Spectrum of single-photon emission and scattering in cavity optomechanics

We present an analytic solution describing the quantum state of a single photon after interacting with a moving mirror in a cavity. This includes situations when the photon is initially stored in a cavity mode as well as when the photon is injected into the cavity. In addition, we obtain the spectrum of the output photon in the resolved-sideband limit, which reveals spectral features of the single-photon strong-coupling regime in this system. We also clarify the conditions under which the phonon sidebands are visible and the photon-state frequency shift can be resolved.Comment: 5 pages, 5 figure

Data Assimilation: A Mathematical Introduction

These notes provide a systematic mathematical treatment of the subject of data assimilation

Non-Markovian master equation for a damped oscillator with time-varying parameters

We derive an exact non-Markovian master equation that generalizes the previous work [Hu, Paz and Zhang, Phys. Rev. D {\bf 45}, 2843 (1992)] to damped harmonic oscillators with time-varying parameters. This is achieved by exploiting the linearity of the system and operator solution in Heisenberg picture. Our equation governs the non-Markovian quantum dynamics when the system is modulated by external devices. As an application, we apply our equation to parity kick decoupling problems. The time-dependent dissipative coefficients in the master equation are shown to be modified drastically when the system is driven by $\pi$ pulses. For coherence protection to be effective, our numerical results indicate that kicking period should be shorter than memory time of the bath. The effects of using soft pulses in an ohmic bath are also discussed

Do early-life exposures explain why more advantaged children get eczema? Findings from the U.K. Millennium Cohort Study

Background: Atopic dermatitis (eczema) in childhood is socially patterned, with higher incidence in more advantaged populations. However, it is unclear what factors explain the social differences. Objectives: To identify early-life risk factors for eczema, and to explore how early-life risk factors explain any differences in eczema. Methods: We estimated odds ratios (ORs) for ever having had eczema by age 5 years in 14 499 children from the U.K. Millennium Cohort Study (MCS), with a focus on maternal, antenatal and early-life risk factors and socioeconomic circumstances (SECs). Risk factors were explored to assess whether they attenuated associations between SECs and eczema. Results: Overall 35·1% of children had ever had eczema by age 5 years. Children of mothers with degree-level qualifications vs. no educational qualifications were more likely to have eczema (OR 1·52, 95% confidence interval 1·31–1·76), and there was a gradient across the socioeconomic spectrum. Maternal atopy, breastfeeding (1–6 weeks and ≥ 6 months), introduction of solids under 4 months or cow's milk under 9 months, antibiotic exposure in the first year of life and grime exposure were associated with an increased odds of having eczema. Female sex, Pakistani and Bangladeshi ethnicity, smoking during pregnancy, exposure to environmental tobacco smoke and having more siblings were associated with reduced odds for eczema. Controlling for maternal, antenatal and early-life characteristics (particularly maternal smoking during pregnancy, breastfeeding and number of siblings) reduced the OR for eczema to 1·26 (95% confidence interval 1·03–1·50) in the group with the highest educational qualifications compared with the least. Conclusions: In a representative U.K. child cohort, eczema was more common in more advantaged children. This was explained partially by early-life factors including not smoking during pregnancy, breastfeeding and having fewer siblings