Discontinuous conductance of bichromatically ac-gated quantum wires

We study the electron transport through a quantum wire under the influence of external time-dependent gate voltages. The wire is modeled by a tight-binding Hamiltonian for which we obtain the current from the corresponding transmission. The numerical evaluation of the dc current reveals that for bichromatic driving, the conductance depends sensitively on the commensurability of the driving frequencies. The current even possesses a discontinuous frequency dependence. Moreover, we find that the conductance as a function of the wire length oscillates with a period that depends on the ratio between the driving frequencies.

Figure 1

Transmission for a wire with $N=5$ sites with equal onsite energies ${\epsilon }_0=0$ as a function of the frequency ${\omega }_1$ for the driving amplitudes (a) ${\Delta }_2=4$ and (b) ${\Delta }_2=2$. The tunnel coupling between two neighboring wire sites is $V=4$, while the other driving parameters read ${\omega }_2=2$, ${\Delta }_1=4$. Solid lines are obtained from expression (18) which is valid for incommensurable frequencies only, while the dashed lines and the circles are obtained from Eq. (19) and, thus, are valid for all driving frequencies ${\omega }_1$, ${\omega }_2$. The fractions in panel (b) mark selected frequency ratios ${\omega }_2/{\omega }_1$.

Figure 2

Average current for the chemical potentials ${\mu }_L=-{\mu }_R=0.1$ as a function of the equal onsite energies ${\epsilon }_0$ for a wire with $N=3$ sites and tunneling matrix elements $V=4$. The first driving field is kept constant [${\Delta }_1=4$ with frequency ${\omega }_1=2$ (a) and ${\omega }_1=4$ (b)]. The second field has driving amplitude ${\Delta }_2=4$, while its frequency is ${\omega }_2=4$ (thin solid lines) and ${\omega }_2=3.99$ (thick solid lines), respectively. The dashed line in panel (a) marks the undriven case which corresponds to vanishing driving amplitudes, i.e., ${\Delta }_1={\Delta }_2=0$.

Figure 3

Transmission as a function of the wire length $N$ for ${\Delta }_1={\Delta }_2=6$, ${\epsilon }_0=0$, $V=4$. (a) the driving frequencies are ${\omega }_1=\sqrt{2}V$, ${\omega }_2=2V$ (circles) and ${\omega }_1=V$, ${\omega }_2=2V$ (triangles) correspond to the oscillation periods $M=4$ and $M=6$, respectively. (b) ${\omega }_1=2V\text{cos}\left(\pi /5\right)$, ${\omega }_2=2V$ (circles) and ${\omega }_1=V$, ${\omega }_2=\sqrt{2}V$ (triangles) corresponding to $M=10$ and $M=12$, respectively. The lines serve as a guide to the eye.

Figure 4

Transmission as a function of the wire length $N$ for commensurable frequencies (${\omega }_1={\omega }_2=4$, triangles) and incommensurable frequencies (${\omega }_2=0.99{\omega }_1$, ${\omega }_1=4$, circles) and the driving amplitudes (a) ${\Delta }_1={\Delta }_2=\Delta =4$ and (b) ${\Delta }_1={\Delta }_2=\Delta =8$. The other parameters are ${\epsilon }_0=0$, $V=4$. The lines are a guide to the eye.