Bloch oscillation

In a single isolated energy band, a wave packet subject to a constant force satisfies $\hslash \dot{k}=F$. Its group velocity is periodic in $k$ with reciprocal-lattice period $2\pi /a$, so its position oscillates with period $2\pi \hslash /(|F|a)$ when its mean velocity over a band is zero. For $E(k)=E_0-2J\cos ka$, integrating the sinusoidal velocity gives the oscillation explicitly. This ideal description neglects scattering and interband tunnelling.

For $ϵ(k)={ϵ}_0-2J\cos ka$ and electron force $-eE$, $k(t)=k_0-eEt/\hslash$. Integrating the group velocity gives $x(t)-x(0)=2J[\cos (k_{0}a-{\omega }_{B}t)-\cos (k_{0}a)]/(eE)$, with ${\omega }_B=eEa/\hslash$ for positive $E$. The amplitude about the mean is $2|J|/(e|E|)$ and the peak-to-peak extent is twice that.

In a 1996 experiment, ultracold cesium atoms were loaded into the lowest band of an optical standing wave made with $852\mathrm{nm}$ light. Chirping the frequency difference of its counterpropagating beams accelerated the lattice, producing a constant inertial force in its frame. Switching off the lattice after different evolution times allowed momentum-distribution measurements. The observed oscillations tested the periodic band velocity underlying a Bloch oscillation; the lattice period and imposed acceleration controlled the oscillation frequency.^[ref]