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Precision magnetometry exploiting excited state quantum phase transitions
by Qian Wang, Ugo Marzolino
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Submission summary
Authors (as registered SciPost users):  Ugo Marzolino 
Submission information  

Preprint Link:  https://arxiv.org/abs/2306.01126v3 (pdf) 
Date submitted:  20231012 14:27 
Submitted by:  Marzolino, Ugo 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
Critical behaviour in phase transitions is a resource for enhanced precision metrology. The reason is that the function, known as Fisher information, is superextensive at critical points, and, at the same time, quantifies performances of metrological protocols. Therefore, preparing metrological probes at phase transitions provides enhanced precision in measuring the transition control parameter. We focus on the LipkinMeshkovGlick model that exhibits excited state quantum phase transitions at different magnetic fields. Resting on the model spectral properties, we show broad peaks of the Fisher information, and propose efficient schemes for precision magnetometry. The LipkinMeshkovGlick model was first introduced for superconductivity and for nuclear systems, and recently realised in several condensed matter platforms. The above metrological schemes can be also exploited to measure microscopic properties of systems able to simulate the LipkinMeshkovGlick model.
Author comments upon resubmission
Reviewer:
It is not clear if the different colors have a specific meaning in Figure 1a. If yes, can the authors please explain it?
Answer:
The colours in figure 1(a) do not have a specific meaning, but are intended to make the figure clearer for small magnetic fields when the curves become closer.
Reviewer:
Can please the author comment which techniques they use to compute the spectrum, even in the sector $s=N/2$ with dimension $N+1$ for the considered systems with a big number of spins, e.g., $N=12000$?
Answer:
We have computed the spectrum of the Hamiltonian in the sectors with $s=N/2$ and fixed parity (either even or odd) using exact diagonalization using MatLab 2018b and the command “eig”. The time required for exact diagonalization is of the order of several minutes even for N=12000. We have added the following sentence five lines after equation (1): “These symmetries allow us to compute numerical exact diagonaliziation of the LMG Hamiltonian in the orthogonal subspaces with $s=N/2$ and even or odd parity”.
Reviewer:
Reorganize the citations, avoid multiple citations to facilitate the readability. For example at the end of the second paragraph of page 4, the citation [50, 27, 51, 52, 23, 53, 47, 54, 27] can be rearranged to have [23, 27, 47, 5054].
Answer:
The document class and the bibtex style in the previous version were responsible of the order of citations. After using the SciPost template, the citations are organized as the reviewer suggested.
Reviewer:
Is it possible to give the analytical expression of the vertical critical lines in the caption of Figure 2?
Answer:
We have added the analytical expression of vertical the lines in the caption of figure 2. These lines correspond to the critical fields $h_c^k$ of different eigenstates ($E_k\rangle$ and $\tilde E_k\rangle$), namely the field value such that $E_k=E_c=Nh_c^k/2$. Therefore, $h_c^k=2E_k/N$. The analytical expression of the eigenvalues $E_k$ is not known in general.
Reviewer:
I think there is an extra (d) In the second line in the caption of Figure 2, and that the expression critical fields for two adjacent excited states’’ should be explicated.
Answer:
We thank the reviewer for noticing this typo. Indeed, the first of the (d) has been replaced with (b). The expression “critical fields for two adjacent excited states” appeared in the paragraph starting with “We further compare…” at page 6. We have rewritten that sentence. We have also replaced the term “adjacent eigenenergy gap $\Delta E=E_{k+1}E_{k+1}$” with “eigenenergy gap $\Delta E=E_{k+1}E_{k+1}$” in the caption of figure 2.
Reviewer:
I would suggest to rewrite subsection 4.1.1 about probe preparation, it is less clear than the rest of the manuscript.
Answer:
We have rewritten subsection 4.1.1. In particular, we slightly changed the first paragraph in order to be a little bit more explicit concerning the reduction to the subsystem with $s=N/2$, and we have added several details on the phase estimation algorithm used for probe preparation in the subsequent paragraphs.
Reviewer:
I would also like to point out that (ground state and excited states) phase transitions of such type of critical systems are nowadays feasibly simulated on NISQhardware, see e.g., Phys. Rev. E 107, 024113 (2023).
Answer:
We thank the referee for bringing the interesting paper Phys. Rev. E 107, 024113 (2023) to our attention. It is indeed a promising study of simultations of the LMG eigenstates with measurements of the corresponding phase transitions on NISQ device. We have introduced very shortly the context and the terminology of NISQ technologies sentence in the introduction. We have then mentioned in the conclusions that the paper Phys. Rev. E 107, 024113 (2023), together with experimental platforms for realizing the LMG model, may inspire new implementations with NISQ devices.
List of changes
 We have added the following sentence five lines after equation (1): “These symmetries allow us to compute numerical exact diagonaliziation of the LMG Hamiltonian in the orthogonal subspaces with s=N/2 and even or odd parity”.
 We have used the SciPost template, and the citations are organized as the reviewer suggested.
 We have added the analytical expression of vertical the lines in the caption of figure 2.
 We have rewritten the sentence starting with “We further compare…” at page 6.
 We have rewritten subsection 4.1.1. In particular, we slightly changed the first paragraph in order to be a little bit more explicit concerning the reduction to the subsystem with s=N/2, and we have added several details on the phase estimation algorithm used for probe preparation in the subsequent paragraphs.
 We have added the following sentence in the introduction “It is also desirable to investigate metrological schemes suited for several physical platforms and using different physical phenomena in the search for feasible implementations, so called noisy intermediatescale quantum (NISQ) technologies [15,16]” in order to introduce the context and the terminology. We have also and added the following sentence in the conclusions “Eingenstates of the LMG model at small size can be simulated also with variational hybrid quantumclassical algorithms with low depth on superconducting transmon qubits, and signatures of their phase transitions can be computed [112]” after mentioning other realizations of the LMG model, and conluded with “The robustness of magnetometric protocols based on the ESQPT and the aforementioned variety of platforms for realizing or simulating the LMG model make these protocols good candidates for NISQ devices”.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 3) on 2024116 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2306.01126v3, delivered 20240116, doi: 10.21468/SciPost.Report.8413
Strengths
1) Detailed presentation of the excitedstate structure of the LMG, and analysis thereof through quantum metrological concepts.
2) Presentation of new protocols allowing to prepare states showing superextensive sensitivity
Weaknesses
Unclear whether the superextensivity really comes from the presence of the ESQPT: see attached report.
Report
I believe this work should be suitable for publication in Scipost, but I believe some of its claims may have to be amended; in particular, I am not sure whether the superextensive scaling reported here is really a signature of an ESQPT, or simply a generic properties of highlyexcited spin models (see attached report for details).
Once these points have been adressed, I will be happy to recommend this work for publication
Requested changes
See attached report.
Author: Ugo Marzolino on 20240305 [id 4337]
(in reply to Report 2 on 20240116)We would like to thank very much the reviewer first of all for her/his accurate and stimulating comments, and then also for the overall positive assessment of our results and of their importance. We have resubmitted the manuscript amended according to all the reviewer’s suggestions.
Attachment:
reply_to_the_second_report_otUTb3p.pdf