Characterizations of prethermal states in periodically driven many-body systems with unbounded chaotic diffusion

We introduce well-defined characterizations of prethermal states in realistic periodically driven many-body systems with unbounded chaotic diffusion of the kinetic energy. These systems, interacting arrays of periodically kicked rotors, are paradigmatic models of many-body chaos theory. We show that the prethermal states in these systems are well described by a generalized Gibbs ensemble based essentially on the average Hamiltonian. The latter is the quasiconserved quantity in the prethermal state and the ensemble is characterized by the temperature of the state. An explicit exact expression for this temperature is derived. Using also arguments based on chaos theory, we demonstrate that the lifetime of the prethermal state is exponentially long in the inverse of the temperature, in units of the driving frequency squared. Our analytical results, in particular those for the temperature and the lifetime of the prethermal state, agree well with numerical observations.

Figure 1

Proposed realization of the systems (1) using bosonic Josephson junctions created by Bose-Einstein condensates in optical lattices [29]. See Ref. [30] for details.

Figure 2

Time evolution of the average kinetic energy per rotor, $E_{\mathrm{kin}}\left(t\right)=(1/N){\sum }_{j=1}^N\langle p_j^2\left(t\right)/2\rangle , N=400$, where the average is over initial conditions with $p_j(t=0)=0$ and 100 values of ${\varphi }_j(t=0)$ homogeneously distributed between 0 and $2\pi$. The short-time dynamics is followed by a prethermal plateau, whose lifetime grows exponentially with $1/K, K=\kappa \tau$. The plots in this figure appear to be almost insensitive to the number $N$ of rotors, for sufficiently large $N$.

Figure 3

Time evolution of the distribution function of $p_j\left(t\right)$ for $N=400$ and $K=0.2$, starting from initial conditions with $p_j(t=0)=0$ and 5000 values of ${\varphi }_j(t=0)$ homogeneously distributed between 0 and $2\pi$. In the prethermal state, the distribution is Gaussian (“$+$” points).

Figure 4

Average kinetic energy per rotor (in units of ${\tau }^2$) as a function of $K$ for $\tilde{p}=0$ and three values of $t/\tau$. Dashed line: theoretical prediction, $E_{\mathrm{kin}}^{*}{\tau }^2={\tau }^2{T}^{*}/2=0.469K$, where $T^{*}$ is given by Eq. (11). A good agreement with the theoretical prediction is observed in the region of $(K,t/\tau )$ values corresponding to prethermal states. This region extends to larger values of $t/\tau$ as $K$ decreases.

Figure 5

Upper panel: Diffusion exponent $\alpha$ [obtained by fitting the function $E_{\mathrm{kin}}\left(t\right)$ to $C{t}^{\alpha }]$ plotted versus $K$ for different values of $t/\tau$. Lower panel: The small-$K$ behaviors collapse when $\alpha$ is plotted versus $K\ln \left[t/\left(A\tau \right)\right]$. Here $A$ is determined from the best fit of the lifetimes $t^{*}/\tau$, at which $\alpha =0.05$, as a function of $K$ (see inset). The dashed line in the inset is the exponential fit $t^{*}/\tau =Aexp(B/K)$ with $A=2.04$ and $B=2.08$.