Coherence recovery mechanisms of quantum Hall edge states

This paper is motivated by an unexpected observation in a recent experiment [S. Tewari et al., Phys. Rev. B 93, 035420 (2016).], where a robust coherence recovery, starting from a certain energy, was detected for an electron injected into a quantum Hall edge at a filling factor of 2. After passing through a quantum dot, the electron then tunnels into the edge with a subsequent propagation towards a symmetric Mach-Zender interferometer (MZI). Afterwards, the visibility of Aharonov-Bohm (AB) oscillations is measured. An earlier study, based on the bosonization framework with a linear spectrum of the edge excitations, predicted a decay of the visibility with increasing energy. We associate this result with the destructive interference of the two quasiparticles (charge and neutral modes), formed at the edge out of the incoming electron wave packet (WP). However, in reality, it might be suppressed due to an imbalance between the quasiparticles, for instance, in the presence of dispersion and/or dissipation. This idea could also be explored further experimentally by applying a periodic potential to the arms of the MZI and thus creating the imbalance. Yet another possibility to restore phase coherence, also based on dispersion and dissipation, accounts for a drop in the energy density of the electron WP by the time it arrives at the interferometer, leading to a significantly smaller dephasing inside it. We then show that the energy density is defined by a parameter completely independent of the injected energy, which naturally explains the emergence of threshold energy in the experiment.

Figure 1

Sketch of the MZ interferometer, comprising the two up and down edge channels of a QH edge system at $\nu =2$. The channels are coupled by tunnel junctions characterized by amplitudes ${\tau }_{\mathcal{L}}$ and ${\tau }_{\mathcal{R}}$. An electron with energy ${\varepsilon }_0$ tunnels to the $1u$ channel from the QD with amplitude ${\tau }_d$.

Figure 2

Comparison of derivatives for the energy distribution functions in two cases: (1) in the presence of dissipation (solid line), according to (42), and (2) the simplest case of linear dispersion [14] (dashed line), where $\partial f_v/\partial \varepsilon \propto -1/{\varepsilon }^2$. These plots reflect the existence of an energy cutoff $\sim 1/2\gamma$ in the former situation, which is found from the assumption that the initial WP energy is large enough, ${\varepsilon }_0\gamma \gg 1$. Thus, by the time the WP reaches the interferometer, it loses its energy, which is defined by the strength of dissipation $\gamma$.

Figure 3

Visibility as a function of the initial wave-packet energy in the case of strong dissipation in both modes, calculated from (50). Here, ${\gamma }_1=2.38\eta , {\gamma }_2=5.68\eta$.

Figure 4

The plot of $\mathcal{F}(u,v)\equiv F(u,v)-1$ (dashed line) for $\gamma \to 0$ and its real part (solid line) in the case $\gamma \ll \eta$ as a function of $t$.The region is divided into five numbered areas over which the integration is performed. Parameter $t_0$ is chosen such that $\gamma \ll t_0\ll \eta$.