# Leveraging collective effects for thermometry in waveguide quantum
electrodynamics

Aleksei Sharafiev1 Mathieu Juan2 Marco Cattaneo3 Gerhard Kirchmair1,4
<EMAIL_ADDRESS>1Institute for Quantum Optics and Quantum
Information, Austrian Academy of Sciences, 6020 Innsbruck, Austria  2Institut
Quantique and Département de Physique, Université de Sherbrooke, Sherbrooke,
Québec, J1K 2R1, Canada  3QTF Centre of Excellence, Department of Physics,
University of Helsinki, P.O. Box 43, FI-00014 Helsinki, Finland  4Institute
for Experimental Physics, University of Innsbruck, 6020 Innsbruck, Austria

###### Abstract

We report a proof-of-principle experiment for a new method of temperature
measurements in waveguide quantum electrodynamics (wQED) experiments, allowing
one to differentiate between global and local baths. The method takes
advantage of collective states of two transmon qubits located in the center of
a waveguide. The Hilbert space of such a system forms two separate subspaces
(bright and dark) which are coupled differently to external noise sources.
Measuring transmission through the waveguide allows one to extract separately
the temperatures of the baths responsible for global and local excitations in
the system. Such a system would allow for building a new type of primary
temperature sensor capable of distinguishing between local and global baths.

††preprint: APS/123-QED

Characterizing and controlling the coupling between quantum systems and
environmental degrees of freedom is one of the central problems in modern
quantum systems engineering [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. In some cases,
this coupling might be useful as in quantum sensing [11], quantum thermometry
[12] or quantum engines [13], while in others it is harmful e.g. for quantum
computation protocols, leading to an additional dephasing as well as excessive
qubit population. The coupling of one quantum system to multiple baths
attracted significant attention during the last years both on theoretical and
experimental sides, including the field of superconducting quantum circuits
[14, 15, 16, 17, 18, 19, 20, 21].

Thermalization of a many body system coupled to various baths is especially
relevant in the context of superconducting devices whose effective
temperatures are known to plateau at 30-50 mK, leading in particular to excess
qubit population [22, 23, 24]. Various possible reasons for this behaviour
have been revealed, including poorly thermalized attenuators in the input
lines of the cryostat [25, 26] as well as nonequilibrium quasiparticles,
perhaps produced by stray infrared radiation [27] or cosmic rays [28, 29].
Continuing research in this area during the last few years resulted in several
proposed and realized protocols for qubit temperature measurements [23, 30,
24] and new techniques for its reduction starting to appear [31].

To understand the thermal baths a superconducting circuit is coupled to one
has to perform thermometry on the circuits. Several protocols have been
recently proposed [32, 33, 34, 35] including measuring the temperature of the
environmental photonic bath through qubit population [36]. The field of
thermometry recently expanded to the quantum regime, where entanglement can be
used to overcome fundamental limitations of classical thermometry [12]. For
instance, exploiting bath-induced correlations may enhance thermometry
precision [37].

Figure 1: (a) Experimental setup: a rectangular waveguide (global bath) with
two transmon qubits inside; a pin is insterted on one side of the waveguide
and connected to a sideport (local bath). (b) Simplified level scheme of the
two transmons with the corresponding wavefunctions (normalization omitted).
Solid arrows correspond to allowed transitions in the global regime, while
dashed lines to prohibited ones. Notations: $\beta<0$ \- anharmonicity of the
transmons, $g>0$ \- capacitive coupling between the transmons.

This work introduces a new method for the individual extraction of temperature
associated with two noise sources affecting a system of two superconducting
transmons, and demonstrates it through a proof-of-principle experiment.
Leveraging the inherent symmetry of the system, our approach employs multi-
probe spectroscopy to distinguish between a collective, global bath and a more
localized noise source. In this context, we will employ the term global to
denote a thermal reservoir symmetrically coupled to both qubits, whereas local
refers to a noise source that is more strongly coupled to one qubit, thereby
inducing an asymmetry in the dissipation channels. The latter scenario
encompasses purely localized baths extensively examined in theoretical studies
[38, 17, 39, 16, 40], wherein noise sources exclusively affect a single qubit.

Our method introduces a significant contribution to the toolbox of existing
cryogenic thermometry protocols, offering a novel capability of distinguishing
between local and global noise source. Therefore, our approach holds the
potential to emerge as an indispensable tool for detecting and discerning the
different noise sources influencing the behavior of superconducting qubits.
Such discernment is paramount for the strategic design of possible
countermeasures and better understand the complex thermalization of extended
circuits.

Our experimental setup is shown in the Fig. 1(a) where two capacitively
coupled flux tunable transmon qubits ($\text{Q}_{1}$ and $\text{Q}_{2}$) are
located in the center of a rectangular cross-section waveguide (the setup is
similar to what was used in [41, 42]). The dimensions of the waveguide
determine the frequency range over which the qubits can be tuned while
remaining in the single mode regime of the waveguide. For the experiment, the
system was equipped with two electromagnets thermally anchored to a base plate
of a dilution refrigerator (not shown), allowing for an independent control of
the qubits frequencies. The transmons were fabricated with a standard
Al/AlOx/Al bridge-free technique [43] on 330$\mu$m-thick sapphire substrate.

Owing to the symmetry of the system, we separate the two-qubit Hilbert space
into a dark or subradiant subspace, which is effectively decoupled from the
noise coming through the waveguide, and a bright or superradiant subspace [9,
14, 44] (see Fig. 1(b)). In addition, a pin mounted on the side all of the
waveguide weakly couples to both qubits. This line is used to engineer a
controlled thermal bath acting on the qubits, whose effective temperature can
be varied in a broad range (roughly from 100 mK to 60 K with the noise source
we used). Crucially, owing to the asymmetry of the evanescent field of the
pin, this bath couples to both dark and bright subspaces.

In the following, the bath associated with the pin line will be referred to as
local in contrast to the bath associated with the fundamental mode of the
waveguide, which we will call global. An alternative attempt to engineer a
local bath through a resistive heater can be found in Appendix A. The system
was designed in a way to have unequal couplings to different baths:
$\gamma_{loc1}<\gamma_{loc2}\ll\gamma_{glob}$. A more strongly coupled global
bath corresponds to a typical experimental situation in waveguide QED when
local baths can be associated with internal losses. Having the qubit pair
coupled strongly to the global bath of a continuous mode environment makes the
system more sensitive to its temperature, in turn leading to a higher residual
temperature for the qubits. For this proof-of-principle experiment this
situation is acceptable as this residual temperature, associated with the
microwave photons in the input line, can be easily included in the model.

The system illustrated in Fig. 1(a) has 3 ports: input and output of the
waveguide, connected correspondingly to standard 50 $\Omega$ input and output
lines, and the side pin line, connected to a second standard 50 $\Omega$ input
line. All input lines in our setup are equipped with 60 dB attenuation, while
the output line has a 40 dBm high-electron-mobility transistor (HEMT)
amplifier along with 2 circulators between the HEMT and the waveguide, to
prevent possible noise propagation from the HEMT to the qubits. To couple our
system to a bath (local or global) we apply controlled white noise to the
corresponding port, produced by amplifying the Johnson noise of a 50 $\Omega$
load with a room temperature amplifier chain combined with a digital
attenuator (see Appendix B for details). By changing the settings of the
attenuator we effectively regulate the noise temperature.

As we can send noise separately to the different input ports, we are able to
independently study the global and local baths. The effect of each bath is
obtained by measuring the transmission through the waveguide which depends on
the steady state of the qubits [45], and thus their temperatures. Due to the
presence of a dark subspace with respect to the global case, the transmission
coefficient in this scenario will be crucially different from the local bath
case where the dark state is broken. Through a suitable fit of these
coefficients we can then extract the temperatures of the baths independently.

For a theoretical description of our model (see details in Appendix C), we
will model the effect of the baths on the two-qubit system through a Bloch-
Redfield master equation in a partial secular approximation [46, 17], whose
Markovianity is justified by the approximately flat spectral density of the
baths (see Appendix D for some related considerations) and by the weak
coupling between qubits and reservoirs, combined with input-output formalism
[45].

Figure 2:  Transmission measurements (dashed red) and corresponding fitting
curves (solid blue) for: (a) single qubit, only global bath applied, (b) two
qubits in resonance, only global bath applied, and (c) two qubits in
resonance, only local bath applied. The main plots correspond to the
transitions $\ket{g}\rightarrow\ket{e}$ or $|00>\rightarrow\ket{B}$ while the
insets to $\ket{e}\rightarrow\ket{f}$ and
$\ket{D}\rightarrow\ket{D^{\prime}}$. (d) Bright-dark diagram obtained from
(b) and (c). Dashed red (blue): transmission measurements with a global
(local) bath. The color of the solid lines (theoretical predictions) reflects
the temperature of the corresponding bath.

We can understand the behavior of the transmission coefficients in the global
and local scenarios by looking at the simplified level scheme in Fig. 1(b) (a
more detailed scheme can be found in Appendix D): under the action of a
perfectly symmetric global bath, the levels $\ket{D}$ and $\ket{D^{\prime}}$
are decoupled from the other levels as the transitions are prohibited by
symmetry. These levels therefore form a dark subspace and will not be
populated by a global bath coupled only to the bright subspace, regardless of
the temperature of the bath. On the other hand, a bath which is coupled only
to one qubit (or asymmetrically to both qubits) breaks the symmetry and
therefore can populate the $\ket{D}$ state.

Note that the $\ket{D}\rightarrow\ket{D^{\prime}}$ transition is allowed even
under the action of a symmetric global bath, and therefore can be probed
through transmission measurements. The visibility of this transistion depends
directly on the steady state population of the state $\ket{D}$. As a result,
the transmission through the waveguide for frequencies corresponding to the
$\ket{00}\rightarrow\ket{B}$ (in the bright subspace) and
$\ket{D}\rightarrow\ket{D^{\prime}}$ (in the dark subspace) transitions can be
used to extract independently the temperatures of both global and local baths.
Remarkably, the argument still holds in non-ideal scenarios. In our
experiment, the anharmonicities of the qubits are non identical [47],
therefore the coupling to the global bath is not perfectly symmetric. However,
we will show that the independent temperature estimation is possible even for
such a non-ideal real-life setup.

First, we calibrate the power reaching the qubits from the 2 input ports,
using a simple Autler-Towns experiment (see Appendix E). For this, we apply a
coherent tone to the corresponding input in resonance with one of the qubits
while the other qubit is detuned below the cutoff of the waveguide. As a
result, the qubit frequency splits and the width of the splitting is
proportional to the Rabi frequency. Then, we fit the effects of the noise
source power on one of the qubits ($\text{Q}_{2}$) to calibrate the effective
temperature of the bath. The model has 2 fitting parameters: residual
temperature $T_{res}$ and conversion coefficient $\alpha$ between the power of
the source and added noise temperature, i.e. $T=T_{res}+\alpha\cdot P$, where
$P$ is the power we send into the fridge. We estimate these values by getting
the best possible fit of our theoretical model to the experimental
transmission coefficients as a function of temperature of the global bath for
the main transition $\ket{g}\rightarrow\ket{e}$ (see Appendix C) for a bath
with temperature $T$. The results of the experiment and their best theoretical
fit are shown in Fig. 2(a). The extracted value of $T_{res}$ is 95 mK with a
base temperature of the fridge of 14 mK. We associate this relatively high
value with the influence of a residual global bath - photons coming from the
input port of the waveguide. This result is expected in waveguide QED
experiments, where the qubits are placed in a continuous 50 $\Omega$
environment.

Then, we compute the transmission coefficients of different transitions for
the single and two qubit system as a function of temperature using our
theoretical model with the fit values for $T_{res}$ and $\alpha$, and compare
them with the experimental results. The good agreement between the
experimental and theoretical lines shows that our theoretical model is a good
description of the experiment, as can be seen for instance in the inset in
Fig. 2(a) for the transmission coefficients of the $\ket{e}\rightarrow\ket{f}$
transition or Fig. 2(b) for the $\ket{00}\rightarrow\ket{B}$ transition.

Next, we set both qubits in resonance at 7.8 GHz and vary the noise
temperature of both global and local baths. In this case we are interested in
2 transitions: $\ket{00}\rightarrow\ket{B}$ and
$\ket{D}\rightarrow\ket{D^{\prime}}$, see Fig. 1(a). We start by varying the
temperature of the global bath and monitoring the transmission at the
$f_{00B}$ and $f_{DD^{\prime}}$ frequencies with a weak probe. The result of
the experiment and the corresponding theoretical lines are shown in Fig. 2(b).

In a perfect device, the global bath temperature does not affect the
transmission through the waveguide at the $\ket{D}\rightarrow\ket{D^{\prime}}$
frequency $f_{DD^{\prime}}$, as the global bath is decoupled from the dark
subspace. Nevertheless a dependence on temperature appears both in the
experiment and the theoretical fit, as demonstrated in the inset of Fig. 2(b).
However, the change in intensity is much lower than for the
$\ket{00}\rightarrow\ket{B}$ transition. There are two reasons for this
dependence. First, the spectrum of allowed transitions in the bright subspace
is richer than the one shown in the Fig. 1(b), and it includes transitions
which are close to $f_{DD^{\prime}}$ in higher excitation manifolds, see
Appendix D. Second, no physical system displays perfect symmetry. In our
experiment the symmetry is broken by the imprecise location of the qubit pair
inside the waveguide and by slightly different anharmonicities of the
transmons $\beta$. In our setup, the qubits have anharmonicities
$\beta_{1}\sim-225$ MHz and $\beta_{1}\sim-232$ MHz respectively for
$\text{Q}_{1}$ and $\text{Q}_{2}$, i.e. their difference is of the order of
$\sim 2.6\%$. Spectroscopic lines appearing in the system with up to 3
excitations and the difference in $\beta$ between the qubits have been taken
into account in the model. As one can see from the figure, the observed
transmission at $f_{DD^{\prime}}$ is well described by the curve obtained with
the theoretical model.

Finally we use the side pin to apply a local bath, affecting the qubits
asymmetrically. Performing an Autler-Towns experiment, but applying the pump
tone from the side pin, we were able to extract the ratio of the couplings
between the qubits and the side pin, obtaining
$\gamma_{loc2}/\gamma_{loc1}=1.78$. This ratio was used in the model, while
the coupling itself was kept as a fitting parameter. To extract the coupling
we again fit only single-qubit data, which in this case is fitted
simultaneously for both qubits. For the details on this procedure see Appendix
F. After extracting the coupling, we plot the experimental and theoretical
results for the two qubits and local bath experiment in Fig. 2(c). As before,
the theoretical lines are obtained with the values of $T_{res}$ and $\alpha$
fitted in single-qubit experiments.

Comparing Fig. 2(b) with Fig. 2(c) one can see that indeed the effect on the
$\ket{D}\rightarrow\ket{D^{\prime}}$ transition is more noticeable in the
local bath case: the transmission drop is deeper and it occurs when the
$\ket{00}\rightarrow\ket{B}$ transmission is far from reaching saturation.
This can be better illustrated with what we call bright-dark diagram, see Fig.
2(d), where the axes correspond to the transmission values at the frequencies
$f_{00B}$ and $f_{DD^{\prime}}$. Experimental frequency sweeps in Appendix A
further illustrate the difference between the effects of global and local
baths on the transmission.

From this diagram one can see that local and global bath saturate the
transitions $\ket{D}\rightarrow\ket{D^{\prime}}$ and
$\ket{00}\rightarrow\ket{B}$ very differently. Thus, a combined measurement
allows us to assess the influence of a local bath on a temperature measurement
using the $\ket{00}\rightarrow\ket{B}$ as a calibrated sensor. In our proof of
principle experiment, where both baths have been calibrated, we can actually
determine the temperature of both uniquely by measuring the depth of both
spectroscopy peaks. This works for a temperature range of 50 mK to about 500
mK for the global and 100 mK to about 2 K for the local baths. In a real world
scenario one would not be able to determine the temperature of the local bath
as the coupling would not be known. In this case our method would still allow
to extract its influence on the measurement of the global temperature (where
the system still acts as a calibrated sensor) and correct for it.

To illustrate the measurement process further we have run simulations of a
system based on our experiment that is suitable to detect the desired low
input mode temperatures ($\approx 20mK$) of a circuit quantum electrodynamics
system. The results of this simulation can be seen in Fig. 3. Here we change
the global bath temperature from 10 to 300 mK for a pair of 2 GHz transmons
inside a waveguide. We assume a local bath coupling of 1 kHz with other
parameters corresponding to our current experiment. Again, a combined
measurement of both transitions allows to determine the temperature of the
global bath and extract the influence of the local bath.

From the level diagram (see Fig. 1(b)), one can see that the local bath has a
very strong influence on the temperature measurement using the transmission on
the $\ket{00}\rightarrow\ket{B}$ leading to a wrong inference of the global
bath temperature when only this transition is used. This is the same situation
for thermometry methods based on a single qubit: local and global bath cannot
be differentiated making it impossible to definitively identify the origin of
excess temperature in the system. From the measurement of the
$\ket{D}\rightarrow\ket{D^{\prime}}$ one can determine the residual excitation
of the qubits due to a local baths and correct for it in the global bath
measurement. In other words for a well calibrated two-qubit sensor with a
verified model, each “line” in the dark-bright diagram can be uniquely
associated with a combined measurement of the
$\ket{D}\rightarrow\ket{D^{\prime}}$ and $\ket{00}\rightarrow\ket{B}$ lines.
Two dashed lines in Fig. 3 corresponding to global bath temperature 14 and 30
mK serve as guide to the eye and correspond to typical base plate temperature
of a dilution refrigerator and a lowest residual transmon frequency typically
observed experimentally. Some more details are provided in Appendix G.

Figure 3:  Theoretical bright-dark diagrams for an experiment with low
frequency (2 GHz) transmons. All lines correspond to $T_{glob}$ change from 10
to 300 mK. The shape of the curves allows us to reveal a weakly coupled local
bath even when the residual temperature of the qubits is defined by the global
bath (see the main text).

In conclusion, we have demonstrated a proof-of-principle setup that allows to
distinguish between two types of baths coupled to a quantum sensor by
measuring two transitions of the coupled system. We achieve this by exploiting
system symmetries combined with a good calibration of the system and a working
theory model. Developing this proof-of-principle experiment into a useful
thermometer that would accurately measure the input mode temperature of a
system and be able to correct for undesired local bath couplings would require
switching to a planar CPW architecture as manufacturing tolerances leading to
asymmetries and impedance matching could be improved. We believe the approach
presented in this paper opens a promising avenue for a new type of primary
temperature sensor capable of distinguishing between local and global baths
for temperatures as low as $\sim$20-30 mK enabling more accurate temperature
measurements.

Acknowledgments.–The authors would like to thank Roberta Zambrini for useful
comments on the theoretical model employed in this work. MC would like to
thank the members of the PICO group at Aalto University for interesting
discussions and suggestions. AS and GK acknowledge funding by the European
Research Council (ERC) under the European Unions Horizon 2020 research and
innovation program (714235). This research was funded in part by the Austrian
Science Fund (FWF) DOI 10.55776/F71. MC acknowledges funding from the Research
Council of Finland Centre of Excellence program grant 336810 and from COQUSY
project PID2022-140506NB-C21 funded by MCIN/AEI/10.13039/501100011033. MLJ
acknowledges funding by the Canada First Research Excellence Fund.

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##  Appendix A Adding a heater

A naive way to create a local bath would be adding a physical heater to the
chip. With the aim to compare this approach with our model, we originally
designed the setup with the qubits located on 2 different and rather long - 25
mm - chips. The chips protrude outside of the waveguide and the heater (SMD
resistor) was varnish-glued on the lower end of one of them, as shown in Fig.
A.1LABEL:sub@fig:wg_cross_section. The chips were only connected mechanically
on the upper end with a copper clamp. The resistor was wired to a current
source to regulate its temperature. As the resistor was located around 1 cm
away from the SQUID loops of the qubits, we could be confident that the small
heating current (up to 100$\mu$A) would not affect qubit frequencies. It is
worth to note that this experiment is different from an approach used in some
other works, e.g. [19], where resistor was used just as a capacitively coupled
bath (similarly to our side pin experiment). It is also different from heating
the base plate of the refrigerator, used e.g. in [36], as it effectively leads
to heating the attenuators in the input lines and therefore would change the
temperature of the global bath.

The experiment with the heater revealed that our model was not capable of
predicting the systems behaviour mainly because of significant change of the
junctions critical currents. In turn, it caused a qubit frequency change which
complicated extracting the effect of the qubits populations on the
transmission and would require a more complex model. The Fig.
A.1LABEL:sub@fig:fr_shift shows the change of the lowest transmission point in
a single qubit experiments for a local bath delivered by the side pin, local
bath as a heater and prediction of our model. The discrepancy in the low
attenuation (high temperature) area between the model and the experiment in
our case was around 4 MHz and probably associated with the fact that only 4
levels of the transmon were considered in the theory. The discrepancy for the
heater bath due to critical current change was around 40 MHz, i.e. larger than
the qubits coupling to the waveguide. Moreover, thermal isolation of the
qubits turned out to be worse than we expected, and heating effect could be
simultaneously seen on both qubits. In this case the fitting procedure with a
fixed probe frequency used in the main text could not be reasonably applied.

Nevertheless, to visualize the difference between three baths (global, local-
pin and local-heater) we conducted a frequency sweep of qubit $\text{Q}_{1}$
while keeping qubit $\text{Q}_{2}$ fixed, see Fig. A.2. In all three cases the
bath temperature has been chosen to maximize the effect on the
$\ket{D}\rightarrow\ket{D^{\prime}}$ transition, while the color scale is the
same. The difference between the effects of global and local-pin baths can be
clearly seen from Fig. A.2LABEL:sub@fig:global_sweep_3D and Fig.
A.2LABEL:sub@fig:pin_sweep_3D.

(a)

(b)

Figure A.1:  (a) Cross-section of the waveguide with 2 qubits ($\text{Q}_{1}$
and $\text{Q}_{2}$) inside and a resistive heater attached to one of them. (b)
The effect of local bath heating on the lowest transmission frequency
corresponding to $\ket{g}\rightarrow\ket{e}$ transition. Horizontal axis is
the value of the transmission.

(a)

(b)

(c)

Figure A.2:  Transmission measurements with a weak signal through the
waveguide while sweeping the frequency of the transmon $\text{Q}_{1}$, keeping
transmon $\text{Q}_{2}$ still for 3 different situations: (a) only global bath
is on; (b) only local bath generated by a side pin is on; (c) only local bath
generated by a heater is on.

##  Appendix B Details of the experimental setup

The experimental setup schematic is shown in the Fig.
B.1LABEL:sub@fig:setup_scheme. We use 2 input and 1 output lines inside a
dilution refrigerator. The input lines have 60 dB attenuation and the output
line is equipped with a low-noise HEMT amplifier and a room temperature
amplifier, providing +40 dB amplification each. Noise was applied from a room
temperature amplifier terminated with 50 $\Omega$ on the input, through a
variable attenuator, effectively regulating the noise temperature. A probe
tone was applied and measured with a standard Vector Network Analyzer (VNA).
Current sources were used to independently tune qubits frequencies with
magnetic field and apply current through the heating resistor.

Both qubits inside the waveguide (WG) were flux tunable approximately between
4 and 8 GHz, and were fabricated with a standard Al/AlOx/Al bridge-free
technique [43] on 330$\mu$m-thick sapphire substrate.

The spectrum of the applied noise was measured separately with a spectrum
analyzer and shown with a blue line in the Fig.
B.1LABEL:sub@fig:att_test_and_noise. The noise spectral density was normalized
to its value at the $\ket{00}\rightarrow\ket{B}$ transition frequency.

As we learned, many variable attenuator models suffer from substantial ripples
especially at large attenuation values. To make sure the ripples are not
significant in our case, we checked how flat the attenuation spectrum is. Red
lines in the Fig. B.1LABEL:sub@fig:att_test_and_noise correspond to the
accuracy of the attenuation settings, i.e. difference between set and measured
values. One can see that the noise spectral density as well as the
imperfections of the attenuator do not cause major deviations of the noise
from being relatively white at least at 2 frequencies of interest
corresponding to the $\ket{00}\rightarrow\ket{B}$ and
$\ket{D}\rightarrow\ket{D^{\prime}}$ transitions.

(a)

(b)

Figure B.1: (a) Experimental setup. WG is the waveguide with the qubits
inside. Noise sources (room temperature amplifiers terminated with 50 $\Omega$
at the input) are connected to the waveguide through variable attenuators. (b)
Ripples of the variable attenuator and the normalized spectrum of the noise
source in the frequency range of interest. The noise spectrum was normalized
to its value at the $\ket{00}\rightarrow\ket{B}$ transition. Both
$\ket{00}\rightarrow\ket{B}$ and $\ket{D}\rightarrow\ket{D^{\prime}}$
frequencies are shown with vertical dashed lines.

##  Appendix C Theoretical model

We consider a model of two weakly anharmonic transmon qubits with frequency
$\omega_{1}$ and $\omega_{2}$. The frequency of qubit $\text{Q}_{2}$ can be
varied by applying a suitable flux bias. The qubits interact through a weak
coupling capacitor and the qubit-qubit coupling energy is $g$. They are also
weakly coupled to two different thermal environments, which we term “global”
and “local”, with temperatures respectively $T_{\text{glob}}$ and
$T_{\text{loc}}$. The difference between the global and local baths lies in
how they interact with the qubits: while the global bath is coupled with the
same strength to each qubit (i.e., the coupling is symmetric), the local bath
still interacts with both qubits but is more strongly coupled to one of them,
say to qubit $\text{Q}_{2}$. Finally, the system is probed with a coherent
drive tone with a frequency $\omega_{\text{drive}}$.

The free Hamiltonian of each transmon qubit as a weakly anharmonic oscillator
is [47, 48]:

$H_{q_{\alpha}}=\hbar\omega_{\alpha}a_{\alpha}^{\dagger}a_{\alpha}+\hbar\beta_{\alpha}a_{\alpha}^{\dagger}a_{\alpha}^{\dagger}a_{\alpha}a_{\alpha},\text{
with }\alpha=1,2.$ (1)

$\beta_{\alpha}$ is the anharmonicity of the qubit $\alpha$, and to take into
account thermal effects we will consider multiple transmon levels. The free
system Hamiltonian (including the probe) is:

$H_{S}(t)=\sum_{\alpha=1,2}H_{q_{\alpha}}+H_{\text{int}}+H_{\text{probe}}(t),$
(2)

with

$\begin{split}&H_{\text{int}}=g(a_{1}^{\dagger}a_{2}+a_{1}a_{2}^{\dagger}),\qquad
H_{\text{probe}}(t)=E_{1}a_{1}^{\dagger}e^{-i\omega_{\text{drive}}t}+E_{2}a_{2}^{\dagger}e^{-i\omega_{\text{drive}}t}+H.c.\end{split}$
(3)

The free Hamiltonian of the thermal baths is given by:

$H_{B}=\sum_{j=\text{g},\text{l}}\sum_{k}\Omega_{\alpha,k}b_{j,k}^{\dagger}b_{j,k},$
(4)

where $j=\text{g},\text{l}$, labels the frequencies and operators of
respectively the global and the local bath. Finally, assuming that the qubits
and the baths are coupled through their quadratures, giving rise to
dissipation [18], the interaction Hamiltonian is given by:

$H_{qb}=\mu\left[\sum_{k}\zeta_{\text{g},k}\sum_{\alpha=1,2}(a_{\alpha}^{\dagger}b_{\text{g},k}+a_{\alpha}b_{\text{g},k}^{\dagger})+\sum_{k}\zeta_{\text{l},k}\sum_{\alpha=1,2}\lambda_{\alpha}(a_{\alpha}^{\dagger}b_{\text{l},k}+a_{\alpha}b_{\text{l},k}^{\dagger})\right].$
(5)

$\mu$ is the magnitude of the system-environment interaction. The weak
coupling limit reads $\mu\ll\hbar\omega_{\alpha}$, i.e., the interaction
energy is much lower than the qubit energies. Both $\zeta_{j,k}$ and
$\lambda_{\alpha}$ are dimensionless parameters. The coefficients
$\zeta_{j,k}$ define the spectral density of the baths $J(\omega)$ [18, 9],
which, for simplicity and according to experimental observations (see the
considerations in Appendix D), we take as perfectly flat for both local and
global baths. The global bath is (ideally) coupled to each qubit with the same
weight. In contrast, the local bath is more strongly coupled to qubit
$\text{Q}_{2}$ than to qubit $\text{Q}_{1}$. In the main text we have
introduced the ratio between the local decay rates of the two qubits, which
can be written as

$k=\left(\frac{\lambda_{2}}{\lambda_{1}}\right)^{2},$ (6)

with $\lambda_{2}>\lambda_{1}$. The case with $k\rightarrow 1$ corresponds to
the limit of a global symmetric bath, while $k\rightarrow+\infty$ is the limit
of perfectly local bath, that is, the local environment is completely
decoupled from qubit $\text{Q}_{1}$. Finally, the total Hamiltonian of
system+environment is:

$H=H_{S}(t)+H_{B}+H_{qb}.$ (7)

Note that the we can switch off the local and/or global bath by setting the
coefficients respectively $\zeta_{l,k}$ and $\zeta_{g,k}$ to zero.

The dependence on time in Eq. (7) can be eliminated through a suitable unitary
transformation to write the Hamiltonian in a frame rotating with the driving
frequency [16]. Then, we can derive a Markovian master equation for the state
of the two qubits in the rotating frame. The Markovianity of the evolution is
justified by the weak coupling between the qubits and the bath, and by the
flat spectral density that leads to a vanishing “memory time” of the
environment [46]. Moreover, the master equation is local, i.e., the jump
operators are local operators acting on a single qubit only [17, 39]. This is
justified by the fact that the qubit-qubit coupling energy and the driving
energy are much lower than the system energy [17, 16, 40, 39],
$g,E_{1},E_{2}\ll\hbar\omega_{1},\hbar\omega_{2}$, and that the
autocorrelation functions of the environment are a Dirac’s delta in time due
to the flat spectral density [16].

Transmon qubits are only weakly anharmonic, thus their higher levels can be
easily populated as well at high enough temperature, making them qudits
instead of proper qubits. For simplicity, let us consider the first $d$ levels
of the transmon and truncate the rest, such that the eigenstates of the free
transmon Hamiltonian are $\\{\ket{j}_{\alpha}\\}_{j=0}^{d-1}$ with energies
$e_{0}^{(\alpha)},\ldots,e_{d-1}^{(\alpha)}$. The energies of each transmon
level depend on the qubit frequency and on the anharmonicity according to Eq.
(1). Then, we can introduce the lowering operators:

$\sigma_{j,k}^{(\alpha)}=\sqrt{k}\ket{j}_{\alpha}\\!\bra{k},\text{ such that
}a_{\alpha}=\sum_{j=0}^{d-2}\sigma_{j,j+1}^{(\alpha)}.$ (8)

$\sigma_{j,j+1}^{(\alpha)}$ and $\sigma_{j+1,j}^{(\alpha)}$ are the jump
operators [46, 17] of the local master equation, respectively associated with
the jump frequencies $\omega_{j}^{(\alpha)}$ and $-\omega_{j}^{(\alpha)}$. The
jump frequencies are simply the difference in frequency between two adjacent
energy levels, that is,

$\omega_{j}^{(\alpha)}=e_{j+1}^{(\alpha)}-e_{j}^{(\alpha)}.$ (9)

Finally, the master equation reads:

$\frac{d}{dt}\rho_{S}(t)=-i[\tilde{H}_{q_{\alpha}}+H_{int}+H_{probe}(0),\rho_{S}(t)]+\mathcal{D}_{\text{tot}}[\rho_{S}(t)],$
(10)

with
$\tilde{H}_{q_{\alpha}}=(\omega_{\alpha}-\omega_{\text{drive}})a_{\alpha}^{\dagger}a_{\alpha}+\beta_{\alpha}a_{\alpha}^{\dagger}a_{\alpha}^{\dagger}a_{\alpha}a_{\alpha}$,
and dissipator given by:

$\mathcal{D}_{\text{tot}}[\rho]=\sum_{\alpha,\alpha^{\prime}=1,2}\sum_{j,k=0,\ldots,d-2}\left[\gamma_{\alpha
j,\alpha^{\prime}k}^{\downarrow}\mathcal{D}(\sigma_{j,j+1}^{(\alpha)},\sigma_{k,k+1}^{(\alpha^{\prime})})[\rho]+\gamma_{\alpha
j,\alpha^{\prime}k}^{\uparrow}\mathcal{D}(\sigma_{j+1,j}^{(\alpha)},\sigma_{k+1,k}^{(\alpha^{\prime})})[\rho]\right].$
(11)

We have introduced the notation $\mathcal{D}(a,b)[\rho]=a\rho
b^{\dagger}-\frac{1}{2}\\{b^{\dagger}a,\rho\\}$ and the coefficients:

$\gamma_{\alpha
j,\alpha^{\prime}k}^{\downarrow\uparrow}=\left[\Gamma^{\downarrow\uparrow}(\omega_{j}^{(\alpha)},T_{\text{glob}})+\Gamma^{\downarrow\uparrow}(\omega_{k}^{(\alpha^{\prime})},T_{\text{glob}})\right]+\lambda_{\alpha}\lambda_{\alpha^{\prime}}\left[\Gamma^{\downarrow\uparrow}(\omega_{j}^{(\alpha)},T_{\text{loc}})+\Gamma^{\downarrow\uparrow}(\omega_{k}^{(\alpha^{\prime})},T_{\text{loc}})\right],$
(12)

and we have employed the quantity

$\Gamma^{\downarrow\uparrow}(\omega,T)=\frac{\pi\gamma_{0}}{2}\left(\coth(\frac{\omega}{2k_{B}T})\pm
1\right).$ (13)

The $+$ sign in the brackets corresponds to $\Gamma^{\downarrow}$ and
analogously for $\Gamma^{\uparrow}$. $\gamma_{0}$ is a decay constant with the
units of the inverse of time, which is proportional to the square of the
coupling strenght $\mu^{2}$ and depends on the spectral density of the baths
$J(\omega)$, which is flat. How to find $\gamma_{0}$ in capacitively coupled
transmon qubits, for instance, we refer the reader to Refs. [9, 18]. We note
that the structure of the master equation in Eq. (11) is the same for both the
global and local bath, while the expression for the contributions to the decay
rates is different, as given by Eq. (12). While probing the global bath only,
we need to set $\lambda_{\alpha}=0$ to remove the local contribution.

We note that the master equation in Eq. (10) is in partial secular
approximation [17, 49, 50], i.e., it couples jump operators associated with
slightly different jump frequencies. This is true even if we tune the flux
bias on qubit $\text{Q}_{2}$ such that its frequency matches that of qubit
$\text{Q}_{1}$. Indeed, the anharmonicities $\beta_{1}$ and $\beta_{2}$ are
never exactly the same, so the jump frequencies between higher transmon levels
are slightly different for the two qubits. The master equation in partial
secular approximation takes these small differences into account and is
obtained from the Bloch-Redfield equation by removing the fast-rotating terms
only, instead of removing all the non-resonant terms (“full secular”) as
typically done in the standard derivation of the Lindblad master equation
[46]. The drawback of the partial secular approximation is that it may
generate dynamics that are not exactly completely positive due to small
negativity issues. However, recent works have shown that complete positivity
is often recovered in the partial secular regime [51, 52] and, if small
negativities appear, then they are not a real issue because they are of the
order of the accuracy of the master equation [17], see for instance Ref. [53].

Finally, we employ the steady state of Eq. (10) to estimate the amplitude of
the transmission measurements in the waveguide according to standard input-
output theory [54]. More specifically, we compute the transmission coefficient
derived in Sec. II B of Ref. [45] for an ensemble of inhomogeneous atoms in
waveguide QED.

If only the symmetric global bath was present, then there would exist a dark
subspace (or decoherence-free subspace [55]) corresponding to a subradiant
transition between the qubits [9]. The subspace would not be populated in
ideal and symmetric conditions. In contrast, the local bath breaks the
symmetry generating the dark subspace and it can induce transitions thereto.

(a)

(b)

Figure C.1:  (a) Level scheme and analytical wave functions for 2 coupled
transmons of the same frequency $\omega_{q}$ and anharmonicity $\beta<0$ with
coupling energy $\hbar g$ up to 3 excitations. (b) Numerically calculated
spectroscopy lines with varying qubit $\text{Q}_{1}$ frequency, while the
frequency of qubit $\text{Q}_{2}$ is fixed at 7.81 GHz. The size of the dots
is proportional to corresponding transitional dipole moment. Transparent lines
correspond to the prohibited transitions when the qubit frequencies coincide.
Diamond marks are the analytical values extracted from (a). Red diamonds
correspond to transitions within the bright subspace, while blue diamonds are
used for dark subspace transitions. Labels in the legend reflect mode numbers
between which the transition occurs. In the resonant case these numbers
coincide with levels numbering in (a).

##  Appendix D Considerations on the spectral density

Before describing our approach to the bath modelling we focus on the noise
spectrum, as it is important to understand how the spectral density of the
environment may affect the system dynamics.

In the main text we briefly discussed the energy levels of the coupled
transmons up to 2 excitations, with a simplified scheme in Fig. 1(b). The same
scheme up to 3 excitations is shown in Fig.
C.1LABEL:sub@fig:level_scheme_full. Here we assume identical transmons of the
same frequency $\omega_{q}$ and anharmonicity $\beta<0$. In the figure we
specify the analytical energy values along with the corresponding
wavefunctions. The normalization coefficients $A_{\pm}$, $B_{\pm}$ and
$C_{\pm}$ are also specified on the top of the figure. Allowed (with the
waveguide fundamental mode) transitions are depicted with solid arrows while
prohibited ones correspond to dashed arrows. The scheme shows a clear
separation between bright (on the left) and dark (on the right) subspaces. In
the main text we are using the 2-0 and 4-1 transitions as
$\ket{00}\rightarrow\ket{B}$ and $\ket{D}\rightarrow\ket{D^{\prime}}$
respectively.

The frequencies of our transmons can be tuned independently and the transition
lines seen in the experiment depend on the frequency detuning between the
qubits. Fig. C.1LABEL:sub@fig:levels shows a plethora of lines one can expect
to see in the experiment even when only 4 levels per transmon are taken into
account. In this figure we model a scenario where qubit $\text{Q}_{2}$ has a
stable frequency (dashed horizontal line) and qubit $\text{Q}_{1}$ frequency
is changed by applying magnetic flux. In total the plot has 20 lines, 10 of
them to prohibited transitions if the qubits are in resonance(transparent
color, dashed arrows in the Fig. C.1LABEL:sub@fig:level_scheme_full) and
another 10 allowed transitions (solid color, solid arrows in the Fig.
C.1LABEL:sub@fig:level_scheme_full). Red and blue diamonds are the analytical
values which were only calculated for the resonant case. Red diamonds
correspond to the transitions within the bright subspace, while blue ones are
for dark subspace transitions. For all transitions, marker sizes are
proportional to the corresponding transitional dipole element. One can see
that in the resonant case we expect all allowed transitions to be within 900
MHz between 7.05 GHz and 7.95 GHz. The highest and lowest frequency
transitions (9-3 and 7-5) are only weakly allowed and correspond to population
transfer between the 2-excitation and the 3-excitation manifolds. The lines of
interest (2-0 and 4-1) connect the 1-excitation and 2-excitation manifolds and
are separated only by the anharmonicity $\beta$=-230 MHz.

Figure D.1:  Normalized spectral density of an infinite 3D rectangular cross
section waveguide.

Having this in mind we can now consider whether taking into account a specific
spectral density of the environment affects the results significantly. Fig.
D.1 shows the normalized spectral density of modes inside the infinitely long
waveguide with a cross section dimensions corresponding to our experimental
case. One can see that the lines which we are using in the main text for the
bright-dark diagrams can feel only a few % variation in the spectral density
of the environment. In general terms, the dynamics of the total system may
depend on all the possible transition lines, and within our model the mismatch
between the highest and lowest frequency lines due to the spectral density
might be stronger (around 30%). Nevertheless, we argue that these 2 lines only
play a role in the dynamics at high temperatures where we do not expect
precise result from our model due to not taken into account higher excitation
manifolds. Other lines are closer to each other in frequency and therefore
feel almost the same $J(\omega)$.

Another important consideration is that the line shown in Fig. D.1 diverges
when the frequency approaches the cutoff frequency (around 6.55 GHz in our
case) - this effect being relevant only for an infinitely long waveguide. In
practice we are using commercial couplers connecting our waveguide with a
standard 50 $\Omega$ coaxial lines. These couplers do not provide perfect
impedance matching at the frequencies close to the cutoff and therefore the
effective density of states seen by the qubits does not diverge at the cutoff
and depends on details of the experimental setup. To take into account the
effect of this mismatch (together with other imperfections in the transmission
inevitably appearing in the input and output lines) on $J(\omega)$ seen by the
qubits is a non-trivial task. This is aggravated by the fact that the couplers
are non-identical and originally meant for room temperature usage. For an on-
chip device utilizing a coplanar waveguide architecture this problem does not
appear and one can assume an Ohmic spectral density whenever it is important.
For this proof-of-principle device, in order to avoid additional assumptions
and complications in the theoretical model, we decided to assume a flat
spectral density (frequency independent), as it seems to describe the
experimental results well enough.

One of the difficulties of extracting temperature from transmission
measurements is the necessity to remove the background of the transmission
from the measurements. In experiments with 3D waveguides one has to use
impedance matching elements at both ends of the waveguide, where the standard
50 $\Omega$ coaxial lines are connected to the waveguide. Because of imperfect
impedance matching, the waveguide has low-Q resonant modes interacting with
the strongly coupled qubits. As a result, the background one wishes to
subtract depends on the qubit frequencies and populations. We attribute the
mismatch between the best fit and the experimental data for the qubit
$\text{Q}_{1}$ to this issue, which can be again be alleviated using a planar
CPW architecture.

##  Appendix E Power calibration

In our theoretical model, described in Appendix C, we rely on the input-output
theory [45] which requires to establish the relation between power and Rabi
frequency. For our experimental setup we derive this relation from Autler-
Townes experiments, i.e. by applying a coherent drive (through the waveguide
or the side pin). The Hamiltonian of this drive can be written as

$H_{drive}=\frac{\Omega}{2}(\hat{a}^{\dagger}+\hat{a}).$ (14)

Input-output theory yields the expression for the same Hamiltonian

$\displaystyle H_{drive}=$ $\displaystyle E\hat{a}^{\dagger}+E^{\ast}\hat{a},$
(15a) $\displaystyle E(t)=$
$\displaystyle-\textnormal{i}\sqrt{\frac{\gamma_{qubit}\cdot\omega_{\text{drive}}}{2\omega_{qubit}}}De^{-\textnormal{i}\omega_{\text{drive}}t}$
(15b) $\displaystyle D=$
$\displaystyle\textnormal{calibration}\sqrt{10^{\textnormal{drive[dBm]}/10}}$
(15c)

The driving strength, $D$, is the fit parameter when treating the power sweep.
In resonance ($\omega_{\text{drive}}=\omega_{qubit}$) we obtain the following
relation between the fit parameter $D$ and the Rabi frequency $\Omega$ defined
by Eq. (14)

$\Omega=2\sqrt{\gamma_{qubit}}D.$ (16)

Extracting the calibration for 3 different cases (global bath, side pin qubit
$\text{Q}_{1}$ and side pin qubit $\text{Q}_{2}$) is shown in Fig. E.1.

(a)

(b)

(c)

Figure E.1:  Power calibration utlizing Autler-Townes splitting with the
signal applied (a) through the waveguide mode (coupled symmetrically to both
transmons), (b) through the side pin for the transmon $\text{Q}_{1}$, and (c)
through the side pin for the transmon $\text{Q}_{2}$. The colour corresponds
to transmission through the waveguide measured with a weak probe signal.

##  Appendix F Fitting single qubit experiments with local bath

We discuss now our technique for extracting independently side pin coupling
and local bath temperature. The Autler-Towns experiment alone does not allow
for the extraction of the coupling coefficient of a qubit to the drive line.
For the global bath the situation is simpler as the probe tone couples with
the same coupling constant as the drive.

In principle, the calibration coefficient from the equation (15c) can be
extracted separately from $\sqrt{\gamma_{qubit}}$, but in practice it requires
a rather rigorous calibration of all impedance mismatches in input and output
lines, including mismatches on the input and output of the waveguide. These
mismatches can be frequency and temperature dependent and calibrating them
with the necessary precision is a possible but tedious task. Instead in this
proof-of-principle experiment we chose a different strategy. From the Autler-
Towns experiments with the side pin for qubits $\text{Q}_{1}$ and
$\text{Q}_{2}$ we can extract the ratio $\sqrt{\gamma_{loc1}/\gamma_{loc2}}$.
Using the side pin as a source for a local bath we follow both
$\ket{g}\rightarrow\ket{e}$ and $\ket{e}\rightarrow\ket{f}$ transitions (see
Fig. F.1). The curves for the $ket{g}\rightarrow\ket{e}$ transitions were
fitted jointly for both qubits. Two fitting parameters were used in the
procedure: $\alpha_{loc}$ and $\gamma_{loc1}$. Cable attenuation and impedance
mismatches between cables are included in $\alpha_{loc}$, which was only
allowed to vary around $\alpha_{glob}$, which in turn we already extracted
during the global bath fitting. The second fitting parameter $\gamma_{loc1}$,
which includes the large impedance mismatch between the coax cable with the
pin and the waveguide, was allowed to vary within 3 orders of magnitude around
the value expected from the device geometry. The local bath temperature values
$T_{loc}=T_{res}+\alpha_{loc}\cdot P$ obtained from the fit correspond to what
we already expect from the setup cabling and the considerations above. Results
for the$\ket{e}\rightarrow\ket{f}$ transitions were not used in the fit and
fitting curves in Fig. F.1LABEL:sub@fig:local_2q_fit_ef just show how they
appear in the model with all parameters fixed after the
$\ket{g}\rightarrow\ket{e}$ fit. The values of $\alpha_{loc}$ and
$\gamma_{loc1}$ obtained through this procedure were used to produce the 2
qubit local bath theory lines for Figs. 2(c) and 2(d) of the main text.

(a)

(b)

Figure F.1:  Result of a single qubit experiments with a side pin as a local
bath fitted together (see the text): (a) $\ket{g}\rightarrow\ket{e}$
transition and (b) $\ket{e}\rightarrow\ket{f}$.

##  Appendix G Further theoretical results

In this section we briefly discuss some relevant results one can obtain with
the theoretical model, which were not included in the main text.

Figure G.1:  Modelled bright-dark diagram for different coupling asymmetry of
the side pin. The curves corresponding to the side pin coupled only to one of
the qubits ($k\ll 1$ or $k\gg 1$) are distinguishable from the perfectly
global case $k=1$. Points correspond to $T_{loc}$ changing from 0 to 200 K.

One question appearing from Fig. 2(d) of the main text is how having a purely
local (coupled only to one of the qubits) bath would change our bright-dark
diagrams. Fig. G.1 shows these diagrams for different balance parameter
$k=\gamma_{loc2}/\gamma_{loc1}$. One can see that the
$\ket{D}\rightarrow\ket{D^{\prime}}$ transmission dip is the shallowest when
$k=1$, i.e. in a purely global case, while it is the deepest when
$k\rightarrow 0$ or $\infty$. Anyway, the dark-bright diagram for the latter
scenario does not differ dramatically in comparison with our experimental
situation $k=1.78$. The reason why the $k=1$ line is not perfectly flat is
that there are allowed transitions between higher levels corresponding to
frequencies close to $\ket{D}\rightarrow\ket{D^{\prime}}$ (see line 9-5 from
Fig. C.1a).

(a)

(b)

Figure G.2:  Result of single qubit experiments with a side pin as a local
bath fitted jointly(see the text): (a) $\ket{g}\rightarrow\ket{e}$ transition
and (b) $\ket{e}\rightarrow\ket{f}$.

Fig. G.2(a) is an addition to Fig. 3 in the main text and shows how varying
the transmon frequency changes the bright-dark diagram. As before, the global
bath temperature is being swept from 1 to 300 mK and $T_{loc}$ is fixed at 30
mK, while the other parameters correspond to the experimental case discussed
in the main text, i.e. $\gamma_{loc1}=\gamma_{loc2}$ and
$\gamma_{loc1}/2\pi=1$ kHz. Green circles on the lines correspond to
$T_{glob}=14$ mK and gray circles are at $T_{glob}=30$ mK. One can see that
lower frequency qubits react to the presence of the local bath even if it is
weakly coupled and relatively cold.

Fig. G.2LABEL:sub@fig:fq_cuts_no_T_loc corresponds to the same situation but
the local bath temperature is fixed at a very low value of 1 mK. We depict
here only 2 lines in comparison with Fig. G.2LABEL:sub@fig:fq_cuts, as they
mostly overlap. In this case the local bath effectively acts as a weakly
coupled relaxation channel, and regardless of the transmons frequency the
bright-dark diagram looks qualitatively the same.

We expect that measuring these lines in an experiment we should allow to shed
some light on the source of the residual excitation in a modern transmon
systems.