Damping of Bloch oscillations: Variational solutions of the Boltzmann equation beyond linear response

Variational solutions of the Boltzmann equation usually rely on the concept of linear response. We extend the variational approach for tight-binding models at high entropies to a regime far beyond linear response. We analyze both weakly interacting fermions and incoherent bosons on a lattice. We consider a case where the particles are driven by a constant force, leading to the well-known Bloch oscillations, and we consider interactions that are weak enough not to overdamp these oscillations. This regime is computationally demanding and relevant for ultracold atoms in optical lattices. We derive a simple theory in terms of coupled dynamic equations for the particle density, energy density, current, and heat current, allowing for analytic solutions. As an application, we identify damping coefficients for Bloch oscillations in the Hubbard model at weak interactions and compute them for a one-dimensional toy model. We also approximately solve the long-time dynamics of a weakly interacting, strongly Bloch-oscillating cloud of fermionic particles in a tilted lattice, leading to a subdiffusive scaling exponent.

Figure 1

Sketch of typical momentum distribution functions $f\left(k\right)$ in equilibrium $[f^0$; (blue) line with crosses] and out of equilibrium $[f^0+\delta f$; solid (purple) line]. (a) A typical linear response setup at low temperature or low entropy. The nonequilibrium distribution is slightly displaced to positive momenta, e.g., due to an external force. (b) The regime of interest in this paper (high entropies and large fields). Here, the displacement to positive momenta is large. Nevertheless, the difference $\delta f$ between the equilibrium distribution $f^0$ and the nonequilibrium distribution still remains small. In both situations, linearization of the Boltzmann equation is possible.

Figure 2

Overdamped Bloch oscillations of the kinetic energy at filling $1/2$. We compare a full numerical simulation [dashed (red) curve] of the Boltzmann equation, (38), with our analytic result from Eq. (35) (solid black curve). The parameters are $U/J=4$ (yielding $\tau =0.25/J)$ and $F=0.4J$ such that $F\tau =0.1$.

Figure 3

Underdamped Bloch oscillations of the kinetic energy at filling $1/2$. Dashed (red) curve: numerical simulation of the Boltzmann equation, (38), for the parameters $U/J=1$ (yielding $\tau =4/J)$ and $F=2.5J$, such that $F\tau =10$ gives rise to the regime of weak damping. Solid black curve: analytic result from Eq. (36).

Figure 4

Marginally damped Bloch oscillations of the kinetic energy at filling $1/2$. Dashed (red) curve: numerical simulation of the Boltzmann equation, (38), for the parameters $U/J=1$ (yielding $\tau =4/J)$ and $F=0.25J$, such that $F\tau =1$ gives rise to the marginal case. Solid black curve: analytic result from Eq. (37).

Figure 5

Sketch of an interacting, Bloch oscillating cloud of atoms in a tilted lattice. Without interactions, individual particles are Bloch localized due to energy conservation and the bounded kinetic energy. This localization can, however, be lifted by interparticle scattering. As the scattering rate increases with the particle density, we can expect fast diffusion in the bulk and slow diffusion in the tails of the cloud.

Figure 6

Numerical solution of Eq. (53) plotted against the scaling function $G\left[x\right]$. The simulated particle densities $n(x,t)$ were rescaled as $t^{1/3}n(x{t}^{1/3},t)$ and plotted as a function of $x$ for different times $t=1$, 5, 25, and 50. The plot reveals that the numerical solutions $n$ assume the shape of the scaling function $G\left[x\right]$ upon rescaling at long times.

Figure 7

Energy and momentum preserving scattering processes in one dimension. The solid (red) curve shows the kinetic energy as a function of the momentum. Nontrivial scattering processes are possible for pairs of momenta that are symmetrically centered around momentum $\pm \pi /2$ [open (blue) circles]. They can scatter in any pair of final momentum states with the same property (pairs of filled black circles are two examples of a continuum of possibilities).