Normalized two-state truncation of a displaced oscillator ($⟨x{⟩}_{01}=x_0/(1+{\eta }^2)$)

Let $x_{\mathrm{zp}}=\sqrt{\hslash /(2m\omega )}$ and $\eta =x_0/(2x_{\mathrm{zp}})$. The normalized first-order truncation is $(|0⟩+\eta |1⟩)/\sqrt{1+{\eta }^2}$. Its mean displacement is $2\eta x_{\mathrm{zp}}/(1+{\eta }^2)$, agreeing with $x_0$ only at small $|\eta |$. The exact translated ground state is a coherent state whose expansion contains all number states and has mean occupation ${\eta }^2$. Correct linear response does not justify extrapolating a truncated state to large fields.