Peer review of: Electric field control of radiative heat transfer in a superconducting circuit (2020)

<b>REVIEWER COMMENTS</B> 

Reviewer #1 (Remarks to the Author): 

This paper reports a realization of purely electrostatic photonic heat valve in a superconducting circuit, 
where the thermal conductance of a single photonic channel between two reservoirs can be tuned 
close to its quantum limit. The observed temperature modulation and hence that of the thermal 
conductance show 2e-periodicity as a function of the gate charge up to a certain temperature limit 
below which the quasiparticle poisoning can be neglected. The authors explain the observed heat flow 
oscillations due to the competition between Cooper-pair tunneling and Coulomb repulsion in a gate-
controlled Cooper pair transistor with a simple model. By neglecting the junction asymmetry and 
quasiparticle contributions and by approximating the Josephson junction as a gate-tunable LC 
resonator, the authors explain their measurements with no free parameters simply based on the 
Landauer scattering theory picture and the heat balance equation incorporating the phonon bath. They 
also justify their approximations in the Supplementary Information. As the gate charge can be varied 
down to a single charge level, the heat valve can be closed or opened at will paving the way for the 
magnetic-field-free control of electron-photon relaxation mechanisms. 

I find the presentation of the manuscript succinct, informative and well-written. The motivation, 
experimental details and the theoretical description to support their observation are explicit and nicely 
presented throughout the paper. The experimental and theoretical methodologies seem pertinent and 
sound. The expected ramification of this work is also summarized in the concluding remarks followed 
by the Methods section which details the fabrication of the device and the analysis in the paper. The 
Supplementary Information further addresses various details of the experiment and the theory. I think 
that this paper will be influential in several cutting edge fields of nano science in strong relevance to 
quantum thermodynamics, superconducting coherent caloritronics and quantum thermoelectrics. In 
view of the above points, I would recommend this manuscript for publication in Nature 
Communications. 

Nevertheless, I wonder if there were any precedent theoretical and/or experimental works related to 
the authors' findings regarding the electric field control of heat even if they may not be of photonic 
nature or might have been studied in different systems, as the references are completely lacking in 
the present manuscript. It will be helpful for the readers if a brief sketch of the previous efforts can be 
provided on which the present work is based. In this way, I believe that the significance of the 
authors' work filling the preexisting gaps will be more clearly visible. 



Reviewer #2 (Remarks to the Author): 

The authors report a gate-modulated thermal conductivity between two reservoirs using a Cooper pair 
transistor as a valve. This is an interesting experiment and, I think with some discussion & possible 
modification, it could be suitable for publication in Nature Communications. 

I read and re-read parts of this manuscript to try to re-cast it into a picture that is closer to how I 
think about these things. Part of why I felt the need to do this is that it seems the manuscript is 
written to make this sound maybe a little more profound than I think it needs to, or maybe should be. 
It could also be that I really am failing to grasp what this is showing but, I think I understand enough 
to ask some basic questions: 

1. Equation 1 indicates that heat flow can be cast in a Landauer picture, as the integral of the 
difference of the equilibrium mode occupancies of two reservoirs, weighted by a single-channel 
transmission factor $\tau(\omega)$. 

Invoking Landauer here, is, I guess, a way to connect with a mesoscopic picture of transport, but I 
really don't think that it's necessary. They write down an expression for what this transmission 
coefficient is, $\tau(\omega) = 4 R_1 R_2/|Z_{tot}(\omega)|^2$. In their circuit model, this would 
correspond to what you would derive for the rf/microwave transmission coefficient with port 
impedances $R_1$ and $R_2$. As such, what they are saying, and what I think the paper is about, is 
really how 1D black body radiation propagates through a gate-tunable filter, the Cooper pair transistor. 
This heat transport problem requires integrating over a broad power spectral density that is mostly 
thermal up to the 'thermal bandwidth' cutoff, $k_B T/\hbar$, rather than a narrow bandwidth centered 
somewhere in the quantum limit $\omega\gg k_B T/\hbar$. 

This is all a long-winded way to ask whether the authors have purposely chosen to not discuss this as 
a 1D blackbody + filter picture explicitly. It seems much simpler to frame the discussion this way, 
rather than as about approaching a quantum of thermal conductance or invoking Landauer. Although 
the latter sounds more exciting to some people, I think it oversells this, especially when we're talking 
about one scattering element. 

I will note that I sat down and estimated the transmission based on some simple assumptions to 
understand the series impedance of the Cooper pair transistor. Since the plasma frequency is probably 
very high (I estimate 30+ GHz), this should be well-approximated by the gate-modulated Josephson 
inductance. With nanoampere-scale critical currents (shown from the Joyez-type 3-state calculation in 
the Supplementary), the impedance looks to be something like a few hundred Ohms, modulating by a 
few hundred Ohms. However, just plugging into their equation for the transmission (again, just what 
one would calculate for in general for two different port impedances), I see a gate-modulated low-pass 
filter response with a 3 dB cutoff that varies between 1.5 to 3.5 GHz. When looked at in this way, it's 
pretty clear that heat conveyed via noise propagating at these frequencies will be modulated by 
scattering from the CPT impedance. I have attached a PDF of a Jupyter notebook showing what these 
back-of-the-envelope estimates look like to me. I hope that by including this, you can tell me more 
precisely whether I'm misinterpreting the basic picture. 

2. Temperature scales, Other temperature choices 

Apologies if I missed it, but I'm not sure what motivates the choice for the reservoir & cryostat 
temperatures. Presumably they could all be much lower. How does the contrast in the temperature 
oscillations vary with this temperature? Not in a theoretical sense, but in practice? The measurements 
in Fig 3b are taken at relatively high temperatures compared to the base temperature of their dilution 
refrigerator. 

3. Quasiparticle poisoning 

This is a much smaller comment, but important nonetheless. In the Supplementary, the authors 
discuss how they think about quasiparticle poisoning. It's a little bit of an outlier today as it judges the 
degree of poisoning from an averaged switching current. This is a very old way (Joyez thesis) to 
assess the parity. It gives some weighted measure of the average parity, but taking the average 
removes any hints of the fast dynamics that underlie all aluminum single-charge and even transmon 
devices. To be specific, even with a 2e periodic switching current curve, there can be significant 
quasiparticle poisoning. This is more clearly shown in histogrammed switching current statistics, 
shown in work from the mid 2000s on Cooper pair transistors from NIST, UNSW, and others. It seems 
that Joyez is the only reference for quasiparticles in these devices, although the literature is pretty 

deep at this point. I believe the authors' own group has published other CPT work that is much better-
referenced in this regard. 

3. Device limitations/Improvements. 

It would be nice if the authors could discuss how this device might be improved, or if its performance 
(contrast) can be improved. If this picture of gate-modulated low-pass filtering is ok, then one can 
imagine producing a device with more contrast in the minimum/maximum critical currents. One could 
also impedance match the lead resistances to the CPT impedance. There's some practical limit to this, 
but more generally, is there some practical benefit (research or applied) to improving the performance? 

Summary 

This is an interesting paper, although I think that some of the language is a little dramatic for my 
taste (although I acknowledge that some of this seems necessary for certain journals), and while I'm 
not suggesting it for publication right away, I'm hoping that my comments/questions can help clarify 
the picture in the paper so that it's more instructive to a wider audience. I think the bigger questions 
they're asking are interesting in general, but I think that the experimental discussion could be made 
clearer. 


Reviewer #3 (Remarks to the Author): 

In this work, Maillet et al present the realization of heat flow modulation using a fully electric knob, 
i.e., a Cooper pair transistor. The authors succeed in opening and closing a photonic heat channel 
between two normal metal reservoirs by playing with the Cooper pair tunneling and Coulomb blockade 
effect. To do so they use a nice lockin technique to monitor temperature variations, they demonstrate 
(supp material) that the photonic channel is the main path for heat transport and, finally, they 
demonstrate that temperature modulation is 2e-periodic, univocal proof of the relationship between 
heat flow and 2e quantization in the transistor. There is a sufficient amount of data presented along 
the paper, figures are extremely simple but really evoking of the phenomenon under investigation. 
Theoretical analysis of the results is also very clearly explained, rigorous and convincing. To 
summarize, the work presented by Maillet et al is very beautiful and interesting for a broad scientific 
community and I recommend its publication in Nat Commun. 
In the following I pose very few questions/suggestions for the authors: 

1) In the introduction (abstract) part the authors comment on the flux-charge duality. This statement 
is confusing to me as they want to avoid any reference to a magnetic flux controlling knob. Would it 
be better to talk about charge-phase conjugation? 
2) Along the paper I didn’t get what factors determine the amplitude of the temperature oscillations 
that the authors are observing. Is it possible to increase this amplitude by, e.g., increasing the 
thermal gradient or decreasing the cryostat base temperature? 
3) I find that the definition of the maximum achieved contrast a bit confusing. Max contrast = 1 
means simply that G_gamma,min = 0, am I right? 
4) What do you mean by “negligible retarded response” (methods section) and how do you observe it 
experimentally? 
5) What do you mean by “the issue of finite frequency behavior” (conclusions section)? 
6) Finally, I have a fundamental question. What is exactly the role played by the superconductor? As 
far as I understand it, the role of the superconducting part is to block heat currents mediated by 
quasiparticles. If this is the case, could it be possible to demonstrate a similar effect using a 
conventional normal-metal single-electron transistor stacked between superconducting leads or am I 
missing something? Is that the equivalent effect that you are observing at large temperatures where 
you attribute the 1e-periodicity to quasiparticle poisoning (supp material)? If this is the case, has this 
effect been observed experimentally previously?