Mach-Zehnder interferometry with periodic voltage pulses

We investigate theoretically a Mach-Zehnder interferometer driven by a time-dependent voltage. Motivated by recent experiments, we focus on a train of Lorentzian voltage pulses which we compare to a sinusoidal and a constant voltage. We discuss the visibilities of Aharonov-Bohm oscillations in the current and in the noise. For the current, we find a strikingly different behavior in the driven as compared to the static case for voltage pulses containing multiple charges. For pulses containing fractional charges, we find a universality at path-length differences equal to multiples of the spacing between the voltage pulses. These observations can be explained by the electronic energy distribution of the driven contact. In the noise oscillations, we find additional features which are characteristic to time-dependent transport. Finite electronic temperatures are found to have a qualitatively different influence on the current and the noise.

Figure 1

Mach-Zehnder interferometer driven by a time-dependent voltage $V\left(t\right)$. The interferometer consists of two paths of lengths $L_u$ and $L_d$ which enclose a magnetic flux $\varphi$. Two quantum point contacts $A$ and $B$ allow the single-particle states to be delocalized over both paths leading to interference contributions in current and noise measured at the outputs.

Figure 2

Energy distribution of a driven contact. The filling is plotted for a contact driven by Lorentzian (blue) and sinusoidal (red) voltage pulses as well as a constant bias (green). The three panels (a)–(c) show different pulse charges $q$ for zero (solid) and finite (dotted) temperatures. For voltages applied at $\Omega =2\pi \times 6$ GHz, the dotted curves correspond to $T\approx 14$ mK. The filling shows plateaus of width $\hslash \Omega$ for all charges $q$. For fractional charges, the steps are shifted by $e{V}_{dc}=q\hslash \Omega$ with respect to $\mu =0$. The widths of the pulses $\Gamma$ are chosen to best visualize the steplike behavior of the filling.

Figure 3

Visibility of the current oscillations for different voltage pulses. In all cases the visibility shows an oscillatory behavior which falls off as $1/\tau$. For $q=l/p$, the visibility goes to zero if $\tau$ is an integer multiple of $p\mathcal{T}$ in the driven and of $p\mathcal{T}/l$ in the static case. A frequency $\Omega =2\pi \times 6$ GHz corresponds to a period $\mathcal{T}\approx 0.15$ ns which translates into a path-length difference of $\Delta L\approx 1.5\mu \mathrm{m}$. (a) For single-charge pulses, the visibility of the driven MZI is reduced with respect to the static case but looks qualitatively similar. (b) For multicharge pulses, the visibility behaves strikingly different in the driven case. (c), (d) At $\tau =j\mathcal{T}$, the visibility is independent of the pulse shape (marked by circles). All these observations are readily explained by the filling of the driven contact (see text).

Figure 4

Visibility of the noise oscillations with period $2\pi$ for different voltage pulses. In all cases the visibility shows an oscillatory behavior which falls off as $1/\tau$. The period of the oscillations is temperature dependent and only at $T=0$ the same as for the current visibility. Additional features due to the interference noise can be seen for fractional pulses, panels (c) and (d). They are illustrated with the help of the inset in panel (c) which shows the visibility due to the transport noise alone. Here ${\mathcal{D}}_A={\mathcal{D}}_B=0.75$. For voltages applied at $\Omega =2\pi \times 6$ GHz, the dotted curves correspond to $T\approx 35$ mK.

Figure 5

Visibility of the noise oscillations with period $\pi$ for different voltage pulses. In all cases the visibility shows an oscillatory behavior which falls off as $1/\tau$. The period of the oscillations is half the period of ${\mathcal{V}}_{\varphi }$ and is temperature dependent. Additional features due to the interference noise can be seen for all charges. These are illustrated with the help of the inset in panel (a) which shows the visibility due to the transport noise alone. In contrast to the current visibility, the noise visibilities thus encode information about the coherences and cannot be explained by the filling alone. Here ${\mathcal{D}}_A={\mathcal{D}}_B=0.75$.