Strongly-correlated phases in trapped-ion quantum simulators

We study quantum (T = 0) phases of strongly-correlated matter, and their possible implementation in a quantum simulator. We focus on the non-perturbative regimes of 1D spin-boson models. As a reference physical system we consider trapped-ion chains. We realize complex many-body states, such as a ground state exhibiting magnetic frustration, a lattice gauge theory, and a topological insulator. The exquisite control over these phases offered by a quantum simulator opens up exciting possibilities for exploring the exotic phenomena emerging in these systems, such as enhanced fluctuations and correlations. We address the non-perturbative regimes of the phase diagrams by means of mean-field theories and the numerical algorithm DMRG. We have established the universality class of the continuous transition in the spin-boson chain, the existence of a first order phase transition when the system is endowed with a gauge symmetry, and the possibility of probing topological states of matter in these systems. Our results show that some of the most exotic phases of quantum matter can be readily realized in trapped-ion quantum simulators. This offers the possibility of exploring these physical models beyond their original realm of applicability, which may provide us with new insights on both theoretical and applied fields of physics, ranging from high-energy processes to low-energy cooperative phenomena

Lindley-type recursions

In dit proefschrift staat de volgende Lindley-achtige recursie centraal: Wn+1 = max{0,Bn+1 - An -Wn}. (1) Deze "niet-stijgende" recursie is belangrijk in de analyse van systemen waarbij een bediende alterneert tussen twee bedieningsstations. Een station biedt ruimte voor ´e´en klant. De bediende alterneert tussen beide stations en bediend ´e´en klant per keer. Aangenomen wordt dat voortdurend bij beide stations klanten staan te wachten. Zodra een wachtende klant een station betreed, begint de eerste fase van zijn bediening, die bestaat uit een voorbereidende fase. De bediende is hier niet bij betrokken: pas nadat de voorbereidende fase is afgerond kan een klant aan de tweede fase van zijn bediening beginnen, welke wordt uitgevoerd door de bediende. Dus de eigenlijke bediening bestaat alleen uit de tweede fase. Het kan voorkomen dat de bediende moet wachten totdat de voorbereiding van de volgende klant is afgelopen. We zijn dan ook ge¨interesseerd in de wachttijd van de bediende. Als Bn de voorbereidingstijd is voor de n-de klant en An de bedieningstijd is van de n-de klant, dan kan de wachttijd van de bediende voor de (n + 1)-ste klant beschreven worden door middel van Recursie (1). Een belangrijke observatie is dat deze recursie vrijwel identiek is aan Lindley’s recursie. Het enige verschil is het min-teken voor Wn. Dit model is gemotiveerd door diverse toepassingen waarvan er twee worden besproken in Hoofdstuk 1. De eerste toepassing betreft oog-operaties. De tweede toepassing is gerelateerd aan carousel systemen. Dit soort systemen zijn uitgebreid bestudeerd; Sectie 1.3 geeft een literatuuroverzicht. Verderop in dit hoofdstuk geven we een gedetailleerde modelbeschrijving en noemen we enkele verschillen tussen de analyse van dit model en het standaard wachtrijmodel. Hoofdstuk 2 bestudeert enkele algemene eigenschappen van Recursie (1), zoals de stabiliteit van het systeem, existentie van een evenwichtsverdeling, convergentie naar deze verdeling als n naar oneindig gaat en het staartgedrag en de covariantie functie van de verdeling van de wachttijd van de bediende. Een rode draad in dit proefschrift is de afleiding van de evenwichtsverdeling van de wachttijd van de bediende. In de volgende drie hoofdstukken leiden we deze verdeling af onder diverse aannames over de verdeling van de voorbereidingstijd en bedieningstijd van een generieke klant. We bestuderen gevallen die analoog zijn aan de klassieke M/G/1, G/PH/1 en PH/P/1 wachtrijmodellen, waarbij "P" staat voor polynomiale verdelingen. Ge¨inspireerd door de toepassingen van ons model, bekijken we enkele prestatiematen voor dit systeem, zoals de doorzet. Dit maakt een vergelijk met de prestatie van niet-alternerende systemen mogelijk. In Hoofdstuk 6 onderzoeken we methoden om de wachttijdverdeling te benaderen door de verdeling van de voorbereidingstijd of bedieningstijd te benaderen met een verdeling die exacte berekeningen mogelijk maakt. We beschrijven hoe zo’n verdel- ing kan worden gevonden en we geven een bovengrens voor de fout tussen de werkelijke wachttijdverdeling en zijn benadering. In alle voorgaande hoofdstukken hebben we aangenomen dat alle voorbereidingstijden en bedieningstijden onafhankelijk van elkaar zijn. In Hoofdstuk 7 laten we deze aanname vallen. We onderzoeken twee specifieke vormen van afhankelijkheid tussen deze variabelen. Voor beide vormen leiden we opnieuw de limietverdeling af van de wachttijd van de bediende. Hoofdstuk 8 analyseert een recursie welke een uitbreiding is van zowel Lindley’s recursie als (1). We bekijken, namelijk, de recursie Wn+1 = max{0,Bn+1 - An + YnWn}, met Yn een stochastische variabele die zowel de waarde 1 als -1 kan aannemen. Voor deze recursie onderzoeken we stabiliteit, en we berekenen de limietverdeling in twee specifieke gevallen, waarmee we de bestaande theorie voor Lindley’s recursie en Recursie (1) generaliseren. De analyse maakt duidelijk dat de technieken voor het analyseren van (1) en voor het analyseren Lindley’s recursie moeten worden gecombineerd. Diverse methoden om Lindley’s recursie te analyseren zijn ook nuttig voor de analyse van (1). Wanneer we aannemen dat de voorbereidingstijd een fase-type verdeling heeft, dan reduceert de analyse van (1) tot de analyse van een Markovketen met eindige toestandsruimte. Ook kunnen Laplace-transformaties of Wiener- Hopf technieken in diverse gevallen worden toegepast (cf. Sectie 1.6). In andere gevallen moet een niet-standaard differentiaalvergelijking worden opgelost, of moet uitgeweken worden naar een iteratieve benadering van de wachttijdverdeling. In Hoofdstuk 5 dient ook een speciale klasse van verdelingen ge¨introduceerd te worden die het mogelijk maakt om een Fredholm vergelijking op te lossen. In de meeste gevallen zijn de resultaten expliciet of kunnen worden weergegeven in termen van de oplossing van een lineair stelsel vergelijkingen, zie bijvoorbeeld Stelling 4.8. Het proefschrift wordt afgesloten met enkele afsluitende opmerkingen en diverse suggesties voor verder onderzoek

Discrete Breathers in One- and Two-Dimensional Lattices

Discrete breathers are time-periodic and spatially localised exact solutions in translationally invariant nonlinear lattices. They are generic solutions, since only moderate conditions are required for their existence. Closed analytic forms for breather solutions are generally not known. We use asymptotic methods to determine both the properties and the approximate form of discrete breather solutions in various lattices. We find the conditions for which the one-dimensional FPU chain admits breather solutions, generalising a known result for stationary breathers to include moving breathers. These conditions are verified by numerical simulations. We show that the FPU chain with quartic interaction potential supports long-lived waveforms which are combinations of a breather and a kink. The amplitude of classical monotone kinks is shown to have a nonzero minimum, whereas the amplitude of breathing-kinks can be arbitrarily small. We consider a two-dimensional FPU lattice with square rotational symmetry. An analysis to third-order in the wave amplitude is inadequate, since this leads to a partial differential equation which does not admit stable soliton solutions for the breather envelope. We overcome this by extending the analysis to higher-order, obtaining a modified partial differential equation which includes known stabilising terms. From this, we determine regions of parameter space where breather solutions are expected. Our analytic results are supported by extensive numerical simulations, which suggest that the two-dimensional square FPU lattice supports long-lived stationary and moving breather modes. We find no restriction upon the direction in which breathers can travel through the lattice. Asymptotic estimates for the breather energy confirm that there is a minimum threshold energy which must be exceeded for breathers to exist in the two-dimensional lattice. We find similar results for a two-dimensional FPU lattice with hexagonal rotational symmetry

Convex Geometry and its Applications (hybrid meeting)

The geometry of convex domains in Euclidean space plays a central role in several branches of mathematics: functional and harmonic analysis, the theory of PDE, linear programming and, increasingly, in the study of algorithms in computer science. The purpose of this meeting was to bring together researchers from the analytic, geometric and probabilistic groups who have contributed to these developments

Numerical Study of Flow and Heat Transfer in Rotating Microchannels

Investigation of fluid flow and heat transfer in rotating microchannels is important for centrifugal microfluidics, which has emerged as an advanced technique in biomedical applications and chemical separations. The centrifugal force and the Coriolis force, arising as a consequence of the microchannel rotation, change the flow pattern significantly from the symmetric profile of a non-rotating channel. A successful design of a centrifugal microfluidic device depends on effectively regulating these forces in rotating microchannels. Although a large number of experimental studies have been performed in order to demonstrate the applications of centrifugal microfluidics in various fields, a systematic study on the effect of rotation, channel aspect ratio, and wall boundary conditions on the fluid flow and heat transfer phenomena in rotating microchannels has not yet been conducted. During the present study, pressure-based finite volume solvers in both staggered and collocated grids were developed to solve steady and unsteady, incompressible Navier-Stokes equations. The serial solver in collocated grid was parallelized using a Message Passing Interface (MPI) library. In order to accelerate the convergence of the collocated finite volume solver, a non-linear multi-grid method was developed. The parallel performances of the single and multi-grid solvers were tested on a two-dimensional lid driven cavity flow. High fidelity benchmark solution to a lid driven cavity flow problem in a 1024 x 1024 grid was presented for a range of Reynolds numbers. Parallel multigrid speedup as high as three orders of magnitude was achieved for low Reynolds number flows. In addition, the optimal multigrid efficiency was validated. The fluid flow in a rotating microchannel was modeled as a steady, laminar in compressible flow with no slip and slip boundary conditions. For no slip boundary condition, critical values of parameters that determine the extent of the centrifugal force and the Coriolis force were identified. The critical aspect ratio (=width/height) that causes the optimal mixing of two liquids was found to be 1.0. For liquid slip boundary condition, the effect of rotation on liquid slip flow in rotating microchannels with hydrophobic and superhydrophobic surfaces was studied. New correlations for friction relation (fRe) as a function of slip length (λ) and rotational Reynolds number (Reω) were proposed. It was also found that, the liquid slip can increase or decrease the heat transfer depending on the secondary flow effect and the aspect ratio of the microchannel. The microscale effects, such as surface tension and contact angle boundary condition, were included in the modeled problem. A level set method was applied to incorporate these microscale effects, which will enable us to investigate the unsteady nature of the liquid meniscus during two-phase flow simulations

Experimental & Numerical Investigation of Pool Boiling on Engineered Surfaces with Integrated Thin-flim Temperature Sensors

The objective of this investigation is to measure and analyze surface temperature fluctuations in pool boiling. The surface temperature fluctuations were recorded on silicon surfaces with and without multi-walled carbon nanotubes (MWCNT). Novel Thin Film Thermocouples (TFT) are micro-fabricated on test substrates to measure surface temperatures. A dielectric liquid refrigerant (PF-5060) is used as test fluid. Both nucleate and lm boiling regimes are investigated for the silicon test substrates. Dynamics of nucleate boiling is investigated on the CNT coated substrates. High frequency temperature fluctuation data is analyzed for the presence of determinism using non-linear time series analysis techniques in TISEAN(copyright) software. The impact of subcooling and micro/nano-scale surface texturing using MWCNT coatings on the dynamics of pool boiling is assessed. Dynamic invariants such as correlation dimensions and Lyapunov spectrum are evaluated for the reconstructed attractor. A non-linear noise reduction scheme is employed to reduce the level of noise in the data. Previous investigations in pool boiling chaos, reported in literature were based on temperature measurements underneath the test surface consisting of single or few active nucleation sites. Previous studies have indicated the presence of low-dimensional behavior in nucleate boiling and high-dimensional behavior in CHF and film boiling. Currently, there is no study detailing the effects of multiple nucleation sites, subcooling and surface texturing on pool boiling dynamics. The investigation comprises of four parts: i) in situ micro-machining of Chromelalumel (K-type) TFT, ii) calibration of these sensors, iii) utilizing these sensors in pool boiling experiments iv) analysis of these fluctuations using techniques of nonlinear time series analysis. Ten TFT are fabricated on a rectangular silicon surface within an area of ~ 3.00 cm x 3.00 cm. The sensing junctions of the TFT measure 50 mm in width and 250 nm in depth. Surface temperature fluctuations of the order of i) 0.65-0.93 degrees C are observed near ONB ii) 2.3-6.5 degrees C in FDNB iii) 2.60-5.00 degrees C at CHF and iv) 2.3-3.5 degrees C in film boiling. Investigations show the possible presence of chaotic dynamics near CHF and in film-boiling in saturated and subcooled pool boiling. Fully-developed nucleate boiling (FDNB) is chaotic. No clear assessment of the dynamics could be made in the onset of nucleate boiling (ONB) and partial nucleate boiling (PNB) regimes due to the effects of noise. However, the frequency spectra in these regimes appear to have two independent frequencies and their integral combinations indicating a possible quasiperiodic bifurcation route to chaos. The dimensionality in FDNB, at CHF and in film-boiling is lower in saturated pool boiling as compared to values in corresponding regimes in subcooled pool boiling. Surface temperature fluctuations can damage electronic components and need to be carefully controlled. Understanding the nature of these fluctuations will aid in deciding the modeling approach for surface temperature transients on an electronic chip. Subsequently, the TFT signals can be employed in a suitable feedback control loop to prevent the occurrence of hotspots

Efficiency in audio processing : filter banks and transcoding

Audio transcoding is the conversion of digital audio from one compressed form A to another compressed form B, where A and B have different compression properties, such as a different bit-rate, sampling frequency or compression method. This is typically achieved by decoding A to an intermediate uncompressed form, and then encoding it to B. A significant portion of the involved computational effort pertains to operating the synthesis filter bank, which is an important processing block in the decoding stage, and the analysis filter bank, which is an important processing block in the encoding stage. This thesis presents methods for efficient implementations of filter banks and audio transcoders, and is separated into two main parts. In the first part, a new class of Frequency Response Masking (FRM) filter banks is introduced. These filter banks are usually characterized by comprising a tree-structured cascade of subfilters, which have small individual filter lengths. Methods of complexity reduction are proposed for the scenarios when the filter banks are operated in single-rate mode, and when they are operated in multirate mode; and for the scenarios when the input signal is real-valued, and when it is complex-valued. An efficient variable bandwidth FRM filter bank is designed by using signed-powers-of-two reduction of its subfilter coefficients. Our design has a complexity an order lower than that of an octave filter bank with the same specifications. In the second part, the audio transcoding process is analyzed. Audio transcoding is modeled as a cascaded quantization process, and the cascaded quantization of an input signal is analyzed under different conditions, for the MPEG 1 Layer 2 and MP3 compression methods. One condition is the input-to-output delay of the transcoder, which is known to have an impact on the audio quality of the transcoded material. Methods to reduce the error in a cascaded quantization process are also proposed. An ultra-fast MP3 transcoder that requires only integer operations is proposed and implemented in software. Our implementation shows an improvement by a factor of 5 to 16 over other best known transcoders in terms of execution speed