Error Exponents of Low-Density Parity-Check Codes on the Binary Erasure Channel

We introduce a thermodynamic (large deviation) formalism for computing error exponents in error-correcting codes. Within this framework, we apply the heuristic cavity method from statistical mechanics to derive the average and typical error exponents of low-density parity-check (LDPC) codes on the binary erasure channel (BEC) under maximum-likelihood decoding.Comment: 5 pages, 4 figure

A Model for the Generation and Transmission of Variations in Evolution

The inheritance of characteristics induced by the environment has often been opposed to the theory of evolution by natural selection. Yet, while evolution by natural selection requires new heritable traits to be produced and transmitted, it does not prescribe, per se, the mechanisms by which this is operated. The mechanisms of inheritance are not, however, unconstrained, since they are themselves subject to natural selection. We introduce a general, analytically solvable mathematical model to compare the adaptive value of different schemes of inheritance. Our model allows for variations to be inherited, randomly produced, or environmentally induced, and, irrespectively, to be either transmitted or not during reproduction. The adaptation of the different schemes for processing variations is quantified for a range of fluctuating environments, following an approach that links quantitative genetics with stochastic control theory

Evolution of sparsity and modularity in a model of protein allostery

The sequence of a protein is not only constrained by its physical and biochemical properties under current selection, but also by features of its past evolutionary history. Understanding the extent and the form that these evolutionary constraints may take is important to interpret the information in protein sequences. To study this problem, we introduce a simple but physical model of protein evolution where selection targets allostery, the functional coupling of distal sites on protein surfaces. This model shows how the geometrical organization of couplings between amino acids within a protein structure can depend crucially on its evolutionary history. In particular, two scenarios are found to generate a spatial concentration of functional constraints: high mutation rates and fluctuating selective pressures. This second scenario offers a plausible explanation for the high tolerance of natural proteins to mutations and for the spatial organization of their least tolerant amino acids, as revealed by sequence analyses and mutagenesis experiments. It also implies a faculty to adapt to new selective pressures that is consistent with observations. Besides, the model illustrates how several independent functional modules may emerge within a same protein structure, depending on the nature of past environmental fluctuations. Our model thus relates the evolutionary history and evolutionary potential of proteins to the geometry of their functional constraints, with implications for decoding and engineering protein sequences

Gallium phosphide photonic crystal nanocavities in the visible

Photonic crystal nanocavities at visible wavelengths are fabricated in a high refractive index (n>3.2) gallium phosphide membrane. The cavities are probed via a cross-polarized reflectivity measurement and show resonances at wavelengths as low as 645 nm at room temperature, with quality factors between 500 and 1700 for modes with volumes 0.7(λ/n)^3. These structures could be employed for submicron scale optoelectronic devices in the visible, and for coupling to emitters with resonances in the visible such as nitrogen vacancy centers, and biomolecules and organic molecules

Cavity-enhanced second harmonic generation via nonlinear-overlap optimization

We describe an approach based on topology optimization that enables automatic discovery of wavelength-scale photonic structures for achieving high-efficiency second-harmonic generation (SHG). A key distinction from previous formulation and designs that seek to maximize Purcell factors at individual frequencies is that our method not only aims to achieve frequency matching (across an entire octave) and large radiative lifetimes, but also optimizes the equally important nonlinear--coupling figure of merit $\bar{\beta}$, involving a complicated spatial overlap-integral between modes. We apply this method to the particular problem of optimizing micropost and grating-slab cavities (one-dimensional multilayered structures) and demonstrate that a variety of material platforms can support modes with the requisite frequencies, large lifetimes $Q > 10^4$, small modal volumes $\sim (\lambda/n)^3$, and extremely large $\bar{\beta} \gtrsim 10^{-2}$, leading to orders of magnitude enhancements in SHG efficiency compared to state of the art photonic designs. Such giant $\bar{\beta}$ alleviate the need for ultra-narrow linewidths and thus pave the way for wavelength-scale SHG devices with faster operating timescales and higher tolerance to fabrication imperfections

Multiply resonant high quality photonic crystal nanocavities

We propose and experimentally demonstrate a photonic crystal nanocavity with multiple resonances that can be tuned nearly independently. The design is composed of two orthogonal intersecting nanobeam cavities. Experimentally, we measure cavity quality factors of 6,600 and 1000 for resonances separated by 382 nm; we measure a maximum separation between resonances of 506 nm. These structures are promising for enhancing efficiency in nonlinear optical processes such as sum/difference frequency and stimulated Raman scattering.Comment: 3 figure