Quantum phase-field model: vortices and THz-induced gap dynamics in superconductors

The ability to simulate the spatial and temporal ordering dynamics of quantum phases in inhomogeneous systems, particularly in the low-temperature regime, is crucial for understanding condensate physics and for enabling emerging device technologies. However, fully microscopic kinetic approaches are often computationally prohibitive for macroscopic spatiotemporal simulations in realistic device geometries, while the widely used phenomenological Ginzburg-Landau formulation is, in principle, valid only near the critical transition temperature $T_c$ and becomes inaccurate in the technologically relevant low-temperature regime. To describe the dynamics of quantum phase ordering, we propose a phase-field formulation derived from a microscopic many-body description using the superconducting order as an example. It leads to a compact dynamical evolution equation for the superconducting order parameter, enabling real-space and real-time simulations of phase ordering dynamics. The simulations successfully capture key dynamical phenomena, including vortex nucleation and motion under static magnetic fields, as well as ultrafast gap oscillations driven by THz fields, over the full temperature range from 0 K up to the critical transition temperature. Beyond superconductivity, this method can be extended to a broad class of quantum condensates and ordered systems, providing a practical computational approach to studying low-temperature ordering dynamics, topological defect evolution, and ultrafast electromagnetic responses of quantum phases in realistic geometries.