Improve the Floquet solver, with the possibility to get also the time-dependent components

[albertomercurio]

The current solver for finding the Floquet steady-state uses the matrix continued fractions (see, e.g., Supplementary Materials of ¹ to obtain an effective Liouvillian, and then compute the steady-state to it.

Another method would solve instead the full linear system at once:

$$ ( \mathcal{L}_0 - i n \omega_d ) \hat{\rho}_n + \mathcal{L}_1 \hat{\rho}_{n-1} + \mathcal{L}_{-1} \hat{\rho}_{n+1} = 0 $$

This is a tridiagonal linear system of the form

$$ \mathbf{A} \cdot \mathbf{b} = 0 $$

where

$$ \mathbf{A} = \begin{pmatrix} \mathcal{L}_0 - i (-n_{\text{max}}) \omega_{\textrm{d}} & \mathcal{L}_{-1} & 0 & \cdots & 0 \\ \mathcal{L}_1 & \mathcal{L}_0 - i (-n_{\text{max}}+1) \omega_{\textrm{d}} & \mathcal{L}_{-1} & \cdots & 0 \\ 0 & \mathcal{L}_1 & \mathcal{L}_0 - i (-n_{\text{max}}+2) \omega_{\textrm{d}} & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & \mathcal{L}_0 - i n_{\text{max}} \omega_{\textrm{d}} \end{pmatrix} $$

and

$$ \mathbf{b} = \begin{pmatrix} \hat{\rho}_{-n_{\text{max}}} \\ \hat{\rho}_{-n_{\text{max}}+1} \\ \vdots \\ \hat{\rho}_{0} \\ \vdots \\ \hat{\rho}_{n_{\text{max}}-1} \\ \hat{\rho}_{n_{\text{max}}} \end{pmatrix} $$

This will allow to simultaneously obtain all the $\hat{\rho}_n$.

The steadystate_floquet function has already the argument lf_solver that can be used to specify which method to use to get the floquet steadystate.

References

Footnotes

1. Macrì, Vincenzo, et al. "Spontaneous scattering of Raman photons from cavity-QED systems in the ultrastrong coupling regime." Physical Review Letters 129.27 (2022): 273602. ↩

[samu-sys]

This is a comment to allow the Unitary Hack bots to find the PR above ( #153 )