[Paper Review] Vortex Nucleation in a Dissipative Variant of the Nonlinear Schrödinger Equation under Rotation

R. Carretero-González, P. G. Kevrekidis+1 more|arXiv (Cornell University)|Dec 1, 2014

Cold Atom Physics and Bose-Einstein Condensates9 references4 citations

TL;DR

This paper investigates vortex nucleation in a dissipative variant of the nonlinear Schrödinger equation under rotation, showing that the most unstable mode scales as $\mu^{2/3}$ with chemical potential $\mu$. Through asymptotic analysis and numerical simulations, it demonstrates that rotational instability leads to peripheral vortex necklace formation, followed by symmetry-breaking pattern selection that sequentially recruits a few vortices into a stable, symmetric core configuration, relevant for finite-temperature Bose-Einstein condensates.

ABSTRACT

In the present work, we motivate and explore the dynamics of a dissipative variant of the nonlinear Schr{ö}dinger equation under the impact of external rotation. As in the well established Hamiltonian case, the rotation gives rise to the formation of vortices. We show, however, that the most unstable mode leading to this instability scales with an appropriate power of the chemical potential $μ$ of the system, increasing proportionally to $μ^{2/3}$. The precise form of the relevant formula, obtained through our asymptotic analysis, provides the most unstable mode as a function of the atomic density and the trap strength. We show how these unstable modes typically nucleate a large number of vortices in the periphery of the atomic cloud. However, through a pattern selection mechanism, prompted by symmetry-breaking, only few isolated vortices are pulled in sequentially from the periphery towards the bulk of the cloud resulting in highly symmetric stable vortex configurations with far fewer vortices than the original unstable mode. These results may be of relevance to the experimentally tractable realm of finite temperature atomic condensates.

Motivation & Objective

• To understand vortex nucleation dynamics in a dissipative variant of the Gross-Pitaevskii equation under rotation, particularly in the context of finite-temperature atomic condensates.
• To identify the most unstable mode driving vortex formation and determine its dependence on chemical potential $\mu$ and trap strength.
• To explain how symmetry-breaking mechanisms select a few stable vortices from a large peripheral vortex necklace.
• To connect theoretical predictions with experimentally tractable parameters, such as the thermal coupling parameter $\gamma$, for diagnostics in real BEC experiments.

Proposed method

• Derives the most unstable mode via asymptotic analysis of the linearized dissipative Gross-Pitaevskii equation under rotation, identifying its scaling with $\mu^{2/3}$.
• Uses a perturbative approach to decompose the eigenvalue problem into two Sturm-Liouville operators $L_1$ and $L_2$, both proven to be negative definite.
• Applies Sturm's oscillation theorem and spectral analysis to establish the existence of a minimum in the critical rotation frequency $\Omega_0(m_0)$ as a function of angular momentum quantum number $m_0$, confirming instability onset.
• Performs numerical simulations to track the evolution from a vortex-less state to a peripheral vortex necklace, followed by sequential inward migration of vortices.
• Employs a pattern selection mechanism based on symmetry breaking to explain the emergence of stable, symmetric vortex configurations with few vortices in the core.
• Validated results through numerical computation of the eigenvalue problem and curve analysis of $\Omega_0(m_0)$, showing blow-up at $m_0 \to 0^+$ and $m_0 \to \infty$, confirming a minimum.

Experimental results

Research questions

• RQ1 How does the most unstable mode responsible for vortex nucleation scale with the chemical potential $\mu$ in a dissipative, rotating nonlinear Schrödinger system?
• RQ2 What is the role of symmetry-breaking in selecting a few stable vortices from a large peripheral vortex necklace?
• RQ3 How does the thermal coupling parameter $\gamma$ influence the transition from unstable vortex nucleation to stable vortex configurations?
• RQ4 Can the dissipative Gross-Pitaevskii equation model predict experimentally observable vortex patterns in finite-temperature Bose-Einstein condensates?
• RQ5 What determines the final number of vortices in the stable core configuration, and how does this depend on $\gamma$, $\mu$, and trap strength?

Key findings

• The most unstable mode for vortex nucleation scales as $\mu^{2/3}$, derived through asymptotic analysis of the linearized system.
• Vortex nucleation under rotation leads to the formation of a large peripheral vortex necklace, which is unstable and prone to symmetry-breaking.
• A pattern selection mechanism based on symmetry breaking sequentially recruits a few vortices from the periphery into the core, forming a stable, symmetric vortex configuration.
• The final stable configuration contains far fewer vortices than the initial unstable necklace, indicating a selection process rather than random aggregation.
• The critical rotation frequency $\Omega_0(m_0)$ has a minimum at a finite, positive $m_0$, confirming the existence of a dominant unstable mode.
• Numerical results confirm that $\Omega_0 \sim 0.397/m_0$ as $m_0 \to 0^+$ and $\Omega_0 \sim m_0$ as $m_0 \to \infty$, proving the existence of a minimum in the instability curve.