Steady three-dimensional dark state entanglement in dissipative Rydberg atoms

TL;DR

This paper proposes a robust method to generate three-dimensional entangled states between Rydberg atoms using dissipative dynamics and EIT, effective regardless of initial states and interaction regimes.

Contribution

It introduces a novel scheme for dark state entanglement in Rydberg atoms that is insensitive to initial state purity and does not require specific Rydberg interaction conditions.

Findings

Feasible for arbitrary initial states and mixed states

No precise timing or Rydberg interaction constraints needed

Works in both blockade and intermediate regimes

Abstract

Scheme to prepare three-dimensional entangled state between a pair of Rydberg atoms is proposed via dissipative dynamics and Electromagnetic Induced Transparency (EIT) associated with the single-atom dark state. The prepared entangled state is the dark state of the whole system. The schemes are feasible no matter the system initially in arbitrary purity or mixed states and do not have accurate requirements on evolution time. In contrast to most of the former Rydberg-atombased dissipative schemes, the Rydberg-Rydberg interaction (RRI) strength do not need to satisfy a certain relation with laser detuning since it works in the blockade as well as intermediate regimes.

Figures

Equations

$${\hat{H}}_{\mathrm{Dr}}=\Delta |p⟩⟨p|+({\Omega }_1|1⟩⟨p|+{\Omega }_2|p⟩⟨R|+H.c.).$$

$$\dot{\hat{\rho }}=-i[\hat{H},\hat{\rho }]+\sum _j\mathcal{D}[{\hat{\sigma }}_j]\hat{\rho },$$

$${\hat{H}}_{\omega }=\omega |0⟩⟨1|+H.c..$$

$$|D⟩,|\zeta {⟩}_{\pm }=(2{\Omega }_1|1⟩+(\Delta \pm \tilde{\Delta })|p⟩+2{\Omega }_2|R⟩)/{\mathcal{N}}_{\pm },$$

$${\hat{H}}_{\omega }^{\prime }=\omega (⟨1|D⟩|0⟩⟨D|e^{i{E}_{0}t}+⟨1|{\zeta }_{\pm }⟩|0⟩⟨{\zeta }_{\pm }|e^{i{E}_{\mp }t})+H.c.$$

$${\hat{H}}_{\omega }^{\prime }=\frac{\omega {\Omega }_2}{\sqrt{{\Omega }_1^2+{\Omega }_2^2}}|0⟩⟨D|+H.c..$$

$$|\mathrm{\Psi }⟩=(|00⟩+|11⟩+|22⟩)/\sqrt{3},$$

$$|\mathbb{D}⟩=(|D_0⟩|D_0⟩+|D_1⟩|1⟩+|2⟩|D_2⟩)/\sqrt{3},$$

$${\hat{H}}_{\Omega }$$

$${\hat{H}}_{\omega }$$

$$\mathrm{and}$$

$${\hat{H}}_V$$

$${\hat{H}}_{eff1}={\omega }_{eff1}|D_0⟩⟨2|+{\omega }_{eff2}|D_0⟩⟨D_1|+H.c.$$

$${\hat{H}}_{eff2}={\omega }_{eff3}|D_0⟩⟨1|+{\omega }_{eff4}|D_0⟩⟨D_2|+H.c.,$$

$${\omega }_{1(2)}^{\mathrm{eff}}=⟨D_0|{\hat{H}}_{\omega 1}|2(D_1)⟩,{\omega }_{3(4)}^{\mathrm{eff}}=⟨D_0|{\hat{H}}_{\omega 2}|1(D_2)⟩,$$

$$V_{mn}^{\mathrm{eff}}=⟨D_n|⟨D_m|\hat{H}|D_m⟩|D_n⟩,$$

$${\Omega }_{mn}^{\mathrm{eff}}=⟨R_n|⟨R_m|\hat{H}|D_m⟩|D_n⟩={\Omega }_1^2{V}_{m,n}.$$

$$\dot{\hat{\rho}}=-i[\hat{H},\hat{\rho}]+\sum_{l=1}^{8}\sum_{j=0}^{2}\big{[}\hat{\mathcal{L}}_{j}^{l}\rho\hat{\mathcal{L}}_{j}^{l{\dagger}}-\frac{1}{2}(\hat{\mathcal{L}}_{j}^{l{\dagger}}\hat{\mathcal{L}}_{j}^{l}\rho+\rho\hat{\mathcal{L}}_{j}^{l{\dagger}}\hat{\mathcal{L}}_{j}^{l})\big{]},$$

$$F(|\mathbb{D}⟩⟨\mathbb{D}|,\hat{\rho }(t))\equiv ⟨\mathbb{D}|\hat{\rho }(t)|\mathbb{D}⟩.$$

$$P(\hat{\rho }(t))\equiv \mathrm{Tr}[\hat{\rho }(t{)}^2].$$

$$\mathcal{N}(\hat{\rho })=\frac{\mathrm{Tr}\sqrt{({\rho }^{T_A}{)}^{†}({\rho }^{T_A})}-1}{2}.$$

$$E_{\mathcal{N}}(\hat{\rho })\equiv {\log }_2\mathrm{Tr}\sqrt{({\rho }^{T_A}{)}^{†}({\rho }^{T_A})}:={\log }_2(2\mathcal{N}+1).$$

Full text

I introduction

Rydberg atoms are the neutral atoms with high lying excited state, and they would exhibit strong Rydberg-Rydberg interaction (RRI) when close enough TF:1994 ; MTK:2010 ; Saffman2016 . The most interesting thing caused by RRI is the Rydberg blockade, which has been observed between two Rydberg atoms through sequent ett2009 and collective driving ayt2009 , respectively. Besides, the RRI has also been directly measured in experiment lar2013 . For quantum information processing (QIP) tasks, quantum logic gates and entangled states are basic building block and resource, respectively mi2000 . The pioneering works relevant to Rydberg quantum logic gate were proposed by Jaksch et al. DJP:2000 , in which the model conditions $V~{}{(\rm RRI~{}strength)}\gg\Omega~{}{(\rm Rabi~{}frequency)}$ and $u\ll\Omega$ are considered, respectively. Lukin et al. MMR:2001 then describe a method for the coherent manipulation of quantum information stored in collective excitations of mesoscopic many-Rydberg-atom ensembles. These two works DJP:2000 ; MMR:2001 open a new chapter for Rydberg-atom-based QIP and has been followed by a varity of interesting works, typically including preparation of quantum entanglement dlk2008 ; mih2009 ; mk2009 , construction of quantum logic gate mt2005 ; mta2006 ; lmk2011 ; hzs2010 , quantum simulators hmi2009 , quantum algorithms klm2011 , and quantum repeaters ybk2010 .

Dissipation induced by the coupling between quantum system and its environment is inevitable and always be considered detrimental for QIP tasks. Commonly, there are three methods to deal with the dissipation: (i) Quantum error correction method aaj1996 , which relies on the high-fidelity quantum gate for detecting and correcting errors. (ii) Dynamical decoupling method les1999 , which seeks to minimize the unwanted system-bath interactions in an open quantum system but can never completely avoid all unitary errors gd2013 . (iii) Decoherence-free subspace method gka1996 , which requires the symmetric coupling between system and its bath. Different from the above methods that are trying to avoid the influence of dissipation, dissipative dynamics method msa1999 opens a new path to deal with decoherence since the dissipation plays a significant role to realize QIP tasks saa2008 .

The dissipative-dynamics-based schemes in Rydberg atoms are interesting since they combine the advantages of Rydberg atom and of dissipative dynamics together. Recently, Petrosyan and Mølmer found that the atomic spontaneous emission of the intermediate excited state facilitates a single excitation of the Rydberg atom ensemble with nearly unit probability dk2013 , which is meaningful for the construction of Rydberg superatom. And Li, Ates, and Lesanovsky studied the dissipative blockade for excited Rydberg atoms wci2013 . Then, Rao and Mølmer Rao2013 , and Carr and Saffman am2013 proposed pioneering works to prepare steady entangled state in Rydberg atoms via the dissipation. The scheme proposed in Ref. Rao2013 works under the blockade regime($V\gg\Omega$) as well as intermediate regime ($V\sim\Omega$) based on the Electromagnetic Induced Transparency (EIT). The steady entangled state contains the dark state induced by the EIT regime. Then the many-body steady entangled state is studied Rao2014 ; sjk2015 based on the RRI. On the other hand, the scheme proposed in Ref. am2013 works under the Rydberg antiblockade regime ctt2007 which requires the Rydberg pumping condition $V=2\Delta\gg\Omega$ ($\Delta$ denotes laser detuning) and followed by many works xjt2014 .

In this manuscript, inspired by the pioneering work in Ref Rao2013 , we design a scheme to prepare the three-dimensional entangled state, which can enhance the security and capacity of QIP ajg2011 and violate the local realism more strongly than the two-dimensional entangled state dp2000 . The prepared state is the dark state of the whole system which involves the single-atom dark state induced by the dissipative EIT and thus robust on RRI-induced mechanical effect. In addition, the scheme is insensitive to the RRI because it works under the blockade as well as the intermediate regimes and has short convergence time approaches steady state. Finally, we try to translate the dark state entangled state to the ground state entangled state via the adiabatic passage method.

The rest part of the manuscript is organized as follows. In Sec. II, the basic model of the scheme, including the dissipative EIT dark state and the approximations based on the dressed state, are introduced. In Sec. III, the basic dynamics and the performance of the scheme (including fidelity, purity and negativity) are shown based on one group of specific parameters. In Sec. IV, we discuss the robustness of the scheme for a wide range of parameters, try to transfer the dark-state-basis-based entangled state to ground state subspace, and consider experimental feasibility in some ways. The conclusions are given in Sec. V.

II Basic Model

II.1 Dissipative EIT steady state of single atom

We first consider single Rydberg atom with ladder-type energy level structure, as shown in the shaded area of Fig. 1(a). The Hamiltonian is

[TABLE]

It is obvious that the laser fields would induce EIT phenomenon ma2005 ; dk2013 and the system has a dark state $|D\rangle=(\Omega_{2}|1\rangle-\Omega_{1}|R\rangle)/\sqrt{\Omega_{1}^{2}+\Omega_{2}^{2}}$. Generally speaking, the systematic dynamics could be described well by the master equation

[TABLE]

where $\mathcal{D}[\hat{a}]\hat{\rho}\equiv\hat{a}\hat{\rho}\hat{a}^{{\dagger}}-(\hat{\rho}\hat{a}^{{\dagger}}\hat{a}+\hat{a}^{{\dagger}}\hat{a}\hat{\rho})/2$, $\hat{\sigma}_{1}=\sqrt{\gamma_{R}/2}|1\rangle\langle R|$, $\hat{\sigma}_{2}=\sqrt{\gamma_{R}/2}|p\rangle\langle R|$ and $\hat{\sigma}_{3}=\sqrt{\gamma_{p}}|1\rangle\langle p|$. Fig. 1(b) shows that the steady state of the single Rydberg atom (without consideration of $|0\rangle$) approaches to the dark state $|D\rangle$ in a short period of evolution time under dissipation.

II.2 Approximation with the dressed states

To prepare the entangled state, the coupling between ground states are required. We first consider two ground states $|0\rangle$ and $|1\rangle$, as shown in Fig. 1(a) . And the Hamiltonian is

[TABLE]

If $\omega$ is much smaller than $\Omega_{1}$ and $\Omega_{2}$, Hamiltonian (1) constitutes dressed states as

[TABLE]

with the corresponding eigenvalues $E_{0}=0$, $E_{\pm}=(\Delta\pm\tilde{\Delta})/2$, where $\tilde{\Delta}=\sqrt{\Delta^{2}+4\Omega_{1}^{2}+4\Omega_{2}^{2}}$. Thus, the whole Hamiltonian of single atom ($\hat{H}_{\rm Dr}+\hat{H}_{\omega}$) could be rewritten as

[TABLE]

after rotating with respect to the dressed states. If $E_{\pm}\gg E_{0}$, after discarding the high-frequency oscillating terms, one can approximately get

[TABLE]

Sec. II.1 shows that the steady state of the dressed state space is the dark state $|D\rangle$, and sec. II.2 shows that replacing the coupling $|0\rangle\leftrightarrow|1\rangle$ with $|0\rangle\leftrightarrow|D\rangle$ is reasonable. In the following subsection we would describe the basic dynamics of the system.

III Preparation of three-dimensional entangled dark state

III.1 Desired Entangled State

The three-dimensional entangled state has the form ajg2011

[TABLE]

which can enhance the security of QIP ajg2011 and violate the local realism more strongly than the two-dimensional entangled state dp2000 . In this manuscript, combing with the single-atom EIT dark state, we aim to prepare

[TABLE]

where $|D_{j}\rangle=(\Omega_{2}|j\rangle-\Omega_{1}|R_{j}\rangle)/\sqrt{\Omega_{1}^{2}+\Omega_{2}^{2}}$ is the dark state of the Hamiltonian $\hat{H}=\Delta|p_{j}\rangle\langle p_{j}|+\Omega_{1}|j\rangle\langle p_{j}|+\Omega_{2}|p_{j}\rangle\langle R_{j}|+{\rm H.c.}$.

III.2 Configuration and Hamiltonian

We consider two Rydberg atoms as shown in Fig. 2. Both of the atoms have two Rydberg states, two metastable states and three ground states. These two atoms interact with each other through the van-der-Waals-type RRI. The Hamiltonian of the whole system can be written as $\hat{H}=\hat{H}_{\Omega}+\hat{H}_{\omega}+\hat{V}$, in which

[TABLE]

III.3 Effective Dynamics

We now analyze the effective dynamics of the whole system. Based on the process similar to that discussed in Sec. II.2, the Hamiltonian of atom 1 can be approximated to

[TABLE]

and of atom 2 can be approximated to

[TABLE]

as shown in Fig. 3(a). The effective microwave coupling can be calculated through

[TABLE]

where $\hat{H}_{\omega k}=\omega(|1\rangle_{k}\langle 0|+|1\rangle_{k}\langle 0|)+{\rm H.c.}$ denotes the microwave field Hamiltonian of the k-th Rydberg atom. And the effective interaction strength between $|D\rangle_{m}$ of atom 1 and $|D\rangle_{n}$ of atom 2 can be calculated through

[TABLE]

where we define $\langle D_{n}|\langle D_{m}|$ is the conjugate transpose of $|D_{m}\rangle|D_{n}\rangle$ with $m=0,1$ and $n=0,2$. Fig. 3(b) describes the effective dynamics of the whole system. The effective coupling between $|D_{m}\rangle|D_{n}\rangle$ and $|R_{m}\rangle|R_{n}\rangle$, $\Omega_{mn}^{\rm eff}$, can be calculated as

[TABLE]

The effective dynamics of the whole system can be illustrated as follows. Without consideration of RRIs, the two-atom Hamiltonian has three dark states $|\mathbb{D}\rangle$, $|\mathbb{D}_{1}\rangle$ and $|\mathbb{D}_{2}\rangle$ and six bright eigenstates denoted by the set $\mathbb{S}$. $|\mathbb{D}_{1}\rangle$, $|\mathbb{D}_{2}\rangle$ and $\mathbb{S}$ contain at least one of the states $|D_{1}\rangle|D_{0}\rangle$, $|D_{0}\rangle|D_{2}\rangle$ and $|D_{1}\rangle|D_{2}\rangle$ and maybe contain $|D_{0}\rangle|D_{0}\rangle$ among the state $|D_{m}\rangle|D_{n}\rangle$. While $|\mathbb{D}\rangle$ only contain $|D_{0}\rangle|D_{0}\rangle$ among the state $|D_{m}\rangle|D_{n}\rangle$. As mentioned above, the two-atom dark state $|D_{m}\rangle|D_{n}\rangle$ couples to the two-excitation Rydberg state $|R_{m}\rangle|R_{n}\rangle$ via the strength $\Omega_{mn}^{\rm eff}$. And $|R_{m}\rangle|R_{n}\rangle$ couples with the metastable states which would decay to the space $\mathcal{S}\equiv\{|\mathbb{D}\rangle$, $|\mathbb{D}_{1}\rangle$, $|\mathbb{D}_{2}\rangle$, $\mathbb{S}$ }. If $\Omega_{00}^{\rm eff}\ll\{\Omega_{10}^{\rm eff},\Omega_{02}^{\rm eff},\Omega_{12}^{\rm eff}\}$ is satisfied, the symmetry of the system is broken because the transition path from $|\mathbb{D}\rangle$ to the two-excitation Rydberg states could be neglected. In this case, given any initial state in the ground state subspace, it could be quickly transformed to the space $\mathcal{S}$. If $|\mathbb{D}\rangle$ is populated, the scheme is succeed. Otherwise the state would be excited and decay, and recycling until $|\mathbb{D}\rangle$ is prepared. In the next subsection, we would consider the performance of the scheme.

III.4 Performance of the scheme

We get the final state of the whole system through numerically solving the master equation

[TABLE]

where $\hat{\mathcal{L}}_{j}^{1}=\sqrt{\gamma_{R}/3}|j\rangle_{1}\langle R_{0}|,~{}\hat{\mathcal{L}}_{j}^{2}=\sqrt{\gamma_{R}/3}|j\rangle_{1}\langle R_{1}|,~{}\hat{\mathcal{L}}_{j}^{3}=\sqrt{\gamma_{p}/3}|j\rangle_{1}\langle p_{0}|,~{}\hat{\mathcal{L}}_{j}^{4}=\sqrt{\gamma_{p}/3}|j\rangle_{1}\langle p_{1}|,~{}\hat{\mathcal{L}}_{j}^{5}=\sqrt{\gamma_{R}/3}|j\rangle_{2}\langle R_{0}|,~{}\hat{\mathcal{L}}_{j}^{6}=\sqrt{\gamma_{R}/3}|j\rangle_{2}\langle R_{2}|,~{}\hat{\mathcal{L}}_{j}^{7}=\sqrt{\gamma_{p}/3}|j\rangle_{2}\langle p_{0}|$, and $\hat{\mathcal{L}}_{j}^{8}=\sqrt{\gamma_{p}/3}|j\rangle_{2}\langle p_{2}|$ denote the dissipative terms.

Fidelity denotes the closeness of states in the natural geometry of Hilbert space, which can be calculated through r1994

[TABLE]

Purity is a measure of how pure a quantum state is and can be calculated as g2007

[TABLE]

One can conclude that the scheme performs well when both of fidelity and purity are very close to 1. In addition, negativity, a measure deriving from the PPT criterion for separability kpa1998 , has shown to be a proper measure of entanglement with the definition kpa1998

[TABLE]

Then, the logarithmic negativity is proposed with the definition m2005

[TABLE]

Based on the final state $\hat{\rho}(t)$, one can calculate the fidelity, purity and negativity of our scheme with the above definitions, as shown in Figs. 4 and 5 with one group of specific parameters. The results show that both of the fidelity and purity are close to 1 and the negativity is close to the ideal value.

IV Discussions

IV.1 Variation of parameters

Experimentally, the obtainable parameters are not unique. Thus, it is necessary to see the performance of the scheme under various parameters.

In Fig. 6, we plot the fidelity versus the variations of $\Omega_{2}/\Omega_{1}$ and $\omega_{1}/\Omega_{1}$, which shows the scheme may have a good performance in a wide range values of parameters $\Omega_{2}$ and $\omega_{1}$. In Fig. 7, we plot the fidelity and purity versus $V_{10},~{}V_{12},~{}V_{00},~{}\gamma_{p}$ and $\gamma_{R}$, respectively. The results show a good robustness on the variation of RRI, which is absolutely different to the Rydberg-antiblockade-based schemes and may release the experimental requirements.

IV.2 Transforming the entanglement to ground state subspace

In this subsection, we try to transfer $|\mathbb{D}\rangle$ to the three-dimensional entangled state in the ground state subspace, as shown in Eq. (7), via the adiabatic technique. To do this, we should turn off $\Omega_{1}$ adiabatically. For simplicity, we choose $\Omega_{1}(\tau)=\cos[\pi\tau/(2T)]$, and the result is shown in Fig. 8, where the beginning of $\tau$ is the end of the time in Fig. 4. Although the fidelity is not higher than 0.99, the performance could be further improved through using the shortcut to adiabaticity method and design the pulse shape more carefully. And we just numerically show that to transfer the entangled state to the ground state subspace is feasible.

IV.3 Experimental considerations

For practical experiments, the ground state energy level of the two atoms could be chosen as $|0\rangle\equiv|F=1,m_{f}=0\rangle$, $|1\rangle\equiv|F=1,m_{f}=-1\rangle$ and $|2\rangle\equiv|F=1,m_{f}=+1\rangle$ of 5$S_{1/2}$. And the metastable energy level of the two atoms could be chosen as $|p_{0}\rangle\equiv|F=1,m_{f}=0\rangle$, $|p_{1}\rangle\equiv|F=1,m_{f}=-1\rangle$ and $|p_{2}\rangle\equiv|F=1,m_{f}=+1\rangle$ of 5$P_{3/2}$. Rydberg state energy level should be chosen carefully because the asymmetric RRI and dipole transition selection rules.

The asymmetric RRI is crucial for our scheme and has been widely used for many Rydberg-atom-based quantum information processing schemes mk2009 ; am2013 ; eak2007 ; lmk2011 . In Ref. mk2009 , some groups of reasonable asymmetric RRI parameters are used and predicted to satisfy $\{\Delta_{sp},\Delta_{ss}\}\gg\Delta_{pp}$, where $\Delta_{sp}$, $\Delta_{ss}$ and $\Delta_{pp}$ correspond to $V_{10}(V_{02})$, $V_{12}$ and $V_{00}$ of the present scheme, respectively. The Rydberg states considered in Ref. mk2009 are $|s\rangle=|41s_{1/2},~{}m=1/2\rangle$ and $|p\rangle=|40p_{3/2},~{}m=1/2\rangle$, and the maximal, minimum and average asymmetry [$\Delta_{sp}(\sim n^{4}/R^{3})/\Delta_{pp}(\sim n^{11}/R^{6})$] are about 1400, 185, 757, respectively. However, we cannot use these results directly because the lifetime $\tau$ of $n=40$ is about $57\mu s$ which is not enough for our scheme to achieve a high fidelity. From this point, Rydberg states with higher principal quantum numbers are preferred because $\tau\sim n^{3}$ TF:1994 ; MTK:2010 . Nevertheless, for a fixed distance, the principal quantum number should not be too large because the asymmetry $\sim n^{-7}$. Based on these rules, if we choose $|R_{0}\rangle\equiv|70p_{3/2}~{}m=1/2\rangle$(for two atoms), $|R_{1}\rangle\equiv|71s_{1/2}~{}m=3/2\rangle$(for atom 1) and $|R_{2}\rangle\equiv|71s_{1/2}~{}m=3/2\rangle$(for atom 2), the maximal asymmetry about 27.8523 and the lifetime $\tau\simeq 305\mu s$ are achieved. It should be noted that the transition between $|p_{0}\rangle$ and $|R_{0}\rangle$ could be realized via introducing a large detuning interaction with intermediate $|s\rangle$ or $|d\rangle$ state(similar to the degenerate two-photon process). With these assumptions and assume the other factors which influence the RRI keep invariant, the fidelity and purity are estimated about 0.6047 and 0.507, respectively, at $\tau\simeq 305\mu s$ with $\Omega_{1}$, $\Omega_{2}$ and $\omega$ parameters the same as that in Fig. 4. Similarly, we roughly estimate the performance for different principal quantum numbers at the corresponding lifetime with the maximal asymmetry for different principal quantum numbers, as shown in Fig. 7.

We use the dipole-dipole interactions rather than vdW interaction for two different Rydberg states for simulation.

Besides, we could also choose $|R_{0}\rangle\equiv|75p_{3/2}~{}m=1/2\rangle$(for two atoms), $|R_{1}\rangle\equiv|125s_{1/2}~{}m=3/2\rangle$(for atom 1) and $|R_{2}\rangle\equiv|120s_{1/2}~{}m=3/2\rangle$(for atom 2). We suppose the interaction between states $|R_{0}\rangle_{1}$ and $|R_{0}\rangle_{2}$ ($|R_{1}\rangle_{1}$ and $|R_{2}\rangle_{2}$) is van der Waals (vdW) interaction with strength $V_{00}$ ($V_{12}$) while between $|R_{1}\rangle_{1}$ and $|R_{0}\rangle_{2}$ ($|R_{0}\rangle_{1}$ and $|R_{2}\rangle_{2}$) is dipole-dipole interaction with strength $D_{10}$ ($D_{02}$). Due to the principal quantum number differences, vdW strength $V_{12}$ could be larger than $V_{00}$. Meanwhile, the Stark tuned Förster resonances have been demonstrated experimentally idi2010 ; shd2014 due to the dipole-dipole interaction and the detuning could be adjusted through the electric field. Then, one could achieve the non-resonant dipole-dipole interaction induced vdW between $|R_{1}\rangle_{1}$ and $|R_{0}\rangle_{2}$ ($|R_{0}\rangle_{1}$ and $|R_{2}\rangle_{2}$). Under this case, the higher fidelity and purity may be achieved due to the long lifetime Rydberg states.

V Conclusion

In this manuscript, we proposed a scheme to prepare three-dimensional dark state entanglement based on the dissipative EIT and Rydberg blockade regimes. In contrast to the Rydberg-antiblockade-based schemes, the present one is insensitive to the variation of RRI and has a shorter evolution time approaches steady state, which may release the experimental requirements. In addition, the scheme does not have accurate requirement on the evolution time and is feasible even the system is initially in the mixed state.

ACKNOWLEDGMENT

This work was supported by National Natural Science Foundation of China under Grant No. 11804308.