Abstract:
We investigate the finite temperature momentum distribution of bosonic noncondensate particles inside a 3D optical lattice near the superfluid to Mott insulator transition point, treating the quantum fluctuation and thermal fluctuation effects on equal footing. We explicitly address the different momentum ($q$) dependence of quasi-particles excitations resulted from thermal and quantum origin: the former scales as $|\bfq|^{-2}$ and hence is dominant in the small momentum region, while the later scales as $|\bfq|^{-1}$ and hence dominant in the large momentum limit. Analytic and semi-analytic results are derived, providing a unique method to determine the temperature, condensate density, coherent length and/or single particle gap etc. inside the optical lattice. Our results also agree with the scaling theory of a quantum $XY$ model near the transition point. Experimental implication of the TOF measurement is also discussed.

Abstract:
We investigate the quantum phase transitions of bosonic polar molecules in a two-dimensional double layer system. We show that an interlayer bound state of dipoles (dimers) can be formed when the dipole strength is above a critical value, leading to a zero-energy resonance in the interlayer s-wave scattering channel. In the positive detuning side of the resonance, the strong repulsive interlayer pseudopotential can drive the system into a maximally entangled state, where the wave function is a superposition of two states that have all molecules in one layer and none in the other. We discuss how the zero-energy resonance, dimer states, and the maximally entangled state can be measured in time-of-flight experiments.

Abstract:
We develop a strong-coupling theory for the superfluidity of fermion pairing phase in a Bose-Fermi mixture. Dynamical screening, self-energy renormalization, and a pairing gap function are included self-consistently within the adiabatic limit (i.e., the phonon velocity is much smaller than the Fermi velocity). An analytical solution for the transition temperature (Tc) is derived within reasonable approximations. Using typical parameters of a 40K-87Rb mixture, we find that the calculated Tc is several times larger than that obtained in the weak coupling theory, and can be up to several percents of the Fermi temperature.

Abstract:
We derive a general effective many-body theory for bosonic polar molecules in strong interaction regime, which cannot be correctly described by previous theories within the first Born approximation. The effective Hamiltonian has additional interaction terms, which surprisingly reduces the anisotropic features of dipolar interaction near the shape resonance regime. In the 2D system with dipole moment perpendicular to the plane, we find that the phonon dispersion scales as $\sqrt{|\bfp|}$ in the low momentum ($\bfp$) limit, showing the same low energy properties as a 2D charged Bose gas with Coulomb ($1/r$) interactions.

Abstract:
We systematically investigate the properties of the quenched disorder potential in an atomic waveguide, and study its effects to the dynamics of condensate in the strong disorder region. We show that even very small wire shape fluctuations can cause strong disorder potential along the wire direction, leading to the fragmentation phenomena as the condensate is close to the wire surface. The generic disorder potential is Gaussian correlated random potential with vanishing correlations in both short and long wavelength limits and with a strong correlation weight at a finite length scale, set by the atom-wire distance. When the condensate is fragmentized, we investigate the coherent and incoherent dynamics of the condensate, and demonstrate that it can undergo a crossover from a coherent condensate to an insulating Bose-glass phase in strong disorder (or low density) regime. Our numerical results obtained within the meanfield approximation are semi-quantitatively consistent with the experimental results.

Abstract:
We analyze theoretically the collective mode dispersion in multi-layer stacks of two dimensional dipolar condensates and find a strong enhancement of the roton instability. We discuss the interplay between the dynamical instability and roton softening for moving condensates. We use our results to analyze the decoherence rate of Bloch oscillations for systems in which the s-wave scattering length is tuned close to zero using Feshbach resonance. Our results are in qualitative agreement with recent experiments of Fattori {\it et al.} on $^{39}$K atoms.

Abstract:
We revisit dipolar motion of condensate atoms in one-dimensional optical lattices and harmonic magnetic traps including quantum fluctuations within the truncated Wigner approximation. In the strong tunneling limit we reproduce the meanfield results with a sharp dynamical transition at the critical displacement. When the tunneling is reduced, on the contrary, strong quantum fluctuations lead to finite damping of condensate oscillations even at infinitesimal displacement. We argue that there is a smooth crossover between the chaotic classical transition at finite displacement and the superfluid-to-insulator phase transition at zero displacement. We further analyze the time dependence of the density fluctuations and of the coherence of the condensate and find several nontrivial dynamical effects, which can be observed in the present experimental conditions.

Abstract:
We develop a generic theory for the resonant inelastic light (Raman) scattering by a conduction band quantum plasma taking into account the presence of the filled valence band in doped semiconductor nanostructures within a generalized resonant random phase approximation (RPA). Our generalized RPA theory explicitly incorporates the two-step resonance process where an electron from the filled valence band is first excited by the incident photon into the conduction band before an electron from the conduction band falls back into the valence band emitting the scattered photon. We show that when the incident photon energy is close to a resonance energy, i.e. the valence-to-conduction band gap of the semiconductor structure, the Raman scattering spectral weight at single particle excitation energies may be substantially enhanced even for long wavelength excitations, and may become comparable to the spectral weight of collective charge density excitations (plasmon). Away from resonance, i.e. when the incident photon energy is different from the band gap energy, plasmons dominate the Raman scattering spectrum. We find no qualitative difference in the resonance effects on the Raman scattering spectra among systems of different dimensionalities (one, two and three) within RPA. This is explained by the decoherence effect of the resonant interband transition on the collective motion of conduction band electrons. Our theoretical calculations agree well (qualitatively and semi-quantitatively) with the available experimental results, in contrast to the standard nonresonant RPA theory which predicts vanishing long wavelength Raman spectral weight for single particle excitations.

Abstract:
The essential quantum many-body physics of an ultracold quantum gas relies on the single-particle Green's functions.\ We demonstrate that it can be extracted by the spectrum of electromagnetically induced transparency (EIT).\ The single-particle Green's function can be reconstructed by the measurements of frequency moments in EIT spectroscopy.\ This optical measurement provides an efficient and nondestructive method to reveal the many-body properties, and we propose an experimental setup to realize it.\ Finite temperature and finite size effects are discussed, and we demonstrate the reconstruction steps of Green's function for the examples of three-dimensional Mott-insulator phase and one-dimensional Luttinger liquid.

Abstract:
We develop a general theory to study the electromagnetically induced transparency (EIT) in ultracold quantum gases, applicable for both Bose and Fermi gases with arbitrary inter-particle interaction strength. We show that, in the weak probe field limit, the EIT spectrum is solely determined by the single particle Green's function of the ground state atoms, and reflects interesting quantum many-body effects when atoms are virtually coupled to the low-lying Rydberg states. As an example, we apply our theory to 1D Luttinger liquid, Bose-Mott insulator state, and the superfluid state of two-component Fermi gases, and show how the many-body features can be observed non-destructively in the unconventional EIT spectrum.