the adiabatic potentials $V^{\pm}({x})=(-\delta_i\pm\sqrt{\delta_i^{2}+\Omega_i(x)^{2}})/2$. These are shown schematically in Fig. \[fig:levelstructure\]b). The complete wavefunction can then be expanded in a basis of these adiabatic eigenstates, which play the role of Born-Oppenheimer channel functions, $$|\Phi(t)\rangle=c_+ (x,t) |\Psi_{+}\rangle+c_- (x,t) |\Psi_{-}\rangle,$$ resulting in the equation $$i\hbar\frac{\partial}{\partial t}c_{\pm}(x,t)=[H_M+V^{\pm}(x)]c_{\pm}(x,t)+H_{M}^{\pm}c_{\mp}(x,t),$$ where $H_{M}^{\pm}=\langle\Psi_{\pm}(x,t)|H_M|\Psi_{\mp}(x,t)\rangle$ gives the non-adiabatic couplings between the dressed states due to the motion of the atom. Provided these latter terms are small, the two equations decouple and the atoms remain in a single dressed state. In our case, there will be no non-adiabatic loss of atoms in this sense, provided that they are loaded into low energy states of the lower of the two adiabatic potentials ($V^-$). If atoms are loaded into the higher energy dressed potential, loss of atoms into the continuum states of the $V^-$ potential can occur, however this is exponentially suppressed as the separation between adiabatic potentials is increased, with the loss rate $\Gamma_l \sim \Gamma_0 \exp(-\delta_i/\omega)$ [@yi08], where $\Gamma_0$ is a prefactor that we do not compute in detail here, and $\omega$ is the trap frequency in an individual site of the dressed potential.

Dressed potentials for independent states
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When these dressed potentials are created independently for different $m_I$ 