Optimal Control Technique for Many-Body Quantum Dynamics

We present an efficient strategy for controlling a vast range of nonintegrable quantum many-body one-dimensional systems that can be merged with state-of-the-art tensor network simulation methods such as the density matrix renormalization group. To demonstrate its potential, we employ it to solve a major issue in current optical-lattice physics with ultracold atoms: we show how to reduce by about 2 orders of magnitude the time needed to bring a superfluid gas into a Mott insulator state, while suppressing defects by more than 1 order of magnitude as compared to current experiments [T. Stöferle et al., Phys. Rev. Lett. 92, 130403 (2004)]. Finally, we show that the optimal pulse is robust against atom number fluctuations.

Figure 1

(a) An initial guess pulse $c^0(t)$ is used as a starting point. (b) The function $\mathcal{F}(\overrightarrow{a})$ for the case $\overrightarrow{a}=\{a_1,a_2\}$ and the initial polytope (white triangle) are defined and moved “downhill” [darker gray (red) triangles] until convergence is reached. (c) The final point is recast as the optimal pulse $c(t)$.

Figure 2

Scheme of the Mott-superfluid transition in the homogeneous system for average occupation number $⟨n⟩=1$: increasing the lattice depth $V$ (black line) the atom’s superfluid wave functions (upper) localize in the wells (lower). If the transition is not adiabatic—or optimized—defects appear (here represented by a hole and a doubly occupied site).

Figure 3

Initial guess (dashed black line) and optimal ramp (solid red line) $V(t)$ for the Bose-Hubbard model in the presence of the trap with $N=30$ sites, total time evolution $T\simeq 3\mathrm{ms}$. Inset: Populations $⟨n_i⟩$ (empty black symbols) and fluctuations $⟨\Delta n_i^2⟩$ (full red symbols) at time $t=T$ for the exponential initial guess (circles) and optimal ramp (squares) for $N=10$.

Figure 4

Residual defect density $\rho$ for $N=10,20,30,40$, $T\simeq 3\mathrm{ms}$, ${\rho }_c=0.001$ for the homogeneous system (green squares) and in the presence of the trap (gray circles). The gray (red) region highlights the typical $\rho$ for different initial ramp shapes (see text). Inset: Final $\rho$ computed applying the pulse optimized for system size $N=20$ to different system sizes $\Delta N=-4,\dots ,4$ (at constant filling). The results are almost independent from the truncation error for $m>50$.