Statistical periodicity in driven quantum systems: General formalism and application to noisy Floquet topological chains

Much recent experimental effort has focused on the realization of exotic quantum states and dynamics predicted to occur in periodically driven systems. But how robust are the sought-after features, such as Floquet topological surface states, against unavoidable imperfections in the periodic driving? In this paper, we address this question in a broader context and study the dynamics of quantum systems subject to noise with periodically recurring statistics. We show that the stroboscopic time evolution of such systems is described by a noise-averaged Floquet superoperator. The eigenvectors and -values of this superoperator generalize the familiar concepts of Floquet states and quasienergies and allow us to describe decoherence due to noise efficiently. Applying the general formalism to the example of a noisy Floquet topological chain, we rederive and corroborate our recent findings on the noise-induced decay of topologically protected end states. These results follow directly from an expansion of the end state in eigenvectors of the Floquet superoperator.

Figure 1

Sketch of the model considered in Sec. 5. Lattice sites form doublets (gray dashed ellipses) and are labeled with the plaquette number $j$ and a doublet index $s=\pm$. The driving protocol comprises four steps $i=1,2,3,4$ during which hopping along only certain “active” bonds is allowed, as indicated with the thick blue lines. As a result of the particular driving, localized states, indicated by dashed black circles, form at the ends of the ladder at lattice sites $(0,-)$ and $(L,+)$.

Figure 2

Survival probability of the end mode computed using Eq. (43) as a function of the number of noisy driving cycles, $n$. We use a ladder consisting of 50 rungs and a noise strength $\tau /T=1/80$. Away from the resonant driving point ($JT=5.8$, orange curve), $P_{\mathrm{s}}$ decays exponentially with $n$. The decay slows down to a diffusive one either when fine-tuning the system to $JT=2\pi$ (blue), or by including disorder ($VT/4=0.4$, yellow). The disordered curve is obtained by averaging over 80 simultaneous noise-disorder realizations, with error bars smaller than the linewidth. The solid black line shows an exponential decay with rate $2J^2{\tau }^2\simeq 0.0105$ as predicted in Eq. (52), the dotted line is a fit to a diffusive decay $\sim t^{-1/2}$, whereas the dashed line is the analytic result of Eq. (62).

Figure 3

Real and imaginary parts of the eigenvalues of the Floquet superoperator $\mathcal{F}$ [Eq. (41)], obtained numerically for a topological chain consisting of 20 rungs, with a noise strength $\tau /T=0.1$. Points inside the unit circle correspond to nonstationary states, which have negative imaginary parts, $\mathrm{Im}\left(\lambda \right)<0$, and the color scale denotes the squared overlap of each superket with the edge density matrix, $|{\langle \alpha |e\rangle }_{♯}{|}^2$. At the resonant driving point ($JT=2\pi$, right panel), there is a larger number of states with a significant overlap than away from resonant driving ($JT=5.8$, left panel).

Figure 4

The squared overlap of the eigenoperators of $\mathcal{F}$ with the edge superket is plotted as a function of the imaginary part of the corresponding eigenvalues. Away from the resonant driving point (top panel), only a few eigenoperators have large values of $|{\langle \alpha |e\rangle }_{♯}{|}^2$, leading to an exponential decay. At resonant driving however (bottom panel), many states have a large and equal overlap with the edge, leading to a diffusive decay of the survival probability. We use a ladder of 50 rungs and $\tau /T=1/40$ in both panels. The inset contains only those eigenoperators for which $|{\langle \alpha |e\rangle }_{♯}{|}^2>0.005$, using the data from the bottom panel. They are sorted in decreasing order of $\mathrm{Im}\left[\lambda \right]$ as ${\lambda }_0>{\lambda }_1>...>{\lambda }_{N-1}$ and plotted as a function of $k=\pi j/N$, where $j\in \{0,N-1\}$. For comparison, the solid curve shows the predicted behavior of ${\lambda }_{\rho }\left(k\right)$ from Eqs. (57).

Figure 5

Probability distribution of $\mathrm{Im}\left[\lambda \right]$, $\mathcal{P}\left(\mathrm{Im}\right[\lambda \left]\right)$, corresponding to all eigenoperators of $\mathcal{F}$ for which the squared overlap with the edge superket is greater than ${10}^{-4}$. We use an on-site disorder strength $VT/4=0.4$ and a noise strength $\tau /T=1/80$. The three histograms represent different system sizes: 40 rungs (red), 50 rungs (green), and 60 rungs (blue). They are obtained from 150 simultaneous disorder-noise realizations in the case of 40 and 50 rung systems, and by using 50 realizations for the largest system size. The probability distribution is peaked towards $\lambda \to 0$, with a peak hight that increases with system size. The black curve shows a square root singularity, which by Eq. (43) leads to a diffusive decay of the end mode survival probability.