Decoherence in a double-dot Aharonov-Bohm interferometer: Numerical renormalization group study

Coherence in electronic interferometers is typically believed to be restored fully in the limit of small voltages, frequencies, and temperatures. However, it is crucial to check this essentially perturbative argument by nonperturbative methods. Here we use the numerical renormalization group to study ac transport and decoherence in an experimentally realizable model interferometer, a parallel double quantum dot coupled to a phonon mode. The model allows us to clearly distinguish renormalization effects from decoherence. We discuss finite-frequency transport and confirm the restoration of coherence in the dc limit.

Figure 1

Double-dot Aharonov-Bohm interferometer. Decoherence suppresses the amplitude of $AB$ oscillations (dashed line). At destructive interference, ${\varphi }_{AB}={\Phi }_0/2$, only incoherent transport processes contribute in a symmetric setup. Mapping the Hamiltonian to (anti)symmetric combinations of the dot levels allows for a transparent exploitation of this cancellation of coherent transport.

Figure 2

Mapping the Hamiltonian to (anti)symmetric combinations of the dot levels: (a) the mapped electron-phonon interaction vertex restricts (b) the number and type of diagrams in a perturbative calculation of ac conductance, reflecting the destructive interference of all coherent contributions. (Solid lines denote the electron Green's functions and wiggly lines stand for phonon propagators.)

Figure 3

Plot of the ac conductance through the destructive ABI. In the limit of weak electron-phonon coupling the NRG and perturbative calculations match. For stronger electron-phonon coupling $g/{\omega }_{\mathrm{ph}}\approx 1$, additional side peaks at ${\omega }_{\mathrm{ac}}=\epsilon +n{\omega }_{\mathrm{ph}}(n\in \mathbb{N})$ appear and the first resonance is shifted to lower frequency. Here ${\mathcal{G}}_{LR}({\omega }_{\mathrm{ac}}=0)\equiv 0$ is found also for strong electron-phonon coupling. The parameters are ${\omega }_{\mathrm{ph}}=0.05,\epsilon =0.025,\Gamma =2\pi ({\rho }_L+{\rho }_R){\left|V_t\right|}^2=0.02$ in units of the bandwidth, and conductances in units of $e^2/h$ and we consider the zero-temperature limit and no dc bias.