Analytical result on the supercurrent through a superconductor/quantum-dot/superconductor junction

We present an analytical result for the supercurrent across a superconductor/quantum-dot/superconductor junction. By converting the current integration into a special contour integral, we can express the current as a sum of the residues of poles. These poles are real and give a natural definition of the Andreev bound states. We also use the exact result to explain some features of the supercurrent transport behavior.Comment: 8 pages, 2 figure

Higher-Order Representations for Visual Recognition

In this thesis, we present a simple and effective architecture called Bilinear Convolutional Neural Networks (B-CNNs). These networks represent an image as a pooled outer product of features derived from two CNNs and capture localized feature interactions in a translationally invariant manner. B-CNNs generalize classical orderless texture-based image models such as bag-of-visual-words and Fisher vector representations. However, unlike prior work, they can be trained in an end-to-end manner. In the experiments, we demonstrate that these representations generalize well to novel domains by fine-tuning and achieve excellent results on fine-grained, texture and scene recognition tasks. The visualization of fine-tuned convolutional filters shows that the models are able to capture highly localized attributes. We present a texture synthesis framework that allows us to visualize the pre-images of fine-grained categories and the invariances that are captured by these models. In order to enhance the discriminative power of the B-CNN representations, we investigate normalization techniques for rescaling the importance of individual features during aggregation. Spectral normalization scales the spectrum of the covariance matrix obtained after bilinear pooling and offers a significant improvement. However, the computation involves singular value decomposition, which is not computationally efficient on modern GPUs. We present an iteration-based approximation of matrix square-root along with its gradients to speed up the computation and study its effect on fine-tuning deep neural networks. Another approach is democratic aggregation, which aims to equalize the contributions of individual feature vector into the final pooled image descriptor. This achieves a comparable improvement, and can be approximated in a low-dimensional embedding unlike the spectral normalization. Therefore, this approach is friendly to aggregating higher-dimensional features. We demonstrate that the two approaches are closely related, and we discuss their trade-off between performance and efficiency

Andreev reflection through a quantum dot coupled with two ferromagnets and a superconductor

We study the Andreev reflection (AR) in a three terminal mesoscopic hybrid system, in which two ferromagnets (F$_1$ and F$_2$) are coupled to a superconductor (S) through a quantum dot (QD). By using non-equilibrium Green function, we derive a general current formula which allows arbitrary spin polarizations, magnetization orientations and bias voltages in F$_1$ and F$_2$. The formula is applied to study both zero bias conductance and finite bias current. The current conducted by crossed AR involving F$_1$, F$_2$ and S is particularly unusual, in which an electron with spin $\sigma$ incident from one of the ferromagnets picks up another electron with spin $\bar{\sigma}$ from the other one, both enter S and form a Cooper pair. Several special cases are investigated to reveal the properties of AR in this system.Comment: 15 pages, 7 figures, LaTe

Probing Spin States of Coupled Quantum Dots by dc Josephson Current

We propose an idea for probing spin states of two coupled quantum dots (CQD), by the dc Josephson current flowing through them. This theory requires weak coupling between CQD and electrodes, but allows arbitrary inter-dot tunnel coupling, intra- and inter- dot Coulomb interactions. We find that the Coulomb blockade peaks exhibit a non-monotonous dependence on the Zeeman splitting of CQD, which can be understood in terms of the Andreev bound states. More importantly, the supercurrent in the Coulomb blockade valleys may provide the information of the spin states of CQD: for CQD with total electron number N=1,3 (odd), the supercurrent will reverse its sign if CQD becomes a magnetic molecule; for CQD with N=2 (even), the supercurrent will decrease sharply around the transition between the spin singlet and triplet ground states of CQD.Comment: 10 pages, 3 figure

Theory of Nonequilibrium Coherent Transport through an Interacting Mesoscopic Region Weakly Coupled to Electrodes

We develop a theory for the nonequilibrium coherent transport through a mesoscopic region, based on the nonequilibrium Green function technique. The theory requires the weak coupling between the central mesoscopic region and the multiple electrodes connected to it, but allows arbitrary hopping and interaction in the central region. An equation determining the nonequilibrium distribution in the central interacting region is derived and plays an important role in the theory. The theory is applied to two special cases for demonstrations, revealing the novel effects associated with the combination of phase coherence, Coulomb interaction, and nonequilibrium distribution.Comment: 10 Pages, 5 figure