Analysis and design of large space structures with nonlinear joints

Issued as Final report, Project no. E-25-62

Quantum simulation in strongly correlated optical lattices

An outstanding problem in physics is how to understand strongly interacting quantum many-body systems such as the quark-gluon plasma, neutron stars, superfluid 4He, and the high-temperature superconducting cuprates. The physics approach to this problem is to reduce these complex systems to minimal models that are believed to retain relevant phenomenology. For example, the Hubbard model — the focus of this thesis — describes quantum particles tunneling between sites of a lattice with on-site interactions. The Hubbard model is conjectured to describe the low-energy charge and spin properties of high-temperature superconducting cuprates. Thus far, there are no analytic solutions to the Hubbard model, and numerical calculations are difficult and even impossible in some regimes (e.g., the Fermi-Hubbard model away from half-filling). Therefore, whether the Hubbard model is a minimal model for the cuprates remains unresolved. In the face of these difficulties, a new approach has emerged — quantum simulation. The premise of quantum simulation is to perform experiments on a quantum system that is well-described by the model we are trying to study, has tunable parameters, and is easily probed. Ultracold atoms trapped in optical lattices are an ideal candidate for quantum simulation of the Hubbard models. This thesis describes work on two such systems: a 87Rb (boson) optical lattice experiment in the group of Brian DeMarco at the University of Illinois to simulate Bose-Hubbard physics, and a 40K (fermion) optical lattice experiment in the group of Joseph Thywissen at the University of Toronto to simulate Fermi-Hubbard physics. My work on the 87Rb apparatus focuses on three main topics: simulating the Bose-Hubbard (BH) model out of equilibrium, developing thermometry probes, and developing impurity probes using a 3D spin-dependent lattice. Theoretical techniques (e.g., QMC) are adept at describing the equilibrium properties of the BH model, but the dynamics are unknown — simulation is able to bridge this gap. We perform two experiments to simulate the BH model out of equilibrium. In the first experiment, published in Ref. [1], we measure the decay rate of the center-of-mass velocity for a Bose-Einstein condensate trapped in a cubic lattice. We explore this dissipation for different Bose-Hubbard parameters (corresponding to different lattice depths) and temperatures. We observe a decay rate that asymptotes to a finite value at zero temperature, which we interpret as evidence of intrinsic decay due to quantum tunneling of phase slips. The decay rate exponentially increases with temperature, which is consistent with a cross-over from quantum tunneling to thermal activation. While phase slips are a well-known dissipation mechanism in superconductors, numerous effects prevent unambiguous detection of quantum phase slips. Therefore, our measurement is among the strongest evidence for quantum tunneling of phase slips. In a second experiment, published in Ref. [2] with theory collaborators at Cornell University, we investigate condensate fraction evolution during fast (i.e., millisecond) ramps of the lattice potential depth. These ramps simulate the BH model with time-dependent parameters. We determine that interactions lead to significant condensate fraction redistribution during these ramps, in agreement with mean-field calculations. This result clarifies adiabatic timescales for the lattice gas and strongly constrains bandmapping as an equilibrium probe. Another part of this thesis work involves developing thermometry techniques for the lattice gas. These techniques are important because the ability to measure temperature is required for quantum simulation and to evaluate in-lattice cooling schemes. In work published in Ref. [3], we explore measuring temperature by directly fitting the quasimomentum distribution of a thermal lattice gas. We attempt to obtain quasimomentum distributions by bandmapping, a process in which the lattice depth is reduced slowly compared to the bandgap but fast with respect to all other timescales. We find that these temperature measurements fail when the thermal energy is comparable to the bandwidth of the lattice. This failure results from two main causes. First, the quasimomentum distribution is an insensitive probe at high temperatures because the band is occupied (i.e., additional thermal energy cannot be accommodated in the kinetic energy degrees of freedom). Second, the bandmapping process does not produce accurate quasimomentum distributions because of smoothing at the Brillouin zone edge. We determine that measuring temperature using the in-situ width overcomes these issues. The in-situ width does not asymptote to a finite value as temperature increases, and the in-situ width can be measured directly without using a mapping procedure. In a second experiment, we investigate using condensate fraction (obtained from the time-of-flight momentum distribution) as an indirect means to measure temperature in the superfluid regime of the BH model. Since no standard fitting procedure exists for the lattice time-of-flight distributions, we define and test a procedure as part of this work. We measure condensate fraction for a range of lattice depths varying from deep in the superfluid regime to lattice depths proximate to the Mott-insulator transition. We also vary the entropy per particle, which is measured in the harmonic trap before adiabatically loading into the lattice. As expected, the condensate fraction increases as entropy decreases, and the condensate fraction decreases at high lattice depths (due to quantum depletion). We compare our experimental results to condensate fraction predicted by the non-interacting, Hartree-Fock-Bogoliubov-Popov, and site-decoupled-mean-field theories. Theory and experiment disagree, which motivates several future extensions to this work, including calculating condensate fraction (and testing our fit procedure) using quantum Monte Carlo numerics, and experimentally and theoretically investigating the dynamics of the lattice load process (for the finite-temperature strongly correlated regime). Finally, we develop impurity probes for the Bose-Hubbard model by employing a spin-dependent lattice. A primary accomplishment of this thesis work was to develop the first 3D spin-dependent lattice in the strongly correlated regime (published in Ref. [4]). The spin-dependent lattice depth is proportional to |gFmF|, enabling the creation of mixtures of atoms trapped in the lattice (nonzero mF) co-trapped with atoms that do not experience the lattice (mF = 0). We use the non-lattice atoms as an impurity probe. We investigate using the impurity to probe the lattice temperature, and we determine that thermalization between the impurity and lattice gas is suppressed for larger lattice depths. Using a comparison to a Fermi’s golden rule calculation of the collisional energy exchange rate, we determine that this effect is consistent with suppression of energy-exchanging collisions by a mismatch between the impurity and lattice gas dispersion. While this result invalidates the concept of an impurity thermometer, it paves the way for a unique cooling scheme that relies on inter-species thermal isolation. We also explore impurity transport through the lattice gas. In other preliminary measurements, we also identify the decay rate of the center-of-mass motion as a prospective impurity probe. A separate aspect of this thesis work is the design and construction of a new 40K apparatus for single-site imaging of atoms to simulate the 2D Fermi-Hubbard model. The main component of this apparatus is high resolution fluorescence imaging on the 4S-5P transition of K at 404.5nm. Fluorescence imaging using this transition has two advantages over imaging on the standard D2 transition at 767nm: a smaller wavelength and therefore higher resolution, and a lower Doppler temperature limit which enables longer imaging times. To validate this approach, we demonstrate the first 40K magneto-optical trap (MOT) using the 404.5nm transition