Many-Body Dynamical Localization in a Kicked Lieb-Liniger Gas

The kicked rotor system is a textbook example of how classical and quantum dynamics can drastically differ. The energy of a classical particle confined to a ring and kicked periodically will increase linearly in time whereas in the quantum version the energy saturates after a finite number of kicks. The quantum system undergoes Anderson localization in angular-momentum space. Conventional wisdom says that in a many-particle system with short-range interactions the localization will be destroyed due to the coupling of widely separated momentum states. Here we provide evidence that for an interacting one-dimensional Bose gas, the Lieb-Liniger model, the dynamical localization can persist at least for an unexpectedly long time.

Figure 1

Main plot. Upper solid blue curve: variance of the momentum density $n(\lambda ,t)$ in the kicked Lieb-Liniger gas as a function of time relative to the initial variance: $\mathrm{var}[n(\lambda ,t)]-\mathrm{var}[n(\lambda ,0)]$. It saturates with time, signaling at least transient dynamical localization in sharp contrast to the classical diffusion (heating) under kicking. Lower dotted red curve: scaled variance of $\overline{n}(p,t)$. Insets: the same data for $\mathrm{var}[n(\lambda ,t)]-\mathrm{var}[n(\lambda ,0)]$ as a function of time at shorter timescales, which are more relevant to experiments [11, 12, 13]. Parameters: $V=0.5,q=4\pi /L,\gamma =10,\mathcal{N}=200$. At low enough kicking strength, both variances are well saturated, as we demonstrate in the Supplemental Material [59].

Figure 2

Main plot: decimal logarithms of the momentum densities at the end of the evolution. For $\overline{n}(p)$, all odd momentum components are zero, because we start with the uniform-density initial state and $q=2$ (in units of $2\pi /L$). Only even components of $\overline{n}(p)$ are plotted therefore. Parameters: $V=0.5,q=4\pi /L,\gamma =10,\mathcal{N}=200$. Inset: normalized momentum density $n(\lambda )/(2\pi )$ in the linear scale. The Fermi momentum at our parameter choice is ${\lambda }_F=100$.