Tunneling between Two Luttinger Liquids Induced by a Driving Field

Tunneling between two Luttinger liquids driven by a time-dependent field with a frequency $f$ is investigated using the zero-mode bosonization. We show that inclusion of the zero modes is essential in order to obtain correct results in the limit $L_T/L\gg 1$ ( $L$ is the channel length and $L_T$ is the thermal length). We find that the tunneling current is quantized in units of “ $\mathrm{ef}$” and takes the form $I\sim \mathrm{ef}\Sigma {n=1\mathrm{}}^{\infty }\delta [{\mu }_F-\frac{2\pi }{L}vħ(n-\frac{1}{2}\left)\right]$.