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Mapping as a probe for heating suppression in periodically driven quantum manybody systems
by Etienne Wamba, Axel Pelster, James R. Anglin
Submission summary
Authors (as registered SciPost users):  James Anglin 
Submission information  

Preprint Link:  https://arxiv.org/abs/2108.07171v1 (pdf) 
Date submitted:  20210817 14:14 
Submitted by:  Anglin, James 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
Experiments on periodically driven quantum systems have effectively realized quasiHamiltonians, in the sense of Floquet theory, that are otherwise inaccessible in static condensed matter systems. Although the Floquet quasiHamiltonians are timeindependent, however, these continuously driven systems can still suffer from heating due to a secular growth in the expectation value of the timedependent physical Hamiltonian. Here we use an exact spacetime mapping to construct a class of manybody systems with rapid periodic driving which we nonetheless prove to be completely free of heating, by mapping them exactly onto timeindependent systems. The absence of heating despite the periodic driving occurs in these cases of harmonically trapped dilute Bose gas because the driving is a certain periodic but anharmonic modulation of the gas's twobody contact interaction, at a particular frequency. Although we prove that the absence of heating is exact within full quantum manybody theory, we then use meanfield theory to simulate 'Floquet heating spectroscopy' and compute the heating rate when the driving frequency is varied away from the critical value for zero heating. In both weakly and strongly nonlinear regimes, the heating rate as a function of driving frequency appears to show a number of Fano resonances, suggesting that the exactly proven absence of heating at the critical frequency may be explained in terms of destructive interferences between excitation modes.
Current status:
Reports on this Submission
Strengths
1 Selfcontained and clear presentation of the research background and new results
2 Interesting idea of connecting timedependent and independent problems
3 Complete numerical and theoretical investigation of a specific problem
Weaknesses
1 The studied model seems to be quite special and to have a limited range of applicability.
2 Lack of implications for Floquet engineering and Floquet heating in generic systems
Report
In this paper, the authors theoretically study the heating problem in periodically driven quantum manybody systems. The authors devise a special class of driven Hamiltonians using the spacetime mapping technique that the authors developed in their previous studies. Under these driven Hamiltonians, the quantum system does not heat up, unlike under generic Hamiltonians. The authors discuss how to construct such Hamiltonians and address why the heating does not happen. These Hamiltonians never lead to heating because they are, by construction, equivalent to a timeindependent Hamiltonian.
Although the idea is interesting and the analysis is welldone, I am worried about how this study contributes to understanding Floquet engineering or Floquet heating, which are emphasized to motivate their research in the Introduction.
(1) Floquet engineering: If an experimentalist realizes a (spacetimemapped) timedependent Hamiltonian, is it useful? I think this is the Floquet engineer's viewpoint, and no heating is not enough.
(2) Floquet heating: I am not convinced that the heating studied in this work is related to Floquet heating in the general context. As is well known (e.g., in Ref. [3]), the Floquet heating is characteristic to periodically driven *manybody* Hamiltonians. However, the authors use the meanfield approximation, which approximates the original manybody Hamiltonian to a onebody one. Is it possible to talk about the Floquet heating within the onebody approximation? I am skeptical about this because I do not see the prominent exponentially slow heating [Phys. Rev. Lett. 115, 256803] in authors' Figs. 5 and 6.
Unless either (1) or (2) is resolved, the implications of the authors' results would be quite limited. Of course, I am sure that they scrutinize the specific model very carefully and professionally.
Considering the highstandard acceptance criteria of SciPost Physics, I find neither of the four expectations met while general acceptance criteria are met. Thus, I recommend that the authors send their manuscript to another specialized journal after making appropriate revisions.
Requested changes
1 "quasiHamiltonian" in Abstract might not be a standard terminology and could be replaced by a Floquet effective Hamiltonian as they use in the paper.
2 Is r in Eq.(2) defined? Is it $r\equiv\mathbf{r}$?
Report 1 by Thomas Bilitewski on 2021102 (Invited Report)
Strengths
1 establishes an exact mapping proving absence of heating in a class of interacting periodically driven quantum systems
2 establishes (numerically) the suppression of heating not only at the exact mapping point, but for deviations in the driving, and at a number of resonance far from this point
Weaknesses
1 While the mapping is exact it does applies to a rather special class of models only. It is not immediately obvious that many of the interesting interacting manybody phases of interest in Floquet engineering could benefit/be realised within the constraints of being able to be mapped to a static system.
2 While the authors try to connect the suppression of heating observed in the exact mapping to wider classes of periodically driven systems, and interpret the resonances found as "Fano" resonances potentially active more generally, the evidence for this seems to be mainly a good fit to the numerically observed resonance features.
Report
The manuscript "Mapping as a probe for heating suppression in periodically driven quantum manybody systems" discusses the use of spacetime mapping identities applied to the case of harmonically trapped contact interacting Bose gases to establish the absence of secular longtime heating for a specific class of periodic modulation of system parameters.
In addition to discussing the exact mapping, the authors further establish the suppression of heating close to the exact mapping point, and at a number of further resonances they interpret as Fano resonances in numerical simulations of the mean field GP equation.
The field of Floquet engineering of interacting manybody phases is highlyactive, and exact results on the absence of heating in interacting systems are certainly of great interest. The work appears perfectly technically sound, and seems to indicate some applicability beyond the exactly mappable cases.
However, I am not fully convinced that the results established here will easily translate to more general situations relevant for the realisation of nontrivial phases. In addition, while the application to periodically driven situations is novel, the mappings themselves have been established by the authors in prior work (Ref [32,33]).
The work satisfies all general acceptance criteria, I am not fully certain if it satisfies the expectation criteria. Given that the general mapping has been identified before I don't see it as groundbreaking or a breakthrough. I believe the strongest argument could be made for it opening up a new way of studying periodically driven systems, with some potential for future work, for which I am willing to recommend publication.
Requested changes
1 I believe a more thorough discussion of avoiding heating in Floquet systems in the introduction might be beneficial. The introduction seems to skip over the field of prethermalisation, including the rigorous bounds on heating rates, and the field of MBL and disorder which are two of the dominant ways to engineer interacting nontrivial Floquet phases.
2 The manuscript uses "trapunits" /"trap length" for space, time and energy at various points, e.g. in all figures. I couldn't find an explicit definition of those, and if those refer to trap parameters for experiment A or B separately. A definition before they are first used might be useful, in particular since for timedependent trap parameters it's not immediately obvious what those would be.
3 The GP equation (Eq 14) as a nonlinear equation requires a normalisation for $\int dx \psi^2$. Since there are two common conventions of either $\int dx \psi^2 = N$, or $\int dx \psi^2 = 1$ absorbing the N factor into the definition of g, the authors should specify which convention they are using.
4 I'd like some clarification on a small point in the argument in section 5.2 on equivalence of longterm heating between experiment A and B. The authors discuss nongrowth of $\int dx \psi_A^2$, and claim this implies nongrowth of Eq.(21). That doesn't seem to follow without additional assumptions on the timedependence since $\frac{d}{dt_A} \int dx \psi_A(x,t_A)^2$ can grow without bound even if $\int dx \psi_A^2$ is bounded.
I assume the situation of periodicallydriven systems might impose additional constraints that make the statement correct, in which case that might be worth clarifying.
Minor changes:
5 The use of "quasiHamiltonian" in the abstract seems uncommon to me. Most of the literature seems to rather use effective Hamiltonian
6 Eq (11) uses $\tan^{1}$, and Eq (12) uses $\arctan$
7 Caption of Fig 5 refers to "less than the sharp resonance frequency $\nu \approx 1.9$". I assume the authors refer to the resonance found in Fig. 6. However at this point in the text this resonance hasn't been mentioned, and would need some context.