† Corresponding author. E-mail:
Project supported by the National Natural Science Foundation of China (Grant No. 11464046).
Combining adiabatic passage and Rydberg antiblockade, we propose a scheme to implement a two-qubit phase gate between two Rydberg atoms. Detuning parameters between frequencies of atomic transitions and those of the corresponding driving lasers are carefully chosen to offset the blockade effect of two Rydberg atoms, so that an effective Hamiltonian, representing a single-photon detuning Λ-type three-level system and concluding the quantum state of two Rydberg atoms excited simultaneously, is obtained. The adiabatic-passage technique, based on the effective Hamiltonian, is adopted to implement a two-atom phase gate by using two time-dependent Rabi frequencies. Numerical simulations indicate that a high-fidelity two-qubit π-phase gate is constructed and its operation time does not have to be controlled accurately. Besides, owing to the long coherence time of the Rydberg state, the phase gate is robust against atomic spontaneous emission.
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
With rotating-wave approximation, the Hamiltonian of the whole system can be written as (assuming ħ = 1):
![](cpb_27_10_100307/cpb_27_10_100307_ieqn3.gif)
![](cpb_27_10_100307/cpb_27_10_100307_ieqn4.gif)
![](cpb_27_10_100307/cpb_27_10_100307_ieqn5.gif)
![](cpb_27_10_100307/cpb_27_10_100307_ieqn6.gif)
![](cpb_27_10_100307/cpb_27_10_100307_ieqn7.gif)
![](cpb_27_10_100307/cpb_27_10_100307_ieqn8.gif)
Hamiltonian Heff in Eq. (
First of all, we change Ω1 and Ω2 to time-dependent Ω1(t) and Ω2(t), respectively, while other parameters keep time-independent. Hence, Ωa, Ωb, and δrr become time-dependent Ωa(t), Ωb(t), and δrr(t), respectively. The instantaneous eigenstates of Hamiltonian Heff in Eq. (
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 |Φn(t)⟩ (n = +,0,−) initially, then at an arbitrary time the state of the effective system will be[47]
![](cpb_27_10_100307/cpb_27_10_100307_ieqn9.gif)
![](cpb_27_10_100307/cpb_27_10_100307_ieqn10.gif)
In order to meet the boundary conditions θ(0) = 0 and θ(tf) = π, the Rabi frequencies Ω1(t) and Ω2(t) can be chosen as
![](cpb_27_10_100307/cpb_27_10_100307_ieqn11.gif)
The transformation |11⟩ → −|11⟩ is a key to the implementation of the π-phase gate, and in Fig.
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 eiϕ0. For the scheme proposed for a π-phase gate, we desire that if the initial state is eiϕ0|00⟩, eiϕ0|01⟩, or eiϕ0|10⟩, neither the state nor the phase will change at all. For the initial state eiϕ0|11⟩, however, after adiabatic passage, the state will keep unchanged but the phase will change with π. In Fig.
![]() | 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. (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. |
![]() | 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
![](cpb_27_10_100307/cpb_27_10_100307_ieqn17.gif)
![](cpb_27_10_100307/cpb_27_10_100307_ieqn18.gif)
![](cpb_27_10_100307/cpb_27_10_100307_ieqn19.gif)
![](cpb_27_10_100307/cpb_27_10_100307_ieqn20.gif)
![](cpb_27_10_100307/cpb_27_10_100307_ieqn21.gif)
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.
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