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Abstract
The multipartite GHZ states are useful resources for quantum information processing. Here we put forward a scalable way to adiabatically prepare the multipartite GHZ states in a chain of Rydberg atoms. Building on the ground-state blockade effect of Rydberg atoms and the stimulated Raman adiabatic passage (STIRAP), we suppress the adverse effect of the atomic spontaneous emission, and obtain a high fidelity of the multipartite GHZ states without requirements on the operational time. After investigating the feasibility of the proposal, we show a 3-qubit GHZ state can be generated in a wide range of relevant parameters and a fidelity above $98\%$ is achievable with the current experimental technologies.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Entanglement is an elementary static resource of quantum mechanics. It provides the possibilities to test quantum mechanics against local hidden theory [1,2], and performs various applications in quantum information processing, such as superdense coding [3,4], quantum teleportation [5–7], quantum fingerprinting [8,9], and direct characterization of quantum dynamics [10]. While the entangled states of bipartite systems have been studied substantially [11,12], the multipartite entangled states are of more great interest for quantum information and quantum computation [13–26], which can allow us to research the richer and more complex structures of a many-body quantum system [27]. However, the correlations they exhibit cannot be regarded as a trivial generalization of bipartite entanglement. As the number of particles for an entangled state exceeding two, the evaluation of the entanglement will be more difficult.
For the numerous multipartite entangled states, Greenberger- Horne-Zeilinger (GHZ) states [28] represent a paradigmatic example, which play an important role in fundamental tests of quantum mechanics. They not only manifest a conflict with local realism for non-statistical predictions of quantum mechanics [28], but also serve as a reference in quantum estimation theory yielding the Heisenberg scaling [13]. Therefore, the preparation of multipartite GHZ states is an indispensable part and many schemes have been proposed to create the multipartite GHZ states via a single- or multi-step process [29–31]. Among the more powerful and spectacular strategies are adiabatic passage schemes, in which an interference-induced dark state provides a possibility for the efficient transfer of population from the source to target state. It has some advantages because the spontaneous emission from excited states can be suppressed and it is robust against experimental parameter errors. The most famous examples of adiabatic passages are the stimulated Raman adiabatic passage (STIRAP) [32–34]. It is also an extensively used tool in quantum information processing tasks besides preparations of entanglement [35–37].
Neutral atoms excited to high principal quantum number states are now emerging as a reliable tool for investigating the few- and many-body physics, since they can interact through strong and long-range dipole-dipole or van der Waals interaction [30]. After ingeniously integrating the interaction with other techniques, a lot of intriguing effects have been given rise to, where the two notable instances are Rydberg blockade [38] and antiblockade [39]. The former is capable of suppressing the simultaneous excitation for multiple Rydberg atoms. On the contrary, the latter will facilitate the simultaneous excitation of multiple Rydberg atoms and significantly inhibit the Rydberg blockade effect. Benefiting from these attractive effects, a variety of proposals were designed theoretically and experimentally for quantum computation [40–50], entanglement generation [51–63], quantum algorithms [64], quantum simulators [65], and quantum repeaters [66].
By virtue of a strong Rydberg antiblockade effect and a weak Raman transition, our group [67] further put forward another effect, the ground-state blockade of Rydberg atoms. Differing from the Rydberg blockade and antiblockade, this mechanism will prevent the double occupation of a certain ground state. It can maintain the nonlinear Rydberg-Rydberg interaction, and avoid the adverse effect of atomic spontaneous emission by restraining the evolution of the system into the subspace spanned by the ground states. We have applied the effect to obtain a Bell state and a $W$ state. Quite recently, Ostmann et al. [58] presented three different protocols with a multistep process in chains of Rydberg atoms, which can be used to prepare an antiferromagnetic GHZ state and a class of matrix product states. But the scheme is not good at resisting against atomic spontaneous emission and a concrete control on time of evolution is demanded. It inspires us to exploit the ground-state blockade to generate multipartite GHZ states in chains of Rydberg atoms.
In this paper, we propose a novel scheme to prepare a $N$-qubit GHZ state ($N=3,4,5\dots$) in a chain of four-level Rydberg atoms via the combination of the ground-state blockade effect and the STIRAP, where the atoms all consist of two ground states $|0\rangle ,|1\rangle$ (encoded quantum qubits), one excited state $|p\rangle$, and one Rydberg state $|r\rangle$. To intuitively explain the operational principle, we take the case of the three-qubit GHZ state $(|010\rangle +|101\rangle )/\sqrt {2}$ as an example. Due to the ground-state blockade effect, the initial state $|000\rangle$ will evolve into $(|010\rangle +|100\rangle )/\sqrt {2}$, which will further transform into the target state $(|010\rangle +|101\rangle )/\sqrt {2}$. First, there needs no accurate control over the operational time as the technique of STIRAP. Furthermore, the present scheme is robust against the atomic spontaneous emission and need not precisely tailored Rabi frequencies, Rydberg-mediated interaction and the detuning of the driving fields. The designs of other relevant parameters are also flexible. These will be demonstrated in the following in detail. Additionally, we thoroughly investigate the experimental feasibility with the state-of-the-art technology and a high fidelity above $98\%$ can be acquired.
2. Principle of the present scheme
To prepare the $N$-qubit GHZ state, our scheme needs a $(N-1)$-step process, which can be divided into two groups: Step 1 and the other steps (Step $n$, $n=2,3,\ldots ,N-1$). The concrete operations of the two groups and the corresponding atomic energy levels of $N$ identical Rydberg atoms have been shown in Fig. 1. For the Step 1, only the first two atoms will be addressed individually by three lasers. The ground states $|0\rangle$ and $|1\rangle$ of the first two atoms are dispersively coupled to the excited states $|p\rangle$ by the lasers with Rabi frequencies $\Omega _a(t)$ and $\Omega _b(t)$, detunings $-\Delta _p$, respectively. In order to avoid the precise termination of the operation, we integrate the STIRAP into our proposal by introducing the laser pulses in possession of Rabi frequencies,
where $T$, $t_c$ and $\tau$ are the pulse period, the pulse duration and the delay between the pulses, respectively. Moreover, the transitions $|1\rangle \leftrightarrow |r\rangle$ of the two atoms are driven by the lasers with Rabi frequencies $\Omega _c$, detunings $-\Delta _r$. In the interaction picture, the Hamiltonian can be written as
Fig. 1. The total scheme preparing the $N$-qubit GHZ state composes of $N-1$ steps, which can be divided into two groups: Step 1 and Step $n~(n=2,3,\ldots ,N-1)$. For the Step 1, only the first two atoms will be addressed individually by three lasers, respectively. As for the Step $n$, we only coupled the $n$-th atom to one laser and the $(n+1)$-th atom to three lasers, respectively.
After performing a rotation with respect to $\exp (-it\Delta _p|p\rangle \langle p|-it\Delta _r|r\rangle \langle r|)$, the Hamiltonian can be simplified as
According to the principle of Rydberg ground-state blockade, we diagonalize the terms, $\Omega |11\rangle _{12}\langle rr|+\textrm {H.c.}$, to rewritten the Eq. (5) as,
While the Step 1 is finished, the other steps summarized as Step $n$ ($n=2,3,4\cdots N$-1) will be executed successively. As shown in Fig. 1, for the Step $n$, only the $n$-th and $(n+1)$-th atoms interact with external lasers. The $n$-th atom couples with a single laser (Rabi frequency $\Omega _c$, detuning $\Delta _r$) to realize the transition $|1\rangle _n\leftrightarrow |r\rangle _n$, and the manipulations of $(n+1)$-th atom are the same as those of the atoms in Step 1. The associated Hamiltonian is
Owing to the previous steps, the initial state of Step $n$ will be the state $|\psi _\textrm {even}\rangle$ ($n$ is an even number) or $|\psi _\textrm {odd}\rangle$ ($n$ is an odd number) with $|\psi _\textrm {even}\rangle =(|0101\cdots 01\rangle _{1234\cdots n-1}\otimes |00\rangle _{n,n+1}\otimes |0\cdots 0\rangle _{n+2\cdots N}+|1010\cdots 10\rangle _{1234\cdots n-1}\otimes |10\rangle _{n,n+1}\otimes |0\cdots 0\rangle _{n+2\cdots N})/\sqrt {2}$ or $|\psi _\textrm {odd}\rangle =(|0101\cdots 010\rangle _{1234\cdots n-3,n-2,n-1}\otimes |10\rangle _{n,n+1}\otimes |0\cdots 0\rangle _{n+2\cdots N}+|1010\cdots 101\rangle _{1234\cdots n-3,n-2,n-1}\otimes |00\rangle _{n,n+1}\otimes |0\cdots 0\rangle _{n+2\cdots N})/\sqrt {2}$. And there are only the $n$-th and $(n+1)$-th atoms in action for Step $n$. Therefore, referring to the Eq. (11), the state $(|01\rangle _{n,n+1}+|10\rangle _{n,n+1})/\sqrt {2}$ can be achieved from $(|00\rangle _{n,n+1}+|10\rangle _{n,n+1})/\sqrt {2}$. It is worth mentioning that there will be a relative phase between the terms of $|01\rangle _{n,n+1}$ and $|10\rangle _{n,n+1}$, which can be eliminated by a single qubit logical gate operated on the subspace $\{|0\rangle ,|1\rangle \}$.
In order to certify the feasibility of our proposal and the validity of the effective system, we show the shapes of pulses to prepare the $N$-qubit GHZ states, and plot the fidelity of the corresponding GHZ states governed by the original Hamiltonian and the effective Hamiltonian in Fig. 2. The aim of Figs. 2(a) and 2(c) is to prepare the 3-qubit GHZ state $|\varphi _2\rangle =(|010\rangle +|101\rangle )/\sqrt {2}$. We first apply the Step 1 to evolve the system from the initial state $|000\rangle$ to the state $|\varphi _1\rangle =(|010\rangle +|100\rangle )/\sqrt {2}$ at the timescale $\Omega _ct\in [0,6000]$. The fidelity of $|\varphi _1\rangle$ arrives at $99.65\%$ in the end of Step 1. In the Step 2 ($\Omega _ct\in (6000,14000]$), the state $|\varphi _1\rangle$ is transferred to the 3-qubit GHZ state $|\varphi _2\rangle$, and the target state owns a high fidelity of up to $99.57\%$. Similarly, in Figs. 2(b) and 2(d), we intend to prepare the 4-qubit GHZ state $|\phi _3\rangle =(|0101\rangle +|1010\rangle )/\sqrt {2}$. For the Step 1, the state $|\phi _1\rangle =(|0100\rangle +|1000\rangle )/\sqrt {2}$ comes from the initial state $|0000\rangle$ as $\Omega _ct\in [0,6000]$. Next, the Step 2 transfers the state $|\phi _1\rangle$ to the state $|\phi _2\rangle =(|0100\rangle +|1010\rangle )/\sqrt {2}$ at $\Omega _ct\in (6000,14000]$. Ultimately, the 4-qubit GHZ state $|\phi _3\rangle$ is successfully generated with a high fidelity $99.49\%$ in the Step 3 ($\Omega _ct\in (14000,22000]$). In addition, the empty circles of Figs. 2(c) and 2(d) are the fidelity of $|\varphi _2\rangle$ and $|\phi _3\rangle$ governed by the effective Hamiltonian. Their behaviors are identical to those of corresponding solid lines. Thus the feasibility of the original system and the validity of the reduced system are adequately proven. The tendencies of the original system can be accurately predicted by the reduced system.
Fig. 2. (a) The shapes of pulses to prepare the 3-qubit GHZ state. For Step 1 ($\Omega _c t\in [0,6000]$), we set $t_c=6000/\Omega _c$. For Step 2 ($\Omega _c t\in (6000,14000]$), we set $t_c=8000/\Omega _c$. (b) The shapes of pulses to prepare the 4-qubit GHZ state. For Step 1 ($\Omega _c t\in [0,6000]$), we set $t_c=6000/\Omega _c$. For Steps 2 and 3 ($\Omega _c t\in (6000,14000]$ and $(14000,22000]$), we set $t_c=8000/\Omega _c$. (c) and (d) are the fidelity of the $3$- and $4$-qubit states governed by the original Hamiltonian and the effective Hamiltonian, where the fidelity of state $\rho _i=|i\rangle \langle i|$ is defined as $F=\textrm {Tr}\sqrt {\rho _i^{1/2}\rho (t)\rho _i^{1/2}}$ and $\rho (t)$ is the density matrix of system at time $t$. The other relevant parameters are all chosen as: $\Omega _a=\Omega _b=\sqrt {0.05}\Omega _c$, $\Delta _p=20\Omega _c$, $\Delta _r=20\Omega _c$, $T=0.15t_c$, and $\tau =0.1t_c$.
3. Analysis on the influences of relevant parameters
As is well known, the heart of the Rydberg antiblockade is a rigorous relation between the strength of Rydberg-mediated interaction and the detuning of the driving fields [39]. It obviously increases the experimental complexity. However, the Rydberg antiblockade effect of our scheme is easier to achieve. Because there is no rigorous requirement on the strength of Rydberg-mediated interaction and the detuning of the driving fields. In the previous derivation, we assume $U_{n,n+1}=U_{12}=2\Delta _r-\Omega$ for simplicity, which is not necessary. In actual fact, there can be a difference $\delta$, i.e.,$U_{n,n+1}=U_{12}=2\Delta _r-\Omega +\delta$. The Hamiltonian of Eq. (5) can be represented as
In Fig. 3, we investigate the dynamical evolution of the fidelity for the 3-qubit GHZ state with different $\delta$. When the $\delta$ is $0.01\Omega _c$, $0.03\Omega _c$, and $0.06\Omega _c$, the ratios of $\tilde \Omega _+$ to $\Omega _a\Omega _b\sin \theta /\Delta _p$ and $\tilde \Omega _-$ to $\Omega _a\Omega _b\cos \theta /\Delta _p$ are respectively equal to $43.14,~50.33,~63.68$ and $37.13,~32.15,~26.23$. Consequently, the fidelities of the target state are all as great as $99.57\%$ at $\Omega _ct=14000$. And the appearances fully affirm the experimental flexibility of our scheme.
Fig. 3. The dynamical evolution of the fidelity for the 3-qubit GHZ state with different $\delta$. For the Step 1 and Step 2, we set $t_c=6000/\Omega _c$ and $t_c=8000/\Omega _c$, respectively. The other relevant parameters are $\Omega _{a}=\Omega _b=\sqrt {2}\Omega _c$, $\Delta _p=800\Omega _c$, $\Delta _r=20\Omega _c$, $T=0.15t_c$, and $\tau =0.1t_c$.
In Fig. 4, we exhibit the fidelity of the 3-qubit GHZ state as functions of time with different $\Delta _r$. While the $\Delta _r$ is respectively chosen as $10\Omega _c$, $100\Omega _c$, and $400\Omega _c$, the fidelity of the target state will arrive at $99.05\%$, $99.76\%$, and $98.22\%$ in succession, which means the quality of the target state is not a linear growth with the values of $\Delta _r$. It can be interpreted by the limiting conditions $\Delta _r\gg \Omega _c$ and $\Omega =2\Omega _c^2/\Delta _r\gg \max [\Omega _a(t)\Omega _b(t)/\Delta _p]$, where a little $\Delta _r$ or a large one will destroy the former or the latter. Although there may be an optimal value of $\Delta _r$ for the preparation of GHZ states, it still has a wide range for $\Delta _r$ to ensure the target state with a high fidelity above $99\%$.
Fig. 4. The dynamical evolution of the fidelity for the 3-qubit GHZ state with different $\Delta _r$. For the Step 1 and Step 2, we set $t_c=6000/\Omega _c$ and $t_c=8000/\Omega _c$, respectively. The other relevant parameters are $\Omega _{a}=\Omega _b=\sqrt {2}\Omega _c$, $\Delta _p=800\Omega _c$, $\delta =0$, $T=0.15t_c$, and $\tau =0.1t_c$.
Besides the outstanding flexibility, our scheme is also immune to the atomic spontaneous emission, because the adiabatic elimination of the excited states can be perfectly executed by the suitable regulation on $\Delta _p$, $\Omega _a$, and $\Omega _b$.
The original master equation of the $j$-th step ($j=1,2,\ldots ,N-1$) can be written as
whereIn the Fig. 5, we illustrate the fidelity of the 3-qubit GHZ state by the Eq. (15) with the different $\Delta _p$ and a fixed ratio $\Omega _a\Omega _b/\Delta _p$. When $\Delta _p$ is selected as $20\Omega _c,$ $200\Omega _c$, $400\Omega _c$, and $1600\Omega _c$ in turn, the corresponding fidelity of the 3-qubit GHZ state is equal to $74.96\%$, $96.10\%$, $97.55\%$ and $98.45\%$ at $\Omega _ct=14000$. These performances soundly indicate the robustness of the scheme against atomic spontaneous emission is strengthened via increasing $\Delta _p$ (for a fixed ratio $\Omega _a\Omega _b/\Delta _p$). It is also in accordance with the character of the ground-state blockade effect [67].
Fig. 5. The fidelity of the $3$-qubit GHZ state governed by the original master equation. The decay rates are $\gamma _p=3\Omega _c$ and $\gamma =0.001\Omega _c$. The other relevant parameters are the same as those of Fig. 2(c).
In an experiment, one of the key elements for our scheme is the single site addressability of the Rydberg atoms. Referring to [58,69–73], arraying a group of Rydberg atoms into various geometries has been an available experimental technology. Particularly, Nogrette et al. [69] demonstrated single-atom trapping in two-dimensional arrays of microtraps with arbitrary geometries. And Endres et al. [72] took advantage of atom-by-atom assembly to implement a platform for the deterministic preparation of regular one-dimensional arrays of individually controlled cold atoms. In addition, based on the techniques of optical tweezer arrays with Rydberg atoms, Ostmann et al. [58] also devised GHZ and matrix product states engineering and quantum state transport in a one-dimensional geometry. Another core technology of our scheme is the ground-state blockade effect of Rydberg atoms, the experimental feasibility of which has been fleshed out in more detail in [67]. And the Rabi laser frequencies of $\Omega _a$ and $\Omega _b$ can be altered continuously between $2\pi \times (0,100)$ MHz [61,74]. As for the transition $|1\rangle \leftrightarrow |r\rangle$, it can be realized by two indirect transitions with Rabi frequencies $\Omega _1, \Omega _2$ and a common detuning $\Delta$ [53,75,76], and the $\Omega _c$ can be equivalent to $\Omega _1\Omega _2/\Delta$. Consequently, the desired value of $\Omega _c$ can be obtained by experimentally tuning $\Delta$ at will. In [50,67,74,77,78], we can acquire a group of experimental parameters for the atomic decay rates in the ${}^{87}$Rb atom with $(\gamma _e,\gamma )=2\pi \times (3,0.001)$ MHz, where the excited state and the Rydberg state respectively correspond to the $5p_{3/2}$ atomic state and the $97d_{5/2}$ Rydberg state. According to the decay rates and the other experimental parameters $(\Omega _c,\Omega _a,\Omega _b,\Delta _p,\Delta _r)=2\pi \times (1,1.4,1.4,800,20)$ MHz, we numerically calculate the fidelity of the 3-qubit GHZ state, which reaches $98.42\%$, where the $t_c$ for the Step 1 and Step 2 are $6000/\Omega _c$ and $8000/\Omega _c$, $T$ is $0.15t_c$, and $\tau$ is $0.1t_c$. All of the analyses reliably confirm the experimental feasibility of our scheme.
4. Summary
In summary, we elaborately achieve a scheme to adiabatically prepare multipartite GHZ state in a chain of Rydberg atoms. Employing the ground-state blockade of Rydberg atoms and the technique of the STIRAP, the scheme simultaneously possesses the robustness against atomic spontaneous emission and the flexibility for experimental operations. Moreover, there is no rigorous requirement on the relevant parameters and they can be varied in wide ranges during the course of an experiment. We reasonably investigate the influences of relevant parameters, scientifically explore the feasibility of the scheme by the current experimental parameters, and finally obtain a high fidelity of 3-qubit GHZ state above $98\%$. We believe our scheme supplies a new prospect with regard to the generation of entangled states.
Funding
National Natural Science Foundation of China (NSFC) (11774047).
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