Squeezing of light via reflection from a silicon micromechanical resonator
Amir H. Safavi-Naeini,
1, 2,
∗
Simon Gr ̈oblacher,
1, 2,
∗
Jeff T. Hill,
1, 2,
∗
Jasper Chan,
1
Markus Aspelmeyer,
3
and Oskar Painter
1, 2, 4,
†
1
Kavli Nanoscience Institute and Thomas J. Watson, Sr.,
Laboratory of Applied Physics, California Institute of Technology, Pasadena, CA 91125
2
Institute for Quantum Information and Matter,
California Institute of Technology, Pasadena, CA 91125
3
Vienna Center for Quantum Science and Technology (VCQ),
Faculty of Physics, University of Vienna, A-1090 Wien, Austria
4
Max Planck Institute for the Science of Light, G ̈unther-Scharowsky-Straße 1/Bldg. 24, D-91058 Erlangen, Germany
(Dated: February 26, 2013)
We present the measurement of squeezed light generation using an engineered optomechanical
system fabricated from a silicon microchip and composed of a micromechanical resonator coupled to a
nanophotonic cavity. Laser light is used to measure the fluctuations in the position of the mechanical
resonator at a measurement rate comparable to the free dynamics of the mechanical resonator, and
greater than its thermal decoherence rate. By approaching the strong continuous measurement
regime we observe, through homodyne detection, non-trivial modifications of the reflected light’s
vacuum fluctuation spectrum. In spite of the mechanical resonator’s highly excited thermal state
(10
,
000 phonons), we observe squeezing at the level of 4
.
5
±
0
.
5% below that of shot-noise over
a few MHz bandwidth around the mechanical resonance frequency of 28 MHz. This squeezing is
interpreted as an unambiguous quantum signature of radiation pressure shot-noise.
Monitoring a mechanical object’s motion, even with the
gentle touch of light, fundamentally alters its dynamics.
The experimental manifestation of this basic principle of
quantum mechanics, its link to the quantum nature of
light, and the extension of quantum measurement to the
macroscopic realm have all received extensive attention
over the last half century [1, 2]. The use of squeezed
light, with quantum fluctuations below that of the vac-
uum field, was proposed nearly three decades ago [3] as a
means for beating the standard quantum limits in preci-
sion displacement and force measurements. Conversely, it
has also been proposed that a strong continuous measure-
ment of a mirror’s position with light, may itself give rise
to squeezed light [4, 5]. In this Letter, we present such a
continuous position measurement using an engineered op-
tomechanical system fabricated from a silicon microchip
and composed of a micromechanical resonator coupled to
a nanophotonic cavity. Laser light is used to measure the
fluctuations in the position of the mechanical resonator
at a measurement rate comparable to the free dynamics
of the mechanical resonator, and greater than its thermal
decoherence rate. By approaching the strong continuous
measurement regime we observe, through homodyne de-
tection, non-trivial modifications of the reflected light’s
vacuum fluctuation spectrum. In spite of the mechanical
resonator’s highly excited thermal state (10
4
phonons),
we observe squeezing at the level of 4
.
5
±
0
.
5% below that
of shot-noise over a few MHz bandwidth around the me-
chanical resonance frequency of
ω
m
/
2
π
≈
28 MHz. This
∗
These authors contributed equally to this work.
†
Electronic address: opainter@caltech.edu; URL:
http://copilot.
caltech.edu
squeezing is interpreted as an unambiguous quantum sig-
nature of radiation pressure shot-noise [6].
The generation of states of light with fluctuations be-
low that of vacuum has been of great theoretical inter-
est since the 1970s [3, 7–9]. Early experimental work
demonstrated squeezing of a few percent in a large vari-
ety of different nonlinear systems, such as neutral atoms
in a cavity [10], optical fibers [11], and crystals with bulk
optical nonlinearities [12, 13]. This initial research was
mainly pursued as a strategy to mitigate the effects of
shot-noise given the possibility of improved optical com-
munication [7] and better sensitivity in gravitational wave
detectors [3, 8]. In recent years, in addition to being in-
stalled in gravitational wave detectors [14], squeezed light
has enhanced metrology in more applied settings [15].
The vacuum fluctuations arising from the quantum na-
ture of light determine our ability to optically resolve
mechanical motion and set limits on the perturbation
caused by the act of measurement [16]. A well-suited
system to experimentally study quantum measurement
is that of cavity-optomechanics, where an optical cav-
ity’s resonance frequency can be designed to be sensi-
tive to the position of a mechanical system. By moni-
toring the phase and intensity of the reflected light from
such a cavity, a continuous measurement of mechanical
displacement can be made. Systems operating on this
simple premise have been realized in a variety of experi-
mental settings, such as in large-scale laser gravitational
wave interferometers [17], microwave circuits with elec-
tromechanical elements [18, 19], mechanical elements in-
tegrated with or comprising Fabry-P ́erot cavities [20–22],
and on-chip nanophotonic cavities sensitive to mechani-
cal deformations [23, 24]. Conceptually, the same basic
optomechanical interaction appears when the collective
motion of an ultracold gas of atoms shifts the resonance
arXiv:1302.6179v1 [quant-ph] 25 Feb 2013
2
frequency of an optical cavity, and recent gas-phase ex-
periments have shown a variety of optomechanical effects,
including squeezed light generation [25].
The simplest cavity-optomechanical system consists of
two modes, an optical and a mechanical resonance, and
is parameterized by their respective resonance frequencies
(
ω
o
,
ω
m
) and quality factors (
Q
o
,
Q
m
), and the coupling
rate of intracavity light to the resonant mechanical mo-
tion (
g
0
). The Hamiltonian describing the interaction be-
tween light and mechanics is
H
int
=
~
g
0
ˆ
a
†
ˆ
a
ˆ
x/x
zpf
, with
the amplitude of the mechanical resonator motion being
given by ˆ
x
=
x
zpf
·
(
ˆ
b
†
+
ˆ
b
), where
x
zpf
=
√
~
/
2
m
eff
ω
m
is
the zero-point fluctuation and
m
eff
the effective motional
mass of the resonator. Here ˆ
a
(ˆ
a
†
) and
ˆ
b
(
ˆ
b
†
) are the
annihilation (creation) operators of optical and mechan-
ical excitations, respectively. The optical cavity decay
rate,
κ
=
ω
o
/Q
o
, is the loss rate of photons from the
cavity and the rate at which optical vacuum fluctuations,
or shot-noise, is coupled into the optical resonance [26].
Similarly, the mechanical damping rate
γ
i
=
ω
m
/Q
m
is
the rate at which thermal bath fluctuations couple into
the mechanical system. In all experimental realizations of
optomechanics to date, including that presented here, the
optomechanical coupling rate
g
0
has been much smaller
than the cavity decay rate
κ
. As such, without a strong
coherent drive, the interaction of the vacuum fluctuations
with the mechanics is negligible.
Under the effect of a coherent laser drive, the cavity is
populated with a mean intracavity photon number
〈
n
c
〉
,
and one considers the optical fluctuations about the clas-
sical steady-state, ˆ
a
→
√
〈
n
c
〉
+ ˆ
a
. This modifies the
optomechanical interaction resulting in a linear coupling
between the fluctuations of the intracavity optical field
(
ˆ
X
0
= ˆ
a
+ ˆ
a
†
) and the position fluctuations of the me-
chanical system ˆ
x
:
H
int
=
~
G
ˆ
X
0
ˆ
x/x
zpf
. The paramet-
ric linear coupling occurs at an effective interaction rate
of
G
≡
√
〈
n
c
〉
g
0
. Through this interaction, the inten-
sity fluctuations of the vacuum field
ˆ
X
(in)
θ
=0
(
t
) entering the
cavity impart a force on the mechanical system,
ˆ
F
BA
(
t
) =
~
·
√
Γ
meas
x
zpf
ˆ
X
(in)
θ
=0
(
t
)
,
(1)
which is the radiation pressure shot-noise (RPSN) force.
The mechanical motion in turn is recorded in the phase
of the light leaving the cavity,
ˆ
X
(out)
θ
(
t
) =
−
ˆ
X
(in)
θ
(
t
)
−
2
√
Γ
meas
x
zpf
ˆ
x
(
t
)
·
sin(
θ
)
,
(2)
where
θ
is the quadrature angle of the detected optical
field, with
θ
= 0 (
π/
2) referring to the intensity (phase)
quadrature. The latter can be interpreted as the infor-
mational aspect of measurement, with the optical cavity
playing the role of the position detector (measuring ob-
servable ˆ
x
at a rate Γ
meas
≡
4
G
2
/κ
), while the former is
the quantum measurement back-action imposing a fluc-
tuating noise force onto the mechanical system [2]. In ad-
dition to the back-action noise, thermal fluctuations from
the bath also drive the mechanical motion, with their
magnitudes becoming comparable as Γ
meas
approaches
the thermalization rate Γ
thermal
(
ω
)
≡
γ
i
̄
n
(
ω
). The bath
occupation number ̄
n
(
ω
) is equivalent to its high temper-
ature limit
k
B
T
b
/
~
ω
in these measurements.
The auto-correlation of the optical output field has con-
tributions due to the auto-correlation of the optical in-
put field and the mechanical fluctuations, as well as their
cross-correlations. Due to the back-action force on the
mechanical system, the motion has non-zero correlations
with the RPSN (
〈
ˆ
x
(
t
)
ˆ
F
BA
(
t
′
)
〉
), which can be negative
for some values of
θ
. Under certain conditions this can
lead to the squeezing of the optical output field. The
spectral density of a balanced homodyne measurement of
the output field, normalized to shot-noise, is given in the
quasi-static limit for frequencies
ω
ω
m
(see the Ap-
pendix):
̄
S
out
II
(
ω
) = 1 + 4(Γ
meas
/ω
m
) sin(2
θ
)
+4
Γ
meas
ω
m
̄
n
(
ω
)
Q
m
(1 + cos(2
θ
))
.
(3)
Measurement of the spectrum below shot-noise (squeez-
ing) is therefore attributable to cross-correlations of mea-
surement back-action noise due the optical vacuum fluc-
tuations and position fluctuations of the mechanical res-
onator.
The primary technical hurdle to observing such squeez-
ing, as in many quantum measurements, is the strong cou-
pling of a preferred detection channel (the optical probe)
simultaneous with the minimization of unwanted environ-
mental perturbations of the mechanical system. As indi-
cated in Eq. (11), fluctuations from the thermal bath limit
squeezing of the optical probe field to a regime in which
̄
n
(
ω
)
< Q
m
. This requirement is equivalent to having a
Q
-frequency product,
Q
m
ω
m
> k
B
T
b
/
~
. Equation (11)
also indicates that squeezing becomes appreciable only as
the measurement rate of the mechanics by the light field
approaches the mechanical frequency. In some respects,
both of these challenges have been overcome in recent op-
tomechanics work [19, 24, 27], whereby thermal bath cou-
pling has been made smaller than the optically-induced
cooling of the mechanical resonator. However, significant
squeezing over an appreciable spectral bandwidth requires
not only a large cooperativity between light and mechan-
ics, as represented by
C
= Γ
meas
/γ
i
and realized in cool-
ing experiments, but the more stringent requirement that
the effective measurement back-action force be compa-
rable to all forces acting on the mechanics, including the
elastic restoring force of the mechanical structure. A more
comprehensive model (see Appendix), including the reso-
nant response of the mechanical system to RPSN, shows
that the strict requirements of the quasi-static model are
somewhat relaxed, and that the squeezing scales approx-
imately as Γ
meas
/δ
, where
δ
is the effective bandwidth of
squeezing around the mechanical resonance frequency.
In order to meet the requirements of strong measure-
ment and efficient detection, we designed a zipper-style
3
1
0
−1
0
1
−12
1
a
b
c
5
μ
m
1
μ
m
FIG. 1:
Optomechanical device. a
, Scanning electron mi-
croscope image of a waveguide-coupled zipper optomechanical
cavity. The waveguide width is adiabatically tapered along its
length and terminated with a photonic crystal mirror next to
the cavity. The tapering of the waveguide allows for efficient
input/output coupling while the photonic crystal termination
makes the coupling to the cavity single-sided. Two zipper cav-
ities are coupled above and below the waveguide, each with a
slightly different optical resonance frequency allowing them to
be separately addressed.
b
, (left) Close-up of the coupling re-
gion between one of the cavities and the waveguide. (right)
Finite element method (FEM) simulation of the cavity field
leaking into the waveguide (log scale). Note that the field
does not leak into the mirror region of the waveguide.
c
, (top)
FEM simulation showing the in-plane electrical field of the
fundamental optical cavity mode. (bottom) FEM simulation
of the displacement of the fundamental in-plane differential
mode of the structure with frequency
ω
m
/
2
π
= 28 MHz. The
mechanical motion, modifying the gap between the beams,
shifts the optical cavity frequency leading to optomechanical
coupling.
optomechanical cavity [23] with a novel integrated waveg-
uide coupler fabricated from the 220 nm thick silicon de-
vice layer of a silicon-on-insulator microchip (see Fig. 1a).
The in-plane differential motion of the two beams at a
fundamental frequency of
ω
m
/
2
π
= 28 MHz strongly
modulates the co-localized fundamental optical resonance
of the cavity with a theoretical vacuum coupling rate of
g
0
/
2
π
= 1 MHz. As shown in Fig. 1b, we use a silicon
waveguide with a high reflectivity photonic crystal end-
mirror to efficiently excite and collect light from the op-
tical cavity. Coupling light from the silicon waveguide to
a single-mode optical fiber is performed using an optical
fiber taper and a combination of adiabatic mode coupling
and transformation (see Fig. 1b).
The experimental setup used to characterize the zipper
cavity system and measure the optomechanical squeezing
of light is shown in Fig. 2a. The silicon sample is placed
in a continuous flow
4
He cryostat with a cold finger tem-
perature of 10 K. A signal laser beam is used to probe
the optomechanical system and measure the mechanical
motion of the zipper cavity. A wavelength scan of the re-
flected signal from the cavity is plotted in Fig. 2c, showing
an optical resonance with a linewidth
κ/
2
π
= 3
.
42 GHz
at a wavelength of
λ
c
= 1540 nm. Inefficiencies in the
collection and detection of light correspond to additional
uncorrelated shot-noise in the signal and can reduce the
squeezing to undetectable levels. For the device studied
here, the cavity coupling efficiency, corresponding to the
percentage of photons sent into the cavity which are re-
flected, is determined to be
η
k
= 0
.
54. The fiber-to-chip
coupling efficiency is measured at
η
CP
= 0
.
90. A homo-
dyne detection scheme [28] allows for high efficiency de-
tection of arbitrary quadratures of the optical signal field.
Characterization and optimization of the efficiency of the
entire optical signal path and homodyne detection sys-
tem (see Appendix for details) results in an overall setup
efficiency of
η
setup
= 0
.
48, corresponding to a total signal
detection efficiency of
η
tot
=
η
setup
η
κ
= 0
.
26.
Figure 2c shows the noise spectrum of the thermal mo-
tion of the mechanical resonator obtained by positioning
the laser frequency near the cavity resonance and tuning
the relative local oscillator (LO) phase of the homodyne
detector,
θ
lock
, to measure the quadrature of the reflected
signal in which mechanical motion is imprinted (roughly
the phase quadrature for near-resonance probing). The
mechanical spectrum is seen to contain the in-plane dif-
ferential mode of interest at
ω
m
/
2
π
= 28 MHz, as well
as several other more weakly coupled mechanical reso-
nances of the nanobeams and coupling waveguide (the
in-plane differential mode peak appears reduced relative
to the other modes in this plot due to the limited resolu-
tion bandwidth of the measurement). A high-resolution,
narrowband spectrum of the in-plane differential mode is
displayed as an inset to Fig. 2c, and shows a linewidth of
γ
i
/
2
π
= 172 Hz, corresponding to a mechanical
Q
-factor
of
Q
m
= 1
.
66
×
10
5
. The vacuum coupling rate of the
in-plane differential mode, measured from the detuning
dependence of the optical spring shift and damping (see
Appendix), is determined to be
g
0
/
2
π
= 750 kHz, in good
correspondence with theory. From the calibration of the
noise power under the Lorentzian in Fig. 2c, the in-plane
differential mode is found to thermalize (at low optical
probe power) to a temperature of
T
b
∼
16 K, correspond-
ing to a phonon occupancy of
〈
n
〉∼
1
.
2
×
10
4
. This yields
a ratio,
Q
m
~
ω
m
/k
B
T
b
≈
13, well within the regime where
coherent motion and squeezing are possible.
In order to systematically and accurately study the
noise properties of the reflected optical signal from the
cavity we make a series of measurements to character-
ize our laser and detection setup. Figure 2d shows the
measured noise power spectral density (PSD) of the bal-
anced homodyne detector (dark current subtracted) for
ω
≈
ω
m
as a function of LO power (signal blocked),
indicating a linear dependence on power and negligible
added noise above shot-noise. In the measured squeezing
data to follow, a LO power of 3 mW is used. Calibra-
tion of the laser intensity and frequency noise over the
frequency range of interest (
ω/
2
π
= 1–40 MHz) is mea-
4
Frequency (MHz)
40
35
30
25
20
BH Relative PSD
1.00
0.99
1.01
R
2
= 0.9999
0
1.0
2.0
3.0
1.0
2.0
3.0
0
LO power (mW)
BH PSD (pW/Hz)
4.0
dark current
c
d
e
-60
-120
-100
-80
-2
-1
0
1
2
(
ω
-
ω
m
)/2
π
(kHz)
10
1
-130
-120
-110
-100
-90
-80
-70
PSD dBm/Hz
Frequency (MHz)
Normalized reection
Normalized detuning,
∆
/
κ
0
1
1540.01
1540.09
1540.17
Wavelength (nm)
κ/
2π
=
3.42
GHz
0
0.1
0.2
0.3
0.4
0.5
10
0
10
-1
10
-2
b
∆
lock
≅
0.044
κ
IM
VOA
88 MHz
~
EDFA
FPC
FPC
RSA
ENA
λ
~
1540
nm
λ
-meter
-
BHD
LHe cryostat
FPC
IM
FPC
FPC
PM
PD1
PD2
1
2
3
AOM
RSA
FS
a
Tap
er
FPC
LPF
VC
η
12
=85%
η
23
=88%
η
3H
=92%
η
HD
=66%
η
CP
=90%
FIG. 2:
Experimental setup and device characterization. a
A tunable external-cavity diode laser, actively locked to
a wavemeter (
λ
-meter), is used to generate a strong local oscillator (LO) and the measurement signal. A portion of each
field is split off, detected, and used to individually stabilize the intensity in each arm using intensity modulators (IM). Fiber
polarization controllers (FPC) adjust the polarization of the LO and signal. A variable optical attenuator (VOA) is used to
set the signal power and an acousto-optic modulator (AOM) is used to generate a tone of known amplitude for calibration (see
Appendix). The signal beam is fed into a fiber taper that is mounted inside a
4
He continuous flow cryostat and is coupled to
the zipper cavity. The reflected signal beam from the cavity is separated using a circulator, and is switched between one of
three detection paths: a power meter (PM) for power calibration, a photodetector (PD1) for optical spectroscopy of the cavity,
and an erbium-doped fiber amplifier (EDFA) for spectroscopy of the coupled cavity-optomechanical system on a real-time
spectrum analyzer (RSA) or network analyzer (ENA). For the measurement of squeezing, the reflected cavity signal is sent to
a fourth path containing a variable coupler (VC), where it is recombined with the strong local oscillator and detected on a
balanced homodyne detector (BHD). The relative phase between the LO and the signal is set using a fiber stretcher (FS).
b
(top) Reflected signal from the optical cavity at low optical power (
n
c
≈
10, linewidth
κ/
2
π
= 3
.
42 GHz). (bottom) High-
power (
n
c
= 790) reflected signal, showing the cavity-laser detuning (dashed line) locked to during squeezing measurements.
c
Homodyne noise PSD of the reflected signal showing the transduced thermal Brownian motion of the zipper cavity at
T
b
= 16 K
(green curve;
n
c
= 80). The red curve is the shot noise level and the black curve is the detector’s dark noise. The inset shows
a zoom-in of the fundamental in-plane differential mechanical mode of the zipper cavity (linewidth
γ
i
/
2
π
= 172 Hz).
d
Mean
value of the PSD of the balanced homodyne detector as a function of the LO power (signal blocked). The filled data point
indicates the LO power used in the squeezing measurements. The red and dashed black curves correspond to a linear fit to
the data and the detector dark current level, respectively.
e
Noise PSD as a function of
θ
lock
with the signal detuned far
off-resonance at ∆
/κ
≈
30 (shades green to red), referenced to the noise level with the signal blocked (blue).
sured by direct photodetection of the laser, pre- and post-
transmission through a fiber Mach-Zehnder interferome-
ter with a known frequency response. The laser intensity
noise is measured to be shot-noise dominated over this fre-
quency range, while the laser frequency noise is measured
to be roughly flat at a level of
S
ωω
∼
5
×
10
3
rad
2
·
Hz.
Laser phase noise on the signal beam can be converted
to intensity noise by reflection from the dispersive cavity
or due to frequency dependent components in the optical
train, and can add to the detected noise floor. Due to
the broad linewidth of the zipper cavity resonance, the
effects of the laser phase noise are found to be negligi-
ble. The full suite of noise and calibration measurements
performed are described in detail in the Appendix.
Measurement of the noise in the reflected optical signal
from the cavity as a function of quadrature angle, fre-
quency, and signal power is presented in Figs. 3 and 4.
These measurements are performed for laser light on res-
onance with the optical cavity and for input signal powers
varying from 252 nW to 3
.
99
μ
W in steps of 2 dB, with the
maximum signal power corresponding to an average intra-
cavity photon number of
〈
n
c
〉
= 3
,
153. Positioning of the
laser at the appropriate cavity detuning for each signal
power is performed by scanning the wavelength across
the cavity resonance while monitoring the reflection, and
then stepping the laser frequency towards the cavity from
5
d
0.95
0.99
0.98
0.97
0.96
1.0
10
0
10
1
10
2
10
3
0.6
0.4
0.8
0.2
a
b
c
35
30
25
20
15
10
Frequency (MHz)
θ
lock
/π
0.4
0.2
0.6
0.8
θ
lock
/π
0.95
1
1.05
1.1
10
20
30
0.95
1
1.05
1.1
Normalized PSD
Frequency (MHz)
Normalized PSD
0.90
0.90
FIG. 3:
Optomechanical squeezing of light. a
, Theoretical model. Density plot of the predicted reflected signal noise
PSD, as measured on a balanced homodyne detector and normalized to shot-noise, for a simplified model of the optomechanical
system (see Appendix). Areas below shot-noise are shown in blue shades on a linear scale. Areas with noise above shot-noise
are shown in orange shades on a log-scale. The solid white line is a contour delineating noise above and below shot-noise.
b
,
Experimental data. Density plot of the measured reflected signal noise PSD for
n
c
= 790 normalized to the measured shot-noise
level.
c
, Slice of the measured density plot in
b
taken at
θ
lock
/π
= 0
.
23.
d
, Slice of the measured density plot in
b
taken at
θ
lock
/π
= 0
.
16. In
c
and
d
, the black curve corresponds to the measured data slice extracted from
b
. The dark blue traces are
several measurements of the shot-noise level (average shown in light blue). Also indicated is a model of the squeezing in the
absence of thermal noise (red dashed curve), the same model with thermal noise included (solid red curve), and a full noise
model including additional phenomenological noise sources (solid green curve).
the red side until the reflection matches the level that cor-
responds to a detuning of ∆
lock
/κ
≈
0
.
04 (see Fig. 2b).
The laser is locked to this frequency using a wavemeter
with a frequency resolution of
±
0
.
0015
κ
. Drift of the op-
tical cavity resonance over the timescale of a single noise
spectrum measurement (minutes) is found to be negligi-
ble. An estimate of the variance of ∆
lock
is determined
from the dependence of the transduction of the mechani-
cal motion on the quadrature phase, indicating that from
one lock to another ∆
lock
/κ
= 0
.
044
±
0
.
006.
In Fig. 3 we plot the theoretically predicted and mea-
sured noise PSD versus quadrature angle for a signal
power corresponding to
〈
n
c
〉
= 790 photons.
Each
quadrature spectrum is the average of 150 traces taken
over 20 seconds, and after every other spectrum, the sig-
nal arm is blocked and the shot-noise PSD is measured.
The shot-noise level, which represents the noise of the
electromagnetic vacuum on the signal arm, is used to nor-
malize the spectra. We find at certain quadrature angles,
and for frequencies a few MHz around the mechanical res-
onance frequency, that the light reflected from the zipper
cavity shows a noise PSD below that of vacuum. The den-
sity plot of the theoretically predicted noise PSD (Fig. 3a)
shows the expected wideband squeezing due to the strong
optomechanical coupling in these devices, as well as a
change in the phase angle where squeezing is observed at
below and above the mechanical frequency. This change is
due to the change in sign of the mechanical susceptibility
and the corresponding change in phase of the mechanical
response to RPSN. The measured noise PSD density plot
(Fig. 3b) shows the presence of several other mechanical
noise peaks and a reduced squeezing bandwidth, yet the
overall phase- and frequency-dependent characteristics of
the squeezing around the strongly-coupled in-plane me-
chanical mode are clearly present. In particular, Figs. 3c
and d show two slices of the noise PSD density plot which
show the region of squeezing change from being below to
above the mechanical resonance frequency.
In Fig. 4a we show the measured noise PSD (grey cir-
cles) as a function of quadrature angle for a frequency
slice at
ω/
2
π
= 27
.
9 MHz of the data shown in Fig. 3b.
The measured squeezing (anti-squeezing) is seen to be
smaller (larger) than expected from a model of the op-
tomechanical cavity without thermal noise. We also plot
in Fig. 4b the maximum measured and modeled squeez-
ing as a function of signal power. The simple theory pre-
dicts a squeezing level (blue curve) which monotonically
increases with signal power, whereas the measured max-
imum squeezing saturates at a level of 4
.
5
±
0
.
5% below
the shot-noise at an intracavity power
〈
n
c
〉
= 1
,
984 pho-
tons. In order to understand the noise processes that limit
the bandwidth and magnitude of the measured squeez-
ing, we plot in Fig. 4c the noise PSD (shot-noise sub-
tracted) for phase quadratures that maximize (left plot)
and minimize (right plot) the transduction of the me-
chanical mode peak at
ω
m
/
2
π
= 28 MHz. Along with
6
b
2,000
0
1.0
0.88
0.92
0.96
Normalized PSD
c
ω
>
ω
m
∼ω
-1
∼ω
-1/2
Intracavity photon number,
〈
n
c
〉
2,000
0
ω
<
ω
m
a
0.2
0.4
0.6
0.8
1
0.1
0.2
0.9
1
0
2
3
θ
lock
/
π
Normalized PSD (shot-noise subtracted)
Frequency (MHz)
1
10
30
3
10
-6
10
-4
10
-2
10
0
10
2
10
4
1
10
30
3
θ
lock
/
π
= 0.80
θ
lock
/
π
= 0.26
Normalized PSD
FIG. 4:
Spectral and power dependence of noise. a
,
Measured (filled circles) balanced homodyne noise power of
the reflected signal at
ω/
2
π
= 27
.
9 MHz versus quadrature
angle (∆
lock
/κ
= 0
.
044 and
n
c
= 790). The red curve cor-
responds to the full noise model. The solid blue curve is for
a model including the response of the mechanical mode in
the absence of thermal noise, i.e., driven by RPSN only (the
dashed blue curve shows the thermal noise component).
b
,
Measured (filled circles) minimum noise PSD normalized to
shot-noise versus
n
c
. The left plot is the maximum squeezing
for
ω < ω
m
and the right for
ω > ω
m
. Also shown is the
single-mode noise model (blue curve) and the full noise model
(red curve).
c
, Balanced homodyne noise PSD of the reflected
cavity signal for ∆
lock
/κ
= 0
.
052 and
n
c
= 3
,
153. Left (right)
plot shows phase quadrature corresponding to maximum (min-
imum) transduction of mechanical motion. The black curve is
the measured data with the shot-noise level subtracted. Also
shown are modeled laser phase noise (green curve), the single-
mode noise model (blue curve), and the full noise model (red
curve).
the measured data (black curve), we also plot the esti-
mated noise due to phase noise of the signal laser (green
curve) and that for a single mechanical mode (blue curve)
assuming a thermal bath temperature of
T
b
= 16 K and
a frequency-independent damping rate. The single-mode
noise model greatly underestimates the background noise
level, especially in the quadrature minimizing transduced
motion in which we measure a
ω
−
1
/
2
(as opposed to
ω
−
1
)
frequency dependence to the low frequency noise. As de-
scribed in the Appendix, the additional measured noise is
thought to arise from a combination of the thermal noise
tails of higher frequency mechanical modes and fluctu-
ations in the optical cavity damping rate, along with a
small amount of heating due to optical absorption in the
silicon cavity. The red curves in each of the plots in Fig. 4
show the full noise model incorporating these added phe-
nomenological terms.
Somewhat surprisingly, these measurements show that
by reflecting light off a thin-film mechanical resonator un-
dergoing large amplitude thermal motion, light that is in
certain respects quieter than vacuum can be obtained.
This is found to result from radiation-pressure fluctua-
tions being strongly imprinted on and modified by the
motion of the mechanical resonator, thus demonstrating
a fundamentally quantum aspect of displacement mea-
surement. The devices in this work utilize lithographic
patterning at the nanoscale to transform silicon into a
material with an effective quantum optical nonlinearity at
an engineerable optical wavelength. The modest level of
squeezing realized in this work is predominantly limited
by thermal noise, but also by the efficiency with which
the reflected light can be collected by external optics.
The effects of thermal noise may be substantially reduced
by working with materials of higher intrinsic
Q
-frequency
product, such as diamond and silicon carbide [29]. Given
the microchip form of the devices studied here, and the
potential for device integration, it is interesting to con-
sider whether new squeezed light applications might arise.
For example, squeezed light generated by one device could
be directly sent into another device for use as an optical
probe. Such an on-chip squeezer and detector could be
used as a quantum-enhanced micro-mechanical displace-
ment and force sensor [30]. More generally, we expect
future experiments with feedback and strong measure-
ment of the dynamics of a mechanical system to be within
reach. In addition, using quantum light as an input we
expect to be able to generate entangled states of mechan-
ics and light with similar devices.
The authors would like to thank K. Hammerer and A.
A. Clerk for valuable discussions. This work was sup-
ported by the DARPA/MTO MESO program, Institute
for Quantum Information and Matter, an NSF Physics
Frontiers Center with support of the Gordon and Betty
Moore Foundation, and the Kavli Nanoscience Institute
at Caltech. ASN and JC gratefully acknowledge support
from NSERC. SG acknowledges support from the Euro-
pean Commission through a Marie Curie fellowship.
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Appendix
Contents
I. Theory
8
A. Approximate quasi-static theory
9
B. The effect of dynamics and correlation between
RPSN and position
10
C. General derivation of squeezing
11
1. The effect of imperfect optical coupling and
inefficient detection
12
II. Experiment
12
A. Measurement of losses
12
B. Data collection procedure
13
C. Relation between detuning and quadrature
14
III. Sample Fabrication and Characterization
14
A. Fabrication
14
B. Optical Characterization
15
C. Mechanical Characterization
15
IV. Noise Spectroscopy Details
15
A. Homodyne measurement with laser noise
15
B. Measurement and characterization of laser
noise
15
C. Linearity of detector with local oscillator
power
17
D. Detected noise level with unbalancing
18
E. Estimating added noise in the optical train
18
F. The effect of laser phase noise
19
G. Phenomenological dispersive noise model: the
effect of structural damping
19
H. Phenomonological absorptive noise model
19
I. Comparing measured spectra to theoretically
predicted spectra
20
V. Summary of Noise Model
21
VI. Mathematical Definitions
22
References
24
I. THEORY
Optomechanical systems can be described theoretically
with the Hamiltonian (see main text)
H
=
~
ω
o
ˆ
a
†
ˆ
a
+
~
ω
m0
ˆ
b
†
ˆ
b
+
~
g
0
ˆ
a
†
ˆ
a
(
ˆ
b
†
+
ˆ
b
)
,
(A1)
where ˆ
a
and
ˆ
b
are the annihilation operators for pho-
tons and phonons in the system, respectively. Generally,
the system is driven by intense laser radiation at a fre-
quency
ω
L
, making it convenient to work in an interaction
frame where
ω
o
is replaced by ∆ in the above Hamilto-
nian with ∆ =
ω
o
−
ω
L
. To quantum mechanically de-
scribe the dissipation and noise from the environment, we
use the quantum-optical Langevin differential equations
(QLEs) [26, 31, 32],
̇
ˆ
a
(
t
) =
−
(
i
∆ +
κ
2
)
ˆ
a
−
ig
0
ˆ
a
(
ˆ
b
†
+
ˆ
b
)
−
√
κ
e
ˆ
a
in
(
t
)
−
√
κ
i
ˆ
a
in
,
i
(
t
)
,
̇
ˆ
b
(
t
) =
−
(
iω
m0
+
γ
i
2
)
ˆ
b
−
ig
0
ˆ
a
†
ˆ
a
−
√
γ
i
ˆ
b
in
(
t
)
,
which account for coupling to the bath with dissipation
rates
κ
i
,
κ
e
, and
γ
i
for the intrinsic cavity energy decay
rate, optical losses to the waveguide coupler, and total
mechanical losses, respectively. The total optical losses
are
κ
=
κ
e
+
κ
i
. These loss rates are necessarily accom-
panied by random fluctuating inputs ˆ
a
in
(
t
), ˆ
a
in
,
i
(
t
), and
ˆ
b
in
(
t
), for optical vacuum noise coming from the coupler,
optical vacuum noise coming from other optical loss chan-
nels, and mechanical noise (including thermal).
The study of squeezing is a study of noise propagation
in the system of interest and as such, a detailed under-
standing of the noise properties is required. The equa-
tions above are derived by making certain assumptions
about the noise, and are generally true for the case of
an optical cavity, where thermal noise is not present, and
where we are interested only in a bandwidth of roughly
10
8
smaller than the optical frequency (0 – 40 MHz band-
width of a 200 THz resonator). For the mechanical sys-
tem, where we operate at very large thermal bath oc-
cupancies (
10
3
) and are interested in the broadband
properties of noise sources (0 – 40 MHz for a 30 MHz
resonator), a more detailed understanding of the bath is
required, and will be presented in the section on thermal
noise.
At this point, we linearize the equations assuming a
strong coherent drive field
α
0
, and displace the annihi-
lation operator for the photons by making the transfor-
mation ˆ
a
→
α
0
+ ˆ
a
. This approximation, which neglects
terms of order ˆ
a
2
is valid for systems such as ours where
g
0
κ
, i.e. the
vacuum weak coupling
regime. We are
then left with a parametrically enhanced coupling rate
G
=
g
0
|
α
0
|
. Using the relations given in the mathemati-
cal definitions section (VI) of this document, we write the
solution to the QLEs in the Fourier domain as
ˆ
a
(
ω
) =
−
√
κ
e
ˆ
a
in
(
ω
)
−
√
κ
i
ˆ
a
in
,
i
(
ω
)
−
iG
(
ˆ
b
(
ω
) +
ˆ
b
†
(
ω
))
i
(∆
−
ω
) +
κ/
2
,
ˆ
b
(
ω
) =
−
√
γ
i
ˆ
b
in
(
ω
)
i
(
ω
m0
−
ω
) +
γ
i
/
2
−
iG
(ˆ
a
(
ω
) + ˆ
a
†
(
ω
))
i
(
ω
m0
−
ω
) +
γ
i
/
2
.
(A2)
Finally we note that by manipulation of these equa-
tions, the mechanical motion can be expressed as a (renor-
malized) response to the environmental noise and the op-
tical vacuum fluctuations incident on the optical cavity
through the optomechanical coupling