A strongly-coupled $\Lambda$-type micromechanical system

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by  Hajime Okamoto

A strongly-coupled Λ-type micromechanical system
Hajime Okamoto,1,2 Ryan Schilling,1 Hendrik Schu¨tz,1 Vivishek Sudhir,1
Dalziel J. Wilson,1 Hiroshi Yamaguchi,2 and Tobias J. Kippenberg1
1Institute of Condensed Matter Physics, E´cole Polytechnique
Fe´de´rale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
2NTT Basic Research Laboratories, Nippon Telegraph and Telephone Corporation, Atsugi 243-0198, Japan
(Dated: January 22, 2016)
We study a classical Λ-type three-level system based on three high-Q micromechanical beam
resonators embedded in a gradient electric field. By modulating the strength of the field at the
difference frequency between adjacent beam modes, we realize strong dynamic two-mode coupling,
viathedielectricforce. Drivingadjacentpairssimultaneously,weobservetheformationofapurely
mechanical‘dark’stateandanall-phononicanalogofcoherentpopulationtrapping—signaturesof
6 strongthree-modecoupling. TheΛ-typemicromechanicalsystemisanaturalextentionofpreviously
1 demonstrated ‘two-level’ micromechanical systems and offers new perspectives on the architecture
0 of all-phononic micromechanical circuits and arrays.
2

n
The ability to control phonon transport in mi-
a
J cromechanical systems has important consequences
1 for applications such as low-power mechanical fil-
2 ters, switches, routers, memories, and logic gates
[1–9]. As a basic starting point, considerable effort

] has been aimed towards achieving tunable coupling
l
l between two modes of a micromechanical resonator,
a
h using a combination of piezoelectric [10, 11], pho-
- tothermal [12, 13], and dielectric [14] forces. Real-
s
izationofstrongcoupling—thatis,coherentenergy
e
m exchange between two modes at a rate (g) larger
thantheirmechanicaldissipation(γ)—hasrecently
.
t enabledclassicalanalogsof‘two-levelsystem’coher-
a
m entcontrol,includingRabiflops,Ramseyfringesand
Hahn echo [10, 15]. These demonstrations provide
-
d a dress rehearsal for phonon transport in future mi-
n cromechanical circuits and arrays [16, 17].
o
c In this Letter, we demonstrate strong coupling
FIG.1: Overview of the device. (a)False-colorscan-
[ between three radiofrequency micromechanical res-
ning electron micrograph of the device; yellow, Au; red,
onatorsformingaclassicalanalogofaΛ-typethree-
1 Si N ; grey, Si substrate. (b) Magnified scanning elec-
level system (two nearly degenerate, low energy lev- 3 4
v tron micrograph, highlighting the region surrounded by
3 els and single high energy level). The inclusion of black squares in (a). (c) Two sketches of the sample in
2 a third ‘level’ opens the door to a rich variety of cross-section,correspondingtohorizontalcutsalongthe
6 physics not accessible to two-level systems, such as pink and red dotted line in (b). The thickness of Si N
3 4
5 analogs of EIT (electromagnetically induced trans- and metal layers are 100 and 45 nm, respectively. The
0
parency [18]) and CPT (coherent population trap- horizontal gap between adjacent beams is 80 nm.
.
1 ping [19]). To our knowledge, a fully micromechan-
0 ical Λ-type system has not been previously imple-
respectively. The middle beam has a fundamental
6 mented. By contrast, V-type three-level systems
1 formed by a pair of optical cavity modes coupled mode frequency of ωM ∼ 2π·4.50 MHz. To couple
: these modes, a voltage is applied between two elec-
v to a common mechanical resonator mode have been
trodes patterned on the outer beams. The resulting
i extensivelyexploredinthecontextofcavityoptome-
X electric field produces a dielectric potential, which,
chanics[20–22],enablingdemonstrationsofoptome-
r owing to the spatial dependence of the field, gives
a chanicalstatetransfer[20]andoptomechanicaldark
risetoastaticintermodalcoupling[15],analogousto
states [21]. In analogy to the latter, we show that
the strain-mediated coupling of two beams sharing
bydrivingbothpairsoftheΛ-systemmicromechan-
a non-rigid anchor [10]. We wish to emphasize that,
ical system simultaneously, destructive interference
contrarytothelattercase,dielectricintermodecou-
leads to the formation of a mechanical dark state
pling can be switched completely off by grounding
and a phononic analog of CPT.
the electrodes. In this sense, the mechanical modes
The micromechanical system, shown in Fig. 1, may be considered to reside on three physically iso-
consists of a planar stack of three high-stress Si N lated mechanical resonators. Though conceptually
3 4
microbeams. Theouterleftandrightbeamspossess thesameasthreemodesofasingleresonator,physi-
fundamental out-of-plane flexural modes with fre- calseparationisadvantageousforbuildingmicrome-
quenciesω ∼2π·1.67MHzandω ∼2π·1.72MHz, chanical circuits/arrays, as it enables the resonators
L R
2
FIG. 3: Energy level diagram. (a) Energy level dia-
gram of the Λ-type micromechanical system in the ab-
senceofdynamiccoupling. (b,c)Energyleveldiagramin
thepresenceofpump(probe)tone∆ =ω −ω ,
L(R) M L(R)
exhibiting normal mode splitting of the left (right) and
FIG. 2: Dielectric tuning of mechanical eigenfre-
middle beam modes. (d) Energy level diagram of the
quencies and optical readout. (a-c)Thermalmotion
three-levelsysteminthepresenceofsimultaneouspump
of the middle beam (mode M), right beam (mode R),
andprobetones. Allthreelevelsexhibitmodesplitting.
andleftbeam(modeL),readoutusingalensed-fiberin-
Notably, one of the dressed states is ‘dark’ in the sense
terferometer. For these measurements no DC voltage is
that involves no motion of the center beam.
applied across the metal electrodes. Displacement noise
spectral densities, S , are expressed in units scaled to
x
the twice the peak zero point spectral density, Sxzp(ωm) “strong coupling”.
(i.e., the displacement noise produced by single thermal
Experimental characterization of the device is
phonon). In these units, the peak spectral density is
shown in Fig. 2. Mechanical spectra are recorded
equal to with the thermal occupation, n ≈k T/(cid:126)ω ,
th B m used a lensed fiber-based interferometer [24] with a
where ω is the mechanical frequency. (d) Dependence
m
spot size (∼ 4µm) large enough to simultaneously
ofmodefrequenciesontheDCvoltageappliedacrossthe
electronics. Modeshapescomputedbythefiniteelement record the thermal displacement of each beam. The
method are shown on the left. opticalfield,suppliedbya780nmTi:sapphirelaser,
is attenuated until photothermal effects are negligi-
tobefictionalizedforindependentpurposes. Forin- ble. Sample and fiber are embedded in a vacuum
stance, we envision that the central beam in Fig. 1 chamber at 10−4 mbar. When no voltage is applied
may be integrated into the a near-field of an optical across the electrodes, the fundamental out-of-plane
cavity [23], for precision readout and actuation. modes of the middle (M), right (R), and left (L)
Applying a DC voltage produces static coupling beamsresideattheirnaturalfrequencieswithenergy
between the beams; however, energy transfer be- dissipation rates of γM =2π·100 Hz, γR =2π·170
tween vibrational modes remains weak because of Hz, and γL = 2π · 140 Hz, respectively, shown in
their different eigenfrequencies. To overcome this descending order in Fig. 2a-c. The identity of each
non-degeneracy, an AC voltage may be applied at mode is verified by finite element simulation (using
the difference frequency between modes of adjacent COMSOL, see Fig. 2d). When a DC voltage is ap-
beams. The resulting modulated static coupling plied to the electrodes, the mechanical frequencies
strength (see Eq. 1) leads to energy flow between are blue-shifted (Fig. 2d). The frequency shift is
the two modes — hereafter referred to as dynamic nearlyquadratic, consistentwiththedielectricforce
intermode coupling. The magnitude of the dynamic in a linear field gradient [14].
coupling rate is proportional to the product of the To realize dynamic intermode coupling, an AC
AC voltage, the DC voltage, and a geometric factor voltage is applied across the electrodes at one of
proportional to the cross derivative of the dielectric the two difference frequencies, ∆ =ω −ω .
L(R) M L(R)
potential. When sufficiently large, the dynamically (Following [22, 25, 26], we refer to these as the
coupled modes exhibit normal mode splitting. The ‘pump’ (∆ ) and ‘probe’ (∆ ) tones. See Fig. 3.)
L R
magnitudeofthissplittingcorrespondstotheircou- For sufficiently large AC voltages, normal mode
pling rate. In our system, owing to the high room splitting is observed in the thermal displacement
temperature quality factor of the tensily stressed noise spectrum. Shown in Fig. 4 is the displace-
beams (Q ∼ 104), the normal mode splitting can ment noise spectrum plotted versus AC voltage am-
be made much larger than that of mechanical dis- plitude V for a fixed DC offset of V = 10
L(R) DC
sipation (i.e. energy decay) rate, corresponding to V. The magnitude of the mode splitting is pro-
3
Here x is the generalized displacement
L(R,M)
of the left(right,middle) beam mode, g =
L(R)
Ω cos(∆ t) is the modulated intermode cou-
L(R) L(R)
pling rate with strength Ω and F is a
L(R) L(R,M)
generalized external force. The normal modes of
√
Eq. 1 are: x =(g x +g x ±g x )/( 2g ) and
± L L R R 0 M 0
(cid:112)
x = (−g x +g x )/g , where g = g2 +g2.
D R L L R 0 0 L R
x corresponds to the ‘dark’ mode, which contains
D
no contribution from x . To obtain Fig. 5d-f we
M
solvedEq.1numericallyusingtherotatingwaveap-
proximation [10], viz., assuming
(cid:34) ∞ (cid:35)
(cid:88)
x (t)=Re a (t)ei(ω+m∆L+n∆R)t (2)
L m,n
m,n=−∞
FIG.4: Dynamicintermodecouplingbydielectric
frequency modulation. Left column: Noise spectrum (and similar expressions for x and x ). Models
R M
ofmodeM(a),modeR(b),andmodeL(c)versuspump in Fig. 5d-f are produced using the first-order
amplitude (V ) with pump frequency tuned to ∆ =
L L approximation m=n=1.
ω −ω , V = 10 V, and no probe voltage applied.
M L DC
Right column: Noise spectrum of mode M (d), mode R
(e), and mode L (f) versus probe amplitude (V ) with
p In summary, we have demonstrated strong dy-
probe frequency tuned to ∆ =ω −ω , V = 10 V,
R M R DC namic coupling of three Si N micromechanical
and no pump voltage applied. 3 4
resonators via dielectric forces, enabling formation
of a mechanical ‘dark’ state and a classical analog
portional to the AC voltage amplitude, and cor-
of coherent phonon population trapping. It bears
responds to twice the energy coupling rate g
L(R) emphasis that experiments were performed at room
between the left(right) and middle beam modes.
temperature and that the device (included me-
Strong coupling, g > {γ ,γ }, is achieved
L(R) L(R) M chanical and electrical elements) is fully integrated
for V > 1 V in both cases. At the largest AC
L(R) pp on a chip, allowing for robust control. Pulsed
voltage amplitudes, the cooperativities achieved are
switching of the gradient field is available, thus
C =4g2 /(γ γ )>1×104.
L(R) L(R) L(R) M two-level control techniques (albeit here in the
When the pump and probe tones are applied si-
entirely classical analog) such as Rabi oscillation
multaneously, strong coupling is induced among all
and Ramsey fringe measurements [10, 15]) and
three beams. This coupling is akin to a resonant
three-level techniques like pulsed EIT [25] are
Raman interaction, and features the formation of a
available. This control might be extended, for
dynamically decoupled ‘dark’ state due to destruc-
example, to nodes of micromechanical resonator
tive interference (see Fig. 3d). To study this effect,
circuits and arrays [16, 17]. We also remark
the displacement noise spectrum is recorded versus
that, by design, the three-beam system shown in
probe strength (V ) for a fixed pump strength of
R Fig. 1 is readily incorporated in the evanescent
V = 10V and a fixed DC offset of V = 10
L pp DC near-field of a optical microdisk cavity; this plat-
V. As the probe strength is increased, a trio of
form has been shown to enable state-of-the-art
dressed states emerges near ω (Fig. 5b) and ω
R L interferometric displacement sensitivities as well as
(Fig.5c),reflectingtheonsetofstrongdynamiccou-
efficient radiation pressure force actuation [23, 27],
pling between all three beams. Notably, only two
offering compelling opportunities for coherent cou-
peaksappearnearω evenwhenthethreemechan-
M pling of electrical fields and optical fields [28–31].
ical resonators are strongly coupled (Fig. 5a). The
energetically allowed third peak appears ‘dark’ be-
cause of destructive interference between modes L, We thank NTT for granting and supporting the
R, and M. This effect is analogous to coherent pop- sabbaticalleaveofH.O.. Weacknowledgenanofabri-
ulationtrapping,asexploredinoptomechanicalsys- cation advice from M. Zervas and A. Feofanov. The
tems [22, 25, 26]. samplewasfabricatedattheCMi(CenterforMicro-
The experimental results shown in Fig. 5a-c are Nanotechnology) at EPFL. Research was funded
reproduced in Fig. 5d-f using a classical coupled- by an ERC Advanced Grant (QuREM), the Marie
resonator model: Curie Initial Training Network for Cavity Quan-
tum Optomechanics (cQOM), and partial support
x¨L+γLx˙L+ωL2xL =FL−gLxM (1a) from MEXT KAKENHI (Grant No. 15H05869 and
x¨ +γ x˙ +ω2 x =F −g x −g x (1b) 15K21727). D.J.W. acknowledges support from the
M M M M M M L L R R
European Commission through a Marie Skodowska-
x¨ +γ x˙ +ω2x =F −g x . (1c)
R R R R R R R M Curie Fellowship (IIF project 331985).
4
FIG. 5: Analog of coherent population trapping among three dynamically coupled micromechanical
beams. Probevoltage(V )dependenceofmodeM(a),modeR(b),andmodeL(c)forV =10Vwhenthepump
R DC
frequencyistunedto∆ =ω −ω andtheprobefrequencyistunedto∆ =ω −ω . Thepumpvoltageishere
L M L R M R
fixedatV =10V . Simulatedresultsfor(a), (b), and(c)areshownin(d), (e), (f), respectively. Cross-sectionsof
L pp
(a), (b), and (c) at V =10V are shown in (g), (h), and (i).
R pp
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