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Bose-Einstein condensates in PT-symmetric potentials

Daniel Haag

Published Jan 1, 2017• 0 citations
Publisher

Abstract

PT symmetry, alongside its applications and possible realizations, has established itself as a hot topic in recent years. Next to its possible impact on the fundamental nature of quantum physics, it has proved to be useful in the research on open systems of all kind. While PT physics has already been very successful in the field of optics, this thesis presents numerical calculations that show how PT symmetry can be applied to the mean-field theory of ultra-cold gases, where particles are added and removed coherently from a Bose-Einstein condensate. As long as the eigenvalues of a system are continuous functions with respect to such an exchange rate of particles, PT symmetry is preserved for weak perturbations. Therefore, a prediction of these changes for high excitation numbers is incredibly valuable to ensure stable stationary realizations. Not only does the present work test such rigorous estimates with regard to their validity, it also goes beyond analytically accessible regimes and presents numerical estimations for the nonlinear description of interacting particles. With this in mind, only low excited states have to be considered when studying non-trivial modifications to the PT-symmetric double-well potential. Not only do the results at hand show that this simplest implementation of PT-symmetric concepts is extremely stable with respect to experimental inaccuracies, but such modifications give rise to possible applications. Weakening the barrier inside the double well by introducing another particle channel from the gain to the loss well has a non-trivial effect on the net transport: In general, the net current does not profit from this modification, specific configurations even allow its complete suppression. Further, not only static systems, but rotating condensates are considered. It is shown, that the net particle transport is not only possible in the presence of the rotation; the vortices in the rotating wave functions even generate the necessary current by moving in such a way that their circular current adds to the net current.

DOI
10.18419/opus-9054

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PT symmetry, alongside its applications and possible realizations, has established itself as a hot topic in recent years. Next to its possible impact on the fundamental nature of quantum physics, it has proved to be useful in the research on open systems of all kind. While PT physics has already been very successful in the field of optics, this thesis presents numerical calculations that show how PT symmetry can be applied to the mean-field theory of ultra-cold gases, where particles are added and removed coherently from a Bose-Einstein condensate. As long as the eigenvalues of a system are continuous functions with respect to such an exchange rate of particles, PT symmetry is preserved for weak perturbations. Therefore, a prediction of these changes for high excitation numbers is incredibly valuable to ensure stable stationary realizations. Not only does the present work test such rigorous estimates with regard to their validity, it also goes beyond analytically accessible regimes and presents numerical estimations for the nonlinear description of interacting particles. With this in mind, only low excited states have to be considered when studying non-trivial modifications to the PT-symmetric double-well potential. Not only do the results at hand show that this simplest implementation of PT-symmetric concepts is extremely stable with respect to experimental inaccuracies, but such modifications give rise to possible applications. Weakening the barrier inside the double well by introducing another particle channel from the gain to the loss well has a non-trivial effect on the net transport: In general, the net current does not profit from this modification, specific configurations even allow its complete suppression. Further, not only static systems, but rotating condensates are considered. It is shown, that the net particle transport is not only possible in the presence of the rotation; the vortices in the rotating wave functions even generate the necessary current by moving in such a way that their circular current adds to the net current.
PT\mathcal{PT}PT-symmetric quantum mechanics allows finding stationary states in mean-field systems with balanced gain and loss of particles. In this work we apply this method to rotating Bose-Einstein condensates with contact interaction which are known to support ground states with vortices. Due to the particle exchange with the environment transport phenomena through ultracold gases with vortices can be studied. We find that even strongly interacting rotating systems support stable PT\mathcal{PT}PT-symmetric ground states, sustaining a current parallel and perpendicular to the vortex cores. The vortices move through the nonuniform particle density and leave or enter the condensate through its borders creating the required net current.
The understanding of nonlinear PT symmetric quantum systems, arising for example in the theory of Bose–Einstein condensates in PT symmetric potentials, is widely based on numerical investigations, and little is known about generic features induced by the interplay of PT symmetry and nonlinearity. To gain deeper insights it is important to have analytically solvable toy models at hand. In the present paper the stationary states of a simple toy model of a PT symmetric system previously introduced in [1, 2] are investigated. The model can be interpreted as a simple description of a Bose–Einstein condensate in a PT symmetric double well trap in a two-mode approximation. The eigenvalues and eigenstates of the system can be explicitly calculated in a straightforward manner; the resulting structures resemble those that have recently been found numerically for a more realistic PT symmetric double delta potential. In addition, a continuation of the system is introduced that allows an interpretation in terms of a simple linear matrix model. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Quantum physics with non-Hermitian operators’.
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-$\ensuremath{\delta}$ 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\mathcal{PT}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 $\ensuremath{\delta}$-shaped potential used in our model.
In this thesis, I explore the behavior of rotating ultra-cold Bose gases, by diagonalizing the Hamiltonian. This method has the advantage of being exact in the limit of weak interactions, and thus is a useful complement to the more commonly used Gross-Pitaevskii (GP) equation, which relies on a mean-field approximation. It is generally known that an ultra-cold Bose gas shows some remarkable properties under rotation, such as quantized vortices and persistent currents, and these are one main reason why we choose to study rotation. Abstract A repulsively interacting Bose-Einstein condensate (BEC) in a harmonic trapping potential will form singly quantized vortices when rotated. Papers I-V are examples of different kinds of generalizations to this basic system, each showing its particular non-trivial new features. Abstract In Paper I we make the trapping potential weakly anharmonic, and confirm the existence of multiply quantized vortices in both repulsively and attractively interacting condensates, which were previously predicted in Gross-Pitaevskii calculations. Abstract In Papers II and III, we generalize to a two-component system in a harmonic trap. The vortices that form under rotation will in general be coreless, i.e. one component will form a vortex and rotate around the other component. In Paper II we give some analytical expressions for the dispersion relation and occupation numbers at low angular momenta, and in Paper III we show how multiply quantized vortices can be stable in these systems. Abstract Finally we investigate bosons in potentials which can support persistent currents, i.e. rotating states that do not decay in a finite amount of time. These are possible if there is an energy barrier separating the rotating state from the non-rotating ground state. In Paper IV, we have a two-component condensate in a one-dimensional potential and we calculate among other things the minimum interaction strength where persistent currents can occur. In Paper V, we study an experimentally relevant two-dimensional annulus. Again we find that persistent currents are possible above a certain interaction strength, and we find this strength as a function of the width of the annulus, as well as how it depends on the relative population of the two species.
We investigate vortex excitations in dilute Bose-Einstein condensates in the presence of complex PT\mathcal{PT}PT-symmetric potentials. These complex potentials are used to describe a balanced gain and loss of particles and allow an easier calculation of stationary states in open systems than in a full dynamical calculation including the whole environment. We examine the conditions under which stationary vortex states can exist and consider transitions from vortex to nonvortex states. In addition, we study the influences of PT\mathcal{PT}PT symmetry on the dynamics of nonstationary vortex states placed at off-center positions.
The observation of PT\mathcal{PT}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 PT\mathcal{PT}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 PT\mathcal{PT}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 PT\mathcal{PT}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 PT\mathcal{PT}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 PT\mathcal{PT}PT-symmetric double wells.