Chan Chak Ming = 二元玻色-愛因斯坦凝聚態之硏究 / 陳澤明. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2001. / Includes bibliographical references (leaves [100]-104). / Text in English; abstracts in English and Chinese. / Chan Chak Ming = Er yuan Bose-Aiyinsitan ning ju tai zhi yan jiu / Chen Zeming. / Abstract --- p.i / Acknowledgments --- p.ii / Contents --- p.iii / List of Figures --- p.vi / Chapter Chapter 1. --- Introduction --- p.1 / Chapter Chapter 2. --- Theory of Bose-Einstein Condensate (BEC) --- p.4 / Chapter 2.1 --- Trapped Ideal Bose Gas --- p.5 / Chapter 2.2 --- Bogoliubov Theory of Weakly Interacting Bosons --- p.7 / Chapter 2.2.1 --- One-component BEC --- p.7 / Chapter 2.2.2 --- Two-component BEC --- p.12 / Chapter Chapter 3. --- Condensate Wavefunctions and Collective Excitations --- p.16 / Chapter 3.1 --- Properties of Condensate Wavefunctions --- p.16 / Chapter 3.2 --- Collective Excitations --- p.21 / Chapter 3.3 --- Appendix: Numerical Methods and Practical Procedures --- p.26 / Chapter 3.3.1 --- Gradient Descent Method --- p.27 / Chapter 3.3.2 --- Iterative Diagonalization Method --- p.28 / Chapter 3.3.3 --- Practical Procedures --- p.30 / Chapter Chapter 4. --- Noncondensate Atoms in Homogeneous BEC --- p.33 / Chapter 4.1 --- Noncondensate Atoms in One-Component BEC --- p.33 / Chapter 4.2 --- Bogoliubov Theory for Two-species Homogeneous BEC --- p.35 / Chapter 4.3 --- Same Mass System: m1= m2 --- p.37 / Chapter 4.4 --- Unequal Mass System: m1 ≠ m2 --- p.48 / Chapter 4.5 --- Summary --- p.54 / Chapter Chapter 5. --- Noncondensate Atoms in a Trapped BEC --- p.55 / Chapter 5.1 --- Case I: The Noncondensate Atoms in the Mixture of Two Spin States of 87Rb --- p.57 / Chapter 5.2 --- Case II: The Noncondensate Atom in the Mixture of 87Rb and 23Na --- p.61 / Chapter 5.3 --- Summary --- p.64 / Chapter Chapter 6. --- Two-component BEC in Relative Motion --- p.65 / Chapter 6.1 --- Bogoliubov Theory for Motional Two BEC --- p.65 / Chapter 6.2 --- Stability Analysis --- p.69 / Chapter 6.2.1 --- Dynamical Stability Analysis --- p.69 / Chapter 6.2.2 --- Anomalous Mode Analysis --- p.75 / Chapter 6.2.3 --- "Critical Velocity, Anomalous Modes Critical Velocity and Sound Velocities" --- p.78 / Chapter 6.3 --- Motional Two-component BEC in a Ring --- p.80 / Chapter 6.4 --- Two-component BEC of the Same Species --- p.85 / Chapter 6.4.1 --- Moving Particles in Momentum Space --- p.88 / Chapter 6.4.2 --- Moving Particles in Real Space --- p.93 / Chapter 6.4.2.1 --- Strong coupling regime: (g > k02/2) --- p.93 / Chapter 6.4.2.2 --- Weak-coupling regime: (g《<k02/2) --- p.95 / Chapter 6.5 --- Summary --- p.95 / Chapter Chapter 7. --- Conclusion --- p.93 / Bibliography --- p.100

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Ng Ho Tsang = 玻色-愛因斯坦凝聚態在一個雙井形位勢的量子糾纏的理論 / 吳浩錚. / "1st November 2004." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (p. 159-167) / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. Ann Arbor, MI : ProQuest Information and Learning Company, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese. / Ng Ho Tsang = Bose-Aiyinsitan ning ju tai zai yi ge shuang jing xing wei shi de liang zi jiu chan de li lun / Wu Haozheng.

Estudamos transições de fases quânticas em gases bosônicos ultrafrios aprisionados em redes óticas. A física desses sistemas é capturada por um modelo do tipo Bose-Hubbard que, no caso de um sistema sem desordem, em que os átomos têm interação de curto alcance e o tunelamento é apenas entre sítios primeiros vizinhos, prevê a transição de fases quântica superfluido-isolante de Mott (SF-MI) quando a profundidade do potencial da rede ótica é variado. Num primeiro estudo, verificamos como o diagrama de fases dessa transição muda quando passamos de uma rede quadrada para uma hexagonal. Num segundo, investigamos como a desordem modifica essa transição. No estudo com rede hexagonal, apresentamos o diagrama de fases da transição SF-MI e uma estimativa para o ponto crítico do primeiro lobo de Mott. Esses resultados foram obtidos usando o algoritmo de Monte Carlo quântico denominado Worm. Comparamos nossos resultados com os obtidos a partir de uma aproximação de campo médio e com os de um sistema com uma rede ótica quadrada. Ao introduzir desordem no sistema, uma nova fase emerge no diagrama de fases do estado fundamental intermediando a fase superfluida e a isolante de Mott. Essa nova fase é conhecida como vidro de Bose (BG) e a transição de fases quântica SF-BG que ocorre nesse sistema gerou muitas controvérsias desde seus primeiros estudos iniciados no fim dos anos 80. Apesar dos avanços em direção ao entendimento completo desta transição, a caracterização básica das suas propriedades críticas ainda é debatida. O que motivou nosso estudo, foi a publicação de resultados experimentais e numéricos em sistemas tridimensionais [Yu et al. Nature 489, 379 (2012), Yu et al. PRB 86, 134421 (2012)] que violam a lei de escala $\\phi= u z$, em que $\\phi$ é o expoente da temperatura crítica, $z$ é o expoente crítico dinâmico e $ u$ é o expoente do comprimento de correlação. Abordamos essa controvérsia numericamente fazendo uma análise de escalonamento finito usando o algoritmo Worm nas suas versões quântica e clássica. Nossos resultados demonstram que trabalhos anteriores sobre a dependência da temperatura de transição superfluido-líquido normal com o potencial químico (ou campo magnético, em sistemas de spin), $T_c \\propto (\\mu-\\mu_c)^\\phi$, estavam equivocados na interpretação de um comportamento transiente na aproximação da região crítica genuína. Quando os parâmetros do modelo são modificados de maneira a ampliar a região crítica quântica, simulações com ambos os modelos clássico e quântico revelam que a lei de escala $\\phi= u z$ [com $\\phi=2.7(2)$, $z=3$ e $ u = 0.88(5)$] é válida. Também estimamos o expoente crítico do parâmetro de ordem, encontrando $\\beta=1.5(2)$. / In this thesis, we have studied phase transitions in ultracold atoms trapped in optical lattices. The physics of these systems is captured by Bose-Hubbard-like models, which predicts a quantum phase transition (the so called superfluid-Mott insulator, or SF-MI) when varying the potential depth of the optical lattice in a system without disorder, where atoms have short range interactions, and tunneling is allowed only between nearest neighbors. Our studies followed two directions, one is concerned with the influence of the geometry of the lattice namely, we study the changes in the phase diagram of the SF-MI phase transition when the optical lattice is hexagonal. A second direction is to include disorder in the original system. In our study of the hexagonal lattice, we obtain the phase diagram for the SF-MI transition and give an approximation for the critical point of the first Mott lobe, using a quantum Monte Carlo algorithm called Worm. We also compare our results with the ones from the squared lattice and obtained using mean-field approximation. When disorder is included in the system, a new phase emerge in the ground-state phase diagram intermediating the superfluid and Mott-insulator phases. This new phase is called Bose-glass (BG) and the quantum phase transition SF-BG was the subject of many controversies since its first studies in the late 80s. Though many progress towards its thorough understanding were made, basics characterization of critical proprieties are still under debate. Our study was motivated by the publication of recent experimental and numerical studies in three-dimensional systems [Yu et al. Nature 489, 379 (2012), Yu et al. PRB 86, 134421 (2012)] reporting strong violations of the key quantum critical relation, $\\phi= u z$, where $\\phi$ is the critical-temperature exponent, $z$ and $ u$ are the dynamic and correlation length critical exponents, respectively. We addressed this controversy numerically performing finite-size scaling analysis using the Worm algorithm, both in its quantum and classical scheme. Our results demonstrate that previous work on the superfluid-to-normal fluid transition-temperature dependence on chemical potential (or magnetic field, in spin systems), $T_c \\propto (\\mu-\\mu_c)^\\phi$, was misinterpreting transient behavior on approach to the fluctuation region with the genuine critical law. When the model parameters are modified to have a broad quantum critical region, simulations of both quantum and classical models reveal that the $\\phi= u z$ law [with $\\phi=2.7(2)$, $z=3$, and $ u = 0.88(5)$] holds true. We also estimate the order parameter exponent, finding $\\beta=1.5(2)$.

In this thesis we develop Bragg scattering as a tool for probing and manipulating ultra-cold atoms. Our approach is based on a mean-field treatment of degenerate quantum gases. Bose-Einstein condensates are described by the Gross-Pitaevskii equation and degenerate Fermi gases are described by the Bogoliubov-de-Gennes equations. Our work is presented in three inter-related topics. In Part I we investigate Bose-Einstein condensation in a time-averaged orbiting potential trap by deriving solitary-wave dynamical eigenstates of the system. We invoke the quadratic average approximation in which the dynamic effects of the time-dependent potential can be described simply, even when accounting for atomic collisions. By deriving the transformation to the translating frame, dynamical eigenstates of the system are defined and those states are solitary-wave solutions in the laboratory frame, with a particular circular centre-of-mass motion independent of the strength of the collisional interactions. Our treatment in the translating frame is more general than previous treatments that use the rotating frame to define system eigenstates, as the use of the rotating frame restricts eigenstates to those that are cylindrically symmetric about their centre of mass. In Part II we describe Bragg spectroscopy of a condensate with solitary-wave motion. Our approach is based on a momentum space two-bin approximation, derived by Blakie et al. [Journal of Physics B 33:3961, 2000] to describe Bragg scattering of a stationary condensate. To provide an analytic treatment of Bragg scattering of a solitary-wave condensate we use the translating frame, in which the time dependence of the system is described entirely by a time-dependent optical potential. We derive a simplified treatment of the two-bin approximation that provides a physical interpretation of the Bragg spectrum of a solitary-wave condensate. Our methods are applied to Bragg spectroscopy of a condensate in a time-averaged orbiting potential trap, which accelerates as a solitary wave as derived in Part I. The time-averaged orbiting potential trap system is ideal for testing our approximate analytic methods because the micromotion velocity is large compared to the condensate momentum width. In Part III we present a theoretical treatment of Bragg scattering of an ultra-cold Fermi gas. We give the first non-perturbative numerical calculations of the dynamic behaviour of a degenerate Fermi gas subjected to an optical Bragg grating. We observe first order Bragg scattering, familiar from Bragg scattering of stationary Bose-Einstein condensates, and at lower Bragg frequencies we predict scattering of Cooper pairs into a correlated spherical shell of atoms. Correlated-pair scattering is associated with formation of a grating in the pair potential. We give an analytic treatment of Bragg scattering of a homogeneous Fermi gas, and develop a model that reproduces the key features of the correlated-pair Bragg scattering. We discuss the effect of either a trapping potential or finite temperature on the correlated-pair Bragg scattering.

Thesis (Ph. D.)--University of Texas at Austin, 2001. / Vita. Includes bibliographical references (leaves 126-136). Available also in a digital version from Dissertation Abstracts.

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