6 FIG. 6: Transition probability out of the M = 0 state as a function of the reduced position in the decelerator. The po- sition dependence is symmetric about θ = − 90 $^{◦}$. The switch rise and fall time is τ = 500 ns, and the electric field profile is the one shown in Fig. 1(b). Solid line: LiH in N = 1. Dashed line: CaF in N = 4. FIG. 5: (Color online) Evolution of the populations in the various sublevels of the N = 4 manifold of CaF as a function of time since throwing the switch, and at three different re- duced positions in the decelerator. 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). In discussing the degree of molecule loss to be ex- pected from these unwanted transitions, we need to dis- tinguish molecules that are phase-stable from those that are not. The phase-stable molecules lie within the longi- tudinal phase-space acceptance of the decelerator; they are the ones being decelerated. The phase angle of these molecules, called φ and meaning the value of θ at the mo- ment of switching, is constrained by the action of the de- celerator to lie within the boundaries of the phase-space acceptance. These boundaries are shown in Fig. 1(c) for various values of the synchronous phase angle, φ$_{0}$ . This figure shows that if the decelerator is operated at large phase angles, say φ$_{0}$ > 45 $^{◦}$, corresponding to strong de- celeration, the phase-stable molecules cannot reach the region centred on θ = − 90 ◦ where transitions are likely to occur. The group of molecules being decelerated are largely protected from transitions driven by the switch- ing field. If we plot a molecule’s phase-space position every time the decelerator switches we find that the phase-stable molecules move on closed trajectories cen- tred on the synchronous molecule in the phase-space plot of Fig. 1(c). Thus, even when the decelerator is operated at low phase angle, there will be a sizeable fraction of phase-stable molecules that never enter the regions where transitions are likely. By contrast, the molecules that are not phase-stable tend to explore all possible values of φ as they make their way through the decelerator. Since the region where transitions occur is sizeable, and since in a typical deceleration experiment the field switches ting progressively slower and the population being con- fined more and more to the low | M | states. At θ = − 50 ◦ all the population remains in the M = 0 and | M | = 1 states. Again, this is consistent with our expectations from Fig. 3(b) and Eq. (8). It is also interesting to study the situation at θ = 90 $^{◦}$, where it is necessary to find the solution up to t ∼ 3 µ s so as to capture any dynamics that may occur when ω and ω$_{ni}$ pass through their point of closest approach (see Fig. 3(f)). We find the population transfer to be negligible here. Figure 6 summarizes the results we have obtained. It shows the transition probability for LiH | 1 , 0 〉 and CaF | 4 , 0 〉 molecules as a function of the position in the decel- erator at the moment of switching. Transitions occur in a zone centred on θ = − 90 $^{◦}$. For CaF the maximum tran- sition probability is 84% and the zone has a full width at half maximum of about 60 $^{◦}$. For LiH, the maximum probability is 11% and the zone has a full width at half maximum of about 20 $^{◦}$. Obviously, faster switching or lower applied voltages will tend to increase the transi- tion probabilities and extend the width of the zone. B. Loss 0 20 40 60 80 100 - 90 90 45 - 45 0 Reduced position,  θ  (degrees) Transition probability (%) 0.0 0.2 0.4 0.6 0.8 1.0 20 0 40 60 80 100 Population (a) -90º 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 20 0 40 60 80 100 20 0 40 60 80 100 (b) -70º (c) -50º Time (ns) Time (ns) Time (ns) Population Population 0 1 2 3 4 0 1 2 3 0 1