Dissipative preparation of a Floquet topological insulator in an optical lattice via bath engineering

In this manuscript, the authors study periodically driven fermions in an optical lattice coupled to an engineered bosonic bath. This setup is motivated by quantum-gas experiments with a Bose-Fermi mixture, where bosonic atoms can act as a bath for fermionic atoms. Specifically, they consider a situation where fermions are confined in a two-dimensional honeycomb lattice and bosons form quasi-1D Bose-Einstein condensates (BECs) at each lattice site. While the fermions are heated up by periodic driving, this heating is suppressed by the coupling to the bosonic bath. The authors find that the cooling is efficient due to suppression of the spectral density of the bath at high energies if one uses quasi-1D BECs as the bath and the mass of bosonic atoms is sufficiently larger than that of fermionic atoms. As a result, Floquet topological insulator phases of fermions are stabilized and show quantized responses, which are confirmed by numerical simulations.

Floquet engineering is now a useful tool in quantum-gas experiments, but the associated heating is often problematic. The proposal in this manuscript is interesting and appealing since they consider a concrete and realistic setting for the implementation of a bath with bosonic atoms such as 133Cs. Below I enclose my questions and comments on the manuscript:

(1) In Fig. 4(a), the authors show the Chern number of the lowest Floquet band, but the system size is small (it contains only 16 unit cells). How did the authors extract the Chern number from such small systems? Is there any finite-size effect on this calculation?

(2) The authors claim that the anomalous Floquet topological insulator is also stabilized by this scheme. However, the authors only show that the pumped charge drops to zero in the anomalous Floquet topological phase. Since a trivial insulator also shows zero pumped charge, it is not enough for claiming that this is an anomalous Floquet topological insulator. Can the authors show a decisive signature of this anomalous topological phase, e.g., the existence of edge states, in this dissipation-engineered Floquet system?

(3) The effective temperature in Fig. 6(c) shows that the cooling seems inefficient in the anomalous Floquet topological phase. Can the authors explain the origin of this behavior? Is it related to the gap closing associated with the topological phase transition?

(4) In Sec. II A, the authors write that the model is described by a Hubbard-Holstein Hamiltonian. This is a little misleading since the system is non-interacting fermions and does not have a Hubbard interaction.

(5) In Eq. (7), the system-bath Hamiltonian is time-dependent, while the right-hand side appears to be time-independent.

(6) Below Eq. (22), the authors write "with the lattice-trapped bath (red line), with the ohmic bath (green line)". It seems that the colors do not correspond to those in Fig. 7(a).

In conclusion, this manuscript shows a promising way to stabilizing Floquet topological phases and provides a guide for experiments in the near future.

1- the paper proposes a solution to the outstanding challenge of preparing a genuine 2D topological insulator in optical lattices
2- the proposal uses a combination of existing experimental techniques, including a possible observable for the topological insulator (quantised pumping)
3- it compares concrete alternatives (different combinations of species and experimental settings) and suggests an optimal configuration (large mass imbalance, 1D bath tubes)

1- the required physical ingredients of the proposal lead to a high overall experimental complexity
2- there are some open questions regarding the ratio of polarisabilities for the two atomic species
3- the method for detecting the topological insulator (quantised pumping) appears a bit convoluted

The manuscript tackles one of the outstanding challenges in the field of quantum simulation, namely the preparation of a genuine 2D topological insulator. Two main challenges must be met to prepare a 2D topological insulator. On the one hand, a band gap must necessarily close during the transition from trivial to topological band. On the other hand, the Floquet driving, necessary for creating topological bands, can lead to unwanted heating out of the topologically insulating state.

The authors elegantly solve both problems by engineering a very specific bath configuration, namely one-dimensional tubes along the orthogonal direction of the 2D topological-insulator lattice. These tubes contain heavy bosonic atoms, which interact weakly with the lighter fermions of the lattice.

The density-of-states of the bosons must be suitably narrow in energy, ideally smaller than the drive frequency. In this configuration, the fermions automatically form a near-perfect topological insulator as the non-equilibrium steady state.

In general, the manuscript meets all the necessary criteria for SciPost Physics (well-written, contains abstract, citations, specific details of derivations). Furthermore, it addresses an important, currently unsolved problem in the field of quantum simulation of topological matter. Therefore, the paper has the potential to be published in SciPost Physics, following some minor revisions.

1- The authors quickly converge towards establishing the mass imbalance of the two species as relevant parameter for successful bath engineering. While a large momentum transfer in system-bath collisions certainly makes sense intuitively, the aspect of lattice depths due to different polarisabilities (how near-resonant the lattice is) for the two species remains largely unexplored. As mentioned by the authors in the context of tube-versus-2D confinement, it is possible to find tune-out wavelengths in which one species does not feel the confinement of the other species.

I think the authors may have overlooked this aspect in terms of the main lattice potential. As I understand it, the most important criterion for the bath engineering is that the tubes of bosons are very disconnected, while the lattice remains dispersive for the fermions. The disconnected nature of tubes leads to the correct density-of-states in the bath (Fig. 5a).

Here, I would suggest taking a closer look at the lattice wavelength with respect to the atomic transitions. The proposed optimal scenario (lithium-6, cesium-133, lambda = 740nm) may turn out to be unrealistic, unless I have overlooked something, because the wavelength 740nm is mainly repulsive for cesium (D-lines around 850-890nm) while it is attractive for lithium-6 (D-lines at 671nm). In this case, the bath tubes would lie on the bonds of the lattice, not at the lattice sites.

2- Verifying a 2D topological insulator experimentally is in itself quite a challenging task. When measuring a quantised bulk-Hall response in finite (box) system, for instance, one would find an accumulation of charge at the boundary and then a reflection of the quantised current via edge states and the upper band with C = -1.

The difficulty of detection is apparent, as even in the ideal (T = 0) system the response is only ~90% 'quantised' (Fig. 7a). Thus, the chosen method for detecting the topological insulator appears sub-optimal.

In addition, I would like to see how the Peierls phases are generated concretely. I don't yet understand how adding a step-potential leads to the stripe of Peierls phases. Traditionally, I would have expected typical quantum Hall response, for example, by switching on a linear gradient along the x-direction leading to a quantised bulk current along y.

Evaluating the current operator across a single bond may provide another observable, instead of summing the entire charge in the upper half of the system.