Bose-Einstein condensation (BEC) has been observed in a dilute gas of sodium atoms. A Bose-Einstein condensate consists of a macroscopic population of the ground state of the system, and is a coherent state of matter. In an ideal gas, this phase transition is purely quantum-statistical. The study of BEC in weakly interacting systems which can be controlled and observed with precision holds the promise of revealing new macroscopic quantum phenomena that can be understood from first principles.

Bose-Einstein condensation in a gas of sodium atoms

In 1998, Bender and Boettcher found that a wide class of Hamiltonians, even though non-Hermitian, can still exhibit entirely real spectra provided that they obey parity-time requirements or PT symmetry. Here we demonstrate experimentally passive PT-symmetry breaking within the realm of optics. This phase transition leads to a loss induced optical transparency in specially designed pseudo-Hermitian guiding potentials.

Observation of PT-Symmetry Breaking in Complex Optical Potentials

In recent years, notions drawn from non-Hermitian physics and parity–time (PT) symmetry have attracted considerable attention. In particular, the realization that the interplay between gain and loss can lead to entirely new and unexpected features has initiated an intense research effort to explore non-Hermitian systems both theoretically and experimentally. Here we review recent progress in this emerging field, and provide an outlook to future directions and developments. This Review Article outlines the exploration of the interplay between parity–time symmetry and non-Hermitian physics in optics, plasmonics and optomechanics.

Non-Hermitian physics and PT symmetry

Annals of physics

The bulk-boundary correspondence is among the central issues of non-Hermitian topological states. We show that a previously overlooked "non-Hermitian skin effect" necessitates redefinition of topological invariants in a generalized Brillouin zone. The resultant phase diagrams dramatically differ from the usual Bloch theory. Specifically, we obtain the phase diagram of the non-Hermitian Su-Schrieffer-Heeger model, whose topological zero modes are determined by the non-Bloch winding number instead of the Bloch-Hamiltonian-based topological number. Our work settles the issue of the breakdown of conventional bulk-boundary correspondence and introduces the non-Bloch bulk-boundary correspondence.

Edge States and Topological Invariants of Non-Hermitian Systems.

The observation of $\mathcal{PT}$ symmetry in a coupled optical waveguide system that involves a complex refractive index has been demonstrated impressively in the experiment by R\"uter et al. [Nat. Phys. 6, 192 (2010)]. This is, however, only an optical analog of a quantum system, and it would be highly desirable to observe the manifestation of $\mathcal{PT}$ symmetry and the resulting properties also in a real, experimentally accessible, quantum system. Following a suggestion by Klaiman et al. [Phys. Rev. Lett. 101, 080402 (2008)], we investigate a $\mathcal{PT}$-symmetric arrangement of a Bose-Einstein condensate in a double-well potential, where in one well cold atoms are injected while in the other particles are extracted from the condensate. We investigate, in particular, the effects of the nonlinearity in the Gross-Pitaevskii equation on the $\mathcal{PT}$ properties of the condensate. To study these effects we analyze a simple one-dimensional model system in which the condensate is placed into two $\mathcal{PT}$-symmetric $\ensuremath{\delta}$-function traps. The analysis will serve as a useful guide for studies of the behavior of Bose-Einstein condensates in realistic $\mathcal{PT}$-symmetric double wells.

Model of a PT -symmetric Bose-Einstein condensate in a δ -function double-well potential

We report the existence of exceptional points for the hydrogen atom in crossed magnetic and electric fields in numerical calculations. The resonances of the system are investigated and it is shown how exceptional points can be found by exploiting characteristic properties of the degeneracies, which are branch point singularities. A possibility for the observation of exceptional points in an experiment with atoms is proposed.

Exceptional Points in Atomic Spectra

Non-Hermitian systems with $\mathcal{PT}$ symmetry can possess purely real eigenvalue spectra. In this work two one-dimensional systems with two different topological phases, the topological nontrivial phase (TNP) and the topological trivial phase (TTP), combined with $\mathcal{PT}$-symmetric non-Hermitian potentials are investigated. The models of choice are the Su-Schrieffer-Heeger (SSH) model and the Kitaev chain. The interplay of a spontaneous $\mathcal{PT}$-symmetry breaking due to gain and loss with the topological phase is different for the two models. The SSH model undergoes a $\mathcal{PT}$-symmetry breaking transition in the TNP immediately with the presence of a nonvanishing gain and loss strength $\ensuremath{\gamma}$, whereas the TTP exhibits a parameter regime in which a purely real eigenvalue spectrum exists. For the Kitaev chain the $\mathcal{PT}$-symmetry breaking is independent of the topological phase. We show that the topologically interesting states---the edge states---are the reason for the different behaviors of the two models and that the intrinsic particle-hole symmetry of the edge states in the Kitaev chain is responsible for a conservation of $\mathcal{PT}$ symmetry in the TNP.

Relation between PT -symmetry breaking and topologically nontrivial phases in the Su-Schrieffer-Heeger and Kitaev models

We investigate dissipative extensions of the Su-Schrieffer-Heeger model with regard to different approaches of modeling dissipation. In doing so, we use two distinct frameworks to describe the gain and loss of particles: One uses Lindblad operators within the scope of Lindblad master equations, and the other uses complex potentials as an effective description of dissipation. The reservoirs are chosen in such a way that the non-Hermitian complex potentials are $\mathcal{PT}$-symmetric. From the effective theory we extract a state which has similar properties as the nonequilibrium steady state following from Lindblad master equations with respect to lattice site occupation. We find considerable similarities in the spectra of the effective Hamiltonian and the corresponding Liouvillian. Further, we generalize the concept of the Zak phase to the dissipative scenario in terms of the Lindblad description and relate it to the topological phases of the underlying Hermitian Hamiltonian.

Topological invariants in dissipative extensions of the Su-Schrieffer-Heeger model

The time-independent nonlinear Schrodinger equation is solved for two attractive delta-function-shaped potential wells where an imaginary loss term is added in one well, and a gain term of the same size but with opposite sign in the other. We show that for vanishing nonlinearity the model captures all the features known from studies of parity–time-(-) symmetric optical waveguides, e.g., the coalescence of modes at an exceptional point at a critical value of the loss/gain parameter, and the breaking of symmetry beyond. With the nonlinearity present, the equation is a model for a Bose–Einstein condensate with loss and gain in a double-well potential. We find that the nonlinear Hamiltonian picks as stationary eigenstates exactly such solutions which render the nonlinear Hamiltonian itself symmetric, but observe coalescence and bifurcation scenarios different from those known from linear -symmetric Hamiltonians.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Quantum physics with non-Hermitian operators'.

https://iopscience.iop.org/article/10.1088/1751-8113/45/44/444008/pdf

Holger Cartarius

Papers

Model of a PT -symmetric Bose-Einstein condensate in a δ -function double-well potential

Exceptional Points in Atomic Spectra

Relation between PT -symmetry breaking and topologically nontrivial phases in the Su-Schrieffer-Heeger and Kitaev models

Topological invariants in dissipative extensions of the Su-Schrieffer-Heeger model

Nonlinear Schrödinger equation for a ${\mathcal{PT}}$-symmetric delta-function double well