High-fidelity Generation of Bell and W States in Giant Atom System via Bound State in the Continuum

Quantum entanglement lies at the core of the quantum information and plays a significant role in quantum communication and quantum networks RH2009 ; LA2015 , such as quantum key distribution AK1991 ; FG2003 , quantum teleportation CH1993 ; JJ2003 , and quantum-computing technology DP2000 . Among the various types of entangled states, Bell states represent the maximally entangled bipartite states PG1995 ; LQ1999 , while W states are a prominent class of many-qubit entangled states known for their exceptional resilience to particle loss WD2000 ; MK2000 . To date, the Bell state and W state have been extensively studied HY2020 ; MA2009 ; MN2010 ; JP2021 and their suitability for applications in quantum information protocols has driven their preparation in various systems FH2014 ; AA2013 ; CL2015 ; BF2019 .

In the early days, the preparation of entangled states primarily focused on photonic and ionic systems RP2009 ; WW2009 ; HH2005 . The Rydberg ensemble has emerged as a promising alternative for achieving scalable entanglement among neutral atoms VM2021 ; AO2019 ; YJ2020 ; MG2015 ; JLi2019 ; CW2022 . Currently, the advancement of quantum information in solid-state systems has motivated researchers to explore effective schemes for realizing entangled states in artificial systems, such as semiconductor quantum dots WB2012 and superconducting qubits JQ2011 . Especially, superconducting circuits are regarded as one of the most promising solid-state systems for implementing quantum computing. In contrast to natural atoms, the superconducting qubits can be engineered to couple with the waveguide via multiple points to acts as giant artificial atoms MV2014 ; RM2017 ; AM2021 . The interference effects arising from these multiple coupling points BK2020 ; YT2022 ; XL2022 ; AF2018 ; LD2023 ; DC2020 ; AC2020 ; AS2022 can lead to the emergence of bound states in the continuum (BIC) LL2024 ; ER2024 . The so called BIC locates in the continual band of the system in the frequency domain, but the wave function exhibits localized character in the real space. Nowadays, the BIC is widely applied in laser technology KI2019 ; KK2017 and quantum information processing XZ2023 .

In this paper, we propose a scheme for generating maximally entangled states in a giant atom system that is coupled to a coupled resonator waveguide BS2018 ; MZ2019 ; JL2019 ; JS2020 ; SP2021 ; KF2013 . We demonstrate that the BIC in our system exhibits robustness against both onsite frequency and hopping strength disorder of the waveguide. This robustness provides a distinctive mechanism for entangled state generation. Actually, in giant atom system, the maximum entangled state can also be prepared by leveraging the adjustable decay processes and interference processes of giant atoms in waveguides XL2023 ; AC2023 . Alternatively, the Bell and W states generation via steady state is also an effective approach, but it requires sophisticated designs for the coupling and dissipation terms GZ2023 ; GZa2023 . In contrast, in our scheme based on the BIC, we only need to find the optimal driving duration to achieve the maximum entangled state, which can be maintained after undergoes a long time evolution.

For the two-giant atom setup, we demonstrate that the Bell state can be realized with a fidelity higher than $98\%$ in braided, separate and nested configurations. These findings challenge the notion that only the decoherence free interaction (DFI) AF2018 ; LD2023 , which is exclusively engineered in the braided setup, is useful in quantum information processing. Furthermore, we extend our analysis to a configuration involving three giant atoms, to obtain the high-fidelity W states with the assistance of BIC.

The model in our consideration is composed by two giant atoms which couple to a coupled resonator waveguide in braided configuration, as illustrated in Fig. 1. The giant atoms are designed to couple with the waveguide through two separate coupling sites $x_{i}$ and $x_{i}+n$ $(i=1,2)$. The coupling space between the left leg of two giant atoms is $\Delta x=x_{2}-x_{1}$ in Fig. 1. The total Hamiltonian, with the rotating wave approximation, contains three parts of $H=H_{a}+H_{c}+H_{I}$ ($\hbar=1$), where

defines the free Hamiltonian of the two giant atoms. $\Omega$ is the transition frequency between the excited state $|e\rangle_{i}$ and the ground state $|g\rangle_{i}$ of the giant atom $i$. The operator $\sigma_{i}^{+}(\sigma_{i}^{-})$ is the raising (lowering) operator of the giant atom. The coupled resonator waveguide is modeled by the Hamiltonian

Here, $a_{j}^{\dagger}(a_{j})$ is the creation (annihilation) operator of the field in the waveguide on site $j$. $\omega_{c}$ is the resonators’ frequency and $\xi$ is the nearest-neighbor hopping strength. Finally, the coupling between the giant atoms and the waveguide field is described by the Hamiltonian

where $g$ is the coupling strength.

Now, we can Fourier transform $a_{x}=\sum_{k}e^{-ikx}a_{k}/\sqrt{N_{c}}$ with $N_{c}\rightarrow\infty$ being the number of the resonators, so that Eq. (2) becomes $H_{c}=\sum_{k}\omega_{k}a_{k}^{\dagger}a_{k}$, with the dispersion relation of the waveguide satisfying $\omega_{k}=\omega_{c}-2\xi\cos k$. In Fourier space, the interaction Hamiltonian reads

Here, $g_{ik}=g(e^{-ikx_{i}}+e^{-ik(x_{i}+n)})/\sqrt{N_{c}}$ is the coupling strength, which depends on the coupling sites. When the giant atom is resonant with the bare resonator of the waveguide, that is $\Omega=\omega_{c}$, we plot the energy spectrum of the entire system in the single excitation subspace as a function of the coupling strength $g$ as shown in Fig. 2(a). Here, the system is characterized by a continual energy band centered at $\omega_{c}$ with a total width of $4\xi$. Except for the continual band, the interaction between the giant atoms and the waveguide breaks the translational symmetry of the waveguide, leading to bound states outside of the continuum (BOC). They locate above and below the continual band, and are indicated by red and blue curves in Fig. 2(a). The BOC has been identified in various systems with structured environments, such as photonic crystals and atom-waveguide coupled systems SJ1990 ; LZ2013 ; WZ2020 . Furthermore, the giant atoms also give rise to the BIC, which are situated within the continuous band, as represented by the black line in Fig. 2(a). When the giant atom is coupled to the 2D coupled resonator array, people also find the BIC, which bound the photon inside the atomic regime ER2024 .

The above results based on the master equation imply that the initial Bell state is immune to the dissipation induced by the coupling to the waveguide, which serves as the environment. In other words, we can drive one of the giant atom, to prepare the two giant atom system into the Bell state. Then, we turn off the driving, the Bell state can be always alive with the assistance of the BIC. In such scheme, the driving Hamiltonian is expressed as

with $\eta$ being the Rabi frequency and $\omega_{d}$ is the frequency of the driving field. $\Theta(t_{0}-t)$ is the Heaviside function and then $t_{0}$ is the duration of driving to the first giant atom. In superconducting qubits system, the driving can be realized with independent circuits applied to the giant atoms RB2014 ; PK2019 . Now the dynamics of the two giant atoms is still governed by the master equation in Eq. (LABEL:masterequation), but $H_{a}$ is replaced by $H_{a}^{(d)}=H_{a}+H_{d}$. For a weak driving field, the Lindblad form of Eq. (LABEL:masterequation) dose not change.

At the initial time $t=0$, we prepare the two giant atoms in the ground state $|gg\rangle=|g\rangle_{1}|g\rangle_{2}$. By numerically solving the master equation, we plot the time evolution of the fidelity

For long-time driving $(t_{0}\rightarrow\infty)$, we observe periodic oscillations of the fidelity, as depicted in Fig. 4(a). The first maximum value, $\mathcal{F}_{\rm max}\approx 1$, occurs at the time $\xi t_{\rm max}=223$. This implies that the system reaches the Bell state at this moment.

This process can be intuitively decomposed as the following two steps. In the first step, the classical field drives the two atom from state $|gg\rangle$ into $|eg\rangle$. In the second step, governed by the effective Hamiltonian in Eq. (9), the interaction induces an excitation exchange between the two atoms, leading to the dressed entangled Bell states $|\Psi_{\pm}\rangle=(|eg\rangle\pm|ge\rangle)/\sqrt{2}$. The symmetric Bell state $|\Psi_{+}\rangle$ coincides the BIC state of the atom-waveguide coupled state and is maintained during subsequent time evolution. Meanwhile, the anti-symmetry Bell state $|\Psi_{-}\rangle$ will further decay back to the ground state and is subsequently re-driven. These two steps repeat multiple times, leading to the periodical oscillation in Fig. 4(a).

Due to the aforementioned circle, it is impossible to generate the steady Bell state with a continual driving field as $t_{0}\rightarrow\infty$. To address this issue, we turn off the driving field at the moment of $t_{\rm max}$ and allow the system to evolve governed by the master equation in Eq. (LABEL:masterequation). With the protection of the BIC, we can achieve a fidelity of $\mathcal{F}_{\rm Bell}$ as high as $100\%$ with the weak driving strength of $\eta=0.01\xi$, as shown in Fig. 4(b). For a stronger driving strength of $0.05\xi$, the fidelity remains above $95\%$, but the required time is significantly shortened.

In the above section, we have investigated the Bell state generation scheme in the braided configuration. Since our scheme relies only on the BIC in the atom-waveguide coupled system, it is not limited in the braided setup. As a natural generation, we also consider the separate and nested configurations as shown in Fig. 5, which support one BIC, respectively. Using the similar strategies as before, we drive the first giant atom with an optimized during time, the corresponding Bell state preparation fidelity is demonstrated in the lower panels of Fig. 5. Again, we achieve a fidelity higher than $98\%$.

It is intuitively that the DFI cannot be over emphasized in the entangled state generation XL2023 and such DFI can be only realized in the braided configuration AF2018 ; LD2023 . Our work challenges this conventional view by the relaxing conditions for the existence of BIC. Even in our braided configuration, the effective Hamiltonian in Eq. (9) still contains the dissipation term which does not work with the DFI framework, but the BIC plays a key role to the Bell state generation. In the separate and nested configuration, we also can not find the DFI to realize a steady Bell state in the long time limit.

Our scheme can be further extended to three giant atoms systems for the generation of three-particle W states, $|W\rangle=(|e,g,g\rangle+|g,e,g\rangle+|g,g,e\rangle)/\sqrt{3}$. As depicted in Fig. 6(a), we consider three giant atoms coupled to the coupled resonator waveguide. The two coupled sites of each giant atom are $x_{i}$ and $x_{i}+n$ $(i=1,2,3)$. The space of neighboring giant atoms is described as $\Delta x=x_{2}-x_{1}$ and $\Delta s=x_{3}-(x_{1}+n)$. When the first giant atom is driven with Rabi frequency $\eta$, the total Hamiltonian of the system is $H=H_{a}+H_{c}+H_{I}$, where

Under the Markovian approximation, the effective dynamics of the giant atoms is governed by the master equation

with the Lindblad superoperator

where $A_{ij}$ is same as Eq. (8).

At the initial time $t=0$, the system of three giant atoms is set in the ground state $|g,g,g\rangle$. With the master equation, we plot the time evolution of the fidelity $\mathcal{F}_{W}$ of the W state in Fig. 6(b) where the fidelity is defined as $\mathcal{F}_{W}={\rm Tr}\sqrt{\sqrt{\rho_{W}}\rho\sqrt{\rho_{W}}}$. Under the driving field on the first giant atom, the fidelity $\mathcal{F}_{W}$ increases monotomically to its maximum value at the optimal time $t_{\rm max}$, after which the driving field is turned off, and the W state fidelity approach $90\%$ in the long time limit, and is nearly independent of the driving strength as shown in Fig. 6(c). Also, the scheme can be employed in the separate and nested configuration.

In conclusion, we have proposed an effective scheme for generating the maximally entangled state in the giant atom waveguide system via BIC, which is robust to disorder in the waveguide. Using an external classical field, we first drive the system into the BIC, where the atomic part corresponds to an entangled state. Once the driving field is turned off, the BIC decouples from the other states of the atom-waveguide coupled system, resulting in an infinite lifetime for the atomic maximally entangled state. For the two giant atom setup, we go beyond the framework of the DFI AF2018 ; LD2023 to generate a Bell state in braided, separated, and nested configurations of the giant atoms, with the fidelity exceeding $98\%$. For the three giant atom setup, we can realize the generation of a W state with a fidelity approaching $90\%$.

The giant atom can be implemented using superconducting transmon qubits BK2020 ; GA2019 ; ZW2022 , and the coupled resonator waveguides have also been fabricated with tens of site in superconducting circuits. By expanding the capacitively coupled lumped element, nearest-neighbor hopping strengths in the range of $100$–$200$ MHz have been achieved PR2017 ; XYZ2023 . Consequently, the results shown in Figs. 4, 5, and 6 can be realized, even with bare qubit decay times of $T_{1}=10\,{\rm\mu s}$ MK2020 . Our study extends beyond the DFI mechanism, providing a useful approach for constructing quantum networks and enabling quantum information processing.

Note During the preparation of this work, we find an investigation to generate the entangled states in the waveguide QED setup with the assistance of BIC via single photon scattering GM2025 .