Macroscopic excitations in confined Bose-Einstein condensates, searching for quantum turbulence

Abstract: We present a survey of macroscopic excitations of harmonically confined Bose-Einstein condensates (BEC), described by Gross-Pitaevskii (GP) equation, in search of routes to develop quantum turbulence. These excitations can all be created by phase imprinting techniques on an otherwise equilibrium Bose-Einstein condensate. We analyze two crossed vortices, two parallel anti-vortices, a vortex ring, a vortex with topological charge Q = 2, and a tangle of 4 vortices. Since GP equation is time-reversal invariant, we are careful to distinguish time intervals in which this symmetry is preserved and those in which rounding errors play a role. We find that the system tends to reach stationary states that may be widely classified as having either an array of vortices with collective excitations at different length scales or an agitated state composed mainly of Bogoliubov phonons.

• The paper simulates various macroscopic excitations in confined Bose-Einstein condensates using the Gross-Pitaevskii equation to identify potential routes to quantum turbulence.
• Numerical simulations show diverse behaviors, from stable crossed vortices exhibiting Kolmogorov-like spectra to vortex collisions rapidly leading to phonon-dominated states.
• The findings suggest energy cascades can occur in quantum fluids and highlight the intricate balance of stability and chaos, motivating further research into thermal effects and larger simulations.

Understanding Macroscopic Excitations and Quantum Turbulence in Confined Bose-Einstein Condensates

The paper "Macroscopic excitations in confined Bose-Einstein condensates, searching for quantum turbulence" presents a comprehensive exploration of dynamic phenomena within Bose-Einstein Condensates (BECs) confined by an external harmonic potential. The research leverages the Gross-Pitaevskii (GP) equation as a framework to simulate and understand various macroscopic excitations that may lead to quantum turbulence. This analysis is pertinent to the field of quantum fluid dynamics, where turbulence remains an elusive and complex phenomenon in both classical and quantum regimes.

Key Excitations and Methodology

The authors investigate several specific excitations through numerical simulations using the GP equation. They focus on:

1. Two crossed vortices.
2. Two parallel anti-vortices.
3. An off-center vortex ring.
4. A vortex with topological charge $Q = 2$.
5. A tangle of four vortices.

These excitations offer a range of dynamic behaviors and potential routes to quantum turbulence within the condensate. Each scenario begins with a phase-imprinted initial state superimposed on an equilibrium BEC to observe dynamic evolutions and eventual transitions to stationary states.

Numerical Insights and Results

The simulations demonstrate that while the GP equation is time-reversal invariant, rounding errors in numerical computations can lead to the appearance of irreversible evolution in practice. The authors quantify this aspect by examining the fidelity of the system over time, revealing how different excitations tend towards stability or evolve into complex states.

Key Findings Include:

• Two Crossed Vortices: These maintain their time-reversal invariance over long simulation times, and their energy spectrum shows characteristics akin to the Kolmogorov $k^{-5/3}$ law, suggesting a cascading energy transfer among different scales.
• Vortex Anti-Collision: This case rapidly transitions to a non-linear, agitated state dominated by Bogoliubov phonons, with less resemblance to classical turbulence signatures.
• Off-Center Vortex Ring: The vortex ring evolves in and out of the condensate, breaking up and contributing to a phonon-dominated stationary state.
• Charge $Q=2$ Vortex: Known instability of high-charge vortices is confirmed, with the vortex decaying into two unit-charge vortices that orbit each other, suggesting energy dynamics similar to the crossed vortices scenario.
• Four Vortex Tangle: Rapid tangle dynamics leads to a temporally localized turbulent state before settling into a stationary configuration with characteristic vortex activities and possible collective modes.

Implications and Future Work

The exploration of macroscopic excitations in confined BECs elucidates the complexities of quantum turbulence and supports the hypothesis that even in quantum fluids, an energy cascade among scales is possible. The transient alignment of some excitation spectra with the Kolmogorov law highlights potential analogies with classical turbulence, although the intricacies of quantum systems introduce distinct behaviors such as phonon dominance.

Theoretical and Practical Implications:

• Incremental understanding of quantum turbulence provides insights for developing new quantum fluid technologies and contributes fundamental knowledge relevant to quantum computation environments.
• The delineation between stationary states chequered by vortex dynamics versus phonon excitations necessitates further exploration of energy transfer mechanisms under varying condensate conditions and interactions.

Future Research Directions:

• Incorporating thermal effects and dissipation within BEC models could bridge towards more realistic scenarios akin to experimental settings.
• Extending numerical studies to large-scale, higher-dimensional simulations may yield further insights into the isotropic and anisotropic nature of excitations and turbulence in quantum fluids.
• Experimental validation of these simulated behaviors in controlled BEC setups would provide tangible evidence supporting theoretical predictions.

In summary, this paper contributes a detailed numerical exploration of dynamics inside harmonically trapped BECs and highlights the intricate balance of stability and chaos in the field of quantum turbulence. The findings necessitate further investigation into the diverse phenomena observed, underscoring the fascinating complexity of macroscopic quantum states and their potential deterministic routes to turbulence.