Topological superposition

Table 1. Pairwise comparison of the topology and primary sequence of members of the short spacer family. The alpha carbon atoms defining the zinc protease fold (orange segment. Fig. 3) have been used in the topological superposition [56]. The distances refer to the root mean square deviations of this fold between pairs of structures. The corresponding pairwise primary sequence homology is also shown.

Diederichs, K. Structural superposition of proteins with unknown alignment and detection of topological similarity using a six-dimensional search algorithm. Proteins Struc., Func., Genet. 1995, 23, 187-195. 

The same is true for low-dimensional systems, d — 1 and 2. The point is not only that for such systems the better statistics could be achieved accompanied with reasonable computational time spent for it. Another circumstance is that we can expect here that the superposition approximation gives greater errors. For example, for one-dimensional contact recombination the so-called bus effect is known [17] given particles A and B can react only after particles separating them disappear during reaction. This topological effect is not foreseen by the superposition approximation but can affect considerably the reaction rate. 

We study the various superpositions of states that can be created by adiabatic passage in a robust way with respect to variations of the field amplitude, using the topological analysis with resonances of Section V.D (see also Section IV.B.3 for the case of one laser). We assume that one starts (at time t = ti) with a coherent state for the photon field and in the atomic state 1). We study here the A-system. Our results are easily extended to the other system (ladder and V), using the appropriate signs accompanying the field frequencies. We study the creation of a superposition of states at the final time t = tf. 

The topological analysis thus shows that with two quasi-resonant delayed lasers it is not possible to end in a superposition of states between the lowest states 1) and 13) in a robust way. We can remark that in Ref. 76, it has been shown that one can create such a superposition—however, in a nonrobust way but still by adiabatic passage, by modifying the end of the STIRAP process (with the counterintuitive sequence), maintaining a fixed ratio of Stokes and pump pulse amplitudes. 

---