<p>The development of ultracold atom physics enables people to study fundamental questions in quantum mechanics within this highly-tunable platform. This dissertation focuses on several topics of the dynamical evolution of quantum systems.</p><p>Chapter 2 and 3 talk about Loschmidt echo, a simple quantity that reveals many hidden properties of a system’s time evolution. Chapter 2 looks for vanishing Loschmidt echo in the complex plane of time and the corresponding dynamical quantum phase transitions (DQPT) in the thermodynamic limit. For a two-site Bose-Hubbard model consisting of weakly interacting particles, DQPTs reside at the time scale inversely proportional to the interaction, where highly entangled pair condensates also show up. Chapter 3 discusses the revival of Loschmidt echo in a discrete time crystal, a Floquet system whose discrete temporal transition symmetry is spontaneously broken. We propose a new design and demonstrate its robustness against the fluctuations in the driving field. It can also be used in precision measurement to go beyond the Heisenberg limit. Experimental schemes are presented.</p><p>Out-of-time-order correlator (OTOC) is a more complicated variant of Loschmidt echo. Experimentally it requires reversing the time evolution. In Chapter 4, by exploiting the SU(1,1) symmetry of a weakly interacting BEC and connecting its quantum dynamics to a hyperbolic space, we obtain a geometric framework that enables experimentalists to manipulate the evolution with great freedom. Backward evolution is then realized effectively to measure OTOC of such SU(1,1) systems.</p><p>Chapter 5 discusses the decoherence of a spin impurity immersed in a spinor BEC. Our calculations show that by looking at the dynamics of the impurity’s reduced density matrix, the phase of the spinor BEC can be detected.</p>

This dissertation introduces some new properties of both superfluid phases of fermions with spin-orbit coupling (SOC) and superradiant phases of photons in an optical cavity. The effects of SOC on the phase transition between normal and superfluid phase are revealed; an unconventional crossover driven by SOC from the Bardeen-Cooper-Schrieffer (BCS) state to the Bose-Einstein condensate (BEC) state is verified in three different systems; and two kinds of excitations, a Goldstone mode and a Higgs mode, are demonstrated to occur in a quantum optical system. We investigate the BCS superfluid state of two-component atomic Fermi gases in the presence of three kinds of SOCs. We find that SOC drives a class of BCS to BEC crossover that is different from the conventional one without SOC. Here, we extend the concepts of the coherence length and Cooper-pair size in the absence of SOC to Fermi systems with SOC. We study the dependence of chemical potential, coherence length, and Cooper-pair size on the SOC strength and the scattering length in three dimensions (3D) (or the twobody binding energy in two dimensions (2D)) for three attractively interacting Fermi gases with 3D Rashba, 3D Weyl, and 2D Rashba SOC respectively. By adding a population imbalance to a Fermi gas with Rashba-type SOC, we also map out the finite-temperature phase diagram. Due to a competition between SOC and population imbalance, the finite-temperature phase diagram reveals a large variety of new features, including the expanding of the superfluid state regime and the shrinking of both the phase separation and the normal regimes. We find that the tricritical point moves toward a regime of low temperature, high magnetic field, and high polarization as the SOC strength increases. Besides Fermi fluids, this dissertation also gives a new angle of view on the superradiant phase in the Dicke model. Here, we demonstrate that Goldstone and Higgs modes can be observed in an optical system with only a few atoms inside a cavity. The model we study is the U(1)/Z2 Dicke model with N qubits (two-level atoms) coupled to a single photon mode.

My dissertation, In the Beginning was the Sign, examines the entangled histories of literary modernism and mathematical modernity and revisits their claim to a radical rupture with the past. Informed by Lacanian psychoanalysis, media theory, and deconstruction, I trace how the interplay of literary and mathematical form transformed classical imaginations of the human. Authors like Carl Einstein, Robert Musil or Ernst Cassirer challenge the organic concept of subject formation as Bildung with a new and purely symbolic kind of mathematical abstraction that informs their writing on both thematic and formal levels. In the tradition of Plato’s Meno, they adduce these new forms of mathematical knowledge to find genuinely modern answers to the classical question of the good life. Paradoxically, in striving to portray their own time as a radical novelty that was able to break with its cultural heritage, these authors summon the canon at its most canonical. The mathematician Hilbert, for instance, rewrites the opening of the Gospel of John, translating logos as ‘sign’ rather than ‘word.’ Analyzing literary, philosophical, and mathematical texts in German, English, and French, I show that the questioning of the logical foundations of thought in the so-called foundational crisis in mathematics was re-mediated through a new genealogical exploration of the foundations of European rationality in the texts of classical antiquity.

We show that Einstein’s gravitational field equations for the Friedmann- Robertson-Lemaître-Walker (FRLW) and for two conformal versions of the Bianchi I and Bianchi V perfect fluid scalar field cosmological models, can be equivalently reformulated in terms of a single equation of either generalized Ermakov-Milne- Pinney (EMP) or (non)linear Schrödinger (NLS) type. This work generalizes or presents an alternative to similar reformulations published by the authors who inspired this thesis: R. Hawkins, J. Lidsey, T. Christodoulakis, T. Grammenos, C. Helias, P. Kevrekidis, G. Papadopoulos and F.Williams. In particular we cast much of these authors’ works into a single framework via straightforward derivations of the EMP and NLS equations from a simple linear combination of the relevant Einstein equations. By rewriting the resulting expression in terms of the volume expansion factor and performing a change of variables, we obtain an uncoupled EMP or NLS equation that is independent of the imposition of additional conservation equations. Since the correspondences shown here present an alternative route for obtaining exact solutions to Einstein’s equations, we reconstruct many known exact solutions via their EMP or NLS counterparts and show by numerical analysis the stability properties of many solutions.

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A prescription is given to obtain some exact results for certain external potentials �� (r) of the time-independent Gross-Pitaevskii and Schrodinger equations. The study motivation is the ability to program �� (r) experimentally in cold atom Bose-Einstein condensates. Rather than derive wavefunctions that are solutions for a given �� (r), we ask which �� (r) will have a given pdf (probability density function) �� (r). Several examples in 1 dimension (1D), 2 dimensions (2D), and 3 dimensions (3D) are presented for well-known pdfs in the position space. Exact potentials with zero, one and two walls are obtained and explained in detail. Apart from position space, the method is also applicable to obtain exact solutions for the Time-independent Schr¨odinger equation (TISE) and Gross-Pitaevskii equation (GPeq) for pdfs in momentum space. For this, we derived the potentials which are generated from the pdfs of the hydrogen atom in the real space as well as in the momentum space. However, the method was also extended for the time-dependent case. The prescription is also applicable to solve time-dependent pdfs. The aim is to find the ��(r, ��) which generates the pdf ��(r, ��). As a special case, we tested our method by studying the well known case for the Gaussian wave packet in 1D with zero potential ��(��, ��) = 0.

No description available.

In this doctoral thesis, we theoretically investigate the propagation of sound waves in dilute Bose gases, in both the collisionless and hydrodynamic regimes. The study of sound wave is a topic of high relevance for the understanding of dynamical properties of any fluid, classical or quantum, and further provides insightful information about the equation of state of the system. In our work, we focus in particular on the two-dimensional (2D) Bose gas, in which the sound wave is predicted to give useful information about the nature of the superfluid phase transition. Recently, experimental measurement of sound wave in a uniform 2D Bose gas has become available, and we show that the measured data are quantitatively well explained by our collisionless theory. Finally, we study the mixtures of weakly interacting Bose gases, by developing a beyond mean-field theory, which includes the effects of thermal and quantum fluctuations in both the density and spin channels. Our new theory allows for the investigation of sound dynamics, as well as the fundamental problem of phase- separation.

Understanding molecular and ion dynamics in soft materials used for fuel cell, battery, and drug delivery vehicle applications on multiple time and length scales provides critical information for the development of next generation materials. In this dissertation, new insights into transport and kinetic processes such as diffusion coefficients, translational activation energies (Ea), and rate constants for molecular exchange, as well as how these processes depend on material chemistry and morphology are shown. This dissertation also aims to serve as a guide for material scientists wanting to expand their research capabilities via nuclear magnetic resonance (NMR) techniques. By employing variable temperature pulsed-field-gradient (PFG) NMR diffusometry, which can probe molecular transport over nm – μm length scales, I first explore transport and morphology on a series of ion-conducting materials: an organic ionic plastic crystal, a proton-exchange membrane, and a polymer-gel electrolyte. These studies show the dependencies of small molecule and ion transport on modulations to material parameters, including thermal or magnetic treatment, water content, and/or crosslink density. I discuss the fundamental significance of the length scale over which translational Ea reports on these systems (~ 1 nm) and the resulting implications for using the Arrhenius equation parameters to understand and rationally design new ion-conductors. Next, I describe how NMR spectroscopy can be utilized to investigate the effect of loading a small molecule into the core of a spherical block copolymer micelle (to mimic, e.g., drug loading) on the hydrodynamic radius (rH) and polymer chain dynamics. In particular, I present spin-lattice relaxation (T1) results that directly measure single chain exchange rate kexch between micelles and diffusion results that inform on the unimer exchange mechanism. These convenient NMR methods thus offer an economical alternative (or complement) to time-resolved small angle neutron scattering (TR-SANS). / Ph. D.

In 1981, Unruh suggested the possibility of simulating the dynamics of quantum fields in curved spacetimes using sound-waves propagating in moving fluids: a supersonic flow would indeed influence the dynamics of sound similarly to what happens to light when it’s dragged by the spacetime geometry in strong gravity environments. This simple yet groundbreaking observation has lead to the beginning of a whole new field of research, nowadays known as Analog Gravity. Due to their superfluid character, intrinsic quantum nature and impressive experimental tunability, Bose-Einstein condensates represent one of the most promising platforms to realize analog spacetimes, including black-hole geometries with horizons and ergoregions, as well as of time-dependent configurations relevant to cosmology. In this Thesis we go beyond the standard single-component BEC and focus on two-component mixtures of atomic condensates, possibly in the presence of a coherent coupling between the two-components: the availability of various branches of elementary excitations with different sound speed and effective mass may in fact lead to advantages in the implementation of interesting geometries and, eventually, to the exploration of a broader spectrum of physical processes. We first consider black-hole related phenomena (Hawking radiation and rotational superradiance) that have already been analysed with single-component systems, generalising the results to mixtures; we then proceed to tackle a problem (the decay from the false vacuum) which instead requires the additional degrees of freedom that only a mixture displays.