# Three questions about John von Neumann

Melvyn B. Nathanson  Department of Mathematics
Lehman College (CUNY)
Bronx, NY 10468<EMAIL_ADDRESS>
John von Neumann was one of the greatest scientists of the 20th century and
one of the most complicated. His research career began in mathematics and
mathematical physics. Before, during, and after World War II he was deeply
involved in military work and was a central figure in the design and
construction of atomic and hydrogen bombs. He was one of the creators of game
theory and its use in economics and he was (perhaps the most significant of
his achievements) the dominant figure in the early development of computer
hardware and software. He was also an influential US government policy maker
on defense matters and a strong proponent of building intercontinental
ballistic missiles and designing the small powerful nuclear weapons that would
become the warheads of ICBMs.

Ananyo Bhattacharya’s book, _The Man from the Future: The Visionary Life of
John von Neumann_ [1], is in part a biography of the man, but it is more a
magisterial survey of von Neumann’s work, a series of essays that follow, in
roughly chronological order, the different fields in which von Neumann worked.
The first chapters are on mathematical logic and the foundations of quantum
mechanics, then chapters on his work on nuclear weapons, on computer design,
and on mathematical economics. Finally, there are chapters on his consulting
for the US government and for the RAND corporation and on his work on cellular
automata. This is a book worth reading.

Studying the life of von Neumann, one is led to ask three related questions:

  1. (1)

Von Neumann was a great scientist. Readers of the _Notices of the American
Mathematical Society_ might ask: Was he a great mathematician?

  2. (2)

Von Neumann not only strongly and successfully lobbied for the construction of
horrendous weapons of mass destruction and their efficient delivery on
presumptive enemies, but he also advocated a pre-emptive nuclear strike on the
Soviet Union before that country had the ability to retaliate. Afterwards, he
forcefully promoted the doctrine of Mutual Assured Destruction. Was von
Neumann evil?

  3. (3)

Von Neumann was one of the first permanent professors at the Institute for
Advanced Study (IAS), viewed by many mathematicians as mathematical heaven,
but, as he moved away from “pure mathematics,” he felt increasingly unwelcome
and uncomfortable there. He resigned in 1956. Why did he quit?

##  1\. Was von Neumann a great mathematician?

John von Neumann was born Neumann János Lajos in Budapest on December 28,
1903. He attended an excellent high school in Budapest, but in math and
science he far outpaced his fellow students and teachers. His father hired
private tutors, first Gábor Szegö and then Michael Fekete, to teach him
advanced mathematics. Von Neumann’s first paper, published in 1922, was a
joint paper with Fekete on the zeros of polynomials. In 1923 and 1925 he
published important papers in set theory. In 1926 he received a Ph.D. in
mathematics from the University of Budapest and also a degree in chemical
engineering from ETH Zurich.

Between 1922 and 1932, von Neumann published 45 papers on mathematics and
mathematical physics. He wrote foundational papers on set theory and logic,
ergodic theory, game theory, and functional analysis. These were the topics
_du jour_. He also wrote papers on physics, several with his Hungarian high
school friend Eugene Wigner. Perhaps his most significant work in this decade
was _Mathematische Grundlagen der Quantenmechanik_ , published in Berlin by
Springer in 1932 and reprinted (in German) in the United States by Dover
Publications in 1943. Freeman Dyson [3] wrote,

> Johnny’s book was the first exposition of quantum mechanics that made the
> theory mathematically respectable…. [But] I was surprised to discover that
> nobody in the physics journals ever referred to Johnny’s book. So far as the
> physicists were concerned, Johnny did not exist….

Stan Ulam [11, p. 8], von Neumann’s closest friend in America, wrote:

> Mathematicians, at the outset of their creative work, are often confronted
> by two conflicting motivations: the first is to contribute to the edifice of
> existing work – it is there that one can be sure of gaining recognition
> quickly by solving outstanding problems – the second is the desire to blaze
> new trails and to create new syntheses. This latter course is a more risky
> undertaking, the final judgment of value or success appearing only in the
> future. In his early work, Johnny chose the first of these alternatives.

Johnny had wanted to become famous quickly and he succeeded, but he had doubts
about the importance of his work. He wrote influential papers on mathematical
logic, but in 1931 Gödel (citing a 1927 paper of von Neumann) proved the great
incompleteness theorem. Von Neumann proved the mean ergodic theorem in 1931
(published in 1932), but Birkhoff (aware of von Neumann’s result) proved and
published the more fundamental individual ergodic theorem in 1931.

As World War II approached, von Neumann continued to write papers in
mathematics, including a series of papers “On rings of operators” that fills
the entire third volume of his _Collected Works_ [13], but he became
increasingly involved with military-related research in many areas, including
ballistics and the development of nuclear weapons. He was a consultant to many
defense installations and an important figure on the Manhattan Project in Los
Alamos. The weapons work required extensive numerical calculations, and he
soon became the key contributor to the theoretical framework and actual
building of the first digital computers. He participated in the development of
the ENIAC at the University of Pennsylvania and led the design and
construction of the MANIAC computer at IAS.

In a frequently quoted passage from the essay “The mathematician,” reprinted
in the first volume of his _Collected Works_ [13], von Neumann wrote

> As a mathematical discipline travels far from its empirical source, or still
> more, if it is a second and third generation only indirectly inspired by
> ideas coming from “reality” it is beset with very grave dangers. It becomes
> more and more purely aestheticizing, more and more purely l’art pour l’art….
> [W]henever this stage is reached, the only remedy seems to me to be the
> rejuvenating return to the source: the re-injection of more or less directly
> empirical ideas.

This describes a path that mathematicians might choose, and the path that von
Neumann chose. As Bhattacharya notes,

> Von Neumann constantly sought new practical fields to which he could apply
> his mathematical genius, and he seemed to choose each one with an unerring
> sense of its potential to revolutionize human affairs.

Bhattacharya follows this with Freeman Dyson’s summary evaluation of von
Neumann’s career:

> As he moved from pure mathematics to physics to economics to engineering, he
> became steadily less deep and steadily more important.

Mark Kac [7, p. 125] believed that von Neumann was “one of the greatest
mathematicians of our century.” In the academic world we have not only grade
inflation but also adjective inflation. _De mortuis nil nisi bonum_.
Obituaries often call distinguished dead professors “great.” There is, of
course, a distinction between doing great work in mathematics and being “one
of the greatest” mathematicians. In the 20th century in “pure mathematics”
there were Hilbert, Poincaré, Weyl, Gödel, Banach, Weil, Erdős, Gel’fand,
Grothendieck, …. Was von Neumann in that class?

##  2\. Was von Neumann evil?

During World War II, von Neumann worked enthusiastically on the design and
construction of atomic bombs and after the war he strongly encouraged the
development of hydrogen bombs. During the war, the fear that Germany might try
to manufacture nuclear weapons led many American physicists and engineers to
work on the Manhattan Project, though at least one major physicist quit when
it became clear that the Germans could not produce a bomb. The pursuit of
fusion weapons was less widely supported. On October 30, 1949, the General
Advisory Committee of the Atomic Energy Commission wrote:

> [The hydrogen bomb] is not a weapon which can be used exclusively for the
> destruction of material installations of military or semi-military purposes.
> Its use therefore carries much further than the atomic bomb itself the
> policy of extermination of civilian populations. We all hope that by one
> means or another, the development of these weapons can be avoided…. Its use
> would involve a decision to slaughter a vast number of civilians.

Von Neumann was a super-hawk. Bhattacharya writes (p. 208),

> By 1946, von Neumann was predicting that devastating nuclear war was
> imminent…. His answer was preventive war - a surprise attack that would wipe
> out the Soviet Union’s nuclear arsenal (and a good number of its people too)
> before the country was able to retaliate. ‘If you say why not bomb them
> tomorrow, I say why not today?’ he reportedly said in 1950. ‘If you say
> today at 5 o’clock, I say why not one o’clock?’111Von Neumann’s quote is
> from his obituary in Life magazine [2].

Von Neumann abandoned the effort to launch a pre-emptive strike when it became
obvious that the Soviet Union had enough nuclear weapons to retaliate.

Was von Neumann wrong to have worked on the atomic bomb or on the hydrogen
bomb? Should he be criticized for striving to belong to the scientific and
policy-making elite that developed intercontinental ballistic missiles and
American nuclear policy? Was he wrong to have forsaken “pure mathematics” and
mathematic physics?

There is the dual question. Are we wrong to pursue pure mathematics and not to
apply at least some of our intellectual skills to help solve problems plaguing
this planet? There is no shortage of “real world” problems to try to solve. Is
the mantra that even the purest mathematics is eventually useful only self-
serving self-justification?

The answer, of course, is that there is more than one ethical choice. It is
easy to condemn von Neumann for some of the work he did, but he chose
honorably to think about and work on the hardest political and military policy
problems of his time. We should be grateful that we are free to choose the
problems we want to think about.

##  3\. Why did von Neumann quit the Institute?

Von Neumann wanted to be rich. Money was important to him. He grew up in an
extremely wealthy family in Budapest, and, according to his friends and to his
biographers, he was most comfortable with people who were also from wealthy
families. He enjoyed the company of generals and admirals, senators and
cabinet secretaries, corporate bigwigs.

He and Eugene Wigner were young, poorly paid academics in Germany in 1931 when
they received job offers from Oswald Veblen, who was then recruiting for the
Princeton mathematics department. Wigner wrote,

> One day I received a cable offering a visiting professorship at about eight
> times the salary which I had at the Institute of Technology in Berlin. I
> thought this was an error in transmission. John von Neumann received the
> same cable, so we decided maybe it was true, and we accepted. The pay was
> $3,000 for the semester, with $1,000 for travel–a small fortune at the time.

In 1932, the Institute for Advanced Study announced the creation of its first
school, the School of Mathematics. Its first appointment was Oswald Veblen.
The following year, the Institute hired James Wadell Alexander, Albert
Einstein, von Neumann, and Hermann Weyl. In 1935, Marston Morse joined the
faculty. The Minutes of the IAS, dated January 28, 1933, record:

> RESOLVED, That Professor John von Neumann be appointed as a Professor in the
> School of Mathematics upon the following terms: that his appointment as
> Professor in the School of Mathematics date from April 1, 1933; that his
> salary be fixed at Ten thousand Dollars ($10,000.00) per annum….

During World War II, many mathematicians and physicists at the Institute did
war work for the government, but it was expected that after the war they would
resume their “pure” scientific research. Von Neumann not only did not forsake
applied science, but, when he returned to Princeton, began the Electronic
Computer Project (ECP) at the Institute, a government-funded project with a
large staff of engineers and hardware specialists working to construct the
next generation (at the time, actually, the prototypical) computer. His wife
Klári von Neumann wrote,

> There he clearly stunned, or even horrified, some of his mathematical
> colleagues of the most erudite abstraction, by openly professing his great
> interest in other mathematical tools than the blackboard and chalk or pencil
> and paper. His proposal to build an electronic computing machine under the
> sacred dome of the Institute, was not received with applause, to say the
> least.

Much of the programming on the computer was devoted to solving problems
related to the development of hydrogen bombs. According to Theodore Taylor, a
Los Alamos bomb designer, “The objective was pretty specifically to be able to
do the coupled hydrodynamics and radiation flow necessary for H-bombs.”

Most of the mathematics faculty at IAS were opposed to both the ECP and to
weapons research. George Dyson [4, p. 82] writes,

> At the Institute for Advanced Study in early 1946, even applied mathematics
> was out of bounds. Mathematicians who had worked on applications during the
> war were expected to leave them behind. Von Neumann, however, was hooked.

According to Freeman Dyson,

> The mathematicians certainly knew there was classified work going on. They
> may not have known it was hydrogen bombs, but it was pretty obvious. And
> they were certainly opposed to that.

The humanities faculty at IAS also opposed the ECP. “In spirit we
mathematicians at the Institute would cast our lot in with the humanists,”
said Marston Morse.

After the war, von Neumann was more and more involved with high level
consulting with the government, and was soon living in Washington and serving
as a Commissioner on the Atomic Energy Commission. He seems to have felt more
and more unwelcome in Princeton. According to his wife Klári:

> The lines were drawn and after the first flurry of excitement it became
> clear that we did not belong in Princeton any more.…. [On May 16, 1954]
> while in Los Angeles on air force strategic missile business…, von Neumann
> met with Paul A. Dodd, dean of letters and sciences at UCLA, who offered him
> a special interdisciplinary position, with no teaching responsibilities, as
> professor at large.

The UCLA job would give von Neumann unlimited opportunities to consult with
IBM and other companies and replenish his bank account, which had been
depleted during his years of government service.

On August 2, 1955, von Neumann was diagnosed with advanced, metastasizing
cancer. He entered Walter Reed Hospital in Washington, DC, in March 1956. On
March 19, 1956, he wrote to Oppenheimer:

> After thinking matters over, I have decided to proceed in a direction which
> will lead me away from the Institute. In order to make matters precise and
> definite, I plan to resign from the Institute’s faculty, probably as of
> December 31. I will write you on this subject in due time, but I thought
> that I ought to inform you of the nature of my plans without delay.
>
> This does not mean that I am not considering the 22 years I spent at the
> Institute the most fruitful ones of my life and scientific career. The
> association with you and my other colleagues at the Institute has been a
> great experience.
>
> Please consider this matter strictly confidential until we communicate
> further.

Bhattacharya writes,

> Cancer had come at a particularly cruel time. The truth was that von Neumann
> had been unhappy at the IAS for several years before his death. ‘Von
> Neumann, when I was there at Princeton, was under extreme pressure’ says
> Benoît Mandelbrot, …‘from mathematicians, who were despising him for no
> longer being a mathematician; by the physicists, who were despising him for
> never having been a real physicist; and by everyone for having brought to
> Princeton this collection of low-class individuals called “programmers”….
> Von Neumann was simply being shunned. And he was not a man to take it.’

Von Neumann never got to California. He died in Walter Reed Hospital on
February 8, 1957.

Bhattacharya’s book is the best introduction to the life and work of von
Neumann.

##  4\. Other reading

The literature on von Neumann is enormous. There is a fine biography by Norman
Macrae, _John von Neumann_ [8], and also books by his brother Nicholas A.
Vonneumann [14] and his daughter Marina von Neumann Whitman [15]. Ulam’s
autobiography [12] contains many stories about von Neumann.

There are serious studies on almost every aspect of von Neumann’s work. For
the history of computers, there are books by Goldstine [5] and Dyson [4]. On
the history of the development of nuclear weapons, see Rhodes [9, 10]. An
issue of the _Bulletin of The American Mathematical Society_ (volume 64:3
(1958)) contains surveys of von Neumann’s mathematical work, beginning with a
valuable article by Ulam [11]. The essys by Freeman Dyson [3] and Paul Halmos
[6] are also excellent.

I thank Caitlin Rizzo, archivist at the Shelby White and Leon Levy Archives
Center of the Institute for Advanced Study, for providing access to its papers
on von Neumann.

## References

  * [1] Ananyo Bhattacharya, _The Man from the Future: The Visionary Life of John von Neumann_ , W. W. Norton & Company, New York, 2021.
  * [2] Clay Blair, Jr., Passing of a great mind, _Life_ , 25 February 1957, p. 96.
  * [3] Freeman Dyson, A walk through Johnny von Neumann’s garden, Notices Amer. Math. Soc. 60:2 (2013), 154–161.
  * [4] George Dyson, _Turing’s Cathedral_ , Pantheon Books, New York, 2012.
  * [5] Herman H. Goldstine, _The Computer from Pascal to von Neumann_ , Princeton University Press, Princeton, 1972.
  * [6] P. R. Halmos, The legend of John von Neumann, Amer. Math.Monthly, 80:4 (1973), 382-394.
  * [7] Mark Kac, _Enigmas of Chance_ , HarperCollins, New York, 1985.
  * [8] Norman Macrae, _John von Neumann_ , Pantheon books, New York, 1992.
  * [9] Richard Rhodes, _The Making of the Atomic Bomb_ , Simon & Schuster, New York, 1987.
  * [10] Richard Rhodes, _Dark Sun: The Making of the Hydrogen Bomb_ , Simon & Schuster, New York, 1995.
  * [11] Stanislaw M. Ulam, John von Neumann 1903–1957, Bull. Amer. Math. Soc. 64:3 (1958), 1 – 49.
  * [12] Stanislaw M. Ulam, _Adventures of a Mathematician_ , Charles Scribner’s Sons, New York, 1976.
  * [13] John von Neumann, John, _Collected Works_ , volumes I – VI, edited by A. H. Taub, Pergamon Press, New York-Oxford-London-Paris, 1961–1963.
  * [14] Nicholas A. Vonneumann, _John von Neumann as Seen by His Brother_ , P. O. Box 3097, Meadowbrook, PA.
  * [15] Marina von Neumann Whitman, _The Martian’s Daughter_ , University of Michigan Press, Ann Arbor, 2012.


††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.††thanks: These authors contributed
equally to this work.

# Hot Carrier Thermalization and Josephson Inductance Thermometry in a
Graphene-based Microwave Circuit

Raj Katti  Department of Physics, California Institute of Technology,
Pasadena, California 91125, USA  Harpreet Arora  T. J. Watson Laboratory of
Applied Physics, California Institute of Technology, 1200 East California
Boulevard, Pasadena, California 91125, USA  Institute for Quantum Information
and Matter, California Institute of Technology, Pasadena, California 91125,
USA  Olli-Pentti Saira  Department of Physics, California Institute of
Technology, Pasadena, California 91125, USA  Brookhaven National Laboratory,
Upton NY 11973, USA  Kenji Watanabe  National Institute for Materials Science,
Namiki 1-1, Tsukuba, Ibaraki 305 0044, Japan  Takashi Taniguchi  National
Institute for Materials Science, Namiki 1-1, Tsukuba, Ibaraki 305 0044, Japan
Keith C. Schwab  Department of Physics, California Institute of Technology,
Pasadena, California 91125, USA  T. J. Watson Laboratory of Applied Physics,
California Institute of Technology, 1200 East California Boulevard, Pasadena,
California 91125, USA  Michael Roukes  Department of Physics, California
Institute of Technology, Pasadena, California 91125, USA  T. J. Watson
Laboratory of Applied Physics, California Institute of Technology, 1200 East
California Boulevard, Pasadena, California 91125, USA  Stevan Nadj-Perge
Correspondence<EMAIL_ADDRESS>T. J. Watson Laboratory of Applied
Physics, California Institute of Technology, 1200 East California Boulevard,
Pasadena, California 91125, USA  Institute for Quantum Information and Matter,
California Institute of Technology, Pasadena, California 91125, USA

###### Abstract

Due to its exceptional electronic and thermal properties, graphene is a key
material for bolometry, calorimetry, and photon detection. However, despite
graphene’s relatively simple electronic structure, the physical processes
responsible for the transport of heat from the electrons to the lattice are
experimentally still elusive. Here, we measure the thermal response of low-
disorder graphene encapsulated in hexagonal boron nitride (hBN) by integrating
it within a multi-terminal superconducting device coupled to a microwave
resonator. This technique allows us to simultaneously apply Joule heat power
to the graphene flake while performing calibrated readout of the electron
temperature. We probe the thermalization rates of both electrons and holes
with high precision and observe a thermalization scaling exponent consistent
with cooling dominated by resonant electron-phonon coupling processes
occurring at the interface between graphene and superconducting leads. The
technique utilized here is applicable for wide range of semiconducting-
superconducting interface heterostructures and provides new insights into the
thermalization pathways essential for the next-generation thermal detectors.

Graphene provides a tantalizing opportunity for the design and development of
bolometric detectors, due to its exceedingly small heat capacity [1, 2], much
smaller compared to traditionally synthesized thin films. In addition, the
thermal conductivity of graphene can be greatly changed by coupling it to
superconducting or normal electrodes or placing it on different substrates.
Moreover, when graphene is contacted using superconducting electrodes, the
resulting Josephson coupling and the corresponding supercurrents are highly
dependent on electron temperature [3]. Accordingly, graphene-based Josephson
junctions (gJJs) are particularly promising for detecting ultra-small thermal
responses at milli-Kelvin temperatures. In turn, gJJs can be tuned in many
ways, as graphene couples well with a variety of superconductors to form
highly transparent junctions, enabling supercurrents to persist over several
microns [4, 5]. Using different superconductors, junction geometry, and
operation at different carrier densities allows, in principle, for a range of
specific optimizations needed for detecting small heat and optical signals. To
achieve the highest sensitivity, for example, one can choose to operate at the
lowest temperatures and employ superconductors with a small superconducting
gap, similar to the approach that is taken in conventional superconducting
nanowire-based detectors. If a large dynamic range is required, tuning the
critical currents in graphene junctions by controlling carrier density can
provide additional flexibility in design.

Figure 1: Graphene Josephson junction and the characterization of the
resonator circuit. (a), Optical image showing a top-down view of the graphene
flake encapsulated in hexagonal boron nitride (blue-green) contacted by
superconducting electrodes (light blue). The top contact is placed in close
proximity to the ground wires to form the graphene Josephson junction (gJJ).
The bottom contact placed far from the ground electrodes is used to apply
Joule heating. The Inset shows the partial cross-section across the gJJ.
Tuning the global carrier density in the graphene flake is achieved by
applying a DC voltage $V_{BG}$ to a graphite backgate. (b), Simplified
electrical circuit schematic (for full schematic see Appendix B). A
superconducting Niobium Titanium Nitride (NbTiN) resonator is coupled to the
external microwave line via a coupling capacitor and terminated by the gJJ.
The gJJ is electrically modeled as the parallel sum of a dissipationless
branch of inductance $L_{J}=\frac{\Phi_{0}}{2\pi I_{c}}$ and a dissipative
branch of resistance $R_{SG}$. A dedicated heater port allows application of
Joule heat to the graphene flake. (c), $|S_{21}|$ vs. $V_{BG}$ shows the
evolution of the resonance feature. Near the charge neutrality point (CNP;
$V_{CNP}=-0.3\,\rm{V}$), the gJJ maximally loads the resonator and,
consequently, minimizes the resonant frequency. Far from the CNP, the gJJ acts
as a low-impedance termination and maximizes the resonant frequency. On the
hole-side ($V_{BG}<V_{CNP}$), Fabry-Perot type oscillations are visible due to
formation of the regions of different doping in the bulk graphene (hole
doping; p-type) and in vicinity of contacts (electron doping; n-type) [6].

Despite the significant progress in integrating graphene with superconducting
nanoelectronic devices, the present understanding of the thermalization of
electrons and holes in these systems is still incomplete. In most transport
measurements performed to date, thermalization in gJJs is thought to be
primarily driven by the electron-phonon interaction in graphene bulk [7], as
the diffusion of unpaired electrons into the metallic leads is suppressed due
to the superconducting gap. However, in the case where graphene is
encapsulated within boron nitride (hBN), deduced values of electron-phonon
coupling from the experimental thermalization rates [8] are typically orders-
of-magnitude larger than theoretical predictions. Such a discrepancy is not
expected for materials with a simple band structure such as graphene, where
both the electronic and phonon spectrum can be readily calculated. Further,
recent scanning SQUID experiments, which provide spatially resolved thermal
imaging of graphene [9] have revealed that, when electronic transport in
graphene is ballistic, signatures of electron thermalization are present only
near physical edges, local defects, and close to metallic contacts. However,
such signatures of such boundary-mediated thermalization have not so far been
evident in transport measurements. Here we present thermal measurements of a
device architecture in which graphene temperature is measured via changes in
Josephson inductance [10] caused by heating. In contrast to typical critical
current measurements that involve switching between superconducting and
resistive states, this approach allows to continuously monitor thermal
response with high precision that, in principle, depends only on the
measurement integration time. Surprisingly, for both electron and hole doping,
we observe temperature dependence of the thermal conductance consistent with a
resonant electronic scattering mechanism [11, 12] that occurs at the interface
between graphene and superconducting leads.

Figure 1 shows a schematic of the device architecture and basic
characterization measurements. A gJJ is integrated into a graphene flake of
approximate area $\mathrm{A}=25\,\mu\rm{m}^{2}$ (Fig. 1(a)). The gJJ consists
of a central superconducting contact separated from two symmetrically placed
superconducting contacts shorted to the ground plane. Connection is made at
the other end of the flake to a heater port used for thermal characterization
(see Appendix A and B for details of device fabrication and the measurement
architecture). Superconducting aluminum is used for all contacts as it has a
small gap relative to other elemental superconductors; we expect this will
maximize temperature sensitivity in the sub-Kelvin temperature range of our
measurements. To probe the response of the gJJ supercurrent to changes in
electron density and temperature, we couple it to an on-chip resonator [6, 13]
(Fig. 1(b)). Since the gJJ acts as an additional inductive element, it
modifies the resonant frequency, which we monitor through microwave
reflectometery. The parameters characterizing the gJJ, the Josephson
inductance $L_{J}=\frac{\Phi_{0}}{2\pi I_{c}}$ and subgap resistance $R_{SG}$,
depend strongly on electron density (see also Appendix E). Accordingly, the
resonant frequency and spectral width are both highly dependent on the back
gate voltage $V_{BG}$ [6] (Fig. 1(c)). Note that we can resolve the resonance
over a large range of gate voltages; this allows us to study phenomena arising
from electron and hole doping as well as near charge neutrality
($\mathrm{V_{BG}}\approx-0.4\,\rm{V}$). For hole doping
($\mathrm{V_{BG}}<-0.4\,\rm{V}$), Fabry-Perot-type oscillations indicate that
carrier transport is ballistic in our high-quality graphene sample.

Figure 2:  Resonance Fits and Extracted Parameters. (a), Representative
$|S_{21}|$ data and fits for electron and hole doping. Color and labels denote
backgate voltage $V_{BG}$. Arrows show the direction of resonant frequency
shifts as $V_{BG}$ is swept from positive voltage (electron doped) to negative
voltage (hole doped) through charge-neutrality. Green arrow shows the resonant
frequency approaching $500\,\rm{MHz}$ as electron-doping decreases to charge
neutrality. Blue arrow shows that the resonant frequency increasing away from
$500\,\rm{MHz}$ as hole-doping increases. (b), $f_{0}$ as a function of
$V_{BG}$ for representative $T_{mxc}$ = $160$, $210$, $260$, $310$, $360$,
$410$, $460\,\rm{mK}$. Extracted resonant frequency $f_{0}$ as a function of
$V_{BG}$ shows a characteristic lineshape consistent with Fig. 1(c). As
$T_{mxc}$ increases, $f_{0}$ decreases for all backgate voltages. (c), $Q_{i}$
as a function of $V_{BG}$ and $T_{mxc}$. (d, e), RSJ model parameters $I_{c}$
(d), and $R_{SG}$ (e) as a function of $V_{BG}$ and $T_{mxc}$. $I_{c}$ and
$R_{SG}$ are determined using a numerical impedance model of the resonator/gJJ
device with resonance parameters ($f_{0}$, $Q_{i}$) as inputs (see Fig. 7).

In addition to the electrostatic doping, the circuit resonance is also
strongly dependent upon temperature (Fig. 2). When the device temperature
increases, the resonance dip shifts to lower frequencies and broadens,
reflecting increased losses occurring within the junction. Importantly, the
observed shape of the resonance can be fitted using a standard four-parameter
Lorentzian fit function at all accessible carrier densities ($2.2\times
10^{12}\,\mathrm{holes/cm}^{2}<n_{carrier}<5.5\times
10^{11}\,\mathrm{electrons/cm}^{2}$) and temperatures
($\mathrm{160mK}<T_{mxc}<\mathrm{480mK}$) (see also Appendix C). The high
level of agreement between data and the fit (Fig. 2(a)) allows us to relate
the deduced resonance parameters to the physical properties of the junction.
In particular, shifts of resonant frequency $f_{0}$ and the overall resonance
shape, which is set by internal quality factor $Q_{i}$, can be related to
parameters of the resistively-shunted junction (RSJ) junction model [14], the
gJJ critical current $I_{c}$ and sub-gap resistance $R_{SG}$ [6] (see Fig.
1(b) and Appendix E). These quantities determine the small-signal electrical
response of the junction at any temperature and doping level. We note that an
estimate of microwave losses in the junction is not accessible from the
switching current measurements that have typically been employed in gJJ
threshold detection schemes. Fitting the temperature dependence of $I_{c}(T)$
allows estimation of an induced superconducting gap $\Delta\sim 80\,\rm{\mu
eV}$ (see Appendix F). Finally, since we expect the resonator ringdown time
$\tau$ to be the limiting time constant in our device, we estimate from the
fitted resonance parameters that $\tau<150\,\rm{ns}$ for all backgate voltages
(see Fig. 7).

To characterize the thermal properties of the gJJ device, we employ a
measurement configuration in which the gJJ is heated internally by applying a
DC current $I_{heater}$ to the heater port (Fig. 3). The port electrode is
placed sufficiently far from the ground electrodes to preclude supercurrent
flow. This configuration allows us to accurately monitor the input power
delivered to the graphene flake while simultaneously monitoring the resonance
frequency. For different device temperatures and doping, representative
changes in the $S_{21}$ resonance dip are shown in Fig. 3(a-c) and Fig.
3(f-h). By increasing the stage temperature from $170\,\rm{mK}$ to
$400\,\rm{mK}$, we observe a decrease in the resonant frequency of
$27\,\rm{MHz}$ for holes, compared to $6\,\rm{MHz}$ for electrons. This is
consistent with greater inductive loading (lower $I_{c}$) in the hole regime
(see Appendix E). By applying a heater current $I_{heater}$, the internal
flake temperature $T$ is increased above $T_{mxc}$, decreasing the resonant
frequency. Combined with the measurements taken at different temperatures for
calibration (Fig. 3(e,j)) the power vs. temperature characterization and,
consequently, the thermal conductivity $G_{th}$ of the graphene flake can be
determined. We use this approach to investigate thermal properties for both
electron and hole doping regimes.

Figure 3:  Electron-side and Hole-side Heating and Calibration. (a-c),
Electron-side ($V_{BG}=1.1\,\rm{V}$) and (f-h) hole-side
($V_{BG}=-1.8\,\rm{V}$) $|S_{21}(f)|$ vs. $I_{heater}$ for three
representative mixing stage temperatures (a, f) $T_{mxc}=170\,\rm{mK}$ (b, g)
$T_{mxc}=280\,\rm{mK}$ (c, h) $T_{mxc}=400\,\rm{mK}$. Applying a DC heater
current $I_{heater}$ to the designated heater port decreases the resonant
frequency of the device. As expected, the shifts are symmetric with respect to
the polarity of $I_{heater}$. (d), Electron-side and (i) hole-side $I_{c}$ as
a function of $I_{heater}$ for different mixing chamber temperatures. Fitting
(a-c) and (f-h) allows extraction of resonance parameters ($f_{0}$, $Q_{i}$)
and junction parameters ($I_{c}$, $R_{SG}$) (see also Appendix D). The dashed
line at $I_{heater}=0\,\rm{nA}$ corresponds to the data cut plotted in (e) and
(j). (e) Electron-side and (j), hole-side calibration curve, the unheated
$I_{c}$ as a function of $T_{mxc}$. Since $I_{c}$ monotonically decreases with
increasing mixing chamber temperature $T_{mxc}$, there is a one-to-one
correspondence between $I_{c}$ and graphene flake temperature.

The data we have acquired is consistent with a power law $P_{heater}=\Sigma
A(T^{n}-T_{mxc}^{n})$, with electron temperature $T$, stage temperature
$T_{mxc}$, scaling exponent $n$ and the electron-phonon coupling prefactor
$\Sigma A$ (see also Appendix G). We plot $\partial P/\partial
T=G_{th}=n\Sigma AT^{n-1}$ (Fig. 4(c)) which shows that the scaling exponents
for hole and electron doping are consistent with $n=5$. We note that our
fitting procedures produce only comparably small errors for each of the
individual data points and, accordingly, the uncertainty of the extracted
scaling exponent is much less than $1$. This enables us to clearly distinguish
that the exponent obtained here is _not_ consistent with the $n=3$ or $n=4$
scaling predicted for bulk electron-phonon coupling in reduced dimensions [15,
16]. While an $n=5$ scaling exponent is expected for the electron-phonon
coupling of a 3D electron gas [17], these considerations do not apply for our
graphene device in which the electron and phonon density-of-states are 2D.
Also, we note that the mechanism where hot electrons (or holes) diffuse into
the superconducting aluminum leads before thermalization, while in principle
possible, is not consistent with our observations (see Appendix H for more
detailed discussion).

Measurements of hBN-encapsulated graphene performed previously [5, 8] reveal
that $G_{th}$ (scaled by the area) is about three orders-of-magnitude larger
than predictions by simple bulk electron-phonon coupling theory. The magnitude
of $G_{th}\sim 5-300\,\rm{pW/K}$ in our measurements is consistent with these
observations. Due to enhanced mobility, hBN-encapsulated graphene is typically
in the ballistic scattering limit, in which the carrier mean free path
$l_{mfp}$ is limited by the device dimension ($L_{device}\approx 5\,\rm{\mu
m}$ in our sample). This observation has led to the hypothesis that the
enhanced $G_{th}$ may arise from “resonant supercollisions” [11, 12] a
scenario consistent with the spatially resolved measurements [18, 9]. In this
scenario, defects located at edge of the graphene flake locally enhance
electron-phonon interactions and open a thermalization pathway that dominates
over electron-phonon coupling in the bulk. Spatially-resolved scanning SQUID
measurements show an enhancement of surface phonon temperature at graphene
edges and close to metal contacts. Theory formulated to explain these results
[12] suggests that an $n=5$ scaling exponent should hold down to milli-Kelvin
temperatures ($T<T_{BG}$) in the limit of strong scattering ($\delta\sim 1$).
In this context, our high precision measurements provide the first clear
evidence that an $n=5$ scaling exponent signaling that resonance
supercollisions indeed dominate the thermalization in graphene at sub-Kelvin
temperatures. We note that further exploration of the device parameter space
(e.g. sample size, aspect ratio, disorder) maybe needed to disentangle
relations between different microscopic thermalization mechanisms in general.

Figure 4: Power-Temperature Curves. (a) Electron-side flake temperature as a
function of heat power. From the injected DC current $I_{heater}$ and measured
voltage drop $V$ across the heater port, the injected heater power can be
determined $P_{heater}=I_{heater}\times V$. From the $I_{c}$ vs. $I_{heater}$
traces in Fig. 3(d,i) and the $I_{c}$ vs. $T_{mxc}$ calibration in Fig.
3(e,j), flake temperature can be determined as a function of applied
$P_{heater}$. Color corresponds to the mixing chamber stage temperature. (b)
$G_{th}$ vs. $T_{mxc}$. Taking the numerical derivative $\frac{\partial
P}{\partial T}$ of Fig. 4(a) allows the data to be plotted on a single line.
Fit line is to the power law $G_{th}=\partial P/\partial T=n\Sigma AT^{n-1}$
where $n$ is the scaling exponent and $\Sigma A$ (in units of
$\rm{W}/\rm{K^{5}}$) is the multiplicative factor. (c) Electron and hole
$G_{th}$ vs. $T_{mxc}$ (log-log scale). Hole and electron doping both show
power law scaling with an $n=5$ exponent.

We note that $G_{th}$ exhibits a power law consistent with $n=5$ for both
electron and hole doping, indicating that this mechanism remains dominant in
both regimes. Interestingly, the electron- and hole-side prefactors differ by
a factor of approximately two (see Fig. 4(c)). A possible explanation for this
difference arises from the energy distribution of resonant scattering centers
[9]. In this scenario, different scattering centers are activated when the
chemical potential of the flake is shifted by the back-gate. Therefore, it is
possible that the difference in the prefactors can be attributed to different
populations of activated scatterers. Additionally, we note that, in the case
of hole doping, the intrinsic p-n junction formed between the graphene region
close to the Al contacts (which is always intrinsically n-doped) and the
p-doped bulk may also play a role. In this regime, holes from the bulk must
pass across the p-n junction in order to efficiently thermalize via resonant
scattering centers. Since the p-n junction has a finite transmission
probability, it may therefore reduce the overall thermalization rate. We note
that attaining an accurate calculation of the thermalization prefactor from
first principles is difficult due to effects outlined above and further
theoretical work is needed for quantitive comparisons.

In the context of detector technologies, graphene is argued to be a promising
platform for future scalable far-infrared or microwave detector-arrays [8,
19]. Its utility for this purpose is typically evaluated on the basis of
optimization of several key attributes including response time, responsivity,
thermal insulation and multiplexing that, in turn, require simultaneous
optimization of multiple device parameters. The hBN-encapsulated graphene
devices studied here provide large supercurrents and sub-microsecond response
times that allow for continuous monitoring of thermal response, and
integration of the resonator readout that permits straightforward frequency-
division multiplexing of many devices on a single feedline [20, 21]. Moreover,
in our scheme the presence of a separate heater port can be employed for
broad-spectrum energy detection. We note that a thermal insulation of the
architecture employed here can be achieved at the expense of lowering the
mobility in graphene by, for example, placing it directly on the oxide
substrate [19] instead of hBN.

Finally, we briefly compare the inductance readout scheme employed here with
graphene detectors based on junction switching [8, 22] (between the zero and
finite voltage state) as their potential applications may significantly
differ. The latter type of detectors register a “count” when the incident
photon energy is above a given threshold, and therefore forfeit the
possibility of energy spectroscopy provided by the linear, resonantly-coupled
graphene detector architecture pursued in this work. Further, threshold
detectors intrinsically provide slower response, which is limited by the
cooling and resetting of the junction after a photon absorption event. While
this type of detector may be a desirable option in the experiments where
photon energy and arriving time is known or controlled, the inductance readout
detection scheme is more suitable for novel spectroscopy applications of
unknown sources [23], including dark matter detection [24, 25, 26, 27] and
photon and phonon counting [28] where linear response and ability to fully
evaluate detection performance is important (see Appendix I for noise
equivalent power characterization).

Acknowledgments: We acknowledge useful discussions with Sophie Li, Matt
Matheney, Ewa Rej, and Jonas Zmuidzinas. Funding: This work was supported by
NSF through program CAREER DMR-1753306 and Gist-Caltech memorandum of
understanding. S.N.-P. also acknowledges the support of DOE-QIS program (DE-
SC0019166), IQIM (NSF funded physics frontiers center) and the Sloan
foundation. M.L.R acknowledges support from NSF grant NSF-DMR-1806473.

##  APPENDIX A Fabrication

Fabrication of the superconducting resonator and coupling capacitor proceeds
by sputtering a few hundred nanometers of Niobium Titanitum Nitride (NbTiN) on
an undoped silicon wafer with 300 nanometers of thermal oxide. Typical
superconducting transition temperatures are $\sim$14 K. Subsequently, the
resonator and coupling capacitor are patterned by electron beam lithography
followed by an $\,\rm{SF_{6}}$ wet etch and Ar reactive ion etch. The graphene
heterostructure is assembled using standard exfoliation and stamping methods
and dropped on the resonator chip. 1-D edge contacts between the
superconducting metal and graphene heterostructure are patterned by electron
beam lithography followed by an Ar reactive ion etch and an electron beam
evaporation of the titanium adhesion layer and aluminum contacts.

Figure 5: $S_{21}$ Circuit Diagram. The circuit diagram shows both the
$S_{21}$ readout of the resonance feature and the application and readout of
$P_{heater}$. In the $S_{21}$ measurement, a signal is sourced from the
PicoVNA2 vector network analyzer (VNA) and passes through a series of
attenuators down to the resonator/graphene device held at $T_{mxc}$. The
transmitted portion of the signal is amplified by an amplification chain with
a first-stage 4K LNA and read out by the VNA. Application of $P_{heater}$
proceeds by sourcing a current $I$ generated by a voltage sourced by the
Agilent 33221A AWG and dropped over a $1\,\rm{M}\Omega$ ballast resistor.
After passing through a two stages of filters, the sourced current flows
through the normal resistance $R_{heater}$ of the heater port of the graphene
sample and dissipates Joule heat power in the flake. The voltage drop $V$
across graphene heater is amplified by as SR560 preamp and read out by a
32201A digital multimeter (DMM). In this way, the $P_{heater}=I\times V$
delivered to the graphene flake is measured in a 4-wire measurement.

##  APPENDIX B $S_{21}$ Measurement Circuit

A standard $S_{21}$ transmission measurement is performed in which a swept
microwave tone is sent out of Port 1 of a PicoVNA 2 vector network analyzer
(VNA) and down through attenuators and stages of the dilution refrigerator.
The impedance of the resonator/gJJ device loads the line and scatters the
incoming microwave tone. The transmitted portion of the microwave signal is
amplified by a first-stage $T_{n}=4\,\rm{K}$ CIT low noise amplifier, and then
by three room temperature amplifiers, where it is detected by Port 2 of the
VNA.

To improve DC isolation between the device and the VNA, we include inner/outer
DC blocks on the ports of the VNA. To vary readout power incident upon to the
device, we vary the room temperature attenuation between $-50\,\rm{dB}$ and
$-80\,\rm{dB}$. The attenuation at the fridge stages ensures the $300\,\rm{K}$
noise at room temperature is attenuated below the noise floor of the mixing
chamber. In the diagram, the attenuators and amplifiers are positioned
immediately under the fridge stage to which they are thermally anchored.

The heater measurements in Fig. 3 and Fig. 4 of the main text are performed by
applying a DC heater current $I_{heater}$ to the heater port of the graphene
flake and reading out the corresponding voltage drop in a 4-wire measurement.
To source $I_{heater}$, an Agilent 33210A AWG outputs a DC voltage for the DC
heating measurements of the main text and an AC voltage for the noise
equivalent power [29] measurements of section I. Since the ballast resistor
$R_{ballast}=1\,\rm{M\Omega}$ is 3 orders-of-magnitude larger than the heater
port resistance $R_{heater}\approx 1\,\rm{k\Omega}$, the series combination of
the AWG and $R_{ballast}$ can be well-approximated as a current source
$I_{heater}$. The $I_{heater}$ current travels down PhBr twisted-pair lines to
the heater port where it Joule heats the graphene flake. Outside of the
fridge, the shield on the twisted pair lines is held at fridge ground. The
return line of the twisted pair is grounded through a $100\,\Omega$ resistor
to a breakout box (not shown) which is also held at fridge ground. The return
line terminates at the negative terminal of the AWG. We note that the possible
ground loop introduced by the grounding of the twisted pair return line
through the $100\,\Omega$ resistor does not have an appreciable effect on the
measurement.

##  APPENDIX C $S_{21}$ Fitting Procedure

Fitting of the resonance feature follows the procedure in Ref. 30. Background-
subtracted $S_{21}$ transmission data is fit to a four-parameter fitting
function

$S_{21}=1-\frac{Q_{0}/Q_{c}-2iQ_{0}\frac{\delta\omega}{\omega_{0}}}{1+2iQ_{0}\frac{\omega-\omega_{0}}{\omega_{0}}}$

Extracted fit parameters include resonant frequency $\omega_{0}$, internal
quality factor $Q_{i}$, coupling quality factor $Q_{c}$, and asymmetry
parameter $\delta\omega_{0}$. Total quality factor is defined as the parallel
sum of the dissipation channels
$\frac{1}{Q_{0}}=\frac{1}{Q_{i}}+\frac{1}{Q_{c}}$. Error bars in Fig. 2b-d
correspond to the 95% (2$\sigma$) confidence level calculated from the
covariance matrix of the fits. An asymmetry in the resonance circle can cause
the diameter of the resonance circle to occur off of the real axis. Such an
asymmetry may arise from a non-negligible line inductance or mismatched
input/output impedance.

###  1 Resonance Dependence on $V_{BG}$

Figure 1(c) shows how the resonance changes as a function of $V_{BG}$. The
maximal tuning of resonance frequency $f_{0}$ with $V_{BG}$ occurs in the
range $[V_{CNP}$, $V_{CNP}+0.3\,\rm{V}]$, where the $\frac{\partial
f_{0}}{\partial V_{BG}}\approx\frac{670\,\rm{MHz}}{1V}$. Assuming a parallel-
plate capacitance of hBN ($\epsilon_{r}=3$) and a separation $d=30\,\rm{nm}$
between the graphene flake and backgate, $\frac{\partial f_{0}}{\partial
n_{carrier}}\approx\frac{1.21\,\rm{GHz}}{10^{12}/\,\rm{cm}^{2}}$. Since we
estimate the area of our graphene flake to be $A=25\mu m^{2}$, the maximum
sensitivity of our device used as an electrometer is $\frac{\partial
f_{0}}{\partial N_{carrier}}=\frac{4.84\,\rm{kHz}}{1\,\rm{e^{-}}}$.

##  APPENDIX D Fitting Procedure for Extraction of RSJ Parameters

To deduce the physical parameters of the gJJ from the fit parameters of the
$S_{21}$ resonance feature, we employ an electrical impedance model of our
device which takes the inputs ($f_{0}$, $Q_{i}$) and numerically solves for
junction parameters ($I_{c}$, $R_{SG}$).
$Sonnet\textsuperscript{\textregistered}15.53$ is used to estimate the
physical parameters of the NbTiN transmission line resonator [31] (See Table
1). The coupling capacitance $C_{C}$ is estimated by fitting a set of
resonances at $V_{BG}=-1.9\,\rm{V}$, numerically solving for $C_{C}$, and
creating a histogram of extracted $C_{C}$ values with mode
$C_{c}=0.243\,\,\rm{pF}$ and standard deviation of approximately
$\sigma_{C_{c}}=0.02\,\rm{pF}$.

Figure 6: Impedance Model. The electrical impedance model of the resonator-graphene device consists of the graphene Josephson junction in the RSJ model, a NbTiN transmission line resonator characterized by parameters in Table 1, a coupling capacitor $C_{c}$, and 50 $\Omega$ microwave ports. $C_{C}$ | Coupling capacitor | $0.243\,\rm{pF}$
---|---|---
$l$ | TLR length | $4989\,\mu\rm{m}$
$C^{\prime}$ | TLR capacitance per length | $3515\,\rm{pF/m}$
$L^{\prime}$ | TLR inductance per length | $1130\,\rm{nH/m}$
$Z_{0}^{\prime}$ | TLR characteristic impedance | $17.9\,\Omega$
$v_{ph}$ | TLR phase velocity | $1.575\times 10^{7}\,\rm{m/s}$
$Z_{0}$ | Reference characteristic impedance | $50\,\Omega$
$Z_{out}$ | Parallel two-port impedance | $25\,\Omega$
Table 1: Coupling Capacitor, Transmission Line Resonator (TLR), and Microwave
Port Parameters.

##  APPENDIX E Discussion of Extracted Parameters from Resonance Fits and RSJ
Model

As shown in Fig. 7, our fitting and modeling procedure allows several fit and
junction parameters to be plotted as a function of backgate voltage $V_{BG}$
and flake temperature $T_{mxc}$.

Figure 7: Extracted Parameters from Resonance Fits and Impedance Model. (a)
$Q_{i}$ vs. $V_{BG}$. The internal quality factor $Q_{i}$ is extracted from
the $S_{21}$ fit function in C. (b) $Q_{i}/Q_{c}$ vs. $V_{BG}$. Ratio of
internal quality factor $Q_{i}$ and coupling quality factor $Q_{c}$ (also
extracted from the $S_{21}$ fit function) shows that the device is in the
undercoupled limit for all backgate voltages. (c) $\omega L_{J}/R_{SG}$ vs.
$V_{BG}$. Ratio of the inductive branch impedance to resistive branch
impedance in the RSJ model. (d) $\tau=Q_{0}/\omega_{i}$ vs. $V_{BG}$. The
resonator time constant $\tau$ is expected to set the system time constant for
all measured backgate voltages and temperatures. (e) $L_{J}$ vs. $V_{BG}$. The
Josephson inductance $L_{J}=\frac{\Phi_{0}}{2\pi I_{c}}$. (f) $f_{0}$ vs.
$V_{BG}$. The red line corresponds to the projected unloaded
($L_{J}=0\,\rm{nH}$) resonance frequency. $\Delta f$ corresponds to the loaded
($L_{J}\neq 0\,\rm{nH}$) resonance frequency at $T_{mxc}=160\,\rm{mK}$.
$\delta f$ corresponds to further shift in the resonance frequency due to the
increase in flake temperature.

Figure 7(a) shows a dip in $Q_{i}$ at $V_{BG}=-2\rm{V}$, which is propagated
to the other plots Fig. 7(b-d). This dip arises from an asymmetry in the
$S_{21}$ parameter which rotates the resonance circle off the real axis. Such
rotations can arise from line impedance mismatches and parasitic couplings
[30]. Since $R_{SG}$ is determined primarily by $Q_{i}$, $R_{SG}$ is sensitive
to dissipation in the graphene flake as well as the electromagnetic
environment of the flake/resonator assembly. By contrast, $f_{0}$ and $I_{c}$
are largely insensitive to these effects, so our thermometry based upon the
dispersive shifts of the resonance is also largely insensitive to these
effects.

Figure 7(b) shows that our device for all backgate voltages is in the
undercoupled limit ($Q_{i}<Q_{c}$), where dissipation occurs primarily within
device instead of via the coupling to the microwave lines. The variation of
the coupling quality factor $Q_{c}$ is consistent with the circuit model and a
constant coupling capacitor $C_{c}=0.243$ pF.

The dispersive shifts of the resonance can be understood from the impedance
model shown in Fig. 6, which consists of a transmission line resonator
terminated by the junction impedance. This model predicts an unloaded
($L_{J}=0\,\rm{nH}$) resonant frequency of $f_{unload}=774.75\,\rm{MHz}$ as
indicated by the solid red line in Fig. 7(f). When a finite inductance $L_{J}$
loads the transmission line resonator, the resonant frequency decreases. This
occurs because a change in the terminating impedance alters the boundary
condition at the terminating end of the resonator. In the case of the unloaded
resonator, i.e. a $\lambda/4$ resonator, the termination is a short-to-ground,
which fixes the boundary voltage at $V=0$. This enforces the resonance
condition that the length of the resonator equals one quarter of the resonant
wavelength, i.e. $\lambda/4=l$. However, terminating the transmission line
resonator in an inductance alters the boundary condition such that the
boundary voltage amplitude is fixed at some $V=V_{0}>0$. This has the effect
of enforcing the resonance condition that a quarter-wavelength is larger than
the resonator length, i.e. $\lambda/4>l$, or, analogously, that the resonant
frequency is decreased relative to the unloaded case. The larger the
terminating impedance, i.e. the larger $L_{J}$, the lower the resonant
frequency [6, 31].

Due to higher contact transparency, electron doping should exhibit a larger
supercurrent than hole doping. It follows that the electron side should
exhibit a smaller $L_{J}$ than the hole side, and, correspondingly, the
electron side should exhibit a smaller decrease in resonant frequency relative
to $f_{unload}$ than the hole side. This is consistent with Fig. 7(f) for
electron and hole doping, i.e. $\Delta f_{electron}<\Delta f_{hole}$ where
$\Delta f$ is defined as the resonant frequency decrease at
$T_{mxc}=160\,\rm{mK}$.

Increasing the flake temperature further increases $L_{J}$ and decreases the
resonant frequency. A rough estimate of the further decrease of the resonant
frequency $\delta f$ due to increased temperature is as follows:

$\frac{|\delta f|}{|\Delta f|}\approx\frac{|\delta I_{c}|}{|\Delta I_{c}|}$

As discussed in the section F, $I_{c}$ typically decreases by 20-30% as the
flake temperature is increased from $160\,\rm{mK}$ to $400\,\rm{mK}$. From the
main text Fig. 3,

Hole Side:

$\frac{|\delta f_{hole}|}{|\Delta
f_{hole}|}=\frac{26\,\rm{MHz}}{110\,\rm{MHz}}\approx 24\%$

Electron Side:

$\frac{|\delta f_{electron}|}{|\Delta
f_{electron}|}=\frac{5.9\,\rm{MHz}}{18.6\,\rm{MHz}}\approx 32\%$

The change in resonant frequency is therefore consistent with the typical
change in $I_{c}(T)$. We conclude that the greater magnitude of frequency
decrease on the hole side relative to the electron side follows as a
straightforward result of the greater inductive loading of the transmission
line resonator.

As shown in Fig. 7(c), $\frac{\omega_{0}L_{J}}{R_{SG}}$ is a common figure-of-
merit for RF-driven Josephson junctions [32]. It compares the impedance of the
dissipationless supercurrent branch to the dissipative resistive branch. A
smaller value of $\frac{\omega_{0}L_{J}}{R_{SG}}$ denotes a less dissipative
device. At $T_{mxc}=160\,\rm{mK}$, $\frac{\omega_{0}L_{J}}{R_{SG}}\approx
1.5\%$ within a factor of 2. As the temperature rises to
$T_{mxc}=400\,\rm{mK}$, $\frac{\omega_{0}L_{J}}{R_{SG}}$ increases to 3%. This
is consistent with decreases in $I_{c}$ raising the impedance of the
dissipationless branch and driving more current through the dissipative
branch, as indicated by the degrading quality factor with increasing flake
temperature (see Fig. 2(c)).

##  APPENDIX F $I_{c}$ vs. $T_{mxc}$ Fits and Extraction of Induced
Superconducting Gap

Figure 8: (a) $I_{c}$ vs. $T$. An example fit of $I_{c}$ vs. $T$ for
$V_{BG}=-2.01\,\rm{V}$ with extracted fit parameters $I_{c}(0)$ and $\Delta$.
(b) Fit parameter $I_{c}(0)$ vs. $V_{BG}$. $I_{c}(0)$ fit parameter is shown
for both electron and hole doping. (c) Fit parameter $\Delta$ vs. $V_{BG}$. A
coarse estimate of induced gap $\Delta\approx 80\,\rm{\mu V}$. Fine features
are discussed in the text. (d) Hole side $I_{c}$ vs. $V_{BG}$. Blue trace is
hole side $I_{c}$ data for $T_{mxc}=160\,\rm{mK}$. Red trace is the slowly-
varying background as fit to a $7^{\rm{th}}$-order polynomial. (e) Background-
subtracted $\Delta I_{c}$ vs. $k_{F}$. $\Delta I_{c}$ is obtained by
subtracting the two traces in Fig. 8(d). (f) Power spectral density of $\Delta
I_{c}$. The large peak is consistent with an effective Fabry-Perot cavity
length of $L_{cav}=361.51\,\rm{nm}$.

Due to the measurement architecture employed here, we cannot perform 4-wire
measurements directly on the gJJ to estimate the induced superconducting gap
$\Delta_{0}$. Instead, we perform a fitting procedure based upon the
temperature dependence of the critical current $I_{c}(T)$.

The $I_{c}(T)$ vs. $V_{BG}$ data in Fig. 2(d) is fit to extract physical
parameters. The fit function we employ describes the supercurrent that arises
from thermally populating the Andreev bound states (ABS) in a ballistic
junction [33].

$I_{c}(T)=I_{c}(0)\tanh\Big{(}\frac{\Delta}{2k_{B}T}\Big{)}$

The two fit parameters correspond to the physical parameters $I_{c}(0)$, the
zero-temperature critical current, and $\Delta$, the induced superconducting
gap. An example fit is shown in Fig. 8(a).

In Fig. 8(b), modulation of the fit parameter $I_{c}(0)$ with $V_{BG}$ on the
hole side is consistent with $pnp$-type Fabry-Perot interference as discussed
in the main text and Fig. 1(c). Following the standard method for determining
Fabry-Perot cavity length in ballistic graphene, we subtract the slowly
varying background with a fit to a $7^{\rm{th}}$-order polynomial (see Fig.
8(d-e)) and take the power spectral density (see Fig. 8(f)). The large peak in
the power spectral density is consistent with a Fabry-Perot cavity length of
$L_{cav}=361.51\,\rm{nm}$. Structure on the electron side could be caused by
an $nn^{\prime}n$-type Fabry-Perot cavity [34].

From Fig. 8(c), we can make a coarse estimate of the induced superconducting
gap $\Delta\approx 80\,\rm{\mu V}$. However, further measurements are needed
to determine whether the finer structure of Fig. 8(c) is due to the physics of
the S-G-S junction or an artifact of the fitting procedure. Toward this end,
it would be useful to perform simultaneous RF characterization and DC
multiple-Andreev reflection measurements on a gJJ sample [6].

##  APPENDIX G Power vs. Temperature Fitting Procedure

To obtain the $G_{th}=\frac{\partial P}{\partial T}$ vs. $T$ in Fig. 4(c), we
first perform piece-wise linear fits of the $P-T$ curves of Fig. 4(a).
Subsequently, we perform a nonlinear least squares fit of the $G_{th}$ vs. $T$
to the fitting function

$G_{th}=n\Sigma AT^{n-1}$ (1)

with free fit parameters $n$ the scaling exponent and $\Sigma A$ the electron-
phonon coupling. The errors in the free fit parameters correspond to the
$2\sigma$ (95%) errors obtained from the nonlinear least squares fit. On the
electron side, we include an exclusion criteria at the limit of the
temperature resolution of our device. This criteria does not appreciably
change the extracted $n$ or $\Sigma A$. Without the exclusion criteria the
extracted fit parameters are $n=5.04\pm 0.2$ and $\Sigma A=(25.25\pm
6.89)\times 10^{-10}\,\rm{W/K}^{5}$. With the exclusion criteria, the
extracted fit parameters are $n=4.92\pm 0.14$ and $\Sigma A=(20.73\pm
3.90)\times 10^{-10}\,\rm{W/K}^{5}$.

Figure 9: Noise Equivalent Power (NEP) characterization. (a) Schematic of
Measurement Chain. First panel shows a pure carrier tone sent down the
microwave line. Second panel shows that an applied $\omega$ heater current and
subsequent $2\omega$ modulation of the heat power (and temperature of the
graphene flake) yields a $2\omega$ modulation of the transmission function
($S_{21}$ parameter) between unheated (blue) and heated (red) states. The pure
tone (dashed line) is placed within the bandwidth of the transmission function
and amplitude modulated at $2\omega$ with a modulation index that depends on
the magnitude of the $S_{21}$ dip. Third panel shows the amplitude-modulated
signal with sidebands at $2\omega$ as it appears on the spectrum analyzer. The
measured signal-to-noise ratio of the sideband is used to determine the $NEP$.
(b) Circuit Diagram. A continuous-wave carrier tone at $\omega_{c}$ is sent
down a microwave line to the graphene device, amplified, and read out by a
spectrum analyzer. An AC heater current at frequency $\omega=2\pi\times
337\,\rm{Hz}$ injects a $2\omega$ heat power $P_{heater}$ in the graphene
flake and produces $2\omega$ amplitude modulation of the carrier tone, as
discussed in (a). (c) Representative spectrum at output of measurement chain.
Spectrum as read out by spectrum analyzer ($\rm{RBW}=1\,\rm{Hz}$) for applied
heat power off (blue) and on (red). The primary effect of the applied heat is
to produce sidebands spaced at $2\omega$ from the the carrier tone. Other
peaks in the spectrum exist at multiples of the line frequency. A peak at
$\omega$ is consistent with a DC offset in the applied heat power. Inset shows
the $2\omega$ sideband. (d) Sideband Power vs. $P_{heater}$. In the
low-$P_{heater}$ linear-response regime, the sideband voltage $V_{sb}\propto
P_{heater}$. Since the spectrum analyzer reads out the sideband power,
$P_{sb}\propto P_{heater}^{2}$, which is consistent with the slope at low
$P_{heater}$. (e) $NEP$ vs. $P_{heater}$. The linear-response regime is
characterized by a regime of constant $NEP$, before rising as the amplitude
modulation saturates to its maximal value. The $NEP$ plotted in (g,f)
corresponds to the linear response regime (green dashed line). (f, g) $NEP$
vs. carrier power $P_{c}$ and carrier frequency $f_{c}$ for (f) electron-side
($V_{BG}=1.0\,\rm{V}$) and (g) hole-side ($V_{BG}=-2.75\,\rm{V}$). Minimal
$NEP$ occurs near the resonance dip minimum where amplitude modulation is
largest. As carrier power $P_{c}$ is increased, the resonance dip downshifts
to lower frequencies and is driven into nonlinearity, as characterized by an
asymmetric resonance lineshape with steep falling edge and shallow rising
edge. The minimum $NEP$ tracks the steep falling edge where amplitude
modulation is greatest.

##  APPENDIX H Other thermalization pathways

In this section, we briefly discuss alternative thermalization pathways that
can occur in our experimental geometry. While they indeed occur, we note that
the thermal conductance corresponding to these alternative pathways are all
too small to explain our measurements.

Thermalization via bulk phonons

The bulk phonons are often invoked as the main source of electron
thermalization in graphene. However, besides having a different exponent
($n=3$ or $n=4$ not agreeing with our data, see Fig. 10), the cooling rate via
bulk graphene phonons is too small to explain experimental findings. As
discussed in previous literature, the typical thermal conductance expected
from bulk phonons is two orders of magnitude smaller than the measured data.
We note that, in this context of the overall cooling rate, our measurements
are roughly in line with previous graphene-hBN experiments.

Figure 10: The fits for the electron and hole doping in different temperature
ranges. Electron doping: Blue dashed line is the original full fit from 180 mK
to 400 mK. Red is 300 mK to 400mK. Green is 330 mK to 400 mK. Hole doping: Red
dashed line is the full data fit. Cyan is 300 mK-390 mK. Magenta is 330 mK-390
mK. For both electron and hole doping the n decreases (but stays well above
n¿4) when part of the data is used for fitting. This indicates that $n=4$or
$n=3$ power exponents are inadequate to describe our data.  Figure 11: The
optical image showing the area covered in Aluminum and NbTiN. Considering that
Al thickness is 100nm the total volume of Aluminum part is $V=1.5\times
10^{-16}$ m3 approximately. This small Al volume limits the amount of
thermalization that can be achieved through contacts.

Thermalization in Aluminum leads: Another possibility is that the hot
electrons enter Al leads. While the tunneling of electrons (or holes) in a
superconductor is expected to be suppressed due to the existence of a finite
single-particle gap, previous work found that this process can still be
sizable when using Aluminum electrodes. The critical distinction between our
study and previous work using Aluminum contacts (see, for example, Ref. [35]),
is that we used Al only as immediate contact to graphene (Fig. 11), and beyond
that, the electrodes used in our experiment are made from Niobium Titanium
Nitride (NbTiN) which has a much larger superconducting gap ($\Delta\approx 1$
meV). In this context, the amount of heat that could “leak” into Aluminum
contacts is much lower compared to measured values of thermal conductance. A
total volume of Aluminum in our device is only approximately $V=150\times
10^{-18}$ m3. By considering the established value for electron-phonon
coupling in Aluminum ($\Sigma=0.3\times 10^{9}$ W/K5m3, see Ref. [36]) the
corresponding P. vs T dependence is expected to follow $P\approx
0.98\times\Sigma\times V\times T^{5}\times exp(-\Delta/(kT))$. Here
$\Delta\approx 170-200$ $\micro$eV is Aluminum superconducting gap measured in
our experiments and a numerical pre-factor of 0.98 is estimated in Ref. [37].
At $T=200$ mK this rate corresponds to $G=dP/dT=0.25\times 10^{-12}$ W,
approximately two orders of magnitude smaller than the observed thermalization
rates at our lowest temperatures.

##  APPENDIX I Noise Equivalent Power

###  1 Theory

A key figure-of-merit for linear power detectors is noise-equivalent power
($NEP$). A power-to-voltage detector has a responsivity $\mathcal{R}$, such
that

$V_{out}=\mathcal{R}(P_{in})$

In the linear-response regime, i.e. for small applied power, this expression
simplifies to

$\delta V_{out}\approx\bigg{(}\frac{\partial V_{out}}{\partial
P_{in}}\bigg{|}_{\delta P_{in}=0}\bigg{)}*\delta P_{in}$

In this regime, the $NEP$ of a power-to-voltage detector (in units of
$\frac{W}{\sqrt{Hz}}$) can be defined as that power spectral density at the
device input which produces the measured voltage spectral density
$\sqrt{S_{V}}$ at the output:

$NEP\equiv\frac{\sqrt{S_{V}}}{\partial V_{out}/\partial
P_{heater}\rvert_{\delta P_{heater}=0}}$

The above expression suggests two immediate ways to measure the $NEP$. One is
to measure the voltage spectral density $\sqrt{S_{V}}$ at the output and the
device responsivity $\partial V_{out}/\partial P_{heater}$. Another is to
measure the applied power at the input $\delta P_{in}$, and the $SNR$ at the
output as suggested by rearranging the above expression

$NEP\equiv\frac{\sqrt{S_{V}}}{\delta V_{out}}*\delta P_{in}=\frac{\delta
P_{in}}{SNR}=\frac{\delta P_{heater}}{4\times SNR}$

In the above equation, the $SNR$ is in units of $\frac{V}{V/\sqrt{Hz}}$ and
$P_{in}=P_{heater}/4$. The latter expression is true since we have implicitly
assumed $P_{in}$ is that input power that produces the measurable $V_{out}$
signal. In our case, only one quarter of the heat power $P_{heater}$ injected
at the heater port produces the measured sideband signal.

###  2 Experimental Design

To measure the $NEP$, we use the measurement setup in Fig. 9(a,b) and perform
the following procedure:

  * •

We apply a carrier tone on the microwave line (Fig. 9(a), first panel). The
$S_{21}$ parameter is the transfer function which determines the magnitude and
phase of the signal at the output. Thus, a carrier tone at the resonant
frequency, i.e., at the maximal dip of the $S_{21}$ parameter will have a
smaller transmitted magnitude than a carrier tone placed off-resonance.

  * •

Measurement of $P_{heater}(\omega)=I_{heater}(\omega)\times
V_{heater}(\omega)$ is achieved by sourcing a current $I_{heater}(\omega)$ to
the heater port and measuring the voltage drop
$V_{heater}(\omega)=I_{heater}(\omega)R_{heater}$ over the heater port in a
4-wire lock-in measurement. Since we apply an AC heater current
$I_{heater}(\omega)\propto\cos{\omega t}$, it follows that
$P_{heater}(\omega)\propto\cos^{2}{\omega t}=\frac{1}{2}(1+\cos(2\omega t))$.
Only the $2\omega$ term in the final expression contributes to the $V_{out}$
sideband signal.

  * •

Applying an AC heat power $P_{heater}$ to the heater port modulates the
$S_{21}$ parameter between unheated and heated states (Fig. 9(a), second
panel). Consistent with the heating measurements performed in the main text,
the heated state has a lower resonant frequency and lower quality factor than
the unheated state. The $2\omega$ component of the input power $P_{heater}$
modulates the flake temperature at $2\omega$. Thus, modulation of the $S_{21}$
resonance feature will occur at $2\omega$.

  * •

Placing the frequency of the carrier tone within the bandwidth of the
modulated $S_{21}$ resonance feature will amplitude modulate the carrier,
producing sidebands spaced at $2\omega$ from the carrier (Fig. 9(a), second
and third panel). Provided that the device is in the linear-response regime,
the voltage of the sidebands will increase in proportion to applied heat
power, i.e. $V_{sb}\propto P_{heater}$. It follows that the power of the
sidebands will increase as $P_{sb}\propto P_{heater}^{2}$.

  * •

The amplitude-modulated carrier is read out by a spectrum analyzer (Fig. 9(a),
third panel). The signal-to-noise ratio of the sideband is used to calculate
the $NEP$. We note that only one sideband is used in the $NEP$ measurement.

###  3 Sideband Spectrum

In Fig. 9(c), we see that application of an AC heater current of magnitude
$I_{heater}=20$ nA results in sidebands at 2$\omega$ offset from the carrier,
where $\omega=2\pi\times 337\,\rm{Hz}$. In addition to the 2$\omega$
sidebands, sidebands at multiples of the $60\,\rm{Hz}$ line frequency
frequency are present. Additionally, there are sidebands at $\omega$
approximately $10\,\rm{dB}$ down from the 2$\omega$ sidebands. This can be
explained by a small DC offset in the heater current.

With increasing heater power, the magnitude of the sidebands saturates at a
value consistent with expectations. It is straightforward to show that a
resonance dip of $3\,\rm{dB}$ generates a maximum amplitude modulation index
$m=17\%$, which should produce sidebands $21\,\rm{dB}$ lower than the carrier.
This is in agreement with the measured sideband magnitude that is
$23\,\rm{dB}$ lower than the carrier.

###  4 Sideband Scaling

In the linear response regime, $V_{sb}\propto\delta P_{heater}$. Therefore,
the sideband signal as measured on the spectrum analyzer (in power units)
should scale as $P_{sb}\propto P^{2}_{heater}$, or by $20\,\rm{dB/decade}$.
This is seen in Fig. 9(d) for applied heat $P_{heater}$ in the range
$-120\,\rm{dBm}$ to $-105\,\rm{dBm}$, where the slope of fit at
low-$P_{heater}$ is consistent with a scaling exponent $n=2$. This confirms
that our measurement is in linear-response regime at low $P_{heater}$. For
greater applied $P_{heater}$, the sideband power saturates as the amplitude
modulation reaches the full maximum of the resonance dip.

###  5 $\mathbf{NEP}$ vs. $\mathbf{P_{heater}}$

In the linear response regime, the $NEP$ is constant with respect to
$P_{heater}$ since $V_{sb}\propto P_{heater}$. This is shown Fig. 9e for
$P_{heater}<-105\,\rm{dBm}$. As stated above, the $NEP$ rises for
$P_{heater}>-105\,\rm{dBm}$ as the $SNR$ saturates while $P_{heater}$
continues to increase.

###  6 $\mathbf{NEP}$ vs. Carrier Frequency and Carrier Power

To explore the $NEP$ as a function of the carrier tone, we generate a heat map
with swept carrier frequency $f_{c}$ and carrier power $P_{c}$ (Fig. 9(f,g)).
For the lowest carrier powers, the $NEP$ is minimized for carrier frequencies
close to the resonance minimum, where the responsivity of the resonance to
applied heater power is greatest and therefore the amplitude modulation of the
carrier is greatest. As the carrier power $P_{c}$ is increased, the junction
is driven to nonlinearity, resulting in a resonance dip with a steep falling
edge and a shallow rising edge. This has the effect of enhancing the $NEP$ on
the falling edge and reducing it on the rising edge. For carrier powers
$P_{c}>-98\,\rm{dBm}$, the quality factor of the resonance feature is degraded
to such an extent that the amplitude modulation of sideband is reduced and the
$NEP$ increases. The $NEP$ reaches a minimum value of $7\times
10^{-17}\,\rm{W/}\sqrt{\,\rm{Hz}}$ for a carrier power $P_{carrier}=-102$ dBm
and carrier frequency $f_{carrier}=753.5$ MHz.

###  7 Detection Limits

The measured minimum noise-equivalent power $NEP_{min}\approx 7\times
10^{-17}\,\rm{W/}\sqrt{\rm{Hz}}$. It is limited by the noise of the 4K cryoamp
and is $\sim 20\times$ larger than the thermal fluctuation-limited
$NEP=\sqrt{4k_{B}T^{2}G_{th}}$ at $T_{mxc}=200\,\rm{mK}$. At
$T_{mxc}=58\,\rm{mK}$ , the projected thermal fluctuation-limited
$NEP_{proj}\approx 1\times 10^{-19}\,\rm{W/\sqrt{Hz}}$, assuming that the
$T^{4}$ dependence of $G_{th}$ holds down to these temperatures [29, 38]. The
corresponding thermal fluctuation-limited energy resolution $\delta
E=NEP_{proj}\sqrt{\tau_{th}}\approx h\times 65\,\rm{GHz}$, assuming the
projected thermal time constant $\tau_{th}=\frac{C_{th}}{G_{th}}\approx
170\,\rm{ns}$, $n_{carrier}=\frac{10^{12}}{cm^{2}}$, $A=25\,\mu m^{2}$.

## References

  * [1] Fong, K. C. & Schwab, K. C.  Ultrasensitive and Wide-Bandwidth Thermal Measurements of Graphene at Low Temperatures.  _Physical Review X_ 2, 031006 (2012).
  * [2] Fong, K. C. _et al._ Measurement of the Electronic Thermal Conductance Channels and Heat Capacity of Graphene at Low Temperature.  _Physical Review X_ 3, 041008 (2013).
  * [3] Borzenets, I. V. _et al._ Ballistic Graphene Josephson Junctions from the Short to the Long Junction Regimes.  _Physical Review Letters_ 117, 237002 (2016).
  * [4] Calado, V. E. _et al._ Ballistic Josephson junctions in edge-contacted graphene.  _Nature Nanotechnology_ 10, 761–764 (2015).
  * [5] Draelos, A. W. _et al._ Supercurrent Flow in Multiterminal Graphene Josephson Junctions.  _Nano Letters_ 19, 1039–1043 (2019).
  * [6] Schmidt, F. E., Jenkins, M. D., Watanabe, K., Taniguchi, T. & Steele, G. A.  A ballistic graphene superconducting microwave circuit.  _Nature Communications_ 9, 4069 (2018).
  * [7] Borzenets, I. V. _et al._ Phonon Bottleneck in Graphene-Based Josephson Junctions at Millikelvin Temperatures.  _Physical Review Letters_ 111, 027001 (2013).
  * [8] Lee, G.-H. _et al._ Graphene-based Josephson junction microwave bolometer.  _Nature_ 586, 42–46 (2020).
  * [9] Halbertal, D. _et al._ Imaging resonant dissipation from individual atomic defects in graphene.  _Science_ 358, 1303–1306 (2017).
  * [10] Saira, O.-P., Zgirski, M., Viisanen, K. L., Golubev, D. S. & Pekola, J. P.  Dispersive Thermometry with a Josephson Junction Coupled to a Resonator.  _Physical Review Applied_ 6, 024005 (2016).
  * [11] Kong, J. F., Levitov, L., Halbertal, D. & Zeldov, E.  Resonant electron-lattice cooling in graphene.  _Physical Review B_ 97, 245416 (2018).
  * [12] Tikhonov, K. S., Gornyi, I. V., Kachorovskii, V. Y. & Mirlin, A. D.  Resonant supercollisions and electron-phonon heat transfer in graphene.  _Physical Review B_ 97, 085415 (2018).
  * [13] Wang, J. I.-J. _et al._ Coherent control of a hybrid superconducting circuit made with graphene-based van der Waals heterostructures.  _Nature Nanotechnology_ 14, 120–125 (2019).
  * [14] Tinkham, M.  _Introduction to Superconductivity_ (Publisher: Dover Publications, 2004).
  * [15] Viljas, J. K. & Heikkilä, T. T.  Electron-phonon heat transfer in monolayer and bilayer graphene.  _Physical Review B_ 81, 245404 (2010).
  * [16] Chen, W. & Clerk, A. A.  Electron-phonon mediated heat flow in disordered graphene.  _Physical Review B_ 86, 125443 (2012).
  * [17] Roukes, M. L., Freeman, M. R., Germain, R. S., Richardson, R. C. & Ketchen, M. B.  Hot electrons and energy transport in metals at millikelvin temperatures.  _Physical Review Letters_ 55, 422–425 (1985).
  * [18] Halbertal, D. _et al._ Nanoscale thermal imaging of dissipation in quantum systems.  _Nature_ 539, 407–410 (2016).
  * [19] Kokkoniemi, R. _et al._ Bolometer operating at the threshold for circuit quantum electrodynamics.  _Nature_ 586, 47–51 (2020).
  * [20] Day, P. K., LeDuc, H. G., Mazin, B. A., Vayonakis, A. & Zmuidzinas, J.  A broadband superconducting detector suitable for use in large arrays.  _Nature_ 425, 817–821 (2003).
  * [21] Wandui, A. _et al._ Thermal kinetic inductance detectors for millimeter-wave detection.  _Journal of Applied Physics_ 128, 044508 (2020).
  * [22] Walsh, E. D. _et al._ Graphene-Based Josephson-Junction Single-Photon Detector.  _Physical Review Applied_ 8, 024022 (2017).
  * [23] Lara-Avila, S. _et al._ Towards quantum-limited coherent detection of terahertz waves in charge-neutral graphene.  _Nature Astronomy_ 3, 983–988 (2019).
  * [24] Hochberg, Y., Kahn, Y., Lisanti, M., Tully, C. G. & Zurek, K. M.  Directional detection of dark matter with two-dimensional targets.  _Physics Letters B_ 772, 239–246 (2017).
  * [25] Kim, D., Park, J.-C., Fong, K. C. & Lee, G.-H.  Detecting keV-Range Super-Light Dark Matter Using Graphene Josephson Junction.  _arXiv:2002.07821 [cond-mat, physics:hep-ex, physics:hep-ph]_ (2020).  eprint 2002.07821.
  * [26] McAllister, B. T. _et al._ The ORGAN experiment: An axion haloscope above 15 GHz.  _Physics of the Dark Universe_ 18, 67–72 (2017).
  * [27] Baracchini, E. _et al._ PTOLEMY: A Proposal for Thermal Relic Detection of Massive Neutrinos and Directional Detection of MeV Dark Matter.  _arXiv:1808.01892 [astro-ph, physics:hep-ex, physics:physics]_ (2018).  eprint 1808.01892.
  * [28] Roukes, M. L.  Yoctocalorimetry: Phonon counting in nanostructures.  _Physica B: Condensed Matter_ 263–264, 1–15 (1999).
  * [29] Mather, J. C.  Bolometer noise: Nonequilibrium theory.  _Applied Optics_ 21, 1125–1129 (1982).
  * [30] Geerlings, K. _et al._ Improving the quality factor of microwave compact resonators by optimizing their geometrical parameters.  _Applied Physics Letters_ 100, 192601 (2012).
  * [31] M. Pozar, D.  _Microwave Engineering, 4th Edition_ (Wiley, 2011).
  * [32] Van Duzer, T. & W Turner, C.  _Principles of Superconductive Devices and Circuits_ (Pearson, 1998).
  * [33] Lee, G.-H., Kim, S., Jhi, S.-H. & Lee, H.-J.  Ultimately short ballistic vertical graphene Josephson junctions.  _Nature Communications_ 6, 6181 (2015).
  * [34] Nanda, G. _et al._ Current-Phase Relation of Ballistic Graphene Josephson Junctions.  _Nano Letters_ 17, 3396–3401 (2017).
  * [35] Voutilainen, J. _et al._ Energy relaxation in graphene and its measurement with supercurrent.  _Physical Review B_ 84, 045419 (2011).
  * [36] Giazotto, F., Heikkilä, T. T., Luukanen, A., Savin, A. M. & Pekola, J. P.  Opportunities for mesoscopics in thermometry and refrigeration: Physics and applications.  _Reviews of Modern Physics_ 78, 217–274 (2006).
  * [37] Timofeev, A. V. _et al._ Recombination-Limited Energy Relaxation in a Bardeen-Cooper-Schrieffer Superconductor.  _Physical Review Letters_ 102, 017003 (2009).
  * [38] Moseley, S. H., Mather, J. C. & McCammon, D.  Thermal detectors as x-ray spectrometers.  _Journal of Applied Physics_ 56, 1257–1262 (1984).