We show that the bulk-boundary correspondence for topological insulators can be modified in the presence of non-Hermiticity. We consider a one-dimensional tight-binding model with gain and loss as well as long-range hopping. The system is described by a non-Hermitian Hamiltonian that encircles an exceptional point in momentum space. The winding number has a fractional value of 1=2. There is only one dynamically stable zero-energy edge state due to the defectiveness of the Hamiltonian. This edge state is robust to disorder due to protection by a chiral symmetry. We also discuss experimental realization with arrays of coupled resonator optical waveguides.

Quantum phase transitions are usually studied in terms of Hermitian Hamiltonians. However, cold-atom experiments are intrinsically non-Hermitian because of spontaneous decay. Here, we show that nonHermitian systems exhibit quantum phase transitions that are beyond the paradigm of Hermitian physics. We consider the non-Hermitian XY model, which can be implemented using three-level atoms with spontaneous decay. We exactly solve the model in one dimension and show that there is a quantum phase transition from short-range order to quasi-long-range order despite the absence of a continuous symmetry in the Hamiltonian. The ordered phase has a frustrated spin pattern. The critical exponent ν can be 1 or 1=2. Our results can be seen experimentally with trapped ions, cavity QED, and atoms in optical lattices.

The van der Pol oscillator is the prototypical self-sustained oscillator and has been used to model nonlinear behavior in biological and other classical processes. We investigate how quantum fluctuations affect phase locking of one or many van der Pol oscillators. We find that phase locking is much more robust in the quantum model than in the equivalent classical model. Trapped-ion experiments are ideally suited to simulate van der Pol oscillators in the quantum regime via sideband heating and cooling of motional modes. We provide realistic experimental parameters for 171Yb+ achievable with current technology.

We study a driven-dissipative system of atoms in the presence of laser excitation to a Rydberg state and spontaneous emission. The atoms interact via the blockade effect, whereby an atom in the Rydberg state shifts the Rydberg level of neighboring atoms. We use mean-field theory to study how the Rydberg population varies in space. As the laser frequency changes, there is a continuous transition between the uniform and antiferromagnetic phases. The nonequilibrium nature also leads to a novel oscillatory phase and bistability between the uniform and antiferromagnetic phases. The behavior of matter far from equilibrium is a fascinating area of study. The presence of driving and dissipation can lead to remarkable phenomena that are not possible in equilibrium. This has motivated much research on nonequilibrium physics in classical systems, such as fluids, chemical reactions, and biological media [1,2]. An interesting question is: what novel phases appear when a quantum system is driven far from equilibrium? Recent cold-atom experiments have studied equilibrium quantum systems in great detail, but they are also a natural setting to study nonequilibrium quantum systems due to the tunability of driving and dissipation [3][4][5][6][7][8].In this paper, we study a nonequilibrium many-body quantum system interacting via Rydberg blockade. A Rydberg atom is one whose electron is excited to a high energy level n. The van der Waals interaction between two atoms in identical Rydberg levels scales as n 11 , and this leads to a blockade effect for large n: when one atom is excited to the Rydberg state, it prevents nearby atoms from being excited. This is the basis for quantum information processing schemes with Rydberg atoms [9][10][11][12][13][14] and a variety of novel phenomena [15][16][17][18][19][20][21][22]. In these schemes, spontaneous emission should be minimized, since it destroys quantum information. On the other hand, spontaneous emission as a source of dissipation may lead to interesting physics, and it can actually be tuned by using different Rydberg levels.We study a lattice of atoms continuously excited to the Rydberg state and spontaneously decaying back to the ground state. Consider the Rydberg population of each atom; that is, the fraction of time it spends in the Rydberg state. What is the spatial distribution of the Rydberg population in steady state? Using mean-field theory, we show that, as the laser frequency is varied, the system undergoes a continuous transition between a phase with spatially uniform population and a phase with higher population on every other atom. We call the latter the antiferromagnetic phase, since a two-level atom is formally equivalent to a spin-1/2 particle (ground and excited states correspond to down and up spins, respectively) [23]. The nonequilibrium nature also leads to an oscillatory phase, in which the Rydberg population oscillates periodically in time, and to bistability between the uniform and antiferromagnetic phases. Simulations of the full quantum model in one dimensi...