QUANTUM CORRELATIONS OF LIGHTS IN MACROSCOPIC ENVIRONMENTS

This dissertation presents a detailed study in exploring quantum correlations of lights in macroscopic environments. We have explored quantum correlations of single photons, weak coherent states, and polarization-correlated/polarization-entangled photons in macroscopic environments. These included macroscopic mirrors, macroscopic photon number, spatially separated observers, noisy photons source and propagation medium with loss or disturbances. We proposed a measurement scheme for observing quantum correlations and entanglement in the spatial properties of two macroscopic mirrors using single photons spatial compass state. We explored the phase space distribution features of spatial compass states, such as chessboard pattern by using the Wigner function. The displacement and tilt correlations of the two mirrors were manifested through the propensities of the compass states. This technique can be used to extract Einstein-Podolsky-Rosen correlations (EPR) of the two mirrors. We then formulated the discrete-like property of the propensity Pb(m,n), which can be used to explore environmental perturbed quantum jumps of the EPR correlations in phase space. With single photons spatial compass state, the variances in position and momentum are much smaller than standard quantum limit when using a Gaussian TEM00 beam. We observed intrinsic quantum correlations of weak coherent states between two parties through balanced homodyne detection. Our scheme can be used as a supplement to decoy-state BB84 protocol and differential phase-shift QKD protocol. We prepared four types of bipartite correlations ±cos2(θ12) that shared between two parties. We also demonstrated bits correlations between two parties separated by 10 km optical fiber. The bits information will be protected by the large quantum phase fluctuation of weak coherent states, adding another physical layer of security to these protocols for quantum key distribution. Using 10 m of highly nonlinear fiber (HNLF) at 77 K, we observed coincidence to accidental-coincidence ratio of 130±5 for correlated photon-pair and Two-Photon Interference visibility >98% entangled photon-pair. We also verified the non-local behavior of polarization-entangled photon pair by violating Clauser-Horne-Shimony-Holt Bell’s inequality by more than 12 standard deviations. With the HNLF at 300 K (77 K), photon-pair production rate about factor 3(2) higher than a 300 m dispersion-shifted fiber is observed. Then, we studied quantum correlation and interference of photon-pairs; with one photon of the photon-air experiencing multiple scattering in a random medium. We observed that depolarization noise photon in multiple scattering degrading the purity of photon-pair, and the existence of Raman noise photon in a photon-pair source will contribute to the depolarization affect. We found that quantum correlation of polarization-entangled photon-pair is better preserved than polarization-correlated photon-pair as one photon of the photon-pair scattered through a random medium. Our findings showed that high purity polarization-entangled photon-pair is better candidate for long distance quantum key distribution.

THREE HUNDRED YEARS OF THE ST. PETERSBURG PARADOX

The St. Petersburg Paradox was first presented by Nicholas Bernoulli in 1713. It is related to a gambling game whose mathematical expected payoff is infinite, but no reasonable person would pay more than $25 to play it. In the history, a number of ideas in different areas have been developed to solve this paradox, and this report will mainly focus on mathematical perspective of this paradox. Different ideas and papers will be reviewed, including both classical ones of 18th and 19th century and some latest developments. Each model will be evaluated by simulation using Mathematica.

FOSSIL PLANTS AND PALYNOMORPHS FROM THE LATE PALEOZOIC CUTLER FORMATION, EASTERN PARADOX BASIN

The late Paleozoic Cutler Formation, where exposed near the modern-day town of Gateway, Colorado, has traditionally been interpreted as the product of alluvial fan deposition within the easternmost portion of the Paradox Basin. The Paradox Basin formed between the western margin of the Uncompahgre Uplift segment of the Ancestral Rocky Mountains and the western paleoshoreline of the North American portion of Pangea. The Paradox Basin region is commonly thought to have experienced semi-arid to arid conditions and warm temperatures during the Pennsylvanian and Permian. Evidence described in this paper support prior interpretations regarding paleoclimate conditions and the inferred depositional environment for the Cutler Formation near Gateway, Colorado. Plant fossils collected from the late Paleozoic Cutler Formation in The Palisade Wilderness Study Area (managed by the U.S. Department of the Interior, Bureau of Land Management) of western Colorado include Calamites, Walchia, Pecopteris, and many calamitean fragments. The flora collected is interpreted to have lived in an arid or semi-arid environment that included wet areas of limited areal extent located near the apex of an alluvial fan system. Palynological analysis of samples collected revealed the presence of the common Pennsylvanian palynomorphs Thymospora pseudothiessenii and Lophotriletes microsaetosus. These fossils suggest that warm and at least seasonally and locally wet conditions existed in the area during the time that the plants were growing. All evidence of late Paleozoic plant life collected during this study was found along the western margin of the Uncompahgre Uplift segment of the Ancestral Rocky Mountains. During the late Paleozoic, sediment was eroded from the Uncompahgre Uplift and deposited in the adjacent Paradox Basin. The preservation of plant fossils in the most proximal parts of the Paradox Basin is remarkable due to the fact that much of the proximal Cutler Formation consists of conglomerates and sandstones deposited as debris flow and by fluvial systems. The plants must have grown in a protected setting, possibly an abandoned channel on the alluvial fan, and been rapidly buried in the subsiding Paradox Basin. It is likely that there was abundant vegetation in and adjacent to low-lying wet areas at the time the Cutler Formation was deposited.