Phase of the transmission amplitude for a quantum dot embedded in the arm of an electronic Mach-Zehnder interferometer

We investigate an electronic Mach-Zehnder interferometer with high visibility in the quantum Hall regime. The superposition of the electrostatic potentials from a quantum point contact (QPC) and the residual disorder potential from doping impurities frequently results in the formation of inadvertent quantum dots (QDs) in one arm of the interferometer. This gives rise to resonances in the QPC transmission characteristics. While crossing the QD resonance in energy, the interferometer gains a phase shift of $\pi$ in the interference pattern.

Figure 1

Scanning electron microscopy image of Mach-Zehnder interferometer with the scheme of paths for nonequilibrium current. The transmission of QPC1 and QPC2 is set to 0.5. QPC0 transmits the outer edge channel and reflects the inner one in case the filling factor being more than 1. The MG is used to shift the phase.

Figure 2

(Color) (a) Transmission characteristic of QPC2 as a function of the QPCs gate voltages at $B=4.8\text{T}$. The horizontal line denotes half transmission of the QPC which should in a single-particle picture correspond to the highest visibility. (b) Visibility of Aharonov-Bohm oscillations, experimentally measured (thick red line) and calculated from transmission of QPC2 (thin black line) versus QPC2 gate voltage [$T_{\text{QPC}1}=0.5$, see Eq. (2)]. The regions of discrepancy are marked by arrows. (c) Interferometer current for different modulation gate voltages, i.e., different AB phases $\left(T_{\text{QPC}1}=0.5\right)$.

Figure 3

(a) Scheme for probing transmission characteristic of QPC2 with QPC1 opened, and that (b) for interference experiment with both QPCs partially transmitted. The circle at the QPC2 symbolizes the quasibound state of quantum dot. (c) Oscillatory transmission of outermost edge channel as a function of magnetic field when transmission of QPC2 is less than 1. The arrow marks the point where the additional frequency (with almost half period) sets in.

Figure 4

(Color online) (a) The resonance C (dots), shown in Fig. 2c, fitted by model of an interferometer with quantum dot in one arm (full line) for different voltages of modulation gate $V_{\text{MG}}=0.4$, 0.8, 1.2, and 2.0 mV changing from bottom to top. (b) The same for the peak B in Fig. 2c. The curves in (a) and (b) shifted for clarity except lowest one. (c) The phase shift between interferometer arms, found from the fit to the resonances B and C, plotted as a function of $V_{\text{MG}}$. The full lines in (c) are expected slopes resulted from AB period $\Delta V_{\text{MG}}=1.6\text{mV}$ in $V_{\text{MG}}$. The phase evolution for the resonance A was evaluated by procedure described in the text.