No description available.

In this thesis we numerically compute the scattering lengths and bound states for sodium-rubidium collisions at low energy. This work was motivated by experiments which aim to produce Bose-Einstein condensates (BEC) mixtures of sodium-rubidium. Elastic collision properties are important for the rethermalization of the atoms during the evaporative cooling process. Inelastic processes, which we also discuss to some extent, cause trap losses in those experiments. In order to reach the required temperature and density the elastic collision rates should be sufficiently large compared to the inelastic rates. The scattering lengths, which completely specify the elastic collision parameters at low energy, determine the miscibility and phase diagram of the sodium-rubidium condensate mixture. We calculate the scattering lengths approximately and find agreement with previous calculations indicating that miscible phases of sodium and rubidium condensates do not appear to be feasible in the absence of external fields.

Detection of explosives has always been a challenging issue all over the world. Different analytical techniques and instrumentation methods have been explored to obtain a 100% fail proof detector. Some technologies have matured and have been deployed in the field already. However, active research is still being pursued to make the ultimate explosive detection device. The present thesis broadly addresses the development of Raman spectroscopy based techniques for the detection of explosives. Although Raman spectroscopy has technologically developed and has become a regular tool for chemical identification, its use in the field of detection of explosives has been limited. Two aspects of detection were addressed in this thesis. The first part consists of the detection of minute quantities or traces of explosives using a Raman based method. In order to approach this problem, surface enhanced Raman spectroscopy (SERS), an offshoot of Raman spectroscopy was explored. Chapters 2-4 deal with developing efficient SERS substrates. In this endeavour, the first and the most obvious choice as SERS substrates were silver (Ag) nanoparticles (NPs). However, we were exploring methods that could be simple one-pot synthesis methods, cost-effective and without employing strong reducing agents (green). Therefore, Ag NPs were synthesized using biosynthetic route. These nanoparticles were used to study their SERS efficiency. Sub-nano molar concentration of dye as well explosive like trinitrotoluene (TNT) and hexanitrohexaazaisowurtzitane (CL-20) could be obtained for both the clove reduced as well as pepper Ag nanoparticles. Hence Ag NPs are very efficient SERS substrates. In the second part of the work on SERS, bimetallic nanoparticles with core-shell (Agcore-Aushell) architecture were synthesized, characterized and tested for SERS activity. After successful synthesis and characterization of the bimetallic nanoparticles, these were tested for their SERS activities using a dye molecule and an explosive molecule. SERS spectra could be obtained for the bimetallic nanoparticles. It was observed that the sensitivity of these NPs were almost at par with the mono-metallic Ag NPs. In order to bring SERS from laboratory to field, a more practical approach was to prepare solid SERS substrates or SERS substrates on solid platform. In the next chapter, we ventured into the most abundant material which forms the backbone of the organic world, carbon. Various carbonaceous materials ranging from chemically synthesized graphene, graphene oxide, multi-walled carbon nanotube (MWCNT), graphite and activated charcoal were explored as potential substrates for surface enhanced Raman spectroscopic applications. The analytes chosen for this particular study were some fluorescent molecules such as rhodamine B (RB), rhodamine 6G (R6G), crystal violet (CV), Nile blue A (NBA) and a non-fluorescent molecule acetaminophen, commonly known as paracetamol. Enhanced Raman signals were observed for the fluorescent molecules, especially for the molecules whose absorbance maxima are near the excitation wavelength of the laser (514.5 nm). The most interesting outcome of this work was obtaining enhanced Raman signals of nanomolar concentration of R6G on activated charcoal. However, for the non-fluorescent molecule, paracetamol, Raman spectra could not be observed beyond -5 10M concentration for all the carbon substrates including chemically synthesized graphene and MWCNT. This study was crucial in our quest for an ideal SERS substrate. Our observations let us to conclude that chemically synthesized graphene was not the only candidate for the preparation of SERS substrates. Since carbon materials efficiently adsorb and also provide a separate channel for energy decay (fluorescence quenching), even activated charcoal could be employed as a SERS platform. However, carbon alone could not provide an effective solution for the preparation of SERS substrates. Therefore, combining the plasmonic effect of the metal nanoparticles with the efficient adsorption and fluorescence quenching of carbon materials would be ideal. In the next part of the carbon studies, graphene-Ag composites which were either prepared by in situ reduction process or physically mixed were studied for SERS activity. An ideal SERS substrate should possess the following properties: (i) Support plasmon, thereby provide SERS enhancement (ii) Easy to fabricate or synthesize (large scale/bulk) (iii) Ensure high reproducibility and sensitivity (iv) Low false alarm from matrix chemicals (v) Cost effective (vi) Solid substrate (in the form of chip, pellet, slide etc.) Hence, as a final study, carbon silver based composites were explored. R6G was chosen as an analyte again and SERS experiments were conducted. Raman signals at low concentration could be obtained for the carbon-Ag composites as well. In addition, feasibility experiments were also conducted for an explosive molecule, FOX-7. From these preliminary experiments we observed that carbon-metal NP composites can be efficient, cost-effective SERS substrates that will overcome the current issue. The previous chapters dealt with the trace detection of explosives. The next part of the thesis deals with the development of the Raman spectroscopic methods for non-invasive detection of concealed objects. Chapters 4 and 5 primarily focus on explosives detection. Spatially offset Raman spectroscopy (SORS) instrumentation was developed in the laboratory for non-invasive detection solid and liquid explosives. Several experiments were carried out to detect concealed materials inside high density polyethylene (HDPE) containers, coloured glass bottles, envelopes etc. with this technique, Raman signals of materials could be retrieved even within 4 mm thick outer-layer. SORS imaging experiments were also performed on bilayered compounds, tablets etc. However, while performing the SORS experiments, it was observed that due to the restriction in geometry imposed by the method, the signals from the inner-layers could be obtained only up to a certain depth. This posed a serious limitation of SORS for practical scenarios, where the thickness of the outer layer may be tens of mm. In such situation, SORS may not be an effective method. We then performed Raman experiments using a transmission geometry using a series of samples. The transmission Raman (TR) experiments yielded better SNR for the inner (concealed) material as compared to the outer material. Although transmission Raman experiments yielded better signal but these experiments were again geometry dependent, hence, less flexible and TR experiments did not provide information about the position of the underlying materials. In order to obtain complete information, it was necessary to understand photon migration in a multiple scattering medium. It is known that a photon in a multiple scattering medium may be approximated to undergo a random-walk. Statistically, the photon that undergoes multiple scattering in a medium loses its sense of origin (direction), hence, there is a finite probability to observe the exiting photon in any direction. Rayleigh and NIR based imaging modalities have been conducted using this model. Diffuse optical tomographic (DOT) measurements also deal with measuring the photons that have exited the sample after undergoing multiple scattering in a turbid medium. If it was possible to collect the Rayleigh photons or the diffuse photons in DOT experiments, in principle, Raman photons could also be collected from several directions. It was then proposed that if Rayleigh scattered photons can exit at 4π solid angle from a sample, then it can be assumed that some Rayleigh photons may convert to Raman photons, which in turn, shall have a finite probability to exit the sample from all the sides (4π solid angles). This idea of collecting Raman photons has never been discussed before! Thus, as expected based on the above principles, we were able to record Raman scattered photons at all angles and on all sides. This new technique has been termed as ‘Universal Multiple Angle Raman Spectroscopy (UMARS)’. Monte Carlo simulation studies were also performed to understand the distribution of photons in a multiple scattering medium. Simulation studies also revealed that Raman photons exited from all sides of the medium at varying percentages. Hence, several fiber optic probes were designed for illumination and collection to perform the UMARS experiments for samples concealed at depths beyond 20 mm. UMARS was not only applied successfully for the detection of concealed explosives, but also for biologically relevant samples as well. In fact a pharmaceutical tablet as thick as 7 mm was also tested with UMARS and signals could be successfully obtained. Since the UMARS signals were obtained from all possible angles, imaging experiments were also conducted to obtain sample specific information. Frequency-specific images of bilayer materials could be obtained. In the case where one material was concealed within another, the reconstruction of the frequency-specific intensities in a contour plot revealed the position of the concealed layer. One of the most challenging and exciting studies that was conducted was to use UMARS to obtain shapes of hidden materials. Several shapes such as dumbbell, ellipsoid etc were fabricated (made of glass) and were filled with a test chemical, trans-stilbene (TS). This shape was placed inside an outer material like ammonium nitrate (AN) that was taken in a glass beaker. The diameter of the beaker was varied from 25 mm to 60 mm. A series of UMARS measurement was carried out with 10 collection fiber optic probes. The spatial resolution (vertical) was varied from 200 μm to 1 mm. Series of UMARS images were obtained which were then processed and the intensity of the individual fibers were averaged (CCD row pixels) based on the image of the individual fiber on the CCD. The frequency specific intensity of the materials was utilized to reconstruct 2D or a 3D shape. The shapes of the objects could be clearly discerned using UMARS imaging. This marks a major step for the development of UMARS as a 3D imaging modality. UMARS experiments conducted so far have affirmed our belief that this technology can be used as an effective technique for screening solid and liquid samples at airports, railway stations and other entry points. 3D imaging for biomedical diagnostics will provide molecular information in addition to the location and shape of an object inside a tissue such as calcified masses and bones. In the final part of the thesis, 2D Raman correlation spectroscopic method was applied to understand the dynamics of a system that was subjected to external perturbation. In the field of explosive processing and formulations, large batches are generally prepared. However, it is very difficult to ascertain the molecular or structural changes that occur during the processing of these formulations in situ. Analytical methods to monitor the changes online are limited. Raman spectroscopy can be an effective technique for such measurements. This process however, generates a large number of spectra. In such cases, it becomes cumbersome to handle such large number of data and obtain meaningful information. 2D correlation spectroscopy can be applied under such situations. 2D correlation analysis generates essentially two maps, synchronous and asynchronous. In this study, 2D Raman correlation spectroscopy was applied to ammonium nitrate that was subjected to temperature variations. 2D maps were constructed to obtain information about the structural changes associated with temperature. The synchronous map reveals the overall similarity of the intensity changes. Whereas, the 2D asynchronous maps provide the sequence of changes that occur. Based on the set of well defined rules proposed by Isao Noda, the synchronous and the asynchronous correlation maps were analysed. Hence, generalized 2D correlation spectroscopy can be extended to any kind of perturbation and will prove useful in understanding the structural dynamics. The objective of the thesis was to explore various facets of Raman spectroscopy that would be useful in the field of high energy materials specifically in the detection of explosives. Attempts were made for the development of trace detection of explosives using Raman based technique, SERS. In addition, bulk detection of concealed explosives was performed non-invasively using SORS and UMARS. In the field of high energy materials, these techniques will find immense applications. Raman spectroscopy, as we saw is a very important technique that can be used as a stand-alone method and can also be interfaced with other analytical or imaging modalities. This treatise is an example where the strength of this powerful spectroscopic method has been explored to some extent.

Nous étudions les champs de Maxwell à l'extérieur de trous noirs de Reissner-Nordstrom-de Sitter. Nous commençons par étudier la géométrie de ces espaces-temps : nous donnons une condition sous laquelle la métrique admet trois horizons puis dans ce cadre nous construisons l'extension analytique maximale d'un trou noir de Reissner-Nordstrom-de Sitter. Nous donnons ensuite une description générale des champs de Maxwell en espace-temps courbe, de leur décomposition en composantes spinorielle ainsi que de leur énergie. La première étude analytique établit la décroissance ponctuelle de champs de Maxwell à l'extérieur d'un trou noir de Reissner-Nordstrom-de Sitter ainsi que la décroissance uniforme de l'énergie sur un hyperboloïde qui s'éloigne dans le futur. Ce chapitre utilise des méthodes de champs de vecteurs (estimations d'énergie géométriques) dans l'esprit des travaux de Pieter Blue. Enfin nous construisons une théorie du scattering conforme pour les champs de Maxwell à l'extérieur du trou noir. Ceci consiste en la résolution du problème de Goursat pour les champs de Maxwell à la frontière isotrope de l'extérieur du trou noir, constituée des horizons du trou noir et horizons cosmologiques futurs et passés. Les estimations de décroissance uniforme de l'énergie sont cruciales dans cette partie. / We study Maxwell fields on the exterior of Reissner-Nordstrom-de Sitter black holes. We start by studying the geometry of these spacetimes: we give the condition under which the metric admits three horizons and in this case we construct the maximal analytic extension of the Reissner-Nordstrom-de Sitter black hole. We then give a general description of Maxwell fields on curves spacetimes, their decomposition into spin components, and their energies. The first result establishes the pointwise decay of the Maxwell field in the exterior of a Reissner-Nordstrom-de Sitter black hole, as well as the uniform decay of the energy flux across a hyperboloid that recedes in the future. This chapter uses the vector fields methods (geometric energy estimates) in the spirit of the work of Pieter Blue. Finally, we construct a conformal scattering theory for Maxwell fields in the exterior of the black hole. This amounts to solving the Goursat problem for Maxwell fields on the null boundary of the exterior region, consisting of the future and past black hole and cosmological horizons. The uniform decay estimates of the energy are crucial to the construction of the conformal scattering theory.

This thesis is devoted to improve our knowledge on the theory of polarized line formation in a magneto-turbulent medium, and in a scattering dominated magnetized medium, where partial redistribution (PRD) effects become important. Thus the thesis consists of two parts. In the first part we carry out a detailed investigation on the effect of random magnetic fields on Zeeman line radiative transfer. In the second part we develop the theory of polarized line formation in the presence of arbitrary magnetic fields and with PRD. We present numerical methods of solution of the relevant transfer equation in both part-I and II. In Chapter I we give a general introduction, that describes the basic physical concepts required in both parts of the thesis. Chapters 2-6 deal with the part-I, namely stochastic polarized Zeeman line formation. Chapters 7-10 deal with part –II, namely the theory and numerics of polarized line formation in scattering media. Chapter II is devoted to the future outlook on the problems described in part-I and II of the thesis. Appendices are devoted to additional mathematical details. Part-I of the Thesis: Stochastic polarized line formation in magneto-turbulent media Magneto-convection on the Sun has a size spectrum that spans several orders of magnitudes and hence develops turbulent elements or eddies the sizes of which are much smaller than the spatial resolution of current spectro-polarimeters (about 0.2 arcsec or 150km at the photospheric level). We were thus strongly motivated to consider the Zeeman effect in a medium where the magnetic field is random with characteristic scales of variation comparable to the radiative transfer characteristic scales. In Chapter 2, we consider the micro-turbulent limit and study the mean zeeman absorption matrix in detail. The micro-turbulent limit refers to the case when the scales of fluctuations of the random field are much smaller than the photon mean free paths associated to the line formation. The ‘mean’ absorption and anomalous dispersion coefficients are calculated for random fields with a non-Zero mean value - isotropic or anisotropic Gaussian distributions that are azimuthally invariant about the direction of the mean field. The averaging method is described in detail, and fairly explicit expressions for the mean coefficients are established. A detailed numerical investigation of the mean coefficients illustrates two simple effects of the magnetic field fluctuations: (i) broadening of the components by fluctuations of the field strength, leaving the π-components unchanged, and (ii) averaging over the angular dependence of the π and components. Angular averaging can modify the frequency profiles of the mean coefficients quite drastically, namely, the appearance of an unpolarized central component in the diagonal absorption coefficient, even when the mean field is in the direction of the line-of-sight. For isotropic fluctuations, the mean coefficients can be expressed in terms of generalized Voigt and Faraday-Voigt functions, which are related to the derivatives of the Voigt and Faraday-Voigt functions. In chapter 3, we study these functions in detail. Simple recurrence relations are established and used for the calculation of the functions themselves and of their partial derivatives. Asymptotic expansions are also derived. In Chapter 4, we consider the Zeeman effect from a magnetic field which has a finite correlation length(meso-turbulence) that can be varied from zero to infinity and thus made comparable to the photon mean free-path. The random vector magnetic field B is modeled by a Kubo-Anderson process – a piecewise constant Markov process characterized by a correlation length and a probability distribution function(PDF) for the random values of the magnetic field. The micro- and macro-turbulent limits are recovered when the correlation length goes to zero or infinity respectively. Mean values and rms fluctuations around the mean values are calculated numerically for a random magnetic field with isotropic Gaussian fluctuations. The effects of a finite correlation length are discussed in detail. The rms fluctuations of the Stokes parameters are shown to be very sensitive to the correlation length of the magnetic field. It is suggested to use them as a diagnostic tools to determine the scale of unresolved features in the solar atmosphere. In Chapter 5, using statistical approach, we analyze the effects of random magnetic fields on Stokes line profiles. We consider the micro and macro-turbulent regimes, which provide bounds for more general random fields with finite scales of variations. The mean Stokes parameters are obtained in the micro-turbulent regime, by first averaging the Zeeman absorption matrix Φ over the PDF P(B) of the magnetic field and then solving the concerned radiative transfer equation. In the macro-turbulent regime, the mean solution is obtained by averaging the emergent solution over P(B). In this chapter, we consider the same Gaussian PDFs that are used to construct (Φ) in chapter 2. Numerical simulations of magneto-convection and analysis of solar magnetograms provide the empirical PDF for the magnetic field line-of-sight component on the solar atmosphere. In Chapter 6, we explore the effects of different kinds of PDFs on Zeeman line formation. We again consider the limits of micro and macro-turbulence. The types of PDFs considered are: (a) Voigt function and stretched exponential type PDFs for fields with fixed direction but fluctuating strength. (b) Cylindrically symmetrical power law for the angular distribution of magnetic fields with given field strength. (c) Composite PDFs accounting for randomness in both strength and direction obtained by combining a Voigt function or a stretched exponential with an angular power law. The composite PDF proposed has an angular distribution peaked about the vertical direction for strong fields and is nearly isotropically distributed for weak fields, which could mimic solar surface random fields. We also describe how the averaging technique for a normal Zeeman triplet may be generalized to the more common case of anomalous Zeeman splitting patterns. Part-II of the Thesis: Polarized line formation in scattering media-Theory and numerical methods Many of the strongest and most conspicuous lines in the Second Solar Spectrum are strong lines that are formed rather high, often in the chromosphere above the temperature minimum. From the standard, unpolarized and non-magnetic line-formation theory such lines are known to be formed under the conditions that are very far from local thermodynamic equilibrium. They are characterized by broad damping wings surrounding the line core. Doppler shifts in combination with collisions cause photons that are absorbed at a given frequency to be redistributed in frequency across the line profile in a complex way during the scattering process. Two idealized, limiting cases to describe this redistribution are “frequency coherence” and “complete redistribution” (CRD), but the general theory that properly combines these two limiting cases goes under the name “partial frequency redistribution” (PRD). Resonance lines which are usually strong can be properly modeled only when PRD is taken into account. To use these strong lines for magnetic field diagnostics we need a line scattering theory of PRD in the presence of magnetic fields of arbitrary strength. In the second part of the thesis we develop such a theory and derive the polarized PRD matrices. These matrices are then used in the polarized line transfer equation to compute the emergent Stokes parameters. Polarized scattering in spectral lines is governed by a 4 x 4 matrix that describes how the Stokes vector is scattered in all directions and redistributed in frequency within the line. In Chapter 7, using a classical approach we develop the theory for this redistribution matrix in the presence of magnetic fields of arbitrary strength and direction, and for a J = 0 → 1 → 0 transition. This case of arbitrary magnetic fields is called the Hanle-Zeeman regime, since it covers both the partially overlapping weak and strong-field regimes, in which the Hanle and Zeeman effects respectively dominate the scattering polarization. In this general regime the angle-frequency correlations that describe the so-called PRD are intimately coupled to the polarization properties. We also show how the classical theory can be extended to treat atomic and molecular scattering transitions for any combinations of J quantum numbers. In chapter 8 , we show explicitly that for a J = 0 → 1 → 0 scattering transition there exists an equivalence between the Hanle-Zeeman redistribution matrix that is derived through quantum electrodynamics(Bommier 1997b) and the one derived in Chapter 7 starting from the classical, time-dependent oscillator theory of Bommier & Stenflo (1999). This equivalence holds for all strengths and directions of the magnetic field. Several aspects of the Hanle-Zeeman redistribution matrix are illustrated, and explicit algebraic expressions are given, which are of practical use for the polarized line transfer computations. In chapter 9, we solve the polarized radiative transfer equation numerically, taking into account both the Zeeman absorption matrix and the Hanle-Zeeman redistribution matrix. We compute the line profiles for arbitrary field strengths, and scattering dominated line transitions. We use a perturbation method (see eg. Nagendra et al. 2002) to solve the Hanle-Zeeman line transfer problem. The limiting cases of weak field Hanle scattering and strong field Zeeman true absorption are retrieved. The ilntermediate regime, where both Zeeman absorption and scattering effects are important, is studied in some detail. Numerical method used to solve the Hanle-Zeeman line transfer problem in Chapter 9 is computationally expensive. Hence it is necessary to develop fast iterative methods like PALI (Polarized Approximate Lambda Iteration). As a first step in this direction we develop such a method in Chapter 10 to solve the transfer problem with weak field Hanle scattering. We use a ‘redistribution matrix’ with coupling between frequency redistribution and polarization and no domain decomposition. Such a matrix is constructed by angle-averaging the frequency dependent terms in the exact weak field Hanle redistribution matrix for a two-level atom with unpolarized ground level (that can be obtained by taking the weak field limit of the Hanle-Zeeman redistribution matrix). In the past, the PALI technique has been applied to redistribution matrices in which frequency redistribution is ‘decoupled’ from scattering polarization, the decoupling being achieved by an adequate decomposition of the frequency space into several domains. In this chapter, we examine the consequences of frequency space decomposition, and the resulting decoupling between the frequency redistribution and polarization, on the solution of the polarized transfer equation for the Stokes parameters.

No description available.

No description available.

No description available.

This thesis presents the work done on electronic transport in various interacting and non-interacting systems in one and two dimensions. The systems under study are: an interacting quantum dot [1], a non-interacting quantum wire and a ring in which time-dependent potentials are applied [2], an interacting quantum wire and networks of multiple quantum wires with resistive regions [3, 4], one-dimensional edge stages of a two-dimensional topological insulator [5], and a hybrid system of two-dimensional surface states of a three-dimensional topological insulator and a superconductor [6]. In the first chapter, we introduce a number of concepts which are used in the rest of the thesis, such as scattering theory, Landauer conductance formula, quantum wires, bosonization, topological insulators and superconductor. In the second chapter, we study transport through a quantum dot with interacting electrons which is connected to two reservoirs. The quantum dot is modeled by two sites within a tight-binding model with spinless electrons. Using the Lippman-Schwinger method, we write down an exact two-particle wave function for the dot-reservoir system with the interaction localized in the region of the dot. We discuss the phenomena of two-particle resonance and rectification. In the third chapter, we study pumping in two kinds of one-dimensional systems: (i) an infinite line connected to reservoirs at the two ends, and (ii) an isolated ring. The infinite line is modeled by the Dirac equation with two time-independent point-like backscatterers that create a resonant barrier. We demonstrate that even if the reservoirs are at the same chemical potential, a net current can be driven through the channel by the application of one or more time-dependent point-like potentials. When the left-right symmetry is broken, a net current can be pumped from one reservoir to the other by applying a time-varying potential at only one site. For a finite ring, we model the system by a tight-binding model. The ring is isolated in the sense that it is not connected to any reservoir or environment. The system is driven by one or more time-varying on-site potentials. We develop an exact method to calculate the current averaged over an infinite amount of time by converting it to the calculation of the current carried by certain states averaged over just one time period. Using this method, we demonstrate that an oscillating potential at only one site cannot pump charge, and oscillating potentials at two or more sites are necessary to pump charge. Further we study the dependence of the pumped current on the phases and the amplitudes of the oscillating potentials at two sites. In the fourth chapter, we study the effect of resistances present in an extended region in a one-dimensional quantum wire described by a Tomonaga-Luttinger liquid model. We combine the concept of a Rayleigh dissipation function with the technique of bosonization to model the dissipative region. In the DC limit, we find that the resistance of the dissipative patch adds in series to the contact resistance. Using a current splitting matrix M to describe junctions, we study in detail the conductances of: a three-wire junction with resistances and a parallel combination of resistances. The conductance and power dissipated in these networks depend in general on the resistances and the current splitting matrices that make up the network. We also show that the idea of a Rayleigh dissipation function can be extended to couple two wires; this gives rise to a finite transconductance analogous to the Coulomb drag. In the fifth chapter, we study the effect of a Zeeman field coupled to the edge states of a two-dimensional topological insulator. These edge states form two one-dimensional channels with spin-momentum locking which are protected by time-reversal symmetry. We study what happens when time-reversal symmetry is broken by a magnetic field which is Zeeman-coupled to the edge states. We show that a magnetic field over a finite region leads to Fabry-P´erot type resonances and the conductance can be controlled by changing the direction of the magnetic field. We also study the effect of a static impurity in the patch that can backscatter electrons in the presence of a magnetic field. In the sixth chapter, we use the Blonder-Tinkham-Klapwijk formalism to study trans-port across a line junction lying between two orthogonal topological insulator surfaces and a superconductor (which can have either s-wave or p-wave pairing). The charge and spin conductances across such a junction and their behaviors as a function of the bias voltage applied across the junction and various junction parameters are studied. Our study reveals that in addition to the zero conductance bias peak, there is a non-zero spin conductance for some particular spin states of the triplet Cooper pairs. We also find an unusual satellite peak (in addition to the usual zero bias peak) in the spin conductance for a p-wave symmetry of the superconductor order parameter.

Robust states exist at the interfaces between topologically trivial and nontrivial phases of matter. These boundary states are expression of the nontrivial bulk properties through a connection dubbed the bulk-boundary correspondence. Whether the bulk is topological or not is determined by the value of a topological invariant. This quantity is defined with respect to symmetries and dimensionality of the system, such that it takes only quantized values. For static topological phases that are realized in ground-states of isolated, time-independent systems, the topological invariant is related to the properties of the Hamiltonian operator. In contrast, Floquet topological phases that are realized in open systems with periodical pumping of energy are topologically characterized with a unitary Floquet operator i.e., the time-evolution operator over the entire period. Topological phases of matter can be distinguished by the dimensionality of robust boundary states with respect to the protecting bulk. This dissertation concerns recently discovered higher-order topological phases where the difference between dimensionalities of bulk and boundary states is larger than one. Using analytical and numerical single-particle techniques, we focus on instances where static higher-order topology can be understood with insights from the mature field of Floquet topology. Namely, even though static systems do not admit a Floquet description, we find examples of higher-order systems to which certain unitary operators can be attributed. The understanding of topological characteristics of these systems is therefore conditioned by the knowledge on topological properties of unitary operators, among which the Floquet operator is well-known. The first half of this thesis concerns toy models of static higher-order topological phases that are topologically characterized in terms of unitary operators. We find that a class of these systems called quadrupole topological insulators exhibit a wider range of topological phases than known previously. In the second half of this dissertation, we study reflection matrices of higher-order topological phases and show that they can exhibit the same topological features as Floquet systems. Our findings suggest a new route to experimental realizations of Floquet systems, the one that avoids noise-induced decoherence inevitable in many other experimental setups.