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 twowell 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.In quantum mechanics an observable is described by an Hermitian operator. This is true in particular for the energy, which is represented by the Hamiltonian. The Hermicity is sufficient for purely real eigenvalues, but is this really a necessary condition? Bender and Boettcher found that for non-Hermitian Hamiltonians with a weaker condition, namely PT symmetry, there exist parameters for which the energy eigenvalue spectrum is purely real [1], where PT stands for a combined action of parity P (x → −x, p → −p), and time reversalDue to the close analogy between the Schrödinger equation and the equations describing the propagation of light in structured wave guides, a PT -symmetric optical system could be visualized [2] and experimentally investigated [3,4]. The necessary complex potential corresponds to a complex refractive index, which is realized by balanced gain and loss of light in the wave guide. Several other systems with PT symmetry have been suggested and partially realized, including lasers [5-7], electronics [8-10], microwave cavities [11], and quantum field theories [1,12]. But up to date, a quantum mechanical realization of a PT -symmetric system is still missing.It was proposed [2] that a system similar to complex refractive indices in wave guides could be realized with Bose-Einstein condensates (BECs) in double-well potentials, where particles are injected in one well and removed from the other one. BECs in PT -symmetric double-well potentials have been investigated in the Bose-Hubbard model and the mean-field approximation [13][14][15][16][17]. In these Refs., the PT symmetry is given by a complex potential which fulfills the condition V * (x) = V (−x) and describes the coherent in-and outcoupling of atoms into and from the system. It has already been shown that a bidirectional coupling between two BECs is possible and at the same time particles may be continuously ejected * Manuel.Kreibich@itp1.uni-stuttgart.de Figure 1. (a) PT -symmetric two-well system with tunneling rate J and complex potential ±iΓ, which describes c...

We discuss the possibility of realizing a non-Hermitian, i. e. an open two-well system of ultra-cold atoms by enclosing it with additional time-dependent wells that serve as particle reservoirs. With the appropriate design of the additional wells PT -symmetric currents can be induced to and from the inner wells, which support stable solutions. We show that interaction in the mean-field limit does not destroy this property. As a first method we use a simplified variational ansatz leading to a discrete nonlinear Schrödinger equation. A more accurate and more general variational ansatz is then used to confirm the results.

A PT -symmetric Bose-Einstein condensate can theoretically be 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 are implemented by an accelerating Bose-Einstein condensate in a titled optical lattice. A defect consisting of two wells at the same energy level then acts as a PT -symmetric double-well if the tilt in the energy offsets of all further wells in the lattice is varied in time. We map the time-dependence of the amplitudes of a frozen Gaussian variational ansatz to a matrix model and increase the system size step by step starting with a six-well setup. In terms of this simple matrix model we derive conditions under which two wells of the Hermitian multi-well system behave exactly as the two wells of the PT -symmetric system.

We study the relation between the eigenfrequencies of the Bogoliubov excitations of Bose-Einstein condensates and the eigenvalues of the Jacobian stability matrix in a variational approach that maps the Gross-Pitaevskii equation to a system of equations of motion for the variational parameters. We do this for Bose-Einstein condensates with attractive contact interaction in an external trap and for a simple model of a self-trapped Bose-Einstein condensate with attractive 1/r interaction. The stationary solutions of the Gross-Pitaevskii equation and Bogoliubov excitations are calculated using a finite-difference scheme. The Bogoliubov spectra of the ground and excited state of the self-trapped monopolar condensate exhibit a Rydberg-like structure, which can be explained by means of a quantum-defect theory. On the variational side, we treat the problem using an ansatz of time-dependent coupled Gaussian functions combined with spherical harmonics. We first apply this ansatz to a condensate in an external trap without long-range interaction and calculate the excitation spectrum with the help of the time-dependent variational principle. Comparing with the full-numerical results, we find good agreement for the eigenfrequencies of the lowest excitation modes with arbitrary angular momenta. The variational method is then applied to calculate the excitations of the self-trapped monopolar condensates and the eigenfrequencies of the excitation modes are compared.

We investigate thermally excited, dipolar Bose-Einstein condensates. Quasi-particle excitations of the atomic cloud cause density fluctuations which can induce the collapse of the condensate if the inter-particle interaction is attractive. Within a variational approach, we identify the collectively excited stationary states of the gas which form transition states on the way to the BEC's collapse. We analyze transition states with different m-fold rotational symmetry and identify the one which mediates the collapse. The latter's symmetry depends on the trap aspect ratio of the external trapping potential which determines the shape of the BEC. Moreover, we present the collapse dynamics of the BEC and calculate the corresponding decay rate using transition state theory. We observe that the thermally induced collapse mechanism is important near the critical scattering length, where the lifetime of the condensate can be significantly reduced. Our results are valid for an arbitrary strength of the dipole-dipole interaction. Specific applications are discussed for the elements 52 Cr, 164 Dy and 168 Er with which dipolar BECs have been experimentally realized.