The superconducting clock-circuit: Improving the coherence of Josephson radiation beyond the thermodynamic uncertainty relation

1- links stochastic thermodynamics (specifically TURs and their breakdown) to the mesoscopic physics of superconducting devices

2-describes the physics of a device that is amenable to practical implementation and can have an impact on current quantum technology

1- some important details are relegated to appendices or missing

I think the paper meets the expectation criterium of "providing a novel and synergetic link between different research areas": stochastic thermodynamic and mesoscopic physics of superconducting devices.

After revision I expect the paper to meet all the general acceptance criteria. See requested changes below

1- While the paper is clear and well written, it can improve further in regard to better integrate the content of the appendix in the main text. Maybe the appendices can be inserted in the main text? (optional)

2- How is the entropy production rate sigma calculated? Since the main message of the ms is that the device beats the TUR bound, which is expressed in terms of entropy production rate, it is crucial that the estimation of the entropy production rate be detailed. I could not find that in the text/appendices

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1—Given a direct connection between a well-known limitation in the coherence of the AC Josephson effect and the Thermodynamic Uncertainty Relation (TUR).
2— Provided a minimal setup for the realization of a superconducting circuit able to produce coherent radiation beyond standard TUR limits.
3—The introduction is well-written and clearly defines the context and goal of the paper.

1- The current presentation in Section 4 (and partially in Section 5) may be a little obscure due to some details only present in the Appendices - see also the report for some suggestions.

The authors theoretically investigate a superconducting circuit for the on-chip generation of coherent Josephson radiation from a purely DC input bias. The AC Josephson effect naturally converts a DC voltage bias into an oscillating signal, but the coherence of such radiation is limited by the Thermodynamic Uncertainty Relation (TUR). Inspired by recent work on TUR violation in a classical pendulum clock [Ref. 30], the authors propose a superconducting clock circuit in which the violation of TUR determines a lower bound on the linewidth of the emitted radiation.

More precisely, the authors consider a minimal model for a pendulum clock and investigate a circuit with two degrees of freedom. In their setup, a Josephson junction with a real voltage bias (counter) is coupled to an underdamped resonator (oscillator) via an identical Josephson junction. Using an external magnetic field in the loop, such a configuration at half-flux quantum mimics the escapement potential of the pendulum clock necessary to overcome the TUR limitation in the overdamped dynamics of the counter.

On the technical side, the authors solve the quantum dynamics of the counter-oscillator system using a Lindblad equation for the oscillator and a Langevin approach for the counter. The details of the model are discussed in Appendix A, and some analytical approximations and classical limits are provided in the main text. The authors identify a synchronization strength parameter $\beta$, which provides a figure of merit for the generation of coherent radiation beyond the TUR limit for a realistic voltage bias Josephson junction.

This work can stimulate further theoretical research on this topic and possibly be relevant for the experimental implementation of on-chip coherent radiation in high-impedance environments. As such, I would recommend this article for publication in SciPost after addressing the comments below.

1— In section 3, the escapement potential follows from Eq.(4) at half flux quantum in the limit of small oscillation amplitude. The cancellation of the $\cos(Y)$ term at the leading order expansion in $X\ll 2\pi$ requires the two junctions to be identical. How robust is the setup to the asymmetry between the two junctions possibly arising from fabrication? Does the asymmetry generally tighten the bounds on the quality factor of the radiation? It would be good to mention these points in the discussion.

2—To improve readability for a broader audience and make the article more self-contained, I would consider expanding Sec. II concerning the classical model of a pendulum clock. This choice could give more insight, for instance, of the ansatz $Y(t)=\omega_0 t+\theta(t)$ made in Sec. IV for the quantum regime. Moreover, the direct derivation of Adler-type equations (9) from Eq.(3) could be provided in some Appendix or in alternative some relevant citation could be added on this point. On another note, I would add some details on the continuous model for the escapement potential and maybe provide (even only as a footnote or in the appendix) the explicit connection of the potential $V_C(X, Y)$ and the contents of Ref. [30].

3—Section 4 (and similarly Section 5) is somewhat harder to follow compared to the rest of the main text since some of the results heavily depend on the derivation given in the Appendix. One solution would be to break down Appendices A and B in some subsections and add more references to them when quoting some of the relevant results of Section 4. Moreover, I would make more explicit which set of equations is solved when doing ''simulation ... for the full quantum mechanical model" compared to analytics. On a final note: is there some intuition on the increasing size of the voltage plateaux by increasing the light-matter coupling $r$?

4— Some technical comments: does the dephasing rate take the form of Eq. (1) only under the assumption $k_B T\gg eV$? If so, it would be good to specify it. After Eq.(23), should it be $\dot\theta\ll \omega_0$ rather than $\dot\theta\lesssim \omega_0$? Finally, I would provide some references (even to relevant books) for the path-integral discussion and other technical points arising after Eq.(18) in Appendix A. This could help the non-expert reader.

5— For completeness, add explicitly the definition of some symbols and
notation in the text:
-define the electric charge $e$ and Planck's constant $h$ (and/or the
reduced one $\hbar$) in the Introduction when these symbols are first
introduced in the paper;
-define the notation on the brackets: $\langle\langle Y^2\rangle\rangle$
for the variance and $\langle Y \rangle$ for the expectation value (here
both concerning time averaging, I think);
-define $\kappa$ as the coupling parameter in section 2.
-add a reference to Appendix A when discussing the small junction
capacitance limit before Eq.(6) since they appear only in the schematic of
Fig.4 and not in Fig.1 of the main text
-The definition of $Z_0$ slightly differs in the main text and in Appendix A
after Eq. (17), where the latter also contains the junction capacitance.

6— In Fig.2 and Fig.3, I would consider using a double axis for the x- coordinate adding the corresponding value of the current (in units of $I_c$ or $I_0$). What is the value of $\beta$ for the dotted curves in Figs. 3b and 3c? Morevoer I would add "(not shown)" after "Off resonance, $Q$ is of the order
of $10^{-9}$..." in the caption of Fig.3.

7—Few typos:
-a factor $4\pi$ is likely missing in the denominator of the quality factor
$Q_0$ - see paragraph following Eq. (1) in the Introduction;
-in the text above Eq. (18), I think one should replace $Z\rightarrow Z_0$ in
the definition of the coupling parameter $r$;
-"light-mater" $\rightarrow$"light-matter" in the text before Eq. (29);
-Again, in the text before Eq. (29), a sentence seems broken: "for the
expectation values of.";
-after Eq.(30), there is likely a typo in $r n_{coh}=
\kappa/(4\omega_0\gamma)$, where $\omega_0$ and $\gamma$
should be squared according to Eq. (8).

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