High-Q nested resonator in an actively stabilized optomechanical cavity

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by  Frank M. Buters

High-Q nested resonator in an actively stabilized optomechanical cavity
F.M. Buters,1,∗ K. Heeck,1 H.J. Eerkens,1 M.J. Weaver,2 F. Luna,2 S. de Man,1 and D. Bouwmeester1,2
1Huygens-Kamerlingh Onnes Laboratorium, Universiteit Leiden, 2333 CA Leiden, The Netherlands
2Department of Physics, University of California, Santa Barbara, California 93106, USA
(Dated: January 24, 2017)
Experiments involving micro- and nanomechanical resonators need to be carefully designed to
reduce mechanical environmental noise. A small scale on-chip approach is to add a resonator to
the system as a mechanical low-pass filter. However, the inherent low frequency of the low-pass
filter causes the system to be easily excited mechanically. We solve this problem by applying
activefeedbacktotheresonator,therebyminimizingthemotionwithrespectthefrontmirrorofan
optomechanical cavity. Not only does this method actively stabilize the cavity length, but it also
7 retains the on-chip vibration isolation.
1
0
2 I. INTRODUCTION resonatorwithrespecttothefrontmirrorofthecavity,all
length variations of the cavity are eliminated, including
n
a Micro- and nanomechanical systems are widely used themotionofthelowfrequencyresonatorwhileretaining
J for precision measurements across a variety of research the on-chip vibration isolation.

3 topics, see Ref. [1] for an overview. In cavity optome-
2 chanicsforexample, electromagneticradiationisusedto
perform displacement measurements that beat the stan- II. EXPERIMENTAL DETAILS
]
s dard quantum limit [2, 3]. Similarly, in atomic force mi-
c croscopy (AFM) forces as small as 10−18 N are measur-
For the actuation we make use of the dielectric force.
i
t able [4]. This is convenient as any dielectric body in the presence
p
Acommonchallengewhenusingmechanicalresonators
o of an electric field gradient experiences a dielectric force
is achieving a high mechanical quality factor (Q-factor)
. [20]. Recently this method was also used to demonstrate
s and shielding from vibrational noise. Recently, clamp-
c control of a micro- and nanomechanical resonators [21,
inglosses,orcouplingtoexternalmechanicalmodes,has
i 22].
s been identified as a major source of loss in mechanical
y TheenergyofadipoleinanelectricfieldisU =−p·E.
systems[5–7],affectingboththemechanicalQ-factorand
h TheforcethedipoleexperiencesisF=−∇U =(p·∇)E.
p the amount of vibrational noise entering the system. A
With the dipole moment p = αE, this can be written
[ solutiontothisproblemistointroducephononiccrystals
as F = (αE·∇)E. The strength of the dielectric force
[8,9]andlowfrequencymechanicalresonators[10–12]to
2 depends therefore on the applied electric field, its gradi-
isolate the device from the environment.
v ent and the polarizability of the medium. Although the
Surrounding the resonator of interest with an addi-
2 nested resonator is largely made from silicon, which is
1 tional low frequency resonator has a severe drawback.
weakly polarizable, the experiment is carried out in vac-
2 The additional low frequency resonator itself can be me-
uum, so relatively large electric fields can be generated.
4 chanicallyexcitedbytheenvironment. Theobvioussolu-
For example, a back-of-the-envelope calculation using a
0 tion would be to fixate theadditional resonator, but this
. charged sphere placed ∼µm distance behind our sample
1 will remove the effect of the mechanical low-pass filter.
[23] shows that forces up to ∼µN are feasible.
0 Typically this trade-off is circumvented by reducing the
We realize an electric field gradient by placing a small
7 motion of the resonator using active feedback cooling.
1 ring electrode behind the nested resonator, as depicted
Activefeedbackcooling,alsocalledcolddamping,uses
: in Fig. 1. The typical distance between electrode and
v therealtimedisplacementofthemechanicalresonatorto
resonator is 500 µm. Both a bias voltage and a modula-
i applyasuitablefeedbacksignaltoanactuator(mechan-
X tion voltage are applied. Since the electric field is linear
ical, optical or electrical) [13–19]. Increasing the gain of
r in applied voltage, the dielectric force can be written as
the feedback signal to the actuator results in more feed-
a F =βV2 =β(V +V )2 ≈βV2 +2βV V ,inthe
backcoolinguptothepointwheretheamplifiedread-out d.c. a.c. d.c. d.c. a.c.
limit when the applied bias V is much larger than the
noise causes the mechanical motion to increase. d.c.
modulationV ,withβ beingsomeconstant. Notethat
Inthisarticlewewillfirstdemonstrateactivefeedback a.c.
theforceisnow linear inV , therefore amodulationat
cooling of our nested trampoline resonator [12]. Because a.c.
frequency f will result in a force at frequency f .
thisresonatorispartofanoptomechanicalcavity,amore 0 0
Theread-outofthemechanicalmotionisdoneinthree
elegant approach to solving this problem is possible. We
different ways. A single-mode (SM) fiber is located ap-
will show how, by actively stabilizing the position of the
prox. 500 µm behind the mass of the outer resonator
to form a fiber interferometer [24], as is shown in Fig.
1. The interference signal created by the light reflect-
∗ [email protected] ing off the outer resonator and the light reflected off the
2
106
Ring s) 103 Power (arb.units)111000--063
electrode t
uni 0
MM fiber r (arb. 100 Phase (deg.) 11505000
e
ow10-3 Frequ2e9n2c.y5 (kHz)
SM fiber P
10-6
L=5 cm
103 104 105
Frequency (Hz)
Figure 1. Schematic overview of the geometry for apply-
ing an electric field gradient. A ring electrode, single-mode Figure 2. The mechanical transfer function of the nested
(SM)fiberandmulti-mode(MM)fiberareplacedbehindthe resonator design measured by applying an electrostatic force
resonator. The resonator itself is part of an optomechanical to the outer resonator and reading out the response of the
Fabry-Perot cavity. Inset: optical image of the nested res- inner resonator via the cavity. Inset: Driven response of the
onator which consists of an inner trampoline resonator sur- inner resonator.
rounded by a larger outer resonator.
III. RESULTS
fiberfacetallowsforasensitiveread-outoftheresonators
A. Active feedback cooling
motion. Our fiber interferometer using a standard dis-
tributed feedback laser diode at 1550 nm reaches a read-
√
out sensitivity of approx. 300 fm/ Hz around 3.5 kHz. The main point of the addition of a dielectric force is
not to drive the outer resonator, but to reduce its mo-
Forthesecond read-outmethoda lowfinesse (F=300)
tion. To do so, we use the method of active feedback
Fabry-Perot cavity, 5 cm long operating around 980 nm,
cooling [13]. The displacement signal from the fiber in-
is used. The transmitted cavity light is collected using a
terferometer is sent through a differentiating circuit and
multi-mode (MM) fiber placed in the center behind the
isamplifiedwithavariablegainamplifier. Togetherwith
sample as is shown in Fig. 1.
aDCvoltagethesignalis thensent to theringelectrode
The third method uses the same Fabry-Perot cavity to provide a damping force for the motion of the outer
togetherwithawavelengthof1064nmtocreateahighfi- resonator. To avoid difficulty in interpreting the data
nesse(F≈17000)cavity. ThePound-Drever-Hallmethod [19], weuseanout-of-loop measurementprovided bythe
[25] is used to read out the displacement of this cavity low finesse cavity to read out the effect of the feedback.
fromthelightreflectingoffthecavity. Finally,thewhole From the mechanical noise spectrum obtained via the
set-up resides in a vibration isolated vacuum chamber cavity read-out, the effective mechanical Q-factor of the
withabasepressureof10−3 mbartoeliminatetheeffect outerresonatorisdeterminedtogetherwiththetotaldis-
of gas damping on the mechanical properties. placement (cid:10)x2(cid:11) by fitting a Lorentzian. Via the equipar-
As a demonstration of the actuation via the dielec- tition theorem the effective temperature can be calcu-
tric force, the mechanical transfer function of the nested lated using T = (cid:10)x2(cid:11)MΩ2m/kb with kb the Boltzmann
resonator is measured by varying the frequency of the constant, M = 7×10−7 kg, and Ωm = 2π×3488 rad/s
applied force to the outer resonator and reading out the for this particular sample.
responseoftheinnerresonatorviathelowfinesseoptical With the feedback circuit activated, the motion of the
cavity. The data (blue points) of Fig. 2 follow the the- resonator will be both damped and cooled. Increasing
ory curve for two coupled harmonic oscillators. At the the gain results in more damping and cooling of the me-
frequency of the inner resonator, which for this sample chanical motion up to the point where the amplified de-
is 292.5 kHz, more than 60 dB of isolation is provided tection noise becomes comparable to the thermal noise.
via the nested resonator design. This is consistent with The read-out noise from the feedback is transferred to
previousfindings[12]. Theinsetshowsthattheinnerres- the mechanical motion of the resonator, causing the ef-
onator can be driven in this way as well. Note that this fective temperature to increase again. We have observed
measurement assumes a constant force excitation which, precisely this behavior, as is shown in Fig. 3. Note that
judging by the agreement between experiment and the- this is visible because of the out-of-loop measurement.
ory of Fig. 2, is valid. Therefore the bandwidth of this Measuring the mechanical noise spectrum via the inter-
method of actuation is at least 100 kHz. ferometer, within the feedback loop, will result in noise
3
10 2.0
(a) 1.5 (b)
m)
5 1.0
Signal (a.u.) 05 splacement (n 0010....0505
Di
PDH error 1.5
Voltage source
10 2.0
0 10 20 30 40 50 0 10 20 30 40 50
Time (ms) Time (ms)
Figure 3. Active feedback cooling of the outer resonator. Figure 4. Scanning of the cavity length (a) Varying the
When the gain is increased, both the effective temperature position of the outer resonator results in the typical PDH
and the Q-factor decreases. For large gains, the amplified error signal. (b) The displacement of the outer resonator is
read-out noise actually drives the outer resonator, increas- monitored using the fiber interferometer.
ing the effective temperature. (a) Selection of power spectra
demonstrating cooling of the mechanical mode. This out-of-
Inner resonator via PDH Outer resonator via interferometer
lNoootpemtheaatsuarsemmaelnltaudsdeistitohnealreraedso-onuatncveiaisthveisilbowle.fi(nbe)ssEeffceacvtiitvye. 10-1 OClpoesne dlo loopop (a) 103 OClpoesne dlo loopop (b)
mode temperature as a function of mechanical Q. Hz)10-2 Hz) 102
/ /
squashing [19]. 2D (pm10-3 2D (pm 101
WeareabletoreducetheintrinsicQ-factorof90000to PS10-4 PS 100
about 20. The mechanical mode temperature is reduced
10-1
to about 15 K using this form of cooling. To check if 10-5
286288290292294296 3.47 3.49
the limiting factor is indeed the noise floor of the inter- Frequency (kHz) Frequency (kHz)
ferometric read-out, the data is fitted using the theory
derivedbyPoggioetal. [19]. Foractivefeedbackcooling Figure 5. Active stabilization of the outer resonator. When
the effective temperature is given by: the feedback loop is closed, the cavity is resonant with the
laser, making the motion of the inner resonator visible as is
T kΩ g2 shownin(a). Ontheotherhand,themotionoftheouterres-
T = bath + m S (1)
eff 1+g 4k Q 1+g xn onatorisnolongervisibleviatheinterferometer,asisshown
b 0
in (b).
with T the environmental temperature, g the feed-
bath
back gain, k the spring constant, Q the intrinsic qual-
0
ity factor and S the read-out noise power. In our the position of the outer resonator using the dielectric
xn
case: k = 340 N/m, T = 293 K, and Q = 90000. force,whiletrackingthecavityresonanceviathePound-
bath 0
Theonlyfreeparameteristheread-outnoisepowerS . Drever-Hall (PDH) method [25]. In this way the nested
xn
From the fit (red curve Fig.3(b)) we obtain a value for resonator design not only mechanically decouples the in-
√
(cid:112)S = 380±10 fm/ Hz, which is close to the noise ner resonator from the environment, but also stabilizes
xn
floor of our interferometric read-out. the cavity length with respect to the laser frequency.
Usingactivefeedbackthetotalrmsdisplacementisre- In Fig. 4(a) the typical PDH error signal (blue) is ob-
duced from 6 pm to 0.8 pm, which for most applications tainedbyscanningthecavitylengthusingahighvoltage
is sufficient. However in our case, we require further re- amplifier (red). Note that to ensure a linear response,
duction of the motion. To achieve this, the interferomet- a DC bias voltage (not shown here) is also added. The
ric read-out is replaced with a more sensitive read-out displacement of the outer resonator is observed via the
method using a high finesse optical cavity. interferometer, as is shown in Fig. 4(b). A displacement
of ±1.5 nm provides sufficient range to keep the cavity
resonant with the laser.
B. Active stabilization We typically require a feedback bandwidth of about
10 kHz to keep laser and cavity resonant. However, the
The use of a high finesse optical cavity typically re- mechanical resonance at f = 3488 Hz provides a very
quires a means to keep laser and cavity resonant. Usu- rapidπphaseshiftinthetransferfunction. Anotchfilter
ally, the laser frequency is continuously adjusted to keep is placed at f =3.4 kHz in the feedback loop to smooth
itresonantwiththecavitymode. Analternativemethod this transition. A second notch filter at f = 10.2 kHz
would be to adjust the cavity length to keep the cav- is used to compensate the first higher order mechanical
ity resonant with the laser. For the set-up depicted in mode.
Fig. 1, the cavity length L can be varied by changing When the feedback loop is closed, the cavity should
4
remain resonant with the laser. Therefore the motion of effects [26] present in an optomechanical cavity.
the inner resonator, which occurs at a frequency beyond
thefeedbackbandwidth,shouldbevisibleinthePDHer-
ror signal. Fig. 5(a) shows the Fourier transform of this IV. CONCLUSION
error signal and indeed, with a closed loop, the thermal
motionoftheinnerresonatorisvisible. Thethermalmo- In conclusion, we have demonstrated how to solve the
tion of the outer resonator has been mostly eliminated, problem of fixating an on-chip mechanical low-pass filter
and is no longer visible via the interferometric read-out, while retaining the mechanical isolation. By making use
see Fig. 5(b). of an optomechanical cavity, the motion of the resonator
can be referenced to the front mirror. Not only does
this stabilize the cavity with respect to the laser, it also
FromFig. 5itisclearthatthefeedbacknotonlyworks, stiffens the resonator, thereby significantly reducing its
butalsothattheon-chipisolationstillworksasevidenced motion. Wehavemadeuseofanoptomechanicalsystem,
by the clean spectrum of Fig. 5(b). However, what has butinprinciplethetechniquespresentedinthisworkcan
happened to the motion of the outer resonator? Effec- be applied to any system as long as a suitable reference
tively, with active stabilization, a very strong electrical can be chosen.
springisplacedbetweentheouterresonatorandthefront
mirror. The only way for the outer resonator to move at
a particular frequency, is if the front mirror also moves ACKNOWLEDGMENT
at this frequency. This stiffening of the outer resonator
explains why, with a closed loop, the mechanical motion The authors acknowledge the useful discussions with
isnolongervisibleinFig. 5(b). Theadditionalelectrical W.Loeffler. TheauthorswouldalsoliketothankH.van
spring also helps to prevent any unwanted optical spring der Meer for technical assistance and support.
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