Given external radiation, produced from an electric field travelling from the left and an electric field travelling from the right, producing a standing wave $$E_0e^{i(kx - \omega t)} + E_0e^{-i(kx - \omega t)},$$the Stark energy shift of a two level system under the influence of this electric field is $$\Delta E(x) = \hbar \Delta \omega_0 (x) = \frac{\hbar \Omega^{2}(x)}{4 \delta} = \hbar \bigg( \frac{\vec{d} \cdot \vec{E}_0}{ \hbar} \bigg)^2/ 4 \delta,$$ where $\Omega$ is the Rabi frequency and $\delta$ is the detuning $\omega_0 - \omega$.
How does it follow that the force on the particle depends on the gradient of the electric field at the position of the particle?
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