Quantum Adiabatic Doping with Incommensurate Optical Lattices

Quantum simulations of Fermi-Hubbard models have been attracting considerable effort in the optical lattice research, with the ultracold antiferromagnetic atomic phase reached at half filling in recent years. An unresolved issue is to dope the system while maintaining the low thermal entropy. Here we propose to achieve the low temperature phase of the doped Fermi-Hubbard model using incommensurate optical lattices through adiabatic quantum evolution. In this theoretical proposal, we find that one major problem about the adiabatic doping is atomic localization in the incommensurate lattice, potentially causing an exponential slowing down of the adiabatic procedure. We study both one- and two-dimensional incommensurate optical lattices, and find that the localization prevents efficient adiabatic doping in the strong lattice regime for both cases. With density matrix renormalization group calculation, we further show that the slowing down problem in one dimension can be circumvented by considering interaction induced many-body delocalization, which is experimentally feasible using Feshbach resonance techniques. This protocol is expected to be efficient as well in two dimensions where the localization phenomenon is less stable.

Figure 1

Quantum adiabatic doping of one-dimensional lattice with free fermions. (a) The schematic illustration of the adiabatic evolution of the lattice. The “dashed” lines in (a) illustrate the evolution of a typical fermion density profile during the quantum adiabatic doping (see more results in Supplemental Material [38]). The lattice potential is periodic at both initial and final stage (shown by the top and bottom panels), but the intermediate lattice (the middle panel) is unavoidably incommensurate. (b) The wave function overlap of the final dynamical state with the ground state of the final Hamiltonian. (c) The excitation energy (per particle) of the final state as compared to the ground state of the final Hamiltonian (see Supplemental Material [38]), where the energy scale $J$ is the single-particle tunneling in the final lattice, and $N$ is the total particle number. (c) shares the same legend as shown in (b). As we increase the adiabatic time $T$, the overlap in (b) systematically increases and the excitation energy in (c) decreases. In the adiabatic evolution, we set $V=V^{\prime }$ [see Eqs. (2) and (3)]. (d), the inverse participation ratio (IPR) [39, 40]. The “dashed” lines in (d) correspond to the adiabatic Hamiltonian paths calculated in (b) and (c). The number of periods $L$ (see the main text) is set to be 55 here. See the main text for definition of wave function overlap, adiabatic time ($T$), and recoil energy ($E_R$).

Figure 2

Quantum adiabatic doping of two-dimensional lattice with free fermions. (a) The final state wave function overlap [see Eq. (6)] following the two-dimensional adiabatic doping. (b) The excitation energy (per particle) of the final state. (See the Supplemental Material [38] for a detailed description.) Like in the one-dimensional case, as we increase the adiabatic time $T$, we find the systematic increase of the wave function overlap and the decrease in the excitation energy. In the adiabatic evolution, we set $V=V^{\prime }$ [see Eqs. (2) and (3)]. (b) shares the same legend as shown in (a). In (b), the energy scale $J$ is the single-particle tunneling in the final lattice, and $N$ is the total particle number. In the two-dimensional incommensurate lattice, the localization problem that prevents efficient adiabatic doping becomes worse as compared to the one-dimensional case. Here we simulate 25 unit cells (five periods along each dimension) of the original lattice potential. The recoil energy $E_R$ is introduced in the main text.

Figure 3

Performance of quantum adiabatic doping with one-dimensional interacting fermions. We simulate the adiabatic procedure with density matrix renormalization group (DMRG) calculation, taking the Hamiltonian in Eq. (5). The inset shows the time sequence of the adiabatic evolution. The interaction is adiabatically turned on before tuning the lattice potential strengths, then held constant, and adiabatically turned off after the final desired lattice potential is reached. To show how interaction affects the adiabatic evolution, we carry out DMRG calculation in the regime $t/T\in [0,1]$, where the interaction is held constant. The lattice potential strengths are $V=V^{\prime }=8E_R$ for which the intermediate regime for the adiabatic evolution is strongly localized in absence of interaction. The adiabatic time is set to be $T=200$ in the unit of $\hslash /E_R$. It is evident that introducing interaction effects dramatically improve the final state wave function overlap [see Eq. (6)] shown by the star s and the excitation energy by the squares following the adiabatic doping, and that increasing interaction strength systematically improves the performance of the procedure. Here the energy scale $J$ is the single-particle tunneling in the final lattice, and $N$ is the total particle number. See the Supplemental Material [38] for details on the method.