Mott insulating ultracold gases possess a unique whole-atom exchange interaction which enables large quantum fluctuations between the Zeeman sublevels of each atom. By strengthening this interaction---either through the use of large-spin atoms or by tuning the particle-particle interactions via optical Feshbach resonance---one may enhance fluctuations and facilitate the appearance of the long-sought-after quantum spin liquid phase---all in the highly tunable environment of cold atoms. To illustrate the relationship between the spin magnitude, interaction strength, and resulting magnetic phases, we present and solve a mean-field theory for bosons optically confined to the one-particle-per-site Mott state, using both analytic and numerical methods. We find on square and triangular lattices for bosons of hyperfine spin $f>2$ that making the repulsive $s$-wave scattering length through the singlet channel small---relative to the higher-order scattering channels---accesses a short-range resonating valence bond (s-RVB) spin liquid phase.
Mott insulating ultracold gases possess a unique whole-atom exchange interaction which enables large quantum fluctuations between the Zeeman sublevels of each atom. By strengthening this interaction---either through the use of large-spin atoms or by tuning the particle-particle interactions via optical Feshbach resonance---one may enhance fluctuations and facilitate the appearance of the long-sought-after quantum spin liquid phase---all in the highly tunable environment of cold atoms. To illustrate the relationship between the spin magnitude, interaction strength, and resulting magnetic phases, we present and solve a mean-field theory for bosons optically confined to the one-particle-per-site Mott state, using both analytic and numerical methods. We find on square and triangular lattices for bosons of hyperfine spin f>2 that making the repulsive s-wave scattering length through the singlet channel small---relative to the higher-order scattering channels---accesses a short-range resonating valence bond (s-RVB) spin liquid phase.
This thesis investigates the plausibility of producing a quantum spin liquid (QSL) with ultracold bosonic atoms optically confined to the Mott insulating state. QSLs have received a great deal of attention for being an antiferromagnetic groundstate with many exotic properties, including the absence of local order, long-range entanglement, and fractionalized excitations. However, the identification and characterization of these phases in solid state systems remains a great challenge. Here we outline an alternate route to uncovering the QSL phase, which from the nature of spin angular momentum for ultracold atoms encounters many properties unique to these systems along the way. This proposal is possible because the magnetic exchange interactions for Mott insulating ultracold atoms are mediated by the hopping of whole atoms. Whole-atom exchange—a unique property of cold atoms—allows large fluctuations between the quantized Zeeman sublevels of each atomic spin. As we demonstrate, these fluctuations increase when large-spin atoms are used, or when interactions are tuned via optical Feshbach resonance (OFR). These strong quantum spin fluctuations inhibit classical magnetic ordering, and lead to a QSL ground state. To illustrate the relationship between the spin magnitude, interaction strength, and QSL ground state, we present two distinct approaches to solving the relevant Hamiltonian. With mean field theory we find that for large spin (f > 2), and strong scattering through the spin-singlet channel, that magnetically-ordered Bose-Einstein condensates are unstable to the formation of a QSL. We then utilize Rayleigh-Schodinger perturbation theory to derive an effective Hamiltonian for our system in the Hilbert space of nearest-neighbor singlet coverings. At large spin this Hamiltonian produces a type of quantum dimer model (QDM), which are known to possess QSL phases. We derive the QDM parameters t, t′ and V as a function of spin on several lattices, finding they scale with the inverse number of Zeeman sublevels. We then determine the proximity of the physically accessible states to the QSL phase, and discuss how other regions of the phase diagram may be accessed. We then conclude by highlighting several advantages to studying QSLs and QDMs with ultracold bosons.
Feshbach resonances-namely, resonances between an unbound two-body (atomic) state and a bound (molecular) state, differing in magnetic moment-are a unique tool to tune the interaction properties of ultracold atoms. Here we show that the spin-changing interactions, coherently coupling the atomic and molecular states, can act as a novel mechanism to stabilize an insulating phase-the Feshbach insulator-for bosons in an optical lattice close to a narrow Feshbach resonance. Making use of quantum Monte Carlo simulations and mean-field theory, we show that the Feshbach insulator appears around the resonance, preventing the system from collapsing when the effective atomic scattering length becomes negative. On the atomic side of the resonance, the transition from condensate to Feshbach insulator has a characteristic first-order nature, due to the simultaneous loss of coherence in the atomic and molecular components. These features appear clearly in the ground-state phase diagram of, e.g., ^{87}Rb around its 414 G resonance, and they are therefore directly amenable to experimental observation.
An important new development in the field of ultracold atomic gases is the study of the properties of these gases in a so-called optical lattice. An optical lattice is a periodic trapping potential for the atoms that is formed by the interference pattern of a few laser beams. A reason for the interest in these systems is that the effects of the interatomic interactions can be strongly enhanced. More specifically, it has been shown in a beautiful experiment by Greiner et al. in 2002 that by loading a Bose-Einstein condensate into an optical lattice it is possible for the system to undergo a quantum phase transition to a new quantum phase of matter, the so-called Mott insulator phase. Within this Mott insulator phase each lattice site is occupied by exactly one atom. This makes the Mott insulator phase especially well suited for applications in the field of quantum computation and quantum information processing. We have theoretically investigated the above mentioned quantum phase transition and our formalism allows for a description of the Mott insulator phase at nonzero temperatures. Another important experimental development in the field of ultracold atomic gases is the use of Feshbach resonances to control the interatomic interactions. Such a resonance occurs whenever two colliding atoms form a long-lived molecule for some time. The crucial point of a Feshbach resonance is that the above mentioned molecule has a magnetic moment that is not equal to twice the magnetic moment of the atom. As a consequence the energy difference between the two atoms and the molecule and hence the interactions between the atoms can be controlled by using an external magnetic field. By combining these two techniques, i.e, by trapping ultracold atomic gases in an optical lattice and by tuning a magnetic field near a Feshbach resonance there can be a new quantum phase transition between two superfluid phases. We gave derived the the theory for the description of these Feshbach resonances in optical lattices and applied it to various systems. To be a bit more precise, if we tune the external magnetic field such that the energy difference between a molecule and two atoms is sufficiently negative, then the gas consists of a Bose-Einstein condensate of molecules. In contrast, if the energy difference is large enough and positive we have a gas that consists primarily of a Bose-Einstein condensate of atoms. It turns out that these two limits are separated by an Ising-like quantum phase transition. By using atomic Bose gases near a Feshbach resonance detailed experimental studies which test the theoretical predictions of the statistical and dynamical properties of these quantum phase transitions can be made.
This thesis studies bosonic and fermionic quantum gases. In the first part, a variational many-body theory is used to explain a Bragg scattering experiment carried out on a bosonic quantum gas [S.B. Papp et al. Bragg Spectroscopy of a Strongly Interacting 85Rb Bose-Einstein Condensate. Phys. Rev. Lett. 101, 135301 (2008)]. A key feature in the experiment was the use of a Feshbach resonance, which made large values of the scattering length accessible. Due to the large values of the scattering length, existing models such as the Beliaev model could not be used to explain the observations, and therefore the experiment was a particularly interesting one to analyze. In our first approach, we constructed ad hoc potentials that fitted the observed excitation spectrum, and later we improved our approach by using T-matrix formalism to describe the Feshbach resonant system. All in all, the phenomenological model we developed fits the observed excitation spectrum and yields correct molecular Feshbach resonance state energies in certain cases. The second part of this thesis studies fermionic quantum gases. We focus on studying a gas of spin-1/2 particles confined to a spin-dependent optical lattice. The lattice geometry is such that the up-spin component is loaded in a honeycomb lattice, and the down-spin component is confined to the underlying triangular lattice. We considered attractive on-site and nearest-neighbor interactions, and formulated the nearest-neighbor interaction term in such a way that it takes into account the possibility of spontaneous time-reversal symmetry breaking. Furthermore, we took into account the possibility of Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase, which breaks spatial symmetry. Within a mean-field approximation, we showed that the FFLO state is the ground state of the system in many instances. In addition, we found out that the system spontaneously breaks time-reversal symmetry if the nearest-neighbor interaction strength is large. Due to the time-reversal symmetry breaking, the system has topologically non-trivial phases characterized by nonzero Chern numbers. Finally, there were also cases where the time-reversal symmetry was broken in the FFLO phase, and thus we found a phase where spatial and time-reversal symmetries are simultaneously broken.
We investigate magnetic properties of Mott-insulating phases of ultracold Bose and Fermi spinor gases in optical lattices. We consider in particular the F = 2 Bose gas, and the F = 3/2 and 5/2 Fermi gases. We derive effective spin Hamiltonians for one and two atoms per site and discuss the possibilities of manipulating the magnetic properties of the system using optical Feshbach resonances. We discuss low temperature quantum phases of a Rb-87 gas in the F = 2 hyperfine state, as well as possible realizations of high spin Fermi gases with either Li-6 or Cs-132 atoms in the F = 3/2 state, and with Yb-173 atoms in the F = 5/2 state.
We investigate magnetic properties of Mott-insulating phases of ultracold Bose and Fermi spinor gases in optical lattices. We consider in particular the F = 2 Bose gas, and the F = 3/2 and 5/2 Fermi gases. We derive effective spin Hamiltonians for one and two atoms per site and discuss the possibilities of manipulating the magnetic properties of the system using optical Feshbach resonances. We discuss low temperature quantum phases of a 87 Rb gas in the F = 2 hyperfine state, as well as possible realizations of high spin Fermi gases with either 6 Li or 132 Cs atoms in the F = 3/2 state, and with 173 Yb atoms in the F = 5/2 state.