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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]

Gregor Jotzu and collaborators realized an artificial Haldane model in 2014. They loaded spin-polarized potassium-40 fermions into a laser-made honeycomb lattice. Piezoelectric mirrors shook the optical lattice along two axes; their relative phase controlled time-reversal symmetry breaking. Optical phase control adjusted the sublattice offset.^[ref]

A magnetic field gradient drove Bloch oscillations. Band mapping measured interband transfer to locate valley-gap closures. Opposite-force drift measurements distinguished cancelling from reinforcing Berry curvature contributions; reversing shaking handedness reversed the drift. Interband transfer and harmonic confinement affected this readout, including drift beyond ideal transition lines. This neutral-atom Floquet Hamiltonian realization did not measure electrical sheet conductivity. Magnetic fields were used for preparation and forcing.^[ref]

The normalized Bloch states $|k⟩=N^{-1/2}{\sum }_{n=1}^N{e}^{ikna}|n⟩$ satisfy periodicity when $k=2\pi j/(Na)$. Acting with the hopping terms shifts the coefficients by one site, givingThe first Brillouin zone is a fundamental cell of reciprocal space, for example $[-\pi /a,\pi /a)$; wavevectors differing by $2\pi /a$ label the same state. It contains exactly $N$ distinct allowed wavevectors and hence $N$ orbital states. The written Hamiltonian has no spin index; if electron spin degeneracy is included there are $2N$ states.

For a narrow wave packet, group velocity and band-curvature effective mass areFor the usual $J>0$, the mass is negative in the outer half of the zone, $\pi /(2a)<|k|\le \pi /a$, taking one endpoint representative. At $|k|=\pi /(2a)$ the inverse mass vanishes and the mass diverges. For arbitrary nonzero $J$, the sign criterion is $J\cos ka<0$; $J=0$ is a flat band.

For electron charge $-e$ with $e>0$ in a constant electric field $\mathcal{E}$, the single-band semiclassical equations are $\hslash \dot{k}=-e\mathcal{E}$, $\dot{x}=v(k)$. Thus $k(t)=k_0-e\mathcal{E}t/\hslash$, and for $\mathcal{E}\ne 0$,These are Bloch oscillations of period $2\pi \hslash /(|e\mathcal{E}|a)$. The zero-field limit is $x-x_0=v(k_0)t$. This assumes a narrow packet and neglects scattering and interband transitions.

For many electrons obeying the Pauli exclusion principle, a partially filled band has available nearby states and can carry current under a displaced occupation distribution: it describes a metal within band theory. A completely filled band has cancelling velocities over the zone, so its current is zero; if separated from empty higher bands by a nonzero gap, it describes an insulator at sufficiently low temperature and weak electric field. With spin degeneracy this isolated band is filled at two electrons per site. The single-electron Hamiltonian alone does not specify the filling or a gap to a next band, so those extra physical assumptions are required for classifying a material.

In an isolated energy band, semiclassical motion satisfies $\hslash \dot{k}=F$ and $v={\hslash }^{-1}dE/dk$. The band is periodic in crystal momentum, so a constant force sweeps the state through equivalent points of the Brillouin zone. Its velocity repeats rather than increasing without bound. For an electron in a constant electric field, $F=-e\mathcal{E}$, and the Bloch oscillation period isFor $E(k)=E_0-2J\cos ka$, $v=(2Ja/\hslash )\sin ka$ and integration gives a bounded position oscillation of amplitude $2J/|F|$ about its mean. A filled band has no corresponding net transport oscillation after summing uniformly over all $k$.

Observation requires scattering times longer than $T_B$ and negligible interband tunnelling. Ordinary crystals often scatter too rapidly before a full zone traversal; larger-period structures and clean atom-optical lattices help. In the cold-atom Bloch oscillation experiment, accelerating a standing-wave lattice supplied a controlled force, and releasing the atoms at different times revealed an oscillating momentum distribution.^[ref]

For an electron wavepacket in one isolated band under a uniform electric field,Because $ϵ(k+2\pi /a)=ϵ(k)$, acceleration through successive Brillouin zones makes the velocity periodic rather than increasing indefinitely. For $ϵ={ϵ}_0-2J\cos (ka)$,for a positive field along the chosen axis. Integration givesThe excursion amplitude of a Bloch-oscillating wavepacket is $2|J|/(e|E|)$ about its mean and peak-to-peak extent $4|J|/(e|E|)$. The turning of velocity corresponds semiclassically to Bragg reflection near a zone boundary.

Coherent Bloch oscillations require scattering/dephasing times long compared with $T_B$ and a field weak enough to avoid substantial interband tunnelling. Ordinary metals usually scatter too rapidly; semiconductor superlattices and cold atoms in periodic optical potentials provide cleaner realizations. A completely filled band has canceling velocities, so its net current does not oscillate simply because individual wavepackets can.^[ref]