\pi )= 500\text{ MHz}$

The fidelity of the two-qubit swap gate in the ideal case is $F=\langle \psi _{{\mathrm{ideal}}} |\rho (t)| \psi _{{\mathrm{ideal}}} \rangle =99.98\%$ in the evolution period $\tau =2.267~\mu\text{s}$ for the time-dependent control case, while for the time-independent driving case, the fidelity is $F=99.97\%$ at the evolution period $\tau '=5~\mu\text{s}$, where the initial state is $(|00\rangle + |01\rangle - |10\rangle + |11\rangle )/2$ and the ideal final state is $(|00\rangle - |01\rangle +|10\rangle + |11\rangle )/2$. Hence, by employing the time-dependent control, one can obtain a quantum gate with higher fidelity in a shorter time. Figure 3 shows the fidelity of the swap gate versus the parameters $\Omega _{m}$ and $\Omega _{S}$. An inspection of the plot reveals that the soft-coupling approach results in much higher fidelities in a wide range of parameters, even for strong coupling regimes and a wide range of evolution times. Thus, we can say in the SRP mechanism based on the technique of soft quantum control that the two-qubit swap gate is implemented with high and stable fidelity in a shorter time frame.

Figure 3

Fidelity versus the parameters $\Omega _{m}$ and $\Omega _{S}$ to the target evolution without unwanted coupling by using a Gaussian soft coupling $\Omega (t)=\Omega _{m}\exp [-(t-2T)^{2}/T^{2}]$. The other parameters are $\Delta /(2\pi ) =500 \sqrt{2}\text{ MHz}$, $J/(2\pi )= 500\text{ MHz}$, and $T=\sqrt{\pi}/(\mathrm{Erf}[2]\Omega _{m})$

3 Performance of the holonomic swap gate

3.1 Fluctuations of relevant parameters

In the above process, we have assumed the detuning $\Delta =\sqrt{2}J$, which can be challenging to achieve precisely in experiments. To assess the impact of deviations from the desired dipole-dipole interaction on the SRP mechanism, we consider $J/(2\pi )=(500+\Delta J)\text{ MHz}$ with the detuning parameter $\Delta /(2\pi ) =500\sqrt{2}\text{ MHz}$. In Fig. 4(a), the fidelity of the swap gate is plotted against the deviation ΔJ. Interestingly, the current SRP mechanism is shown to be insensitive to fluctuations in coupling strength J between two atoms, since the fidelity of the gate consistently remains above 99% in the continuous range of the coupling strength from $\Delta J=-1.78$ to $\Delta J=1.88$. Furthermore, in Fig. 4(b) we explore another practical scenario involving the presence of the Förster defect. In this case, the dipole-dipole coupling between the two Rydberg atoms in Eq. (1) is modified as $H_{dd}=J|dd\rangle (\langle pf_{1}|+\langle f_{1}p|)+\mathrm{H.c.}+ \delta (|pf_{1}\rangle \langle pf_{1}|+|f_{1}p\rangle \langle f_{1}p|)$, where δ represents the Förster defect. Surprisingly, the fidelity of the swap gate remains unaffected by the Förster defect, staying above 92% throughout the continuous range of $\delta /(2\pi )=-15$ to $\delta /(2\pi )=15\text{ MHz}$.

Figure 4

The effect of deviation ΔJ (a) and the Förster defect δ (b) on the fidelity of the swap gate based on the technique of soft quantum control. The time-dependent parameter is $\Omega (t)=\Omega _{m}\exp [-(t-2T)^{2}/T^{2}]$ with $\Omega _{m}/(2\pi )=0.5\text{ MHz}$ and $T=\sqrt{\pi}/(\mathrm{Erf}[2]\Omega _{m})$. The other parameters are $\Omega _{S} /(2\pi ) =5\text{ MHz}$, $\Delta /(2\pi )=500\sqrt{2}\text{ MHz}$, and $J/(2\pi )=(500+\Delta J)\text{ MHz}$

3.2 Influence of spontaneous emission of Rydberg states

When we consider the spontaneous emission of the Rydberg states, the Markovian master equation of the system in Lindblad form reads:

$$ \dot{\rho}=-i [H_{\mathrm{full}},\rho ]+\sum _{k=1}^{2}\sum_{l=d,p,f_{1}} \Biggl\{ \sum_{m=0}^{1} \gamma _{l}^{m} \mathcal{D}\bigl[ \vert m\rangle _{k} \langle l \vert \bigr] +\sum_{n=a_{1}}^{a_{n}} \gamma _{l}^{n}\mathcal{D}\bigl[ \vert n \rangle _{k} \langle l \vert \bigr]\Biggr\} , $$

(13)

where $\mathcal{D}[|m\rangle \langle l|]=[|m\rangle \langle l|\rho |l \rangle \langle m|-1/2(|l\rangle \langle l|\rho +\rho |l\rangle \langle l|)]$ and $\{|a_{1}\rangle ,\ldots , |a_{n}\rangle \}$ denotes the subspace consists of the external leakage levels out of $\{|0\rangle , |1\rangle \}$ [87, 88]. $\gamma _{l}^{m}$ is the branching ratio of the spontaneous decay rate from the state $|l\rangle$ to $|m\rangle$, which satisfies $\gamma _{l}=(\sum_{m=0}^{1} \gamma _{l}^{m}+\sum_{n=a_{1}}^{a_{n}} \gamma _{l}^{n})=1/\tau _{l}$. For the sake of simplifying the calculations, we let $\gamma _{l}^{m}=\gamma _{l}^{n}=\gamma _{l}/8$. The effective lifetimes of the Rydberg states $|p\rangle =|61 P_{1/2},m_{J}=1/2\rangle$, $|d\rangle =|59 D_{3/2},m_{J}=3/2\rangle$, and $|f_{1}\rangle =|57 F_{5/2},m_{J}=5/2\rangle$ of two ⁸⁷Rb atoms are $\tau _{p}=0.527\text{ ms}$, $\tau _{d}=0.215\text{ ms}$, and $\tau _{f_{1}}=0.127\text{ ms}$, respectively. In this case, the fidelity for the two-qubit swap gate is $F=\langle \psi _{{\mathrm{ideal}}} |\rho (t)| \psi _{{\mathrm{ideal}}} \rangle =99.85\%$, where the initial state is $(|00\rangle + |01\rangle - |10\rangle + |11\rangle )/2$ and the ideal final state is $(|00\rangle - |01\rangle +|10\rangle + |11\rangle )/2$. Accordingly, the swap gate is shown to be robust against decoherence.

4 Holonomic three-qubit controlled-swap gate

In this section, we implement a holonomic controlled-swap gate by introducing an additional atom as the control atom, which has many known applications, such as preparation of the quantum state [89–91], quantum switches [92–94], and a variational quantum algorithm [95, 96].

The interaction between two neutral atoms i and j separated by a distance $R_{ij}$ can be expressed, to the leading order, through the dipole-dipole interaction [97–101]:

$$ V_{dd} = \frac{1}{4\pi \varepsilon _{0}} \frac{\mathbf{d}_{i}\cdot \mathbf{d}_{j}-3(\mathbf{d}_{i}\cdot \mathbf{n}_{ij})(\mathbf{d}_{j}\cdot \mathbf{n}_{ij})}{R_{ij}^{3}}, $$

(14)

where $\mathbf{d}_{i}=(d_{x},d_{y},d_{z})$ is the electric dipole moment operator of atom i, and $\mathbf{n}_{ij} = \mathbf{R}_{ij}/R_{ij}$ is the unit vector connecting the two atoms from atoms i to j. We denote the quantization axis with z, and the angle between z and $\mathbf{n}_{ij}$ with $\theta _{ij}$. In the spherical basis, it is convenient to use the spherical dipole operators: $d_{i,0}=d_{i,z}$ and $d_{i,\pm}=\mp (d_{i,x}\pm id_{i,y})/\sqrt{2}$. The operator $d_{i,0}$ conserves the magnetic quantum number $m_{j}$, whereas the operators $d_{i,\pm}$ change $m_{j}$ by one ($\Delta m_{j}=\pm 1$). In the spherical basis, the dipole-dipole interaction can be written as:

$$\begin{aligned} V_{dd} =& \frac{1}{2}\sum _{i\neq j} \frac{1}{4\pi \varepsilon _{0}R_{ij}^{3}} \bigl[\mathcal{A}_{1}( \theta _{ij}) (d_{i,+}d_{j,-}+d_{i,-}d_{j,+}+2d_{i,z}d_{j,z})+ \mathcal{A}_{2}(\theta _{ij}) (d_{i,+}d_{j,z} \\ &{}-d_{i,-}d_{j,z}+d_{i,z}d_{j,+}-d_{i,z}d_{j,-})- \mathcal{A}_{3}( \theta _{ij}) (d_{i,+}d_{j,+}+d_{i,-}d_{j,-}) \bigr]. \end{aligned}$$

(15)

The operator $V_{dd}$ in Eq. (15) contains three terms with angular prefactors $\mathcal{A}_{1}(\theta _{ij})=(1-3\cos ^{2}\theta _{ij})/2$, $\mathcal{A}_{2}(\theta _{ij})=3\sin \theta _{ij} \cos \theta _{ij}/ \sqrt{2}$, and $\mathcal{A}_{3}(\theta _{ij})=3\sin ^{2}\theta _{ij}/2$, which couple pair states where the total magnetic quantum number $M=m_{1}+m_{2}$ changes by $\Delta M=0$, $\Delta M=\pm 1$, and $\Delta M=\pm 2$, respectively. Here, we consider the situation where no magnetic field is applied to the atoms. Coming back to Eq. (15), terms with the angular prefactor $\mathcal{A}_{1}(\theta _{ij})$ couple $|dd\rangle$ and the symmetric state $|pf_{1}\rangle _{\mathrm{s}}=(|pf_{1}\rangle +|f_{1}p\rangle )/\sqrt{2}$, with $|f_{1}\rangle =|57 F_{5/2},m_{J}=5/2\rangle$ ($\Delta M=0$). Terms with the angular prefactor $\mathcal{A}_{2}(\theta _{ij})$ couple $|dd\rangle$ and $|pf_{2}\rangle _{\mathrm{s}}=(|pf_{2}\rangle +|f_{2}p\rangle )/\sqrt{2}$, with $|f_{2}\rangle =|57 F_{5/2},m_{J}=3/2\rangle$ ($\Delta M=-1$). Finally, terms with the angular prefactor $\mathcal{A}_{3}(\theta _{ij})$ couple $|dd\rangle$ and $|pf_{3}\rangle _{\mathrm{s}}=(|pf_{3}\rangle +|f_{3}p\rangle )/\sqrt{2}$, with $|f_{3}\rangle =|57 F_{5/2},m_{J}=1/2\rangle$ ($\Delta M=-2$). Thus, for the case of three atoms, each atom consists of five long-lived Rydberg states $|d\rangle$, $|p\rangle$, $|f_{1}\rangle$, $|f_{2}\rangle$, and $|f_{3}\rangle$, and two ground states $|0\rangle$ and $|1\rangle$ shown in Fig. 5(b). Furthermore, the ground state $|0\rangle$ of the control atom is coupled to the excited state $|d\rangle$ by a laser field with Rabi frequency $\Omega _{2}$ (which has a red detuning Δ). In the absence of a magnetic field, we thus expect three resonances between $|dd\rangle$ and the states $|pf_{1}\rangle _{\mathrm{s}}$, $|pf_{2}\rangle _{\mathrm{s}}$, and $|pf_{3}\rangle _{\mathrm{s}}$. The dipole-dipole interaction for the three-atom model has a concise form

$$\begin{aligned} V_{dd} =& \sum_{i\neq j} \frac{1}{\sqrt{2}R_{ij}^{3}} \bigl[- \mathcal{A}_{1}(\theta _{ij})C_{3} \vert dd\rangle _{ij} \langle pf_{1} \vert _{ \mathrm{s}}+\mathcal{A}_{2}(\theta _{ij})C'_{3} \vert dd\rangle _{ij} \langle pf_{2} \vert _{ \mathrm{s}} \\ &{}+\mathcal{A}_{3}(\theta _{ij})C''_{3} \vert dd\rangle _{ij} \langle pf_{3} \vert _{ \mathrm{s}} \bigr]+\mathrm{H.c.}, \end{aligned}$$

(16)

where $C'_{3}/(2\pi )=1.61~{\text{GHz $\mu$m}}^{3}$ and $C''_{3}/(2\pi )=0.8~{\text{GHz $\mu$m}}^{3}$ [102, 103]. The arrangement of control atom 1 and target atoms 2 and 3 is shown in Fig. 5(a). We denote atoms 2 and 3 along the quantization z axis, so the angle between z and $\mathbf{n}_{23}$ is $\theta _{23}=0$. Meanwhile, the angle between z and $\mathbf{n}_{13}$ is $\theta _{13}=\theta$, and the angle between z and $\mathbf{n}_{12}$ is $\theta _{12}=\pi -\theta$. We can fix the angle $\theta \approx 55.80^{\circ}$ to make the eigenvalues of the electric dipole-dipole interaction between any pair of three atoms equal, given that $4\cos ^{3}\theta \sqrt{C^{2}_{3}(3\cos ^{2}\theta -1)^{2}+18C^{\prime \,2}_{3} \sin ^{2}\theta \cos ^{2}\theta +9C^{\prime\prime \,2}_{3}\sin ^{4}\theta}/C_{3}=1$.

Figure 5

Schematic view of the atomic-level configuration. (a) The arrangement of the control atom 1 and target atoms 2 and 3. (b) One laser field with the Rabi frequency $\Omega _{2}$ is applied to drive the transition $|0\rangle \leftrightarrow |d\rangle$ of the first atom (control atom) with a red detuning Δ. Three types of laser fields are applied to drive the second and third atoms. One laser field with the Rabi frequency $\Omega _{1}$ is applied to drive the transition $|0\rangle \leftrightarrow |d\rangle$ with a blue detuning Δ. Meanwhile, the ground state $|1\rangle$ is dispersively coupled to the excited state $|d\rangle$ by one laser field with Rabi frequency $\Omega _{2}$ (which has a red detuning Δ) and another resonant laser field with Rabi frequency Ω, simultaneously

The Hamiltonian of the three-atom system in the interaction picture reads

$$ H'_{\mathrm{full}} =\Omega _{2}e^{-i\Delta t} \vert 0\rangle _{1}\langle d \vert + \sum _{k=2}^{3} \bigl[\Omega _{1}e^{i\Delta t} \vert 0\rangle _{k}\langle d \vert +\bigl( \Omega + \Omega _{2}e^{-i\Delta t}\bigr) \vert 1\rangle _{k}\langle d \vert \bigr]+ \mathrm{H.c.}+{V}_{dd}. $$

(17)

Due to the strong dipole-dipole interaction strength between the Rydberg atoms, we make a rotation with respect to

$$ U_{\mathrm{rot}} =\exp \Biggl[-\sqrt{2}iJt\sum _{k=1}^{6}\bigl( \vert E_{k+} \rangle \langle E_{k+} \vert - \vert E_{k-}\rangle \langle E_{k-} \vert \bigr) \Biggr], $$

(18)

where

$$\begin{aligned}& |E_{1\pm}\rangle =\frac{1}{\sqrt{2}} \bigl( \vert 0dd \rangle \pm \vert 0pf_{1} \rangle _{\mathrm{s}} \bigr), \\& |E_{2\pm}\rangle =\frac{1}{\sqrt{2}} \biggl[|d0d\rangle \pm \frac{2}{C_{3}} \bigl(\mathcal{B}_{1} \vert p0f_{1} \rangle _{\mathrm{s}}+ \mathcal{B}_{2} \vert p0f_{2} \rangle _{\mathrm{s}}+\mathcal{B}_{3}|p0f_{3} \rangle _{\mathrm{s}} \bigr) \biggr], \\& |E_{3\pm}\rangle =\frac{1}{\sqrt{2}} \biggl[|dd0\rangle \pm \frac{2}{C_{3}} \bigl(\mathcal{B}_{1} \vert pf_{1}0 \rangle _{\mathrm{s}}- \mathcal{B}_{2} \vert pf_{2}0 \rangle _{\mathrm{s}}+\mathcal{B}_{3}|pf_{3}0 \rangle _{\mathrm{s}} \bigr) \biggr], \\& |E_{4\pm}\rangle =\frac{1}{\sqrt{2}} \bigl( \vert 1dd\rangle \pm \vert 1pf_{1} \rangle _{\mathrm{s}} \bigr), \\& |E_{5\pm}\rangle =\frac{1}{\sqrt{2}} \biggl[|d1d\rangle \pm \frac{2}{C_{3}} \bigl(\mathcal{B}_{1} \vert p1f_{1} \rangle _{\mathrm{s}}+ \mathcal{B}_{2} \vert p1f_{2} \rangle _{\mathrm{s}}+\mathcal{B}_{3}|p1f_{3} \rangle _{\mathrm{s}} \bigr) \biggr], \\& |E_{6\pm}\rangle =\frac{1}{\sqrt{2}} \biggl[|dd1\rangle \pm \frac{2}{C_{3}} \bigl(\mathcal{B}_{1} \vert pf_{1}1 \rangle _{\mathrm{s}}- \mathcal{B}_{2} \vert pf_{2}1 \rangle _{\mathrm{s}}+\mathcal{B}_{3}|pf_{3}1 \rangle _{\mathrm{s}} \bigr) \biggr], \end{aligned}$$

are the eigenstates of the Rydberg dipole-dipole interaction with the eigenvalues being $E_{k\pm}=\pm \sqrt{2}J$ ($k=1,2,\ldots ,6$) respectively, where $J=C_{3}/R^{3}_{23}$. The symmetric states are $|0pf_{1}\rangle _{\mathrm{s}}=(|0pf_{1}\rangle +|0f_{1}p\rangle )/\sqrt{2}$, $|p0f_{l}\rangle _{\mathrm{s}}=(|p0f_{l}\rangle +|f_{l}0p\rangle )/\sqrt{2}$, $|pf_{l}0\rangle _{\mathrm{s}}=(|pf_{l}0\rangle +|f_{l}p0\rangle )\sqrt{2}$, $|1pf_{1}\rangle _{\mathrm{s}}=(|1pf_{1}\rangle +|1f_{1}p\rangle )/\sqrt{2}$, $|p1f_{l}\rangle _{\mathrm{s}}=(|p1f_{l}\rangle +|f_{l}1p\rangle )/\sqrt{2}$, and $|pf_{l}1\rangle _{\mathrm{s}}=(|pf_{l}1\rangle +|f_{l}p1\rangle )\sqrt{2})/ \sqrt{2}$ with $l=1,2,3$. The coefficients are $\mathcal{B}_{1}=2C_{3}\cos ^{3}\theta (3\cos ^{2}\theta -1)$, $\mathcal{B}_{2}=6\sqrt{2}C'_{3}\sin \theta \cos ^{4}\theta$, and $\mathcal{B}_{3}=6C''_{3}\sin ^{2}\theta \cos ^{3}\theta$. The transformed Hamiltonian takes the following form

$$\begin{aligned}& H'_{\mathrm{full}}=H'_{1}+H'_{2}, \\& \begin{aligned} H'_{1}&=\Omega _{S} \vert 000\rangle \bigl(\langle 0d0 \vert +\langle 00d \vert \bigr)e^{i \Delta t}+ \vert 001\rangle \bigl[\langle 00d \vert \bigl(\Omega +\Omega _{S}e^{-i \Delta t}\bigr)+\langle 0d1 \vert \Omega _{S}e^{i\Delta t} \bigr] \\ &\quad{} + \vert 010\rangle \bigl[\langle 0d0 \vert \bigl(\Omega + \Omega _{S}e^{-i\Delta t}\bigr) + \langle 01d \vert \Omega _{S}e^{i\Delta t} \bigr] \\ &\quad{} + \vert 011\rangle \bigl(\langle 01d \vert + \langle 0d1 \vert \bigr) \bigl(\Omega +\Omega _{S}e^{-i\Delta t}\bigr) \\ &\quad{} +\sqrt{2}\Omega _{S} \vert 100\rangle \bigl\langle T'_{0} \bigr\vert e^{i\Delta t}+ \frac{1}{\sqrt{2}} \vert 101\rangle \bigl[\bigl(\bigl\langle T'_{0} \bigl\vert -\bigl\langle S'_{0} \bigr\vert \bigr) \bigl( \Omega +\Omega _{S}e^{-i\Delta t}\bigr)+\langle 1d1|\sqrt{2} \Omega _{S}e^{i \Delta t} \bigr] \\ &\quad{}+\frac{1}{\sqrt{2}} \vert 110\rangle \bigl[\bigl(\bigl\langle T'_{0} \bigr\vert + \bigl\langle S'_{0} \bigr\vert \bigr) \bigl( \Omega +\Omega _{S}e^{-i\Delta t}\bigr) + \langle 11d|\sqrt{2}\Omega _{S}e^{i\Delta t} \bigr] \\ &\quad{}+\sqrt{2}\bigl(\Omega +\Omega _{S}e^{-i\Delta t}\bigr) \vert 111\rangle \bigl\langle T'_{1} \bigr\vert + \mathrm{H.c.} , \end{aligned} \\& H'_{2}=\frac{1}{\sqrt{2}}\Omega _{S} \vert 00d\rangle \bigl[\langle E_{1+} \vert e^{i( \Delta -\sqrt{2}J)t}+ \langle E_{1-} \vert e^{i(\Delta +\sqrt{2}J)t}+ \langle E_{2+} \vert e^{-i(\Delta +\sqrt{2}J) t} \\& \hphantom{H'_{2}}\quad{} + \langle E_{2-} \vert e^{-i(\Delta -\sqrt{2}J) t} \bigr]+ \frac{1}{\sqrt{2}} \Omega _{S} \vert 0d0\rangle \bigl[ \langle E_{1+} \vert e^{i(\Delta -\sqrt{2}J)t}+ \langle E_{1-} \vert e^{i(\Delta +\sqrt{2}J)t} \\& \hphantom{H'_{2}}\quad{}+\langle E_{3+} \vert e^{-i(\Delta +\sqrt{2}J) t} + \langle E_{3-} \vert e^{-i(\Delta -\sqrt{2}J) t} \bigr] +\frac{1}{{\sqrt{2}}} \vert 01d\rangle \bigl[ \langle E_{1+} \vert \bigl( \Omega e^{-i\sqrt{2}J t} \\& \hphantom{H'_{2}}\quad{}+\Omega _{S}e^{-i(\Delta +\sqrt{2}J) t}\bigr)+\langle E_{1-} \vert \bigl(\Omega e^{i \sqrt{2}J t}+\Omega _{S}e^{-i(\Delta -\sqrt{2}J) t} \bigr)+\langle E_{5+} \vert \Omega _{S}e^{-i(\Delta +\sqrt{2}J) t} \\& \hphantom{H'_{2}}\quad{}+\langle E_{5-} \vert \Omega _{S}e^{-i(\Delta -\sqrt{2}J) t} \bigr]+ \frac{1}{\sqrt{2}} \vert 0d1\rangle \bigl[\langle E_{1+} \vert \bigl(\Omega e^{-i\sqrt{2}J t}+\Omega _{S}e^{-i(\Delta +\sqrt{2}J) t}\bigr) \\& \hphantom{H'_{2}}\quad{}+\langle E_{1-} \vert \bigl(\Omega e^{i\sqrt{2}J t}+\Omega _{S}e^{-i(\Delta - \sqrt{2}J) t}\bigr)+\langle E_{6+} \vert \Omega _{S}e^{-i(\Delta +\sqrt{2}J) t} \\& \hphantom{H'_{2}}\quad{}+\langle E_{6-} \vert \Omega _{S}e^{-i(\Delta -\sqrt{2}J) t} \bigr]+\Omega _{S} \bigl\vert T'_{0} \bigr\rangle \bigl[\langle E_{4+} \vert e^{i(\Delta -\sqrt{2}J)t}+\langle E_{4-} \vert e^{i( \Delta +\sqrt{2}J)t} \bigr] \\& \hphantom{H'_{2}}\quad{}+ \bigr\vert T'_{1}\bigr\rangle \bigl[\langle E_{4+} \vert \bigl(\Omega e^{-i\sqrt{2}J t}+ \Omega _{S}e^{-i(\Delta +\sqrt{2}J) t}\bigr)+\langle E_{4 -} \vert \bigl(\Omega e^{i \sqrt{2}J t}+\Omega _{S}e^{-i(\Delta -\sqrt{2}J) t}\bigr) \bigr] \\& \hphantom{H'_{2}}\quad{}+\mathrm{H.c.}, \end{aligned}$$

(19)

where $\Omega _{1}=\Omega _{2}=\Omega _{S}$ for simplicity. $|T'_{0}(S'_{0})\rangle =(|1d0\rangle \pm |10d\rangle )/\sqrt{2}$ and $|T'_{1}\rangle =(|1d1\rangle +|11d\rangle )/\sqrt{2}$. As is the case for two-atom, we also consider the large detuning case (e.g. $\Delta \gg \{\Omega _{S},\Omega \}$ and $\Delta =\sqrt{2}J$), the Hamiltonian can be reduced as

$$\begin{aligned} H' =&\Omega \bigl[ \vert 001\rangle \langle 00d \vert + \vert 010\rangle \langle 0d0 \vert +|011 \rangle \bigl(\langle 01d \vert +\langle 0d1 \vert \bigr) \bigr]+ \frac{1}{\sqrt{2}}\Omega \bigl( \vert 110 \rangle - \vert 101 \rangle \bigr)\bigl\langle S'_{0} \bigr\vert \\ &{} +\frac{1}{\sqrt{2}}\Omega \bigl( \vert 110\rangle + \vert 101\rangle \bigr)\bigl\langle T'_{0} \bigr\vert + \sqrt{2}\Omega \vert 111\rangle \bigl\langle T'_{1} \bigr\vert +\frac{1}{\sqrt{2}}\Omega _{S} \vert 00d \rangle \bigl( \langle E_{1+} \vert +\langle E_{2-} \vert \bigr) \\ &{} +\frac{1}{\sqrt{2}}\Omega _{S} \vert 0d0\rangle \bigl( \langle E_{1+} \vert + \langle E_{3-} \vert \bigr)+\frac{1}{\sqrt{2}}\Omega _{S}|01d\rangle \bigl(\langle E_{1-} \vert + \langle E_{5-} \vert \bigr) \\ & +\frac{1}{\sqrt{2}}\Omega _{S}|0d1\rangle \bigl(\langle E_{1-} \vert + \langle E_{6-} \vert \bigr) +\Omega _{S} \bigl\vert T'_{0}\bigr\rangle \langle E_{4+} \vert +\Omega _{S} \bigl\vert T'_{1} \bigr\rangle \langle E_{4-} \vert +\mathrm{H.c.}. \end{aligned}$$

(20)

Similarly, we employ the soft quantum control $\Omega (t)=\Omega _{m}\exp [-(t-2T)^{2}/T^{2}]$ to the three-atom system. The Hamiltonian $H'$ in Eq. (20) can be divided into two parts

$$ H' =H'_{\mathrm{S}}+H'_{\mathrm{int}}, $$

(21)

where

$$\begin{aligned}& \begin{aligned} H'_{\mathrm{S}} &=\frac{1}{\sqrt{2}}\Omega _{S} \vert 00d\rangle \bigl(\langle E_{1+} \vert + \langle E_{2-} \vert \bigr) +\frac{1}{\sqrt{2}}\Omega _{S} \vert 0d0\rangle \bigl(\langle E_{1+} \vert + \langle E_{3-} \vert \bigr) \\ &\quad{} +\frac{1}{\sqrt{2}}\Omega _{S} \bigl\vert 01d\rangle \bigl( \langle E_{1-} \vert +\langle E_{5-} \vert \bigr) +\frac{1}{\sqrt{2}}\Omega _{S} \vert 0d1\rangle \bigl( \langle E_{1-} \vert +\langle E_{6-} \vert \bigr) \\ &\quad{} +\Omega _{S} \bigl\vert T'_{0}\bigr\rangle \langle E_{4+} \vert +\Omega _{S} \bigl\vert T'_{1} \bigr\rangle \langle E_{4-} \vert +\mathrm{H.c.}, \end{aligned} \\& \begin{aligned} H'_{\mathrm{int}} &= \Omega \bigl[ \vert 001\rangle \langle 00d \vert + \vert 010 \rangle \langle 0d0 \vert +|011 \rangle \bigl(\langle 01d \vert +\langle 0d1 |\bigr) \bigr]+ \frac{1}{\sqrt{2}}\Omega \bigl( \vert 110\rangle - \vert 101\rangle \bigr)\bigl\langle S'_{0} \bigr\vert \\ &\quad{} +\frac{1}{\sqrt{2}}\Omega (t) \bigl( \vert 110\rangle + \vert 101 \rangle \bigr)\bigl\langle T'_{0} \bigr\vert + \sqrt{2} \Omega (t)\vert 111\rangle \bigl\langle T'_{1} \bigr\vert +\mathrm{H.c.}. \end{aligned} \end{aligned}$$

Using the above same method from Eq. (4) to Eq. (7), the corresponding average Hamiltonian of Eq. (20) is

$$\begin{aligned} \bar{H}' =&\frac{g}{\sqrt{2}}\bigl( \vert 110\rangle - \vert 101\rangle \bigr)\bigl\langle S'_{0} \bigr\vert + \frac{g}{\sqrt{2}} \bigl\vert S'_{0}\bigr\rangle \bigl(\langle 110 \vert -\langle 101 \vert \bigr) \\ &{} +\frac{1}{4T} \int _{0}^{4T}\sqrt{\Omega (t)^{2}+\Omega _{S}^{2}} \,dt \bigl( \vert \phi _{1+}\rangle \langle \phi _{1+} \vert - \vert \phi _{1-}\rangle \langle \phi _{1-} \vert \bigr) \\ &{} +\frac{1}{4T} \int _{0}^{4T}\sqrt{2\Omega (t)^{2}+\Omega _{S}^{2}} \,dt \bigl( \vert \phi _{2+}\rangle \langle \phi _{2+} \vert - |\phi _{2-}\rangle \langle \phi _{2-}) \vert \\ &{} +\frac{1}{4T} \int _{0}^{4T}\sqrt{ \frac{2\Omega (t)^{2}+\Omega _{S}^{2}}{2}} \,dt \bigl( \vert \phi _{3+}\rangle \langle \phi _{3+} \vert - \vert \phi _{3-}\rangle \langle \phi _{3-} \vert \bigr) \\ &{} +\frac{1}{4T} \int _{0}^{4T}\sqrt{ \frac{2\Omega (t)^{2}+3\Omega _{S}^{2}}{2}} \,dt \bigl( \vert \phi _{4+}\rangle \langle \phi _{4+} \vert - \vert \phi _{4-}\rangle \langle \phi _{4-} \vert \bigr) \\ &{} +\frac{1}{4T} \int _{0}^{4T}\frac{\Omega _{S}}{\sqrt{2}} \,dt \bigl( \vert \phi _{5+} \rangle \langle \phi _{5+} \vert - \vert \phi _{5-}\rangle \langle \phi _{5-} \vert \bigr) \\ &{} +\frac{1}{4T} \int _{0}^{4T} \sqrt{ \frac{4\Omega (t)^{2}+3\Omega _{S}^{2}}{2}} \,dt \bigl( \vert \phi _{6+}\rangle \langle \phi _{6+} \vert - \vert \phi _{6-}\rangle \langle \phi _{6-} \vert \bigr), \end{aligned}$$

(22)

where $g=\sqrt{\pi}\Omega _{m}\mathrm{Erf}[2]/4$. $|\phi _{1\pm}\rangle =(|T'_{0}\rangle \pm |E_{4+}\rangle )/\sqrt{2}$, $|\phi _{2\pm}\rangle =(|T'_{1}\rangle \pm |E_{4-}\rangle )/\sqrt{2}$, $|\phi _{3\pm}\rangle =[|E_{3-}\rangle -|E_{2-}\rangle \pm (|0d0 \rangle -|00d\rangle )]/2$, $|\phi _{4\pm}\rangle =[|E_{3-}\rangle +|E_{2-}\rangle +2|E_{1+} \rangle \pm \sqrt{3}(|0d0\rangle +|00d\rangle )]/2\sqrt{3}$, $|\phi _{5\pm}\rangle =[|E_{6-}\rangle -|E_{5-}\rangle \pm (|0d1 \rangle -|01d\rangle )]/2$, and $|\phi _{6\pm}\rangle =[|E_{6-}\rangle +|E_{5-}\rangle +2|E_{1-} \rangle \pm \sqrt{3}(|0d1\rangle +|01d\rangle )]/2\sqrt{3}$ are the eigenstates of $H'_{\mathrm{S}}$ governed by $\Omega _{S}$.

The propagator $U'=e^{-4i\bar{H}' T}$ with the evolution period $\tau =4T$ can generate a high-fidelity three-qubit controlled-swap gate. The evolution operator $U'$ in the basis $\{|S'_{0}\rangle , |000\rangle ,|001\rangle ,|010\rangle ,|011 \rangle ,|100\rangle ,|101\rangle ,|110\rangle ,|111\rangle \}$ takes the following form

$$\begin{aligned} U'(\tau )= \begin{pmatrix} \cos \lambda _{\tau}& 0 & 0& 0 & 0 & 0 &\frac{\sqrt{2}i}{2}\sin \lambda _{\tau} &-\frac{\sqrt{2}i}{2}\sin \lambda _{\tau} &0 \\ 0 &1 & 0 & 0& 0 &0 &0 &0 &0 \\ 0& 0 &1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 &1 & 0 & 0 & 0 &0 &0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 &0 & 0 \\ 0 & 0& 0& 0 & 0& 1& 0& 0& 0 \\ \frac{\sqrt{2}i}{2}\sin \lambda _{\tau} & 0& 0 &0 & 0 &0 & \frac{1}{2}(1+\cos \lambda _{\tau}) &\frac{1}{2}(1-\cos \lambda _{ \tau}) & 0 \\ -\frac{\sqrt{2}i}{2}\sin \lambda _{\tau} & 0 & 0 & 0& 0 &0 & \frac{1}{2}(1-\cos \lambda _{\tau}) & \frac{1}{2}(1+\cos \lambda _{ \tau}) &0 \\ 0 & 0& 0& 0& 0 & 0 &0 &0 & 1 \end{pmatrix}, \end{aligned}$$

where $\lambda _{\tau}=4Tg$. Setting $\lambda _{\tau}=4Tg=\pi$, the final effective evolution operator $U'(\tau )$ reads

$$\begin{aligned} U'(\tau )= \begin{pmatrix} -1& 0& 0& 0 &0 & 0 &0 &0 & 0 \\ 0& 1& 0& 0 &0 & 0 &0 &0 & 0 \\ 0& 0& 1& 0 &0 & 0 &0 &0 & 0 \\ 0& 0& 0& 1 &0 & 0 &0 &0 & 0 \\ 0& 0& 0& 0 &1 & 0 & 0 &0 & 0 \\ 0& 0& 0& 0 &0& 1& 0& 0& 0 \\ 0& 0& 0& 0 &0 &0 & 0 &1 & 0 \\ 0& 0& 0& 0 &0 &0 & 1 & 0 &0 \\ 0& 0& 0& 0 &0 & 0 &0 &0 & 1 \end{pmatrix}, \end{aligned}$$

(23)

which is a three-qubit controlled-swap gate, in which the two target qubits swap their information $|01\rangle _{23}\Longleftrightarrow |10\rangle _{23}$ if and only if the control qubit is in $|1\rangle _{1}$, on the computational subspace $\mathbf {S}'=\mathrm{Span}\{|000\rangle ,|001\rangle ,|010\rangle ,|011 \rangle ,|100\rangle ,|101\rangle ,|110\rangle ,|111\rangle \}$ as follows

$$\begin{aligned} U'_{\mathrm{cswap}}= \begin{pmatrix} 1& 0& 0 &0 & 0 &0 &0 & 0 \\ 0& 1& 0 &0 & 0 &0 &0 & 0 \\ 0& 0& 1 &0 & 0 &0 &0 & 0 \\ 0& 0& 0 &1 & 0 & 0 &0 & 0 \\ 0& 0& 0 &0& 1& 0& 0& 0 \\ 0& 0& 0 &0 &0 & 0 &1 & 0 \\ 0& 0& 0 &0 &0 & 1 & 0 &0 \\ 0& 0& 0 &0 & 0 &0 &0 & 1 \end{pmatrix}. \end{aligned}$$

(24)

Using the same method as the holonomic proof for two-qubit swap gate, one can also confirm that both conditions $(\mathrm{I)}$ and $(\mathrm{II)}$ in Eq. (12) are satisfied. Therefore, $U'(\tau )$ is a holonomic three-qubit controlled-swap gate in subspace $\mathbf {S}'$. Through the temporal evolution of all ground states obtained from the full Hamiltonian Eq. (17) depicted in Fig. 6, it is shown that the controlled-swap gate $U'(\tau )$ within the subspace $\mathbf {S}'$ is pure holonomic. Additionally, the figure illustrates that Gaussian time-dependent soft-control enables efficient rotating-wave approximation across a wide parameter range.

Figure 6

The temporal evolution of populations for different ground states governed by the full Hamiltonian in Eq. (17). The time-dependent parameter is $\Omega (t)=\Omega _{m}\exp [-(t-2T)^{2}/T^{2}]$ with $\Omega _{m}/(2\pi )=0.5\text{ MHz}$ and $T=\sqrt{\pi}/(\mathrm{Erf}[2]\Omega _{m})$. The other parameters are $\Omega _{S} /(2\pi ) =5\text{ MHz}$, $\Delta /(2\pi ) =500\sqrt{2}\text{ MHz}$, and $J/(2\pi )= 500\text{ MHz}$

The fidelity for the three-qubit controlled-swap gate in the ideal case is $F=\langle \psi '_{{\mathrm{ideal}}}|\rho (t) | \psi '_{{\mathrm{ideal}}} \rangle =99.93\%$ after the same evolution period $\tau =2.267~\mu\text{s}$ as for the two-qubit case. Here, the initial state is $(|000\rangle +|001\rangle +|010\rangle +|011\rangle +|100\rangle + |111 \rangle +\sqrt{3}|101\rangle -\sqrt{3}|110\rangle )/(2\sqrt{3})$ and the ideal final state is $(|000\rangle +|001\rangle +|010\rangle +|011\rangle +|100\rangle + |111 \rangle -\sqrt{3}|101\rangle +\sqrt{3}|110\rangle )/(2\sqrt{3})$.

We then discuss the practical situation in which Förster defects exist. The pair states $|pf_{l}\rangle$ and $|f_{l}p\rangle$ are degenerate with $l=1,2,3$, and the Förster defect between $|dd\rangle$ and $|pf_{l}\rangle$ ($|f_{l}p\rangle$) is $\delta /(2\pi )=8.5$ MHz in the absence of an electric field [82]. In this case, the dipole-dipole coupling between the three-atom in Eq. (17) is modified as $H'_{dd}=V_{ dd}+\delta [|pf_{1}\rangle _{23}\langle pf_{1}|+|f_{1}p \rangle _{23}\langle f_{1}p|+\sum_{l=1,2,3}(|pf_{l}\rangle _{12} \langle pf_{l}|+|f_{l}p\rangle _{12}\langle f_{l}p|+|pf_{l}\rangle _{13} \langle pf_{l}|+|f_{l}p\rangle _{13}\langle f_{l}p|)]$. By making slight adjustments to the laser fields in Fig. 5, for the control atom 1 we set the laser field with the Rabi frequency $\Omega _{2}$ possess a red detuning $\Delta _{2}=(\sqrt{8J^{2}+\delta ^{2}}-\delta )/2$. Meanwhile, for target atoms 2 and 3 we set one laser field with the Rabi frequency $\Omega _{1}$ has a blue detuning $\Delta _{1}=(\sqrt{8J^{2}+\delta ^{2}}+\delta )/2$, and the other laser field with the Rabi frequency $\Omega _{2}$ has a red detuning $\Delta _{2}=(\sqrt{8J^{2}+\delta ^{2}}-\delta )/2$. Through numerical simulations, it is found that the fidelity of the quantum gate is still maintained at 99.93%. In the following, to assess the effect of deviations from the expected dipole-dipole interactions in the same Förster defect, we consider $J/(2\pi )=(500+\Delta J)\text{ MHz}$ with the detuning parameters $\Delta _{1}/(2\pi )= (\sqrt{8\cdot 500^{2}+8.5^{2}}+8.5)/2=711.37\text{ MHz}$ and $\Delta _{2}/(2\pi ) = (\sqrt{8\cdot 500^{2}+8.5^{2}}-8.5)/2=702.87\text{ MHz}$. In Fig. 7, the fidelity of the controlled-swap gate is plotted against the deviation ΔJ, which consistently remains above 90% in the continuous range of the coupling strength from $\Delta J=-2.6$ to $\Delta J=2.6$. Therefore, the current SRP mechanism is also insensitive to fluctuations in the coupling strength J for three-qubit case.

Figure 7

The effect of deviation ΔJ on the fidelity of the controlled-swap gate in the presence of Förster defect $\delta /(2\pi ) =8.5\text{ MHz}$. The time-dependent parameter is $\Omega (t)=\Omega _{m}\exp [-(t-2T)^{2}/T^{2}]$ with $\Omega _{m}/(2\pi )=0.5\text{ MHz}$ and $T=\sqrt{\pi}/(\mathrm{Erf}[2]\Omega _{m})$. The other parameters are $\Omega _{S} /(2\pi ) =5\text{ MHz}$, $\Delta _{1}/(2\pi ) =711.37\text{ MHz}$, $\Delta _{2}/(2\pi ) =702.87\text{ MHz}$, and $J/(2\pi )=(500+\Delta J)\text{ MHz}$

We also consider the spontaneous emission of the Rydberg states in the same Förster defect. Based on the Markovian master equation of the system in Lindblad form Eq. (13), however, one needs to consider two extra Rydberg states $|f_{2}\rangle$ and $|f_{3}\rangle$ when the ploar angle $\theta _{ij}\neq 0$. The fidelity for the three-qubit controlled-swap gate is $F=\langle \psi '_{{\mathrm{ideal}}} |\rho (t)| \psi '_{{\mathrm{ideal}}} \rangle =99.78\%$. Consequently, the three-qubit controlled-swap gate is implemented with a robust fidelity in a shorter time frame.

5 Conclusion

In conclusion, we have demonstrated a rapid implementation of holonomic swap and controlled-swap gates for neutral atoms using SRP. By incorporating time-dependent control which enables highly selective coupling between different on-resonance constituents of composite quantum systems, within the SRP mechanism, we achieved an average Hamiltonian that achieves a higher and more stable population in a shorter time, leading to an efficient rotating-wave approximation across a broad parameter range. Our approach accelerates the synthesis of a robust two-qubit swap gate, which is robust against variations in the dipole-dipole interaction, Förster defect fluctuation, and spontaneous emission of Rydberg states. Furthermore, our mechanism readily extends to the direct implementation of a holonomic three-qubit controlled-swap gate by introducing a control atom and selecting an appropriate angle θ between the interatomic axis and the quantization axis z and an appropriate driving field. Combining the robustness against control imprecisions and high-speed evolution of nonadiabatic HQC, we hope our work may provide an alternative approach toward fault-tolerant quantum computation.

Appendices

Appendix A: The terms oscillating with high frequencies

Under the conditons $\Delta \gg \{\Omega _{S},\Omega (t)\}$ and $\Delta =\sqrt{2}J$, the Hamiltonian in Eq. (2) can be written in the follwing form

$$\begin{aligned}& H_{\mathrm{full}} = H_{\mathrm{res}}+H_{\mathrm{dis}}, \\& \begin{aligned} H_{\mathrm{res}}&=\frac{1}{\sqrt{2}}\Omega (t) \bigl( \vert 10\rangle - \vert 01\rangle \bigr) \langle S_{0} \vert +\frac{1}{\sqrt{2}} \Omega (t) \bigl( \vert 10\rangle + \vert 01 \rangle\bigr) \langle T_{0} \vert +\sqrt{2}\Omega (t) \vert 11\rangle \langle T_{1} \vert \\ &\quad{} +\Omega _{S} \vert T_{0}\rangle \langle E_{+} \vert +\Omega _{S} \vert T_{1} \rangle \langle E_{-} \vert +\mathrm{H.c.}, \end{aligned} \\& \begin{aligned} H_{\mathrm{dis}}&=\sqrt{2}\Omega _{S} \vert 00\rangle \langle T_{0} \vert e^{i\Delta t} +\frac{1}{\sqrt{2}} \Omega _{S}e^{-i\Delta t}\bigl( \vert 10\rangle - \vert 01 \rangle \bigr) \langle S_{0} \vert \\ &\quad{} +\frac{1}{\sqrt{2}}\Omega _{S}e^{-i\Delta t}\bigl( \vert 10 \rangle + \vert 01 \rangle \bigr)\langle T_{0} \vert + \vert 01\rangle \langle d1 \vert \Omega _{S}e^{i\Delta t}+ \vert 10 \rangle \langle 1d \vert \Omega _{S}e^{i\Delta t} \\ &\quad{}+\sqrt{2}\Omega _{S}e^{-i\Delta t} \vert 11\rangle \langle T_{1} \vert +\Omega _{S} \vert T_{0} \rangle \langle E_{-} \vert e^{i2\Delta t}+|T_{1} \rangle \bigl[\langle E_{+} \vert \bigl( \Omega e^{-i\Delta t} +\Omega _{S}e^{-i2\Delta t}\bigr) \\ &\quad{}+\langle E_{-} \vert \Omega e^{i\Delta t} \bigr] + \mathrm{H.c.}. \end{aligned} \end{aligned}$$

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The terms in $H_{\mathrm{dis}}$ are the dispersive interaction and their actions are equal to the Stark shifts of atomic levels, and these Stark shifts can be canceled by introducing the other ancillary levels and laser fields to induce the opposite Stark shifts. Thus, the Hamiltonian in Eq. (2) can be evaluated explicitly

$$\begin{aligned} H=H_{\mathrm{res}} =&\frac{1}{\sqrt{2}}\Omega (t) \bigl( \vert 10\rangle - \vert 01\rangle \bigr) \langle S_{0} \vert + \frac{1}{\sqrt{2}}\Omega (t) \bigl( \vert 10\rangle + \vert 01\rangle \bigr) \langle T_{0} \vert +\sqrt{2}\Omega (t) \vert 11 \rangle \langle T_{1} \vert \\ &{} +\Omega _{S} \vert T_{0}\rangle \langle E_{+} \vert +\Omega _{S} \vert T_{1} \rangle \langle E_{-} \vert +\mathrm{H.c.}. \end{aligned}$$

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Appendix B: The average Hamiltonian

The Hamiltonian $H_{\mathrm{S}}$ in Eq. (4) has the eigenstates

$$\begin{aligned}& \bigl|\psi ^{D}_{1}\bigr\rangle =\frac{1}{\sqrt{2}}\bigl( \vert T_{0}\rangle + \vert E_{+} \rangle \bigr), \\& \bigl|\psi ^{D}_{2}\bigr\rangle =\frac{1}{\sqrt{2}}\bigl( \vert T_{0}\rangle - \vert E_{+} \rangle \bigr), \\& \bigl|\psi ^{D}_{3}\bigr\rangle =\frac{1}{\sqrt{2}}\bigl( \vert T_{1}\rangle + \vert E_{-} \rangle \bigr), \\& \bigl|\psi ^{D}_{4}\bigr\rangle =\frac{1}{\sqrt{2}}\bigl( \vert T_{1}\rangle - \vert E_{-} \rangle \bigr), \\& \bigl|\psi ^{D}_{5}\bigr\rangle =\frac{1}{\sqrt{2}} \biggl[|S_{0}\rangle + \frac{1}{\sqrt{2}}\bigl( \vert 10\rangle - \vert 01\rangle \bigr)\biggr], \\& \bigl|\psi ^{D}_{6}\bigr\rangle =\frac{1}{\sqrt{2}} \biggl[|S_{0}\rangle - \frac{1}{\sqrt{2}}\bigl( \vert 10\rangle - \vert 01\rangle \bigr)\biggr], \\& \bigl|\psi ^{D}_{7}\bigr\rangle =\frac{1}{\sqrt{2}} \bigl( \vert 10\rangle + \vert 01 \rangle \bigr), \\& \bigl|\psi ^{D}_{8}\bigr\rangle =|11\rangle . \end{aligned}$$

The corresponding eigenvalues are $\omega _{1}=\Omega _{S}$, $\omega _{2}=-\Omega _{S}$, $\omega _{3}=\Omega _{S}$, $\omega _{4}=-\Omega _{S}$, $\omega _{5}=\omega _{6}=\omega _{7}=\omega _{8}=0$ and the corresponding projection operators are $\mathbb{P}(\omega _{j})=|\psi ^{D}_{j}\rangle \langle \psi ^{D}_{j}|$ ($j=1,2,\ldots,8$) respectively.

The whole Hamiltonian in Eq. (4) under the projection operators $\mathbb{P}(\omega _{j})$ is written as $H^{1}=H_{{\mathrm{S}}}^{1}+H_{\mathrm{int}}^{1}$, and its form in the basis {$|\psi ^{D}_{j}\rangle$} is

$$ H^{1}= \begin{pmatrix} \Omega _{S} & 0 & 0 &0 & 0 &0&\frac{1}{\sqrt{2}}\Omega (t)&0 \\ 0& -\Omega _{S}& 0& 0& 0 &0&\frac{1}{\sqrt{2}}\Omega (t)&0 \\ 0 &0 & \Omega _{S}&0& 0& 0 &0 & \Omega (t) \\ 0 &0 & 0&-\Omega _{S}& 0& 0 &0 & \Omega (t) \\ 0 &0 & 0&0& \Omega (t)& 0 &0 & 0 \\ 0 &0 & 0&0& 0& -\Omega (t) &0 & 0 \\ \frac{1}{\sqrt{2}}\Omega (t) &\frac{1}{\sqrt{2}}\Omega (t) & 0&0& 0& 0 &0 & 0 \\ 0 &0 & \Omega (t)&\Omega (t)& 0& 0 &0 & 0 \end{pmatrix}. $$

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The corresponding eigenvalues are $E_{1}=\sqrt{\Omega (t)^{2}+\Omega _{S}^{2}}$, $E_{2}=-\sqrt{\Omega (t)^{2}+ \Omega _{S}^{2}}$, $E_{3}= \sqrt{2\Omega (t)^{2}+\Omega _{S}^{2}}$, $E_{4}=- \sqrt{2\Omega (t)^{2}+\Omega _{S}^{2}}$, $E_{5}=\Omega (t)$, $E_{6}=- \Omega (t)$, $E_{7}=E_{8}=0$. Based on Eq. (6), one can find that $\phi _{j}(4T)=\int _{0}^{4T}E_{j}\,dt$. Substituting the results in Eq. (7), the corresponding average Hamiltonian of Eq. (4) is obtained as

$$\begin{aligned} \bar{H} =&\frac{1}{4T} \int _{0}^{4T}\sqrt{\Omega (t)^{2}+\Omega _{S}^{2}} \,dt \bigl( \bigl\vert \psi ^{D}_{1}\bigr\rangle \bigl\langle \psi ^{D}_{1} \bigr\vert - \bigl\vert \psi ^{D}_{2} \bigr\rangle \bigl\langle \psi ^{D}_{2} \bigr\vert \bigr) \\ &{}+\frac{1}{4T} \int _{0}^{4T}\sqrt{2\Omega (t)^{2}+\Omega _{S}^{2}} \,dt \bigl( \bigl\vert \psi ^{D}_{3}\bigr\rangle \bigl\langle \psi ^{D}_{3} \bigr\vert - \bigl\vert \psi ^{D}_{4}\bigr\rangle \bigl\langle \psi ^{D}_{4} \bigr\vert \bigr) \\ &{}+\frac{1}{4T} \int _{0}^{4T}\Omega (t)\,dt\bigl( \bigl\vert \psi ^{D}_{5}\bigr\rangle \bigl\langle \psi ^{D}_{5} \bigr\vert - \bigl\vert \psi ^{D}_{6}\bigr\rangle \bigl\langle \psi ^{D}_{6} \bigr\vert \bigr) \\ =&\frac{1}{4T} \int _{0}^{4T}\sqrt{\Omega (t)^{2}+\Omega _{S}^{2}} \,dt \bigl( \bigl\vert \psi ^{D}_{1}\bigr\rangle \bigl\langle \psi ^{D}_{1} \bigr\vert - \bigl\vert \psi ^{D}_{2}\bigr\rangle \bigl\langle \psi ^{D}_{2} \bigr\vert \bigr) \\ &{}+\frac{1}{4T} \int _{0}^{4T}\sqrt{2\Omega (t)^{2}+\Omega _{S}^{2}} \,dt \bigl( \bigl\vert \psi ^{D}_{3}\bigr\rangle \bigl\langle \psi ^{D}_{3} \bigr\vert - \bigl\vert \psi ^{D}_{4}\bigr\rangle \bigl\langle \psi ^{D}_{4} \bigr\vert \bigr) \\ &{} +\frac{1}{4T} \int _{0}^{4T}\Omega _{m}\exp \biggl[- \frac{(t-2T)^{2}}{T^{2}} \biggr]\,dt \biggl[\frac{1}{\sqrt{2}}\bigl( \vert 10 \rangle - \vert 01 \rangle \bigr)\langle S_{0} \vert + \frac{1}{\sqrt{2}}|S_{0}\rangle \bigl(\langle 10 \vert - \langle 01 \vert \bigr) \biggr] \\ =&\frac{1}{4T} \int _{0}^{4T}\sqrt{\Omega (t)^{2}+\Omega _{S}^{2}} \,dt \bigl( \bigl\vert \psi ^{D}_{1}\bigr\rangle \bigl\langle \psi ^{D}_{1} \bigr\vert - \bigl\vert \psi ^{D}_{2}\bigr\rangle \bigl\langle \psi ^{D}_{2} \bigr\vert \bigr)\ \\ &{}+\frac{1}{4T} \int _{0}^{4T}\sqrt{2\Omega (t)^{2}+\Omega _{S}^{2}} \,dt \bigl( \bigl\vert \psi ^{D}_{3}\bigr\rangle \bigl\langle \psi ^{D}_{3} \bigr\vert - \bigl\vert \psi ^{D}_{4}\bigr\rangle \bigl\langle \psi ^{D}_{4} \bigr\vert \bigr) \\ &{} +\frac{g}{\sqrt{2}}\bigl( \vert 10\rangle - \vert 01\rangle \bigr)\langle S_{0} \vert + \frac{g}{\sqrt{2}}|S_{0} \rangle \bigl(\langle 10 \vert -\langle 01 \vert \bigr), \end{aligned}$$

where $g=\int _{0}^{4T}\Omega _{m}\exp[-\frac{(t-2T)^{2}}{T^{2}}]\,dt/(4T)= \sqrt{\pi}\Omega _{m}\mathrm{Erf}[2]/4$.