Feedback cooling of fermionic atoms in optical lattices

Zhao et al explore dissipative approaches to engineering topological band structures with ultracold fermions in optical lattices. This manuscript addresses a fundamental limitation of established methods based on adiabatic passage: a critical point has to be crossed in any topological phase transition at which there is no adiabatic timescale leading to imperfect quantum state engineering.

The explored approach, based on engineering dissipation using Markovian feedback control such that the desired final state is a dark state of the Lindbladian jump operator, builds on existing theoretical frameworks (e.g. the Wiseman-Milburn equation) and related proposals. Indeed, feedback control offers significant possibilities for cold-atom experiments given the feasibility of controlling millisecond dynamics using control actuators with bandwidths exceeding 1-10 kHz.

The submitted manuscript is well written, and the scientific content is well explained and employs sound methodologies. I particularly appreciate the authors' consideration of experimental limitations (namely, locality of measurement and feedback around few neighbouring sites) in designing approximate schemes that are more feasible for real-world implementation. While the manuscript does not address technical implementation of the proposed scheme, this does not detract from the contribution of the work which provides useful insight into the limits of the proposed approaches.

I believe this manuscript is appropriate for the target journal of SciPost Physics, provided the relatively minor comments and questions raised below can be addressed in a revised submission.

Specific questions and comments:

1. gamma = 1e-4 is presumably chosen to satisfy the condition that the feedback Hamiltonian can be neglected. What is the error associated with the neglect of this term? Could it be, for example, estimated and compared to the infidelity predictions of the approximate method? If the error is on the order of 1%, for example, then comparison of infidelities at the 1e-3 level (as in Fig 3b) may not be appropriate.

2. I would appreciate further elaboration on the control actuation, i.e. how the feedback operators can be implemented in practice. The authors mention, for example, that accelerating the optical lattice can be used to tune the complex tunneling amplitude between sites. Has this been demonstrated experimentally? What are the limits of this control, e.g. what limits how much support the control actuation on site A has from very distant sites on the lattice?

3. For large systems, the authors adopt a mean-field approach (as opposed to direct numerical simulation of the master equation) to treat large systems in the Haldane model. The quality of the protocol is quantified by the deviation of the mean occupation numbers from the target ground state. I am curious as to how well the topological structure can be characterized by mode occupation alone, which does not capture coherences between different modes. Could the authors provide further clarification on the validity of characterizing topological insulator regimes using occupations alone? Furthermore, is there a relationship between the deviation D and the fidelity F that could be explored in simple cases where an exact analytical or numerical solution is tractable?

4. At present, the validity of the mean-field approximation <n_k n_q> = <n_k><n_q> is not sufficiently addressed. Could the validity of the mean-field approach be validated by direct comparison to exact numerics for small system sizes? I expect that for the small system sizes the two will have a notable deviation, but in order to trust the mean-field approximation (i.e. neglecting non-trivial correlations) it would be useful to show that the two approaches converge for increasing system size.

Ask for minor revision