Boundary Time Crystals as AC sensors: enhancements and constraints

The manuscript addresses an innovative and relevant topic, proposing the use of boundary time crystals (BTCs) for metrological purposes, specifically as quantum sensors for AC fields. This novel application of BTCs is likely to inspire further investigations in quantum metrology, as indicated by the manuscript’s existing 7 citations, which underscores its resonance with specialists in the field. Additionally, identifying entropy production as a limitation in achieving the Heisenberg limit adds valuable insights into the interplay of dissipation and metrological performance.

The primary weakness of the manuscript is its lack of self-contained explanations in certain sections, which could hinder accessibility for a broader audience. Expanding some discussions and providing additional references would enhance the work's clarity and comprehensibility. Furthermore, inconsistencies in normalization conventions across figures and a limited discussion of certain assumptions weaken the overall presentation.

The manuscript is well-structured, presenting a comprehensive numerical investigation of the boundary time crystal (BTC) model and exploring its potential as a sensor for AC fields. It demonstrates the feasibility of BTCs for quantum-enhanced sensing, leveraging their long-lived coherence and multipartite correlations. However, the results indicate that while BTC sensors surpass classical limits, they fall significantly short of achieving the Heisenberg limit, with entropy production identified as the primary factor limiting performance. By connecting these findings to the broader context of quantum metrology, the manuscript offers valuable insights into the role of dissipation and coherence in open quantum systems.
I recommend a minor revision to address these concerns. After these revisions are done I believe the present work will meet the journal's acceptance criteria.

1. Model Explanation:
For readers unfamiliar with time crystals, it is unclear why this model constitutes a time crystal. The statement "Specifically this is known..." following Equation (1) is vague. I suggest expanding this section with additional references or providing a brief derivation in an appendix to substantiate the claim.

2. Discussion of Quantum Fisher Information (QFI):
Extending the discussion of QFI in Equation (4) to include the expected scales and their implications for the sensor’s performance would be welcome. For instance, explaining the significance of F/N > 1 as a quantum enhancement and why F/N \approx N indicates the Heisenberg limit. Although some of this discussion is already present in the introduction, I believe that moving and expanding it would benefit the coherence of the manuscript. I also think that the introduction of the operator $\hat{L}_g$ seemed unnecessary since it was not used in practice to obtain the results.

3. Normalization in Figures:
I would recommend using consistent normalization of the figures. For example, in Figure 1 F was normalized to N^2, while Figures 2 and 5 normalize F to N. Alternatively the authors could justify, in each figure, their choice for the given normalization.

4. Choice of External Field Orientation:
The external field is assumed to be oriented in the z-direction, as defined in Equation (3). Clarify whether this choice is standard or made for convenience. If it is standard, cite appropriate literature. If not, discuss the implications of considering arbitrary driving directions.

5. Entropy Suppression:
Does the parameter \omega_0 (coherent driving strength) influence entropy production? Could modifying the driving Hamiltonian, e.g., introducing Heisenberg interactions, suppress entropy production and improve performance?

6. Code Availability:
To ensure reproducibility of the results I would suggest, not require, the authors to publish also simulation codes on a public repository such as GitHub. This aligns with the transparency encouraged by the SciPost community and will benefit future researchers working on related topics.

7. Symbol Usage:
The same symbol `S` is used for total spin length and entropy. This could confuse readers. Consider using a distinct symbol for the spin operator.

8. Typo:
On page 6. "... they were were generated ... " has too many "were" -s.

Ask for minor revision

The authors investigate the application of boundary time crystals for the purpose of AC field quantum sensing. In this model, a number of two level systems (spins) evolve according to a specific, permutation-symmetric master equation with a collective drive and dissipation. This system has a non-trivial phase in the thermodynamic limit, depending on the ratio of the drive and decay parameters. If the driving is strong enough, permanent oscillations occur asymptotically. In a recent paper by some of the current authors, it has been demonstrated that Floquet time crystals may be employed for the same purpose. In analogy to the case of Floquet time crystals, the quantum Fisher information measure (QFI) is used as a tool to assess the possibility of enhanced sensing.

A numerical analysis of the QFI behavior (considering the resonant case and optimal phase corresponding to the thermodynamic limit) reveals that it is possible to fit a power-exponential curve as a function of time, for various system sizes N and frequencies. The authors the examine various BTC properties and identify the role of entropy in this context. Namely, the entropy grows and reaches some finite value in the thermodynamic limit, thus the system is not in a pure quantum state (entropic cost). The entropic cost may be decreased by decreasing dissipation. The authors also compare the results of a mean field calculation, providing some insight how on larger time-scales the exact results deviates from the MF calculations, although on small time-scales they both coincide.

The presented calculations lead to the conclusion that BTC sensors possibly provide a moderate quantum enhancement, however it cannot be exculded that around the critical point of the system, higher performance can be achieved. I wonder, why and how much the sensitivity could be enhanced around the critical point, since a brief explanation would add a lot to the analysis of the paper. This is just an optional question. It is possible that the answer is complicated and could be addressed in another publication.

In conclusion, I find the manuscript very well written, interesting, about a timely and important topic, thus I suggests its publication in SciPost Physics.

Publish (easily meets expectations and criteria for this Journal; among top 50%)