Using the notation c$_{m}$ = 〈 m | ψ 〉 we obtain the following equation for the time evolution of the coefficients c$_{m}$ : Using this result in Eq. (A2) we obtain Thus, for m ′ = m we get write the Schr¨ odinger equation for the state vector | ψ 〉 as Taking the derivative of ˆ H$_{1}$ = ∑ m ′′ ϵ$_{m}$ ′′ | m $^{′′}$〉〈 m $^{′′}$| with respect to time we get Finally, writing c$_{m}$ = a$_{m}$e − iϵ$_{m}$$^{t/$_{ℏ}$}$, and ϵ$_{m}$ − ϵ$_{m}$ ′ = $_{ℏ}$ω$_{mm}$ ′ , we arrive at This way of expressing the time evolution of the state vector makes it clear what operator is responsible for in- ducing transitions between the instantaneous eigenstates, which of the eigenstates are coupled by the time-varying Hamiltonian, and what condition must hold for the evo- lution to be adiabatic. The last term in this equation can be put into a more convenient form by noting that It follows that for m ′ = m 12 〈 m $^{′}$| ∂ ˆ H$_{1}$ ∂t | m 〉 = ( ϵ$_{m}$ − ϵ$_{m}$ ′ ) 〈 m $^{′}$| ∂ ∂t | m 〉 . (A6) ∑ m 〈 m $^{′}$| ˆ H$_{1}$ + ˆ H$_{2}$ | m 〉〈 m | ψ 〉 = i$_{ℏ}$ ∑ m 〈 m $^{′}$| ∂ ∂t ( | m 〉〈 m | ψ 〉 ) = i$_{ℏ}$ ∑ m ( 〈 m $^{′}$| ∂ ∂t | m 〉〈 m | ψ 〉 + 〈 m $^{′}$| m 〉 d dt 〈 m | ψ 〉$^{)}$ . (A1) ϵ$_{m}$ ′ c$_{m}$ ′ + ∑ m 〈 m ′ ∣ ∣ ∣ ˆ H$_{2}$ − i$_{ℏ}$ ( ϵ$_{m}$ − ϵ$_{m}$ ′ ) ∂ ˆ H$_{1}$ ∂t ∣ ∣ ∣ m$^{〉}$ c$_{m}$ = i$_{ℏ}$ dc$_{m}$ ′ dt . (A7) ∑ m 〈 m $^{′}$| ˆ H$_{1}$ + ˆ H$_{2}$ − i$_{ℏ}$ ∂ ∂t | m 〉 c$_{m}$ = i$_{ℏ}$ dc$_{m}$ ′ dt . (A2) ∂ ˆ H$_{1}$ ∂t = ∑ m ′′ $^{[(}$dϵ$_{m}$ ′′ dt ) | m $^{′′}$〉〈 m $^{′′}$| + ϵ$_{m}$ ′′ $^{(}$∂ | m $^{′′}$〉 ∂t ) 〈 m $^{′′}$| + ϵ$_{m}$ ′′ | m $^{′′}$〉 $^{(}$∂ 〈 m $^{′′}$| ∂t )] . (A3) ∑ m 〈 m ′ ∣ ∣ ∣ ˆ H$_{2}$ − i ω$_{mm}$ ′ ∂ ˆ H$_{1}$ ∂t ∣ ∣ ∣ m$^{〉}$ a$_{m}$e − iω$_{mm}$ ′ t = i$_{ℏ}$ da$_{m}$ ′ dt . (A8) 〈 m $^{′}$| ∂ ˆ H$_{1}$ ∂t | m 〉 = ϵ$_{m}$ 〈 m $^{′}$| ∂ ∂t | m 〉 + ϵ$_{m}$ ′ $^{(}$∂ 〈 m $^{′}$| ∂t ) | m 〉 . (A4) ∂ ∂t ( 〈 m $^{′}$| m 〉 ) = 0 = $^{(}$∂ 〈 m $^{′}$| ∂t ) | m 〉 + 〈 m $^{′}$| ∂ ∂t | m 〉 . (A5) [18] T. E. Wall, J. F. Kanem, J. J. Hudson, B. E. Sauer, D. Cho, M. G. Boshier, E. A. Hinds and M. R. Tarbutt, Phys. Rev. A 78 , 062509 (2008). [17] S. K. Tokunaga, J. O. Stack, J. J. Hudson, B. E. Sauer, E. A. Hinds and M. R. Tarbutt, J. Chem. Phys. 126 , 124314 (2007). [16] W. J. Childs and L. S. Goodman, Phys. Rev. A 21 , 1216 (1980). [15] J. J. Sakurai, Modern quantum mechanics , (Addison- Wesley, Reading, 1994). [14] H. L. Bethlem, G. Berden, A. J. A. van Roij, F. M. H. Crompvoets and G. Meijer, Phys. Rev. Lett. 84 , 5744 (2000). [13] M. Kajita, Eur. Phys. J. D 38 , 315 (2006). [12] M. Kajita, T. Suzuki, H. Odashima, Y. Moriwaki and M. Tachikawa, Jpn. J. Appl. Phys. 40 , L1260 (2001). [11] M. Kirste, B. G. Sartakov, M. Schnell and G. Meijer, Phys. Rev. A 79 , 051401(R) (2009). [10] S. A. Meek, H. Conrad and G. Meijer, Science 324 , 1699 (2009). [1] S. Y. T. van de Meerakker, H. L. Bethlem and G. Meijer, in Cold Molecules , edited by R. V. Krems, W. C. Stwalley and B. Friedrich (CRC Press, Boca Raton 2009), chap. 14. [2] H. L. Bethlem, G. Berden and G. Meijer, Phys. Rev. Lett. 83 , 1558 (1999). [3] S. Y. T. van de Meerakker, N. Vanhaecke and G. Meijer, Annu. Rev. Phys. Chem. 57 , 159 (2006). [4] S. Hoekstra, M. Mets¨ al¨ a, P. C. Zieger, L. Scharfenberg, J. J. Gilijamse, G. Meijer and S. Y. T. van de Meerakker, Phys. Rev. A 76 , 063408 (2007). [5] M. Lara, B. L. Lev and J. L. Bohn, Phys. Rev. A 78 , 033433 (2008). [6] S. K. Tokunaga, J. M. Dyne, E. A. Hinds and M. R. Tarbutt, New J. Phys. 11 , 055038 (2009). [7] K. Wohlfart, F. Filsinger, F. Gr¨ atz, J. K¨ upper and G. Meijer, Phys. Rev. A 78 , 033421 (2008). [8] O. Bucicov, M. Nowak, S. Jung, G. Meijer, E. Tiemann and C. Lisdat, Eur. Phys. J. D 46 , 463 (2008). [9] A. Schwettmann, J. Franklin, R. Overstreet and J. P. Shaffer, J. Chem. Phys. 123 , 194305 (2005).