$\mathcal{PT}$-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 $\mathcal{PT}$-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.
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-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.
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.
We investigate vortex excitations in dilute Bose-Einstein condensates in the presence of complex 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 symmetry on the dynamics of nonstationary vortex states placed at off-center positions.
Bose-Einstein condensates with balanced gain and loss can support stationary states despite the exchange of particles with the environment. In the mean-field approximation this is described by the PT-symmetric Gross-Pitaevskii equation with real eigenvalues. In this work we study the role of stationary states in the appropriate many-particle description. It is shown that without particle interaction there exist two nonoscillating trajectories which can be interpreted as the many-particle equivalent of the stationary PT-symmetric mean-field states. Furthermore, the system has a nonequilibrium steady state which acts as an attractor in the oscillating regime. This steady state is a pure condensate for strong gain and loss contributions if the interaction between the particles is sufficiently weak.
Balanced gain and loss renders the mean-field description of Bose-Einstein condensates PT symmetric. However, any experimental realization has to deal with unbalancing in the gain and loss contributions breaking the PT symmetry. We will show that such an asymmetry does not necessarily lead to a system without a stable mean-field ground state. Indeed, by exploiting the nonlinear properties of the condensate, a small asymmetry can stabilize the system even further due to a self-regulation of the particle number.
This thesis contains theoretical studies of weakly interacting, trapped Bose-Einstein condensed gases. The main emphasis is on the rotational properties of these systems. A typical Bose-Einstein condensate reacts to external rotation by forming singly quantized vortices. By the word ”typical” we refer to a condensate with repulsive interparticle interactions and with a frozen spin degree of freedom. However, there are some atomic species for which the low-energy interactions are attractive. Moreover, the tuning of different condensate properties is relatively easy. For example, there are ways to experimentally change the effective interaction from repulsive to attractive. For an attractively interacting condensate the vortex is not the only possible rotational state but the sample may also experience rotation through its center of mass. In the first part of the thesis, we analyze some properties of a rotating attractive Bose gas at the zero temperature limit. First, we note that in order to maintain energetical stability, the gas has to be rotated in an external trap which is steeper than a harmonic trap. We study the system in an anharmonic quadratic + quartic potential. The rotating ground state phase-space is determined with analytical and numerical methods. If the interactions are strong enough, the rotating ground state appears to be a state with the rotating center of mass. On the other hand, at the limit of weak interactions the system experiences rotation by forming a multiply quantized vortex. We also determine the spectrum of elementary excitations for these quantum many-particle states. We find that the broken symmetry of the center of mass rotating state induces an additional zero-energy (Goldstone) mode and that the small oscillations of the ground state exhibit the decoupling between the center of mass and relative motions. If the spin of the atoms is allowed to change freely, a rich variety of topological excitations in condensates can be studied. External rotation is one of the tools for examining the excitations in spinor condensates. In a spinless scalar condensate the appearance of a quantized vortex is manifested by the vortex line in which the condensate density vanishes. In the plane perpendicular to the vortex line, one speaks of the vortex core for the density minimum. We study the properties of slowly rotated F = 1 spinor condensates and show that for ferromagnetic interactions the ground state is a coreless vortex. While the individual spin components may carry ordinary vortices the overall density of the system is coreless. We also study a method to create a point-like monopole structure in a ferromagnetic condensate by suitably imprinting vortices and solitons for different spin components of the condensate.
Since its realization in dilute bosonic atomic gases [], [], Bose-Einstein condensation of alkali atoms and hydrogen has been produced and studied extensively in the laboratory [], and has permitted an intriguing glimpse into the macroscopic quantum world. In view of potential applications [], [], [], the study of quantized vortices, which are well-known signatures of superfluidity, is one of the key issues. In fact, bulk superfluids are distinguished from normal fluids by their ability to support dissipationless flow. Such persistent currents are intimately related to the existence of quantized vortices, which are localized phase singularities with integer topological charge []. The superfluid vortex is an example of a topological defect that is well known in superconductors [] and in liquid helium []. The occurrence of quantized vortices in superfluids has been the focus of fundamental theoretical and experimental work []. Different research groups have obtained quantized vortices in Bose-Einstein condensates (BECs) experimentally, e.g., the JILA group [], [], the ENS group [] and the MIT group [], []. Currently, there are at least two typical ways to generate quantized vortices from a BEC ground state: (i) impose a laser beam rotating with an angular velocity on the magnetic trap holding the atoms to create a harmonic anisotropic potential [], [], []; or (ii) add to the stationary magnetic trap a narrow, moving Gaussian potential, representing a far-blue detuned laser [], [], [], [], [], [].