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decelerated. Note that these low-field seeking states all turn over to become high-field seekers when the field is large, and that the turning point shifts to higher electric field for larger values of N and B . 

At time t = 0 the molecule is in the particular eigen- state |N$_{p}$ , M$_{p}$ 〉$_{E}$$_{1}$ . We can calculate the time evolu- tion of the state vector | α 〉 in the z -fixed basis | NM 〉 , 

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by solving the Schr¨ odinger equation for the coefficients c$_{NM}$ = 〈 NM | α 〉 , 

with this initial condition and with ˆ H ( t ) given by Eq. (1). Having found the time evolution of | α 〉 , we can then cal- culate the probability of finding the molecule in any one of the states |N , M 〉$_{E}$$_{(}$$_{t}$$_{)}$ as a function of t . 

Although this calculation is straightforward, the in- tegration of Eq. (2) is slow because the basis set needs to include a large number of N states and because the timescale of the switch is vastly larger than 1 /B , which sets the scale for the time step in the integration. For- tunately, a few simple transformations provide us with an equation that is much faster to solve and which also makes it obvious that it is the rotation of the electric field vector that drives the transitions. 

If the coordinate system were rotated so that the field at time t pointed along the local z $^{′}$-axis, the instanta- neous Hamiltonian would be ˆ H ′ = hB ˆ N 2 − µE ( t ) ˆ C (1) 0 , E ( t ) being the electric field magnitude at time t . Hamil- tonian (1) is related to this one by a rotation about the y -axis through an angle β ( t ), ˆ H = ˆ D − 1 ˆ H ′ ˆ D . Here ˆ D is the relevant rotation operator, ˆ D = exp( − i ˆ N$_{y}$β/$_{ℏ}$ ). The evolution of the state vector | α 〉 is governed by the Schr¨ odinger equation: i$_{ℏ}$ ∂ $_{∂t}$| α 〉 = ˆ H ( t ) | α 〉 . Apply- ing the rotation operator to both sides of this equation, and introducing the state vector in the rotating frame, | α $^{′}$〉 = ˆ D | α 〉 , we obtain 

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Using 

and the fact that ˆ D − 1 commutes with ˆ N$_{y}$ , we obtain the 

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