Measurement of Phase and Magnitude of the Reflection Coefficient of a Quantum Dot

Electron Phase Shift at the Zero-Bias Anomaly of Quantum Point Contacts

The Kondo effect is the many-body screening of a local spin by a cloud of electrons at very low temperature. It has been proposed as an explanation of the zero-bias anomaly in quantum point contacts where interactions drive a spontaneous charge localization. However, the Kondo origin of this anomaly remains under debate, and additional experimental evidence is necessary. Here we report on the first phase-sensitive measurement of the zero-bias anomaly in quantum point contacts using a scanning gate microscope to create an electronic interferometer. We observe an abrupt shift of the interference fringes by half a period in the bias range of the zero-bias anomaly, a behavior which cannot be reproduced by single-particle models. We instead relate it to the phase shift experienced by electrons scattering off a Kondo system. Our experiment therefore provides new evidence of this many-body effect in quantum point contacts.

The Friedel sum rule at Fano resonances

If a mesoscopic system is non-chaotic or non-ergodic then the thermodynamic and transport properties do not depend on the impurity averaged density of states. We show that the partial density of states as well as the density of states of a given system with a given impurity configuration can be determined exactly from the asymptotic wavefunction (or scattering matrix) at the resonances. The asymptotic wavefunction can be determined experimentally without any knowledge about the quantum mechanical potential (including electron-electron interaction) or wavefunction in the interior of the system. Some counterintuitive relations derived here allow this.

Phase rigidity and avoided level crossings in the complex energy plane

We consider the effective Hamiltonian of an open quantum system, its biorthogonal eigenfunctions phi(lambda), and define the value r(lambda)=(phi(lambda)|phi(lambda))/<phi(lambda)|phi(lambda)> that characterizes the phase rigidity of the eigenfunctions phi(lambda). In the scenario with avoided level crossings, r(lambda) varies between 1 and 0 due to the mutual influence of neighboring resonances. The variation of r(lambda) is an internal property of an open quantum system. In the literature, the phase rigidity rho of the scattering wave function Psi(C)(E) is considered. Since Psi(C)(E) can be represented in the interior of the system by the phi(lambda), the phase rigidity rho of the Psi(C)(E) is related to the r(lambda) and therefore also to the mutual influence of neighboring resonances. As a consequence, the reduction of the phase rigidity rho to values smaller than 1 should be considered, at least partly, as an internal property of an open quantum system in the overlapping regime. The relation to measurable values such as the transmission through a quantum dot, follows from the fact that the transmission is, in any case, resonant at energies that are determined by the real part of the eigenvalues of the effective Hamiltonian. We illustrate the relation between phase rigidity rho and transmission numerically for small open cavities.

Do intradot electron-electron interactions induce dephasing?

We investigate the degree of coherence of electronic transport through a quantum dot (QD) in the presence of an intradot electron-electron interaction. By using an open multiterminal Aharonov-Bohm (AB) setup, we find that the intradot interaction does not induce any dephasing effect and the electron transport through the QD is fully coherent. We also observe that the asymmetric amplitude of the AB oscillation in the conductance through the two-terminal AB setup originates from the interplay between the confined structure and the electron-electron interaction. Thus, one cannot associate a dephasing process with this asymmetric amplitude, as has been done in previous studies.

Magnetic-field-dependent transmission phase of a double-dot system in a quantum ring

The Aharonov-Bohm effect is measured in a four-terminal open ring geometry. Two quantum dots are embedded in the structure, one in each of the two interfering paths. The number of electrons in the two dots can be controlled independently. The transmission phase is measured as electrons are added to or taken away from the individual quantum dots. Although the measured phase shifts are in qualitative agreement with theoretical predictions, the phase evolution exhibits unexpected dependence on the magnetic field. Phase lapses are found only in certain ranges of the magnetic field.

Effective Hamiltonian and unitarity of the S matrix

The properties of open quantum systems are described well by an effective Hamiltonian H that consists of two parts: the Hamiltonian H of the closed system with discrete eigenstates and the coupling matrix W between discrete states and continuum. The eigenvalues of H determine the poles of the S matrix. The coupling matrix elements W(cc')(k) between the eigenstates k of H and the continuum may be very different from the coupling matrix elements W(cc')(k) between the eigenstates of H and the continuum. Due to the unitarity of the S matrix, the W(cc')(k) depend on energy in a nontrivial manner. This conflicts with the assumptions of some approaches to reactions in the overlapping regime. Explicit expressions for the wave functions of the resonance states and for their phases in the neighborhood of, respectively, avoided level crossings in the complex plane and double poles of the S matrix are given.

Broken unitarity and phase measurements in Aharonov-Bohm interferometers

Aharonov-Bohm mesoscopic solid-state interferometers yield a conductance which contains a term cos(phi+beta), where phi relates to the magnetic flux. Experiments with a quantum dot on one of the interfering paths aim to relate beta to the dot's intrinsic Friedel transmission phase alpha(1). For closed systems, which conserve the electron current (unitarity), the Onsager relation requires that beta = 0 or pi. For open systems, we show that in general beta depends on the details of the broken unitarity. Although it gives information on the resonances of the dot, beta is generally not equal to alpha(1). A direct relation between beta and alpha(1) requires specific ways of opening the system, which are discussed.

Fano resonances as a probe of phase coherence in quantum dots

In the presence of direct trajectories connecting source and drain contacts, the conductance of a quantum dot may exhibit resonances of the Fano type. Since Fano resonances result from the interference of two transmission pathways, their line shape (as described by the Fano parameter q) is sensitive to dephasing in the quantum dot. We show that under certain circumstances the dephasing time can be extracted from a measurement of q for a single resonance. We also show that q fluctuates from level to level, and we calculate its probability distribution for a chaotic quantum dot. Our results are relevant to recent experiments by Göres et al. [Phys. Rev. B 62, 2188 (2000)].