Günter Wunner

Hydrogen atoms in arbitrary magnetic fields: I. Energy levels and wavefunctions?

The energy values of many low-lying states of the one-electron problem in the presence of a homogeneous magnetic field of arbitrary strength (0<B<or=4.7*108 T) are calculated with high numerical accuracy for a sufficiently dense mesh of B. The wavefunctions are expanded either in terms of spherical harmonics (weak and moderate fields) or in terms of Landau states (strong and very strong fields), with r- or z-dependent expansion functions that are determined with the use of an adopted version of the MCHF code of Froese Fischer (1978). At intermediate field strengths up to 24 expansion terms are included. The structural change of the wavefunctions with magnetic field is discussed quantitatively for a few representative states. As an application, the splittings of the components of the Lyman- alpha , beta , and the Balmer- alpha lines of the hydrogen atom are presented (including the effects of the finite proton mass) as continuous functions of the field strength over the whole range of B considered.

Eigenvalue structure of a Bose–Einstein condensate in a

We study a Bose–Einstein condensate in a -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 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 analyse 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.

\mathcal {PT}

A PT-symmetric Bose-Einstein condensate can be theoretically described using a complex optical potential, however, the experimental realization of such an optical potential describing the coherent in- and outcoupling of particles is a nontrivial task. We propose an experiment for a quantum mechanical realization of a PT-symmetric system, where the PT-symmetric currents of a two-well system are implemented by coupling two additional wells to the system, which act as particle reservoirs. In terms of a simple four-mode model we derive conditions under which the two middle wells of the Hermitian four-well system behave exactly as the two wells of the PT-symmetric system. We apply these conditions to calculate stationary solutions and oscillatory dynamics. By means of frozen Gaussian wave packets we relate the Gross-Pitaevskii equation to the four-mode model and give parameters required for the external potential, which provides approximate conditions for a realistic experimental setup.

-symmetric double well

The authors define a new statistic, the parametric number variance, which measures the correlation of fluctuations in energy levels as a parameter external to the system is varied. A semiclassical formula is obtained and regimes of universal and system dependent behaviour are predicted. Numerical calculations of the PNV in two model systems are found to be in good agreement with the semiclassical theory.

Hermitian four-well potential as a realization of a PT-symmetric system

For pt.I see ibid., vol.17, p.29-52 (1984). Wavelengths, dipole strengths, oscillator strengths and transition probabilities for electromagnetic transitions between low-lying states of the hydrogen atom embedded in a magnetic field of arbitrary strength (0<B<or=4.70*108T) are calculated with high numerical accuracy for a sufficiently dense mesh of B values using the wavefunctions and energy values determined in paper I by expanding the wavefunctions in terms of spherical harmonics (weak and moderate fields), or in terms of Landau states (strong and very strong fields). Effects caused by the finiteness of the proton mass are taken into account, and the scaling laws are quoted by the aid of which the results presented can be utilised to cover quantitatively the whole hydrogenic sequence in intense magnetic fields; finally, the quality of the results of previous calculations is checked.

The parametric number variance

We show how non-Hermitian potentials used to describe probability gain and loss in effective theories of open quantum systems can be achieved by a coupling of the system to an environment. We do this by coupling a Bose-Einstein condensate (BEC) trapped in an attractive double-δ potential to a condensate fraction outside the double well. We investigate which requirements have to be imposed on possible environments with a linear coupling to the system. This information is used to determine an environment required for stationary states of the BEC. To investigate the stability of the system we use fully numerical simulations of the dynamics. It turns out that the approach is viable and possible setups for the realization of a PT-symmetric potential for a BEC are accessible. Vulnerabilities of the whole system to small perturbations can be attributed to the singular character of the simplified δ-shaped potential used in our model.

Hydrogen atoms in arbitrary magnetic fields. II. Bound-bound transitions

Bose?Einstein condensates are described in a mean-field approach by the nonlinear Gross?Pitaevskii equation and exhibit phenomena of nonlinear dynamics. The stationary states can undergo bifurcations in such a way that two or more eigenvalues and the corresponding wavefunctions coalesce at critical values of external parameters. For example, in condensates without long-range interactions a stable and an unstable state are created in a tangent bifurcation when the scattering length of the contact interaction is varied. At the critical point, the coalescing states show the properties of an exceptional point. In dipolar condensates fingerprints of a pitchfork bifurcation have been discovered by Rau et?al (2010 Phys. Rev.A 81 031605). We present a method to uncover all states participating in a pitchfork bifurcation, and investigate in detail the signatures of exceptional points related to bifurcations in dipolar condensates. For the perturbation by two parameters, namely the scattering length and a parameter breaking the cylindrical symmetry of the harmonic trap, two cases leading to different characteristic eigenvalue and eigenvector patterns under cyclic variations of the parameters need to be distinguished. The observed structures resemble those of three coalescing eigenfunctions obtained by Demange and Graefe (2012 J. Phys. A: Math. Theor. 45 025303) using perturbation theory for non-Hermitian operators in a linear model. Furthermore, the splitting of the exceptional point under symmetry breaking in either two or three branching singularities is examined. Characteristic features are observed when one, two or three exceptional points are encircled simultaneously.

Coupling approach for the realization of a PT -symmetric potential for a Bose-Einstein condensate in a double well

Non-Hermitian systems with

Bifurcations and exceptional points in dipolar Bose–Einstein condensates

\mathcal{PT}

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

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