Cooling and control of a cavity opto-electromechanical system
Kwan H. Lee,1Terry G. McRae,1,2Glen I. Harris,1Joachim Knittel,1and Warwick P. Bowen1
1Department of Physics, University of Queensland, St Lucia, Queensland 4072, Australia
2MacDiarmid Institute, Physics Department, University of Otago, Dunedin, New Zealand
(Dated: February 11, 2010)
We implement a cavity opto-electromechanical system integrating electrical actuation capabilities of na-
noelectromechanical devices with ultrasensitive mechanical transduction achieved via intra-cavity opto-
mechanical coupling. Electrical gradient forces as large as 0.40 µN are realized, with simultaneous mechanical
transduction sensitivity of 1.5×10−18m Hz−1/2representing a three orders of magnitude improvement over any
nanoelectromechanical system to date. Opto-electromechanical feedback cooling is demonstrated, exhibiting
strong squashing of the in-loop transduction signal. Out-of-loop transduction provides accurate temperature
calibration even in the critical paradigm where measurement backaction induces opto-mechanical correlations.
Mechanical oscillators are predicted to exhibit striking
quantum behavior; enabling experimental tests of long-
standing scientific problems such as quantum gravity
and quantum nonlinear dynamics[3–5], as well as far-
reaching applications in metrology and quantum infor-
mation systems. Rapid progress towards this quantum
regime is underway in both cavity optomechanical systems
(COMS) and nanoelectromechanical systems (NEMS)[9,
10]. COMS enable ultrasensitive transduction of the mechan-
ical motion, presenting a solution to the key challenge of re-
solving the oscillators quantum zero-point fluctuations.
To date, however, mechanical actuation in COMS has been
achieved via radiation pressure[11–13], which is inherently
weak and severely constrained in the quantum regime by heat-
tuation of NEMS, by comparison, can be orders of magnitude
stronger and is far less prone to heating[10, 14, 15]; providing
access to nonlinear mechanical behavior, as well as greater
scope for quantum control and cooling[10, 14, 16].
Recently, a non-dissipative electrical actuation technique
using localized gradient forces has been developed for di-
electric NEMS. In this Letter we report a cavity opto-
electromechanical system (COEMS) which extends this tech-
nique to COMS based on silica microtoroids on a silicon chip.
The microtoroid structure integrates high quality optical and
mechanical resonances; while the dielectric nature of silica
is naturally suited to gradient force actuation. Electri-
cal gradient forces as large a 0.40 µN are achieved, enabling
strong mechanical actuation without observable heating ef-
fects. Simultaneously, ultrasensitive optical transduction is
implemented at the level of 1.5×10−18m Hz−1/2close to
the mechanical zero-point motion and surpassing the current
state-of-the-art in NEMS by three orders of magnitude.
Electo-mechanical actuation and opto-mechanical trans-
duction, when combined within a feedback loop, allow
immediate control of the state of the mechanical oscilla-
tor; with the capacity to facilitate, for example, feedback
cooling or heating, electro-optic spring effects, and
phonon lasing. Here, feedback cooling is implemented
as a demonstration. All previous COMS and NEMS feed-
back cooling experiments have used a single in-loop trans-
ducer for both cooling and characterization of the mechani-
FIG. 1: (color online) Experimental schematic including electronic
locking method for in-loop probe. The out-of-loop probe was ther-
mally locked. FPC: fibre polarization controller. Inset: electric field
distribution between electrodes. All experiments were carried out at
room temperature (T = 300 K) and atmospheric pressure.
cal oscillator[10, 12, 14]. However, anti-correlations are es-
tablished between the mechanical motion and the transduc-
tion noise, which cause squashing and an over-estimate of the
achieved cooling. Critically, the in-loop transduction sig-
nal predicts a homogeneously decreasing temperature with
increasing gain, when in fact transduction noise imprinted
on the mechanical motion causes net heating at high gain.
Here, we implement for the first time a second out-of-loop
transducer, providing accurate temperature characteriza-
tion even in the presence of strong squashing. This allows
the first direct observation of the transduction noise limit of
feedback cooling. Out-of-loop transduction is important in
any circumstance where the mechanical motion becomes cor-
related to the transduction noise. Particularly critical is the
quantum paradigm where such correlations are inherently es-
tablished by radiation pressure induced backaction.
arXiv:0909.0082v4 [quant-ph] 11 Feb 2010
The capacity to strongly electrically actuate COMS repre-
sents an enabling step towards the experimental realization of
mechanical nonlinear dynamics; with nanofabrication tech-
niques providing the means to engineer the nonlinear prop-
erties. Such dynamics are traditionally the realm of NEMS,
where arrays of coupled mechanical oscillators allow, for
example, oscillation synchronization, enhanced sensing
architechtures, sub-Heisenberg limit metrology, and
mechanical quantum state engineering. COMS, however,
present the advantage of superior transduction sensitivity, af-
fording the possibility of achieving the new regime of quan-
tum nonlinear dynamics. Furthermore, integration into ultra-
cold superconducting circuits has the potential to unify cavity
opto-mechanics and the new and burgeoning field of circuit
A schematic of our experimental setup is shown in Fig. 1.
To achieve gradient force actuation a radio-frequency (RF)
voltage was applied to a sharp stainless steel electrode with
2 µm tip diameter positioned 15 µm vertically above a mi-
crotoroid with major and minor diameters of 60 and 6 µm,
respectively, and a 10 µm undercut. The height was chosen to
balance the stronger gradient forces achievable close to the
microtoroid, with the potential for chemical contamination
and structural damage due to physical contact. A grounded
flat electrode was mounted beneath the silicon chip with the
combination of electrodes forming a capacitor. The applied
voltage induced a point-like charge build-up on the sharp elec-
trode tip, with a corresponding sheet of opposite charge on the
flat electrode, as shown in the inset in Fig. 1 and confirmed
through finite element modeling. Since the microtoroid was in
close vicinity to the sharp electrode, the electric field it expe-
rienced was well approximated by that of a point charge. The
strong gradient of such a field, combined with surface charge
induced static electric fields which polarize the microtoroid,
enabled large gradient forces to be exerted.
1550 and 980 nm respectively provided in- and out-of-
loop optical probe fields. Both fields were evanescently
coupled into a microtoroid using a tapered optical fiber
held in close proximity to the microtoroid surface, and
were frequency detuned to the full-width-half-maximum of
whispering gallery modes with relatively high coupled optical
quality factors of Q ≈ 106. Mechanical motion altered the
cavity’s optical path length, and as a result was transferred to
the amplitude of the out-coupled in- and out-of-loop optical
probes. The out-coupled probes were split, and detected
on Si and InGaAs photodiodes; respectively providing in-
and out-of-loop electronic transduction signals.
The transduction spectrum obtained by spectral analysis
of the in-loop transduction signal without gradient force ac-
tuation or feedback is shown by the grey curve in Fig. 2A
for 1 mW of incident probe power. The absolute mechani-
cal displacement amplitude was calibrated by observing the
response of the optical transmission to a known reference
tral peaks due to Brownian motion of microtoroid mechani-
cal modes. Finite-element modeling identified these modes
to be the lowest order flexural mode, and the two lowest or-
der crown modes; with the peak frequencies agreeing with the
model to within 5%.
To observe the quantum behavior of a mechanical oscil-
lator the transduction sensitivity must be sufficient to re-
solve the oscillators zero-point motion. Both the transduc-
tion sensitivity and the amplitude of zero-point motion of
our COEMS can be determined from a fit to the transduced
spectral density S(ω) =?3
= 2kBTΓjmj|χj(ω)|2is the spectral density of Brown-
ian motion of mechanical mode j with effective mass
mj, damping rate Γj, and mechanical susceptibility χj(ω) =
1.5×10−18m Hz−1/2was established from the fit, three or-
ders of magnitude better than any NEMS to date; and the
mode effective masses and damping rates were (m1,m2,m3)=
(280,410,33) µg and (Γ1,Γ2,Γ3)/2π=(9.5,11.5,6.8) kHz, re-
spectively. The peak of the zero point motion spectral den-
sity can then be calculated as S(j)
three modes. The transduction sensitivity is, therefore, within
two orders of magnitude of the mechanical zero-point mo-
tion. Techniques have recently been developed to substan-
tially improve both the microtoroid transduction sensitivity
x(ω)+SN; where SN is the
transduction noise due in our case to laser phase noise, and
m,j−ω2−iΓjω)]−1. A transduction sensitivity of S1/2
zp= ?/mjΓjωm,j, to give
zp)1/2= (1.4,1.1,4.6)×10−20m Hz−1/2for the
FIG. 2: (color online) Gradient force actuation of a COEMS. A
Square root transduction spectra. Black curve: measured spectra;
red curve: theoretical model including interference between the me-
chanical modes; grey curve: Brownian motion spectra; dash-dotted
line: transduction sensitivity. Insets: (from left to right) finite ele-
ment models of the lowest order crown (j = 1), lowest order radial
flexural (j = 2), and second order crown (j = 3) modes. Resolution
bandwidth: ΓRBW/2π=3 kHz. B Square root peak transduction spec-
tra as a function of RF drive amplitude. Green triangles: j = 1, red
circles: j=2, black squares: j=3.
through improved optical quality and shot noise limited ho-
modyne detection; and the mechanical mode damping
rate and effective mass through nanofabrication, cryo-
genic cooling, and evacuation of the media surrounding
the microtoroid. These techniques are fully compatible
with the gradient force actuation scheme demonstrated here,
and should enable sub-zero point motion transduction sensi-
tivity in our COEMS architecture.
Gradient force actuation of the COEMS was characterized
by applying the output voltage from a network analyzer to
the sharp electrode, and monitoring the frequency response
of the in-loop transduction signal. Fig. 2A shows the ob-
served mechanical frequency response for a 3 Vrmsnetwork
analyzer output voltage. A large increase in mechanical os-
cillation is observed when the RF drive frequency matches a
mechanical resonance frequency. The maximum oscillation
amplitude observed was S1/2
peak of the second order crown mode (j=3), with an applied
voltage of only 3 Vrms. This corresponds to a peak-to-peak
gradient force of F = 4π−1/2m3ωm,3Γ3Γ1/2
surpassing all cryogenic COMS to date by more than an or-
der of magnitude. The peak oscillation amplitude of each
mechanical mode was found to be linear as a function of ap-
plied voltage, as shown in Fig. 2B. A linear dependence on
DC voltage applied to the sharp electrode was also observed,
confirming gradient forces as the actuation mechanism.
Neither the presence of the sharp electrode, nor the RF drive,
caused any observable degradation of mechanical quality fac-
tor; demonstrating the low-dissipation nature of the actuation.
The criterion T<?ωm/kBmust be met for the quantum be-
havior of a mechanical oscillator to dominate classical ther-
mal fluctuations. For typical mechanical resonance frequen-
cies this imposes the stringent condition of milli- to micro-
is a key concern, placing an upper limit on the sustainable in-
tracavity power. Using radiation pressure, an intracavity
power of P ≈ cF/π = 36 W would be required to achieve the
maximum actuation force observed here. However, for
microtoroids in a cryogenic environment Schliesser et al.
ature increase, precluding operation in the quantum regime.
Hence, the COEMS presented here provides both the unique
capacity to strongly drive mechanical oscillators in the quan-
tum regime; and, transduction sensitivity close to the mechan-
ical zero-point motion. Combined, these attributes provide
new quantum control capabilities for mechanical systems, as
well as an enabling step towards the observation of mechani-
cal quantum nonlinear dynamics.
To demonstrate electro-optic feedback cooling we intro-
duce a feedback loop using gradient force actuation to apply
the in-loop transduction signal back upon the mechanical mo-
tion. Delaying the feedback by a quarter cycle provides a vis-
cous damping force, which both cools and damps the mechan-
ical motion. Including spectrally flat in-loop transduction
noise, the final temperature T of the oscillator under feedback
x,max= 2.4×10−14m Hz−1/2at the
x,max= 0.40 µN,
with gain g can be shown to be
1 + g,
where T0is the initial temperature, and SNR = Sx,0(ωm)/SIL
is the signal-to-noise ratio of the peak of the in-loop trans-
duction spectra without feedback to the in-loop transduction
noise. One sees that feedback induces cooling. However,
competing heating due to transduction noise imprinted on the
mechanical oscillator imposes a minimum achievable temper-
ature of Tmin=2T0[√1+SNR − 1]/SNR at g=√1+SNR − 1,
with the temperature increasing at higher gain.
Figure 3A and B respectively show the effect of the feed-
back on the in- and out-of-loop transduction spectra for a
radial flexural mode at 6.272 MHz with an effective mass
of 30±10 µg and 11.5 kHz damping rate. In both cases a
significant reduction in mechanical noise power, and hence
cooling, is observed with increasing feedback gain. How-
ever, a rapid acceleration of apparent cooling is seen via in-
loop transduction for gains g > 3.5 with eventual inversion
of the mechanical response at g > 10; in stark contrast to
the observations from out-of-loop transduction. This dramatic
squashing of the noise spectra to below the transduction noise
FIG. 3: (color online) Feedback cooling with varying feedback gain.
A and B: In-loop and out-of-loop transduction spectra. C: Temper-
ature inferred using in- and out-of-loop transduction signals. The
red circles ? and blue squares ? respectively denote out- and in-loop
temperature inferences; the solid red curve and dashed blue curve
respectively denote the theoretical predictions of actual mechanical
oscillator temperature and inferred in-loop temperature.
level is the result of feedback induced anti-correlations be-
tween the in-loop transduction noise and mechanical oscilla-
tor motion[10, 12, 14].
When mechanical motion and transduction noise are uncor-
related, the mode temperature is proportional to the integrated
area between the transduction spectra and the transduction
noise. Fig. 3C shows the in- and out-of-loop mechanical
oscillator temperatures inferred in this way as a function of
feedback gain. The out-of-loop temperature is in good agree-
ment with the theoretical prediction of Eq. (1). A minimum
temperature of T = 58 K > Tmin= 53 K was observed for
g=8, limited by the in-loop transduction noise which gave a
signal-to-noise ratio of SNR = 100; with the temperature, as
predicted, observed to increase at higher gains due to feed-
back noise imprinted on the mechanical oscillator. In con-
trast, at high gain anti-correlations between oscillator motion
and transduction noise cause the in-loop temperature infer-
ence to diverge significantly from theory, dropping well be-
low the theoretical limit and passing through 0 K before be-
coming unphysical on inversion of the observed mechanical
response. The result is a paradoxical and erroneous continual
reduction in the in-loop temperature inference with increasing
gain. These results dramatically demonstrate the requirement
of independent temperature verification in feedback cooling,
as first demonstrated here. Independent verification is essen-
tial, not only in the regime of high feedback gain, but also
in the critical quantum paradigm where measurement back-
action itself perturbs and correlates the mechanical oscillator
Although this proof-of-principle demonstration achieved
only a modest final phonon occupation number of ?n? =
kBT/?ωm= 19,000, the technique is fully compatible with
the recent advancements in microtoroid opto-mechanics dis-
cussed earlier[11, 24, 25]. Implementing these techniques
from a cryogenic starting temperature of 1.7 K, a near ground
state final phonon occupation number of ?n? ≈ 0.7 could be
tem which combines the ultrasensitive transduction capabili-
ties of cavity optomechanics with gradient force control capa-
bilities from nanoelectromechanics. Electrical gradient forces
as large as 0.40 µN were achieved, significantly higher than
has been demonstrated with radiation pressure actuation with-
out substantial heating. Simultaneously, a transduction sen-
sitivity of 1.5×10−18m Hz−1/2was observed, less than two
orders of magnitude away from the mechanical zero-point
motion. Electrically actuated, optically transduced feedback
cooling was achieved for the first time as a demonstration of
the control capabilities of our system. The implementation of
an out-of-loop probe allowed the first independent tempera-
ture verification, illustrating both the transduction noise limit
of feedback cooling and the striking effect of squashing on
in-loop temperature inferences. Our results represent impor-
tant progress in the control of mechanical systems at the quan-
tum level; as well an enabling step towards the new regime of
quantum nonlinear mechanics, where strong mechanical driv-
ing of a ground-state cooled mechanical oscillator allows ex-
ploration of nonlinear quantum dynamics.
This research was funded by the Australian Research Coun-
cil grant DP0987146, and performed in part at the Australian
National Fabrication Facility.
helpful advice from T. Kippenberg, G. Milburn, and T. Stace;
and experimental support from A. Rakic and K. Bertling.
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