Statistical mechanics of Floquet systems: The pervasive problem of near degeneracies

Although the statistical mechanics of periodically driven (“Floquet”) systems in contact with a heat bath has some formal analogy with the traditional statistical mechanics of undriven systems, closer examination reveals radical differences. In Floquet systems all quasienergies ${\epsilon }_j$ can be placed in a finite frequency interval $0\le {\epsilon }_j<\omega$ (with $\omega$ the driving frequency and $\hslash =1$). Therefore, if we describe a Floquet system approximately by restricting its available state space to be spanned by a finite number $N$ of basis states, then the number of near degeneracies ($\left|{\epsilon }_j-{\epsilon }_k\right|\le \delta$ for arbitrarily small fixed $\delta$) in this interval grows without limit as $N$ is increased. As we noted in a previous paper, this leads to pathologies, including drastic changes in Floquet states, as $N$ tends to infinity. In earlier work on Floquet systems in contact with a heat bath these difficulties were often put aside by fixing $N$ while taking the coupling to the bath to be smaller than any quasienergy difference. This led to a simple explicit theory for the reduced density matrix, with some major differences from the usual time-independent statistical mechanics. We show that, for weak but finite coupling between system and heat bath, the accuracy of a calculation within the truncated space spanned by the $N$ lowest energy eigenstates of the undriven system is limited, as $N$ increases indefinitely, only by the usual Born-Markov approximation, which neglects bath memory effects. As we seek higher accuracy by increasing $N$, we inevitably encounter quasienergy differences smaller than the system-bath coupling. We therefore derive here the equations for the steady-state reduced density matrix without restriction on the size of quasienergy splittings. In general, this matrix is no longer diagonal in the Floquet states. We analyze, in particular, the behavior near a weakly avoided crossing, where near degeneracies of quasienergies appear. In spite of the Floquet state pathologies, the explicit form of our results for the density matrix gives a consistent prescription for the statistical mechanics of periodically driven systems in the limit as $N$ approaches infinity.

Figure 1

Behavior of quasienergies around an avoided crossing of size $\Delta$. The dashed lines show diabatic (unmixed) states 1,2 and the solid curves show the adiabatic Floquet states $a,b$.

Figure 2

Simple example with a thermal bath coupling neighboring states only (double headed arrows) and flux $F$ introduced by the effective rate $R^{\text{ac}}$ due to an avoided crossing of states $n_1$ and $n_2$.

Figure 3

Dramatic change in occupations $p_n$ with (solid curve) and without (dashed curve) avoided crossing.