Driven quantum dot coupled to a fractional quantum Hall edge

We study a model of a quantum dot coupled to a quantum Hall edge of the Laughlin state, taking into account short-range interactions between the dot and the edge. This system has been studied experimentally in electron quantum optics in the context of single particle sources. We consider driving the dot out of equilibrium by a time-dependent bias voltage. We calculate the resulting current on the edge by applying the Kubo formula to the bosonized Hamiltonian. The Hamiltonian of this system can also be mapped to the spin-boson model and, in this picture, the current can be perturbatively calculated using the noninteracting blip approximation. We show that both methods of solution are in fact equivalent. We present numerics demonstrating that the perturbative approaches capture the essential physics at early times, although they fail to capture the charge quantization (or lack thereof) in the current pulses integrated over long times.

Figure 1

Schematic model of the experimental setup showing a quantum dot (QD) coupled to an FQHE edge state. The edge is coupled to the dot via a quantum point contact (QPC) at $x=0$. The gate voltage can be used to tune the tunneling $\lambda \left(t\right)$ between the dot and the edge. A single energy level $\varepsilon \left(t\right)$ on the quantum dot is controlled via applied bias voltage. We study the occupation number $N\left(t\right)$ of the dot over time.

Figure 2

Sketch showing that the whole of $t$-space can be divided into two regimes which overlap: the $t\gg \frac{a}{v}$ regime and the $t\ll \beta$ regime due to the fact that $\frac{a}{v}\ll \ll \beta$. Since we prove the identity Eq. (24) in both limits, we have proven it for all $t$.

Figure 3

Comparison of the perturbative solution Eq. (17) with two nonperturbative methods. We show the time evolution of the current after a linear ramp $\varepsilon \left(t\right)=\xi (t-t_0)$ with parameters $a=0.005v{\mathrm{\Delta }}^{-1},\xi =4{\mathrm{\Delta }}^2,t_0=5{\mathrm{\Delta }}^{-1}$. We plot $-dN/d\left(t\mathrm{\Delta }\right)$, where $N\left(t\right)=\langle \widehat{N}\left(t\right)\rangle$. The insets show the time evolution of the occupation number $N\left(t\right)$ on the QD. (a) Sketch of the sweep protocol. (b) For $\alpha =1/2$, we compare the exact solution as derived in Appendix pp4 (yellow) with our perturbative result (blue). (c) For $\alpha =0.01$, we compare the result from the GME (yellow) with our perturbative result (blue). Detailed expressions for the GME are given in Ref. [2]. In both cases, the early time agreement between the solutions is excellent.

Figure 4

Comparison of the full solution Eq. (25) with the analytical expression Eq. (29), which is valid in the limit ${\varepsilon }_0\ll \mathrm{\Omega }$. We show the time evolution of the current when the dot is driven with a bias $\varepsilon \left(t\right)={\varepsilon }_{0}cos\mathrm{\Omega }t$. We use the parameters $\mathrm{\Omega }=5\mathrm{\Delta },a=0.01v{\mathrm{\Delta }}^{-1}$, and ${\varepsilon }_0=0.1\mathrm{\Delta }$. The agreement between the solutions is excellent as long as $vt\gg a$.