This thesis presents the results of an experimental study on single-charge effects in nanoscale Josephson junctions and Cooper pair transistors (CPTs). In nanoscale Josephson junctions the charging energy EC becomes significant at sub-Kelvin temperatures and single-charge effects, such as the Coulomb blockade of Cooper pair tunneling, influence the transport properties. In order to observe charging effects in a single Josephson junction, the impedance of the electromagnetic environment surrounding the junction has to be larger than the quantum resistance (RQ=h/4e2≈6.45kΩ). In this work the high impedance environment is obtained by biasing the sample under test (single Josephson junction or CPT) with four one-dimensional Josephson junction arrays having SQUID geometry. The advantage of this configuration is the possibility of tuning in situ the effective impedance of the electromagnetic environment. By applying a magnetic field perpendicular to the SQUID loops, the Josephson energy EJ of the SQUIDs is suppressed, resulting in an increase of the measured zero bias resistance of the arrays of several orders of magnitude (104< R0 (Ω) <109). This bias method enables the measurement of the same sample in environments with different impedance. As the impedance of the environment is increased, the current-voltage characteristics (IVCs) of the single Josephson junction and of the CPT show a well defined Coulomb blockade feature with a region of negative differential resistance, signature of the coherent tunneling of single Cooper pairs. The measured IVCs of a single Josephson junction with SQUID geometry in the high impedance environment show a qualitative agreement with the Bloch band theory as the EJ/EC ratio of the junction is tuned with the magnetic field. We also studied a single nontunable Josephson junction with strong coupling (EJ/EC > 1), where the exact dual of the overdamped Josephson effect is realized, resulting in a dual shape of the IVC, where the roles of current and voltage are exchanged. Here, we make for the first time a detailed quantitative comparison with a theory which includes the effect of fluctuations due to the finite temperature of the environment. The measurements on CPTs in the high impedance environment showed that the Coulomb blockade voltage is modulated periodically by the gate-induced charge. The gate-voltage dependence of the CPT changes from e-periodic to 2e-periodic as the impedance of the environment is increased. The high impedance environment reduces quasiparticle tunneling rates, thereby restoring the even parity of the CPT island. This behavior suggests that high impedance leads can be used to effectively suppress quasiparticle poisoning. / QC 20100928

Josephson junction qubits are promising candidates for a scalable quantum processor. Such qubits are commonly manipulated by means of sequences of rf-pulses and different methods are used to determine their quantum state. The readout should be able to distinguish the two qubit states with high accuracy and be faster than the relaxation time of the qubit. We discuss and experiment with a readout method based on the switching of a Josephson junction from the zero voltage state to a finite voltage state. The Josephson junction circuit has a non-linear dynamics and when it is brought to a bifurcation point, it can be made arbitrarily sensitive to small perturbations. This extreme sensitivity at a bifurcation point can be used to distinguish the two quantum states if the topology of the phase space of the circuit leads to a quick separation into the final states where re-crossings of the bifurcation point are negligible. We optimize a switching current detector by analyzing the phase space of a Josephson junction circuit with frequency dependent damping. A pulse and hold technique is used where an initial current pulse brings the junction close to its bifurcation point and the subsequent hold level is used to give the circuit enough time to evolve until the two states can be distinguished by the measuring instrument. We generate the pulse and hold waveform by a new technique where a voltage step with following linear voltage rise is applied to a bias capacitor. The frequency dependent damping is realized by an on-chip RC-environment fabricated with optical lithography. Josephson junction circuits are added on by means of e-beam lithography. Measurements show that switching currents can be detected with pulses as short as 5 ns and a resolution of 2.5% for a sample directly connected to the measurement leads of the cryostat. Detailed analysis of the switching currents in the RC-environment show that pulses with a duration of 20 us can be explained by a generalization of Kramers' escape theory, whereas switching the same sample with 25 ns pulses occurs out of thermal equilibrium, with sensitivity and speed adequate for qubit readout. / QC 20100924

Fabrication of nanoscale Josephson junctions and Superconducting Quantum Interference Devices (SQUID) is very promising but challenging topic in the superconducting electronics and device technology. In order to achieve best sensitivity of SQUIDs and to reproduce them easily with a straightforward method, new fabrication techniques for realization of nanoSQUIDs needs to be investigated. This study concentrates on investigation of new fabrication methodology for manufacturing nanoSQUIDs with High Temperature Bi-Crystal Grain Boundary Josephson Junctions fabricated onto SrTiO3 bi-crystal substrates using YBa2Cu3O7-δ (YBCO) thin-films. In this process nanoscale patterning of YBCO was realized by using electron beam patterning and physical dry etching of YBCO thin films on STO substrates. YBCO thin films were deposited using RF magnetron sputtering technique in the mixture of Ar and O2 gases and followed by annealing at high temperatures in O2 atmosphere. Structural characterization of YBCO thin films was done by Scanning Electron Microscope (SEM) and Energy Dispersive X-ray Spectroscopy (EDX). Superconducting properties of thin films was characterized by AC magnetic susceptibility measurements. Nanoscale structures on YBCO thin films were fabricated by one E-Beam Lithography (EBL) step followed by Reactive Ion Etching (RIE) and physical dry etching. First SiO2 thin film were deposited on YBCO by RF magnetron sputtering and it was patterned by EBL using Polystyrene (PS) as resist material and RIE. Then SiO2 was used as an etch mask for physical dry etching of YBCO and nanoscale structures on YBCO were formed.

Fabrication of nanoscale Josephson junctions and Superconducting Quantum Interference Devices (SQUID) is very promising but challenging topic in the superconducting electronics and device technology. In order to achieve best sensitivity of SQUIDs and to reproduce them easily with a straightforward method, new fabrication techniques for realization of nanoSQUIDs needs to be investigated. This study concentrates on investigation of new fabrication methodology for manufacturing nanoSQUIDs with High Temperature Bi-Crystal Grain Boundary Josephson Junctions fabricated onto SrTiO3 bi-crystal substrates using YBa2Cu3O7-δ (YBCO) thin-films. In this process nanoscale patterning of YBCO was realized by using electron beam patterning and physical dry etching of YBCO thin films on STO substrates. YBCO thin films were deposited using RF magnetron sputtering technique in the mixture of Ar and O2 gases and followed by annealing at high temperatures in O2 atmosphere. Structural characterization of YBCO thin films was done by Scanning Electron Microscope (SEM) and Energy Dispersive X-ray Spectroscopy (EDX). Superconducting properties of thin films was characterized by AC magnetic susceptibility measurements. Nanoscale structures on YBCO thin films were fabricated by one E-Beam Lithography (EBL) step followed by Reactive Ion Etching (RIE) and physical dry etching. First SiO2 thin film were deposited on YBCO by RF magnetron sputtering and it was patterned by EBL using Polystyrene (PS) as resist material and RIE. Then SiO2 was used as an etch mask for physical dry etching of YBCO and nanoscale structures on YBCO were formed.

Nous utilisons un microscope combinant microscopie à force atomique (AFM) et microscopie à effet tunnel (STM) à très basse température (~100 mK) afin d'étudier des nanocircuits mésoscopiques. Pour effectuer l'AFM dans ces conditions, nous utilisons un diapason en quartz recouvert de deux électrodes sur l'une desquelles est collée la pointe. Lorsque le diapason vibre on peut alors localiser en AFM un échantillon conducteur sur un substrat isolant, puis stopper les vibration pour réaliser des spectroscopies tunnel le long de la partie conductrice. Nous utilisons des pointes de platine-iridium ce qui nous permet de mesurer la densité d'états électronique locale. Nous nous sommes intéressés aux jonctions Josephson hybrides composées d'un îlot d'environ 1 µm de métal normal (cuivre) séparant deux supraconducteurs (aluminium). Ces échantillons sont réalisées par lithographie électronique et évaporation sous angle.Les courbes courant-tension de ces jonctions deviennent hystérétiques à très basse température ce qui est vraisemblablement dû à la dissipation thermique dans la partie normale. Nous avons pu localiser de manière fiable des échantillons uniques et effectuer simultanément des mesures en transport et des spectroscopies locales. Nous avons vu que la densité d'états du supraconducteur varie continuement à proximité du métal normal. Nous avons également observé un chauffage du supraconducteur avec le courant traversant la jonction. La mesure de la densité d'états du supraconducteur permet alors d'estimer la température électronique dans l'échantillon. La comparaison avec notre modèle thermique montre que l'énergie thermique produite dans le métal normal semble être évacuée mieux que prévu. / We use a microscope combining atomic force microscopy (AFM) and scanning tunneling microscopy (STM) at very low temperature (~100 mK) to study mesoscopic nanocircuits. To perform AFM measurements, we use quartz tuning forks covered with metallic electrodes on which we glue the tip. By using the tuning fork as a dynamic force sensor, we can localize the sample. Then, switching off the oscillation, we can perform local spectroscopies along the conductive part. We use platinum-iridium coated tips to measure the local density of states. This work is focused on hybrid Josephson junctions composed of a normal metal (copper) island of approximately 1 µm separating two superconductors (aluminium). These samples are made by electronic lithography and shadow evaporation.The current-voltage characteristics of these junctions become hysteretic at very low temperature because of thermal dissipation in the normal part. We achieved the localization of a unique sample and performed simultaneously transport measurements and local spectroscopies. We observed that the density of states of the superconductor varies continuously close to the normal metal. We also observed heating in the superconductor when the junction is current biased. The measure of the density of states of the superconductors gives an estimation of the electronic temperature in the sample. The comparison with our thermal model shows that the energy produced in the normal metal seems to be evacuated better than expected.

La jonction Josephson (JJ) est le composé de base de nombreux circuits électroniques supraconducteurs (SQUID, détecteurs d’ondes millimétriques, logique RSFQ). Avec la découverte des supraconducteurs HTc (à haute température critique), d’intenses recherches ont été entreprises pour réaliser des JJ fonctionnant à la température de l’azote liquide (77 K) permettant l’utilisation d’une cryogénie compacte. Toutefois, la complexité de ces matériaux a longtemps rendu difficile le développement d’une technologie viable et simple à mettre en œuvre. Parmi les méthodes de fabrication de jonctions, celle utilisant l’irradiation par faisceau d’ions a atteint un niveau de maturité suffisamment important pour pouvoir envisager la production de circuits comportant plusieurs milliers de JJ à HTc.Le but de ma thèse repose sur la fabrication, la caractérisation et l’étude des propriétés électromagnétiques de réseaux de jonctions Josephson réalisés dans des films minces d’YBa2Cu3O7-d. par la méthode d’irradiation ionique. / The Josephson junction (JJ) is the basis of many compound superconducting electronic circuits (SQUID detectors millimeter wave RSFQ logic). With the discovery of HTS superconductors (high critical temperature), intensive research has been undertaken to make JJ operating at the temperature of liquid nitrogen (77 K) allowing the use of a compact cryogenics. However, the complexity of these materials has long hampered the development of a viable technology and simple to implement. Among the junctions manufacturing methods, one using the ion beam irradiation has reached a sufficiently high degree of maturity to be able to envisage the production of circuits having thousands of JJ to HTS.The aim of my thesis is based on the fabrication, characterization and study of the electromagnetic properties of Josephson arrays achieved in thin films of YBa2Cu3O7-d. by the ion irradiation method.

The experimental pieces of evidence for the existence of Majorana states in topo- logical superconductors have so far been inconclusive despite intense research in the past two decades [Zha+20; Kay+20]. Combined with promising applications in quantum computing [Nay+08; Ali+11] and the resulting technological development of society, the elusiveness of Majorana states keeps motivating theoretical and ex- perimental research to this day. Our analytical findings and numerical explorations in new topological superconducting platforms suggest several tools and solutions for their future realisation in condensed matter systems. The planar Josephson junction (pJJ) introduced in 2017 by F. Pientka et al. [Pie+17] and M. Hell et al. [HLF17] is a versatile platform for topological superconductivity. It harnesses the tunability of the superconducting phase difference across the Josephson junction as an external control parameter that switches the pJJ between the trivial and topological phases of matter. The junction between the (trivial) superconductors is quasi-one-dimensional and hosts one new Majorana zero mode at each of its ends following each topological phase transition. However, the creation of a second Majorana zero mode on one end triggers a return to the trivial regime as both zero modes hybridize into a regular non-topological fermion. It is then crucial to identify the model parameters that lead to topological phases with a single Majorana state per end. Our main result on the pJJ establishes the general constraint on its microscopic parameters—including the phase difference and a magnetic field—to cross the topo- logical phase transitions. The identification of sectors in parameter space leading to a single Majorana mode becomes then straightforward. In some limits the pJJ develops a topological sector at small magnetic field for a phase difference close to the value p while it remains trivial at the same field near zero phase difference. Since the phase is sufficient to turn on and off the topology, we call this feature “switchable topology”. Looking for switchable topology is experimentally relevant as it makes the topology easily tunable while keeping intact the proximitized su- perconductivity otherwise jeopardized by the applied field. Concretely, we found switchable topology in three configurations: in wide junctions with a transparent interface with the superconducting regions, in fine-tuned narrow junctions weakly coupled to the superconducting regions, and in junctions with a strong Zeeman energy when they are ultranarrow and transparent. Thanks to our exact analytical results, setups interpolating between these limits can adjust the desired properties at will. The other important finding about the pJJ concerns the stability of its topological phases, by which we mean the presence of a sizable spectral gap in the topological sector. We observed that the Rashba spin-orbit coupling is responsible for strongly decreasing the gap in the relevant topological sector at low Zeeman field, but sym- metry arguments justify that wide, transparent junctions are generically immune to this effect for large enough Rashba coupling. After 2017, other platforms started to use the Josephson superconducting phase difference as a knob to trigger topological superconductivity [Liu+19; JY21]. We introduce here the stacked Josephson junction (sJJ) as a new platform for topological superconductivity, which is made of two non-centrosymmetric superconductors sandwiching a two-dimensional magnet around which chiral Majorana edge modes propagate. Unlike the Majorana zero modes in the pJJ, chiral Majorana modes can add to each other if they propagate in the same direction, as indicated by the integer Chern number of their topological phase. The bulk-edge correspondence, however, only constrains the net number of topological edge states and allows room for other non-topological states to coexist with the chiral Majorana states without interacting with them. We found that the presence of trivial chiral edge modes in the sJJ restricts access to the Majorana states themselves. The symmetry protection of the trivial modes, fortunately, disappears with an in-plane magnetic field applied through the magnet or with superconducting leads different on the top and at the bottom of the stacked junction. The theoretical investigations of topological platforms have currently outnum- bered the experiments with convincing signatures of Majorana edge states. This imbalance calls for new ways to probe the agreement between topological models and laboratory setups. The critical current of a Josephson junction acts as a link between the microscopic description and macroscopic observables. Thermoelectric measurements, which distinguish between supercurrent and quasiparticle current, modify this model-dependent connection, and would provide an electrical probe to estimate the validity of a model like that of the pJJ. We computed the contribution to the thermoelectric coefficient of the bulk states of a uniform superconductor, that has a similar environment to that of the pJJ (i.e., Rashba coupling and in-plane Zeeman field). The results were not conclusive and motivated us to suggest new analytical and numerical approaches to obtain the thermoelectric response of the pJJ, in particular by including the contribution of the Andreev bound states and non-linear effects.:Foreword — how to read this thesis 1 Preamble A popular short story: pencils and lightbulbs 5 Basics and concepts 1 Introduction to Majorana physics 13 1.1 The electrons & their properties 13 1.1.1 Hamiltonian for the planar Josephson junction 17 1.2 The scattering matrix for bound states 19 1.3 Andreev bound states for topology 24 1.4 Topological superconductivity & Majorana edge states 28 1.5 Induced topological superconductivity 34 1.6 Summary 36 Appendices 37 1.A Microscopic dynamics 37 1.A.1 Origin of spin–orbit coupling 37 1.A.2 Bogoliubov-deGennes symmetrization 37 1.A.3 Andreev reflection below the coherence length 38 1.A.4 Proximity-induced superconductivity 40 1.A.5 From s- to p-wave superconductivity 41 1.B Scattering theory for bound states 44 1.B.1 Bound states as trapped waves 44 1.B.2 Scattering theory for an open region 45 1.B.3 Scattering theory for two open regions 46 1.B.4 Bound states recovered from an open region 47 1.B.5 Numerical scattering theory for bound states 48 2 Perspectives on electronic transport 53 2.1 Electric current in a metal 53 2.2 Quantum-mechanical current 54 2.2.1 Expression for the microscopic current 55 2.3 Thermoelectric current 57 2.3.1 The Boltzmann transport equation 61 2.4 Supercurrents and the superconducting coherence phase 64 2.4.1 Josephson currents 67 Appendices 71 2.A Electric current from a potential difference 71 2.B Scattering and current 71 2.C Hole-based current in metals 73 Introduction Introduction to the Research Projects 77 i Topological properties of Josephson junctions 3 Switchable topology in the planar Josephson junction 85 Motivation & Overview of the Study 85 3.1 The planar Josephson junction and the nanowire setup 87 3.1.1 Comparison with the nanowire setup. 89 3.2 Model 92 3.3 General formula for the phase transitions 94 3.3.1 Spin decoupling for the phase transitions 96 3.3.2 Exact reflection coefficients 97 3.3.3 Exact scattering formula and Andreev reflectivity 98 3.3.4 Andreev approximation 100 3.3.5 Dimensionless formulation 101 3.3.6 Numerical and analytical checks 103 3.4 Three regimes for switchable topology 105 3.4.1 Diamond-shape regime 108 3.4.2 V-shape regime 110 3.4.3 Nanowire regime 111 3.4.4 Summary: extent of the topological transitions 114 3.5 Avoiding regimes with a small topological gap 117 3.5.1 Gapless lines as BDI phase transitions 119 3.5.2 Opening the gap in f = p 120 3.5.3 Role of the Rashba coupling 121 3.6 Conclusion 125 Appendices 129 3.A Limiting cases of the pJJ 129 3.A.1 Andreev approximation 129 3.A.2 Small field limit 131 3.A.3 Delta-barrier junction 131 3.A.4 Semiconductor nanowire 132 3.B Normal reflection via surface impurity and surface refraction 134 3.C Symmetry-constrained gap closings 136 3.D Linear deviation of the gapless line near f = p 138 3.E Calculations for the scattering formula 141 3.E.1 Boundary conditions 141 3.E.2 Combinations of scattering coefficients 142 3.E.3 Andreev coefficients for the phase transitions 143 3.E.4 Formula for B > μ 145 4 Topological and trivial chiral states in the stacked Josephson junction 147 Motivation & Overview of the Study 147 4.1 The basics of the stacked Josephson junction 149 4.2 Continuous and lattice models 151 4.3 Topological index 155 4.3.1 Methodology for the Chern number 155 4.3.2 Interpretation of the results 156 4.4 Topological and trivial edge states 162 4.5 BDI phase transitions 167 4.5.1 Dimensional reduction 168 4.5.2 Link between topological invariants 170 4.5.3 Explaining the low-energy sector 171 4.6 Conclusion 174 Appendices 177 4.A Symmetries of the Hamiltonian 177 4.A.1 Class D 177 4.A.2 Class BDI 177 4.A.3 Gapless line in f = p 178 4.A.4 Symmetry around f = p 179 4.B The parity index in 2D TSC 180 ii Transport properties of the planar Josephson junction 5 An approach to thermoelectric effects in the planar Josephson junction 183 Motivation & Overview of the Study 183 5.1 From the Josephson junction to a homogeneous superconductor 185 5.2 Model and Phenomenology 187 5.2.1 Homogeneous superconductor 187 5.2.2 Analytical spectrum and two-surface approximation 188 5.2.3 Magnetoelectric supercurrent: phenomenology 191 5.3 Electric current in a spin–orbit coupled superconductor 194 5.3.1 Formula for the current 196 5.3.2 Zero-temperature current 198 5.3.3 Small perturbations at finite temperature 200 5.4 Thermoelectric current in a spin–orbit coupled superconductor 206 5.4.1 Distribution imbalance under temperature bias 208 5.4.2 Explicit formula for the thermoelectric current 209 5.5 Discussion and Outlook 213 Appendices 219 5.A The Boltzmann equation in temperature-biased superconductors 219 5.A.1 The linear approximation 220 5.A.2 The low-temperature approximation 220 5.A.3 Integral solution of the Boltzmann equation 223 5.B Diagonalisation of the planar superconductor 225 5.B.1 Eigenstates of spin–orbit coupled superconductor 225 5.B.2 Eigenstates with a small Zeeman field 227 Conclusion Majorana quasiparticles in Josephson junctions 233 Extra Material 6 Mathematical details of Scattering theory 241 6.1 Asymmetric quantum well 241 6.2 Scattering theory for an open region 243 6.2.1 Change in potential over a small region 243 6.2.2 Change in spin-orbit coupling over a small region 245 6.2.3 Change in mass over a small region 245 7 Numerical codes for chapter 4 247 7.1 BDI index 247 7.2 Chern number 255 7.3 Spectral gap 257 7.4 Localized edge states 258 8 Short courses 261 8.1 Two formulations of superconductivity 261 8.1.1 The BCS Hamiltonian 261 8.1.2 The Bogoliubov transformation 263 8.1.3 Bogoliubov-de Gennes symmetrization 264 8.1.4 Building the semiconductor representation 266 8.2 Topological band theory 270 8.3 Majorana physics in 1D 274 8.3.1 The SSH chain 275 8.3.2 The Kitaev chain 277 Bibliography 283

No description available.

By exploiting the versatility of ultracold atoms experiments, a variety of many-body phe- nomena can be studied. Ferromagnetism in a trapped ultracold gas of repulsively interacting fermions is considered within a local-density approximation and beyond, where energetic sig- natures indicate its onset. Transport of a Bose-Einstein condensate is explored experimentally in a tunable double-well potential, and a crossover from hydrodynamic to Josephson transport is observed as the barrier between condensates is raised. To add a degree of freedom for the manipulation of ultracold gases, two schemes for species-specific optical lattices are developed theoretically.

We address the excitation of quantum breathers in small nonlinear networks of two and three degrees of freedom, in order to study their properties. The invariance under permutation of two sites of these networks substitutes the translation invariance that is present in nonlinear lattices, where (classical) discrete breathers are time periodic space localized solutions of the underlying classical equations of motion. We do a systematic analysis of the spectrum and eigenstates of such small systems, characterizing quantum breather states by their tunnelling rate (energy splitting), site correlations, fluctuations of the number of quanta, and entanglement. We observe how these properties are reflected in the time evolution of initially localized excitations. Quantum breathers manifest as pairs of nearly degenerate eigenstates that show strong site correlation of quanta, and are characterized by a strong excitation of quanta on one site of the network which perform slow coherent tunnelling motion from one site to another. They enhance the fluctuations of quanta, and are the least entangled states among the group of eigenstates in the same range of the energy spectrum. We use our analysis methods to consider the excitation of quantum breathers in a cell of two coupled Josephson junctions, and study their properties as compared with those in the previous cases. We describe how quantum breathers could be experimentally observed by employing the already developed techniques for quantum information processing with Josephson junctions.