2 two states the field on the beamline rotates by 90 ◦ and changes in magnitude from the value given by the solid line to that given by the dashed line in Fig. 1(b). The switching sequence applied to the decelerator is synchronized to the motion of a molecule having a par- ticular initial position and velocity. By construction, this synchronous molecule has the same value of θ (modulo π ) every time the decelerator switches, and that value is called the synchronous phase angle, φ$_{0}$ . In the coordi- nates of Fig. 1, φ$_{0}$ = 0 corresponds to zero deceleration, and φ$_{0}$ = 90 ◦ corresponds to maximum deceleration. The decelerator exhibits phase stability [14], meaning that it traps a group of molecules centred in phase space on the synchronous molecule. These phase-stable molecules re- main close to the synchronous molecule throughout the deceleration process. The area they occupy in phase space is the longitudinal acceptance of the decelerator and depends on the degree of deceleration applied. Fig- ure 1(c) shows an example of this acceptance for several values of φ$_{0}$ . We will return to this figure to explain how the phase-stable molecules are less prone to nonadiabatic transitions than the non-phase-stable molecules. OH and NH in low-field seeking states, there is good ex- perimental evidence that nonadiabatic transitions do not occur [3, 4]. For the specific case of OH, this is supported by theoretical work [5]. However, for LiH in the first ro- tationally excited state, numerical simulations could only be reconciled with experimental results by including in the simulations the probability of nonadiabatic transi- tions from the low-field to the high-field seeking compo- nent, driven by the rotation of the electric field during the switch [6]. In experiments on the deceleration of OH molecules in high-field seeking states, there was strong loss due to a crossing of hyperfine states which was elim- inated by applying a small bias voltage so that the de- celerator field was always above the field where the two states cross [7]. In experiments on SO$_{2}$, potential loss due to an avoided crossing of the decelerated state was considered, but occurred at too high a field to be im- portant [8]. Loss at an avoided crossing has been dis- cussed in the context of decelerating asymmetric rotors [9]. Loss of molecules due to nonadiabtic transitions was also found to be significant in the recently demonstrated chip decelerator [10], and is important in traps for polar molecules, as observed in [11] and discussed theoretically in [5, 12, 13]. We will first outline the theory of nonadiabatic transi- tions in a Stark decelerator. This leads to a simple adia- baticity criterion and to a simple method for calculating transition probabilities. We then provide some example calculations, showing that the transition probability is a strong function of position in the decelerator, and that molecules in the non-decelerated group are far more likely to be lost by this mechanism than those in the deceler- ated group. Finally, we compare the theoretical results with the results of experiments on the Stark deceleration of LiH and CaF molecules. We consider a rigidly rotating linear molecule of dipole moment µ and rotational constant B in an electric field E which is E$_{1}$ ˆ z before the switch and E$_{2}$ ˆ x after the switch. During the switch, the first field decays with a time con- stant τ$_{1}$ while the second field turns on with time con- stant τ$_{2}$ . The time-dependent Hamiltonian, ˆ H ( t ), that describes this system is given by where ˆ N is the rotational angular momentum operator, and we have written the angular factors in the Hamilto- nian in terms of the components of the rank 1 spherical harmonic operator, ˆ C (1) q = √ 4 π 3 ˆ Y (1) q . In the absence of the electric field, the eigenstates are the angular momentum eigenstates | N,M 〉 , satis- fying ˆ N $^{2}$| N,M 〉 = N ( N + 1) | N,M 〉 and ˆ N$_{z}$ | N,M 〉 = M | N,M 〉 . The electric field mixes states of the same M but different N so that N is no longer a constant of the motion. We use the notation |N ,M 〉$_{E}$ to denote the mixed- N eigenstates in the static field E ˆ z , omitting the subscript E when it is not needed. To identify a specific state we assign a value to N which is equal to N in the limit that E → 0 adiabatically. These eigenstates, and the corresponding eigenvalues, ϵ$_{N}$$_{,M}$ , are calculated by diagonalizing ˆ H (0) for many different values of E$_{1}$ us- ing a suitably large basis set formed from the field-free eigenstates. Figure 2 shows the energies of the lowest- lying states as a function of the electric field. Both the energy and the electric field are written in dimensionless units by dividing the former by the rotational constant hB , and the latter by the ratio of rotational constant to dipole moment hB/µ . The Stark decelerator works best for the states with the largest positive Stark shift since these states are naturally focussed and are most strongly Figure 1(a) illustrates the layout of a typical deceler- ator [2], showing three deceleration stages. Each stage is formed from a pair of parallel rods with the molecu- lar beam passing between them. The axes of the rods are parallel to the z -axis in the odd-numbered stages and parallel to the x -axis in the even-numbered stages. The molecular beam travels along the y -axis and, to ex- press the periodicity of the decelerator, the longitudinal coordinate is written (in radians) as the reduced posi- tion θ = πy/L , L being the distance between successive stages. The decelerator is switched between two states, state 1 which has the odd stages grounded and the even stages charged (as shown in Fig. 1(a)), and state 2 which has the even stages grounded and the odd stages charged. The profile of electric field strength in these two states is shown in Fig. 1(b) for a decelerator that has R = 1 . 5 mm, r$_{0}$ = 1 mm, L = 6 mm and V = 15 kV. On the y -axis, the electric field is directed along z in state 1 and along x in state 2. When the decelerator switches between these II. THEORY ˆ H ( t ) = hB ˆ N $^{2}$− µE$_{1}$e − t/τ$_{1}$ ˆ C (1) 0 − µE$_{2}$ √ 2 (1 − e − $^{t/τ$_{2}$}$)( ˆ C (1) − $_{1}$− ˆ C (1) $_{+1}$) , (1)