No abstract

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. PACS numbers: 32.60.+i, 32.80.Fb The appearance of the coalescence of two eigenstates, so-called exceptional points [1], in physical systems described by non-Hermitian matrices has attracted growing interest [2,3,4,5,6]. Typical systems in which such a phenomenon can occur are open quantum systems with decaying unbound states. One possibility to describe these open quantum systems are non-Hermitian Hamiltonians obtained with the complex rotation method [7]. In this case, the resonances appear as complex eigenvalues whose real and imaginary parts are connected with the energy and the resonance width, respectively. The eigenvectors have not to be orthogonal in contrast to Hermitian Hamiltonians describing bound states in quantum mechanics. In particular, it is possible that the eigenspace for two degenerate eigenvalues is only onedimensional, i.e., there is only one linear independent eigenvector. If the system of interest depends on a complex parameter λ (or two real parameters), a branch point singularity of two eigenstates can appear at critical parameter values λ c , which are called exceptional points. Exceptional points have been discovered in a broad variety of physical systems. Among them are acoustical systems [8], atoms in optical lattices [4,9], and complex atoms in laser fields [10]. Detailed experiments have been carried out with resonances in microwave cavities [5,6,11]. However, up to now exceptional points have not been found in atoms in static external fields. The main reason is that there is only one parameter in the cases studied most intensely, viz. atoms either in a magnetic or in an electric field. For the occurrence of exceptional points, the parameter space has to be at least two-dimensional, i.e., at least two real parameters are required, which can be represented by crossed magnetic and electric fields. Atoms in static external magnetic and electric fields are fundamental physical systems. As real quantum systems they are accessible both with experimental and theoretical methods and have been used for comparisons with semiclassical theories [12,13,14]. They are ideally suited to study the influence of exceptional points on quantum systems. E.g., the occurrence of phenomena like Ericson fluctuations in photoionization spectra has been demonstrated both in numerical studies [15,16] and experiments [17].In this Letter, we investigate numerically the resonances of the hydrogen atom in static magnetic and electric fields and report the first detection of exceptional points in this system. The confirmation of the existence of exceptional points supplements the richness o...

The existence of 𝒫𝒯 symmetric wave functions describing Bose‐Einstein condensates in a one‐dimensional and a fully three‐dimensional double‐well setup is investigated theoretically. When particles are removed from one well and coherently injected into the other the external potential is 𝒫𝒯 symmetric. We solve the underlying Gross‐Pitaevskii equation by way of the time‐dependent variational principle (TDVP) and show that the 𝒫𝒯 symmetry of the external potential is preserved by both the wave functions and the nonlinear Hamiltonian as long as eigenstates with real eigenvalues are obtained. 𝒫𝒯 broken solutions of the time‐independent Gross‐Pitaevskii equation are also found but have no physical relevance. To prove the applicability of the TDVP we compare its results with numerically exact solutions in the one‐dimensional case. The linear stability analysis and the temporal evolution of condensate wave functions demonstrate that the 𝒫𝒯 symmetric condensates are stable and should be observable in an experiment.

Abstract. We study a Bose-Einstein condensate in a PT -symmetric double-well potential where particles are coherently injected in one well and removed from the other well. In mean-field approximation the condensate is described by the Gross-Pitaevskii equation thus falling into the category of nonlinear non-Hermitian quantum systems. After extending the concept of PT symmetry to such systems, we apply an analytic continuation to the Gross-Pitaevskii equation from complex to bicomplex numbers and show a thorough numerical investigation of the four-dimensional bicomplex eigenvalue spectrum. The continuation introduces additional symmetries to the system which are confirmed by the numerical calculations and furthermore allows us to analyze the bifurcation scenarios and exceptional points of the system. We present a linear matrix model and show the excellent agreement with our numerical results. The matrix model includes both exceptional points found in the double-well potential, namely an EP2 at the tangent bifurcation and an EP3 at the pitchfork bifurcation. When the two bifurcation points coincide the matrix model possesses four degenerate eigenvectors. Close to that point we observe the characteristic features of four interacting modes in both the matrix model and the numerical calculations, which provides clear evidence for the existence of an EP4.

Non-Hermitian systems with 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 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 PT -symmetry breaking due to gain and loss with the topological phase is different for the two models. The SSH model undergoes a PT -symmetry breaking transition in the TNP immediately with the presence of a non-vanishing gain and loss strength γ, whereas the TTP exhibits a parameter regime in which a purely real eigenvalue spectrum exists. For the Kitaev chain the PT -symmetry breaking is independent of the topological phase. We show that the topological 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 PT symmetry in the TNP.