Revealing the finite-frequency response of a bosonic quantum impurity

This work deals with the experimental realization and investigation of an archetype bosonic quantum impurity system, the boundary sine-Gordon (BSG) model. Such system is realized in a superconducting platform where a small SQUID is galvanically coupled to a high impedance transmission line. The latter, being composed of 4250 Josephson junctions, simulates an Ohmic bosonic environment with high frequency cut-off wp. The impedance Z of the transmission line is designed to be a sizeable fraction of the quantum of resistance RQ. The SQUID is characterized by a flux-dependent Josephson energy EJ(phi); its internal capacitance CJ provides the second relevant energy scale, the charging energy EC. Finally, properties of the SBG system are evinced from measurements of the finite frequency response of the transmission line in various regimes of parameters.
Indeed major result of this work is the identification of different dynamical regimes of the impurity in a
EJ(phi)/EC vs w/wp plane. To this aim, the authors compare experimental data with analytical predictions for the dissipative response at high frequencies, and for the reactive response in the regime of moderate phase fluctuations. In the latter case a renormalization of the Josephson energy of the SQUID by the high frequency environmental modes is found to be in line with expectations for the ideal SBG model in a large parameter range. For deviations, the authors simply quote the very recent work by Masuka et al., Ref. 22, where a numerical renormalization group analysis is performed for a similar model. However, no comparison with predictions of that work is performed. This should be amended in a revised version of the paper.

In summary, this work represents a relevant and very timely endeavor in the field of quantum simulators. It confirms theoretical predictions on various parameter regimes of the BSG model and identifies open theoretical challenges. As such I highly recommend publication in SciPost. Before this, the authors should critically address the work by Masuka et al. in the context of their experimental results and theoretical analysis.

1. This work reports the results of a difficult experiment performed on a controllable quantum many-body system.

1. Lack of comparisons with the results of several earlier experiments performed with very similar systems.
2. A part of the theoretical treatment is not reliable.
3. Some of the qualitative considerations sound misleading or erroneous.
4. The way several recent papers referred to is highly inadequate.

This paper reports the measurements of the electromagnetic response of a small SQUID immersed in a high-impedance harmonic environment. The latter was constructed of relatively large-area Josephson junctions and may be thought of as a dissipationless transmission line supporting the propagation of microwave photons. The photons quantum fluctuations renormalize the SQUID's properties, bringing quantum many-body effects into the picture.

The main results of the work are represented by the experimental data of Figs. 5 and 6. The former figure shows the renormalized by quantum fluctuations Josephson energy EJ of the SQUID; the latter one details the dissipation of the transmission line photons caused by their decay into photons of smaller frequency.

The manuscript also contains some theoretical considerations focusing mostly on the Josephson energy renormalization in the case of "large" EJ and photon decay in the case of "small" EJ. The EJ-renormalization part contains little new (by authors' admission) but is an integral part of the data analysis. The other part is poorly formulated, contains some ad-hoc assumptions and tweaks performed just to get a better agreement with the experiment. Therefore, I doubt that section qualifies as a theory wielding any predictive power; in my view, it causes more harm than good (in part, because some of the qualitative considerations are misleading or erroneous).

The narrative of the manuscript is passable, except of a couple egregiously misrepresented references.

There are precious few experiments of the type performed by the Authors, so this work worth publishing in some form. However, the Authors must take care of several serious issues which I will address next.

1. The results for the dissipation, Fig. 6, should be compared with the respective results of Ref. [45] (Fig. 3 therein).

2. Quantify the definition of "large" EJ early in Section 4.1. For an unprepared reader, that becomes (sort of) clear only after seeing the in-line expression for the phase fluctuations preceding Eq. (14).

3. Say something about EJ* in the beginning of Sec. 4.2. At the very least, tell it is not the one of Eq. (14).

4. Calculation of the dissipation \gamma to the second order in EJ can be performed by standard means involving no more than the Gaussian averaging of exponentials linear in boson creation-annihilation operators (much like this is done in, e.g., theory of Mossbauer effect). Do that and present the result in a clear form in Sec. 4.2.
Remove or relegate to a remote Appendix all of the currently present stuff involving the use of poorly defined (in my view) diagrammatic technique and tweaks such as "suitable regularization" with little physical meaning. The current content of p. 15 and a part of p. 16 does more harm than good. Here are two examples. The linearity of photon spectrum alluded to on p. 15 as important ingredient, is hardly important in a decay with no momentum conservation (as it is for the processes induced by a quantum impurity). The phase slips at low energy are suppressed (as the system flows towards the superconducting fixed point) and hardly relevant for the dissipation -- in an apparent contradiction to the statement on p. 16.

5. The properly performed evaluation of \gamma should be sensitive to the time-reversal symmetry due to the photon transition selection rules that should be demonstrated to instill some confidence in the results.

6. I am appalled by the way the work of Manucharyan's lab is represented in the manuscript. Clearly, the latter repeats the measurements reported in Ref. [45] on a circuit with somewhat different parameters. This is not spelled out in the manuscript under a lame excuse that Ref. [45] explores the "insulating" regime of BSG (see p. 1 of the manuscript). There are people among the Authors qualified enough to clearly understand: there is no qualitative difference between the "insulating" and "superconducting" regimes at \omega>> EJ*. On top of that, neither of the experiments is accurate enough to see the minute emerging differences between the regimes within the range of explored microwave frequencies. A later related work of the Manucharyan's group, https://doi.org/10.48550/arXiv.2203.17186, is simply not cited.

7. I find the way Ref. [46] cited is misleading and does poor service to the community. Among the references to that paper scattered through the manuscript, the ones on p. 5 and p. 6 are the most damaging. The former one reads: "The presence of the secondary cutoff EC can affect quantitatively the general phase diagram of the BSG model [46]". This actually is an erroneous conclusion of Ref. [46], as a subset of the manuscript Authors explained in their recent comment https://arxiv.org/abs/2210.00742. (In addition to the quantitative arguments of that comment, one may recall that the phase diagram is entirely determined by the infrared physics oblivious to the specifics of an ultraviolet cutoff). The laudatory reference on p. 6, "Only recently [46] has a theoretical understanding started to emerge of the non-trivial physics ..." just helps to proliferate in the community an erroneous idea, which is challenged in the comment by the very manuscript's Authors. In my view, citations to Ref. [46] -- if kept -- should represent the views of the manuscript's Authors.