Electron-photon coupling in mesoscopic quantum electrodynamics

Understanding the interaction between cavity photons and electronic nanocircuits is crucial for the development of mesoscopic quantum electrodynamics (QED). One has to combine ingredients from atomic cavity QED, such as orbital degrees of freedom, with tunneling physics and strong cavity field inhomogeneities, specific to superconducting circuit QED. It is therefore necessary to introduce a formalism which bridges between these two domains. We develop a general method based on a photonic pseudopotential to describe the electric coupling between electrons in a nanocircuit and cavity photons. In this picture, photons can induce simultaneously orbital energy shifts, tunneling, and local orbital transitions. We study in detail the elementary example of a single quantum dot with a single normal metal reservoir, coupled to a cavity. Photon-induced tunneling terms lead to a nonuniversal relation between the cavity frequency pull and the damping pull. Our formalism can also be applied to multiple quantum dot circuits, molecular circuits, quantum point contacts, metallic tunnel junctions, and superconducting nanostructures enclosing Andreev bound states or Majorana bound states, for instance.

Figure 1

Scheme of a loopless nanocircuit with source/drain electrodes (blue) and electrostatic gates (green) embedded in a photonic cavity (purple). The yellow cloud represents the photonic field. The cavity presents some protrusions (purple stripes) fabricated to increase the photonic field inhomogeneities (darker yellow areas).

Figure 2

Scheme of the various contributions to $G_{{\widehat{h}}_{\mathrm{int}}{\widehat{h}}_{\mathrm{int}}}\left(t\right)$. The wavy lines correspond to photonic propagators, the simple dashed lines to bare electronic propagators in the normal metal reservoir, and the double full lines to electronic propagators in the quantum dot dressed by the dot/reservoir tunneling processes.

Figure 3

Reduced $\Delta {\omega }_0$ and $\sqrt{\Delta {\Lambda }_0}$ versus ${\varepsilon }_d$ for different values of $\gamma$. We have used $t/\Gamma =0.001, D=20\Gamma$, and pulsation scales ${\Omega }_0=2{\alpha }^2/\pi \Gamma \hslash$ and ${\Omega }_1=\hslash {\omega }_0{\Omega }_0/\Gamma$.

Figure 4

Ratio $\Theta$ versus ${\varepsilon }_d$ for different values of $\gamma$. We have used the same values of $t, D$, and ${\Omega }_0$ as in Fig. 3.

Figure 5

(a) Effective model for the circuit of Fig. 1, where the nanocircuit reservoirs are decomposed into effective orbital reservoirs (dark gray elements) and effective plasmonic reservoirs (nearby purple elements). (b) Ensemble $\mathcal{C}$ of the perfect conductors considered for the generation of the photonic modes. The photonic field in yellow is inhomogeneous, as represented by darker yellow areas near the cavity protrusions, and by the white (screening) areas between the small purple conductors. (c) Effective orbital nanocircuit $\mathcal{O}$.