Deviation from the normal mode expansion in a coupled graphene-nanomechanical system

0.9 MB·

by  Cornelia Schwarz

Deviation from the normal mode expansion
in a coupled graphene-nanomechanical system
Cornelia Schwarz,1 Benjamin Pigeau,1 Laure Mercier de L´epinay,1 Aur´elien G. Kuhn,1
Dipankar Kalita,1 Nedjma Bendiab,1 La¨etitia Marty,1 Vincent Bouchiat,1 and Olivier Arcizet1
1Institut N´eel, Universit´e Grenoble Alpes - CNRS:UPR2940, 38042 Grenoble, France
We optomechanically measure the vibrations of a nanomechanical system made of a
graphene membrane suspended on a silicon nitride nanoresonator. When probing the
6
thermal noiseof the couplednanomechanical device,we observea significantdeviation
1
from the normal mode expansion. It originates from the heterogeneous character of
0
mechanicaldissipationoverthespatialextensionofcoupledeigenmodes,whichviolates
2
one of the fundamental prerequisite for employing this commonly used description of

n the nanoresonators’ thermal noise. We subsequently measure the local mechanical
a susceptibility and demonstrate that the fluctuation-dissipation theorem still holds and
J permits a proper evaluation of the thermal noise of the nanomechanical system. Since

2 it naturally becomes delicate to ensure a good spatial homogeneity at the nanoscale,
thisapproachisfundamentaltocorrectlydescribethethermalnoiseofnanomechanical

] systems which ultimately impact their sensing capacity.
l
l
a
h
Nanomechanical oscillators are routinely used in
-
s fundamental and applied physics [1, 2] as ultrasensitive
e
force or mass sensors due to their increased sensitivity
m
to their environment. The understanding of dissipation
.
t at the nanoscale is the key ingredient towards extreme
a
m sensitivity operation. Among others, carbon-based
nanoresonators and novel 2D materials [3] have revolu-
-
d tionized the field of nanomechanics [4–14] by pushing
n the oscillator dimensions down to a single atomic layer.
o
The extremesensitivities achievedareultimately limited
c
by the thermal noise of the nanoresonators, which
[
underlines the importance of correctly understanding

1
and describing their Brownian motion. The thermal
v
noiseofa vibratingnanomechanicalsystemis commonly
4
5 described using the normal mode expansion, which as-
1 sumes that each eigenmode is driven by an independent
0 fluctuating Langevin force, presenting no correlation
0
with other eigenmodes. However, this intuitive de-
.
1 scription only holds when the mechanical dissipation is
0 homogeneously distributed in the system [15, 16]. Oth- FIG.1. Theexperimentalsetup. a)SEMofa20×20µm2
6 suspended CVD grown graphene monolayer supported on a
erwise, inhomogeneous damping can create dissipative
1 300nm-thick SiN nanomembrane, as sketched in b). c) Ex-
: coupling between eigenmodes, leading to a violation perimental setup: a balanced homodyne detection measures
v of their assumed independence. Such deviations which the phase fluctuations of the probe laser field reflected by
i
X have been reported on macroscopic devices [17, 18], are the sample and monitor its position fluctuations. A second
expected to be extremely important in nanomechanical counter-propagating pump laser beam can be intensity mod-
r
a systems, since it becomes increasingly difficult to ensure ulatedtooptomechanicallydrivethecouplednanoresonators.
The experiment is performed at pressures below 10−3mbar.
and even measure a good spatial homogeneity over the
d)Modeldescribingtheinertiallycouplednanoresonators. e)
entire nanosystem as its size is decreased. However,
Thermal noise of a graphene membrane, the sharp peaks on
no deviations from the normal mode expansion were each side are weakly coupled SiN eigenmodes.
observed at the nanoscale up to now, despite the large
variety of nanoresonators investigated.
In this article, we report on the deviation from the
normal mode expansion in the optomechanically mea- from the normal mode expansion, we exploit the in-
sured thermal noise of a nanomechanical arrangement ertial coupling between both nanoresonators: under
made of a suspended graphene monolayer coupled to a temperature-controlled tunable hybridization, the
a silicon nitride nanomembrane which supports the coupled eigenmodes become spatially delocalized on the
graphene resonator. To fully explore the deviation two subsystems whose intrinsic mechanical damping
2
rates differ by 2 orders of magnitude. In this situation oscillates around 100kHz. They present larger quality
the damping homogeneity is therefore altered which factors (above 1000) but higher masses, on the order of
results in a pronounceddeviationfromthe normalmode 10−12kg. In the following we investigate the thermal
expansion that we report on and analyze. Then we noise of the coupled system.
provethat the fluctuation-dissipationtheoremstill holds
by measuring the local mechanical susceptibility of the Hybridization of graphene eigenmodes– In order to
coupled nanomechanical system. tune the eigenfrequencies, we exploit the partial absorp-
These considerations are essential to correctly describe tionofasecondlaserbeamat532nmfocusseddowntoan
nanomechanical systems affected by inhomogeneous opticalwaistof 300nm,spatiallysuperimposedonthe
≃
damping and point out the importance of having ac- probe beam and injected from the backside of the sam-
cess to the local mechanical susceptibility to correctly ple. It generates a slight temperature increase which is
estimate the thermal noise of complex nanomechanical almost not detectable in the Brownian temperature (see
systems. Fig.2e) but is sufficient to significantly thermally tune
the graphene eigenfrequency. A clear hybridization be-
The experiment– Our nanomechanical system is a tween both graphene and Si3N4 eigenmodes is shown in
fully suspended single layer graphene sheet deposited Fig.2bwhereapronouncedfrequencyanticrossingcanbe
on a square window opened in a Si N nano-resonator, seen, as well as a modification of the mechanical damp-
3 4
itself supported on a tapped silicon wafer (see Fig.1) ing rates. Such signatures are fingerprints of strongdual
which allows a dual optical access from both sides. mode coupling [25], which canalso affect the force sensi-
It is obtained (see SI) by transfer in liquid phase of tivity [26–28].
monolayer, poly-crystalline graphene grown by CVD The modelisation of our inertially coupled nanomechan-
on Cu [19, 20] and suspended over up to 25x25µm2 ical system is based on cascaded mechanical oscillators
on a pre-etched stoichiometric Si3N4 membrane which [15], as sketched in Fig.1d. Their vibrations δxG,δxS
is 500-nm-thick and 100-µm-wide. A 633nm probe around the rest positions are coupled through δx¨G =
laser is focussed on the graphene resonator with a high −Ω2G(δxG−δxS)−ΓG(δx˙G−δx˙S)+δFG/MG andδx¨S =
numerical aperture objective (≈ 400nm optical waist). −Ω2SδxS−ΓSδx˙S+µΩ2G(δxG−δxS)+µΓG(δx˙G−δx˙S)+
The weak reflected beam constitutes the signal arm of δFS/MS, where ΩG,S/2π (resp. ΓG,S ) are the uncou-
a balanced homodyne detection [21] (see Fig.1c and pled frequencies (resp. damping rates). δFG is an ex-
SI). This interferometer permits a shot-noise limited ternal force applied on the graphene membrane, MG
readout of the membrane’s thermal noise with injected the grapheneeffective massatthe measurementlocation
optical powers ranging from 1 to 100µW. A fast piezo [16], while µ MG/MS parameterizes the hybridiza-
≡
element driving the local oscillator mirror permits a tion strength. Depending on the graphene and Si3N4
robust calibration of the interferometer, insensitive in membrane geometries which govern the vibration mode
particular to spatial drifts or reflectivity variations due spectrum and their spatial profiles, anticrossings with
to non-homogeneous graphene properties (wrinkles or varying strength can be observed (see SI). Intuitively,
grain boundaries). Reflectivities in the 1 10% range if graphene is positioned at a node of the membrane
−
were measured on monolayers depending on the level of eigenmode, their hybridization will be reduced. In the
contaminants. A typical calibrated displacement noise Fourier domain we have δxG = χ[Ω] δFG , using
spectrum is shown in Fig.1. Its reproduction at varying δx [Ω] dteiΩtδx (t). (cid:0)TδhxeS(cid:1)dynamical·(cid:16)mµaδtFriSx(cid:17)χ[Ω]−1
optical powers permits to verify the absence of optical i ≡ R i
being R
backaction (see SI). The uncoupled graphene resonators
presentfundamental eigenmodes in the 1-10MHz range, χ−1 M Ω2 χ−1
G G − G , (1)
with quality factors from 10 to 500 in vacuum and effec- (cid:18)µ M Ω2 χ−1 µ χ−1+χ−1+M Ω2 (cid:19)
tive masses ranging from 10−16 to 10−14kg. Operating (cid:0) G − G (cid:1) (cid:0) S G G (cid:1)
where we used the uncoupled mechanical susceptibili-
with fully transmitting systems permits suppressing
additional cavity effects [22] which could complicate ties χG,S ≡ MG−,1S Ω2G,S−Ω2−iΩΓG,S −1. Diagonaliz-
the noise thermometry. The spatial profile of graphene ing the restoringfo(cid:0)rcematrix M−1χ[0]−(cid:1)1 yields the new
G
eigenmodes can be mapped by probing thermal noise eigenfrequencies Ω /2π of the coupled system:
±
spectra at varying positions on the graphene membrane,
see SI. The slight elliptical structure and the frequency Ω2 (1+µ)Ω2G+Ω2S q(Ω2S−(1+µ)Ω2G)2+4µΩ2GΩ2S.
splitting observed on higher order modes reflects the ± ≡ 2 ± 2
presence of a residual 20 MPa stress along the diagonal (2)
direction [23, 24], attributed to the graphene transfer When µ 1, the minimum relative frequency splitting
≪
process. Also visible on the thermal noise spectrum are amounts to õ, corresponding to a canonically defined
sharp peaks corresponding to higher order eigenmodes coupling strength of g = ΩG√µ [25]. Depending on the
of the Si N nanomembrane, whose fundamental mode samplegeometryalargevarietyofcouplingstrengthscan
3 4
3
Violation of the normal mode expansion– Meanwhile,
a striking feature can be seen in the displacement noise
spectrashowninFig.2: acharacteristicpeakasymmetry
and a sharp noise minimum between both eigenmodes
areclearlyvisible inthe anticrossingregion. Thesespec-
tra cannot be fitted with two independent mechanical
thermalnoisespectra,seeFig.2a,withadeviationlarger
than 10 dB observed in the vicinity of Ω . Therefore
S
the measured thermal noise cannot be described by
two eigenmodes driven with independent Langevin
forces, which reveals the violation of the normal mode
expansion. This is a consequence of the spatial inho-
mogeneity of damping rates across the system: acoustic
vibrations are more efficiently damped in graphene than
in Si N . When the eigenmodes become hybridized,
3 4
their spatial profiles are delocalized over both systems,
see Fig.2c insets, so that mechanical damping becomes
inhomogeneous over the eigenmode spatial extension.
Thus the spatial profile of the vibration pattern cannot
be stationary anymore since it is non-homogeneously
damped and cannot be preserved over time. As such,
dissipation is now able to couple eigenmodes, which
breaks the fundamental hypothesis required to apply
FIG. 2. Thermal noise of the hybridized eigenmodes the normal mode expansion [15, 16]. When ΩG = ΩS,
a)Thermalnoiseofthecouplednanomechanicalsystemwhen the thermal noise spectral density at the minimum noise
tuned to an anticrossing region by adjusting the pump in- frequency is measured at a level 2Γ /Γ times lower
G S
tensity (400µW). Lower traces are obtained after numerical ≈
than the prediction of the normal mode expansion, see
background substraction. Solid lines are thebest fitsderived
SI. The understanding of this deviation is critical to
employing the normal mode expansion. Dashed green lines
patch the normal mode expansion and work out an
are fits using expression (3). b) Spectra measured through
the anticrossing for increasing tuning laser powers. Dashed analytical description of the system fluctuations.
lines are fits using (3) with fitting parameters ΩS,G,ΓS,G re-
ported in c), d) using µ = 0.002. Purple disks represent the Thermal noise of the hybridized nanomechanical
measured coupledeigenfrequencies Ω±/2π andsolid linesare system– To properly describe the nanosystem thermal
deduced from equation (2). d) Similar analysis for damping
noise, it is necessary to return to the original formula-
ratesΓ±/2π. e)RelativeBrowniantemperaturevariationde- tion of the fluctuation-dissipation theorem [15, 30]:
duced from the fits.
2k T
B
S [Ω]= Imχ [Ω] , (3)
δxG Ω | GG |
| |
beobserved,upto200kHz,largelyenteringtheso-called which relates the measured displacement noise spectral
strongcouplingregime(g >ΓS,ΓG). Theexperimentally density to the local mechanical susceptibility χGG. The
measured coupled eigenfrequencies are shown in Fig.2c latter connects the optomechanically measureddeforma-
forincreasingpumplaserpowers. Theycanbewellfitted tions of the graphene membrane δx [Ω] to the external
G
using equation (2) and a linear pump power dependence forceδF appliedonthegraphenemembraneatthemea-
G
for the uncoupled graphene and Si3N4 eigenfrequencies surement point: δxG[Ω] = χGG[Ω]δFG. First we pursue
of 284Hz/µW and 2Hz/µW respectively. The latter theanalysisbasedonthemodelemployedabove. Invert-
− −
corresponds to a maximum static heating of the Si3N4 ing equation (1) we obtain:
nanoresonator estimated at the level of 1K [29]. Us-
ingtheexperimentallymeasuredheatdiff≃usioncoefficient χ−1+M Ω2 2
of 5 10−6m2/s, see SI, the thermal heat resistance of χGG[Ω]−1 =χ−G1− χ−(cid:0)1G+χ−1+GM(cid:1)Ω2. (4)
grap×hene was numerically estimated at 0.25K per µW G S G
absorbed. The effective mechanicaldamping rates Γ of which permits deriving the expected thermal noise (see
±
the coupled modes can be roughly estimated using the SI).Ourexperimentalresultscanbe wellfitted withthis
FWHM of the thermal noise spectra, see Fig.2d, and model,seeFig.2a,2b,usingthefittingparameterswhich
used to extrapolate the uncoupled damping rates (see are reported in Fig.2c,2d and 2e. The magnitude of the
SI). coupling parameter µ = 0.002 is also in agreement with
theratioofbareeffectivemassesofbothnanoresonators.
4
No significant variation in the fitted noise temperature feedback loop in the measurement span, the transfer
could be detected, see Fig.2e, which places an upper function of allphotodetectors employedand the spectral
bound of 10K on the maximum temperature increase response of the AOM. With this, the expected thermal
≃
induced by the tuning laser. This observation is also noise can be properly estimated using equation (3) and
consistent with the estimated thermal resistance given compared to the measured thermal noise spectrum, as
above and allows to neglect the role of temperature shown in Fig.3e. The excellent quantitative agreement
inhomogeneities in our modelisation. found between both measurements all across the hy-
Validity of the fluctuation dissipation theorem in the bridization (see SI) demonstrates the validity of the
coupled nanomechanical system–Verifyingthevalidityof fluctuation-dissipation theorem in our strongly coupled
the fluctuation-dissipation theorem is essential in order nanomechanical arrangement.
to assess that the measured spectra correspond to the The hybridization dramatically modifies the graphene
thermal noise of the system. Following the principles mechanical response and has an impact on the signal-
of linear response theory [30], this requires measuring to-noise ratio (SNR) observed in a force measurement.
the local mechanical susceptibility χ of the coupled For a monochromatic force of amplitude δF ap-
GG G
nanomechanical system. To do so we modulate the plied in the center of the graphene membrane, the
pump beam intensity by means of an acousto-optic SNR can be expressed as SNR[Ω] = |Imχ−G1| where
modulator (AOM) and realize response measurements SNRG |Imχ−GG1|
SNR δF2/2M Γ k T represents the SNR of the
by sweeping the modulation frequency while recording G ≡ G G G B
uncoupled graphene alone. As verified experimentally
the driven displacement. Both laser spots are carefully
and confirmed with the model, see SI, the SNR can
superimposed on the graphene membrane to access to
be improved with respect to the uncoupled graphene
the local susceptibility; it is worth mentioning that this
resonator in narrow frequency bands in the vicinity
measurement cannot be realized with electrostatic gate
of the Si N resonance. As already employed with
or with piezo actuations since their spatial excitation 3 4
macroscopic devices [18], this constitutes a strategy for
profile is not localized on the measurement spot. We
achieving larger sensitivities in hybrid nano-sensors.
first verify the linearity of the actuation, see Fig.3a by
varying the optical modulation depth δP over 2 orders
of magnitude without modifying the mean pump power
(60µW) to ensure a stable graphene frequency, away
Conclusions– We have demonstrated the violation of
from anticrossings. No deviation from linearity were
thenormalmodeexpansioninamultimodenanomechan-
observed in the driven oscillations up to a maximum
ical system and verified that the fluctuation-dissipation
amplitude of 1nm, a few times the monolayer thickness
theoremwelldescribesitsthermalnoisedespitethelarge
(0.3nm), so that we perfectly sit in the linear actua-
mass and damping asymmetries. This work underlines
tion/measurementregime. Atypical actuationefficiency
the importance of measuring the local mechanical sus-
of 17pm/µW is measured, corresponding to an optical
ceptibility of a nanosystem to correctly understand its
force of 540fN/µW. This is significantly larger than the
thermal noise. Since a good sample homogeneity cannot
radiation pressure force contribution of 0.3fN/µW for a
be maintained in extremely down-sized nanomechanical
10%absorptioncoefficient,whichconfirmsthe dominant
devices, we anticipate that these deviations will play
role of thermo-optical forces in the optical actuation of
an important role in the future of nanomechanical
graphene [22]. The backaction noise due to the intensity
sensors. Our observations, realized on inertially coupled
fluctuations of the shot noise limited laser beams can
nanomechanical oscillators have a more general reach
thus be evaluated at the level of 0.1fm/√Hz for
≃ andarealsovalidwhenmechanicalmodesareexternally
P =100µW. This is largely negligible compared to the
0 coupled, such as by optical or electrostatic force field
measured thermal noise so that backaction cancelation
gradients [8, 14, 33–36]. Such a fundamental approach
[31] or classical noise squashing mechanisms [32] can be
could be used for developing new force detection proto-
safely excluded to interpret our results.
cols based on multimodal nanosystems.
Several response measurements were subsequently
performed through the anticrossing in the same mea-
surement conditions as in Fig.2a by progressively
increasing the pump intensity, while maintaining a Acknowledgements— We warmly thank J.Jarreau,
fixed modulation depth (δP/P0 = 30%). The response C.Hoarau, E.Eyraud, D.Lepoittevin, B.Fernandez,
curves shown in Fig.3c permit, once combined with the J.P.Poizat,A.GloppeandB.Besgaforexperimentaland
optical to force conversion factor measured in Fig.3a in technical assistance. This project is supported by the
absence ofhybridization, to determine the complex local ANR FOCUS, the ERC Starting Grant StG-2012-HQ-
mechanical susceptibility, χGG[Ω], as shown in Fig.3d. NOM, the E.U. Graphene Flagship and Lanef (CryOp-
Its proper determination requires taking into account tics). C.S. acknowledges funding from the Nanoscience
the weak residual contribution of the interferometer Foundation.
5
FIG.3. Optomechanical response of the hybridizednanomechanical system. a). Optomechanicalresponseobtained
by modulating the pump intensity for increasing modulation depths δP with a fixed average tuning power (P0 = 60µW).
b) Maximum driven displacement reported as a function of δP. The solid line has a slope of 17µm/W. c) Optomechanical
responsesobtainedforincreasingopticalpumppowerP0(30%modulationstrength). d)Amplitudeandphaseofthemechanical
susceptibility χGG derived for 400µW tuning power. The corresponding thermal noise spectrum expected using equation (3)
is reported in e) (i) and presents a very good agreement with the measured spectrum (ii). A 30% correction was used here
on theoptical to force conversion factor determined in Fig. 3a, to account for a slight modification of the actuation efficiency
between both measurements.
M.C.Gordillo,J.Boronat, andA.Bachtold,Phys.Rev.
Lett. 112, 196103 (2014).
[12] R. M. Cole, G. A. Brawley, V. P. Adiga, R. De Alba,
[1] A. Cleland, Foundations of Nanomechanics (Springer, J.M.Parpia,B.Ilic,H.G.Craighead, andW.P.Bowen,
Heidelberg, 2003). Phys. Rev.Applied 3, 024004 (2015).
[2] K. L. Ekinci and M. L. Roukes, Rev. Sci. Instrum. 76, [13] A. Reserbat-Plantey, K. G. Sch¨adler, L. Gaudreau,
061101 (2005). G. Navickaite, J. Gu¨ttinger, D. Chang, C. Toninelli,
[3] A. Castellanos-Gomez, H. S. J. Singh, V. van der Zant, A. Bachtold, and F. H. Koppens, arXiv:1401.4010
and G. A. Steele, Ann.Phys.(Berlin) 527, 2744 (2015). (2015).
[4] V. Sazonova, Y. Yaish, H. Ustunel, D. Roundy, T. A. [14] C.-H. Liu, I. S. Kim, and L. J. Lauhon, Nano Lett. 15,
Arias, and P. L. McEuen, Nature431, 284 (2004). 6727 (2015).
[5] C. Chen, S.Rosenblatt, K.I. Bolotin, W.Kalb, P. Kim, [15] P. Saulson, Physical Review D 42, 2437 (1990).
I. Kymissis, H. L. Stormer, T. F. Heinz, and J. Hone, [16] M. Pinard, Y. Hadjar, and A. Heidmann,Eur. Phys.J.
Nat Nano 4, 861 (2009). D 7, 107 (1999).
[6] A. M. Van der Zande, R. A. Barton, J. S. Alden, C. S. [17] K. Yamamoto, S. Otsuka, M. Ando, K. Kawabe, and
Ruiz-Vargas, W. S. Whitney, P. H. Q. Pham, J. Park, K. Tsubono, Phys.Lett. A 280, 289 (2001).
J.M.Parpia,H.G.Craighead, andP.L.McEuen,Nano [18] L. Conti, M. De Rosa, F. Marin, L. Taffarello, and
Letters 10, 4869 (2010). M.Cerdonio,JournalofAppliedPhysics93,3589(2003).
[7] J. Chaste, A. Eichler, J. Moser, G. Ceballos, R. Rurali, [19] Z.Han,A.Kimouche,D.Kalita,A.Allain,H.Arjmandi-
and A. Bachtold, NatureNanotech. 7, 300 (2012). Tash, A. Reserbat-Plantey, L. Marty, S. Pairis, V. Re-
[8] J. Moser, J. Guttinger, A. Eichler, J. Esplandiu, E. Liu, ita, N.Bendiab,J. Coraux, andV.Bouchiat, Advanced
M. Dykman, and A.Bachtold, Nat Nano 8,493 (2013). Functional Materials 24, 964 (2014).
[9] P.Weber,J.Gu¨ttinger, I.Tsioutsios, D.E. Chang, and [20] A. Reserbat-Plantey, D. Kalita, Z. Han, L. Ferlazzo,
A.Bachtold, NanoLetters 14, 2854 (2014). S.Autier-Laurent,K.Komatsu,C.Li,R.Weil,A.Ralko,
[10] V.Singh,S.J. Bosman, B. H.Schneider,Y.M.Blanter, L. Marty, S. Gu´eron, N. Bendiab, H. Bouchiat, and
A. Castellanos-Gomez, and G. Steele, Nat Nano 9, 820 V. Bouchiat, NanoLetters 14, 5044 (2014).
(2014). [21] P.F.Cohadon,A.Heidmann, andM.Pinard,Phys.Rev.
[11] A. Tavernarakis, J. Chaste, A. Eichler, G. Ceballos, Lett. 83, 3174 (1999).
6
[22] R. A. Barton, I. R. Storch, V. P. Adiga, R. Sakakibara, (2011).
B.R.Cipriany,B.Ilic,S.P.Wang,P.Ong,P.L.McEuen, [30] R. Kubo,Rep. Prog. Phys.29, 255 (1966).
J.M.Parpia, andH.G.Craighead,NanoLett.12,4681 [31] T. Caniard, P. Verlot, T. Briant, P.-F. Cohadon, and
(2012). A. Heidmann,Phys. Rev.Lett. 99, 110801 (2007).
[23] A. Fartash, I. K. Schuller, and M. Grimsditch, J. Appl. [32] J.Laurent,A.Mosset,O.Arcizet,J.Chevrier,S.Huant,
Phys.71, 4244 (1992). and H.Sellier, Phys. Rev.Lett. 107, 050801 (2011).
[24] M.J.Seitner,K.Gajo, andE.M.Weig,AppliedPhysics [33] A. B. Shkarin, N. E. Flowers-Jacobs, S. W. Hoch, A. D.
Letters 105, 213101 (2014). Kashkanova,C.Deutsch,J.Reichel, andJ.G.E.Harris,
[25] L.Novotny,AmericanJournalofPhysics78,1199(2010). Phys. Rev.Lett. 112, 013602 (2014).
[26] G. Anetsberger, R. Riviere, A. Schliesser, O. Arcizet, [34] J. Gieseler, B. Deutsch, R. Quidant, and L. Novotny,
and T. Kippenberg, Nat Photon 2, 627 (2008). Phys. Rev.Lett. 109, 103603 (2012).
[27] A. J¨ockel, M. T. Rakher, M. Korppi, S. Camerer, [35] T. Faust, J. Rieger, M. J. Seitner, P. Krenn, J. P. Kot-
D. Hunger, M. Mader, and P. Treutlein, Appl. Phys. thaus, and E. M. Weig, Phys. Rev. Lett. 109, 037205
Lett.99, 143109 (2011). (2012).
[28] Y. Tsaturyan, A. Barg, A. Simonsen, L. G. Villanueva, [36] A.Gloppe,P.Verlot,E.Dupont-Ferrier,A.Siria,P.Pon-
S.Schmid,A.Schliesser, andE.S.Polzik,Opt.Express charal,G.Bachelier,P.Vincent, andO.Arcizet,Nature
22, 6810 (2014). Nano. 9, 920 (2014).
[29] T.Larsen,S.Schmid,L.Gr¨onberg,A.Niskanen,J.Has-
sel,S.Dohn, andA.Boisen,AppliedPhysicsLetters98,