Periodic and quasiperiodic revivals in periodically driven interacting quantum systems

Recently it has been shown that interparticle interactions generically destroy dynamical localization in periodically driven systems, resulting in diffusive transport and heating. In this Rapid Communication we rigorously construct a family of interacting driven systems which are dynamically localized and effectively decoupled from the external driving potential. We show that these systems exhibit tunable periodic or quasiperiodic revivals of the many-body wave function and thus of all physical observables. By numerically examining spinless fermions on a one-dimensional lattice we show that the analytically obtained revivals of such systems remain stable for finite systems with open boundary conditions while having a finite lifetime in the presence of static spatial disorder. We find this lifetime to be inversely proportional to the disorder strength.

Figure 1

Driving protocol: The driving is alternated between a hopping term operating over the first half period, $t\in \left[0,\frac{T}{2}\right)$, and an interaction term with no hopping operating in the second half period, $t\in \left[\frac{T}{2},T\right)$. There is also an additional external periodic potential which for concreteness was chosen to have an alternating sign over the two half periods. The figure illustrates the prefactor and the sign of the different terms.

Figure 2

Spectrum of the Floquet operator $\widehat{U}\left(T\right)$ on the complex unit circle. (a) Completely degenerate spectrum, $\widehat{U}\left(T\right)=1$, with trivial stroboscopic dynamics. (b) Partially degenerate spectrum which corresponds to complex unit roots (fifth root here). The system exhibits subharmonic revival, $\widehat{U}\left(5T\right)=1$. (c) A spectrum which corresponds to a generic quasiperiodic dynamics.

Figure 3

Fourier transform of the one particle Green's function, ${\tilde{G}}_i\left(\omega \right)=\mathcal{F}T\left[G_i\left(t\right)\right]$, in a clean open chain of lengths $L=18$ and $L=9$ for different interaction strengths tuned to different subharmonic modes $(1:p)$, using a driving period of $T=4$. The inset shows the stroboscopic dynamics in the real time domain for $p=5$, which is not periodic in $T$, but periodic in $5T$.

Figure 4

Stability of the subharmonic revival mode $(1:3)$ to the introduction of static disorder. Left: Fourier transform of the Green's function, ${\tilde{G}}_i\left(\omega \right)$ for different disorder strengths, $W$, for the subharmonic revival mode $(1:3)$. Here, $T=3$ and $L=18$ ($L=21)$ using 1000 (300) driving periods, respectively. Black arrows indicate the estimate of the width of the revival mode for the $L=18$ data (see text). Right: The width of the spectral peak on the left as a function of the disorder strength. The visible small fluctuations originate both from statistical and systematic sampling errors.