Nonequilibrium quasiparticle distribution in superconducting resonators: effect of pair-breaking photons

1- Quasiparticles are a widespread nuisance in superconducting circuits, but their origin and modelling is still poorly understood, so this work appears timely
2- The work combines numerical solution of rate equations, with simpler analytical insights that could be useful for experimentalists

1- The manuscript will appeal to specialists
2- It is often cryptic, and involves lots of experimental parameters
3- There is strong overlap with a previous publications by the authors (Ref [17])

This work considers the important question, both of fundamental
and technological importance, of the origin of quasiparticle poisoning
in superconducting circuits. The main emphasis is put on the
role of high energy photons, with a combination of analytical
and numerical analysis.

The paper will certainly be useful to theorists and experimentalists
in the field, but is hard to read for non specialists.

I am suggesting below several recommendations to the authors that could improve the readability of their manuscript, as well as asking some needed clarification.

Overall, this is a serious and useful study. It seems that it does not meet
any of the four main "novelty" criteria for SciPost Physics, but it would be
perfectly acceptable as SciPost Core, after revision.

Page 1: the highlighted (in upper case) part of the following sentence
is confusing:
"a large number of (non-pair-breaking) photons can “heat
up” the quasiparticles by pushing them to higher energy
as compared to the phonon temperature; these quasiparticles
can relax by emitting phonons, so that the latter are
also driven out of equilibrium. If the emission process
involves a RELATIVELY LARGE NUMBER OF PAIR-BREAKING PHONONS
with frequency ω > 2∆".
Owing to energy conservation, I do not see how low energy photons
can create enough quasiparticles as to emit a large number of
pair-breaking phonons above the gap.

Page 1: in this sentence
"we present the kinetic equation that determines the quasiparticle
distribution; it extends the previously used kinetic equations describing
the interaction of quasiparticles with photons of energy below the pair
breaking threshold 2∆ [17, 23, 24] by including a contribution from a
mode of energy above the threshold"
Why is the assumption of a single mode justified? Typically, the
photons experienced by a superconducting device will have gone through
a series of complex filtering, but I don't see a reason why it should
be spectrally sharp

Page 2, Eq. (2):
Why is \bar{n} not a function of \omega_0 here?
Why is this not an integral over the the low-energy photon density of states?
Identify the various terms here w.r.t. the bare processes in Fig. 1

Page 2, Eq. (4) and Eq. (5):
Why is \bar{n}_{PB} not a function of \omega_{PB} here?
Identify the various terms here w.r.t. the bare processes in Fig. 1

Page 2, before Eq. (4):
State clearly that pair breaking photons have a definite energy omega_{PB}

Page 2, Fig.1:
1/ why is the diagram for quasiparticle to photon recombination omitted?
2/ are the higher order diagrams g) and h) really useful in the analysis?

Page 3, above Eq. (6):
I understand that Pauli blocking factors can be replaced to unity at low
temperature, but:
1/ is that really legitimate when the non-equilibrium distribution function
is not known before hand?
2/ doesn't that violate detailed balance or other sum rules?

Page 3, Eq. (6):
1/ if the integral over \epsilon starts at zero, why is the BCS term
diverging at \epsilon+\omega=0 and not \epsilon+\omega=\Delta?
2/ St_g^{hon} is not defined
3/ why doesn't the phonon population $n$ enter here explicitely?

Page 3:
Globally, I find this whole section hard to read

Page 3, Eq. (12):
going from the first line to the second seems unjustified,
since the factor U(E) has a divergence at the superconducting gap.
Also, why can it be replaced by unity?

Page 4-5:
the mapping and analysis of the Volterra equation is relatively
well-explained. Can the author clarify the physical difference
between \gamma_\star and \gamma_\star' ?

Page 6, Fig. 4 (top plot):
Indicate "exact" as a black line inside the figure

Page 7:
it would be nice to have a few sentences at the end of this section
that summarizes the main results. In particular, regarding the range
of maximal values for f(\gamma) in Fig. 4, what do they imply experimentally?
Is 10^(-5) a "big" or a "small" number?

Page 10:
It is not clear where Eq. (49) comes from

Page 11, Eq. (54-55) and Fig 8:
1/ Can the extrinsic quality factor Q_{i,ext} be extracted
from this fit, and what could one learn from it?
For instance, if it is due to dielectric losses, its
temperature and frequency dependence is known in the literature
2/ It seems that Fig. 10 fits better without Q_{i,ext}, which
does not make sense. Can the authors clarify?
3/ why don't the properties (e.g. impedance) of the considered
resonator enter the equation?

Page 13, Eq. (59):
can the authors explain briefly where the equation for the energy
shift comes from, is it just Kramers-Kronig?

Ask for minor revision

1. The authors theoretically solve an important problem, which can potentially be useful for improving the quality of low temperature detectors

2. The authors rely on well established formalism and do not introduce additional fit parameters, untested assumptions etc.

1. The paper is rather technical and, therefore, it is difficult to read.

2. The paper is the follow up of Ref. [17]. Therefore, the authors should clearly state what is the difference between the current manuscript and Ref. [17]

The authors theoretically investigate the effect of high frequency
pair-breaking photons on the quality factor and on the frequency of
a superconducting resonator. They extend their previous work [17] on
this subject, and obtain approximate analytical expressions
for the quasiparticle distribution function in presence of both
pair breaking and of low frequency photons. They show that this
function may strongly deviate from equilibrium. Finally, they
use this result to study the dependence of the quality factor
and of the frequency of a superconducting resonator on power
and on temperature. The results reasonably well agree with
the experiment. Thus, the authors provide theoretical support
for the mechanism of non-equilibrium quasiparticle generation
by high frequency photons.

I am sure that the reported results are correct because
they are based on the well established kinetic equation formalism.
Nevertheless, in my opinion, the presentation of these results can be
improved. Indeed, the paper is full of complicated expressions,
which are difficult to follow. I understand that one cannot fully
avoid this and that the kinetic equations naturally lead to that.
However, it would be beneficial for the reader if the authors
would explain in simple terms the physical meaning of obtained
results, and would discuss the general system setup.
For example, the following issues can be additionally clarified:

1. The set of peaks in the distribution function plotted in Fig. 5
are located at energies m\omega_0. Can one interpret this as a result
of multi-photon processes, at least at the qualitative level?
The same question applies to Fig. 6.

2. Another unclear issue is related to the very low values of the distribution
function in Figs. 5 and 6. Namely, does it matter if this function
has a peak with the height 10^{-18}? What are the minimal values
of f(E), which are relevant for the experiment?
Do the peaks in Figs. 5 and 6 affect the quality factor in Figs. 8 and 9?

3. Why do the authors consider only one high frequency mode with the
frequency \omega_0 > 2\Delta? Does it correspond to the experimental
setup? I think assume that, for example, a high quality factor niobium cavity
of a centimeter size should have dense spectrum around 100 GHz frequency,
which corresponds to 2\Delta.

Finally, the authors should also clearly state the difference
between the current manuscript and Ref. [17], which has a lot
of similarities with the manuscript.

In conclusion, I recommend accepting the paper after minor
changes in the presentation, which have been mentioned above.

The requested changes are also mentioned in the report.

1. Clearly state what is new as compared to Ref. [17]

2. Add a bit more qualitative discussion of the results