qdmag : A Python package for simulating nonequilibrium magnetization of molecules using quantum master equations

Understanding the magnetization dynamics of magnetic molecules is essential for applications in quantum information processing, spintronics, and data storage. These molecules are open quantum systems that interact with their environment, particularly through spin-phonon coupling, which drives relaxation and decoherence. While existing software packages can calculate equilibrium magnetic properties or simulate EPR spectra, they often lack the capability to model nonequilibrium magnetization under time-varying external magnetic fields. The authors present qdmag, a new Python utility designed to bridge this gap.

qdmag solves the generalized Lindblad quantum master equation, treating spin-phonon coupling as a dissipation term. The package supports complex spin Hamiltonians, including magnetic exchange interactions and zero-field splitting (ZFS) terms up to the 12th order using extended Stevens operators. To handle the computational demands of high-dimensional spin systems, the software employs two key strategies:

1. Effective Hamiltonian Construction: For large systems, qdmag constructs a reduced-dimension effective Hamiltonian based on thermally relevant states, making the simulation of multinuclear complexes feasible.
2. Staircase Approximation: Instead of traditional integration methods like Runge-Kutta, which require very small time steps to maintain stability, qdmag uses a staircase approximation of the time-varying magnetic field. This allows for significantly larger time steps and efficient long-time evolution of the density matrix over millisecond timescales.

The authors demonstrate the package's capabilities through three case studies: a mononuclear Ho-based complex, a two-spin S=1/2 dimer, and a Mn trimer. These examples showcase the software's ability to handle various magnetic field profiles (linear, piecewise, spline-fitted, and sinusoidal), perform powder averaging for realistic sample modeling, and accurately capture nonequilibrium features like Zeeman level crossings and Rabi-type oscillations.