Steady state of periodically driven quantum systems

TL;DR

The paper addresses how periodic driving and coupling to a low-density thermal gas shape the nonequilibrium steady state (NESS) of an open quantum $N$-level system. It introduces a Floquet scattering framework that yields general Floquet thermalization conditions, constraining the transition rates via unitarity of the Floquet S-matrix and T-matrix elements, and deriving a Pauli-rate description for the Floquet populations. A key finding is that at high temperatures ($\beta\to 0$) the NESS becomes uniform, $\wp_j^{\text{NESS}}=1/N$, independent of driving details, due to unitarity enforcing symmetric rate flows; deviations from Boltzmann behavior at high temperature are suppressed unless extra driving symmetries are broken. At low temperatures the NESS generally deviates from thermal equilibrium, and the authors validate their theory with numerical toy models (driven two- and three-level systems) that compute Floquet transition rates and illustrate the approach to thermalized behavior as temperature increases. Overall, the work provides universal, beyond-Born-Markov constraints on driven open quantum systems and offers a robust toolkit for predicting and testing driven NESS in Floquet-engineered settings.

Abstract

Periodic driving is used to steer physical systems to unique stationary states or nonequilibrium steady states (NESS), producing enhanced properties inaccessible to non-driven systems. For open quantum systems, characterizing the NESS is challenging and existing results are generally limited to specific types of driving and the Born-Markov approximation. Here we go beyond these limits by studying a generic periodically driven $ N$-level quantum system interacting with a low-density thermal gas. Exploiting the framework of Floquet scattering theory, we establish general Floquet thermalization conditions constraining the nature of the NESS and the transition rates. Moreover, we examine theoretically the structure of the NESS in the high temperature limit, and find out that the NESS complies, rather surprisingly, with an uniform probability distribution (predicted by the Boltzmann law) for any driving. Numerical calculations illustrate our theoretical elaborations for a simple toy model.

Figures (7)

• Figure 1: Numerical check of the Floquet thermalization conditions (\ref{['thermalization-Floquet-general-take-2-prelim']}) for a driven three-level system. Details of the used model can be found in section III of the main text and in S.6 of the SI.
• Figure 2: The structure of NESS for a simple toy model of a driven two-level system. The uniform population distribution emerges at $\beta=0$. Details of the used model can be found in section III of the main text and in S.6 of the SI.
• Figure 3: Quantity $\hbar\omega \sum_{j} \sum_{j'} \,\sum_{\nu=-\infty}^{\nu=+\infty} \nu \, a_{jj'}^{\nu}$ plotted against $\beta$ for a simple toy model of a driven two-level system. One can see that the property (\ref{['eq:nocht-prelim']}) holds at $\beta=0$. Interestingly, the green line shows that even $\hbar\omega \sum_{j} \sum_{\nu=-\infty}^{\nu=+\infty} \nu \, a_{jj}^{\nu}$ vanishes at $\beta=0$, rationalization of such behavior is left as an open question for possible further research. Details of the used model can be found in section III of the main text and in S.6 of the SI.
• Figure 4: The structure of NESS plotted as a function of normalized inverse temperature $\beta(E_{20}^{\rm QE}-E_{10}^{\rm QE})$. Note the special behavior of the populations $\wp_j^{\rm NESS}$ in the $\beta \to 0$ and $\beta \to \infty$ limits. Inset:$\left(a_{jj'}^{-\nu} \, e^{-\beta E_{j'\space\nu}^{\rm QE}}\right)\space\mathlarger{\mathlarger{\mathlarger{\mathlarger{\mathlarger{/}}}}}\space\space\left(e^{-\beta E_{j0}^{\rm QE}} a_{j'\space j}^{\nu}\right)$ plotted here against $\beta(E_{20}^{\rm QE}-E_{10}^{\rm QE})$ for the case of $j=1$ and $j'=2$. Showing explicitly that the detailed balance condition does not hold. Specifications of the used model can be found in section III of the main text and in S.6 of the SI. Note that the property (\ref{['rates-extra-symmetry']}) does apply for the present case, ensuring compliance with the Boltzmann law up to ${\cal O}(\beta)$.
• Figure 5: High temperature limit (r.h.s. of (\ref{['david-criterion']})). Here, the $\lambda$-dependence saturates for large values of $\lambda$. Therefore, for the specific system and type of driving used in our calculation, the pertinent NESS is basically thermal for $\beta \ll 0.2$, regardless upon the strength of driving. This saturation phenomenon deserves to be explored in more detail in the future. Specifications of the used three level model complying with (\ref{['rates-extra-symmetry']}) can be found in section III of the main text and S.6 of the SI