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A solver which solves [`steadystate_fourier`](@ref) by first extracting an effective time-independent Liouvillian and then using the `steadystate_solver` to extract the steadystate..
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# Parameters
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- `steadystate_solver::SteadyStateSolver=SteadyStateDirectSolver()`: The solver to use for the effective Liouvillian.
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!!! note
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This solver is only available for [`steadystate_fourier`](@ref).
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`H_p` and `H_m`, where `H_p` oscillates
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as ``e^{i \omega t}`` and `H_m` oscillates as ``e^{-i \omega t}``.
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There are two solvers available for this function:
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- `SSFloquetLinearSolve`: Solves the linear system of equations.
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- `SteadyStateLinearSolver`: Solves the linear system of equations.
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- `SSFloquetEffectiveLiouvillian`: Solves the effective Liouvillian.
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For both cases, `n_max` is the number of Fourier components to consider, and `tol` is the tolerance for the solver.
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In the case of `SSFloquetLinearSolve`, the full linear system is solved at once:
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In the case of `SteadyStateLinearSolver`, the full linear system is solved at once:
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```math
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( \mathcal{L}_0 - i n \omega_d ) \hat{\rho}_n + \mathcal{L}_1 \hat{\rho}_{n-1} + \mathcal{L}_{-1} \hat{\rho}_{n+1} = 0
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In the case of `SSFloquetEffectiveLiouvillian`, instead, the effective Liouvillian is calculated using the matrix continued fraction method.
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!!! note "different return"
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The two solvers returns different objects. The `SSFloquetLinearSolve` returns a list of [`QuantumObject`](@ref), containing the density matrices for each Fourier component (``\hat{\rho}_{-n}``, with ``n`` from ``0`` to ``n_\textrm{max}``), while the `SSFloquetEffectiveLiouvillian` returns only ``\hat{\rho}_0``.
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The two solvers returns different objects. The `SteadyStateLinearSolver` returns a list of [`QuantumObject`](@ref), containing the density matrices for each Fourier component (``\hat{\rho}_{-n}``, with ``n`` from ``0`` to ``n_\textrm{max}``), while the `SSFloquetEffectiveLiouvillian` returns only ``\hat{\rho}_0``.
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!!! note
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`steadystate_floquet` is a synonym of `steadystate_fourier`.
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## Arguments
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- `H_0::QuantumObject`: The Hamiltonian or the Liouvillian of the undriven system.
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- `c_ops::Union{Nothing,AbstractVector} = nothing`: The optional collapse operators.
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- `n_max::Integer = 2`: The number of Fourier components to consider.
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- `tol::R = 1e-8`: The tolerance for the solver.
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- `solver::FSolver = SSFloquetLinearSolve`: The solver to use.
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- `solver::FSolver = SteadyStateLinearSolver`: The solver to use.
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- `kwargs...`: Additional keyword arguments to be passed to the solver.
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