Recently, the identification of non-equilibrium signatures of topology in the dynamics of such systems has attracted particular attention [3][4][5][6] . Here, we experimentally study the dynamical evolution of the wavefunction using time-and momentum-resolved full state tomography for spin-polarized fermionic atoms in driven optical lattices 7 . We observe the appearance, movement and annihilation of dynamical vortices in momentum space after sudden quenches close to the topological phase transition. These dynamical vortices can be interpreted as dynamical Fisher zeros of the Loschmidt amplitude 8 , which signal a so-called dynamical phase transition 9,10 . Our results pave the way to a deeper understanding of the connection between topological phases and non-equilibrium dynamics.The discovery of topological matter has revolutionized our understanding of band theory: not only are the dispersions of the energy bands important, but so is the geometry of the corresponding eigenstates 1 . The non-local nature of the topological invariants characterizing such phases goes beyond the Landau paradigm of local order parameters and leads to topological protection, for example, against disorder. Ultracold quantum gases in optical lattices allow for controlled studies of archetypal topological models [11][12][13][14] . In addition, compared with, for example condensed-matter systems, they also allow for detailed studies of the relation between dynamics and topology as the timescales are experimentally easier to access. Dynamical studies of driven systems have recently attracted attention in terms of their high T c superconductivity 15 . A particular challenge is to identify non-equilibrium signatures of topology in the dynamics of highly excited states 3,4,16 . Here, we observe the time evolution of the wavefunction after a sudden quench in a Haldanelike model and find dynamical vortices as a signature of the topological nature of the underlying ground state.In the experiments described here, the state tomography method allows mapping of the full quantum-mechanical wavefunction of non-interacting ultracold fermionic quantum gases in an optical lattice for any time after a sudden quench of the system close to or into a Chern insulating phase. As a key result, we identify in an intense series of measurements the appearance, movement and annihilation In the initial system, tunnelling J AB between the A and B sites is suppressed by a large energy offset. In the final Floquet system, tunnelling is re-established by means of near-resonant driving. b, At each momentum, the Hamiltonian describes the coupling between the states of the A and B sublattices, and can be visualized on a Bloch sphere. In the initial system, the Hamiltonian for all momenta points to the north pole, whereas in the Floquet system, the Hamiltonian covers a large surface of the Bloch sphere. c, Phase diagram for the Floquet Hamiltonian as a function of shaking amplitude and detuning with respect to the sublattice offset for the case of circular lattice shaking...

Emulation of gauge fields for ultracold atoms provides access to a class of exotic states arising in strong magnetic fields. Here we report on the experimental realisation of tunable staggered gauge fields in a periodically driven triangular lattice. For maximal staggered magnetic fluxes, the doubly degenerate superfluid ground state breaks both a discrete Z 2 (Ising) symmetry and a continuous U (1) symmetry. By measuring an Ising order parameter, we observe a thermally driven phase transition from an ordered antiferromagnetic to an unordered paramagnetic state and textbook-like magnetisation curves. Both the experimental and theoretical analysis of the coherence properties of the ultracold gas demonstrate the strong influence of the Z 2 symmetry onto the condensed phase.Phase transitions in systems with combined continuous and discrete symmetries are fundamentally different from their purely continuous and discrete counterparts. The interplay between different types of excitations in the various degrees of freedom can lead to a complex behaviour and coupling of the associated order parameters [1][2][3][4][5]. A paradigm example is the fully frustrated XY model on a triangular lattice. It combines vector spin-type symmetries with discrete chiral degrees of freedom, which result in the famous spin-chirality coupling at low temperatures [6]. However, experimental studies in solid-state systems are challenging in view of implementing and isolating an XY model Hamiltonian [7][8][9].Ultracold bosonic quantum gases in optical lattices, on the other hand, constitute a highly versatile system with an extraordinary degree of control [10,11]. In particular, the recent experimental realisations of artificial gauge potentials for bulk [12][13][14][15] and optical lattice systems [16][17][18][19] allow for the investigation of new physical regimes, not realisable in condensed matter systems.Here, we demonstrate the realisation of a system with combined U (1) and Z 2 symmetries using ultracold atoms submitted to artificial gauge fields. Our experimental setup consists of an ultracold gas of 87 Rb atoms held in a two-dimensional triangular lattice [20] (see Fig. 1a). At each lattice site j with particle number N j , the weakly interacting superfluid gas can be described by the local order parameter a j = N j e iϕj . As a central aspect, the local phases ϕ j are mapped onto classical XY spins s j = (cos ϕ j , sin ϕ j ), where the tunneling matrix elements between neighbouring lattice sites correspond to the spinspin coupling parameters. Such classical spins possess a continuous degree of freedom. In presence of a long-range order, analogous to the onset of Bose-Einstein condensation (BEC), the order parameter assumes an arbitrary, but fixed phase, thus breaking the continuous U (1) symmetry [21].Beyond that, we experimentally engineer strong staggered gauge fields, which generate an additional discrete Z 2 symmetry in our system. The resulting magnetic flux induces cyclotron-like mass currents around each plaquette. The two poss...

The Dicke model with a weak dissipation channel is realized by coupling a Bose-Einstein condensate to an optical cavity with ultranarrow bandwidth. We explore the dynamical critical properties of the Hepp-Lieb-Dicke phase transition by performing quenches across the phase boundary. We observe hysteresis in the transition between a homogeneous phase and a self-organized collective phase with an enclosed loop area showing power-law scaling with respect to the quench time, which suggests an interpretation within a general framework introduced by Kibble and Zurek. The observed hysteretic dynamics is well reproduced by numerically solving the mean-field equation derived from a generalized Dicke Hamiltonian. Our work promotes the understanding of nonequilibrium physics in open many-body systems with infinite range interactions.dynamical phase transition | critical behavior | Dicke model | quantum gas | cavity QED A lthough equilibrium phases in quantum many-body systems have been explored for a long time with great success, nonequilibrium phenomena in such systems are far less well understood (1). A paradigm for exploring nonequilibrium dynamics is the quench scenario, where a system parameter is subjected to a sudden change between two values associated with different equilibrium phases. Quantum degenerate atomic gases with their unique degree of control are particularly adapted for experimental quench studies (2, 3). For isolated quantum manybody systems a wealth of theoretical and experimental investigations of quench dynamics has appeared recently (4-11). A natural extension of such studies is to consider driven open systems, where dynamical equilibrium states can arise via a competition between dissipation and driving, and nonequilibrium transitions between such phases can occur as a function of some external control parameter (12-15). A nearly ideal experimental platform for this endeavor are quantum degenerate atomic gases subjected to optical high-finesse cavities, where the usual extensive control in cold gas systems can be combined with a precisely engineered coupling to the external bath of vacuum radiation modes (16).Here, we study a dynamical phase transition in the open Dicke model emulated in an atom-cavity system prepared near zero temperature. The Dicke model is a paradigmatic scenario of quantum many-body physics, still subject to intensive research despite a history more than half a century long (17-28). It describes the interaction of N two-level atoms with a common mode of the electromagnetic radiation field. Hepp and Lieb already pointed out in the 1970s that upon varying the coupling strength, this model possesses a second-order equilibrium quantum phase transition between a homogeneous phase, in which each atom interacts separately with the radiation mode, and a collective phase in which all atomic dipoles align to form a macroscopic dipole moment (19,22). It has been early suspected that the critical properties of the externally pumped open Dicke model should give rise to nonlinear hysteretic ...

We use bosonization approach to investigate quantum phases in mixtures of bosonic and fermionic atoms confined in one dimensional optical lattices. The phase diagrams can be well understood in terms of polarons, which correspond to atoms that are "dressed" by screening clouds of the other atom species. For a mixture of single species of fermionic and bosonic atoms we find a charge density wave phase, a phase with fermion pairing, and a regime of phase separation. For a mixture of two species of fermionic atoms and one species of bosonic atoms we obtain spin and charge density wave phases, a Wigner crystal phase, singlet and triplet paired states of fermions, and a phase separation regime. Equivalence between the Luttinger liquid description of polarons and the canonical polaron transformation is established and the techniques to detect the resulting quantum phases are discussed.Mixtures of ultra-cold bosonic and fermionic atoms, that have recently become accessible experimentally, represent a promising new system for studying strongly correlated many-body physics [1]. Bosonic atoms mediate interactions between fermions and allow efficient cooling of the system [2]. Several novel phenomena have been predicted theoretically for Boson-Fermion mixtures (BFM) including pairing of fermions [3], formation of composite particles [4], spontaneous breaking of translational symmetry in optical lattices [5] and appearance of charge density wave (CDW) [6]. Most of these theoretical studies relied on integrating out bosonic degrees of freedom to obtain an effective interaction between fermions, and then using a mean-field approach to investigate many-body states [3]. This approach, however, becomes unreliable in the regime of strong interactions. In particular, it fails in low-dimensional systems due to enhanced fluctuations and non-perturbative effects of interactions.In this paper we use bosonization method [8,9] to investigate one dimensional (1D) BFM. The resulting quantum phases can be understood by introducing polarons, i.e. atoms of one species surrounded by screening clouds of the other species. Such dressed quasiparticles exhibit effective interactions and modified effective masses. In our analysis the polarons emerge as quasiparticles with the slowest decaying correlation functions while quantum phases of the system arise from a competition of various ordering instabilities of such polarons. The phase diagrams we obtain (Figs. 1-3) show a remarkable similarity to the Luttinger liquid phase diagrams of 1D interacting electron systems [11], suggesting that 1D BFM may be understood as Luttinger liquids of polarons.In Fig. 1 we show a phase diagram for a mixture of bosons and spinless fermions as a function of experimentally controlled parameters: the scattering length between bosons and fermions (a bf ) and the strength of the longitudinal optical lattice for bosonic atoms (V b, ) [12]. For relatively weak boson-fermion interactions and slow bosons (i.e. strong optical lattice for bosonic atoms)