# Emergent non-Hermitian skin effect in the synthetic space of (anti-)$\cal PT$-symmetric dimers

### Submission summary

 As Contributors: Ievgen Arkhipov Arxiv Link: https://arxiv.org/abs/2110.15286v4 (pdf) Date submitted: 2022-06-24 19:31 Submitted by: Arkhipov, Ievgen Submitted to: SciPost Physics Academic field: Physics Specialties: Atomic, Molecular and Optical Physics - Experiment Condensed Matter Physics - Theory Quantum Physics Approach: Theoretical

### Abstract

Phase transitions in non-Hermitian systems are at the focus of cutting edge theoretical and experimental research. On the one hand, parity-time- ($\cal PT$-) and anti-$\cal PT$-symmetric physics have gained ever-growing interest, due to the existence of non-Hermitian spectral singularities called exceptional points (EPs). On the other, topological and localization transitions in non-Hermitian systems reveal new phenomena, e.g., the non-Hermitian skin effect and the absence of conventional bulk-boundary correspondence. The great majority of previous studies exclusively focus on non-Hermitian Hamiltonians, whose realization requires an {\it a priori} fine-tuned extended lattices to exhibit topological and localization transition phenomena. In this work, we show how the non-Hermitian localization phenomena can naturally emerge in the synthetic field moments space of zero-dimensional bosonic anti-$\cal PT$ and $\cal PT$-symmetric quantum dimers. This offers an opportunity to simulate localization transitions in low-dimensional systems, without the need to construct complex arrays of, e.g., coupled cavities or waveguides. Indeed, the field moment equations of motion can describe an equivalent (quasi-)particle moving in a one-dimensional (1D) synthetic lattice. This synthetic field moments space can exhibit a nontrivial localization phenomena, such as non-Hermitian skin effect, induced by the presence of highly-degenerate EPs. We demonstrate our results on the example of an exactly solvable non-Hermitian 1D model, emerging in the moments space of an anti-$\cal PT$-symmetric two-mode system. Our results can be directly verified in state-of-the-art optical setups, such as superconducting circuits and toroidal resonators, by measuring photon moments or correlation functions.

###### Current status:
Has been resubmitted

We thank the Referee for his/her careful reading of our work and for the overall appreciation of the manuscript, finding it "very well written and very interesting" and recommending it for publication. The Referee asks two minor changes and poses important and interesting questions. We provide our answers on the Referee's comments (1-3) below.

1. The referee writes: The citation to Fig.1(d) below Eq.(3) should be Fig.1(e). Fig.1(d) should be cited elsewhere in the text.

Our reply: We thanks the Referee for spotting that typo. We have fixed it the revised version of the manuscript.

1. The referee writes: I am a bit puzzled why references [110] and [111] are cited at the end of section VI. To my knowledge, they do not discuss lattices of N coupled cavities.

Our reply: We apologize for any possible confusion. Indeed, [110] and [111] do not treat directly lattice models, but one of the motivations behind their investigation is the emergent non-Hermitian Hamiltonian describing single-particle physics of D-dimensional lattices [110] and Lieb-like lattices [111]. The revised manuscript now reads: Thus, to obtain a NHH akin to those emerging in higher-dimensional lattice architectures [36,110,111] one has to tune the few parameters of the dimer instead of fine-tuning all the parameters of the lattice.

1. The referee writes: The authors focus on bosonic systems in their work. Could they comment on if and, if yes, how their work could be translated to the fermionic case?

Our reply: We thank the Referee for the very interesting question. The use of higher-order moments and the mapping to the single-particle NHH is based on the bosonic operator algebra, and as such a direct extension to the fermionic problem is not possible. Nonetheless, it is an interesting question what is the structure of the moment space of quadratic fermionic systems, and it could constitute an interesting future research direction. We added a corresponding paragraph at the end of Sec. V of the revised manuscript: Another interesting direction of future research will be the investigation of higher-order moments spaces of quadratic fermionic fields, and their possible similar mapping to higher-dimensional lattice systems. A foreseeable challenge towards this extension is the mathematical construction of the fermionic higher-order moments space and the associated mapping to the single-particle NHHs. Indeed, the Kronecker sum algebra, that we employed in the bosonic case, cannot be directly applied to fermionic particles.

### List of changes

1. Reference to the Fig. 1(d) has been changed to Fig. 1(e), below Eq. (3).
2. Second sentence in the second paragraph of Sec. VI has been modified.
3. A new paragraph at the end of Sec. V has been added.

### Submission & Refereeing History

Resubmission 2110.15286v5 on 9 August 2022

Resubmission 2110.15286v4 on 24 June 2022
Submission 2110.15286v3 on 28 January 2022

## Reports on this Submission

### Report

The authors show that effective non-Hermitian Hamiltonians can be simulated by using the synthetic space of field moments of a 0D bosonic system. This has several advantages, such as the possibility of obtaining effective 1D non-Hermitian systems (and their topological properties) without the need to fine-tune individual system parameters across the entire lattice. The authors demonstrate these results using a two mode system.

I enjoyed reading this paper. I think that the results are clearly presented, and the discussion is sufficiently pedagogical to make the paper accessible to nonspecialists.

After checking the citations and the previous works of the authors, however, I became confused about the novelty of these results. Upon first reading, the discussion of the abstract and introduction gave me the impression that the auhors are presenting a novel method of obtaining non-Hermitian Hamiltonians. Now, instead, my impression is that this is an incremental work, which builds on the methods and models that were already introduced by the same authors in 2102.13646 and in 2006.03557 (both published in Phys. Rev. A).

If my understanding is wrong, I would like to ask the authors to please correct me. My understanding is that the idea of simulating effective non-Hermitian systems using the space of field moments (together with its practical advantages) was already covered in these two works. Further, the same model of two incoherently coupled modes is used in all three works. The presence of nth order exceptional points is demonstrated in all three papers, and plotted as a function of the same model parameters (compare Fig. 1 of 2006.03557, Fig. 3 of 2102.13646, and Fig. 1e of this submission). I also note that in 2102.13646, it is explicitly mentioned that the authors are doing a follow-up investigation of 2006.03557. For example, the authors say:
"The model under consideration is the same as in Ref. [68]."
and
"In our previous study [68], we analyzed EPs, up to their third order, of such an anti-PT -symmetric bimodal cavity"
I did not find such statements in the current submission, giving me the mistaken initial impression that these represented novel results.

I do not believe this paper should be published in its current form. I suggest that the authors very clearly and explicitly state which results have already been obtained before (both by themselves and by others). As far as I've been able to understand from looking at all three papers, the current submission is an incremental work building on the same idea and on the same model, and therefore would not fulfill the acceptance criteria of Scipost Physics.

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### Report

The authors have addressed my comments and questions to my satisfaction, and I believe this paper is ready for publication.

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