of 10
1
SUPPLEMENTARY FIGURES
4 K
20 mK
300 K
HEM
T
LNA
-50dB
-20dB
-30dB
SG
VNA
SA
'
nanomembrane
coil
beam
coupler
C
s
C
m
I
c
L
C
l
Supplementary Figure 1:
Experimental setup.
The output tone of a microwave signal generator (SG) and the output tone
of a vector network analyzer (VNA) are combined at room temperature, attenuated, routed to the sample at about 20 mK and
inductively coupled to the LC circuit on the nanomembrane. We detect the reflected output tone after amplification with a high
electron mobility transistor amplifier (HEMT), further amplification with a low noise amplifier (LNA), and switchable pump
tone cancelation (dashed lines). The measurement is done either phase coherently with the VNA, or we detect the incoherent
power spectrum with the spectrum analyzer (SA).
Supplementary Figure 2:
Geometry for high frequency breathing mode simulation.
The simulated structure with 65
nm thick Aluminum (highlighted blue) on 300 nm thick silicon nitride (gray) is shown. We use a symmetric boundary condition
in the center of the beam (bottom left).
G
a
b
,
n
m
^,
n
r
a
in
^
a
out
^
^
·
e
·
i
a
b,r
,
n
b,r
°
m,i
b
b,m
,
n
b,m
^
^
a
b,wg
,
n
b,wg
^
Supplementary Figure 3:
System modes, coupling rates and noise baths.
In the reflective geometry the microwave cavity
mode ˆ
a
is coupled to the coherent waveguide modes ˆ
a
in
and ˆ
a
out
with the external coupling strength
κ
e
. It is also coupled to
a bath of noise photons, ideally at the refrigerator temperature
n
b,r
, with the intrinsic coupling strength
κ
i
. In addition, the
waveguide bath mode ˆ
a
b,
wg
can be populated with thermal noise photons
n
b,
wg
, which also couples with
κ
e
. The mechanical
resonator mode
ˆ
b
is coupled to the microwave resonator with the parametrically enhanced electromechanical coupling strength
G
. In addition, it is coupled to a bath of noise phonons, ideally at the refrigerator temperature
n
b,m
, with the intrinsic coupling
rate
γ
m,i
.
2
SUPPLEMENTARY NOTES
Supplementary Note 1: Circuit properties
Coil simulation
Our device is fabricated and simulated on a 300 nm thick and (777
μ
m)
2
large Si
3
N
4
membrane. The coil wire
is 500 nm wide and 120 nm thick, with a 1
μ
m pitch, 42 turns forming a square with lateral length of only 87
μ
m,
well in the lumped element limit. According to finite element simulations, which includes wire cross-overs, the coil
is inductive up to its self resonance frequency of
ν
srf
= 13
.
38 GHz, where the half wavelength roughly matches the
total wire length of
l
= 7
.
7 mm. We repeat this simulation with a small additional shunt capacitor of known value
(∆
C
= 0
.
1 fF) and extract the new self resonance frequency
ν
srf
,
2
. Solving the two simple relations
ω
srf
= (
LC
l
)
1
/
2
and
ω
srf
,
2
= (
L
(
C
l
+ ∆
C
))
1
/
2
, we extract
L
= 68 nH and
C
l
= 2
.
1 fF. These results are valid close to - but below -
the self resonance frequency of the coil. In this limit we realize a maximum impedance of
Z
0
=
L/C
l
5
.
7 kΩ, far
exceeding the vacuum impedance
Z
vac
377 Ω, and approaching the resistance quantum
R
q
=
h/
(2
e
)
2
6 kΩ.
Full circuit parameters
Knowing the inductance
L
of the fabricated inductor, as well as the actually measured resonance frequency of
ω
r
/
(2
π
) = 7
.
965 GHz, yields a total capacitance of
C
tot
=
C
l
+
C
m
+
C
s
= 5
.
9 fF and a total circuit impedance
of
Z
tot
= 3
.
4 kΩ (see Supplementary Figure 1). The modulated capacitance
C
m
is a function of the capacitor slot
size, which we estimate from numerical simulations of the measured coupling strength to be 62 nm. Numerical finite
element simulations yield a nanobeam capacitance of
C
m
1
.
6 fF for this gap size (see Figs. in main text), which
gives a participation factor of
η
0
.
27.
Using a self resonance frequency simulation of the full electrical circuit including
C
m
, we can attribute the remaining
stray capacitance of
C
s
2
.
2 fF to the coil to capacitor wiring (62 %), the presence of a second resonant circuit
(15 %), the coupling wire (8 %), non-ideal crossovers (8 %). The remaining 0.15 fF (7 %) we attribute to frequency
dependence, packaging and our uncertainty of the relative permittivity of silicon nitride at low temperature

r
7
.
5
8.
As expected, for these full circuit simulations we extract the same inductance
L
as for the coil only simulations. The
value of
L
= 68 nH is consistent with both, the modified Wheeler and the current sheet method [1], to within
±
2 nH.
High frequency mechanical mode
In order to estimate the electromechanical coupling of the high frequency acoustic mode, we consider that the
identical microwave circuit is coupled to both sides of the nanobeam. Here the outer capacitor length is taken to
match the acoustic defect region of 2
×
3 lattice constants, see Supplementary Figure 2. We find a reduced participation
ratio
η
= 0
.
14 due to the reduced
C
m
0
.
7 fF in this case. Further improvements in reducing the circuit’s stray
capacitance will have a big impact for efficient coupling to high frequency modes.
Supplementary Note 2: Device Fabrication
Wafer preparation
After a thorough RCA clean (organic and particle clean, oxide strip, ionic clean) [2], we grow a 300 nm thick film
of stoichiometric Si
3
N
4
at a temperature of 835 C, using low pressure chemical vapor deposition on both sides of a
doubly polished 200
μ
m thick, high resistivity (
>
10 kΩ-cm), Si
100
wafer. After cooldown, the dielectric film has
a stress of
1 GPa due to the differential expansion coefficient. We spin a protective layer of photoresist and dice
the wafer in 10 mm
×
10 mm chips.
3
Membrane patterning
The chips are cleaned using weak sonication in acetone (ACE) and isopropyl alcohol (IPA) and prebaked at 180
C
for 2 min on a hotplate. We then spin the front side with ZEP 520A at 67 Hz (4000 rpm) for protection, bake at
180
C for 2 min, spin the back side with ZEP 520A at 33 Hz (2000 rpm) and bake at 180
C for 2 min. Patterning
of the 16 membrane areas of size 1 mm
×
1 mm each, is done with 100 keV electron beam exposure with a 200 nA
electron beam, 50 nm fracturing size and a dose of 250
μ
C
/
cm
2
on the chip back side. This layer is carefully aligned
to the chip corners. We develop with ZND-N50 for 2.5 min and rinse in methyl isobutyl ketone (MIBK) for 0.5 min.
This is followed by an ICP-RIE etch of the silicon nitride in the developed areas, using a C
4
F
8
(0.57 cm
3
/s = 34 sccm)
/ SF
6
(0.2 cm
3
/s = 12 sccm) plasma, generated with an ICP power of 1000 W, RF power of 30 W and a DC bias of
84 V, at a pressure of 2000 Pa (15 Torr) and a temperature of 25
C for 7 min 15 s. We finish this layer by a thorough
cleaning of the chips using weak sonication in TCE, IPA, ZDMAC, ACE and IPA.
Nanobeam patterning and membrane pre-etching
This layer initially follows the same procedure to pattern the top side of the chip (no resist on the back side)
with the nanobeams, pull-in cuts and the global and pattern alignment markers of size (20
μ
m)
2
, with these process
parameters: 300 pA beam, 2.5 nm fracturing, 275
μ
C
/
cm
2
dose, 7 min 50 s etch time. We then use an o-ring sealed
holder to expose only the back side of the chip to 30 % KOH in water at 85
C with the stir bar at 7 Hz (400 rpm).
This anisotropic Si wet etch is stopped when the wafer becomes semi-transparent (dark orange) in the membrane
area, when illuminated with an LED on the sealed side of the chip. The color indicates a silicon thickness of
5
μ
m
which is usually achieved after 2
.
5 h of etching. After cleaning the chip in ultra-pure deionized water and IPA, we wet
etch both the front and back side of the chip in 30 % KOH in water at 65
C with the stir bar at 1.7 Hz (100 rpm) for
70 s. This partially undercuts (
100 nm) the nanobeams for a clean subsequent inverse shadow evaporation process
[3], used to pattern the small gapped capacitors. The chips are then rinsed in hot water, fresh piranha solution (mix
45 mL H
2
SO
4
with 15 mL H
2
O
2
at 85
C with the stir bar at 5 Hz, 300 rpm) for 8 min followed by a water and IPA
rinse.
Capacitors and ground plane
This layer patterns all of the electrical circuit, except for the coil wires. We start with a prebake at 180
C for 2
min, and spin the front side with ZEP 520A at 33 Hz (2000 rpm), followed by another bake at 180
C for 2 min.
We use 100 keV electron beam lithography to pattern the ground plane and transmission lines (200 nA beam, 50 nm
fracturing, 290
μ
C
/
cm
2
dose with PEC), as well as the capacitor wires, and the wires connecting the capacitors with
the coil end and center (10 nA beam, 10 nm fracturing, 275
μ
C
/
cm
2
dose). This layer is carefully aligned to the etched
negative markers from the previous step. We develop the chips in the same way and use a
O
2
plasma ash process
(0.83 cm
3
/s = 50 sccm O
2
, 0.74 bar, 13.56 MHz, 35 W, 2 min) to descum the surface before deposition of aluminum.
For the deposition we use an electron beam evaporator (0.3 nm/s, 65 nm thickness at 1
·
10
5
Pa to 2
·
10
5
Pa). We
then do a lift-off process in 80
C NMP for
>
1 h and carefully rinse in ACE and IPA.
Scaffolding layer
Now we pattern a scaffolding layer to fabricate the cross-overs. After prebaking, we spin LOR 5B at 50 Hz
(3000 rpm) and bake at 180
C for 5 min, followed by spinning PMMA 950k A2 at 67 Hz (4000 rpm) and baking at
180
C for 5 min. We then beam write the negative pattern of the cross-over support structure using aligned electron
beam lithography (200 nA beam, 25 nm fracturing, 1000
μ
C
/
cm
2
dose). The resist is developed using MIBK:IPA
(1:3) for 1 min, and rinsed in IPA for 30 s. We then wet etch the scaffolding layer using MF-319 for 8 s, followed by a
water rinse and IPA which stops the etch. Finally we remove the remaining PMMA layer with ACE (30 s) and reflow
the LOR cross-over support layer on a hot plate at 200
C for 10 min. This creates a structurally stable arc shaped
cross over scaffolding.
4
Coil wire patterning
In order to pattern the narrow pitch coils, we spin PMMA 495 A8 at 33 Hz (2000 rpm), bake, spin PMMA 950k A2
at 33 Hz (2000 rpm) and bake again. Then we lithographically define the coil wires, which overlap the capacitor wires
(10 nA beam, 10 nm fracturing, 1800
μ
C
/
cm
2
dose) and develop the resist as described previously. Development is
followed by the same plasma ashing, deposition of aluminum (1 nm/s, 120 nm, p
2
·
10
5
Pa) and lift off, during
which the NMP (at 80
C, 3 h) dissolves the LOR scaffolding layer.
DC contact wire
After a careful rinse with ACE and IPA we reproduce the previous layer recipe to pattern a small (500 nm
×
4
μ
m)
DC contact wire that symmetrically covers all overlap regions between capacitor wire and coil wire (two per coil
and capacitor). Here we use an in-situ ion gun etch process (normal incidence with 4 cm gridded Kaufman source,
400 V, 21 mA for 5 min) right before the aluminum deposition of thickness 140 nm, in order to establish reliable
contact. Contact is tested after lift-off on DC test structures of the same contact size located in the center of the chip.
High resistance contacts with low capacitance at microwave frequencies would lead to additional parasitic in-series
capacitances of the fabricated circuit.
Release
For the final release step we prepare a silicon enriched solution of TMAH to selectively etch the silicon without
aluminum corrosion [4, 5]. We use a custom built reactor vessel with thermometer port and a hotplate with magnetic
stir bar to mix 60 g of TMAH (25 %, 6N) and 250 g water, and then add 5.12 g of silicon powder (-325 mesh, 5N)
and stir at 5 Hz (300 rpm). After the chemical reaction calms down we start heating the solution up to 80
C. When
the solution is clear, we wait for 1 h and prepare a clear mixture of 5.21 g of TMAH (25 %, 6N) and 2.11 g of the
oxidizing agent ammonium persulfate in a small beaker. We add the mixture to the solution with the stir bar at 17 Hz
(1000 rpm), wait 10 min to 15 min, reduce the stir speed and add the sample in a vertical position. The sample is
securely clamped, but with the membranes open to a steady flow of solution on the back and front side of the chip.
We keep the solution at 80
C and wait for the membranes to become fully transparent (1 h to 2 h). As a last step
we carefully remove the sample, rinse it throughly in hot water, cold water, IPA, ultra purified IPA, and dry it using
a CO
2
critical point dryer.
Supplementary Note 3: Experimental setup
For the measurements of coherent and incoherent electromechanical response, we combine the output of a vector
network analyzer with up to two microwave sources, feed the microwave tones to the base plate of a cryogen free dilution
refrigerator using UT-085 stainless steel coaxial cables with feedthroughs for thermalization at each temperature stage
and an additional attenuation of 50 dB to suppress room temperature Johnson noise (see Supplementary Figure 1).
We couple to the sample in a reflective geometry using a circulator and a low loss, high dielectric constant, copper
printed circuit board (PCB). On the PCB and chip we use 50 Ω coplanar waveguides to route the microwave tones
all the way to the membrane with very little reflections. On the membrane the center conductor is shorted to ground
through a small wire, which couples the waveguide inductively to the LC resonant circuit.
On the output side we use another circulator for isolating the sample from 4 K noise, otherwise entering in reverse
direction. Crimped niobium titanium superconducting cables are used to connect directly to a low noise, high electron
mobility transistor amplifier (HEMT) at 4 K. From there we use lowloss UT-085 stainless steel - beryllium copper
cables and amplify once more at room temperature. In order to suppress spurious response peaks in high drive power
cooling measurements, we add a phase and amplitude adjusted part of the pump tone to the output signal, as shown
in Supplementary Figure 1. Using a phase shifter we compensate for about 3 m of path length difference and typically
achieve a carrier suppression of
>
40 dB.
5
Supplementary Note 4: Derivation of cavity response functions
Fourier Transform
We use the following convention for the Fourier transform. Given an operator
ˆ
A
we define
ˆ
A
(
t
) =
1
2
π
ˆ
+
−∞
dωe
iωt
ˆ
A
(
ω
)
(1)
ˆ
A
(
ω
) =
1
2
π
ˆ
+
−∞
dte
iωt
ˆ
A
(
t
)
.
(2)
Reflective coupling to a microwave resonator
We consider a resonator mode ˆ
a
at frequency
ω
r
, which is coupled to a single waveguide with coupling strength
κ
e
,
and to the environment with the coupling strength
κ
i
(see Supplementary Figure 3). We follow general input-output
theory [6] to write the time derivative of the annihilation operator
̇
ˆ
a
(
t
) =
(
r
+
κ
2
)
ˆ
a
(
t
)
κ
e
ˆ
a
in
(
t
)
κ
i
ˆ
a
b,r
(
t
)
κ
e
ˆ
a
b,
wg
(
t
)
,
(3)
where
κ
=
κ
e
+
κ
i
is the total resonator linewidth, ˆ
a
in
(
t
) represents the annihilation operator of the coherent input
mode, ˆ
a
b,
wg
(
t
) the waveguide mode operator, and ˆ
a
b,r
(
t
) is the respective field operator of the resonator environment.
We take the Fourier transform to remove the time derivative, and simplify to get the frequency dependence
ˆ
a
(
ω
) =
κ
e
ˆ
a
in
(
ω
)
κ
i
ˆ
a
b,r
(
ω
)
κ
e
ˆ
a
b,
wg
(
t
)
κ/
2 +
i
(
ω
r
ω
)
.
(4)
The resonator output field is defined as
ˆ
a
out
(
ω
) = ˆ
a
in
(
ω
) +
κ
e
ˆ
a
(
ω
)
= ˆ
a
in
(
ω
) +
κ
e
ˆ
a
in
(
ω
)
κ
e
κ
i
ˆ
a
b,r
(
ω
)
κ
e
ˆ
a
b,
wg
(
t
)
κ/
2 +
i
(
ω
r
ω
)
,
(5)
which we can use to calculate the complex scattering parameter as measured by a network analyzer
S
11
(
ω
) =
ˆ
a
out
(
ω
)
ˆ
a
in
(
ω
)
= 1
κ
e
κ/
2 +
i
(
ω
r
ω
)
,
(6)
where the incoherent bath mode terms drop out. We use this function to simultaneously fit the real and imaginary
part of the measured cavity response and extract the intrinsic and extrinsic cavity coupling rates.
Drive photon number
It is useful to define the intra-cavity photon number
n
d
due to a classical coherent drive tone at frequency
ω
d
. We
replace the field operators in Supplementary Equation 4 with the classical amplitudes ˆ
a
(
ω
)
α
(
ω
) and discard the
resonator and waveguide bath modes to get
n
d
=
|
α
d
|
2
=
|
α
in
|
2
4
κ
e
κ
2
+ 4∆
2
r,d
.
(7)
Here we have introduced the resonator drive detuning ∆
r,d
=
ω
r
ω
d
and the input photon flux
|
α
in
|
2
=
P
in
/
( ̄
d
).
The power at the cavity input can be expressed as
P
in
= 10
3
10
(
A
+
P
d
)
/
10
with
P
d
the drive power in dBm and
A
the
total attenuation of the input line in dB.
6
Asymmetric lineshape
Fano line shapes generally originate from interference between a resonant mode and a background mode [7]. Ex-
perimental imperfections, such as leakage or reflections in the feedline circuit, can lead to such asymmetric cavity line
shapes. We can model this effect by introducing a complex valued external resonator to waveguide coupling parameter
̄
κ
e
=
|
κ
e
|
e
iq
, where
q
is a version of the Fano parameter. While small
q
values do not change the magnitude of the
inferred external coupling (or the drive photon number), they correctly model small asymmetries in the Lorentzian
cavity response [8]. For simplicity we define the generalized coupling ̄
κ
e
=
κ
e
iq
and substitute into Supplementary
Equation 6, to get the generalized resonator line shape
S
11
(
ω
) = 1
κ
e
iq
κ/
2
i
(
ω
r
ω
)
.
(8)
Supplementary Note 5: Derivation of cavity electromechanical response functions
In this section we follow previous work [9–12] to calculate the coherent response and the full noise spectrum of the
system. In contrast to earlier treatments we also include thermal noise in the feedline circuit, which gives rise to an
increased mechanical occupation and an asymmetric cavity noise line shape.
System Hamiltonian and equations of motion
The Hamiltonian of the coupled microwave cavity-mechanical system (see Supplementary Figure 3) can be written
as
ˆ
H
= ̄
r
ˆ
a
ˆ
a
+ ̄
m
ˆ
b
ˆ
b
+ ̄
hg
0
ˆ
a
ˆ
a
(
ˆ
b
+
ˆ
b
)
,
(9)
where
ˆ
b
(
ˆ
b
) is the annihilation (creation) operator of the mechanical mode at frequency
ω
m
, and
g
0
is the electrome-
chanical coupling strength, i.e. the resonator frequency shift due to a mechanical displacement corresponding to half
a phonon on average. We excite the microwave resonator mode using a strong drive tone at frequency
ω
d
, detuned
from the resonator frequency by ∆
r,d
=
ω
r
ω
d
. The linearized Hamiltonian in the rotating frame is then given as
ˆ
H
=
̄
h
r,d
ˆ
a
ˆ
a
+ ̄
m
ˆ
b
ˆ
b
+ ̄
hG
a
+ ˆ
a
)(
ˆ
b
+
ˆ
b
)
,
(10)
where
G
=
n
d
g
0
is the parametrically enhanced optomechanical coupling strength. The linearized Langevin equa-
tions are given as
̇
ˆ
a
(
t
) =
(
i
r,d
+
κ
2
)
ˆ
a
(
t
)
iG
(
ˆ
b
(
t
) +
ˆ
b
(
t
))
κ
e
ˆ
a
in
(
t
)
κ
i
ˆ
a
b,r
(
t
)
κ
e
ˆ
a
b,
wg
(
t
)
(11)
̇
ˆ
b
(
t
) =
(
m
+
γ
m,i
2
)
iG
a
(
t
) + ˆ
a
(
t
))
γ
m,i
ˆ
b
b,m
(
t
)
.
(12)
Taking the Fourier transform and simplifying we obtain
χ
1
r
(
ω
a
(
ω
) =
iG
(
ˆ
b
(
ω
) +
ˆ
b
(
ω
))
κ
e
ˆ
a
in
(
ω
)
κ
i
ˆ
a
b,r
(
ω
)
(13)
̃
χ
1
r
(
ω
a
(
ω
) =
iG
(
ˆ
b
(
ω
) +
ˆ
b
(
ω
))
κ
e
ˆ
a
in
(
ω
)
κ
i
ˆ
a
b,r
(
ω
)
(14)
χ
1
m
(
ω
)
ˆ
b
(
ω
) =
iG
a
(
ω
) + ˆ
a
(
ω
))
γ
m,i
ˆ
b
b,m
(
ω
)
(15)
̃
χ
1
m
(
ω
)
ˆ
b
(
ω
) =
iG
a
(
ω
) + ˆ
a
(
ω
))
γ
m,i
ˆ
b
b,m
(
ω
)
,
(16)
where we have introduced the uncoupled susceptibilities of the cavity and the mechanical mode
χ
1
r
(
ω
) =
κ/
2 +
i
(∆
r,d
ω
)
(17)
̃
χ
1
r
(
ω
) =
κ/
2
i
(∆
r,d
+
ω
)
(18)
χ
1
m
(
ω
) =
γ
m,i
/
2 +
i
(
ω
m
ω
)
(19)
̃
χ
1
m
(
ω
) =
γ
m,i
/
2
i
(
ω
m
+
ω
)
.
(20)
7
In the sideband resolved limit
ω
m

κ,G
, and for positive detuning of the drive tone ∆
r,d
ω
m
(red side pumping),
the linearized Langevin equations can be written approximately as,
ˆ
a
(
ω
) =
iGχ
m
χ
r
γ
m,i
ˆ
b
b,m
(
ω
)
χ
r
(
κ
e
ˆ
a
in
(
ω
) +
κ
i
ˆ
a
b,r
(
ω
) +
κ
e
ˆ
a
b,
wg
(
ω
))
1 +
G
2
χ
m
χ
r
(21)
ˆ
b
(
ω
) =
χ
m
γ
m,i
ˆ
b
b,m
(
ω
)
iGχ
m
χ
r
(
κ
e
ˆ
a
in
(
ω
) +
κ
i
ˆ
a
b,r
(
ω
) +
κ
e
ˆ
a
b,
wg
(
ω
))
1 +
G
2
χ
m
χ
r
.
(22)
Now we can calculate the cavity output mode
ˆ
a
out
(
ω
) = ˆ
a
in
(
ω
) +
κ
e
ˆ
a
(
ω
)
= ˆ
a
in
(
ω
)
a
in
(
ω
) + ˆ
a
b,
wg
(
ω
))
κ
e
χ
r
1 +
G
2
χ
m
χ
r
ˆ
a
b,r
(
ω
)
κ
e
κ
i
χ
r
1 +
G
2
χ
m
χ
r
+
ˆ
b
b,m
(
ω
)
iG
κ
e
γ
m,i
χ
m
χ
r
1 +
G
2
χ
m
χ
r
.
(23)
Electromagnetically Induced Transparency
We first calculate the coherent part of the system response using Supplementary Equation 23 and drop incoherent
noise terms to get
S
11
(
ω
) =
ˆ
a
out
(
ω
)
ˆ
a
in
(
ω
)
= 1
κ
e
χ
r
1 +
G
2
χ
m
χ
r
.
(24)
Substituting the bare response of the cavity and the mechanical oscillator we get the coherent EIT response function
valid for small probe drive strengths
S
11
(
ω
) = 1
κ
e
κ/
2 +
i
(∆
r,d
ω
) +
G
2
γ
m,i
/
2+
i
(
ω
m
ω
)
.
(25)
In order to take into account potential interference with a continuum of parasitic modes, we can follow the procedure
outlined above. Substituting
κ
e
κ
e
iq
we get
S
11
,
as
(
ω
) = 1
κ
e
iq
κ/
2 +
i
(∆
r,d
ω
) +
G
2
γ
m,i
/
2+
i
(
ω
m
ω
)
,
(26)
which can be used to fit asymmetric EIT spectra.
Supplementary Note 6: Quantum derivation of observed noise spectra
Using the Fourier transforms defined above, we can write the spectral density of an operator
ˆ
A
as
S
AA
(
t
) =
ˆ
+
−∞
dτe
iωτ
ˆ
A
(
t
+
τ
)
ˆ
A
(
t
)
(27)
S
AA
(
ω
) =
ˆ
+
−∞
ˆ
A
(
ω
)
ˆ
A
(
ω
)
.
(28)
The auto-correlation function of the detected normalized field amplitude (or the photo current) of the output mode
ˆ
I
(
t
) = ˆ
a
out
(
t
) + ˆ
a
out
(
t
) is then given as
S
II
=
ˆ
+
−∞
〈(
ˆ
a
out
(
ω
) + ˆ
a
out
(
ω
)
)(
ˆ
a
out
(
ω
) + ˆ
a
out
(
ω
)
)〉
.
(29)
Substituting ˆ
a
out
(
ω
) and ˆ
a
out
(
ω
) from Supplementary Equation 23 we find a general expression for the single sided
noise spectrum
S
(
ω
) =
n
b,
wg
(
1
κ
e
̃
χ
r
1 +
G
2
̃
χ
m
̃
χ
r
)
2
+
n
b,r
|
κ
e
κ
i
̃
χ
r
|
2
|
1 +
G
2
̃
χ
m
̃
χ
r
|
2
+
n
b,m
κ
e
γ
m,i
G
2
|
̃
χ
m
|
2
|
̃
χ
r
|
2
|
1 +
G
2
̃
χ
m
̃
χ
r
|
2
+ (
n
b,
wg
+ 1)
(
1
κ
e
χ
r
1 +
G
2
χ
m
χ
r
)
2
+ (
n
b,r
+ 1)
κ
e
κ
i
|
χ
r
|
2
|
1 +
G
2
χ
m
χ
r
|
2
+ (
n
b,m
+ 1)
κ
e
γ
m,i
G
2
|
χ
m
|
2
|
χ
r
|
2
|
1 +
G
2
χ
m
χ
r
|
2
.
(30)
8
Here,
n
b,
wg
and
n
b,r
represent the bath of noise photons from the waveguide and the microwave resonator environment
respectively;
n
b,m
corresponds to the phonon bath at the mechanical frequency (see Supplementary Figure 3). We
assume thermal input noise correlations for all input noise terms, i.e.
ˆ
b
b,m
(
ω
)
ˆ
b
b,m
(
ω
)
= (
n
b,m
+ 1)
δ
(
ω
+
ω
),
ˆ
b
b,m
(
ω
)
ˆ
b
b,m
(
ω
)
=
n
b,m
δ
(
ω
+
ω
),
ˆ
a
b,r
(
ω
a
b,r
(
ω
)
= (
n
b,r
+ 1)
δ
(
ω
+
ω
),
ˆ
a
b,r
(
ω
a
b,r
(
ω
)
=
n
b,r
δ
(
ω
+
ω
),
ˆ
a
b,
wg
(
ω
a
b,
wg
(
ω
)
= (
n
b,
wg
+ 1)
δ
(
ω
+
ω
) and
ˆ
a
b,
wg
(
ω
a
b,
wg
(
ω
)
=
n
b,
wg
δ
(
ω
+
ω
).
In the sideband resolved regime and positive detuning (red sideband pump) we can drop the terms proportional
to
̃
χ
m
and
̃
χ
r
. In order to represent a realistic experimental setup, we introduce the fixed gain
G
in units of dB and
the system noise temperature
n
add
in units of resonator quanta and referenced to the cavity output. We can now
write the full expression for the single sided power spectral density as measured by a spectrum analyzer, valid in the
presence of all relevant noise baths
S
(
ω
) = ̄
d
10
G
/
10
[
n
add
+
n
b,
wg
+ (
n
b,
wg
+ 1)
(
1
κ
e
χ
r
1 +
G
2
χ
m
χ
r
)
2
+ (
n
b,r
+ 1)
κ
e
κ
i
|
χ
r
|
2
|
1 +
G
2
χ
m
χ
r
|
2
+ (
n
b,m
+ 1)
κ
e
γ
m,i
G
2
|
χ
m
|
2
|
χ
r
|
2
|
1 +
G
2
χ
m
χ
r
|
2
]
.
(31)
We minimize the number of free parameters by eliminating the resonator bath
n
b,r
using the relation
κn
r
=
κ
e
n
b,
wg
+
κ
i
n
b,r
.
(32)
With the Supplementary Equations 21 and 22 we can calculate [10] the mechanical occupation
n
m
n
m
=
n
b,m
(
γ
m,i
κ
4
G
2
+
κ
2
4
G
2
+
κγ
m,i
)
+
n
r
(
4
G
2
4
G
2
+
κγ
m,i
)
,
(33)
which we use to also replace the mechanical bath occupation
n
b,m
in Supplementary Equation 31.
Thermal waveguide noise
At large drive photon numbers we observe a power dependent increase of the measured noise background. Using
a cavity filter to remove broad band phase noise of the microwave source did not remove this background. Similarly,
the power levels of the observed microwave signals are believed to be far from saturating the HEMT amplifier.
However, we found that the noise figure of the low noise amplifier (LNA) is degrading at higher pump powers. This
effect explains the observed noise measurement background rise, which we therefore absorb into a small increase of
n
add
. Based on the implemented filtering of Johnson noise on the input (attenuators) and output lines (circulators /
isolators) we estimate the broad band black body radiation entering the chip waveguide to be
n
b,
wg
0
.
01. See for
example Ref. [13] for an independent cavity temperature measurement in a similar setup with approximately 10 dB
less attenuation.
Asymmetric noise spectra
In our measurements the cavity noise bath exceeds the waveguide noise bath for all relevant pump powers. In this
case the microwave resonator bath
n
r
manifests itself as a broad band resonator noise peak on top of the background.
This power dependent noise peak shows a small asymmetry for the highest pump powers. We follow a similar procedure
as outlined above and introduce a complex waveguide coupling constant to find better agreement with the measured
data in this limit. We make the substitution
κ
e
κ
e
iq
in the first term proportional to
n
b,
wg
in Supplementary
Equation 31 and expand it. For
κ
2
e

q
2
we can only keep
q
to linear order and simplify the expressions. We can
then write the asymmetric noise power spectrum with two additional terms as
S
as
(
ω
) =
S
(
ω
) + ̄
d
10
G
/
10
(
n
b,
wg
+ 1)
[
2
qG
2
(
ω
m
ω
)
|
χ
m
|
2
|
χ
r
|
2
|
1 +
G
2
χ
m
χ
r
|
2
+
2
q
(∆
r,d
ω
)
|
χ
r
|
2
|
1 +
G
2
χ
m
χ
r
|
2
]
.
(34)
The additional two terms are odd functions with a vanishing integral. This ensures the same fit results as obtained
compared to using the symmetric model Supplementary Equation 31. Defined in this way, the asymmetry scales with
the waveguide noise bath and the Fano parameter
q
, which is independent of any other parameters. We find excellent
9
agreement between this model and the measured broad band noise spectra at high pump powers (see main text).
It is important to point out that only the relevant bath occupancies
n
m
,
n
r
,
n
b,
wg
as well as
q
(in the case of the
highest drive powers) are actual fit parameters, while all other relevant parameters are extracted from - or verified in
- independent (low drive power) measurements.
Relation to the displacement spectrum
In the weak coupling regime we can relate the single sided displacement spectrum
S
x
(
ω
) using
S
(
ω
)
̄
=
S
x
(
ω
)
x
2
zpf
κ
e
κ
Γ
+
(35)
with the photon scattering rate Γ
+
4
G
2
for the optimal detuning ∆
r,d
=
ω
m
, and the factor
κ
e
taking into
account the limited collection efficiency of photons leaving the cavity.
Supplementary Note 7: Low drive power limits
At low drive powers and sufficient shielding from room temperature Johnson noise, it is a very good approximation
to set
n
b,
wg
0. Eliminating the waveguide noise input allows for a significant simplification of the power spectrum
S
(
ω
) = ̄
d
10
G
/
10
(
1 +
n
add
+
4
κ
e
(
n
r
κ
(
γ
2
m,i
+ 4(
ω
m
ω
)
2
) + 4
n
b,m
γ
m,i
G
2
)
|
4
G
2
+ (
κ
+ 2
i
(∆
r,d
ω
))(
γ
m
+ 2
i
(
ω
m
ω
))
|
2
)
,
(36)
with the mechanical noise bath
n
b,m
related to the mechanical occupation
n
m
in Supplementary Equation 33. At low
drive photon numbers we see no indication of TLS, pump phase noise, or waveguide heating of the cavity. The chosen
attenuation and shielding of input and output microwave lines connecting the sample to room temperature Johnson
noise limits the expected cavity occupation to
n
r
0
.
01. Under these assumptions, which are verified by our low
power measurements (constant background noise, no cavity noise peak), we can simplify the power spectrum to the
more standard form used in cavity electro- and optomechnics
S
(
ω
) = ̄
d
10
G
/
10
(
1 +
n
add
+
16
n
b,m
κ
e
γ
m,i
G
2
|
4
G
2
+ (
κ
+ 2
i
(∆
r,d
ω
))(
γ
m
+ 2
i
(
ω
m
ω
))
|
2
)
.
(37)
Without resonator occupation, Supplementary Equation 33 simplifies to
n
m
=
n
b,m
(
γ
m,i
κ
4
G
2
+
κ
2
4
G
2
+
κγ
m,i
)
n
b,m
(
1
C
+ 1
)
,
(38)
where we introduced the cooperativity
C
= 4
G
2
/
(
κγ
m,i
) and assumed moderate coupling strength 4
G
2

κ
2
in the
last step.
Supplementary Note 8: Linear response limit - system calibration
For very small drive powers where
C

1 we can simplify the expected thermal noise spectrum further. Dropping
terms associated with backaction allows to measure the displacement noise in a self-calibrated way. This compact
model is particularly useful to back out
g
0
and the system noise temperature with a minimal number of assumptions.
Starting with Supplementary Equation 37 we can make the replacement
n
b,m
n
m
and drop the backaction term
in the denominator. Both is valid for
C
0. We then insert
G
=
n
d
g
0
with the drive photon number defined in
Supplementary Equation 7. We furthermore assume that the gain of the system is flat over the relevant detuning
such that we can introduce the directly reflected pump power measured at the spectrum analyzer
P
r
= 10
G
/
10
P
out
.
The cavity output power is related to the cavity input power via Supplementary Equation 6
|
S
11
|
2
=
P
out
P
in
=
4∆
2
r,d
+ (
κ
2
κ
e
)
2
4∆
2
r,d
+
κ
2
.
(39)
10
Finally, we can write the measured noise spectrum, normalized by the measured reflected pump tone
S
(
ω
)
P
r
=
O
+
64
n
m
κ
2
e
γ
m,i
g
2
0
(4∆
2
r,d
+ (
κ
2
κ
e
)
2
)(
κ
2
+ 4(∆
r,d
ω
)
2
)(
γ
2
m,i
+ 4(
ω
m
ω
)
2
.
(40)
Only directly measurable system parameters and the temperature of the mechanical mode need to be known to
extract
g
0
without any further assumptions about the particular gain, attenuation or noise temperature of the chosen
measurement setup. Knowing
g
0
, it is easy to accurately back out the input attenuation
A
=
66
.
3 dB and drive
photon number
n
d
(for example from an EIT measurement). Furthermore, from the measured offset
O
= (1 +
n
add
)
4
κ
e
n
d
(4∆
2
r,d
+ (
κ
2
κ
e
)
2
)
(41)
we conveniently infer the system noise temperature in units of photons
n
add
30. The system gain
G ≈
46 dB is
then easily determined from the not-normalized wide band background of the measured noise spectrum
S
(
ω
).
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