DIAMOND OPTOMECHANICAL CRYSTALS
Michael J. Burek
a
, Justin D. Cohen
b
, Seán M. Meenehan
b
, Nayera El-Sawah
a,c
, Cleaven Chia
a
,
Thibaud Ruelle
a,d
, Srujan Meesala
a
, Jake Rochman
a,c
, Haig A. Atikian
a
, Matthew Markham
e
,
Daniel J. Twitchen
e
, Mikhail D. Lukin
f
, Oskar Painter
b
, and Marko Lon
č
ar
a,†
a.
John A. Paulson School of Engineering and Applied
Sciences, Harvard University, 29 Oxford Street,
Cambridge, MA 02138, USA
b.
Kavli Nanoscience Institute, Institute for Quantum
Information and Matter and Thomas J. Watson, Sr.,
Laboratory of Applied Physics, California
Institute of Technology,
Pasadena, CA 91125, USA
c.
University of Waterloo, 200 University Av
enue West, Waterloo,
ON, N2L 3G1, Canada
d.
École Polytechnique Fédérale de Lausa
nne (EPFL), CH-1015 Lausanne, Switzerland
e.
Element Six Innovation, Fermi Avenue, Harwel
l Oxford, Didcot, Oxfordshire OX110QR, UK
f.
Department of Physics, Harvard Univers
ity, 17 Oxford Street, Cambridge, MA 02138, USA
†
Corresponding author contact: E-mail: loncar@s
eas.harvard.edu. Tel: (617) 495-579. Fax: (617) 496-
6404.
2
ABSTRACT
– Cavity-optomechanical systems realized
in single-crystal diamond are poised to
benefit from its extraordinary material properties,
including low mechanical dissipation and a wide
optical transparency window. Diamond is also rich
in optically active defects, such as the nitrogen-
vacancy (NV) and silicon-vacancy (S
iV) centers, which behave as atom-like systems in the solid state.
Predictions and observations of
coherent coupling of the NV
electronic spin to phonon
s via lattice strain
has motivated the development of diamond nanomech
anical devices aimed at
realization of hybrid
quantum systems, in which phonons provide an inte
rface with diamond spins. In this work, we
demonstrate
diamond optomechanical crystals
(OMCs), a device platform to enable such applications,
wherein the co-localization of
~ 200 THz photons and few to 10 GHz phonons in a quasi-periodic
diamond nanostructure leads to coupling of an optical
cavity field to a mechan
ical mode via radiation
pressure. In contrast to other
material systems, diamond OMCs ope
rating in the resolved-sideband
regime possess large intracavity photon capacity (> 10
5
) and sufficient optomechan
ical coupling rates to
reach a cooperativity of ~ 20 at room temperatur
e, allowing for the observation of optomechanically
induced transparency and the realization of la
rge amplitude optomechanical self-oscillations.
3
1.0 INTRODUCTION
Optomechanical crystals (OMCs), first demonstrated in silicon
1
, and later in other materials like
silicon nitride
2,3
, aluminum nitride
4,5
, and gallium arsenide
6,7
, have emerged as a fruitful optomechanics
platform, wherein radiation pressure
effects provide exquisitely sensit
ive optical control of mechanical
vibrations. Such systems have enabled dem
onstrations of quantum ground state cooling
8
,
optomechanically induced transparency (OMIT)
9
, squeezed light
10
, and wavelength conversion
11
. Highly
coherent photon-phonon interactions in OMCs are a direct
result of the ability to
engineer a large single-
photon optomechanical coupling rate (
g
o
), while retaining sufficiently small optical (
κ
) and intrinsic
mechanical (
γ
i
) dissipation rates. Similar structures realized
in single-crystal diam
ond – which features a
unique combination of superior mechan
ical, thermal, and optical properties
12
– are expected to exhibit
pronounced optomechanical interactions, qua
ntified by the cooperativity parameter
i
o
c
g
n
C
2
4
(where
n
c
is the intracavity photon number). Specifi
cally, the wide bandgap
of diamond (~ 5.5 eV)
precludes multi-photon absorption over a wide wavelength range (from visible to infrared). This,
combined with its high thermal conductivity and sma
ll thermal expansion, enables monolithic diamond
optical cavities that can withstand significant optical
power densities, while
avoiding degradation in
optical linewidth or drifts in resonance wavelength
due to thermal lensing. The large intracavity photon
capacity of diamond can thus result in high coopera
tivities necessary for either strong mechanical
driving or effective laser cooling
8
. Moreover, diamond is among the stiffest materials known and
possesses extremely low thermoelastic mechanical damping, with recently demonstrated monolithic
diamond cantilevers exhibiting mechanical
Q
-factors in excess of 10
6
at room temperature
13
. In what
follows, we make use of these features to demons
trate OMCs in single-crystal diamond with unique
performance. Our diamond OMCs support an optical mode at
ω
o
/2
π
~ 200 THz, co-resonant with two
4
localized acoustic phonon modes at
ω
m
/2
π
~ 5.5 GHz and ~ 9.5 GHz. Both
mechanical resonances are
well coupled to the optical cavity, with
vacuum optomechanical coupling rates of
g
o
/2
π
~ 120 kHz and ~
220 kHz, respectively. With a
measured optical linewidth of
κ
/2
π
~ 1.1 GHz, our diamond OMC system
operates in the so-called resolved sideband regime (
ω
m
/
κ
>> 1), necessary for efficient radiation-
pressure driven dynamic backaction. This enables our
diamond OMCs to be optically driven to C >> 1
at room temperature, highlighted
by the observations of “phonon lasing”
14
and OMIT
9
in our structures.
2.0 DIAMOND OPTOMECHANICAL CRYS
TAL DESIGN AND FABRICATION
The OMCs of this work consist of a one dimens
ional nanobeam photonic crystal cavity fabricated in
synthetic single-crystal diamond
15
using previously develope
d ‘angled-etching’ techniques
16,17
. The
nanobeam cavity is based on a diamond waveguide with a
triangular cross-section th
at is perforated with
a periodic lattice of elliptically shaped air hol
es. One unit cell of the waveguide and corresponding
photonic bandstructure are shown in
Figure 1 (a) and Figure 1 (b), resp
ectively. The latte
r includes both
transverse electric (TE-like, solid black lines) and
transverse magnetic (TM-li
ke, dashed blue lines)
guided modes, while the gr
ey shaded region indicates
the continuum of radiation and leaky modes that
exist above the light line for the structure. In th
is work, we focus on TE-like modes (see Figure 1 (b)
inset), near the X-point frequency of
ω
o
/2
π
~ 200 THz (
λ
~ 1550 nm), since they
can lead to the
realization of very high
Q
-factor optical cavities
15
. Importantly, our photonic
crystal waveguide also
supports acoustic guided modes that spatially overla
p with optical modes, a
nd can couple to them via
radiation pressure. The corresponding
mechanical bandstructure (Figure 1 (c)) reveals a rich library of
guided acoustic modes in the few
to ~ 12 GHz frequency range (see
Supplementary
Information
for
extended discussions
18
). The guided modes, categorized by even
(solid black lines) and odd (dashed blue
lines) vector symmetries about the
xz
-plane, yield symmetry based quasi-bandgaps. Following OMC
5
design rules
19,20
, we identified the guided modes derived from the
Γ
-point of the 4
th
and 7
th
y
-symmetric
bands (frequency of
ω
m
/2
π
~ 6.9 GHz and ~ 11.5 GHz) – referred to
hereafter as the “flapping” and
“swelling” acoustic guided modes (F
igure 1 (d) and (e)), respectively – as the mechanical modes of
interest for large optomechanical coupling. To pr
oduce an optimized diamond OMC design, we focus on
the acoustic flapping mode due to the large quasi
-bandgap below its nativ
e band – indicated by the
shaded pink region in Figure 1 (c).
To realize a diamond OMC cavity from the aforemen
tioned OMC waveguide, the lattice of air holes is
chirped
19
such as to transition from a “mirror” region
formed by the base unit cell in Figure 1 (a) to a
“defect” cell. The selected defect
cell dimensions simultaneously raise
and lower the frequencies of the
target optical and mechanical modes, respectivel
y, into their corresponding
quasi-bandgaps. Gradually
reducing the unit cell lattice constant while
also decreasing the ai
r hole aspect ratio (
h
y
/
h
x
) achieves the
necessary band edge tuning (see right and left panels
of Figure 1 (b) and (c), respectively). An optimized
design
18
was determined via numerical optimization me
thods, based on FEM simulations (COMSOL) to
calculate the optical (
ω
o
) and mechanical (
ω
m
) cavity resonance frequencie
s, the optical Q-factor (
Q
o
),
and
g
o
. Both moving boundary (
g
o,MB
) and photo-elastic (
g
o,PE
) contributions to the single-photon
optomechanical coupling rate were considered
18
, with the calculation of
g
o,PE
using the following
photoelastic coefficients of diamond
21
: (
p
11
,
p
12
,
p
44
) = (-0.25, 0.043, -0.172).
Normalized electric field
(
E
y
) and mechanical displacement profiles (
xy
-plane) of the final optimized diamond OMC design are
shown in Figure 1 (f) and (g), respectivel
y. The optimized design – which assumes
x
-axis orientation
aligned with the in-plane diamond
[110] crystallographic direction
– has an optical resonance at
ω
o
/2
π
=
196 THz (
λ
o
= 1529 nm), radiation-limited optical
Q
-factor of 7.4 x 10
5
, mode volume of 0.57(
λ
/n
)
3
,
acoustic flapping mode mechanical resonance at
ω
m
/2
π
= 6.18 GHz, and zero-point motion of
x
zpf
= 3.1
fm. The final coupling rate
for this design was g
o
/2
π
= 136 kHz, and included a moving boundary and
photo-elastic contribution of
g
o,MB
/2
π
= 62 kHz and
g
o,PE
/2
π
= 74 kHz, respectively.
6
With our final diamond OMC design optimized for
the acoustic flapping mode, we also observe a
localization of the previously me
ntioned acoustic swelling mode (dis
placement profile shown in Figure
1 (h)) at a mechanical frequency of
ω
m
/2
π
= 9.01 GHz, with a zero-point motion of
x
zpf
= 2.2 fm. The
simulated optomechanical coupling rate for this design was g
o
/2
π
= 234 kHz, whic
h includes a moving
boundary and photo-elastic contribution of
g
o,MB
/2
π
= 50 kHz and
g
o,PE
/2
π
= 184 kHz, respectively. We
attribute the overall greater optomech
anical coupling rate of the acoustic swelling mode to its cross-
sectional strain profile, which more
favorably overlaps with the TE-lik
e optical mode. While this mode
is better coupled to the localized optical cavity, its predicted mechanical resonance frequency is not
localized within a symmetry-based quasi-bandgap
(see Figure 1 (c)), which may ultimately limit its
mechanical Q-factor in fabricated structures
1,20
.
As previously mentioned, fabrication of di
amond OMCs utilized a
ngled-etching techniques
15-18
(as
illustrated in Figure 2 (a)), which employ anisotropi
c oxygen-based plasma etching at an oblique angle
to the substrate surface resulting
in suspended structures with a tr
iangular cross-section. The final
fabricated structures, displayed
in Figure 2 (b) - (d),
reveal excellent reproduction of the intended
design. A unique feature of angled-etched structur
es is their triangular cross-sectional symmetry
18
. The
high-resolution SEM image shown in Figure 2 (e) re
veals a fabricated diamond OMC (oriented upside
down), with insets displaying a
tilted cross-s
ectional view.
3.0 OPTICAL AND MECHANICAL SPECTROSCOPY
The fiber-optical characterization set up
18
used to perform both optical
and mechanical spectroscopy
of diamond OMCs is schematically di
splayed in Figure 3 (a). Briefly,
light from a tunable laser source
(TLS) was evanescently coupled to
the device under test via a dimple
d fiber taper. A small portion of
laser signal fed to a wavemeter enab
led continuous monitoring of the
laser frequency. An erbium doped
7
fiber amplifier (EDFA) was used in certain experi
ments to increase the maximum input laser power, and
a variable optical attenuator (VOA)
was used to set the final laser
power delivered to the device. The
optical cavity transmission spectrum was collected
by a low-speed (125 MHz) photodetector, while a
high-speed (12 GHz) photor
eceiver monitored the radio frequency
(RF) response of the mechanical
cavity via a real-time spectrum analy
zer (RSA). For OMIT measurements
discussed later in this work,
an electro-optic phase modul
ator (EOPM), placed in the input fibe
r path, was used to create a weak
tunable probe signal on the pump laser control field.
Port 1 of a high frequency
vector network analyzer
(VNA) supplied the RF input to the EOPM, while
port 2 of the VNA collected the RF output of the
high-speed photoreceiver. All measurements were perf
ormed at room temperature and ambient pressure.
A transmission spectrum of a representative diamond
OMC, displayed in Figure 3 (b), reveals the
optical cavity resonance centered at
λ
o
= 1529.2 nm, with a measured
total and intrinsic optical
Q
-factor
of
Q
t
~ 1.76 x 10
5
and
Q
i
~ 2.70 x 10
5
, respectively. The corresponding total cavity decay rate, fiber
taper coupling rate, and intrinsic optical decay rate are
κ
/2
π
= 1.114 GHz,
κ
e
/2
π
= 399 MHz, and
κ
i
/2
π
=
715 MHz, respectively. With the in
put laser slightly detuned from th
e optical cavity, the broadband RF
spectrum of thermally excited motion at room temp
erature (i.e., thermal Brownian motion) reveals a
series of mechanical resonances
18
, as shown in the normalized power
spectral density (NPSD) in Figure
3 (c). Specifically, we attribute the sharp reso
nance observed at ~ 5.5 GHz to the diamond OMC
acoustic flapping mode. A high-resolution RF spectrum (s
hown in Figure 3 (d)) of this feature reveals a
Lorentzian mechanical resonance
of the diamond OMC centered at
ω
m
/2
π
= 5.52 GHz with a room
temperature mechanical
Q
-factor of
Q
m
~ 4100.
Given the measured optical cavity decay rate, our
diamond OMC operates in the resolved sideband
regime, with
ω
m
/
κ
~ 4.86. In this regime, while the input laser is either red- or blue-detuned from the
optical cavity by a mechanical frequency (
Δ
= (
ω
o
–
ω
l
) = ±
ω
m
), mechanical motion of the acoustic
mode phase-modulates the transmitted light, giving rise
to a sideband of the input laser resonant with the
8
optical cavity. The other first-orde
r motional sideband, which is not reso
nant with the optical cavity, is
suppressed in this scenario. As a result, the mech
anical motion produces an intensity modulation in the
radio frequency (RF) power spectrum of the photoreceiver
signal. To observe this
effect directly, a weak
input laser was tuned across the opt
ical cavity at a constant power,
while simultaneously monitoring the
RF spectrum near the diamond OMC acoustic flapping mo
de. Figure 3 (e) displays the collected spectra
as a function of laser detuning, with the simultane
ously collected optical transmission spectrum also
plotted. A clear increase in optomechanical transduc
tion is observed as the lase
r is tuned off-resonance
from the optical cavity by ± ~
45 pm, corresponding to a detuning
of approximately a mechanical
frequency. Additionally, st
rong transduction occurs with the lase
r tuned within the cavity bandwidth,
and a clear optical bistability is present in the op
tical cavity transmission spectrum. We attribute both
observations to non-linear optical absorption (likel
y due to surface contamination), which cause a
thermo-optic red shift in the optic
al resonance wavelength and an incr
ease in thermal Brownian motion
of the mechanical cavity. To mitigate such ther
mal effects, a similar measurement was performed,
however now with the input laser
power continually adjusted via
the VOA to maintain a constant
intracavity photon number at each laser detuning (Figur
es 3 (f) and (g)). From the measured optical
cavity resonance frequency and linewidth,
n
c
is calculated by the relation:
2
2
2
/
2
l
e
i
c
P
n
(1)
where
P
i
is the input laser power set by the VOA. In the resolved sideband limit
22
, optomechanical
backaction causes additional mechanical damping (
γ
OM
) and springing (
δω
m
= |
ω
m
–
ω
m,o
|) rates,
respectively, of:
9
2
1
2
1
Re
2
2
m
m
o
c
OM
i
i
g
n
(2)
and
2
1
2
1
Im
2
m
m
o
c
m
i
i
g
n
(3)
Under optimal detuning, with
Δ
= ±
ω
m
, a maximum optomechanically induced damping rate of
2
4
o
c
OM
g
n
is expected. Figure 3 (f)
and (g) display the experime
ntally derived damping and
springing curves (grey circles)
for the diamond OMC acoustic flappi
ng mode, respectively. A weak
intracavity power, corresponding to
n
c
~ 10,000 photons, was used for this measurement to avoid any
thermal drifts in the cavity resonance. Indeed,
the optomechanically induced damping is maximized
(minimized) when the laser is detuned a mechanical fr
equency red (blue) of the optical cavity. Fits to
these data sets following Eq. (2) and Eq. (3) (so
lid red lines), gave an estimate for the intrinsic
mechanical damping of
γ
i
/2
π
~ 1.37 MHz and the single-photon
optomechanical coupling rate of
g
o
/2
π
~
118 kHz. This estimate differs only slightly from our
design, which we attribut
e to uncertain
ty in the
photo-elastic constants of diamond
at telecom frequencies, as we
ll as fabrication imperfections.
Figure 4 (a) plots the measured
mechanical linewidth of the di
amond OMC acoustic flapping mode,
collected under optimal red- and bl
ue-sideband laser detuning as a
function of input power, up to the
maximum output of the laser (in this case, corresponding to
n
c
~ 63,000 photons). The effects of
backaction are clearly visible,
with the laser red detuned (
Δ
= +
ω
m
;
γ
red
) resulting in damping and the
laser blue detuned (
Δ
= –
ω
m
;
γ
blue
) giving rise to anti-damping of moti
on. From the mean value extracted
from
γ
red
and
γ
blue
data points, the estimated intrinsic mechanical linewidth is
γ
i
/2
π
= 1.41 +/- 0.06 MHz.
10
The inset of Figure 4 (a) displays th
e optomechanically induced damping (
γ
OM
=
γ
red
–
γ
i
, black squares),
plotted versus
n
c
. A linear fit to the
γ
OM
data yields
g
o
/2
π
~ 123 +/- 6 kHz, which agrees well with
simulations, and is consistent with previous estimates
from the data plotted in Fi
gure 3 (f) and (g). With
the laser blue-detuned by a mechanic
al frequency, a threshold where
γ
blue
~ 0 is reached at
approximately
n
c
,
thr
~ 27,000, exciting the diamond OMC mech
anical cavity into large amplitude
optomechanical self-oscillatio
ns, so-called “phonon lasing”
14
. Mechanical spectra of the diamond OMC
taken below, at, and above this phonon lasing threshol
d (shown in Figure 4 (b
)) show an over 70 dB
increase in peak mechanical amplitude (Figure 4 (b) inset).
The optomechanical cooperativity (
C
≡
γ
OM
/
γ
i
) is plotted versus
n
c
in Figure 4 (c). To drive
γ
OM
beyond the level reached with the tunable la
ser output alone (i.e. to enable larger
n
c
), an EDFA was
inserted before the VOA to increase the maximum i
nput laser power. With the laser red-detuned by a
mechanical frequency, a maximum cooperativity of
C
~ 6.6 was reached for the acoustic flapping mode,
as represented by the open squares
in Figure 4 (b). Amplified spontan
eous emission (ASE
) optical noise
output by the EDFA prevented a direct estimate
of the intracavity phot
on number. However from
previous estimates of
κ
,
γ
i
, and
g
o
, a maximum intracavity photon number of
n
c,max
~ 159,000 was
inferred (as indicated by the extrap
olated dashed linear fit). Beyond this
input power level, thermo-optic
bistability shifts made it difficult to achieve precis
e laser detuning equal to a
mechanical frequency. In
relation to previous reported limits, diamond OMCs
have an intracavity photon capacity nearly twice as
large as OMC structures r
ealized in silicon nitride
2,3
.
With the demonstration of C
>> 1, optomechanical transducti
on in our diamond OMC acoustic
flapping mode occurs at a substantially faster ra
te than energy loss of
the system. This enables
observation of the optomechanical analog to elect
romechanically induced transparency, so-called
OMIT
9
. To observe OMIT in our diamond OMC structur
es, the input laser is red-detuned from the
optical cavity and fixed as a
strong driving control field (
ω
c
), while a weak probe field (
ω
p
, realized as
11
sidebands created by an EOPM) is
swept in frequency across the
optical cavity resonance. Under
optimal detuning conditions, whereb
y the control laser detuning equa
ls a mechanical frequency (
Δ
oc
≡
(
ω
o
–
ω
c
) =
ω
m
) and the probe-control detuning satis
fies a two-photon resonance condition (
Δ
pc
≡
(
ω
p
–
ω
c
) =
Δ
oc
), destructive interfer
ence of probe photons with
control photons scatte
red by the mechanical
resonator occurs. This yields a transparency window
on the optical cavity transmission spectra, with its
bandwidth set by the mechanical damping rate. A centr
al requirement for this scattering phenomenon is
that the probe and phonon-scattere
d photons are phase coherent, wh
ich demonstrates a coherent
interaction of the mechanical resona
tor with the optical cavity. As pr
eviously mentioned, OMIT in our
diamond OMC structures
is observed via an |S
21
| measurement with a VNA (Figure 3(a)), where port 1
of the VNA drives the EOPM input
to create the weak probe fiel
d which sweeps across the optical
cavity, and port 2 collects
the RF output of the hi
gh-speed photoreceiver. Fi
gure 4 (d) displays a
representative series of
normalized OMIT spectra (|S
21
|/max{|S
21
|}), collected with the control laser
detuned approximately
Δ
oc
~ [(
ω
m
– 490 MHz),
ω
m
,
(
ω
m
+ 580 MHz)] and an intracavity photon number
of
n
c
~ 59,000. In these broadband OMIT spectra, we obs
erve a clear dip representing the transparency
window (right inset panels of Figure 4 (d) display zoom
ed-in spectra of this fine
feature). Fits to the
normalized OMIT spectra
18
, which followed the methodology reported previously
2,3
, estimate a
cooperativity of C ~ 1.9 for da
ta collected with optimal
Δ
oc
~
ω
m
detuning, in good agreement with the
cooperativity value measured in Figure
4 (c) under similar input laser power.
In addition to the resonance featur
e at ~ 5.5 GHz, two sharp features
are also observed in the diamond
OMC broadband thermal Brownian mo
tion RF spectrum (Figure 3 (c))
near ~ 9.5 GHz. Figure 5 (a)
displays a zoomed-in RF spectrum
around these features, collected with
a weak laser signal slightly
detuned from the optical cavity resonance. Four clea
r resonances are present
in this span, with the
central feature at ~ 9.5 GHz most
strongly transduced by the optical ca
vity field. A high-resolution RF
spectrum (shown in Figure 5 (b)) of
this feature reveals a Lorentzi
an mechanical resonance of the
12
diamond OMC centered at
ω
m
/2
π
= 9.45 GHz with a mech
anical Q-factor of
Q
m
~ 7700. This
corresponds to a
f·Q
product of ~ 7.3 x 10
13
Hz, which is among the highest demonstrated for either a
bulk or small-scale single-crys
tal diamond mechanical oscill
ator at room temperature
23,24
.
As before, we extract the optomechanical coupling
rate for this mode by t
uning the lase
r across the
optical cavity resonance, while maintaini
ng a constant intracavity photon number of
n
c
~ 6000, and
simultaneously monitoring the mechanical resonan
ce at 9.45 GHz. Fitting
Eq. (2) and (3) to the
collected mechanical linewidth and
frequency data (displayed in Figu
re 5 (c) and (d), respectively),
yields an estimate for the intr
insic mechanical damping of
γ
i
/2
π
~ 1.18 MHz and the single-photon
optomechanical coupling rate of
g
o
/2
π
~ 239 kHz. This value, as
well as mechanical resonance
frequency, is in good agreement with simulation
results obtained for diamond OMC acoustic swelling
mode shown in Figure 1 (e). Re
peating similar measurements on the other ~ 9.5 GHz resonances
observed in Figure 5 (a) confir
med the mechanical mode at
ω
m
/2
π
= 9.45 GHz couples most strongly to
the optical cavity. We believe that these additional re
sonances are likely of similar modal character, but
hybridized with guided body modes of the diamond OMC
given the lack of confinement in an acoustic
quasi-bandgap
25
.
Figure 5 (e) plots the measured mechanical linew
idth of the diamond OMC acoustic swelling mode,
collected under optimal
red- and blue-sideband laser detuning (
Δ
= ±
ω
m
) as a function of input power,
up to the maximum output of the laser, as well as with
the amplified laser pump. In this instance, with
the increased side
band resolution of
ω
m
/
κ
~ 8.5, the maximum laser pow
er output corre
sponds to only
n
c
~ 29,000 photons. The mean value extracted from
γ
red
and
γ
blue
data points yields an estimated intrinsic
mechanical linewidth of
γ
i
/2
π
= 1.27 +/- 0.02 MHz. A plot of
optomechanical cooperativity versus
n
c
,
shown in Figure 5 (b), yields a second estim
ate for the optomechanical coupling rate of
g
o
/2
π
~ 217 +/-
12 kHz, which is consistent with
previous estimates for the di
amond OMC acoustic swelling mode.
From Figure 5 (f), the threshold power for th
e observation of optomechan
ical self-oscillations
18
under
13
optimal blue-detuning was only
n
c,thr
~ 7,600. Under optimal red-detune
d laser conditions, the increased
laser power afforded by the input EDFA enabled us
to reach a room temperat
ure mechanical linewidth
γ
red
/2
π
~ 26.7 MHz (Figure 5 (e)), co
rresponding to a maximum observed cooperativity of C ~ 19.9
(Figure 5 (f)). With previous estimates of
κ
,
γ
i
, and
g
o
for this acoustic swel
ling mode, an intracavity
photon number of only
n
c
~ 162,000 was inferred at
this cooperativity leve
l. As before, higher
cooperativities were not observed due to instabilit
ies in the measurement under the high optical input
power. OMIT was also observed
for this acoustic swelling mode
18
.
4.0 CONCLUSIONS
In summary, we have demonstrated resolved
sideband cavity-optomechanics in single-crystal
diamond, operating in the few to ~ 10 GHz range, wh
ere optomechanical coupling via radiation pressure
was sufficient to reach a room temperature coopera
tivity of nearly ~ 20 for an intracavity photon
population on the order of 10
5
. Present devices also offer a promisi
ng platform for reaching much larger
cooperativities when, for instance, operated at cryoge
nic temperatures, where m
echanical Q-factors of
diamond resonators have been
shown to improve significantly
13
. Moreover, incorporating diamond color
centers with monolithic OMCs is an interestin
g route to applications in quantum-nonlinear
optomechanics. Diamond is rich in optically active defe
cts (color centers), such
as the nitrogen-vacancy
(NV) and silicon-vacancy (SiV) center, which be
have as atom-like systems in the solid state
26,27
. Recent
experiments
23,28-34
exploring coherent coupling of the NV
electronic spin to phonons in mechanical
resonators via lattice strain have
demonstrated manipulation of the NV
spin state at large driven
mechanical amplitudes, but remain far below the
strong spin-phonon coupling regime. One way to boost
this interaction would be to engi
neer truly nanoscale resonators, w
ith feature sizes of a few hundred
14
nanometers, and with frequencies in
the hundreds of MHz to few GHz
range – such mechanical modes
would provide a large change
in local strain per phonon
31
. The localized phononic modes of OMCs not
only satisfy these requirements
35
, but also are conveniently actuated an
d transduced with
optical fields in
the well-established telecom wavelength range.
Diamond OMCs with coupl
ed color centers may
ultimately be used to map non-classical spin qubit stat
es as well as quantum states of light onto phonons
and vice-versa
36
, and will enable fundamentally new ways to
prepare, control, and read out the quantum
states of diamond spin qubits. Lastly
, individual diamond OMCs
integrated into la
rger arrays coupled
through phononic waveguides
25
could enable long-range spin-s
pin interactions mediated by phonons
37
.
We note that, parallel to this work, Mitchell
et al.
have demonstrated cavity optomechanics in single
crystal diamond microdisks
38
.
ACKNOWLEDGEMENTS
This work was supported in part by the ONR Quantum Optomechanics MURI (Award No. N00014-
15-1-2761), AFOSR Quantum Me
mories MURI (grant FA
9550-12-1-0025), DARPA QuINESS
program, NSF QOP (grant PHY-
0969816), and NSF CUA (grant
PHY-1125846), the Institute for
Quantum Information and Matter, an NSF Physics
Frontiers Center with support of the Gordon and
Betty Moore Foundation, and the Kav
li Nanoscience Institute at Caltech. M.J. Burek and H.A. Atikian
were supported in part by the Harvard Quantum Optic
s Center (HQOC). C. Chia
was supported in part
by Singapore’s Agency for Science,
Technology and Research (A*STAR)
. T. Ruelle was supported in
part from the Fondation Zdenek and Michaela Bakala. Th
is work was performed in part at the Center for
Nanoscale Systems (CNS), a member of the Nati
onal Nanotechnology Infrastruc
ture Network (NNIN),
which is supported by the National Science F
oundation under NSF award no. ECS-0335765. CNS is
part of Harvard University.
15
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