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arXiv:2401.08393 · Physical Review A

Loop current states and their stability in small fractal lattices of Bose-Einstein condensates

Georg Koch, Anna Posazhennikova

Published Sep 3, 2024• 7 citations
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Abstract

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 ${u}_{c}$. 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.

DOI
10.1103/physreva.110.033301

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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\ensuremathπ/32\ensuremath{\pi}/32\ensuremathπ/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 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 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 ring-shaped array of Bose-Einstein condensed atomic gases can display circular currents if the relative phase of neighboring condensates becomes locked to certain values. It is shown that, irrespective of the mechanism responsible for generating these states, only a restricted set of currents are stable, depending on the number of condensates, on the interaction and tunneling energies, and on the total number of particles. Different instabilities due to quasiparticle excitations are characterized, and possible experimental setups for testing the stability prediction are also discussed.
We study the dynamics of three-dimensional weakly linked Bose-Einstein condensates using a multimode model with an effective interaction parameter. The system is confined by a ring-shaped four-well trapping potential. By constructing a two-mode Hamiltonian in a reduced highly symmetric phase space, we examine the periodic orbits and calculate their time periods both in the self-trapping and Josephson regimes. The dynamics in the vicinity of the reduced phase space is investigated by means of a Floquet multiplier analysis, finding regions of different linear stability and analyzing their implications on the exact dynamics. The numerical exploration in an extended region of the phase space demonstrates that two-mode tools can also be useful for performing a partition of the space in different regimes. Comparisons with Gross-Pitaevskii simulations confirm these findings and emphasize the importance of properly determining the effective on-site interaction parameter governing the multimode dynamics.
Multiple mode couplings in topological coherent modes of Bose–Einstein condensate are considered, by introducing an external alternating (resonating) field in the system. This analysis is based on the analytical solutions of nonlinear Gross–Pitaevskii equation for a trapped Bose gas at nearly absolute zero temperature. The dynamics of fractional populations of the generated coherent modes are analysed, particularly for a three-level system in the limit of small to large detuning of the intermediate state. These coupled topological modes, though nonlinear, are analogous to a resonant atom and exhibit a variety of significant non-trivial phenomena (effects), like: dynamic phase transitions, interference patterns, critical phenomena, mode-locking and chaotic motion.
A Bose-Einstein condensate loaded in a one-dimensional bichromatic optical lattice with constituent sublattices having incommensurate periods is considered. Using the rational approximations for the incommensurate periods, we show that below the mobility edge the localized states are distributed nearly homogeneously in the space and explore the versatility of such potentials. We show that superposition of symmetric and antisymmetric localized states can be used to simulate various physical dynamical regimes, known to occur in double-well and multiwell traps. As examples, we obtain an alternative realization of a bosonic Josephson junction, whose coherent oscillations display beatings or switching in the weakly nonlinear regime, and describe self-trapping and four-mode dynamics, mimicking coherent oscillations and self-trapping in four-well potentials. These phenomena can be observed for different pairs of modes, which are localized due to the interference rather than due to a confining trap. The results obtained using few-mode approximations are compared with the direct numerical simulations of the one-dimensional Gross-Pitaevskii equation. The localized states and the related dynamics are found to persist for long times even in the repulsive condensates. We also described bifurcations of the families of nonlinear modes, the symmetry breaking, and stable minigap solitons.
The dynamics of the three coupled bosonic wells (trimer) containing N bosons is investigated within a standard (mean-field) semiclassical picture based on the coherent-state method. Various periodic solutions (configured as pi-like, dimerlike, and vortex states) representing collective modes are obtained analytically when the fixed points of trimer dynamics are identified on the N=const submanifold in the phase space. Hyperbolic, maximum and minimum points are recognized in the fixed-point set by studying the Hessian signature of the trimer Hamiltonian. The system dynamics in the neighborhood of periodic orbits (associated with fixed points) is studied via numeric integration of trimer motion equations, thus revealing a diffused chaotic behavior (not excluding the presence of regular orbits), macroscopic effects of population inversion, and self-trapping. In particular, the behavior of orbits with initial conditions close to the dimerlike periodic orbits shows how the self-trapping effect of dimerlike integrable subregimes is destroyed by the presence of chaos.