Creation of entangled W states of four two-level atoms in a cavity via quadrapod adiabatic passage

In this paper, we considered four two-level atoms coupled with a microwave cavity via stimulated Raman adiabatic passage in quadrapod linkage pattern. Engineering Rabi frequencies of a system with suitable pulse orders, results in entangled W states of four atoms. We also compared the theoretical results with the numerical solutions of the Schrödinger equation in the adiabatic limit which shows the creation of W states at the end of dynamics.

Quantum entanglement, one of the most striking features of quantum mechanics, has become an essential element in quantum information processing. Two-mode optical entanglement has already been produced experimentally via an optical parametric oscillator in optical crystals (Ou et al. 1992;Zhang et al. 2000;Villar et al. 2005) and four-wave mixing (Ding et al. 2011;Pooser et al. 2009). Stimulated Raman adiabatic passage of STIRAP is a high qualified method used to completely control population transformation between levels at several atomic systems Unanyan et al. 1998;Amniat-Talab et al. 2011).
In this method, the complete population transfer occurs between the ground states, without populating the intermediate excited state. The ground state has a relatively high lifetime of spontaneous emission, so decoherence effects resulted from spontaneous emission of atoms are negligible and this is the advantage of the stimulated adiabatic passage.
There are two entangled states: the first one is GHZ-state (Greenberger et al. 1989) and the second is W-state (Dur et al. 2000).
One of the most striking features of W quantum state (e.g. entanglement state) is: (|0, 1, 0 + |1, 0, 0 + |0, 0, 1 )/ √ 3, If we trace the state of one particle, the state of the system will be remain entangled, or if we measure a particle at {|0 , |1 } other particles would be remain in maximally entangled state.
In this paper, we derive the corresponding propagator in the adiabatic limit and show that it allows the generation of any coherent superposition of the ground states in quadrapod system by adiabatic passage in a single step with the use of pulses of Gaussian forms. In Sect. 2, we determine the effective Hamiltonian of the quadrapod system to an effective system. We show that one can then transfer the population to any superposition of four ground states. In Sect. 3 we introduce time evolution. Section 4 provides a summary of the results.

Construction of the effective Hamiltonian
In this paper, to understanding of basic concept of the cavity-laser interaction, which is achieved between four atoms by excited states, we have to analyze this coupling by four pulsed fields denoted by. G 1 (t), G 2 (t), G 3 (t) and G 4 (t), as shown in Fig. 1.
The initial and final atomic states of the system take the form of: |e, g, g, g, 0 + |g, e, g, g, 0 + |g, g, e, g, 0 + |g, g, g, e, 0 where, subscribes i and j showing the atomic states of the system at before and after of interaction with pulsed fields, respectively, with e and g being atoms excited and ground states, respectively. We assume the fields have a Gaussian form as: where, τ i and T i are the time delay between the pump and stokes pulses, and the pulse duration, respectively. For four identical atoms, the evolution of the coupled atoms-cavity-laser system is described by the Hamiltonian where, the summation in the first term describes four atoms and their interaction with the laser and cavity, and the second term describes the cavity energy, with a and a + being the annihilation and creation operators for a cavity photon acting upon the states, respectively, and ω e being the frequency of the coupling excited states. Assuming: |1 = |e, g, g, g, 0 |2 = |g, e, g, g, 0 |3 = |g, g, e, g, 0 |4 = |g, g, g, e, 0 |5 = |g, g, g, g, 1 effective interaction Hamiltonian of the system becomes Detuning of the frequency has been neglected.

Adiabatic time evolution operator
Diagonalizer matrix of the effective Hamiltonian is where, we defined Hence, the Hamiltonian in the basis of adiabatic states can be written as: In order to obtain the Hamiltonian of the system in terms of adiabatic bases we considered the adiabatic approximation condition as: where, |φ i , |φ j are eigenstates of the Hamiltonian and E i , E j are eigenvalues of energy. So, we have obtained the adiabatic Hamiltonian of the system as: where, = χ 4 . By definition of mixing angles as the time evolution operator is given by Substituting adiabatic Hamiltonian from Eq. 11 into the Eq. 13 gives where, M is a 3 × 3 matrix with matrix elements given by where, γ i can be interpreted as the geometric phase as: and δ is defined as dynamic phase of the system as: Now, by the use of following equations and by means of the initial state of the system and engineering suitable pulse orders (according to Eq. 2), we can calculate the final state of the system. In the W state, the final state of the system should take the form of We have used Gaussian pulses, satisfying the following conditions before and after interaction with the fields, which they are calculated using mixing angles So, we can write geometrical phases in Eqs. 16 as: Then, by substituting results in Eqs. 15, the matrix elements are given At the initial and final interaction times, T(t) can be found by applying pulse conditions as: Substituting results in Eq. 18, we have Finally, by substituting results into Eq. 26, the state vector of the system can be found as Eq. 19. The results of numerical calculations are shown in Fig. 2.

Conclusion
In conclusion, we have studied a stimulated Raman adiabatic passage in quadrapod linkage pattern driven by three pulses by deriving the corresponding propagator in the adiabatic limit. Engineering Rabi frequencies of the system with a suitable pulse orders allows us to transfer the population to a desired superposition of ground states with final equal populations. It can be seen from Fig. 2c that at final state fidelity of desired state explained Eq. 19 equals to one.
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