Unextendible Product Basis for Fermionic Systems

We discuss the concept of unextendible product basis (UPB) and generalized UPB for fermionic systems, using Slater determinants as an analogue of product states, in the antisymmetric subspace \wedge^ N \bC^M. We construct an explicit example of generalized fermionic unextendible product basis (FUPB) of minimum cardinality $N(M-N)+1$ for any $N\ge2,M\ge4$. We also show that any bipartite antisymmetric space \wedge^ 2 \bC^M of codimension two is spanned by Slater determinants, and the spaces of higher codimension may not be spanned by Slater determinants. Furthermore, we construct an example of complex FUPB of $N=2,M=4$ with minimum cardinality $5$. In contrast, we show that a real FUPB does not exist for $N=2,M=4$ . Finally we provide a systematic construction for FUPBs of higher dimensions using FUPBs and UPBs of lower dimensions.Comment: 17 pages, no figure. Comments are welcom

Optimizing Ranking Measures for Compact Binary Code Learning

Hashing has proven a valuable tool for large-scale information retrieval. Despite much success, existing hashing methods optimize over simple objectives such as the reconstruction error or graph Laplacian related loss functions, instead of the performance evaluation criteria of interest---multivariate performance measures such as the AUC and NDCG. Here we present a general framework (termed StructHash) that allows one to directly optimize multivariate performance measures. The resulting optimization problem can involve exponentially or infinitely many variables and constraints, which is more challenging than standard structured output learning. To solve the StructHash optimization problem, we use a combination of column generation and cutting-plane techniques. We demonstrate the generality of StructHash by applying it to ranking prediction and image retrieval, and show that it outperforms a few state-of-the-art hashing methods.Comment: Appearing in Proc. European Conference on Computer Vision 201

Ultra-sensitive YBCO nanoSQUIDs for the investigation of magnetic nanoparticles

Superconducting quantum interference devices (SQUIDs) are used in an impressively large variety of applications requiring sensitive detection of magnetic flux. In recent years, there has been a growing scientific and technological interest in the development of nanoSQUIDs, i.e. strongly miniaturized SQUIDs with lateral size on the sub-micrometer scale, that can be used to detect the magnetization of small spin systems like individual magnetic nanoparticles. The development of nanoSQUIDs is a major research topic at the Physics Institute - Experimental Physics II. In this thesis, we first review the major achievements obtained so far on the development of sensitive nanoSQUIDs in Tübingen, based on Nb and YBa2Cu3O7 (YBCO) as superconductors. This part emphasizes the advantages offered by YBCO nanoSQUIDs, fabricated on bicrystal SrTiO3 (STO) substrates, regarding enhanced ranges of temperature and magnetic field, over which those nanoSQUIDs can be operated. Regarding the application of YBCO nanoSQUIDs fabricated on STO bicrystal substrates, we have studied the occurrence of magnetic-field-driven nucleation and annihilation of magnetic vortices in individual ultrasmall ferromagnetic Co particles by YBCO nanoSQUID magnetometry. We demonstrate that the Co particles reveal bi-stable magnetization states at zero applied field, with the vortex state being the ground state. This topic is important in order to understand the thermal and temporal stability of noncollinear and other nontrivial spin textures, e.g., vortices or skyrmions, confined in ultrasmall ferromagnets. Improving the sensitivity and long-time stability of YBCO nanoSQUIDs are in the focus of the research activities presented within this thesis. A process for the fabrication of YBCO nanoSQUIDs on MgO bicrystal substrates has been developed. The lower dielectric permittivity of MgO, as compared to STO, offers the possibility to realize YBCO nanoSQUIDs without the need of a resistively shunting Au layer on top of the YBCO film. This in turn offers a significant increase of the characteristic voltage of the grain boundary Josephson junctions intersecting the SQUID loop, which should significantly improve the sensitivity of the nanoSQUIDs. We demonstrate that YBCO nanoSQUIDs patterned by focused Ga ion beam (Ga FIB) milling on MgO bicrystals can have non-hysteretic current voltage characteristics (IVCs) at 4.2K even without Au as shunting layer, which shows the high potential to further improve the flux sensitivity. We further clarify the evolution of the electric transport and noise properties at 4.2K of YBCO nanoSQUIDs on bicrystal MgO substrates, upon decreasing the thickness of the Au film used as a resistively shunting layer. Moreover, we compare the performance of YBCO nanoSQUIDs fabricated on STO and MgO bicrystals at 77K and 4.2 K. A new approach based on heteroepitaxially grown superlattices was implemented in order to improve the flux sensitivity of nanoSQUIDs. We report on the fabrication and characterization of nanopatterned dc SQUIDs with grain boundary Josephson junctions based on heteroepitaxially grown YBCO/STO superlattices on STO bicrystal substrates. Nanopatterning is performed by Ga FIB milling. The electric transport properties and thermal white flux noise of superlattice nanoSQUIDs are comparable to single layer YBCO devices on STO bicrystals. However, we find that the superlattice nanoSQUIDs have more than an order of magnitude smaller low-frequency excess flux nois. We attribute this improvement to an improved microstructure at the grain boundaries forming the Josephson junctions in our YBCO nanoSQUDs. Last but not least, we developed a novel weak link in YBCO thin films based on an artificial bottom-up technology, i.e., by using Ga FIB milling to prepare nanogrooves in single crystal STO substrates, prior to YBCO thin film growth. This technique combined with cutting edge equipment like extreme ultraviolet lithography could provide a cost-effective and reliable pathway for scaling up superconducting circuits operating at liquid-nitrogen temperature