Abstract
Motivated by “transitionless quantum driving”, we construct shortcuts to adiabatic passage in a threeatom system to create a singlet state with the help of quantum zeno dynamics and nonresonant lasers. The influence of various decoherence processes is discussed by numerical simulation and the results reveal that the scheme is fast and robust against decoherence and operational imperfection. We also investigate how to select the experimental parameters to control the cavity dissipation and atomic spontaneous emission which will have an application value in experiment.
Introduction
Quantum entanglement is an intriguing property of composite systems. The generation of entangled states for two or more particles is not only fundamental for demonstrating quantum nonlocality^{1,2}, but also useful in quantum information processing (QIP)^{3,4,5,6}, typically the Bell state, the GreenbergerHorneZeilinger (GHZ) state and the W state^{7,8,9,10,11,12,13}. Different entangled state has different advantages. For example, the W state is likely to retain bipartite entanglement when any one of the three qubits is traced out, thus it is robust against qubit loss. The GHZ state is the most entangled state and can maximally violate the Bell inequalities^{7}. Recently, some attention has been paid to a special type of entangled state called the Nparticle (N ≥ 2) Nlevel singlet state^{14}. The form of the Natom singlet state can be expressed as
where are the generalized LeviCivita symbols, {n_{l}} are the permutations, and denote the bases of the qubits^{15}. It has been shown that the singlet state not only is in connection with violations of Bell inequalities^{16}, but also can be used to construct decoherencefree subspace, which is robust against collective decoherence^{17}. Moreover, the singlet state can be used to solve several problems which have no classical solutions, including “N strangers”, “secret sharing”, “liar detection”, and so on^{14,17}. Furthermore, the singlet state also can be used in a scheme designed to probe a quantum gate that can realize an unknown unitary transformation^{18}. In recent years, lots of theoretical schemes have been proposed to generate singlet state in the cavity quantum electrodynamics (CQED) system via different techniques^{17,18,19,20,21,22,23}. Among these techniques^{17,18,19,20,21,22,23}, there are two famous techniques for their robustness against decoherence in proper conditions. One is stimulated Raman adiabatic passage (STIRAP)^{20,21}, the other is Quantum Zeno dynamics (QZD)^{15,22,23}. In general, adiabatic passage technique has been widely used and an advantage of the technique is that can reduce populations of the intermediate excited states. Therefore, the technique would restrain the influence of atomic spontaneous emission on the fidelity. As we know, the adiabatic condition is managed to be slow to make sure each of the eigenstates of the system evolves along itself all the time without transition to other ones. So, the operation time is long in previous schemes^{20,21} via adiabatic passage. Differ from the adiabatic passage, QZD is usually robust against photon leakage but sensitive to atomic spontaneous emission^{15,22,23}. Therefore, some of the researchers introduce detuning between the atomic transition to restrain the influence of atomic spontaneous emission. However, that also increases the operation time. In general, the interaction time for a method is the shorter the better. Otherwise, the method may be useless because the dissipation caused by decoherence, noise, and losses on the target state increases with the increasing of the interaction time^{24}.
In order to solve this problem, in recent years researchers pay more attention to “shortcuts to adiabatic passage (STAP)”^{25,26,27,28} which employs a set of techniques to speed up a slow quantum adiabatic process through a nonadiabatic route. Usually STAP can overcome the harmful effect caused by decoherence, noise and losses during the long operation time. Recently, STAP has been applied in a wide range of system to implement quantum information processing (QIP) in theory and experiment^{25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57}. In order to construct STAP to speed up adiabatic processes effectively, many methods^{25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41} are related. Such as, invariantbased inverse engineering proposed by Muga and Chen^{25,26,27,28,29,30,31}, can achieve the fast population transfer within two internal states of a single Λtype atom. “Transitionless quantum driving” (TQD)^{32,33,34,35} proposed by Berry, provides a very effective method to construct the “counterdiabatic driving” (CDD) Hamiltonian H(t) which can accurately derive the instantaneous eigenstates of H_{0}(t) to speed up adiabatic processes effectively. But it is also found that the designed CDD Hamiltonian is hard to be directly implemented in practice, especially in multiparticle system. In order to solve the problem, many schemes^{29,33,34,45,46} have been put forward. In 2014, by using secondorder perturbation approximation twice under large detuning condition and transitionless quantum driving, Lu et al. have proposed an effective scheme^{45} to implement the fast populations transfer and prepare a fast maximum entanglement between two atoms in a cavity. The idea inspires that using some traditional methods to approximate a complicated Hamiltonian into an effective and simple one first, then constructing shortcuts for the effective Hamiltonian might be a promising method to speed up evolution process of a system. Later, Chen et al.^{46} have proposed a promising method to construct STAP for a threeatom system to generate GHZ states in the cavity QED system in light of QZD and TQD. Their schemes might be useful to realize fast and noiseresistant quantum information processing for multiqubit system in current technology.
In this paper inspired by the schemes^{45,46}, we discuss how to construct STAP to fastly generate a threeatom singlet state in cavity QED system by using the approach of “transitionless tracking algorithm”. Based on quantum Zeno dynamics^{58,59} and large detuning conditon, we can simplify the original Hamiltonian of system and obtain the effective Hamiltonian equivalent to the corresponding CDD Hamiltonian, the evolution process of system can be speeded up, and the STAP can be achieved in experiment easily. What’s more, numerical investigation shows that our scheme is also fast and robust against both cavity decay and atomic spontaneous emission for threeatom singlet state preparation. It will be much useful in dealing with the fast and noiseresistant generation of Natom singlet state.
The paper is organized as follows. In section II, we describe a theoretical model for three atoms which are trapped in a bimodalmode cavity. In section III, we demonstrate how to construct STAP for the system in section II, and use the constructed shortcut to generate a threeatom singlet state. The numerical simulation and experimental discussion about the validity of the scheme are also given. Finally, a summary is given in section IV.
Theoretical Model
The sketch of the experimental setup is shown in Fig. 1. Three identical fourlevel atoms with three ground states , and , and an excited state are trapped in a bimodalmode cavity. The atomic transition is driven resonantly through classical laser field with timedependent Rabi frequency Ω(t), the transition is coupled resonantly to the leftcircularly polarized mode of the cavity with coupling λ_{L}, and transition is coupled resonantly to the rightcircularly polarized mode of the cavity with coupling λ_{R}. Under the rotatingwave approximation (RWA), the interaction Hamiltonian for this system reads (ħ = 1):
where a_{L} and a_{R} are the leftcircularly and the rightcircularly annihilation operators for cavity mode, respectively. We set λ_{L,i} = λ_{R,i} = λ for simplicity. If we assume the initial state of the system is , the system will evolve within a singleexcitation subspace with basis states
Then, we rewrite the Hamiltonian H_{ac} and H_{al} with the eigenvectors of H_{ac}:
with eigenvalues η_{1} = η_{2} = 0, η_{3} = η_{4} = λ, η_{5} = η_{6} = −λ, η_{7} = 2λ, η_{8} = −2λ, , and . We obtain
Through performing the unitary transformation and neglecting the terms with high oscillating frequency by setting the condition Ω_{i} ≪ λ, we obtain an effective Hamiltonian
here we set Ω_{2} = Ω_{3}, and .
We can see Hamiltonian in equation (6) as a simple threelevel system with an excited state and two ground states and . For this effective Hamiltonian, its eigenstates are easily obtained
corresponding eigenvalues ε_{0} = 0, , respectively, where and . When the adiabatic condition is fulfilled, the initial state will follow closely, and when , we can obtain the threeatom singlet state:
However, this process will take quite a long time to obtain the target state, which is undesirable. We will talk in later.
Using STAP to generate a threeatom singlet state
The instantaneous eigenstates (k = 0, ±) for the effective Hamiltonian H_{eff}(t) in equation (6) do not satisfy the Schrodinger equation . According to Berry’s transitionless tracking algorithm^{32}, from H_{eff}(t), one can reverse engineer H(t) which is related to the original Hamiltonian H_{I}(t) and can drive the eigenstates exactly. From refs 45, 52, 53, we learn the simplest Hamiltonian H(t) is derived in the form
Substituting equation (7) into equation (9), we obtain
where . For this threeatom system, the Hamiltonian H(t) is hard or even impossible to be implemented in real experiment^{45}. We should find an alternative physically feasible (APF) Hamiltonian whose effect is equivalent to H(t). Therefore, we consider that the three atoms are trapped in a cavity and the atomic level configuration is shown in Fig. 2. We make all the resonant atomic transitions into nonresonant atomic transitions with detuning Δ. The nonresonant Hamiltonian reads
Then similar to the approximation for the Hamiltonian from equation (2) to equation (6) in section II, we also obtain an effective Hamiltonian for the present nonresonant system^{15}
By adiabatically eliminating the state under the condition , we obtain the final effective Hamiltonian
When we choose and (here Ω′ is a real number), the first two terms can be removed, and the Hamiltonian in equation (13) becomes
That means, as long as , , the Hamiltonian for speeding up the adiabatic dark state evolution governed by under condition has been constructed. Hence, Ω′ is given
We will show the numerical analysis of the creation of a threeatom singlet state governed by . To satisfy the boundary condition of the fractional stimulated Raman adiabatic passage (STIRAP),
the Rabi frequencies Ω_{1}(t) and Ω_{3}(t) in the original Hamiltonian H_{I}(t) are chosen as
where Ω_{0} is the pulse amplitude, t_{f} is the operation time, and t_{0}, t_{c} are some related parameters. In order to create a threeatom singlet state, the finial state should be according to equation (8). Therefore, we have tan α = 2. And choosing parameters for the laser pulses suitably to fulfill the boundary condition in equation (16), the timedependent Ω_{1}(t) and Ω_{3}(t) are gotten as shown in Fig. 3 with parameters t_{0} = 0.14t_{f} and t_{c} = 0.19t_{f}.
Figure 4 shows the relationship between the fidelity of the generated threeatom singlet state (governed by the APF Hamiltonian ) and two parameters Δ and t_{f} when Ω_{0} = 0.2λ, where the fidelity for the threeatom singlet state is given through (ρ(t_{f}) is the density operator of the whole system when t = t_{f}). It’s easy to find that there is a wide range of selectable values for parameters Δ and t_{f} to get a high fidelity. And the fidelity increases with the increasing of t_{f} while decreases with the increasing of Δ. This is easy to understand. If we set , according to equation (17), we can obtain two dimensionless parameters
Therefore, putting equations (17) and (18) into equation (15), we obtain
where
is a dimensionless function. A brief analysis of G tells that the amplitude of G is close to 1. That is, the amplitude of Ω′ is mainly dominated by . In order to satisfy the condition Ω′ ≪ λ and Ω′ ≪ Δ, we can work out Δ/t_{f} ≪ 1 and Δt_{f} ≫ 1. So, long t_{f} can lead to a high fidelity as shown in Fig. 4. When the detuning Δ is smaller or near 0, it’s not meet the condition Δt_{f} ≫ 1, so the fidelity is lower in a short time as shown in Fig. 4. We know , shortening the evolution time implies that relative large laser intensities is required, and this would destroy the Zeno condition. Yet slightly destroying the Zeno condition is also helpful to achieve the target state in a much shorter interaction time^{45,47}.
Next, to comfirm the operation time required for the creation of the threeatom singlet state governed by is much shorter than that governed by H_{I}, we contrast the performances of population transfer from the initial state in Fig. 5. The timedependent population for any state is given by , where ρ(t) is the corresponding timedependent density operator. Figure 5(a) shows time evolution of the populations for the states is the initial state and governed by the APF Hamiltonian with Ω_{0} = 0.2λ, t_{f} = 40/λ and Δ = 3λ. Figure 5(b) shows time evolution of the populations for the states and governed by the original Hamiltonian H_{I} with Ω_{0} = 0.2λ and t_{f} = 1000/λ. The comparison of Fig. 5(a,b) shows that with this set of parameters, the APF Hamiltonian can govern the evolution to achieve a nearperfect threeatom singlet state from state in short interaction time while the original Hamiltonian H_{I} can not. We also plot the fidelities of the evolved states governed by and H_{I} in Fig. 6, with respect to the target threeatom singlet state. As shown in Fig. 6, when the interaction time t_{f} = 40/λ, the fidelity governed by is already 99.98%. While, when t_{f} = 1000/λ, the fidelity governed by H_{I} achieves 99.93%. The interaction time required for creation of the threeatom singlet state via STAP is much shorter than adiabatic passage.
Since most of the parameters are hard to faultlessly achieve in experiment, we need to investigate the variations in the parameters induced by the experimental imperfection. We calculate the fidelity by varying error parameters of the mismatch between the laser amplitude Ω_{0} and the total operation time t_{f}, the detuning Δ and the cavity mode with coupling constant λ, respectively. We define δx = x′ − x as the deviation of x, here x denotes the ideal value and x′ denotes the actual value. Then in Fig. 7(a) we plot the fidelity of the threeatom singlet state versus the variations in total operation time t_{f} and laser amplitude Ω_{0}. In Fig. 7(b) we plot the fidelity of the threeatom singlet state versus the variations in coupling constant λ and the detuning Δ. We find that the scheme is robust against all of these variations. For example, a deviation δΔ/Δ = 10% and δλ/λ = −10% only causes a reduction about 0.66% in the fidelity. In order to have a fair comparison, we show the influence of fluctuations versus total operation time t_{f} and laser amplitude Ω_{0} on the fidelity for the STIRAP in Fig. 8. As we can find, the STIRAP scheme almost perfectly restrain the influence caused by the parameters’ fluctuations without doubt. Nevertheless, in Fig. 7(a) we can find that the fidelity of the target state for the STAP is still higher than 99.5% even when the deviation δΩ_{0}/Ω_{0} = δt_{f}/t_{f} = 10%, so we can say the scheme via STAP is also robust against these variations.
Next, we will analyze the influence of dissipation induced by the atomic spontaneous emission and the cavity decay. The master equation of motion for the density matrix of the whole system can be expressed as
where ρ is the density operator for the whole system, γ_{n,p} is the spontaneous emission rate from the excited state to the ground states (p = g_{0}, g_{1}, g_{2}) of the nth (n = 1, 2, 3) atom. κ_{L} (κ_{R}) is the decay rate of the left(right)circular cavity mode. For simplicity, we assume κ_{L} = κ_{R} = κ and γ_{n,p} = γ. Figure 9(a,b) show the fidelities of the threeatom singlet state governed by the APF Hamiltonian versus κ/λ and γ/λ with {Ω_{0} = 0.2λ, Δ = 3λ, t_{f} = 40/λ} and {Ω_{0} = 0.2λ, Δ = λ and t_{f} = 40/λ}, respectively. We can find the fidelity F decrease slowly with the increasing of cavity decay and atomic spontaneous emission. When κ = γ = 0.05λ, we still can create a threeatom singlet state with a high fidelity 91.03% as shown in Fig. 9(a). By comparing Fig. 9(a,b), we find the effect of the atomic spontaneous emission and cavity field dissipation varies with different parameters values. So, we plot the fidelity of the threeatom singlet state versus κ/λ and Δ/λ with Ω_{0} = 0.2λ, t_{f} = 40/λ, and γ/λ = 0 in Fig. 10(a). Figure 10(b) shows the fidelity of the threeatom singlet state versus γ/λ and Δ/λ with Ω_{0} = 0.2λ, t_{f} = 40/λ, and κ/λ = 0. We find that when κ/λ is nonzero, the fidelity F decreases with the increasing of Δ/λ as shown in Fig. 10(a). When γ/λ is nonzero, the fidelity F increases with the increasing of Δ/λ as shown in Fig. 10(b). The phenomenon can be understood as follows. From equation (19), we know , so the laser Ω′ increases with the increasing of detuning Δ. When Δ is large enough, the Zeno condition Ω′ ≪ λ for the nonresonant system is not ideally fulfilled, then the intermediate states including the cavityexcited states would be populated during the evolution, which would cause the system to be sensitive to the cavity decays. In other words, as long as the detuning Δ is small, the system is robust to the cavity decays as shown in Fig. 10. But substituting equation (19) into the condition Ω′ ≪ Δ, we deduce , it denotes large Δ would be better. So, taking the two conditions into account, when the detuning Δ ≈ 1.5λ, atomic spontaneous emission and cavity field dissipation have an equal influence in the fidelity. According to the sensitivity of experimental apparatus to the atomic spontaneous emission and cavity field dissipation, we can reasonably select different parameters in practical. As we know, in general in order to restrain atomic spontaneous emission in QZD and cavity decay in STIRAP, we introduce detuning between the atomic transition, and that increases the evolution time. However in our scheme we only need to select appropriate parameter to restrain atomic spontaneous emission and cavity decay in a short time. To sum up, it is a better choice for the experimental researchers because the threeatom singlet state is generated much faster in the present shortcut scheme that contributes to the experimental research.
Finally, we present a brief discussion about the basic factors for the experimental realization of a threeatom singlet state. In a real experiment, the cesium atoms which have been cooled and trapped in a small optical cavity in the strong coupling regime^{60,61} can be used in this scheme. The state corresponds to F = 4, m = 3 hyperfine state of the 6^{2}P_{1/2} electronic excited state, the state corresponds to F = 4, m = 3 hyperfine state of the 6^{2}S_{1/2} electronic ground state, the state corresponds to F = 3, m = 2 hyperfine state of the 6^{2}S_{1/2} electronic ground state, and the state corresponds to F = 3, m = 4 hyperfine state of the 6^{2}S_{1/2} electronic ground state, respectively. In recent experimental conditions^{62,63,64}, it is predicted to achieve the parameters λ = 2π × 750 MHz, κ = 2π × 3.5 MHz, and γ = 2π × 2.62 MHz and the optical cavity mode wavelength in a range between 630 and 850 nm. By substituting the ratios κ/λ = 0.0047,γ/λ = 0.0035 into equation (21), we will obtain a high fidelity about 99.05%, which shows our scheme to prepare a threeatom singlet state is relatively robust. Nowadays, according to the literature^{65,66,67,68}, the laser pulse which is used in our scheme can be obtained in a real experiment, so, our scheme is feasible in experiment.
Summary
We have presented a promising method to construct shortcuts to adiabatic passage (STAP) for a threeatom system to generate singlet state in the cavity QED system. We simplify a multiqubit system and choose the laser pulses to implement the fast generation of entangled states in light of quantum zeno dynamics and “transitionless quantum driving”. In comparison to QZD, the significant feature is that we do not need to control the evolution time exactly. As comparing with the STIRAP, the significant feature is the shorter evolution time. When dissipation is considered, we can find that the scheme is robust against the decoherence caused by both atomic spontaneous emission, photon leakage and operational imperfection. In addition, the present scheme only needs to select appropriate parameter to restrain atomic spontaneous emission and cavity decay in a short time. Numerical simulation result shows that the scheme has a high fidelity and may be possible to implement with the current experimental technology. In shorts, the scheme is robust, effective and fast. Actually, the present scheme in section III can be effectively applied to Natom system for preparation of Natom singlet state. We hope our work be useful for the experimental realization of quantum information in the near future.
Additional Information
How to cite this article: Chen, Z. et al. Fast generation of threeatom singlet state by transitionless quantum driving. Sci. Rep. 6, 22202; doi: 10.1038/srep22202 (2016).
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Acknowledgements
This work was supported by the National Natural Science Foundation of China under Grants No. 11575045 and No. 11374054, the Foundation of Ministry of Education of China under Grant No. 212085, and the Major State Basic Research Development Program of China under Grant No. 2012CB921601.
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Affiliations
Department of Physics, Fuzhou University, Fuzhou 350002, China
 Zhen Chen
 , YeHong Chen
 , Yan Xia
 & BiHua Huang
Department of Physics, Harbin Institute of Technology, Harbin 150001, China
 Jie Song
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Contributions
Y.X. and Z.C. came up with the initial idea for the work and performed the simulations for the model. J.S. and B.H.H. performed the calculations for the model. Y.X., Z.C. and Y.H.C. performed all the data analysis and the initial draft of the manuscript. All authors participated in the writing and revising of the text.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to Yan Xia.
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