Peer review of: Observation of topological transport quantization by dissipation in fast Thouless pumps (2020)

REVIEWER COMMENTS
Reviewer #1 (Remarks to the Author): 

This manuscript deals with non-Hermitean Floquet physics, more precisely with a theoretical analysis 
and experimental realisation of time-periodic dissipation-induced enhancing of Thouless pumping 
quantization in the driven Rice-Mele model. 

The experimental realisation consists of arrays of coupled waveguides, with the propagation direction 
z playing the role of time, in the paraxial approximation to the Helmholtz equation. 

The theoretical analysis introduces the concept of time-periodic modulation of dissipation, and 
explores the consequences in a non-Hermitean Floquet framework. 

The combination of theory and experiment, both careful and well described, realised in this manuscript 
is quite convincing, and I would be in favour of publication. 

There is one aspect of the discussion that I think the authors should improve. One of the common 
approaches to open quantum systems, especially in the context of quantum optics, goes through a 
Markovian quantum master equation of the Lindblad form. There, the non-Hermitean terms are 
supplemented by quantum jump terms which, conspiring together, allow, for instance, reaching 
stationary solutions in time-independent cases. 
I now understand that such a LIndblad approach belongs in some way to a ``different community' of 
physicists. Still, I would love to read, in the present manuscript, a discussion of the connections of the 
present non-Hermitean dissipation approach to the other, Lindbladian, framework. (Incidentally, the 
Lindblad approach is much less developed in the explicitly time-dependent case, which would be 
relevant here.) 

Minor corrections/typos: 

Schröderinger in place of Schrödinger on page 1 

qunatization in place of quantization in the caption of Figure 2. 

cite in place of site on pag. 4. 

In the text description of Fig. 7 (b) it is said ``(waveguide A: circles, waveguide B: squares)', but I 
see no squares; rather triangles. 




Reviewer #2 (Remarks to the Author): 

In this work, the authors construct a dissipative variant of Rice-Mele topological pumping that leads to 
quantized center-of-mass displacement in periodically-modulated coupled polariton waveguide array. 
The work is of contemporary interest when topological effects are explored both in time-dependent 
systems, as well as in non-Hermitian systems. The extension of topological pumping and its realization 
in a finite time period alongside dissipation, highlights an important distinction to adiabatic equilibrium 
idealizations. In this sense, I find the work relevant for publication in Nat. Commun. given that the 
authors can address/answer the following points: 
1. The authors heavily rely on the fact that at their input facet, the Wannier orbitals of each band of 
the Rice-Mele model are localized on a single site. Thus, they can excite effectively a single band. Had 
the initial state in the cycle corresponded to another configuration along the pump cycle, this would 
not have worked. I would ask the authors to highlight this point in the paper. 

2. Similar ideas where topological bands are populated by dissipation have been proposed in the past, 
for example, in [Phys. Rev. Lett. 112, 133902 (2014)]. Please refer and discuss the connection to 
such works. 

3. Can the authors discuss the FBBZ SPP also for Figs. 6 and 7. 

4. Can the authors derive an analytic expression for the gap size G (to lowest orders in the Floquet-
Bloch truncation - this should not be complicated) - both for the dissipative and non-dissipative case? 
This would corroborate the numerical analysis and will provide a clear relation between Omega and 
gamma. Correspondingly, in Fig. 2f, why does the gap close at ~0.8 Omega? 

5. In Fig. 3b, the wavepacket excited with site B is scattering back to the other band and moves in the 
same direction as the one excited with site A. What kind of backscattering process leads to this 
behavior? 

6. The Rice-Mele topological pumping is closely related to the adiabatic limit of a Landau-Zener 
process. The setting employed in this work is even more closely related to the dissipative version of 
Landau-Zener. I would like the authors to comment on this relation, and it could assist them also in 
answering the point 5 above. 

7. In the methods section above Eq. (14), why is the constant dissipation assumption required? 

8. I am confused by the quantized transport predicted by equations (14) and (15): commonly the 
topological pump is related to a time-dependent scan over the polarization of the populated band. It is 
obtained similarly to (14), where two contributions arise corresponding to a group velocity term [as 
appearing on the right hand side of (14)], and another geometric term that leads to the topological 
winding (quantized displacement). How does this comply with the presented analysis? Is the transport 
quantized due to topology at all? 

9. Last, the center of mass displacement appears to be relatively quantized. However, the losses lead 
to much reduced intensity. This means that quantized charge transport will be lost. I would ask the 
authors to discuss this point, its prospective technological implications, and its relation to previous 
attempts to obtain a a high-precision metrological estimation of the Ampere, see, for example 
[https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.101.066801] 



Reviewer #3 (Remarks to the Author): 

In this paper, the authors make significant experimental and theoretical progress towards non-
adiabatic charge pumping in the presence of loss. Theoretically, they make what I believe is the first 
prediction of quantized non-adiabatic charge pumping in non-Hermitian systems. They then confirm 
these predictions experimentally using dielectric-loaded surface plasmon-polariton waveguides, which 
enable analogues of Schrodinger equations similar to pioneering work on Floquet Chern insulators, but 
with the presence of controllable “time-dependent” loss. In short, I find the work quite interesting, 
well-written, and novel, and recommend publication in Nature Communications. I have some notes 
below that should be considered to improve the paper, but are not necessary for publication. 

On the theory front, I would appreciate more details of the proof of Eq. 14 and 15. As currently 
written, I take this as an intuitive approximate result, but mathematically there are a number of steps 
missing. For instance, the paper jumps back and forth between the Floquet extended zone treatment 
and the quasienergies. An important example of this is the statement that the norm of the modes 
decays as exp(-Gamma_{k alpha} t). This would only be true in the absence of micromotion, or its 
non-Hermitian equivalent, whereas in practice it will definitely be the case that some points in time 
are lossier than others. It is not clear how this enters into the proof that charge pumping corresponds 
to energy winding. Without a more detailed proof, I think that Eqs. 14 and 15 should be argued as 
approximate statements neglecting k-dependence and t-dependence of the imaginary part of the 
energy, which is consistent with numerics + experiments. 

Regarding the experiments, I am a bit confused by the center of mass plots, which seem to 
consistently show an upturn above linear behavior in most of the charge pumping data. Is this 
understood? If so, a few words about why this happens and why we expect it to plateau to a quantized 
slope for larger number of “Floquet cycles” is merited.