# Low-noise cryogenic microwave amplifier characterization with a calibrated
noise source

M. Malnou<EMAIL_ADDRESS>National Institute of Standards and
Technology, Boulder, Colorado 80305, USA  Department of Physics, University of
Colorado, Boulder, Colorado 80309, USA  T. F. Q. Larson  National Institute of
Standards and Technology, Boulder, Colorado 80305, USA  Department of Physics,
University of Colorado, Boulder, Colorado 80309, USA  J. D. Teufel  National
Institute of Standards and Technology, Boulder, Colorado 80305, USA  F. Lecocq
National Institute of Standards and Technology, Boulder, Colorado 80305, USA
J. Aumentado  National Institute of Standards and Technology, Boulder,
Colorado 80305, USA

###### Abstract

Parametric amplifiers have become a workhorse in superconducting quantum
computing, however research and development of these devices has been hampered
by inconsistent, and sometimes misleading noise performance characterization
methodologies. The concepts behind noise characterization are deceptively
simple, and there are many places where one can make mistakes, either in
measurement or interpretation and analysis. In this article we cover the
basics of noise performance characterization, and the special problems it
presents in parametric amplifiers with limited power handling capability. We
illustrate the issues with three specific examples: a high-electron mobility
transistor amplifier, a Josephson traveling-wave parametric amplifier, and a
Josephson parametric amplifier. We emphasize the use of a 50-$\Omega$ shot
noise tunnel junction (SNTJ) as a broadband noise source, demonstrating its
utility for cryogenic amplifier amplifications. These practical examples
highlight the role of loss as well as the additional parametric amplifier
‘idler’ input mode.

##  I Introduction

Ultra low noise microwave amplifiers have played a critical, often
transformative, role in several areas of scientific study. For example, they
have allowed the widespread dissemination of superconducting quantum sensors
for astronomy and astrophysics observations [1, 2, 3, 4, 5, 6, 7]. They have
also allowed the development of circuit-based superconducting quantum
information and computing research [8, 9, 10]. More recently they have boosted
interest in physics beyond the standard model, via their use in axion dark
matter searches [11, 12, 13].

The majority of these ultra low noise amplifiers consist of superconducting
Josephson junction-based circuits, operating as parametric amplifiers [14, 15,
16, 17], that can be mounted on the mixing chamber stage of a dilution
refrigerator close to the devices-under-test. These are preamplifiers, the
first stage of signal amplification, and have performance that can approach
the so-called quantum limit for added noise, with sufficient gain to overwhelm
the remaining post-amplification chain, usually a cryogenic high electron
mobility transistor (HEMT) amplifier placed at the 3–4 K stage followed by
additional room temperature amplifiers to raise the signal power to suitable
levels for demodulation and processing.

The continued development of these amplifiers requires a suite of
characterization tools that are capable of operating at these low temperatures
with methodology that is not too cumbersome for laboratories that do not
specialize in metrology. There are already methods outlined in the literature
discussing how one can approach this problem for scattering parameter
calibration and characterization [18, 19, 20], yet the problem of
characterizing very low amplifier added noise is comparatively difficult. Much
of the problem originates in the fact that added noise characterization is an
absolute measurement, requiring a known noise power level at the input of the
amplifier, while scattering parameter measurements are relative, _i.e._ , the
ratio of outgoing to incident voltage wave amplitudes.

In this article, we outline the basic definitions and methodology used when
characterizing the noise of nearly-quantum-limited amplifiers. We also shed
light on some common pitfalls encountered in the interpretation of
measurements when using a broadband calibrated noise source. We discuss
critical considerations in designing an amplification chain as well as how to
use a shot noise tunnel junction (SNTJ), a self-calibrating noise source, to
characterize amplifier noise performance. With the SNTJ we measure the noise
of commonly-used amplification chains where the first amplifier is either a
HEMT, a Josephson traveling-wave parametric amplifier (JTWPA), or a Josephson
parametric amplifier (JPA). Although the SNTJ is a relatively recent addition
to the toolbox of noise characterization tools [21, 22] and not as well-known
as other methods, it is a versatile noise source that is well-suited to
cryogenic microwave characterization. We highlight this by comparing its
operation to that of a variable temperature stage (VTS), on which the SNTJ is
mounted.

##  II Theory of noise measurement

###  II.1 Elements in an amplification chain

##### Units and definitions

We focus on linear amplification of a white noise, which we consider as our
“signal” [23]. Randomly fluctuating, it follows a Gaussian statistics with
zero mean. The noise is therefore fully characterized by its associated
standard deviation, whose power spectral density $P$ (in units of W/Hz) can be
measured, in practice, with a spectrum analyzer. Throughout the article we
express the noise in photon-normalized units, $N=P/(h\nu)$, with $h$ the
Planck constant and $\nu$ the photon frequency. Note that $N$ is not a photon-
number operator, in particular $N$ already includes vacuum fluctuations (see
appendix B.1). In the literature the noise is also sometimes normalized to a
temperature $T_{N}=P/k_{B}$ (in K), with $k_{B}$ the Boltzmann constant. For
energies much larger than that of vacuum, $T_{N}$ represents the equivalent
temperature of a resistor that would dissipate the same amount of noise into a
matched circuit (see Eq. 9).

##### Amplification

In the intuitive picture of linear amplification (corresponding to the so-
called phase preserving - or phase-insensitive - case), all the components of
a signal get identically amplified while traveling along an amplification
chain, consisting of a succession of attenuation and amplification stages, see
Fig. 1. Then, the $k$-th amplifier of the chain not only amplifies an input
signal by some gain $G_{k}$, but also adds some unavoidable noise $N_{k}$ on
top of the signal. The input/output relation of that amplifier thus reads

$N_{\mathrm{out},k}=G_{k}(N_{\mathrm{in},k}+N_{k}).$ (1)

Here, $N_{\mathrm{in},k}$, $N_{\mathrm{out},k}$ and $N_{k}$ are all Gaussian
noises, and in fact they all are in photon-normalized units. But
$N_{\mathrm{in},k}$ is (or contains) a known and controlled quantity, while
$N_{k}$ is not. That is why we will we call $N_{\mathrm{in},k}$
($N_{\mathrm{out},k}$) the input (output) signal of the $k$-th amplifier, and
we will call $N_{k}$ the added noise of that amplifier.

##### Attenuation

A loss stage attenuates the signal, and replaces a portion of that signal by
some Johnson noise $N_{T_{k}}$, whose value depends on the temperature $T_{k}$
of the stage (see Eq. 9). More precisely, the $k$-th loss stage acts as a
beamsplitter and transforms an input signal $\tilde{N}_{\mathrm{in},k}$ (equal
to the output of the preceding amplifier, $N_{\mathrm{out},k-1}$) to an output
signal $N_{\mathrm{in},k}$ (which corresponds to the $k$-th amplifier’s input)
following:

$N_{\mathrm{in},k}=\eta_{k}\tilde{N}_{\mathrm{in},k}+(1-\eta_{k})N_{T_{k}},$
(2)

where $0<\eta_{k}<1$ is called the transmission efficiency [24].

##### Friis formula

Combining the $k$-th amplification stage (Eq. 1) with the preceding $k$-th
loss stage (Eq. 2), we can define an effective amplifier, with gain
$\tilde{G}_{k}$ and added noise $\tilde{N}_{k}$, that transforms the input
signal $\tilde{N}_{\mathrm{in},k}$ to an output signal $N_{\mathrm{out},k}$
such that

$\displaystyle N_{\mathrm{out},k}$
$\displaystyle=\tilde{G}_{k}(\tilde{N}_{\mathrm{in},k}+\tilde{N}_{k})$ (3)
$\displaystyle=\tilde{G}_{k}(N_{\mathrm{out},k-1}+\tilde{N}_{k}),$

where

$\displaystyle\tilde{G}_{k}$ $\displaystyle=\eta_{k}G_{k},$ (4)
$\displaystyle\tilde{N}_{k}$
$\displaystyle=\frac{(1-\eta_{k})N_{T_{k}}+N_{k}}{\eta_{k}}.$ (5)

Critically, it becomes apparent that the preceding loss stage degrades the
performance of the amplifier, since $\tilde{G}_{k}\leq G_{k}$ and
$\tilde{N}_{k}\geq N_{k}$.

The whole chain is then defined between two reference planes: an input
reference plane $\mathcal{R}$ (or simply “the reference plane” in the
following) located where the input signal $N_{\mathrm{in}}$ enters the chain,
and an output reference plane $\mathcal{R}_{\mathrm{out}}$, located where we
measure the output signal $N_{\mathrm{out}}$ (usually inside a lab
instrument). Iteratively applying Eq. 3, the whole amplification chain can be
reduced to a single effective amplifier, transforming $N_{\mathrm{in}}$ to
$N_{\mathrm{out}}$ such that

$N_{\mathrm{out}}=G_{\mathrm{sys}}\left(N_{\mathrm{in}}+N_{\mathrm{sys}}\right),$
(6)

where $G_{\mathrm{sys}}=\prod_{k=1}^{n}\tilde{G}_{k}$ is the gain of the whole
amplification chain (also called system gain) and where

$N_{\mathrm{sys}}=\tilde{N}_{1}+\frac{\tilde{N}_{2}}{\tilde{G}_{1}}+\frac{\tilde{N}_{3}}{\tilde{G}_{1}\tilde{G}_{2}}+...$
(7)

is the system-added noise. Equation 7 is the so-called Friis formula [25]. We
must ensure that
$\tilde{N}_{1}\gg\tilde{N}_{2}/\tilde{G}_{1}\gg\mathrm{etc}...$ for the
amplifiers’ gains to overwhelm the following noisy elements. The amplification
chain is then well designed, and $N_{\mathrm{sys}}$ is dominated by
$\tilde{N}_{1}$, which accounts for both the added noise of the first
amplifier and for the transmission efficiency $\eta_{1}$ between the reference
plane and the input of that amplifier.

Figure 1: Schematic of an amplification chain. (a) An input signal
$N_{\mathrm{in}}$ enters the chain at the reference plane $\mathcal{R}$ and
passes through successive stages of attenuation, characterized by their
transmission efficiencies $\eta_{k}$ and temperatures $T_{k}$, with
$k\in\\{1,...,n\\}$, alternating with successive stages of amplification,
characterized by their gain $G_{k}$ and added noise $N_{k}$, with
$k\in\\{1,...,n\\}$. At the output reference plane
$\mathcal{R}_{\mathrm{out}}$, the resulting amplified signal is
$N_{\mathrm{out}}$. (b) Each pair of amplification and loss stage can be
reduced to an effective amplifier, with gain $\tilde{G}_{k}$ and added noise
$\tilde{N}_{k}$. (c) The whole chain itself can be reduced to an effective
amplifier, with gain $G_{\mathrm{sys}}$ and added noise $N_{\mathrm{sys}}$.

###  II.2 Principle of a noise measurement

The goal of a noise measurement consists of obtaining a fair evaluation of the
system-added noise $N_{\mathrm{sys}}$, and possibly also $N_{1}$, the first
amplifier added noise. Both matter, for different reasons. In fact,
$N_{\mathrm{sys}}$ is the only quantity that can be directly measured, and it
is ultimately the quantity that a user cares about. But it is chain-dependent:
in particular, it depends on $\eta_{1}$, which likely differs from system to
system. Inversely, $N_{1}$ is chain-independent, but heavily relies on
calibration measurements. It can be calculated by subtracting out the effect
of $\eta_{1}$ to $N_{\mathrm{sys}}$, i.e. by “moving the reference plane” of
the chain. However, in many cases the question of exactly where the reference
plane should be moved to is debatable: when moved up to the amplifier input,
it may be discarding the loss inserted by standard microwave components
mandatory for its operation. Also, any amplifier packaging inserts loss,
therefore in practice any amplification stage adds some noise.

Measuring a system-added noise always relies on knowing the amount of noise
somewhere within the chain, usually at the input reference plane
$\mathcal{R}$. Then, measuring the output noise $N_{\mathrm{out}}$ while
varying $N_{\mathrm{in}}$, a least-square regression gives both
$G_{\mathrm{sys}}$ and $N_{\mathrm{sys}}$ as shown by Eq. 6, with a caveat
when the first amplifier is parametric: in that case, the y-intercept of the
regression is usually not $N_{\mathrm{sys}}$, see Sec. II.5. This measurement
is called a Y-factor [26], and the noise source generating $N_{\mathrm{in}}$
can be, for example, a hot/cold load [11, 27], a variable temperature stage
(VTS) [28, 29, 30], a shot noise tunnel junction (SNTJ) [31, 21, 22, 29], a
qubit [17], a nanowire [16], or a diode at room temperature in front of a cold
attenuator[32]. Ideally, $N_{\mathrm{in}}$ need only take two values, but in
practice it is desirable to sweep $N_{\mathrm{in}}$ to calculate uncertainties
on $N_{\mathrm{sys}}$ and $G_{\mathrm{sys}}$ and to get rid of experimental
artifact: for example, the chain’s gain may not be perfectly linear over the
variation range, which cannot be diagnosed with only two points.

A variant of the Y-factor, called the ‘noise visibility ratio’ [33, 34] relies
on knowing the system-added noise $N_{\mathrm{sys}}^{\mathrm{off}}$ when the
first amplifier is turned off ($G_{1}=1$) while keeping $N_{\mathrm{in}}$ to a
fixed value (for example $N_{\mathrm{in}}=1/2$). Then, turning the amplifier
on, the output noise rises by a measurable amount $r$ such that

$r=G_{1}\frac{N_{\mathrm{in}}+N_{\mathrm{sys}}^{\mathrm{on}}}{N_{\mathrm{in}}+N_{\mathrm{sys}}^{\mathrm{off}}},$
(8)

from which we can deduce $N_{\mathrm{sys}}^{\mathrm{on}}$ (the system-added
noise with the amplifier on), see appendix C. In general, this technique is
approximate, because it is often difficult to precisely know
$N_{\mathrm{sys}}^{\mathrm{off}}$. In particular it can significantly depart
from $N_{2}$, the intrinsic noise added by the second amplifier (that may be
given in a datasheet) due to the loss before the second amplifier.
Nonetheless, the noise rise technique is useful to obtain a quick estimate of
the system-added noise.

###  II.3 Noise sources

Varying $N_{\mathrm{in}}$ necessitates the use of a noise source, and in the
following, we focus on the two most versatile self-calibrated noise sources -
a resistor and a SNTJ - leaving aside techniques utilizing a qubit (which can
only give a narrow-band noise characterization), a nanowire (which only works
in the very low temperature regime) or a noise diode (whose emitted noise is
in general not tunable, and which relies on a calibrated attenuator).

####  II.3.1 Variable temperature resistor

In the case of the hot/cold load, the noise source consists of two resistors
at two different temperatures. A variable temperature stage (VTS) is a
generalization, where the noise source is a single resistor whose temperature
is made variable (it also evades issues with a possible unequal attenuation
between the two resistors paths and the first amplifier). In such cases, the
noise dissipated by the resistor at temperature $T$ into a matched circuit is
given by the (quantum) Johnson noise formula:

$\displaystyle N_{\mathrm{in}}^{\mathrm{Johnson}}$
$\displaystyle=\frac{1}{e^{\nicefrac{{h\nu}}{{k_{B}T}}}-1}+\frac{1}{2}$ (9)
$\displaystyle=\frac{1}{2}\coth{\left(\frac{h\nu}{2k_{B}T}\right)}\xrightarrow[h\nu\ll
k_{B}T]{}\frac{k_{B}T}{h\nu}.$

When $k_{B}T\ll h\nu$, we have $N_{\mathrm{in}}^{\mathrm{Johnson}}=1/2$.
Conversely, in the classical limit where $h\nu\ll k_{B}T$ (indicated the
“$\rightarrow$”) we obtain the classical Johnson noise power spectral density:
$S_{P}^{\mathrm{Johnson}}=k_{B}T$, expressed in W/Hz (appendix D discusses the
various expressions of noise spectral densities). Therefore in the classical
limit, it is a white noise.

####  II.3.2 The shot-noise tunnel junction

A SNTJ relies on the shot noise associated to the random tunneling of
electrons through a normal-insulator-normal tunnel junction, with
characteristic impedance close to $50$ Ω. Like any resistor, the current-
voltage relation of the SNTJ is linear; however, unlike a typical resistor,
its noise increases with dc bias. A voltage bias $V$ accross the SNTJ shifts
the Fermi levels on both sides of the junction, which influences the tunneling
rate, and with it the shot noise level. More precisely, the noise at a
frequency $\nu$ dissipated by the SNTJ into a matched circuit is equal to [21,
35]

$\displaystyle N_{\mathrm{in}}^{\mathrm{shot}}=$
$\displaystyle\frac{eV+h\nu}{4h\nu}\coth{\left(\frac{eV+h\nu}{2k_{B}T_{e}}\right)}$
(10) $\displaystyle+$
$\displaystyle\frac{eV-h\nu}{4h\nu}\coth{\left(\frac{eV-h\nu}{2k_{B}T_{e}}\right)}$
$\displaystyle\xrightarrow[\\{k_{B}T_{e},h\nu\\}\ll e\lvert
V\rvert]{}\frac{e\lvert V\rvert}{2h\nu},$

where the two hyperbolic cotangent terms account for quantum corrections that
arise at low temperatures, when $k_{B}T_{e}\ll h\nu$, which is the regime we
focus on in this article. Here, $e$ is the electron charge and $T_{e}$ is the
temperature of the electrons. In practice, $T_{e}$ will be a fit parameter.
Note that Eq. 10 presents other limit cases, depending on how $eV$, $k_{B}T$
and $h\nu$ compare with each other, see appendix E.

At zero bias
$N_{\mathrm{in}}^{\mathrm{shot}}=N_{\mathrm{in}}^{\mathrm{Johnson}}$ i.e. we
are in the Johnson noise limit, and the SNTJ behaves like a $50$ Ω load.
Inversely, when $h\nu\ll e\lvert V\rvert$ we are in the classical shot noise
limit, and $N_{\mathrm{in}}^{\mathrm{shot}}=eV/(2h\nu)$ (see also appendix D).
Converting to W/Hz, we obtain the classical shot noise power spectral density:
$S_{P}^{\mathrm{shot}}=e\lvert V\rvert/2$. Therefore in the classical limit,
it is also a white noise.

####  II.3.3 Comparison between VTS and SNTJ

Figure 2 shows how $N_{\mathrm{out}}$ theoretically varies with the
temperature $T$ of a VTS (a) and with the bias voltage $V$ of a SNTJ (b), for
a given system gain $G_{\mathrm{sys}}$ and system-added noise
$N_{\mathrm{sys}}$. Both responses are qualitatively identical. In the simple
case where $N_{\mathrm{out}}$ is related to $N_{\mathrm{in}}$ via Eq. 6, the
slope of the linear response (i.e. at high bias, in the classical limit) is
equal to $G_{\mathrm{sys}}$, and the y-intercept is equal to the output-
referred system-added noise $G_{\mathrm{sys}}\times N_{\mathrm{sys}}$. At zero
bias, the difference between the output noise and the y-intercept gives the
unavoidable half quantum, present within the input signal.

Although both noise sources are conceptually equivalent, using a VTS presents
several advantages: (i) it is somewhat straightforward to fabricate, because
it only necessitates readily available components: a calibrated thermometer, a
heater, and a $50$ Ω load, all mounted on a copper support. The VTS is then
weakly thermally connected to a cryostat plate, for example to the mixing
chamber plate. (ii) The precision on $N_{\mathrm{in}}^{\mathrm{Johnson}}$ only
depends on how well the VTS thermometer has been calibrated, and how matched
the $50$ Ω load is to the readout line; two tasks that have excellent
commercial solutions. In comparison, the SNTJ has not, so far, been easily
accessible to the superconducting circuit community. However, it presents
three major advantages compared to a VTS: (i) it allows for much faster noise
measurements, because varying $N_{\mathrm{in}}^{\mathrm{shot}}$ is performed
with a bias voltage $V$, while varying $N_{\mathrm{in}}^{\mathrm{Johnson}}$
with a VTS implies to change and stabilize its temperature. Thus the noise can
be changed in nanoseconds instead of minutes. (ii) It does not generate
parasitic heating while, depending on the VTS thermal connection to its
anchoring plate, heating up the VTS may affect the cryostat plates’
temperatures, in turn modifying the effective noise inserted by the various
components in the readout chain, thereby falsifying the simple linear
regression of Eq. 6. (iii) A SNTJ has a much higher dynamic range than a VTS,
fundamentally because the noise generated by a SNTJ relies on the charge
constant $e$, while a VTS relies on the Boltzmann constant $k_{B}$, and
$k_{B}/e=86\,\mathrm{\micro V}/\mathrm{K}$. In other words, it is much easier
for a SNTJ to generate a large noise. For example, a SNTJ biased with 1 mV
generates noise equivalent to a VTS at 11.6 K.

Figure 2: Theoretical output noise $N_{\mathrm{out}}$ given by Eq. 6, as a
function of (a) the temperature $T$ of a VTS or (b) the voltage bias $V$ of a
SNTJ, and calculated at $6$ GHz. In the linear (high bias) regime, the curve
is a straight line, with slope $G_{\mathrm{sys}}$ and y-intercept
$G_{\mathrm{sys}}\times N_{\mathrm{sys}}$. We arbitrarily chose
$G_{\mathrm{sys}}=10^{9}$ and $N_{\mathrm{sys}}=1$. When using the SNTJ, the
response is plotted for $T_{e}=0$ K (black), and for $T_{e}=0.1$ K (red),
showing that a non-zero physical temperature as a rounding effect at low
biases.

###  II.4 Description of an ideal parametric amplifier

Parametric amplifiers play a key role in many readout applications, because
their added noise can be, in theory, at the fundamental quantum limit [36], so
when used as the first amplifier in the chain, they allow us to minimize
$N_{\mathrm{sys}}$. During parametric amplification, photons at a ‘pump’
frequency ($\omega_{p}$) are used to amplify a ‘signal’ (frequency
$\omega_{s}$) while also creating ‘idler’ photons
$\omega_{i}=n\omega_{p}-\omega_{s}$, where $n=1$ or $2$ corresponds to the so-
called ‘3-wave’ or ‘4-wave’ mixing operations [37]. A parametric amplifier can
be operated (i) as phase-insensitive (also called “phase-preserving”), where
its gain does not depend on the signal’s phase. This process unavoidably adds
at least half a photon worth of noise. (ii) As phase-sensitive, where the gain
is dependent on the signal’s phase. This process need not add any noise.

Figure 3: Schematic of the field operators transformations, executed by a
parametric amplifier. In the phase-insensitive case, the operators
$a_{\mathrm{in}}$ and $b^{\dagger}_{\mathrm{in}}$ represent the input signal
and idler modes. They transform into $a_{\mathrm{out}}$ and
$b^{\dagger}_{\mathrm{out}}$. In the phase-sensitive case, for each Fourier
component $a_{\mathrm{in}}(\delta)$ we have
$a^{\dagger}_{\mathrm{in}}(-\delta)=b^{\dagger}_{\mathrm{in}}(-\delta)$ and
$a^{\dagger}_{\mathrm{out}}(-\delta)=b^{\dagger}_{\mathrm{out}}(-\delta)$.

These modes of operation are governed by constraints on the commutation
relations on field operators at the input and at the output of the amplifier
[38]. Upon amplification, the input operator $a_{\mathrm{in}}$ transforms into
the output operators $a_{\mathrm{out}}$, and the commutation relation
$[a_{\mathrm{in}},a^{\dagger}_{\mathrm{in}}]=1$ must remain true at the
output: $[a_{\mathrm{out}},a^{\dagger}_{\mathrm{out}}]=1$.

####  II.4.1 Phase-insensitive case

In the phase-insensitive case, we use an ancillary input mode
$b_{\mathrm{in}}$ such that

$a_{\mathrm{out}}=\sqrt{G_{1}}a_{\mathrm{in}}+\sqrt{G_{1}-1}b^{\dagger}_{\mathrm{in}},$
(11)

where $G_{1}\geq 1$ is the signal gain. Equivalently,
$b_{\mathrm{out}}^{\dagger}$ is expressed as a function of
$b_{\mathrm{in}}^{\dagger}$ and $a_{\mathrm{in}}$ [36]. In parametric
amplification, where pump photons (at a frequency $\omega_{p}$) are converted
into signal ($\omega_{s}$) and idler ($\omega_{i}$) photons, this ancillary
mode is at the idler frequency. Recast in terms of photon-normalized signal,
Eq. 11 yields:

$N_{\mathrm{out},1}=G_{1}N_{\mathrm{in}}+(G_{1}-1)N_{\mathrm{in}}^{i},$ (12)

where $N_{\mathrm{in}}=\braket{a^{\dagger}_{\mathrm{in}}a_{\mathrm{in}}}+1/2$,
and
$N_{\mathrm{in}}^{i}=\braket{b^{\dagger}_{\mathrm{in}}b_{\mathrm{in}}}+1/2$
are the photon-normalized inputs at the signal and idler frequencies
respectively, and where
$N_{\mathrm{out},1}=\braket{a^{\dagger}_{\mathrm{out}}a_{\mathrm{out}}}+1/2$
is the photon-normalized amplifier output at the signal frequency, see
appendix B.1. When $G_{1}\gg 1$, Eq. 12 reduces to

$N_{\mathrm{out},1}=G_{1}(N_{\mathrm{in}}+N_{\mathrm{in}}^{i}).$ (13)

The idler input, $N_{\mathrm{in}}^{i}$, therefore appears as some undesirable
noise. Importantly, this noise does not come from an internal degree of
freedom, but rather only depends on the state at the idler’s input, and when
the input is cold the amplifier adds the minimum amount of noise:
$N_{\mathrm{in}}^{i}=1/2$. This is the quantum limit on added noise for a
phase-insensitive amplifier. Note that the amplifier can be at a physically
high temperature and remain quantum-limited, as long as its idler input mode
is cold [34].

As pointed out by Caves, “quantum mechanics extracts its due twice”[36]:
looking at Eq. 13, there is also a minimum of $1/2$ quantum of fluctuations in
$N_{\mathrm{in}}$ which, in a different context than noise measurements, can
be present on top of a displacement. Added to the fluctuations in
$N_{\mathrm{in}}^{i}$, it defines the standard quantum limit (SQL) as 1
quantum worth of noise. It is the minimum amount of total noise that can be
measured at the output of a phase-insensitive amplification chain.

####  II.4.2 Phase-sensitive case

Alternatively, one can exploit the correlations between signal and idler by
looking at their linear combinations. This is the phase sensitive case, where
the signal and idler Fourier components are centered about
$\omega_{0}=(\omega_{s}+\omega_{i})/2$. Therefore, in a frame rotating at
$\omega_{0}$ we can pair each Fourier component of the signal
$a_{\mathrm{in}}(\delta)$ to
$a^{\dagger}_{\mathrm{in}}(-\delta)=b^{\dagger}_{\mathrm{in}}(-\delta)$, where
$\delta$ is the frequency detuning to $\omega_{0}$. It yields:

$a_{\mathrm{out}}(\delta)=\sqrt{G_{1}(\delta)}e^{-i\theta}a_{\mathrm{in}}(\delta)+\sqrt{G_{1}(\delta)-1}e^{i\theta}a^{\dagger}_{\mathrm{in}}(-\delta),$
(14)

where $\theta$ is the phase difference between signal and pump, and
$G_{1}(\delta)$ remains the phase-insensitive gain (that links the input and
output signal at frequency $\omega_{s}=\omega_{0}+\delta$). In the following
we omit the dependence in $\delta$ to keep the equations to a simple form. The
decomposition of $a_{\mathrm{out}}$ onto the axes of the rotating frame
yields:

$\displaystyle X_{\mathrm{out}}$
$\displaystyle=(X_{\mathrm{in}}\cos{\theta}+Y_{\mathrm{in}}\sin{\theta})(\sqrt{G_{1}}+\sqrt{G_{1}-1})$
(15) $\displaystyle Y_{\mathrm{out}}$
$\displaystyle=(Y_{\mathrm{in}}\cos{\theta}-X_{\mathrm{in}}\sin{\theta})(\sqrt{G_{1}}-\sqrt{G_{1}-1}),$
(16)

which are also called the “quadratures” of the signal (and the rotating frame
is also called the quadrature basis). Here,
$X_{k}=(a_{k}+a^{\dagger}_{k})/\sqrt{2}$ and
$Y_{k}=(a_{k}-a^{\dagger}_{k})/(i\sqrt{2})$, with
$k=\\{\mathrm{in},\mathrm{out}\\}$. Rotating the signal quadratures so that
$\theta=0$ and assuming $G_{1}\gg 1$, we obtain

$\displaystyle X_{\mathrm{out}}$ $\displaystyle=X_{\mathrm{in}}2\sqrt{G_{1}}$
(17) $\displaystyle Y_{\mathrm{out}}$
$\displaystyle=\frac{Y_{\mathrm{in}}}{2\sqrt{G_{1}}},$ (18)

respectively the amplified and squeezed quadratures. These equations show that
neither the amplified nor the squeezed quadrature suffer from any added noise:
the phase-sensitive amplification is noiseless.

At the output of the amplifier, the signal is measured along one quadrature
$I_{\mathrm{out},1}$, that forms an angle $\alpha$ with $X_{\mathrm{in}}$.
Therefore we have

$I_{\mathrm{out},1}=X_{\mathrm{in}}2\sqrt{G_{1}}\cos{\alpha}+\frac{Y_{\mathrm{in}}}{2\sqrt{G_{1}}}\sin{\alpha}.$
(19)

In practice, $\alpha$ depends on the pump’s phase of the parametric amplifier.
In photon-normalized units it yields:

$N_{\mathrm{out},1}=\mathcal{G}_{1}(\alpha)N_{\mathrm{in}},$ (20)

with

$\mathcal{G}_{1}(\alpha)=4G_{1}\cos^{2}{\alpha}+\frac{\sin^{2}{\alpha}}{4G_{1}},$
(21)

assuming
$N_{\mathrm{in}}=\braket{X_{\mathrm{in}}^{2}}=\braket{Y_{\mathrm{in}}^{2}}$,
and assuming uncorrelated quadrature inputs i.e.
$\braket{X_{\mathrm{in}}Y_{\mathrm{in}}}=0$. Here, $\mathcal{G}_{1}(\alpha)$
is the effective phase-sensitive gain. When $\alpha=0$,
$N_{\mathrm{out},1}=4GN_{\mathrm{in}}$, and we retrieve the well-known result
that $\mathcal{G}(0)$ is 4 times (or $6$ dB) higher than $G_{1}$. When
$\alpha=\pi/2$, $N_{\mathrm{out},1}=N_{\mathrm{in}}/(4G_{1})$, and the input
noise $N_{\mathrm{in}}$ has been squeezed by the amplifier.

###  II.5 Measuring the noise case of a single-input-port, non-ideal
parametric amplifier

####  II.5.1 Phase-insensitive case

A major pitfall of noise measurements, especially in the context of parametric
amplifiers, consists of misinterpreting Eq. 6 when using a wideband noise
source. In fact, as we have seen in Sec. II.4, a phase-insensitive parametric
amplifier possess two inputs, one at the signal frequency and one at the idler
frequency. But, for many parametric amplifiers, these input modes enter the
amplifier via the same physical input port. Therefore, a wideband noise source
placed at the input port of the amplifier illuminates both the signal and
idler modes inputs, and the amplification chain of Fig. 1 now has to reflect
this ambivalence. In the case where the first amplifier of the chain is
parametric (and when only this amplifier is parametric), the amplification
chain of Fig. 1 can be recast into that of Fig. 4, where the input of the
modes now enter via two different ports.

Also, we now consider the parametric amplifier as non-ideal, therefore (i) the
gain $G_{1}^{i}$ along the idler-to-signal path may not be exactly equal to
$G_{1}-1$ (where $G_{1}$ is the gain along the signal-to-signal path) as it
was the case in Sec. II.4, because of possible loss asymmetries between the
signal and idler frequencies within the amplifier. (ii) We introduce two extra
noise term, $N_{1,\mathrm{ex}}$ and $N_{1,\mathrm{ex}}^{i}$, added by the
amplifier in excess above the quantum limit along the idler-to-signal and
signal-to-signal paths, respectively. They originate from internal non-
idealities, _i.e._ in the ideal-amplifier case
$N_{1,\mathrm{ex}}=N_{1,\mathrm{ex}}^{i}=0$. Rewriting Eq. 12 in this context,
the output of the amplifier at the signal frequency is:

$N_{\mathrm{out},1}=\tilde{G}_{1}\left(N_{\mathrm{in}}+\frac{\tilde{G}_{1}^{i}}{\tilde{G}_{1}}N_{\mathrm{in}}^{i}+\tilde{N}_{1,\mathrm{ex}}\right),$
(22)

where $\tilde{G}_{1}=\eta_{1}G_{1}$ and
$\tilde{G}_{1}^{i}=\eta_{1}^{i}G_{1}^{i}$ are the effective signal-to-signal
and idler-to-signal gains, respectively, and where $\tilde{N}_{1,\mathrm{ex}}$
is the effective signal-to-signal excess noise, referred to the signal input,
that accounts for both $N_{1,\mathrm{ex}}$ and $N_{1,\mathrm{ex}}^{i}$, and
for the noise added by the loss before the amplifier, see appendix B.1.

In principle one would have to know $\tilde{G}_{1}^{i}/\tilde{G}_{1}$ in order
to fit Eq. 22 and extract $\tilde{N}_{1,\mathrm{ex}}$, but knowing this ratio
can be experimentally difficult, because it requires to know the frequency-
dependent internal loss of the amplifier. However, in most practical cases it
is reasonable to assume that both gains don’t vary by more than a factor $2$
of each other, an uncertainty that only has a small effect on
$\tilde{N}_{1,\mathrm{ex}}$ (less than $30$ %), see appendix B.1. Therefore, a
good approximation consists of assuming $\tilde{G}_{1}=\tilde{G}_{1}^{i}$. The
signal at the output of the chain then takes the form:

$N_{\mathrm{out}}=G_{\mathrm{sys}}(N_{\mathrm{in}}+N_{\mathrm{in}}^{i}+N_{\mathrm{sys,ex}}),$
(23)

where $N_{\mathrm{sys,ex}}$ is the system-excess noise at the signal
frequency, i.e. noise above the quantum limit on added noise of
$N_{\mathrm{sys}}=1/2$ quanta. Therefore, $N_{\mathrm{sys,ex}}=0$ if the
readout chain is quantum-limited. The output signal $N_{\mathrm{out}}$ depends
on both the signal and idler input modes, in other words we have separated the
noise intrinsic to the amplification chain, represented by
$N_{\mathrm{sys,ex}}$, to the noise extrinsic to the amplification chain,
represented by $N_{\mathrm{in}}^{i}$. When using a broadband noise source (VTS
or a SNTJ) to characterize the noise of the amplification chain we illuminate
both the input signal and idler ports, i.e. we vary both $N_{\mathrm{in}}$ and
$N_{\mathrm{in}}^{i}$ (and both given either by Eq. 9 in the case of a VTS or
by Eq. 10 in the case of a SNTJ, expressed at the signal and idler frequency
respectively). Therefore, the y-intercept of such a measurement is now
$N_{\mathrm{sys,ex}}$, not $N_{\mathrm{sys}}$ as seen previously (see Sec.
II.3.3). In turn, the total system-added noise is
$N_{\mathrm{sys}}=N_{\mathrm{in}}^{i}+N_{\mathrm{sys,ex}}$, which encompasses
all the amplification and loss in the chain, similarly as in Eq. 7. In the
ideal case where the idler input of a DUT replacing the SNTJ is cold i.e.
$N_{\mathrm{in}}^{i}=1/2$ (and when $\tilde{G}_{1}=\tilde{G}_{1}^{i}$) the
system-added noise would be

$N_{\mathrm{sys}}=\frac{1}{2}+N_{\mathrm{sys,ex}}.$ (24)

Figure 4: Schematic of an amplification chain where the first amplifier is
parametric. Drawn with two input ports and one output port, we separate the
input signal and idler modes. On the path from the signal input to the signal
output (i.e. the signal-to-signal path), the amplifier’s gain is $G_{1}$ and
adds an excess of noise (above the quantum limit) $N_{1,\mathrm{ex}}$. On the
idler-to-signal path the gain is $G_{1}^{i}$ and the excess of noise is
$N_{1,\mathrm{ex}}^{i}$.

Failing to account for the idler input mode, in other words using the model
represented by Eq. 6 to interpret the data instead of using the model
represented by Eq. 23 leads to under-estimating $N_{\mathrm{sys}}$ by more
than a factor two. In fact, when $N_{\mathrm{in}}=N_{\mathrm{in}}^{i}$ (a good
approximation for narrow-band parametric amplifiers), Eq. 23 writes as

$N_{\mathrm{out}}=2G_{\mathrm{sys}}\left(N_{\mathrm{in}}+\frac{N_{\mathrm{sys,ex}}}{2}\right).$
(25)

Therefore, the interpretation of the slope and y-intercept of the output noise
curves in Fig. 2 is, when wrongly using Eq. 6, $G_{\mathrm{sys}}$ and
$G_{\mathrm{sys}}\times N_{\mathrm{sys}}$, while it should be
$2G_{\mathrm{sys}}$ and $2G_{\mathrm{sys}}\times N_{\mathrm{sys,ex}}/2$,
respectively, i.e. the y-intercept is in reality half of the output-referred
system-excess noise.

Note that when $\eta_{1}^{i}=0$ (achieved for example with a filter), the
noise source only illuminates the signal port. In other words, the (phase-
insensitive) parametric amplifier can be thought of as having only one input
port at the signal frequency, while the idler input port receives some noise
whose value depends on the port’s temperature $T$, as given by Eq. 9. When
$k_{B}T\ll h\nu$, this noise is half a quanta. Therefore, in this case the
noise measurement can be carried out using Eqs. 6 and 7, with
$\tilde{N}_{1}\geq 1/2$.

####  II.5.2 Phase-sensitive case

In the phase-sensitive case, the “signal” dwells over both the signal and
idler frequencies: there is no idler mode anymore, and the amplifier can once
again be thought of as having only one input port receiving a signal. In this
case, the input noise $N_{\mathrm{in}}$ is transformed according to Eq. 20.
Accounting for amplifier non-idealities via an excess of noise
$N_{1,\mathrm{ex}}\geq 0$, the output signal along the measurement quadrature
is:

$N_{\mathrm{out},1}(\alpha)=\tilde{\mathcal{G}}_{1}(\alpha)(N_{\mathrm{in}}+\tilde{N}_{1,\mathrm{ex}}),$
(26)

where
$\tilde{N}_{1,\mathrm{ex}}=N_{1,\mathrm{ex}}/\sqrt{\eta_{1}\eta_{1}^{i}}$, and
$\tilde{\mathcal{G}}_{1}(\alpha)=\sqrt{\eta_{1}\eta_{1}^{i}}\mathcal{G}_{1}(\alpha)$,
see appendix B.2.

When $\alpha=\pi/2$, the measurement quadrature is aligned with the
amplifier’s squeezed quadrature. However, only with a following phase-
sensitive parametric amplifier can we measure (not infer) squeezed noise [39,
28]. If however the parametric amplifier is followed by a phase-insensitive
amplifier, say a HEMT, with effective gain $\tilde{G}_{2}$ and added noise
$\tilde{N}_{2}$, the signal at the HEMT output is:

$N_{\mathrm{out},2}(\alpha)=\tilde{G}_{2}\tilde{\mathcal{G}}_{1}(\alpha)\left(N_{\mathrm{in}}+\tilde{N}_{1,\mathrm{ex}}+\frac{\tilde{N}_{2}}{\tilde{\mathcal{G}}_{1}(\alpha)}\right).$
(27)

Therefore, in that situation the (non-ideal) system-added noise is

$N_{\mathrm{sys}}(\alpha)=\tilde{N}_{1,\mathrm{ex}}+\frac{\tilde{N}_{2}}{\tilde{\mathcal{G}}_{1}(\alpha)},$
(28)

neglecting the noise added by further amplification stages. When $\alpha=0$,
the measurement quadrature is aligned with the amplified quadrature and

$N_{\mathrm{out},2}(0)=\tilde{G}_{2}\eta_{1}4G_{1}(N_{\mathrm{in}}+\tilde{N}_{1,\mathrm{ex}}),$
(29)

assuming that $\tilde{N}_{2}/(\eta_{1}4G_{1})\ll\tilde{N}_{1,\mathrm{ex}}$. In
this case, the added noise is dominated by $\tilde{N}_{1,\mathrm{ex}}$.
Inversely, if $\alpha=\pi/2$:

$N_{\mathrm{out},2}(\pi/2)=\tilde{G}_{2}\frac{\eta_{1}}{4G_{1}}\left(N_{\mathrm{in}}+\tilde{N}_{2}\frac{4G_{1}}{\eta_{1}}\right),$
(30)

and the input-referred system-added noise jumps to
$\tilde{N}_{2}4G_{1}/\eta_{1}$. As $\alpha$ varies between $0$ and $\pi/2$,
$N_{\mathrm{sys}}(\alpha)$ takes values between these two extreme cases.

###  II.6 Amplifier saturation

Not only may a real amplifier add some excess of noise above the quantum
limit, but it can also saturate. Saturation manifests as a compression of the
gain: as the input signal increases, the amplifier gain diminishes. In other
words, $G_{1}=G_{1}(N_{\mathrm{in}})$. It is usually not a desirable regime
for the amplifier operate in, because its response ceases to be linear.

Physically, in parametric amplifiers, saturation can originate either from
pump depletion or from the excitation of higher order nonlinearities [40, 28,
41]. One of the metrics conventionally used to characterize saturation is the
input $1$ dB compression point $N_{1\mathrm{dB}}$, defined as the input signal
(here in photon-normalized units) at which the gain has dropped by $1$ dB (the
output $1$ dB compression point is sometimes used, but it is painfully
dependent on the amplifier’s gain). This point is a somewhat arbitrary
criteria, not a hard limit below which the amplifier is perfectly linear, and
above which it is completely nonlinear.

Amplifier saturation may heavily affect the noise curves (obtained by varying
$N_{\mathrm{in}}$) where the system gain is proportional to the local slope.
Therefore, a steadily diminishing gain translates into a steadily diminishing
slope. In Sec. III.3.1 we present a way to circumvent this issue when
performing a noise measurement.

##  III Broadband noise characterization of canonical amplification chains

We illustrate how to perform a noise measurement on several amplification
chains, containing microwave components typically used in superconducting
circuit readout. Our noise source is double: it consists of a SNTJ, itself
mounted on a VTS, see Fig. 5. This configuration offers two features: (i) it
allows us to compare the system-added noise obtained when using the SNTJ to
that obtained when using the VTS, because at zero bias the SNTJ acts as a $50$
Ω load (see Sec. II.3.2). (ii) It allows us to measure the loss inserted by
the components of the chain that are mounted on the VTS, the SNTJ packaging
and the following (commercial) bias tee, mandatory for the SNTJ’s operation as
a noise source. Thus, it allows us to move the reference plane of the noise
measurement using the SNTJ, from the output of the SNTJ (on the chip) to the
output of the bias tee. We measure the noise of three amplifications chains:
(i) when a HEMT amplifier at $4$ K is the first amplifier of the chain, (ii)
when it is preceded by a Josephson Traveling Wave Parametric Amplifier
(JTWPA), or (iii) by a resonant Josephson Parametric Amplifier (JPA), placed
at millikelvin temperatures.

Figure 5: Schematic of the three cryogenic amplification chains, on which we
performed noise measurements. At the input of the chain the noise source is
double: (i) a SNTJ and (ii) a variable temperature stage (VTS), consisting of
of a thermal stage whose temperature can be controlled independently to that
of the mixing chamber plate on which it is anchored. We studied three
configurations: (a) one where the first amplifier is the HEMT, placed at $4$
K, (b) one where it is preceded by a JTWPA, and (c) where it is preceded by a
JPA, both placed at $10$ mK. Their presence necessitates the use of additional
microwave components: circulator, isolator and directional coupler.

###  III.1 The HEMT as first amplifier

Figure 6: Noise characterization of the amplification chain where the HEMT is
the first amplifier. (a) The output noise referred to the input of the chain,
$N_{\mathrm{out}}/G_{\mathrm{sys}}$, is plotted as a function of the input
noise $N_{\mathrm{in}}$. We show measurements using either the SNTJ (black
markers) or the VTS (blue markers). The noise is integrated within a $8$ MHz
window, around $f=6$ GHz. A fit of the responses (gray lines) gives
$N_{\mathrm{sys}}$. It is plotted against frequency in (b), when using the
SNTJ (black line) and when using the VTS (blue line). We also show the HEMT-
added noise specified in its datasheet (purple line). The fit of the response
in (a) also gives $G_{\mathrm{sys}}$ which we plot against frequency in (c),
following the same color coding. Dividing out both system gains, we extract
the transmission efficiency $\eta_{1}$, plotted against frequency in (d). In
the regime of small SNTJ voltage biases $V$, we plot in (e)
$N_{\mathrm{out}}/G_{\mathrm{sys}}$ obtained around $f=6$ GHz, for two VTS
temperatures: $T=34$ mK (black line) and $T=300$ mK (red line). The
y-intercept of the asymptotes (dashed lines) also gives $N_{\mathrm{sys}}$ for
this frequency. A fit of these curves (gray lines) allows us to extract the
electronic temperature $T_{e}$, which we plot against $T$ in (f) (black
markers). $T_{e}$ closely follows $T$ (gray line) over the whole VTS
temperature range.

When the HEMT is the first amplifier of the chain (see Fig. 5a), we can ask:
is the measured system-added noise close to the intrinsic HEMT-added noise,
specified in its datasheet? In Fig. 6a we show the input-referred output noise
$N_{\mathrm{out}}/G_{\mathrm{sys}}$, measured around $6$ GHz with a spectrum
analyzer (see appendix F), as a function of the input noise generated by the
SNTJ (Eq. 10), superimposed with data obtained when using the VTS as a noise
source (Eq. 9). Focusing here on the asymptotic behavior, the saturation of
the HEMT is high enough so that the response remains linear over the whole
excursion, up to $N_{\mathrm{in}}=14$ when using the SNTJ (and
$N_{\mathrm{in}}=7$ when using the VTS), providing a reliable way to extract
the system gain and added noise. A fit of the noise curves (or here, simply
reading the y-intercept), see appendix G, gives $N_{\mathrm{sys}}$. Figure 6b
shows $N_{\mathrm{sys}}$, measured as a function of frequency, for the two
situations (SNTJ and VTS). Both noise sources give an average system-added
noise of $30$ to $50$ quanta between $4$ and $8$ GHz, equivalent to a noise
temperature of $10$ to $12$ K. While it is about twice that of the HEMT-added
noise specified by the datasheet, this is completely expected due to the
inevitable loss between the reference plane and the HEMT itself.

The system-added noise obtained with the VTS is consistently lower to that
obtained with the SNTJ, because the reference plane of the measurement with
the VTS advances to the bias tee output (see Fig. 5). Conversely, the chain’s
gain $G_{\mathrm{sys}}$ is consistently lower with the SNTJ than with the VTS,
see Fig. 6c. Dividing out both gains yields the transmission efficiency of all
the components on the VTS,
$\eta_{1}=G_{\mathrm{sys}}^{\mathrm{SNTJ}}/G_{\mathrm{sys}}^{\mathrm{VTS}}$,
i.e. the SNTJ packaging and bias tee’s insertion loss is
$\mathcal{I}_{1}=-10\log_{10}\eta_{1}$. In Fig. 6d we show $\eta_{1}$ as a
function of frequency. It is between $0.8$ and $0.9$, which means that the
loss is between $0.5$ to $1$ dB between $4$ and $9$ GHz, in agreement with
previous, similar measurements [22, 29].

There is however a noticeable feature around $7$ GHz, where the system gain is
significantly lower than anywhere else in frequency, see Fig. 6c, and where
the system-added noise is significantly higher, in particular when using the
SNTJ, see Fig. 6b. Consequently, the packaging insertion loss is significantly
higher at this frequency, see Fig. 6d. The origin of this feature is unclear,
and its presence was unexpected; it could be due to a variation in the noise
emitted by the SNTJ, or it could come from a strong impedance mismatch at a
defective connection somewhere within the chain (i.e. a broken cable).
Nonetheless, it underlines the importance of wideband noise measurements to
characterize readout chains: we cannot assume the response to be homogeneous
and well-behaved everywhere.

Finally, focusing now on the small bias response, we measure the output noise
at two VTS temperatures around $6$ GHz, shown in Fig. 6e (and see appendix F
for the measurement method). A fit of these curves using Eq. 10 and Eq. 6 (see
appendix G) allows us to retrieve the electronic temperature $T_{e}$, which we
plot against $T$, the VTS physical temperature, in Fig. 6f. The agreement is
excellent, even at the lowest temperature, so the SNTJ can also be used as a
self-calibrated thermometer [21].

###  III.2 Using a JTWPA as a pre-amplifier

We now use the SNTJ to perform a noise measurement on the amplification chain
presented in Fig. 5b, where a JTWPA has been inserted before the HEMT. The
JTWPA is operated to produce $G_{1}=20$ dB of wideband gain, see Fig. 7a.
Then, the wideband system-added noise, shown in Fig. 7b (black trace), is
obtained from the asymptotic behavior of the output response: the SNTJ is
biased at $10$ evenly spaced values between $eV/(2k_{B})=\pm 0.5$ K, and for
each value we record an output spectrum. At each frequency we then fit the
variation of the output noise as a function of the input noise using Eq. 23
(which assumes $\tilde{G}_{1}=\tilde{G}_{1}^{i}$) and calculate
$N_{\mathrm{sys}}$ using Eq. 24. It is the system-added noise, with reference
plane at the SNTJ. From $4$ to $8$ GHz, $N_{\mathrm{sys}}$ varies between $3$
and $5$ quanta, typical for such an amplification chain. The sharp increase of
noise around $6$ GHz comes from resonant features inside the JTWPA, necessary
to obtain phase-matching and exponential gain via a four-wave mixing process
[42, 17].

Knowing $\eta_{1}$, corresponding to the transmission efficiency of all the
components mounted on the VTS (see Fig. 6d) and assuming
$\eta_{1}=\eta_{1}^{i}$, we can move the reference plane to the output of the
VTS, and get the corrected system-excess noise (see appendix B.1):

$N_{\mathrm{sys,ex,corr}}=\eta_{1}N_{\mathrm{sys,ex}}-2(1-\eta_{1})N_{T_{1}},$
(31)

where the factor $2$ comes from the combination of the loss between the signal
and idler frequencies, and where $N_{T_{1}}=0.5$, because when operating the
SNTJ the VTS is kept cold. We then compute the corrected system-added noise
$N_{\mathrm{sys,corr}}=N_{\mathrm{sys,ex,corr}}+1/2$. As shown in Fig. 7a,
$N_{\mathrm{sys,corr}}$ (blue trace) is lower by about $0.5$ quanta compared
to $N_{\mathrm{sys}}$. Some components and connections remain between the
JTWPA and the new reference plane, but these components are integral to the
use of the JTWPA. Therefore they should be considered as part of the amplifier
itself, and as such $N_{\mathrm{sys,corr}}$ constitutes a fair estimate of the
true added noise of the amplifier.

Figure 7: Noise characterization of an amplification chain where the JTWPA is
the first (phase-insensitive) amplifier. (a) The JTWPA gain is measured with
the VNA (ratio pump on/off). (b) The system-added noise $N_{\mathrm{sys}}$
(black line) is a few quanta above the quantum limit on added noise (where
$N_{\mathrm{sys}}=0.5$, dashed line). Knowing the loss of the components
mounted on the VTS, we move the reference plane from the SNTJ output to the
VTS output, and show a corrected added noise (blue line).

###  III.3 Using a JPA as a pre-amplifier

Replacing the JTWPA with a flux-pumped JPA, we now study the amplification
chain presented in Fig. 5c. In principle, the measurement and data analysis
should be similar to that performed when using the JTWPA; it should even be
simpler, because the JPA is a narrow-band amplifier, with a bandwidth of about
$5$ to $10$ MHz at $20$ dB of gain. However the JPA is typically an amplifier
with a very low saturation power, which complicates the noise
characterization. Furthermore, the JPA is also commonly used as a phase-
sensitive amplifier [15, 39, 28, 13]. Below, we experimentally address both
aspects.

####  III.3.1 Phase-insensitive noise

Figure 8: Noise characterization of the chain where the JPA is the first
(phase-insensitive) amplifier. (a) The JPA gain is measured on the VNA (and
fit to a Lorentzian) while varying the SNTJ bias (black line). The dashed
lines indicate the $1$ dB compression point. (b) The output noise as a
function of the SNTJ voltage bias $V$ (black dashed line) saturates at high
biases, and the fit of the curve using a linear model (gray dashed line) is
not appropriate. Renormalizing the noise with the gain, the output noise
regains a linear behavior (black line), and is then fitted with the linear
model (gray line). (c) In the situation of uncorrected noise curve the
residuals of the fit have a clear structure at high $V$. The different curves
correspond to fitting various portion of the output noise. (d) In the
renormalized case, the structure in the residuals dampens. (e) The system-
added noise $N_{\mathrm{sys}}$ as a function of the fitting window’s width
presents two distinct behaviors: (i) when using the uncorrected data
(triangles), $N_{\mathrm{sys}}$ diverges. (ii) When using the corrected data
(circles), the variation of $N_{\mathrm{sys}}$ with the fitted portion of the
curve dampens. When moving the reference plane to the VTS output,
$N_{\mathrm{sys}}$ decreases by about $0.2$ quanta (blue).

In Fig. 8a we plot the JPA gain, measured on a vector network analyzer (VNA)
with a small probe tone, as a function of the SNTJ voltage bias $V$. We
measure $N_{1\mathrm{dB}}=2.1$ quanta (at $6$ GHz, referenced to the SNTJ
output) but obviously the gain steadily drops before that, even at very low
signals, and continues to steadily drop beyond that. Then, in Fig. 8b we show
the output noise as a function of $V$ (dashed black line), which exhibits a
nonlinear behavior at high bias due to amplifier saturation. Fitting this
curve with the model described by Eq. 23, now wrong because it assumes a
constant chain’s gain, fails to produce a consistent result. As shown in Fig.
8c, the residuals of the fit show a systematic deviation that grows with the
portion of the curve upon which the fit is performed, and concurrently the
system-added noise $N_{\mathrm{sys}}$ increases and diverges, see Fig. 8e
(since the fitted gain decreases due to compression).

To correct for this effect, we divide out the gain variation measured with the
VNA from the noise curve, which then retrieves a more linear behavior, see
Fig. 8b. We then fit the output noise with a corrected model, close to that
described by Eq. 23 (see appendix B.1). The systematic deviation dampens.
However the residuals (Fig. 8d) sill have some structure when fitting a wide
portion of the SNTJ noise curve. Similarly, the variation of
$N_{\mathrm{sys}}$ as a function of the fitted portion of the curve dampens,
see Fig. 8e. When fitting a narrow portion we find $N_{\mathrm{sys}}=1.1$
quanta, but the gain is likely to be under-estimated because of the rounding
of the curve around $V=0$. When fitting the whole curve we find
$N_{\mathrm{sys}}=0.85$ quanta but the residuals show a systematic error. The
true system-added noise (at low input signal) lies in between these two
values.

We can, once again, move the reference plane of this measurement up to the VTS
output using Eq. 31, because we know $\eta_{1}$ from previous measurements.
Then, the corrected system-added noise, decreases by about $0.2$ quanta (see
Fig. 8e) , but of course remains above the quantum limit on added noise of
$0.5$ quanta.

####  III.3.2 Phase-sensitive noise

The same cryogenic setup can be used to measure the phase-sensitive system-
added noise $N_{\mathrm{sys}}(\alpha)$, equal to zero if the chain is ideal
(see Eq. 28). At room temperature, a local oscillator (LO), whose frequency
$f_{LO}=6$ GHz is half that of the JPA’s pump, now drives a mixer that
overlaps the signal and idler frequency bands down to a baseband. The baseband
noise is then measured on a spectrum analyzer (see apppendix F) and depends on
$\alpha$, the phase difference between the phase of the LO and that of the
JPA’s amplified quadrature.

Figure 9: Phase sensitive behavior of the amplification chain, where the JPA
is the first amplifier. The system gain $G_{\mathrm{sys}}(\alpha)$ (blue, left
y-axis) and the system-added noise $N_{\mathrm{sys}}(\alpha)$ (black, right
y-axis) are plotted as a function of $\alpha$, the angle between the measured
quadrature and the JPA’s amplified quadrature. The data (points) agrees with a
simple predictive model (lines).

The JPA saturation also affects the phase-sensitive noise. Worse, saturation
is phase-dependent. In fact, in general the amplified quadrature saturates at
higher power than the squeezed quadrature, as the theory and the measurements
of phasor deformation have shown [43, 40, 44, 28, 45]. In comparison,
saturation in phase-insensitive case can be viewed as an effective, average
behavior over all the quadrature angles. To correct for this phasor
deformation, one would have to measure it at various phasor amplitudes, which
falls out of the scope of this paper. Instead, we restricted our fits to a
fairly narrow portion of the noise curve, between $\pm 1.25$ quanta, where the
gain does not excessively compress, and therefore where the obtained gain and
excess of noise should not be too far from their true value (at least when
looking at the amplified quadrature). With that caveat in mind, Fig. 9 shows
the system gain $G_{\mathrm{sys}}(\alpha)$ (which contains
$\mathcal{G}(\alpha)$) and system-added noise $N_{\mathrm{sys}}(\alpha)$,
measured with the SNTJ. Clearly, the behavior of the data (points) as a
function of $\alpha$ makes sense: we can derive a simple predictive model for
$G_{\mathrm{sys}}(\alpha)$ and $N_{\mathrm{sys}}(\alpha)$ using Eq. 27 and
using the experimental HEMT-only system gain and added noise previously
measured in Sec. III.1 (see appendix H). This model (lines) agrees fairly well
with the data. Overall, when $\alpha=0\,\mathrm{mod}(\pi)$, the measured
quadrature is the JPA’s amplified quadrature, therefore the system gain is
maximal while the added noise is minimal, reaching a lowest value of $0.43$
quanta, not too far from the system-excess noise
$N_{\mathrm{sys,ex}}=N_{\mathrm{sys}}-1/2$ found in the phase-insensitive
case.

##  IV Conclusion

Calibrated noise measurements are based on simple theoretical concepts yet
there are several subtle errors that can sneak into both the measurement and
interpretation/analysis, especially when characterizing parametric amplifiers.
One might speculate that this is based on the idea that a ‘good’ parametric
amplifier must be quantum limited, resulting in a literature where the
majority of published noise characterization efforts report quantum-limited
behavior. While an ideal parametric amplifier can be, in principle, quantum-
limited, it does not mean that (i) a real amplification chain will be and (ii)
that one will rigorously measure its noise. In fact, the activation of
spurious parametric processes, dielectric loss, heating, and any other non-
idealities will contribute to decreasing the noise performance. Furthermore, a
practical parametric amplifier requires intervening components to bias and
isolate it from the quantum circuit that it’s measuring. These components
(isolators, circulators, directional couplers, PCBs, connectors, _etc._) add
insertion loss and therefore degrade the effective added noise. Even when this
basic fact is acknowledged, it’s not uncommon to attempt to move the reference
plane to the circuit on-chip by taking manufacturer specification sheet
numbers for these components– a practice that essentially negates all other
attempts at rigor. Meanwhile, the low power handling capability of many
Josephson parametric amplifiers can make noise measurements hard to parse
without also measuring the gain saturation properties carefully.

All of these real-life complications support the need for a more flexible and
reliable noise source. Compared to a traditional hot/cold load (or to its
continuously variable variant, the VTS), the SNTJ enables much faster noise
measurements or, equivalently, allows us to acquire many more points on the
noise curve, due to its bias in voltage and not in temperature. Further
developments, for example engineering an on-chip bias-tee, should make this
remarkable device widely accessible and user-friendly. But the devil is in the
details: when using a wideband noise source to characterize a parametric
amplifier, the idler port contribution to the output noise must be considered
carefully. As shown here, ignorance of this term comes at the cost of
underestimating the added noise contribution, erroneously giving one the sense
that their performance is closer (or even below!) the standard quantum limit.

We have illustrated how to perform a noise measurement with a SNTJ on three
canonical amplification chains, and shown how to move the chain’s reference
plane using a separate noise measurement performed with a VTS. With the HEMT
as first amplifier, the system-added noise, referred to the SNTJ output, is a
few tens of quanta, with a JTWPA it is a few quanta, and with a JPA it is even
better yielding just a fraction of a quanta. These numbers are what an end
user ultimately cares about, because they have been obtained from a realistic
use of these amplifiers, including necessary lossy components. In the same
vein, we strongly support explicitly stating the added noise of the whole
amplification chain, uncorrected from potential noisy contributions due to
loss. Discriminating between the noise coming solely from the chip and the
noise coming from the extraneous, lossy components is of interest for
amplifier designers, and should be carried out with a cold calibration. While
it is obvious to many, it is still worthwhile stating explicitly: no practical
parametric amplifier chain can ever operate at the quantum limit for added
noise when referred to an input that includes required lossy components.
Perhaps, even more to the point, this is totally expected, even with a
perfectly lossless on-chip amplifier circuit.

## Acknowledgment

We gratefully acknowledge Katrina Sliwa and the MIT Lincoln Laboratories, for
providing us with a JTWPA. Certain commercial materials and equipment are
identified in this paper to foster understanding. Such identification does not
imply recommendation or endorsement by the National Institute of Standards and
Technology, nor does it imply that the materials or equipment identified are
necessarily the best available for the purpose.

##  Appendix A List of variables

Table 1 defines the noise-related variables used throughout the article.

variable name | definition
---|---
$N_{\mathrm{in}}$ | signal at the input of the amplification chain i.e. at the reference plane $\mathcal{R}$
$N_{\mathrm{out}}$ | signal at the output of the amplification chain i.e. at the reference plane $\mathcal{R}_{\mathrm{out}}$
$N_{\mathrm{in}}^{i}$ | idler at the input of the amplification chain i.e. at the reference plane $\mathcal{R}$
$N_{\mathrm{in},k}$ | signal at the input of the $k$-th amplifier
$\tilde{N}_{\mathrm{in},k}$ | signal at the input of the effective $k$-th amplifier, equal to $N_{\mathrm{out},k-1}$
$N_{\mathrm{out},k}$ | signal at the output of the $k$-th amplifier
$G_{k}$ | signal gain of the $k$-th amplifier
$\tilde{G}_{k}$ | signal gain of the $k$-th effective amplifier
$G_{\mathrm{sys}}$ | signal gain of the whole amplification chain i.e. system gain
$G_{1}^{i}$ | idler-to-signal gain of the parametric amplifier, first amplifier in the chain
$\tilde{G}_{1}^{i}$ | idler-to-signal gain of the effective parametric amplifier, first amplifier in the chain
$\mathcal{G}_{1}(\alpha)$ | phase-sensitive signal gain of the parametric amplifier, first amplifier in the chain
$\tilde{\mathcal{G}}_{1}(\alpha)$ | phase-sensitive signal gain of the effective parametric amplifier, first amplifier in the chain
$\eta_{k}$ | transmission efficiency of the $k$-th loss stage at the signal frequency
$\eta_{k}^{i}$ | transmission efficiency of the $k$-th loss stage at the idler frequency
$N_{k}$ | added noise of the $k$-th amplifier, referred to its input
$\tilde{N}_{k}$ | added noise of the $k$-th effective amplifier, referred to its input
$N_{\mathrm{sys}}$ | system-added noise, referred to the input of the chain
$N_{\mathrm{sys}}^{\mathrm{on}}$ | system-added noise when the first amplifier is on, referred to the input of the chain
$N_{\mathrm{sys}}^{\mathrm{off}}$ | system-added noise when the first amplifier is off, referred to the input of the chain
$N_{T_{k}}$ | Johnson noise generated by the $k$-th loss stage at a temperature $T_{k}$
$N_{1,\mathrm{ex}}$ | signal-to-signal excess of noise of the first amplifier in the chain, referred to the signal input
$N_{1,\mathrm{ex}}^{i}$ | idler-to-signal excess of noise of the first amplifier in the chain, referred to the idler input
$\tilde{N}_{1,\mathrm{ex}}$ | effective signal-to-signal excess of noise of the first amplifier in the chain, referred to the signal input
$N_{\mathrm{sys,ex}}$ | system-excess noise, referred to the input of the chain
$N_{\mathrm{in}}^{\mathrm{Johnson}}$ | Johnson noise delivered by a resistor to a matched circuit
$N_{\mathrm{in}}^{\mathrm{shot}}$ | Shot noise delivered by a SNTJ to a matched circuit
Table 1: List of the noise-related variables used in the article. All the $N$
variables are noises, expressed in photon-normalized units (see Sec. II.1).
All the $G$ variables are photon-normalized gains.

##  Appendix B Signal transformation with a parametric amplifier

###  B.1 Phase-insensitive case

####  B.1.1 Operators to noise in photon-normalized units

Considering Eq. 11 and its Hermitian conjugate, we calculate the average
output photon number

$\displaystyle\bar{n}_{\mathrm{out}}$
$\displaystyle=\braket{a_{\mathrm{out}}^{\dagger}a_{\mathrm{out}}}$ (32)
$\displaystyle=G_{1}\bar{n}_{\mathrm{in}}+(G_{1}-1)(\bar{n}_{\mathrm{in}}^{i}+1)$
(33)
$\displaystyle=G_{1}\left(\bar{n}_{\mathrm{in}}+\frac{1}{2}\right)+(G_{1}-1)\left(\bar{n}_{\mathrm{in}}^{i}+\frac{1}{2}\right)-\frac{1}{2},$
(34)

where
$\bar{n}_{\mathrm{in}}=\braket{a_{\mathrm{in}}^{\dagger}a_{\mathrm{in}}}$, and
$\bar{n}_{\mathrm{in}}^{i}=\braket{b_{\mathrm{in}}^{\dagger}b_{\mathrm{in}}}$
are the photon-number operators. We then obtain Eq. 12 from Eq. 34 by taking
the $-1/2$ to the left-hand side:

$N_{\mathrm{out},1}=G_{1}N_{\mathrm{in}}+(G_{1}-1)N_{\mathrm{in}}^{i},$ (35)

with $N_{\mathrm{out},1}=\bar{n}_{\mathrm{out}}+1/2$,
$N_{\mathrm{in}}=\bar{n}_{\mathrm{in}}+1/2$ and
$N_{\mathrm{in}}^{i}=\bar{n}_{\mathrm{in}}^{i}+1/2$.

####  B.1.2 Excess of noise

Let us know consider a non-ideal parametric amplifier: it has a gain $G_{1}$
($G_{1}^{i}$) and adds some excess noise $N_{1,\mathrm{ex}}$
($N_{1,\mathrm{ex}}^{i}$) along the signal-to-signal (idler-to-signal) path.
If we also include the loss $\eta_{1}$ ($\eta_{1}^{i}$) along the signal
(idler) path before the amplifier, we obtain at its output:

$\displaystyle N_{\mathrm{out},1}=G_{1}$
$\displaystyle[\eta_{1}N_{\mathrm{in}}+(1-\eta_{1})N_{T_{1}}+N_{1,\mathrm{ex}}]$
(36) $\displaystyle+G_{1}^{i}$
$\displaystyle[\eta_{1}^{i}N_{\mathrm{in}}^{i}+(1-\eta_{1}^{i})N_{T_{1}}+N_{1,\mathrm{ex}}^{i}],$

that is to say

$N_{\mathrm{out},1}=\tilde{G}_{1}\left(N_{\mathrm{in}}+\frac{\tilde{G}_{1}^{i}}{\tilde{G}_{1}}N_{\mathrm{in}}^{i}+\tilde{N}_{1,\mathrm{ex}}\right),$
(37)

where

$\tilde{N}_{1,\mathrm{ex}}=\frac{(1-\eta_{1})N_{T_{1}}+N_{1,\mathrm{ex}}}{\eta_{1}}+\frac{\tilde{G}_{1}^{i}}{\tilde{G}_{1}}\frac{(1-\eta_{1}^{i})N_{T_{1}}+N_{1,\mathrm{ex}}^{i}}{\eta_{1}^{i}}.$
(38)

####  B.1.3 Effect of asymmetric gains

We assume $\tilde{G}_{1}^{i}/\tilde{G}_{1}=1$. In fact, in most practical
cases, we will have $1/2<\tilde{G}_{1}^{i}/\tilde{G}_{1}<2$, i.e. the
effective signal-to-signal and idler-to-signal gains do not differ by more
than $3$ dB (due to loss). Therefore we can ask: how
$\tilde{N}_{1,\mathrm{ex}}$ varies when $\tilde{G}_{1}^{i}/\tilde{G}_{1}$
varies between $1/2$ to $2$? For simplicity, let’s consider
$N_{\mathrm{in}}=N_{\mathrm{in}}^{i}$. Then, from Eq. 37:

$N_{\mathrm{out},1}=\frac{\tilde{G}_{1}}{1+\frac{\tilde{G}_{1}^{i}}{\tilde{G}_{1}}}\left(N_{\mathrm{in}}+\frac{\tilde{N}_{1,\mathrm{ex}}}{1+\frac{\tilde{G}_{1}^{i}}{\tilde{G}_{1}}}\right),$
(39)

and the y-intercept, which corresponds to half the excess noise, varies
between $\tilde{N}_{1,\mathrm{ex}}/3$ and $2\tilde{N}_{1,\mathrm{ex}}/3$.
Therefore, $\tilde{N}_{1,\mathrm{ex}}$ varies between
$2\tilde{N}_{1,\mathrm{ex}}/3$ and $4\tilde{N}_{1,\mathrm{ex}}/3$.

####  B.1.4 Moving the reference plane

If we assume $\tilde{G}_{1}^{i}/\tilde{G}_{1}=1$ and $\eta_{1}=\eta_{1}^{i}$,
Eq. 38 gives

$\tilde{N}_{1,\mathrm{ex}}=\frac{2(1-\eta_{1})+N_{1,\mathrm{ex}}+N_{1,\mathrm{ex}}^{i}}{\eta_{1}},$
(40)

therefore if we know $\eta_{1}$ we can remove its effect to the excess of
noise added by that amplifier. More generally, if we measure a system-excess
noise $N_{\mathrm{sys,ex}}$, we can remove the effect of $\eta_{1}$ and get a
corrected system-excess noise:

$N_{\mathrm{sys,ex,corr}}=\eta_{1}N_{\mathrm{sys,ex}}-2(1-\eta_{1})N_{T_{1}}.$
(41)

####  B.1.5 Filtering out the idler input

Finally, when the calibrated noise entering the idler input is filtered out,
in other words when $\eta_{1}^{i}=0$ Eq. 36 reduces to

$\displaystyle N_{\mathrm{out},1}=$ $\displaystyle
G_{1}[\eta_{1}N_{\mathrm{in}}+(1-\eta_{1})N_{T_{1}}+N_{1,\mathrm{ex}}]$ (42)
$\displaystyle+$ $\displaystyle G_{1}^{i}(N_{T_{1}}+N_{1,\mathrm{ex}}^{i}),$

that is to say

$N_{\mathrm{out},1}=\tilde{G}_{1}\left(N_{\mathrm{in}}+\frac{G_{1}^{i}}{\tilde{G}_{1}}N_{T_{1}}+\tilde{N}_{1,\mathrm{ex}}\right),$
(43)

where

$\tilde{N}_{1,\mathrm{ex}}=\frac{(1-\eta_{1})N_{T_{1}}+N_{1,\mathrm{ex}}}{\eta_{1}}+\frac{G_{1}^{i}}{\tilde{G}_{1}}N_{1,\mathrm{ex}}^{i}.$
(44)

In other words, the parametric amplifier has an excess of noise
$\tilde{N}_{1,\mathrm{ex}}$, and an added noise
$G_{1}^{i}/\tilde{G}_{1}N_{T_{1}}+\tilde{N}_{1,\mathrm{ex}}$. In the high gain
regime and when the idler input is cold, an ideal amplifier is therefore still
limited by the quantum limit because in that case
$G_{1}^{i}/\tilde{G}_{1}N_{T_{1}}=0.5$.

####  B.1.6 Added noise of a saturating parametric amplifier

When a parametric amplifier saturates, its effective gain $\tilde{G}_{1}$
becomes dependent on the input signal. When measuring the noise added by the
JPA using the SNTJ, it means that $\tilde{G}_{1}=\tilde{G}_{1}(V)$, with $V$
the SNTJ voltage bias. We define $\lambda(V)$ such that:

$\tilde{G}_{1}(V)=G_{1}\lambda(V),$ (45)

where $G_{1}$ is the JPA small signal gain. In other words, $\lambda(V)$ is a
quantity that indicates how the JPA gain compresses (when $\lambda(V)<1$) as a
function of $V$. Experimentally, $\lambda(V)$ is given by the curve in Fig.
8a. The signal at the JPA output is then:

$N_{\mathrm{out},1}=\tilde{G}_{1}\lambda(V)(N_{\mathrm{in}}+N_{\mathrm{in}}^{i}+\tilde{N}_{1,\mathrm{ex}}),$
(46)

and the signal at the HEMT output is:

$N_{\mathrm{out},2}=G_{2}\tilde{G}_{1}\lambda(V)\left(N_{\mathrm{in}}+N_{\mathrm{in}}^{i}+\tilde{N}_{1,\mathrm{ex}}+\frac{\tilde{N}_{2}}{\tilde{G}_{1}\lambda(V)}\right).$
(47)

Dividing out the signal at the output of the chain by $\lambda(V)$ we obtain

$\frac{N_{\mathrm{out}}}{\lambda(V)}=G_{\mathrm{sys}}\left(N_{\mathrm{in}}+N_{\mathrm{in}}^{i}+\tilde{N}_{1,\mathrm{ex}}+\frac{\tilde{N}_{2}}{\tilde{G}_{1}\lambda(V)}\right),$
(48)

where we have approximated the system-added noise as

$N_{\mathrm{sys}}=1/2+\tilde{N}_{1,\mathrm{ex}}+\tilde{N}_{2}/(\tilde{G}_{1}\lambda(V)).$
(49)

In general we don’t know $\tilde{N}_{2}$, and furthermore
$\tilde{N}_{2}/(\tilde{G}_{1}\lambda(V))$ cannot be neglected compared to
$\tilde{N}_{1,\mathrm{ex}}$, because ideally $\tilde{N}_{1,\mathrm{ex}}=0$.
But in our situation we have a good approximation of $\tilde{N}_{2}$ as the
system-added noise of the chain where the HEMT is the first amplifier (case
(a) of Fig. 5) obtained using the VTS (_i.e._ blue curve in Fig. 6b). At $6$
GHz we find $\tilde{N}_{2}=33$ quanta. Then, when using the JPA, $G_{1}=23.8$
dB (the small signal gain), see Fig. 8a. Therefore
$\tilde{N}_{2}/\tilde{G}_{1}=0.14$ quanta. We then can fit the corrected noise
curves using Eq. 48 to find $\tilde{N}_{1,\mathrm{ex}}$, from which we obtain
the system-added noise (Eq. 49). It is represented in Fig. 8 (circles) for a
small signal input (_i.e._ for $\lambda(V)=1$).

###  B.2 Phase-sensitive case, with pre-amplifier loss

As mentioned in the main text, in the phase sensitive case the role of the
idler mode $b_{\mathrm{in}}$ is played by the signal mode $a_{\mathrm{in}}$.
Then, loss is modeled as a beamsplitter interaction, transforming
$a_{\mathrm{in}}$ such that

$\displaystyle a_{\mathrm{in}}$
$\displaystyle\rightarrow\sqrt{\eta_{1}}a_{\mathrm{in}}+\sqrt{1-\eta_{1}}\xi^{s}$
(50) $\displaystyle a_{\mathrm{in}}^{\dagger}$
$\displaystyle\rightarrow\sqrt{\eta_{1}^{i}}a_{\mathrm{in}}^{\dagger}+\sqrt{1-\eta_{1}^{i}}{\xi^{i}}^{\dagger},$
(51)

where $\xi^{s}$ and ${\xi^{i}}^{\dagger}$ are ladder operators for two noise
modes, uncorrelated with each other and uncorrelated to to $a_{\mathrm{in}}$
and $a_{\mathrm{in}}^{\dagger}$ (we arbitrarily chose the creation operator
for the idler noise mode, to track it easily). Also, the transmission
efficiencies $\eta_{1}$ and $\eta_{1}^{i}$ seen by $a_{\mathrm{in}}$ and
$a_{\mathrm{in}}^{\dagger}$ respectively are not necessarily equal, because
these two modes are not at the same frequency. At the output of the amplifier
we obtain

$\displaystyle a_{\mathrm{out}}=$
$\displaystyle\sqrt{G_{1}}e^{-i\theta}\left(\sqrt{\eta_{1}}a_{\mathrm{in}}+\sqrt{1-\eta_{1}}\xi^{s}\right)$
(52) $\displaystyle+$
$\displaystyle\sqrt{G_{1}-1}e^{i\theta}\left(\sqrt{\eta_{1}^{i}}a_{\mathrm{in}}^{\dagger}+\sqrt{1-\eta_{1}^{i}}{\xi^{i}}^{\dagger}\right)$
$\displaystyle a_{\mathrm{out}}^{\dagger}=$
$\displaystyle\sqrt{G_{1}}e^{i\theta}\left(\sqrt{\eta_{1}^{i}}a_{\mathrm{in}}^{\dagger}+\sqrt{1-\eta_{1}^{i}}{\xi^{i}}^{\dagger}\right)$
(53) $\displaystyle+$
$\displaystyle\sqrt{G_{1}-1}e^{-i\theta}\left(\sqrt{\eta_{1}}a_{\mathrm{in}}+\sqrt{1-\eta_{1}}\xi^{s}\right),$

from what we can calculate
$X_{\mathrm{out}}=(a_{\mathrm{out}}+a_{\mathrm{out}}^{\dagger})/\sqrt{2}$ and
$Y_{\mathrm{out}}=(a_{\mathrm{out}}-a_{\mathrm{out}}^{\dagger})/(i\sqrt{2})$.
Again, let’s take $\theta=0$ to have $X_{\mathrm{out}}$ be the amplified
quadrature and $Y_{\mathrm{out}}$ be the squeezed quadrature, and let’s work
in the high gain limit, $G_{1}\gg 1$:

$\displaystyle X_{\mathrm{out}}$
$\displaystyle=2\sqrt{G_{1}}\left(\frac{\sqrt{\eta_{1}}a_{\mathrm{in}}+\sqrt{\eta_{1}^{i}}a_{\mathrm{in}}^{\dagger}}{\sqrt{2}}+\frac{\sqrt{1-\eta_{1}}\xi^{s}+\sqrt{1-\eta_{1}^{i}}{\xi^{i}}^{\dagger}}{\sqrt{2}}\right)$
(54) $\displaystyle Y_{\mathrm{out}}$
$\displaystyle=\frac{1}{2\sqrt{G_{1}}}\left(\frac{\sqrt{\eta_{1}}a_{\mathrm{in}}-\sqrt{\eta_{1}^{i}}a_{\mathrm{in}}^{\dagger}}{i\sqrt{2}}+\frac{\sqrt{1-\eta_{1}}\xi^{s}-\sqrt{1-\eta_{1}^{i}}{\xi^{i}}^{\dagger}}{i\sqrt{2}}\right).$
(55)

All the modes being vacuum or thermal states, we then obtain:

$\displaystyle\braket{X_{\mathrm{out}}^{2}}$
$\displaystyle=4G_{1}\sqrt{\eta_{1}\eta_{1}^{i}}\left(\bar{n}_{\mathrm{in}}+\frac{1}{2}\right)$
(56) $\displaystyle\braket{Y_{\mathrm{out}}^{2}}$
$\displaystyle=\frac{\sqrt{\eta_{1}\eta_{1}^{i}}}{4G_{1}}\left(\bar{n}_{\mathrm{in}}+\frac{1}{2}\right),$
(57)

where
$\bar{n}_{\mathrm{in}}=\braket{a_{\mathrm{in}}^{\dagger}a_{\mathrm{in}}}$. We
see that both transmission efficiencies degrade the output quadratures photon
number, and in the limit case where one of them, say $\eta_{1}^{i}$ is zero,
we recover the phase-insensitive case: in particular, when $\eta_{1}=1$ Eq.
LABEL:eq:aoutps is is identical to Eq. 11 (taking the now arbitrary phase
$\theta$ equal to zero). At the output of the amplifier, we measure the signal
in a given orientation $\alpha$:

$\displaystyle N_{\mathrm{out},1}(\alpha)$
$\displaystyle=N_{\mathrm{in}}\sqrt{\eta_{1}\eta_{1}^{i}}\left(4G_{1}\cos^{2}{\alpha}+\frac{\sin^{2}{\alpha}}{4G_{1}}\right)$
(58)
$\displaystyle=N_{\mathrm{in}}\sqrt{\eta_{1}\eta_{1}^{i}}\mathcal{G}_{1}(\alpha),$

where $N_{\mathrm{in}}=\bar{n}_{\mathrm{in}}+1/2$. Finally, if we include some
excess noise $N_{1,\mathrm{ex}}$, added by the amplifier, we get:

$N_{\mathrm{out},1}(\alpha)=\sqrt{\eta_{1}\eta_{1}^{i}}\mathcal{G}_{1}(\alpha)\left(N_{\mathrm{in}}+\tilde{N}_{1,\mathrm{ex}}\right),$
(59)

where
$\tilde{N}_{1,\mathrm{ex}}=N_{1,\mathrm{ex}}/(\sqrt{\eta_{1}\eta_{1}^{i}})$ is
the input-referred excess noise.

##  Appendix C Noise rise theory

A noise rise - also called a noise visibility ratio - measurement allows us to
estimate the first amplifier’s added noise $N_{1}$. It consists of comparing
the noise measured when the first amplifier is on, to when it is off. Such a
measurement assumes some amount of noise $N_{\mathrm{in}}$ at the input of the
chain based on its physical temperature, according to Eq. 9; at $30$ mK and at
gigahertz frequencies, we can assume $N_{\mathrm{in}}=1/2$. When this
temperature is equal to $T_{1}$, i.e. to that of the loss between the chain’s
input and the first amplifier, characterized by $\eta_{1}$ (see Fig.1), the
chain’s input reference plane $\mathcal{R}$ is effectively moved up to the
input of the first amplifier, because this loss replaces noise with noise at
the same power. It unveils a first caveat: a noise rise measurement does not
account for the effect of the lossy elements inserted before the first
amplifier (including packaging and connectors) mandatory to its operation,
therefore it is an optimistic estimate of $N_{1}$.

When the first amplifier is off we assume that it has a gain $G_{1}=1$, and if
it is also at a temperature $T_{1}$, then $\mathcal{R}$ is effectively moved
up to the input of the loss characterized by its transmission efficiency
$\eta_{2}$ and its temperature $T_{2}>T_{1}$. In this case, following Eqs. 6,
4 and 5 the signal at the output of the second amplifier is:

$N_{\mathrm{out},2}^{\mathrm{off}}=\tilde{G}_{2}(N_{\mathrm{in}}+\tilde{N}_{2}),$
(60)

where $\tilde{G}_{2}=\eta_{2}G_{2}$ and
$\tilde{N}_{2}=[(1-\eta_{2})N_{T_{2}}+N_{2}]/\eta_{2}$. One may estimate
$\tilde{G}_{2}$ and $\tilde{N}_{2}$ with various ways, for example by heating
up the whole base-temperature plate of the cryostat, effectively performing a
hot/cold load measurement. But again, it reveals another caveat, because in
many practical situations there is some lossy elements at temperature $T_{1}$
placed on the base-temperature plate, after the first amplifier. This loss
will degrade the effective gain and added noise of the second-stage amplifier,
but it is not accounted for when estimating $\tilde{G}_{2}$ and
$\tilde{N}_{2}$ in such a way.

In comparison, When the first (possibly parametric) amplifier is on:

$\displaystyle N_{\mathrm{out},2}^{\mathrm{on}}$
$\displaystyle=G_{1}\tilde{G}_{2}(N_{\mathrm{in}}+N_{1}+\frac{\tilde{N}_{2}}{G_{1}})$
(61) $\displaystyle\simeq G_{1}\tilde{G}_{2}(N_{\mathrm{in}}+N_{1}),$ (62)

where the second line assumes that $G_{1}$ overwhelms the following loss.
Thus, we can form the ratio

$r=\frac{N_{\mathrm{out},2}^{\mathrm{on}}}{N_{\mathrm{out},2}^{\mathrm{off}}}=G_{1}\frac{N_{\mathrm{in}}+N_{1}}{N_{\mathrm{in}}+\tilde{N}_{2}},$
(63)

which, if the second amplifier’s added noise overwhelms any of the following
noise added by the chain, represents how much the noise rises at the output of
the chain when turning on the first amplifier. The noise rise can be seen on a
spectrum analyzer, for example. In practice we can have a fair estimate of
$G_{1}$ by measuring the gain of the first amplifier on a VNA (ratio on/off
traces). Knowing $\tilde{N}_{2}$ we can then deduce $N_{1}$. In practice, all
the caveats mentioned above make this technique approximate, biased toward
finding a lower noise than what it truly is.

##  Appendix D The various units of a noise spectral density

###  D.1 Johnson noise

From the fluctuation-dissipation theorem, a resistor $R$ at temperature $T$
generates noise. In the classical limit, its power spectral density can be
expressed in $\mathrm{V^{2}/Hz}$ as:

$S_{V}^{\mathrm{Johnson}}=4k_{B}TR.$ (64)

Within a given bandwidth $B$ the root mean squared voltage fluctuations
$V_{\mathrm{RMS}}$ emitted by the resistor are:

$\displaystyle V_{\mathrm{RMS}}$
$\displaystyle=\sqrt{S_{V}^{\mathrm{Johnson}}B}$ (65)
$\displaystyle=\sqrt{4k_{B}TRB}.$

Then, the power dissipated into a matched circuit is

$\displaystyle P^{\mathrm{Johnson}}$
$\displaystyle=\frac{V_{\mathrm{RMS}}^{2}}{4R}$ (66) $\displaystyle=k_{B}TB,$

where the factor $4$ comes from a voltage divider configuration. Thus, the
power spectral density, in W/Hz, dissipated into a matched circuit is:

$S_{P}^{\mathrm{Johnson}}=k_{B}T.$ (67)

###  D.2 Shot noise

In a SNTJ, shot noise originates from the discrete tunneling of electrons. In
the classical limit, the associated power spectral density can be expressed in
$\mathrm{A^{2}/Hz}$ as [23]:

$S_{I}^{\mathrm{shot}}=2e\lvert I\rvert,$ (68)

with $I$ the current flowing through the SNTJ. Then, the power spectral
density, in W/Hz, dissipated into a matched circuit is

$S_{P}^{\mathrm{shot}}=\frac{e\lvert V\rvert}{2},$ (69)

where $V$ is the voltage across the SNTJ.

##  Appendix E The various limit cases of the noise generated by a SNTJ

The SNTJ generates a noise as given by Eq. 10 i.e. whose power spectral
density (PSD), in W/Hz, is:

$\displaystyle P_{\mathrm{in}}^{\mathrm{shot}}=$
$\displaystyle\frac{eV+h\nu}{4}\coth{\left(\frac{eV+h\nu}{2k_{B}T_{e}}\right)}$
(70) $\displaystyle+$
$\displaystyle\frac{eV-h\nu}{4}\coth{\left(\frac{eV-h\nu}{2k_{B}T_{e}}\right)}$

This complicated formula may be hard to grasp, because it simultaneously
depends on $\nu$, $V$, and $T_{e}$. But, we can consider three limit cases
where the formula is simpler.

When $eV=0$ (or when it is much smaller than $k_{B}T_{e}$ and $h\nu$), Eq. 70
simplifies to

$\displaystyle P_{\mathrm{in}}^{\mathrm{shot}}(V=0)=$
$\displaystyle\frac{h\nu}{2}\coth{\left(\frac{h\nu}{2k_{B}T_{e}}\right)}$ (71)
$\displaystyle\xrightarrow[k_{B}T_{e}\ll h\nu]{}\frac{h\nu}{2}$
$\displaystyle\xrightarrow[k_{B}T_{e}\gg h\nu]{}k_{B}T_{e},$

equal to $P_{\mathrm{in}}^{\mathrm{Johnson}}$, the PSD delivered by a resistor
to a matched circuit.

When $h\nu=0$ (or when it is much smaller than $e\lvert V\rvert$ and
$k_{B}T_{e}$), Eq. 70 simplifies to

$\displaystyle P_{\mathrm{in}}^{\mathrm{shot}}(\nu=0)=$
$\displaystyle\frac{eV}{2}\coth{\left(\frac{eV}{2k_{B}T_{e}}\right)}$ (72)
$\displaystyle\xrightarrow[k_{B}T_{e}\ll e\lvert V\rvert]{}\frac{e\lvert
V\rvert}{2}$ $\displaystyle\xrightarrow[k_{B}T_{e}\gg e\lvert
V\rvert]{}k_{B}T_{e},$

equivalent to $P_{\mathrm{in}}^{\mathrm{Johnson}}$, with $eV$ replacing
$h\nu$.

Finally, when $k_{B}T_{e}=0$ (or when it is much smaller than $h\nu$ and
$e\lvert V\rvert$), Eq. 70 simplifies to

$P_{\mathrm{in}}^{\mathrm{shot}}(T_{e}=0)=\begin{cases}\frac{h\nu}{2}&\mathrm{if\hskip
14.22636pt}e\lvert V\rvert\leq h\nu\\\ \frac{e\lvert
V\rvert}{2}&\mathrm{if\hskip 14.22636pt}e\lvert V\rvert\geq h\nu,\end{cases}$
(73)

which is the limit case represented in Fig. 2b, in black. A non-null
temperature smooths the transition between the two regimes, happening at
$e\lvert V\rvert=h\nu$.

##  Appendix F Experimental setup

Figure 10: Schematic of the setup used to perform all the noise measurements.
We distinguish the rack ground (black) from the fridge ground (blue). The VTS
thermometer and heater are connected to a room temperature AC resistance
bridge with twisted-pair cables. On the SNTJ and JPA dc biasing lines, there
are $100$ kΩ resistors followed by $1$ MHz low-pass filters (LPFs) at $4$ K
and $1$ MHz LPFs at $10$ mK. On the rf pump and VNA probe lines, microwave
attenuators are placed at several thermal stages, whose values are indicated
on the schematic. The VNA output tone is sent either to the JTWPA pump line,
or to the JPA probe line, depending on the cryogenic configuration, (b) or
(c). At the output, three switches allow us to route signals onto various
equipment. In particular, the mixer is preceded by a room temperature rf
mplifier and $7.2$ GHz anti-aliasing LPF, and then followed by a baseband
amplifier and $350$ MHz LPF. See the main text for further details.

Figure 10 shows a schematic of the full experimental setup, with various
configurations, depending on which measurement was performed. Common to all
four noise measurements presented in the main text is the VTS and SNTJ biases
(on the left of the figure). An AC resistance bridge allows us to measure the
VTS temperature, and to servo the VTS heater.

The fridge ground and the ground of the room temperature electronics rack only
connect at a well-defined wall ground. However, the SNTJ and JPA dc biasing
electronics is floated with respect to the rack ground, and only connects to
the ground via the fridge ground, because the SNTJ and JPA are shorted to the
fridge ground within their respective packaging. We thus avoid the formation
of ground loops, that can perturb the SNTJ bias.

###  F.1 The VTS

The VTS consists of a copper plate, on which is mounted a calibrated resistor
(ruthenium oxide), a heater (a $1$ kΩ resistor), the SNTJ and the following
bias tee (Anritsu K250). We weakly couple the VTS to the mixing chamber plate
via a stainless steel screw. We then use superconducting niobium microwave
cables for the dc and microwave connections to the VTS, thereby preventing any
thermal connection via these channels.

###  F.2 SNTJ impedance and biasing procedure

In principle, the SNTJ is fabricated to be $50$ Ω; in practice, we measure its
impedance in situ with a voltage tap. More precisely, two branches lead to the
SNTJ: one delivers a current, and the other allows us to measure the voltage
drop across the SNTJ. In Fig. 10, these correspond to the branches connected
to the arbitrary waveform generator (AWG) and to the voltmeter (V),
respectively. The SNTJ biasing procedure is as follows: (i) we set the gain of
the amplifier preceding the voltmeter to $\times 100$ (verified by sending
some small voltage to its input with the AWG, and reading the output with the
voltmeter). (ii) Replacing the AWG with a current source, we send a known
current across the SNTJ, and measure the corresponding voltage drop with the
voltmeter. It gives us the SNTJ’s impedance $Z_{\mathrm{SNTJ}}$ (neglecting
the resistance of the bias tee). In our experiment, we found
$Z_{\mathrm{SNTJ}}=49.3$ Ω. (iii) Using the AWG, we send a known voltage and
measure the voltage drop across the SNTJ. It gives us the division ratio
between the SNTJ’s impedance and that of the biasing resistor placed at $4$ K,
nominally of $100$ kΩ. Such value permits a large division ratio ($1:2001$)
which damps down possible voltage fluctuations coming from the voltage source
(Keysight 33250A). Note that what we measure here is the SNTJ’s dc impedance.
The rf impedance might slightly deviate from the dc value, but the noise
measurements are not very sensitive to such small deviations. In fact, the
noise power transmitted to the transmission line varies only marginally with a
reasonably small mismatch between the SNTJ and the line impedance.

###  F.3 Wiring of the various experiments

The number of additional input lines used to control the cryogenic setup
depends on the cold configuration (a), (b) or (c). When measuring the noise of
with the HEMT as first amplifier (a), no additional input line is used, and
the output line is routed to the spectrum analyzer (SA), after further room
temperature amplification. When using the JTWPA (b), an additional line in the
cryostat is used to deliver the JTWPA pump and a VNA probe tone. At the
output, a microwave switch routes signals either to the SA, or to the
receiving port of the VNA. When using the JPA (c), a dc line allows us to
flux-tune the JPA (operating in a 3-wave mixing mode), and another delivers
the JPA pump [46]. At cryogenic temperatures, the dc and rf pump signals are
combined before entering the pump port of the JPA. Conversely, another line is
used to send a VNA probe tone, which enters via the JPA signal port (together
with the noise emitted by the SNTJ). At the output, signals can be routed with
microwave switches either onto the VNA, or onto the SA directly when
performing a phase-insensitive noise measurement, or onto the SA after being
mixed down when performing the phase-sensitive noise measurement of the
amplification chain containing the JPA. In that case, the local oscillator
(LO) frequency of the mixer is half that of the JPA pump.

###  F.4 Noise acquisition with a spectrum analyzer

When measuring the small bias response of the shot noise curves in the phase-
insensitive case, the AWG outputs a low frequency ramp ($20$ Hz), with an
amplitude symmetric around $0$ V. The SA sweep is then synchronized to the AWG
ramp via its external trigger, with the SA operated in zero span mode. The SA
central frequency determines the frequency around which the noise is measured,
within a window defined by the SA resolution bandwidth (RBW). All our
amplifiers’ bandwidth is large enough, and the noise is “white” enough over a
couple megahertz to have the RBW be $8$ MHz. In fact, a large RBW increases
the signal-to-noise ratio of the measurement. The SA sweep time is then set to
$t_{\mathrm{SA}}=1/20$ sec and we set the number of points per sweep at
$n_{\mathrm{SA}}=501$. Therefore, $t_{\mathrm{SA}}/n_{\mathrm{SA}}$ is the
time spend on each point. To have all the points be independent realisation of
the noise, we must set the SA video bandwidth (VBW) to a value larger than
this time. In practice we use $\mathrm{VBW}=3t_{\mathrm{SA}}/n_{\mathrm{SA}}$.

In the phase-sensitive case (using the JPA), the SA is also configured in zero
span, but this time it is centered around a frequency slightly detuned from
zero ($2.6$ MHz). In fact, in the phase-sensitive case the JPA amplification
bandwidth is effectively “folded” onto itself, the central amplification
bandwidth being referred back to the zero frequency of the SA. We also reduce
the RBW to a small enough value ($1.5$ MHz), which prevents the measured noise
to be contaminated by the $1/f$ response of the SA.

When measuring the wideband noise curves using the SNTJ, we set the AWG to a
dc mode and send discrete biasing voltage values. Equivalently, when using the
VTS we servo it to a given temperature. For each of these values (voltage or
temperature), we set the SA span to cover a desired frequency range ($3.5$ to
$8.5$ GHz) while keeping $n_{\mathrm{SA}}=501$. We can now freely choose the
VBW, as long as it is not bigger than the RBW, which we keep at $8$ MHz (we
don’t expect to have features narrower than that in the spectrum). Increasing
the VBW is equivalent to averaging for a longer time at each point. The
equipment can also average many spectral traces before sending it to the
acquisition computer. In practice, we use both types of averages (point-by-
point and trace-by-trace): we choose $\mathrm{VBW}=15$ kHz, and we choose to
average over $1500$ traces, which takes about $3$ minutes.

##  Appendix G Fit of the shot-noise curves

The fit of the shot noise curves is performed in two main steps: we first fit
the asymptotes, where the the shot noise takes a simple form,
$N_{\mathrm{in}}^{\mathrm{shot}}=e\lvert V\rvert/(2h\nu)$ (see Eq. 10), linear
with the bias voltage $V$ and independent to the electronic temperature
$T_{e}$. When fitting noise coming out of a parametric amplifier, we use Eq.
23, and therefore we have two inputs, at the signal and idler frequency. We
obtain three fitting parameters: an estimate of the system gain
$G_{\mathrm{sys}}$ and added noise $N_{\mathrm{sys}}$ (in the case of the
HEMT) or excess noise $N_{\mathrm{sys,ex}}$ (in the case of a parametric
amplifier), and an offset voltage $V_{\mathrm{offs}}$, close to $0$. In fact,
in principle, the positive and negative portions of the fit crosses at $V=0$,
but in practice we leave it as a fitting parameter $V_{\mathrm{offs}}$,
because the zero bias voltage delivered by the AWG may not be perfectly null.
We set guesses parameters for the least square regression fit:
$V_{\mathrm{offs}}^{\mathrm{g}}=0$, $N_{\mathrm{sys}}=50$ (HEMT) or
$N_{\mathrm{sys,ex}}=2$ (JTWPA) or $N_{\mathrm{sys,ex}}=0$ (JPA), and the
guess for the gain is calculated from the average absolute slope between the
positive and negative biases.

We then fit the full curves using Eq. 10, where we fix $V_{\mathrm{offs}}$ to
what has been found previously, and where we let the chain’s gain and added
noise vary by $\pm 50$ % around the asymptotic values. When fitting the HEMT-
only data, the guess for the electronic temperature $T_{e}$ is the SNTJ’s
physical temperature $T$. When fitting the JTWPA or the JPA data, we fix
$T_{e}=T$, validated by the fits from the HEMT-only data.

##  Appendix H Phase-sensitive noise fitting model

In the phase-sensitive case, we estimate
$M=\max({\mathcal{\tilde{G}}(\alpha))}$ by subtracting
$G_{\mathrm{sys}}^{H}=72$ dB, the chain’s gain at $6$ GHz found when the HEMT
is the first amplifier, see Fig. 6c, to the maximal chain’s gain presented in
Fig. 9. We find $M=19.5$ dB. We can then approximate
$\mathcal{\tilde{G}}(\alpha)\simeq M\cos^{2}{\alpha}+\sin^{2}{\alpha}/M$. It
is exact along the amplified quadrature, and approximate along the squeezed
quadrature, because $M$ includes $(\eta_{1}\eta_{1}^{i})^{1/2}$. In Fig. 9 we
plot $G_{\mathrm{sys}}^{H}{\mathcal{\tilde{G}}(\alpha)}$ (plain blue line).

We estimate $N_{\mathrm{sys}}(\alpha)$ by taking $N_{\mathrm{sys}}^{H}=33$ the
system-added noise at $6$ GHz found when the HEMT is the first amplifier. We
then extract $m=\min{(N_{\mathrm{sys}}}(\alpha))$ from Fig. 9 and plot
$m+N_{\mathrm{sys}}^{H}/\mathcal{\tilde{G}}(\alpha)-\min{(N_{\mathrm{sys}}^{H}/\mathcal{\tilde{G}}(\alpha))}$
(plain black line).

## References

  * Galitzki  _et al._ [2018] N. Galitzki, A. Ali, K. S. Arnold, P. C. Ashton, J. E. Austermann, C. Baccigalupi, T. Baildon, D. Barron, J. A. Beall, S. Beckman, S. M. M. Bruno, S. Bryan, P. G. Calisse, G. E. Chesmore, Y. Chinone, S. K. Choi, G. Coppi, K. D. Crowley, K. T. Crowley, A. Cukierman, M. J. Devlin, S. Dicker, B. Dober, S. M. Duff, J. Dunkley, G. Fabbian, P. A. Gallardo, M. Gerbino, N. Goeckner-Wald, J. E. Golec, J. E. Gudmundsson, E. E. Healy, S. Henderson, C. A. Hill, G. C. Hilton, S.-P. P. Ho, L. A. Howe, J. Hubmayr, O. Jeong, B. Keating, B. J. Koopman, K. Kiuchi, A. Kusaka, J. Lashner, A. T. Lee, Y. Li, M. Limon, M. Lungu, F. Matsuda, P. D. Mauskopf, A. J. May, N. McCallum, J. McMahon, F. Nati, M. D. Niemack, J. L. Orlowski-Scherer, S. C. Parshley, L. Piccirillo, M. S. Rao, C. Raum, M. Salatino, J. S. Seibert, C. Sierra, M. Silva-Feaver, S. M. Simon, S. T. Staggs, J. R. Stevens, A. Suzuki, G. Teply, R. Thornton, C. Tsai, J. N. Ullom, E. M. Vavagiakis, M. R. Vissers, B. Westbrook, E. J. Wollack, Z. Xu, and N. Zhu, “The Simons Observatory: instrument overview,” in  _Millimeter, Submillimeter, and Far-Infrared Detectors and Instrumentation for Astronomy IX_, Vol. 10708, edited by J. Zmuidzinas and J.-R. Gao, International Society for Optics and Photonics (SPIE, 2018) p. 1070804.
  * McCarrick  _et al._ [2021] H. McCarrick, E. Healy, Z. Ahmed, K. Arnold, Z. Atkins, J. E. Austermann, T. Bhandarkar, J. A. Beall, S. M. Bruno, S. K. Choi, J. Connors, N. F. Cothard, K. D. Crowley, S. Dicker, B. Dober, C. J. Duell, S. M. Duff, D. Dutcher, J. C. Frisch, N. Galitzki, M. B. Gralla, J. E. Gudmundsson, S. W. Henderson, G. C. Hilton, S.-P. P. Ho, Z. B. Huber, J. Hubmayr, J. Iuliano, B. R. Johnson, A. M. Kofman, A. Kusaka, J. Lashner, A. T. Lee, Y. Li, M. J. Link, T. J. Lucas, M. Lungu, J. A. B. Mates, J. J. McMahon, M. D. Niemack, J. Orlowski-Scherer, J. Seibert, M. Silva-Feaver, S. M. Simon, S. Staggs, A. Suzuki, T. Terasaki, R. Thornton, J. N. Ullom, E. M. Vavagiakis, L. R. Vale, J. V. Lanen, M. R. Vissers, Y. Wang, E. J. Wollack, Z. Xu, E. Young, C. Yu, K. Zheng, and N. Zhu, “The simons observatory microwave squid multiplexing detector module design,” The Astrophysical Journal 922, 38 (2021).
  * Noroozian  _et al._ [2013] O. Noroozian, J. A. B. Mates, D. A. Bennett, J. A. Brevik, J. W. Fowler, J. Gao, G. C. Hilton, R. D. Horansky, K. D. Irwin, Z. Kang, D. R. Schmidt, L. R. Vale, and J. N. Ullom, “High-resolution gamma-ray spectroscopy with a microwave-multiplexed transition-edge sensor array,” Applied Physics Letters 103, 202602 (2013).
  * Mates  _et al._ [2017] J. A. B. Mates, D. T. Becker, D. A. Bennett, B. J. Dober, J. D. Gard, J. P. Hays-Wehle, J. W. Fowler, G. C. Hilton, C. D. Reintsema, D. R. Schmidt, D. S. Swetz, L. R. Vale, and J. N. Ullom, “Simultaneous readout of 128 x-ray and gamma-ray transition-edge microcalorimeters using microwave squid multiplexing,” Applied Physics Letters 111, 062601 (2017), https://doi.org/10.1063/1.4986222 .
  * Nakashima  _et al._ [2020] Y. Nakashima, F. Hirayama, S. Kohjiro, H. Yamamori, S. Nagasawa, A. Sato, S. Yamada, R. Hayakawa, N. Y. Yamasaki, K. Mitsuda, K. Nagayoshi, H. Akamatsu, L. Gottardi, E. Taralli, M. P. Bruijn, M. L. Ridder, J. R. Gao, and J. W. A. den Herder, “Low-noise microwave squid multiplexed readout of 38 x-ray transition-edge sensor microcalorimeters,” Applied Physics Letters 117, 122601 (2020).
  * Szypryt  _et al._ [2021] P. Szypryt, D. A. Bennett, W. J. Boone, A. L. Dagel, G. Dalton, W. B. Doriese, M. Durkin, J. W. Fowler, E. J. Garboczi, J. D. Gard, G. C. Hilton, J. Imrek, E. S. Jimenez, V. Y. Kotsubo, K. Larson, Z. H. Levine, J. A. B. Mates, D. McArthur, K. M. Morgan, N. Nakamura, G. C. O’Neil, N. J. Ortiz, C. G. Pappas, C. D. Reintsema, D. R. Schmidt, D. S. Swetz, K. R. Thompson, J. N. Ullom, C. Walker, J. C. Weber, A. L. Wessels, and J. W. Wheeler, “Design of a 3000-pixel transition-edge sensor x-ray spectrometer for microcircuit tomography,” IEEE Transactions on Applied Superconductivity 31, 1–5 (2021).
  * Malnou  _et al._ [2023] M. Malnou, J. A. B. Mates, M. R. Vissers, L. R. Vale, D. R. Schmidt, D. A. Bennett, J. Gao, and J. N. Ullom, “Improved microwave SQUID multiplexer readout using a kinetic-inductance traveling-wave parametric amplifier,” Applied Physics Letters 122, 214001 (2023).
  * Arute  _et al._ [2019] F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. S. L. Brandao, D. A. Buell, B. Burkett, Y. Chen, Z. Chen, B. Chiaro, R. Collins, W. Courtney, A. Dunsworth, E. Farhi, B. Foxen, A. Fowler, C. Gidney, M. Giustina, R. Graff, K. Guerin, S. Habegger, M. P. Harrigan, M. J. Hartmann, A. Ho, M. Hoffmann, T. Huang, T. S. Humble, S. V. Isakov, E. Jeffrey, Z. Jiang, D. Kafri, K. Kechedzhi, J. Kelly, P. V. Klimov, S. Knysh, A. Korotkov, F. Kostritsa, D. Landhuis, M. Lindmark, E. Lucero, D. Lyakh, S. Mandrà, J. R. McClean, M. McEwen, A. Megrant, X. Mi, K. Michielsen, M. Mohseni, J. Mutus, O. Naaman, M. Neeley, C. Neill, M. Y. Niu, E. Ostby, A. Petukhov, J. C. Platt, C. Quintana, E. G. Rieffel, P. Roushan, N. C. Rubin, D. Sank, K. J. Satzinger, V. Smelyanskiy, K. J. Sung, M. D. Trevithick, A. Vainsencher, B. Villalonga, T. White, Z. J. Yao, P. Yeh, A. Zalcman, H. Neven, and J. M. Martinis, “Quantum supremacy using a programmable superconducting processor,” Nature 574, 505–510 (2019).
  * Krinner  _et al._ [2022] S. Krinner, N. Lacroix, A. Remm, A. Di Paolo, E. Genois, C. Leroux, C. Hellings, S. Lazar, F. Swiadek, J. Herrmann, G. J. Norris, C. K. Andersen, M. Müller, A. Blais, C. Eichler, and A. Wallraff, “Realizing repeated quantum error correction in a distance-three surface code,” Nature 605, 669–674 (2022).
  * Acharya  _et al._ [2023] R. Acharya _et al._ , “Suppressing quantum errors by scaling a surface code logical qubit,” Nature 614, 676–681 (2023).
  * Brubaker  _et al._ [2017] B. M. Brubaker, L. Zhong, Y. V. Gurevich, S. B. Cahn, S. K. Lamoreaux, M. Simanovskaia, J. R. Root, S. M. Lewis, S. Al Kenany, K. M. Backes, I. Urdinaran, N. M. Rapidis, T. M. Shokair, K. A. van Bibber, D. A. Palken, M. Malnou, W. F. Kindel, M. A. Anil, K. W. Lehnert, and G. Carosi, “First results from a microwave cavity axion search at $24\text{ }\text{ }\mu\mathrm{eV}$,” Phys. Rev. Lett. 118, 061302 (2017).
  * Du  _et al._ [2018] N. Du, N. Force, R. Khatiwada, E. Lentz, R. Ottens, L. J. Rosenberg, G. Rybka, G. Carosi, N. Woollett, D. Bowring, A. S. Chou, A. Sonnenschein, W. Wester, C. Boutan, N. S. Oblath, R. Bradley, E. J. Daw, A. V. Dixit, J. Clarke, S. R. O’Kelley, N. Crisosto, J. R. Gleason, S. Jois, P. Sikivie, I. Stern, N. S. Sullivan, D. B. Tanner, and G. C. Hilton (ADMX Collaboration), “Search for invisible axion dark matter with the axion dark matter experiment,” Phys. Rev. Lett. 120, 151301 (2018).
  * Backes  _et al._ [2021] K. M. Backes, D. A. Palken, S. A. Kenany, B. M. Brubaker, S. B. Cahn, A. Droster, G. C. Hilton, S. Ghosh, H. Jackson, S. K. Lamoreaux, A. F. Leder, K. W. Lehnert, S. M. Lewis, M. Malnou, R. H. Maruyama, N. M. Rapidis, M. Simanovskaia, S. Singh, D. H. Speller, I. Urdinaran, L. R. Vale, E. C. van Assendelft, K. van Bibber, and H. Wang, “A quantum enhanced search for dark matter axions,” Nature 590, 238–242 (2021).
  * Yurke  _et al._ [1989] B. Yurke, L. R. Corruccini, P. G. Kaminsky, L. W. Rupp, A. D. Smith, A. H. Silver, R. W. Simon, and E. A. Whittaker, “Observation of parametric amplification and deamplification in a Josephson parametric amplifier,” Phys. Rev. A 39, 2519–2533 (1989).
  * Castellanos-Beltran  _et al._ [2008] M. A. Castellanos-Beltran, K. D. Irwin, G. C. Hilton, L. R. Vale, and K. W. Lehnert, “Amplification and squeezing of quantum noise with a tunable Josephson metamaterial,” Nature Physics 4, 929–931 (2008).
  * Bergeal  _et al._ [2010] N. Bergeal, F. Schackert, M. Metcalfe, R. Vijay, V. E. Manucharyan, L. Frunzio, D. E. Prober, R. J. Schoelkopf, S. M. Girvin, and M. H. Devoret, “Phase-preserving amplification near the quantum limit with a josephson ring modulator,” Nature 465, 64–68 (2010).
  * Macklin  _et al._ [2015] C. Macklin, K. O’Brien, D. Hover, M. E. Schwartz, V. Bolkhovsky, X. Zhang, W. D. Oliver, and I. Siddiqi, “A near–quantum-limited josephson traveling-wave parametric amplifier,” Science 350, 307–310 (2015).
  * Ranzani  _et al._ [2013] L. Ranzani, L. Spietz, Z. Popovic, and J. Aumentado, “Two-port microwave calibration at millikelvin temperatures,” Review of scientific instruments 84 (2013).
  * Yeh and Anlage [2013] J.-H. Yeh and S. M. Anlage, “In situ broadband cryogenic calibration for two-port superconducting microwave resonators,” Review of Scientific Instruments 84 (2013).
  * Stanley  _et al._ [2022] M. Stanley, S. De Graaf, T. Hönigl-Decrinis, T. Lindström, and N. M. Ridler, “Characterizing scattering parameters of superconducting quantum integrated circuits at milli-kelvin temperatures,” IEEE Access 10, 43376–43386 (2022).
  * Spietz, Schoelkopf, and Pari [2006] L. Spietz, R. J. Schoelkopf, and P. Pari, “Shot noise thermometry down to 10mk,” Applied Physics Letters 89, 183123 (2006).
  * Chang  _et al._ [2016] S.-W. Chang, J. Aumentado, W.-T. Wong, and J. Bardin, “Noise measurement of cryogenic low noise amplifiers using a tunnel-junction shot-noise source,” in  _2016 IEEE MTT-S International Microwave Symposium (IMS)_ (IEEE, 2016) pp. 1–4.
  * Beenakker and Schönenberger [2003] C. Beenakker and C. Schönenberger, “Quantum Shot Noise,” Physics Today 56, 37–42 (2003).
  * Walls and Milburn [2011] D. F. Walls and G. J. Milburn,  _Quantum Optics, 2nd Edition_ (Springer Berlin, Heidelberg, 2011).
  * Friis [1944] H. Friis, “Noise figures of radio receivers,” Proceedings of the IRE 32, 419–422 (1944).
  * Evans and McLeish [1977] G. Evans and C. W. McLeish,  _RF Radiometer Handbook_ (1977) provided by the SAO/NASA Astrophysics Data System.
  * Zobrist  _et al._ [2019] N. Zobrist, B. H. Eom, P. Day, B. A. Mazin, S. R. Meeker, B. Bumble, H. G. LeDuc, G. Coiffard, P. Szypryt, N. Fruitwala, I. Lipartito, and C. Bockstiegel, “Wide-band parametric amplifier readout and resolution of optical microwave kinetic inductance detectors,” Applied Physics Letters 115 (2019), 10.1063/1.5098469, 042601.
  * Malnou  _et al._ [2018] M. Malnou, D. A. Palken, L. R. Vale, G. C. Hilton, and K. W. Lehnert, “Optimal operation of a josephson parametric amplifier for vacuum squeezing,” Phys. Rev. Appl. 9, 044023 (2018).
  * Malnou  _et al._ [2021] M. Malnou, M. Vissers, J. Wheeler, J. Aumentado, J. Hubmayr, J. Ullom, and J. Gao, “Three-wave mixing kinetic inductance traveling-wave amplifier with near-quantum-limited noise performance,” PRX Quantum 2, 010302 (2021).
  * Simbierowicz  _et al._ [2021] S. Simbierowicz, V. Vesterinen, J. Milem, A. Lintunen, M. Oksanen, L. Roschier, L. Grönberg, J. Hassel, D. Gunnarsson, and R. E. Lake, “Characterizing cryogenic amplifiers with a matched temperature-variable noise source,” Review of Scientific Instruments 92 (2021), 10.1063/5.0028951, 034708, https://pubs.aip.org/aip/rsi/article-pdf/doi/10.1063/5.0028951/13865090/034708_1_online.pdf .
  * Spietz  _et al._ [2003] L. Spietz, K. W. Lehnert, I. Siddiqi, and R. J. Schoelkopf, “Primary electronic thermometry using the shot noise of a tunnel junction,” Science 300, 1929–1932 (2003).
  * Cano, Wadefalk, and Gallego-Puyol [2010] J. L. Cano, N. Wadefalk, and J. D. Gallego-Puyol, “Ultra-wideband chip attenuator for precise noise measurements at cryogenic temperatures,” IEEE Transactions on Microwave Theory and Techniques 58, 2504–2510 (2010).
  * White  _et al._ [2015] T. C. White, J. Y. Mutus, I.-C. Hoi, R. Barends, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, J. Kelly, A. Megrant, C. Neill, P. J. J. O’Malley, P. Roushan, D. Sank, A. Vainsencher, J. Wenner, S. Chaudhuri, J. Gao, and J. M. Martinis, “Traveling wave parametric amplifier with josephson junctions using minimal resonator phase matching,” Applied Physics Letters 106, 242601 (2015).
  * Malnou  _et al._ [2022] M. Malnou, J. Aumentado, M. Vissers, J. Wheeler, J. Hubmayr, J. Ullom, and J. Gao, “Performance of a kinetic inductance traveling-wave parametric amplifier at 4 kelvin: Toward an alternative to semiconductor amplifiers,” Phys. Rev. Appl. 17, 044009 (2022).
  * Lecocq  _et al._ [2017] F. Lecocq, L. Ranzani, G. A. Peterson, K. Cicak, R. W. Simmonds, J. D. Teufel, and J. Aumentado, “Nonreciprocal microwave signal processing with a field-programmable josephson amplifier,” Phys. Rev. Appl. 7, 024028 (2017).
  * Caves [1982] C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D 26, 1817–1839 (1982).
  * Boyd [2019] R. W. Boyd,  _Nonlinear Optics_ (Academic press, 2019).
  * Yamamoto and Mukai [1990] Y. Yamamoto and T. Mukai, “Fundamentals of optical amplifiers,” Optical and quantum electronics 21, S1–S14 (1990).
  * Mallet  _et al._ [2011] F. Mallet, M. A. Castellanos-Beltran, H. S. Ku, S. Glancy, E. Knill, K. D. Irwin, G. C. Hilton, L. R. Vale, and K. W. Lehnert, “Quantum state tomography of an itinerant squeezed microwave field,” Phys. Rev. Lett. 106, 220502 (2011).
  * Boutin  _et al._ [2017] S. Boutin, D. M. Toyli, A. V. Venkatramani, A. W. Eddins, I. Siddiqi, and A. Blais, “Effect of higher-order nonlinearities on amplification and squeezing in josephson parametric amplifiers,” Phys. Rev. Appl. 8, 054030 (2017).
  * Remm  _et al._ [2023] A. Remm, S. Krinner, N. Lacroix, C. Hellings, F. m. c. Swiadek, G. J. Norris, C. Eichler, and A. Wallraff, “Intermodulation distortion in a josephson traveling-wave parametric amplifier,” Phys. Rev. Appl. 20, 034027 (2023).
  * O’Brien  _et al._ [2014] K. O’Brien, C. Macklin, I. Siddiqi, and X. Zhang, “Resonant phase matching of josephson junction traveling wave parametric amplifiers,” Phys. Rev. Lett. 113, 157001 (2014).
  * Wustmann and Shumeiko [2013] W. Wustmann and V. Shumeiko, “Parametric resonance in tunable superconducting cavities,” Phys. Rev. B 87, 184501 (2013).
  * Bienfait  _et al._ [2017] A. Bienfait, P. Campagne-Ibarcq, A. H. Kiilerich, X. Zhou, S. Probst, J. J. Pla, T. Schenkel, D. Vion, D. Esteve, J. J. L. Morton, K. Moelmer, and P. Bertet, “Magnetic resonance with squeezed microwaves,” Phys. Rev. X 7, 041011 (2017).
  * Parker  _et al._ [2022] D. J. Parker, M. Savytskyi, W. Vine, A. Laucht, T. Duty, A. Morello, A. L. Grimsmo, and J. J. Pla, “Degenerate parametric amplification via three-wave mixing using kinetic inductance,” Phys. Rev. Appl. 17, 034064 (2022).
  * Peterson [2020] G. A. Peterson,  _Parametric Coupling between Microwaves and Motion in Quantum Circuits: Fundamental Limits and Applications_ , Ph.D. thesis, University of Colorado at Boulder (2020).