Vortex patterns of a two-dimensional Bose-Einstein condensate at the almost critical rotation speed

Technical Summary: Vortex Patterns of a Two-Dimensional Bose–Einstein Condensate at the Almost Critical Rotation Speed

Problem Statement
This study investigates the vortex patterns and ground-state properties of a two-dimensional Bose–Einstein condensate (BEC) rotating at angular frequencies $\Omega$ approaching the centrifugal limit ($\Omega \to 1$). The research addresses the fundamental dichotomy in stability and structure arising from the sign of the contact interaction strength, $\kappa$ (or the effective coupling $G$). Specifically, it contrasts the behavior of repulsive systems ($G > 0$), which are known to form quantized vortex lattices, against attractive systems ($G < 0$), which are prone to collapse. The central challenge lies in accurately modeling these distinct regimes near the critical rotation speed, where the effective trapping potential vanishes, and the dynamics are confined to the Lowest Landau Level (LLL).

Methodology
The authors employ distinct numerical strategies tailored to the specific physical requirements of each interaction regime, explicitly avoiding a single unified framework in favor of methodological precision:

1. Repulsive Regime ($G > 0$):

   • Framework: The study utilizes a variational minimization of the Gross–Pitaevskii (GP) functional restricted to the LLL.
   • Implementation: The condensate wavefunction is expanded in an orthonormal LLL basis. The minimization is performed using GPU-accelerated constrained optimization (specifically the trust-constr algorithm from SciPy) with exact projection onto the LLL.
   • Validation: The method is benchmarked against Thomas–Fermi (TF) theory and the theoretical Abrikosov constant.

2. Attractive Regime ($G < 0$):

   • Framework: The study integrates the full real-space GP equation in imaginary time.
   • Rationale: An LLL truncation is deemed inappropriate here because the collapsing core requires high-momentum components that would be discarded by the projection.
   • Implementation: A time-splitting spectral scheme is used for propagation, with exact handling of the trap, interaction, and rotation terms. The method enforces mass conservation at every step.
   • Special Cases: To study stationary vortex states (giant vortices), the flow is restricted to fixed angular momentum sectors ($L_z = m$) using equivariant radial solvers.

Key Contributions and Results

• Repulsive Interactions and Abrikosov Lattices:

  • The numerical approach successfully reproduces the formation of triangular Abrikosov vortex lattices.
  • The computed density profiles align quantitatively with Thomas–Fermi theory, particularly regarding the cloud radius and the inverted-parabola envelope.
  • Crucially, the study recovers the Abrikosov constant $e_{Ab}(1) \approx 1.1596$ through energy inversion, confirming the analogy between rapidly rotating BECs and type-II superconductors.
  • The number of vortices scales monotonically with interaction strength $G$ and rotation frequency $\Omega$, following the rigid-body Feynman law ($N_v \approx \Omega r^2$) within the bulk.

• Attractive Interactions and Collapse:

  • In stark contrast to the repulsive case, the rotating ground state for attractive interactions carries no vortex lattice at any rotation frequency.
  • The condensate forms a single, nodeless lump that contracts radially as the attraction strength $G$ increases.
  • The system collapses as $G \to -G^*$, where the critical threshold is determined by the Gagliardo–Nirenberg inequality: $G^* = \|Q\|_{L^2}^2 \simeq 11.70$. This collapse occurs essentially independently of the rotation speed $\Omega$.
  • Vortex lattices imprinted on attractive condensates are unstable; imaginary-time relaxation expels the vortices, returning the system to a nodeless state.

• Giant Vortices (Trapped Solitons):

  • The only stationary vortex states found in the attractive regime are "giant vortices" (charge-$m$ Townes solitons), which exist only when the angular momentum is artificially constrained to a fixed sector.
  • These states are not collapse-free; they possess their own charge-dependent collapse thresholds $G^*(m) = \|Q_m\|_{L^2}^2$. The authors numerically confirm these thresholds for $m=0, 1, 2, 3$ (approx. 11.70, 48.3, 89.8, 132.4, respectively).
  • As $G \to -G^*(m)$, the giant vortex contracts onto the rescaled charge-$m$ ground state $Q_m$.

• Numerical Benchmarking:

  • The paper benchmarks four numerical solvers for the repulsive case. It concludes that optimization-based solvers with explicit constraints (like trust-constr) provide the most reliable ground states and accurate vortex counting, whereas split-step imaginary-time evolution without gauge projection can generate spurious vortex proliferation near $\Omega \to 1$.
  • For the attractive case, the imaginary-time scheme demonstrates second-order temporal convergence and exact mass conservation, accurately capturing the collapse profile near the critical threshold.

Significance and Claims
The paper claims to provide a rigorous numerical benchmark for vortex formation and collapse in rotating two-dimensional quantum gases. Its primary significance lies in the clear, methodologically distinct validation of two cornerstone regimes:

1. Vortex Crystallization: Confirming the quantitative agreement between LLL variational minimization and Thomas–Fermi theory for repulsive gases, including the precise recovery of the Abrikosov constant.
2. Critical Blow-up: Establishing that attractive rotating condensates do not support vortex lattices in their ground state, but rather undergo a collapse governed by the Gagliardo–Nirenberg threshold, largely independent of rotation.

The authors explicitly note that their analysis is deterministic and at zero temperature. They acknowledge that thermal fluctuations and noise, which are not included, would likely round the sharp thresholds and independence from $\Omega$ observed here, but they maintain that the qualitative separation between the two interaction signs remains robust. The study does not propose new experimental setups but serves to validate theoretical predictions and numerical methods for existing physical regimes.