Efficient shortcuts to adiabatic passage for three-dimensional entanglement generation via transitionless quantum driving

We propose an effective scheme of shortcuts to adiabaticity for generating a three-dimensional entanglement of two atoms trapped in a cavity using the transitionless quantum driving (TQD) approach. The key point of this approach is to construct an effective Hamiltonian that drives the dynamics of a system along instantaneous eigenstates of a reference Hamiltonian to reproduce the same final state as that of an adiabatic process within a much shorter time. In this paper, the shortcuts to adiabatic passage are constructed by introducing two auxiliary excited levels in each atom and applying extra cavity modes and classical fields to drive the relevant transitions. Thereby, the three-dimensional entanglement is obtained with a faster rate than that in the adiabatic passage. Moreover, the influences of atomic spontaneous emission and photon loss on the fidelity are discussed by numerical simulation. The results show that the speed of entanglement implementation is greatly improved by the use of adiabatic shortcuts and that this entanglement implementation is robust against decoherence. This will be beneficial to the preparation of high-dimensional entanglement in experiment and provides the necessary conditions for the application of high-dimensional entangled states in quantum information processing.


Results
Basic model. We consider a multimode cavity in which two atoms are trapped as shown in Fig. 1(a). The atomic level configuration depicted in Fig. 1(b) was used by Wu et al. 18 . Atom 1 has two excited states |e j 〉 1 ( j = L, R, the same below) and five ground states |1〉 1 , |R〉 1 , |g〉 1 , |L〉 1 , and |0〉 1 , while atom 2 is a five-level system with three ground states |R〉 2 , |g〉 2 , |L〉 2 and two excited states |e j 〉 2 . For atom 1, the transitions |e L 〉 1 ↔ |1〉 1 and |e R 〉 1 ↔ |0〉 1 are driven by the classical fields with the same Rabi frequency Ω 1 (t), and the transition |e j 〉 1 ↔ | j〉 1 is resonantly driven by the corresponding cavity mode a 1j and the coupling strength g 1j . For atom 2, the transitions |e j 〉 2 ↔ | j〉 2 are driven by the classical fields with the same Rabi frequency Ω 2 (t), and the transition |e j 〉 2 ↔ |g〉 2 is resonantly driven by the corresponding cavity mode a 2j with the coupling strength g 2j . The configuration described here can be obtained from the hyperfine structure of cold alkali-metal atoms [48][49][50] . Here we use two 87 Rb atoms that have been cooled and trapped in a small optical cavity. For atom 1, 5 2 S 1/2 ground level |F = 1, m = 2〉 (|F = 1, m = − 2〉 ) can be used as the state |L〉 (|R〉 ) and |F = 2, m = 1〉 (|F = 2, m = − 1〉 ) as |1〉 (|0〉 ), respectively. The 5 2 P 3/2 excited level |F′ = 1, m = 1〉 (|F′ = 1, m = − 1〉 ) can be used as the state |e L 〉 (|e R 〉 ). Other hyperfine levels in the ground-state manifold can be used as |g〉 for atom 1. For atom 2, the 5 2 S 1/2 ground level |F = 1, m = 0〉 , |F = 2, m = 2〉 , |F = 2, m = − 2〉 can be used as states |g〉 , |R〉 , and |L〉 , respectively. The excited level |F′ = 1, m = 1〉 (|F′ = 1, m = − 1〉 ) corresponds to |e L 〉 (|e R 〉 ). The total Hamiltonian in the interaction picture can be written as (ħ = 1) where β is the phase difference between the two time-dependent classical fields and we have assumed β = 3π/2 here; a ij (i = 1, 2; j = R, L) is the annihilation operator for the corresponding cavity modes with R(L)-circular polarization, and g ij (i = 1, 2; j = R, L) is the coupling strength between the corresponding cavity mode and the atom. We now describe an idea for constructing the shortcuts to adiabatic passage to generate the three-dimensional entanglement between the two atoms; this can be written as   For simplicity, we assume g ij = g (i = 1, 2; j = R, L) and assume the condition of weak-driving fields i Then, the eigenstates |ψ n (t)〉 at the instantaneous time t and the corresponding eigenvalues ξ n (t) of H I (t) that obey the equation H I (t)|ψ n (t)〉 = ξ n (t)|ψ n (t)〉 can be derived analytically as    It can be seen that the eigenstate |ψ 1 (t)〉 is a dark state in the subspace with an eigenvalue of ξ 1 = 0. If the adiabatic is fulfilled 51 , the initial state will undergo an evolution determined by |ψ 1 (t)〉 , which neglects the probability of populating the |ϕ 3 〉 state during the entire evolution. Undoubtedly, the adiabatic passage is an effective method for implementing the transformation from the initial state to the final state, but it requires a long time to complete the evolution. This is undesirable due to decoherence. The shortcut to the adiabatic passage is a good choice for the acceleration of the adiabatic evolution in a nonadiabatic manner. The evolutions of the other two initial states will be interpreted later.
Shortcuts for a generating three-dimensional entanglement of two atoms. The instantaneous eigenstates |ψ n (t)〉 for the Hamiltonian H I (t) do not satisfy the Schrodinger equation i∂ t |ψ n (t)〉 = H I (t)|ψ n (t)〉 . According to Berry's general transitionless tracking algorithm, one can reverse engineer a Hamiltonian related to the original Hamiltonian H I (t), but drives the eigenstates exactly 40 . The Hamiltonian can be obtained by using However, because the two atoms with double Λ level configurations in the original system are resonant with the cavity modes as well as with the classical lasers (see Fig. 1(a)), the two excited states of each atom are occupied with a considerable proportion of the population. It is difficult to realize the intended transitions between the ground states within the atoms. Thus, in practice, the direct implementation of the CDD Hamiltonian H(t) is Scientific RepoRts | 6:30929 | DOI: 10.1038/srep30929 still challenging, especially in multi-particle systems. It is necessary for us to construct an alternative physically feasible (APF) Hamiltonian equivalent to H(t).
To construct the APF Hamiltonian, two auxiliary levels must be introduced in each of the atoms described above, as depicted in Fig. 1(c). For atom 1, the 5 2 P 3/2 excited levels |F′ = 2, ± 2〉 of atom 87 Rb can be used as the two auxiliary excited levels ẽ L and , respectively. For atom 2, the excited levels ′ = =  F m 2, 1 of 5 2 P 1/2 can be used as the auxiliary levels ẽ L and ẽ R , respectively. Correspondingly, two additional classical driving fields with Rabi frequencies Ω ∼ t ( ) i (i = 1, 2) and two auxiliary cavity field modes are introduced to drive the relevant transitions. The transition ↔j e j ( j = L, R) of atom 1 and ↔g e R L ( ) of atom 2 are coupled, respectively, to the auxiliary cavity modes with the coupling constant  g ij (i = 1, 2 and j = R, L) and detuning Δ 2 . The two classical laser fields are applied to drive the transition ↔ẽ 0 ( 1 ) and ↔j e j of atoms 1 and 2, respectively, with the same detuning Δ 1 . Under the rotating wave approximation, the auxiliary interaction Hamiltonian is (ħ = 1) . The effective Hamiltonian (10) is equivalent to the CDD Hamiltonian H(t) in Eq. (7) with Hence, the Rabi frequency of the auxiliary laser field contributes to the construction of the APF Hamiltonian and can be determined from the original frequencies Ω 1 (t) and Ω 2 (t) as To satisfy the adiabatic condition

(t) and Ω 2 (t) in the original Hamiltonian H I (t) can be chosen as
where Ω 0 is the pulse amplitude, τ is the time delay, and T is the operating duration. Figure 2 shows Ω 1 (t)/Ω 0 and Ω 2 (t)/Ω 0 plotted as a function of t/T for a fixed value of time delay chosen for the best adiabatic passage. Applying this shortcut to the adiabatic passage, the initial state |0〉 1 |g〉 2 |0〉 c finally evolves to state |R〉 1 |R〉 2 |0〉 c . In contrast, if the initial state is |1〉 1 |g〉 2 |0〉 c , the system is restricted to the subspace spanned by the basis vectors In this case, using the method described above, we can easily obtain the effective Hamiltonian . Finally, the system will evolve to the state |L〉 1 |L〉 2 |0〉 c . Meanwhile, the initial state |g〉 1 |g〉 2 |0〉 c remains unchanged during the evolution due to the absence of excitation.
Considering all of the above cases, we can see that the two atoms in the initial state will evolve into the three-dimensional entangled state in Eq. (2), assisted by a vacuum cavity exploiting the shortcuts to adiabatic passage. All of the cavity modes finally stay in the vacuum states.

Discussion
To prove the efficiency of the shortcuts assisted by the Hamiltonian ∼ H t ( ), we use a contrast between the performance of the population transfer from the initial state to the final state driven by the APF Hamiltonian ∼ H t ( ) and that governed by the original Hamiltonian H I (t), as shown in Fig. 3.
The time-dependent population for any state |ϕ〉 is given by the relationship P = 〈 ϕ|ρ(t)|ϕ〉 , where ρ(t) is the corresponding time-dependent density operator. A comparison of Fig. 3(a,b) shows that the APF Hamiltonian ∼ H t ( ) governs the system to achieve a near-perfect population transfer in a short interaction time, whereas the original Hamiltonian H I (t) does not show such an effect. This can be understood in physics. By introducing the auxiliary levels in each atom and driving the transitions between the auxiliary levels and the ground states with the cavity modes and classical fields, the interaction energy within the system is increased. This enhances the effective transition strength (or coupling strength) between the ground states (  L R 1(0) ( ) 1 1 and  L R g ( ) 2 2 ), thereby greatly increasing the population probabilities of the target states and accelerating the entire process.
We can also compare the fidelities of the entangled states governed by the original Hamiltonian H I (t), H I (t) assisted by the APF Hamiltonian ∼ H t ( ) and those governed by the CDD Hamiltonian H(t). As shown in Fig. 4, as a fast and feasible experimental method, the fidelity for our shortcuts scheme can achieve the same perfect degree as that driven by the CDD Hamiltonian F CDD , with only a slightly longer time. Meanwhile, this process is much faster than the adiabatic passage.
In a realistic implementation, in addition to the operating speed requirements, the robustness of the scheme against the possible decoherence caused by atomic spontaneous emission γ and cavity decay κ should also be considered. Using the Lindblad master equation, we can simulate the fidelity of this scheme defined by with the ρ(t) being the reduced density matrix of the final state. An examination of Fig. 5(a) shows that under the dissipative conditions, the intended entanglement state can be obtained with a high fidelity of more than 90% in the present shortcut scheme. Moreover, the fidelity increases with decreasing γ and κ, e.g., a fidelity 98.18% can be reached with γ/g = 0.014 and κ/g = 0.01. To reveal the effectiveness of the shortcuts, the fidelity of original scheme under the dissipation is shown in Fig. 5(b). Comparison of Fig. 5(a,b) shows that the original fidelity is always lower than that in the present shortcut to adiabatic passage under the same degree of dissipation factors (cavity decay and spontaneous emission). The spontaneous emission in the shortcuts to the adiabatic passage has a smaller influence than does that in the original scheme. Therefore, our present scheme is more robust.
The realistic problem related to the experiment is how to capture the two 87 Rb atoms into the same cavity and control them precisely by using different laser pulses in the same cavity. The optical dipole trap (ODT) is one optimal candidate system for QIP using laser cooling techniques [55][56][57][58][59][60] . A quantum register composed of 5 qubits and the controllable Rabi oscillation for 5 qubits has been realized by using monochromatic microwave field to coherently control these atoms 61 . Kim and Saffman et al. constructed five one-dimensional linear ODTs with a spatial distance on the micron level between each other using diffractive optical elements and successfully realized 5 single-atom qubits with no mutual disturbing between any two qubits 62 . We can also use the technology of an atom conveyor belt 63,64 to implement our present scheme. Chapman's group developed this technique further with dual atom conveyor belts 65,66 so that two single atoms confined in two optical lattices can be transferred to the designated positions in the cavity. These two lattices are sufficiently far apart along the direction perpendicular to the axis of the cavity that the probe beams excite atoms in only one of the lattices. The atoms are loaded simultaneously from the magneto-optical trap, but each lattice has independent translational control. Using strong focusing laser fields and detuning the frequencies, the required transitions can be realized.  ∼ H t ( ) with Δ 1 = 6 g and Δ 2 = 7 g and original Hamiltonian H I (t), respectively. The Rabi frequencies Ω 1 (t) and Ω 2 (t) are defined by Eqs (22) and (23) with Ω 0 = 0.2 g, τ = 0.22 T.

Conclusion
We have constructed a shortcut to the adiabatic passage for the three-dimensional entanglement of two atoms by using the TQD method. We construct a supplemental interaction Hamiltonian that is equivalent to the counter-diabatic Hamiltonian under a large detuning regime. The numerical simulation demonstrates that the scheme is fast and robust against the decoherence caused by atomic spontaneous emission and cavity decay. We have also discussed the feasibility of the scheme in experiment. In view of the high security of high-dimensional entanglement in quantum communication and quantum cryptography, our present scheme is expected to have practical applications in quantum communication.
reflecting Figure 5. Fidelity of the target state as a function of κ/g and γ/g. The Rabi frequencies Ω 1 (t) and Ω 2 (t) are defined by Eqs (22) and (23)  Modelling of decoherence effects. The short evolution time is the striking characteristic of our scheme, but the evolution will inevitably suffer from decoherence. Therefore, we pay attention to the effects of decoherence on our entanglement generation. The main dissipation channels include the spontaneous emission of atoms and cavity decay. Considering all of these factors, the evolution of our scheme is governed by the following master equation  , γ 1 = γ/5, γ 2 = γ/3 for simplicity. In experiments, the cavity QED parameters g/2π ≈ 750 MHz, κ/2π ≈ 2.62 MHz and γ/2π ≈ 3.5 MHz are predictively achievable in ref. 67. For such parameters, the fidelity of our scheme is larger than 99.0%, thus, the present shortcut scheme is robust against both cavity decay and atomic spontaneous emission.