Tilted optical lattices with defects as realizations ofPTsymmetry in Bose-Einstein condensates

Abstract

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.

Balanced gain and loss in spatially extended non- PT -symmetric multiwell potentials

“…Here, V n , Γ n , σ n , and a n are the well depth, the gain-loss parameter, the width, and the position of the center of the n-th well, respectively. Figure 1 shows a sketch of the potential (14) for the case N = 3.…”

“…We want to examine this explicitly for a triple-well potential of the form Eq. (14) with N = 3. Again we choose fixed values for the well widths and distances, that is, Re µ (f)…”

“…Re µ2 Re µ3 Figure 5. (a) Gain and loss parameters Γ 1 , Γ 2 , and Γ 3 of the triple-well potential (14) with σ 1 = σ 2 = σ 3 = 1/ √ 2, a 1 = −a 3 = −3, a 2 = 0, V 1 = −1.8, and V 2 = −2 for balanced gain and loss. (b) Imaginary parts and (c) real parts of the first three energy eigenvalues.…”

“…(a) Values of ∆V (Γ 1 , Γ 2 ) with a real ground state for the complex asymmetric double-well potential(14) with V 1 = −3, σ 1 = σ 2 = 1, and a 1 = −a 2 = −1.5 as well as (b) the corresponding real ground state energy µ 1 (Γ 1 , Γ 2 ). The insets show the corresponding quantities in the matrix model, namely (a) the difference of the onsite energies ε(γ 1 , γ 2 ) and (b) the real ground state energy µ(γ 1 , γ 2 ).…”

Purity oscillations in coupled Bose-Einstein condensates

“…Our starting point is the proposal of Kreibich et al [30,31], which consists of a multi-well trap for the BEC. Some of the inner wells are considered as the system and the outer wells form the environment.…”