SU(N) symmetry can emerge in a quantum system with N single-particle spin states when spin is decoupled from inter-particle interactions. So far, only indirect evidence for this symmetry exists, and the scattering parameters remain largely unknown. Here we report the first spectroscopic observation of SU(N=10) symmetry in 87 Sr using the state-of-the-art measurement precision offered by an ultra-stable laser. By encoding the electronic orbital degree of freedom in two clock states, while keeping the system open to 10 nuclear spin sublevels, we probe the non-equilibrium two-orbital SU(N) magnetism via Ramsey spectroscopy of atoms confined in an array of two-dimensional optical traps. We study the spin-orbital quantum dynamics and determine all relevant interaction parameters. This work prepares for using alkaline-earth atoms as test-beds for iconic orbital models. Keywords: SU(N), Orbital, MagnetismSymmetries play a fundamental role in the laws of nature. A very prominent example is SU(N) symmetry as the source of intriguing features of quantum systems. For instance, the SU(3) symmetry of quantum chromodynamics governs the behavior of quarks and gluons. When generalized to large N, it is anticipated to give rise to a large degeneracy and exotic many-body behaviors. Owing to the strong decoupling between the electronic-orbital and nuclear-spin degrees of freedom [1], alkaline-earth (-like) atoms, prepared in the two lowest electronic states (clock states), are predicted to obey SU(N=2I+1) symmetry with respect to the nuclear spin (I) [2][3][4][5]. Thanks to this symmetry, in addition to their use as ideal time keepers [6] and quantum information processors [7], alkaline earth atoms are emerging as a unique platform for the investigation of high-energy lattice gauge theories [8], for testing iconic orbital models used to describe transition metal oxides, heavy fermion compounds, and spin liquid phases [9], and for the observation of exotic topological phases [5,10]. Progress towards these goals includes the production of quantum degenerate gases for calcium [11] and all stable isotopes of strontium and ytterbium [12,13], the capability of imaging individual spin components via optical Stern-Gerlach methods [14], and control of interactions with optical Feshbach resonances [12,15,16]. Furthermore, the best atomic clock has been produced with lattice confined Sr atoms [6], and many-body spin dynamics have been studied directly in that system [17].However, thus far only indirect evidence for SU(N) symmetry exists, including inference from suppressed nuclear spin-relaxation rates [14], reduced temperatures in a Mott insulator for increased number of spin states [18], and the changing character of a strongly-interacting onedimensional fermionic system as a function of N [19]. Furthermore, these observations are limited to the electronic ground state. The corresponding ground-state swave scattering parameter, a gg , has been determined from photo-association [20] and rovibrational spectroscopy [21], but the excited st...

Topological states of fermionic matter can be induced by means of a suitably engineered dissipative dynamics. Dissipation then does not occur as a perturbation, but rather as the main resource for many-body dynamics, providing a targeted cooling into topological phases starting from arbitrary initial states. We explore the concept of topological order in this setting, developing and applying a general theoretical framework based on the system density matrix that replaces the wave function appropriate for the discussion of Hamiltonian ground-state physics. We identify key analogies and differences to the more conventional Hamiltonian scenario. Differences essentially arise from the fact that the properties of the spectrum and of the state of the system are not as tightly related as in the Hamiltonian context. We provide a symmetry-based topological classification of bulk steady states and identify the classes that are achievable by means of quasi-local dissipative processes driving into superfluid paired states. We also explore the fate of the bulk-edge correspondence in the dissipative setting and demonstrate the emergence of Majorana edge modes. We illustrate our findings in one-and two-dimensional models that are experimentally realistic in the context of cold atoms. 5

We introduce a family of states, the fPEPS, which describes fermionic systems on lattices in arbitrary spatial dimensions. It constitutes the natural extension of another family of states, the PEPS, which efficiently approximate ground and thermal states of spin systems with short-range interactions. We give an explicit mapping between those families, which allows us to extend previous simulation methods to fermionic systems. We also show that fPEPS naturally arise as exact ground states of certain fermionic Hamiltonians. We give an example of such a Hamiltonian, exhibiting criticality while obeying an area law.

We present evidence for Majorana edge states in a number conserving theory describing a system of spinless fermions on two wires that are coupled by pair hopping. Our analysis is based on a combination of a qualitative low energy approach and numerical techniques using the density matrix renormalization group. In addition, we discuss an experimental realization of pair-hopping interactions in cold atom gases confined in optical lattices.

The notion of "paired" fermions is central to important condensed matter phenomena such as superconductivity and superfluidity. While the concept is widely used and its physical meaning is clear there exists no systematic and mathematical theory of pairing which would allow to unambiguously characterize and systematically detect paired states. We propose a definition of pairing and develop methods for its detection and quantification applicable to current experimental setups. Pairing is shown to be a quantum correlation different from entanglement, giving further understanding in the structure of highly correlated quantum systems. In addition, we will show the resource character of paired states for precision metrology, proving that the BCS states allow phase measurements at the Heisenberg limit.

We present a unifying theoretical framework that describes recently observed many-body effects during the interrogation of an optical lattice clock operated with thousands of fermionic alkaline earth atoms. The framework is based on a many-body master equation that accounts for the interplay between elastic and inelastic p-wave and s-wave interactions, finite temperature effects and excitation inhomogeneity during the quantum dynamics of the interrogated atoms. Solutions of the master equation in different parameter regimes are presented and compared. It is shown that a general solution can be obtained by using the so called Truncated Wigner Approximation which is applied in our case in the context of an open quantum system. We use the developed framework to model the density shift and decay of the fringes observed during Ramsey spectroscopy in the JILA 87 Sr and NIST 171 Yb optical lattice clocks. The developed framework opens a suitable path for dealing with a variety of strongly-correlated and driven open-quantum spin systems.arXiv:1310.5248v2 [cond-mat.quant-gas]

The quest for realization of non-Abelian phases of matter, driven by their possible use in fault-tolerant topological quantum computing, has been spearheaded by recent developments in p-wave superconductors. The chiral p x þ ip y -wave superconductor in two-dimensions exhibiting Majorana modes provides the simplest phase supporting non-Abelian quasiparticles and can be seen as the blueprint of fractional topological order. Alternatively, Kitaev's Majorana wire has emerged as an ideal toy model to understand Majorana modes. Here we present a way to make the transition from Kitaev's Majorana wires to twodimensional p-wave superconductors in a system with cold atomic gases in an optical lattice. The main idea is based on an approach to generate p-wave interactions by coupling orbital degrees of freedom with strong s-wave interactions. We demonstrate how this design can induce Majorana modes at edge dislocations in the optical lattice, and we provide an experimentally feasible protocol for the observation of the non-Abelian statistics.

We introduce a one-dimensional system of fermionic atoms in an optical lattice whose phase diagram includes topological states of different symmetry classes with a simple possibility to switch between them. The states and topological phase transitions between them can be identified by looking at their zero-energy edge modes which are Majorana fermions. We propose several universal methods of detecting the Majorana edge states, based on their genuine features: the zero-energy, localized character of the wave functions and the induced non-local fermionic correlations.