Quantum Brownian motion in a quasiperiodic potential

We consider a quantum particle subject to Ohmic dissipation, moving in a bichromatic quasiperiodic potential. In a periodic potential the particle undergoes a zero-temperature localization-delocalization transition as dissipation strength is decreased. We show that the delocalized phase is absent in the quasiperiodic case, even when the deviation from periodicity is infinitesimal. Using the renormalization group, we determine how the effective localization length depends on the dissipation. We show that a similar problem can emerge in the strong-coupling limit of a mobile impurity moving in a periodic lattice and immersed in a one-dimensional quantum gas.

Figure 1

Localization length ${\xi }^{*}$ as a function of dissipation $\alpha$ for quasiperiodic potential with $\gamma =\phi$ (solid green). Inset: Same plot on a log-log scale. As $\alpha$ is decreased, ${\xi }^{*}$ is a piecewise function that changes nonanalytically for $\alpha \sim {\alpha }_n={\phi }^{-2n}$ between successive ${\xi }_n=\frac{q_0}{2\pi }\sqrt{\frac{2{\ell }_n}{\alpha }}$ [see Eq. (9)].