FIG. 4: (Color online) Evolution of the populations in the sublevels of the N = 1 manifold of LiH as a function of time since throwing the switch and at the position θ = − 90 $^{◦}$. The curves are labelled by their value of | M | . The initial state is M = 0, the switch rise and fall time is τ = 500 ns, and the electric field profile is the one shown in Fig. 1(b). FIG. 3: (Color online) The rotation rate, dβ/dt (solid), the angular frequency separation of LiH | 1 , 0 〉 and | 1 , 1 〉 states (dotted), and the angular frequency separation of CaF | 4 , 0 〉 and | 4 , 1 〉 states (dashed), as a function of time since throwing the switch and at various reduced positions ( θ ) within the decelerator. The switch rise and fall time was set to τ = 500 ns, and the electric field profile used in the calculation is the one shown in Fig. 1(b). Note the logarithmic scale on both axes. Figure 4 shows what happens to LiH molecules ini- tially in the | 1 , 0 〉 state, when they are at θ = − 90 ◦ at the moment of switching. Here, the rise and fall times of the switch are 500 ns, and the electric field profile is as shown in Fig. 1(b). As anticipated from the consid- erations above, the evolution is not fully adiabatic. The rapid rotation of the field vector that occurs during the first 10 ns results in some population transfer into the M = ± 1 states. After a few tens of ns the evolution becomes adiabatic and there is no further change in the state amplitudes. Moving away from this θ = − 90 ◦ posi- tion we find that the transition probability drops rapidly to zero. We note that in all cases the amplitudes of the ± M states are identical because the molecule starts in M = 0 and, in our coordinate system, the z -axis lies in the plane of rotation. This symmetry is also seen by setting M = 0 in Eqs. (7) and (8). Figure 5 shows what happens to CaF molecules ini- tially in the | 4 , 0 〉 state, when they are at θ = − 90 $^{◦}$, − 70 ◦ and − 50 $^{◦}$. When θ = − 90 $^{◦}$, the initial popula- tion in the | 4 , 0 〉 state is rapidly transferred to the other M states within N = 4. First, the | M | = 1 state is populated, and from here the | M | = 2 , 3 and 4 states all become populated as time progresses. After about 50 ns the populations stop changing because the evolu- tion changes over from being strongly nonadiabatic to being adiabatic, as anticipated from our consideration of Fig. 3(a). At θ = − 70 ◦ the evolution is slower and less of the population is driven to the higher | M | states. In particular, the | M | = 3 states, which received the largest share of the population at θ = − 90 $^{◦}$, now receive very little, and the | M | = 4 states are never populated at all. These trends continue as θ increases, the evolution get- To calculate the exact time-evolution of the state of the molecule, we set N ′ = N in Eq. (6), having already established that couplings between different N manifolds are negligible. Then we solve numerically the resulting magnitude between the two switch states decreases, and ω$_{ni}$ increases because the minimum field value increases. At θ = − 54 ◦ for example, the maximum value of ω is about 3 times smaller than the minimum value of ω CaF ni , while at θ = − 18 ◦ the former is more than two orders of magnitude smaller than the latter. At all positions, the Stark splitting for LiH is far greater than for CaF meaning that LiH is not as susceptible to nonadiabatic transitions. The situation at θ = 90 ◦ again seems dan- gerous - the field starts out very large along z and ends up very small along x . This means that most of the rota- tion occurs at the very end of the switch time, at which time the frequency separation of the two states is very small. However, because the gradient of the exponen- tial rise and fall is far smaller at late times than at early times, the rotation is not as fast and the maximum value of ω/ω CaF ni is about 1/20. set of differential equations, N ∑ M = −N 〈N , M $^{′}$| ω ( t ) ˆ N$_{y}$ |N , M 〉 a$_{M}$e − iω$_{(}$$_{N}$$_{,M}$$_{)(}$$_{N}$$_{,M}$ ′ $_{)}$t = i$_{ℏ}$ da$_{M}$ ′ dt . (9) 5 20 0 40 60 80 100 Time (ns) 0.0 0.2 0.4 0.6 0.8 1.0 Population |M| = 0 |M| = 1 10 6 10 7 10 8 10 9 10 10 6 10 7 10 8 10 9 10 10 6 10 7 10 8 10 9 10 Angular frequency (rad / s) 10 -9 10 -8 10 -7 10 -6 10 -9 10 -8 10 -7 10 -6 10 -9 10 -8 10 -7 10 -6 10 -9 10 -8 10 -7 10 -6 10 -9 10 -8 10 -7 10 -6 Time (s) (a) -90º (b) -54º (c) -18º (d) 18º (e) 54º (f) 90º 10 -9 10 -8 10 -7 10 -6 10 6 10 7 10 8 10 9 10 10 6 10 7 10 8 10 9 10 10 6 10 7 10 8 10 9 10