Ultracold neutral atoms and quantum degenerate neutral atomic gases have proved to be useful in the simulation of the quantum behavior of a number of models, Hamiltonians, and physical systems. One reason for this is the ability to control the Hamiltonian for the atoms in ways that are difficult in other systems; another is the ability to measure certain quantities, like momentum distributions, that are often not easily measured in other systems. Bose condensation, Fermi degeneracy, Cooper pairing, BCS/BEC crossover, and the behavior of quantum particles in periodic potentials are among the areas where cold atoms exhibit behavior that is difficult to observe in analogous condensed matter systems. One difficulty with neutral atoms is in simulating the response of charged particles to external electric and magnetic fields. One approach to simulating the behavior of charged particles in a magnetic field has been to use rotation, where the Coriolis force mimics the Lorentz force. Because of some limitations of this approach, other techniques have been proposed. We present the successful simulation of a magnetic vector potential for neutral atoms and the experimental demonstration of a synthetic electric field arising from its time derivative. It is also possible to create a synthetic magnetic field arising from the spatial derivative (curl) of such a vector potential.

Geometric potentials can constitute in the future a very important tool for simulating with neutral atoms the dynamics of an electron gas in a large magnetic field. I will present a semi-classical interpretation of the geometric potentials that arise in quantum optics. These potentials are due to Berryâ s phase when an atom moves slowly in a light field, following adiabatically one of the dressed states. The scalar potential corresponds to the kinetic energy of an atomic micro-motion caused by quantum fluctuations of the radiative force. The Lorentz-type force appears as a result of the motion-induced perturbation of the internal atomic state. I will also discuss a practical scheme for implementing these potentials that operates even for a large detuning with respect to the atomic resonance. This makes it applicable to alkali-metal atoms without significant heating due to spontaneous emission. The validity of the adiabatic approximation is tested, by in tegrating the set of coupled Gross-Pitaevskii equations associated with the various internal atomic states. The steady state of the interacting gas indeed exhibits a vortex lattice, as expected from the adiabatic gauge field.

While dilute Bose gases are typically in a weakly interacting regime, they can be driven into regimes of strong correlations by various means. A very interesting possible way to do this is by the use of a "rotating optical lattice". I shall discuss the nature of the groundstates of such systems. The relevant physics involves the interplay between the fractional quantum Hall effect for bosons and the "Hofstadter butterfly" spectrum. I will explain how this interplay can lead to novel strongly correlated phases.

Origins of non-trivial Berry connections are discussed in their relation to the Dirac like dispersion. After describing a generic frame work, several uses of the Berry connections in condensed matter physics are given. They include graphene with/without magnetic field, gapped quantum spins and the BEC-BCS crossover. Non trivial topological structures described by the Berry connections give rise to the existence of edge states which are localized quantum degrees and experimental observables. Their relation to the entanglement is also discussed.

We study the properties of an ultracold Fermi gas loaded in an optical square lattice and subjected to an external and classical non-Abelian gauge field. We show that this system can be exploited as an optical analogue of relativistic quantum electrodynamics, offering a remarkable route to access the exotic properties of massless Dirac fermions with cold atoms experiments.

Over the recent years significant advances have been made in studying the physical properties of ultra-cold atomic gases. These are the systems where the atomic physics meets the condensed matter physics. Yet the atoms are electrically neutral species and there is no direct analogy to the magnetic phenomena of solids, such as the quantum Hall effect. In the initial part of the talk we shall review the schemes enabling to produce the artificial magnetic field for cold atoms using several light beams. We discuss the possibilities to create both the Abelian and also non-Abelian gauge potentials. Subsequently we shall talk on some recent studies of the effects due to the non-Abelian gauge potentials for cold atoms, such as their quasi-relativistic behaviour [1-3] and negative reflection [4].
[1] G. Juzeliunas, J. Ruseckas, M. Lindberg, L. Santos, and P. Öhberg, Phys. Rev. A 77 011802(R) (2008).
[2] J. Y. Vaishnav and C. W. Clark, Phys. Rev. Lett. 100, 153002 (2008) . [3] M. Merkl, F. E. Zimmer, G. Juzeliunas and P. Öhberg, Europhys. Lett. 83, 54002 (2008).
[4] G. Juzeliunas, J. Ruseckas, A. Jacob, L. Santos, P. Öhberg, Phys. Rev. Lett. 100, 200405 (2008).

Ultracold quantum gases in optical lattices offer a wide variety of manipulation techniques in order to control and engineer the interactions between particles on a lattice. By using non-standard lattice configurations e.g. the spin-spin interaction between neighbouring atoms can be engineered to create highly correlated or entangled quantum states in one- or two dimensions. Furthermore, we show how effective multi-particle interactions emerge from within the Hubbard model when higher lattice bands are taken into account. We have determined the interaction strengths of these multi-particle interactions up to the case of six-body interactions. Finally we report on novel experiments with single site and single atom resolution of ultracold atoms on a lattice and outline the novel prospects emerging from such advanced control possibilities.

One of the most serious problems raised by attempts to build devices based on qubits is the decoherence induced by the residual couplings between these small systems and their environment. Instead of trying to eliminate this quantum noise, an alternative strategy has been proposed some years ago by Kitaev, namely to build quantum systems which would be to a large extent insensitive to external perturbations. This is achieved by a deeply non-local coding of the information in terms of many-body wave-functions. I'll present a recent physical implementation of such protected qubits with small Josephson junction arrays. The experimental results obtained at Rutgers by M. Gershenson and his group show clearly that the protection mechanism works as expected for static perturbations. This system provides a direct simulation of a $Z_2$ lattice gauge theory in a strongly deconfined regime.

I will discuss how non-Abelian anyons may occur in nature and how braiding them suitably could lead to fault-tolerant quantum computation without any need for quantum error correction. Both atomic/molecular and solid state systems will be discussed as possible platforms for non-Abelian anyons and topological quantum computation. Experimental prospects for topological quantum computation will be discussed critically.

We present several instances where quantum and classical versions of Lattice Gauge Theories (LGT) play a fundamental role in quantum computation and strongly correlated systems. They include: i/ Random LGTs in Topological Color Codes for quantum computation. ii/ Unifying classical spin models with global and local symmetries; iii/ New phases of quantum matter with non-Abelian Optical Lattices.

Despite some 20 orders of magnitude difference in energy scales, the physics of ultracold strongly interacting cold atom systems and ultradense nuclear matter, in the form of a quark-gluon plasma, share features in common. This talk will review similarities in physics, and how one might learn about the phases of QCD from cold atoms.

Two-dimensional quantum magnets with frustration and competing interactions can sometimes exhibit Spin-liquid ground states with no symmetry breaking but gapless strongly interacting excitations. Among the most exotic are the (Spin) Bose-Metal phases which exhibit singular structure along surfaces in momentum space - ``Bose surfaces" - which are loosely analogous to a Fermi surface but offer no weakly interacting quasiparticle description. Such Bose-metals leave a distinctive fingerprint when studied on quasi-one-dimensional ladders, allowing a systematic approach via controlled numerical and analytical studies. By engineering different Fermi surfaces for the two spin species of cold atoms trapped in an optical lattice, it might be possible to access an exotic ``Cooper-pair Bose-Metal" phase.

The talk will discuss concepts for quantum phases of bosonic and/or fermionic quantum gases in triangular and hexagonal optical lattices.

Motivated by the description of the graphene electronic structure in terms of the relativistic Dirac equation, a generalization to four dimensions yields a strictly local fermion action describing two massless species and possessing an exact chiral symmetry. This is the minimum number of species required by the well known ``no-go'' theorems.

Ultra cold atoms are remarkable systems with a truly unprecedented level of experimental control and one application of this control is engineering the systems hamiltonian. To date this has focused mostly on the real-space potential that the atoms experience for example, multiple-well traps or optical lattice potentials. Here we present our experimental work which tailors the energy-momentum dispersion of the cold atoms. We couple different internal states of rubidium 87 via a momentum-selective Raman transition and load our system into the resulting adiabatic eigenstates. Using this technique we show the controlled modification of the energy-momentum dispersion leads to a synthetic vector potential. With the engineered vector potential in hand, we will see how suitable time dependence leads to synthetic electric fields and how spatial gradients can produce synthetic magnetic fields.

Following Feynman and as elaborated on by Lloyd, a universal quantum simulator (UQS) is a controlled quantum device which efficiently reproduces the dynamics of any other many particle quantum system with short range interactions. This dynamics can refer to both coherent Hamiltonian and dissipative open system evolution. Here we show that laser excited Rydberg atoms in large spacing optical or magnetic lattices provide an efficient implementation of a UQS for spin models involving (high order)n-body interactions. This includes the simulation of Hamiltonians of exotic spin models involving n-particle constraints such a the Kitaev toric code, color code, and string nets. In addition, it provides the ingredients for dissipative preparation of entangled states based on engineering n-particle reservoir couplings. The key basic building block of our architecture is efficient and high-fidelity n-qubit entangling gates via auxiliary Rydberg atoms, including a possible dissipative time step via optical pumping. This allows to mimic the time evolution of the system efficiently by a sequence of fast, parallel and high-fidelity n-particle coherent and dissipative Rydberg gates.

Introduction to the basic concepts of lattice gauge theories and methods to study them. Discussion of possible applications for cold gases.

Our recent success of preparing and manipulating a degenerate gas containing three different spin states of the fermionic 6Li atom will be reported. This three-component system is particularly interesting as the scattering lengths aij between the three states can be tuned widely using three broad and overlapping Feshbach resonances making it possible to create an SU(3)-symmetric gas. In first experiments it turned out that understanding the few-body physics, governed by the Efimov effect is an important prerequisite for the long-term goal of observing many-body physics such as for example a color superconducting phase in our gas. Our ideas on how we believe we can achieve such goals will be discussed.