DoD High Performance Computing Modernization Program Users Group Conference (HPCMP-UGC)
Modeling Mixtures of Different Mass Ultracold Atoms in Optical Lattices: An Illustration of High Efficiency and Linear Scaling on the Cray XT4 via a Capability Applications Project at ERDC
Defense Advanced Research Projects Agency (DARPA) is running a program to create an analog quantum-mechanical simulator for strongly interacting quantum particles. It is called an optical lattice emulator, and it involves ultracold neutral atoms moving in a corrugated periodic potential created by an optical lattice. The atoms interact with each other via collisions. In this work, we show how one can apply inhomogenenous dynamical mean-field theory (IDMFT) as an approximate computational tool to study the behavior of just such a system. Ultimately, conventional computation will be employed to benchmark the optical lattice emulator on a numerically tractable problem (a verification and validation study), and then the emulator will be applied to problems that cannot be solved with conventional computation. Here, we demonstrate an efficient massively parallel implementation of the IDMFT algorithm that is transportable, scales to many thousands of processors, and runs at approximately 50% of the theoretical peak speed of the machine.
Defense Advanced Research Projects Agency (DARPA) is running a program to create an analog quantum-mechanical simulator for strongly interacting quantum particles. It is called an optical lattice emulator, and it involves ultracold neutral atoms moving in a corrugated periodic potential created by an optical lattice. The atoms interact with each other via collisions. In this work, we show how one can apply inhomogenenous dynamical mean-field theory (IDMFT) as an approximate computational tool to study the behavior of just such a system. Ultimately, conventional computation will be employed to benchmark the optical lattice emulator on a numerically tractable problem (a verification and validation study), and then the emulator will be applied to problems that cannot be solved with conventional computation. Here, we demonstrate an efficient massively parallel implementation of the IDMFT algorithm that is transportable, scales to many thousands of processors, and runs at approximately 50% of the theoretical peak speed of the machine.
This work is part of a Defense Advanced Research Projects Agency (DARPA) sponsored project to build an optical lattice emulator, where models of strongly correlated electrons in condensed matter physics are simulated with ultracold atoms moving in an optical lattice. Recently, members of our team have been able to form dense clouds of dipolar fermionic molecules from mixtures of (fermionic)40K and (bosonic)87Rb. Here, we use high performance computing (HPC) resources to find a way to improve the efficiency of molecule formation from the current 20% to almost 100% and thereby show how to create a much denser cloud of dipolar molecules. Our code scales nearly linearly on up to 4,000 (or more) processors, and runs at an efficiency that is almost at 100% of the speed for one arithmetic operation per clock cycle. We have not been able to get the code to run with multiple operations per clock cycle in spite of using highly efficient and optimized libraries for BLAS and LAPACK.
We present a nonequilibrium strong-coupling approach to inhomogeneous systems of ultracold atoms in optical lattices. We demonstrate its application to the Mott-insulating phase of a two-dimensional Fermi-Hubbard model in the presence of a trap potential. Since the theory is formulated self-consistently, the numerical implementation relies on a massively parallel evaluation of the self-energy and the Green's function at each lattice site, employing thousands of CPUs. While the computation of the self-energy is straightforward to parallelize, the evaluation of the Green's function requires the inversion of a large sparse 10(d) × 10(d) matrix, with d > 6. As a crucial ingredient, our solution heavily relies on the smallness of the hopping as compared to the interaction strength and yields a widely scalable realization of a rapidly converging iterative algorithm which evaluates all elements of the Green's function. Results are validated by comparing with the homogeneous case via the local-density approximation. These calculations also show that the local-density approximation is valid in nonequilibrium setups without mass transport.
Two models are discussed here to illustrate the quantum many-body phenomena in mixtures of ultra-cold atoms in optical lattices. The first model describes a mixture of two species of bosonic atoms of equal masses in optical lattices and the second describes a mixture of heavy bosonic atoms and light fermionic atoms in optical lattices. For both models, we assume the trap is present and use parameters typical in experiment. For the first model, the discussion is aimed at providing a thorough description of the collective behavior of the binary mixture in various interaction regions, with emphasis on two many-body phenomena, pairing and anti-pairing, as a result of the inter-species interaction. The pairing leads to a new type of superfluid order, called the paired superfluid (PSF) and the anti-pairing leads to another type of superfluid order, called the counter-flow superfluid (CFSF). In addition, we discuss the coexistence of charge density wave order with the three superfluid orders in the strong interaction region. We use both Luttinger liquid theory and the time evolving block decimation (TEBD) method to study this model in one dimension. The discussion is organized in three parts: the phase diagram and the correlation functions; the noise correlation functions; and the transport properties. Two phase diagrams are constructed to map the different orders in the parameter space. The correlation functions, include noise correlations, are carefully examined for the determination of the orders and for possible detection methods. In the end, the transport properties of the PSF and CFSF orders are studied through the dipole oscillation induced by trap displacement. For the second model, examining a mixture of heavy bosons and light fermions, the discussion is oriented toward determining the thermal properties of the mixture for attractive inter-species interactions. This work is motivated by experiments creating artificial molecules through optical and magnetic control of ultra-cold atoms. We use the strong coupling (SC) expansion method to evaluate the density profile, the onsite inter-species correlations, the density fluctuations and the entropy per particle. Analytical expressions are derived for all the quantities above as well as the partition function. To benchmark the accuracy, the SC calculations are compared with inhomogeneous dynamical mean field theory (IDMFT) and Monte Carlo (MC) simulation. From the calculations, we find that 1) the efficiency of creating pre-formed molecules is significantly increased by confining the mixtures onto optical lattices; 2) the temperature of the mixtures in optical lattices can be reliably estimated through the density gradient and the density fluctuations.
Many outstanding problems in quantum physics, such as high-Tc superconductivity or quark confinement, are still - after decades of research - awaiting commonly accepted explanations. One reason is that such systems are often difficult to control, show an intermingling of several effects, or are not easily accessible to measurement. To arrive at a deeper understanding of the physics at work, researchers typically derive simplified models designed to capture the most striking phenomena of the system under consideration. However, due to the exponential complexity of Hilbert space, even some of the simplest of such models pose formidable challenges to analytical and numerical calculations. In 1982, Feynman proposed to solve such quantum models with experimental simulation on a physically distinct, specifically engineered quantum system [Int. J. Theor.Phys. 21, 467]. Designed to be governed by the same underlying equations as the original model, it is hoped that direct measurements on these so called quantum simulators (QSs) will allow to gather deep insights into outstanding problems of physics and beyond. In this thesis, we identify four requirements that a useful QS has to fulfill, relevance, control, reliability, and efficiency. Focusing on these, we review the state of the art of two popular approaches, digital QSs (i.e., special purpose quantum computers) and analog QSs (devices with always-on interactions). Further, focusing on possibilities to increase control over QSs, we discuss a scheme to engineer quantum correlations between mesoscopic numbers of spinful particles in optical lattices. This technique, based on quantum polarization spectroscopy, may be useful for state preparation and quantum information protocols. Additionally, employing several analytical and numerical methods for the calculation of many-body ground states, we demonstrate the variety of condensed-matter problems that can be attacked with QSs consisting of ultracold ions or neutral atoms in optical lattices. The chosen examples, some of which have already been realized in experiment, include such diverse settings as frustrated antiferromagnetism, quantum phase transitions in exotic lattice geometries, topological insulators, non-Abelian gauge-fields, orbital order of ultracold Fermions, and systems with long-range interactions. The experimental realization of all of these models requires techniques which go beyond standard optical lattices, e.g., time-periodic driving of lattices with exotic geometry, loading ultracold atoms into higher bands, or immersing trapped ions into an optical lattice. The chosen models, motivated by important open questions of quantum physics, pose difficult problems for classical computers, but they may be amenable in the near future to quantum simulation with ultracold atoms or ions. While the experimental control over relevant models has increased dramatically in the last years, the reliability and efficiency of QSs has received considerably less attention. As a second important part of this thesis, we emphasize the need to consider these aspects under realistic experimental conditions. We discuss specific situations where terms that have typically been neglected in the description of the QS introduce systematic errors and even lead to novel physics. Further, we characterize in a generic example the influence of quenched disorder on an analog QS. Its performance for simulating universal behavior near a quantum phase transition seems satisfactory for low disorder. Moreover, our results suggest a connection between the reliability and efficiency of a QS: it works less reliable exactly in those interesting regimes where classical calculations are less efficient. If QSs fulfill all of our four requirements, they may revolutionize our approach to quantum-mechanical problems, allowing to solve the behavior of complex Hamiltonians, and to design nano-scale materials and chemical compounds from the ground up.
We describe a generalization of the local density approximation for ultra-cold atomic systems, which takes into account the gradients of the Green’s functions with respect to position when solving the local Dyson equation and, hence, is sensitive to the local variation of the system within the trap. We use this technique along with dynamical mean-field theory to simulate the Hubbard model data of the Swiss group, which measured the Mott transition in a three-dimensional optical lattice with ultra-cold 40 K atoms. We simulate the same size system as used in experiment, which includes about 8 million lattice sites and up to 300,000 atoms. We describe our computational algorithm in detail, along with modifications needed for the communications to make it work on the Cray XE6. We show strong scaling curves up to approximately 43,500 cores. The code performs remarkably well with perfect linear scaling up to the largest systems we ran on. The results that we present are a comparison of the dynamical mean-field theory results within the generalized gradient approximation to the strong-coupling perturbation theory, which is known to be accurate for high-temperature and large interaction strength. This work is part of the Defense Advanced Research Projects Agency (DARPA) optical lattice emulator program.
This thesis covers most of my work in the field of ultracold atoms loaded in optical lattices. It makes a route through the physics of cold atoms in periodic potentials starting from the simple noninteracting system and going into the many-body physics that describes the strongly correlated Mott insulator regime. Even though this thesis is a theoretical work all the chapters are linked either with experiments already done or with ongoing experimental efforts. This thesis can be divided into four different parts. The first part comprises chapters 1 to 3. In these chapters, after a brief introduction to the field of optical lattices I review the fundamental aspects pertaining to the physics of systems in periodic potentials. The second part deals with the superfluid weakly interacting regime where standard mean field techniques can be applied. This is covered in chapters 4 and 5. Specifically, chapter 4 introduces the discrete nonlinear Schroedinger equation (DNLSE) and uses it to model some experiments. In chapter 5 I go one step further and include the small quantum fluctuations neglected in the DNLSE by studying quadratic approximations of the Bose-Hubbard Hamiltonian. Chapters 6 to 8 can be grouped as the third part of the thesis. In them I adopt an effective action formalism, the so called two particle irreducible effective action (2PI) together with the closed time path (CTP) formalism to study far-from-equilibrium dynamics. The many-body techniques discussed in these chapters systematically include higher order quantum corrections, not included in the quadratic approximations of the Hamiltonian, which we show are crucial for a correct description of the quantum dynamics outside the very weakly interacting regime. Finally, chapter 9 to 11 are devoted to study the Mott insulator phase. In these chapters using perturbation theory I study the Mott insulator ground state and its excitation spectrum, the response of the system to Bragg spectroscopy, and propose a mechanism to correct for the residual quantum coherences inherent to the Mott insulator ground state. Even though small these are not ideal for the use of neutral atoms in optical lattice as a tool for quantum computation.