Nonlinear Radiation Pressure Dynamics in an Optomechanical Crystal
Alex G. Krause,
1,2
Jeff T. Hill,
1,2,3
Max Ludwig,
4
Amir H. Safavi-Naeini,
1,2,3
Jasper Chan,
1,2
Florian Marquardt,
4,5
and Oskar Painter
1,2
,*
1
Kavli Nanoscience Institute and Thomas J. Watson, Sr., Laboratory of Applied Physics,
California Institute of Technology, Pasadena, California 91125, USA
2
Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA
3
Edward L. Ginzton Laboratory, Stanford University, Stanford, California 94305, USA
4
Institute for Theoretical Physics, Universität Erlangen-Nürnberg, 91058 Erlangen, Germany
5
Max Planck Institute for the Science of Light, Günther-Scharowsky-Straße 1/Bau 24, D-91058 Erlangen, Germany
(Received 22 April 2015; published 2 December 2015)
Utilizing a silicon nanobeam optomechanical crystal, we investigate the attractor diagram arising from
the radiation pressure interaction between a localized optical cavity at
λ
c
¼
1542
nm and a mechanical
resonance at
ω
m
=
2
π
¼
3
.
72
GHz. At a temperature of
T
b
≈
10
K, highly nonlinear driving of mechanical
motion is observed via continuous wave optical pumping. Introduction of a time-dependent (modulated)
optical pump is used to steer the system towards an otherwise inaccessible dynamically stable attractor in
which mechanical self-oscillation occurs for an optical pump red detuned from the cavity resonance. An
analytical model incorporating thermo-optic effects due to optical absorption heating is developed and
found to accurately predict the measured device behavior.
DOI:
10.1103/PhysRevLett.115.233601
PACS numbers: 42.50.Wk, 42.50.Pq
Cavity-optomechanical systems involving interactions of
light and mechanical motion in a mechanically compliant
electromagnetic cavity
[1]
are of interest for precision
sensors
[2,3]
, in nonlinear optics
[4,5]
, and in the study of
macroscopic quantum systems
[6,7]
. To lowest order, the
mechanical displacement linearly modulates the frequency
of the optical resonance in a cavity-optomechanical system.
This, however, gives rise to an inherently nonlinear phase
modulation, and through radiation pressure backaction on
the mechanical element, yields nonlinear system dynamics
[8]
. Much of the previous work has focused on the
linearized regime where the interaction with the optical
field still gives rise to a host of interesting phenomena such
as a modified spring constant
[9]
, damping or amplification
of the mechanics
[10]
, and electromagnetically induced
transparency
–
like slow-light effects
[11,12]
. Recently,
several experiments have pushed into the quantum regime
using backaction cooling to bring nanomechanical reso-
nators near their quantum ground state of motion
[13,14]
.
In this Letter, we instead demonstrate new features and
tools in the nonlinear regime of large mechanical oscillation
amplitude. In contrast to the well-known static fixed points
of an optomechanical system
[15]
, we are interested here in
the dynamic multistability associated with the finite-ampli-
tude mechanical limit cycles that result from radiation
pressure dynamic backaction. Previous experimental works
have shown that a blue-detuned laser drive can lead to
stable mechanical self-oscillations
[16
–
20]
, and dynamic
bistability has been observed for a photothermally driven
micromechanical system
[21]
and in the collective density
oscillations of an atomic Bose-Einstein condensate inside a
Fabry-Perot cavity
[22]
. Theoretical predictions, however,
indicate that radiation pressure dynamic backaction can
lead to an even more intricate, multistable attractor diagram
[8]
. In the present Letter we are able to verify the predicted
attractor diagram and, further, utilize a modulated laser
drive to steer the system into an isolated high-amplitude
attractor. This introduces pulsed control of nonlinear
dynamics in optomechanical systems dominated by radi-
ation pressure backaction, in analogy to what has been
shown recently for a system with an intrinsic mechanical
bistability
[23]
.
We employ a one-dimensional optomechanical crystal
(OMC) cavity designed to have strongly interacting optical
and mechanical resonances
[24]
. The OMC structure is
created from a freestanding silicon beam by etching into it a
periodic array of holes which act as Bragg mirrors for both
acoustic and optical waves
[25]
. A scanning electron
micrograph (SEM) of an OMC cavity is shown in
Fig.
1(a)
along with finite-element-method (FEM) simu-
lations of the colocalized optical [Fig.
1(b)
] and mechanical
[Fig.
1(c)
] resonances. To reduce radiation of the mechani-
cal energy into the bulk, the cavity is surrounded by a
periodic
“
cross
”
structure which has a full acoustic band
gap around the mechanical frequency [Fig.
1(a)
, green
overlay]
[26]
.
The experimental setup is shown schematically in
Fig.
1(d)
. The silicon chip containing the device is placed
into a helium flow cryostat where it rests on a cold finger
at
T
≈
4
K (the device temperature is measured to be
T
b
≈
10
K). Laser light is sent into the device via a tapered
optical fiber, which, when placed in the near field of the
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© 2015 American Physical Society
device, evanescently couples to the optical resonance of the
OMC
[27]
. A narrow linewidth, frequency tunable pump
laser is used to excite and measure the optical and
mechanical resonances of the OMC cavity. The transmitted
pump light is sent to a high-bandwidth photodetector (
D
1
),
which is connected to a real-time spectrum analyzer (RSA)
for spectral analysis. Scanning the pump laser frequency
and measuring the time-averaged transmitted light intensity
yields a resonance dip at
λ
c
¼
1542
nm for the fundamen-
tal optical mode of the device under study in this work.
From the spectrum the intrinsic and taper-loaded energy
decay rate of the optical resonance is estimated to be
κ
i
=
2
π
¼
580
MHz and
κ
=
2
π
¼
1
.
7
GHz, respectively.
Mechanical motion modulates the phase of the internal
optical cavity field, scattering the pump light into motional
sidebands which beat with the unscattered pump field
on the photodiode
[13]
. From the microwave spectrum
of the measured photocurrent at low pump power we
find the breathing mechanical mode to be at frequency
ω
m
=
2
π
¼
3
.
72
GHz, with an intrinsic linewidth of
γ
i
=
2
π
¼
24
kHz. These device parameters put our system well into
the sideband resolved regime
κ
=
ω
m
≪
1
.
The interaction between the internal light field and the
mechanical motion is given by the interaction Hamiltonian,
H
int
¼
ℏ
g
0
ˆ
a
†
ˆ
a
ˆ
x
, where
ˆ
a
(
ˆ
x
) is the optical (mechanical)
field amplitude, and
g
0
is the vacuum optomechanical
coupling rate. The mechanical displacement expectation is
given by
x
¼
x
zpf
h
ˆ
x
i
, where the zero-point amplitude of the
resonator is
x
zpf
¼ð
ℏ
=
2
m
eff
ω
m
Þ
1
=
2
¼
2
.
7
fm (estimated
using a motional mass
m
eff
¼
311
fg calculated from
FEM simulation). By calibrating the optically induced
mechanical damping versus pump power
[13]
, we find
that
g
0
=
2
π
¼
941
kHz. In the device studied here, this
vacuum coupling rate is dominated by the photoelastic
component of the radiation pressure force
[24]
.
The classical nonlinear equations of motion for the
mechanical displacement (
x
) and the optical cavity ampli-
tude (
a
¼h
ˆ
a
i
) are
̈
x
ð
t
Þ¼
−
γ
i
_
x
ð
t
Þ
−
ω
2
m
x
ð
t
Þþ
2
ω
m
g
0
x
zpf
j
a
ð
t
Þj
2
;
ð
1
Þ
_
a
ð
t
Þ¼
−
κ
2
þ
i
Δ
L
þ
g
0
x
zpf
x
ð
t
Þ
a
ð
t
Þþ
ffiffiffiffiffi
κ
e
2
r
a
in
;
ð
2
Þ
where
a
in
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
P
in
=
ℏ
ω
L
p
is the effective drive amplitude of
the pump laser (input power
P
in
and frequency
ω
L
),
κ
e
=
2
is
the fiber taper input coupling rate,
ω
c
is the optical
cavity resonance frequency, and
Δ
L
≡
ω
L
−
ω
c
. For self-
sustained oscillations, where the motion of the oscillator is
coherent on time scales much longer than the cavity
lifetime, we can take the mechanical motion to be sinus-
oidal with amplitude
A
,
x
ð
t
Þ¼
A
sin
ω
m
t
. The optical
cavity field is then given by
a
ð
t
Þ¼
ffiffiffiffiffi
κ
e
2
r
a
in
e
i
Φ
ð
t
Þ
X
n
i
n
α
n
e
in
ω
m
t
;
ð
3
Þ
where
Φ
ð
t
Þ¼
−
β
m
cos
ω
m
t
and
α
n
¼
J
n
ð
β
m
Þ
=
½
κ
=
2
þ
i
ð
n
ω
m
−
Δ
L
Þ
. Here,
J
n
is the Bessel function of
the first kind,
n
th order, and its argument is the unitless
modulation strength
β
m
¼ð
Ag
0
Þ
=
ð
x
zpf
ω
m
Þ
.For
β
m
≪
1
only the terms oscillating at the mechanical frequency,
ω
m
, are appreciable, so the interaction can be linearized,
and only the first-order radiation pressure terms are present.
However, for
β
≥
1
the higher harmonic terms at each
n
ω
m
have significant amplitude and backaction force.
The thermal amplitude is too small to enter the nonlinear
regime in our devices
ð
β
th
≈
0
.
01
Þ
; however, backaction
from the pump laser can provide amplification for driving
the mechanical resonator into the high-
β
, nonlinear regime.
The resulting mechanical gain spectrum in the amplitude-
detuning plane (the attractor diagram) can be solved for by
calculating the energy lost in one mechanical cycle
(
P
fric
¼
m
eff
γ
i
h
_
x
2
i
)andcomparingittothatgained(orlost)
from the optical radiation force [
P
rad
¼ð
ℏ
g
0
=x
zpf
Þhj
ˆ
a
j
2
_
x
i
]
[8]
.Figure
2(a)
shows a plot of the gain spectrum for the
parameters of the device studied here with a laser pump
power of
P
in
¼
151
μ
W. Imposing energy conservation,
P
rad
=P
fric
¼þ
1
, yields the steady-state solution contour
lines. Although the entire contour is a physical solution,
(a)
(b)
(c)
(d)
FIG. 1 (color online). (a) SEM of the OMC cavity surrounded
by a phononic shield (green). (b) FEM-simulated electromagnetic
energy density of the first-order optical mode. (c) FEM-simulated
mechanical mode profile (displacement exaggerated). In (b) the
color scale bar indicates large (red) and small (blue) energy
density, whereas in (c) the scale bar indicates large (red) and small
(blue) displacement amplitude. (d) Simplified schematic of the
experimental setup. WM: wave meter,
Δ
φ
: electro-optic phase
modulator,
D
1
: pump light detector,
D
2
: probe detector, VOA:
variable optical attenuator, PM: power meter. Pump and probe
lasers are not mutually coherent to avoid interference effects and
we modulate the probe and monitor the detected tone using a
lock-in amplifier (not shown).
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the equilibrium is only stable when the power ratio
decreases upon increasing the mechanical amplitude,
ð
∂
=
∂
β
Þð
P
rad
=P
fric
Þ
<
0
(i.e., stability is found at the
“
tops
”
of the contours)
[8]
. At higher powers (black
contours) we see that for many laser detunings there
are several stable mechanical-amplitude solutions dem-
onstrating the presence of dynamic multistability.
In the device studied here, there is a thermo-optic
frequency shift of the optical cavity caused by heating
due to intracavity optical absorption. The thermal time
constant of the device structure is slow relative to the
optical cavity coupling rate, but fast compared to the laser
scan speed. Absorption heating can thus be modeled as a
shift of the laser detuning proportional to the average
intracavity photon number (
̄
n
a
),
Δ
L
¼
Δ
L;
0
þ
c
to
̄
n
a
, where
Δ
L;
0
is the bare laser-cavity detuning in the absence of
thermo-optic effects. The per photon thermo-optic fre-
quency shift of the optical cavity is measured to be
c
to
=
2
π
¼
−
216
kHz. Including this effect, the shifted
contours are shown in Fig.
2(b)
as a function of the bare
detuning
Δ
L;
0
. The solid lines with arrows indicate the
expected path traversed by the mechanical resonator during
a slow laser scan from lower to higher laser frequency (left
to right) at each power. The dashed lines are contours that
are either unstable, or unreachable by this adiabatic laser
sweep. Note that while thermo-optic frequency shifts can
be up to
10
ω
m
on resonance, the contours traced out by the
laser sweep are only slightly shifted, as the laser never
reaches the cavity resonance due to the thermo-optic
bistability.
In Figs.
2(c)
–
2(g)
we explore the lowest-lying contour of
the attractor diagram by measuring the optical transmission
as the pump laser is tuned from red to blue across the
optical cavity resonance with different fixed optical input
powers. At low optical input powers (
P
in
<
0
.
3
μ
W), only
a single resonance dip associated with the bare optical
cavity is observed. Upon increasing the laser power,
radiation pressure backaction amplifies the thermal motion
of the mechanical resonator beyond threshold and into a
large amplitude state. When this occurs a large fraction of
the intracavity photons are scattered, resulting in addi-
tional transmission dips near each detuning
Δ
L
¼
n
ω
m
.
Physically, mechanical oscillations at the
n
th sideband
detuning are generated by a multiphoton gain process
involving
n
photon-phonon scattering events. This stair-
step behavior is seen in the measured transmission spec-
trum of both Fig.
2(c)
and Fig.
2(e)
. The theoretically
calculated spectra for our measured device parameters are
shown in Fig.
2(d)
, showing good agreement with the
measured spectra after including thermo-optic effects.
Figure
2(f)
shows the microwave noise power spectrum
at the highest measured input power (
P
in
¼
151
μ
W),
(a)
(b)
(c)
(d)
(e)
(f)
(g)
FIG. 2 (color online). (a) Calculated gain spectrum for the OMC in the amplitude-detuning plane. Color scale indicates
ð
P
rad
=P
fric
−
1
Þ
at
P
in
¼
151
μ
W. Solid line curves indicate power-conserving solution contours at selected input powers:
0
.
65
μ
W (white),
6
.
5
μ
W
(grey),
151
μ
W (black). (b) Same as (a) with contours now shifted by estimated thermo-optic effects (the intensity plot of the gain is left
unshifted for reference). Solid line curves indicate the path taken by the mechanical oscillator during a slow laser frequency sweep. Dashed
lines are contours which are either unstable or unreachable using this method. (c) Image plot of the measured optical transmission spectrum
versus laser detuning and power. The wavelength scan rate (
∼
300
GHz
=
s) is much slower than the internal dynamics of the
optomechanical system. (d) Image plot of the theoretically calculated transmission spectra including thermo-optic shifts and a slow
drift in the optical resonance frequency over the course of the measurement from low to high power. Spectra in (c) and (d) are normalized at
each power level. (e) Plot of the normalized optical transmission from scans in (c) at
P
in
¼
0
.
12
μ
W (top panel),
0
.
65
μ
W (center panel),
151
μ
W (bottom panel). The blue points are measured data and the red curves are the theoretical model. (f) Power spectral density of
transmitted pump photocurrent near the mechanical frequency for
P
in
¼
151
μ
W. (g) Total integrated power of spectra in (f). Measured
data are plotted as green circles, with the theoretical model (up to a scale factor) shown as a solid red curve. The red arrow in each plot
indicates the laser scan direction.
PRL
115,
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PHYSICAL REVIEW LETTERS
week ending
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233601-3