Shortcuts to adiabatic passage for fast generation of Greenberger-Horne-Zeilinger states by transitionless quantum driving

Berry’s approach on “transitionless quantum driving” shows how to set a Hamiltonian which drives the dynamics of a system along instantaneous eigenstates of a reference Hamiltonian to reproduce the same final result of an adiabatic process in a shorter time. In this paper, motivated by transitionless quantum driving, we construct shortcuts to adiabatic passage in a three-atom system to create the Greenberger-Horne-Zeilinger states with the help of quantum Zeno dynamics and of non-resonant lasers. The influence of various decoherence processes is discussed by numerical simulation and the result proves that the scheme is fast and robust against decoherence and operational imperfection.

multiparticle systems. Examples of ways to overcome this problem may be found in refs [34][35][36][37][38]. Also, in a large detuning limit, Lu et al. 6 have found a simplified effective Hamiltonian equivalent to H(t). This idea inspires us that finding an alternative physically feasible (APF) Hamiltonian which is effectively equivalent to H(t). However, the approximation in ref. 6 is too complex to be generalized to N-qubit entanglement cases. It is known to all that, entanglement of more qubits shows more nonclassical effects and is more useful for quantum applications. For example, one of the two kinds of three-qubit entangled states named the Greenberger-Horne-Zeilinger (GHZ) states provide a possibility for testing quantum mechanics against local hidden theory without using Bell's inequality 39,40 . Therefore, great interest has arisen regarding the significant role of the GHZ states in the foundations of quantum mechanics measurement theory and quantum communication. In view of that we wonder if it is possible to use TQD to construct shortcuts for one-step generation of multi-qubit entanglement, i.e., the three-atom GHZ states, without abandoning any of the atoms or using auxiliary levels.
In this scenario, motivated by refs 5-8, we use TQD to construct STAP to generate the three-atom GHZ states effectively and rapidly in one step. It would be a promising idea of applying STAP to realize multi-qubit entanglement generation in cavity QED systems. Different from ref. 6, we use the quantum Zeno dynamics 41,42 to simplify the system first and then under the large detuning conditon, we obtain the effective Hamiltonian which is equivalent to the corresponding CDD Hamiltonian to speed up the evolution process. Therefore, the adiabatic process for a multi-qubit system is speeded up, and the STAP is easy to be achieved in experiment. Comparing with ref. 8, we use TQD in this paper so that the laser pulses are not strongly limited and we do not need to use auxiliary levels or multi-step operations to generate the three-atom GHZ states. Moreover, we find that any quantum system whose Hamiltonian is possible to be simplified into the form in eq. (15), the corresponding APF Hamiltonian can be built and the STAP can be constructed with the same approach presented in this paper. The above advantages mean the present scheme is much more useful in dealing with the fast and noise-resistant generation of multi-qubit entanglement or even other QIP.

Basic theories
Transitionless quantum driving. Consider an arbitrary time-dependent Hamiltonian H 0 (t), with instantaneous eigenstates and energies given by When this system satisfies the adiabatic condition, H 0 (t) will drive the system into  The simplest choice is ζ = 0 n , for which the bare states ϕ ( ) t n , with no phase factors, are driven by 26  42 . It is derived from the quantum Zeno effect which describes a phenomenon that the system can actually evolve away from its initial state while it still remains in the so-called Zeno subspace determined by the measurement when frequently projected onto a multidimensional subspace. According to von Neumann's projection postulate, the quantum Zeno dynamics can be achieved via continuous coupling between the system and an external system instead of discontinuous measurements 42 . In general, we assume that a dynamical evolution process is governed by the Hamiltonian where H obs is the Hamiltonian of the quantum system investigated, K is a coupling constant, and H meas is viewed as an additional interaction Hamiltonian performing the measurement. In the "infinitely strong measurement" limit → ∞ K 41,42 , The Hamiltonian for the whole system is nearly equivalent to Zeno n n obs n n n whit P n being the nth orthogonal projection onto the invariant Zeno subspace ∀ Pn and the eigenspace of KH meas belonging to the eigenvalue ε n , i.e., ε = K H P P meas n n n .

Model
We consider three Λ -type atoms are trapped in a bimodal-mode cavity as shown in Fig. 1 , respectively, and each of them has an excited state e . The atomic transition ↔ f e is driven resonantly through classical laser field with time-dependent Rabi frequency Ω (t), transition | 〉 ↔ | 〉 g e l is coupled resonantly to the left-circularly polarized mode of the cavity with coupling λ l , and transition | 〉 ↔ | 〉 g e r is coupled resonantly to the right-circularly polarized mode of the cavity with coupling λ r . Under the rotating-wave approximation (RWA), the interaction Hamiltonian for this system reads ( = ) ħ 1 : where H AL denotes the coupling between the atoms and the laser pulses, and H AC denotes the coupling between the atoms and the cavities, a l and a r are the left-and right-circularly annihilation operators of the cavity modes, and β means the two Rabi frequencies are β-dephased from each other. If we assume the initial state is | , , 〉 | , 〉 , , f g g 0 0 l r c 1 2 3 , the system will evolve within a single-excitation subspace with basis states  In light of quantum Zeno dynamics, we rewrite the Hamiltonian H I in eq. (11) Here φ 0 , φ 1 , φ 2 , φ 3 , and φ 4 are the eigenvectors of H AC corresponding eigenvalues ε = 0 , and ε = − λ 3 4 , respectively. And we obtain (we set λ = λ = λ) Through performing the unitary transformation = and neglecting the terms with high oscillating frequency by setting the condition Ω / , Ω / λ  3 3 1 3 (the Zeno condition), we obtain an effective Hamiltonian which can be seen as a simple three-level system with an excited state φ 0 and two ground states ψ 1 and ψ 7 . For this effective Hamiltonian, its eigenstates are easily obtained and Ω = ( Ω + Ω ) . When β = π, it shows the most common form: . However, this process will take quite a long time to obtain the target state, which is undesirable.

Using TQD to construct shortcuts to adiabatic passage
The instantaneous eigenstates n k ( = ,±) k 0 for the effective Hamiltonian Substituting eq. (16) in eq. (17), we obtain . Similar to ref. 6, for this three-atom system in a real experiment, the Hamiltonian H(t) is hard or even impossible to be implemented in practice. We should find an APF Hamiltonian whose effect is equivalent to H(t). The model used for the APF Hamiltonian is similar to that in Fig. 1 with three atoms trapped in a cavity, and the atomic level configuration is shown in Fig. 2: the transition ↔ f e is non-resonantly driven by classical field with time-dependent Rabi frequency Ω and detuning Δ , the transition | 〉 (| 〉) ↔ | 〉 g g e l r is coupled non-resonantly to the cavity with coupling λ l (λ r ) and detuning Δ . where β′ is the phase difference between Ω 1 and Ω 3 . Then similar to the approximation for the Hamiltonian from eq. (11) to eq. (15), we also obtain an effective Hamiltonian for the present non-resonant system 43 By adiabatically eliminating the state φ 0 under the condition ∆ Ω / , Ω / ˜3 3 1 3 , we obtain the final effective Hamiltonian Choosing Ω = Ω = Ω( )˜˜t 1 3 , the first two terms of eq. (21) can be removed, and the Hamiltonian becomes  where = , ± , ± , l 0 1 2 .

Fast and noise-resistant generation of the three-atom GHZ states with STAP
We will show that the creation of a three-atom GHZ state governed by ′ H I is much faster than that governed by H I . To satisfy the boundary condition of the fractional stimulated Raman adiabatic passage (STIRAP), 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 three-atom GHZ state, the finial state ψ ( ) according to eq. (16). Therefore, we have tanα = 1. By choosing parameters for the laser pulses suitably to fulfill the boundary condition in eq. (25), the time-dependent Ω ( ) t 1 and Ω ( ) t 3 are gotten as shown in Fig. 3 with parameters = .
t t 0 19 c f . For simplicity, we set β = 0 in the following discussion. Fig. 4 shows the relationship between the fidelity of the generated three-atom GHZ state (governed by the APF Hamiltonian ′ ( )) H t I and two parameters Δ and t f when Ω = . λ 0 2 0 satisfying the Zeno condition, where the fidelity for the three-atom GHZ state is given through is the density operator of the whole system when = t t f ). We find that there is a wide range of selectable values for parameters Δ and t f to get a high fidelity of the three-atom GHZ state. The fidelity increases with the increasing of t f while decreases with the increasing of Δ . It is not hard to understand, putting eq. (26) into eq. (24) and setting = ′ × t t t f , we can find  where Ω′ 0 is the amplitude of Ω( ′) t . That means, in order to satisfy the Zeno condition Ω λ  3 and the large detuning condition Ω ∆  3 , the ratio Δ /t f should be small enough. Moreover, this relationship also explains the phenomenon in Fig. 4 that to achieve a high fidelity with a larger detuning Δ , a longer interaction time t f is required. Then to prove the operation time required for the creation of the three-atom GHZ state governed by ′ H I is much shorter than that governed by H I , we contrast the performances of population transfer from the initial state ψ 1 governed by the APF Hamiltonian ′ H I and that governed by the original Hamiltonian H I in Fig. 5  . The time-dependent population for any state ψ is given by the relationship where ρ(t) is the corresponding time-dependent density operator. The comparison of Fig. 5(a,b) shows that with this set of parameters, the APF Hamiltonian ′ ( ) H t I can govern the evolution to achieve a near-perfect three-atom GHZ state from state ψ 1 in short interaction time while the original Hamiltonian ( ) H t I can not. In fact, through solving the adiabatic condition is a wave function whose amplitude is irrelevant to t f . The result shows when Ω 0 is a constant, the longer the operation time t f is, the better the adiabatic condition is satisfied. This is proved in Fig. 6. Figure 6 reveals the relationship between G(t f ) and λ t f , where ( ) = θ Ω = .  Fig. 7, with respect to the target three-atom GHZ state. Shown in the figure, even with a large laser intensity, say, Ω = . λ 0 5 0 , the interaction time required for creation of the three-atom GHZ state via adiabatic passage is still much longer than that via STAP. Generally speaking, the adiabatic condition is satisfied much better with a relatively larger laser intensity, while, the system would be very sensitive to the decoherence caused by the cavity decay with a relatively large laser intensity. This will be proved in the following. Once the dissipation is considered, the evolution of the system can be modeled by a master equation in Lindblad form, . It turns out that the present shortcut scheme with this set of parameters is much more sensitive to the cavity decays than the spontaneous emissions. Ref. 5 contributes to understanding this phenomenon, in fact, with this set of parameters, the Zeno condition for the non-resonant system is not ideally fulfilled because shortening the time implies an energy cost 12,24 (in this system, the energy cost denotes requiring relative-large laser intensities). Known from ref. 5, destroying the Zeno condition slightly is also helpful to achieve the target state in a much shorter interaction time. However, if the Zeno condition has not been satisfied very well, the intermediate states including the cavity-excited states would be populated during the evolution, which causes that the system is sensitive to the cavity decays. However, we can find in Fig. 8(b) which shows fidelity of the three-atom GHZ in the presence of decoherence, with large laser intensities, the adiabatic scheme is also sensitive to the cavity decays as we mentioned above. The comparison of these two figures drops a result that the present shortcut scheme is almost the same with the adiabatic one in restraining the decoherence.
The robustness against operational imperfection is also a main factor for the feasibility of the scheme because most of the parameters are hard to accurately achieve in experiment. Therefore, we define δ = ′ − x x x as the deviation of any parameter x, where ′ x is the actual value and x is the ideal value. Then in Fig. 9(a) we plot the fidelity of the GHZ state versus the variations in total operation time T ( = . ) T t 1 2 f and laser amplitude Ω′ 0 , and in Fig 9(b) we plot the fidelity of the GHZ state versus the variations in coupling λ and detuning Δ . As shown in the figures, the scheme is robust against all of these variations. Any deviation δ / = % x x 10 ( ∈ , Ω′ , λ, ∆ ) x T { } 0 causes a reduction less than 3% in the fidelity.
In a real experiment, the cesium atoms which have been cooled and trapped in a small optical cavity in the strong-coupling regime 44,45 can be used in this scheme. We take the hyperfine states of / P 6 2 1 2 as the excited sates and the hyperfine states of / S 6 2 1 2 as the ground sates. With a set of cavity QED parameters π λ = × 750 2 MHz, κ π = . × 3 5 2 MHz, and γ π = . × 2 62 2 MHz in strong-coupling regime [46][47][48] , the fidelity of the three-atom GHZ state in this paper is 98.24%. Thus, the scheme is robust and might be promising within the limits of current technology.

Conclusion
We have presented a promising method to construct shortcuts to adiabatic passage (STAP) for a three-atom system to generate GHZ states in the cavity QED system. Through using quantum Zeno dynamics and "transitionless quantum driving", we are free to simplify a complicated Hamiltonian and choose the laser pulses to construct shortcuts in multi-qubit system to implement the fast quantum information processing. Numerical simulation demonstrates that the scheme is fast and robust against the decoherence caused by both atomic spontaneous emission, photon leakage and operational imperfection. The deficiency is that the present scheme might be sensitive to the cavity decays because of some inevitable factors. Compared with the previous shortcut methods, this method obviously works better at entanglement generation in multi-qubit systems. In fact, any quantum system whose Hamiltonian is possible to be simplified into the form in eq. (15), the shortcut can be constructed with the same method presented in this paper. For example, similar to refs 49,50 for the generation of the multiparticle GHZ states in an atom-fiber-cavity combined system, we can shorten the operation time using the same method in the following steps: (1) We consider the Hamiltonian H AL (the Hamiltonian describing the interaction between atoms and lasers) as H obs in eq. (9), and the rest of the total Hamiltonian (the interaction between atoms and cavities, and the interaction between cavities and fibers) as KH meas in eq. (9). Then, choosing the "dark Zeno subspace", that means ε = 0 n in eq. (10), we can obtain an effective Hamiltonian named the Zeno Hamiltonian = ∑ H PH P Zeno n n obs n .
(2) For this effective Hamiltonian, by using TQD, we construct the CDD Hamiltonian H(t) that speeds up the adiabatic process. (3) Similar to the GHZ state generation, we find out the corresponding non-resonant system (the APF Hamiltonian) whose effective Hamiltonian ( ) H t eff has the form in eq. (22). (4) Making ( ) = ( ) H t H t eff , the parameters for the APF Hamiltonian are determined and the shortcut is constructed. Then the APF Hamiltonian would govern the system to achieve the same final result as the adiabatic process governed by the original Hamiltonian with a much shorter operation time. Similar idea can be generalized to generate other multi-qubit quantum entangled states, for example, Bell states, W states, singlet states, and so on. This might lead to a useful step toward realizing fast and noise-resistant quantum information processing for multi-qubit systems in current technology.