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We present a numerical approach to calculate non-equilibrium eigenstates of aperiodically time-modulated quantum system. The approach is based on the use ofa chain of single-step time-independent propagating operators. Each operator istime-specific and constructed by combining the Magnus expansion of thetime-dependent system Hamiltonian with the Chebyshev expansion of an operatorexponent. A construction of a unitary matrix of the Floquet operator, whichevolves a system state over the full modulation period, is performed bypropagating the identity matrix over the period. The independence of theevolutions of basis vectors makes the propagation stage suitable forimplementation on a parallel cluster. Once the propagation stage is completed,a routine diagonalization of the Floquet matrix is performed. Finally, anadditional propagation round, now with the eigenvectors as the initial states,allows to resolve the time-dependence of the Floquet states and calculate theircharacteristics. We demonstrate the accuracy and scalability of the algorithmby applying it to calculate the Floquet states of two quantum models, namely(i) a synthesized random-matrix Hamiltonian and (ii) a many-body Bose-Hubbarddimer, both of the size up to $10^4$ states.

Calculating Floquet states of large quantum systems: A parallelization strategy and its cluster implementation