Shaken harmonic trap ($H^{\prime }(t)=-m{\omega }^2{x}_0(t)x+\frac{1}{2}m{\omega }^2{x}_0(t{)}^{2}I$)

Displacing a one-dimensional quantum harmonic oscillator produces a linear force term and a scalar shift relative to the unshifted Hamiltonian. The scalar adds only a common phase. The linear term couples number states separated by one excitation. Arbitrary linear driving of the ground state creates a coherent state; a first-order expansion of that state contains only the vacuum and first excited state.^[ref]

A sinusoidal trap displacement with amplitude $\epsilon$ and duration $T$ produces a resonant first-order excitation amplitude proportional to $\epsilon \omega T/a_0$, where $a_0$ is the oscillator length. Keeping its square much smaller than one is the long-time condition for weak excitation. The small-displacement condition $\epsilon /a_0\ll 1$ alone does not ensure uniform validity as $T$ grows. Exact linear driving remains solvable as a coherent state after this first-order approximation fails.

The ladder identity $x=\sqrt{\hslash /(2m\omega )}(a+a^{†})$ proves the matrix element in the title. In time-dependent perturbation theory, a linear force excites only $n=1$ directly from $n=0$. Parity alone rules out even excited states; the ladder identity also rules out odd $n>1$. Higher-order evolution can reach other levels, so the driven oscillator is not an exact two-level system.