Feedback cooling of fermionic atoms in optical lattices

TL;DR

This work presents a dissipative route to prepare topological insulator states of fermionic atoms in optical lattices using measurement-based Markovian feedback. By designing jump operators that pump particles from the upper to the lower band, the ground state becomes a dark state of the dissipative dynamics, enabling robust preparation even from arbitrary initial states. The authors develop an exact jump-operator construction and a more experimentally feasible approximate scheme, benchmarking them on 1D SSH/Rice-Mele chains and the 2D Haldane model. They use fidelity calculations for small systems and mean-field kinetic equations for large systems to demonstrate successful ground-state preparation and analyze how performance correlates with band topology and energy gaps, highlighting the practical potential and limitations of dissipative state engineering for quantum simulators.

Abstract

We discuss the preparation of topological insulator states with fermionic ultracold atoms in optical lattices by means of measurement-based Markovian feedback control. The designed measurement and feedback operators induce an effective dissipative channel that stabilizes the desired insulator state, either in an exact way or approximately in the case where additional experimental constraints are assumed. Successful state preparation is demonstrated in one-dimensional insulators as well as for Haldane's Chern insulator, by calculating the fidelity between the target ground state and the steady state of the feedback-modified master equation. The fidelity is obtained numerically through exact diagonalization or via time evolution of the system with moderate sizes. For larger 2D systems, we compare the mean occupation of the single-particle eigenstates for the ground and steady state computed through mean-field kinetic equations.

Figures (14)

• Figure 1: (a) A sketch of the Rice-Mele model in blue color, characterized by a staggered onsite potential, intracell hopping amplitude $J_1$, and intercell hopping amplitude $J_2$. $J_{A}$ and $J_{B}$ denote the next nearest neighbour hopping amplitudes. The potential can be realized by a superlattice, Eq. \ref{['eq:lattice_potential_rice_mele']}, in the experiments. The black dashed line indicates a unit cell. For $E=0$ one obtains the Su-Schrieffer-Heeger model as a filled gray color, which is characterized by a symmetric double-well system. (b) Single-particle energy spectrum for the SSH model with periodic boundary condition, i.e., Eq. \ref{['ssh-spectrum']} for $E=0$ and $J_1=2J_2=2J$. The symmetries related to $\pm k$ and between upper(+)/lower (-) bands are shown.
• Figure 2: The approximate approach for the SSH model. The gray shaded background color indicates the trivial phase and the white background color indicates the topological phase. (a) The minimal overlap of Eq. \ref{['eq:boverlap']} between the state $\ket{b_{\ell,-}}$ of lower band given by Eq. \ref{['blm']} and the upper band $\ket{k_{+}}$ for different small systems with $N$ unit cells as a function of $J_1/J_2$. (b) Fidelity defined in Eq. \ref{['fidelity']} as a function of $J_1/J_2$ with open boundary conditions for different small systems with $N$ unit cells. The steady state $\rho_{\rm ss}$ is obtained by solving Eq. \ref{['me_fbM']} with the jump operator constrained to two sites within one unit cell given by Eq. \ref{['eq:collapse_two_sites']}. The measurement strength is set to $\gamma=0.0001J$ with $J$ the energy unit.
• Figure 3: The approximate method for a Rice-Mele pumping cycle. (a) Hopping amplitudes $J_1$, $J_2$ and onsite potential $E$ as a function of $\theta$ given by Eq. \ref{['RM-parameter']}. (b) The minimal overlap and the fidelity defined in Eq. \ref{['fidelity']} during the pumping cycle as a function of $\theta$ for a small system with 3 unit cells under open boundary condition. The steady state $\rho_{\rm ss}$ is given by Eq. \ref{['me_fbM']} and the jump operator constrained to two sites within one unit cell is given by Eq. \ref{['eq:collapse_two_sites']}. The measurement strength is set to $\gamma=0.0001J$ with $J$ the energy unit. The steady state is obtained by solving Eq. \ref{['me_fbM']}.
• Figure 4: (a) Haldane model with hopping vectors in real space. The region in blue shading shows one unit cell with two sites $A$ and $B$. $J_1$, $J_2$ and $J_3$ are the nearest neighbour hopping amplitudes. The next nearest neighbour (NNN) hopping in a clockwise closed path with magnetic flux $3\phi$ enclosed is shown in orange color and the corresponding NNN hopping acquires a complex phase $e^{i\phi}$, and vice versa, the anticlockwise NNN hopping acquires a complex phase $e^{-i\phi}$. In order to enumerate the unit cells, we consider two directions with labels $m$ and $n$, which take integer values. Each lattice site is described by three parameters: $mn\sigma$, with $mn$ the index of unit cell and $\sigma = A, B$. (b) Single particle spectrum under periodic boundary condition for the Haldane model with $J_\lambda = J$, $t_{A} = t_B = 0.1J$, $\phi=\pi /2$, $\Delta = 0.52J$.
• Figure 5: (a) Fidelity between the steady state given by Eq. \ref{['me_fbM']} with jump operator $C$ of Eq. \ref{['eq:jump_exact']} (for the exact approach) and the ground state of the Haldane model \ref{['eq:HaldaneHamiltonian']} for a small system with four unit cells as a function of the on-site potential $\Delta$ for different values of the complex hopping phase $\phi$. The steady state is obtained by solving Eq. \ref{['me_fbM']}. We use OBC with a small interaction \ref{['eq:Hint']} of strength $0.001J$ to break the degeneracy.