Now we turn to the rotation of the field, i.e. to the matrix elements of ω ˆ N$_{y}$ in Eq. (6). Here, the situation is quite different because this term couples states of dif- ferent M within the same manifold, and these states ap- proach each other very closely when the field is small. The matrix element that we need is only non-zero when M ′ = M ± 1, in which case it is where $_{ℏ}$ω$_{mm}$ ′ = ϵ$_{m}$ − ϵ$_{m}$ ′ , a$_{m}$ = 〈 m | α $^{′}$〉 e iϵ$_{m}$$^{t/$_{ℏ}$}$, m stands for the pair of quantum numbers N , M , and m ′ for N $^{′}$, M $^{′}$. Next, following the algebra set out in App. A, we ex- press this equation in a basis formed from the instanta- neous eigenvectors of H $^{′}$, obtaining where ω ( t ) = dβ $_{dt}$. time evolution of the state in the rotating frame, At low electric field, where transitions are most likely to occur, the mixing of different N states is small and so only one term ( N = N ) makes any significant contribu- tion to the sum. The probability of finding the molecule in the state M ′ = M ± 1 is therefore For low-lying rotational states, which are the ones of most interest to us, the transition probability will always be very small if ω ≪ ω$_{ni}$ - this is the criterion for the rota- tion to be adiabatic. In Eq. (6) the dynamics are expressed in a particularly transparent way. The first operator in the matrix element is due to the rotation of the electric field vector and cou- ples states of different M , while the second arises from the change in the electric field magnitude and can only couple states of the same M belonging to different man- ifolds N . Provided the matrix elements in the sum vary little on the timescale of 1 /ω$_{mm}$ ′ , the terms in the sum oscillate around zero with an angular frequency ω$_{mm}$ ′ . Those amplitudes a$_{m}$ that are initially zero will always remain small unless they grow significantly in a time of order 1 /ω$_{mm}$ ′ . Thus, if the matrix elements are all much smaller than the energy splittings $_{ℏ}$ω$_{mm}$ ′ , the state am- plitudes hardly change meaning that the molecule adi- abatically follows the changing field. This is the situa- tion that we want in the Stark decelerator, so let us now estimate the transition probability arising from the off- diagonal terms. Assuming that these terms are at least small, we can treat them using time-dependent first-order perturbation theory. For a perturbation ˆ V , the probabil- ity P$_{n}$ of being in state | n 〉 , having started out in the initial state | i 〉 , is at most 4 |〈 n | ˆ V | i 〉| $^{2}$/ ($_{ℏ}$ ω$_{ni}$ ) 2 (see [15] for example). We consider two examples, LiH prepared in the ini- tial state | i 〉 = | 1 , 0 〉 , and CaF prepared in the initial state | i 〉 = | 4 , 0 〉 . They move through the decelerator illustrated in Fig. 1(a), with R = 1 . 5 mm, r$_{0}$ = 1 mm, L = 6 mm and V = 15 kV. We take | n 〉 = | 1 , 1 〉 for LiH and | n 〉 = | 4 , 1 〉 for CaF, and we take the rise and fall time of the switch to be τ = 500 ns. Figure 3 shows how the values of ω = dβ/dt (solid lines), ω LiH ni (dotted lines) and ω CaF ni (dashed lines) change as a function of time for several different values of the reduced position θ . When θ = − 90 $^{◦}$, ω exceeds ω CaF ni for the first 15 ns after the switch is thrown. At this position in the decel- erator the field changes from a very small field along z to a very large field along x (see Fig. 1(a,b)), and so the field vector completes almost all of the 90 ◦ rotation in the first few ns, i.e. in a small fraction of τ . As a result, the rotation rate is very large over this interval of time, ω ∼ 10 $^{8}$rad s − $^{1}$. At the same time, the total field is still very small, and so the angular frequency interval between the two M states is also very small, ω CaF ni ∼ 10 $^{7}$rad s − $^{1}$. Taking first the last operator on the right hand side of Eq. (6), we have 〈 n | ˆ V | i 〉 = iµ dE dt 〈 n | ˆ C (1) 0 | i 〉 /ω$_{ni}$. The maximum value of this last matrix element is 1, while that of dE dt is E$_{max}$/τ , E$_{max}$ being the maximum electric field in the decelerator. As shown in Fig. 2, the weak- field seeking states have turning points, and this sets an upper limit to E$_{max}$ . Within manifold N , the state with the largest positive Stark shift is |N , M = 0 〉 , whose turning point is at a field of approximately 4 π$_{ℏ}$B N $^{2}$/µ . For those states that are coupled by the perturbation, the smallest possible value of ω$_{ni}$ occurs at low electric field and is 4 πB N . Putting all this together, we conclude that the probability of a transition induced by the changing electric field magnitude is smaller than 1 / (2 πBτ ) $^{2}$. Most molecules of interest have B > 10 GHz, while the rise and fall time of the switch is typically τ > 100 ns, and so the maximum transition probability, about 10 − $^{8}$, is completely negligible. We see that the rotation is far from adiabatic. A CaF molecule that happens to be at this position when the field switches is very likely to end up in a different state. For LiH the situation is not as severe because, at early times, ω LiH ni exceeds ω by a factor of 3. Nevertheless, since the two frequencies are still of the same order of magnitude, we can expect there to be a significant tran- sition probability. As we move away from this position in the decelerator, ω decreases as the difference in field A. Examples 〈N , M ± 1 | ˆ N$_{y}$ |N , M 〉 = ∓ i$_{ℏ}$ 2 ∑ N $^{√}$( N ∓ M )( N ± M + 1) × 〈N , M ± 1 | N, M ± 1 〉〈 N, M |N , M 〉 . (7) P$_{±}$$_{1}$ ( t ) < ω 2 ω 2 ni ( N ∓ M )( N ± M + 1) . (8) ∑ m 〈 m ′ ∣ ∣ ∣ ω ( t ) ˆ N$_{y}$ − i ω$_{mm}$ ′ ∂ ˆ H ′ ∂t ∣ ∣ ∣ m$^{〉}$ a$_{m}$e − iω$_{mm}$ ′ t = i$_{ℏ}$ da$_{m}$ ′ dt , (6) ( ˆ H ′ + ω ( t ) ˆ N$_{y}$ ) | α $^{′}$〉 = i$_{ℏ}$ ∂ ∂t | α $^{′}$〉 , (5) 4