Articles
https://doi.org/10.1038/s41565-019-0377-2
Quantum electromechanics of a hypersonic
crystal
Mahmoud Kalaee
1,2,4
, Mohammad Mirhosseini
1,2,4
, Paul B. Dieterle
1,2
, Matilda Peruzzo
3
,
Johannes M. Fink
2,3
and Oskar Painter
1,2
*
1
Kavli Nanoscience Institute and Thomas J. Watson, Sr., Laboratory of Applied Physics, California Institute of Technology, Pasadena, CA, USA.
2
Institute
for Quantum Information and Matter, California Institute of Technology, Pasadena, CA, USA.
3
Institute of Science and Technology Austria, Klosterneuburg,
Austria.
4
These authors contributed equally: Mahmoud Kalaee, Mohammad Mirhosseini. *e-mail: opainter@caltech.edu
SUPPLEMENTARY INFORMATION
In the format provided by the authors and unedited.
Nature NaNO
techNOl
Ogy
|
www.nature.com/naturenanotechnology
2
10
12
14
16
360
400
440
480
520
560
8
2
J
(MHz)
d
(μm)
10
1
10
0
10
-1
10
-2
δ(μm)
20
40
60
0
∙
e,-
∙
e,+
Coil1
Coil2
d
δ
a
b
c
∙
e
(MHz)
FIG. S-1.
Planar spiral coil inductor and lumped element LC resonators. a,
A schematic of the full electromechanical
circuit. The microwave circuit is made of aluminum shown in gray color and the silicon membrane is shown in blue. The
two LC microwave resonators consist of a high impedance coil which is capacitively coupled to a high frequency mechanical
beam. Moreover, the microwave resonators are inductively coupled to each other. Tuning the distance between the two coils
d
allows for control over the strength of the coupling between the two LC resonators. Additionally, positioning of the two LC
resonator system with respect to the branching point of the coupler wire allows for fine tuning of coupling of the even and odd
supermodes to the microwave feed line.
b
, Shows the frequency splitting of the even and odd super modes 2
J
=
ω
r,
+
−
ω
r,
−
versus the distance between the coils.
c,
External coupling of even (odd)
κ
e,
+
(
κ
e,
−
) to the microwave feed line versus the
separation between branching point of the coupler wire and the center of the two LC resonators
δ
.
3
10
11
10
10
10
9
10
8
10
7
10
6
10
5
1
2
3
5
4
# of Mirror Periods
Q
m,rad
PML
0
-50
-100
-150
10 log
10
(
½
/
½
0
)
FIG. S-2.
Mechanical radiation-
Q
simulations. a,
FEM Simulation of the mechanical beam resonator surrounded by a
2-period phononic band gap structure, silicon membrane and a perfectly matched layer (PML). Color indicates the normalized
mechanical energy density with red (green) color corresponding to the area with highest (lowest) mechanical energy density.
b,
Simulated radiation-limited mechanical quality factor versus the number of the periods of the phononic band gap shield.
4
G
b
,
n
m
a
in
^
a
out
^
^
a
b,r1
,
n
b,r1
°
i
b
b,m
,
n
b,m
^
^
a
1
,
n
r1
^
a
2
,
n
r2
^
∙
e,2
a
b,r2
,
n
b,r2
^
∙
i,1
∙
e,1
∙
i,2
J
FIG. S-3.
Input-Output Mode Schematic.
In the reflective geometry the microwave cavity mode ˆ
a
1
(ˆ
a
2
) is coupled to the
coherent waveguide modes ˆ
a
in
and ˆ
a
out
with the external coupling strength
κ
e,
1
(
κ
e,
2
). It is also coupled to a bath of noise
photons, ideally at the refrigerator temperature
n
b,r
1
(
n
b,r
2
), with the intrinsic coupling strength
κ
i,
1
(
κ
i,
2
). 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
i
.
5
4 K
7 mK
300 K
HEMT
LNA
-30dB
-20dB
-30dB
SG1
VNA
SA
'
I
c
M
L
c
L
c
C
c
C
c
C
m
C
m
SG2
FIG. S-4.
Experimental setup.
The output tone of a microwave signal generators (SG1 and SG2) and the output tone
of a vector network analyzer (VNA) are combined at room temperature, attenuated, routed to the sample at about 7 mK
and inductively coupled to the LC circuit on the silicon membrane. We detect the reflected output tone which is routed and
isolated with two microwave circulators and after amplification with a high electron mobility transistor amplifier (HEMT) at
4K stage, switchable pump tone cancellation (dashed lines), and further amplification with a low noise amplifier (LNA). The
measurement is done either phase coherently with the VNA, or we detect the incoherent power spectrum with the spectrum
analyzer (SA).
6
t
(ms)
-85
-80
-75
-70
-65
-60
-55
-50
|
S
11
|(
ω
d
=
ω
r,
+
-
ω
m
) (dB)
0
-1
-2
1
2
3
4
5
δ
exc.
read-
out
ω
r,
+
-
ω
m
ω
r,
+
t
0
exp[-
γ
m
t
]
exp[-
κ
+
t
]
FIG. S-5.
Mechanical Ringdown Measurement.
Pulsed excitation and read-out measurement of the ringdown of the
acoustic energy in the nanobeam mechanical resonator. In this measurement the drive tone at 6 dBm power corresponding to
an intra-cavity photon number of
n
d,
−
= 1
.
42
×
10
5
is kept on after the excitation and the decay includes back action damping
due to the drive tone. The initial steep decay results from the leakage of the photons from the microwave cavity. The decay
rate of the mechanical resonator
γ
m
is extracted by fitting an exponential curve to the slow decay section of the signal versus
time.
7
-0.3
-0.2
-0.1
0
0.1
0.2
∆
r+,d
/2π - 424.69 (MHz)
126
128
130
132
134
136
κ
+
(kHz)
-400
-300
-200
-100
0
100
200
300
400
10
-1
10
0
S
11
(
ω
p
- ω
r,
+
)/2
π
(kH
z)
-350
-150
0
150
350
10
-1
10
0
Mechanical response
Upper-frequency
microwave resonance
S
11
(
ω
p
- ω
r,
+
)/2
π
(kH
z)
-20
02
0
0
2
4
6
8
10
12
1/S
11
(
ω
p
- ω
r,
+
)/2
π
(kH
z)
05
10
15
0
0.1
0.2
0.3
0.4
0.5
0.6
S
11
(
ω
p
- ω
r,
+
)/2
π
(kH
z)
a
c
b
d
e
≈
2
G/
2
π
=
16.8 kHz
FIG. S-6.
Modeling of the two-photon response. a,
Reflection spectrum of a weak coherent probe tone at frequency
ω
p
close
to the upper microwave resonance at
ω
r,
+
, in the presence of a strong drive tone at
ω
d
which is detuned by ∆
r
+
,d
≡
ω
r,
+
−
ω
d
.
The mechanical lineshape and the microwave resonance parameters can be identified separately in this measurement where
∆
r
+
,d
6
=
ω
m
. Solid red line shows the theory fit to the microwave resonance assuming a Lorentzian lineshape.
b,
The fitted
value of total linewidth (
κ
) for the upper frequency microwave resonance as a function of the detuning of the strong drive
tone (∆
r
+
,d
).
c,
Fitted EIT-like response for zero two-photon detuning (∆
r
+
,d
≈
ω
m
).
d,
The central part of the reflection
spectrum is used for extracting the jitter in the mechanical frequency. Solid line shows a fit assuming a FWHM jitter of
δω
m
/
2
π
= 1
.
98 kHz.
e,
Inverse of the probe tone reflection as a function of frequency. The values for parametrically enhanced
electromechanical coupling (
G
), and the intrinsic decay rate of the mechanical mode (
γ
i
) are extracted by fitting a theory curve
to the two peaks.
8
SUPPLEMENTARY NOTE 1. CIRCUIT PROPERTIES AND COIL DESIGN
The electromechanical circuit is composed of two identical LC coil resonators, each capacitively coupled to a
mechanically compliant nanobeam supporting a mechanical mode with slightly different frequencies of 420 MHz and
425 MHz. Fig. S-1a shows a schematic of the microwave electromechanical circuit. The coil Al wires are 500 nm
wide and 120 nm thick, with a 1
μ
m pitch and 35 turns forming a square of dimension 74
μ
m
×
74
μ
m. According
to the finite element simulations, the coils have a self resonance frequency at
ω
srf
/
2
π
= 13
.
98 GHz. This simulation
includes the cross-overs and the coupler wires. Using the modified Wheeler formula [1], the self inductance of the coil is
L
= 41
.
8 nH. Using the simple relation
ω
srf
= (
LC
s
)
−
1
2
, we obtain the total stray capacitance of
C
s
=
C
c
+
C
p
= 3
.
1 fF,
where
C
c
is the parasitic capacitance from the coil inductor and
C
p
is the extra parasitic capacitance due to the extra
wiring. The modulating capacitance
C
m
for different gap size of the electrodes is shown in Fig. 1d of the main text.
From the measured
g
0
in our experiment we estimate the capacitor gap to be
s
≈
40 nm, from which we estimate the
modulating capacitance to be
C
m
= 2
.
1 fF. This yields a participation factor of
η
≈
0
.
4 and a bare LC coil resonance
frequency of
ω
r,
0
/
2
π
= 10
.
77 GHz (when adding the estimated motional capacitance to stray capacitance
C
s
of the
coil and extra wiring). Using the simulated value of 2
J
, we estimate
ω
r,
+
= 10
.
9975 GHz and
ω
r,
−
= 10
.
5625 GHz,
very close to the measured values of the supermodes which are 10
.
9930 GHz and 10
.
5788 GHz, respectively.
The two mechanical beams are designed to have the same length and gap size such that the modulating capacitance
and the frequency of both microwave resonators are identical. By placing the two microwave resonator near each
other, the two coils are inductively coupled forming an even and odd symmetry microwave supermodes. By adjusting
the distance between the two inductors, we can control the strength of the mutual inductance hence, controlling the
frequency splitting between the even and odd microwave mode. Figure S-1b shows the frequency splitting between
the two microwave modes as a function of the distance between the coil inductors. For the device presented in this
manuscript, the distance is chosen to be
d
= 12
μ
m yielding a simulated frequency splitting of 2
J/
2
π
= 415 MHz
between the microwave modes. Moreover, the placement of the two inductors with respect the branching point of
the coupler wire denoted by
δ
allows us to independently control the coupling of the microwave supermodes to the
microwave feed line. The symmetric positioning of the two microwave coil resonator with respect to the branching
point of the microwave feed line leads to destructive interference in the external decay path for the even supermode
and constructive interference for the odd supermode, leading to completely bright and dark microwave resonances.
Breaking this symmetry and displacing the two microwave coil resonator allows us to control the coupling of the
two microwave supermodes. Fig. S-1c shows the coupling of the even and odd microwave modes to the microwave
feed line versus the displacement of the coil inductors
δ
. For the chosen
δ
= 16
.
5
μ
m, we expect the coupling of
κ
e,
+
/
2
π
= 102 kHz for the even mode at higher frequency and
κ
e,
−
/
2
π
= 9
.
6 MHz for the odd mode at lower
frequency. All the microwave LC resonator simulations are performed using the SONNET software package [2]. In
these simulations we have assumed a relative permittivity of
Si
= 11
.
7 for the silicon membrane.
9
SUPPLEMENTARY NOTE 2. PHONONIC BANDGAP AND MECHANICAL
Q
SIMULATION
Periodic structures can be engineered to have a phononic bandgap where mechanical energy loss by linear elastic
coupling to the environment is suppressed. Lack of a phononic bandgap in the nanobeam structure means that the
localized mechanical mode of the beam can couple with radiating acoustic modes due to imperfections in the structure
that cause acoustic scattering and break the beam symmetry. To prevent the coupling of the localized mechanical
modes of the nanobeam with the acoustic modes of the surrounding silicon membrane and substrate we have embedded
our mechanical beam resonator in a phononic bandgap ‘cross’ structure (see Fig. 2 in the main text). To model the
damping of the localized breathing mode due to acoustic radiation we include a special circular pad region around
the structure that absorbs all the outgoing acoustic waves. This domain is called a perfectly matched layer (PML).
What distinguishes a PML domain from an ordinary absorbing/lossy material is that all the waves incident upon the
PML from a non-PML media do not reflect at the interface, hence the PML strongly absorbs all outgoing waves from
the interior of the computational region.
An example of the FEM simulation with PML is shown in Fig. S-2a. In this image the nanobeam acoustic resonator
and a 2-period phononic bandgap shield are surrounded by an inner circular membrane of normal Si material and
an outer, larger circular PML domain also made of Si. The color indicates the normalized acoustic energy density
of the breathing mode throughout the structure. As shown, with using just 2 periods for the phononic shield, the
radiated mechanical energy density is suppressed by
≈
100 dB. Fig. S-2b shows the simulated radiation-limited
quality factors of the breathing mode of the mechanical resonator versus the number of phononic shield periods
surrounding the mechanical nanobeam. All the finite element method simulation of the mechanical nanobeam are
done using COMSOL Multiphysics [3]. The mechanical simulation use the full anisotropic elasticity matrix where
(
C
11
,C
12
,C
44
) = (166
,
64
,
80) GPa and assumes a [100] crystalline orientation aligned with the
x
-axis of the nanobeam
(direction along its length).
10
SUPPLEMENTARY NOTE 3. DERIVATION OF TWO-MODE ELECTROMECHANICAL RESPONSE
In this section we follow previous work [4–9] to the calculate coherent response and the noise spectrum of the
electromechanical system.
A. System Hamiltonian of two-mode microwave electromechanical system
The Hamiltonian of the coupled microwave electromechanical system (see Fig. S-3) can be written as
ˆ
H
= ̄
hω
r,
0
ˆ
a
†
1
ˆ
a
1
+ ̄
hω
r,
0
ˆ
a
†
2
ˆ
a
2
+ ̄
hJ
(ˆ
a
†
1
ˆ
a
2
+ ˆ
a
†
2
ˆ
a
1
) + ̄
hω
m
ˆ
b
†
ˆ
b
+ ̄
hg
0
ˆ
a
†
1
ˆ
a
1
(
ˆ
b
†
+
ˆ
b
)
,
(S-1)
where ˆ
a
j
(ˆ
a
†
j
) are the annihilation (creation) operators of the local microwave cavity modes with bare frequency
ω
r,
0
and
ˆ
b
(
ˆ
b
†
) are the annihilation (creation) operator of the mechanical mode.
J
is the coupling between the two
microwave cavity and
g
0
is the single photon coupling between the microwave cavity 1 and the mechanical mode. We
diagonalize the Hamiltonian by introducing the even and odd superposition of the microwave cavity modes of the
form ˆ
a
±
=
ˆ
a
1
±
ˆ
a
2
√
2
.
The new Hamiltonian is now written as
ˆ
H
′
= ̄
hω
r,
+
ˆ
a
†
+
ˆ
a
+
+ ̄
hω
r,
−
ˆ
a
†
−
ˆ
a
−
+ ̄
hω
m
ˆ
b
†
ˆ
b
+ ̄
h
g
0
2
(ˆ
a
†
+
ˆ
a
−
+ ˆ
a
†
−
ˆ
a
+
+ ˆ
a
†
+
ˆ
a
+
+ ˆ
a
†
−
ˆ
a
−
)(
ˆ
b
†
+
ˆ
b
)
,
(S-2)
where the new supermode frequencies are
ω
±
=
ω
r,
0
±
J
. We excite the microwave resonators by using a strong
coherent drive at frequency
ω
d
detuned from the microwave resonators frequency by ∆
r
±
,d
=
ω
r
±
−
ω
d
. Thus we can
write the Hamiltonian in the rotating frame
ˆ
̃
H
=
−
̄
h
∆
r
+
,d
ˆ
a
†
+
ˆ
a
+
−
̄
h
∆
r
−
,d
ˆ
a
†
−
ˆ
a
−
+ ̄
hω
m
ˆ
b
†
ˆ
b
+ ̄
h
g
0
2
(ˆ
a
†
+
ˆ
a
−
+ ˆ
a
†
−
ˆ
a
+
+ ˆ
a
†
+
ˆ
a
+
+ ˆ
a
†
−
ˆ
a
−
)(
ˆ
b
†
+
ˆ
b
)
.
(S-3)
Assuming a strong red detuned drive from the higher frequency microwave cavity and ∆
r
+
,d
≈
ω
m
∆
r
−
,d
we
can linearize the Hamiltonian in the rotating frame to obtain
ˆ
̃
H
=
−
̄
h
∆
r
+
,d
ˆ
a
†
+
ˆ
a
+
−
̄
h
∆
r
−
,d
ˆ
a
†
−
ˆ
a
−
+ ̄
hω
m
ˆ
b
†
ˆ
b
+ ̄
hG
(ˆ
a
−
+ ˆ
a
†
−
+ ˆ
a
†
+
+ ˆ
a
+
)(
ˆ
b
†
+
ˆ
b
)
,
(S-4)
where
G
=
g
0
2
√
n
d,
−
and
n
d,
−
corresponds to intra-cavity photon number of the lower frequency microwave cavity
defined by
n
d,
−
=
P
d
̄
hω
d
4
κ
e,
−
κ
2
−
+ 4∆
2
r
−
,d
,
(S-5)
where
P
d
is the power at the cavity input, expressed by
P
d
= 10
−
3
10
(
A
+
P
in
)
/
10
with
P
in
the drive power in dBm and
A
the total attenuation of the input line in dB. For the Hamiltonian in Eq. (S-4), the linearized Langevin equations
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
)
,
(S-6)
̇
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
)
,
(S-7)
̇
ˆ
b
(
t
) =
−
(
iω
m
+
γ
i
2
)
−
iG
(ˆ
a
+
(
t
) + ˆ
a
†
−
(
t
))
−
√
γ
i
ˆ
b
b
(
t
)
.
(S-8)
Here, ˆ
a
b,r
±
are annihilation operators of the microwave baths coupled to the even and odd microwave cavities. We
define 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
)
,
(S-9)
ˆ
A
(
ω
) =
1
√
2
π
ˆ
+
∞
−∞
dte
iωt
ˆ
A
(
t
)
.
(S-10)
11
Now, taking the Fourier transform of the linearized Langevin equations and simplifying we obtain
χ
−
1
r
−
(
ω
)ˆ
a
−
(
ω
) =
−
iG
(
ˆ
b
(
ω
) +
ˆ
b
†
(
ω
))
−
√
κ
e,
−
ˆ
a
in
(
ω
)
−
√
κ
i,
−
ˆ
a
b,r
−
(
ω
)
,
(S-11)
χ
−
1
r
+
(
ω
)ˆ
a
+
(
ω
) =
−
iG
(
ˆ
b
(
ω
) +
ˆ
b
†
(
ω
))
−
√
κ
e,
+
ˆ
a
in
(
ω
)
−
√
κ
i,
+
ˆ
a
b,r
+
(
ω
)
,
(S-12)
χ
−
1
m
(
ω
)
ˆ
b
(
ω
) =
−
iG
(ˆ
a
−
(
ω
) + ˆ
a
†
−
(
ω
) + ˆ
a
+
(
ω
) + ˆ
a
†
+
(
ω
))
−
√
γ
i
ˆ
b
b
(
ω
)
,
(S-13)
where
χ
r
±
and
χ
m
are the uncoupled susceptibilities of the electrical and the mechanical modes defined by
χ
−
1
r
±
(
ω
) =
κ
±
/
2 +
i
(∆
r
±
,d
−
ω
)
,
(S-14)
χ
−
1
m
(
ω
) =
γ
i
/
2 +
i
(
ω
m
−
ω
)
.
(S-15)
In the sideband-resolved limit
ω
m
κ
±
,G
and for positive detuning of the drive tone with respect to the high
frequency microwave cavity ∆
r
+
,d
≈
ω
m
(red side pumping), we have
χ
m
χ
r
−
χ
m
χ
r
+
. Thus the linearized Langevin
equation can be simplified further and written approximately as
ˆ
a
+
(
ω
) =
iGχ
r
+
χ
m
√
γ
i
ˆ
b
b
(
ω
)
−
χ
r
+
(
√
κ
e,
+
ˆ
a
in
(
ω
) +
√
κ
i,
+
ˆ
a
b,r
+
(
ω
))
1 +
G
2
χ
r
+
χ
m
,
(S-16)
ˆ
a
−
(
ω
) =
−
χ
r
−
(
√
κ
e,
−
ˆ
a
in
(
ω
) +
√
κ
i,
−
ˆ
a
b,r
−
(
ω
))
,
(S-17)
ˆ
b
(
ω
) =
−
χ
m
√
γ
i
ˆ
b
b
(
ω
)
−
iGχ
m
χ
r
+
(
√
κ
e,
+
ˆ
a
in
(
ω
) +
√
κ
i,
+
ˆ
a
b,r
+
(
ω
))
1 +
G
2
χ
r
+
χ
m
,
(S-18)
where we have dropped the terms proportional
χ
m
χ
r
−
. For driving of the odd-symmetry microwave mode, transduc-
tion of the acoustic mechanical mode is observed in the output field of the even-symmetry microwave cavity mode.
Using the Input-Output formalism and Eq. (S-16) we find for the output field near resonance with the even-symmetry
microwave mode
ˆ
a
out
(
ω
) = ˆ
a
in
(
ω
) +
√
κ
e,
+
ˆ
a
+
(
ω
)
,
= ˆ
a
in
(
ω
)
−
κ
e,
+
χ
r
+
ˆ
a
in
(
ω
)
1 +
G
2
χ
m
χ
r
+
−
√
κ
e,
+
κ
i,
+
χ
r
+
ˆ
a
b,r
+
(
ω
)
1 +
G
2
χ
m
χ
r
+
+
ˆ
b
b
(
ω
)
iG
√
κ
e,
+
γ
i
χ
m
χ
r
+
1 +
G
2
χ
m
χ
r
+
.
(S-19)
B. Electromagnetically Induced Transparency
We first calculate the coherent part of the output signal by using Eq. (S-19) and dropping the incoherent terms to
get
S
11
=
〈
ˆ
a
out
(
ω
)
〉
〈
ˆ
a
in
(
ω
)
〉
= 1
−
κ
e,
+
χ
r
+
1 +
G
2
χ
m
χ
r
+
,
(S-20)
and substituting for the bare microwave and mechanical cavity susceptibilities we get the coherent electromechanical
analogue of the electromagnetic induced transparency valid for small probe powers,
S
11
= 1
−
κ
e,
+
κ
+
/
2 +
i
(∆
r
+
,d
−
ω
) +
G
2
γ
i
/
2+
i
(
ω
m
−
ω
)
.
(S-21)
C. Quantum derivation of observed noise spectra
For an operator
ˆ
A
the power spectral density is written as
S
AA
(
t
) =
ˆ
+
∞
−∞
dτe
iωτ
〈
ˆ
A
†
(
t
+
τ
)
ˆ
A
(
t
)
〉
.
(S-22)
Utilizing Fourier transform defined above, we can re-define the power spectral density of an operator
ˆ
A
in terms of
frequency [8] as
S
AA
(
ω
) =
ˆ
+
∞
−∞
dω
′
〈
ˆ
A
†
(
ω
)
ˆ
A
(
ω
′
)
〉
.
(S-23)
12
Thus, for the auto-correlation of the detected normalized field amplitude (or the photo current)
ˆ
I
(
t
) = ˆ
a
out
(
t
)+ˆ
a
†
out
(
t
),
we get
S
II
=
ˆ
+
∞
−∞
dω
′
〈(
ˆ
a
out
(
ω
) + ˆ
a
†
out
(
ω
)
)(
ˆ
a
out
(
ω
′
) + ˆ
a
†
out
(
ω
′
)
)〉
.
(S-24)
Substituting for ˆ
a
out
(
ω
) and ˆ
a
†
out
(
ω
) from Eq. (S-19) and using thermal noise correlation for input noise terms
(
i.e.
〈
ˆ
b
b
(
ω
)
ˆ
b
†
b
(
ω
′
)
〉
= (
n
b,m
+ 1)
δ
(
ω
+
ω
′
),
〈
ˆ
b
†
b
(
ω
)
ˆ
b
b
(
ω
′
)
〉
=
n
b,m
δ
(
ω
+
ω
′
),
〈
ˆ
a
b,r
+
(
ω
)ˆ
a
†
b,r
+
(
ω
′
)
〉
= (
n
b,r
+
+ 1)
δ
(
ω
+
ω
′
),
〈
ˆ
a
†
b,r
+
(
ω
)ˆ
a
b,r
+
(
ω
′
)
〉
=
n
b,r
+
δ
(
ω
+
ω
′
),
〈
ˆ
a
in
(
ω
)ˆ
a
†
in
(
ω
′
)
〉
=
δ
(
ω
+
ω
′
)), the power spectral density is
S
II
(
ω
) =
∣
∣
∣
(
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
i
G
2
|
χ
m
|
2
|
χ
r
+
|
2
|
1 +
G
2
χ
m
χ
r
+
|
2
.
(S-25)
The mechanical occupancy can also be calculated by using Eq. (S-16) and Eq. (S-18) as in Ref. [5]
n
m
=
n
b,m
(
γ
i
κ
+
4
G
2
+
κ
2
+
4
G
2
+
κ
+
γ
i
)
+
n
r,
+
(
4
G
2
4
G
2
+
κ
+
γ
i
)
.
(S-26)
13
SUPPLEMENTARY NOTE 4. EXPERIMENTAL SETUP
For measuring the coherent and incoherent response of our circuit, we combine the output of a vector network
analyzer with two other microwave sources and feed the microwave signals to the base plate of a cryogen free dilution
refrigerator at
T
f
≈
10 mK using coaxial cables with feedthroughs and attenuators for thermalization at multiple
temperature stages yielding total attenuation of
A
= 76 dB and suppressing room temperature Johnson noise (see
Fig. S-4) to about 0
.
05 photons. We couple to the sample in a reflective geometry using a circulator and a low loss
copper printed circuit board (PCB). On the PCB and the chip we use 50 Ω coplanar waveguides (CPW) to route
the microwave tones to the membrane with very little reflections (
<
−
25 dB). Near the LC circuit, we extend the
center conductor of the CPW waveguide and short it to ground with a narrow wire passing near the inductor coils
and inductively couple to the microwave resonators.
On the output side, we use an isolator for isolating the sample from the 4 K stage noise. Niobium titanium
superconducting cables are used to connect the isolator directly to a low noise, high electron mobility transistor
amplifier (HEMT) at 4 K stage. The microwave signal is again amplified with a low noise room temperature amplifier
(LNA). In order to suppress spurious response peaks for high drive power measurements, we cancel the high drive
power tone by adding a phase and amplitude adjusted part of the pump tone to the output signal before the room
temperature amplifier, as shown in Fig. S-4. After the final amplification we use an electronically controlled microwave
switch to distribute the signal to either the spectrum analyzer or the second vector network analyzer port. The total
gain of the system is
G
= 57
.
6 dB.
14
SUPPLEMENTARY NOTE 5. MECHANICAL RINGDOWN MEASUREMENTS
In the ringdown measurements performed in this work we apply, as in the EIT-like spectroscopy, a 100 ms two-tone
pulse to ring up the mechanics with a strong drive tone at
ω
d
=
ω
r,
+
−
ω
m
and a weak probe tone at
ω
p
=
ω
r,
+
.
Detection of the acoustic mode occupancy is performed with a read-out pulse in which the weak probe tone is turned
off and motionally scattered photons from the strong drive tone (still at
ω
d
=
ω
r,
+
−
ω
m
) are detected on a spectrum
analyzer in zero-span mode with center frequency at
ω
r,
+
and resolution bandwidth (RBW) set to 30 kHz (
γ/
2
π
).
A plot of the motionally scattered photons, corresponding the acoustic mode occupancy versus time is shown in
Fig. S-5 for
n
d,
−
= 1
.
42
×
10
5
photons. The initial steep decay results from the leakage of the photons from the
microwave cavity and is followed by a slower decay of the signal due to the mechanical damping. The measured
mechanical damping contains contributions from back-action
γ
em
and intrinsic energy damping of the acoustic mode
γ
i
.
15
SUPPLEMENTARY NOTE 6. MODELING THE TWO-PHOTON RESPONSE AND DETERMINING
COOPERATIVITY
To probe the acoustic properties of the fabricated device, and to determine the degree of coherent coupling between
the microwave and acoustic resonances, we use a two-tone pump and probe scheme. The presence of acoustic frequency
jitter complicates the modeling of the electromechanical response, and so we detail here the methods we employ to
determine the electromechanical cooperative coupling,
C
. In this scheme, described also in the main text, a strong
drive tone (
ω
d
) is applied at a variable detuning from the lower frequency microwave resonance while a weaker probe
tone (
ω
p
) is scanned across the upper frequency microwave resonance. Figure S-6a shows one of the measured spectrum
from Fig. 4d of the main text, where the drive power in units of intra-cavity photons is
n
d,
−
= 2
.
25
×
10
5
. The spectral
lineshape of the reflected signal as a function of the drive and probe detuning can be written as
S
11
= 1
−
κ
e,
+
κ
+
2
+
iδ
r
+
,p
+
2
G
2
γ
i
+2
i
(
δ
r
+
,p
−
(
ω
m
−
∆
r
+
,d
))
,
(S-27)
where
δ
r
+
,p
≡
ω
p
−
ω
r,
+
and ∆
r
+
,d
≡
ω
r,
+
−
ω
d
. Here,
κ
e,
+
,
κ
+
, and
ω
r,
+
denote the parameters of the upper
frequency microwave resonance. We extract these parameters first by fitting a Lorentzian lineshape to the microwave
resonance when the mechanical response is detuned from the center of cavity (∆
r
+
,d
6
=
ω
m
). Figure S-6b shows the
extracted values of
κ
+
as a function of
ω
d
. The external cavity decay rate
κ
e,
+
is fitted to the constant value of
85
.
3 kHz for all the measurements.
Using the extracted values for the microwave resonance we fit the reflection spectrum at the two-photon resonance
(‘transparency’ window) when the pump-probe difference frequency matches that of the capacitively-coupled acoustic
mode frequency,
ω
p
−
ω
d
=
ω
m
(See Fig. S-6c). A particular challenge in modeling the devices of this work is the
presence of frequency jitter in the acoustic frequency. Acoustic frequency jitter tends to blur out the time-averaged
measurement of the two-photon resonance peak, and in doing so can lead to a significant underestimation of the
coherent electro-mechanical coupling when directly applying Eq. (S-27). In order to avoid this fitting problem, we
first extract the statistical variations in
ω
m
(frequency jitter) from the linewidth of the mechanical response at the
center of transparency window (Fig. S-6d). This linewidth contains contributions from electromechanical back-action
(
γ
em
= 4
G
2
/κ
+
), intrinsic energy damping of the acoustic mode (
γ
i
), and any pure dephasing (frequency jitter) of
the acoustic mode. Using the values (
G/
2
π
= 8
.
9 kHz and
γ
i
/
2
π
= 68 Hz) found directly from the time-domain
measurements of mechanical ringdown under the same microwave drive conditions, we back out a FWHM frequency
jitter value of
δω
m
/
2
π
= 1
.
98 kHz in the spectrum of Fig. S-6d.
We next refine our estimates of
G
and
γ
i
by fitting a theory curve to the
inverse
of the measured reflection spectrum.
Figure S-6e shows the spectral shape of the inverse response. The frequency separation between the two peaks in the
inverse spectrum can be shown to be very close to the value of 2
G
at two-photon resonance (in the strong coupling
limit these two peaks correspond to the hybridized eigenmodes of the coupled system of acoustic and electromagnetic
modes). Importantly, the position of these two peaks in the inverse spectrum are relatively insensitive to acoustic
frequency jitter. From this measured frequency separation we find a value of
G/
2
π
= 8
.
4 kHz, which is in good
agreement with the value found from the ring down measurement (8
.
9 kHz). In the final step of our modeling, we find
the value of
γ
i
by fitting the inverse response using our best estimates of the microwave resonance, parametrically
enhanced coupling, and frequency jitter. The value of the best estimate for intrinsic mechanical damping is found
to be
γ
i
/
2
π
= 91 Hz, with a standard deviation of 66 Hz. In this case the uncertainty in
γ
i
is primarily set by the
uncertainty in estimating
κ
+
, which has been extracted from the curve in Fig. S-6b. The slightly elevated value of
γ
i
from two-tone spectroscopy (91 Hz) with respect to the time-domain ‘in-the-dark’ ring-down could be due to the
degradation of the mechanical quality factor in presence of pump-induced heating. Nevertheless, we emphasize that
the difference between the two values is not statistically significant. We should also note that estimating
γ
i
in the
large cooperativity limit from EIT spectra is inherently challenging (even in the absence of frequency jitter) due to
the saturation in the transparency window peak height.
Using the best estimates of the parameters
γ
i
,
G
, and
κ
+
found from the procedure detailed above, we find the
mean value of cooperativity
C
= 4
G
2
/κ
+
γ
i
= 28
.
5, with a standard deviation of
σ
C
= 7
.
3. This value is consistent
with the cooperativity expected from time-domain ringdown measurements.
16
SUPPLEMENTARY NOTE 7. FREQUENCY JITTER
Our analysis of the frequency noise of the breathing mode follows that from Refs. [10, 11]. A main takeaway from the
referenced work is that frequency fluctuations slower than the decay rate of the the mechanical resonance (
δω
γ
m
),
can be considered as slow fluctuations in the mechanical susceptibility leading to slow variation in amplitude and
phase of the driven response. Thus an applied near resonant tone is spectrally broadened around its frequency
ω
F
.
For frequency fluctuations faster than the intrinsic energy decay rate of the mechanical oscillator (
δ
ω
γ
m
), the
driving term quickly loses the memory of the driving frequency, and the mechanical response is broadened over the
entire range of the frequency fluctuations. This results in a similar response spectrum to that of the thermal spectrum,
with the only difference being that the amplitude of the signal is dependent on the driving amplitude rather than the
temperature of the thermal bath coupled to the mechanical mode.
Figure 6a of the main text shows the narrowband (slow) and broadband (fast) spectra of the mechanical frequency
response in the presence of a weak coherent tone applied near resonance. We denote the area of the narrowband and
broadband peaks by
S
nb
and
S
bb
, respectively. We can estimate the broadband component of the frequency jitter
of the mechanical oscillator by measuring the area underneath the narrowband and broadband peaks and using the
relation
̃
γ
m
γ
m
≈
1 +
S
bb
S
δ
(
1
−
S
nb
S
δ
)
,
(S-28)
where ̃
γ
m
is the broadened mechanical linewidth and
S
δ
corresponds to the area underneath the applied coherent
tone in the absence of mechanical interaction, measured by detuning the tone frequency away from the mechanical
resonance. We have extracted the total linewidth of 6.7 kHz by fitting a Lorentzian to the thermometry response of
the mechanical resonator in the absence of any weak drive tones. Using the relation in Eq. (S-28), the broadband (fast)
frequency fluctuations contribute to only 15% of the measured total mechanical linewidth. Back-action cooling
γ
em
and intrinsic decay rate
γ
i
of the mechanical oscillator add up to 29% of the total linewidth, independently measured
using mechanical ring down measurement. We attribute the remaining 58% of the linewidth to low frequency jitter
of the mechanical resonance frequency.
The source(s) of the frequency jitter and anomalous mechanical heating observed in current devices is not well
understood at this point; however, there are a few candidate sources to consider. Two-level tunneling systems
(TLS) [12] found within amorphous surface oxide layers are known to cause excess damping and noise in microwave
superconducting quantum circuits [13]. TLS also couple to strain fields, and the high frequency of the hypersonic
breathing mode may lead to coupling (via nonlinear phonon-phonon or TLS-phonon scattering) to the same TLS bath
as that of the microwave pump. The observed fluctuation in the breathing mode occupancy is also reminiscent of
the bursty nature of quasi-particle (QP) generation measured in thin films of superconducting Al [14, 15] due to high
energy particle impacts. Modelling of the interaction of QPs with sub-gap electromagnetic radiation indicates that
even weak microwave probe fields can lead to non-equilibrium QP and phonon distributions above that of the thermal
background [16]. Both TLS and QP considerations indicate that moving to a different superconducting material with
short QP relaxation time and a clean surface – such as NbTiN [17] – may significantly reduce acoustic mode heating
and frequency jitter. An additional attribute of NbTiN is its large kinetic inductance [18], which can be employed to
further reduce parasitic capacitance and increase the electromechanical coupling.
17
SUPPLEMENTARY NOTE 8. HEATING MODEL
The heating curve observed in Fig. 6(c) of the main text is fit using a phenomenological heating model, first described
in Ref. [19, 20] to treat heating effects due to optical absorption in optomechanical crystals. As mentioned in the
main text and in Note 7, at this point we do not have a microscopic model for the measured anomalous heating. The
phenomenological model assumes coupling between the mechanical oscillator and three different thermal baths. The
first bath is the effectively zero-temperature bath of the red-detuned pump tone which is coupled to the mechanical
oscillator at a rate
γ
em
. The second bath is the ambient fridge bath which is nominally at a temperature of 10 mK,
but which we leave as a fit parameter corresponding to occupancy
n
b,m
. The fridge bath is assumed to be coupled
to the mechanical oscillator at the measured intrinsic damping rate of
γ
i
/
2
π
= 68 Hz. The third bath is the pump
induced hot bath of unknown microscopic origin. We assume that this bath has a pump-power dependent occupancy
(
n
p,m
), and couples to the breathing mode through at a rate
γ
p
, also potentially pump-power dependent. Moreover,
to best capture the dynamics of the heating curve, it is assumed that the hot bath has a finite equilibration time.
Thus the red curve that we fit to the data assumes that a fraction (
δ
b
) of the hot thermal bath turns on almost
instantaneously, while the remainder (1
−
δ
b
) has a slow exponential increase to its steady state value.
From this description of the phenomenological model, we can write a rate equation for the increase in the breathing
mode thermal occupancy versus time in the presence of the red-detuned pump tone,
̇
n
m
=
−
γ
m
n
m
+
γ
i
n
p,m
(1
−
δ
b
e
−
γ
s
t
) +
γ
i
n
b,m
,
(S-29)
where the total mechanical damping rate is
γ
m
=
γ
i
+
γ
em
+
γ
p
,
δ
b
is the slow-growing fraction of
n
p,m
, and
γ
s
is the
exponential rate at which this slow fraction of the pump-induced hot bath turns on. Assuming
γ
p
is time-independent,
this rate equation has a simple solution of the form
n
m
(
t
) =
n
b,m
e
−
γ
m
t
+
n
H
(1
−
e
−
γ
m
t
) +
n
δ
(
e
−
γ
s
t
−
e
−
γ
m
t
)
.
(S-30)
Here,
n
δ
is the back-action cooled thermal occupancy in the absence of the slow-growing fraction of the pump-induced
thermal bath occupancy, given by,
n
δ
=
δ
b
(
γ
i
n
p,m
γ
s
−
γ
m
)
,
(S-31)
and
n
H
is the final steady-state breathing mode thermal occupancy, given by,
n
H
=
γ
p
n
p,m
+
γ
i
n
b,m
γ
m
.
(S-32)
From the fit to the heating curve of Fig. 6(c) of the main text we find that the effective fridge temperature that
the breathing mode thermalizes to in the absence of the red-detuned drive tone (i.e., from the measured occupancy
n
b,m
= 1
.
5 at the very beginning of the drive tone pulse), is
T
f,m
≈
40 mK. For the particular curve of Fig. 6(c)
we also extract a steady-state occupancy of
n
H
= 8
.
9 phonons, a slow bath component exponential constant of
γ
s
/
2
π
= 164 Hz and a slow bath fraction of
δ
b
= 0
.
65, and a total mechanical damping rate
γ
m
/
2
π
= 500 Hz. This
yields from Eq. (S-32) a product
γ
p
n
p,m
= 4348 phonons/s for the pump-induced bath. As the parasitic damping rate
induced by the pump tone is found to be quite small on the scale of
γ
em
(see Supplementary Note 6), it is difficult
to separately determine
γ
p
and
n
p,m
. As a rough estimate, if we take
γ
p
≈
γ
i
, we find a pump-induced hot bath
occupancy of
n
p,m
= 63
.
9 phonons, corresponding to a pump-induced bath temperature of
T
p,m
≈
1
.
24 K. Note that
in general there need not be any relation between
γ
p
and
γ
i
, as the pump-induced bath is likely an internal source
of heating within the phononic shield, whereas the coupling of the breathing mode to the ambient fridge bath occurs
through the phononic shield.
We plot in Fig. 6(c) of the main text a back-action cooling curve assuming a fixed bath temperature of
T
b
= 500 mK
and a fixed intrinsic breathing mode damping rate of
γ
i
/
2
π
= 68 Hz. This is mainly as a guide to the eye, indicating
how different the measured cooling curve is from a conventional back-action cooling curve. In general we expect a
nonlinear power dependence of both
γ
p
and
n
p,m
, which can lead to non-monotonic cooling curves as measured in
Refs. [19, 20] for optical heating in optomechanical crystals.