# Microwave quantum diode

Rishabh Upadhyay1 Pico group, QTF Centre of Excellence, Department of Applied
Physics, Aalto University School of Science, P.O. Box 13500, 00076 Aalto,
Finland  Dmitry S. Golubev1 Pico group, QTF Centre of Excellence, Department
of Applied Physics, Aalto University School of Science, P.O. Box 13500, 00076
Aalto, Finland  Yu-Cheng Chang1 Pico group, QTF Centre of Excellence,
Department of Applied Physics, Aalto University School of Science, P.O. Box
13500, 00076 Aalto, Finland  George Thomas 1 Pico group, QTF Centre of
Excellence, Department of Applied Physics, Aalto University School of Science,
P.O. Box 13500, 00076 Aalto, Finland  Andrew Guthrie1 Pico group, QTF Centre
of Excellence, Department of Applied Physics, Aalto University School of
Science, P.O. Box 13500, 00076 Aalto, Finland  Joonas T. Peltonen1 Pico group,
QTF Centre of Excellence, Department of Applied Physics, Aalto University
School of Science, P.O. Box 13500, 00076 Aalto, Finland  Jukka P. Pekola1 Pico
group, QTF Centre of Excellence, Department of Applied Physics, Aalto
University School of Science, P.O. Box 13500, 00076 Aalto, Finland

###### Abstract

The fragile nature of quantum circuits is a major bottleneck to scalable
quantum applications. Operating at cryogenic temperatures, quantum circuits
are highly vulnerable to amplifier backaction and external noise. Non-
reciprocal microwave devices such as circulators and isolators are used for
this purpose. These devices have a considerable footprint in cryostats,
limiting the scalability of quantum circuits. We present a compact microwave
diode architecture, which exploits the non-linearity of a superconducting flux
qubit. At the qubit degeneracy point we experimentally demonstrate a
significant difference between the power levels transmitted in opposite
directions. The observations align with the proposed theoretical model. At -99
dBm input power, and near the qubit-resonator avoided crossing region, we
report the transmission rectification ratio exceeding $90\%$ for a $50$ MHz
wide frequency range from $6.81$ GHz to $6.86$ GHz, and over $60\%$ for the
$250$ MHz range from $6.67$ GHz to $6.91$ GHz. The presented architecture is
compact, and easily scalable towards multiple readout channels, potentially
opening up diverse opportunities in quantum information, microwave read-out
and optomechanics.

Introduction:

Quantum engineering, a dynamic discipline bridging the fundamentals of quantum
mechanics and established engineering fields has developed significantly in
the past few decades. Two-level systems such as superconducting quantum bits
are the building blocks of quantum circuits. Qubits of this type are the most
researched and promising candidates for the realization of quantum information
processing [1, 2, 3, 4, 5]. The characteristics of the superconducting qubits
such as eigen energies, non-linearity, coupling strengths etc. can be tailored
more easily by adjusting the design parameters [6, 7] than some other two-
level microscopic quantum systems [8, 9, 10, 11, 12, 13]. Qubits are
engineered to have large non-linearity, which makes it possible to selectively
address and control them [1, 3, 7, 14]. This dynamic property makes
superconducting qubits a strong candidate for plethora of applications.

Quantum devices operate at low temperatures and require good isolation from
external noises. Microwave devices, such as circulators and isolators, protect
quantum circuits by unidirectionally routing the output signal, whilst
simultaneously isolating noise from the output channel back to the quantum
circuit. Their non-reciprocal character relies on the properties of ferrites
[15, 16, 17]. Ferrite-based non-reciprocal devices are bulky [15, 16, 17], and
they cannot be positioned near the quantum circuit because they require strong
magnetic fields. This limits the scalability of cryogenic quantum circuits
[15, 16, 18, 19]. Various ferrite-free alternatives have been proposed in
literature. These proposals rely on the properties of noble materials [18],
non-linear behavior of artificial atoms [16], dc superconducting quantum
interference devices (dc-SQUID) [20], and arrays of Josephson junctions (JJ’s)
[19, 21, 22, 23]. Our device is a proof of concept (PoC), potentially useful
in the applications relevant to microwave read-out components in the field of
superconducting quantum circuits.

In this work, we present a robust and simple on-chip microwave diode
demonstrating transmission rectification based on a superconducting flux qubit
[14]. The concept of the device is shown in Fig. 1 (a). The flux qubit is
inductively coupled to two superconducting resonators of different lengths
with different coupling strengths. The design details are reported in Section
I. Probing the qubit at the half-flux (degeneracy point) with one tone-
spectroscopy, we observe identical patterns of transmission coefficient for
signals propagating in the opposite directions, which are shifted by 5 dB in
power. This power induces the jump from the low energy to the high energy
state of the qubit. This shift indicates the non-reciprocal behaviour in our
device, expressed in terms of transmission rectification ratio (R) in this
article. We study the transmission rectification ratio, $R$, under different
injected microwave powers and in a wide range of frequencies and magnetic
fluxes applied to the qubit. Due to its strong non-reciprocity, our device
could potentially be utilized as a ferrite free on-chip isolator in a
microwave readout scheme [20, 19, 18, 23]. The strong non-reciprocal behaviour
observed in the reported device is relevant in the field of circuit quantum
thermodynamics (c-QTD) [24, 25] to facilitate and manage the heat flow in
superconducting quantum circuits [26, 27, 28, 29, 30, 31, 32, 30, 33].
Moreover, its compact size makes it suitable for multiple read-out channels.
The possibility to control the transmission rectification ratio with tiny
magnetic fields provides an additional advantage.

Figure 1: (a) Conceptual representation of the device as an artificial atom
coupled to two resonators. The photons pass easily from the right side (green
arrows) to the left side, whereas those coming from left side (red arrow) are
mainly reflected back. (b) Optical microscope image of the device bonded on
gold plated copper sample stage. The RF signal enters either via port 1 or
port 2. Port 3 and Port 4 are the output ports. The black arrows indicate the
direction of signal propagation. (c) Circuit model of the device. Here, $\Phi$
is the external magnetic flux threading through the qubit loop.  Figure 2: (a)
The layout of the device exhibiting the two resonators and the three-junction
superconducting flux qubit at the center. The right and the left resonators
are shown by yellow and by dark-blue colors respectively. The areas enclosed
with white dotted lines at the top-right and bottom-left corner show the
locations of the capacitors coupling the right and the left resonators with
the feedlines. In panels (b) and (f) we show the magnified images of these
capacitors. The gray shaded area close to the center in panel (a) shows a
three-terminal flux qubit. Its zoomed image is shown in panel (d). The qubit
is coupled to both resonators via the local inductances to the left and right,
highlighted with different colors. (c) An enlarged electron micrograph of the
two big junctions of the flux qubit. (e) Electron microscope image of the
smaller qubit junction.  Figure 3: Measurement setup. (a) Different
temperature stages of the fridge with respective attenuation at each stage.
The attenuation in the input lines 1 and 2 are nominally identical. (b) An
enlarged image of the sample setup at the mixing chamber. Input 1 connects to
the port 1 and input 2 — to the port 2. The output ports 3 and 4 are connected
to the low-temperature coaxial microwave switch. The switch is driven by DC
(V) bias. It connects one of its input ports (input 4 in the figure) to the
output and, at the same time, terminates the other input (input 3 in the
figure) at 50 $\mathrm{\Omega}$.

##  I Design and measurement scheme

Design. An optical microscope image of the device is shown in Fig. 1 (b), and
the circuit diagram in Fig. 1 (c). The core element of the device is a three-
junction superconducting flux qubit [14, 34]. Its magnified image is presented
in Fig. 2 (d). The superconducting loop of the flux qubit contains Josephson
junctions. The areas of junctions numbered as 2 and 3 are nominally the same,
whereas the junction 1 is smaller. Hence, the critical currents of junctions 2
and 3 are nominally equal, $I_{C2}=I_{C3}=I_{C}$, and that of the first
junction is given by $I_{C1}=\alpha I_{C}$, where we estimate $\alpha=0.632$.
The flux qubit has two superconducting islands, which we number as the island
1 and the island 2. The total capacitance of island 1 is given by
$C_{G1}=C_{1}+C_{2}+C_{g1}$, where $C_{1}$, $C_{2}$ are the capacitances of
the junctions 1, 2 and $C_{g1}$ the capacitance to the ground plane.
Analogously, the total capacitance of the second island is
$C_{G2}=C_{2}+C_{3}+C_{g2}$. The flux qubit is inductively coupled to the
resonators 1 and 2 via the inductances $L_{1}$ and $L_{2}$, realized as a
superconducting aluminum wire divided in two parts by the grounding electrode,
see Fig. 2 (d). The corresponding coupling constants are proportional to the
inductances, $g_{j}\propto L_{j}$ ($j=1,2$) [35, 36, 37, 38]. The left
resonator (resonator 1) having the designed frequency $f_{1}=6.5$ GHz, is more
weakly coupled to the qubit, while the right resonator (resonator 2), with a
designed frequency of $f_{2}=7.5$ GHz, has stronger coupling, i.e.
$g_{1}<g_{2}$. The frequencies $f_{1}$ and $f_{2}$ cited above are the nominal
values expected in the limit of $\lambda/4$-resonators, which is achieved at
$L_{1},L_{2}\to 0$. At finite qubit resonator coupling they may shift
downwards by an unknown value. Both resonators are coupled to the transmission
lines via nominally equal coupling capacitances as shown in Fig. 2 (b),(f).
Further details of device and its fabrication are reported in Section IV.

Measurement scheme. The device is wire-bonded to a gold-coated printed circuit
board as shown in Fig. 1 (b). The RF spectroscopy measurements have been
performed in a cryo-free dilution refrigerator at the base temperature of 15
mK. The measurement setup is schematically shown in Fig. 3 (a). We have used
two input lines with identical attenuation at different temperature stages of
the fridge. At the output end, we have used a commercial coaxial microwave
switch, which allowed us to connect either port 3 or port 4 of the device to a
single output channel. Under applied DC bias this coaxial microwave switch
connects the output channel to the desired port and terminates the other port
to $50~{}\Omega$ to avoid reflections, as shown in Fig. 3 (b). This setup
eliminates unwanted differences in microwave transmission through output ports
3 and 4, which might otherwise affect the measurements. The signal then
follows an output line that includes two isolators embedded at the mixing
chamber. At the 4 K stage the signal is amplified by a low noise HEMT
amplifier by 42 dB. Furthermore, the signal is amplified at room temperature
by 52 dB. For the flux bias of the qubit we have used a DC-driven global
magnet coil at the mixing chamber, as shown in Fig. 3 (a). We have
characterized the device using one tone and two tone-spectroscopy methods,
briefly discussed in Section II and in Appendix C.

Figure 4: Measured transmission coefficient of the device as a function of the
injected microwave power and frequency. (a) Transmission coefficient
$|S_{31}|^{2}$. (b) Transmission coefficient $|S_{42}|^{2}$. The dotted lines
are the theoretically expected positions of the peak maxima given by Eq. (2).

##  II Results and discussion

In this section we report on one tone-spectroscopy measurements where we study
the qubit-resonator interaction under small applied magnetic fields, and at
various injected microwave powers. Transmission rectification manifests itself
as the difference between the transmission coefficients $S_{31}$ and $S_{42}$.
The transmission coefficient $S_{XY}$ is defined as the ratio of the signal
amplitude coming out of the port X and of that going into the port Y. We tune
the qubit to the degeneracy point, corresponding to half a flux quanta
threading the qubit loop, $\Phi=0.5\Phi_{0}$.

We sweep the probe signal frequency at different probing powers. To minimise
possible errors caused by the attenuation in the input lines, we have applied
an in-situ calibration method described in Section IV under background
calibration subsection. In Figs. 4 (a) and 4 (b) we plot the transmission
coefficient $|S_{31}|^{2}$ and $|S_{42}|^{2}$ obtained in this way.

We explore the range of frequencies around the frequency of the hybrid mode of
the two resonators $f_{h}=6.761$ GHz. This mode is formed because the
resonators 1 and 2 are coupled not only to the qubit, but also to each other
via the inductances $L_{1},L_{2}$. At small microwave powers the frequency of
this mode is shifted upwards dispersively induced by its coupling to the
qubit, where $\chi=22$ MHz, and the transmissions $S_{31}$ and $S_{42}$
exhibit resonances at the frequency $f_{h}+\chi=6.784$ GHz. At high powers the
hybrid mode decouples from the qubit and the resonance frequency moves from
$f_{h}+\chi$ to $f_{h}$. At intermediate power levels the system shows
strongly non-linear behavior typical for the quantum Duffing oscillator [39,
40, 41]. The lowest power, required to drive the system in the strongly non-
linear regime, is observed at resonance frequency $f_{h}+\chi$. We denote such
power as $P^{\rm nl}_{j}$, where the index $j$ indicates the resonator through
which the driving power is injected. In Fig. 4 (a) we indicate the power
$P^{\rm nl}_{1}=-112$ dBm by a solid black arrow, and in Fig. 4 (b) we mark
$P^{\rm nl}_{2}=-117$ dBm by a dotted black arrow. The observed power
difference equals $5$ dB, or $4.3$ fW. It provides power scale of transmission
rectification in our device.

We define the transmission rectification ratio (R) [42, 43] as

$\begin{split}R=\left|\frac{{|S_{42}|^{2}}-{|S_{31}|^{2}}}{{|S_{42}|^{2}}+{|S_{31}|^{2}}}\right|\end{split}$
(1)

In Figs. 5 (a,d,g) we plot this ratio as a function of frequency and magnetic
flux for three different levels of the injected microwave power to show the
increasing trend in transmission rectification ratio with the injected power.
To suppress background noise, in these figures we retain only such data points
where the sum $|S_{42}|^{2}+|S_{31}|^{2}$ exceeds certain threshold value
provided in the label above the graphs. We observe stronger transmission
rectification close to the resonance frequencies corresponding to the hybrid
modes of the resonators and the qubit. These modes are also revealed by the
usual one tone-spectroscopy measurements, see Fig. 8 in Appendix C. To
illustrate the transmission rectification effect further, in Figs. 5
(b,c,e,f,h,i) we plot the transmission coefficient $|S_{42}|^{2}$ and
$|S_{31}|^{2}$ at the same three levels of the microwave power and at two
selected values of the flux, $\Phi/\Phi_{0}=0.45$ and $\Phi/\Phi_{0}=0.5$. At
the lowest power, $-134$ dBm, the transmission rectification happens only very
close to the resonance frequencies, see Fig. 5 (a,b,c). In contrast, at high
power -99 dBm (see Fig. 5 (g,h,i)) the transmission rectification ratio
exceeds $60\%$ in the wide band of $\approx 250$ MHz near the avoided crossing
points, and between $6.81$ GHz to $6.86$ GHz, it exceeds over $90\%$ for a 50
MHz wide range. At the intermediate power -114 dBm, we observe splitting of
the single line in the transmission coefficient $S_{42}$ into two, whereas the
transmission coefficient $S_{31}$ still maintains one line as it does at low
microwave powers (see Fig. 5 (f)).

Figure 5: (a,d,g) The transmission rectification ratio R of Eq.(1) measured at
three different levels of the injected microwave power P, -134 dBm, -114 dBm
and -99 dBm. Here, $\Phi$ is the external magnetic flux and $\Phi_{0}$
$(=h/2e)$ is the magnetic flux quantum. (b,e,h) Transmission coefficient
$|S_{31}|^{2}$ and $|S_{42}|^{2}$ for the same three levels of power and at
the flux value $\Phi/\Phi_{0}=0.45$. (c,f,i) Transmission coefficient
$|S_{31}|^{2}$ and $|S_{42}|^{2}$ at the flux value $\Phi/\Phi_{0}=0.5$.

The origin of the diode effect in our device is the non-linearity of the
qubit. For this reason, single Lorentzian lines in the transmission
coefficients $|S_{31}|^{2}$ and $|S_{42}|^{2}$ centered at frequency
$f_{h}+\chi$ split into two lines above the threshold values of the input
powers $P_{1}^{*}$ and $P_{2}^{*}$. In our sample $P_{1}^{*}>P_{2}^{*}$ (see
Fig. 4). The positions of the two peaks after the splitting, i.e. for
$P>P_{j}^{*}$, are given by the expression

$\displaystyle
f_{\pm}(P_{j})=f_{h}+\chi\pm\frac{\sqrt{2}}{3}\frac{\kappa_{h}}{2\pi}\sqrt{\frac{P_{j}^{\rm
in}}{P_{j}^{*}}-1}.$ (2)

Here $\kappa_{h}/(2\pi)=1.1$ MHz is the line-width of the peak at low power.
The derivation of Eq. (2) is provided in Appendix B. In Fig. 4 we indicate the
peak positions (2) by white dashed lines. They agree well with the
experimentally observed ones. In the experiment we observe the ratio
$P_{1}^{*}/P_{2}^{*}=3.2$, which corresponds to a 5 dB difference in power on
the log-scale. A strong transmission rectification effect with $R\sim 1$
occurs at powers $P\gtrsim P_{2}^{*}/3$, at which the peaks in $|S_{31}|^{2}$
and $|S_{42}|^{2}$ overlap weakly. Theory model of Appendix B predicts that
the ratio $P_{1}^{*}/P_{2}^{*}$ scales as
$P_{1}^{*}/P_{2}^{*}\propto\kappa_{h2}/\kappa_{h1}$, where $\kappa_{h1}$ and
$\kappa_{h2}$ are the partial contributions to the damping rate of the hybrid
mode due to the leakage of the energy via the capacitors $C_{K1}$ and
$C_{K2}$. Therefore, one can further enhance the diode effect in our system by
making these capacitors unequal.

##  III Conclusion

We propose a flux tunable on-chip microwave diode architecture. It is based on
a superconducting flux qubit inductively coupled to two superconducting
resonators. Using one tone-spectroscopy, tuning the qubit to the degeneracy
point by applying the half-flux quantum to it, and performing two separate
measurements with microwave signals coming either through resonator 1 or
resonator 2, we obtain $5$ dB difference between the powers needed to drive
the qubit to the strongly non-linear regime. Furthermore, near the qubit-
resonator avoided crossing region we observe high transmission rectification
ratio $R>90\%$ for a narrow frequency bandwidth of $50$ MHz, and $R>60\%$ for
a wider bandwidth of $250$ MHz, at $-99$ dBm input microwave power. The
transmission rectification ratio is flux tunable and strongly depends on the
input microwave power. Based on the reported resonator-qubit-resonator
geometry, the future goal is to realize tunable photonic quantum heat valve
[5, 33] and quantum heat rectifier [30, 32], in the field of circuit quantum
electrodynamics. Furthermore, the reported strong non-reciprocal behaviour
remains relevant for potential applications in quantum information [5, 25,
24], microwave read-out components [16, 20, 22] and optomechanics [48, 49].

##  IV Methods

### Fabrication

The device was fabricated on a highly resistive 675 $\mu$m thick Si wafer
covered by 30 nm thick dielectric layer of Al2O3, which has been made using
atomic layer deposition (ALD). In the next step, a 200 nm thick layer of
superconducting niobium (Nb) has been sputtered using DC magnetron sputtering
technique. This step has been followed by patterning of the feed lines,
resonators and ground-planes on a 300 nm thick positive electron beam resist.
Post baking is done at 150 ∘C for 5 minutes, followed by the reactive ion
etching (RIE) using CF4 $+$ O2 chemistry. Post baking step allows the resist
to improve the adhesion with the substrate thus minimizing the chances of
etching of unexposed Nb parts. In the second lithography step, a bi-layer
PMMA/MMA resist has been used to write Josephson junctions using the standard
Dolan bridge technique [44]. The exposed substrate has been developed in
Methyl-Isobutyl-Ketone (MIBK): Isopropanol alcohol (IPA) developer solution,
and Methyl-glycol-Methanol solution, respectively. Metal deposition has been
performed using an e-beam evaporator. In-situ argon plasma milling has been
performed on the sample surface to mill the native oxide before the
evaporation. It has been done to obtain a clean contact between Nb layers and
Al layers. Afterwards, a 30 nm thick aluminium metal layer has been evaporated
at $+18^{\circ}$ and it has been oxidized in-situ to create the tunnel barrier
in the junctions. Subsequently, a second 30 nm thick aluminium layer is
evaporated at $-18^{\circ}$. To strip the deposited metal from the unexposed
areas, samples are immersed in hot acetone for 40 minutes. The sample is
visually examined under electron microscopy before proceeding to dicing and
measurements. The room temperature resistance of the test SQUID fabricated on
the same chip to mimic the real device has been measured to be $R\approx
2.6~{}\mathrm{k\Omega}$.

Figure 6: Wide-band transmission coefficient graphs. (a). Transmission
coefficient plots for sides S41 and S32, reconstructed to obtain transmission
baseline for background calibration. (b,c). Calibrated transmission
coefficient plots obtained from Eq. (7) and Eq. (8), measured across the
device at -134 dBm and -74 dBm microwave power.

### Background calibration

To ensure the accuracy of the measurements, we perform an in-situ calibration
as described below. Let us consider the measured transmission coefficient
$S_{O_{4}I_{1}}$ from input 1 ($I_{1}$) to output 4 $(O_{4})$ (see Fig. 3
(b)). It can can be expressed as

$\begin{split}\tilde{S}_{O_{4}I_{1}}=\tilde{S}_{1I_{1}}+\tilde{S}_{41}+\tilde{S}_{O_{4}4}.\end{split}$
(3)

Here $\tilde{S}_{ij}=10\log_{10}(|S_{ij}|^{2})$ is the transmission
coefficient in decibel (dB), $\tilde{S}_{1I_{1}}$ is the attenuation factor
from the input line 1 to the port 1 of the device, $\tilde{S}_{41}$ is the
transmission coefficient between the ports 1 and 4, which we are looking for,
and $\tilde{S}_{O_{4}4}$ is the attenuation between the port 4 of the device
and the input port of the microwave switch.

Similarly, we can write the transmission from the input 2 ($I_{2}$) to the
output 3 $(O_{3})$ as

$\begin{split}\tilde{S}_{O_{3}I_{2}}=\tilde{S}_{2I_{2}}+\tilde{S}_{32}+\tilde{S}_{{O_{3}3}},\end{split}$
(4)

and the transmissions across the device as

$\begin{split}\tilde{S}_{O_{3}I_{1}}=\tilde{S}_{1I_{1}}+\tilde{S}_{31}+\tilde{S}_{O_{3}3},\end{split}$
(5)
$\begin{split}\tilde{S}_{O_{4}I_{2}}=\tilde{S}_{2I_{2}}+\tilde{S}_{42}+\tilde{S}_{O_{4}4}.\end{split}$
(6)

We connect the ports 3 and 4 of the device to the input ports $O_{3}$ and
$O_{4}$ of the microwave switch with the identical relatively short wires.
Therefore, we could assume that $\tilde{S}_{O_{3}3}$ $=$
$\tilde{S}_{O_{4}4}=0$.

Next, we perform spectroscopy at off-resonance frequencies at different flux
points. Due to the strong qubit-resonator coupling the change in the
dispersive shift under the applied magnetic flux is larger than the line-width
of the resonance. Therefore, by sweeping the magnetic flux, we can tune the
device to the off-resonance regime with $\tilde{S}_{41}=\tilde{S}_{32}=0$ for
each frequency and in this way can measure the background transmission
coefficient $\tilde{S}_{O_{4}I_{1}{\rm
bg}}=\tilde{S}_{1I_{1}}+\tilde{S}_{O_{4}4}$ and $\tilde{S}_{O_{3}I_{2}{\rm
bg}}=\tilde{S}_{2I_{2}}+\tilde{S}_{O_{3}3}$ for the entire range of
frequencies relevant for the experiment. They are plotted in Fig. 6 (a).
Subtracting Eq. (3) from Eq. (5), Eq. (4) from Eq. (6), and recalling that
$\tilde{S}_{O_{3}3}$ $=$ $\tilde{S}_{O_{4}4}=0$, we obtain the calibrated
transmission coefficients of our system as

$\tilde{S}_{31}=\tilde{S}_{O_{3}I_{1}}-\tilde{S}_{O_{4}I_{1}{\rm bg}},$ (7)

$\tilde{S}_{42}=\tilde{S}_{O_{4}I_{2}}-\tilde{S}_{O_{3}I_{2}{\rm bg}}.$ (8)

The same method has been used to calibrate the transmissions $S_{41},S_{32}$.

We also note, that even without calibration the maximum observed difference
between the background transmissions is less than $10\%$, while the observed
difference in characteristic input powers $P_{1}^{*}$ and $P_{2}^{*}$ is more
than $50\%$. The calibrated transmission coefficients obtained from Eq. (7)
and Eq. (8) at two different microwave powers are plotted in Fig. 6 (b), (c).
In Fig. 6 (c), corresponding to the high microwave power ($-74$ dBm) applied
to the device at the magnetic flux $\Phi=0.5\Phi_{0}$, we observe that
$\tilde{S}_{31}$ and $\tilde{S}_{42}$ almost coincide in the frequency range
from $4$ GHz to $5.2$ GHz, the difference between them does not exceed 1.5 dB.
In contrast, in the range from $5.2$ GHz to $7.8$ GHz we observe significant
difference, i.e. the diode effect, which reaches 15 dB near the frequency of
the hybrid mode.

##  V Acknowledgment

We acknowledge the financial support from Academy of Finland grants (grant
number 297240, 312057 and 303677), and from the European Union’s Horizon 2020
research and innovation programme under the European Research Council (ERC)
programme (grant number 742559) and Marie Sklodowska-Curie actions (grant
agreements 766025). We sincerely recognize the provision of facilities by
Micronova Nanofabrication Centre, and OtaNano - Low Temperature Laboratory of
Aalto University which is a part of European Microkelvin Platform EMP (grant
number. 824109), to perform this research. We thank and acknowledge VTT
Technical Research Center for provision of high quality sputtered Nb films.

##  VI Author contributions

The conceptual idea was drafted by R.U. and G.T. The device design and
fabrication was done by R.U. The measurements were carried by R.U. The data
analysis is conducted by R.U., Y-C.C. and D.S.G. The theoretical model was
conceived by D.S.G., in collaboration with G.T. The microwave simulations were
carried by Y-C.C. and A.G. In this work D.S.G. and Y-C.C. contributed equally.
The technical support was provided by J.T.P. The work was supported and
supervised by J.P.P. The manuscript was written by R.U., with important
contributions from all the authors.

##  Appendix A Simulations

We used lumped elements circuit simulator (QUCS), and high-frequency RF/MW
electromagnetic analysis (SONNET), to simulate the two meandering wire shunted
resonators to ensure the resonances we measured are reasonable, as shown in
Fig. 7. The lengths of the CPW waveguide part of the resonators are 4620 um on
the left and 4007 um on the right. The corresponding inductance of the left
($L_{1^{\prime}}$) and right ($L_{2^{\prime}}$) meandering wire is $\approx$
0.1 nH and $\approx$ 0.25 nH [37]. From finite element analysis simulator
(COMSOL Multiphysics), we calculate the coupling capacitance value of 6.98 fF
between the transmission line and the resonator ($C_{left}$ and $C_{right}$),
which we use in simulations. The simulated resonance frequencies from QUCS
were 6.072 GHz and 6.394 GHz. Simulations from Sonnet reveals the frequencies
of the center lines at 5.311 GHz and 6.130 GHz. Since in the reported device
we use air bridges to connect the ground planes around the circuit, therefore
we now repeat the simulation in Sonnet using the air bridges. In
superconducting quantum devices, air bridges [45, 46] holds the balance of the
ground planes around the central lines, reduces the possibility of microwave
loss due to mode mixing, and avoids pseudo resonances. The simulated designs
with and without air bridges are reported in Fig. 7 (b,c). After adding the
air bridges to mimic the reported device, the simulated frequencies rise to
5.706 GHz and 6.182 GHz. Both of these frequencies are lower than the designed
bare frequencies of the left and right resonators, this could be due to the
imperfect isolation of the propagating microwave. We observed that the
simulated current-field distribution at the resonance frequency at one side of
the device shows a part of the field leaking across the other side because of
the imbalance of the grounding around the meandering shunt that could make the
wave to propagate along the ground surrounded by the centre line of the off-
resonance resonator. This imbalance issue could be verified when we placed a
perfect conductive block as shown in Fig. 7 (d), between the meandering wire
to connect the upper (ground 1) and lower (ground 2) ground planes of the
device, and as a result we observed that the frequencies rise to the values
very close to QUSC. The experimentally observed lower frequency (6.027 GHz) is
within a reasonable range obtained from Sonnet and QUCS simulations, but the
higher frequency (6.762 GHz) is higher than the simulated frequency from QUCS.
As shown in Fig. 7 (a), adding an inductor with around 0.5 nH between the
floating sides of the meandering wires to mimic the Josephson junction could
increase the frequency to 6.7 GHz. Hence, both of our measured frequencies are
under the reasonable range and are used in Appendix B for calculations.

Figure 7: (a) Simulated circuit design image from QUCS, multiports network
analysis. (b) Simulated design image using Sonnet with no bonding wires. (c)
Al Bonding wires mimicking the real device in panel to connect upper (ground
1) and lower (ground 2) grounds. (d). A metallic block to connect the upper
and lower ground planes.

##  Appendix B Theory

In this Appendix we briefly present the theory of the transmission
rectification effect in our device. The system is described by the Hamiltonian

$\displaystyle H=H_{\rm res}+H_{\rm loop}+H_{\rm int}.$ (9)

The Hamiltonian of the two resonators is given by

$\displaystyle H_{\rm res}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{2}\left[\hbar\omega_{j}a_{j}^{\dagger}a_{j}+\sqrt{\frac{\hbar\kappa_{cj}Z_{0}}{\omega_{j}^{3}}}\frac{dI_{j}(t)}{dt}(a_{j}^{\dagger}+a_{j})\right]$
(10) $\displaystyle-\,\hbar
g_{12}(a_{1}^{\dagger}+a_{1})(a_{2}^{\dagger}+a_{2}),$

where $\omega_{j}=2\pi f_{j}$ are the angular frequencies of the fundamental
modes of the resonators 1 and 2 (in our sample $\omega_{2}>\omega_{1}$),
$a_{j}$ are the corresponding ladder operators, $\kappa_{c1},\kappa_{c2}$ are
the damping rates of the resonators due to their capacitive coupling to the
transmission lines,

$\displaystyle\kappa_{cj}=\frac{2\omega_{j}^{3}Z_{0}^{2}C_{Kj}^{2}}{\pi},$
(11)

$Z_{0}$ is the resonator impedance, $C_{K1},C_{K2}$ are the capacitors
connecting the resonators to the transmission lines (see Fig. 1c),
$I_{1}(t),I_{2}(t)$ are the input microwave currents which are related to the
incoming powers as $P_{j}^{\rm in}=Z_{0}\overline{I_{j}^{2}(t)}$ (here bar
implies the time averaging), and $g_{12}$ is the coupling strength between the
two resonators. The Hamiltonian of the superconducting loop $H_{\rm loop}$ is
expressed in terms of the ladder operators $b,b^{\dagger}$ describing its low-
frequency mode, and the flux dependent qubit frequency $\omega_{0}(\Phi)=2\pi
f_{01}(\Phi)$,

$\displaystyle H_{\rm
loop}=\hbar\omega_{0}(\Phi)b^{\dagger}b-\frac{E_{C}}{12}(b^{\dagger}+b)^{4}.$
(12)

Here $E_{C}$ is the effective charging energy of the low frequency mode of the
loop, which determines the anharmonicity of the qubit. From the spectroscopic
measurements presented in Fig. 8 (e,f) we estimate $E_{C}/(2\pi\hbar)\approx
200$ MHz. Finally, the last term in the Hamiltonian (13) describing the
interaction between the loop and the resonators has the form

$H_{\rm int}=-\hbar g_{1}(a_{1}^{\dagger}+a_{1})(b^{\dagger}+b)-\hbar
g_{2}(a_{2}^{\dagger}+a_{2})(b^{\dagger}+b),$ (13)

where $g_{1}$ and $g_{2}$ describe the coupling between the qubit and the
corresponding resonator.

We diagonalize the Hamiltonian of the two resonators (10) and introduce the
hybrid modes with frequencies

$\displaystyle\omega_{h,l}=\sqrt{\frac{\omega_{1}^{2}+\omega_{2}^{2}\pm\sqrt{(\omega_{2}^{2}-\omega_{1}^{2})^{2}+16g_{12}^{2}\omega_{1}\omega_{2}}}{2}}.$
(14)

In the transmission rectification experiment we probe the range of frequencies
close to $f_{h}=\omega_{h}/(2\pi)=6.762$ GHz. For this reason, we leave only
the high frequency mode and write the Hamiltonian of the resonators (10) and
the interaction term (13) in the form

$\displaystyle H_{\rm res}$ $\displaystyle=$
$\displaystyle\hbar\omega_{h}a_{h}^{\dagger}a_{h}+\left(\frac{\sqrt{\kappa_{h1}}}{\omega_{1}^{2}}\frac{dI_{1}(t)}{dt}\right.$
(15)
$\displaystyle\left.+\,\frac{\sqrt{\kappa_{h2}}}{\omega_{2}^{2}}\frac{dI_{2}(t)}{dt}\right)\sqrt{\hbar\omega_{h}Z_{0}}(a_{h}^{\dagger}+a_{h}),$
$\displaystyle H_{\rm int}$ $\displaystyle=$ $\displaystyle-\hbar
g_{h}(a_{h}^{\dagger}+a_{h})(b^{\dagger}+b).$ (16)

Here $a_{h},a_{h}^{\dagger}$ are the ladder operators of the hybrid mode with
the angular frequency $\omega_{h}=2\pi f_{h}$, $\kappa_{h1}$ and $\kappa_{h2}$
are the partial contributions to the total damping rate of this mode coming
from the coupling of the resonators 1 and 2 to the transmission lines,

$\displaystyle\kappa_{h1}=\frac{\omega_{1}}{\omega_{h}}\sin^{2}\theta\,\kappa_{c1},\;\;\;\kappa_{h2}=\frac{\omega_{2}}{\omega_{h}}\cos^{2}\theta\,\kappa_{c2},$
(17)

and the angle $\theta$ is determined by

$\displaystyle\sin
2\theta=\frac{4g_{12}\sqrt{\omega_{1}\omega_{2}}}{\sqrt{(\omega_{2}^{2}-\omega_{1}^{2})^{2}+16g_{12}^{2}\omega_{1}\omega_{2}}}.$
(18)

The total damping rate of the hybrid mode is
$\kappa_{h}=\kappa_{h1}+\kappa_{h2}+\kappa_{hi}$, where $\kappa_{hi}$
describes the internal damping in the resonators.

Based on (15) we observe that the input currents $I_{1}^{\rm in}$ and
$I_{2}^{\rm in}$ have different pre-factors. This difference is the origin of
the asymmetry in our system. From this pre-factors we can determine the ratio
of the powers $P_{1}^{*}$ and $P_{2}^{*}$, above which single transmission
lines in the coefficients $|S_{31}|^{2}$ and $|S_{42}|^{2}$ split into two
lines, without solving the problem. Namely, we find

$\displaystyle\frac{P_{1}^{*}}{P_{2}^{*}}=\frac{\omega_{1}^{4}\kappa_{h2}}{\omega_{2}^{4}\kappa_{h1}}.$
(19)

The powers $P_{1}^{*}$ and $P_{2}^{*}$ are not equal because the frequencies
of the two resonators $f_{1}$ and $f_{2}$ and/or the couplings between the
hybrid mode and the two transmission lines, $\kappa_{h1}$ and $\kappa_{h2}$,
differ from each other. This makes our device a non-linear system with
asymmetric coupling to the two ports. According to the theory, exactly these
properties are required for transmission rectification, see e.g. Ref. [47]. In
our device the coupling capacitors $C_{K1}$ and $C_{K2}$ are nominally equal,
$C_{K1}=C_{K2}$. Therefore the ratio (19) can be simplified to
$P_{1}^{*}/P_{2}^{*}=\cot^{2}\theta$. In the experiment we find
$P_{1}^{*}/P_{2}^{*}=3.2$, see Fig. 4, which corresponds to $\theta=0.51$. To
obtain this value from Eq. (18) and, at the same time, to reproduce the
experimentally observed frequencies of the modes $f_{h}=6.762$ GHz and
$f_{l}=6.026$ GHz from Eq. (14), we choose $f_{1}=6.209$ GHz, $f_{2}=6.595$
GHz and $g_{12}/(2\pi)=313$ MHz. The simulations reported in Appendix A show
that these parameters are reasonable. To verify our model further, we have
estimated the values of the coupling capacitors in COMSOL and found
$C_{K1}=C_{K2}=7$ fF. Adopting this value and the parameters given above, from
Eq. (11) we obtain $\kappa_{c1}/(2\pi)=732.6$ kHz, $\kappa_{c2}/(2\pi)=878$
kHz, and from Eq. (17) we find $\kappa_{h1}/(2\pi)=160$ kHz,
$\kappa_{h2}/(2\pi)=653$ kHz. The damping rates $\kappa_{h1},\kappa_{h2}$ can
be independently estimated by fitting the transmission coefficients
$|S_{41}|^{2}$ and $|S_{32}|^{2}$ at zero magnetic flux, where the qubit is
decoupled from the resonator, to the expressions resulting from our model
Hamiltonian (10):

$\displaystyle|S_{41}|^{2}$ $\displaystyle=$ $\displaystyle
1-\frac{\omega_{h}^{4}}{\omega_{1}^{4}}\frac{\kappa_{h1}^{2}+2\kappa_{h1}\kappa_{h2}}{4(\omega-\omega_{h})^{2}+\kappa_{h}^{2}},$
$\displaystyle|S_{32}|^{2}$ $\displaystyle=$ $\displaystyle
1-\frac{\omega_{h}^{4}}{\omega_{2}^{4}}\frac{\kappa_{h2}^{2}+2\kappa_{h1}\kappa_{h2}}{4(\omega-\omega_{h})^{2}+\kappa_{h}^{2}}.$
(20)

Such fitting procedure gives $\kappa_{h1}/(2\pi)=160$ kHz,
$\kappa_{h2}/(2\pi)=430$ kHz and $\kappa_{h}/(2\pi)=787$ kHz. While
$\kappa_{h1}$ agrees with the theoretical estimate given above, the
experimental rate $\kappa_{h2}$ is a bit lower than the theoretical
prediction. With these parameters the ratio of the powers (19) becomes $2.1$,
which is still not very far from the result of the measurements. Thus, we have
confirmed that our observations reasonably well agree with the model.

Figure 8: One tone and two tone-spectroscopy for -134 dBm input probing power.
Panels (a) and (b) show the one tone-spectroscopy data, and panel (c) — two
tone-spectroscopy data, with probe signal coming through the port 1.
Analogously, panels (d) and (e) show one tone-spectroscopy and (f) — two tone-
spectroscopy, with the probe signal arriving through the port 2. In panels (c)
and (f) the pump tone power (Rabi power) is -94 dBm.

The dependence of the transmission coeffcients $|S_{31}|^{2}$ and
$|S_{42}|^{2}$ on power shown in Fig. 4 can be understood as follows. At
sufficiently low power we can approximately replace the two resonators and the
qubit by a single non-linear system with the Hamiltonian

$\displaystyle H$ $\displaystyle=$
$\displaystyle\hbar\omega_{r}a^{+}a+\frac{\hbar K}{6}(a^{\dagger}+a)^{4}$ (21)
$\displaystyle+\,2\hbar(\epsilon_{1}+\epsilon_{2})\cos\omega
t(a^{\dagger}+a).$

Here $\omega_{r}=\omega_{h}+2\pi\chi$ is the frequency of the hybrid mode
shifted due to the interaction with the qubit, $K$ is the Kerr non-linearity
of the combined system and

$\displaystyle\epsilon_{j}=\frac{\omega_{h}^{2}}{\omega_{j}^{2}}\sqrt{\frac{\kappa_{hj}P_{j}^{\rm
in}}{2\hbar\omega_{h}}}.$ (22)

It has been experimentally shown that this approximation well describes
systems similar to ours [39, 40]. To find the the transmission coefficient
$|S_{31}|^{2}$ at low power we put $\epsilon_{2}=0$ and constract the
following ratio:

$\displaystyle|S_{31}|^{2}=\frac{P_{3}}{P_{1}^{\rm
in}}=\frac{\kappa_{h2}\hbar\omega_{r}}{2P_{1}^{\rm in}}\langle
a^{\dagger}a\rangle.$ (23)

Here $P_{3}$ is the power coming to the port 3 and the factor 2 in the
denominator accounts for the equal sptitting of the power coming out of the
resonator 2 between the ports 2 and 3, i.e. we put
$\kappa_{h2}\hbar\omega_{r}\langle a^{\dagger}a\rangle=P_{2}+P_{3}$ and assume
that $P_{2}=P_{3}$. The average value $\langle a^{\dagger}a\rangle$ for the
Hamiltonian (21) has been evaluted in Ref. [41]. Based on this result we
obtain the expression for the transmission coefficient in the form

$\displaystyle|S_{31}|^{2}\approx\frac{\kappa_{h1}\kappa_{h2}}{4(\omega-\omega_{r})^{2}+\kappa_{h}^{2}}$
$\displaystyle\times\,\frac{\left|{}_{0}F_{2}\left(1-\frac{\omega-\omega_{r}-i\frac{\kappa_{h}}{2}}{K},-\frac{\omega-\omega_{r}+i\frac{\kappa_{h}}{2}}{K},\frac{2\kappa_{h}^{2}P_{1}^{\rm
in}}{9K^{2}P_{1}^{*}}\right)\right|^{2}}{\left|{}_{0}F_{2}\left(-\frac{\omega-\omega_{r}-i\frac{\kappa_{h}}{2}}{K},-\frac{\omega-\omega_{r}+i\frac{\kappa_{h}}{2}}{K},\frac{2\kappa_{h}^{2}P_{1}^{\rm
in}}{9K^{2}P_{1}^{*}}\right)\right|^{2}},\;\;\;\;\;\;\;$ (24)

where ${}_{0}F_{2}(x)$ is the generalized hypergeometric function and

$\displaystyle
P_{1}^{*}=\frac{2}{9}\frac{\hbar\kappa_{h}^{2}\omega_{1}^{4}}{\kappa_{h1}\omega_{h}^{3}}$
(25)

is the power at which the Lorentzian peak in $|S_{31}|^{2}$ splits into two.
The transmission coefficient $|S_{42}|^{2}$ and the power $P_{2}^{*}$ are
given by the same experssions with the interchanged indexes 1 and 2. Analyzing
the expression (24), one can show that in the limit $\kappa_{h}\ll K$ the
maxima of the two peaks appearing at $P_{1}^{\rm in}>P_{1}^{*}$ occur at
frequencies

$\displaystyle\omega_{\pm}=\omega_{r}\pm\frac{\sqrt{2}\kappa_{h}}{3}\sqrt{\frac{P_{1}^{\rm
in}}{P_{1}^{*}}-1}.$ (26)

Inverting this formula, we find that for a given probe frequency $\omega$ the
peak in the transmisssion coefficient occurs at the power

$\displaystyle P_{\rm
peak,1}=P_{1}^{*}\left(1+\frac{9(\omega-\omega_{r})^{2}}{2\kappa_{h}^{2}}\right).$
(27)

In Fig. 4 we show the powers $P_{\rm peak,1}$ and $P_{\rm peak,2}$ by the
white dashed lines. We have used the damping rate $\kappa_{h}=1.1$ MHz, which
has been obtained by fitting the transmission coefficients (20) at low powers
and at the flux value $\Phi=0.5\Phi_{0}$, at which the data shown in Fig. 4
have been gathered. The threshold powers $P_{1}^{*}=-112$ dBm and
$P_{2}^{*}=-117$ dBm have been treated as fitting parameters. From the fits we
also estimate the anharmonicity as $K/(2\pi)\approx-11.5$ MHz.

Finally, we note that at sufficently high power the assumption about weak non-
linearity of the qubit becomes insufficient, and one should consider full
sinusoidal current-phase relation for the three Joesephon junctions of the
qubit. Here we do not consider this regime. It is well known, however, that in
this limit the resonator becomes decoupled from the qubit, and the
transmission lines both in $|S_{31}|^{2}$ and in $|S_{42}|^{2}$ shift to the
bare frequency of the hybrid mode $f_{h}$, see Fig. 4.

##  Appendix C Two tone-Spectroscopy

To determine the qubit transition frequencies we have performed the two-tone
spectroscopy as follows. For every value of magnetic flux we choose the probe
frequency of a continuous weak microwave signal ‘probe tone’ (tone one) using
a vector network analyzer (VNA). Once the flux specific probe frequency is
chosen, using a separate microwave signal generator a ‘pump tone’ (tone two)
is applied to excite the qubit energy levels. The combined results of the one
tone and the two tone spectroscopies are presented in Fig. 8.

To fit the obtained spectra numerically and to find the coupling constants
between the qubit and the resonators, we adapted the following procedure.
First, we diagonalize the Hamiltonian of the flux qubit using two dimensional
plane waves as described in article [37]. In this way, we find the dependence
of the transition frequency $f_{01}(\Phi)$ between the lowest and the first
excited states of the qubit decoupled from the resonators on the magnetic flux
$\Phi$. Fitting the obtained $f_{01}(\Phi)$ dependence to the results of
spectroscopy away from the anti-crossing points, we estimate the parameters of
the flux qubit. Namely, we find the asymmetry parameter $\alpha=0.632$ and the
Josephson energies of the two bigger junctions of the SQUID loop,
$E_{J}/(2\pi\hbar)=I_{C}/4\pi e=37.5$ GHz. Next, we use the equation
describing two coupled oscillators,

$\displaystyle
f_{r}=\sqrt{\frac{f_{i}^{2}+f_{01}^{2}\pm\sqrt{\left(f_{i}^{2}-f_{01}^{2}\right)^{2}+16g_{i}^{2}f_{i}f_{01}}}{2}},$
(28)

to fit every avoided crossing between the hybrid modes of the resonator
(indicated by the index $i$) and the flux dependent qubit frequency in Fig. 8.
In the range of frequencies shown there we observe two hybrid resonator modes,
low frequency mode (index $l$) and high frequency mode (index $h$). The
frequencies of these modes are $f_{l}=6.027$ GHz and $f_{h}=6.762$ GHz, and
the corresponding couplings are $g_{l}\approx 1$ MHz and $g_{h}\approx 175$
MHz.

## References

  * [1] Clarke, J. & Wilhelm, F.K. Superconducting quantum bits. Nature 453, 1031 (2008).
  * [2] Wendin, G., & Shumeiko, V.S. Superconducting Quantum Circuits, Qubits and Computing. arXiv:cond-mat/0508729 (2005).
  * [3] Devoret, M.H., Wallraff, A. & Martinis, J.M. Superconducting Qubits: A Short Review. arXiv:cond-mat/0411174 (2004).
  * [4] Gu, X., Kockum, A.F., Miranowicz, A., Liu, Y. & Nori, F. Microwave photonics with superconducting quantum circuits. Physics Reports 718, 1 (2017).
  * [5] Ronzani, A., Karimi, B., Senior, J., Chang, Y.-C., Peltonen, J.T., Chen, C. & Pekola, J.P. Tunable photonic heat transport in a quantum heat valve. Nat. Phys. 14, 991 (2018).
  * [6] Rasmussen, S.E., Christensen, K.S., Pedersen, S.P., Kristensen, L.B., Bækkegaard, T., Loft, N.J.S., & Zinner, N.T. Superconducting circuit companion—an introduction with worked examples. PRXQuantum 2 , 040204 (2021).
  * [7] Orlando, T. P., Mooij, J. E., Tian, L., Van der Wal, C. H., Levitov, L. S., Lloyd, S. & Mazo, J. J. A superconducting persistent-current qubit. Phys. Rev. B 60, 15398 (1999).
  * [8] Loss, D. & DiVincenzo, D.P., Quantum computation with quantum dots. Phys. Rev. A 57, 120 (1998).
  * [9] Kane, B.E., A silicon-based nuclear spin quantum computer.  Nature 393, 133 (1998).
  * [10] Goto, A., Hashi, K., Ohki, S. & Shimizu, T. Optically induced nuclear spin–spin couplings in GaAs manifested by spin echo decays under optical pumping. Quantum Inf 8, 59 (2022).
  * [11] Petta, J. R., Johnson, A. C., Taylor, J. M., Laird, E.A., Yacoby, A., Lukin, M. D., Marcus, C. M., Hanson,M. P. & Gossard, A. C. Coherent manipulation of coupled electron spins in semiconductor quantum dots. Science 309, 2180 (2005).
  * [12] Wang, Z., Yuan, Y., Liu, X., Muruganathan, M., Mizuta, H. & Sun, J. Double quantum dot-like transport in controllably doped graphene nanoribbon. Appl. Phys. Lett. 118, 083105 (2021).
  * [13] Hanson, R., Kouwenhoven, L. P., Petta, J. R., Tarucha, S. & Vandersypen, L. M. K. Spins in few-electron quantum dots. Rev. Mod. Phys. 79, 1217 (2007).
  * [14] Mooij, J. E., Tian, L., Van der Wal, C. H., Levitov, L. S., Lloyd, S. & Mazo, J. J. Josephson persistent-current qubit. Science 285, 1036 (1999).
  * [15] Pozar, D. M. Microwave Engineering, 4th ed. Wiley, Hoboken, NJ, (2012).
  * [16] Hamann, A. R., Müller, C., Jerger, M., Zanner, M., Combes, J., Pletyukhov, M., Weides, M., Stace, T. M. & Fedorov, A. Nonreciprocity Realized with quantum nonlinearity. Phys. Rev. Lett. 121, 123601 (2018).
  * [17] Kodera, T., Sounas, D.L. & Caloz, C. Magnetless Nonreciprocal Metamaterial (MNM) technology: Application to microwave components. IEEE Transactions on Microwave Theory and Techniques 61, 1030 (2018).
  * [18] Viola, G. & DiVincenzo, D.P. Hall effect gyrators and circulators. 4, 021019 (2014).
  * [19] Ranzani, L., Kotler, S., Sirois, A.J., DeFeo, M. P., Beltran, M. C., Cicak, K., Vale, L. R. & Aumentado, J. Wideband isolation by frequency conversion in a Josephson-junction Transmission Line. Phys. Rev. Applied 8, 054035 (2017).
  * [20] Kamal, A., Clarke, J. & Devoret, M. H. Gain, directionality, and noise in microwave SQUID amplifiers: Input-output approach.  Phys. Rev. B 86, 144510 (2012).
  * [21] Yaakobi, O., Friedland, L., Macklin, C. & Siddiqi, I. Parametric amplification in Josephson junction embedded transmission lines.  Phys. Rev. B 87, 144301 (2013).
  * [22] Lecocq, F., Ranzani, L., Peterson, G. A., Cicak, K., Simmonds, R. W., Teufel, J. D. & Aumentado, J. Nonreciprocal microwave signal processing with a field-programmable Josephson amplifier. Phys. Rev. Applied 7, 024028 (2017).
  * [23] O’Brien, K., Macklin, C., Siddiqi, I. & Zhang, X. Resonant phase matching of Josephson junction traveling wave parametric amplifiers.  Phys. Rev. Lett 113, 157001 (2014).
  * [24] Vinjanampathy, S. & Anders, J. Quantum thermodynamics. Contemporary Physics 57, 545 (2016).
  * [25] Kosloff, R. & Amikam, L. Quantum heat engines and refrigerators: continuous devices. Annu. Rev. Phys. Chem. 65, 365 (2014).
  * [26] Erdman, P. A., Cavina, V., Fazio, R., Taddei, F. & Giovannetti, V. Maximum power and corresponding efficiency for two-level heat engines and refrigerators: optimality of fast cycles. New J. Phys. 21, 103049 (2019).
  * [27] Menczel, P., Pyhäranta, T., Christian, F.& Kay, B. Two-stroke optimization scheme for mesoscopic refrigerators. Phys. Rev. B 3, 224306 (2019).
  * [28] Giazotto, F. & Martínez-Pérez, M. The Josephson heat interferometer. Nature 492, 401–405 (2012).
  * [29] J. O. Miranda, J. O., Joulain, K., Meneses, D., Ezzahri, Y. & Drevillon, J. Photonic thermal diode based on superconductors. Journal of Applied Physics 122, 093105 (2017).
  * [30] Roßnagel, J., Dawkins, S. T., Tolazzi, K. N., Abah, O., Lutz, E., Kaler, F. S. & Singer, K. A single-atom heat engine. Science. 352, 325 (2016).
  * [31] Karimi, B. & Pekola, J.P. Otto refrigerator based on a superconducting qubit: Classical and quantum performance. Phys. Rev. B 94, 184503 (2016).
  * [32] Senior, J., Gubaydullin, A., Karimi, B., Peltonen, J. T., Ankerhold, J. & Pekola,J. P. Heat rectification via a superconducting artificial atom. Commun Phys 3, 40 (2020).
  * [33] Lu, Y ., Lambert, N., Kockum, A. F., Funo. K., Bengtsson, A., Gasparinetti. S., Nori, F., & Delsing, P. Steady-State Heat Transport and Work With a Single Artificial Atom Coupled to a Waveguide: Emission Without External Driving. PRX Quantum 3, 020305 (2022).
  * [34] Yan, F., Gustavsson, S., Kamal, A. et al. The flux qubit revisited to enhance coherence and reproducibility. Nat Commun 7, 12964 (2016).
  * [35] Abdumalikov, A. A., Astafiev, O., Nakamura, Y., Pashkin, Yu. A. & Tsai, J. Vacuum Rabi splitting due to strong coupling of a flux qubit and a coplanar-waveguide resonator. Phys. Rev. B 78, 180502(R) (2008).
  * [36] Bourassa, J., Gambetta, J. M., Abdumalikov, A. A. Jr., Astafiev, O., Nakamura, Y. & Blais, A. Ultrastrong coupling regime of cavity QED with phase-biased flux qubits. Phys. Rev. A 80, 032109 (2009).
  * [37] Upadhyay, R., Thomas, G., Chang, Y-C., Golubev, D.S., Guthrie, A., Gubaydullin, A., Peltonen, J. T. & Pekola, J.P. Robust Strong-Coupling Architecture in Circuit Quantum Electrodynamics. Phys. Rev. Applied 16, 044045 (2021).
  * [38] Miyanaga, T., Tomonaga, A., Ito, H., Mukai, H. & Tsai, J.S. Ultrastrong tunable coupler between superconducting LC resonators. Phys. Rev. Applied 16, 064041 (2021).
  * [39] Yamamoto, T., Inomata, K., Koshino, K., Billangeon, P.-M., Nakamura, Y. & Tsai, J. S. Superconducting flux qubit capacitively coupled to an LC resonator. New J. Phys. 16, 015017 (2014).
  * [40] Yamaji, T., Kagami, S., Yamaguchi, A., Satoh, T., Koshino, K., Goto, H., Lin, Z. R., Nakamura, Y., & Yamamoto, T. Spectroscopic observation of the crossover from a classical Duffing oscillator to a Kerr parametric oscillator. Phys. Rev. A 105, 023519 (2022).
  * [41] Drummond, P. D., and Walls, D. F. Quantum theory of optical bistability. I: Nonlinear polarisability model. J. Phys. A: Math. Gen. 13, 725 (1980).
  * [42] Fratini, F., Mascarenhas, E., Safari, L., Poizat,J-Ph., Valente, D., Auffèves, A., Gerace, D.& Santos, M.F. Fabry-Perot interferometer with quantum mirrors: Nonlinear light transport and rectification. Phys. Rev. Lett. 24, 243601 (2014).
  * [43] Marcos, A.V., Lopez, C.J., Ruiz, M.G.& Sanchez, R. Thermal rectification with interacting electronic channels: Exploiting degeneracy, quantum superpositions, and interference. PhysRevB. 98, 035414 (2018).
  * [44] G.J.Dolan. Offset masks for lift‐off photoprocessing. Appl. Phys. Lett. 31, 337 (1977).
  * [45] Abuwasib, M., Krantz, P.& Delsing, P. Fabrication of large dimension aluminum air-bridges for superconducting quantum circuits. Journal of Vacuum Science & Technology B 31, 031601 (2013).
  * [46] Janzen, N., Kononenko, M.,Ren, s. & Lupascu, A. Aluminum air bridges for superconducting quantum devices realized using a single-step electron-beam lithography process. Phys. Lett. 121, 094001 (2022).
  * [47] Segal, D. & Nitzan, A. Spin-Boson Thermal Rectifier. Phys. Rev. Lett. 94, 034301 (2005).
  * [48] Pirkkalainen, JM., Cho, S., Massel, F., Tuorila. J., Heikkilä. T.T., Hakonen. P.J. & Sillanpää, M.A. Cavity optomechanics mediated by a quantum two-level system. Nat Commun 6, 6981 (2015).
  * [49] Barzanjeh, S., Xuereb, A., Gröblacher, S., Paternostro, M., Regal A. C. & Weig. M. E. Optomechanics for quantum technologies. Nat. Phys. 18, 15–24 (2022).


# Bi-functional metamaterial based on Helmholtz resonators for sound and heat
insulation

Brahim Lemkalli<EMAIL_ADDRESS>Zine El Abiddine Fellah
Sébastien Guenneau  Khalid Lamzoud  Abdellah Mir  Younes Achaoui  Laboratory
of Optics, Information Processing, Mechanics, Energetics and Electronics,
Department of Physics, Moulay Ismail University, B.P. 11201, Zitoune, Meknes,
Morocco  Laboratoire de Mécanique et d’Acoustique, LMA-UMR 7031 Aix-Marseille
University-CNRS-Centrale Marseille, F-13453 Marseille CEDEX 13, France  The
Blackett Laboratory, Physics Department, Imperial College London, SW7 2AZ,
London, UK  UMI 2004 Abraham de Moivre-CNRS, Imperial College London, SW7 2AZ,
London, UK  Department of Physics, Moulay Ismail University, B.P. 11201,
Zitoune, Meknes, Morocco

###### Abstract

Over the last few decades, both heat and broadband sound reduction have become
increasingly significant as a result of concerns about the environment and
noise pollution. In order to address this challenge, we provide a finite
element analysis study of an acoustic metamaterial panel consisting of a unit
cell made of two Helmholtz Resonators with a guide in between. These panels
can attenuate and control both sound propagation and heat flux. For noise
pollution in a building, we first determine the geometric dimension that
corresponds to the operative frequency range. Furthermore, we investigate the
sound transmission loss of the proposed panel as a function of the periodicity
of an array of unit cells. Additionally, we investigate the thermal flux
induced by the panel, especially within a $24$-hour period. The simulation
results show that the proposed panel provides a level of sound attenuation
within the frequency range of $400$ $\mathrm{H}\mathrm{z}$ to $2.5$
$\mathrm{k}\mathrm{H}\mathrm{z}$, as well as interesting heat protection. The
structure is compared to panels made up of a homogeneous medium and without
Helmholtz resonators.

###### keywords:

Acoustic metamaterial panels , Sound insulation , Thermal insulation.

##  1 Introduction

Climate change and noise pollution have emerged as major global issues in
recent decades [1, 2]. This has generated acoustic and thermal insulation
challenges, which usually raise the issue of comfort inside buildings [3, 4].
This also results in a substantial increase in energy consumption, which
induces further reliance on construction materials [5]. Different methods
based on common building material applications have been examined in order to
optimize overall acoustic and thermal performance [6]. These materials include
organic, inorganic, transparent, and bio-insulation materials, as well as
recycled materials [7, 8, 9, 10, 6, 11]. Based on the aforementioned
materials, it is incredibly difficult to maximize both the acoustic and
thermal performances of a building at the same time. This is due to the
multiple physical processes associated with these two fields of physics.

Recent advances have proven that artificial man-made materials, called
”metamaterials, can successfully control sound wave and heat flux propagation.
The optimization of these structures can provide novel functionalities based
on distinctive physical properties not found in nature [12, 13, 14]. Thanks to
their recent advances, metamaterials have recently sparked the interest of
academics all over the world.

Acoustic metamaterials have provided unprecedented noise reduction solutions,
especially with regards to mitigating noise propagation. Several variants of
metamaterials, such as membranes [15, 16, 17, 18], space coiling [19, 20],
Fabry-Perot structures [21], quarter-wavelength resonators [22], and Helmholtz
resonators [23, 24, 25], have different characteristics and advantages to
enhance sound attenuation. Besides, thermal metamaterials have demonstrated
that heat flow may be controlled at will through artificial composite
structures [13, 26]. Heat flux manipulation can be divided into two categories
based on the geometric size of thermal metamaterials. The first is that the
heat flow could be macroscopically steered throughout thermodynamic
conversion, which has been shown to be an effective way to manage heat energy
[27, 28]. The second is that nanoscale structures can control the wavelength
of the phonons propagation [29, 30]. Besides the previously described acoustic
and thermal metamaterials, the concept of multifunctional metamaterials, which
can be used in several physical domains, was also recently proposed [31, 32,
33].

Figure 1: Schematic details of metamaterial based on Helmholtz resonators,
including a 3D panel (left) and a cross section in the $xy$-plane (right). The
periodicity constants are $a$ along the $y$-axis and $(b+e)$ along the
$x$-axis, with $b=2a$ and $e=0.06a$. $N$ is the number of cells along the
$y$-axis. The neck size is $d=0.13a$, the length of the small HR is $l=0.33a$,
the length of the enormous HR is $L=1.16a$, and $c=0.16a$.

In this paper, we propose to use Helmholtz resonators (HRs), which are highly
classic acoustic systems that have been intensively explored for decades [34],
in order to attenuate sound propagation and confine heat spreading. We suggest
panels composed of two symmetrical HRs with periodicity constants ; $a$ along
the $y$-direction and $(b+e)$ along the $x$-direction, where $N$ is the cell
number corresponding to the $y$-direction. The schematic of the acoustic
metamaterial is shown in Fig. 1. Each unit cell is made up of two Helmholtz
resonators with two square-shaped necks, as illustrated in the exploded view.
We analyze the acoustic and thermal behaviors of these panels using the finite
element method through two studies: the first aims to determine the sound
transmission loss response, and the second involves a temperature analysis
that entails evaluating the thermal resistance of these HRs-based panels. We
compare the acoustic and thermal properties of the proposed panels to
classical plates composed of homogeneous medium and hollowed panels, i.e., HRs
without necks.

##  2 Numerical Methods

Numerical models of the acoustic and thermal behaviors of panels are based on
2D calculations. This is due to the symmetry of HRs-based panels in the
$z$-direction. In this context, we conduct two studies: the sound transmission
loss at low frequencies and the thermal resistance within a $24$-hour period,
taking behavior properties numerically within the variation of the number of
cells ($N$) in the 2D panel from one to five, and comparing the responses of
these panels based on HRs with classical panels as well as hollowed panels
(closed cavities). In Table 1, we describe the mechanical and thermal
properties of the materials used in computations.

Table 1:  Mechanical and thermal material properties | Concrete |
---|---|---
Young modulus ($E$) |  |  $25\times 10^{9}$ $\mathrm{P}\mathrm{a}$
Poisson’s ratio ($\nu$) |  | $0.2$
Density ($\rho_{s}$) |  |  $2300$ $\mathrm{k}\mathrm{g}\mathrm{/}\mathrm{m}^{3}$
Thermal conductivity ($k_{s}$) |  |  $1.8$ $\mathrm{W}\mathrm{/}\mathrm{(}\mathrm{m}\mathrm{K}\mathrm{)}$
Specific heat ($c_{s}$) |  |  $880$ $\mathrm{J}\mathrm{/}\mathrm{(}\mathrm{k}\mathrm{g}\mathrm{K}\mathrm{)}$
| Air |
Sound speed ($C$) |  |  $343$ $\mathrm{m}\mathrm{/}\mathrm{s}$
Density ($\rho_{a}$) |  |  $1.21$ $\mathrm{k}\mathrm{g}\mathrm{/}\mathrm{m}^{3}$
Thermal conductivity ($k_{a}$) |  |  $0.025$ $\mathrm{W}\mathrm{/}\mathrm{(}\mathrm{m}\mathrm{K}\mathrm{)}$
Specific heat ($c_{a}$) |  |  $1005.3$ $\mathrm{J}\mathrm{/}\mathrm{(}\mathrm{k}\mathrm{g}\mathrm{K}\mathrm{)}$

A multiphysics model is used to solve the acoustic wave propagation equations
in air and solids for the sound transmission analysis. Sound waves in air are
governed by the equation 1 for the differential pressure $p$. The solid is
considered an isotropic linear elastic material, and the acoustic wave
propagation is determined by solving the weak formalism of the time-harmonic
Navier equation 2. Air-solid interface boundary conditions satisfy the
equations corresponding to interface type in the multiphysics coupling; for
instance, for the air-solid interface, satisfy equation 3, and for the solid-
air interface, satisfy equation 4, where solid tractions equal the pressure
[35].

$\nabla.(\frac{1}{\rho_{a}}\nabla
p)+\frac{\omega^{2}}{\rho_{a}C^{2}}p=0,\;in\;air$ (1)
$\nabla.\sigma=-\omega^{2}\rho_{s}\textbf{u},\;in\;solid$ (2)
$n\frac{1}{\rho_{a}}\nabla p=n\textbf{u},\;in\;air-solid\;interface$ (3)
$n\sigma=np,\;in\;solid-air\;interface$ (4)

where $p$ is the acoustic pressure, $\textbf{u}=(u_{x},u_{y})$ is the
displacement vector, $\rho_{a}$ and $\rho_{s}$ are the densities of the air
and solid, respectively.
$\sigma=\frac{E}{2(1+\nu)}(\frac{1}{(1-2\nu)}\nabla.\textbf{u}+\nabla\textbf{u})$
is the stress tensor for the isotropic linear elastic material, $C$ is the
sound speed in the air, $\omega$ is the angular frequency of the acoustic
wave, $E$ is the Young’s modulus, $\nu$ is the Poisson’s ratio, and $n$ is the
outward unit normal to the air or solid.

In order to simulate the normal incident sound wave, a plane wave radiation
boundary condition was utilized, and Perfectly Matched Layers (PMLs) were
placed at both extremities of the medium to limit spurious reflections, as
depicted in Fig. 2(a). Furthermore, the Floquet-Bloch periodic condition is
applied in the $x$-direction. We calculated the sound transmission loss (STL)
in the frequency range $20$ $\mathrm{H}\mathrm{z}$ to $10$
$\mathrm{k}\mathrm{H}\mathrm{z}$ for each cell with or without HRs and for the
homogeneous plates, using the equation (5).

$STL=-20\times log_{10}\Big{(}\frac{|p_{t}|}{|p_{i}|}\Big{)},$ (5)

where $p_{i}$ and $p_{t}$ are the incident and transmitted pressures,
respectively.

Figure 2: Numerical models. (a) The acoustic model consists of exciting the
unit cells with a plane acoustic wave in the frequency range of $20$
$\mathrm{H}\mathrm{z}$ to $10$ $\mathrm{k}\mathrm{H}\mathrm{z}$. The
transmission is then calculated as the ratio of transmitted pressure to
incident pressure. (b) The thermal model involves subjecting the unit cells to
a sinusoidal outer temperature $T_{outer}$ and measuring the inner temperature
$T_{inner}$ during a $24$ $\mathrm{h}$.

For the thermal analysis, the heat transfer in solids and fluids model was
utilized to characterize the thermal insulation through the panels. By solving
the transient heat equation (6), the temperature distribution within the unit
cells is calculated. The methodology is based on applying a sinusoidal
temperature within a $24$-hour period ranging between $0$
$\mathrm{\SIUnitSymbolDegree}\mathrm{C}$ and $70$
$\mathrm{\SIUnitSymbolDegree}\mathrm{C}$ and determining the inner temperature
of unit cells, as shown in Fig. 2(b). To accomplish this, the terminal
boundaries of the configuration are exposed to ambient temperature, which is
set to $T_{amb}=25$ $\mathrm{\SIUnitSymbolDegree}\mathrm{C}$. The outer
boundary of cells is vulnerable to the outer temperature $T_{out}(t)$, which
describes the temperature variation over a $24$-hour period with the
expression $T_{out}=35sin(\frac{2\pi}{24}t)+35$
$\mathrm{\SIUnitSymbolDegree}\mathrm{C}$. Furthermore, the periodicity
conditions are applied in the $x$-direction.

$k_{i}\Delta T=\rho_{i}c_{i}\frac{\partial T}{\partial t},$ (6)

where $i$ indicates air or solid, $k_{i}$, $\rho_{i}$ and $c_{i}$ are the
thermal conductivity, the density, and the specific heat, respectively.

Figure 3: The transmission loss of several configurations is as follows:
homogeneous medium on the left, Fabry-Perot in the middle, and quarter-
wavelength on the right. (a) Without a cavity. (b) With a cavity. (c) With
HRs. Figure 4: Sound transmission as a function of geometric parameters. (a)
The periodicity $a$ size variation. (b) The neck size variation.

To gain a better understanding of the resonator mechanism utilized in the
panel’s design, we compared the transmission loss of various acoustic
resonators to the mass law. Fig. 3 shows the geometrical designs on the left
and their corresponding transmission losses on the right. Fig. 3(a) depicts
the responses of three configurations: homogeneous medium, Fabry-Perot
resonator, and quarter-wavelength. It is evident that the Fabry-Perot
resonator exhibits resonance frequencies at $n\lambda/2$, and lower
transmission loss than the mass law. On the other hand, the quarter-wavelength
structure has two resonance peaks with high transmission loss, albeit
comparable to the mass law. In Fig. 3(b), the main modification to the
geometries is the incorporation of cavities into the homogeneous medium. It is
evident that the introduction of holes led to the appearance of some resonance
peaks. However, the overall impact was not significant when compared to the
mass law for the Fabry-Perot and quarter-wavelength resonators. Moreover, the
addition of Helmholtz resonators (HRs) to the structures altered the
transmission loss curves. The HRs with an open guide exhibited lower
transmission loss than the mass law. However, as depicted in Fig. 3(c), the
combination of HRs and the quarter-wavelength structure led to higher
transmission loss than the mass law over a broad range of normalized
frequencies. These findings suggest that incorporating HRs into acoustic
panels can enhance their sound absorption properties. Additionally, the
selection of the resonator geometry was demonstrated to have a significant
impact on the effectiveness of sound transmission loss. After identifying the
optimal resonator design, we proceeded to determine the geometrical
characteristics of the acoustic panel shown in Fig. 1.

To promote the use of resonators in the low-frequency range, it is imperative
to determine the geometric parameters that allow for the operation of such
resonators. This is before we can examine the performance of the insulation in
the panels. In the following step, we calculated the sound transmission loss
for one unit cell ($N=1$) and varied the periodicity constant $a$ from $2$
$\mathrm{c}\mathrm{m}$ to $10$ $\mathrm{c}\mathrm{m}$ and neck size $d$ from
$0.1$ $\mathrm{c}\mathrm{m}$ to $1.5$ $\mathrm{c}\mathrm{m}$. Then the sound
transmission loss was calculated in the frequency range from $20$
$\mathrm{H}\mathrm{z}$ to $6$ $\mathrm{k}\mathrm{H}\mathrm{z}$. The results of
the study are illustrated in Fig. 4.

It was found that as $a$ increases, the bandwidth decreases, and thus the
diffraction limit descends in frequency (Fig. 4(a)). As a consequence, the
selected periodicity $a$ has a significant impact on acoustic wave
attenuation. The influence of neck size on the two resonators was investigated
(Fig. 4(b)); the findings reveal that as the neck size increases, the width of
the band gap stays invariant and attenuation begins to assume significant
values. Therefore, we have selected a periodicity of $3$
$\mathrm{c}\mathrm{m}$ and a neck size of $0.4$ $\mathrm{c}\mathrm{m}$ for
which the diffraction limit begins at $4$ $\mathrm{k}\mathrm{H}\mathrm{z}$.
This covers a frequency range from $20$ $\mathrm{H}\mathrm{z}$ to $4$
$\mathrm{k}\mathrm{H}\mathrm{z}$, i.e. the same frequency range employed in
building sound insulation.

##  3 Results and discussion

###  3.1 Acoustic characteristics

Figure 5: Sound transmission loss for panels with and without HRs in red and
blue, and a mass law dotted line in black.

Fig. 5 depicts the simulated sound transmission loss (STL) curve for a panel
composed of one unit cell with HRs (in red). Moreover, the STL provided by the
mass law is presented for comparison (dotted line in black). The operative
sound insulation of the panel with HRs defies the mass law in multiple
frequency regions: $(418,533)$ $\mathrm{H}\mathrm{z}$, $(905,2211)$
$\mathrm{H}\mathrm{z}$, and $(2421,2683)$ $\mathrm{H}\mathrm{z}$. It assumes
that the proposed metamaterial panel can be used to shield low-frequency
sound. Fig. 5 also portrays the panel’s STL curve without HRs (in blue). It is
worth noting that the STL of a panel without HRs exhibits the same behavior as
the mass law in the low-frequency regime $(20,2500)$ $\mathrm{H}\mathrm{z}$.
As a result, the proposed panel has multiple local resonance mechanisms.

Figure 6: Total pressure screenshots of the panel with HRs. (a) At the
frequency of $f_{A}=502.6$ $\mathrm{H}\mathrm{z}$. (b) At the frequency of
$f_{B}=1033$ $\mathrm{H}\mathrm{z}$. (c) At the frequency of $f_{C}=2190$
$\mathrm{H}\mathrm{z}$. (d) At the frequency of $f_{D}=2451$
$\mathrm{H}\mathrm{z}$.

Fig. 6 depicts the total pressure of a panel composed of one unit cell at
frequencies $f_{A}=502.6$ $\mathrm{H}\mathrm{z}$, $f_{B}=1033$
$\mathrm{H}\mathrm{z}$ $f_{C}=2190$ $\mathrm{H}\mathrm{z}$ and $f_{D}=2451$
$\mathrm{H}\mathrm{z}$ to gain further insight into the local resonance
behaviors. However, within the broad band generated by the existence of HRs,
there is a resonance peak corresponding to a single frequency. Since noise is
formed by a chirp with a wide frequency range, this peak has no effect on
noise insulation. Fig. 7 portrays the STL of the panel made up of $N$ unit
cells containing the Helmholtz resonators. As previously discussed, the STL
reduces the amplitude of sound transmission loss over a wide frequency range
for $N=1$. Moreover, panels with two $N=2$ unit cells have a higher STL and a
broader bandwidth than the previous one, whereas panels with three $N=3$ unit
cells, four $N=4$ unit cells, and five $N=5$ unit cells all have improved
acoustic performance as the number of cells grows. As a consequence, as the
number of cells increases, sound transmission reduces.

To summarize, the panel with HRs outperforms its counterpart made of
homogeneous medium and lacking HRs in terms of sound insulation. In other
words, the numerical results of the acoustic analysis show that cells with
Helmholtz resonators have a significantly higher level of sound isolation than
cells without Helmholtz resonators. This means that, in the frequency range of
$20$ to $3000$ $\mathrm{H}\mathrm{z}$, HRs confine the incoming sound wave
more effectively than ordinary panels.

Figure 7: Sound transmission loss as a function of the number of cells $N$ in
the panel with HRs.

###  3.2 Thermal characteristics

Figure 8: The inner temperature fluctuations as a function of exposure time
for panels with and without HRs in red and blue, respectively, and a
homogeneous medium dotted line in black.

Fig. 8 represents the temperature evolution as a function of time for panels
composed of one unit cell. For $24$ hours, the panels are exposed to a
sinusoidal outer temperature (colored in black) varying between $0$ and $70$
$\mathrm{\SIUnitSymbolDegree}\mathrm{C}$ on their outer side. The evolution of
the inner temperature of the panel with HRs (colored in red) is not the same
as the inner temperatures of the other two panels composed of the homogeneous
medium (dotted line in black) and without HRs (colored in blue). Thus, the
inner temperature changes between the three panels and is therefore not
identical. In this case, it is possible to infer that the panels have
different thermal resistances. As a result, the panel containing HRs has
stronger resilience than the other cells. Indeed, cells containing HRs take
longer time to reach the maximum and minimum temperatures than cells without
HRs, which is related to the slots presented in the HRs. Screenshots taken
after four hours of exposure to the temperature distribution are used to gain
a better understanding of the thermal resistance of the three panels, as shown
in Fig. 9. It is clear that the distribution of the counter temperature is not
the same for all three cases; the presence of the HRs makes the panel more
resilient and increases its heat transfer mechanism.

Figure 9: The screenshots of temperature counters after 4 hours of exposure:
homogeneous medium (A), a panel without HRs (B), and a panel with HRs (C).

The inner temperature evolution of the panel of $N=2$ becomes more resilient
than that of the panel composed of a single unit cell with HRs. Additionally,
the variation of the inner temperature as a function of time changes with the
number of cells $N=3$, $N=4$, and $N=5$, as illustrated in Fig. 10. The
thermal resistance of the panels increases with the number of cells in the
$y$-direction.

It is noteworthy that the presence of HRs in the panels functioned as barriers
to reduce heat propagation. This demonstrates, in particular, that the
improved panels provide higher thermal insulation than conventional panels
consisting of homogeneous medium or cells without HRs. The presence of
Helmholtz resonators in unit cells influences their thermal properties,
resulting in extrathermal change processes. As a result, thermal transmission
occurs through three complementary mechanisms in the unit cell: conduction
through the solid, radiation, and convection by air within and outside the
cavities.

Figure 10: The inner temperature of N unit cells with HRs as a function of
time.

##  4 Conclusion

To conclude, we demonstrated a panel based on Helmholtz resonators in order to
reduce both sound and heat transmission. A series of simulations were
performed using the finite element method to characterize the physical
characteristics of the proposed panels in the context of determining their
physical characteristics. According to the sound transmission loss analysis
performed on the proposed metamaterial, the metamaterial’s resonance elements
(HRs) perfectly reflect the incoming energy by trapping it by its anti-
resonance mechanism in the frequency range corresponding to the noise inside
buildings, which ranges in three regions: $(418,533)$ $\mathrm{H}\mathrm{z}$,
$(905,2211)$ $\mathrm{H}\mathrm{z}$, and $(2421,2683)$ $\mathrm{H}\mathrm{z}$.
The results of the heat transfer analysis of panels with HRs indicate that, in
comparison to homogeneous panels and panels without HRs, heat propagation
decreases with exposure time. In other words, the presence of HRs mitigates
the spread of heat fluxes. Conversely, when Helmholtz resonators are involved,
it is as if a thermal barrier is created that prevents thermal fluxes from
spreading outward. Thus, we were able to develop a bi-functional metamaterial
that may be used in the building sector to alleviate noise and heat concerns.

## Acknowledgements

We would like to thank the Moroccan Ministry of Higher Education, Scientific
Research and Innovation and the OCP Foundation who funded this work through
the APRD research program.

## References

  * [1] S. Matos, E. Viardot, B. K. Sovacool, F. W. Geels, Y. Xiong, Innovation and climate change: A review and introduction to the special issue, Technovation (2022) 102612doi:10.1016/j.technovation.2022.102612.
  * [2] R. Thompson, R. B. Smith, Y. B. Karim, C. Shen, K. Drummond, C. Teng, M. B. Toledano, Noise pollution and human cognition: An updated systematic review and meta-analysis of recent evidence, Environment international 158 (2022) 106905\.  doi:10.1016/j.envint.2021.106905.
  * [3] N. M. Aly, H. Seddeq, K. Elnagar, T. Hamouda, Acoustic and thermal performance of sustainable fiber reinforced thermoplastic composite panels for insulation in buildings, Journal of Building Engineering 40 (2021) 102747.  doi:10.1016/j.jobe.2021.102747.
  * [4] C. Buratti, E. Belloni, E. Lascaro, F. Merli, P. Ricciardi, Rice husk panels for building applications: Thermal, acoustic and environmental characterization and comparison with other innovative recycled waste materials, Construction and Building Materials 171 (2018) 338–349.  doi:10.1016/j.conbuildmat.2018.03.089.
  * [5] J. Nurzyński, Is thermal resistance correlated with sound insulation?, Energy Procedia 78 (2015) 152–157.  doi:10.1016/j.egypro.2015.11.131.
  * [6] Y. Dong, J. Kong, S. Mousavi, B. Rismanchi, P.-S. Yap, Wall insulation materials in different climate zones: A review on challenges and opportunities of available alternatives, Thermo 3 (1) (2023) 38–65.  doi:10.3390/thermo3010003.
  * [7] J. Pokornỳ, R. Ševčík, J. Šál, L. Fiala, L. Zárybnická, L. Podolka, Bio-based aggregate in the production of advanced thermal-insulating concrete with improved acoustic performance, Construction and Building Materials 358 (2022) 129436.  doi:10.1016/j.conbuildmat.2022.129436.
  * [8] A. Patnaik, M. Mvubu, S. Muniyasamy, A. Botha, R. D. Anandjiwala, Thermal and sound insulation materials from waste wool and recycled polyester fibers and their biodegradation studies, Energy and Buildings 92 (2015) 161–169.  doi:10.1016/j.enbuild.2015.01.056.
  * [9] H. Gao, H. Liu, L. Liao, L. Mei, G. Lv, L. Liang, G. Zhu, Z. Wang, D. Huang, Improvement of performance of foam perlite thermal insulation material by the design of a triple-hierarchical porous structure, Energy and Buildings 200 (2019) 21–30.  doi:10.1016/j.enbuild.2019.07.010.
  * [10] H. Binici, O. Aksogan, C. Demirhan, Mechanical, thermal and acoustical characterizations of an insulation composite made of bio-based materials, Sustainable Cities and Society 20 (2016) 17–26.  doi:10.1016/j.scs.2015.09.004.
  * [11] I. V. Fornés, D. Vaičiukynienė, D. Nizevičienė, G. Tamošaitis, D. Pupeikis, The improvement of the thermal and acoustic insulation properties of phosphogypsum specimens by adding waste wood fibre, Construction and Building Materials 331 (2022) 127341.  doi:10.1016/j.conbuildmat.2022.127341.
  * [12] J. Fan, L. Zhang, S. Wei, Z. Zhang, S.-K. Choi, B. Song, Y. Shi, A review of additive manufacturing of metamaterials and developing trends, Materials Today (2021).  doi:10.1016/j.mattod.2021.04.019.
  * [13] T. Han, X. Bai, J. T. Thong, B. Li, C.-W. Qiu, Full control and manipulation of heat signatures: cloaking, camouflage and thermal metamaterials, Advanced Materials 26 (11) (2014) 1731–1734.  doi:10.1002/adma.201304448.
  * [14] C. Lim, et al., From photonic crystals to seismic metamaterials: A review via phononic crystals and acoustic metamaterials, Archives of Computational Methods in Engineering 29 (2) (2022) 1137–1198.  doi:10.1007/s11831-021-09612-8.
  * [15] J. Mei, G. Ma, M. Yang, Z. Yang, W. Wen, P. Sheng, Dark acoustic metamaterials as super absorbers for low-frequency sound, Nature communications 3 (1) (2012) 1–7.  doi:10.1038/ncomms1758.
  * [16] M. Yang, C. Meng, C. Fu, Y. Li, Z. Yang, P. Sheng, Subwavelength total acoustic absorption with degenerate resonators, Applied Physics Letters 107 (10) (2015) 104104.  doi:10.1063/1.4930944.
  * [17] H. Zhang, S. Chen, Z. Liu, Y. Song, Y. Xiao, Light-weight large-scale tunable metamaterial panel for low-frequency sound insulation, Applied Physics Express 13 (6) (2020) 067003.  doi:10.35848/1882-0786/ab916b.
  * [18] J.-Y. Jang, C.-S. Park, K. Song, Lightweight soundproofing membrane acoustic metamaterial for broadband sound insulation, Mechanical Systems and Signal Processing 178 (2022) 109270.  doi:10.1016/j.ymssp.2022.109270.
  * [19] Y. Li, B. M. Assouar, Acoustic metasurface-based perfect absorber with deep subwavelength thickness, Applied Physics Letters 108 (6) (2016) 063502.  doi:10.1063/1.4941338.
  * [20] X. Cai, Q. Guo, G. Hu, J. Yang, Ultrathin low-frequency sound absorbing panels based on coplanar spiral tubes or coplanar helmholtz resonators, Applied Physics Letters 105 (12) (2014) 121901.  doi:10.1063/1.4895617.
  * [21] T. Wu, T. Cox, Y. Lam, From a profiled diffuser to an optimized absorber, The Journal of the Acoustical Society of America 108 (2) (2000) 643–650.  doi:10.1121/1.429596.
  * [22] C. Shen, Y. Liu, L. Huang, On acoustic absorption mechanisms of multiple coupled quarter-wavelength resonators: Mutual impedance effects, Journal of Sound and Vibration 508 (2021) 116202.  doi:10.1016/j.jsv.2021.116202.
  * [23] J. Li, W. Wang, Y. Xie, B.-I. Popa, S. A. Cummer, A sound absorbing metasurface with coupled resonators, Applied Physics Letters 109 (9) (2016) 091908.  doi:10.1063/1.4961671.
  * [24] N. Jiménez, W. Huang, V. Romero-García, V. Pagneux, J.-P. Groby, Ultra-thin metamaterial for perfect and quasi-omnidirectional sound absorption, Applied Physics Letters 109 (12) (2016) 121902.  doi:10.1063/1.4962328.
  * [25] N. Jiménez, V. Romero-García, V. Pagneux, J.-P. Groby, Rainbow-trapping absorbers: Broadband, perfect and asymmetric sound absorption by subwavelength panels for transmission problems, Scientific reports 7 (1) (2017) 13595.  doi:10.1038/s41598-017-13706-4.
  * [26] Y. Li, W. Li, T. Han, X. Zheng, J. Li, B. Li, S. Fan, C.-W. Qiu, Transforming heat transfer with thermal metamaterials and devices, Nature Reviews Materials 6 (6) (2021) 488–507.  doi:10.1038/s41578-021-00283-2.
  * [27] K. P. Vemuri, P. R. Bandaru, Geometrical considerations in the control and manipulation of conductive heat flux in multilayered thermal metamaterials, Applied Physics Letters 103 (13) (2013) 133111.  doi:10.1063/1.4823455.
  * [28] R. Schittny, M. Kadic, S. Guenneau, M. Wegener, Experiments on transformation thermodynamics: molding the flow of heat, Physical review letters 110 (19) (2013) 195901.  doi:10.1103/PhysRevLett.110.195901.
  * [29] H. Honarvar, M. I. Hussein, Two orders of magnitude reduction in silicon membrane thermal conductivity by resonance hybridizations, Physical Review B 97 (19) (2018) 195413.  doi:10.1103/PhysRevB.97.195413.
  * [30] H. Honarvar, L. Yang, M. I. Hussein, Thermal transport size effects in silicon membranes featuring nanopillars as local resonators, Applied Physics Letters 108 (26) (2016) 263101.  doi:10.1063/1.4954739.
  * [31] Z. Li, L. Wu, H. Zhang, Y. Li, H. P. Lee, Dual-functional metamaterial with vibration isolation and heat flux guiding, Journal of Sound and Vibration 469 (2020) 115122.  doi:10.1016/j.jsv.2019.115122.
  * [32] L. Zhang, Z. Bai, Y. Chen, Dual-functional hierarchical mechanical metamaterial for vibration insulation and energy absorption, Engineering Structures 271 (2022) 114916.  doi:10.1016/j.engstruct.2022.114916.
  * [33] J. Fan, B. Song, L. Zhang, X. Wang, Z. Zhang, S. Wei, X. Xiang, X. Zhu, Y. Shi, Structural design and additive manufacturing of multifunctional metamaterials with low-frequency sound absorption and load-bearing performances, International Journal of Mechanical Sciences 238 (2023) 107848.  doi:10.1016/j.ijmecsci.2022.107848.
  * [34] L. Rayleigh, The theory of the helmholtz resonator, Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 92 (638) (1916) 265–275.
  * [35] G. H. Yoon, J. S. Jensen, O. Sigmund, Topology optimization of acoustic–structure interaction problems using a mixed finite element formulation, International journal for numerical methods in engineering 70 (9) (2007) 1049–1075.  doi:10.1002/nme.1900.