Compared with a classical computer, a quantum computer possesses greater capability and faster speed to solve some enormous problems and process massive amounts of information; therefore, it has attracted attention from researchers in recent decades.^[1–3] As is well known, all gate operations in quantum computation can be constructed by a series of elementary one-qubit unitary gates and two-qubit logical gates.^[4,5] To put quantum computing into practice, therefore, the quantum gate has been studied theoretically and experimentally in the past several decades.^[6–15] A lot of schemes of implementing quantum gates have been proposed by using diverse physical systems, such as the ion-trap system,^[16] linear optical system,^[17] cavity-QED system,^[18–20] NMR system,^[21] and superconducting system.^[22,23]

As a promising candidate for implementing quantum computation, neutral atoms are of particular interest because of the long coherence time of internal atomic states. Their stable hyperfine energy states are especially suited for encoding logic qubits, and are easily controllable and measurable by using a resonant laser pulse.^[24] On the other hand, neutral atoms exhibit large dipole moments when they are excited to Rydberg states, which induces powerful and long-range van der Waals or dipole–dipole interaction. This interaction between the excited Rydberg atoms can lead to Rydberg blockade suppressing resonant optical excitation of over-one Rydberg atoms.^[25,26] Many protocols of quantum information processing have been proposed via Rydberg–Rydberg interaction (RRI).^[27–33] Instead, when a certain relation between Rydberg interaction strength and the detuning between frequencies of atomic transitions and those of corresponding driving lasers is satisfied, the atoms can also be collectively excited to Rydberg states, which results in the so-called Rydberg antiblockade. In the last several years, the Rydberg antiblockade mechanism has been widely applied in quantum information processing and quantum computation.^[34–43] For example, Su et al. implemented quantum gates in the presence of Rydberg antiblockade;^[36–38] Shao et al. came up with the Rydberg ground-state antiblockade regime;^[39,40] and then Ji et al.^[41] and Zhao et al.^[42] proposed schemes to fuse entanglement and generate entanglement, respectively, via Rydberg ground-state antiblockade.

In this paper, we propose a scheme for the implementation of a quantum phase gate between two atoms by combining adiabatic passage and Rydberg antiblockade. We carefully design the detuning between frequencies of atomic transitions and those of corresponding driving lasers so as to compensate the blockade effect of two Rydberg atoms. An effective Hamiltonian, which denotes a single-photon detuning Λ-type three-level system and involves the quantum state of two atoms excited simultaneously, is obtained. In the context of the Λ-type three-level effective Hamiltonian, the adiabatic-passage technique is employed to implement a two-qubit phase gate by adopting two time-dependent Rabi frequencies. Adiabatic passage is widely used by slowly varying time-dependent parameters to drive the evolution of a quantum system along its certain eigenstate, generally a dark state with zero eigenenergy, to achieve the desired population transfer. The successful implementation of the two-qubit phase gate by means of adiabatic passage means that the access to the high-fidelity phase gate does not require the precise control of operation time, and that the scheme is robust against atomic spontaneous emission with a very low Rydberg-state dissipative rate.

Figure 1 shows the physical model for implementing the two-qubit phase gate. There are two Rydberg atoms trapped in two separated microscopic dipole traps. Atom 1 possesses one high-lying Rydberg state |r⟩ and two hyperfine ground states |0⟩ and |1⟩, while atom 2 has one more ground state |g⟩ than atom 1. For atom 1, transitions |1⟩₁ ↔ |r⟩₁ and |0⟩₁ ↔ |r⟩₁ are driven by classical lasers with Rabi frequencies Ω₁ and and corresponding red detuning Δ₁ and blue detuning Δ₂, respectively. For atom 2, transitions |1⟩₁ ↔ |r⟩₁ and |g⟩₁ ↔ |r⟩₁ are driven by classical lasers with Rabi frequencies and Ω₂ and corresponding blue detuning Δ₂ and red detuning Δ₁, respectively. The quantum information is coded in quantum states |0⟩ and |1⟩ of two atoms, and the desired transformation for the target π-phase gate is

Fig. 1.

	Figure Option ViewDownloadNew Window
	Fig. 1. (color online) Sketch of atomic level structures and transitions.

With rotating-wave approximation, the Hamiltonian of the whole system can be written as (assuming ħ = 1):

Hamiltonian H_eff in Eq. (7) does not involve |00⟩, |01⟩, and |10⟩, which indicates that |00⟩ → |00⟩, |01⟩ → |01⟩, and |10⟩ → |10⟩ are accessible readily. If |11⟩ → −|11⟩ is accomplished, the target two-qubit π-phase gate will be implemented. In this section, we next show how to achieve |11⟩ → −|11⟩ by using adiabatic passage.

First of all, we change Ω₁ and Ω₂ to time-dependent Ω₁(t) and Ω₂(t), respectively, while other parameters keep time-independent. Hence, Ω_a, Ω_b, and δ_rr become time-dependent Ω_a(t), Ω_b(t), and δ_rr(t), respectively. The instantaneous eigenstates of Hamiltonian H_eff in Eq. (7) are given by

Based on the theory of adiabatic approximation,^[45] slowly varying parameters of the quantum system will constrain the transitions between different eigenstates.^[46] If the effective system is in an eigenstate |Φ_n(t)⟩ (n = +,0,−) initially, then at an arbitrary time the state of the effective system will be^[47]

In order to meet the boundary conditions θ(0) = 0 and θ(t_f) = π, the Rabi frequencies Ω₁(t) and Ω₂(t) can be chosen as

Fig. 2.

	Figure Option ViewDownloadNew Window
	Fig. 2. (color online) (a) Time-dependence of Ω1(t) and Ω2(t). (b) Time-dependence of Ωa(t) and Ωb(t). (c) Time-dependence of θ(t) = arccos[Ωb(t)/Ω(t)]. Parameters used here: tc = 3000/Ω0, T = 0.18tc, τ = 0.12tc, , , and Δ2 = 10Ω0.

The transformation |11⟩ → −|11⟩ is a key to the implementation of the π-phase gate, and in Fig. 3(a), thus with |11⟩ being the initial state, we give a comparison between the population transfer governed by the total Hamiltonian H₀ in Eq. (2) and that governed by the effective Hamiltonian H_eff in Eq. (7), for which we have defined P_k = |C_k(t)|² (k = 11,rr,0g) and . More concretely, in Fig. 3(b), we plot the comparison between the variation of C₁₁(t) governed by H₀ and that of governed by the H_eff. Figure 3(a) assisted by Fig. 3(b) proves that the evolution governed by Hamiltonian H_eff gives the perfect population transfer of implementing the ideal transformation |11⟩ → −|11⟩. Apparently, all curves in Fig. 3 originating from H₀, up to slight oscillations, coincide well with those originating from H_eff, which declares that the approximation from H₀ to H_eff is effective and predicts that the implementation of the two-qubit π-phase gate with H₀ would be successful.

Fig. 3.

	Figure Option ViewDownloadNew Window
	Fig. 3. (color online) (a) Comparison between the population transfer governed by the total Hamiltonian H0 in Eq. (2) and that governed by the effective Hamiltonian Heff in Eq. (6). (b) Comparison between the variation of C11(t) governed by H0 and that of governed by the Heff. Parameters used here are the same as in Fig. 2.

For a phase gate, the evolution of the phase of a state is a crucial investigated aspect. Therefore, we explore the evolution of the phase for the four different initial states with a complex phase e^iϕ₀. For the scheme proposed for a π-phase gate, we desire that if the initial state is e^iϕ₀|00⟩, e^iϕ₀|01⟩, or e^iϕ₀|10⟩, neither the state nor the phase will change at all. For the initial state e^iϕ₀|11⟩, however, after adiabatic passage, the state will keep unchanged but the phase will change with π. In Fig. 4, we give the time-dependence of the phase for the initial state e^iϕ₀|M⟩ (M = 00,01,10,11) whose evolution is dominated by Hamiltonian Eq. (2), for which we have chosen ϕ₀ = π/4 for simplicity. We extract the phase of the state |M⟩ by ϕ_M = −i ln(C_M/|C_M|).^[48] As shown in Fig. 4, ϕ₀₀, ϕ₀₁, and ϕ₁₀ keep roughly unchanged, i.e., π/4, except slight fluctuations, while ϕ₁₁ is reduced by π with an acceptable approximation after the adiabatic-passage operation, which fits well with expectations. In a word, figure 4 proves that a complex phase involved in the initial state has no influence on the performance of the scheme.

Fig. 4.

	Figure Option ViewDownloadNew Window
	Fig. 4. (color online) Time-dependence of phases for the four different initial states with a complex phase eiπ/4.

We consider a random initial state read as

Fig. 5.

	Figure Option ViewDownloadNew Window
	Fig. 5. (color online) Fidelity F(t) = |⟨Ψideal|Ψ(t)⟩|2 with different pairs of α and β. Parameters used here are the same as in Fig. 2.

Table 1.

Table 1.

Table 1. Samples of the final fidelity F(tf) with different values of α and β. . α β F(tf) π/16 π/16 0.9703 π/16 19π/16 0.9507 5π/8 π/16 0.9923 5π/8 19π/16 0.9818 19π/16 π/16 0.9801 19π/16 19π/16 0.9779 7π/4 π/16 0.9892 7π/4 19π/16 0.9768 π/16 5π/8 0.9516 π/16 7π/4 0.9429 5π/8 5π/8 0.9569 5π/8 7π/4 0.9651 19π/16 5π/8 0.9634 19π/16 7π/4 0.9702 7π/4 5π/8 0.9572 7π/4 7π/4 0.9746

Table 1. Samples of the final fidelity F(tf) with different values of α and β. .

In fact, a sufficient standard of judging the successful implementation of a quantum gate is the average fidelity defined by

Fig. 6.

	Figure Option ViewDownloadNew Window
	Fig. 6. (color online) Time-dependent variation of the average fidelity with different atomic total spontaneous emission rates. Parameters used here are the same as in Fig. 2.

We have accomplished the implementation of the two-qubit phase gate between two Rydberg atoms by combining adiabatic passage and Rydberg antiblockade. An effective single-photon detuning Λ-type three-level Hamiltonian is obtained in the Rydberg antiblockade regime, and a desired two-qubit π-phase gate is able to be executed by using the adiabatic-passage technique. The total Hamiltonian can give a relatively perfect result, which is pretty similar to that given by the effective Hamiltonian. Numerical simulations show that the fidelity with a specified initial state can always be over 95%. The average fidelity with actual experimental parameters is up to 98.7%, which proves that the proposed scheme for implementing the two-qubit phase gate is robust against atomic spontaneous emission.

In the past several years, a lot of methods or techniques, usually called “shortcuts to adiabatic passage” (STAP), have been proposed to speed up an adiabatic passage,^[51–57] with which many schemes have proposed to achieve quantum information processing and quantum computation.^[11,58–63] In addition, the combinations between RRI and STAP have been accomplished.^[33,40,42] For prospects, therefore, the current scheme is supposed to be accelerated by STAP to construct a fast π-phase gate.