Using two-dimensional optical lattices, it has recently become possible to split up clouds of ultra cold atoms into collections of parallel one-dimensional tubes in which the dynamics is restricted to the direction along their axis. Since the Feshbach-resonance interactions among atoms are short ranged, they can be approximated as contact interactions. It is thus possible to realize optical-trap versions of some classic integrable systems first studied by Lieb, Gaudin, Yang and others. Such systems can display exotic physics like spin-charge separation and unusual paired states. We focus on the study of systems of fermions in one-dimension. At finite temperatures, regimes such as the spin-coherent and the spin-incoherent Luttinger liquids can be realized by tuning the inter-atomic interaction strength and trap parameters. We identify the noise correlation of density fluctuations as a robust observable to discriminate between these two regimes. We also address the effects of spin imbalance for the case of attractive interactions, and study the one-dimensional analog of spatially modulated (FFLO) superconducting states. Our focus is on how the temperature affects the density profiles that are being measured in the experiments. The theoretical study of temperature effects in these systems is of experimental relevance for the problem of thermometry.
Using two-dimensional optical lattices, it has recently become possible to split up clouds of ultra cold atoms into collections of parallel one-dimensional tubes in which the dynamics is restricted to the direction along their axis. Since the Feshbach-resonance interactions among atoms are short ranged, they can be approximated as contact interactions. It is thus possible to realize optical-trap versions of some classic integrable systems first studied by Lieb, Gaudin, Yang and others. Such systems can display exotic physics like spin-charge separation and unusual paired states. We focus on the study of systems of fermions in one-dimension. At finite temperatures, regimes such as the spin-coherent and the spin-incoherent Luttinger liquids can be realized by tuning the inter-atomic interaction strength and trap parameters. We identify the noise correlation of density fluctuations as a robust observable to discriminate between these two regimes. We also address the effects of spin imbalance for the case of attractive interactions, and study the one-dimensional analog of spatially modulated (FFLO) superconducting states. Our focus is on how the temperature affects the density profiles that are being measured in the experiments. The theoretical study of temperature effects in these systems is of experimental relevance for the problem of thermometry.
Despite the fact that by now one dimensional and three dimensional systems of interacting particles are reasonably well understood, very little is known on how to go from the one dimensional physics to the three dimensional one. This is in particular true in a quasi-one dimensional geometry where the hopping of particles between one dimensional chains or tubes can lead to a dimensional crossover between a Luttinger liquid and more conventional high dimensional states. Such a situation is relevant to many physical systems. Recently cold atoms in optical traps have provided a unique and controllable system in which to investigate this physics. We thus analyze a system made of coupled one dimensional tubes of interacting fermions. We explore the observable consequences, such as the phase diagram for isolated tubes, and the possibility to realize unusual superfluid phases in coupled tubes systems
Recent experiments demonstrate the ability to construct cold-atom mixtures with species-selective optical lattices. This allows for the possibility of a mixed-dimension system, where one fermionic atomic species is confined to a two-dimensional lattice, while another species is confined to a three-dimensional lattice that contains the two-dimensional one. We show that by tuning the density of an arbitrary number of three-dimensional atomic species, we can engineer an arbitrary, rotationally symmetric, density-density, effective interaction for the two-dimensional particles. This possibility allows for an effective interaction that favors triplet pairing for two-dimensional, SU(2) symmetric particles. Using a functional renormalization-group analysis for the two-dimensional particles, we derive and numerically confirm that the critical temperature for triplet pairing depends exponentially on the effective interaction strength. We then analyze how the stability of this phase is affected by the particle densities and the fine tuning of interaction parameters. We conclude by briefly discussing experimental considerations and the potential to study triplet-pairing physics, including Majorana fermions and spin textures, with cold atoms on optical lattices.
Abstract We present a theoretical study of confinement-induced quantum phase transitions in optically trapped atomic gases confined to rod-shaped quasi-one-dimensional geometries, and identify their signatures in experimentally accessible optical observables. The rod aspect ratio η = L/(2R) provides a geometric knob for tuning the effective one-dimensional coupling parameter γ ∝ η: increasing the rod length at fixed atom number reduces the linear density and drives the system from the weakly interacting regime through the Tonks–Girardeau (TG) fermionization limit for bosons, and from a Fermi liquid into a strongly correlated Tomonaga–Luttinger liquid (TLL) phase for fermions. Using the Lieb–Liniger model extended to finite temperature via Yang–Yang thermodynamics, and TLL bosonisation for fermions, we compute the phase diagram in the (γ, T/TF ) plane, first- and second-order spatial correlation functions g (1) and g (2) , the isothermal compressibility κ(μ, T ), and the local density fluctuations ⟨(δˆn) 2 ⟩. We show that the zero-distance second-order coherence g (2) (0) provides an unambiguous optical signal distinguishing all relevant phases: g (2) (0) ≈ 1 for the quasicondensate, g (2) (0) ≈ 0 in the TG limit, and g(2)(0) ≈ K c (γ)/2 < 1/2 in the TLL regime, where K c is the Luttinger parameter. These predictions are measurable via Hanbury Brown–Twiss atom interferometry, single-atom fluorescence imaging, and matter-wave interference in optical rod traps. Our results establish the rod aspect ratio as a design parameter for engineering correlated quantum phases and demonstrate optical coherence as their natural experimental readout.
Ultracold atoms have become an essential tool in studying condensed matter phenomena. The advantage of atomic physics experiments is that they provide an easily tunable system. This experiment uses the lowest two ground state hyperfine levels of fermionic lithium. Having two different states creates a pseudo-spin-1/2 system and allows us to emulate electronic systems, such as superconductors and crystal lattices. In our experiment, we can control the ratio between these two states resulting in either a spin-balanced or a spin-imbalanced gas. Imposing an imbalance is analogous to applying a magnetic field to a superconductor which causes the electrons in the material to align to the field (thus breaking the electron pairs which cause superconductivity). This motivates us to understand the phases created when a spin-imbalance is created and the effect of changing the atomic interactions. In a 3D system, we find where superfluidity is suppressed throughout the BEC to BCS crossover. Using phase separation as a guide, we probe the dimensional crossover between 1D and 3D. The phase separation in 1D is inverted from that in 3D, which provides a unique characteristic to distinguish between the dimensions. By varying the tunneling between tubes and the atomic interactions in a 2D optical lattice, we control whether the system is 1D, 3D, or in between. Using the properties of a 3D gas as a guide, we directly observe when the gas has crossed over from being dominated by 1D-like behavior to 3D. In this way, we have found a universal value for the dimensional crossover. The 1D-3D crossover paves the way to search for the exotic FFLO (Fulde-Ferrell-Larkin-Ovchinnikov) superconductor. While most superconductors do not coexist with magnetism, the FFLO phase requires large magnetic fields to support its pairing mechanism. Additionally, this phase is more likely to be found in lower dimensional systems. However, at low dimensions, the effect of temperature fluctuations on the phase is destabilizing, but these temperature effects are reduced with higher dimensionality. Thus, the quasi-1D regime is the optimal region of parameter space to find this phase. The search for direct evidence of FFLO continues in this regime.
Recent success in manipulating ultracold atomic systems allows one to probe different strongly correlated regimes in one dimension. Regimes such as the (spin-coherent) Luttinger liquid and the spin-incoherent Luttinger liquid can be realized by tuning the interatomic interaction strength and trap parameters. We identify the noise correlations of density fluctuations as a robust observable (uniquely suitable in the context of trapped atomic gases) to discriminate between these two regimes. Finally, we address the prospects of realizing and probing these phenomena experimentally using optical lattices.
In this thesis, we discuss quantum phase transitions in low-dimensional optical lattices, namely one- and two-dimensional lattices. The dimensional confinement is realized in experiments by suppressing the hopping in the extra dimensions through a deep potential barrier that prevents the atoms to tunnel across dimensions. The dimensionality considerably affects the physical behavior of the system. For instance, in one dimension, the celebrated Fermi-liquid theory does not describe the many-body behavior of a fermionic system. There are no single particle excitations, but instead collective behavior, and spin and charge degrees of freedom are decoupled . In the first part of this thesis, we consider one-dimensional fermionic gases under the effect of time-dependent driving forces. First, we describe a proposal to realize non-standard Hubbard models that display correlated-hopping terms by modulating interactions through a time-dependent magnetic field. The extra terms describe hopping processes that depend on the particle-density difference in neighboring sites. The effective model is obtained in arbitrary spatial dimensions, but only in one-dimension it is known to be exactly solvable for specific values of the parameters. Moreover, an SU(2) symmetry in the charge sector allows to construct non-trivial superconducting eigenstates (eta-states), discovered by Yang as excited states of the Hubbard model. Then, we study a one-dimensional bipartite optical lattice with fermionic atoms at half-filling. The lattice potential barriers have heights alternating in magnitude. As a consequence, the hopping amplitudes also alternate in magnitude and the elementary unit cell has two sites. This system is known to display Peierls dimerization at half-filling: the fermions form dimers in neighboring sites that inhibit transport and open a gap in the spectrum. After applying shaking on this system we discover that one can dynamically induce a phase transition to an unconventional metallic state with four Fermi-points. The second part of this thesis describes bosonic systems in two-dimensional optical lattices. First, we consider a square optical lattice with alternating deep-shallow wells arranged in a checkerboard pattern. This lattice is realized in the experimental laboratory of Prof. Hemmerich in Hamburg. In the experiment, the visibility is extracted from the time-of-flight imaging to quantify the phase coherence of the bosonic gas. We develop a mean-field theory and determine the phase diagram of this model. We interpret the experimental data by assuming that at large detuning, only the deeper wells are populated and form an imbalanced Mott insulator with empty shallow wells and integer density on the deeper wells. We support this hypothesis by large-scale Gutzwiller calculations. Then, we investigate a system of bosons in the Lieb lattice, which is the lattice geometry of the Cu-O plane in cuprates . We consider configurations of the optical potential where s and p orbitals sitting on neighboring wells become resonant and hybridize. The interest in this problem comes from the possibility to realize time-reversal broken phases, usually known as Varma phases for electronic systems, which have been proposed to explain the pseudo-gap regime of cuprate superconductors. We show the conditions under which the symmetry breaking in a four-band model can lead to states with loop currents. Moreover, the lowest branch of the excitation spectrum has a non-vanishing Chern number and therefore supports topological excitations.