Equilibrium states of generic quantum systems subject to periodic driving

When a closed quantum system is driven periodically with period $T$, it approaches a periodic state synchronized with the drive in which any local observable measured stroboscopically approaches a steady value. For integrable systems, the resulting behavior is captured by a periodic version of a generalized Gibbs ensemble. By contrast, here we show that for generic nonintegrable interacting systems, local observables become independent of the initial state entirely. Essentially, this happens because Floquet eigenstates of the driven system at quasienergy ${\omega }_{\alpha }$ consist of a mixture of the exponentially many eigenstates of the undriven Hamiltonian, which are thus drawn from the entire extensive undriven spectrum. This is a form of equilibration which depends only on the Hilbert space of the undriven system and not on any details of its Hamiltonian.

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Figure 1

Example of the EEV dependence on the quasienergy ${\omega }_f$ for $u/\hslash \omega =1$ and system size and particle number $L=14$, $N=7$ for a Hilbert space dimension $D_H=3432$, with parameters $u=V_1=V_2=J$ and driving frequency $\hslash \omega =h/T=J/4$. Points indicate expectation value of the density at site $i=8$ in an eigenstate $\left|\alpha \right(\epsilon )\rangle$ of $H_{\mathrm{eff}}\left(\epsilon \right)$ versus the state's quasienergy ${\omega }_{\alpha }$ at two different times. The black line indicates $\mathrm{tr}\left(b_8^{†}{b}_8\right)=N/L=0.5$.

Figure 2

Fitted exponent $\alpha$ vs driving amplitude $u$, extracted for the observable $b_3^{†}{b}_4$ as well as the density at site $i=3$ for the Hamiltonian of Eq. (5) for driving period $\omega =2\pi /T=1$. The upturn at small $u$ is a finite-size effect.

Figure 3

Left: Dynamical evolution of instantaneous value of the energy at the beginning of each period. The blue, gold red, and gold data points (three lower lines) correspond to different states, selected from different parts of the band of a Hamiltonian (which is different from the Hamiltonian used during the driving) and for for $L=12$, $N=6$ so $D_H=924$. The three top lines show time evolution of the same states, but for for $L=14$, $N=7$ so $D_H=3003$. In both cases, $u/\hslash \omega =5$. Right: Similar results are obtained for other observables, such as $b_3^{†}{b}_4$. The bottom three lines are for $L=12$, $N=6$ while the top three lines are for $L=14$, $N=7$, and are offset vertically for clarity (in reality, they also oscillate about 0).

Figure 4

Participation ratios for a number of system sizes.

Figure 5

Width of bandwidth of $H_S$ participating in each eigenstate of $H_{\mathrm{eff}}\left(0\right)$, averaged over all its eigenstates for the superlattice Hamiltonian described in the text. The solid line indicates the result for the limit in which all states participate equally.

Figure 6

Best fit and data points for $u=0.5$. This figure corresponds to a single data point in Fig. 2. It is obtained by varying $L$, the system size, and $N$, the number of particles.

Figure 7

Stroboscopic observation of the density at site $i=3$ for a breathing trap as a function of period. The initial states are prepared as described in the main text, and the observations are made at the beginning of the period, $\epsilon =0$.

Figure 8

Eigenstate expectation values for the density at site $i=3$ for a breathing trap for $\epsilon =0$.

Figure 9

Snapshot of final density for a “breathing trap” potential, corresponding to the last point of the blue line of Fig. 7. Note the spatially uniform density despite the strong DC component in the quadratic “trapping” potential.