Tunable Fermi-Edge Resonance in an Open Quantum Dot

Resonant tunneling in an open mesoscopic quantum dot is proposed as a vehicle to realize a tunable Fermi-edge resonance with variable coupling strength. We solve the x-ray edge problem for a generic nonseparable scatterer and apply it to describe tunneling in a quantum dot. The tunneling current power law exponent is linked to the $S$ matrix of the dot. The control of scattering by varying the dot shape and coupling to the leads allows us to explore a wide range of exponents. The sensitivity of mesoscopic coherence to the Wigner-Dyson ensemble symmetry is replicated in the Fermi-edge singularity.

Figure 1

(a) An open quantum dot weakly coupled to a small closed dot which holds a localized electron that can tunnel into the open dot. (b) Relation of the open dot $S$ matrix $S$ and an auxiliary matrix $\tilde{S}$ is illustrated. The latter describes the dot with an extra open channel added to incorporate backscattering on the small dot charge state.

Figure 2

Schematic forward and backward scattering time evolution $e^{-i{\widehat{h}}_1\tau }{e}^{i{\widehat{h}}_0\tau }$, with ballistic transport outside the scattering region. The compound $S$ matrix $R=S_1{S}_0^{-1}$ accounts for sequential scattering described by the $S$ matrices $S_{0,1}$ corresponding to the Hamiltonians ${\widehat{h}}_{0,1}$.

Figure 3

The dependence (4, 5) of the tunneling exponent $\alpha$ on $\mathrm{a}\mathrm{r}\mathrm{g}(r)$ for $|r|=0,0.1,\dots 0.9$; Inset: The exponent $\alpha$ vs $|r|$ for several values $\mathrm{a}\mathrm{r}\mathrm{g}(r)$ marked by arrows $a$–$e$.