Nonlinear optics and wavelength translation via cavity-optomechanics

The field of cavity-optomechanics explores the interaction of light with sound in an ever increasing array of devices. This interaction allows the mechanical system to be both sensed and controlled by the optical system, opening up a wide variety of experiments including the cooling of the mechanical resonator to its quantum mechanical ground state and the squeezing of the optical field upon interaction with the mechanical resonator, to name two.

In this work we explore two very different systems with different types of optomechanical coupling. The first system consists of two microdisk optical resonators stacked on top of each other and separated by a very small slot. The interaction of the disks causes their optical resonance frequencies to be extremely sensitive to the gap between the disks. By careful control of the gap between the disks, the optomechanical coupling can be made to be quadratic to first order which is uncommon in optomechanical systems. With this quadratic coupling the light field is now sensitive to the energy of the mechanical resonator and can directly control the potential energy trapping the mechanical motion. This ability to directly control the spring constant without modifying the energy of the mechanical system, unlike in linear optomechanical coupling, is explored.

Next, the bulk of this thesis deals with a high mechanical frequency optomechanical crystal which is used to coherently convert photons between different frequencies. This is accomplished via the engineered linear optomechanical coupling in these devices. Both classical and quantum systems utilize the interaction of light and matter across a wide range of energies. These systems are often not naturally compatible with one another and require a means of converting photons of dissimilar wavelengths to combine and exploit their different strengths. Here we theoretically propose and experimentally demonstrate coherent wavelength conversion of optical photons using photon-phonon translation in a cavity-optomechanical system. For an engineered silicon optomechanical crystal nanocavity supporting a 4 GHz localized phonon mode, optical signals in a 1.5 MHz bandwidth are coherently converted over a 11.2 THz frequency span between one cavity mode at wavelength 1460 nm and a second cavity mode at 1545 nm with a 93% internal (2% external) peak efficiency. The thermal and quantum limiting noise involved in the conversion process is also analyzed and, in terms of an equivalent photon number signal level, are found to correspond to an internal noise level of only 6 and 4 times 10x^-3 quanta, respectively.

We begin by developing the requisite theoretical background to describe the system. A significant amount of time is then spent describing the fabrication of these silicon nanobeams, with an emphasis on understanding the specifics and motivation. The experimental demonstration of wavelength conversion is then described and analyzed. It is determined that the method of getting photons into the cavity and collected from the cavity is a fundamental limiting factor in the overall efficiency. Finally, a new coupling scheme is designed, fabricated, and tested that provides a means of coupling greater than 90% of photons into and out of the cavity, addressing one of the largest obstacles with the initial wavelength conversion experiment.

The combinatorics of transcriptional regulation

The ability to regulate gene expression is of central importance for the adaptability of living organisms to changes in their internal and external environment. At the transcriptional level, binding of transcription factors (TFs) in the vicinity of promoters can modulate the rate at which transcripts are produced, and as such play an important role in gene regulation. TFs with regulatory action at multiple promoters is the rule rather than the exception, with examples ranging from TFs like the cAMP receptor protein (CRP) in E. coli that regulates hundreds of different genes, to situations involving multiple copies of the same gene, such as on plasmids, or viral DNA. When the number of TFs heavily exceeds the number of binding sites, TF binding to each promoter can be regarded as independent. However, when the number of TF molecules is comparable to the number of binding sites, TF titration will result in coupling ("entanglement") between transcription of different genes. The last few decades have seen rapid advances in our ability to quantitatively measure such effects, which calls for biophysical models to explain these data. Here we develop a statistical mechanical model which takes the TF titration effect into account and use it to predict both the level of gene expression and the resulting correlation in transcription rates for a general set of promoters. To test these predictions experimentally, we create genetic constructs with known TF copy number, binding site affinities, and gene copy number; hence avoiding the need to use free fit parameters. Our results clearly prove the TF titration effect and that the statistical mechanical model can accurately predict the fold change in gene expression for the studied cases. We also generalize these experimental efforts to cover systems with multiple different genes, using the method of mRNA fluorescence in situ hybridization (FISH). Interestingly, we can use the TF titration affect as a tool to measure the plasmid copy number at different points in the cell cycle, as well as the plasmid copy number variance. Finally, we investigate the strategies of transcriptional regulation used in a real organism by analyzing the thousands of known regulatory interactions in E. coli. We introduce a "random promoter architecture model" to identify overrepresented regulatory strategies, such as TF pairs which coregulate the same genes more frequently than would be expected by chance, indicating a related biological function. Furthermore, we investigate whether promoter architecture has a systematic effect on gene expression by linking the regulatory data of E. coli to genome-wide expression censuses.