Periodic Thermodynamics of Isolated Quantum Systems

The nature of the behavior of an isolated many-body quantum system periodically driven in time has been an open question since the beginning of quantum mechanics. After an initial transient period, such a system is known to synchronize with the driving; in contrast to the nondriven case, no fundamental principle has been proposed for constructing the resulting nonequilibrium state. Here, we analytically show that, for a class of integrable systems, the relevant ensemble is constructed by maximizing an appropriately defined entropy subject to constraints, which we explicitly identify. This result constitutes a generalization of the concepts of equilibrium statistical mechanics to a class of far-from-equilibrium systems, up to now mainly accessible using ad hoc methods.

Figure 1

Characterization of the synchronized steady state. Left: Stroboscopic momentum distribution, $\widehat{n}(k)=L^{-1}\sum _{i,j}{\widehat{b}}_i^{†}{\widehat{b}}_j\exp [-2\pi ik(i-j)L^{-1}]$, demonstrating the wide range of behavior that occurs for varying parameters. The points correspond to snapshots of the dynamical evolution at late times ($t=490T$) for $L=200$, while the continuous lines correspond to the PGE prediction. From top to bottom at the extreme left end of the plot, the amplitudes of the superlattice potential, frequency, and filling factor ($\mathrm{\Delta }$, $\delta J$, $\omega$, $\nu$) are (0.6, 0.5, 1.6, $3/4$) (black, dot-dashed), (4, 0.5, 1.5, $1/3$) (yellow, dashed), (4, 0.75, 2, $1/3$) (cyan, full), (0.6, 0.5, 2, $1/4$) (magenta, dotted), and $\epsilon =0$. The next two panels correspond to the parameters for the cyan full line. Center: Expectation value of the momentum distribution $\widehat{n}(k)$ of the bosons during a single period in the synchronized state as a function of the time in the period $\epsilon$. The three lines on the time-momentum plane indicate the times $\epsilon /T=0$, 0.15, 0.25 for which density distributions are shown in the rightmost panel. The momentum distribution undergoes qualitative changes: at some points of the period, it has a single maximum at $k=0$ while at others it acquires double maxima at the edges of the Brillouin zone. Right: Each trace shows the expectation value of the density of the bosons ${\widehat{n}}_i^b={\widehat{b}}_i^{†}{\widehat{b}}_i$ at the time indicated in the middle panel by the line of the same color, for a lattice size $L=100$ and offset for better visibility. The black line indicates the time average of the applied potential; the density peaks at the edges despite the potential being highest there indicating a strongly nonequilibrium situation.

Figure 2

Main plot: Stroboscopic approach to equilibrium with time for the full momentum distribution of the bosons ${\widehat{n}}^{(b)}$, corresponding to the heavy black line in Fig. 1 and for a system size $L=200$ sites. Note the brief initial transient period, followed by small oscillations around a well-defined limit. Inset: Same as the main plot, but for a single component of the momentum distribution. In this plot, $d_{\pi /2}(m)=[{\widehat{n}}^{(b)}(k=\pi /2,mT)-{\widehat{n}}_{\mathrm{PGE}}^{(b)}(k=\pi /2)]/{\widehat{n}}_{\mathrm{PGE}}^{(b)}(k=\pi /2)$ measures the deviation of the actual value from the prediction of the PGE. The heavy blue lines show the average of the deviations after discarding the first 50 periods, which approximates the long-time average. These plots demonstrate that the expectation value of the operator approaches, then oscillates about, a value that is very close (within a few percent) to the PGE prediction. Both the deviation of the average from the PGE prediction and the relative magnitude of the fluctuations about the mean value are shown to scale to zero with system size in Fig. 3.

Figure 3

Approach to equilibrium with system size. The Hamiltonian and color coding is the same as in Fig. 1. Here, $\overline{d}$ measures distance from the PGE prediction, $\overline{d}=(LN{)}^{-1}\sum _{m=n}^{n+N}\sum _k|\widehat{n}(k,mT)-{\widehat{n}}_{\mathrm{PGE}}(k)|$. We take $n=40L$ and $N=20L$, large enough so that the results are insensitive to further increase. The dashed green line is a plot of $\overline{d}\propto L^{-1}$ to guide the eye. These results strongly suggest that the distance of the long-time behavior of the system from our prediction at the thermodynamic limit falls off as a power law.