Strongly correlated dynamics in multichannel quantum RC circuits

We examine dissipation effects in a multichannel quantum RC circuit, comprising a cavity or single-electron box capacitively coupled to a gate and connected to a reservoir lead via several conducting channels. Depending on the engineering details of the quantum RC circuit, the number of channels contributing to transport varies, as does the form of the interchannel couplings. For low-frequency ac transport, the charge-relaxation resistance ($R_q$) is a nontrivial function of the parameters of the system. However, in the vicinity of the charge-degeneracy points and for weak tunneling, we find as a result of cross-mode mixing or channel asymmetry that $R_q$ becomes universal for a metallic cavity at low temperatures, and equals the unit of quantum resistance. To prove this universality, we map the system to an effective one-channel Kondo model, and construct an analogy with the Coulomb gas. Next, we probe the opposite regime of near-perfect transmission using a bosonization approach. Focusing on the two-channel case, we study the effect of backscattering at the lead-dot interface, more specifically, the role of an asymmetry in the backscattering amplitudes, and make a connection with the weak-tunneling regime near the charge-degeneracy points.

Figure 1

Schematic of a multichannel quantum RC circuit with $\mathcal{M}=3$ and $\mathcal{N}=2$, where $\mathcal{M}$ denotes the number of channels in the lead and $\mathcal{N}$ denotes the number of channels in the dot. All the possible lead-dot couplings are depicted by black lines. The ac drive is capacitively coupled to the single-electron box. The geometric capacitance $C_g$ is to be distinguished from the mesoscopic capacitance $C_0$ of the quantum RC circuit.

Figure 2

Renormalization of the dc conductance for a multichannel quantum RC circuit ($\mathcal{M}=\mathcal{N}=5$) with decreasing temperature, at the charge-degeneracy point. We consider randomly generated couplings (unrestricted) and eliminate progressively the interchannel couplings, starting with the farthest channels (four-channel mixing) until all the interchannel couplings are removed (no channel mixing). The random couplings are normalized according to the degree of channel mixing, such that $g\left(0\right)=0.5$. The dotted curve corresponds to the unitary limit of the Kondo model.