The Pancharatnam–Berry phase was discovered by Pancharatnam in studies of polarized light and introduced by Berry as a topological phase for matter wave functions. For light, the Pancharatnam–Berry phase is measured in laser interferometers and exploited in optical elements. Excitons are matter waves that directly transform to photons inheriting their coherence and polarization. This makes excitons a unique interface between matter and light and a unique system for exploring the Pancharatnam–Berry phase for matter waves by light interference experiments. Recent studies led to the discovery of polarization textures in light emission of indirect excitons and exciton–polaritons. This connection of the Pancharatnam–Berry phase to polarization makes it an intrinsic phenomenon for polarization textures. An IX is a bound pair of an electron and a hole confined in spatially separated layers. IXs are realized in coupled quantum well structures. Due to their long lifetimes IXs can cool below the temperature of quantum degeneracy and form a condensate in momentum space. IX condensation is detected by measurement of IX spontaneous coherence with a coherence length much larger than in a classical gas. The large coherence length observed in an IX condensate, reaching ~10 μm, square flower bucket indicates coherent IX transport with suppressed scattering, in agreement with theory.

A cold IX gas is realized in the regions of the external ring and localized bright spot rings in the IX emission. These rings form on the boundaries of electron-rich and hole-rich regions created by current through the structure and optical excitation, respectively; see ref. and references therein. An LBS is a stable, well defined, and tunable source of cold IXs, thus an ideal system for studying coherence and polarization phenomena. Different LBS offer IX sources of different strength and spatial extension; furthermore, these parameters can be controlled by optical excitation and voltage. This variability gives the opportunity to measure correlations between coherence and polarization. Here, we explore LBS to uncover the Pancharatnam–Berry phase in a condensate of IXs.The experiment shows that the phase shifts correlate with the polarization pattern of IX emission and onset of IX spontaneous coherence. The correlation between the phase shift and the polarization change identifies the phase as the Pancharatnam–Berry phase acquired in a condensate of IXs. This phenomenon is discussed below. The spatial separation of an electron and a hole in an IX reduces the overlap of the electron and hole wave functions suppressing the spin relaxation mechanism due to electron–hole exchange. In a classical IX gas, spin transport in the studied structure is limited by 1−2 μm due to Dyakonov–Perel spin relaxation. As a result, for uncondensed IXs at r < rcoh, the spin relaxation is fast and coherent spin precession is not observed. However, the suppression of scattering in IX condensate results in the suppression of the Dyakonov–Perel and Elliott–Yafet mechanisms of spin relaxation enabling long-range coherent spin transport in IX condensate.

Therefore, IX condensation at r > rcoh dramatically enhances the spin relaxation time leading to coherent spin precession and, in turn, precession of the polarization state of IX emission. This precession generates the evolving Pancharatnam–Berry phase of IXs, which is detected as the shift of interference fringes. Figure 4d shows that no decay of the evolving Pancharatnam–Berry phase is observed over macroscopic lengths exceeding 10 μm. This indicates the achievement of macroscopic long-range coherent spin transport in the IX condensate.To demonstrate a concept of the Pancharatnam– Berry phase acquired due to the IX spin precession, we simulate the polarization evolution within the model of IX spin precession. We use the electron and hole spin–orbit interaction constants and splittings between four IX states obtained to fit the IX polarization patterns. The initial polarization in the simulations is taken as horizontal to follow the experiment . The simulated S1 component of Stokes’ vector corresponding to linear polarization of IX emission is presented in Fig. 5a, b. The polarization shows an oscillatory behaviour. Its long-scale component is responsible for the polarization pattern shown in Fig. 1e and studied earlier in ref.. The short-scale component has the spatial period ~0.3 μm and is not resolved with 1.5 μm optical resolution in the imaging experiment. However, these fast changes of the polarization state generate the evolving Pancharatnam–Berry phase of IXs. Figure 5c shows the simulated IX polarization state on the Poincaré sphere for one fast polarization oscillation cycle in Fig. 5a. The IX polarization state goes over a nearly closed contour on the Poincaré sphere.

The Pancharatnam–Berry phase acquired by IXs over this contour can be estimated by connecting the initial and final points and calculating half the solid angle subtended by the obtained contour at the centre of the sphere, Ω/2 . In turn, a momentum kPB associated with the acquired Pancharatnam–Berry phase can be estimated as ~Ω/ where l is the IX path passed during the polarization cycle. For Ω/2 ~ π/2 and l ~ 0.3 μm , this gives kPB ~ 5 μm−1, in qualitative agreement with the jump in IX momentum , which occurs when the coherent spin precession generating the evolving Pancharatnam–Berry phase starts in IX condensate. In summary, shift-interferometry and polarization imaging show that the phase shifts of interference fringes correlate with the polarization pattern of IX emission and onset of IX spontaneous coherence, demonstrating the Pancharatnam–Berry phase in a condensate of IXs. The measured Pancharatnam–Berry phase indicates long-range coherent spin transport.This dissertation presents three essays on agricultural productivity and its relationship with farm size and poverty. Chapter 1 addresses the relationship between farm size and productivity, a recurrent topic in development economics. We clarify the common productivity measures used in this literature, their relationships, and their advantages and limitations. Second, we argue that total factor productivity, not land productivity, is the appropriate indicator for most policy questions. Lastly, using a pseudo-panel of Brazilian farms spanning the period 1985-2006, we provide new evidence on the inverse relationship between farm size and productivity. The inverse relationship between size and land productivity is alive and well. The relationship between total factor productivity and size, in contrast, has evolved with modernization during this period. An inverse relationship between farm size and land productivity is neither necessary nor sufficient for an inverse relationship between farm size and total factor productivity. The hypothesis of a dynamic farm size – productivity relationship is extended to the context of Mexico in Chapter 2, identifying the relationship in a panel of family farms drawn from the Mexican Family Life Survey . We find a time invariant inverse relationship between farm size and both land productivity and total factor productivity. Stochastic frontier analysis reveals that, while technical change is expanding the frontier and technical inefficiency is growing for the entire sample, these changes are more pronounced for larger farms. An inverse relationship along the productivity frontier is disappearing in the wake of Mexico’s NAFTA-era reforms to agricultural policy, black flower bucket yet this change has not affected the farm size – total factor productivity relationship due to growing technical inefficiency. Chapter 3 conducts a counterfactual analysis of the contribution of changing land productivity to poverty alleviation on the farm. Stochastic frontier analysis enables a parametric decomposition of changes to the land productivity distribution in a panel of Mexican family farms. Using the decompositions, the contribution of productivity channels to poverty alleviation are estimated.

The counterfactual analysis suggests that raising land productivity through intensification and technical change would be a more pro-poor approach than through increases in technical efficiency.The relationship between farm size and productivity is a recurrent topic in development economics, almost as old as the discipline itself. John Stuart Mill observed an inverse relationship as early as 1848, later positing that this had changed due to increasing capital intensity of farming . The issue appeared in the works of Marx, resurfaced with Lenin and Chayanov in the early 20th century, and has captivated modern agricultural and development economists for over fifty years. Debate around the nature and causes of this relationship continues despite a mountain of empirical analysis, posing a puzzling question for 21st century researchers . Conventional economic wisdom expects resources to be allocated such that returns to land are equalized across farms; however, the empirical research on developing countries contradicts this and frequently identifies an inverse relationship. Policy-makers in developing countries have engaged the debate, as an inverse relationship between farm size and productivity indicates a role for small farms in development strategies and the potential for land reform to simultaneously generate improvements in equity and efficiency. Harnessing such a relationship to inform policy requires accurate interpretation of the empirical evidence as well as an understanding of its causes, the channels through which it operates, and the factors that condition its strength. Theoretical explanations for2 this phenomenon often result from household heterogeneity and/or market failures, for example Sen’s dual labor market hypothesis, Eswaran and Kotwal’s model of household endowments with credit constraints, and Feder’s model of moral hazard and costly monitoring of hired labor. Risk aversion and plot-level behavioral and agronomic issues provide alternative explanations. Measurement error and omitted variables, such as soil quality , are two empirical issues that could lead to a spuriously observed inverse relationship. Attempts to sort out the relative importance of these mechanisms have been mixed. Adding to the confusion is the variety of productivity measures and empirical approaches that have been used. As with Sen , Deolalikar , Assunção and Braido , Barrett et al. , Deininger et al. , Dillon et al. , and Abay et al. , much of the early literature used land productivity—output per unit of land—as a measure of performance. 2 Conditioning land productivity on input use by estimating a production function is a second commonly used approach that generates an alternative measure of performance . Controlling for a partial set of inputs is distinct from estimating a full production function. Still others employ value added per unit of land , profit per unit of land , profit , or technical efficiency . Despite the recognition that partial measures such as land productivity are problematic , they continue to be used, often alongside alternative productivity measures, and are frequently discussed synonymously with a more general notion of productivity. Where multiple productivity measures are used, the distinctions between the relationships being estimated are seldom addressed directly. As Barrett notes, this literature “habitually, perhaps cavalierly,” uses physical yields and productivity synonymously. Conceptual clarity is needed on how these measures relate to each other and to farm size, and which is most relevant from a policy perspective. We do not attempt to explain the IR, as do many of the contributions in this field. Rather, we seek to clarify the relationships between the various productivity measures used in this literature and explore the implications of the choice of measure. We show that an inverse relationship between farm size and a partial productivity measure, such as land productivity, is neither necessary nor sufficient for an inverse relationship between farm size and a comprehensive measure of productivity, such as total factor productivity. As such, these measures are not generally comparable. An inverse relationship may be observed when using land productivity, but not necessarily when using a comprehensive measure of productivity. Where comprehensive measures of productivity are more relevant and of interest, a focus on land productivity effectively introduces omitted variable bias by not controlling for the intensity with which other inputs are used. In fact, Bardhan , Berry and Cline , Carter , and Heltberg are all examples where, in the presence of an inverse relationship between farm size and land productivity, the use of more comprehensive productivity measures leads to an attenuated, if not direct, farm size – productivity relationship. This highlights the importance of how productivity is measured when assessing its relationship with farm size, and for drawing policy conclusions and recommendations from these relationships.The lack of an explicit focus on total factor productivity is a curious feature of the inverse relationship literature, especially given the early and widespread acknowledgement of its superiority over partial measures. From a policy perspective, total factor productivity is likely the most relevant measure where poverty alleviation, equity and the productive use of all resources are pressing concerns.