Bloch Oscillations & Wannier-Stark Localization

Electrons in a tilted periodic lattice oscillate in real space rather than accelerating — a quantum interference effect

Parameters

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1.00
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0.15
Bloch Oscillations

E(k) = −2J cos(ka). Under field F, crystal momentum evolves as ℏ dk/dt = F (acceleration theorem). Period: T_B = h/Fa.

The wavepacket mean position oscillates as:
x(t) = −(2J/F) cos(Ft/ℏ − k₀a)

Wannier-Stark ladder: eigenstates become localized with energy spacing ΔE = Fa (electric field × lattice constant).

Color = phase of k-space amplitude. The wavepacket sweeps the Brillouin zone, Bragg-reflects at ±π/a, and returns — no DC transport!

T_B = —