Nonidentical atoms, collective states

The collective states (35) are eigenstates of the system of two identical atoms. If the atoms are not identical, the situation becomes more complicated and we will discuss here some consequences of the fact that the atoms could have different transition frequencies and/or different spontaneous emission rates. When the atoms are nonidentical with different transition frequencies, the states (35) are no longer the eigenstates of the Hamiltonian (32). The diagonalization of the matrix (34) with A / 0 leads to the following eigenstates [43]... 

We follow exactly the same route as in the preceding section, and rewrite the master equation (31) in terms of the collective operators Ay = i) ( j, where now the collective states are given in Eq. (40). First, we find that in the case of nonidentical atoms the atomic and collective operators are related by... 

The choice of the collective states (40) as a basis leads to a complicated master equation whose physical properties are tractable only for very specific values of the parameters involved. A different choice of basis collective states is proposed here, which allows us to obtain a simple master equation of the system of two nonidentical atoms. Moreover, we will show that it is possible to create a maximally entangled state in the system of two nonidentical atoms that can be decoupled from the external environment and, at the same time, the state exhibits a strong coherent coupling with the remaining states. 

These equations are quite different from Eqs. (114) that in the case of nonidentical atoms the symmetric and antisymmetric states are equally populated independent of the interatomic separation. These are, however, similar to the steady-state solutions for the small sample model that for small interatomic separations pM = paa 0 and then only the collective ground and the upper states are populated. 

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