How to Measure the Transmission Phase through a Quantum Dot in a Two-Terminal Interferometer

Measurement of the transmission phase through a quantum dot (QD) embedded in an arm of a two-terminal Aharonov-Bohm (AB) interferometer is inhibited by phase symmetry, i.e., the property that the linear response conductance of a two-terminal device is an even function of the magnetic field. It is demonstrated that in a setup consisting of an interferometer with a QD in each of its arms, with one of the QDs capacitively coupled to a nearby quantum point contact (QPC), phase symmetry is broken when a finite voltage bias is applied to the QPC. The transmission phase via the uncoupled QD can then be deduced from the amplitude of the odd component of the AB oscillations.

Figure 1

Schematics of which-path interferometer studied in this Letter. The Aharonov-Bohm interferometer contains a quantum dot in each arm. In order to measure the transmission phase through one of the dots (QD1), a quantum point contact is coupled electrostatically to the other dot (QD2), in order to break the phase symmetry.

Figure 2

(a) Linear response conductance as a function of magnetic flux (horizontal axis) and the energy of the level in the reference arm ${ϵ}_1$ (vertical axis) for $V_{\mathrm{QPC}}=0\mu \mathrm{eV}$, and (b) its change when the QPC bias is $V_{\mathrm{QPC}}=400\mu \mathrm{eV}$. (Other parameters are ${\Gamma }_{11}^L={\Gamma }_{22}^L=1\mu \mathrm{eV}$, ${\Gamma }_{11}^R={\Gamma }_{22}^R=5\mu \mathrm{eV}$, $g_c=1.32\times {10}^{-4}$, ${ϵ}_2=1.5\mu \mathrm{eV}$, ${ϵ}_F=0\mu \mathrm{eV}$.) (c) Phase of AB oscillations as a function of ${ϵ}_1$ for different values of QPC bias; since ${ϵ}_2>{ϵ}_F$, the transmission coefficient is not symmetric in respect to the Fermi level and the phase jump is shifted to ${ϵ}_1=-0.7\mu \mathrm{eV}$. Inset: dephasing rate, $\gamma ({ϵ}_F)$ as a function of the QPC bias, $V_{\mathrm{QPC}}$; different traces correspond to (from left to right) ${ϵ}_2=1.5\mu \mathrm{eV}$, and $300\mu e\mathrm{V}$; (d) odd part of the AB conductance at $V_{\mathrm{QPC}}=400\mu \mathrm{eV}$.

Figure 3

(a),(d) Transmission, (b),(e) real part of the transmission coefficient and odd AB conductance (normalized to its maximum), and (c),(f) cosine of the transmission phase and its value extracted from $G^{\mathrm{odd}}$ [Eq. (2)] for QD1 with two levels of the same parity (left) or different parity (right) (see parameters in the text).