Jörg Main

Three novel high-resolution nonlinear methods for fast signal processing

Three novel nonlinear parameter estimators are devised and implemented for accurate and fast processing of experimentally measured or theoretically generated time signals of arbitrary length. The new techniques can also be used as powerful tools for diagonalization of large matrices that are customarily encountered in quantum chemistry and elsewhere. The key to the success and the common denominator of the proposed methods is a considerably reduced dimensionality of the original data matrix. This is achieved in a preprocessing stage called beamspace windowing or band-limited decimation. The methods are decimated signal diagonalization (DSD), decimated linear predictor (DLP), and decimated Pade approximant (DPA). Their mutual equivalence is shown for the signals that are modeled by a linear combination of time-dependent damped exponentials with stationary amplitudes. The ability to obtain all the peak parameters first and construct the required spectra afterwards enables the present methods to phase correc...

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

Abstract Harmonic inversion is introduced as a powerful tool for both the analysis of quantum spectra and semiclassical periodic orbit quantization. The method allows one to circumvent the uncertainty principle of the conventional Fourier transform and to extract dynamical information from quantum spectra which has been unattainable before, such as bifurcations of orbits, the uncovering of hidden ghost orbits in complex phase space, and the direct observation of symmetry breaking effects. The method also solves the fundamental convergence problems in semiclassical periodic orbit theories – for both the Berry–Tabor formula and Gutzwillers trace formula – and can therefore be applied as a novel technique for periodic orbit quantization, i.e., to calculate semiclassical eigenenergies from afinite set of classical periodic orbits. The advantage of periodic orbit quantization by harmonic inversion is the universality and wide applicability of the method, which will be demonstrated in this work for various open and bound systems with underlying regular, chaotic, and even mixed classical dynamics. The efficiency of the method is increased, i.e., the number of orbits required for periodic orbit quantization is reduced, when the harmonic inversion technique is generalized to the analysis of cross-correlated periodic orbit sums. The method provides not only the eigenenergies and resonances of systems but also allows the semiclassical calculation of diagonal matrix elements and, e.g., for atoms in external fields, individual non-diagonal transition strengths. Furthermore, it is possible to include higher-order terms of the ℏ expanded periodic orbit sum to obtain semiclassical spectra beyond the Gutzwiller and Berry–Tabor approximation.

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

We present and compare three generically applicable signal processing methods for periodic orbit quantization via harmonic inversion of semiclassical recurrence functions. In a first step of each method, a band-limited decimated periodic orbit signal is obtained by analytical frequency windowing of the periodic orbit sum. In a second step, the frequencies and amplitudes of the decimated signal are determined by either decimated linear predictor, decimated Pade approximant, or decimated signal diagonalization. These techniques, which would have been numerically unstable without the windowing, provide numerically more accurate semiclassical spectra than does the filter diagonalization method.

Use of Harmonic Inversion Techniques in Semiclassical Quantization and Analysis of Quantum Spectra

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.

Decimation and harmonic inversion of periodic orbit signals

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