# 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/2110.15286v3 (pdf) Date submitted: 2022-01-28 11:52 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

### Submission & Refereeing History

Resubmission 2110.15286v4 on 24 June 2022

Submission 2110.15286v3 on 28 January 2022

## Reports on this Submission

### Strengths

1. The paper is very clearly written and the images support the text very well
2. The subject of the paper is very interesting and presents novel ideas
3. The authors propose a clear path towards realising non-Hermitian phenomena in truly quantum setups

### Report

The authors introduce what they call a non-Hermitian quantum simulator, which is a Lindbladian system that can reproduce the dynamics of non-Hermitian lattices. The advantage of this method is that it requires no fine-tuning. They illustrate their method with a minimal example of an anti-PT-symmetric bosonic dimer coupled to a Markovian bath consisting of two quantum fields. They show that the synthetic space of the dimer higher-order field moments is equivalent to the non-Hermitian Hamiltonian of a one-dimensional PT-symmetric chain, and identify a localisation transition induces by the presence of an exceptional point.

This paper is very well written and very interesting. It presents a novel method to simulate non-Hermitian dynamics on a quantum level, while circumventing techniques like post-selection. I believe this paper meets all the criteria for publishing in SciPost. As such, I recommend this paper for publication.

### Requested changes

1. The citation to Fig.1(d) below Eq.(3) should be Fig.1(e). Fig.1(d) should be cited elsewhere in the text.
2. 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.
3. 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?

• validity: high
• significance: high
• originality: high
• clarity: high
• formatting: excellent
• grammar: excellent

### Author:  Ievgen Arkhipov  on 2022-06-23  [id 2606]

(in reply to Report 2 on 2022-06-14)
Category:
correction

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.

### Author:  Ievgen Arkhipov  on 2022-06-23  [id 2605]

(in reply to Report 2 on 2022-06-14)
Category:
correction

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.

**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.

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

**2. 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.

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].
*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.*

**3. 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?