]These authors contributed equally to this work.

]These authors contributed equally to this work.

# Radiatively-cooled quantum microwave amplifiers

Mingrui Xu  [  Department of Electrical Engineering, Yale University, New
Haven, Connecticut 06520, USA  Yufeng Wu  [  Department of Electrical
Engineering, Yale University, New Haven, Connecticut 06520, USA  Wei Dai
Department of Applied Physics, Yale University, New Haven, Connecticut 06520,
USA  Hong X. Tang<EMAIL_ADDRESS>Department of Electrical Engineering,
Yale University, New Haven, Connecticut 06520, USA

###### Abstract

Superconducting microwave amplifiers are essential for sensitive signal
readout in superconducting quantum processors. Typically based on Josephson
Junctions, these amplifiers require operation at milli-Kelvin temperatures to
achieve quantum-limited performance. Here we demonstrate a quantum microwave
amplifier that employs radiative cooling to operate at elevated temperatures.
This kinetic-inductance-based parametric amplifier, patterned from a single
layer of high-$T_{\mathrm{c}}$ NbN thin film, maintains a high gain and
meanwhile enables low added noise of 1.3 quanta when operated at 1.5 Kelvin.
Remarkably, this represents only a 0.2 quanta increase compared to the
performance at a base temperature of 0.1 Kelvin. By uplifting the parametric
amplifiers from the mixing chamber without compromising readout efficiency,
this work represents an important step for realizing scalable microwave
quantum technologies.

Superconducting parametric amplifiers are critical components for high-
sensitivity readout of superconducting quantum processors Aumentado (2020). To
achieve fast, high-fidelity qubit readout Jeffrey  _et al._ (2014); Walter
_et al._ (2017a), the first-stage amplifier must operate with near quantum-
limited noise performance Caves (1982). Traditionally, Josephson Junction-
based amplifiers operated at milli-Kelvin temperatures have been used for this
purpose Yurke  _et al._ (1989); Yamamoto  _et al._ (2008); Castellanos-Beltran
_et al._ (2008); Abdo  _et al._ (2011). However, with the increasing number of
superconducting qubits Arute  _et al._ (2019); Kim  _et al._ (2023), the
demand for an increasing number of readout lines poses significant challenges
in terms of space occupation and power consumption at the mixing chamber (MXC)
stage of dilution fridges Meenehan  _et al._ (2014, 2015); Hornibrook  _et
al._ (2015); Midolo  _et al._ (2018). Despite the fact that the amplifier
itself may have a small footprint, the magnetic shield, isolators, and
circulators associated with it still occupy considerable space. Similarly,
while the power required to drive the parametric amplifier may be
insignificant, the attenuators necessary for reducing background noise
contribute to significant power dissipation. These factors collectively pose
challenges to the scalability of microwave quantum technologies.

An ideal superconducting parametric amplifier should maintain quantum-limited
noise regardless of its material temperature. However, current state-of-the-
art parametric amplifiers are typically limited to operation at milli-Kelvin
temperatures due to several reasons. First, the superconducting transition
temperature sets a hard ceiling to the operating temperature of the devices.
Second, the intrinsic loss within the amplifier introduces excessive
fluctuations to the signal by thermalizing the mode temperature to the
material temperature of the device. The latter is further exacerbated by
thermal quasiparticle population as $k_{B}T$ gets comparable to
superconducting bandgap $\hbar\Delta$. Therefore, the use of Josephson-
junction-based amplifiers is restricted to a temperature well below 1K, the
critical temperature of aluminum from which they are fabricated. Hence,
developing microwave parametric amplifiers using materials with high
superconducting transition temperature and high intrinsic quality factor
within the desired temperature range would be preferable.

Kinetic inductance nonlinearity based on single layer NbN, NbTiN, and granular
Al materials has drawn increasing attention in recent years Vissers  _et al._
(2015); Samkharadze  _et al._ (2016); Maleeva  _et al._ (2018). By utilizing a
nanowire design, amplifiers predicated on kinetic inductance have demonstrated
high gain with quantum-limited noise Parker  _et al._ (2022); Xu  _et al._
(2023); Khalifa and Salfi (2023). Traveling-wave amplifiers have also been
demonstrated with excellent performance compared with their Josephson junction
counterparts Ho Eom  _et al._ (2012); Chaudhuri  _et al._ (2017); Esposito
_et al._ (2021); Malnou  _et al._ (2021, 2022). Notably, these materials
possess high critical temperatures, typically around 10 K for NbN and NbTiN
Cheng  _et al._ (2019); Sowa  _et al._ (2017); Malnou  _et al._ (2022). This
characteristic facilitates the operation of the kinetic-inductance traveling-
wave parametric amplifier (KI-TWPA) at 4 K Malnou  _et al._ (2022). As a
result, amplifiers that incorporate these materials can sustain
superconductivity even at higher temperatures.

In this paper, we introduce the nanobridge kinetic-inductance parametric
amplifier (NKPA) Xu  _et al._ (2023) that leverages the radiative-cooling
concept Xu  _et al._ (2020); Wang  _et al._ (2021); Albanese  _et al._ (2020)
to achieve ultralow-added-noise microwave signal amplification. We are the
first to demonstrate that, with radiative cooling in effect, the NKPA achieves
near quantum-limited amplification performance with added noise of
$n_{\mathrm{add}}=1.33\pm 0.04$ at an operating temperature of 1.5 K. These
findings not only solidify our understanding of added noise versus physical
temperature of parametric amplifiers, but also introduce a new operating
regime for superconducting parametric amplifiers. This advancement paves the
way for enhanced scalability in superconducting quantum computing and many
sensing applications.

Figure 1:  (a) Principle of radiatively-cooled amplification. To achieve
radiative cooling for the amplifier installed in a hot enviroment with thermal
occupancy $n_{\mathrm{i}}$, the amplifier is connected to a cooling bath
through external coupling channel with thermal occupancy $n_{\mathrm{e}}$. The
decay rate of the amplifier resonance through intrinsic and external coupling
channels are denoted as $\kappa_{\mathrm{i}}$ and $\kappa_{\mathrm{e}}$,
respectively. (b) Schematic of the experimental setup. The cooling bath at
temperature $T_{\mathrm{src}}$ is implemented as a 30 dB attenuator (near
perfect 50 ohm load) anchored on a variable temperature stage (VTS1). The
amplifier is mounted on a second variable temperature stage (VTS2) that
defines the device material temperature $T_{\mathrm{dev}}$.

In the context of parametric amplification, radiative cooling could help
reduce the added noise of a hot amplifier device ($T_{\rm{dev}}$) by using a
cooling channel connected to a cold thermal bath ($T_{\rm{src}}$), as
illustrated in Fig. 1 (a). The thermal occupancy of the cavity mode
$n_{\mathrm{mode}}$ is related to the thermal occupancy of the physical bath
$n_{\mathrm{i}}$ and cold source $n_{\mathrm{e}}$, each corresponding to
$T_{\rm{dev}}$ and $T_{\rm{src}}$ through Bose–Einstein distribution. The mode
occupancy could be expressed as Xu  _et al._ (2020)

$n_{\mathrm{mode}}=\frac{\kappa_{\mathrm{e}}}{\kappa}n_{\mathrm{e}}+\frac{\kappa_{\mathrm{i}}}{\kappa}n_{\mathrm{i}}$
(1)

[floatfix] where $\kappa_{\mathrm{e}}$ and $\kappa_{\mathrm{i}}$ are the
external coupling rate and the internal loss rate respectively, and
$\kappa=\kappa_{\mathrm{e}}+\kappa_{\mathrm{i}}$ is the total loss rate, and
$n_{\mathrm{i}}$ and $n_{\mathrm{e}}$ are related to $T_{\rm{dev}}$ and
$T_{\rm{src}}$ through Bose-Einstein distribution
$n_{\mathrm{s}}(T)=1/\left(\exp(\hbar\omega_{s}/k_{B}T)-1\right)$. In order to
minimize the thermalization of the mode with its physical environment, our
strategy is to maximize the ratio of the external coupling rate (to the cable
connected to the cold source) over the internal loss rate (to the warmer
physical bath). This radiative cooling scheme Xu  _et al._ (2020); Wang  _et
al._ (2021); Albanese  _et al._ (2020) has been utilized to prepare a
superconducting resonator near its quantum ground state in the presence of
environmental thermal excitations.

Figure 2: Device characterization and amplifier performance with varied device
operating temperature. (a) The magnitude and (b) the phase response of the
NKPA in the linear regime when the device material temperature increases from
0.1 K to 2.7 K. (c) Signal-to-noise ratio improvement with NKPA turned off
(black line) and on (colored lines) in the phase-preserving mode with device
operating temperature from 0.1 K to 2.1 K. The gain for the probe tone,
detuned from the center amplification frequency by 100 kHz, is maintained at
27 dB for different temperatures. The power spectrum, measured with a
resolution bandwidth of 1.8 kHz, is artificially offset to maintain a
consistent power level for the output probe tone. The insert illustrates the
gain profile while the NKPA was functioning as an amplifier.

Our selected amplifier, the NKPA, is patterned from a high $T_{\mathrm{c}}$
NbN film on a silicon substrate. It has previously been demonstrated to be
resilient to magnetic fields of up to 0.5 Tesla Xu  _et al._ (2023). The
device features a nanobridge with a cross-section of $80\,\text{nm}\times
4\,\text{nm}$, which provides a single photon Kerr nonlinearity of
approximately $10\,\text{kHz}$. The resonance frequency of the device is
$7.151\,\text{GHz}$. At base temperature of $0.13\,\text{K}$, the external
coupling rate is found to be around $2\pi\times 65\,\text{MHz}$ from the
reflection spectrum fitting McRae  _et al._ (2020). However, because of the
intentionally designed strong overcoupling, we could not extract the accurate
internal loss rate from the reflection spectrum in the presence of background
ripples in the frequency spectrum as shown in Fig. 1(a). Through radiative
cooling performance reported later, we infer that the internal loss rate is
below $2\pi\times 3$ MHz.

This particular amplifier is mounted on a variable temperature stage (VTS2,
see Fig. 1) consisting of a heater and a calibrated thermometer. VTS2 is
connected to the mixing chamber of a dilution refrigerator through a weak
thermal link made of a stainless steel post, and allows the device operating
temperature to vary from 130 mK to above 3 K while the rest of the mixing
chamber maintains at 50 mK. Another variable temperature stage (VTS1), which
serves as a reference thermal noise source Xu  _et al._ (2023), is implemented
with a 30 dB attenuator mounted on it. The diagram of the setup is shown in
Fig. 1(b). After the NKPA, we use a High Electron Mobility Transistor (HEMT)
amplifier as a second-stage amplifier, with several stages of circulators in
between. The output signal is then monitored by a vector network analyzer
(VNA) and a spectrum analyzer (SA).

By probing the linear reflection spectrum of NKPA while varying the device
operating temperature, it is confirmed that the device maintains very
overcoupled condition from milli-Kelvin temperatures up to 2.7 Kelvin, owing
to the high critical temperature of NbN. No significant frequency shift is
observed until 2 K, as shown in Fig. 2(a) and (b). We use a two-tone drive
scheme to operate the amplifier, with each drive tone detuned by over 105 MHz
from the NKPA’s center frequency. Phase-preserving amplification up to 45 dB
is recorded at up to 1.5 K as discussed in detail in Appendix. Upon turning on
the amplifier in the phase-preserving amplification mode, we observed a 13.47
dB improvement in signal-to-noise ratio using a weak coherent signal as a
reference, shown in Fig. 2(c). As we increase the device’s material
temperature to 1.5 K, the noise floor increased only by 0.74 dB, indicating
the amplifier performance was only marginally affected. The excellent noise
performance is maintained up to 2.1 K, at which point the noise floor
increased more dramatically by 3.55 dB.

###  .1 Added Noise Analysis

To calibrate the amplifier added noise, we sweep the temperature of VTS1 to
generate a reference thermal noise to feed to the NKPA Malnou  _et al._
(2018); Xu  _et al._ (2020). As illustrated in Fig.3 (a), results show the
total added noise of NKPA (referred to the input) is at $1.12\pm 0.03$ quanta
when operated at 130 mK, which is 0.62 quanta above (about twice of) the
quantum limit. The excessive noise likely originates from unaccounted
nonlinear processes in NKPA. Remarkably, the added noise of NKPA does not
significantly increase while the operating temperature of the device increases
to 1.5 K. Even at an operating temperature of 1 K, the added noise remains
below 1.2 quanta.

Figure 3:  Added noise results from noise thermometry calibration as a
function of the NKPA’s operating temperature $T_{\mathrm{dev}}$. (a) The added
noise in quanta referred to the output of the VTS, where the quantum-limited
added noise of an ideal parametric amplifier is at 0.5. (b) The excessive
added noise when NKPA is operated at elevated temperatures compared with the
base temperature (0.13 K). The solid lines represent the expected excessive
added noise level with different external coupling ratios
$\kappa_{\mathrm{e}}/\kappa$.

To further investigate the influence of device operating temperature on the
NKPA added noise, we attempt to understand the excessive added noise of NKPA
as a function of the NKPA operating temperatures. Note that throughout this
experiment, the temperature of the VTS1 remains at 130 mK, so the observed
increase in added noise should be attributed to the resonator’s thermalization
to its material temperature through phonon dissipation. In the high-gain
limit, the excessive noise due to increased device material temperature could
be expressed as:

$\Delta
n_{\mathrm{add}}=2\frac{k_{\mathrm{i}}}{k_{\mathrm{e}}}n_{\mathrm{dev}}=2\left(\frac{\kappa}{k_{\mathrm{e}}}-1\right)n_{\mathrm{dev}}$
(2)

where $\kappa=\kappa_{e}+\kappa_{i}$ represents the resonator linewidth and
$n_{\mathrm{dev}}$ denotes the thermal photon occupation of device material
temperature controlled by VTS2. The results and predicted curves based on
various external coupling ratios $\kappa_{\mathrm{e}}/\kappa$ are shown in
Fig. 3 (b). We found that the data fit relatively well to the predicted
results with $\kappa_{\mathrm{e}}/\kappa=0.98$, indicating very overcoupled
resonance and a decent radiative cooling effect. We can further infer the
internal Q of the resonance of approximately 2000. In this case, we suspect
the internal loss of NKPA is not likely limited by quasiparticles Lutchyn  _et
al._ (2005); Turneaure  _et al._ (1991).

When we further increase the device operating temperature beyond 2 K, the
added noise increases drastically in a fashion that is not explainable by our
linear model, which used both internal and external dissipation rates
reflected in the reflective spectra of NKPA in the linear regime.
Coincidentally, the increase of noise is accompanied by a pronounced shift in
resonance frequency as the temperature increases, similar to the result
reported by Grünhaupt et al. Grünhaupt  _et al._ (2018) from a thin-film
granular Aluminum resonator. Hence, we believe that spurious effects due to
thermal quasiparticles come into play as temperature increases, which results
in a drastic added noise increase to NKPA at around 2K.

###  .2 Prospective application

Figure 4:  The perspective of qubit readout using radiatively cooled NKPA. (a)
Experimental scheme of qubit readout using parametric amplifier at 1-K/4-K
Plate: the qubit is placed on the mixing chamber (MXC), followed by a
circulator to shield against reflected pump power. The NKPA can be installed
on either the 1-K or the bottom of the 4-K plate. A directional coupler is
used to apply the drive power to the NKPA. (b) Theoretical estimation of
readout line noise $n_{\mathrm{sys}}$ referred to the output of Qubit as a
function of the internal coupling ratio $\kappa_{\mathrm{i}}/\kappa$ of
radiatively cooled NKPA. The power transmission coefficient is assumed as
$\alpha_{1}=0.9$ and $\alpha_{2}=0.8$. The gray dashed line indicates the
quantum limit of the system noise ($n_{\mathrm{sys}}=1$) with a phase-
preserving parametric amplifier.

The radiatively-cooled parametric amplifier demonstrated in this study could
potentially address some of the scalability challenges faced by
superconducting quantum computers, particularly the spatial and cooling power
constraints of dilution refrigerators. To better understand the perspective of
using radiatively-cooled microwave amplifiers, in the following, we attempt to
estimate the performance of a readout line incorporating radiatively cooled
NKPA installed at 1-K or 4-K plates. We acknowledge that for the particular
above-mentioned NKPA device discussed in this manuscript, the radiative
cooling model only applies when the operating temperature is below 2 K.
However, a NbTiN-based KI-TWPA is recently reported to exhibit satisfactory
performance operating at a 4 K Malnou  _et al._ (2022), suggesting that the
issue of excessive added noise above 2 K for NKPA device discussed above could
be addressed by improving the material choice and device engineering.

A proposed configuration of utilizing radiatively cooled NKPA for quantum
signal readout is depicted in Fig. 4(a). The primary benefit of shifting the
NKPA and necessary circulators to the 1-K plate is that it liberates precious
refrigeration resources within the mixing chamber. To feed drive power to the
NKPA device, we propose to use a directional coupler thermalized at the 1-K
plate, as opposed to mixing chamber for most junction-based ampliifers. Since
the drive power does not need to be routed through the mixing chamber, its
active heat load to the mixing chamber is thus estimated to be 0.1 $\mu$W per
device. For practical applications such as qubit readout, typically, an
isolator is required after the qubit to prevent any backactions from the
first-stage amplifier. Due to the insertion loss of each component between the
signal source and the amplifier, the signal-to-noise ratio of the entire
output line will be inevitably degraded. Here, we denote the power
transmission coefficient for the readout pulse at the mixing chamber and 1-K
plate as $\alpha_{1}$ and $\alpha_{2}$. Thus, we can express the anticipated
added noise of the system as follows:

$\begin{split}n_{\mathrm{sys}}=&\frac{1}{\alpha_{1}\alpha_{2}}\left[\frac{\kappa_{\mathrm{i}}}{\kappa_{\mathrm{e}}}\left(2n_{\mathrm{s}}(T_{\mathrm{dev}})+\frac{1}{2}\right)+n_{\mathrm{exc}}+1\right]\\\
+&\frac{\kappa}{\kappa_{e}}\left[2\frac{1-\alpha_{1}}{\alpha_{1}}\left(n_{\mathrm{s}}(0.01K)+1\right)+2\frac{1-\alpha_{2}}{\alpha_{1}\alpha_{2}}\left(n_{\mathrm{s}}(1K)+1\right)\right],\end{split}$
(3)

where $n_{\mathrm{exc}}$ denotes the excessive noise of the amplifier. From
this expression, we can evaluate the system noise impacted by the attenuation
of each component. We assume a loss of 0.5 dB ($\alpha_{1}\approx 0.9$) for
the microwave output line in the mixing chamber, and a loss of 1 dB
($\alpha_{2}\approx 0.8$) in the 1 K plate Walter  _et al._ (2017b); Parker
_et al._ (2022); Simbierowicz  _et al._ (2022). To give a realistic
estimation, we retained the excessive noise value from the noise calibration,
$n_{\mathrm{exc}}=0.62$. The simulated system noise referred to the output of
the signal source (e.g. qubit readout pulse) is illustrated in Fig. 4(b) with
device operating temperatures of 1-K and 4-K. Our findings suggest that for
the same level of ratiative cooling effectiveness demonstrated in this work,
i.e. $\kappa_{\mathrm{e}}/\kappa=0.98$ (or $\kappa_{\mathrm{i}}/\kappa=0.02$),
the total output line noise is at 4.6 quanta when the device is at the 1-K
plate. This value marginally increases to 5.1 quanta at the 4-K plate. This
performance is competitive with that of the state-of-the-art Josephson
Traveling-Wave Parametric Amplifiers (JTWPA) operating at the mixing chamber
Macklin  _et al._ (2015); White  _et al._ (2015).

To maintain overcoupling at high temperatures, e.g. 4 K, we believe it would
be helpful to implement NKPA made from superconducting films with higher
$T_{\mathrm{c}}$ to mitigate the TLS and quasiparticle loss McRae  _et al._
(2020). Materials such as NbTiN Yen  _et al._ (1967); Malnou  _et al._ (2022)
or MBE-grown NbN Katzer  _et al._ (2020) can be explored.

In conclusion, we demonstrate a radiatively-cooled superconducting parametric
amplifier that achieves noise performance close to the quantum limit even at
operating temperatures above 1 K. This advance is made possible by employing
the kinetic-inductance nanobridge amplifier technology with
high-$T_{\mathrm{c}}$ NbN films. Our results not only provide valuable
insights into the impact of device material temperature on the excessive noise
observed in parametric amplifiers but also hold tremendous potential for
enabling rapid single-shot qubit readout for large-scale quantum computers by
reducing heat load and space requirements within the mixing chamber.
Ultimately, these findings contribute to enhancing the scalability of
superconducting quantum computing devices.

###  .3 Acknowledgments

The authors would like to thank Professor Michel Devoret, Dr. Gangqiang Liu,
Dr. Alessandro Miano, for useful discussions. We thank Dr. Yong Sun, Dr.
Lauren McCabe, Mr. Kelly Woods, Dr. Michael Rooks, and Dr Sihao Wang for
assistance in device fabrication. We acknowledge funding support from the
Office of Naval Research on the development of nitride-based superconductors
(under Grant No. N00014-20-1-2126). The part of the research that involves
cryogenic instrumentation is supported by the DOE Office of Science, National
Quantum Information Science Research Centers, Co-design Center for Quantum
Advantage (C2QA), Contract No. DE-SC0012704.

##  I Appendix

###  I.1 Added-noise calibration

Figure 5: (a) Noise thermometry calibration setup diagram. The reference
thermal noise sourced from the VTS is preamplified by the NKPA, which operates
at a 27 dB gain, and then further amplified by a high-electron-mobility
transistor (HEMT). The noise power spectrum is measured by a spectrum analyzer
(SA). The cyan line represents superconducting cables. (b) Output noise power
spectrum in quanta as a function of the thermal source temperature
$T_{\mathrm{src}}$.

To calibrate the added noise of the NKPA, we show the noise source and the
detection chain in Fig. 5 (a). The intracavity photon occupancy can be
represented as
$n_{\textrm{mode}}=(\kappa_{\mathrm{i}}n_{\mathrm{i}}+\kappa_{\mathrm{e}}n_{\mathrm{e}})/\kappa$,
incorporating both environmental noise and externally coupled noise. Here,
$n_{\mathrm{i}}$ and $n_{\mathrm{e}}$ are dictated by the Bose-Einstein
distribution
$n_{\mathrm{s}}(T)=1/\left(\exp(\hbar\omega_{s}/k_{B}T)-1\right)$, with
corresponding temperatures being $T_{\mathrm{dev}}$ and $T_{\mathrm{src}}$. In
the phase-preserving amplification mode, the output field is related to the
input signal field $a_{\mathrm{S,mode}}$ and input idler field
$a_{\mathrm{I,mode}}$ as

$a_{\mathrm{out,S}}=\sqrt{G_{\mathrm{N}}}a_{\mathrm{S,mode}}+\sqrt{G_{\mathrm{N}}-1}a_{\mathrm{I,mode}}^{{\dagger}},$
(4)

where $G_{\mathrm{N}}$ is the NKPA gain. As we operate the NKPA in the high-
gain limit, i.e., $G_{\mathrm{N}}\gg 1$, the added noise of the subsequent
amplifiers such as HEMT and Room-temperature amplifier is negligible. With the
assumption that the idler mode occupancy is the same as the signal mode, the
output power spectral density is expressed as

$\begin{split}\frac{S_{\mathrm{out}}}{BW}&=\hbar\omega
G(2n_{\mathrm{mode}}+1),\\\ &=\hbar\omega
G\frac{\kappa_{\mathrm{e}}}{\kappa}(2n_{\mathrm{e}}(T_{\mathrm{src}})+2\frac{\kappa_{\mathrm{i}}}{\kappa_{\mathrm{e}}}n_{\mathrm{i}}(T_{\mathrm{dev}})+\frac{\kappa}{\kappa_{\mathrm{e}}}),\end{split}$
(5)

where $S_{\mathrm{out}}$ is the power spectrum, $BW$ represent resolution
bandwidth of the power spectrum,
$G=G_{\mathrm{R}}G_{\mathrm{H}}G_{\mathrm{N}}$ is the total system gain, and
$G_{\mathrm{R}}$ and $G_{\mathrm{H}}$ correspond to the gain of the room-
temperature (RT) amplifier and HEMT. We fix the device physical temperature
$T_{\mathrm{dev}}=130$ mK, while varying the temperature of the VTS1
$T_{\mathrm{src}}$, during which we take measurements of the power spectrum.
By fitting the normalized power spectrum in quanta
($S_{\mathrm{out}}\kappa/BW\cdot\hbar\omega G\kappa_{\mathrm{e}}$) as a
function of VTS1 temperature $T_{\mathrm{src}}$, we are able to determine the
added noise referred to the amplifier input:

$n_{\mathrm{add}}=2\frac{\kappa_{\mathrm{i}}}{\kappa_{\mathrm{e}}}n_{\mathrm{i}}(T_{\mathrm{dev}})+\frac{\kappa}{\kappa_{\mathrm{e}}}-\frac{1}{2}+n_{\mathrm{exc}},$
(6)

where we subtract the $0.5$ quantum limit of noise quanta accompanying the
input signal, and $n_{\mathrm{exc}}$ is the excess added noise not captured by
the radiatively-cooled amplifier model. The above expression can be rewritten
as

$n_{\mathrm{add}}=\frac{\kappa_{\mathrm{i}}}{\kappa_{\mathrm{e}}}(2n_{\mathrm{i}}(T_{\mathrm{dev}})+1)+0.5+n_{\mathrm{exc}}$
(7)

The base temperature (130 mK) calibration is shown in Fig. 5 (b). Eq. (7)
suggests that with the same intrinsic and external coupling condition, the
incremental added noise due to the increased device physical temperature can
be expressed as

$\Delta
n_{\mathrm{add}}=2\frac{\kappa_{\mathrm{i}}}{\kappa_{\mathrm{e}}}\left[n_{\mathrm{i}}(T_{\mathrm{dev,high}})-n_{\mathrm{i}}(T_{\mathrm{dev,low}})\right].$
(8)

This equation shows that the elevated added noise is linearly dependent on the
difference noise quanta and the internal/external coupling ratio. This
contribution remains small with a very overcoupled device.

###  I.2 Device performance at higher temperatures

Figure 6: Amplifier gain characteristics at 150 mK (solid line) and 1.5 K
(dashed line). Phase-preserving amplification as a function of the signal
frequency detuning from the center frequency of the two pumps. More than 45 dB
gain is observed with -68.90 dBm on-chip pump power at both temperatures.
Figure 7: Device characteristics as a function of device physical temperature.
(a) Added noise in quanta up to 2.12 K. (b) External Q and loaded Q extracted
from the reflection spectrum fitting. (c) Phase-locked loop (PLL) tracking of
frequency shift as a function of device physical temperature
$T_{\mathrm{dev}}$ at 7.1509 GHz. The legend indicates the temperature
sweeping direction.

In the main text, we demonstrate that the amplifier’s added noise remains
quantum-limited up to a physical temperature of 1.5 K. Impressively, the gain
performance does not show any noticeable change, shown in Fig. 6, where we
plot the device gain as a function of detuning under various pump powers. When
we compare the gain performance at 150 mK and 1.5 K, it becomes apparent from
the figure that the gain profiles closely overlap under the same pump power.
This result shows an impressive gain above 45 dB, achieved at both operating
temperatures of 150 mK and 1.5 K.

The device performance, however, becomes less consistent for the temperature
above 2 K, evident from the power spectrum result shown in Fig. 2 in the main
text, where a dramatic increase in the noise floor is observed. This NKPA
performance degradation becomes even more evident with the added noise results
from the NKPA, measured at operating temperatures up to 2.1 K, shown in Fig.
6(a). The added noise rises steeply from 1.3 quanta at 1.5 K to more than 2.8
quanta at 2.1 K.

When the device temperature is increased to above 2 K, we also observe the
deviation of the quality factor (Fig. 7 (b)) and resonance frequency shift
(Fig. 7 (c)). Similar results have been reported in previous literature Zhou
_et al._ (2014); Grünhaupt  _et al._ (2018). Both the frequency shift and the
change of internal Q could be attributed to the temperature-dependent
dielectric constant due to TLSs in surface oxidation Zhou  _et al._ (2014) or
the effect of thermal quasiparticle Grünhaupt  _et al._ (2018). The increase
in the external Q is likely a result of the diminished capacitive coupling
arising from elevated temperatures.

In summary, the concurrent shift in added noise, quality factor, and frequency
around 2 Kelvin indicates the emergence of spurious effects at elevated
temperatures, due to either TLS or thermal quasiparticle. Higher $T_{c}$
superconductor materials such as NbTiN Yen  _et al._ (1967); Malnou  _et al._
(2022) or MBE-grown NbN Katzer  _et al._ (2020) can be explored to mitigate
these effects, and potentially make near-quantum-limited amplifiers that work
at even higher temperatures.

###  I.3 System noise analysis for perspective applications

Figure 8: (a) The output line model for perspective application. The model
includes 3 stages of amplifiers (NKPA, HEMT, and room-temperature (RT)
amplifier), and imperfect transmission coefficients in the mixing chamber and
1-K plate before the amplifers, denoted as $\alpha_{1}$ and $\alpha_{2}$,
respectively. (b) Color contour plot of the system added noise as a function
of attenuations at mixing chamber and 1-K stage. The contour lines show the
system added noise in quanta.

In perspective applications, where radiatively cooled NKPA is used for quantum
signal readout, illustrated in Fig. 4, the additional insertion loss between
the signal source (e.g. the qubit) and the first-stage amplifier could
compromise the readout performance. Here we quantify the total system noise
referred to the signal source at mixing chamber stage. The schematic diagram
of the detection chain is shown in Fig. 8 (a). The input field
$a_{\mathrm{in}}$ from MXC is transmitted to the NKPA installed at 1-K or 4-K
plate, where the signal is being amplified. The signal then proceeds to a HEMT
situated at the 4-K plate before reaching a room-temperature amplifier. We
presume the NKPA’s substantial gain can saturate the added noise from
subsequent amplifiers. Therefore, we neglect added noise from the output line
after NKPA.

We use $\alpha_{1}$ and $\alpha_{2}$ to represent the power transmission
coefficients in the signal line at the MXC and the 1-K or 4-K stage
respectively. In a practical minimal setup, $\alpha_{1}$ should include the
loss of 4 connectors, and an isolator. $\alpha_{2}$ should include the loss of
6 connectors, a directional coupler, and a circulator. The attenuation thus
couples the corresponding thermal fields $h_{1}$ and $h_{2}$ into the output.
Then the input quanta to the NKPA is

$a_{\mathrm{e}}=\sqrt{\alpha_{2}}\left(\sqrt{\alpha_{1}}a_{\mathrm{in}}+\sqrt{1-\alpha_{1}}h_{1}^{\dagger}\right)+\sqrt{1-\alpha_{2}}h_{2}^{\dagger}.$
(9)

where $a_{\mathrm{in}}$ is the input signal at MXC. The equilibrium noise
photon occupation $\langle
h_{\mathrm{1}}^{\dagger}h_{\mathrm{1}}\rangle=n_{\mathrm{s}}\mathrm{(0.01K)}$
and $\langle
h_{\mathrm{2}}^{\dagger}h_{\mathrm{2}}\rangle=n_{\mathrm{s}}\mathrm{(1K)}$.
The NKPA device could be mounted on either the 1-K plate or the backside of
the 4-K plate, as needed by the perspective applications. Therefore, the
intra-cavity photon number of the NKPA can be expressed as

$n_{\mathrm{mode}}=\frac{\kappa_{\mathrm{e}}}{\kappa}\langle
a_{\mathrm{e}}^{\dagger}a_{\mathrm{e}}\rangle+\frac{\kappa_{\mathrm{i}}}{\kappa}n_{\mathrm{s}}(T_{\mathrm{dev}})$
(10)

Plugin Eq. (10) and Eq. (9) into Eq. (5) for both signal and idler modes, we
can derive the system noise as

$\begin{split}n_{\mathrm{sys}}=&\frac{1}{\alpha_{1}\alpha_{2}}\left[\frac{\kappa_{\mathrm{i}}}{\kappa_{\mathrm{e}}}\left(2n_{\mathrm{s}}(T_{\mathrm{dev}})+\frac{1}{2}\right)+n_{\mathrm{exc}}+1\right]\\\
+&\frac{\kappa}{\kappa_{e}}\left[2\frac{1-\alpha_{1}}{\alpha_{1}}\left(n_{\mathrm{s}}(0.01K)+1\right)+2\frac{1-\alpha_{2}}{\alpha_{1}\alpha_{2}}\left(n_{\mathrm{s}}(1K)+1\right)\right]\end{split}$
(11)

The terms inside the first square bracket represent the amplifier noise, with
a prefactor of $1/\alpha_{1}\alpha_{2}$ meaning it is effectively amplified
due to attenuated signal. The two terms inside the second square bracket are
the coupled environment noise from MXC and 1-K plate due to loss. With
$\alpha_{1}\rightarrow 1$ and $\alpha_{2}\rightarrow 1$, Eq. 11 reduce to Eq.
7 plus 0.5 to account for the vacuum noise accompanied by the signal.

In figure 8b, we illustrate the system noise as a function of attenuations in
MXC and 1-K plate ($1-\alpha_{1}$ and $1-\alpha_{2}$), while maintaining a
fixed ration between coupling rates $\kappa_{\mathrm{e}}/\kappa=0.98$. The
dense contour line along the y-axis implies a higher sensitivity of system
noise to the loss at the 1-K plate. The system noise remains under 6 quanta if
the losses at MXC and 1-K are each below 1 dB ($1-\alpha_{1}<0.1$,
$1-\alpha_{2}<0.1$). If both attenuations reach 1.5 dB, the system noise would
exceed 8 quanta.

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#  High-$z$ Sudoku: A diagnostic tool for identifying robust (sub)mm redshifts

Tom J. L. C. Bakx1,2 and Helmut Dannerbauer3,4
1 Division of Particle and Astrophysical Science, Graduate School of Science,
Nagoya University, Aichi 464-8602, Japan.
2 National Astronomical Observatory of Japan, 2-21-1, Osawa, Mitaka, Tokyo
181-8588, Japan.
3 Instituto Astrofísica de Canarias (IAC), E-38205 La Laguna, Tenerife, Spain.
4 Universidad de la Laguna, Dpto. Astrofísica, E-38206 La Laguna, Tenerife,
Spain E-mail<EMAIL_ADDRESS>(Nagoya University)

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We present methods to (i) graphically identify robust redshifts using emission
lines in the (sub)mm regime, (ii) evaluate the capabilities of different
(sub)mm practices for measuring spectroscopic redshifts, and (iii) optimise
future (sub)mm observations towards increasing the fraction of robust
redshifts. Using this publicly-available code
(https://github.com/tjlcbakx/redshift-search-graphs), we discuss scenarios
where robust redshifts can be identified using both single- and multiple-line
detections, as well as scenarios where the redshift remains ambiguous, even
after the detection of multiple lines. Using the redshift distribution of
(sub)mm samples, we quantify the efficiencies of various practices for
measuring spectroscopic redshifts, including interferometers, as well as
existing and future instruments specifically designed for redshift searches.
Finally, we provide a method to optimise the observation strategy for future
(sub)mm spectroscopic redshift searches with the Atacama Large
Millimetre/submillimetre Array, where 2 mm proves indispensable for robust
redshifts in the $z=2-4$ region.

###### keywords:

methods: observational – techniques: spectroscopic – infrared: galaxies –
galaxies: high-redshift

††pubyear: 2021††pagerange: High-$z$ Sudoku: A diagnostic tool for identifying
robust (sub)mm redshifts–High-$z$ Sudoku: A diagnostic tool for identifying
robust (sub)mm redshifts

##  1 Introduction

Since their discovery in the 1990’s (e.g., Smail et al., 1997; Hughes et al.,
1998), the extreme star-formation rates of sub-millimetre galaxies (SMGs) have
been difficult to square with our picture of galaxy evolution (see e.g., Blain
et al., 2002; Casey et al., 2014). Their strong emission in sub-mm wavelengths
implied star-formation rates in excess of 1000 M⊙/yr, approaching or exceeding
the limits of stable galaxy systems through excessive feedback (Rowan-Robinson
et al., 2016), while their strong dust-obscuration poses serious challenges to
large-scale unbiased follow-up attempts across all wavelengths (Ivison et al.,
2016; Casey et al., 2018; Donevski et al., 2018; Duivenvoorden et al., 2018;
Bakx et al., 2020a; Reuter et al., 2020). Such large-scale unbiased follow-up
is crucial, as studies find SMGs to consist of a diverse population (e.g.,
Fardal et al., 2001; Baugh et al., 2005; Lacey et al., 2008; González et al.,
2011; Narayanan et al., 2015).

A first step towards such follow-up requires precise and robust redshift
measurements. Unfortunately, photometric redshifts from sub-mm colours remain
only indicative of a redshift range (e.g., Casey et al., 2012; Pearson et al.,
2013; Ivison et al., 2016; Bakx et al., 2018; Casey, 2020) due to the
degeneracy between dust temperature and redshift (Blain et al., 2003).
Redshifts derived from spectroscopic lines break this degeneracy. The first
redshift searches used optical spectral lines, however they struggled with the
positional uncertainty of SMGs and with a limited bandwidth. Radio
observations improved the positions, but provide a strong bias towards lower
redshifts, and are not workable for large, complete redshift samples (e.g.,
Chapman et al., 2005). Instead, redshift searches using (sub)millimetre
spectroscopy are unhindered by dust extinction and can be related
unambiguously to the (sub)millimetre source, offering a far better alternative
to optical spectroscopy and to the imprecise sub-mm photometric method (e.g.,
Weiß et al., 2009). The (sub)mm spectroscopic method has only become
competitive with the increased bandwidths of the receivers operating at mm and
sub-mm facilities, as for example demonstrated by Vieira et al. (2013) and
Weiß et al. (2013), who presented the first redshift survey for 23 strongly-
lensed dusty star-forming galaxies selected from the South Pole Telescope
(SPT) 2500 deg2 survey using the Atacama Large Millimetre/submillimetre Array
(ALMA) at 3 mm.

Figure 1: The graphical method reveals the full redshift space (x-axis) probed
by the frequency coverage (y-axis) of the (sub)mm observations. Assuming the
observed bandwidths (1., shown on the left-hand side, see Sec. 3.4), the
filled regions show redshifts where single and multiple lines would be
detected (2., orange and blue fill, resp.). The horizontal lines (3.) mark the
observed lines, indicating the potential redshift solutions where they cross
the observable lines (4., circles) with the fill matching the underlying
colour. The top arrows (5.) indicate the position of the potential redshift
solution of each line, where the arrows line up in the case of a multi-line
redshift. The photometric redshift (black errorbar) can help guide the
redshift solution. In this example, the two observed CO line transitions,
CO(3-2) and CO(5-4), result in a robust redshift identification at $z=2.7$,
since we can exclude all but one other solution where the two arrows do not
line up. This other solution at $z\approx 6.4$ can be excluded through the
lack of emission at 109.040 and 140.195 GHz.  Note that the horizontal lines
at $\sim 101$ GHz and $\sim 151$ GHz are small (0.5 GHz) gaps in the frequency
coverage.

The large uncertainties of photometric redshifts and the lack of discerning
spectral features of (sub)mm spectral lines (i.e., unlike the resonant,
a-symmetric emission from Ly$\alpha$) complicate the spectroscopic redshift
determination. Instead, the process of robust redshift determination at
(sub)mm wavelengths, in particular using rotational emission lines from
carbon-monoxide (CO), involves the careful comparison between the potential
redshift solutions of individual lines, and then selecting the solution where
multiple lines agree. This method is far from ideal, as it leads to
confirmation bias for solutions close to the $z_{\rm phot}$ (Yanai & Lercher,
2020), and fails to draw attention to redshift degeneracies. Similarly,
sometimes redshift solutions can be excluded if we fail to detect additional
spectral lines within the bandwidth. In this paper, we expand on the graphical
method from Bakx et al. (2020c) which provides a more user-friendly method for
identifying robust redshifts. This method is capable of highlighting redshift
degeneracies, and enables the easy removal of untenable redshift solutions,
similar to graphical-numerical puzzles, such as the Sudoku111While invented in
the USA, the late Maki Kaji (鍜治 真起) popularized the game as Sūji wa dokushin
ni kagiru (数字は独身に限る), roughly translating to the digits must be single.
puzzle. We describe the method in detail in Section 2, where we explain
different scenarios and note important caveats. In Section 3, we quantify the
efficiency of various methods for finding spectroscopic redshifts, and provide
a method for finding the optimum tunings for future redshift searches in
Subsection 3.4. The python-based code and example scripts for each of the
graphs in this paper are publicly available at
https://github.com/tjlcbakx/redshift-search-graphs, and can be easily and
swiftly adapted for the reader’s own needs.

##  2 Graphical method

Robust redshifts require the exclusion of any other potential redshift
interpretations of the line emission except one. Numerically solving this
cross-comparison is a tedious approach, prone to errors and overlooked
options. Instead, we use the following graphical method for identifying the
potential spectroscopic redshifts for each line. We further detail several
redshift identification examples, and list several caveats with this method
specifically and redshift searches in general.

###  2.1 Explanation of the graphical method

In Figure 1, we show this graphical method for a fiducial $z_{\rm phot}=3$
example source observed with an efficient ALMA tuning setup (see Section 3.4).
Here, the observed frequency range (y-axis) is shown against the redshift
range (x-axis) using the following approach:

  1. 1

Observed bandwidths: Indicate the observed bandwidths, in this case shown as
both the LSB and USB of the ALMA tunings.

  2. 2

Single- and multiple-line detection: The observation is expected to detect the
CO-lines (black lines), and might detect the [C i] and H2O lines (dashed blue
lines; see Section 2.3.2). Orange and blue fill indicate where the
observations would be able to observe one or more CO spectral lines,
respectively.

  3. 3

Detected spectral lines: Each actually-observed spectral line is indicated
with a horizontal line at the observed frequency.

  4. 4

Potential line solutions: Where the observed lines cross the redshifted
spectral lines, a circle marks the potential spectroscopic redshift, with the
circle’s fill representing the underlying colour. At the top of the graph,
arrows indicate each potential spectroscopic redshift solution.

  5. 5

Robust redshift solution: The top arrows line up when multiple lines agree on
a robust redshift, identifying the spectroscopic redshift.

This final check is important, because CO-line redshift surveys can suffer
from potential multiple redshift solutions (see Section 2.3.1 and Figure 4),
since the CO-ladder increases linearly with each transition Jup. For example,
the observation of Jup = 2 and 4 CO-line transitions of a z = 2 galaxy (76.7
and 153 GHz, resp.) could also be interpreted as the Jup = 3 and 6 CO-line
transitions of a z $\approx$ 3.5 galaxy. This mis-identification is a non-
negligible possiblity given the large 13% uncertainties in sub-mm derived
photometric redshifts of around $\Delta z$ $\sim$ 1 (defined as $\Delta
z/(1+z)$, e.g. Pearson et al. 2013; Ivison et al. 2016; Bakx et al. 2018),
although Bendo et al. in prep. finds a sharp reduction in uncertainty down to
$\sim 7$%.

###  2.2 Robust redshifts

####  2.2.1 Robust redshifts from CO lines

Figure 1 shows the derivation of a spectroscopic redshift of $z_{\rm
spec}=2.7$, with two lines at 93.463 and 155.772 GHz, corresponding to CO(3-2)
and CO(5-4) respectively. A high-$z$ redshift solution ($z_{\rm spec}=6.4$)
exists because of the linear spacing between CO line transitions and their
isotopologues, but this solution would require additional lines at 109.04 and
140.195 GHz. The lack of such emission can be used to exclude this redshift
possibility.

####  2.2.2 Robust redshifts from CO and ancillary lines

Figure 2: Same as Fig. 1, showing a robust redshift identification at $z_{\rm
spec}=2.5$ using the efficient ALMA configuration through a combination of CO
and [C i] emission (98.804 and 140.607 GHz resp.). Because non-CO lines do not
follow the linear scaling of CO emission lines, any combination of a redshift
identification using CO and such additional lines is robust, since it removes
the possibility of inconclusive redshifts due to degeneracy in the CO
transitions.

Non-CO lines do not scale linearly with CO line emission, and thus always
offer a robust derivation of the spectroscopic redshift. These lines are
typically fainter than the CO emission, however they do offer the ability of
near-guaranteed robust redshift identifications (see Sec. 2.3.2). This is
because the high fidelity of the central frequency (typically in excess of
$f/\Delta f$ $\approx 10^{2}-10^{4}$) leaves no room for multiple
interpretations. In Figure 2, we show this scenario for a fiducial $z_{\rm
spec}=2.45$ galaxy with observed CO(3-2) and CI(1-0) at 98.804 and 140.607 GHz
respectively.

####  2.2.3 Robust redshifts from single-line detections

Figure 3: Same as Fig. 1, showing a robust redshift at $z=3.0$ using the
efficient ALMA configuration through just a single spectral line. Here, we
require that the single spectral line is observed at high significance
($>$8$\sigma$), with no detectable emission at any of the potential line
solutions. The low-redshift solution can often times be excluded through
optical or near-infrared follow-up (see Section 2.3.2), and equation 2 tells
us the $z=0.6$ solution is 4.5$\sigma$ [$\approx(5-2)/(0.13*5)$] removed using
just the sub-mm $z_{\rm phot}$.

These graphs allow us to visualize the bandwidth, and associated redshift
possibilities that our observations have covered. As shown in Figure 3, if we
detect a convincingly-bright line (see Section 2.3.2) at 144.089 GHz using our
efficient ALMA tuning (see Section 3.4), we can exclude most neighbouring
redshift solutions in the same vein as the exclusion of the $z_{\rm spec}=6.4$
solution in Fig. 1. We would have detected lines at other frequencies for all
but the lowest-redshift solution ($z_{\rm spec}=0.6$). Using the photometric
redshift as a guide, we can exclude this possibility, leaving only $z_{\rm
spec}=3.0$ as a solution, with the detected line being CO(5-4). We note that
the $z_{\rm spec}=2.41$ solution (where the detected line would be [C i])
would still require a CO line to be detected at 101.251 GHz.

###  2.3 Caveats

####  2.3.1 Inconclusive redshifts from multiple CO lines

Figure 4: Same as Fig. 1, showing an inconclusive redshift from multiple
detected CO lines. The detected lines at 107.229 and 160.844 GHz can
correspond equally-well to $z_{\rm spec}=1.15$ or 3.30. Such scenarios appear
when a single scalar multiplied with both CO line transitions results in other
integer possibilities.

Unfortunately, not all multi-line solutions can be excluded through the non-
detection of other lines (such as in Fig. 1). We show such a scenario in
Figure 4, where two redshift solutions of two lines at 107.229 and 160.844 GHz
can correspond to $z_{\rm spec}=1.15$ or 3.30. The low-redshift scenario would
detect CO(2-1) and CO(3-2), while the higher-redshift solution involves
CO(4-3) and CO(6-5). Such scenarios occur when the ratio of the upper
transition (Jup) of both CO line transitions can be multiplied to form another
set of integer numbers. In this case, the common multiplier is two, however
extensive possibilities exist. We can quantify the change in redshift between
a scenario where we either detected CO($a$, $a-1$) for one redshift solution
or CO($b$, $b-1$) for the other redshift solution as follows

$\frac{\Delta z}{1+z}=\frac{b-a}{a}.$ (1)

The 1$\sigma$ uncertainty in $\Delta z/(1+z)$ is typically on the order of 13
% (e.g., Pearson et al., 2013, although this is shown to decrease to only
$\sim 7$% with deep continuum detections at 2 mm by Bendo et al. in prep.). In
case the photometric redshift of a line agrees with a solution of CO($a$,
$a-1$), the photometric redshift can thus identify how many standard-
deviations ($N$) we are offset from a solution at CO($b$, $b-1$) using

$N[\sigma]\approx\frac{b-a}{a\times{}0.13}.$ (2)

In case of multiple detected lines, we can identify the nearest combination of
lines that can be confused for one-another. Mathematically, this multiplier
can be identified using the greatest common divisor, the $GCD$. The nearest
solutions are the multiplication of the remaining components with $GCD\pm 1$.
In the case of a tentative redshift based on CO(4-3) and CO(6-5), the largest
common multiplier of the CO transitions (J${}_{\rm up}=4$ and 6, resp.) is 2,
leaving additional redshift solutions using CO(2-1) and CO(3-2), as well as
CO(6-5) and CO(9-8), typically offset by $\sim 3.8\sigma$
[$\approx(6-3)/(0.13\times{}6)$]. Note that equation 1 and 2 do not differ on
whether one takes the lower- or higher-frequency line, nor the lower- or
higher-redshift possibility.

####  2.3.2 Excluding redshift regimes

We can exclude redshift regions where we would have seen lines in a deep
enough survey. These are typically restricted to CO lines, since the
correlation between CO lines and [C i] emission strongly depends on the
particular ISM conditions (see e.g., Valentino et al. 2018). CO lines
transitions, instead, are strongly correlated to one-another through
transitional partition functions following so-called spectral line energy
distributions (SLEDs). The Milky Way CO SLED has a sharp drop-off in velocity-
integrated fluxes for Jup transitions greater than 3, however for star-
bursting galaxies, these SLEDs can approach the maximum thermalized profile,
where velocity-integrated fluxes increase with $J^{2}$ (see Carilli & Walter
2013).

In the case of Fig. 1, we know the undetected CO lines should be at least on
the same order of brightness as the detected lines, as their transitions are
in between the detected emission lines. For the single-line case in Fig. 3,
the $z_{\rm spec}=2.2$ case would have, in the most unfortunate case, a
CO(3-2) line with an integrated flux of $\sim 56$% (= 32/42) fainter than the
detected line. Similarly, for the higher-redshift solution at $z_{\rm
spec}=3.8$, the CO(5-4) would be $\sim 70$% (= 52/62) as bright as the
detected line.  We show this analysis in graphical form in Figure 5.

Figure 5: We compare the CO SLEDs of the alternative multi-line redshift
solutions shown in Figure 3, with the lower-redshift $z=2.2$ case in the top
panel and the higher-redshift $z=3.8$ case in the bottom panel. These graphs
inform us whether we can exclude redshift possibilities based on the upper-
limits from deep observations. In the lower-redshift scenario, we would have
detected CO(4-3) at 8$\sigma$ and not detected CO(3-2) with a similar
observation depth. In the higher-redshift scenario, we would have detected
CO(6-5) at 8$\sigma$ and not detected CO(4-3). We compare against various
SLEDs. The CO SLEDs from Harrington et al. (2021) in grey fill represents the
average SLED from bright Planck-selected galaxies. The Milky Way inner disk CO
SLED from Fixsen et al. (1996) represents a typical low-excitation CO SLED.
Finally, the thermalized CO SLED represents the typical maximum expected
excitation. In both scenarios, we find the 3$\sigma$ upper limit (grey marker)
below the thermalized profile. This indicates that we can exclude these
redshift scenarios, as even a thermalized profile would not be able to produce
the flux observed at the higher-J side of the SLED.

Taking into account measurement errors, as well as this line ratio, the
exclusion method thus requires relatively-high signal-to-noise ($>8\sigma$)
detections to be robust. In Figure 5 we show that such a 8$\sigma$ detection
of a single spectral line is enough to exclude the potential redshift
solution. We show the CO SLED of typical bright DSFGs from Planck-selected
sources (Harrington et al., 2021), a CO SLED from the Milky Way (Fixsen et
al., 1996), and the maximum expected excitation from a thermalized profile. We
choose the Planck-selected CO SLED because it traces a large range of CO-
transtions, from CO(1-0) to CO(12-11), while still being broadly
representative of the population of DFSGs. The Milky Way CO SLED represents
typical low-excitation gas, with a steep drop-off, while the thermalized
profile reflects the maximum expected excitation. For both the high- and low-
redshift possibilities, the upper-limit lies below the thermalized profile. As
such, no typical SLED would be able to produce both a non-detection at a
low-J, and a detection at a higher-J CO line.

####  2.3.3 Disqualifying low-redshift interlopers

As a general rule, redshift deserts become more prevalent at lower redshifts
(see the discussion in Section 3). Observed frequencies of lines lie far
apart, and relatively-narrow bandwidths cover only small $\Delta z$. For
example, at $z=0.5$, CO(1-0) is currently not accessible with ALMA and CO(3-2)
requires band 6 ($\sim 345$ GHz) observations, while at $z=4$ both CO(4-3) and
CO(5-4) are covered in band 3 (3 mm; $\sim 100$GHz). As such, low-redshift
interlopers can only be excluded directly using large observed bandwidths.
Another solution involves multi-wavelength observations, which become capable
of detecting even heavily dust-obscured galaxies themselves and provide much
more accurate photometric redshifts through optical and near-infrared spectral
template fitting.

##  3 Redshift searches

Table 1: Instruments used for redshift searches

Instrument | Observing bandwidth
---|---
| [GHz] | / | [mm]
ALMA: |  $89.1-112.0$ & | / |  $3.37-2.68$ &
B3 & B4 | $139.9-162.7$ | $2.14-1.84$
Spectral scan: |  $84.2-114.9$ & | / |  $3.56-2.61$ &
B3 & B4 | $125-159.25$ | $2.40-1.88$
ALMA B3 fill | $84.2-114.9$ | / | $3.56-2.61$
RSR | $73-111$ | / | $4.11-2.70$
Zpectrometer | $25.6-37.7$ | / | $11.7-7.9$
Z-Spec | $185-305$ | / | $1.62-0.98$
DESHIMA | $220-440$ | / | $1.36-0.68$
Superspec | $200-300$ | / | $1.50-1.00$
OST (Band 6) | $509-893$ | / | $0.59-0.34$

Notes: The top two rows together show the optimised tuning from Figure 8. The
third and fourth rows together show the spectral scan set-up, assuming five
stacked tunings at both 3 and 2 mm.  Note that there exist a small gap in
frequency coverage ($\sim 0.5$ GHz) around $\sim 101$ GHz and $\sim 151$ GHz
in the optimum ALMA Band 3 & 4 configuration.

The graphical method facilitates the analysis once the observations are done.
This section complements this by providing a figure of merit for past,
existing and future redshift searches (Sections 3.1, 3.2, and 3.3,
respectively). This method can then be used to optimise future redshift
searches (see Section 3.4), and guide the design of instruments for future
redshift surveys.

Figure 6 shows the probability of a robust redshift of various configurations
and instruments that are often used for redshift searches between $z=0$ and 8,
and for two different DSFG samples (HerBS; Bakx et al. 2018; Bakx et al.
2020b, South Pole Telescope; Reuter et al. 2020 , see Fig. 7). The average
redshift of the Herschel-selected HerBS sample is $\bar{z}\approx 2.5$, while
the SPT sample is selected at longer wavelengths and has an average
$\bar{z}=3.5$. Redshift searches on other DSFG samples, such as ALESS (da
Cunha et al., 2015) or the Ultrared sources (Ivison et al., 2016), can then be
scaled accordingly. The distributions are smoothed by $dz\approx 0.5$ to
prevent small-number variations in the final efficiency metrics. The details
of each instrument are shown in Table 1. For the calculation in Figure 6, we
remove regions that are opaque under normal observing conditions (assuming
precipitable water vapour $>0.5$ mm, with transmission greater than 80% of the
peak). We show the total number of sources detected robustly with multiple
lines and with single lines in blue fill and hatches, respectively. The orange
fill and hatches show where the redshift remained ambiguous with one or two
lines, while the white region provides a measure of the redshift desert.  We
define a robust detection by either a detection in CO where all adjacent
solutions within 5$\sigma$ (eq. 2) can be excluded, or through the detection
of a non-CO line.

Figure 6: The redshift identification probability as a function of redshift
between $z=0$ and 8, and for smoothed redshift distributions based on two
samples (i.e., HerBS; Bakx et al. 2018; Bakx et al. 2020b, SPT; Reuter et al.
2020). Orange bars indicate the fraction of sources for which one would detect
a single spectral line, while blue bars indicate the fraction where two or
more spectral lines are detected. Hatched blue fill indicates the cases where
one can identify the redshift robustly with even a single spectral line, while
hatched orange fill indicates the situation where redshift degeneracies remain
down to a 5$\sigma$ uncertainty in $z_{\rm phot}$. The grey dotted lines
indicate the robust (single + multiple lines) redshifts when one expects to
detect [C i], and the black dashed lines indicate the robust (single +
multiple lines) if we only assume the improved $z_{\rm phot}$ uncertainties
shown in Bendo et al. in prep. The top panels indicate the redshift
capabilities of various ALMA set-ups, showing an optimized Band 3 & 4 set-up
(see Table 1 and Figure 8), a linescan Band 3 & 4 set-up, and the filled Band
3 tuning from Weiß et al. (2013). The middle panels show known redshift
receivers (RSR; Erickson et al. 2007, Z-Spec; Naylor et al. 2003,
Zpectrometer; Harris et al. 2007), and the bottom panels show the redshift
capabilities of future instruments that are currently under development
(DESHIMA 2.0; Taniguchi et al. 2021; Rybak et al. 2021, SuperSpec; Redford et
al. 2021, Origin Space Telescope (band 6); Bradford et al. 2021).  Figure 7:
The redshift distribution of HerBS (Bakx et al. 2018; Bakx et al. 2020b) and
the South Pole Telescope (Reuter et al. 2020) are used to estimate the
efficiency of various observing strategies, shown in dashed lines. We smooth
the reported redshift distribution by around $dz\approx 0.5$ (shown in filled
histograms) in order to avoid any small-number variations in our final
efficiency estimations. Each distribution is normalized to a total area of 1.

###  3.1 Interferometric redshift searches

Interferometers are important tools for finding spectroscopic redshifts (e.g.,
Vieira et al., 2013; Weiß et al., 2013; Neri et al., 2020; Chen et al., 2021;
Urquhart et al., 2022). By combining multiple spectral windows,
interferometers can cover large redshift ranges while delivering high-
resolution imaging of both line and continuum emission. However, the telescope
array with the largest collecting area, ALMA, only offers a modest spectral
coverage (up to $\sim$7.75 GHz per tuning). While the necessary integration
time to detect a line from, for example, a bright Herschel source is only
several minutes, an ALMA-based redshift search could thus result in large
overheads ($\sim 30$ min.) due to the need to switch tunings. Generally, there
are two ways around the large overheads. Firstly, ALMA offers the ability to
observe sources close on the sky (typically $\sim 10$ degrees of separation)
without the need to recalibrate. Observations can then be carried out in
batches of sources, reducing the overheads per source (see e.g., Urquhart et
al., 2022). Secondly, ALMA recently started to
offer222https://almascience.eso.org/proposing/proposers-guide spectral scans
that calibrate all spectral windows on the calibrator, cutting down
significantly on slewing overheads, which can amount to half or more of the
total observing time. If the targeted number of sources within a 10 deg. field
is modest (typically $\lesssim 3$), the spectral scan is preferred. Above
this, the observation in batches outperforms the spectral scan.

####  3.1.1 Filling the 3 mm band

The initial observations of SPT sources using the full 3 mm band of ALMA
successfully demonstrated the interferometer’s redshift search capabilities
(Vieira et al., 2013; Weiß et al., 2013; Strandet et al., 2016; Reuter et al.,
2020). These observations use five tunings with ALMA, starting at 84.2 GHz up
to 114.9 GHz with overlapping coverage of around one tuning between
$96.2-102.8$ GHz. Figure 6 shows that, while these observations are able to
guarantee a single-line detection above $z>2$, this tuning provides only
moderate ability to derive robust, unambiguous redshifts on the CO emission
alone. For this, 2 mm observations are needed (see e.g., Bakx et al. 2020c).
[C i] and H2O emission, as well as improved photometric redshifts can vastly
improve the ability of this configuration to estimate the redshift,
particularly for samples biased towards high redshifts. Improved uncertainties
in the photometric redshift (black dashed lines) do not improve the redshift
capabilities significantly. Meanwhile, these observations target the CO
emission up to the CO(6-5) rotational transition (with one exception up to
CO(7-6); Marrone et al. 2018). It is good to note that these emission lines
correlate with the far-infrared luminosity – which is often a direct selection
function for high-redshift DSFGs – and these observations are thus less likely
to bias observations. Instead, observations aiming at higher-J transitions
might be particularly-sensitive to the warmer and denser molecular gas
typically found in AGN (e.g., Xu et al. 2014).

####  3.1.2 Combined 3 and 2 mm tunings

In Section 3.4 and Figure 8 we describe how to find the optimum tuning for
spectroscopic redshifts using ALMA. Figure 6 shows the effect of including
band 4 (2 mm). While there exists some regions where multiple lines fail to
provide robust redshifts assuming the Herschel-based photometry (e.g., Pearson
et al., 2013), the refined photometric redshifts from the 2 mm continuum
(black dashed lines) result in high ($>70$%) robust redshifts beyond $z>2$, at
the limited cost of including an additional tuning ($\sim 20$% longer
observation). The benefit is particularly stark at the low-redshift end of the
galaxy distribution, without needing to rely on the unpredictable [C i] or H2O
lines in redshift searches. These results are in line with the impressive
redshift search efficiency shown in Urquhart et al. (2022) (as well as in
subsequent follow-up; P.I.: Bakx, 2019.2.00155.S), showing a robust redshift
ratio of 72%, and even promising up to 93% with deeper observations.

####  3.1.3 ALMA spectral scans

Recent development at ALMA allowed for efficient observations of multiple (up
to five) spectral windows for a single source. By calibrating all spectral
windows in one go, this observing strategy cuts down significantly on the
overheads. We note here that a scan of just band 3 results in similar redshift
coverage issues as seen in Figure 6, as well as the large overlapping coverage
seen in e.g., Weiß et al. (2013). The coverage issue, however, can be
mitigated by including a spectral scan in band 4, which vastly improves the
redshift coverage across a wide range of redshifts. If this is executed as a
spectral scan, the overheads will be only relatively minor, however this
method will be difficult to execute using batched observations similar to
Urquhart et al. (2022), requiring a doubling in observation time relative to
the 3 mm band method.

####  3.1.4 NOEMA redshift searches

Unlike ALMA, NOrthern Extended Millimeter Array (NOEMA; e.g., Neri et al.
2020) and the IRAM 30m (e.g., Bakx et al. 2020c) offer the ability to adjust
the observing frequencies dynamically, based on observed spectral lines in
other tunings. This is a significant advantage when looking for redshifts,
further aided by the wider bandwidths (up to $\sim 16$ GHz in one tuning, with
possible expansion in the future) that allow for a wide 3 mm scan in just two
tunings. In the 3 mm window, this would amount to an observation similar to
the 3 mm scan in Figure 6, where one typically requires additional
observations in the 2 mm band. Follow-up observations in the 2 mm band do not
lend themselves for easy generalization, which is why we were not able to
generate a similar plot for a NOEMA 3- and 2 mm redshift search, although
simply covering the 3- and 2 mm windows similar to the Band 3 & 4 linescan
suggest a near-complete redshift coverage of the HerBS and SPT samples alike
using four tunings. Nevertheless, the graphical method of this paper (i.e.,
Fig. 1) can offer significant aid in determining additional observations at
both 2 mm and other bands, while also highlighting redshift degeneracies.

###  3.2 Dedicated redshift receivers

The advent of spectrometers with instantaneous wide-band enabled the first
large-scale redshift surveys. While the first (sub)mm redshift searches
involved heterodyne instruments that require multiple tunings (e.g., Weiß et
al. 2009), the notable instruments that performed these dedicated redshift
searches are the Redshift Search Receiver (RSR; Erickson et al. 2007), Z-Spec
(Naylor et al., 2003), and Zpectrometer (Harris et al., 2007) on single-dish
telescopes.

####  3.2.1 Redshift Search Receiver

The Large Millimeter Telescope (LMT)-based observations with the RSR (Erickson
et al., 2007) combine the instantaneous coverage of the 3 mm window with the
powerful 32 to 50 m diameter of the telescope to reveal the spectral lines of
galaxies out to high redshifts (e.g., Zavala et al. 2015, 2018). At lower
redshifts particularly, these observations require ancillary data – such as
with IRAM or $8-10$ m telescopes – to confirm the redshifts robustly. The
addition of a 2 mm spectroscopic ability at the LMT would significantly
increase the telescope’s ability to determine redshifts robustly (e.g., Kawabe
et al. in prep).

####  3.2.2 Z-Spec

The wide bandwidth capability of Z-Spec (Naylor et al., 2003) at 1.0 to 1.6 mm
enabled the near-guaranteed ability to detect more than one spectral line. As
seen in Lupu et al. (2012), the higher-frequency transitions (often of CO
lines) are not guaranteed. In this case, clever stacking of the line emission
as a function of redshift enabled robust redshifts even though individual
lines might be detected only at a few sigma. It is good to note that, unlike
the RSR and Zpectrometer, the diameter of the Caltech Submillimeter
Observatory (CSO) where the telescope operated was modest at 10 m, since high
surface accuracy was needed given the higher-frequency observations.

####  3.2.3 Zpectrometer

The Zpectrometer receiver (Harris et al., 2007) operated on the 100 m Green
Bank Telescope, and principally targeted CO(1-0) and CO(2-1). This large
collecting area is crucial, since the expected emission from these low-
excitation rotational transitions is faint (Harris et al., 2010).
Observationally, the GBT/Zpectrometer set-up had another advantage to increase
its efficiency. While performing on-off chopping to remove background and
atmospheric noise, the telescope chopped between two galaxy positions. The
emission line from the second (’off’) source would then be seen as a negative
signal (Frayer et al., 2011; Harris et al., 2012). This method allows for the
easy identification of redshifts between 2 and 3.5, as well as above 5.
Interestingly, the typical uncertainty on photometric redshifts increases with
redshift, going as $\sim 0.13\times{}(1+z)$, the lower redshifts are basically
robust (hatched blue), while the higher redshifts might be mis-interpreted
CO(2-1) line emission that is instead CO(1-0).

###  3.3 Future instrumentation

New instrumentation is underway to explore the high-redshift Universe using
broad-band spectrometers, in particular using Microwave Kinetic Inductance
Detectors (MKIDs; Day et al. 2003; Endo et al. 2019) technology, and on-board
of the Origin Space Telescope (OST; Battersby et al. 2018).

####  3.3.1 DESHIMA

The DEep Spectroscopic HIgh-redshift MApper instrument combines an integrated
super-conducting spectrometer with a read-out based on MKIDs to enable ultra-
wideband ($>100$ GHz) spectroscopy at medium resolution ($f/\Delta f\approx
500$). We carefully exclude frequency regions with strong atmospheric
absorption from our analysis in Figure 6. As shown in detail in Rybak et al.
(2021), the wide bandwidth at 1.4 to 0.7 mm means that DESHIMA will be
sensitive to mid- to higher-J transitions for sources at $z<3.3$. These
emission lines vary strongly from source to source, however can be brought out
through similar stacking as seen in Z-Spec (e.g., Lupu et al. 2012). Beyond
$z>3.3$, the [C ii] emission line will dominate the emission and will be
readily detected. We note that the robust redshift coverage of DESHIMA at even
the lowest redshifts is high, and DESHIMA could thus additionally be well-
suited towards low-$z$ spectroscopic redshift searches.

####  3.3.2 SuperSpec

The SuperSpec instrument aims to use a lithographically-patterned filterbank
(similar to DESHIMA) to probe the high-redshift Universe, primarily through
atomic lines of high-redshift targets between $z=4-8$ (Redford et al., 2021).
The instrument is designed for operation on the LMT, whose large collecting
area allows for deep integration on even faint targets. Similar to DESHIMA,
this instrument also has the ability to rapidly assess the redshifts of
lower-$z$ targets through mid-J emission lines.

####  3.3.3 Origin Space Telescope

We chose to only model the longest-wavelength band of the medium-resolution
spectroscopic instrument on the OST (band 6; $0.34-0.59$ mm, Battersby et al.
2018) since it is more sensitive to lower-frequency lines such as CO. We note
that additional bands (which are expected to be able to observe
simultaneously) improve the redshift capabilities further by picking up
ancillary atomic lines which further aid in the determination of robust
redshifts. Beyond $z>1.1$, OST will be able to target [C ii] emission, leaving
only a little gap ($2.73<z<2.81$) where at band 6 is unable to pick up either
[O iii]88µm or [C ii]158µm, although [N ii]122µm could also be observed.

###  3.4 Optimising ALMA tunings for spectroscopic redshifts

Figure 8: The distribution in redshift search capability of ALMA for different
band 3 and 4 (3 and 2 mm, resp.) configurations for HerBS targets (Bakx et
al., 2018). As a figure of merit, we take the robust redshift probability plus
50% of the non-robust redshift probability. Here, we assume a ’stacked’
configuration of three spectral windows in each of the respective bands in
order to form two near-contiguous frequency ranges starting at the reported
value. The optimum observing frequencies for such a combined band 3 and 4
observation with ALMA lie in the middle-frequency of the 3 mm window and at
the upper-frequency end of the 2 mm window, as marked with the red-and-blue
star. The frequency set-up in Urquhart et al. (2022) was derived using a
similar method, and they found an efficiency in line with our findings ($\sim
72$%, and up to $\sim 92$% with deeper observations).

We test the ability to include 2 mm spectral windows to improve the robust
redshift coverage of ALMA. The proposed configuration uses a total of six
tunings to generate two near-contiguous frequency coverages at both 3 and 2
mm, which removes the overlapping region of the 3 mm scan. The total figure of
merit is determined not solely by robust redshifts (either through single- or
multiple-lines), but we choose to additionally add non-robust redshifts,
weighted down by 50%. Figure 8 shows the figure of merit for each possible
combination of 3 and 2 mm regions on a smoothed redshift distribution similar
to the HerBS targets (Bakx et al., 2018; Bakx et al., 2020b), assuming a low
uncertainty in the $z_{\rm phot}$ of $\Delta z/(1+z)\approx 7$%. Note that
both the figure of merit, as well as the choice for the redshift distribution
depend on the science case, and can be adjusted by the reader in the scripts
provided online. The figure of merit varies between 0.83 to 0.94 for different
combinations of the 3 and 2 mm window, with the optimum lying at the middle
frequencies in band 3 (3 mm) and at higher frequencies in band 4 (2 mm), with
the precise configuration shown in Table 1. The frequencies of the ALMA 12m
array observations in Urquhart et al. (2022) were derived using a similar
method. They report redshift search efficiencies similar to the ones suggested
by Figure 8, with a robust redshift fraction of 72%, and promising to go up to
93% with deeper observations. While we demonstrate the efficacy of this method
on ALMA, any redshift search with tunable (sub)mm receivers can be optimised
using this method through the publicly-available code.

##  4 Conclusions

This work documents new and efficient methods for (sub)mm redshift searches of
high-$z$ SMGs. The tools discussed in this paper are publicly available at
https://github.com/tjlcbakx/redshift-search-graphs. This paper reports on:

  * $\blacksquare{}$

An efficient graphical method for identifying redshifts, removing the need to
manually cross-compare redshift options. Aside from a description of the
method, we provide a comprehensive overview of caveats in deriving redshifts
using (sub)mm spectral lines in the high-$z$. The method provides highlights
the explored redshift-space, enabling the removal of untenable redshift
options, while also pointing out potential redshift degeneracies.

  * $\blacksquare{}$

A method to calculate the capabilities of various instruments towards deriving
robust spectroscopic redshifts. This is useful, not only to evaluate past
missions, but also to guide future instruments.

  * $\blacksquare{}$

A method to optimise tunings towards robust spectroscopic redshifts (and other
figures of merit) for ALMA and other tunable (sub)mm receivers. Here, we
stress the need for both 3 and 2 mm observations to identify redshifts
robustly.

## Acknowledgements

TB thanks the helpful contributions towards this publication from Dr. Ewoud
A.I. Pool, who provided both a wider scope on the project and improved the
graph descriptions.  We also thank the helpful comments of the referee, which
increased the scientific scope of the paper and the associated code. TB
acknowledges funding from NAOJ ALMA Scientific Research Grant Numbers 2018-09B
and JSPS KAKENHI No. 17H06130. HD acknowledges financial support from the
Spanish Ministry of Science, Innovation and Universities (MICIU) under the
2014 Ramón y Cajal program RYC-2014-15686, from the Agencia Estatal de
Investigación del Ministerio de Ciencia e Innovación (AEI-MCINN) under grant
(La evolución de los cíumulos de galaxias desde el amanecer hasta el mediodía
cósmico) with reference (PID2019-105776GB-I00/DOI:10.13039/501100011033) and
acknowledge support from the ACIISI, Consejería de Economía, Conocimiento y
Empleo del Gobierno de Canarias and the European Regional Development Fund
(ERDF) under grant with reference PROID2020010107. This paper makes use of the
following ALMA data: ADS/JAO.ALMA 2016.2.00133.S, 2018.1.00804.S, 2019.1.01147
and 2019.2.00155.S.

## Data Availability

All code used to derive the graphs in this paper are available at
https://github.com/tjlcbakx/redshift-search-graphs in order to assist in the
robust classification of redshifts in ongoing, future and past programs.

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