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Physical Review A

Nonlinear enhancement of the fractal structure in the escape dynamics of Bose-Einstein condensates

Kevin Mitchell, Boaz Ilan

Published Oct 8, 2009• 6 citations
Publisher

Abstract

We consider the escape dynamics of an ensemble of Bose-Einstein-condensed atoms from an optical-dipole trap consisting of two overlapping Gaussian wells. Earlier theoretical studies (based on a model of quantum evolution using ensembles of classical trajectories) predicted that self-similar fractal features could be visible in this system by measuring the escaping flux as a function of time for varying initial conditions. Here, direct numerical quantum simulations show the clear influence of quantum interference on the escape data. Fractal features are still evident in the data, albeit with interference fringes superposed. Furthermore, the nonlinear influence of atom-atom interactions is also considered, in the context of the $(2+1)$-dimensional Gross-Pitaevskii equation. Of particular note is that an attractive nonlinear interaction enhances the resolution of fractal structures in the escape data. Thus, the interplay between nonlinear focusing and dispersion results in dynamics that resolve the underlying classical fractal more faithfully than the noninteracting quantum (or classical) dynamics.

DOI
10.1103/physreva.80.043406

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We consider the escape dynamics of an ensemble of Bose-Einstein-condensed atoms from an optical-dipole trap consisting of two overlapping Gaussian wells. Earlier theoretical studies (based on a model of quantum evolution using ensembles of classical trajectories) predicted that self-similar fractal features could be visible in this system by measuring the escaping flux as a function of time for varying initial conditions. Here, direct numerical quantum simulations show the clear influence of quantum interference on the escape data. Fractal features are still evident in the data, albeit with interference fringes superposed. Furthermore, the nonlinear influence of atom-atom interactions is also considered, in the context of the (2+1)(2+1)(2+1)-dimensional Gross-Pitaevskii equation. Of particular note is that an attractive nonlinear interaction enhances the resolution of fractal structures in the escape data. Thus, the interplay between nonlinear focusing and dispersion results in dynamics that resolve the underlying classical fractal more faithfully than the noninteracting quantum (or classical) dynamics.
We consider the dynamic escape of a small packet of ultracold atoms launched from within an optical dipole trap. Based on a theoretical analysis of the underlying nonlinear dynamics, we predict that fractal behavior can be seen in experimental escape data. These data can be collected by measuring the time-dependent escape rate for packets launched over a range of angles. This fractal pattern is particularly well resolved below the Bose-Einstein transition temperature---a direct result of the extreme phase-space localization of the condensate. We predict that several self-similar layers of this novel fractal should be measurable, and we explain how this fractal pattern can be predicted and analyzed with recently developed techniques in symbolic dynamics.
In the last decades, the experimental research on Bose–Einstein interferometry has received much attention due to promising technological implications. This has thus motivated the development of numerical simulations aimed at solving the time-dependent Gross–Pitaevskii equation and its reduced one-dimensional version to better understand the development of interference-type features and the subsequent soliton dynamics. In this work, Bohmian mechanics is considered as an additional tool to further explore and analyze the formation and evolution in real time of the soliton arrays that follow the merging of two condensates. An alternative explanation is thus provided in terms of an underlying dynamical velocity field, directly linked to the local phase variations undergone by the condensate along its evolution. Although the reduced one-dimensional model is considered here, it still captures the essence of the phenomenon, rendering a neat picture of the full evolution without diminishing the generality of the description. To better appreciate the subtleties of free versus bound dynamics, two cases are discussed. First, the soliton dynamics exhibited by a coherent superposition of two freely released condensates is studied, discussing the peculiarities of the underlying velocity field and the corresponding flux trajectories in terms of both the peak-to-peak distance between the two initial clouds and the addition of a phase difference between them. In the latter case, an interesting correspondence with the well-known Aharonov-Bohm effect is found. Then, the recurrence dynamics displayed by the more general case of two condensates released from the two opposite turning points of a harmonic trap is considered in terms of the distance between such turning points. In both cases, it is presumed that the initial superposition state is generated by splitting adiabatically a single condensate with the aid of an optical lattice, which is then turned off. Nonetheless, although the lattice does not play any active role in the simulations, the parameters defining the initial states are in compliance with it, which helps in the interpretation and understanding of the results observed.
The visibility of the density interference pattern of two Bose-Einstein condensates, which are produced in traps and overlap after the trapping potential is switched off, is investigated. Coherent wave packets are used to describe the order parameter in a second-quantization formalism. This results in a decrease of the visibility of the interference fringes with increasing time delay between the formation of the condensates and the observation of interference. In the two limiting cases of ideal and very dense gases the correlation time increases, τ →∞, and the result is identical to that obtained using an approach based on the Gross-Pitaevskii equation. Under the conditions of the experiment performed by M. R. Andrews, C. G. Townsend, H. J. Miesner et al. , Science 275 , 6367 (1997), the computed correlation time τ ≈0.2 s is much longer than the confinement time of the condensate, and it is possible to observe the predicted decrease of visibility of the interference fringes of the density of the atoms.
We adopt a nonlinear two-mode model to investigate the Rosen-Zener-St\"uckelberg (RZS) interferometry of an interacting Bose-Einstein condensate trapped in double-well potential with a periodically modulating barrier. For symmetric potential, we analyze the properties of interference patterns and derive the interference conditions. For asymmetric potential, we explore the resonance structures caused by destructive interference at the weakly coupling limit and obtain the resonance conditions. In particular, we find that the chaotic motions in the asymmetric case with low-frequency modulation, can destroy the formation of interference fringes. In the symmetric case with high-frequency modulation, we see Fano-line-like resonance structures and find that the reso- nance positions exactly correspond to the boundary of macroscopic quantum self-trapping. We show that the nonlinearity can significantly alter both the shape and position of interference fringes and resonance structures. We discuss the experimental feasibility by comparing the solution of the Gross-Pitaevskii equation with both the solution of the two-mode model and the truncated Wigner simulation as well. The results suggest important applications of nonlinear RZS interferometry both in precisely controlling the dynamics and in accurately measuring the parameters of interacting many-body systems.
We studied the evolutionary patterns of two-dimensional Bose-Einstein condensates incorporating higher-order nonlinear interactions in harmonic potential. Using the Gross-Pitaevskii equation model with higher-order nonlinear corrections, we derived the analytical solitary vortex solutions via the variational method. The impact of the higher-order nonlinear interaction on the vortex dynamics is quantitatively analyzed, showing its key nonlinear feature contribution in the asymmetric vortex evolution with more precise evolutionary pattern generated. We found that, for the circular symmetric solution, if the nonlinear strength is not high, the higher-order nonlinear corrections essentially have only a tiny perturbative effect on the system’s quasi-static oscillation state, whereas for asymmetric evolution of the solitary vortex, incorporating higher-order corrections will generate an evolution pattern that better matches the results of numerical simulation. The theoretical results derived here can be used to guide relevant experimental studies of higher-order nonlinear effects in ultracold atomic systems.
The mean-field interaction in a Bose condensate provides a nonlinearity which can allow stable structures to exist in the mean-field wavefunction. We discuss a number of examples where condensates, modelled by the one-dimensional Gross - Pitaevskii equation, can produce grey solitons and we consider in detail the case of two identical condensates colliding in a harmonic trap. Solitons are shown to form from dark interference fringes when the soliton structure, constrained in a defined manner, has lower energy than the interference fringe and an analytic expression is given for this condition.