Solitons in Exciton-Polariton System: Reduced Modelling, Analysis, and Numerical Studies

In this dissertation, we study the behavior of exciton-polariton quasiparticles in semiconductor microcavities, under the sourceless and lossless conditions.

First, we simplify the original model by removing the photon dispersion term, thus effectively turn the PDEs system to an ODEs system,

and investigate the behavior of the resulting system, including the equilibrium points and the wave functions of the excitons and the photons.

Second, we add the dispersion term for the excitons to the original model and prove that the band of the discontinuous solitons now become dark solitons.

Third, we employ the Strang-splitting method to our sytem of PDEs and prove the first-order and second-order error bounds in the $H^1$ norm and the $L_2$ norm, respectively.

Using this numerical result, we analyze the stability of the steady state bright soliton solution. This solution revolves around the $x$-axis as time progresses

and the perturbed soliton also rotates around the $x$-axis and tracks closely in terms of amplitude but lags behind the exact one. Our numerical result shows orbital

stability but no $L_2$ stability.

Bound on quantum computation time: Quantum error correction in a critical environment

We obtain an upper bound on the time available for quantum computation for a given quantum computer and decohering environment with quantum error correction implemented. First, we derive an explicit quantum evolution operator for the logical qubits and show that it has the same form as that for the physical qubits but with a reduced coupling strength to the environment. Using this evolution operator, we find the trace distance between the real and ideal states of the logical qubits in two cases. For a super-Ohmic bath, the trace distance saturates, while for Ohmic or sub-Ohmic baths, there is a finite time before the trace distance exceeds a value set by the user. © 2010 The American Physical Society.

Structural manifestation of the delocalization error of density functional approximations: C(4N+2) rings and C(20) bowl, cage, and ring isomers.

The ground state structure of C(4N+2) rings is believed to exhibit a geometric transition from angle alternation (N < or = 2) to bond alternation (N > 2). All previous density functional theory (DFT) studies on these molecules have failed to reproduce this behavior by predicting either that the transition occurs at too large a ring size, or that the transition leads to a higher symmetry cumulene. Employing the recently proposed perspective of delocalization error within DFT we rationalize this failure of common density functional approximations (DFAs) and present calculations with the rCAM-B3LYP exchange-correlation functional that show an angle-to-bond-alternation transition between C(10) and C(14). The behavior exemplified here manifests itself more generally as the well known tendency of DFAs to bias toward delocalized electron distributions as favored by Huckel aromaticity, of which the C(4N+2) rings provide a quintessential example. Additional examples are the relative energies of the C(20) bowl, cage, and ring isomers; we show that the results from functionals with minimal delocalization error are in good agreement with CCSD(T) results, in contrast to other commonly used DFAs. An unbiased DFT treatment of electron delocalization is a key for reliable prediction of relative stability and hence the structures of complex molecules where many structure stabilization mechanisms exist.

Error recovery in cyberphysical digital microfluidic biochips

Droplet-based digital microfluidics technology has now come of age, and software-controlled biochips for healthcare applications are starting to emerge. However, today's digital microfluidic biochips suffer from the drawback that there is no feedback to the control software from the underlying hardware platform. Due to the lack of precision inherent in biochemical experiments, errors are likely during droplet manipulation; error recovery based on the repetition of experiments leads to wastage of expensive reagents and hard-to-prepare samples. By exploiting recent advances in the integration of optical detectors (sensors) into a digital microfluidics biochip, we present a physical-aware system reconfiguration technique that uses sensor data at intermediate checkpoints to dynamically reconfigure the biochip. A cyberphysical resynthesis technique is used to recompute electrode-actuation sequences, thereby deriving new schedules, module placement, and droplet routing pathways, with minimum impact on the time-to-response. © 2012 IEEE.

Cross-Domain Multitask Learning with Latent Probit Models

Learning multiple tasks across heterogeneous domains is a challenging problem since the feature space may not be the same for different tasks. We assume the data in multiple tasks are generated from a latent common domain via sparse domain transforms and propose a latent probit model (LPM) to jointly learn the domain transforms, and the shared probit classifier in the common domain. To learn meaningful task relatedness and avoid over-fitting in classification, we introduce sparsity in the domain transforms matrices, as well as in the common classifier. We derive theoretical bounds for the estimation error of the classifier in terms of the sparsity of domain transforms. An expectation-maximization algorithm is derived for learning the LPM. The effectiveness of the approach is demonstrated on several real datasets.

Hybrid error correction and de novo assembly of single-molecule sequencing reads.

Single-molecule sequencing instruments can generate multikilobase sequences with the potential to greatly improve genome and transcriptome assembly. However, the error rates of single-molecule reads are high, which has limited their use thus far to resequencing bacteria. To address this limitation, we introduce a correction algorithm and assembly strategy that uses short, high-fidelity sequences to correct the error in single-molecule sequences. We demonstrate the utility of this approach on reads generated by a PacBio RS instrument from phage, prokaryotic and eukaryotic whole genomes, including the previously unsequenced genome of the parrot Melopsittacus undulatus, as well as for RNA-Seq reads of the corn (Zea mays) transcriptome. Our long-read correction achieves >99.9% base-call accuracy, leading to substantially better assemblies than current sequencing strategies: in the best example, the median contig size was quintupled relative to high-coverage, second-generation assemblies. Greater gains are predicted if read lengths continue to increase, including the prospect of single-contig bacterial chromosome assembly.