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Quantum Brownian motion in a Landau level

Motivated by questions about the open-system dynamics of topological quantum matter, we investigated the quantum Brownian motion of an electron in a homogeneous magnetic field. When the Fermi length $l_F=\hbar/(v_Fm_{\text{eff}})$ becomes much longer than the magnetic length $l_B=(\hbar c/eB)^{1/2}$, then the spatial coordinates $X,Y$ of the electron cease to commute, $[X,Y]=il_B^2$. As a consequence, localization of the electron becomes limited by Heisenberg uncertainty, and the linear bath-electron coupling becomes unconventional. Moreover, because the kinetic energy of the electron is quenched by the strong magnetic field, the electron has no energy to give to or take from the bath, and so the usual connection between frictional forces and dissipation no longer holds. These two features make quantum Brownian motion topological, in the regime $l_F\gg l_B$, which is at the verge of current experimental capabilities. We model topological quantum Brownian motion in terms of an unconventional operator Langevin equation derived from first principles, and solve this equation with the aim of characterizing diffusion. While diffusion in the noncommutative plane turns out to be conventional, with the mean displacement squared being proportional to $t^α$ and $α=1$, there is an exotic regime for the proportionality constant in which it is directly proportional to the friction coefficient and inversely proportional to the square of the magnetic field: in this regime, friction helps diffusion and the magnetic field suppresses all fluctuations. We also show that quantum tunneling can be completely suppressed in the noncommutative plane for suitably designed metastable potential wells, a feature that might be worth exploiting for storage and protection of quantum information.