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arXiv:2203.14625 · Physical Review A

Chaos onset in large rings of Bose-Einstein condensates

Damian Wozniak, Johann Kroha, Anna Posazhennikova

Published Sep 16, 2022• 3 citations
arXivHTMLPDF

Abstract

We consider large rings of weakly coupled Bose-Einstein condensates, analyzing their transition to chaotic dynamics and loss of coherence. Initially, a ring is considered to be in an eigenstate, i.e., in a commensurate configuration with equal site fillings and equal phase differences between neighboring sites. Such a ring should exhibit a circulating current whose value will depend on the initial, nonzero phase difference. The appearance of such currents is a signature of an established coherence along the ring. If phase difference falls between $\ensuremath{\pi}/2$ and $3\ensuremath{\pi}/2$ and interparticle interaction in condensates exceeds a critical interaction value ${u}_{c}$, the coherence is supposed to be quickly destroyed because the system enters a chaotic regime due to inherent instabilities. This is, however, only part of the story. It turns out that chaotic dynamics and resulting averaging of circular current to zero are generally offset by a critical timescale ${t}_{c}$, which is almost two orders of magnitude larger than the one expected from the linear stability analysis. We study the critical timescale in detail in a broad parameter range.

DOI
10.1103/physreva.106.033316

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We consider large rings of weakly coupled Bose-Einstein condensates, analyzing their transition to chaotic dynamics and loss of coherence. Initially, a ring is considered to be in an eigenstate, i.e., in a commensurate configuration with equal site fillings and equal phase differences between neighboring sites. Such a ring should exhibit a circulating current whose value will depend on the initial, nonzero phase difference. The appearance of such currents is a signature of an established coherence along the ring. If phase difference falls between \ensuremathπ/2\ensuremath{\pi}/2\ensuremathπ/2 and 3\ensuremathπ/23\ensuremath{\pi}/23\ensuremathπ/2 and interparticle interaction in condensates exceeds a critical interaction value uc{u}_{c}uc​, the coherence is supposed to be quickly destroyed because the system enters a chaotic regime due to inherent instabilities. This is, however, only part of the story. It turns out that chaotic dynamics and resulting averaging of circular current to zero are generally offset by a critical timescale tc{t}_{c}tc​, which is almost two orders of magnitude larger than the one expected from the linear stability analysis. We study the critical timescale in detail in a broad parameter range.
A periodically kicked ring of a Bose-Einstein condensate is considered as a nonlinear generalization of the quantum kicked rotor, where the nonlinearity stems from the mean-field interactions between the condensed atoms. For weak interactions, periodic motion (antiresonance) becomes quasiperiodic (quantum beating) but remains stable. There exists a critical strength of interactions beyond which quasiperiodic motion becomes chaotic, resulting in an instability of the condensate manifested by exponential growth in the number of noncondensed atoms. In the stable regime, the system remains predominantly in the two lowest energy states and may be mapped onto a spin model, from which we obtain an analytic expression for the beat frequency and discuss the route to instability. We numerically explore a parameter regime for the occurrence of instability and reveal the characteristic density profile for both condensed and noncondensed atoms. The Arnold diffusion to higher energy levels is found to be responsible for the transition to instability. Similar behavior is observed for dynamically localized states (essentially quasiperiodic motions), where stability remains for weak interactions but is destroyed by strong interactions.
A periodically kicked ring of a Bose-Einstein condensate is considered as a nonlinear generalization of the quantum kicked rotor. For weak interactions between atoms, periodic motion (antiresonance) becomes quasiperiodic (quantum beating) but remains stable. There exists a critical strength of interactions beyond which quasiperiodic motion becomes chaotic, resulting in an instability of the condensate manifested by exponential growth in the number of noncondensed atoms. Similar behavior is observed for dynamically localized states (essentially quasiperiodic motions), where stability remains for weak interactions but is destroyed by strong interactions.
We consider a model of interacting Bose-Einstein condensates on small Sierpi\ifmmode \acute{n}\else \'{n}\fi{}ski gaskets. We study eigenstates which are characterized by cyclic supercurrents for each triangular plaquette (loop states). For noninteracting systems we find at least three classes of loop eigenmodes: standard, chaotic, and periodic. Standard modes are those inherited from the basic three-site ring of condensates with phase differences locked to $2\ensuremath{\pi}/3$. Standard modes become unstable in the interacting system but only when the interaction exceeds a certain critical value uc{u}_{c}uc​. Chaotic modes are characterized by very different circular currents per plaquette, so the usual symmetry of loop currents is broken. Circular supercurrents associated with chaotic modes become chaotic for any finite interaction, signaling the loss of coherence between the condensates. Periodic modes are described by alternating populations and two different phase differences. The modes are self-similar and are present in all generations of Sierpi\ifmmode \acute{n}\else \'{n}\fi{}ski gasket. When the interaction is included, the circular current of such a mode becomes periodic in time with the amplitude growing linearly with the interaction. Above a critical interaction the amplitude saturates, signaling a transition to a macroscopic self-trapping state originally known from a usual Bose-Josephson junction. We perform a systematic analysis of this rich physics.
We numerically investigate the spatiotemporal dynamics of a circular flow mode in a ring array of Bose-Einstein condensates, which consists of MMM identical Bose-Einstein condensates that couple to the nearest-neighbor sites by tunneling. Introducing a rotational potential in the Gross-Pitaevskii equation, a series of circular flow modes is resonantly excited. For the initial conditions with uniform density and one-round phase difference of $2\ensuremath{\pi}\ensuremath{\ell}\phantom{\rule{0.3em}{0ex}}(l=0,1,\dots{},M\ensuremath{-}1)$, two circular flow modes are excited for each circulation-quantized number lll. It is shown that these circular flow modes periodically appear in time in the presence of the rotational field and persistently survive with a constant shape even after the external potential is removed, the transitions between the circular flow modes with different quantized number lll can be realized by applying the phase-imprinting optical pulses, and the rotational velocity of the circular flow mode can be continuously modulated by adiabatically tuning the scattering length in a magnetic field, i.e., by use of the Feshbach resonance.
Abstract We investigate the chaotic dynamics of a coupled three-level atom–molecular Bose–Einstein condensate system composed by one molecular mode and two atomic modes. With the increase of atom–molecular coupling strength, we reveal the emergence of chaotic oscillations of the relative population difference between two atomic modes, which can be proven by the broad windows with a huge number of frequencies in spectral density and the chaotic trajectories in phase space diagrams. The different effects of initial states on atomic population oscillations are revealed, where for more particles in the initial state of the molecular model, chaos appears in the larger parameter region of system dynamics. Furthermore, we find that strong intermolecular interaction strength can suppress chaos resulting from strong atom–molecule coupling. This is due to the difficult transformation between atomic and molecular modes, as well as the relatively independent dynamic evolution of atoms and molecules.
Within the past five years Albert Einstein's concept of a dilute atomic Bose Condensate has been realized in many experimental laboratories. Temperatures in the nano-Kelvin regime have been achieved using magnetic and optical trapping of laser and evaporatively cooled atoms. At such temperatures the relative de Broglie wavelengths of the gaseous trapped atoms can become long compared to their mean spacing, and through a process of bosonic amplification a "quantum phase transition" takes place involving 10 3 to 10 11 atoms, most of which end up in identical single particle quantum states, whose length scales are determined by the external trap. Rb, Na, Li and atomic H have been trapped at variable densities of the order of 10 13 /cm 3 , and in traps of varying geometries. Of these all but Li have an effective repulsive atomic pair interaction, but utilization of molecular Feshbach resonances allows the interactions of other species to be tuned over wide ranges of strengths, including control of the sign of the effective atomic pair interactions. Fully quantum and macroscopic systems at such low densities are a theorist's dream: simple Hartree type mean field theory provides a startlingly accurate description of density profiles, low energy excitation frequencies, and such a description, commonly called the Non-linear Schrödinger Equation (NLSE) or the Gross-Pitaevskii (GP) Equation, will be explored here. The NLSE appropriate for attractive atomic interactions is known to lead to chaotically unstable dynamics, and eventual implosion of the condensate should the local number density exceed a critical value. In this work we illustrate this type of chaotic collapse for attractive condensates, and then explore the types of chaotic dynamics of solitons and vortices, which are the signatures of dynamical non-linearity in the repulsive case. Finally the implications of the symmetry-breaking associated with phase rigidity are explored in model simulations of repulsive condensates: condensates with repulsive atomic interactions break into phase domains when subject to weak shocks and, perhaps surprisingly, break into chaotic "laser-speckle" type patterns as the shock level increases. The fully quantum mechanical NLSE thus displays a full range of chaotic types of motion: from particle-like chaotic collisions of solitons and vortices to fully developed time dependent wave chaos.