of 4
LETTER
doi:10.1038/nature14349
Phonon counting and intensity interferometry of a
nanomechanical resonator
Justin D. Cohen
1
*
,Sea
́
n M. Meenehan
1
*
, Gregory S. MacCabe
1
, Simon Gro
̈
blacher
1,2
{
, Amir H. Safavi-Naeini
1,3
, Francesco Marsili
4
,
Matthew D. Shaw
4
& Oskar Painter
1
In optics, the ability to measure individual quanta of light
(photons) enables a great many applications, ranging from
dynamic imaging within living organisms
1
to secure quantum
communication
2
. Pioneering photon counting experiments, such
as the intensity interferometry performed by Hanbury Brown and
Twiss
3
to measure the angular width of visible stars, have played a
critical role in our understanding of the full quantum nature of
light
4
. As with matter at the atomic scale, the laws of quantum
mechanics also govern the properties of macroscopic mechanical
objects, providing fundamental quantum limits to the sensitivity
of mechanical sensors and transducers. Current research in cavity
optomechanics seeks to use light to explore the quantum prop-
erties of mechanical systems ranging in size from kilogram-mass
mirrors to nanoscale membranes
5
, as well as to develop technolo-
gies for precision sensing
6
and quantum information processing
7,8
.
Here we use an optical probe and single-photon detection to study
the acoustic emission and absorption processes in a silicon nano-
mechanical resonator, and perform a measurement similar to that
used by Hanbury Brown and Twiss to measure correlations in the
emitted phonons as the resonator undergoes a parametric instab-
ility formally equivalent to that of a laser
9
. Owing to the cavity-
enhanced coupling of light with mechanical motion, this effective
phonon counting technique has a noise equivalent phonon sens-
itivity of 0.89
6
0.05. With straightforward improvements to this
method, a variety of quantum state engineering tasks using meso-
scopic mechanical resonators would be enabled
10
, including the
generation and heralding of single-phonon Fock states
11
and the
quantum entanglement of remote mechanical elements
12,13
.
Measurement of the properties of mechanical systems in the
quantum regime typically involves heterodyne detection of a coupled
optical or electrical field, yielding a continuous signal proportional to
the displacement amplitude
14
. An alternative method, particularly sui-
ted to optical read-out, is to utilize photon counting as a means to
probe the quantum dynamics of the coupled optomechanical sys-
tem
15,16
. Photon counting can be readily adapted to study intensity
correlations in an optical field, and has been used not only in the
astronomical studies of thermal light using the technique of
Hanbury Brown and Twiss (HBT), but also in early studies of the
photon statistics of laser light and single-atom fluorescence
4,17
.In
the field of photon-correlation spectroscopy, such intensity interfero-
metry techniques have found widespread application in the measure-
ment of particle and molecular motion in materials
18
. More recently,
photon counting of Raman scattering events in diamond has heralded
and verified the quantum entanglement of a terahertz phonon shared
between two separate bulk diamond crystals
13
. In the case of engi-
neered cavity optomechanical systems, much longer phonon coher-
ence times are attainable, albeit at lower mechanical frequencies
(megahertz to gigahertz), which limit the temperature of operation
and the optical power handling capability of such structures.
Quantum optical schemes for manipulation of the quantum state of
motion in cavity optomechanical systems thus rely on a large per-
phonon scattering rate and efficient detection of scattering events.
Here we embed a high-
Q
, gigahertz-frequency mechanical resonator
inside an optical nanocavity, greatly enhancing the phonon–photon
coupling rate and channelling optical scattering into a preferred optical
mode for collection. Single-photon detection of this scattered light
then allows for a precise counting of single-phonon emission or
absorption events, effectively phonon counting (although this ter-
minology should not be confused with Fock state detection or
quantum non-demolition measurement of phonon number). The
highly engineered and optimized nature of this optomechanical res-
onator furthermore yields a sub-phonon-level counting sensitivity of
the intracavity mechanical resonator occupancy.
A conceptual schematic of the phonon counting experiment is
shown in Fig. 1a. The device consists of a patterned silicon nanobeam
which forms an optomechanical crystal (OMC)
19,20
able to co-localize
acoustic (mechanical) and optical resonances at frequencies
v
m
and
v
c
, respectively. Finite-element-method simulations of the acoustic
and optical resonances are shown at the top of Fig. 1a. The
Hamiltonian describing the interaction between the acoustic and
optical modes is given by
^
H
int
~
B
g
0
^
a
{
^
a
ð
^
b
z
^
b
{
Þ
,where
^
a
ð
^
b
Þ
is the
annihilation operator for the optical (acoustic) mode, and
g
0
is the
optomechanical coupling rate, physically representing the optical fre-
quency shift due to the zero-point motion of the acoustic resonator.
This interaction modulates a laser probe with frequency
v
l
to produce
sidebands at frequencies
v
l
6
v
m
, analogous to the anti-Stokes and
Stokes sidebands in Raman scattering and corresponding to phonon
absorption or emission, respectively. For a system in the resolved side-
band limit, where
v
m
?
k
(
k
is the linewidth of the optical resonance),
the density of states of the optical cavity can be used to resonantly
enhance either scattering process for an appropriately detuned pump.
In particular, applying a large coherent pump red (blue) detuned
from the optical cavity resonance by
D
5
v
c
2
v
l
5
v
m
(
D
52
v
m
)
results in an effective interaction Hamiltonian of the form
^
H
int
~
B
G
ð
^
a
{
^
b
z
^
a
^
b
{
Þ
(
^
H
int
~
B
G
ð
^
a
^
b
z
^
a
{
^
b
{
Þ
), where
G
~
g
0
ffiffiffiffiffi
n
c
p
is the
parametrically enhanced optomechanical coupling rate (
n
c
is the intra-
cavity photon number at frequency
v
l
due to the pump laser). In this
case, the output field annihilation operator
^
a
out
can be shown to consist
of a coherent component at frequency
v
l
as well as a component
at frequency
v
c
which is proportional to
^
b
ð
^
b
{
Þ
(ref. 21). Sending the
cavity output through a series of narrowband optical filters centred on
the cavity resonance, as shown in Fig. 1a, suppresses the pump so that
photon counting events will correspond directly to counting phonon
absorption (emission) events
22
. Subsequently directing the filter output
to an HBT set-up in order to measure the second-order photon cor-
relation function
g
(2)
(
t
) (ref. 3) will then result in a direct measurement
*
These authors contributed equally to this work.
1
Institute for Quantum Information and Matter and Thomas J. Watson Senior Laboratory of Applied Physics, California Institute of Technology, Pasade
na, California 91125, USA.
2
Vienna Center for
Quantum Science and Technology (VCQ), Faculty of Physics, University of Vienna, A-1090 Wien, Austria.
3
Edward L. Ginzton Laboratory, Stanford University, Stanford, California 94305, USA.
4
Jet
Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109, USA.
{
Present address: Kavli Institute of Nanoscience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft,
The Netherlands.
G
2015
Macmillan Publishers Limited. All rights reserved
522 | NATURE | VOL 520 | 23 APRIL 2015
of the normally (anti-normally) ordered second-order phonon correla-
tion function. Although this work deals with measurements of a par-
ticular cavity optomechanical system, the nanobeam OMC, numerous
other geometries possess the requisite optomechanical coupling
strength and high mechanical frequency necessary to implement this
phonon counting scheme
5
.
As described in ref. 20, the nanobeam is patterned in such a way as
to support a ‘breathing’ acoustic resonance at
v
m
/2
p
5
5.6 GHz as
well as a fundamental optical resonance at a free-space wavelength near
1,550 nm, with a theoretical vacuum coupling rate of
g
0
/2
p
5
860 kHz.
All measurements presented here are performed at room temperature
and pressure. The thermal Brownian motion of the acoustic resonance
manifests as a Lorentzian response centred around
v
m
in the noise
power spectral density (NPSD),
S
(
v
), of the cavity reflection photo-
current, as shown in Fig. 1b. The linewidth of this Lorentzian is
c
5
c
i
1
c
OM
,where
c
i
is the intrinsic acoustic energy damping rate
and
c
OM
56
4
G
2
/
k
is the optomechanically induced damping rate due
to dynamical back action when pumping on the red or blue sideband,
respectively
21
. By measuring this linewidth as a function of
n
c
for both
red and blue detuning, we extract
c
i
/2
p
5
3 MHz and
g
0
/2
p
5
645 kHz.
The optical cavity reflection spectrum shown in Fig. 1c reveals a total
optical energy decay rate of
k
/2
p
5
817 MHz and a decay rate into the
detection channel of
k
e
/2
p
5
425 MHz.
To determine the feasibility of the phonon counting scheme for
measurements of the OMC in the quantum regime, one must deter-
mine the signal-to-noise ratio (SNR). While the signal level depends on
the above-measured optomechanical cavity parameters as well as the
overall efficiency of photon detection, any detection events that do not
correspond to Raman-scattered sideband photons will contribute as
noise. These noise counts originate in dark counts of the single-photon
detectors (SPDs) and photodetection of light at frequencies other than
v
c
, most notably in the unscattered portion of the laser pump. Two
cascaded tunable, commercially available Fabry–Perot filters (Micron
Optics, FFP-TF2), with bandwidths of 50 MHz and free spectral ranges
of 20 GHz, are used to attenuate the laser pump by a factor
A
.
80 dB
relative to the peak transmission at
v
c
(see Fig. 1d). The transmission
of the filters is then detected in the HBT apparatus by WSi-based
superconducting nanowire SPDs
23
operating at a system detection
efficiency of
g
SPD
<
70% and a dark count rate of
C
dark
5
4 Hz.
A useful parameterization of these quantities is the amount of noise
(in units of mechanical occupation quanta) as a proportion of the
signal generated by a single phonon in the OMC. Alternatively, this
noise-equivalent phonon number
n
NEP
can be interpreted as the
mechanical occupation which would produce an SNR of 1. We obtain
n
NEP
then by dividing the noise count rates by the per-phonon side-
band photon count rate
C
SB,0
5
g
j
c
OM
j
, where
g
is the total efficiency
of the set-up, including the system efficiency of the SPDs as well as
optical insertion loss along the path from cavity to detector. For a
coherent pump, this yields:
n
NEP
~
k
2
C
dark
4
gk
e
g
2
0
n
c
z
A
kv
m
2
k
e
g
0

2
ð
1
Þ
The above equation makes clear the benefits of large cavity-enhanced
optomechanical coupling
g
0
, both in terms of the low power sensitivity
limited by detector dark counts and the high power sensitivity limited
by pump bleed-through. Further details about the detection set-up,
optical and mechanical spectroscopy, and derivation of
n
NEP
can be
found in the Supplementary Information.
In Fig. 2a we show the measured sensitivity of the phonon counting
set-up for
D
5
v
m
(filled grey circles), as well as the expected theor-
etical curve (dashed line) given by equation (1). The noise count rate is
measured with the pump beam detuned far off-resonance from the
optical cavity,
D
?
v
m
, which eliminates the signal due to motional
sideband photons but does not change the reflected pump signal or
SPD dark counts. To determine
C
SB,0
, the sideband count rate is mea-
sured at low
n
c
where back action is negligible and the mechanical
10
2
10
1
10
0
10
–1
n
c
10
2
10
1
10
0
10
–1
10
–2
10
–3
n
NEP
filter
(GHz)
4
4.5
5.5
6.5
6
5
filter
=
w
m
a
b
10
2
10
1
10
0
n
NEP
Δ
Δ
Figure 2
|
Phonon counting sensitivity. a
, Noise equivalent phonon number
n
NEP
versus intracavity photon number
n
c
calculated using the measured signal
and noise count rates for our current set-up (filled grey circles). Solid lines
indicate the theoretically expected contributions due to dark counts (red) and
pump bleed-through (blue), based on the measured system efficiency and pump
suppression, with the sum of the two contributions displayed as a purple dashed
line. Error bars show one s.d. determined
from the measured count rates, assuming
Poissonian counting statistics.
b
,
n
NEP
versus filter–pump detuning
D
filter
for
n
c
<
65, with (red) and without (grey) an additional C-band band-pass filter
inserted. The vertical green line indicates the detuning corresponding to the data
from
a
, and the horizontal black line indicates the expected limiting sensitivity.
5.576
5.588
5.6
S
(
w
) (dBm Hz
–1
)
R
c
–10
–5
0
5
10
/
(2
π
) (GHz)
A
(dB)
d
0.4
0.2
0
–40
–80
0
w
/(2
π
) (GHz)
b
a
–100
–120
–140
SPD
2
SPD
1
Filter
start
stop
HBT with
TCSPC
Δ
Figure 1
|
Phonon counting and device characterization. a
, Schematic of the
phonon counting measurement. The finite element method simulations depict
the displacement field of the acoustic resonance (top, blue bar) and the electric
field of the optical resonance (next down, bar with no colour filling) of the
nanobeam structure. Pump light at optical detuning
D
56
v
m
is indicated by
the red arrows, while the optomechanically scattered sideband light is
represented by black arrows. The optical cavity output is filtered to reject the
pump, and then detected in a Hanbury Brown and Twiss (HBT) set-up using
two superconducting single photon detectors (SPD
1,2
). The detector outputs
are used as start/stop pulses in a time-correlated single photon counting
module (TCSPC), yielding the second-order phonon correlation function.
b
, Measured power spectral density
S
(
v
) of the acoustic resonance.
c
, Normalized optical cavity reflection spectrum,
R
. Pump detunings of
D
56
v
m
/(2
p
)
56
5.6 GHz are indicated by the red and blue dashed lines,
respectively.
d
, Transmission spectrum of the first (purple) and second
(orange) optical filter, with total filter transmission plotted in black.
A
is the
pump attenuation factor.
LETTER
RESEARCH
G
2015
Macmillan Publishers Limited. All rights reserved
23APRIL2015|VOL520|NATURE|523
mode occupancy
Æ
n
æ
is equal to the room temperature thermal bath
occupancy of
n
b
<
1,100. Since
C
SB,0
scales linearly with
n
c
, we can
determine
C
SB,0
for all
n
c
from this single measurement without rely-
ing on calibration of the optomechanical back action. We can then
compute
n
NEP
by dividing this noise count rate by
C
SB,0
at each value of
n
c
. The measured sensitivity follows the expected curve at low power
due to detector dark counts (solid red curve), but at high
n
c
saturates to
a value several times larger than expected for the filter suppression of
the pump (solid blue curve). In order to better understand this excess
noise, Fig. 2b shows measurements of the
n
NEP
as a function of filter–
pump detuning,
D
filter
, at a high power where the pump transmission
dominates the noise (
n
c
<
65). A strong dependence on
D
filter
is
observed, with a peak in the noise at 5 GHz and a secondary peak at
6.1 GHz, consistent with the phase noise of our pump laser
21
. With the
addition of a C-band bandpass filter before the SPD to remove broad-
band spontaneous emission from the pump laser, and at frequencies
far from the laser phase-noise peaks, the measured
n
NEP
agrees well
with the theoretical predictions based on the filter pump suppression
(horizontal dashed curve). At the relevant detuning of
D
filter
5
v
m
(vertical dashed curve), we measure a limiting sensitivity of
n
NEP
5
0.89
6
0.05. While this sensitivity is directly measured at
n
c
<
65,
n
NEP
is observed in Fig. 2a to be pump-limited for
n
c
0
1, implying
that our current set-up achieves
n
NEP
,
1 for
n
c
of order unity.
In what follows, we focus on measurements made with a blue-
detuned pump (
D
52
v
m
), in which the optomechanical back action
results in instability and self-oscillation of the acoustic resonator
9,21
.
The Stokes sideband count rate detected on a single SPD, shown versus
n
c
in Fig. 3a, displays a pronounced threshold, with an exponential
increase in output power beginning at
n
c
<
1,200, where
C
;
j
c
OM
j
/
c
i
<
0.8, in agreement with the expected onset of instability around
C
5
1(
c
5
0). This sharp oscillation threshold can also be observed
from the measured noise power spectral density (NPSD; Fig. 3b), in
which the amplitude of the mechanical spectrum is seen to rapidly
increase with a simultaneous reduction in linewidth, and in plots of the
in-phase and in-quadrature components of the photocurrent fluctua-
tions, which show a transition from thermal noise to a large-amplitude
sinusoidal oscillation. Also shown in Fig. 3a is the inferred phonon
occupancy
Æ
n
æ
. Below threshold, the photon count rate is related to
Æ
n
æ
via the simple linear relation
C
tot
5
g
j
c
OM
j
(
Æ
n
æ
1
1). At and
above threshold, as detailed in the Supplementary Information, self-
consistent determination of the oscillation amplitude indicates that
even at our highest pump power the mechanical amplitude remains
small enough that this linear approximation remains valid.
The statistical properties of the resonator near the self-oscillation
threshold can also be characte
rized by measuring photon cor-
relations using an HBT set-up as shown in Fig. 1. As noted
earlier, blue-detuned pumpin
g produces anti-normally ordered
phonon correlations. In this case
g
(2)
(
t
) refers to the anti-normally
ordered second-order phonon correlation function, defined
by
g
2
ðÞ
t
ðÞ
~
h
^
b
0
ðÞ
^
b
t
ðÞ
^
b
{
t
ðÞ
^
b
{
0
ðÞi
=
h
^
b
0
ðÞ
^
b
{
0
ðÞi
2
. For measurements
made with a red-detuned pump, as shown in Fig. 4b,
g
(2)
(
t
)
refers to the normally ordered phonon correlation function,
g
2
ðÞ
t
ðÞ
~
h
^
b
{
0
ðÞ
^
b
{
t
ðÞ
^
b
t
ðÞ
^
b
0
ðÞi
=
h
^
b
{
0
ðÞ
^
b
0
ðÞi
2
.Inthecaseoftheclas-
sical states measured here, there i
s no observable difference between
a
n
c
10
6
10
2
10
3
10
8
10
5
5.576
5.588
5.6
–135
–115
–95
NPSD (dBm Hz
–1
)
–75
w
/
(2
π
) (GHz)
50
–50
Q
(
μ
V)
Photon count rate (Hz)
–50
0
–25
0
25
50
25
–25
I
(
μ
V)
10
7
10
9
10
10
bc
10
3
10
4
10
5
10
7
n
10
6
Figure 3
|
Phonon lasing. a
, Phonon count rate (blue: left-hand vertical axis)
and inferred phonon occupancy
Æ
n
æ
(red: right-hand vertical axis) as a function
of intracavity photon number
n
c
for
D
52
v
m
. Dashed lines indicate points
below (blue), at (green) and above (magenta) threshold.
b
, Noise power spectral
densities (NPSD) corresponding to the dashed lines in
a
. The small satellite
peaks in the thermal emission background of the above-threshold spectrum
correspond to beating of the phonon laser line with low-frequency modes of the
nanobeam structure.
c
, Phase plots of the in-phase (
I
) and in-quadrature (
Q
)
amplitudes of the optical heterodyne signal for each of the dashed lines in
a
, acquired in a 36 MHz span around 5.588 GHz over a 60 s time interval.
0
0.4
0.8
0
1
2
0
2
4
1
1.5
2
a
n
c
= 1,700
n
c
= 2,400
g
(2)
(
t
)
b
g
/(2
π
) (MHz)
c
n
c
t
(
μ
s)
n
c
= 1,200
1
1.2
1.4
1.6
1.8
2
g
(2)
(0)
Thermal
DTS
2,500
2,000
1,500
1,000
500
3,000
0
10
–1
10
1
10
–2
10
0
d
2
1.5
1
0.5
0
–0.5
F
(
×
10
4
)
Figure 4
|
Phonon intensity correlations. a
,Normalizedanti-normally
ordered second-order intensity correlation function
g
(2)
(
t
)for
D
52
v
m
,
shown below, at and above threshold (left, middle and right, respectively).
Green lines show a simple exponential fit, while black lines indicate the expected
theoretical curve using decay rates measured from fitting the NPSD linewidth.
b
, Phonon correlation at zero time delay versus
n
c
for
D
52
v
m
(blue) and
D
5
v
m
(red). The top and bottom dashed lines indicate the expected values for
purely thermal or displaced thermal states (DTS), respectively. Error bars show
one s.d. determined from the fit value of
g
(2)
(0).
c
, Mechanical decay rate versus
n
c
for
D
52
v
m
, determined from the measured linewidth of the NPSD
(circles) and from the exponential fit to
g
(2)
(
t
) (diamonds).
d
,Fanofactorversus
n
c
. Error bars show one s.d. determined from the measured count rates,
assuming Poissonian counting statistics, and the fit value of
g
(2)
(0).
RESEARCH
LETTER
G
2015
Macmillan Publishers Limited. All rights reserved
524 | NATURE | VOL 520 | 23 APRIL 2015
the normally and anti-normally ordered correlation functions. As the
oscillation threshold is crossed, the state of the acoustic resonator will
transition from a thermal state into a displaced thermal state (DTS),
and the normalized phonon intensity correlation function near
t
5
0
should show a transition from bunching (
g
(2)
(0)
.
1) to Poissonian
statistics (
g
(2)
(
t
)
5
1forall
t
). Plots of
g
(2)
(
t
) below, at, and above thres-
hold are shown in Fig. 4a. Below threshold, bunching is clearly
visible, with
g
(2)
(0)
5
2 as expected for a purely thermal state. In Fig. 4b
g
(2)
(0) is plotted versus
n
c
for both blue- and red-detuned pump light.
For a blue-detuned pump, a smooth decrease from
g
(2)
(0)
5
2to
g
(2)
(0)
5
1 is observed in the threshold region, while for a red-detuned
pump, the oscillator is observed to remain in a thermal state through
threshold and beyond. The decay rate of the acoustic resonator, mea-
sured from both the linewidth of the NPSD and from an exponential fit
to
g
(2)
(
t
) below threshold, is plotted in Fig. 4c. The decay rate as
measured from the NPSD, which includes both phase and amplitude
fluctuations, is seen to increase around threshold before continuing to
decrease. This behaviour is commonly observed in semiconductor
lasers where a coupling exists between the gain and the cavity refractive
index, and a similar effect arises in optomechanical oscillators due
to the optical spring effect
24
. The decay rate measured from
g
(2)
(
t
),
on the other hand, which measures intensity fluctuations, begins
to deviate from the measured linewidth in the vicinity of threshold.
Thermal phonon emission dictates a strict correspondence between
the second-order and first-order coherence functions
4
;however,
above threshold where the phonon statistics are no longer purely
thermal, such a deviation is possible, and in fact predicted for self-
sustaining oscillators
25
. The Fano factor, defined as
F
5
(
D
n
)
2
/
Æ
n
æ
5
1
1
Æ
n
æ
(
g
(2)
(0)
2
1), provides additional statistical information about
the fluctuations of the oscillator, and is useful for defining a precise
oscillator threshold
26
as well as distinguishing between states that may
have similar or identical values of
g
(2)
(0) (for example, a coherent state
versus a DTS)
24
. The Fano factor of our mechanical oscillator, com-
puted from the measured
g
(2)
(0)andtheinferredvaluesof
Æ
n
æ
,is
displayed in Fig. 4d and shows the expected increase and peak in
fluctuations at threshold. Above threshold, the Fano factor drops again
due to saturation in the optomechanical gain, approaching a measured
value consistent with that expected for a DTS (
F
<
2
n
b
1
1).
Although we have emphasized the analogy between the optomecha-
nical oscillator studied here and a laser, there are unique differences
which arise owing to the intrinsically nonlinear nature of the radiation
pressure interaction in an optomechanical cavity. Recent theoretical
studies
15,24,27
indicate that a laser-driven optomechanical oscillator will
enter a non-classical mechanical state with anti-bunched phonon stat-
istics (
F
,
1), and under slightly more restrictive conditions, strongly
negative Wigner density. Surprisingly, this is predicted to be observable
even for classical parameters, that is, outside the single-photon strong-
coupling regime (
g
0
/
k
,
1), and in the presence of thermal noise.
Beyond phonon correlation spectroscopy of optomechanical oscillators,
it is envisioned that sensitive photon counting of the filtered motional
sidebands may be used in the preparation and heralding of non-
Gaussian quantum states of a mechanical resonator
10
.FortheOMC
cavities of this work, with their large optomechanical coupling rate
and near millisecond-long thermal decoherence time at temperatures
less than 1 K (ref. 28), the phonon addition and subtraction processes of
ref. 10 should be realizable with high fidelity and at rates approaching a
megahertz. Whether for studies of the quantum behaviour of meso-
scopic mechanical objects or in the context of proposed quantum
information processing architectures using phonons and photons
8
,such
photon counting methods are an attractive way of introducing a
quantum nonlinearity into the cavity optomechanical system.
Received 4 October 2014; accepted 10 February 2015.
1. Hoover, E. E. & Squier, J. A. Advances in multiphoton microscopy technology.
Nature Photon.
7,
93–101 (2013).
2. Hadfield, R. H. Single-photon detectors for optical quantum information
applications.
Nature Photon.
3,
696–705 (2009).
3. Hanbury Brown, R. & Twiss, R. Q. A test of a new type of stellar interferometer on
Sirius.
Nature
178,
1046–1048 (1956).
4. Glauber, R. J. The quantum theory of optical coherence.
Phys. Rev.
130,
2529–2539 (1963).
5. Aspelmeyer,M.,Kippenberg,T.J.&Marquardt,F.Cavityoptomechanics.
Rev. Mod.
Phys.
86,
1391–1452 (2014).
6. Suh, J.
et al.
Mechanically detecting and avoiding the quantum fluctuations of a
microwave field.
Science
344,
1262–1265 (2014).
7. Stannigel, K., Rabl, P., Sørensen, A. S., Lukin, M. D. & Zoller, P. Optomechanical
transducers for quantum-information processing.
Phys. Rev. A
84,
042341
(2011).
8. Stannigel,K.
et al.
Optomechanicalquantuminformationprocessingwithphotons
and phonons.
Phys. Rev. Lett.
109,
013603 (2012).
9. Grudinin, I. S., Lee, H., Painter, O. & Vahala, K. J. Phonon laser action in a tunable
two-level system.
Phys. Rev. Lett.
104,
083901 (2010).
10. Vanner,M.R., Aspelmeyer,M.& Kim,M.S.Quantum state orthogonalizationand a
toolsetforquantumoptomechanicalphononcontrol.
Phys. Rev. Lett.
110,
010504
(2013).
11. Galland, C., Sangouard, N., Piro, N., Gisin, N. & Kippenberg, T. J. Heralded single-
phononpreparation,storage,andreadoutincavityoptomechanics.
Phys. Rev. Lett.
112,
143602 (2014).
12. Børkje, K., Nunnenkamp, A. & Girvin, S. M. Proposal for entangling remote
micromechanical oscillators via optical measurements.
Phys. Rev. Lett.
107,
123601 (2011).
13. Lee, K. C.
et al.
Entangling macroscopic diamonds at room temperature.
Science
334,
1253–1256 (2011).
14. Clerk, A. A., Devoret, M. H., Girvin, S. M., Marquardt, F. & Schoelkopf, R. J.
Introduction to quantum noise, measurement, and amplification.
Rev. Mod. Phys.
82,
1155–1208 (2010).
15. Qian, J., Clerk, A. A., Hammerer, K. & Marquardt, F. Quantum signatures of the
optomechanical instability.
Phys. Rev. Lett.
109,
253601 (2012).
16. Kronwald, A., Ludwig, M. & Marquardt, F. Full photon statistics of a light beam
transmitted through an optomechanical system.
Phys. Rev. A
87,
013847
(2013).
17. Kimble, H. J., Dagenais, M. & Mandel, L. Photon antibunching in resonance
fluorescence.
Phys. Rev. Lett.
39,
691–695 (1977).
18. Pike,R.50
th
anniversaryofthe laser.
J. Eur. Opt. Soc. Rapid Publ.
5,
10047s (2010).
19. Chan,J.
et al.
Lasercoolingofananomechanicaloscillatorintoitsquantumground
state.
Nature
478,
89–92 (2011).
20. Chan, J., Safavi-Naeini, A. H., Hill, J. T., Meenehan, S. & Painter, O. Optimized
optomechanical crystal cavity with acoustic radiation shield.
Appl. Phys. Lett.
101,
081115 (2012).
21. Safavi-Naeini, A. H.
et al.
Laser noise in cavity-optomechanical cooling and
thermometry.
New J. Phys.
15,
035007 (2013).
22. Fetter, A. L. Intensity correlations in Raman scattering.
Phys. Rev.
139,
A1616–A1623 (1965).
23. Marsili, F.
et al.
Detecting single infrared photons with 93% system efficiency.
Nature Photon.
7,
210–214 (2013).
24. Rodrigues, D. A. & Armour, A. D. Amplitude noise suppression in cavity-
driven oscillations of a mechanical resonator.
Phys. Rev. Lett.
104,
053601
(2010).
25. Lax, M. Classical noise. V. Noise in self-sustained oscillators.
Phys. Rev.
160,
290–307 (1967).
26. Rice,P.R.&Carmichael,H.J.Photonstatisticsofacavity-QEDlaser:acommenton
the laser-phase-transition analogy.
Phys. Rev. A
50,
4318–4329 (1994).
27. Lo
̈
rch, N., Qian, J., Clerk, A., Marquardt, F. & Hammerer, K. Laser theory for
optomechanics: limit cycles in the quantum regime.
Phys. Rev. X
4,
011015
(2014).
28. Meenehan, S. M.
et al.
Silicon optomechanical crystal resonator at millikelvin
temperatures.
Phys. Rev. A
90,
011803 (2014).
Supplementary Information
is available in the online version of the paper.
Acknowledgements
We thank F. Marquardt and A. G. Krause for discussions, and
V. B. Verma, R. P. Miriam and S. W. Nam for their help with the single-photon detectors
used in this work. This work was supported by the DARPA ORCHID and MESO
programmes, the Institute for Quantum Information and Matter, an NSF Physics
Frontiers Center with the support of the Gordon and Betty Moore Foundation, and the
Kavli Nanoscience Institute at Caltech. Part of the research was carried out at the Jet
Propulsion Laboratory, California Institute of Technology, under a contract with NASA.
A.H.S.-N. acknowledges support from NSERC. S.G. was supported by a Marie Curie
International Out-going Fellowship within the 7th European Community Framework
Programme.
Author Contributions
O.P., S.M.M., J.D.C., S.G. and A.H.S.-N. planned the experiment.
J.D.C., S.G., G.S.M., S.M.M. and A.H.S.-N. performed the device design and fabrication.
F.M. and M.D.S. provided the single-photon detectors along with technical support for
their installation and running. J.D.C., S.M.M., G.S.M. and O.P. performed the
measurements, analysed the data and wrote the manuscript.
Author Information
Reprints and permissions information is available at
www.nature.com/reprints. The authors declare no competing financial interests.
Readers are welcome to comment on the online version of the paper. Correspondence
and requests for materials should be addressed to O.P. (opainter@caltech.edu).
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